♥ T❡♥ts

❡rt✉r❜❛t✐✈❡ ❛♥❞ ♥♦♥✲

♥ S✉❝❤

✵✳ ❈♦♥t❡♥ts ✸✳✸ •❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✹ ◗❑ ♠❛♥✐❢♦•❞ ✇✐t❤ s②♠♠❡tr✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✺ ❊✈❛•✉❛t✐♦♥ ♦❢ t❤❡ ♠❡tr✐❝ ♦♥ t❤❡ ◗❑ ♠❛♥✐❢♦•❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✹ ◗✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s t♦ t❤❡ ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡ ❛♥❞ ❉✲ ✐♥st❛♥t♦♥s ✸✽ ✹✳✶ ❖♥❡ •♦♦♣ ❝♦rr❡❝t❡❞ ♠❡tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✹✳✶✳✶ ❚✇✐st♦r ❞❡s❝r✐♣t✐♦♥ ♦❢ ♦♥❡✲•♦♦♣ ❝♦rr❡❝t❡❞ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ✳ ✸✾ ✹✳✷ ❉✲✐♥st❛♥t♦♥s ✐♥ t✇✐st♦r s♣❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✷✳✶ |❡•❛t✐♦♥s t♦ ✇❛••✲❝r♦ss✐♥❣ ♣❤❡♥♦♠❡♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✸ ❊①♣•✐❝✐t ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ♠❡tr✐❝ ✐♥ t❤❡ ♠✉t✉❛••② •♦❝❛• ❝❛s❡ ✳ ✳ ✳ ✳ ✹✹ ✹✳✹ ❯♥✐✈❡rs❛• ❤②♣❡r♠✉•t✐♣•❡t ❛♥❞ ❚♦❞❛ ♣♦t❡♥t✐❛• ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✹✳✺ ❈✉r✈❛t✉r❡ s✐♥❣✉•❛r✐t② ✐♥ ♣r❡s❡♥❝❡ ♦❢ ❉✲✐♥st❛♥t♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✺ ▼✐rr♦r s②♠♠❡tr② ❛t t❤❡ q✉❛♥t✉♠ •❡✈❡• ✺✻ ✺✳✶ ❉✲❜r❛♥❡s ❛♥❞ ❝❤❛r❣❡ q✉❛♥t✐③❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✺✳✷ ❙✲❞✉❛•✐t② ✐♥ t❤❡ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✺✳✸ ▼♦♥♦❞r♦♠② ✐♥✈❛r✐❛♥❝❡ ❛♥❞ s♣❡❝tr❛• ✢♦✇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✺✳✹ ❙✲❞✉❛•✐t② ✐♥ t❤❡ t✇✐st♦r s♣❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✺✳✹✳✶ ▲✐❢t ♦❢ t❤❡ SL(2, R) ❢♦r tr❡❡ •❡✈❡• ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ✳ ✳ ✳ ✳ ✳ ✻✶ ✺✳✹✳✷ ▼♦❞✉•❛r ❝♦♥str❛✐♥t ♦♥ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✺✳✹✳✸ ▼♦❞✉•❛r ❛❝t✐♦♥ ♦♥ t❤❡ t✇✐st♦r ✜❜❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✺✳✹✳✹ ❱✐rt✉❡ ♦❢ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s ✿ ❛♥ ❡①❛♠♣•❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✻ ❚❤❡ s②♠♠❡tr② ❣r♦✉♣ ♦❢ M HM ✼✵ ✻✳✶ |❡✈✐s✐t✐♥❣ ❝•❛ss✐❝❛• ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✻✳✷ ◗✉❛♥t✉♠ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ❛♥❞ ✐ts s②♠♠❡tr✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✻✳✸ ❋❛✐•✉r❡ ♦❢ t❤❡ ❣r♦✉♣ •❛✇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✻✳✹ ❈♦♥str✉❝t✐♥❣ ❣r♦✉♣ r❡♣r❡s❡♥t❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✻✳✹✳✶ ❈♦♠♠✉t❛t♦rs ♦❢ ❞✐✛❡r❡♥t ❡•❡♠❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✻✳✺ ❙♦♠❡ r❡♠❛r❦s ♦♥ ♠♦❞✐✜❝❛t✐♦♥s ♦❢ t❤❡ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ✳ ✼✻ ✻✳✺✳✶ ❋r❡❡❞✲❲✐tt❡♥ ❛♥♦♠❛•② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✻✳✺✳✷ ❈♦♥♥❡❝t✐♦♥ ♦❢ t❤❡ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ✭✻✳✶✵✮ ✇✐t❤ ❋r❡❡❞✲❲✐tt❡♥ ❛♥♦♠❛•② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✼ ▼✐rr♦r s②♠♠❡tr② ✐♥ ❛❝t✐♦♥ ✿ ✐♥st❛♥t♦♥s ✐♥ t②♣❡ ■■❇ ❢♦r♠✉•❛t✐♦♥ ✽✶ ✼✳✶ ❉✭✲✶✮✲❉✶ ✐♥st❛♥t♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✼✳✷ ❙✉❜t•❡t✐❡s ♦❢ ◆❙✺✲❜r❛♥❡ ❝♦♥tr✐❜✉t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✼✳✸ ❋✐✈❡❜r❛♥❡ ❝♦rr❡❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✼✳✸✳✶ •❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ❞✉❡ t♦ ❉✭✲✶✮✲❉✶ ✐♥st❛♥t♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✼✳✸✳✷ ❉✺✲✐♥st❛♥t♦♥s ❛❢t❡r ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻ ✼✳✸✳✸ ❲❛••❝r♦ss✐♥❣ ❛♥❞ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼ ✼✳✸✳✹ ❋✐✈❡❜r❛♥❡ ✐♥st❛♥t♦♥s ❢r♦♠ ❙✲❞✉❛•✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾ ✐✐ ❈♦♥t❡♥ts✳ ✵✳✵ ✼✳✹ ▼♦♥♦❞r♦♠② ❛♥❞ ❍❡✐s❡♥❜❡r❣ ✐♥✈❛r✐❛♥❝❡ ♦❢ ✜✈❡❜r❛♥❡ ✐♥st❛♥t♦♥ ❝♦r✲ r❡❝t❡❞ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ✼✳✹✳✶ ❚❤❡ ❝❤❛r❛❝t❡r ε(g) ❛♥❞ ❉❡❞❡❦✐♥❞ s✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷ ✼✳✹✳✷ ■♥✈❛r✐❛♥❝❡ ✉♥❞❡r ♠♦♥♦❞r♦♠② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸ ✼✳✹✳✸ ■♥✈❛r✐❛♥❝❡ ✉♥❞❡r ❍❡✐s❡♥❜❡r❣ tr❛♥s❢♦r♠❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ S P [h, X] = - 1 4πα ′ Σ dσdτ √ -h G µν h ab ∂ a X µ ∂ b X ν ,
S P = - 1 4πα ′ Σ dσdτ √ -h G µν h ab ∂ a X µ ∂ b X ν + α ′ νR . ✭✶✳✻✮ ❋♦r ❝♦♠♣•❡t❡♥❡ss✱ ♦♥❡ ❤❛s t♦ ❛•s♦ ♣✉t s♦♠❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✐♥ t❤❡ ❝❛s❡ ♦❢ ♦♣❡♥ str✐♥❣s✱ ❢♦r ✇❤✐❝❤ ◆❡✉♠❛♥♥ ❝♦♥❞✐t✐♦♥ ✐s ✐♠♣♦s❡❞ ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ✇♦
T ab = - 2π √ -h δS P δh ab = - 2 α ′ ∂ a X µ ∂ b X µ - 1 2 h ab ∂ c X µ ∂ c X µ . ✭✶✳✶✵✮ ❚❤❡ ✐♥✈❛r✐❛♥❝❡ ✉♥❞❡r ✐♥✜♥✐t❡s✐♠❛• ❲❡②• tr❛♥s❢♦r♠❛t✐♦♥ ✐♠♣•✐❡s t❤❛t ❝•❛ss✐❝❛••② h ab δS P δh ab = 0 =⇒ T a a = 0. M Y = M C (Y) × M K (Y).
K = -log i(X Λ FΛ (X) -XΛ F Λ (X)) . ✭✷✳✷✮ ❯s✐♥❣ t❤❡ ♣r♦•❡❝t✐✈❡ ❝♦♦r✐♥❛t❡s X Λ /X 0 = (1, z a ) ♦♥❡ ❝❛♥ ✜♥❞ t❤❡ ♠❡tr✐❝ ❢r♦♠ t❤❡ ❑ä❤•❡r ♣♦t❡♥t✐❛• ❜② g a b = ∂ a ∂b K(z, z). ✭✷✳✸✮ ❚❤❡ ❤♦•♦♠♦r♣❤✐❝ t❤r❡❡ ❢♦r♠ Ω ❛♥❞ t❤❡ ❝♦♠♣•❡①✐✜❡❞ ❑ä❤•❡r (1, 1)✲❢♦r♠ ♣r♦✈✐❞❡ ♥❛t✉r❛• ❝❛♥❞✐❞❛t❡s ❢♦r t❤❡ ♠❡tr✐❝✱ t❤❡ ❛❜♦✈❡ ❞❡✜♥❡❞ ❝♦♦r❞✐♥❛t❡s ❛♥❞ t❤❡ ♣r❡♣♦✲ t❡♥t✐❛•✳ ❲❡ ❞✐s❝✉ss t❤❡s❡ ❛❜♦✈❡ t✇♦ ❝❛s❡s s❡♣❛r❛t❡•② ❛♥❞ ✐♥tr♦❞✉❝❡ ❛ ❢❡✇ ❢♦r♠✉•❛s ♦♥ t❤❡ ✇❛②✳ ✷✳✶✳✶ ❈♦♠♣•❡① str✉❝t✉r❡ ♠♦❞✉•✐ ❚❤❡ ❤♦•♦♠♦r♣❤✐❝ t❤r❡❡ ❢♦r♠ Ω ❣✐✈❡s r✐s❡ t♦ t❤❡ ❢♦••♦✇✐♥❣ ❑ä❤•❡r ♣♦t❡♥t✐❛• ♦♥ M C (Y) ❣✐✈❡♥ ❜②✱ K = -log i Y Ω ∧ Ω. ✭✷✳✹✮ ✶✸ ✷✳ ❈♦♠♣❛❝t✐✜❝❛t✐♦♥ ♦❢ t②♣❡ ■■ str✐♥❣ t❤❡♦r✐❡s ♦♥ ❈❛•❛❜✐✲(❛✉ t❤r❡❡❢♦•❞s ❛♥❞ t❤❡✐r ♠♦❞✉•✐ s♣❛❝❡s ❲❡ ♥♦✇ ✇❛♥t t♦ ✐♥tr♦❞✉❝❡ ❤♦•♦♠♦r♣❤✐❝ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ♦♥ M C (Y)✳ ❋♦r ❞♦✐♥❣ s♦✱ •❡t ✉s ❝♦♥s✐❞❡r t❤❡ ❢✉•• ❝♦❤♦♠♦•♦❣② ❣r♦✉♣ H 3 = H 3,0 ⊕ H 2,1 ⊕ H 1,2 ⊕ H 0,3 . ✭✷✳✺✮ ❚❤✐s ✐s ❞✉❛• t♦ H 3 (Y) ✇❤❡r❡ ✇❡ ❛ ❝❤♦♦s❡ ❛ s②♠♣•❡❝t✐❝ ♠❛r❦✐♥❣ A Λ , B Λ s✉❝❤ t❤❛t A Λ #B Σ = δ Λ Σ ✳ •✐✈❡♥ s✉❝❤ ❛ ❜❛s✐s✱ •❡t ✉s ❞❡✜♥❡ X Λ = A Λ Ω, F Λ = B Λ Ω. ✭✷✳✻✮ ❈•❡❛r•② t❤✐s ✐s ❛ ❤✐❣❤•② r❡❞✉♥❞❛♥t ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✭ = 2(h 2,1 + 1✮✮✱ ❛s t❤❡ s♣❛❝❡ ♦❢ ❝♦♠♣•❡① str✉❝t✉r❡ ❞❡❢♦r♠❛t✐♦♥s ✐s ♦♥•② h 2,1 ❞✐♠❡♥s✐♦♥❛•✳ ❆s ❛ r❡s✉•t✱ ♦♥❡ ❝❛♥ ❛•✇❛②s ❝❤♦♦s❡ ❛ ❜❛s✐s ♦❢ ❝②❝•❡s ❢♦r ✇❤✐❝❤ F Λ ❜❡❝♦♠❡s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ F (X) ✇✐t❤ r❡s♣❡❝t t♦ X Λ ✳ ▲❡t ✉s s❤♦✇ ✐t ❡①♣•✐❝✐t•②✳ ❋♦r t❤❛t ✇❡ ♥❡❡❞ t♦ ✉s❡ t❤❡ |✐❡♠❛♥♥ ❜✐•✐♥❡❛r ✐❞❡♥t✐t②✱ Y ψ ∧ χ = Λ A Λ ψ B Λ χ - A Λ χ B Λ ψ ✭✷✳✼✮ ❙✐♥❝❡ t❤❡ ❞❡❢♦r♠❛t✐♦♥ ♦❢ Ω ❝❛♥ ❜❡ ❛t ♠♦st ❛ (2, 1)✲❢♦r♠ 0 = Y Ω ∧ ∂ Λ Ω = Y X Σ B Σ + F Σ A Σ ∧ B Λ + ∂ Λ F Σ A Σ . ✭✷✳✽✮ ❚❤❡♥ ✉s✐♥❣ t❤❡ ❜✐•✐♥❡❛r ✐❞❡♥t✐t② ♦♥❡ ✐♠♠❡❞✐❛t❡•② ✜♥❞s 0 = F Λ -X Σ ∂ Λ F Σ = 2 F Λ - 1 2 ∂ Λ X Σ F Σ . ✭✷✳✾✮ ■♥t❡❣r❛t✐♥❣✱ ✇❡ ✜♥❞ t❤❛t ♣r❡♣♦t❡♥t✐❛• ✐s ❛ s❡❝♦♥❞ ❞❡❣r❡❡ ❤♦♠♦❣❡♥❡♦✉s ❢✉♥❝t✐♦♥ ❛s F = 1 2 X Λ F Λ ✳ ❚❤✐s ❞❡✜♥❡s t❤❡ ♣r❡♣♦t❡♥t✐❛• ❢♦r t❤❡ ❝♦♠♣•❡① str✉❝t✉r❡ ♠♦❞✉•✐ ❛♥❞ ✐♥ ♣❛rt✐❝✉•❛r✱ t❤❡ ❑ä❤•❡r ♣♦t❡♥t✐❛• ✭✷✳✹✮ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ♦♥❡ ❢♦••♦✇✐♥❣ ❢r♦♠ ✭✷✳✷✮✳ ✷✳✶✳✷ ❑ä❤•❡r ♠♦❞✉•✐ ■♥ t❤✐s ❝❛s❡✱ ♦♥❡ s❤♦✉•❞ ❝♦♥s✐❞❡r ✐♥st❡❛❞ t❤❡ ❡✈❡♥ ❝♦❤♦♠♦•♦❣② ❣r♦✉♣✳ H even = H 0 ⊕ H 2 ⊕ H 4 ⊕ H 6 . ✭✷✳✶✵✮ ▲❡t ✉s ❝❤♦♦s❡ ❛ ❜❛s✐s ✐♥ t❤✐s s♣❛❝❡ ♣❛r❛♠❡tr✐③❡❞ ❜② ω I = (1, ω i ) ❛♥❞ ω I = (ω Y , ω i )✱ ✇❤❡r❡ ω Y ✐s t❤❡ ✈♦•✉♠❡ ❢♦r♠✱ s✉❝❤ t❤❛t ω i ∧ ω j = δ j i ω Y , ω i ∧ ω j = κ ijk ω k ✭✷✳✶✶✮ ✶✹ ❋✐❡•❞ ❝♦♥t❡♥t ♦❢ t②♣❡ ■■ s✉♣❡r❣r❛✈✐t✐❡s ✐♥ ❢♦✉r ❞✐♠❡♥s✐♦♥s✳ ✷✳✷ ❛♥❞ t❤❡ ✐♥❞✐❝❡s I = (0, i) = 0, 1, ..., h 1,1 (Y)✳ ❚❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥ ❞❡✜♥❡s t❤❡ tr✐♣•❡ ✐♥t❡rs❡❝t✐♦♥ ♥✉♠❜❡rs✳ ❈❤♦♦s✐♥❣ t❤❡ ❞✉❛• ❜❛s✐s ♦❢ t✇♦ ❝②❝•❡s γ i ❛♥❞ ❛ ❜❛s✐s ♦❢ ❢♦✉r ❝②❝•❡s γ i ✱ t❤❡ tr✐♣•❡ ✐♥t❡rs❡❝t✐♦♥ ♥✉♠❜❡r ❝❛♥ ❜❡ ❢♦✉♥❞ ❛s κ ijk = Y ω i ∧ ω j ∧ ω k = γ i , γ j , γ k . ✭✷✳✶✷✮ ❚❤❡♥ t❤❡ ♥❛t✉r❛• ♠❡tr✐❝ ♦♥ M K (Y) ✐s ❣✐✈❡♥ ❜② g i j = 1 4V Y ω i ∧ * ω j = ∂ i ∂j(-log 8V), ✭✷✳✶✸✮ ✇❤❡r❡ V = 1 6 Y J ∧ J ∧ J ✐s t❤❡ ❈❛•❛❜✐✲(❛✉ ✈♦•✉♠❡✳ ❚❤❡♥ t❤❡ ❤♦•♦♠♦r♣❤✐❝ ❝♦♦r❞✐♥❛t❡s ♦♥ M K (Y) ❛r❡ v i = b i + it i = Y (B + iJ). ✭✷✳✶✹✮ ▲❡t ✉s ✐♥tr♦❞✉❝❡ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❝♦♦r❞✐♥❛t❡s X I = (1, v i )✳ ■♥ t❡r♠s ♦❢ X I t❤❡ ♣r❡♣♦t❡♥t✐❛• ❢♦r t❤❡ ♠♦❞✉•✐ s♣❛❝❡ ♦❢ ❝♦♠♣•❡①✐✜❡❞ ❑ä❤•❡r ❝•❛ss ❞❡❢♦r♠❛t✐♦♥s t❛❦❡s t❤❡ ❢♦r♠✱ F (X) = - 1 6 κ ijk X i X j X k X 0 ,
♥ ❝❤♦♦s❡ ❞✉❛• ❢♦r♠s t♦ A Λ ❛♥❞ B Λ ❝②❝•❡s ✐♥tr♦❞✉❝❡❞ ✐♥ s❡❝t✐♦♥ ✷✳✶✳✶ t♦ ❜❡ α Λ ❛♥❞ β Λ r❡s♣❡❝t✐✈❡•②✱ s✉❝❤ t❤❛t A Λ α Σ = δ Σ Λ , A Λ β Σ = 0, B Λ α Σ = 0, B Λ β Σ = -δ Λ Σ . ✭✷✳✶✻✮ ❚❤❡ ♣r♦❝❡❞✉r❡ ♦❢ ❝♦♠♣❛❝t✐✜❝❛t✐♦♥ t♦ ❢♦✉r ❞✐♠❡♥s✐♦♥s ♦♥ ❈❛•❛❜✐✲(❛✉ ✐s ❞❡s❝r✐❜❡❞ ✐♥ ❞❡t❛✐•s ✐♥ ❬✷( ❛♥❞ ❛ r❡•❡✈❛♥t r❡✈✐❡✇ ✐s ♣r❡s❡♥t ✐♥ ❬✹(✳ ■♥ t❤❡ ❢♦••♦✇✐♥❣ ✇❡ ✇✐•• ♦♥•② ❞✐s❝✉ss t❤❡ ✜❡•❞ ❝♦♥t❡♥ts ✐♥ t❤❡ t✇♦ ❝❛s❡s ♦❢ ✐♥ ❢♦✉r ❞✐♠❡♥s✐♦♥s✳ ✷✳✷✳✶ ❚②♣❡ ■■❆ ❚❤❡ ❜♦s♦♥✐❝ s❡❝t♦r ♦❢ t②♣❡ ■■❆ s✉♣❡r❣r❛✈✐t② ✐♥ 10✲❞✐♠❡♥s✐♦♥s ✐s ❞❡s❝r✐❜❡❞ ❜② t❤❡ ❢♦••♦✇✐♥❣ ❛❝t✐♦♥ ❬✷( ✿ S IIA = 1 2 e -2 φ R * 1 + 4d φ ∧ * φ - 1 2 Ĥ3 ∧ * Ĥ3 -F2 ∧ * F2 + F4 ∧ * F4 -B2 ∧ d Â3 ∧ d Â3 ,
(ĝ XY ) 10d → g µν , t i , z a 4d → g µν , t i , z a 4d-dual , ( B2

) 10d → B 2 + b i ω i 4d → σ, b i 4d-dual , φ10d → φ 4d → φ 4d-dual , ✭✷✳✶✾✮ ✇❤❡r❡ t❤❡ ✜rst ❛rr♦✇ ✐♥❞✐❝❛t❡s ❞✐♠❡♥s✐♦♥❛• r❡❞✉❝t✐♦♥ ❢r♦♠ ✶✵ t♦ ✹ ❞✐♠❡♥s✐♦♥s ❛♥❞ t❤❡ s❡❝♦♥❞ ❛rr♦✇ ✐♥❞✐❝❛t❡s ❞✉❛•✐③❛t✐♦♥ ✐♥ ✹ ❞✐♠❡♥s✐♦♥s✳ ❼ ❚❤❡♥ ❢♦r t❤❡ || s❡❝t♦r✱ ( Â1 ) 10d → (A 0 1 ) 4d → (A 0 1 ) 4d-dual , ( Â3 ) 10d → A 3 + A i 1 ω i + ζ Λ α Λ + ζΛ β Λ 4d → constant, A i 1 , ζ Λ , ζΛ 4d-dual . ✭✷✳✷✵✮ ❚❤✉s✱ ✐♥ ❢♦✉r ❞✐♠❡♥s✐♦♥s t❤❡ ✜❡•❞s ♦r❣❛♥✐③❡ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ♠✉•t✐♣•❡ts ✿ ✶✻ ❋✐❡•❞ ❝♦♥t❡♥t ♦❢ t②♣❡ ■■ s✉♣❡r❣r❛✈✐t✐❡s ✐♥ ❢♦✉r ❞✐♠❡♥s✐♦♥s✳ ✷✳✷ ❼ ❣r❛✈✐t❛t✐♦♥❛• ♠✉•t✐♣•❡t (g µν , A 0 1 )✱ ❼ t❡♥s♦r ♠✉•t✐♣•❡t (B 2 , φ, ζ 0 , ζ0 )✳✱ ❼ h 2,1 ❤②♣❡r♠✉•t✐♣•❡ts (z a , ζ a , ζa ) ✱ ❼ h 1,1 ✈❡❝t♦r ♠✉•t✐♣•❡ts (A i 1 , v i = b i + it i )✳ A 3 ✐s ♥♦t ✐♥❝•✉❞❡❞ ❜❡❝❛✉s❡ ✐♥ ❢♦✉r ❞✐♠❡♥s✐♦♥s ✐t ❝❛♥ ❜❡ ❞✉❛•✐③❡❞ t♦ ❛ ❝♦♥st❛♥t ✇❤✐❝❤ ✇❡ ♣✉t t♦ ③❡r♦ ❛s ✐t ❞♦❡s ♥♦t ❝❛rr② ❛♥② ❞❡❣r❡❡ ♦❢ ❢r❡❡❞♦♠✳ ❍♦✇❡✈❡r ✐♥ ❬✺( ✐t ✇❛s s❤♦✇♥ t❤❛t ✐t ❝❛♥ ♣•❛② ❛♥ ✐♠♣♦rt❛♥t r♦•❡ ❜② ✐♥❞✉❝✐♥❣ ❛ ❣❛✉❣❡ ❝❤❛r❣❡ ❢♦r t❤❡ ◆❙✲❛①✐♦♥ σ✱ ✇❤❡r❡ σ ❛♣♣❡❛rs ❞✉❡ t♦ ❞✉❛•✐③❛t✐♦♥ ♦❢ t❤❡ ✜❡•❞ B 2 ✳ ❆❢t❡r t❤✐s ❞✉❛•✐③❛t✐♦♥ t❤❡ t❡♥s♦r ♠✉•t✐♣•❡t ❣❡ts ❝♦♥✈❡rt❡❞ ✐♥t♦ ❛♥♦t❤❡r ❤②♣❡r♠✉•t✐♣•❡t ❛♥❞ t❤❡r❡ ❛r❡ ♥♦✇ h 2,1 + 1 ❤②♣❡r♠✉•t✐♣•❡ts ✐♥ t♦t❛•✳ ❚❤❡ ❤②♣❡r♠✉•t✐♣•❡t t❤❛t ❝♦♠❡s ❢r♦♠ t❤❡ ❞✉❛•✐③❛t✐♦♥ ♦❢ t❡♥s♦r ♠✉•t✐♣•❡t ✐s ❛•✇❛②s ♣r❡s❡♥t ❛♥❞ ✐s ❝❛••❡❞ t❤❡ ✉♥✐✈❡rs❛• ❤②♣❡r♠✉•t✐♣•❡t✳ ❲❤❡♥ t❤❡ ❝♦♠♣❛❝t✐✜❝❛t✐♦♥ ♠❛♥✐❢♦•❞ ✐s ❝❤♦s❡♥ t♦ ❜❡ ❛ r✐❣✐❞ ❈❛•❛❜✐✲ (❛✉ t❤r❡❡❢♦•❞✱ ❜② ✇❤✐❝❤ ♦♥❡ ♠❡❛♥s t❤❛t t❤❡r❡ ✐s ♥♦ ❝♦♠♣•❡① str✉❝t✉r❡ ♠♦❞✉•✐ ✐✳❡✳ h 2,1 = 0✱ t❤✐s ✐s t❤❡ ♦♥•② ❤②♣❡r♠✉•t✐♣•❡t t❤❛t r❡♠❛✐♥s ✐♥ t❤❡ s♣❡❝tr✉♠✳ ✷✳✷✳✷ ❚②♣❡ ■■❇ ❚❤❡ ❜♦s♦♥✐❝ s❡❝t♦r ♦❢ t②♣❡ ■■❇ s✉♣❡r❣r❛✈✐t② ✐♥ 10✲❞✐♠❡♥s✐♦♥s ✐s ❞❡s❝r✐❜❡❞ ❜② t❤❡ ❢♦••♦✇✐♥❣ ❛❝t✐♦♥ ❬✷( ✿ S IIA = 1 2 e -2 φ R * 1 + 4d φ ∧ * φ - 1 2 Ĥ3 ∧ * Ĥ3 -F1 ∧ * F1 + F3 ∧ * F3 + 1 2 F3 ∧ * F3 -Â4 ∧ d Ĥ3 ∧ d Â2 , ✭✷✳✷✶✮ ❛♥❞ Ĥ3 = d B2 , F1 = d Â0 , F3 = d Â2 -Â0 ∧ Ĥ3 F5 = d Â4 - 1 2 Â2 ∧ Ĥ3 + 1 2 B2 ∧ d Â2 . ✭✷✳✷✷✮ ▲❡t ✉s ✜♥❞ ♦✉t t❤❡ s✐t✉❛t✐♦♥ ❛❢t❡r ❝♦♠♣❛❝t✐✜❝❛t✐♦♥✳ ❼ ■♥ t❤❡ ◆❙✲◆❙ s❡❝t♦r (ĝ XY ) 10d → g µν , t i , z a 4d → g µν , t i , z a 4d-dual , ( B2 ) 10d → B 2 + b i ω i 4d → ψ, b i 4d-dual , φ10d → φ 4d → φ 4d-dual . ✭✷✳✷✸✮ ❼ ❚❤❡♥ ❢♦r t❤❡ || s❡❝t♦r✱ ( Â0 ) 10d → (c 0 ) 4d → (c 0 ) 4d-dual , ( Â2 ) 10d → A 2 + c i ω i 4d → c 0 , c i 4d-dual ( Â4 ) 10d → D i 2 ω i + Di ω i + A λ 1 α Λ + Ã1,Λ β Λ 4d → c i , A 0 1 , A a S = 1 2 R * 1 + S ❱▼ + S ❍▼ . ✭✷✳✷✻✮ ❚❤❡ ❢♦r♠ ♦❢ S ❱▼ ✐s ✇❡•• ❦♥♦✇♥✱ s❡❡ ❢♦r ❡①❛♠♣•❡ ❬✻(✳ ■t ✐s ❛s t❤❡ ❢♦••♦✇✐♥❣✱ S ❱▼ = - 1 2 Im N IJ F I ∧ * F J + 1 2 Re N IJ F I ∧ F J + K i j (v, v)dv i ∧ dv j . ✭✷✳✷✼✮ ❚❤❡ ❣r❛✈✐♣❤♦t♦♥ ✜❡•❞ str❡♥❣t❤ F 0 ❤❛s ❜❡❡♥ ❝♦♠❜✐♥❡❞ ✇✐t❤ t❤❡ ♦t❤❡r ❣❛✉❣❡ ✜❡•❞ str❡♥❣t❤s t♦ F I ✬s✳ N = 2 s✉♣❡rs②♠♠❡tr② ❞✐❝t❛t❡s t❤❡ ❢♦r♠s ♦❢ t❤❡ ♠❛tr✐❝❡s K i j ❛♥❞ N IJ ✳ ❇♦t❤ ♦❢ t❤❡♠ ❛r❡ ❞❡t❡r♠✐♥❡❞ ❜② ❛ ❤♦•♦♠♦r♣❤✐❝ ♣r❡♣♦t❡♥t✐❛• ♦♥ t❤❡ s♣❡❝✐❛• ❑ä❤•❡r ♠❛♥✐❢♦•❞ s♣❛♥♥❡❞ ❜② t❤❡ ♠♦❞✉•✐ v i ✳ ❚❤❡ ❢♦r♠❡r ✐s •✉st t❤❡ ♠❡tr✐❝ ✶✽ ▲♦✇ ❡♥❡r❣② ❡✛❡❝t✐✈❡ ❛❝t✐♦♥s✳ ✷✳✸ ♦♥ t❤✐s s♣❛❝❡ ❛♥❞ ✐s ❣✐✈❡♥ ❜② t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❑ä❤•❡r ♣♦t❡♥t✐❛•✳ ❚❤❡ •❛tt❡r ✐s ❞❡✜♥❡❞ ❛s N IJ (v, v) = FIJ -i N IK N JL v K v L N XY v X v Y , ✭✷✳✷✽✮ ✇❤❡r❡ N IJ = i(F IJ -FIJ ), v I = (1, v i )✳ ❚❤❡ ♠❛tr✐① N IJ ❤❛s ❛ ♣❤②s✐❝❛• ✐♥t❡r♣r❡t❛✲ t✐♦♥ ❛s ❛ ♠❛tr✐① ♦❢ ❣❛✉❣❡ ❝♦✉♣•✐♥❣s✳ ▼♦r❡♦✈❡r ✉♥•✐❦❡ N IJ ✱ ✇❤✐❝❤ ❤❛s ❛ ▲♦r❡♥t③✐❛♥ s✐❣♥❛t✉r❡✱ t❤❡ ♠❛tr✐① N IJ ✐s ♥❡❣❛t✐✈❡ ❞❡✜♥✐t❡ ❛♥❞ ❤❡♥❝❡ t❤❡ ❦✐♥❡t✐❝ t❡r♠ ✐s ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✱ ❛s ✐t ♠✉st ❜❡✳ ◆❡①t ✇❡ t✉r♥ t♦ t❤❡ ❍▼ ♣❛rt ♦❢ t❤❡ ❛❝t✐♦♥✳ ■t t❛❦❡s t❤❡ ❢♦r♠ ♦❢ ❛ ♥♦♥✲•✐♥❡❛r σ✲♠♦❞❡•✱ S ❍▼ = d 4 xg αβ (q)∂ µ q α ∂ µ q β . ✭✷✳✷✾✮ ❍❡r❡ α = 1, ..., 4n H ✱ ✇❤❡r❡ n H ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❍▼✬s ❛♥❞ q α ❞❡♥♦t❡s ❛•• ♦❢ t❤❡ s❝❛•❛rs✳ ❆❣❛✐♥ g αβ (q) ❝❛♥ ♥♦t ❜❡ ❛r❜✐tr❛r②✱ N = 2 s✉♣❡rs②♠♠❡tr② r❡str✐❝ts ✐t t♦ ❜❡ q✉❛t❡r♥✐♦♥ ❑ä❤•❡r ✭◗❑✮ ❬✼(✳ ❲❡ ✇✐•• ❞✐s❝✉ss ❛❜♦✉t t❤❡s❡ t②♣❡s ♦❢ ❣❡♦♠❡tr✐❡s ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✳ ❚❤✐s s❤♦✇s t❤❛t t❤❡ ▲❊❊❆ ♦❢ ❈❛•❛❜✐✲(❛✉ ❝♦♠♣❛❝t✐✜❝❛t✐♦♥s ♦❢ t②♣❡ ■■ str✐♥❣ t❤❡♦r✐❡s ✐s ❝♦♠♣•❡t❡•② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ ❱▼ ❛♥❞ ❍▼ ♠♦❞✉•✐ s♣❛❝❡s✳ ▼♦r❡♦✈❡r ❛t t❤❡ t✇♦ ❞❡r✐✈❛t✐✈❡ •❡✈❡• t❤❡ ❢✉•• ♠♦❞✉•✐ s♣❛❝❡ ❢❛❝t♦r✐③❡s ✱ M 4❞ = M ❱▼ × M ❍▼ , ✭✷✳✸✵✮ ✇❤❡r❡ M VM ❛♥❞ M HM ❛r❡ ✈❡❝t♦r ♠✉•t✐♣•❡t ❛♥❞ ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡s✱ r❡s♣❡❝t✐✈❡•②✳ ❙✐♥❝❡ t❤❡ ❛❜♦✈❡ st❛t❡♠❡♥ts ❛r❡ ❝♦♥s❡q✉❡♥❝❡s ♦❢ N = 2 s✉♣❡rs②♠♠❡✲ tr②✱ t❤❡② s❤♦✉•❞ ❝♦♥t✐♥✉❡ t♦ ❤♦•❞ ❡✈❡♥ ✐♥ ♣r❡s❡♥❝❡ ♦❢ ❛•• q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s✳ ❆s ♠❡♥t✐♦♥❡❞ ❡❛r•✐❡r✱ t❤❡ ❢♦r♠❡r ✐s ❛ s♣❡❝✐❛• ❑ä❤•❡r ✭❙❑✮ ♠❛♥✐❢♦•❞ ❛♥❞ t❤❡ •❛tt❡r ✐s ❛ ◗❑ ♠❛♥✐❢♦•❞✳ ❚❤❡ ❤♦•♦♠♦r♣❤✐❝ ♣r❡♣♦t❡♥t✐❛• ❞❡s❝r✐❜✐♥❣ t❤❡ ✈❡❝t♦r ♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡ ✐s tr❡❡✲•❡✈❡• ❡①❛❝t✱ ♠♦r❡ ♣r❡❝✐s❡•②✱ ✐t ❞♦❡s ♥♦t r❡❝❡✐✈❡ ❛♥② ❝♦rr❡❝t✐♦♥ ✐♥ g s ❛♥❞ ❝❛♥ ❜❡ ❡①♣•✐❝✐t•② ❝♦♠♣✉t❡❞✳ ■♥ ♣❛rt✐❝✉•❛r✱ ❢♦r t❤❡ t②♣❡ ■■❇ t❤❡♦r②✱ ✇❤❡r❡ t❤❡ ✈❡❝t♦r ♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ s♣❛❝❡ ♦❢ ❝♦♠♣•❡① str✉❝t✉r❡ ❞❡❢♦r♠❛✲ t✐♦♥s M VM = M C ( Ŷ)✱ t❤❡ ♣r❡♣♦t❡♥t✐❛• ✐s ❛•s♦ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ α ′ ✲❝♦rr❡❝t✐♦♥s✳ ❚❤❡r❡❢♦r❡ ✐♥ t❤✐s ❝❛s❡ t❤❡ ❱▼✬s ♦❜t❛✐♥❡❞ ❜② ❑❛•✉③❛✲❑•❡✐♥ r❡❞✉❝t✐♦♥ ✐s ❡①❛❝t ❛♥❞ ❤❡♥❝❡ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✭✷✳✹✮✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t❤❡ M VM ✐♥ t❤❡ t②♣❡ ■■❆ s✐❞❡ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ♠♦❞✉•✐ s♣❛❝❡ ♦❢ ❝♦♠♣•❡①✐✜❡❞ ❑ä❤•❡r ❝•❛ss ❞❡❢♦r♠❛t✐♦♥s M VM = M K (Y)✳ ❍♦✇❡✈❡r✱ t❤❡ ♣r❡♣♦t❡♥t✐❛• ✭✷✳✶✺✮ ✐s ♦♥•② t❤❡ •❛r❣❡ ✈♦•✉♠❡ ❛♣♣r♦①✐♠❛t✐♦♥✳ ■ts t❤❡ ❝♦rr❡❝t ❢♦r♠ ❝❛♥ ❜❡ ❞❡r✐✈❡❞ ❜② ❡①♣•♦✐t✐♥❣ ♠✐rr♦r s②♠♠❡tr②✳ ■♥ ♣❛rt✐❝✉•❛r✱ s✐♥❝❡ t❤❡ ♠♦❞✉•✐ s♣❛❝❡s ♦❢ t②♣❡ ■■❆ ❛♥❞ t②♣❡ ■■❇ t❤❡♦r✐❡s s❤♦✉•❞ ❝♦✐♥❝✐❞❡ ♣r♦✈✐❞❡❞ t❤❡② ❛r❡ ❝♦♠♣❛❝t✐✜❡❞ ♦♥ ❛ ♠✐rr♦r ♣❛✐r ♦❢ ❈❛•❛❜✐✲(❛✉s✱ ✇❡ ❤❛✈❡ M A/B ❱▼ (Y) = M B/A ❱▼ ( Ŷ), M A/B ❍▼ (Y) = M B/A ❍▼ ( Ŷ). ✭✷✳✸✶✮ ✶✾ ✷✳ ❈♦♠♣❛❝t✐✜❝❛t✐♦♥ ♦❢ t②♣❡ ■■ str✐♥❣ t❤❡♦r✐❡s ♦♥ ❈❛•❛❜✐✲(❛✉ t❤r❡❡❢♦•❞s ❛♥❞ t❤❡✐r ♠♦❞✉•✐ s♣❛❝❡s ❚❤❡r❡❢♦r❡✱ t❤❡ ♣r❡♣♦t❡♥t✐❛• ♦♥ t❤❡ t②♣❡ ■■❆ s✐❞❡ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❢r♦♠ t❤❡ ❡①❛❝t ♣r❡♣♦t❡♥t✐❛• ♦♥ t❤❡ t②♣❡ ■■❇ s✐❞❡✳ ❚❤❡ r❡s✉•t ♦❢ s✉❝❤ ❝♦♠♣✉t❛t✐♦♥ t✉r♥s ♦✉t t♦ ❜❡ F (X) = -κ ijk X i X j X k 6X 0 + A IJ X I X J + χ Y ζ(3)(X 0 ) 2 2(2πi) 3 - (X 0 ) 2 (2πi) 3 k i γ i ∈H + 2 (Y) n (0) k i Li 3 (e 2πik i X i /X 0 ). ✭✷✳✸✷✮ ❚❤❡ •❛st t✇♦ t❡r♠s ❛r✐s❡ ❞✉❡ t♦ α ′ ✲❝♦rr❡❝t✐♦♥s ❛♥❞ ❝❛♥ ♥♦t ❜❡ ❞❡r✐✈❡❞ ❜② ❑❛•✉③❛✲ ❑•❡✐♥ r❡❞✉❝t✐♦♥s✳ ❚❤❡ t❡r♠ ✐♥✈♦•✈✐♥❣ ζ(3) ❝♦♠❡s ❢r♦♠ ♣❡rt✉r❜❛t✐✈❡ ❝♦rr❡❝t✐♦♥s ❞✉❡ t♦ α ′ ❛♥❞ t❤❡ •❛st t❡r♠ ❝♦♠❡s ❢r♦♠ ❝♦rr❡❝t✐♦♥s ❞✉❡ t♦ ✇♦r•❞s❤❡❡t ✐♥st❛♥t♦♥s✳ ❚❤❡ Li 3 ❛r✐s❡s ❞✉❡ t♦ ♠✉•t✐❝♦✈❡r✐♥❣ ❡✛❡❝ts ❛♥❞ n (0) k i ❛r❡ ❣❡♥✉s ③❡r♦ •♦♣❛❦✉♠❛r✲❱❛❢❛ ✭•❱✮ ✐♥✈❛r✐❛♥ts✳ ❚❤❡ ♣r♦❝❡❞✉r❡ ✐s ❞❡s❝r✐❜❡❞ ✐♥ ❬✽✱ ✾(✳ ❚❤❡ s❡❝♦♥❞ t❡r♠ ✇❤✐❝❤ ✐s ❛ q✉❛❞r❛t✐❝ ♦♥❡ ✇✐t❤ r❡❛• s②♠♠❡tr✐❝ ♠❛tr✐① A IJ ✱ ✐s ✉s✉❛••② ♦♠✐tt❡❞ ✐♥ t❤❡ •✐t❡r❛t✉r❡s ❜❡❝❛✉s❡ ✐t ❞♦❡s ♥♦t ❛✛❡❝t t❤❡ ❑ä❤•❡r ♣♦t❡♥t✐❛•✳ ❍♦✇❡✈❡r ✐♥ •❛t❡r st❛❣❡s ✐t ✇✐•• ❜❡ ❝r✉❝✐❛• ❢♦r ♦✉r ❞✐s❝✉ss✐♦♥ ❛♥❞ ✇❡ ✐♥❝•✉❞❡ ✐t ❢r♦♠ t❤❡ ✈❡r② ❜❡❣✐♥♥✐♥❣✳ ❚❤❡ ✈❡❝t♦r ♠✉•t✐♣•❡t s✐❞❡ ✐s ❝♦♠♣•❡t❡•② ✉♥❞❡r ❝♦♥tr♦• ♦♥❝❡ t❤❡ ♣r❡♣♦t❡♥t✐❛• ✐s ❞❡t❡r♠✐♥❡❞✱ ❜❡❝❛✉s❡ t❤❡ ♠♦❞✉•✐ s♣❛❝❡ ✐s ❛ s♣❡❝✐❛• ❑á❤•❡r ♠❛♥✐❢♦•❞s✱ ❛s ❤❛❞ ❜❡❡♥ ♠❡♥t✐♦♥❡❞ ❜❡❢♦r❡✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t❤❡ s✐t✉❛t✐♦♥ ❢♦r t❤❡ ❤②♣❡r♠✉•t✐♣•❡ts ✐s ♥♦t ❛s s✐♠♣•❡✳ M HM ❝❛♥ ❤❛✈❡ ❝♦rr❡❝t✐♦♥s ✐♥ g s ❬✶✵( ❛♥❞ ❢✉rt❤❡r♠♦r❡✱ t❤❡ ◗❑ ❣❡♦♠❡tr② ✐s ♠✉❝❤ ♠♦r❡ ❝♦♠♣•✐❝❛t❡❞✳ ■♥ ❝❤❛♣t❡r ✸✱ ✇❡ ✇✐•• ❛❞❞r❡ss t❤✐s ✐ss✉❡ ✉s✐♥❣ t✇✐st♦r t❡❝❤♥✐q✉❡s✳ ✷✳✹ ❍②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡ ❚❤✐s s❡❝t✐♦♥ ✐s ❞❡✈♦t❡❞ t♦ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❝•❛ss✐❝❛• ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ❛♥❞ ✐ts s②♠✲ ♠❡tr✐❡s✳ ❲❡ ✇✐•• ❛•s♦ ❞❡s❝r✐❜❡ ❤♦✇ t❤❡ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ❜r❡❛❦ t❤❡ ❝♦♥t✐♥✉♦✉s s②♠♠❡tr✐❡s t♦ ❞✐s❝r❡t❡ ♦♥❡s✳ ✷✳✹✳✶ ❈•❛ss✐❝❛• ❤②♣❡r♠✉•t✐♣•❡t ♠❡tr✐❝ M HM ❛•✇❛②s ❝♦♥t❛✐♥s ✜❡•❞ ♣❛r❛♠❡tr✐③✐♥❣ ❡✐t❤❡r M C ♦r M K ❞❡♣❡♥❞✐♥❣ ♦♥ ✇❤❡t❤❡r ✇❡ ❛r❡ ♦♥ t②♣❡ ■■❆ ♦r t②♣❡ ■■❇ s✐❞❡ r❡s♣❡❝t✐✈❡•②✳ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ❝•❛ss✐❝❛• ♠✐r✲ r♦r s②♠♠❡tr② t❤❡ ♠❡tr✐❝s ♦❢ ❜♦t❤ s✐❞❡s s❤♦✉•❞ ❝♦✐♥❝✐❞❡✳ ■t ✐s ❛❝t✉❛••② ❦♥♦✇♥ t❤❛t t♦ ❡✈❡r② ♣r♦•❡❝t✐✈❡ ✭•♦❝❛•✮ s♣❡❝✐❛• ❑ä❤•❡r ♠❛♥✐❢♦•❞ ♦♥❡ ❝❛♥ ❛ss♦❝✐❛t❡ ❛ q✉❛t❡r♥✐♦♥ ❑ä❤•❡r ♠❛♥✐❢♦•❞ ❬✻✱ ✶✷(✱ s♦ t❤❛t t❤❡ •❛tt❡r ❛r❡ st✐•• ❞❡s❝r✐❜❡❞ ♦♥•② ❜② ♣r❡♣♦t❡♥t✐❛•✳ ❚❤✐s ❝♦♥str✉❝t✐♦♥ r❡❛•✐③❡s t❤❡ ❝•❛ss✐❝❛• ♠❡tr✐❝ ♦♥ M HM ✳ ❚❤❡ ♣r♦❝❡❞✉r❡ ✐s ❦♥♦✇♥ ❛s ❝✲♠❛♣ ❛♥❞ ❝♦rr❡s♣♦♥❞✐♥❣•② t❤❡ ♠❡tr✐❝ ✭✷✳✸✺✮ ✐s s❛✐❞ t♦ ❜❡ ✐♥ t❤❡ ✐♠❛❣❡ ♦❢ ✐t✳ ❲❡ ♥♦✇ t✉r♥ t♦ ✉♥❞❡rst❛♥❞✐♥❣ t❤❡ ♣❤②s✐❝❛• ♦r✐❣✐♥ ♦❢ t❤✐s ❝✲♠❛♣✳ ▲❡t ✉s ❝♦♥s✐❞❡r t②♣❡ ■■❆ ❛♥❞ t②♣❡ ■■❇ t❤❡♦r✐❡s ❝♦♠♣❛❝t✐✜❡❞ ♦♥ t❤❡ s❛♠❡ ❈❛•❛❜✐✲(❛✉ t❤r❡❡❢♦•❞ t✐♠❡s ❛ ❝✐r❝•❡✳ ❚✲❞✉❛•✐t② ❛•♦♥❣ t❤❡ ❝✐r❝•❡ ✐♠♣•✐❡s t❤❛t IIA/(Y × S 1 R ) ∼ = IIB/(Y × S 1 1/R ). ✭✷✳✸✸✮ ✷✵ ❍②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡✳ ✷✳✹ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t❤❡✐r •♦✇ ❡♥❡r❣② ❞❡s❝r✐♣t✐♦♥s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ✐❢ ♦♥❡ ❝♦♠♣❛❝t✲ ✐✜❡s t❤❡ ❢♦✉r ❞✐♠❡♥s✐♦♥❛• t❤❡♦r② ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s s✉❜s❡❝t✐♦♥ ♦♥ ❛ ❝✐r❝•❡✳ ❚❤✐s •❡❛❞s t♦ t✇♦ ♠❛ss•❡ss s❝❛•❛rs ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❡❛❝❤ ♦❢ t❤❡ ❣❛✉❣❡ ✜❡•❞s✳ ❖♥❡ ♦❢ t❤❡♠ ❝♦♠❡s ❢r♦♠ t❤❡ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❣❛✉❣❡ ✜❡•❞ ❛•♦♥❣ t❤❡ ❝✐r❝•❡ ❛♥❞ t❤❡ ♦t❤❡r ❛♣♣❡❛rs ❜② ❞✉❛•✐③❛t✐♦♥ ♦❢ t❤❡ r❡♠❛✐♥✐♥❣ t❤r❡❡ ❞✐♠❡♥s✐♦♥❛• ❝♦♠♣♦♥❡♥t ♦❢ t❤❛t ❣❛✉❣❡ ✜❡•❞✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✐t ♠❡❛♥s t❤❛t ❛ ✈❡❝t♦r ♠✉•t✐♣•❡t ✐♥ ❢♦✉r ❞✐♠❡♥s✐♦♥s •❡❛❞s t♦ ❛ ❤②✲ ♣❡r♠✉•t✐♣•❡t ✐♥ 3d ❛❞❞✐t✐♦♥❛••② t♦ t❤❡ ❤②♣❡r♠✉•t✐♣•❡t ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ♠❡tr✐❝✳ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t❤r❡❡ ❞✐♠❡♥s✐♦♥❛• ❡✛❡❝t✐✈❡ ❛❝t✐♦♥ ✐s t❤❡♥ ❛ ♥♦♥✲•✐♥❡❛r σ✲♠♦❞❡• ✇❤♦s❡ t❛r❣❡t s♣❛❝❡ ✐s ❛ ♣r♦❞✉❝t ♦❢ t✇♦ ◗❑ s♣❛❝❡s✳ ❋♦r ❡✐t❤❡r ♦❢ t②♣❡ ■■❆ ♦r ■■❇ t❤❡♦r✐❡s t❤❡ ❢✉•• t❤r❡❡ ❞✐♠❡♥s✐♦♥❛• ♠♦❞✉•✐ s♣❛❝❡ ✐s ❣✐✈❡♥ ❜②✱ M 3d = Q C (Y) × Q K (Y), ✭✷✳✸✹✮ ✇❤✐❝❤ ❛r❡ ❡①t❡♥s✐♦♥s ♦❢ ❝♦♠♣•❡① str✉❝t✉r❡ ❛♥❞ ❝♦♠♣•❡①✐✜❡❞ ❑á❤•❡r ❝•❛ss ♠♦❞✉•✐ s♣❛❝❡s ♦❢ t❤❡ ❈❛•❛❜✐✲(❛✉✱ r❡s♣❡❝t✐✈❡•②✳ ❖♥❡ ♦❢ t❤❡ ❢❛❝t♦r ❝♦♠❡ ❢r♦♠ t❤❡ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ M HM ❛♥❞ t❤❡ ♦t❤❡r ❝♦♠❡s ❢r♦♠ ✈❡❝t♦r ❛♥❞ ❣r❛✈✐t❛t✐♦♥❛• ♠✉•t✐♣•❡t ✐♥ t❤❡ ✇❛② ✇❤✐❝❤ ✇❡ ❤❛✈❡ •✉st ❞❡s❝r✐❜❡❞ ❛❜♦✈❡✳ ❚❤❡♥ ✭✷✳✸✸✮ s✐♠♣•② ❡①❝❤❛♥❣❡s t❤❡ t✇♦ ❢❛❝t♦rs✳ ❆t t❤❡ ❝•❛ss✐❝❛• •❡✈❡•✱ ✇❤❡r❡ ♦♥❡ ✐❣♥♦r❡s ✐♥st❛♥t♦♥ ❝♦rr❡❝t✐♦♥s ❝♦♠✐♥❣ ❢r♦♠ t❤❡ ✇✐♥❞✐♥❣ ♠♦❞❡s ♦❢ t❤❡ ❝✐r❝•❡✱ t❤❡ ❑❛•✉③❛✲❑•❡✐♥ r❡❞✉❝t✐♦♥ ✐s s✉✣❝✐❡♥t✳ ❚❤❡♥ ✐t ✐s ♣♦ss✐❜•❡ t♦ ✜♥❞ t❤❡ ♠❡tr✐❝ ♦♥ M A/B HM ❢r♦♠ t❤❡ ♠❡tr✐❝ ♦♥ M B/A VM ✱ ❡①♣•❛✐♥✐♥❣ ✇❤② t❤❡ ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡s ❛r❡ st✐•• ❣✐✈❡♥ ❜② t❤❡ s❛♠❡ ❤♦•♦♠♦r♣❤✐❝ ♣r❡♣♦t❡♥t✐❛•s✳ ■♥ t❤❡ ❢♦••♦✇✐♥❣ ✇❡ ♣r❡s❡♥t t❤❡ ❝✲♠❛♣ ♠❡tr✐❝ ✉s✐♥❣ t❤❡ ✜❡•❞s ❛❞❛♣t❡❞ t♦ t❤❡ t②♣❡ ■■❆ s✐❞❡ ♦♥•②✳ ❚♦ ✜♥❞ t❤❡ ♠❡tr✐❝ ♦♥ t❤❡ t②♣❡ ■■❇ s✐❞❡✱ ✇❤❛t ♦♥❡ ♥❡❡❞s ✐s ❛ s✐♠♣•❡ ❛♣♣•✐❝❛t✐♦♥ ♦❢ ❝•❛ss✐❝❛• ♠✐rr♦r s②♠♠❡tr② ✇❤✐❝❤ ✇❡ ✇✐•• ❡①♣•❛✐♥ ❡•❛❜♦r❛t❡•② ✐♥ ❝❤❛♣t❡r ✺✳ ❚❤❡ r❡s✉•t ♦♥ t❤❡ t②♣❡ ■■❆ s✐❞❡ r❡❛❞s ❛s ❢♦••♦✇s✱ ds 2 = dr 2 r 2 - 1 2r ( Im N ) ΛΣ d ζΛ -N ΛΛ ′ dζ Λ ′ d ζΣ -N ΣΣ ′ dζ Σ ′ + 1 16r 2 dσ + ζΛ dζ Λ -ζ Λ d ζΛ 2 + 4K a bdz a dz b, ✭✷✳✸✺✮ ✇❤❡r❡ r = e φ ∝ 1/g 2 (4) ❛♥❞ t❤❡ ❑ä❤•❡r ♣♦t❡♥t✐❛• ✐s ❣✐✈❡♥ ✐♥ t❡r♠s ♦❢ ✭✷✳✹✮✳ ▲❡t ✉s ♣r❡s❡♥t t❤❡ t♦♣♦•♦❣✐❝❛• str✉❝t✉r❡ ♦❢ t❤❡ s♣❛❝❡ ❞❡s❝r✐❜❡❞ ❜② t❤❡ ❝•❛ss✐❝❛• ♠❡tr✐❝ ♣r❡s❡♥t❡❞ ❛❜♦✈❡✳ ❖♥ t♦♣ ♦❢ t❤❡ ❝♦♠♣•❡① str✉❝t✉r❡ ♠♦❞✉•✐ s♣❛❝❡ M C (Y) ♣❛r❛♠❡tr✐③❡❞ ❜② z a ✱ t❤❡r❡ ✐s ❛ t♦r✉s ❜✉♥❞•❡ ♦♥ t❤❡ ✐♥t❡r♠❡❞✐❛t❡ ❏❛❝♦❜✐❛♥✱ J C (Y) ✇✐t❤ t❤❡ ✜❜❡r ♣❛r❛♠❡tr✐③❡❞ ❜② t❤❡ || s❝❛•❛rs✱ ❛s T ζ, ζ = H 3 (Y, R)/H 3 (Y, Z)✳ ❖♥❡ ❝❛♥ ❝❤♦♦s❡ •r✐✣t❤s ♦r ❲❡✐• ❝♦♠♣•❡① str✉❝t✉r❡ ♦♥ t❤❡ ✐♥t❡r♠❡❞✐❛t❡ ❏❛❝♦❜✐❛♥✱ ✇❤❡r❡ t❤❡ ♣❡r✐♦❞s ❛r❡ r❡•❛t❡❞ ❜② ✭✷✳✷✽✮✳ ❖♥ t♦♣ ♦❢ ✐t✱ t❤❡r❡ ✐s st✐•• ❛ ❝✐r❝•❡ ❜✉♥❞•❡ ♣❛r❛♠❡tr✐③❡❞ ❜② t❤❡ ◆❙✲❛①✐♦♥ σ ❛♥❞ t❤❡ ❞✐•❛t♦♥ ❛♣♣❡❛rs ❛s ❛♥ ♦✈❡r❛•• ❢❛❝t♦r✳ ❚❤❡ ✷✶ ✷✳ ❈♦♠♣❛❝t✐✜❝❛t✐♦♥ ♦❢ t②♣❡ ■■ str✐♥❣ t❤❡♦r✐❡s ♦♥ ❈❛•❛❜✐✲(❛✉ t❤r❡❡❢♦•❞s ❛♥❞ t❤❡✐r ♠♦❞✉•✐ s♣❛❝❡s str✉❝t✉r❡ t❤❡♥ ✐s t❤❛t ♦❢ ❛ t✇♦✲st❛❣❡❞ ✜❜r❛t✐♦♥✱ R + r ×       S 1 σ -→ C(r) ↓ T ζ, ζ -→ J c (Y) ↓ M C (Y)       . ✭✷✳✸✻✮ ✷✳✹✳✷ ❙②♠♠❡tr✐❡s ♦❢ t❤❡ ❝•❛ss✐❝❛• ♠❡tr✐❝ ❇❡❢♦r❡ ✇❡ ❡♠❜❛r❦ ♦♥ t♦ t❤❡ st✉❞② ♦❢ t❤❡ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s t♦ t❤❡ ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡✱ ✐t ✐s ✐♠♣♦rt❛♥t t♦ ♣♦♥❞❡r ♦✈❡r t❤❡ s②♠♠❡tr✐❡s ❛t t❤❡ ❝•❛ss✐❝❛• •❡✈❡•✳ ■♥ t❤❡ ❛❜s❡♥❝❡ ♦❢ ❡①♣•✐❝✐t ♠✐❝r♦s❝♦♣✐❝ ❝♦♠♣✉t❛t✐♦♥s✱ t❤❡s❡ s②♠♠❡tr✐❡s ❛r❡ t❤❡ ♦♥•② t♦♦•s t❤❛t ❛••♦✇ ✉s t♦ ❛❝❝❡ss t❤❡ q✉❛♥t✉♠ ❝♦rr❡❝t❡❞ ♠❡tr✐❝✳ ❆t t❤✐s st❛❣❡✱ •❡t ✉s r❡♠✐♥❞ t❤❡ r❡❛❞❡r ❢♦r ❝♦♥✈❡♥✐❡♥❝❡ t❤❡ ✜❡•❞ ❝♦♥t❡♥t ✐♥ t❤❡ ❤②♣❡r♠✉•t✐♣•❡t s❡❝t♦rs ♦❢ t❤❡ ❝❛s❡s ❢♦r t②♣❡ ■■❆ ❛♥❞ t②♣❡ ■■❇ t❤❡♦r✐❡s✳
Type IIA : z a , φ, σ and ζ Λ , ζΛ . k,p + 2π 2 (h

Type IIB : v i ≡ b i + it i , φ, ψ and c 0 , c i , c i , c 0 . ✭✷✳✸✼✮ ❼ ❙②♠♣•❡❝t✐❝ ✐♥✈❛r✐❛♥❝❡ ❚②♣❡ ■■❆ ❝♦♥str✉❝t✐♦♥ ❤❛s ♠❛♥✐❢❡st s②♠♣•❡❝t✐❝ ✐♥✈❛r✐❛♥❝❡ ❜❡❝❛✉s❡ ❛•• t❤❡ ❢♦✉r ❞✐♠❡♥s✐♦♥❛• ✜❡•❞s ❛r❡ ❞❡✜♥❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ❛ s②♠♣•❡❝t✐❝ ❜❛s✐s ♦❢ ✸✲ ❝②❝•❡s (A Λ , B Λ )✳ ❆•t❤♦✉❣❤ ✭✷✳✸✺✮ ✐s ✇r✐tt❡♥ ❞♦✇♥ ✇❤❡♥ ❛ ♣❛rt✐❝✉•❛r ❝❤♦✐❝❡ ♦❢ s✉❝❤ ❝②❝•❡s ✐s ♠❛❞❡✱ t❤❡r❡ ✐s ♥♦ ❝❛♥♦♥✐❝❛• ✇❛② ♦❢ ❞♦✐♥❣ ✐t✳ ■t ✐s ❛•✇❛②s ♣♦s✲ s✐❜•❡ t♦ ♣❡r❢♦r♠ ❛ s②♠♣•❡❝t✐❝ r♦t❛t✐♦♥ ❜② ❛ ♠❛tr✐① O ∈ Sp(2h 2,1 + 2, Z) ❛♥❞ ❛s t❤❡ ♣❤②s✐❝❛• r❡s✉•ts ♠✉st ♥♦t ❜❡ ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ s②♠♣•❡❝t✐❝ ♠❛r❦✐♥❣✱ ✐♥ ♣❛rt✐❝✉•❛r t❤❡ ♠❡tr✐❝✱ t❤❡② s❤♦✉•❞ ❜❡ ♠❛♥✐❢❡st•② ✐♥✈❛r✐❛♥t ✉♥❞❡r s✉❝❤ tr❛♥s✲ ❢♦r♠❛t✐♦♥s✳ ❚❤❡ s②♠♣•❡❝t✐❝ r♦t❛t✐♦♥ ❛❝ts ❛s t❤❡ ❢♦••♦✇✐♥❣ ♦♥ ❛•• s②♠♣•❡❝t✐❝ ✈❡❝t♦rs✱ O : A Λ B Λ → a b c d A Λ B Λ , ✭✷✳✸✽✮ ✇❤❡r❡ a, b, c, d ❛r❡ (h 2,1 + 1) × (h 2,1 + 1) ♠❛tr✐❝❡s ♦❜❡②✐♥❣ a T c -c T a = 0 = b T d -d T b, a T d -c T b = 1. ✭✷✳✸✾✮ ❚❤❡ s❝❛•❛rs r ❛♥❞ σ ❛r❡ ✐♥✈❛r✐❛♥t ✉♥❞❡r s②♠♣•❡❝t✐❝ tr❛♥s❢♦r♠❛t✐♦♥✳ ❚♦ s❡❡ t❤❛t t❤❡ ♠❡tr✐❝ ✐s s②♠♣•❡❝t✐❝❛••② ✐♥✈❛r✐❛♥t✱ •❡t ✉s st❛rt ❜② ❝♦♥s✐❞❡r✐♥❣ t❤❡ s❡❝♦♥❞ t❡r♠✳ ◆♦✇ ♥♦t✐❝✐♥❣ ❬✶✸(✱ N → (c + dN )(a + bN ) -1 , Im N → (a + bN ) -T Im N (a + b N ) -1 , ✭✷✳✹✵✮ ✷✷ ❍②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡✳ ✷✳✹ ✇❡ ✜♥❞ t❤❛t t❤✐s t❡r♠ ✐s ✐♥❞❡❡❞ ✐♥✈❛r✐❛♥t✳ ❚❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ z a ✐s ✐♥❞✉❝❡❞ ❜② s②♠♣•❡❝t✐❝ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ✈❡❝✲ t♦r (X Λ , F Λ )✱ ❜✉✐•t ❢r♦♠ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❝♦♦r❞✐♥❛t❡ ❛♥❞ t❤❡ ♣r❡♣♦t❡♥t✐❛•✳ ❚❤✐s ❢❛❝t ❛••♦✇s ♦♥❡ ✈❡r✐❢② ❡①♣•✐❝✐t•② t❤❛t t❤❡ •❛st t❡r♠ ✐♥ ✭✷✳✸✺✮ ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r s②♠♣•❡❝t✐❝ tr❛♥s❢♦r♠❛t✐♦♥s✳ ❆•t♦❣❡t❤❡r✱ t❤❡s❡ ❡♥s✉r❡ t❤❛t t❤❡ ♠❡tr✐❝ ✐s s②♠♣•❡❝t✐❝❛••② ✐♥✈❛r✐❛♥t✳ ❼ ❙✲❞✉❛•✐t② ▼♦r❡♦✈❡r t❤❡ t②♣❡ ■■❇ str✐♥❣ t❤❡♦r② ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r ❙✲❞✉❛•✐t② ❬✶✹(✳ ■♥ ❛❜s❡♥❝❡ ♦❢ α ′ ♦r g s ✲❝♦rr❡❝t✐♦♥s ❛t •❛r❣❡ ✈♦•✉♠❡ ❛♥❞ s♠❛•• str✐♥❣ ❝♦✉♣•✐♥❣ •✐♠✐t✱ t❤✐s s②♠♠❡tr② ✐s r❡❛•✐③❡❞ ❜② ❝♦♥t✐♥✉♦✉s ❛❝t✐♦♥ ♦❢ SL(2, R) ❣r♦✉♣✳ ■ts ❛❝t✐♦♥ ♦♥ t❤❡ ❝♦♦r❞✐♥❛t❡s ❛❞❛♣t❡❞ t♦ t❤❡ t②♣❡ ■■❇ s✐❞❡ ✭❢♦r ♠❛♥✐❢❡st ✐♥✈❛r✐❛♥❝❡ ✉♥❞❡r SL(2, R)✮ ✐s✱ SL(2, R) ∋ g : τ → aτ + b cτ + d , t a → t a |cτ + d| , c a → c a , c a b a → a b c d c a b a , c 0 ψ → d -c -b a c 0 ψ , ✭✷✳✹✶✮ ✇✐t❤ ad -bc = 1✱ ✇❤❡r❡ t❤❡ r❡•❛t✐♦♥s ❜❡t✇❡❡♥ ✜❡•❞s ✐♥ t②♣❡ ■■❆ ❛♥❞ t②♣❡ ■■❇ ❛r❡ ❣✐✈❡♥ ❜② ✶ ✱ z a = b a + it a , ζ 0 = τ 1 , ζ a = -(c a -τ 1 b a ) , ζa = c a + 1 2 κ abc b b (c c -τ 1 b c ) , ζ0 = c 0 - 1 6 κ abc b a b b (c c -τ 1 b c ) , σ = -2(ψ + 1 2 τ 1 c 0 ) + c a (c a -τ 1 b a ) - 1 6 κ abc b a c b (c c -τ 1 b c ) , ✭✷✳✹✷✮ ❍♦✇❡✈❡r✱ ❛❢t❡r ✐♥❝♦r♣♦r❛t✐♦♥ ♦❢ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ♦♥❡ ❡①♣❡❝ts ✐t t♦ ❜❡ ❜r♦❦❡♥ ❞♦✇♥ t♦ ❛ ❞✐s❝r❡t❡ s✉❜❣r♦✉♣ ♦❢ ❙▲✭✷✱ R✮✳ ❲❡ ✇✐•• ❣❡t ❜❛❝❦ t♦ ✐ts ❞✐s❝✉ss✐♦♥ ✐♥ ❝❤❛♣t❡r ✺✱ ❛♥❞ ✐t ✇✐•• ❜❡ ♦✉r ❣✉✐❞✐♥❣ ♣r✐♥❝✐♣•❡ ❢♦r ❞❡r✐✈✐♥❣ r❡s✉•ts ❢♦r t❤❡ ✏✜✈❡❜r❛♥❡ ✐♥st❛♥t♦♥s✧✳ ❼ ❍❡✐s❡♥❜❡r❣ s②♠♠❡tr② ❚❤❡ || s❝❛•❛rs ❛♥❞ t❤❡ ◆❙ s❝❛•❛r ✐♥ t❤❡ ♠❡tr✐❝ ✭✷✳✸✺✮ ❛r❡ ❛❝t✉❛••② ♦❜t❛✐♥❡❞ ❛s ❡✐t❤❡r ♣❡r✐♦❞s ♦r ❞✉❛•s t♦ ❣❛✉❣❡ ✜❡•❞s ♣r❡s❡♥t ✐♥ t❡♥ ❞✐♠❡♥s✐♦♥❛• s✉♣❡rstr✐♥❣ t❤❡♦r✐❡s✳ ❚❤✐s ✐s t❤❡ ♦r✐❣✐♥ ♦❢ t❤❡ ❍❡✐s❡♥❜❡r❣ s②♠♠❡tr②✱ ❛s ✐t ✐s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❣❛✉❣❡ s②♠♠❡tr✐❡s✳ ❬✶✺( ❞✐s❝✉ss❡s t❤✐s s✐t✉❛t✐♦♥ ❛♥❞ t❤❡② ❛r❡ ❡①❛♠♣•❡s ♦❢ t❤❡ s♦ ❝❛••❡❞ P❡❝❝❡✐✲◗✉✐♥♥ s②♠♠❡tr✐❡s✳ ■ts ❛❝t✐♦♥ ✐s ♠♦st ❝♦♥✈❡♥✐❡♥t•② r❡❛•✐③❡❞ ✐♥ t❡r♠s ♦❢ t②♣❡ ■■❆ ✜❡•❞s ✿ T H,κ : (ζ Λ , ζΛ , σ) → (ζ Λ + η Λ , ζΛ + ηΛ , σ + 2κ -ηΛ ζ Λ + η Λ ζΛ ). ✭✷✳✹✸✮ ✶ ❲❤❡♥ ✇❡ ✉s❡ a t♦ •❛❜❡• t❤❡ ✐♥❞✐❝❡s ♦❢ t②♣❡ ■■❇ ✜❡•❞s✱ ✇❡ t❛❝✐t•② ❛ss✉♠❡ t❤❛t ✇❡ ❛r❡ ✇♦r❦✐♥❣ ♦♥ t❤❡ ♠✐rr♦r ❈❛•❛❜✐✲(❛✉✱ s♦ t❤❛t h 1,1 ( Ŷ) = h 2,1 (Y)✱ ❛♥❞ ❤❡♥❝❡ ✐s ♥♦t ❛ ❝♦♥tr❛❞✐❝t♦r② ♥♦t❛t✐♦♥✳ ✷✸ ✷✳ ❈♦♠♣❛❝t✐✜❝❛t✐♦♥ ♦❢ t②♣❡ ■■ str✐♥❣ t❤❡♦r✐❡s ♦♥ ❈❛•❛❜✐✲(❛✉ t❤r❡❡❢♦•❞s ❛♥❞ t❤❡✐r ♠♦❞✉•✐ s♣❛❝❡s ❋✉rt❤❡r♠♦r❡✱ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ s✉❝❤ tr❛♥s❢♦r♠❛t✐♦♥s ✐s ❣✐✈❡♥ ❜② T H 1 ,κ 1 • T H 2 ,κ 2 = T H 1 +H 2 ,κ 1 +κ 2 + 1 2 H 1 ,H 2 . ✭✷✳✹✹✮ ❍❡r❡ H = (η Λ , ηΛ ) ✐s ❛ s②♠♣•❡❝t✐❝ ✈❡❝t♦r ❛♥❞ t❤❡ s②♠♣•❡❝t✐❝ ✐♥✈❛r✐❛♥t ✐♥♥❡r ♣r♦❞✉❝t ✐s ❣✐✈❡♥ ❜② H, H ′ = ηΛ η ′ Λ -η Λ η′ Λ . ❆t t❤✐s st❛❣❡ H ∈ R b 3 ❛♥❞ κ ∈ R✳ ■♥ t❤❡ ❢♦••♦✇✐♥❣ s❡❝t✐♦♥ ✇❡ ✇✐•• ❞✐s❝✉ss ❤♦✇ ♥♦♥✲♣❡rt✉r❜❛t✐✈❡ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ❞✉❡ t♦ ✐♥st❛♥t♦♥s ❜r❡❛❦ t❤❡♠ ❞♦✇♥ t♦ ✐♥t❡❣❡rs✳ ❼ ❙②♠♠❡tr② ❛r♦✉♥❞ t❤❡ •❛r❣❡ ✈♦•✉♠❡ ♣♦✐♥t ❚❤❡r❡ ✐s ❛♥♦t❤❡r ❦✐♥❞ ♦❢ P❡❝❝❡✐✲◗✉✐♥♥ s②♠♠❡tr② ✇❤✐❝❤ ❝❛♥ ❜❡ ❡❛s✐•② ✉♥❞❡r✲ st♦♦❞ ❢r♦♠ t❤❡ t②♣❡ ■■❇ s✐❞❡✳ ❲❤❡♥ t❤❡ s❝❛•❛rs b a ❞✉❛• t♦ t❤❡ t✇♦ ❢♦r♠ ✜❡•❞ B 2 ✐s ✐♥ t❤❡ ❤②♣❡r♠✉•t✐♣•❡r s❡❝t♦r✱ t❤✐s s②♠♠❡tr② ✐s ✐♥❤❡r✐t❡❞ ❢r♦♠ t❤❡ ❣❛✉❣❡ s②♠♠❡tr② ♦❢ t❤❡ •❛tt❡r✳ ■♥ t❤❡ ♣r❡♣♦t❡♥t✐❛• ✭✷✳✸✷✮ ♦♥❡ ❝❛♥ ❛❝t✉❛••② ❞r♦♣ t❤❡ •❛st t✇♦ t❡r♠s ✐♥ t❤❡ •❛r❣❡ ✈♦•✉♠❡ •✐♠✐t✱ ✇❤❡r❡ α ′ ✲❝♦rr❡❝t✐♦♥s ❛r❡ ❛❜s❡♥t✳ ■♥ t❤❛t ❝❛s❡ ✐t ✐s ❛ ❝♦♥t✐♥✉♦✉s s②♠♠❡tr②✱ ✇❤❡r❡❛s ✐♥ ♣r❡s❡♥❝❡ ♦❢ α ′ ✲❝♦rr❡❝t✐♦♥s ✐t ❜❡❝♦♠❡s ❞✐s❝r❡t❡✳ ❋✉rt❤❡r♠♦r❡ s✉❝❤ s❤✐❢t s❤♦✉•❞ ❜❡ ❛❝❝♦♠♣❛♥✐❡❞ ❜② tr❛♥s✲ ❢♦r♠❛t✐♦♥ ♦❢ t❤❡ || s❝❛•❛rs ❛s t❤❡ ❢♦••♦✇✐♥❣✱ M ǫ a : b a → b a + ǫ a , ζ a → ζ a + ǫ a ζ 0 , ζa → ζa -κ abc ζ b ǫ c - 1 2 κ abc ǫ b ǫ c ζ 0 , ζ0 → ζ0 -ζa ǫ a + 1 2 κ abc ζ a ǫ b ǫ c + 1 6 κ abc ǫ a ǫ b ǫ c ζ 0 . ✭✷✳✹✺✮ ❼ ▼✐rr♦r s②♠♠❡tr② ❆s ✇❡ ❤❛✈❡ ❛•r❡❛❞② s❡❡♥✱ t❤❡ ♠♦❞✉•✐ s♣❛❝❡s ♦❢ t②♣❡ ■■❆ ❛♥❞ t②♣❡ ■■❇ s✐❞❡s ❛r❡ ✐❞❡♥t✐❝❛•✱ ♣r♦✈✐❞❡❞ t❤❡② ❛r❡ ♦❜t❛✐♥❡❞ ❜② ❝♦♠♣❛❝t✜❝❛t✐♦♥ ♦♥ ♠✐rr♦r ❈❛•❛❜✐✲ (❛✉ t❤r❡❡❢♦•❞s✳ ❚❤❡s❡ ❞✐✛❡r❡♥t ❢❛❝❡ts ♦❢ t❤❡ s❛♠❡ t❤❡♦r② ❛r❡ ✉s❡❢✉• t♦ ❞❡❛• ✇✐t❤ ✈❛r✐♦✉s t②♣❡s ♦❢ s②♠♠❡tr✐❡s ✇❤✐❝❤ ❛r❡ ♠♦st ♥❛t✉r❛••② r❡❛•✐③❡❞ ✐♥ ♦♥❡ ♦r t❤❡ ♦t❤❡r✳ ❋♦r ❡①❛♠♣•❡✱ ✇❤❡r❡❛s ❙✲❞✉❛•✐t② ✐s ❛ ♠❛♥✐❢❡st s②♠♠❡tr② ✐♥ t❤❡ t②♣❡ ■■❇ s✐❞❡✱ t②♣❡ ■■❆ ❢♦r♠✉•❛t✐♦♥ ✐s ✐❞❡❛••② s✉✐t❡❞ ❢♦r s②♠♣•❡❝t✐❝ ✐♥✈❛r✐❛♥❝❡✳ ❚❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ ♠✐rr♦r s②♠♠❡tr② ✐s t❤❛t ✐t ❛••♦✇s t♦ ♣❛ss ❢r♦♠ ♦♥❡ t♦ t❤❡ ♦t❤❡r✳ ✷✳✺ ◗✉❛•✐t❛t✐✈❡ ♥❛t✉r❡ ♦❢ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥ ■♥ str✐♥❣ t❤❡♦r② t❤❡r❡ ❛r❡ t✇♦ s♦✉r❝❡s ♦❢ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s t♦ t❤❡ r❡s✉•ts ♦❜✲ t❛✐♥❡❞ ❜② ❑❛•✉③❛✲❑•❡✐♥ r❡❞✉❝t✐♦♥ ✿ t❤❡② ❝❛♥ ❝♦♠❡ ❢r♦♠ t✇♦ ❝♦✉♣•✐♥❣ ❝♦♥st❛♥ts✱ α ′ ❛♥❞ g s ✱ ❛♥❞ ❝❛♥ ❜❡ ❜♦t❤ ♣❡rt✉r❜❛t✐✈❡ ❛♥❞ ♥♦♥✲♣❡rt✉r❜❛t✐✈❡✳ ❚❤❡ α ′ ✲❝♦rr❡❝t✐♦♥s ❝❛♥ ❛♣♣❡❛r ♦♥•② ✇❤❡r❡ t❤❡ ❑ä❤•❡r ♠♦❞✉•✐ ♣r❡s❡♥t✱ ❢♦r ❡①❛♠♣•❡ t❤✐s ✐s t❤❡ r❡❛s♦♥ ✇❤② t②♣❡ ■■❆ ✈❡❝t♦r ♠✉•t✐♣•❡ts r❡❝❡✐✈❡ s✉❝❤ ❝♦rr❡❝t✐♦♥s✳ ❉✉❡ t♦ t❤❡ ❝✲♠❛♣ t❤❡ s✐t✉❛t✐♦♥ ✐s r❡✈❡rs❡❞ ❢♦r t❤❡ ❤②♣❡r♠✉•t✐♣•❡t s❡❝t♦r✳ ❲❤❡r❡❛s M HM ✐s α ′ ✲❡①❛❝t✱ ✐t r❡❝❡✐✈❡s ❝♦rr❡❝t✐♦♥s ❢r♦♠ t❤❡♠ ♦♥ t❤❡ t②♣❡ ■■❇ s✐❞❡✳ ❍♦✇❡✈❡r✱ ❢♦r ❜♦t❤ ♦❢ t❤❡s❡ ❝❛s❡s ✐♥ ✷✹ ◗✉❛•✐t❛t✐✈❡ ♥❛t✉r❡ ♦❢ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥✳ ✷✳✺ t❤❡ •♦✇❡st ♦r❞❡r ✐♥ g s t❤❡ ♠❡tr✐❝ ✐s ❞❡t❡r♠✐♥❡❞ ❝♦♠♣•❡t❡•② ❜② t❤❡ ♣r❡♣♦t❡♥t✐❛• ❜② ✭✷✳✸✺✮✳ ❚❤❡ g s ❝♦rr❡❝t✐♦♥s ❛♣♣❡❛r ✐♥ ❜♦t❤ M A HM ❛♥❞ M B HM ❛s t❤❡ ❞✐•❛t♦♥ ✐s ❛ ♣❛rt ♦❢ ❤②♣❡r♠✉•t✐♣•❡t ✐♥ ❜♦t❤ t②♣❡ ■■❆ ❛♥❞ t②♣❡ ■■❇ t❤❡♦r✐❡s✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞ ❢♦r t❤❡ ✈❡❝t♦r ♠✉•t✐♣•❡ts t❤❡ ♠❡tr✐❝ ✐s tr❡❡ •❡✈❡• ❡①❛❝t ✐♥ g s ✳ ❇❡•♦✇✱ ✇❡ ❞✐s❝✉ss q✉❛•✐t❛t✐✈❡ ❡✛❡❝ts ♦❢ t❤❡ g s ❝♦rr❡❝t✐♦♥s t♦ t❤❡ ❣❡♦♠❡tr② ♦❢ M HM ❜♦t❤ ♣❡rt✉r❜❛t✐✈❡•② ❛♥❞ ♥♦♥✲♣❡rt✉r❜❛t✐✈❡•② ❢♦r t❤❡ ❝❛s❡ ♦❢ ❤②♣❡r♠✉•t✐♣•❡ts✳ ❼ P❡rt✉r❜❛t✐✈❡ ❝♦rr❡❝t✐♦♥s P❡rt✉r❜❛t✐✈❡ ❝♦rr❡❝t✐♦♥s s❤♦✉•❞ ❜❡ ♦❜t❛✐♥❡❞ ❜② ❡①♣❛♥❞✐♥❣ ✐♥ ♣♦✇❡rs ♦❢ ❞✐•❛✲ t♦♥ ✇❤✐❝❤ ✐s t❤❡ ❝♦✉♥t✐♥❣ ♣❛r❛♠❡t❡r ❢♦r t❤❡ str✐♥❣ •♦♦♣ ❡①♣❛♥s✐♦♥✳ ■t ✐s ✐♠♣♦rt❛♥t t❤❛t ✐♥ t❤❡ ♣❡rt✉r❜❛t✐✈❡ ❛♣♣r♦①✐♠❛t✐♦♥ t❤❡ ♠❡tr✐❝ ♠✉st r❡t❛✐♥ t❤❡ ❝♦♥t✐♥✉♦✉s ❍❡✐s❡♥❜❡r❣ s②♠♠❡tr② ✭✷✳✹✸✮ ❬✶✻✱ ✶✼(✳✳ ❼ ◆♦♥✲♣❡rt✉r❜❛t✐✈❡ ❝♦rr❡❝t✐♦♥s ◆♦♥✲♣❡rt✉r❜❛t✐✈❡ ❝♦rr❡❝t✐♦♥s ❡♠❛♥❛t❡ ❢r♦♠ ❊✉❝•✐❞❡❛♥ ❜r❛♥❡s ✇r❛♣♣✐♥❣ ❞✐❢✲ ❢❡r❡♥t ♥♦♥✲❝♦♥tr❛❝t✐❜•❡ ❝②❝•❡s ♦❢ t❤❡ ❈❛•❛❜✐✲②❛✉ t❤r❡❡❢♦•❞ ❬✶✵(✳ ❙✉❝❤ ❝♦rr❡❝✲ t✐♦♥s ❛r❡ ❝❛••❡❞ ✐♥st❛♥t♦♥s ❜❡❝❛✉s❡ t❤❡s❡ ♦❜•❡❝ts s❡❡♠ t♦ ❜❡ ♣♦✐♥t✲•✐❦❡ ♦❜•❡❝ts ❢r♦♠ t❤❡ ❢♦✉r ❞✐♠❡♥s✐♦♥❛• ♣❡rs♣❡❝t✐✈❡✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ♦❢ str✐♥❣ t❤❡♦r② t✇♦ ♦❢ s✉❝❤ ♦❜ •❡❝ts ❣❡♥❡r❛t❡ ✐♥st❛♥t♦♥ ❝♦♥tr✐❜✉t✐♦♥s✱ ✕ ❚❤❡ ✜rst ♦❢ t❤❡♠ ❛r❡ s♦ ❝❛••❡❞ ❉✲✐♥st❛♥t♦♥ ❡✛❡❝ts ❝♦♠✐♥❣ ❢r♦♠ ❉✲❜r❛♥❡s✳ P❤②s✐❝❛••② t❤❡② ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❛r✐s✐♥❣ ❢r♦♠ ❉ p✲❜r❛♥❡s ✇✐t❤ (p + 1)✲❞✐♠❡♥s✐♦♥❛• ✇♦r•❞✈♦•✉♠❡✱ ✇r❛♣♣✐♥❣ ❛♣♣r♦♣r✐❛t❡ ❝②❝•❡s ✐♥ t❤❡ ❈❛•❛❜✐✲ (❛✉✳ ❋♦r t②♣❡ ■■❆✱ p ✐s ❡✈❡♥✱ ✇❤❡r❡❛s ❢♦r t②♣❡ ■■❇ p ✐s ♦❞❞✳ ❙✐♥❝❡ ❛ ❈❛•❛❜✐✲(❛✉ t❤r❡❡❢♦•❞ ❞♦❡s ♥♦t ❛❞♠✐t ❛♥② ♥♦♥✲tr✐✈✐❛• ♦♥❡ ♦r ✜✈❡ ❝②❝•❡✱ t❤❡r❡ ❛r❡ ♦♥•② ❉✷✲✐♥st❛♥t♦♥s ✐♥ t❤❡ t②♣❡ ■■❆ s✐❞❡ ✇r❛♣♣✐♥❣ ✸✲ ❞✐♠❡♥s✐♦♥❛• ❝②❝•❡s ✐♥ Y✳ ❚❤❡② ❛r❡ •❛❜❡•❡❞ ❜② t❤❡ ❝❤❛r❣❡ ✈❡❝t♦r γ = (p Λ , q Λ )✱ r❡♣r❡s❡♥t✐♥❣ t❤❡ ❝②❝•❡ ✇r❛♣♣❡❞✱ q Λ A Λ + p Λ B Λ ∈ H 3 (Y)✳ ■♥ t❤❡ ❝❛s❡ ♦❢ t②♣❡ ■■❇ s✐❞❡ ❛•• ✐♥st❛♥t♦♥s ✇r❛♣♣✐♥❣ ❞✐✛❡r❡♥t ❡✈❡♥✲ ❞✐♠❡♥s✐♦♥❛• ❝②❝•❡s ❛r❡ ♣r❡s❡♥t✳ ❚❤❡ ♦♥❡s t❤❛t ❝♦♠❡ ❢r♦♠ ✇r❛♣♣✐♥❣ 0 ❛♥❞ 2-❞✐♠❡♥s✐♦♥❛• ❝②❝•❡s ❛r❡ ❝❛••❡❞ ❉✭✲✶✮ ❛♥❞ ❉✶ ✐♥st❛♥t♦♥s✳ ❚❤r♦✉❣❤ ♠✐rr♦r s②♠♠❡tr② t❤❡② ❛r❡ r❡•❛t❡❞ t♦ t❤❡ ❆✲❉✷ ✐♥st❛♥t♦♥s ✭ q Λ = 0, p Λ = 0 ❝♦♠✐♥❣ ❢r♦♠ ❉✲❜r❛♥❡s ✇r❛♣♣✐♥❣ A✲❝②❝•❡s ✐♥ t②♣❡ ■■❆ s✐❞❡✮✳ ❚❤❡ ♦t❤❡r t✇♦ ❉✸ ❛♥❞ ❉✺ ✐♥st❛♥t♦♥s ❛r❡ ♠❛♣♣❡❞ t♦ t❤❡ ❇✲❉✷ ✐♥st❛♥t♦♥s ✭❝♦♠✐♥❣ ❢r♦♠ ❉✲❜r❛♥❡s ✇r❛♣♣✐♥❣ B✲❝②❝•❡s ✐♥ t②♣❡ ■■❆✮ ✇✐t❤ ♥♦♥✲✈❛♥✐s❤✐♥❣ p a ❛♥❞ p 0 r❡s♣❡❝t✐✈❡•②✳ ❙❝❤❡♠❛t✐❝❛••② t❤❡ ❉✲✐♥st❛♥t♦♥ ❝♦rr❡❝t✐♦♥s t❛❦❡ t❤❡ ❢♦••♦✇✐♥❣ ❢♦r♠s ❬✶✵(✱ δds 2 | ❉✲✐♥st ∼ Ω(γ; z) e -2π|Zγ |/gs-2πi(q Λ ζ Λ -p Λ ζΛ ) , ✭✷✳✹✻✮ ✇❤❡r❡ ✐t ❞❡♣❡♥❞s ♦♥ s♦♠❡ ♠♦❞✉•✐ ❞❡♣❡♥❞❡♥t ❢✉♥❝t✐♦♥s ❣✐✈✐♥❣ str❡♥❣t❤s t♦ t❤❡ ✐♥st❛♥t♦♥s✱ ❦♥♦✇♥ ❛s ❝❡♥tr❛• ❝❤❛r❣❡s Z γ ❛♥❞ ❛r❡ ❣✐✈❡♥ ❜② Z γ = q Λ z Λ -p Λ F Λ . ✭✷✳✹✼✮ ✷✺ ✷✳ ❈♦♠♣❛❝t✐✜❝❛t✐♦♥ ♦❢ t②♣❡ ■■ str✐♥❣ t❤❡♦r✐❡s ♦♥ ❈❛•❛❜✐✲(❛✉ t❤r❡❡❢♦•❞s ❛♥❞ t❤❡✐r ♠♦❞✉•✐ s♣❛❝❡s ❚❤❡ ❇P❙ ✐♥❞✐❝❡s Ω(γ; z) ❛r❡ s♦♠❡ ♠♦❞✉•✐ ❛♥❞ ❝❤❛r❣❡ ❞❡♣❡♥❞❡♥t ❝♦❡❢✲ ✜❝✐❡♥ts✳ ❚❤❡② ❣✐✈❡ r✐s❡ t♦ t❤❡ ✐♥st❛♥t♦♥ ♠❡❛s✉r❡s ❛♥❞ ✐t ✐s ✇❡•• ❦♥♦✇♥ t❤❛t t❤❡② ❛r❡ r❡•❛t❡❞ t♦ t❤❡ ❝♦✉♥t✐♥❣ st❛t❡s ❢♦r t❤❡ ❝❛s❡ ♦❢ ❜•❛❝❦ ❤♦•❡s✳ ❚❤❡② ❛r❡ ♣✐❡❝❡✇✐s❡ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ♠♦❞✉•✐ z a ✱ ✇❤❡r❡ t❤❡ •✉♠♣s ♦❝❝✉r ❛❝r♦ss ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ s✉r❢❛❝❡s✱ t❤❡ s♦ ❝❛••❡❞ ✇❛••s ♦❢ ♠❛r❣✐♥❛• st❛❜✐•✐t② ❛❜♦✉t ✇❤✐❝❤ ✇❡ ✇✐•• ❞✐s❝✉ss ✐♥ ❝❤❛♣t❡r ✹✳ ■t ✐s ❝•❡❛r t❤❛t t❤❡ ||✲✜❡•❞s ✭ζ Λ , ζΛ ✮ ✱ ✐♥ t❤❡ ❉✲✐♥st❛♥t♦♥ ❛❝t✐♦♥ ❜r❡❛❦ t❤❡ ❝♦♥t✐♥✉♦✉s ❍❡✐s❡♥❜❡r❣ s②♠♠❡tr✐❡s ✭✷✳✹✸✮ t♦ ❞✐s❝r❡t❡ ♦♥❡s✳ ❆❢t❡r ✐♥❝♦r♣♦r❛t✐♦♥ ♦❢ ❛•• s♦rts ♦❢ ❉✲✐♥st❛♥t♦♥s ❛•• s❤✐❢t s②♠♠❡tr✐❡s ❛•♦♥❣ t❤❡ ||✲✜❡•❞s ❜❡❝♦♠❡ ❞✐s❝r❡t❡✳ ❋♦r t❤❡ t②♣❡ ■■❇ s✐❞❡✱ t❤❡ ❡✛❡❝t ✐s s✐♠✐•❛r✱ t❤❡ t❤❡t❛✲❛♥❣•❡s ❛r❡ ♥♦✇ t❤❡ ✜❡•❞s c 0 , c i , c 0 , c i ✳ ✕ ❚❤❡r❡ ✐s ❛♥♦t❤❡r ♥♦♥✲♣❡rt✉r❜❛t✐✈❡ ♦❜❡•❛❝t ❝❛••❡❞ ◆❙✺✲❜r❛♥❡s✱ t❤❛t ❛r❡ ♠❛❣♥❡t✐❝ ❞✉❛• t♦ ❢✉♥❞❛♠❡♥t❛• str✐♥❣s✳ ❙✐♥❝❡ t❤❡② ❤❛✈❡ s✐① ❞✐♠❡♥s✐♦♥❛• ✇♦r•❞✈♦•✉♠❡ t❤❡② ❝❛♥ ✇r❛♣ t❤❡ ❡♥t✐r❡ ❈❛•❛❜✐✲(❛✉ ♠❛♥✐❢♦•❞✳ ❚❤❡② ❛r❡ r❡s♣♦♥s✐❜•❡ ❢♦r t❤❡ ❜r❡❛❦✐♥❣ ❞♦✇♥ ♦❢ t❤❡ ✜♥❛• ❝♦♥t✐♥✉♦✉s s②♠♠❡tr② ✇❤✐❝❤ ✐s ❝♦♥t✐♥✉♦✉s tr❛♥s•❛t✐♦♥ ❛•♦♥❣ ◆❙ ❛①✐♦♥ σ✳ ❙❝❤❡♠❛t✐❝❛••②✱ t❤❡✐r ❛❝t✐♦♥ ✐s ❣✐✈❡♥ ❜②✱ δds 2 | ◆❙✺ ∼ e -2π|k|V/g 2 s -πikσ . ✭✷✳✹✽✮ ❚❛❦✐♥❣ ❛•• ♥♦♥✲♣❡rt✉r❜❛t✐✈❡ ❡✛❡❝ts ✐♥t♦ ❛❝❝♦✉♥t✱ •❡❛❞s t♦ t❤❡ ❤②♣❡r♠✉•✲ t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡ ❞❡✈♦✐❞ ♦❢ ❛•• ❝♦♥t✐♥✉♦✉s ✐s♦♠❡tr✐❡s✱ ✇❤✐•❡ ♦♥•② ❛ ❞✐s❝r❡t❡ s✉❜❣r♦✉♣ ♦❢ t❤❡♠ ✐s r❡t❛✐♥❡❞✳ ✷✳✻ Pr❡s❡♥t st❛t✉s |❡❝❡♥t ②❡❛rs ❤❛✈❡ ♠❛r❦❡❞ ❛ •♦t ♦❢ ♣r♦❣r❡ss ✐♥ t❤❡ ✉♥❞❡rst❛♥❞✐♥❣ t❤❡ ❤②♣❡r♠✉•t✐♣•❡t ♠❡tr✐❝ ❢♦r t❤❡ t②♣❡ ■■ str✐♥❣ t❤❡♦r✐❡s✳ ❲❡ ♣r❡s❡♥t•② ❤❛✈❡ ❛ ♥❡❛t❡r ❞❡s❝r✐♣t✐♦♥ ♦❢ ◗❑ ❣❡♦♠❡tr✐❡s ✉s✐♥❣ t✇✐st♦r t❡❝❤♥✐q✉❡s✱ ✉♥❞❡rst❛♥❞✐♥❣ ✇❤✐❝❤ ✇✐•• ❜❡ t❤❡ ♠❛✐♥ ❢♦❝✉s ♦❢ t❤❡ ❢♦••♦✇✐♥❣ ❝❤❛♣t❡r✳ ❚❤❡ ✐♥❝•✉s✐♦♥ ♦❢ ❉✲✐♥st❛♥t♦♥s ♦♥ t②♣❡ ■■❆ s✐❞❡ ❝♦♥s✐st❡♥t•② ✇✐t❤ t❤❡ s②♠♣•❡❝t✐❝ ✐♥✈❛r✐❛♥❝❡ ❤❛❞ ❜❡❡♥ ❛•r❡❛❞② ❛❝❤✐❡✈❡❞ ✐♥ ❬✶✽✱ ✶✾(✳ ❚❤❡ ❉✭✲✶✮ ❛♥❞ ❉✶ ✐♥st❛♥t♦♥s ♦♥ t❤❡ t②♣❡ ■■❇ s✐❞❡ ❤❛❞ ❛•s♦ ❜❡❡♥ ❛❝❝♦✉♥t❡❞ ❢♦r ✐♥ ❬✷✵(✳ ❋✉rt❤❡r♠♦r❡ ✐♥✐t✐❛• st❡♣s ♦❢ ✐♥❝•✉❞✐♥❣ ◆❙✺ ✐♥st❛♥t♦♥s ❤❛✈❡ ❛•s♦ ❜❡❡♥ t❛❦❡♥ ✐♥ ❬✷✶(✱ ✇❤❡r❡ ❛•s♦ s❡✈❡r❛• ✐♥t❡r❡st✐♥❣ ❛♥❞ ❝♦♥♥❡❝t❡❞ ♣r♦❜•❡♠s ❛r❡ ❛❞❞r❡ss❡❞✱ ❢♦r ❡①❛♠♣•❡ s♦♠❡ r❡•❛t✐♦♥ t♦ t♦♣♦•♦❣✐❝❛• str✐♥❣s✳ ❚❤❡ q✉❡st✐♦♥ ♦❢ ❉✸✲✐♥st❛♥t♦♥s ♦♥ t②♣❡ ■■❇ ❤❛s ❛•s♦ ❜❡❡♥ ❛tt❡♠♣t❡❞ ✐♥ ❬✷✷( ❛•t❤♦✉❣❤ ✐ts ✉♥❞❡rst❛♥❞✐♥❣ st✐•• r❡♠❛✐♥s ✐♥❝♦♠♣•❡t❡ ❛s t❤❡ ❛♥❛•②s✐s ❤❛s ❜❡❡♥ ❞♦♥❡ ✐♥ t❤❡ •❛r❣❡ ✈♦•✉♠❡ •✐♠✐t ❛♥❞ ♦♥•② ✉♣t♦ ✜rst ♦r❞❡r ✐♥ ✐♥st❛♥t♦♥s✳ ❍❡r❡ ✇❡ ❣♦ ❜❡②♦♥❞ t❤❡s❡ r❡s✉•ts✳ ■♥ ♣❛rt✐❝✉•❛r✱ ♥♦✇ ✇❡ ❤❛✈❡ ✉♥❞❡rst♦♦❞ t❤❡ ◆❙✺ ✐♥st❛♥t♦♥s ♦♥ t❤❡ t②♣❡ ■■❇ s✐❞❡ ♠♦st•②✱ ❜❛rr✐♥❣ ❛ ❢❡✇ s✉❜t•❡t✐❡s✱ ❬✷✸✱ ✷✹(✳ ❋✉rt❤❡r♠♦r❡ ❛♥ ❡①♣•✐❝✐t ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ♠❡tr✐❝ ❤❛s ❜❡❡♥ ❝❛•❝✉•❛t❡❞ ✐♥ ❬✶✶( ✐♥ ♣r❡s❡♥❝❡ ♦❢ ❆✲❉✷ ✐♥st❛♥t♦♥s ♦♥ t❤❡ t②♣❡ ■■❆ s✐❞❡✳ ❲❡ r❡✐t❡r❛t❡ t❤❛t ❛•• ♦❢ t❤❡s❡ ✇❡r❡ ❛❝❤✐❡✈❡❞ ❜② ✐♥✈♦❦✐♥❣ ❞✐✛❡r❡♥t s②♠♠❡tr✐❡s ♦❢ t❤❡ t②♣❡ ■■❆ ✷✻ Pr❡s❡♥t st❛t✉s ✳ ✷✳✻ ❛♥❞ ■■❇ ❢♦r♠✉•❛t✐♦♥s✳ ❇❡❢♦r❡ ❝♦♠✐♥❣ t♦ t❤❡s❡ ♣♦✐♥ts ✐♥ •❛t❡r ❝❤❛♣t❡rs✱ ✇❡ ♥❡❡❞ t♦ ✜rst r❡✈✐❡✇ t❤❡ t✇✐st♦r t❡❝❤♥✐q✉❡s ❢♦r t❤❡ ◗❑ ♠❛♥✐❢♦•❞s✱ ✇❤✐❝❤ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ❞♦ ♥❡①t✳ ✷✼ ❈❍❆P❚❊| ✸ ❚✇✐st♦r✐❛• ❞❡s❝r✐♣t✐♦♥ ♦❢ ◗✉❛t❡r♥✐♦♥ ❑ä❤•❡r ♠❛♥✐❢♦•❞s ■♥ t❤❡ ♣r❡❝❡❞✐♥❣ ❝❤❛♣t❡r ✇❡ ❢♦✉♥❞ t❤❛t ❝♦♠♣❛❝t✐✜❝❛t✐♦♥s ♦❢ t②♣❡ ■■ str✐♥❣ t❤❡✲ ♦r✐❡s ♦♥ ❈❛•❛❜✐✲(❛✉ t❤r❡❡❢♦•❞s •❡❛❞ ✉s t♦ st✉❞② ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡s ♦❢ ❢♦✉r ❞✐♠❡♥s✐♦♥❛• N = 2 s✉♣❡r❣r❛✈✐t✐❡s✳ ▼♦r❡♦✈❡r✱ ✇❡ ❛•s♦ s❛✇ t❤❛t t❤❡s❡ ❤②♣❡r✲ ♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡s ❛r❡ q✉❛t❡r♥✐♦♥✲❑ä❤•❡r ♠❛♥✐❢♦•❞s✳ ❚❤❡ ♣❤②s✐❝❛• ♣r♦❜•❡♠ t❤❛t ✇❡ ✇❛♥t t♦ ❛❞❞r❡ss ✐s ❞❡s❝r✐❜✐♥❣ t❤❡s❡ ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡s ✇❤❡♥ ❛•• ♣♦ss✐❜•❡ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ❛r❡ ♣r❡s❡♥t✳ ❇✉t ✐t ✐s ♥♦t ❡❛s② t♦ ❞❡❛• ✇✐t❤ ❣❡♥❡r❛• ◗❑ ♠❛♥✐❢♦•❞s✳ ❚❤❡ ❞✐✣❝✉•t② ♦❢ ❤❛♥❞•✐♥❣ t❤❡♠ st❡♠s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t t❤❡② ❛r❡ r❛t❤❡r ❝♦♠♣•✐❝❛t❡❞ ❣❡♦♠❡tr✐❝ ♦❜•❡❝ts✳ ❚❤✐s ❢❛❝t r❡♥❞❡rs t❤❡ ♣r♦❜•❡♠ ♦❢ ✐♥❝•✉❞✲ ✐♥❣ ✐♥st❛♥t♦♥ ❝♦rr❡❝t✐♦♥s ❞✐r❡❝t•② ❛t t❤❡ •❡✈❡• ♦❢ ♠❡tr✐❝ t♦ ❜❡ ❡①tr❡♠❡•② ❞✐✣❝✉•t✳ ❋♦rt✉♥❛t❡•② t❤❡r❡ ✐s ❛ ✇❛② t♦ ❝✐r❝✉♠✈❡♥t t❤✐s ♣r♦❜•❡♠ ✳ ▲❡t ✉s r❡❝❛•• ✇❤② s♣❡❝✐❛• ❑ä❤•❡r ♠❛♥✐❢♦•❞s ❛r❡ ❝♦♠♣❛r❛t✐✈❡•② ❡❛s✐❡r ❣❡♦♠❡tr✐❝ ♦❜•❡❝ts✳ ❚❤❡ r❡❛s♦♥ ✐s t❤❛t t❤❡r❡ ❡①✐sts ❛ ❤♦•♦♠♦r♣❤✐❝ ♣r❡♣♦t❡♥t✐❛• ✇❤✐❝❤ ❝♦♠✲ ♣•❡t❡•② ❡♥❝♦❞❡s t❤❡✐r ❣❡♦♠❡tr②✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ◗❑ ♠❛♥✐❢♦•❞s ✇❡ ✇♦✉•❞ ❛•s♦ •✐❦❡ t♦ ❤❛✈❡ s✉❝❤ ❛ ♥✐❝❡✱ ❡•❡❣❛♥t ❛♥❞ s✉❜st❛♥t✐❛••② ♠♦r❡ ❝♦♥✈❡♥✐❡♥t ❞❡s❝r✐♣t✐♦♥✳ ❚❤❡ t✇✐st♦r ❝♦♥str✉❝t✐♦♥ t❤❛t ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ♣r❡s❡♥t ✐♥ t❤✐s ❝❤❛♣t❡r ❛❝t✉❛••② ❞♦❡s t❤❡ •♦❜✳ ■t t✉r♥s ♦✉t t❤❛t s✉❝❤ ❝♦♥str✉❝t✐♦♥ •❡❛❞s t♦ s✐♠✐•❛r ♦❜•❡❝ts t❤r♦✉❣❤ ✇❤✐❝❤ t❤❡ ❣❡♦♠❡tr② ♦❢ ◗❑ s♣❛❝❡ ❝❛♥ ❜❡ ❝♦♠♣•❡t❡•② ❡♥❝♦❞❡❞✳ ■♥ t❤❡ ❢♦••♦✇✐♥❣ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ r❡❝❛♣✐t✉•❛t❡ t❤❡ ❡ss❡♥❝❡ ♦❢ t❤❡s❡ t❡❝❤♥✐q✉❡s ❛♥❞ ✇❡ ✇✐•• ✜♥❞ ♦✉t ❤♦✇ ❝❡rt❛✐♥ ❤♦•♦♠♦r♣❤✐❝ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ♦♥ t❤❡ t✇✐st♦r s♣❛❝❡ ✇✐•• s❡r✈❡ ♦✉r ♣✉r♣♦s❡✳ ✸✳✶ ◗✉❛t❡r♥✐♦♥ ❑ä❤•❡r ♠❛♥✐❢♦•❞s ❛♥❞ t❤❡✐r t✇✐st♦r s♣❛❝❡s ❖✉r ❞✐s❝✉ss✐♦♥ ❝•♦s❡•② ❢♦••♦✇s ❬✷✺( ❛♥❞ t❤❡ r❡✈✐❡✇ ❬✹(✳ ❆ ◗❑ s♣❛❝❡ M ✐s ❛ 4n r❡❛• ❞✐♠❡♥s✐♦♥❛• |✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦•❞ ✇✐t❤ t❤❡ ❤♦•♦♥♦♠② ❣r♦✉♣ ❝♦♥t❛✐♥❡❞ ✐♥ U Sp(n)× ✷✽ ◗✉❛t❡r♥✐♦♥ ❑ä❤•❡r ♠❛♥✐❢♦•❞s ❛♥❞ t❤❡✐r t✇✐st♦r s♣❛❝❡s✳ ✸✳✶ SU (2) ❬✷✻(✳ ❚❤❡ n = 1 ❝❛s❡ ✐s t❤❡ s✐♠♣•❡st ❛♥❞ s♣❡❝✐❛• ✐♥ t❤❡ s❡♥s❡ t❤❛t t❤❡ ◗❑ ♠❛♥✐❢♦•❞ ✐s t❤❡♥ ❞❡✜♥❡❞ ❛s ❛ s❡•❢✲❞✉❛• ❊✐♥st❡✐♥ ♠❛♥✐❢♦•❞✳ ❚❤❡ ◗❑ s♣❛❝❡ ❝❛rr✐❡s ❛ tr✐♣•❡t ♦❢ ♥♦♥✲✐♥t❡❣r❛❜•❡ ❛•♠♦st ❝♦♠♣•❡① str✉❝t✉r❡s J i s❛t✐s❢②✐♥❣ t❤❡ q✉❛t❡r♥✐♦♥✐❝ ❛•❣❡❜r❛✳ ❚❤❡② ❞❡✜♥❡ t❤❡ q✉❛t❡r♥✐♦♥✐❝ t✇♦✲❢♦r♠s ❜② ω i (X, Y ) = g(J i X, Y )✳ ❚❤❡s❡ t✇♦ ❢♦r♠s ❛r❡ ❝♦✈❛r✐❛♥t•② ❝•♦s❡❞ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ SU (2) ♣❛rt p ♦❢ t❤❡ ▲❡✈✐✲❈✐✈✐t❛ ❝♦♥♥❡❝t✐♦♥ ❛♥❞ ❛r❡ ♣r♦♣♦rt✐♦♥❛• t♦ t❤❡ ❝✉r✈❛t✉r❡ ♦❢ p✱ d ω + p × ω = 0, d p + 1 2 p × p = ν 2 ω, ✭✸✳✶✮ ✇❤❡r❡ ν ✐s r❡•❛t❡❞ t♦ t❤❡ |✐❝❝✐ ❝✉r✈❛t✉r❡ ν = R 4n(n+2) ✳ ❚❤❡ ♠❛•♦r ♣r♦❜•❡♠ ✇✐t❤ s✉❝❤ s♣❛❝❡s ✐s t❤❛t t❤❡② ❛r❡ ♥♦t ❡✈❡♥ ❝♦♠♣•❡① ♠❛♥✐❢♦•❞s✳ ❍♦✇❡✈❡r✱ ✐t ✐s st✐•• ♣♦ss✐❜•❡ t♦ ❞❡s❝r✐❜❡ t❤❡ ❣❡♦♠❡tr② ♦❢ ◗❑ s♣❛❝❡s ❛♥❛•②t✐❝❛••② ❜② ♣❛ss✐♥❣ t♦ ✐ts t✇✐st♦r s♣❛❝❡✳ ❚❤❡ t✇✐st♦r s♣❛❝❡ Z ♦❢ t❤❡ ◗❑ ♠❛♥✐❢♦•❞ M ✐s t❤❡ t♦t❛• s♣❛❝❡ ♦❢ ❛ CP 1 ❜✉♥❞•❡ ♦♥ M✳ ❚❤✐s CP 1 ✐s t❤❡ s♣❤❡r❡ ❞❡✜♥❡❞ ❜② t❤❡ tr✐♣•❡t ♦❢ ❛•♠♦st ❝♦♠♣•❡① str✉❝t✉r❡s✳ ■♥ ❝♦♥tr❛st t♦ t❤❡ ◗❑ s♣❛❝❡✱ t❤✐s t✇✐st♦r s♣❛❝❡ ✐s ❛ ❝♦♠♣•❡① ♠❛♥✐❢♦•❞✳ ❋✉rt❤❡r♠♦r❡ ✐t ❝❛rr✐❡s ❛♥ ✐♥t❡❣r❛❜•❡ ❝❛♥♦♥✐❝❛• ❝♦♠♣•❡① ❝♦♥t❛❝t str✉❝t✉r❡ ❣✐✈❡♥ ❜② t❤❡ ❦❡r♥❡• ♦❢ t❤❡ O(2)✲t✇✐st❡❞ ✭✶✱✵✮✲❢♦r♠ Dt = dt + p + -ip 3 t + p -t 2 , ✭✸✳✷✮ ✇❤❡r❡ p ± = -1 2 (p 1 ∓ ip 2 ) ❛♥❞ t ✐s t❤❡ st❡r❡♦❣r❛♣❤✐❝ ❝♦♦r❞✐♥❛t❡ ♦❢ CP 1 ✳ ❚❤❡r❡ ✐s ❛ ♠♦r❡ ❝♦♥✈❡♥✐❡♥t ✇❛② t♦ ❞❡s❝r✐❜❡ t❤✐s ❝♦♠♣•❡① ❝♦♥t❛❝t str✉❝t✉r❡ ✈✐❛ ❛ ❤♦•♦♠♦r♣❤✐❝ ♦♥❡✲❢♦r♠ X ✱ ❦♥♦✇♥ ❛s ❝♦♥t❛❝t ♦♥❡✲❢♦r♠✳ ▲♦❝❛••② t❤❡r❡ ❡①✐sts ❛ ❢✉♥❝t✐♦♥✱ t❤❡ s♦ ❝❛••❡❞ ❝♦♥t❛❝t ♣♦t❡♥t✐❛• Φ [i] ✱ t❤❛t ❛••♦✇s ✉s t♦ ❝♦♥str✉❝t ✐t ❢r♦♠ t❤❡ (1, 0)✲❢♦r♠ Dt✱ X [i] = 4 it e Φ [i] Dt, ✭✸✳✸✮ ✇❤❡r❡ t❤❡ ✐♥❞❡① ✐♥❞✐❝❛t❡s t❤❡ ❢❛❝t t❤❛t ✇❡ ❛r❡ ✇♦r❦✐♥❣ ♦♥•② •♦❝❛••② ✐♥ ❛ ♣❛t❝❤ U i ⊂ Z✳ ❚❤✐s ❝♦♥t❛❝t ♣♦t❡♥t✐❛• ✐s ❛ q✉❛♥t✐t② t❤❛t ✇✐•• ♣•❛② ❛ ✈❡r② ✐♠♣♦rt❛♥t r♦•❡ ✐♥ ❢✉t✉r❡✳ ❋♦r ✐♥st❛♥❝❡ ✐t ❞❡t❡r♠✐♥❡s t❤❡ ❑ä❤•❡r ♣♦t❡♥t✐❛• ♦♥ Z ❜② K [i] Z = log 1 + t t |t| + Re Φ [i] . ✭✸✳✹✮ ❚❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ❝♦♥t❛❝t ♦♥❡✲❢♦r♠ X [i] ❜❡❝♦♠❡s ♠❛♥✐❢❡st ✇❤❡♥ ♦♥❡ ✐♥tr♦✲ ❞✉❝❡s ❤♦•♦♠♦r♣❤✐❝ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s✳ ◆❛♠❡•②✱ ✐t ✐s ❛•✇❛②s ♣♦ss✐❜•❡ t♦ ❝❤♦♦s❡ ❝♦♠♣•❡① ❝♦♦r❞✐♥❛t❡s (ξ Λ [i] , ξ[i] Λ , α [i] ) ♦♥ ❛ ♣❛t❝❤ U i ♦♥ t❤❡ t✇✐st♦r s♣❛❝❡ s✉❝❤ t❤❛t X [i] ❝❛♥ ❜❡ •♦❝❛••② tr✐✈✐❛•✐③❡❞ ❛s ❬✷✺✱ ✷✼( X [i] = dα [i] + ξ Λ [i] d ξ[i] Λ . ✭✸✳✺✮ ❚❤❡ ❝♦♥t❛❝t str✉❝t✉r❡ ✐s ❝♦♠♣•❡t❡•② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s r❡•❛t✐♥❣ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ✐♥ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t✇♦ ❞✐✛❡r❡♥t ♣❛t❝❤❡s U i ∩ U j ✳ ❚❤❡s❡ ✷✾ ✸✳ ❚✇✐st♦r✐❛• ❞❡s❝r✐♣t✐♦♥ ♦❢ ◗✉❛t❡r♥✐♦♥ ❑ä❤•❡r ♠❛♥✐❢♦•❞s ✏❝♦♥t❛❝t♦♠♦r♣❤✐s♠s✧ s❤♦✉•❞ ♣r❡s❡r✈❡ t❤❡ ❝♦♥t❛❝t ♦♥❡✲❢♦r♠ ✉♣ t♦ ❛ ❤♦•♦♠♦r♣❤✐❝ ❢❛❝t♦r✱ X [j] = f 2 [ij] X [i] . ✭✸✳✻✮ ❚❤❡ ❝♦♥t❛❝t♦♠♦r♣❤✐s♠s ❞❡✜♥❡❞ ❛❜♦✈❡ ❝❛♥ ❜❡ ♣❛r❛♠❡tr✐③❡❞ ❜② t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s H [ij] ∈ H 1 (Z, O(2))✱ ✇❤✐❝❤ ❛r❡ ❤♦•♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ♦❢ ❉❛r❜♦✉① ❝♦✲ ♦r❞✐♥❛t❡s ξ Λ [i] ♦♥ ♦♥❡ ♣❛t❝❤ ❛♥❞ ξ[j] Λ , α [j] ♦♥ t❤❡ ♦t❤❡r✳ ❚❤❡ ❣•✉✐♥❣ ❝♦♥❞✐t✐♦♥s t❤❛t r❡•❛t❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ♦♥ t✇♦ ❞✐✛❡r❡♥t ♣❛t❝❤❡s t✉r♥ ♦✉t t♦ ❜❡ ξ Λ [j] = ξ Λ [i] -∂ ξ[j] Λ H [ij] + ξ Λ [j] ∂ α [j] H [ij] , ξ[j] Λ = ξ[i] Λ + ∂ ξ Λ [i] H [ij] , α [j] = α [i] + H [ij] -ξ Λ [i] ∂ ξ Λ [i] H [ij] , ✭✸✳✼✮ ✉s✐♥❣ ✇❤✐❝❤ t❤❡ ❤♦•♦♠♦r♣❤✐❝ r❡s❝❛•✐♥❣ ❢❛❝t♦rs ✐♥ ✭✸✳✻✮ ❜❡❝♦♠❡s f 2 [ij] = 1 -∂ α [j] H [ij] , ✭✸✳✽✮ ❛♥❞ ✐s ❝♦♠♣•❡t❡•② ❞❡t❡r♠✐♥❡❞ ❜② H [ij] ✳ ❚❤❡s❡ ❣•✉✐♥❣ ❝♦♥❞✐t✐♦♥s ❛♣♣❡♥❞❡❞ ✇✐t❤ ❛♣♣r♦♣r✐❛t❡ r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥s ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s ❛ s❡t ♦❢ ✐♥t❡❣r❛• ❡q✉❛t✐♦♥s ❢♦r t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ❬✷✺(✳ ❚❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ❤❡r❡ ❛r❡ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ◗❑ ❝♦♦r❞✐♥❛t❡s (x µ ) ❛s ✇❡•• ❛s t❤❡ CP 1 ✲✜❜❡r ❝♦♦r❞✐♥❛t❡ t✳ ❲❡ r❡q✉✐r❡ t❤❛t t❤❡ ❝♦♦r❞✐♥❛t❡ ξ Λ [i] ❤❛s ♦♥•② s✐♠♣•❡ ♣♦•❡s ❛t t = 0, ∞✱ ❛♥❞ ❛••♦✇ ❢♦r t❤❡ ❝♦♦r❞✐♥❛t❡ α [i] t♦ ❤❛✈❡ •♦❣❛r✐t❤♠✐❝ s✐♥❣✉•❛r✐t✐❡s ❛t t = 0, ∞ ✳ ✶ ξ Λ [i] (x µ , t) = A Λ + t -1 Y Λ -t Ȳ Λ - 1 2 j C j dt ′ 2πit ′ t ′ + t t ′ -t ∂ ξ[j] Λ H [ij] -ξ Λ [j] ∂ α [j] H [ij] , ξ[i] Λ (x µ , t) = B Λ + 1 2 j C j dt ′ 2πit ′ t ′ + t t ′ -t ∂ ξ Λ [i] H [ij] , α [i] (x µ , t) = B α -2ic α log t + 1 2 j C j dt ′ 2πit ′ t ′ + t t ′ -t H [ij] -ξ Λ [i] ∂ ξ Λ [i] H [ij] . ✭✸✳✾✮ ❍❡r❡ t ∈ Û [i] ✱ ✇❤❡r❡ Û [i] ✐s t❤❡ ♣r♦ •❡❝t✐♦♥ ♦❢ U [i] ♦♥ t❤❡ CP 1 ❛♥❞ C i ✬s ❛r❡ ❝♦♥t♦✉rs t❤❛t s✉rr♦✉♥❞ Û [i] ✳ ❚❤❡ ❝♦♠♣•❡① ✈❛r✐❛❜•❡s Y Λ ❛♥❞ t❤❡ r❡❛• ✈❛r✐❛❜•❡s A Λ , B Λ , B α ♣•❛② t❤❡ r♦•❡s ♦♥ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ◗❑ ♠❛♥✐❢♦•❞✳ ❈♦✉♥t✐♥❣ t❤❡ ♥✉♠✲ ❜❡r ♦❢ ♣❛r❛♠❡t❡rs ✇❡ ✜♥❞ 4n + 1✱ ✇❤❡r❡ n ✐s t❤❡ q✉❛t❡r♥✐♦♥✐❝ ❞✐♠❡♥s✐♦♥ ♦❢ M✳ ❚❤❡② ❝♦♥t❛✐♥ ♦♥❡ ♣❛r❛♠❡t❡r ❡①tr❛ ❜❡❝❛✉s❡ t❤❡ ♦✈❡r❛•• ♣❤❛s❡ r♦t❛t✐♦♥ ♦❢ Y Λ ❝❛♥ ❜❡ ✶ ❆❝t✉❛••② ♦♥❡ ❝❛♥ ❛••♦✇ ❢♦r •♦❣❛r✐t❤♠✐❝ s✐♥❣✉•❛r✐t✐❡s ❢♦r ❜♦t❤ ξ[i] Λ ❛♥❞ α [i] ✳ ❚❤❡② ❣✐✈❡ r✐s❡ t♦ t❤❡ s♦ ❝❛••❡❞ ✏❛♥♦♠❛•♦✉s ❞✐♠❡♥s✐♦♥s✧ c Λ ❛♥❞ c α ❬✷✺(✳ ■♥ ✇❤❛t ❢♦••♦✇s✱ c Λ ✬s ❛r❡ ♥♦t r❡•❡✈❛♥t ❢♦r ♦✉r ❝♦♥s✐❞❡r❛t✐♦♥s✳ ❇✉t c α ✐s✱ ❜❡❝❛✉s❡ ✐t ❡♥❝♦❞❡s t❤❡ ♦♥❡ •♦♦♣ ❝♦rr❡❝t✐♦♥ ✐♥ g s ✳ ✸✵ ❈♦♥t❛❝t ❜r❛❝❦❡t ❢♦r♠❛•✐s♠✳ ✸✳✷ ❛❜s♦r❜❡❞ ❜② r❡❞❡✜♥✐♥❣ t❤❡ CP 1 ❝♦♦r❞✐♥❛t❡ t✳ ■♥ ♠❛t❤❡♠❛t✐❝❛• ♣❛r•❛♥❝❡ t❤❡ ♣r♦✲ ❝❡❞✉r❡ ♦❢ s♦•✈✐♥❣ t❤❡s❡ ✐♥t❡❣r❛• ❡q✉❛t✐♦♥s ✐s ❦♥♦✇♥ ❛s ✏♣❛r❛♠❡tr✐③✐♥❣ t❤❡ t✇✐st♦r •✐♥❡s✧✳ ❚❤❡r❡ ✐s ❛♥ ✐♠♣♦rt❛♥t t❤✐♥❣ t♦ ♥♦t✐❝❡ ✐♥ t❤❡ ❛❜♦✈❡ ✐♥t❡❣r❛• ❡q✉❛t✐♦♥s ✿ t❤❡ ✐♥❞❡① i ✐♥s✐❞❡ t❤❡ ✐♥t❡❣r❛• ❝❛♥ ❜❡ r❡♣•❛❝❡❞ ✇✐t❤ ❛♥② ♦t❤❡r ✐♥❞❡① ❢♦r ❛♥② ♦t❤❡r ♣❛t❝❤ ✐♥ t❤❡ t✇✐st♦r s♣❛❝❡ ❛s t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s s❛t✐s❢② ❝❡rt❛✐♥ ❝♦❝②❝•❡ ❝♦♥❞✐t✐♦♥s✳ ❚❤✐s ❢❛❝t ✐s ✐♠♣♦rt❛♥t ✐♥ ❝♦♠♣✉t❛t✐♦♥s ♣r❡s❡♥t❡❞ •❛t❡r✳ ❋✐♥❛••② ♦♥❡ ❝❛♥ ✜♥❞ t❤❡ ❣•✉✐♥❣ ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ❝♦♥t❛❝t ♣♦t❡♥t✐❛•✱ Φ [i] -Φ [j] = log(1 -∂ α [j] H [ij] ). ✭✸✳✶✵✮ ❚❤✐s ❝❛♥ ❜❡ s♦•✈❡❞ t♦ ✜♥❞ ❬✷✺( Φ [i] = φ - 1 2 j C j dt ′ 2πit ′ log(1 -∂ α [j] H [ij] ), ✭✸✳✶✶✮ ✇❤❡r❡ t❤❡ ❝♦♥st❛♥t ♣❛rt ✐s e φ = e Re [Φ [+] (t=0)] = 1 8π j C j dt t t -1 Y Λ -t Ȳ Λ ∂ ξ Λ [i] H [ij] + c α 2 cos 1 4π j C j dt t log(1 -∂ α [j] H [ij] ) , ✭✸✳✶✷✮ ❛♥❞ t❤❡ ✐♥❞❡① [+] ❞❡♥♦t❡s t❤❡ ♣❛t❝❤ s✉rr♦✉♥❞✐♥❣ t❤❡ ♥♦rt❤ ♣♦•❡ ✭ t = 0✮ ♦♥ CP 1 ✳ ❖♥❡ ♠♦r❡ ✐♠♣♦rt❛♥t ❢❛❝t t♦ ♥♦t❡ t❤❛t ❛•• t❤❡ q✉❛♥t✐t✐❡s s✉❝❤ ❛s ❝♦♥t❛❝t ♦♥❡✲ ❢♦r♠✱ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s✱ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ❛♥❞ ❝♦♥t❛❝t ♣♦t❡♥t✐❛• s❛t✐s❢② r❡❛•✲ ✐t② ❝♦♥❞✐t✐♦♥s ✉♥❞❡r t❤❡ ❝♦♠❜✐♥❡❞ ❛❝t✐♦♥ ♦❢ ❝♦♠♣•❡① ❝♦♥•✉❣❛t✐♦♥ ❛♥❞ t❤❡ ❛♥t✐♣♦❞❛• ♠❛♣✱ ς[t] = -1/ t✱ ♣r♦✈✐❞❡❞ t❤❛t t❤❡ ❝♦✈❡r✐♥❣ ♦❢ Z ❜② ♣❛t❝❤❡s {U i } r❡♠❛✐♥s ✐♥✲ ✈❛r✐❛♥t✱ ✐✳❡✳ U i ✐s ♠❛♣♣❡❞ t♦ Uī ❛♥❞ ❜❛❝❦✳ ■♥ ♣❛rt✐❝✉•❛r✱ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s s❛t✐s❢② ς[H [ij] ] = H [ īj ] . ✭✸✳✶✸✮ ✸✳✷ ❈♦♥t❛❝t ❜r❛❝❦❡t ❢♦r♠❛•✐s♠ ❚❤❡r❡ ✐s ❛ ❢✉rt❤❡r s✐♠♣•✐✜❝❛t✐♦♥ ✐❢ ♦♥❡ ✐♥tr♦❞✉❝❡s ❝❡rt❛✐♥ ♦❜•❡❝ts ❝❛••❡❞ ✏❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s✧✳ ❚❤❡② ✇❡r❡ ✐♥tr♦❞✉❝❡❞ ✐♥ ❬✷✸✱ ✷✹(✳ ❚❤❡ ❞✐✣❝✉•t② ♦❢ ♣❛r❛♠❡tr✐③✐♥❣ t❤❡ ❝♦♥t❛❝t tr❛♥s❢♦r♠❛t✐♦♥s ✇✐t❤ ✉s✉❛• tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ❛r✐s❡s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t H [ij] ❛r❡ ❢✉♥❝t✐♦♥s ♦❢ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ✐♥ t✇♦ ❞✐✛❡r❡♥t ♣❛t❝❤❡s✳ ❉✉❡ t♦ t❤✐s t❤❡ ❛♣♣❛r❡♥t•② s✐♠♣•❡ ❣•✉✐♥❣ ❝♦♥❞✐t✐♦♥s ♠❛② ❜❡ ❣❡♥❡r❛t❡❞ ❜② ❝♦♠♣•✐❝❛t❡❞ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s✳ ■♥ ♣❛rt✐❝✉•❛r ✇❤❡♥ ♦♥❡ ❞✐s❝✉ss❡s s②♠♠❡tr✐❡s t❤✐s ❝r❡❛t❡s ❛ ❜✐❣ ♦❜st❛❝•❡✳ ❚②♣✐❝❛••② s✉❝❤ s②♠♠❡tr✐❡s ❞❡✜♥❡ s♦♠❡ ❛❝t✐♦♥ •♦❝❛••② ❛t ❡❛❝❤ ♣❛t❝❤ U i ♦♥ t❤❡ t✇✐st♦r s♣❛❝❡ ❜② t❤❡✐r ❛❝t✐♦♥ ♦♥ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s✳ ❚❤❡s❡ ❛❝t✐♦♥s ❝❛♥ t❛❦❡ ❛ ❤✐❣❤•② ♥♦♥•✐♥❡❛r ❢♦r♠ ✐❢ ♦♥❡ tr✐❡s t♦ r❡❛•✐③❡ s✉❝❤ s②♠♠❡tr✐❡s ✐♥ t❡r♠s ♦❢ H [ij] ✳ ❲❡ ✇✐•• s❡❡ ❡①❛♠♣•❡s ♦❢ s✉❝❤ s✐t✉❛t✐♦♥s •❛t❡r✳ ✸✶ ✸✳ ❚✇✐st♦r✐❛• ❞❡s❝r✐♣t✐♦♥ ♦❢ ◗✉❛t❡r♥✐♦♥ ❑ä❤•❡r ♠❛♥✐❢♦•❞s ❇❡❢♦r❡✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥✱ ✇❡ ✇♦✉•❞ •✐❦❡ t♦ r❡❝❛♣✐t✉•❛t❡ ❝❡rt❛✐♥ ❢❛❝ts ❛❜♦✉t ❝♦♥t❛❝t ❣❡♦♠❡tr②✱ t❤❡ r❡•❡✈❛♥t ❣❡♦♠❡tr② ❢♦r t❤❡ t✇✐st♦r s♣❛❝❡s✳ ▲❡t ✉s ❝♦♥s✐❞❡r ❛ ❝♦♥t❛❝t ✈❡❝t♦r ✜❡•❞ X h ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ❢✉♥❝t✐♦♥ h✳ ■t ✐s ❞❡t❡r♠✐♥❡❞ ❜② ι X h dX = -dh + R(h)X , ι X h X = h. ✭✸✳✶✹✮ ❍❡r❡ ι X ❞❡♥♦t❡s t❤❡ ❝♦♥tr❛❝t✐♦♥ ✇✐t❤ ❛ ✈❡❝t♦r ✜❡•❞ ❛♥❞ R ✐s t❤❡ s♦ ❝❛••❡❞ |❡❡❜ ✈❡❝t♦r ❞❡t❡r♠✐♥❡❞ ❛s t❤❡ ✉♥✐q✉❡ ❡•❡♠❡♥t ♦❢ t❤❡ ❦❡r♥❡• ♦❢ dX s❛t✐s❢②✐♥❣ X (R) = 1✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❞❡✜♥❡ t❤❡ ❝♦♥t❛❝t ❜r❛❝❦❡t ✇❤✐❝❤ ✐s ❛♥ ❡①t❡♥s✐♦♥ ♦❢ P♦✐ss♦♥ ❜r❛❝❦❡t ✐♥ t❤❡ r❡❛•♠ ♦❢ ❝♦♥t❛❝t ❣❡♦♠❡tr②✳ ■ts ❞❡✜♥✐t✐♦♥ ❢♦r t✇♦ ♦❜ •❡❝ts µ 1 ∈ O(2m) ❛♥❞ µ 2 ∈ O(2n) ✐s ❛s ❛ ♠❛♣ t♦ O(2(m + n -1)) ❣✐✈❡♥ ❜② ❬✶✽(✱ {µ 1 , µ 2 } m,n = ∂ ξ Λ µ 1 ∂ ξΛ µ 2 + mµ 1 -ξ Λ ∂ ξ Λ µ 1 ∂ α µ 2 -∂ ξ Λ µ 2 ∂ ξΛ µ 1 -nµ 2 -ξ Λ ∂ ξ Λ µ 2 ∂ α µ 1 . ✭✸✳✶✺✮ ❋r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ t❤❡ s❦❡✇✲s②♠♠❡tr② ✐s ♦❜✈✐♦✉s✱ ✐♥ ♣❛rt✐❝✉•❛r {µ, µ} m,n = (m -n)µ∂ α µ. ✭✸✳✶✻✮ ✭✸✳✶✺✮ ❛•s♦ s❛t✐s✜❡s t❤❡ ▲❡✐❜♥✐t③ r✉•❡ ❛♥❞ t❤❡ ❏❛❝♦❜✐ ✐❞❡♥t✐t②✳ ❋♦r s♦♠❡ s❡❝t✐♦♥ µ 3 ∈ O(2k) t❤❡ ▲❡✐❜♥✐t③ r✉•❡ ❜❡❝♦♠❡s {µ 1 µ 2 , µ 3 } m+n,k = µ 1 {µ 2 , µ 3 } n,k + µ 2 {µ 1 , µ 3 } m,k . ✭✸✳✶✼✮ ❚❤❡ ❏❛❝♦❜✐ ✐❞❡♥t✐t② t✉r♥s ♦✉t t♦ ❜❡ t❤❡ ❢♦••♦✇✐♥❣ r❡•❛t✐♦♥✱ {µ 1 , {µ 2 , µ 3 } n,k } m,n+k-1 -{µ 2 , {µ 1 , µ 3 } m,k } n,m+k-1 -{{µ 1 , µ 2 } m,n , µ 3 } m+n-1,k = 0. ✭✸✳✶✽✮ ❙✐♥❝❡ ✇❡ ✇✐•• ♠♦st•② ♥❡❡❞ t❤❡ ❝❛s❡ ♦❢ {, } 1,0 ❜❡•♦✇✱ t♦ r❡❞✉❝❡ t❤❡ ❝✉♠❜❡rs♦♠❡✲ ♥❡ss ♦❢ t❤❡ ♥♦t❛t✐♦♥ ✇❡ ✇✐•• ❞r♦♣ t❤❡ ✐♥❞✐❝❡s ✐❢ ♥♦ ❝♦♥❢✉s✐♦♥ ❛r✐s❡s✳ ■t ♣r♦✈✐❞❡s t❤❡ ❛❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡•❞ X µ 1 ✇✐t❤ t❤❡ ✭❣❡♥❡r❛•✐③❡❞✮ ♠♦♠❡♥t ♠❛♣ µ 1 ♦♥ ❛ •♦❝❛• ❝♦♠♣•❡① ❢✉♥❝t✐♦♥ µ 2 ❬✷✽(✳ ■♥ ♣❛rt✐❝✉•❛r ❣✐✈❡♥ s✉❝❤ µ 1 ❛♥❞ µ 2 ✱ t❤❡ ❝♦♥t❛❝t ❜r❛❝❦❡t ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r ✜❡•❞ ❛ss♦❝✐❛t❡❞ t♦ µ 1 ❛s {µ 1 , µ 2 } = X µ 1 µ 2 . ✭✸✳✶✾✮ ❙❡tt✐♥❣ µ 1 = h ❛♥❞ µ 2 t♦ ❜❡ ♦♥❡ ♦❢ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s✱ ♦♥❡ ❡①♣•✐❝✐t•② ✜♥❞s {h, ξ Λ } = -∂ ξΛ h + ξ Λ ∂ α h, {h, ξΛ } = ∂ ξ Λ h, {h, α} = h -ξ Λ ∂ ξ Λ h. ✭✸✳✷✵✮ ■❢ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥ h ✐s ✉s❡❞ t♦ r❡♣r❡s❡♥t s❡❝t✐♦♥s ♦❢ t✇♦ ❞✐✛❡r❡♥t ❜✉♥❞•❡s O(2) ❛♥❞ O(0)✱ ♦♥❡ ✜♥❞s t❤❛t t❤❡ ❜r❛❝❦❡t ✐s ♥♦t ❛♥t✐s②♠♠❡tr✐❝✱ t❤❡ ❢❛❝t ✇❤✐❝❤ ❢♦••♦✇s ❢r♦♠ ✭✸✳✶✻✮✳ ■♥st❡❛❞ ✐t s❛t✐s✜❡s✱ {h, h} = h∂ α h = hR(h), ✭✸✳✷✶✮ ✸✷ •❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥✳ ✸✳✸ ✇❤✐❝❤ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜② ❛♣♣•②✐♥❣ ι X h ♦♥ t❤❡ ✜rst ❡q✉❛t✐♦♥ ♦❢ ✭✸✳✶✹✮ ❛♥❞ ♦❢ ❝♦✉rs❡✱ ✇✐t❤♦✉t •♦♦s✐♥❣ ❣❡♥❡r❛•✐t② ✇❡ ❝❛♥ ❝❤♦♦s❡ t❤❡ |❡❡❜ ✈❡❝t♦r ✜❡•❞ R(h) = ∂ α h✳ ❆ ❝r✉❝✐❛• ♣r♦♣❡rt② ♦❢ t❤❡ ❝♦♥t❛❝t ❜r❛❝❦❡t ❢♦••♦✇✐♥❣ ❢r♦♠ ✐ts ❞❡✜♥✐t✐♦♥ ✐s t❤❛t ✐t ❣❡♥❡r❛t❡s ✐♥✜♥✐t❡s✐♠❛• ❝♦♥t❛❝t tr❛♥s❢♦r♠❛t✐♦♥✳ ❚❤✐s ♠❡❛♥s L X h X = dι X h X + ι X h dX = R(h)X . ✭✸✳✷✷✮ ❚❤✉s ❡①♣♦♥❡♥t✐❛t✐♥❣ t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ ❝♦♥t❛❝t ❜r❛❝❦❡t ♦♥❡ ❣❡♥❡r❛t❡s ❛ ✜♥✐t❡ ❝♦♥t❛❝t tr❛♥s❢♦r♠❛t✐♦♥✳ ❚❤✐s ✐♠♣•✐❡s t❤❛t t❤❡ ❣•✉✐♥❣ ❝♦♥❞✐t✐♦♥s ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s✱ Ξ [j] = e {h [ij] ,•} Ξ [i] , ✭✸✳✷✸✮ ❢♦r s♦♠❡ ❢✉♥❝t✐♦♥ h [ij] ✳ ❚❤❡ ❛❝t✐♦♥ ♦♥ ❡❛❝❤ ♦❢ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ✐s ❣✐✈❡♥ ❜② ξ Λ [j] = e {h [ij] , • } ξ Λ [i] = ξ Λ [i] -∂ ξ[j] Λ H [ij] + ξ Λ [j] ∂ α [j] H [ij] , ξ[j] Λ = e {h [ij] , • } ξ[i] Λ = ξ[i] Λ + ∂ ξ Λ [i] H [ij] , α [j] = e {h [ij] , • } α [i] = α [i] + H [ij] -ξ Λ [i] ∂ ξ Λ [i] H [ij] . ✭✸✳✷✹✮ ❋r♦♠ t❤❡ ❛❜♦✈❡ ✇❡ ❝❛♥ ❛•s♦ ✜♥❞ ❛♥ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ✉s✉❛• tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ H [ij] ✐♥ t❡r♠s ♦❢ h [ij] ❛s H [ij] = e {h [ij] , • } -1 α [i] + ξ Λ [i] e {h [ij] , • } -1 ξ[i] Λ . ✭✸✳✷✺✮ ❚❤❡ ✐♠♣♦rt❛♥t ♣♦✐♥t ❤❡r❡ ✐s t❤❛t r❡✇r✐t✐♥❣ t❤❡ ❣•✉✐♥❣ ❝♦♥❞✐t✐♦♥s ❜② ✭✸✳✷✹✮ ❛✈♦✐❞s t❤❡ ❝♦♠♣•✐❝❛t✐♦♥ ♦❢ ❤❛✈✐♥❣ t♦ ❝♦♠♣✉t❡ ❞❡r✐✈❛t✐✈❡s ♦♥•② ❛❢t❡r tr❛♥s❢❡rr✐♥❣ t♦ t❤❡ ♣❛t❝❤❡s U j ❢♦r t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ξ[j] Λ ❛♥❞ α [j] ✱ ✇❤✐❝❤ ❛r❡ ❛r❣✉♠❡♥ts ♦❢ H [ij] ✳ ■♥❞❡❡❞✱ ✐❢ ♦♥❡ ✉s❡s t❤❡ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s✱ t❤❡ ❞❡r✐✈❛t✐✈❡s t❤❛t ❡♥t❡r t❤❡ ❣•✉✐♥❣ ❝♦♥❞✐t✐♦♥s ❝❛♥ ❜❡ t❛❦❡♥ tr❡❛t✐♥❣ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ♦♥ t❤❡ s❛♠❡ ♣❛t❝❤ ❛s ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜•❡s✳ ❚❤✐s s✐♠♣•✐✜❝❛t✐♦♥ ✇✐•• ♣•❛② ❛♥ ✐♠♣♦rt❛♥t r♦•❡ ✇❤❡♥ ✇❡ ✇✐•• ❞✐s❝✉ss ◗❑ ♠❛♥✐❢♦•❞s ✇✐t❤ s②♠♠❡tr✐❡s ❛♥❞ t❤❡✐r t✇✐st♦r s♣❛❝❡s✳ ❆•s♦ ✐t ✐s ✉s❡❢✉• t♦ ♥♦t❡ t❤❛t ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ H [ij] ✐s ❞❡♣❡♥❞❡♥t ♦♥ ξ Λ [i] ♦♥•② t❤❡ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥ h [ij] ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✱ H [ij] (ξ Λ [i] ) = h [ij] (ξ Λ [i] )✳ ✸✳✸ •❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ❚❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ❛r❡ ❛❝t✉❛••② ♥♦t ✉♥✐q✉❡•② ❞❡t❡r♠✐♥❡❞✳ ■t ✐s st✐•• ♣♦ss✐❜•❡ t♦ ♣❡r❢♦r♠ •♦❝❛• ❝♦♥t❛❝t tr❛♥s❢♦r♠❛t✐♦♥s ♦♥ ❡❛❝❤ ♦❢ t❤❡ ♣❛t❝❤❡s ♦❢ t❤❡ t✇✐st♦r s♣❛❝❡ U i ∈ Z ❬✷✺(✳ ❙✐♥❝❡ t❤❡s❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥s ❛r❡ ❝♦♥t❛❝t tr❛♥s❢♦r♠❛t✐♦♥s ❛❢t❡r ❛••✱ t❤❡ ❢♦r♠❛•✐s♠ ♦❢ t❤❡ ❝♦♥t❛❝t ❜r❛❝❦❡ts ❝❛♥ ❜❡ ✉t✐•✐③❡❞ ✐♥ t❤✐s ❝❛s❡ ❛s ✇❡•• ❬✷✹(✳ ❲❡ ✇✐•• ❝❛•• t❤❡ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s t❤❛t ❣❡♥❡r❛t❡ t❤❡s❡ •♦❝❛• ❝♦♥t❛❝t tr❛♥s❢♦r♠❛t✐♦♥s ❛s ✏❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s✧✳ ❆ ❝r✉❝✐❛• ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ✸✸ ✸✳ ❚✇✐st♦r✐❛• ❞❡s❝r✐♣t✐♦♥ ♦❢ ◗✉❛t❡r♥✐♦♥ ❑ä❤•❡r ♠❛♥✐❢♦•❞s tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s t❤❛t ❣❡♥❡r❛t❡ ❝♦♥t❛❝t tr❛♥s❢♦r♠❛t✐♦♥s ❜❡t✇❡❡♥ t✇♦ ❞✐✛❡r❡♥t ♣❛t❝❤❡s ❛♥❞ t❤❡ ❣❡♥❡r❛t♦rs ♦❢ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥s ✐s t❤❛t t❤❡ •❛tt❡r ❤❛s t♦ ❜❡ r❡❣✉•❛r ✐♥ t❤❡ ♣❛t❝❤❡s ♦♥ ✇❤✐❝❤ t❤❡② ❛❝t ❡♥s✉r✐♥❣ t❤❛t t❤❡ r❡❣✉•❛r✐t② ♦❢ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ♦♥ t❤❡✐r ♣❛t❝❤❡s ♦❢ ❞❡✜♥✐t✐♦♥ ✐s ♥♦t s♣♦✐•❡❞✳ ■♥ t❤❡ ❢♦••♦✇✐♥❣✱ ✇❡ ❞❡♥♦t❡ t❤❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ❜② g [i] ❛♥❞ ❛s ❡①♣•❛✐♥❡❞ t❤❡② ❛r❡ r❡❣✉•❛r ✐♥ U i ✳ ❚❤❡ ❡✛❡❝t ♦❢ ❛ ❣❡♥❡r❛• ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❝❛♣t✉r❡❞ ❜② t❤❡ ❢♦••♦✇✐♥❣ ❢♦r✲ ♠✉•❛✱ exp X h [ij] g = e -X g [i] exp X h [ij] e X g [j] , ✭✸✳✷✻✮ ❛♥❞ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ ♣r✐♥❝✐♣•❡ ❜② ✉s✐♥❣ t❤❡ ❇❛❦❡r✲❈❛♠♣❜❡••✲❍❛✉s❞♦r✛ ❢♦r♠✉•❛✳ ▲❛t❡r ✇❡ ✇✐•• ✜♥❞ t❤❛t t❤❡ r❡•❡✈❛♥t ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s t❛❦❡ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥❛• ❢♦r♠ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ♣❛t❝❤❡s ❛♥❞ ✇✐•• ❞❡♣❡♥❞ ♦♥ ♦♥•② t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ξ Λ [i] ✱ ✐✳❡✳ g [i] = g(ξ [i] )✳ ❚❤❡ ❏❛❝♦❜✐ ✐❞❡♥t✐t② ✭✸✳✶✽✮ ✐♥ t❤✐s ❝❛s❡ ✐♠♣•✐❡s [X g , X h ] = X {g,h} 1,1 . ✭✸✳✷✼✮ ❯s✐♥❣ t❤✐s ✇❡ ❝❛♥ ✜♥❞ ❡①♣•✐❝✐t•② ❤♦✇ t❤❡ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥ h [ij] ❝❤❛♥❣❡s t♦ h [ij] g ❞✉❡ t❤❡ ❣❛✉❣❡ tr❛♥❢♦r♠❛t✐♦♥ ✐♥❞✉❝❡❞ ❜② g [i] ❛s h [ij] g = e -{g, • } 1,1 • h [ij] , ✭✸✳✷✽✮ ❚❤❡♥ t❤❡ ❛❜♦✈❡ ❡♥t❛✐•s ✐♥t♦ t❤❡ ❢♦••♦✇✐♥❣ s✐♠♣•❡ r❡•❛t✐♦♥✱ h [ij] g = h [ij] ξ Λ , ξΛ -∂ ξ Λ g , α -g + ξ Λ ∂ ξ Λ g , ✭✸✳✷✾✮ ❛✛❡❝t✐♥❣ t❤❡ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥ h [ij] ❜② ♦♥•② •✐♥❡❛r s❤✐❢ts ✐♥ t❤❡ ❛r❣✉♠❡♥ts✳ ✸✳✹ ◗❑ ♠❛♥✐❢♦•❞ ✇✐t❤ s②♠♠❡tr✐❡s ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✇✐•• ❞✐s❝✉ss t❤❡ ❣❡♥❡r❛• ♣r♦❝❡❞✉r❡ ❢♦r ❞❡❛•✐♥❣ ✇✐t❤ ◗❑ ♠❛♥✐❢♦•❞s ✇✐t❤ s②♠♠❡tr✐❡s✳ ❆s ✇❛s ♠❡♥t✐♦♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡r✱ ✐t ✐s ♥♦t ♣♦ss✐❜•❡ t♦ ♣❡r❢♦r♠ ❞❡t❛✐•❡❞ ♠✐❝r♦s❝♦♣✐❝ ❝♦♠♣✉t❛t✐♦♥s ❡①♣•✐❝✐t•② ✐♥ ♠❛♥② ♦❝❝❛s✐♦♥s✳ ■♥ s✉❝❤ s✐t✉❛t✐♦♥s t❤❡ ♦♥•② ❣✉✐❞✐♥❣ ♣r✐♥❝✐♣•❡s ❛r❡ t❤❡ s②♠♠❡tr✐❡s ♦❢ t❤❡ ♠❡tr✐❝ ♦♥ t❤❡ ♠♦❞✉•✐ s♣❛❝❡✳ |❡♠❛r❦❛❜•②✱ t❤❡ s②♠♠❡tr✐❡s t✉r♥ ♦✉t t♦ ❜❡ s✉✣❝✐❡♥t t♦ ❞❡t❡r♠✐♥❡ t❤❡ ♥♦♥✲♣❡rt✉r❜❛t✐✈❡ str✉❝t✉r❡ ♦❢ M HM ❡♥t✐r❡•②✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ ✐♥ t❡r♠s ♦❢ s♦♠❡ ❈❛•❛❜✐✲(❛✉ ❞❛t❛ s✉❝❤ ❛s ❉♦♥❛•❞s♦♥✲❚❤♦♠❛s ✐♥✈❛r✐❛♥ts✱ ✐♥t❡rs❡❝t✐♦♥ ♥✉♠❜❡rs ❡t❝✳ t❤❡ s②♠♠❡tr✐❡s ✐♠♣♦s❡ str♦♥❣ ❡♥♦✉❣❤ r❡q✉✐r❡♠❡♥ts ❢r♦♠ ✇❤✐❝❤ ✐♥ ♣r✐♥❝✐♣•❡ t❤❡ ♠❡tr✐❝ ✐s ❝♦♠♣✉t❛❜•❡✳ ❋♦r ❤❛✈✐♥❣ ❜❡tt❡r ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ s✉❝❤ s②♠♠❡tr✐❡s ✇❡ ✇✐•• ❛❣❛✐♥ ✉s❡ t❤❡ t✇✐st♦r ❢♦r♠✉•❛t✐♦♥✳ ■t ✐s ✇❡•• ❦♥♦✇♥ t❤❛t ❛•• s②♠♠❡tr✐❡s ♦❢ ◗❑ s♣❛❝❡s ❝❛♥ ❜❡ •✐❢t❡❞ t♦ t❤❡✐r t✇✐st♦r s♣❛❝❡s ❬✷✻(✳ ❆♣❛rt ❢r♦♠ t❤❡ s②♠♠❡tr✐❡s t❤❡r❡ ✐s ♥♦✇ ❛♥ ❡①tr❛ ❛❞✈❛♥t❛❣❡ ♦❢ ❤♦•♦♠♦r♣❤✐❝✐t② ✇❤♦s❡ ✈✐rt✉❡ ✇✐•• ❜❡ ❡①t❡♥s✐✈❡•② ✉s❡❞ •❛t❡r✳ ❚❤❡ •✐❢t ♦❢ t❤❡ s②♠♠❡tr✐❡s t♦ t❤❡ t✇✐st♦r s♣❛❝❡ Z ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❛❝t✐♦♥ ♦♥ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s✳ ◆❛t✉r❛••②✱ s✐♥❝❡ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ❞❡♣❡♥❞ ♦♥ t❤❡ ✸✹ ◗❑ ♠❛♥✐❢♦•❞ ✇✐t❤ s②♠♠❡tr✐❡s ✳ ✸✳✹ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s✱ t❤❡② ❝❛♥ ♥♦t ❜❡ ❛r❜✐tr❛r② ✐♥ ♣r❡s❡♥❝❡ ♦❢ s✉❝❤ s②♠♠❡tr✐❡s✳ ■♥st❡❛❞ t❤❡ ❞❡✜♥✐♥❣ ❛❝t✐♦♥ ♦♥ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ✇✐•• ❣✐✈❡ r✐s❡ t♦ ❝❡rt❛✐♥ ❝♦♥str❛✐♥ts ♦♥ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s✳ ❋✉rt❤❡r♠♦r❡ ❢♦r ❛ ♣r♦♣❡r •✐❢t✱ t❤❡ ♠❡tr✐❝ ♦♥ t❤❡ t✇✐st♦r s♣❛❝❡ s❤♦✉•❞ ❛•s♦ r❡♠❛✐♥ ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ s②♠♠❡tr②✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ st❡r❡♦❣r❛♣❤✐❝ ❝♦♦r❞✐♥❛t❡ t ♠✉st ❛•s♦ tr❛♥s❢♦r♠ ✐♥ ❛ s♣❡❝✐✜❝ ✇❛②✳ ■♥❞❡❡❞✱ t❤✐s ✐s t❤❡ ❝❛s❡ ❜❡❝❛✉s❡ ✐❢ ✇❡ •♦♦❦ ❛t t❤❡ ✐♥t❡❣r❛• ❡q✉❛t✐♦♥s ❢♦r t❤❡ ❉❛r❜♦✉① ❝♦♦r✲ ❞✐♥❛t❡s✱ ✇❡ ✜♥❞ t❤❛t t❤❡r❡ ❛r❡ ❡①♣•✐❝✐t ❞❡♣❡♥❞❡♥❝❡s ♦♥ t✳ ■❢ t❤❡ •✐❢t ✐s ❝♦♥s✐st❡♥t✱ ♦♥❡ s❤♦✉•❞ ❜❡ ❛❜•❡ t♦ r❡❞❡r✐✈❡ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❉❛r❜♦✉① ❝♦✲ ♦r❞✐♥❛t❡s✱ ♣r♦✈✐❞❡❞ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ✜❜❡r ❝♦♦r❞✐♥❛t❡✳ ❋✐♥❛••②✱ s✐♥❝❡ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ❛r❡ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ♣❛t❝❤❡s ❝♦✈❡r✐♥❣ t❤❡ t✇✐st♦r s♣❛❝❡✱ ❝♦♥✲ str❛✐♥ts ♦♥ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ❛•s♦ ✜① ❤♦✇ t❤❡ ♣❛t❝❤❡s s❤♦✉•❞ r❡s❤✉✤❡✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ s②♠♠❡tr✐❡s✱ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡ ♣❛t❝❤❡s t♦ ✇❤✐❝❤ t❤❡② ❛r❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤✱ tr❛♥s❢♦r♠ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ❝♦✈❡r✐♥❣ ♦❢ Z r❡♠❛✐♥s ✐♥✈❛r✐❛♥t✳ ❆t t❤✐s st❛❣❡✱ ✐t ✐s ❛•s♦ ✐♠♣♦rt❛♥t t♦ ♥♦t❡ t❤❛t t❤❡ s②♠♠❡tr✐❡s r❡♥❞❡r t❤❡ ❝♦♥t❛❝t str✉❝t✉r❡ ♦♥ Z ✐♥✈❛r✐❛♥t✳ ■♥ ♣❛rt✐❝✉•❛r✱ t❤❡ ❝♦♥t❛❝t ♦♥❡✲❢♦r♠ ❣❡ts ♦♥•② r❡s❝❛•❡❞ ❜② ❛ ❤♦•♦♠♦r♣❤✐❝ ❢❛❝t♦r ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ s②♠♠❡tr✐❡s✳ ▲❡t ✉s ✐♥✈❡st✐❣❛t❡ t❤❡ ❡✛❡❝t ♦❢ t❤✐s ❛ ❜✐t ❡•❛❜♦r❛t❡•②✳ ❉❡t❛✐•❡❞ ❞✐s❝✉ss✐♦♥s ♦❢ t❤❡s❡ •✐❢t✐♥❣ ♣r♦❜•❡♠s ❛•r❡❛❞② ❡①✐st ✐♥ t❤❡ •✐t❡r❛t✉r❡s ❬✶✽✱ ✷✶✱ ✷✷✱ ✷✽✕✸✵(✳ ❆s ✇❛s r❡♠❛r❦❡❞ ❜❡❢♦r❡ t❤❛t ❝r✉❝✐❛• s✐♠♣•✐✜❝❛t✐♦♥s ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ✐❢ ♦♥❡ ❢♦r♠✉•❛t❡s ❝♦♥str✉❝t✐♦♥ ❡①♣•♦✐t✐♥❣ t❤❡ ❝♦♥t❛❝t ❜r❛❝❦❡t ❢♦r♠✉•❛t✐♦♥✱ ✇❡ ✇✐•• ❢♦❝✉s ♦♥ t❤✐s ♦♥•② ❤❡r❡ ❢♦••♦✇✐♥❣ ❬✷✸✱ ✷✹(✳ ❲❡ ✇❛♥t t♦ ✜♥❞ ❤♦✇ t❤❡ ❝♦♥t❛❝t ❜r❛❝❦❡t✱ ✭✸✳✷✵✮ ❜❡❤❛✈❡s ✉♥❞❡r ❝♦♥t❛❝t tr❛♥s❢♦r♠❛t✐♦♥ ̺ : X → λX ✱ ✇❤❡r❡ λ ✐s ❛ ❤♦•♦♠♦r♣❤✐❝ ❢❛❝t♦r✳ ❋♦r t❤✐s ✇❡ ✇✐•• ♣r♦✈❡ t❤❡ ❢♦••♦✇✐♥❣✱ ̺ • X h = X λ -1 ̺•h . ✭✸✳✸✵✮ ▲❡t ✉s ❞❡♥♦t❡ t❤❡ tr❛♥s❢♦r♠❡❞ q✉❛♥t✐t✐❡s ❜② ♣r✐♠❡s✳ ❲❡ ❤❛✈❡ ✭✸✳✶✹✮ ❛t ♦✉r ❞✐s♣♦s❛•✳ ■♥st❡❛❞ ♦❢ ♣r♦✈✐♥❣ ✭✸✳✸✵✮✱ ✇❡ ✇✐•• ♣r♦✈❡ t❤❡ ❢♦••♦✇✐♥❣✱ ι X λ -1 h ′ d (λX ) = -dh ′ + R ′ (h ′ )λX , ι X λ -1 h ′ (λX ) = h ′ . ✭✸✳✸✶✮ ❚♦ s❤♦✇ ✭✸✳✸✵✮ ✐t ✐s ❡♥♦✉❣❤ t♦ ♣r♦✈❡✱ λR(λ -1 h ′ )X = λι X λ -1 h ′ dX + λd(λ -1 h ′ ) = λι X λ -1 h ′ dX + dh ′ -(ι X λ -1 h ′ X )dλ, ✭✸✳✸✷✮ ❢r♦♠ ✭✸✳✶✹✮✳ ❚❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥ ✐♥ ✭✸✳✸✶✮ ✐s s❛t✐s✜❡❞ tr✐✈✐❛••②✳ ❚❤❡ ✜rst ♦♥❡ ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ✉s✐♥❣ t❤❡ ❛❜♦✈❡ ❛s t❤❡ ❢♦••♦✇✐♥❣✱ ι X λ -1 h ′ dλ X -ι X λ -1 h ′ X dλ + λ ι X f λ -1 h ′ dX + dh ′ -λR ′ (h ′ )X = λ ι X λ -1 h ′ d log λ + R(λ -1 h ′ ) -R ′ (h ′ ) X = 0. ✭✸✳✸✸✮ ❆♣♣•②✐♥❣ t❤❡ ♠❛♣ ̺ t♦ ✭✸✳✶✹✮ ✇❡ ♦❜t❛✐♥ X (R ′ ) = λ -1 , ι R ′ d (λX ) = (ι R ′ dλ) X -d log λ + λ ι R ′ dX = 0. ✭✸✳✸✹✮ ✸✺ ✸✳ ❚✇✐st♦r✐❛• ❞❡s❝r✐♣t✐♦♥ ♦❢ ◗✉❛t❡r♥✐♦♥ ❑ä❤•❡r ♠❛♥✐❢♦•❞s ❈♦♥tr❛❝t✐♥❣ t❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥ ✇✐t❤ X λ -1 h ′ ✇❡ ❣❡t λ -1 h ′ R ′ (λ) -ι X λ -1 h ′ d log λ -λ ι R ′ ι X λ -1 h ′ dX = R ′ (h ′ ) -ι X λ -1 h ′ d log λ -R(λ -1 h ′ ) = 0, ✭✸✳✸✺✮ ❝♦♠♣•❡t✐♥❣ t❤❡ ♣r♦♦❢ ♦❢ ✭✸✳✸✸✮✳ ❚❤✉s ✇❡ ❦♥♦✇ ❤♦✇ ρ ❛❝ts ♦♥ t❤❡ ❝♦♥t❛❝t ❜r❛❝❦❡ts ❢r♦♠ ✭✸✳✸✵✮✳ ■t ✐s ❣✐✈❡♥ ❜② ̺ • {h, f } = {λ -1 ̺ • h, ̺ • f }. ✭✸✳✸✻✮ ❚❤✐s ♣r♦♣❡rt② ❣❡♥❡r❛•✐③❡s t❤❡ ❢❛♠✐•✐❛r ✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ P♦✐ss♦♥ ❜r❛❝❦❡t ✉♥❞❡r ❝❛♥♦♥✐❝❛• tr❛♥s❢♦r♠❛t✐♦♥s t♦ t❤❡ r❡❛•♠ ♦❢ ❝♦♥t❛❝t ❣❡♦♠❡tr②✳ ❚❤❡ ✉s❡❢✉•♥❡ss ♦❢ ♦❢ t❤❡ ♣r♦♣❡rt② ✐♥ ✭✸✳✸✻✮ ✇✐•• ❜❡ r❡✈❡❛•❡❞ ✐♥ ❞✉❡ ❝♦✉rs❡ ♦❢ t❤❡ t❤❡s✐s✱ ✐♥ ♣❛rt✐❝✉•❛r✱ ✇❤❡♥ ✇❡ ❤❛✈❡ t♦ ❞❡❛• ✇✐t❤ ❙✲❞✉❛•✐t② s②♠♠❡tr② ♦♥ t❤❡ t②♣❡ ■■❇ s✐❞❡✳ ✸✳✺ ❊✈❛•✉❛t✐♦♥ ♦❢ t❤❡ ♠❡tr✐❝ ♦♥ t❤❡ ◗❑ ♠❛♥✐❢♦•❞s ❚❤✐s s❡❝t✐♦♥ ✐s ❞❡✈♦t❡❞ t♦✇❛r❞s ❡①♣•❛✐♥✐♥❣ t❤❡ ♣r♦❝❡❞✉r❡ ♦❢ ❝♦♠♣✉t✐♥❣ t❤❡ ♠❡tr✐❝ ♦♥ ◗❑ s♣❛❝❡ st❛rt✐♥❣ ❢r♦♠ t❤❡ ❞❛t❛ ♦♥ ✐ts t✇✐st♦r s♣❛❝❡✳ ❚❤❡ ❞✐s❝✉ss✐♦♥ ✐s ❛•♦♥❣ t❤❡ •✐♥❡s ♦❢ ❬✶✶✱ ✷✺(✳ ■♥ t❤❡ ♥❡①t ❝❤❛♣t❡r ✇❡ ♣r❡s❡♥t t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛♥ ❡①♣•✐❝✐t ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ♠❡tr✐❝ t♦ ❛•• ♦r❞❡rs ✐♥ ✐♥st❛♥t♦♥ ❡①♣❛♥s✐♦♥ ❢♦r ❛ s♣❡❝✐❛• ❝❛s❡✳ ❙✐♥❝❡ •❛t❡r ✇❡ ✇✐•• ❞❡r✐✈❡ t❤❡ ♠❡tr✐❝ ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s H [ij] ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ α✱ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ❢♦r♠✉•❛s ✇❡ ❞r♦♣ t❤✐s ❞❡♣❡♥❞❡♥❝❡ ❢r♦♠ t❤❡ ✈❡r② ❜❡❣✐♥♥✐♥❣✳ ❋♦r t❤✐s ❝❛s❡ ♠♦r❡♦✈❡r t❤❡ ❝♦♥t❛❝t ♣♦t❡♥t✐❛• ❜❡❝♦♠❡s ❣•♦❜❛••② ✇❡•• ❞❡✜♥❡❞✳ ❖✉r ✐♥t❡r❡st •✐❡s ✐♥ t❤❡ ❝❛s❡ ✇❤❡♥ t❤❡ ❝✉r✈❛t✉r❡ ν ✐s ♥❡❣❛t✐✈❡✳ ❍❡♥❝❡ ✇❡ s❡t ν = -1✱ ❛s ✐t ❝❛♥ ❜❡ ❛❞•✉st❡❞ ❜② ❝❤❛♥❣✐♥❣ t❤❡ ✈❛•✉❡ ♦❢ t❤❡ ❝♦s♠♦•♦❣✐❝❛• ❝♦♥st❛♥t ❛✛❡❝t✐♥❣ t❤❡ ♠❡tr✐❝ ❜② ❛♥ ♦✈❡r❛•• ❢❛❝t♦r ♦♥•②✳ ❋✐rst ♦♥❡ ♥❡❡❞s t♦ ❡①♣❛♥❞ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ❛r♦✉♥❞ t❤❡ ♥♦rt❤ ♣♦•❡ ✐♥ t❤❡ CP 1 ✱ ξ Λ [+] = ξ Λ,-1 [+] t -1 + ξ Λ,0 [+] + O(t), ξ[+] Λ = ξ[+] Λ,0 + O(t), α [+] = -2ic α log t + α [+] 0 + O(t), ✭✸✳✸✼✮ ❋r♦♠ ✭✸✳✸✮ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ ▲❡✈✐✲❈✐✈✐t❛ ♣❛rt ♦❢ t❤❡ SU (2) ❝♦♥♥❡❝t✐♦♥✱ p + = i 4 e -φ ξ Λ,-1 [+] d ξ[+] Λ,0 , p 3 = - 1 4 e -φ dα [+] 0 + ξ Λ,0 [+] d ξ[+] Λ,0 + ξ Λ,-1 [+] d ξ[+] Λ,1 , ✭✸✳✸✽✮ ❢r♦♠ ✇❤✐❝❤ ✇❡ ❞❡t❡r♠✐♥❡ t❤❡ ✷✲❢♦r♠ ω 3 ❜② t❤❡ ❢♦••♦✇✐♥❣✱ ω 3 = -2dp 3 + 4ip + ∧ p -. ✭✸✳✸✾✮ ✸✻ ❊✈❛•✉❛t✐♦♥ ♦❢ t❤❡ ♠❡tr✐❝ ♦♥ t❤❡ ◗❑ ♠❛♥✐❢♦•❞s✳ ✸✳✺ ❚❤❡ ♥❡①t ❛✐♠ ✐s t♦ r❡❡①♣r❡ss t❤✐s ω 3 ✐♥ t❡r♠s ♦❢ ❛ ❜❛s✐s ♦❢ (1, 0)✲❢♦r♠s✳ ❙✉❝❤ ❛ ❜❛s✐s ✇❛s ♣r♦✈✐❞❡❞ ✐♥ ❬✶✶✱ ✷✺( ❛♥❞ r❡❛❞s ❛s π a = d ξ a,-1 [+] /ξ 0,-1 [+] , πΛ = d ξ[+] Λ,0 , πα = 1 2i dα [+] 0 + 2c d log ξ 0,-1 [+] . ✭✸✳✹✵✮ ❚❤✐s ❜❛s✐s s♣❡❝✐✜❡s t❤❡ ❛•♠♦st ❝♦♠♣•❡① str✉❝t✉r❡ J 3 ✳ ❋✐♥❛••② t❤❡ ♠❡tr✐❝ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❢r♦♠ g(X, Y ) = ω 3 (X, J 3 Y )✳ ❚♦ ❞♦ t❤✐s ✐♥ ♣r❛❝t✐❝❡✱ ♦♥❡ s❤♦✉•❞ r❡✇r✐t❡ ω 3 ✱ ❝♦♠♣✉t❡❞ ❜② ✭✸✳✸✾✮ ✐♥ t❡r♠s ♦❢ ❞✐✛❡r❡♥t✐❛•s ♦❢ ✭❣❡♥❡r✐❝❛••② r❡❛•✮ ❝♦♦r❞✐♥❛t❡s ♦♥ M✳ ❚❤❡♥ ✐♥ t❡r♠s ♦❢ t❤❡ ❜❛s✐s π X = (π a , πΛ , πα ) ❣✐✈❡♥ ✐♥ ✭✸✳✹✵✮✳ ❚❤❡ ✜♥❛• r❡s✉•t s❤♦✉•❞ •♦♦❦ •✐❦❡ ω 3 = ig X Ȳ π X ∧ πY , ✭✸✳✹✶✮ ✇❤✐❝❤ ✐s ❡①♣•✐❝✐t•② ♦❢ ✭✶✱✶✮ ❉♦•❜❡❛✉•t t②♣❡✳ ❋r♦♠ ✭✸✳✹✶✮ t❤❡ ♠❡tr✐❝ r❡❛❞✐•② ❢♦••♦✇s ❛s ds 2 = 2g X Ȳ π X ⊗ πY ✳ ■♥ t❤❡ ♥❡①t ❝❤❛♣t❡r ✇❡ ✇✐•• ❞✐s❝✉ss ✐♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ t❤✐s ♣r♦❝❡❞✉r❡ ❡①♣•✐❝✐t•② ✐♥ s♦♠❡ s♣❡❝✐❛• ❝❛s❡s✳ ✸✼ ❈❍❆P❚❊| ✹ ◗✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s t♦ t❤❡ ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡ ❛♥❞ ❉✲✐♥st❛♥t♦♥s ■♥ t❤✐s ❝❤❛♣t❡r ✇❡ ✐♥✐t✐❛t❡ t❤❡ ❛♥❛•②s✐s ♦❢ ✐♥❝•✉❞✐♥❣ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s t♦ t❤❡ ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡ M HM ✳ ❆s ✇❛s ♠❡♥t✐♦♥❡❞✱ t❤❡s❡ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ❝❛♥ ❜❡ ♦❢ t✇♦ t②♣❡s✱ ♣❡rt✉r❜❛t✐✈❡ ❛♥❞ ♥♦♥✲♣❡rt✉r❜❛t✐✈❡✳ ❋✐rst✱ ✇❡ ✇✐•• ✜♥❞ ♦✉t ❤♦✇ M HM ❣❡ts ❞❡❢♦r♠❡❞ ❜② ✐♥❝•✉s✐♦♥ ♦❢ ♣❡rt✉r❜❛t✐✈❡ ❝♦rr❡❝t✐♦♥s ✐♥ str✐♥❣ ❝♦✉♣•✐♥❣ ❝♦♥st❛♥t g s ✳ ❚❤❡♥ ✇❡ ✇✐•• ♣r♦❝❡❡❞ ♦♥ t♦ ✜♥❞✐♥❣ ✐ts ❞❡s❝r✐♣t✐♦♥ ✇❤❡♥ ❉✲✐♥st❛♥t♦♥s ❛r❡ ♣r❡s❡♥t r❡str✐❝t✐♥❣ t♦ t❤❡ t②♣❡ ■■❆ ❢♦r♠✉•❛t✐♦♥✳ ❲❡ ✇✐•• ❞♦ t❤✐s ❜② ❡①♣•♦✐t✐♥❣ t❤❡ t✇✐st♦r ❢♦r♠❛•✐s♠ t❤❛t ❤❛s ❜❡❡♥ ❞✐s❝✉ss❡❞ ✐♥ ❝❤❛♣t❡r ✸✳ ❆❢t❡r t❤❛t ✇❡ ✇✐•• ❝♦♠♣✉t❡ ❛♥ ❡①♣•✐❝✐t ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ♠❡tr✐❝ ✇❤❡♥ ❛ ♣❛rt✐❝✉•❛r s✉❜s❡t ♦❢ t❤❡ ❉✲✐♥st❛♥t♦♥s ✐s ♣r❡s❡♥t ❬✶✶(✳ ❲❡ ✇✐•• ❛•s♦ ❝♦♥s✐❞❡r ✐♥ ❞❡t❛✐• t❤❡ ❢♦✉r ❞✐♠❡♥s✐♦♥❛• ♠♦❞✉•✐ s♣❛❝❡ r❡s✉•t✐♥❣ ❢r♦♠ t❤❡ ❝♦♠♣❛❝t✐✜❝❛t✐♦♥ ♣❡r❢♦r♠❡❞ ♦♥ ❛ r✐❣✐❞ ❈❛•❛❜✐✲(❛✉ ♠❛♥✐❢♦•❞ ❛♥❞ ✐ts r❡•❛t✐♦♥ ✇✐t❤ ❚♦❞❛ ❡q✉❛t✐♦♥✳ ❲❡ ✇✐•• ✜♥✐s❤ t❤✐s ❝❤❛♣t❡r ❜② ❞✐s❝✉ss✐♥❣ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❝✉r✈❛t✉r❡ s✐♥❣✉•❛r✐t② ❛♣♣❡❛r✐♥❣ ✐♥ t❤❡ ♣❡rt✉r❜❛t✐✈❡ ♠❡tr✐❝ ✭✹✳✶✮✱ ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❉✲✐♥st❛♥t♦♥s✳ ✹✳✶ ❖♥❡ •♦♦♣ ❝♦rr❡❝t❡❞ ♠❡tr✐❝ ❆t t❤❡ ♦♥❡✲•♦♦♣ •❡✈❡• t❤❡ ♠❡tr✐❝ ♦♥ M HM ✐s ✇❡•• ❦♥♦✇♥ ❬✶✻✱ ✶✼✱ ✸✶✕✸✸(✳ ■t ❤❛s ❛•s♦ ❜❡❡♥ r❡♣r♦❞✉❝❡❞ ✉s✐♥❣ t✇✐st♦r✐❛• ❛♥❛•②s✐s ✐♥ ❬✷✺(✳ ❚❤❡ r❡s✉•t ✐s✱ ds 2 = r + 2c r 2 (r + c) dr 2 - 1 r N ΛΣ - 2(r + c) rK z Λ zΣ d ζΛ -F ΛΛ ′ dζ Λ ′ d ζΣ -FΣΣ ′ dζ Σ ′ + r + c 16r 2 (r + 2c) dσ + ζΛ dζ Λ -ζ Λ d ζΛ + 8cA K 2 + 4(r + c) r K a bdz a dz b , ✭✹✳✶✮ ✸✽ ❖♥❡ •♦♦♣ ❝♦rr❡❝t❡❞ ♠❡tr✐❝✳ ✹✳✶ ✇❤❡r❡ ❛s ✇❛s ✐♥tr♦❞✉❝❡❞ ❜❡❢♦r❡✱ N ΛΣ = -2 Im F ΛΣ ✐s ❛♥ ✐♥✈❡rt✐❜•❡ ♠❛tr✐① ✇✐t❤ ✐♥✲ ❞❡✜♥✐t❡ s✐❣♥❛t✉r❡ ✭N ΛΣ ❜❡✐♥❣ t❤❡ ✐♥✈❡rs❡✮ ♦♥ t❤❡ s♣❡❝✐❛• ❑ä❤•❡r ❜❛s❡ ♣❛r❛♠❡tr✐③❡❞ ❜② z a ✱ A K ✐s t❤❡ ❑ä❤•❡r ❝♦♥♥❡❝t✐♦♥ A K = i 2 (K a dz a -K ādz ā)✳ ❚❤❡ ♦♥❡✲•♦♦♣ ❝♦r✲ r❡❝t✐♦♥ ✐s ♣❛r❛♠❡tr✐③❡❞ ❜② c ✇❤✐❝❤ ✐s c = - χ Y 192π ✱ ✇❤❡r❡ χ Y = 2 (h 1,1 (Y) -h 2,1 (Y)) ✐s t❤❡ ❊✉•❡r ❝•❛ss ❢♦r t❤❡ ❈❛•❛❜✐✲(❛✉ t❤r❡❡❢♦•❞ Y✳ ❚♦♣♦•♦❣✐❝❛••② t❤❡ ♦♥❡✲•♦♦♣ ❞❡✲ ❢♦r♠❡❞ ♠❡tr✐❝ st✐•• ❤❛s t❤❡ str✉❝t✉r❡ ♦❢ ✭✷✳✸✻✮✳ ■t ✐s ✇❡•• ❦♥♦✇♥ t❤❛t t❤❡ ♣❡rt✉r❜❛t✐✈❡ ♠❡tr✐❝ ❞♦❡s ♥♦t ❛••♦✇ ❝♦rr❡❝t✐♦♥s ❤✐❣❤❡r t❤❛♥ ♦♥❡ •♦♦♣ ✐♥ g s ✳ ❚❤❡ ❡✈✐❞❡♥❝❡ ❢♦r ✐t ❝❛♥ ❜❡ ✉♥❞❡rst♦♦❞ ❜② ❛♥❛•②s✐s ♦❢ t❤❡ str✉❝t✉r❡ ♦❢ ✭✹✳✶✮✳ ❲❡ ♦❜s❡r✈❡ t❤❛t t❤❡ ♦♥❡✲•♦♦♣ ♣❛r❛♠❡t❡r c ❝♦rr❡❝ts ✈❛r✐♦✉s t❡r♠s ✇❤✐❝❤ ❞❡♣❡♥❞ ♦♥ t❤❡ ❞✐•❛t♦♥ r ❛s ✇❡•• ❛s t❤❡ ❝♦♥♥❡❝t✐♦♥ ♦❢ t❤❡ σ ❝✐r❝•❡ ❜✉♥❞•❡ ❜② ✐♥tr♦❞✉❝✐♥❣ ❛ t❡r♠ ♣r♦♣♦rt✐♦♥❛• t♦ t❤❡ ❑ä❤•❡r ❝♦♥♥❡❝t✐♦♥ A K ✱ Dσ = dσ + ζΛ dζ Λ -ζ Λ d ζΛ + 8cA K . ✭✹✳✷✮ ❚❤❡ ❝✉r✈❛t✉r❡ ♦❢ t❤❡ ❝✐r❝•❡ ❜✉♥❞•❡ ♦❢ t❤❡ ❛①✐♦♥ σ ✐♥ ✭✷✳✸✻✮ t❛❦❡s t❤❡ ❢♦••♦✇✐♥❣ ❢♦r♠ d( 1 2 Dσ) = ω T - χ Y 48π dA K , ✭✹✳✸✮ ✇❤❡r❡ ω T = d ζΛ ∧dζ Λ ✐s t❤❡ ❝✉r✈❛t✉r❡ ❢♦r♠ ♦❢ t❤❡ t♦r✉s T ζ, ζ = H 3 (Y, R)/H 3 (Y, Z) ♣❛r❛♠❡tr✐③✐♥❣ t❤❡ ✜❜❡r ♦❢ t❤❡ ✐♥t❡r♠❡❞✐❛t❡ ❏❛❝♦❜✐❛♥ J c (Y)✳ P❡rt✉r❜❛t✐✈❡ ❝♦rr❡❝✲ t✐♦♥s ❜❡②♦♥❞ ♦♥❡✲•♦♦♣ ✇✐•• ♣r❡s✉♠❛❜•② •❡❛❞ t♦ ♣r❡s❡♥❝❡ ♦❢ ♣♦s✐t✐✈❡ ♣♦✇❡r t❡r♠s ✐♥ r ✐♥ t❤❡ ❝♦♥♥❡❝t✐♦♥ ♦❢ t❤❡ σ ❝✐r❝•❡ ❜✉♥❞•❡ ❝♦♥✢✐❝t✐♥❣ ✇✐t❤ t❤❡ q✉❛♥t✐③❛t✐♦♥ ❝♦♥❞✐✲ t✐♦♥ ♦♥ t❤❡ ✜rst ❈❤❡r♥ ❝•❛ss ❬✷✶(✳ ■♥ ♣❛rt✐❝✉•❛r✱ t❤✐s ❛r❣✉♠❡♥t ✐♥❞✐❝❛t❡s ✇❤② ❤✐❣❤❡r •♦♦♣ ❝♦rr❡❝t✐♦♥s t❤❛♥ ♦♥❡✲•♦♦♣ ❛r❡ ❢♦r❜✐❞❞❡♥✳ ✹✳✶✳✶ ❚✇✐st♦r ❞❡s❝r✐♣t✐♦♥ ♦❢ ♦♥❡✲•♦♦♣ ❝♦rr❡❝t❡❞ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ▲❡t ✉s ♥♦✇ ❣✐✈❡ t❤❡ t✇✐st♦r✐❛• ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♦♥❡✲•♦♦♣ ❝♦rr❡❝t❡❞ ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡ ❞❡s❝r✐❜❡❞ ❜② t❤❡ ♠❡tr✐❝ ✭✹✳✶✮✳ ❆s ✇❡ ❤❛✈❡ ❛•r❡❛❞② •❡❛r♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡r✱ ✐t ✐s s✉✣❝✐❡♥t t♦ ♣r♦✈✐❞❡ ❛ ❝♦✈❡r✐♥❣ ❛♥❞ t❤❡ s❡t ♦❢ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ❛ss♦❝✐❛t❡❞ t♦ ✐t✳ ■♥ ♣r❛❝t✐❝❡✱ ✇❡ ✇✐•• ♦♥•② ✇♦r❦ •♦❝❛••② ✐♥ t❤❡ ♠♦❞✉•✐ s♣❛❝❡✳ ❚❤✐s ♠❡❛♥s t❤❛t ✇❡ s❤♦✉•❞ ♦♥•② ❝♦♥s✐❞❡r t❤❡ ❝♦✈❡r✐♥❣ ♦❢ t❤❡ CP 1 ✜❜❡r ❛♥❞ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ❝❛♥ ❞❡✈❡•♦♣ s✐♥❣✉•❛r✐t✐❡s ♦♥•② ✐♥ t✳ ❚❤✉s ✇❡ ❤❛✈❡ t♦ ❝♦♥s✐❞❡r t❤❡ |✐❡♠❛♥♥ s♣❤❡r❡ ♦❢ t❤❡ CP 1 ✳ ▲❡t ✉s ❝♦✈❡r ✐t t❤r❡❡ ♣❛t❝❤❡s U + , U -❛♥❞ U 0 ✱ ✇❤❡r❡ t❤❡ ✜rst t✇♦ ❝♦✈❡r t❤❡ ♥♦rt❤ ♣♦•❡ ❛♥❞ t❤❡ s♦✉t❤ ♣♦•❡ r❡s♣❡❝t✐✈❡•② ❛♥❞ U 0 ❝♦✈❡rs t❤❡ ❡q✉❛t♦r✐❛• r❡❣✐♦♥✳ ❈•❡❛r•② ♦♥❡ r❡q✉✐r❡s t✇♦ ✐♥❞❡♣❡♥❞❡♥t tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ❢♦r t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✐♥t❡rs❡❝t✐♦♥s ♦❢ U 0 ✇✐t❤ U ± ✳ ■t t✉r♥s ♦✉t t❤❛t ✐❢ ✇❡ ❝❤♦♦s❡ t❤❡ ❢♦••♦✇✐♥❣ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s✱ H [+0] = F (ξ), H [0-] = F (ξ), ✭✹✳✹✮ ♦♥❡ r❡❝♦✈❡rs t❤❡ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ❝•❛ss✐❝❛• ♠❡tr✐❝ ✭✷✳✸✺✮✳ ❚♦ ✐♥❝♦r♣♦r❛t❡ ♦♥❡✲ •♦♦♣ ❝♦rr❡❝t✐♦♥✱ ♦♥❡ s❤♦✉•❞ ❛❞❞ t❤❡ ❛♥♦♠❛•♦✉s ❞✐♠❡♥s✐♦♥s c α = -2c✳ ❚❤❡♥ ✐t ✐s str❛✐❣❤t❢♦r✇❛r❞ t♦ r❡❝♦✈❡r ✭✹✳✶✮ ❢♦••♦✇✐♥❣ t❤❡ ♣r♦❝❡❞✉r❡ ♣r❡s❡♥t❡❞ ✐♥ ✸✳✺✳ ✸✾ ✹✳ ◗✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s t♦ t❤❡ ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡ ❛♥❞ ❉✲✐♥st❛♥t♦♥s ❚❤✐s ✐s ❛♥ ❡①❛♠♣•❡ ♦❢ t♦r✐❝ ◗❑ ♠❛♥✐❢♦•❞ ✭t❤❛t ❤❛s n + 1 ❝♦♠♠✉t✐♥❣ ✐s♦♠❡tr✐❡s✮ ❢♦r ✇❤✐❝❤ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡ ξ Λ ✐s ❣•♦❜❛••② ✇❡••✲❞❡✜♥❡❞✳ ❋♦r t❤✐s r❡❛s♦♥ ✇❡ ❛❜st❛✐♥ ❢r♦♠ ✇r✐t✐♥❣ t❤❡ ♣❛t❝❤ ✐♥❞❡① ❢♦r t❤❡♠ ✐♥ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥✳ ❲❡ ❛•s♦ ✇r✐t❡ ❞♦✇♥ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ♦♥ t❤❡ ♣❛t❝❤ U 0 ✱ ξ Λ [0] = ζ Λ + R(t -1 z Λ -tz Λ ), ξ[0] Λ = ζΛ + R(t -1 F Λ (z) -t FΛ (z)), α[0] = σ + R(t -1 W (z) -t W (z)) -8ic log t, ✭✹✳✺✮ ✇❤❡r❡ α[0] = -2α [0] -ξ Λ [0] ξ[0] Λ ✐s ❛ s②♠♣•❡❝t✐❝ ✐♥✈❛r✐❛♥t ♦❜ •❡❝t ❝♦♥str✉❝t❡❞ ❢r♦♠ α [0] ✱ ❛♥❞ W (z) = F Λ (z)ζ Λ -z Λ ζΛ . ✭✹✳✻✮ ❋✉rt❤❡r♠♦r❡ R ✐s r❡•❛t❡❞ t♦ t❤❡ ❞✐•❛t♦♥✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ t❤❡ ❞✐•❛t♦♥ ✐s ✐❞❡♥t✐✜❡❞ ✇✐t❤ t❤❡ ❝♦♥t❛❝t ♣♦t❡♥t✐❛• ✇❤✐❝❤ ❢♦r t❤✐s ❝❛s❡ ✐s ❣•♦❜❛••② ✇❡••✲❞❡✜♥❡❞ ❛s ✇❡ ❛r❡ ✇♦r❦✐♥❣ ✇✐t❤ ❛ t♦r✐❝ ◗❑ ♠❛♥✐❢♦•❞✱ e Φ = e φ = R 2 4 K(z, z) -c. ✭✹✳✼✮ ✹✳✷ ❉✲✐♥st❛♥t♦♥s ✐♥ t✇✐st♦r s♣❛❝❡ ❚❤❡ ♠❡tr✐❝ ✭✹✳✶✮ ❣✐✈❡♥ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥ ♣r♦✈✐❞❡s t❤❡ ❝♦♠♣•❡t❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ ♣❡rt✉r❜❛t✐✈❡•② ❝♦rr❡❝t❡❞ ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡✳ ❚❤❡ ♥❡①t st❡♣ ✐s t♦ ✐♥❝•✉❞❡ ✐♥st❛♥t♦♥s✳ ❲❡ ❦♥♦✇ ❛•r❡❛❞② ❢r♦♠ ❝❤❛♣t❡r ✷ t❤❛t t❤❡② ❝❛♥ ❜❡ ♦❢ t❤❡ t✇♦ t②♣❡s ✿ ❉✲ ✐♥st❛♥t♦♥s ❛♥❞ ◆❙✺✲✐♥st❛♥t♦♥s✳ ❚❤❡ •❛tt❡r ❛r❡ ❡①♣♦♥❡♥t✐❛••② s✉♣♣r❡ss❡❞ ❝♦♠♣❛r❡❞ t♦ t❤❡ ❢♦r♠❡r ✭✇❡ ❛ss✉♠❡ t❤❛t ✇❡ ❛r❡ ✇♦r❦✐♥❣ ✐♥ t❤❡ r❡❣✐♦♥ ♦❢ t❤❡ ♠♦❞✉•✐ s♣❛❝❡ ✇❤❡r❡ t❤❡ str✐♥❣ ❝♦✉♣•✐♥❣ ✐s s♠❛••✮ ❛♥❞ ✇✐•• ❜❡ ✐❣♥♦r❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✳ ❇❡s✐❞❡s✱ ✐♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✇✐•• ❜❡ ✇♦r❦✐♥❣ ✐♥ t❤❡ t②♣❡ ■■❆ s✐❞❡✳ ❚❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ t✇✐st♦r s♣❛❝❡ ✐♥ ♣r❡s❡♥❝❡ ♦❢ ❉✲✐♥st❛♥t♦♥s ❤❛❞ ❜❡❡♥ ❢♦✉♥❞ ✐♥ ❬✶✽✱ ✶✾(✳ ❙✐♥❝❡ ✇❡ ❛r❡ ✇♦r❦✐♥❣ ♦♥ t❤❡ t②♣❡ ■■❆ s✐❞❡✱ t❤❡ ❉✲✐♥st❛♥t♦♥s ❝❛♥ ♣♦ss✐❜•② ✇r❛♣ ♦♥•② ♥♦♥✲❝♦♥tr❛❝t✐❜•❡ t❤r❡❡ ❝②❝•❡s✳ ■t s❤♦✉•❞ ❜❡ ❛ s♣❡❝✐❛• ▲❛❣r❛♥❣✐❛♥ s✉❜♠❛♥✐❢♦•❞ ✐♥ t❤❡ ❤♦♠♦•♦❣② ❝•❛ss q Λ γ Λ -p Λ γ Λ ∈ H 3 (Y, Z)✱ ✇❤❡r❡ t❤❡ ✈❡❝t♦rs γ = (p Λ , q Λ ) ❛r❡ t❤❡ ❝❤❛r❣❡s ♦❢ t❤❡ ❉✲❜r❛♥❡s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡♠✳ ❖✉r ❣♦❛• ❤❡r❡ ✐s t♦ ❞❡s❝r✐❜❡ ❝♦rr❡❝t✐♦♥s t❤❛t ❛♣♣❡❛r t♦ M HM ♦✇✐♥❣ t♦ t❤❡ ♣r❡s❡♥❝❡ ♦❢ s✉❝❤ ❇P❙ ♦❜•❡❝ts✳ ❖✉r ❝♦♥str✉❝t✐♦♥ ❝r✉❝✐❛••② ❤✐♥❣❡s ♦♥ t❤❡ t❡❝❤♥✐q✉❡s ❞❡✈❡•♦♣❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡r✳ ■♥tr♦❞✉❝t✐♦♥ ♦❢ t❤❡ ❉✲✐♥st❛♥t♦♥s ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s ❛ r❡✜♥❡♠❡♥t ♦❢ t❤❡ ❝♦✈❡r✐♥❣ ♦❢ t❤❡ t✇✐st♦r s♣❛❝❡ ❝♦♠♣❛r❡❞ t♦ t❤❡ ♦♥❡✲•♦♦♣ ❝♦rr❡❝t❡❞ ❝❛s❡✳ ❖♥❝❡ t❤❡ ✜♥❛• ❞❡s❝r✐♣t✐♦♥ ✐s ♦❜t❛✐♥❡❞✱ ✐t ✐s ✐♥ ♣r✐♥❝✐♣•❡ ♣♦ss✐❜•❡ t♦ ❝♦♠♣✉t❡ t❤❡ ♠❡tr✐❝ ♦♥ t❤❡ ❉✲✐♥st❛♥t♦♥ ❝♦rr❡❝t❡❞ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ❢♦••♦✇✐♥❣ t❤❡ ♣r♦❝❡❞✉r❡ ✐♥ ✸✳✺✳ ❙♦ t❤❡ ✜rst t❛s❦ ✐s t♦ ❞❡s❝r✐❜❡ t❤✐s ♥❡✇ ❝♦✈❡r✐♥❣✳ ❚♦ ❜❡❣✐♥ •❡t ✉s ✜① ♣❛rt✐❝✉•❛r ✈❛•✉❡ ❢♦r t❤❡ ❝♦♠♣•❡① str✉❝t✉r❡ ♠♦❞✉•✐ z a ✳ ❚❤❡♥ t❤❡ ♣❤❛s❡ ♦❢ t❤❡ ❝❡♥tr❛• ❝❤❛r❣❡ ✹✵ ❉✲✐♥st❛♥t♦♥s ✐♥ t✇✐st♦r s♣❛❝❡✳ ✹✳✷ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❞❡t❡r♠✐♥❡ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ s♦ ❝❛••❡❞ ❇P❙ r❛②s ℓ γ ✇❤✐❝❤ ❛r❡ •✐♥❡s ♦♥ t❤❡ CP 1 t❤❛t r✉♥ ❢r♦♠ t❤❡ s♦✉t❤ t♦ t❤❡ ♥♦rt❤ ♣♦•❡ ❞❡✜♥❡❞ ❜②✱ ℓ γ = {t : Z γ /it ∈ iR -}. ✭✹✳✽✮ ❋♦r ❡❛❝❤ ℓ γ t❤❡r❡ ✐s ❛♥ ❛♥t✐✲❜r❛♥❡ ❇P❙ r❛② ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❝❤❛r❣❡ -γ✳ ℓ -γ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜② t❛❦✐♥❣ ❛♥t✐♣♦❞❛• ✐♠❛❣❡ ♦❢ t❤❡ ❢♦r♠❡r✳ ❚♦❣❡t❤❡r t❤❡② ❢♦r♠ ❛ ❝✐r❝•❡ ✇❤✐❝❤ ❞✐✈✐❞❡s t❤❡ ❡q✉❛t♦r✐❛• ♣❛t❝❤ U 0 ✐♥t♦ t✇♦ ♣❛rts✳ ■♥tr♦❞✉❝t✐♦♥ ♦❢ ❛•• ♣♦ss✐❜•❡ ❉✲✐♥st❛♥t♦♥s t❤❡♥ ❛♠♦✉♥ts t♦ s✉❜❞✐✈✐❞✐♥❣ U 0 ❜② ✐♥tr♦❞✉❝✐♥❣ ❛ ❝♦✉♥t❛❜•② ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ s✉❝❤ ❇P❙ r❛②s ✐♥❞❡①❡❞ ❜② t❤❡✐r ❛ss♦❝✐❛t❡❞ ❝❤❛r❣❡s✳ ❆❢t❡r t❤✐s ♦♥❡ ♥❡❡❞s t♦ ❣✐✈❡ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ❢♦r t❤❡s❡ r❛②s✳ ❚❤❡② ❛r❡ ❣✐✈❡♥ ❜② H [ij] = H γ (Ξ [i] γ ) - 1 2 q Λ p Λ (H ′ γ (Ξ [i] γ )) 2 , ✭✹✳✾✮ ✇❤❡r❡ H γ (Ξ γ ) = Ω(γ) 4π 2 Li 2 σ D (γ) e -2πiΞγ , ✭✹✳✶✵✮ ❛♥❞ Ξ [i] γ = q Λ ξ Λ [i] -p Λ ξ[i] Λ ✱ σ D (γ) ❛♥❞ Ω(γ) ❛r❡ t❤❡ s♦ ❝❛••❡❞ q✉❛❞r❛t✐❝ r❡✜♥❡♠❡♥ts ❛♥❞ t❤❡ ❉♦♥❛•❞s♦♥✲❚❤♦♠❛s ✐♥✈❛r✐❛♥t r❡s♣❡❝t✐✈❡•②✱ ✇❤✐❝❤ ✇❡ ❞❡✜♥❡ ❜❡•♦✇✳ ❚❤❡ q✉❛❞r❛t✐❝ r❡✜♥❡♠❡♥t ✐s ❛ ♣❤❛s❡ ❛ss✐❣♥♠❡♥t t♦ t❤❡ ❝❤❛r❣❡ •❛tt✐❝❡ Γ r❡q✉✐r❡❞ ❢♦r t❤❡ ❝♦♥s✐st❡♥❝② ✇✐t❤ t❤❡ ✇❛••✲❝r♦ss✐♥❣ ✇❤✐❝❤ ✇❡ s❤❛•• ❞✐s❝✉ss s❤♦rt•②✳ ■t s❤♦✉•❞ s❛t✐s❢② t❤❡ ❢♦••♦✇✐♥❣ r❡•❛t✐♦♥✱ σ D (γ + γ ′ ) = (-1) γ,γ ′ σ D (γ)σ D (γ ′ ), ✭✹✳✶✶✮ ✇❤❡r❡ , ❞❡♥♦t❡s t❤❡ st❛♥❞❛r❞ s②♠♣•❡❝t✐❝ ♣❛✐r✐♥❣✳ ❚❤❡ ♠♦st ❣❡♥❡r❛• s♦•✉t✐♦♥ t♦ t❤❡ ❛❜♦✈❡ ❝♦♥❞✐t✐♦♥ r❡❛❞s ❛s σ D (γ) = exp 2πi - 1 2 q Λ p Λ + q Λ θ Λ D -p Λ φ D Λ , ✭✹✳✶✷✮ ✇❤❡r❡ (θ Λ D , φ D Λ ) ❛r❡ t❤❡ s♦ ❝❛••❡❞ ❝❤❛r❛❝t❡r✐st✐❝s✳ ■❢ t❤❡② ❛r❡ ❤❛•❢ ✐♥t❡❣❡rs✱ t❤❡ q✉❛❞r❛t✐❝ r❡✜♥❡♠❡♥t ❜❡❝♦♠❡s •✉st ❛ s✐❣♥ ❢❛❝t♦r✳ ❚❤❡ ♥✉♠❡r✐❝❛• ❢❛❝t♦rs Ω(γ)✬s ❛r❡ s♦♠❡ t♦♣♦•♦❣✐❝❛• ✐♥✈❛r✐❛♥ts ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❈❛•❛❜✐✲(❛✉ ♠❛♥✐❢♦•❞s✳ ❲❤❡♥ t❤❡ ♠❛❣♥❡t✐❝ ❝❤❛r❣❡s ❛r❡ ❛❜s❡♥t✱ ✐✳❡✳ p Λ = 0✱ t❤❡② r❡❞✉❝❡ t♦ t❤❡ ❣❡♥✉s ③❡r♦ •♦♣❛❦✉♠❛r✲❱❛❢❛ ✐♥✈❛r✐❛♥ts t❤❛t ❛r❡ ♣r❡s❡♥t ✐♥ t❤❡ ✐♥ t❤❡ ✐♥st❛♥t♦♥ ♣❛rt ♦❢ t❤❡ ❤♦•♦♠♦r♣❤✐❝ ♣r❡♣♦t❡♥t✐❛• ✐♥ ✭✷✳✸✷✮✱ Ω(0, q Λ ) = n (0) qa , Ω(0, 0, q 0 ) = χ Y . ✭✹✳✶✸✮ ❖♥❡ s❤♦✉•❞ ❛•s♦ ✇♦rr② ❛❜♦✉t t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ♣❛t❝❤❡s t❤❛t s✉rr♦✉♥❞ t❤❡ ♥♦rt❤ ❛♥❞ s♦✉t❤ ♣♦•❡s U ± ✱ H [0+] = F (ξ [+] ) + G γ , H [0-] = F (ξ [-] ) -G γ . ✭✹✳✶✹✮ ✹✶ ✹✳ ◗✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s t♦ t❤❡ ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡ ❛♥❞ ❉✲✐♥st❛♥t♦♥s ❚❤❡ ❡①♣•✐❝✐t ❡①♣r❡ss✐♦♥ ♦❢ G γ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✶✾(✳ ❍❡r❡ ✇❡ •✉st ✇❛♥t t♦ ♠❡♥t✐♦♥ t❤❛t G γ ❤❛s t✇♦ ❜r❛♥❝❤ ❝✉ts ❢r♦♠ t = 0 ❛♥❞ t = ∞ t♦ t❤❡ t✇♦ r♦♦ts ♦❢ Ξ [i] γ (t) = 0✱ s✉❝❤ t❤❛t t❤❡ •✉♠♣s ✐♥ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ❣❡t ❝❛♥❝❡••❡❞ ❜② t❤❡ ♦♥❡s ✐♥ H [0+] ❛♥❞ H [-0] •❡❛❞✐♥❣ t♦ r❡❣✉•❛r ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ✐♥ [±] ♣❛t❝❤❡s✳ ❆t t❤✐s st❛❣❡ ✇❡ ✇♦✉•❞ •✐❦❡ t♦ s❤♦✇ ❛♥ ✐♥st❛♥❝❡ ♦❢ t❤❡ ✈✐rt✉❡ ♦❢ t❤❡ ❝♦♥t❛❝t ❜r❛❝❦❡t ❢♦r♠✉•❛t✐♦♥ t❤❛t ✇❛s ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡r✳ ❚❤❡ ❉✲✐♥st❛♥t♦♥ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ ✐♥ ✭✹✳✾✮ ❝❛♥ ❜❡ r❡♣r♦❞✉❝❡❞ ✉s✐♥❣ ✭✸✳✷✺✮ ✐❢ ♦♥❡ ❝❤♦♦s❡s t❤❡ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s t♦ ❜❡ h [γ] (ξ, ξ) = H γ (Ξ γ ). ✭✹✳✶✺✮ ◆♦t✐❝❡ t❤❛t ❛•r❡❛❞② ❢♦r t❤✐s r❡•❛t✐✈❡•② s✐♠♣•❡r ❝❛s❡ ✇❡ s❡❡ t❤❛t ✇❡ ❝❛♥ ❣❡t r✐❞ ♦❢ t❤❡ ♥♦♥✲•✐♥❡❛r t❡r♠ ✐♥ ✭✹✳✾✮ ✉s✐♥❣ ♦✉r ♥❡✇ ❢♦r♠❛•✐s♠ ♦❢ ❝♦♥t❛❝t ❜r❛❝❦❡ts✳ ❲❡ ❛•s♦ ✇r✐t❡ ❞♦✇♥ t❤❡ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ❤❡r❡ ❛s t❤❡② ✇✐•• ❜❡ r❡q✉✐r❡❞ ❢♦r ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ♠❡tr✐❝ ✇❤✐❝❤ ✇❡ ♣r❡s❡♥t •❛t❡r✱ ξ Λ [γ] = ζ Λ + R t -1 z Λ -t zΛ + 1 8π 2 γ ′ Ω(γ ′ )p ′Λ J (1) γ ′ (t), ξ[γ] Λ = ζΛ + R t -1 F Λ -t FΛ + 1 8π 2 γ ′ Ω(γ ′ )q ′ Λ J (1) γ ′ (t), α [γ] = 4ic log t - 1 2 σ - R 2 t -1 W -t W + R 16π 2 γ ′ Ω(γ ′ ) t -1 Z γ ′ + t Zγ ′ J (1) γ ′ (0) - i 16π 3 γ ′ Ω(γ ′ ) ℓ γ ′ dt ′ t ′ t + t ′ t -t ′ L σ D (γ ′ ) e -2πiΞγ (t ′ ) - 1 2 ξ Λ [γ] (t) ξ[γ] Λ (t), ✭✹✳✶✻✮ ✇❤❡r❡ W (z) ≡ F Λ (z)ζ Λ -z Λ ζΛ ✭✹✳✶✼✮ ❛♥❞ ✇❡ ✐♥tr♦❞✉❝❡❞ t✇♦ ❢✉♥❝t✐♦♥s ✶ L ǫ (z) = Li 2 (ǫz) + 1 2 log z log(1 -ǫz), J (1) γ (t) = ℓγ dt ′ t ′ t + t ′ t -t ′ log 1 -σ D (γ)e -2πiΞγ (t ′ ) . ✭✹✳✶✽✮ ✹✳✷✳✶ |❡•❛t✐♦♥s t♦ ✇❛••✲❝r♦ss✐♥❣ ♣❤❡♥♦♠❡♥❛ ❚✐•• t❤✐s ♣♦✐♥t ✇❡ ❝♦♥✜♥❡❞ ♦✉r ❛tt❡♥t✐♦♥ t♦ ❛ ✜①❡❞ ✈❛•✉❡ ♦❢ t❤❡ ❝♦♠♣•❡① str✉❝t✉r❡ ♠♦❞✉•✐✳ ▲❡t ✉s ♥♦✇ ✈❛r② ✐t s•♦✇•② ❛❝r♦ss t❤❡ ♠♦❞✉•✐ s♣❛❝❡✳ ❆s ❛ r❡s✉•t✱ t❤❡ ❝❡♥tr❛• ❝❤❛r❣❡s Z γ (z) ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❝❤❛r❣❡s ❝❤❛♥❣❡ ❛♥❞ ❝♦♥s❡q✉❡♥t•② ❛t ✶ ❚❤❡ ✜rst ❢✉♥❝t✐♦♥ ✐s ❛ ✈❛r✐❛♥t ♦❢ t❤❡ |♦❣❡r ❞✐•♦❣❛r✐t❤♠ ✇❤✐❝❤ s❛t✐s✜❡s t❤❡ ❢❛♠♦✉s ♣❡♥t❛❣♦♥ ✐❞❡♥t✐t② ❛♥❞ ♣•❛②s ❛♥ ✐♠♣♦rt❛♥t r♦•❡ ✐♥ ✐♥t❡❣r❛❜✐•✐t② ❬✸✹(✳ ✹✷ ❉✲✐♥st❛♥t♦♥s ✐♥ t✇✐st♦r s♣❛❝❡✳ ✹✳✷ s♦♠❡ ♣♦✐♥t ❢♦r t✇♦ ❞✐✛❡r❡♥t ❝❤❛r❣❡s t❤❡② ❛•✐❣♥ ✭✐✳❡✳ t❤❡✐r ♣❤❛s❡s ❜❡❝♦♠❡ t❤❡ s❛♠❡✮ ❛❢t❡r ✇❤✐❝❤ t❤❡② ❡①❝❤❛♥❣❡ t❤❡✐r r❡•❛t✐✈❡ ♣♦s✐t✐♦♥s✳ ❇✉t t❤❡ ❝♦♥t❛❝t♦♠♦r♣❤✐s♠s ❛❝r♦ss s✉❝❤ ❇P❙ r❛②s ❞♦ ♥♦t ❝♦♠♠✉t❡ ♣r♦✈✐❞❡❞ t❤❡ ❝❤❛r❣❡s ❛r❡ ♠✉t✉❛••② ♥♦♥✲ •♦❝❛•✱ γ, γ ′ = 0✳ ❚❤✉s ✐t ♠❡❛♥s t❤❛t t✇♦ ❞✐✛❡r❡♥t ♦r❞❡rs ♦❢ ❇P❙ r❛②s •❡❛❞ t♦ t✇♦ ❞✐✛❡r❡♥t s♦•✉t✐♦♥s ❢♦r t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s •❡❛❞✐♥❣ t♦ t✇♦ ❞✐✛❡r❡♥t ♠❡tr✐❝s ❢♦r t❤❡♠✳ ❚❤✐s ✐s ❛♥ ❛♣♣❛r❡♥t ❝♦♥tr❛❞✐❝t✐♦♥ ❛s t❤❡ ♠❡tr✐❝ ♦♥ t❤❡ ♠♦❞✉•✐ s♣❛❝❡ ✐s s✉♣♣♦s❡❞ t♦ ❜❡ s♠♦♦t❤ ❡✈❡r②✇❤❡r❡ ❢r♦♠ ♣❤②s✐❝❛• ❡①♣❡❝t❛t✐♦♥s✳ ❚♦ r❡❝♦♥❝✐•❡ ✇✐t❤ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ♠❡tr✐❝✱ ♦♥❡ ♠✉st t❛❦❡ ✐♥t♦ ❝♦♥s✐❞❡r❛t✐♦♥ t❤❡ ✇❛••✲❝r♦ss✐♥❣ ♣❤❡♥♦♠❡♥❛✳ ■♥❞❡❡❞✱ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❣❛✉❣❡ t❤❡♦r✐❡s ❛♥❞ s✉♣❡r✲ ❣r❛✈✐t② ✐t ✐s ✇❡••✲❦♥♦✇♥ t❤❛t t❤❡r❡ ❡①✐st ❝♦❞✐♠❡♥s✐♦♥ ✶ •✐♥❡s ♦❢ ♠❛r❣✐♥❛• st❛❜✐•✐t② ✭▲▼❙✮ ✐♥ t❤❡ ♠♦❞✉•✐ s♣❛❝❡ ❬✸✺✱ ✸✻(✳ ❆❝r♦ss t❤❡ ▲▼❙ t❤❡ s♣❡❝tr✉♠ ♦❢ ❇P❙ st❛t❡s •✉♠♣s ❞✐s❝♦♥t✐♥✉♦✉s•②✳ ❚❤✐s ♠❡❛♥s t❤❛t ❝r♦ss✐♥❣ t❤❡ ✇❛•• ❡✐t❤❡r t❤❡ ❇P❙ st❛t❡s ❞❡❝❛② ♦r r❡❝♦♠❜✐♥❡ t♦ ❢♦r♠ ❛ ❜♦✉♥❞ st❛t❡✳ ❋♦r ❡①❛♠♣•❡✱ ✐t ✐s ❦♥♦✇♥ t❤❛t t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❝❡♥t❡rs ♦❢ ❛ t✇♦✲❝❡♥t❡r❞ ❜•❛❝❦ ❤♦•❡ ✐s ❣✐✈❡♥ ❜② ❬✸✻( r 12 = 1 2 γ 1 , γ 2 |Z γ 1 + Z γ 2 | Im (Z γ 1 Zγ 2 ) . ✭✹✳✶✾✮ ❚❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤❡ ❞❡♥♦♠✐♥❛t♦r ✐♠♣•✐❡s t❤❛t t❤❡ ♠♦❞✉•✐ s♣❛❝❡ ✐s s♣•✐t ✐♥t♦ t✇♦ ❝❤❛♠❜❡rs ❜② t❤❡ ❝♦❞✐♠❡♥s✐♦♥ ✶ ▲▼❙ ❢♦r ✇❤✐❝❤ ❝❡♥tr❛• ❝❤❛r❣❡s ♦❢ γ 1 ❛♥❞ γ 2 ❛•✐❣♥✳ ❚❤❡s❡ t✇♦ r❡❣✐♦♥s ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❝❛s❡s r 12 > 0 ❛♥❞ r 12 < 0✳ ❙✐♥❝❡ ❛ ♥❡❣❛t✐✈❡ ❞✐st❛♥❝❡ ✐s ♠❡❛♥✐♥❣•❡ss ♣❤②s✐❝❛••②✱ ✐t ✐♥❞✐❝❛t❡s t❤❛t t❤❡ t✇♦✲❝❡♥t❡r❡❞ s♦•✉t✐♦♥ ✐s ✉♥st❛❜•❡ ❛♥❞ ✐t ❞❡❝❛②s✳ ❚❤✉s ✇❡ s❡❡ t❤❛t ▲▼❙ ❛♣♣❡❛rs ✇❤❡♥ t❤❡ ❝❡♥tr❛• ❝❤❛r❣❡s ❛•✐❣♥✳ ■♥ t❤❡ •❛♥❣✉❛❣❡ ♦❢ t✇✐st♦r ❝♦♥str✉❝t✐♦♥ t❤✐s ♠❡❛♥s t❤❛t t❤❡ ❇P❙ r❛②s ℓ γ ✬s ❛•✐❣♥✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ s✐♥❣•❡ ♣❛rt✐❝•❡ s♣❡❝tr✉♠ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ ❇P❙ ✐♥❞✐❝❡s Ω(γ) ❛♥❞ t❤❡② ❛♣♣❡❛r ❡①♣•✐❝✐t•② ✐♥ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s✳ ❚❤❡② ❛•s♦ ❞❡♣❡♥❞ ♦♥ t❤❡ ♠♦❞✉•✐ z I ✱ ❜✉t s✐♥❝❡ t❤❡② ❛r❡ ♣✐❡❝❡✇✐s❡ ❝♦♥st❛♥t ✇❡ ❞r♦♣ t❤✐s ❞❡♣❡♥❞❡♥❝❡ ✐❢ ♥♦ ❝♦♥❢✉s✐♦♥ ❛r✐s❡s✳ ■t t✉r♥s ♦✉t t❤❛t t♦ t❛❦❡ •✉♠♣s ♦❢ t❤❡ ❇P❙ ✐♥❞✐❝❡s ✐♥t♦ ❛❝❝♦✉♥t ✐s ✐♠♣♦rt❛♥t✳ ❇❡❝❛✉s❡ t❤✐s t❤❡ ♦♥•② ♣♦ss✐❜✐•✐t② t❤r♦✉❣❤ ✇❤✐❝❤ ✇❡ ❝❛♥ ❝❛♥❝❡• t❤❡ ❞✐s❝♦♥t✐♥✉✐t② ❛r✐s✐♥❣ ❢r♦♠ r❡♦r❞❡r✐♥❣ ♦❢ t❤❡ ❇P❙ r❛②s✳ ❚❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ❝❛s❡ ❤❛s ❜❡❡♥ ❡st❛❜•✐s❤❡❞ ✐♥ ❬✸✼(✳ ◆♦t❡ t❤❛t t❤❡ ♦♣❡r❛t♦rs e X h [γ] = e {h [γ] ,•} ❣❡♥❡r❛t✐♥❣ t❤❡ ❝♦♥t❛❝t tr❛♥s❢♦r♠❛✲ t✐♦♥s ✐♥❞✉❝❡❞ ❜② ✭✹✳✶✺✮ ❛r❡ ♥♦t❤✐♥❣ ❡•s❡ ❜✉t ❛ •✐❢t t♦ t❤❡ ❝♦♥t❛❝t ❣❡♦♠❡tr② ♦❢ t❤❡ ❑♦♥ts❡✈✐❝❤✲❙♦✐❜❡•♠❛♥ ✭❑❙✮ ♦♣❡r❛t♦rs U Ω(γ) γ s❛t✐s❢②✐♥❣ t❤❡ ✇❛•• ❝r♦ss✐♥❣ ❢♦r✲ ♠✉•❛ ❬✸✽(✳ ■t ❞✐❝t❛t❡s ❤♦✇ t❤❡ ❉❚ ✐♥✈❛r✐❛♥ts ❝❤❛♥❣❡ ❛❢t❡r ❝r♦ss✐♥❣ ❛ ✇❛•• ♦❢ ♠❛r❣✐♥❛• st❛❜✐•✐t② ✐♥ t❤❡ ♠♦❞✉•✐ s♣❛❝❡ ♦❢ t❤❡ ❝♦♠♣•❡① str✉❝t✉r❡ ❞❡❢♦r♠❛t✐♦♥s ♣❛r❛♠❡tr✐③❡❞ ❜② z a ❛♥❞ ❡♥s✉r❡s t❤❡ s♠♦♦t❤♥❡ss ♦❢ t❤❡ ♠♦❞✉•✐ s♣❛❝❡ ♠❡tr✐❝ ❛❝r♦ss t❤❡ ✇❛••s ❬✸✼(✳ Pr♦✈✐❞❡❞ Γ(z) ✐s ❛ s❡t ♦❢ ❝❤❛r❣❡s ❢♦r ✇❤✐❝❤ Z γ (z) ❜❡❝♦♠❡ ❛•✐❣♥❡❞ ❛t ♣♦✐♥t z a ❛♥❞ Ω ± (γ) ❛r❡ t❤❡ ❉❚ ✐♥✈❛r✐❛♥ts ♦♥ t❤❡ t✇♦ s✐❞❡s ♦❢ t❤❡ ✇❛••✱ t❤❡ ❑❙ ❢♦r♠✉•❛ st❛t❡s t❤❛t γ∈Γ(z) U Ω -(γ) γ = γ∈Γ(z) U Ω + (γ) γ , ✭✹✳✷✵✮ ✹✸ ✹✳ ◗✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s t♦ t❤❡ ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡ ❛♥❞ ❉✲✐♥st❛♥t♦♥s ✇❤❡r❡ t❤❡ t✇♦ ♣r♦❞✉❝ts ❛r❡ t❛❦❡♥ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ♦r❞❡r✳ ✭■♥ ❜♦t❤ ❝❛s❡s t❤❡ ♦r❞❡r ❝♦rr❡s♣♦♥❞s t♦ ❞❡❝r❡❛s✐♥❣ t❤❡ ♣❤❛s❡ ♦❢ Z γ ❛t ❛ ❣✐✈❡♥ ♣♦✐♥t ✐♥ t❤❡ ♠♦❞✉•✐ s♣❛❝❡✳✮ ❚❤❡ ♦♣❡r❛t♦rs U γ ❛r❡ ❞❡✜♥❡❞ ❛s U γ = exp ∞ n=1 e nγ n 2 , ✭✹✳✷✶✮ ❛♥❞ e γ s❛t✐s✜❡s t❤❡ ❛•❣❡❜r❛ ♦❢ t❤❡ ❝♦♠♣•❡① t♦r✉s [e γ , e γ ′ ] = (-1) γ,γ ′ γ, γ ′ e γ+γ ′ . ✭✹✳✷✷✮ ■❞❡♥t✐❢②✐♥❣ e γ = σ D (γ)e -2πIΞγ ✱ ♦♥❡ r❡❝♦❣♥✐③❡s ✐♥ t❤❡ ♦♣❡r❛t♦rs U Ω(γ) γ ❛s t❤❡ ❡①✲ ♣♦♥❡♥t✐❛t❡❞ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s H γ ❛♥❞ t❤❡ ❛❞•♦✐♥t ❛❝t✐♦♥ ♦♥ e γ ′ ❣❡♥❡r❛t❡s t❤❡ s②♠♣•❡❝t♦♠♦r♣❤✐s♠s✳ ❆❝t✉❛••② ❢♦r t❤✐s ❝❤❛♣t❡r t❤❡ ❛❜♦✈❡ ❞✐s❝✉ss✐♦♥ s✉✣❝❡s ❛s t❤❡ ❉✲✐♥st❛♥t♦♥s ✐♥❞✉❝❡ s②♠♣•❡❝t♦♠♦r♣❤✐s♠ ✐♥ t❤❡ (ξ, ξ) s✉❜s♣❛❝❡✳ ❍♦✇❡✈❡r✱ ❢♦r t❤❡ ♠♦st ❣❡♥❡r❛• ❝❛s❡✱ ♦♥❡ ♥❡❡❞s t♦ st✐•• ❡①t❡♥❞ t❤❡ ❑❙ ❢♦r♠✉•❛ t♦ t❤❡ r❡❛•♠ ♦❢ ❝♦♥t❛❝t ❣❡♦♠❡tr② ✇❤✐❝❤ ✇❛s ❛❝❤✐❡✈❡❞ ✐♥ ❬✸✾( ❛♥❞ ✇❛s ❞♦♥❡ t❤r♦✉❣❤ ✉s❡ ♦❢ ❝❡rt❛✐♥ ❞✐•♦❣❛r✐t❤♠ ✐❞❡♥t✐t✐❡s✳ ❚❤✐s ❜r✐♥❣s ♦✉t t❤❡ ❢❛❝t t❤❛t t❤❡ s♠♦♦t❤♥❡ss ♦❢ t❤❡ ♠♦❞✉•✐ s♣❛❝❡ M HM ❝r✉✲ ❝✐❛••② ❞❡♣❡♥❞s ♦♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ •✐♥❡s ♦❢ ♠❛r❣✐♥❛• st❛❜✐•✐t②✳ ❆❝r♦ss t❤❡♠ t❤❡ s♣❡❝tr✉♠ ♦❢ ❇P❙ ♣❛rt✐❝•❡s ❝❤❛♥❣❡ ❛♥❞ t❤❡ ✇❛② ✐t ❞♦❡s ✐s ❞✐❝t❛t❡❞ ❜② t❤❡ ❑❙ ✇❛••✲❝r♦ss✐♥❣ ❢♦r♠✉•❛ ✭✹✳✷✵✮✳ ❚❤❡ r❡❛•✐③❛t✐♦♥ ♦❢ t❤✐s ✐♥ t❤❡ t✇✐st♦r✐❛• •❛♥❣✉❛❣❡ t❤❡♥ ❤❛s t❤❡ s✐♠♣•❡ ✐♥t❡r♣r❡t❛t✐♦♥ t❤❛t ❝r♦ss✐♥❣ t❤❡ ✇❛•• •❡❛❞s t♦ r❡s❤✉✤✐♥❣ ♦❢ t❤❡ ❝♦♥t♦✉rs ♦♥ t❤❡ CP 1 ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ❞✐s❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ❛♥❞ t❤❡ •✉♠♣s ♦❢ t❤❡ ❇P❙ ✐♥❞✐❝❡s ♣r❡❝✐s❡•② ❝❛♥❝❡• ❡❛❝❤ ♦t❤❡r✳ ❚❤✐s ✐s ❛❝t✉❛••② ❛ r❡st❛t❡♠❡♥t ♦❢ t❤❡ ❑❙ ❢♦r♠✉•❛ ✐♥ ♦✉r ❝♦♥t❡①t✳ ✹✳✸ ❊①♣•✐❝✐t ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ♠❡tr✐❝ ✐♥ t❤❡ ♠✉t✉✲ ❛••② •♦❝❛• ❝❛s❡ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✇✐•• ❝♦♥str✉❝t ❛♥ ❡①♣•✐❝✐t ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ♠❡tr✐❝ ❢♦r t❤❡ ❝❛s❡ ♦❢ ♠✉t✉❛••② •♦❝❛• ❝❤❛r❣❡s γ, γ ′ ✳ ■♥ t❤❡ ♠♦st ❣❡♥❡r❛• ❝❛s❡ ✐♠♣•❡♠❡♥t✐♥❣ t❤❡ ♣r♦❝❡❞✉r❡ ♦❢ s❡❝t✐♦♥ ✸✳✺ ❛♠♦✉♥ts t♦ s♦•✈✐♥❣ t❤❡ ✐♥t❡❣r❛• ❡q✉❛t✐♦♥s ❢♦r t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ❡①♣•✐❝✐t•②✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❛tt❛✐♥❡❞ ♦r❞❡r ❜② ♦r❞❡r ♦♥•②✳ ❋♦r t❤❡ ❝❛s❡ ♦❢ ❉✲✐♥st❛♥t♦♥ ❝♦rr❡❝t❡❞ ♠❡tr✐❝ ♥♦♥ ♠✉t✉❛••② •♦❝❛• ❝❤❛r❣❡s ✜♥❞✐♥❣ ❛♥ ❡①♣r❡ss✐♦♥ t❤❛t ✐s ✈❛•✐❞ ❢♦r ❛•• ♦r❞❡rs ✐♥ t❤❡ ✐♥st❛♥t♦♥ ❡①♣❛♥s✐♦♥ ✐s ✐♠♣♦ss✐❜•❡ ❛s t❤❡ ❡q✉❛t✐♦♥s t❛❦❡ t❤❡ ❢♦r♠ ♦❢ ❛ s❡t ♦❢ ❚❇❆ ❡q✉❛t✐♦♥s ❬✹✵(✱ ✇❤✐❝❤ ❛r❡ ❦♥♦✇♥ t♦ ❜❡ ✐♠♣♦ss✐❜•❡ t♦ s♦•✈❡ ❡①❛❝t•②✳ ❉✉❡ t♦ t❤✐s r❡❛s♦♥✱ ✇❡ r❡str✐❝t t♦ t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ ❉✲✐♥st❛♥t♦♥ ❝❤❛r❣❡s s❛t✐s❢② γ, γ ′ = 0✳ ❚❤❡ ✜rst st❡♣ ✐s t♦ ❡①tr❛❝t t❤❡ r❡•❡✈❛♥t ▲❛✉r❡♥t ❝♦❡✣❝✐❡♥ts ❢♦r t❤❡ ❉❛r❜♦✉① ✹✹ ❊①♣•✐❝✐t ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ♠❡tr✐❝ ✐♥ t❤❡ ♠✉t✉❛••② •♦❝❛• ❝❛s❡✳ ✹✳✸ ❝♦♦r❞✐♥❛t❡s✱ ξ Λ,-1 [+] = Rz Λ , ✭✹✳✷✸✮ ξ Λ,0 [+] = ζ Λ - 1 8π 2 γ Ω(γ)p Λ J (1) γ , ✭✹✳✷✹✮ ξ[+] Λ,0 = ζΛ -F ΛΣ ζ Σ - 1 8π 2 γ Ω(γ)V γΛ J (1) γ , ✭✹✳✷✺✮ ξ[+] Λ,1 = -iRz Σ N ΛΣ - 1 2R F ΛΣΘ ζ Σ ζ Θ - 1 4π 2 γ Ω(γ) V γΛ J (1,+) γ - 1 2R F ΛΣΘ p Σ ζ Θ J (1) γ + 1 32π 2 R F ΛΣΘ p Σ γ ′ Ω(γ ′ )p ′Θ J (1) γ J (1) γ ′ , ✭✹✳✷✻✮ α [+] 0 = - 1 2 σ + ζ Λ ζΛ -F ΛΣ ζ Λ ζ Σ + 2i (r + c) ✭✹✳✷✼✮ - 1 8π 2 γ Ω(γ) 1 2πi ℓγ dt ′ t ′ Li 2 σ D (γ)e -2πiΞγ (t ′ ) -V γΛ ζ Λ J (1) γ -RZ γ J (1,+) γ + 1 16π 2 p Λ J (1) γ γ ′ Ω(γ ′ )V γ ′ Λ J (1) γ ′ , ✭✹✳✷✽✮ ✇❤❡r❡ ✇❡ ✉s❡❞ t❤❡ ❢♦••♦✇✐♥❣ ❛❜❜r❡✈✐❛t✐♦♥s ❢♦r t❤❡ ✐♥t❡❣r❛•s J (1) γ = ℓγ dt t log 1 -σ D (γ)e -2πiΞγ (t) , ✭✹✳✷✾✮ J (2) γ = ℓγ dt t 1 σ D (γ)e 2πiΞγ (t) -1 , ✭✹✳✸✵✮ J (1,±) γ = ± ℓγ dt t 1±1 log 1 -σ D (γ)e -2πiΞγ (t) , ✭✹✳✸✶✮ J (2,±) γ = ± ℓγ dt t 1±1 1 σ D (γ)e 2πiΞγ (t) -1 . ✭✹✳✸✷✮ ❲❡ ❛•s♦ ✐♥tr♦❞✉❝❡❞ t✇♦ ✉s❡❢✉• ♥♦t❛t✐♦♥s Ξ γ (t) = q Λ ζ Λ -p Λ ζΛ + R(t -1 Z γ -t Zγ ) ≡ Θ γ + R(t -1 Z γ -t Zγ ) ❛♥❞ V γΛ = q Λ -F ΛΣ p Σ ✳ ▲❛st•② t❤❡ ❝♦♥t❛❝t ♣♦t❡♥t✐❛• ❜❡❝♦♠❡s r = R 2 4 K(z, z) -c - iR 32π 2 γ Ω(γ) ℓγ dt t t -1 Z γ -t Zγ log 1 -σ D (γ)e -2πiΞγ (t) . ✭✹✳✸✸✮ ◆♦✇ •❡t ✉s ❝♦♥str✉❝t t❤❡ ❜❛s✐s ♦❢ ♦♥❡✲❢♦r♠s ✉s✐♥❣ ✭✸✳✹✵✮✳ ❚❤✐s ❜❛s✐s ❝❛♥ ❝❡r✲ ✹✺ ✹✳ ◗✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s t♦ t❤❡ ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡ ❛♥❞ ❉✲✐♥st❛♥t♦♥s t❛✐♥•② ❜❡ ❢✉rt❤❡r s✐♠♣•✐✜❡❞ ❛♥❞ ❛r❡ ❣✐✈❡♥ ❜②✱ dz a , Y Λ = d ζΛ -F ΛΣ dζ Σ - 1 8π 2 γ Ω(γ) q Λ -p Σ F ΛΣ dJ (1) γ , Σ = dr + 2c d log R - i 16π 2 γ Ω(γ) RZ γ dJ (1,+) γ -J (1,-) γ d R Zγ + i 4 dσ + ζΛ dζ Λ -ζ Λ d ζΛ . ✭✹✳✸✹✮ ❆s ✇❛s ♠❡♥t✐♦♥❡❞ t❤❡ ❣♦❛• ✐s t♦ r❡✇r✐t❡ t❤❡ q✉❛t❡r♥✐♦♥✐❝ ✷✲❢♦r♠ ω 3 ❛s ✭✸✳✹✶✮ ✉s✐♥❣ t❤❡ (1, 0)✲❢♦r♠s ✐♥ ✭✹✳✸✹✮✳ ❋♦r t❤✐s ✇❡ ❝♦♠♣✉t❡ t❤❡ SU (2) ♣❛rts ♦❢ t❤❡ ▲❡✈✐✲ ❈✐✈✐t❛ ❝♦♥♥❡❝t✐♦♥ ✜rst✳ ❯s✐♥❣ ✭✸✳✸✽✮ ✐t ✐s ❢♦✉♥❞ t♦ ❜❡ p + = i 4r Rz Λ d ζΛ -F ΛΣ dζ Σ - R 8π 2 γ Ω(γ)Z γ dJ (1) γ , p 3 = 1 8r dσ + ζΛ dζ Λ -ζ Λ d ζΛ + 2R 2 KA K - R 4π 2 γ Ω(γ) J (1,+) γ dZ γ -J (1,-) γ d Zγ . ✭✹✳✸✺✮ ❚❤❡♥ ω 3 ❜❡❝♦♠❡s ω 3 = 1 4r 2 dr ∧ dσ + ζΛ dζ Λ -ζ Λ d ζΛ - R 4π 2 γ Ω(γ) J (1,+) γ dZ γ -J (1,-) γ d Zγ + R 2 K 2r d log r R 2 ∧ A K + 1 2r dζ Λ ∧ d ζΛ -iR 2 N ΛΣ dz Λ ∧ dz Σ + iR 2 2r z Λ zΣ Y Λ ∧ ȲΣ + 1 16π 2 r γ Ω(γ) dJ (1,+) γ ∧ d (RZ γ ) -dJ (1,-) γ ∧ d R Zγ . ✭✹✳✸✻✮ ❲❡ ✇✐•• ♥♦t r❡♣r♦❞✉❝❡ t❤❡ ❢✉•• ❝❛•❝✉•❛t✐♦♥ ❤❡r❡✳ ❲❡ r❡❢❡r t❤❡ r❡❛❞❡r t♦ ❛♣♣❡♥❞✐① ❇ ♦❢ ❬✶✶( ❢♦r t❤✐s✳ ❲❡ ♦♥•② ❣✐✈❡ t❤❡ ✜♥❛• ❡①♣r❡ss✐♦♥ ❢♦r ω 3 ❤❡r❡ ✇r✐tt❡♥ ✐♥ t❤❡ ❢♦r♠ ♦❢ ✭✸✳✹✶✮ ✹✻ ❊①♣•✐❝✐t ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ♠❡tr✐❝ ✐♥ t❤❡ ♠✉t✉❛••② •♦❝❛• ❝❛s❡✳ ✹✳✸ ω 3 = i Σ ∧ Σ 4r 2 1 -2r R 2 U + iR 2 4r 2 z Λ zΣ Y Λ ∧ ȲΣ - i 2r M ΛΣ Y Λ + iR 2π γ Ω(γ)V γΛ J (2,+) γ dZ γ -U -1 Z γ ∂K ∧ ȲΣ - iR 2π γ ′ Ω(γ ′ ) Vγ ′ Σ J (2,-) γ ′ d Zγ ′ -U -1 Zγ ′ ∂K + i 2rU Y Λ M ΛΣ vΣ - iR 2π γ Ω(γ)W γ dZ γ ∧ v Λ ′ M Λ ′ Σ ′ ȲΣ ′ + iR 2π γ ′ Ω(γ ′ ) Wγ ′ d Zγ ′ + iR 2 K 2r K ab dz a ∧ dz b - 1 K 2 U 2 1 2π γ Ω(γ)|Z γ | 2 J (2) γ -v Λ M ΛΣ vΣ 2 ∂K ∧ ∂K + 1 2πK γ Ω(γ)J (2) γ dZ γ -U -1 Z γ ∂K ∧ d Zγ -U -1 Zγ ∂K . ✭✹✳✸✼✮ ❋r♦♠ t❤✐s t❤❡ ♠❡tr✐❝ ❝❛♥ ❜❡ ❡❛s✐•② r❡❛❞ ♦✛ t❤r♦✉❣❤ ω 3 (X, Y ) = g(X, J 3 Y )✳ ❚❤❡ ♦♥❡✲❢♦r♠ Y Λ ✐♥tr♦❞✉❝❡❞ ✐♥ ✭✹✳✸✹✮ ✐s r❡✇r✐tt❡♥ ❛s Y Λ = d ζΛ -F ΛΣ dζ Σ - i 4π γ Ω(γ)V γΛ J (2) γ dΘ γ + R J (2,+) γ dZ γ + J (2,-) γ d Zγ + 2πR U v γ 4 dr R 2 -dK - γ ′ Ω(γ ′ ) v γ ′ R dΘ γ ′ - J (2) γ ′ 2π Zγ ′ dZ γ ′ + Z γ ′ d Zγ ′ . ✭✹✳✸✽✮ ❆♥ ✐♠♣♦rt❛♥t ❢❡❛t✉r❡ ♦❢ t❤✐s r❡s✉•t ✐s t❤❛t ✐t s❤♦✇s t❤❛t ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ✐♥st❛♥✲ t♦♥s Y Λ ❤❛s ❛ ♥♦♥✲✈❛♥✐s❤✐♥❣ ♣r♦•❡❝t✐♦♥ ❛•♦♥❣ dr✳ ❚❤❡ ♦t❤❡r ♦♥❡✲❢♦r♠ Σ ❜❡❝♦♠❡s Σ = 2 1 - 2r R 2 U dr + i 4 dσ + ζΛ dζ Λ -ζ Λ d ζΛ + V , ✭✹✳✸✾✮ ❚♦ ❜❡ ❝♦♥❝✐s❡ ✇❡ ✐♥tr♦❞✉❝❡❞ s❡✈❡r❛• ♥♦t❛t✐♦♥s ♦♥❝❡ ❛❣❛✐♥✿ ❼ ❚❤❡ ✜rst ♦♥❡ ✐s ❛♥ ✐♥✈❡rt✐❜•❡ ♠❛tr✐① M ΛΣ ✇❤✐❝❤ ✐s ❛♥ ✐♥st❛♥t♦♥ ❝♦rr❡❝t❡❞ ✈❡rs✐♦♥ ♦❢ N ΛΣ ✳ ■t ✐s ❣✐✈❡♥ ❜② M ΛΣ = N ΛΣ - 1 2π γ Ω(γ)J (2) γ VγΛ V γΣ . ✭✹✳✹✵✮ ✹✼ ✹✳ ◗✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s t♦ t❤❡ ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡ ❛♥❞ ❉✲✐♥st❛♥t♦♥s ❼ ❲❡ ❛•s♦ ✉s❡❞ t❤❡ ❢♦••♦✇✐♥❣ s❡t ♦❢ ❝♦♥✈❡♥✐❡♥t ♥♦t❛t✐♦♥s ✿ v γ = 1 4π Z γ J (2,+) γ + Zγ J (2,-) γ , v Λ = γ Ω(γ)v γ V γΛ . ✭✹✳✹✶✮ ❼ ❆ ♣♦t❡♥t✐❛• U ✇❤✐❝❤ ✐s U = K - 1 2π γ Ω(γ)|Z γ | 2 J (2) γ + v Λ M ΛΣ vΣ , ✭✹✳✹✷✮ ❛♥❞ ❛♥♦t❤❡r ♣♦t❡♥t✐❛• •❛❜❡•❡❞ ❜② ❝❤❛r❣❡s ✐s ❣✐✈❡♥ ❜② W γ = Zγ J (2) γ -J (2,+) γ v Λ M ΛΣ VγΣ . ✭✹✳✹✸✮ ❼ ❲❡ ❤❛✈❡ ❛•r❡❛❞② s❡❡♥ t❤❛t t❤❡ ♦♥❡✲•♦♦♣ ❝♦rr❡❝t✐♦♥ ✐♥tr♦❞✉❝❡s ❛ t❡r♠ ♣r♦✲ ♣♦rt✐♦♥❛• t♦ t❤❡ ❑ä❤•❡r ❝♦♥♥❡❝t✐♦♥ t♦ t❤❡ σ ❝✐r❝•❡ ❜✉♥❞•❡✳ ■♥ ♣r❡s❡♥❝❡ ♦❢ ✐♥st❛♥t♦♥ ❝♦rr❡❝t✐♦♥s✱ t❤✐s t❡r♠ ✐s ♣r♦♠♦t❡❞ t♦ V = 2R 2 K 1 - 4r R 2 U A K + 8r RU γ Ω(γ) v γ + 1 2π J (2) γ V γΛ M ΛΣ vΣ C γ + 2r πiU γ Ω(γ) W γ + RU 8πir J (1,+) γ dZ γ ✭✹✳✹✹✮ -Wγ + RU 8πir J (1,-) γ d Zγ , ✇❤❡r❡ ✇❡ ✐♥tr♦❞✉❝❡❞ t❤❡ ❢♦••♦✇✐♥❣ •✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛•s ♦❢ t❤❡ ||✲✜❡•❞s C γ = N ΛΣ q Λ -Re F ΛΞ p Ξ d ζΣ -Re F ΣΘ dζ Θ + 1 4 N ΛΣ p Λ dζ Σ . ✭✹✳✹✺✮ ✹✽ ❊①♣•✐❝✐t ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ♠❡tr✐❝ ✐♥ t❤❡ ♠✉t✉❛••② •♦❝❛• ❝❛s❡✳ ✹✳✹ ❋✐♥❛••② •❡t ✉s ✇r✐t❡ ❞♦✇♥ t❤❡ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ♠❡tr✐❝ ✐♥ t❤✐s ❝❛s❡✱ ds 2 = 2 r 2 1 - 2r R 2 U (dr) 2 + 1 32r 2 1 -2r R 2 U dσ + ζΛ dζ Λ -ζ Λ d ζΛ + V 2 + R 2 2r 2 |z Λ Y Λ | 2 + 1 rU Y Λ M ΛΣ vΣ - iR 2π γ Ω(γ)W γ dZ γ 2 - 1 r M ΛΣ Y Λ + iR 2π γ Ω(γ)V γΛ J (2,+) γ dZ γ -U -1 Z γ ∂K × ȲΣ - iR 2π γ ′ Ω(γ ′ ) Vγ ′ Σ J (2,-) γ ′ d Zγ ′ -U -1 Zγ ′ ∂K + R 2 K r K a bdz a dz b - 1 (2πKU) 2 γ Ω(γ)Z γ W γ 2 |∂K| 2 + 1 2πK γ Ω(γ)J (2) γ dZ γ -U -1 Z γ ∂K 2 . ✭✹✳✹✻✮ ❆❝t✉❛••②✱ t❤❡ r❡s✉•t ✐♥ ✭✹✳✹✻✮ ✐s ✈❛•✐❞ ❢♦r t✇♦ ❝❛s❡s✳ ❖♥❡ ♦❢ t❤❡♠ ✐s t❤❡ ❝❛s❡ ✇❤❡♥ ✇❡ r❡str✐❝t t♦ t❤❡ s❡t ♦❢ ♠✉t✉❛••② •♦❝❛• ❝❤❛r❣❡s✱ ❢♦r ✇❤✐❝❤ ✐t ✐s ❡①❛❝t✳ ❚❤❡ ♦t❤❡r ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✐♥❝•✉s✐♦♥ ♦❢ ❛•• ❉✲✐♥st❛♥t♦♥ ❝❤❛r❣❡s ✭✹✳✹✻✮✱ ❜✉t r❡str✐❝t❡❞ t♦ t❤❡ ♦♥❡✲✐♥st❛♥t♦♥ ❛♣♣r♦①✐♠❛t✐♦♥✳ ❚❤✐s ❡✛❡❝t✐✈❡•② ♠❡❛♥s t♦ tr✉♥❝❛t❡ t❤❡ ✐♥st❛♥t♦♥ s❡r✐❡s ❡①♣❛♥s✐♦♥ ❛t t❤❡ •✐♥❡❛r ♦r❞❡r ✐♥ ❉♦♥❛•❞s♦♥✲❚❤♦♠❛s ✭❉❚✮ ✐♥✈❛r✐❛♥ts Ω(γ)✳ ❋✉rt❤❡r s✐♠♣•✐✜❝❛t♦♥s ♦❝❝✉r ✐♥ t❤❡ •❛tt❡r ❝❛s❡✳ ❖♥❡ ❝❛♥ ❡①♣•✐❝✐t•② ❝♦♥str✉❝t t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ♠❛tr✐① M ΛΣ ❢r♦♠ ✭✹✳✹✵✮ ❛♥❞ ♦♥❡ ❝❛♥ s♦•✈❡ ❢♦r R ✐♥ t❡r♠s ♦❢ ♦t❤❡r ❝♦♦r❞✐♥❛t❡s t❤r♦✉❣❤ t❤❡ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ❝♦♥t❛❝t ♣♦t❡♥t✐❛• ✭✹✳✸✸✮✳ ❖♥❡ ♠♦r❡ ❝♦♠♣•✐❝❛t✐♦♥ ❛r✐s❡s ❞✉❡ t♦ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ✐♥st❛♥t♦♥ ❝♦rr❡❝t✐♦♥s✳ ❚❤❡ ♥✐❝❡ t✇♦✲st❛❣❡❞ ✜❜r❛t✐♦♥ str✉❝t✉r❡ ♦❢ t❤❡ ♦♥❡✲•♦♦♣ ❝♦rr❡❝t❡❞ s♣❛❝❡ ✭✷✳✸✻✮ ✐s ♥♦✇ ❜r♦❦❡♥ ❞♦✇♥✳ ❋✐rst•②✱ t❤❡ ❞✐•❛t♦♥ ✐s ♥♦t ❢❛❝t♦r✐③❡❞ ♥♦✇✳ ■♥st❡❛❞ ✐t ❡♥t❡rs ♥♦♥✲tr✐✈✐❛••② ❛•♦♥❣ ✇✐t❤ z a ❛♥❞ (ζ Λ , ζΛ ) ✐♥ t❤❡ ❡①♣r❡ss✐♦♥ ❢♦r Y Λ ❞✉❡ t♦ ♣r❡s❡♥❝❡ ♦❢ t❡r♠s ♣r♦♣♦rt✐♦♥❛• t♦ dr✳ ❙❡❝♦♥❞•②✱ t❤❡ ♠❡tr✐❝ ♦♥ t❤❡ s✉❜s♣❛❝❡ ♣❛r❛♠❡tr✐③❡❞ ❜② t❤❡ ❝♦♠♣•❡① str✉❝t✉r❡ ♠♦❞✉•✐ ❛❝q✉✐r❡s ❛ ❝♦♠♣•✐❝❛t❡❞ ❞❡♣❡♥❞❡♥❝❡ ♦♥ t❤❡ ♦t❤❡r ❝♦♦r❞✐♥❛t❡s✳ ❚❤❡ ♦♥•② ✜❜r❛t✐♦♥ t❤❛t s✉r✈✐✈❡s ✐s t❤❡ ❝✐r❝•❡ ❜✉♥❞•❡ ♦❢ t❤❡ ◆❙✲❛①✐♦♥ σ s♦ t❤❛t t❤❡ t♦t❛• s♣❛❝❡ ❤❛s t❤❡ ❢♦••♦✇✐♥❣ t♦♣♦•♦❣✐❝❛• ♦❜str✉❝t✐♦♥ S 1 σ -→ M ❍▼ ↓ B inst r,z,ζ, ζ . ✭✹✳✹✼✮ ❚❤❡ ❝♦♥♥❡❝t✐♦♥ ♦♥ t❤✐s ❝✐r❝•❡ ❜✉♥❞•❡ ❝❛♥ ❜❡ ❡❛s✐•② r❡❛❞ ♦✛ ❢r♦♠ ✭✹✳✹✻✮ ❛♥❞ ✐s ❣✐✈❡♥ ❜②✱ ζΛ dζ Λ -ζ Λ d ζΛ + V✱ ✇❤❡r❡ V ✐s ❣✐✈❡♥ ❜② ✭✹✳✹✺✮✳ ❋✐♥❛••②✱ s✐♥❝❡ ✇❡ ✇♦r❦✐♥❣ ✐♥ t②♣❡ ■■❆ ❢r❛♠❡✇♦r❦ t❤❡ ♠❡tr✐❝ ♠✉st ❜❡ s②♠♣•❡❝t✐❝ ✐♥✈❛r✐❛♥t✳ ❍♦✇❡✈❡r✱ ✐t ✐s ❤❛r❞ t♦ s❡❡ ✐t ✐♥ t❤❡ ❡①♣r❡ss✐♦♥ ✭✹✳✹✻✮✳ ❚❤❛t ✐t ✐s ✐♥❞❡❡❞ ♣r❡s❡♥t ❤❛s ❜❡❡♥ ✈❡r✐✜❡❞ ❡①♣•✐❝✐t•② ✐♥ ❛♣♣❡♥❞✐① ❇✳✹ ✐♥ ❬✶✶(✳ ✹✾ ✹✳ ◗✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s t♦ t❤❡ ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡ ❛♥❞ ❉✲✐♥st❛♥t♦♥s ✹✳✹ ❯♥✐✈❡rs❛• ❤②♣❡r♠✉•t✐♣•❡t ❛♥❞ ❚♦❞❛ ♣♦t❡♥t✐❛• ❖♥❡ ✈❡r② ✐♥t❡r❡st✐♥❣ ❝❛s❡ ✐♥ ♦✉r ❝♦♥t❡①t ❛r✐s❡s ✇❤❡♥ t❤❡ ❝♦♠♣❛❝t✐✜❝❛t✐♦♥ ✐s ❞♦♥❡ ♦♥ ❛ ✏r✐❣✐❞ ❈❛•❛❜✐✲(❛✉✧ t❤r❡❡❢♦•❞✱ ✇❤✐❝❤ ❞♦❡s ♥♦t ❤❛✈❡ ❛♥② ❝♦♠♣•❡① str✉❝t✉r❡ ♠♦❞✉•✐ ❛s ❢♦r t❤❡♠ h 2,1 (Y) = 0✳ ❚❤❡ ✈❛♥✐s❤✐♥❣ ♦❢ t❤❡ ❍♦❞❣❡ ♥✉♠❜❡r ❢♦r t②♣❡ ■■❆ str✐♥❣ t❤❡♦r② ❝♦♠♣❛❝t✐✜❡❞ ♦♥ Y t❤❡r❡❢♦r❡ ✐♠♣•✐❡s t❤❛t t❤❡r❡ ✐s ♦♥•② ♦♥❡ ❤②♣❡r♠✉•t✐♣•❡t ✇❤✐❝❤ ✐♥ •✐t❡r❛t✉r❡s ✐s ❦♥♦✇♥ ❛s ✉♥✐✈❡rs❛• ❤②♣❡r♠✉•t✐♣•❡t ❬✹✶(✳ ❚❤❡ ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡ ✐♥ t❤✐s s✐t✉❛t✐♦♥ ✐s t❤❡♥ ❛ ❢♦✉r ❞✐♠❡♥s✐♦♥❛• ◗❑ ♠❛♥✐❢♦•❞✳ ■♥ t❤✐s s♣❡❝✐❛• ❝❛s❡ t❤❡ ◗❑ ❣❡♦♠❡tr② ❛••♦✇s ❢♦r ❛ ♠♦r❡ ❡①♣•✐❝✐t ❞❡s❝r✐♣t✐♦♥ ❛s ✐t ✐s ❞❡✜♥❡❞ ❛s ❛♥ ❊✐♥st❡✐♥ ♠❛♥✐❢♦•❞ ✇✐t❤ ♥♦♥✲✈❛♥✐s❤✐♥❣ ❝♦s♠♦•♦❣✐❝❛• ❝♦♥st❛♥t ❛♥❞ s❡•❢✲❞✉❛• ❲❡②• ❝✉r✈❛t✉r❡✳ ■❣♥♦r✐♥❣ ◆❙✺✲❜r❛♥❡ ✐♥st❛♥t♦♥s✱ ❛s ✇❡ ❤❛✈❡ ❜❡❡♥ ❞♦✐♥❣ t❤r♦✉❣❤♦✉t t❤✐s ❝❤❛♣t❡r✱ ✐♠♣•✐❡s t❤❛t t❤❡r❡ ✐s ❛t •❡❛st ♦♥❡ ❝♦♥t✐♥✉♦✉s ✐s♦♠❡tr② ♣r❡s❡r✈❡❞✱ t❤❡ tr❛♥s•❛t✐♦♥ ❛•♦♥❣ t❤❡ ◆❙✲❛①✐♦♥ σ ❢♦r t❤❡ ❍▼ ♠♦❞✉•✐ s♣❛❝❡✳ ❲✐t❤ ❛ ♣r♦♣❡r ❝❤♦✐❝❡ ♦❢ ❝♦♦r❞✐♥❛t❡ ♠❡tr✐❝ ♦♥ s✉❝❤ ❢♦✉r✲❞✐♠❡♥s✐♦♥❛• ◗❑ ♠❛♥✐❢♦•❞s ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ t❤❡ ❢♦r♠ ♦❢ ❚♦❞ ❛♥s❛t③ ❬✹✷( ♣❛r❛♠❡tr✐③❡❞ ❜② ♦♥❡ r❡❛• ❢✉♥❝t✐♦♥✱ ds 2 = - 3 Λ P ρ 2 dρ 2 + 4e T dzdz + 1 P ρ 2 (dθ + Θ) 2 , ✭✹✳✹✽✮ ✇❤❡r❡ T ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ (ρ, z, z) ❛♥❞ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ θ ♣❛r❛♠❡tr✐③✐♥❣ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ✐s♦♠❡tr②✳ ❋✉rt❤❡r♠♦r❡✱ P = 1 - 1 2 ρ∂ ρ T, ✭✹✳✹✾✮ dΘ = i (∂ z P dz -∂ z P dz) ∧ dρ -2i∂ ρ (P e T )dz ∧ dz, ✭✹✳✺✵✮ ✇❤❡r❡❛s t❤❡ ❊✐♥st❡✐♥ s❡•❢✲❞✉❛•✐t② ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ ♠❡tr✐❝ ✐s ❡♥❝♦❞❡❞ ✐♥ t❤❡ ❚♦❞❛ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥ t♦ ❜❡ s❛t✐s✜❡❞ ❜② t❤❡ ❢✉♥❝t✐♦♥ T ✱ ∂ z ∂ z T + ∂ 2 ρ e T = 0. ✭✹✳✺✶✮ ❚❤✐s ❞❡s❝r✐♣t✐♦♥ ✐s ♠♦r❡ ❝♦♥✈❡♥✐❡♥t ❜❡❝❛✉s❡ ❛•• ✐♥st❛♥t♦♥ ❝♦rr❡❝t✐♦♥s ❛r❡ ❡♥✲ ❝♦❞❡❞ ✐♥ •✉st ♦♥❡ ❢✉♥❝t✐♦♥✱ t❤❡ ❚♦❞❛ ♣♦t❡♥t✐❛• T ✇❤✐❝❤ ✐s ❝♦♥str❛✐♥❡❞ ❜② t❤❡ ❚♦❞❛ ❡q✉❛t✐♦♥ ✭✹✳✺✶✮✳ ❍❡♥❝❡ t♦ ❡①tr❛❝t t❤❡♠ ✐t s✉✣❝❡s t♦ ✜♥❞ ❛♣♣r♦♣r✐❛t❡ s♦•✉✲ t✐♦♥ ♦❢ t❤❡ ❚♦❞❛ ❡q✉❛t✐♦♥✳ ❚❤✐s str❛t❡❣② ✇❛s ✈❡r② s✉❝❝❡ss❢✉• ❛t t❤❡ ♦♥❡✲✐♥st❛♥t♦♥ •❡✈❡• ❬✹✸✕✹✼( ✷ ✳ ❍♦✇❡✈❡r✱ t❤❡ r❡s✉•ts ❜❡②♦♥❞ t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ ❛r❡ ♦❢t❡♥ ♥♦t r❡•✐✲ ❛❜•❡ ❜❡❝❛✉s❡✱ t♦ ✜① t❤❡ ❛♠❜✐❣✉✐t✐❡s ♦❢ ✐♥t❡❣r❛t✐♦♥ t②♣✐❝❛••② s♦♠❡ ❛♥s❛t③ ❢♦r T ✐s ♠❛❞❡✳ ❉✉❡ t♦ s✉❝❤ ✉♥•✉st✐✜❡❞ s✐♠♣•✐✜❝❛t✐♦♥s t❤♦s❡ r❡s✉•ts ❝❛♥ ♥♦t ❜❡ tr✉st❡❞ ✐♥ ❢✉•• ❣❡♥❡r❛•✐t②✳ ❇✉t ♥♦✇ t❤❛t ✇❡ ❤❛✈❡ t❤❡ t✇✐st♦r ❝♦♥str✉❝t✐♦♥✱ ✇❡ ❞♦ ♥♦t ♥❡❡❞ t♦ ✜♥❞ t❤❡ ❚♦❞❛ ♣♦t❡♥t✐❛• ❢r♦♠ t❤❡ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥✳ ■♥st❡❛❞ ✐t s❡r✈❡s •✉st ❛s ❛ ❝♦♥s✐st❡♥❝② ✷ ❆ s✐♠✐•❛r str❛t❡❣② ❝❛♥ ❜❡ ❛♣♣•✐❡❞ t♦ ❞❡r✐✈❡ ◆❙✺✲❜r❛♥❡ ✐♥st❛♥t♦♥s ❛s ✇❡•• ❬✹✽(✱ ❜❡❝❛✉s❡ ❣❡♥❡r✐❝ ✹❞ ◗❑ ♠❛♥✐❢♦•❞s ❝❛♥ ❜❡ ♣❛r❛♠❡tr✐③❡❞ ❜② s♦•✉t✐♦♥s ♦❢ ❛♥♦t❤❡r✱ ♠♦r❡ ❝♦♠♣•✐❝❛t❡❞ ♥♦♥✲•✐♥❡❛r ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥✱ ✇❤✐❝❤ r❡♣•❛❝❡s t❤❡ ❚♦❞❛ ❡q✉❛t✐♦♥ ✐♥ t❤❡ ❛❜s❡♥❝❡ ♦❢ t❤❡ ✐s♦♠❡tr② ❬✹✾(✳ ✺✵ ❈✉r✈❛t✉r❡ s✐♥❣✉•❛r✐t② ✐♥ ♣r❡s❡♥❝❡ ♦❢ ❉✲✐♥st❛♥t♦♥s✳ ✹✳✺ ❝❤❡❝❦✱ t❤❛t ✐s ♦♥❝❡ ❛ ♣r♦♣❡r ✐❞❡♥t✐✜❝❛t✐♦♥ ♦❢ ❝♦♦r❞✐♥❛t❡s ❛♥❞ t❤❡ ❚♦❞❛ ♣♦t❡♥t✐❛• ✐s ♠❛❞❡ ♦♥❡ ❤❛s t♦ ✈❡r✐❢② ✐❢ t❤❡ ❚♦❞❛ ❡q✉❛t✐♦♥ s❤♦✉•❞ ❜❡ s❛t✐s✜❡❞ ❛✉t♦♠❛t✐❝❛••②✳ ■♥❞❡❡❞✱ t❤✐s ✐s t❤❡ ❝❛s❡ ✐❢ ♦♥❡ ✐❞❡♥t✐✜❡s ❬✺✵(✱ ρ = e φ , z = i 2 ξ[+] 0 , θ = - 1 8 σ, T = 2 log(R/2). ✭✹✳✺✷✮ ❖♥❡ ❝❛♥ ♥♦✇ ❛❝t✉❛••② ❝♦♠♣✉t❡ t❤❡ ♠❡tr✐❝ M HM ❜② r❡str✐❝t✐♥❣ ✭✹✳✹✻✮ t♦ ❢♦✉r ❞✐✲ ♠❡♥s✐♦♥s✳ ❖♥❡ ❝❛♥ ❛•s♦ ❡①♣•✐❝✐t•② ✈❡r✐❢② t❤❛t T s❛t✐s✜❡s ❚♦❞❛ ❡q✉❛t✐♦♥✳ ❋♦r ❞❡t❛✐•s ♦❢ t❤❡s❡ ❝❤❡❝❦s ❛♥❞ r❡s✉•t ❢♦r t❤❡ ♠❡tr✐❝ ✇❡ r❡❢❡r t❤❡ r❡❛❞❡r t♦ ❬✶✶(✳ ❙✐♥❝❡ t❤❡r❡ ✐s ♥♦ ❝♦♠♣•❡① str✉❝t✉r❡ ♠♦❞✉•✐ ✐♥ t❤✐s ❝❛s❡ t❤❡ ♦♥•② ♣♦ss✐❜•❡ ✈❛•✉❡ t❤❡ ✐♥❞✐❝❡s Λ, Σ ❝❛♥ t❛❦❡ ❤❡r❡ ✐s 0✳ ❲❡ t❤❡r❡❢♦r❡ ❞r♦♣ t❤❡ ✐♥❞❡① ❛•t♦❣❡t❤❡r✳ ▼♦r❡✲ ♦✈❡r t❤❡ ♣r❡♣♦t❡♥t✐❛• ❢♦r t❤❡ r✐❣✐❞ ❈❛•❛❜✐✲(❛✉s ✐s ♥❡❝❡ss❛r✐•② q✉❛❞r❛t✐❝ ❛s ✐t ✐s ❛ ❞❡❣r❡❡ t✇♦ ❤♦♠♦❣❡♥❡♦✉s ❢✉♥❝t✐♦♥ ♦❢ ♦♥•② ♦♥❡ ❛r❣✉♠❡♥t ❬✺✼(✱ F (X) = λ 2 X 2 , λ ≡ λ 1 -iλ 2 = B Ω A Ω , ✭✹✳✺✸✮ ✇❤❡r❡ λ ✐s ❛ ✜①❡❞ ❝♦♠♣•❡① ♥✉♠❜❡r✱ ❣✐✈❡♥ ❜② t❤❡ r❛t✐♦ ♦❢ ♣❡r✐♦❞s ♦❢ t❤❡ ❤♦•♦♠♦r✲ ♣❤✐❝ ✸✲❢♦r♠ Ω ∈ H 3,0 (Y) ♦✈❡r ❛♥ ✐♥t❡❣r❛• s②♠♣•❡❝t✐❝ ❜❛s✐s (A, B) ♦❢ H 3 (Y, Z)✱ ✇✐t❤ λ 2 > 0✳ ❈❤♦♦s✐♥❣ ❡•❡❝tr✐❝ ❢r❛♠❡ ❢♦r t❤❡ ♠✉t✉❛• •♦❝❛• ❝❤❛r❣❡s ✇❡ ❝❛♥ ❡①♣•✐❝✐t•② ❝♦♥✲ str✉❝t t❤❡ s♦•✉t✐♦♥s ❢♦r t❤❡ ❚♦❞❛ ❝♦♦r❞✐♥❛t❡s ❬✶✶( ❣✐✈❡♥ ❜② ❛ s❡t ♦❢ tr❛♥s❝❡♥❞❡♥t❛• ❡q✉❛t✐♦♥s✱ e T = 1 2λ 2 (ρ + c) - e T /2 4π 2 λ 2 q>0 Ω((0, q))q cos 2πqζ K 1 8πqe T /2 , z = i 2 ( ζ -λζ) - 1 4π 2 q>0 Ω((0, q))q sin 2πqζ K 0 8πqe T /2 , ✭✹✳✺✹✮ ✇❤❡r❡ ✇❡ ❤❛✈❡ ✐♥tr♦❞✉❝❡❞ t❤❡ r❛t✐♦♥❛• ❉❚ ✐♥✈❛r✐❛♥ts ❞❡✜♥❡❞ ❜② Ω(γ) = d|γ Ω(γ) d 2 . ✭✹✳✺✺✮ ■♥ t❤❡ ❣❡♥❡r❛• ❝❛s❡ ✇❤❡r❡ ♥♦ r❡str✐❝t✐♦♥ ♦♥ ❝❤❛r❣❡s ✐s ✐♠♣♦s❡❞ t❤❡ ❛❜♦✈❡ ❡q✉❛✲ t✐♦♥s ❜❡❝♦♠❡ ♠♦r❡ ✐♥✈♦•✈❡❞ ♦✇✐♥❣ t♦ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❝♦♠♣•✐❝❛t❡❞ t✇✐st♦r ✐♥t❡❣r❛•s ✇❤✐❝❤ ✐♥ ❣❡♥❡r❛• ❝❛♥ ♥♦t ❜❡ r❡❞✉❝❡❞ t♦ ❛ ♣r♦❞✉❝t ♦❢ ❇❡ss❡• ❢✉♥❝t✐♦♥s ✉♥•✐❦❡ t❤❡ s✐t✉❛t✐♦♥ ❛❜♦✈❡✳ ✹✳✺ ❈✉r✈❛t✉r❡ s✐♥❣✉•❛r✐t② ✐♥ ♣r❡s❡♥❝❡ ♦❢ ❉✲✐♥st❛♥t♦♥s ❏✉st ❜② ♠❡r❡ ✐♥s♣❡❝t✐♦♥ ♦r ❛ ♠♦r❡ ❞✐•✐❣❡♥t ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ q✉❛❞r❛t✐❝ ❝✉r✈❛t✉r❡ ✭|✐❡♠❛♥♥ t❡♥s♦r sq✉❛r❡❞✮ ♦❢ t❤❡ ♠❡tr✐❝ ✭✹✳✶✮ ♦♥❡ ❝❛♥ ✜♥❞ t❤❛t t❤❡r❡ ✐s ❛ ❝✉r✈❛t✉r❡ ✺✶ ✹✳ ◗✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s t♦ t❤❡ ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡ ❛♥❞ ❉✲✐♥st❛♥t♦♥s s✐♥❣✉•❛r✐t② ❢♦r χ Y > 0 ❛t r = -2c✳ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✇♦✉•❞ •✐❦❡ t♦ ❛❞❞r❡ss t❤❡ q✉❡st✐♦♥ t❤❛t ✇❤❛t ❤❛♣♣❡♥s t♦ t❤✐s s✐♥❣✉•❛r✐t② ✇❤❡♥ ❉✲✐♥st❛♥t♦♥s ❛r❡ ✐♥❝•✉❞❡❞✳ ▲♦♦❦✐♥❣ ❛t t❤❡ ❦✐♥❡t✐❝ t❡r♠s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❞✐•❛t♦♥ ❛♥❞ t❤❡ ◆❙✲❛①✐♦♥ σ ✐♥ ✭✹✳✹✻✮ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡♠ ✇✐t❤ s✐♠✐•❛r t❡r♠s ✐♥ t❤❡ ♦♥❡✲•♦♦♣ ❝♦rr❡❝t❡❞ ♠❡tr✐❝ ✭✹✳✶✮ ✇❡ ✜♥❞ t❤❛t t❤❡ ❢❛❝t♦r r + 2c ✐s ♣r♦♠♦t❡❞ ✐♥ ♣r❡s❡♥❝❡ ♦❢ t❤❡ ✐♥st❛♥t♦♥ ❝♦rr❡❝t✐♦♥s t♦ 1 2 R 2 U -r. ✭✹✳✺✻✮ ❚❤❡r❡❢♦r❡ ✐t ✐s ♥❛t✉r❛• t♦ ❡①♣❡❝t t❤❛t s❡tt✐♥❣ t❤❡ ❛❜♦✈❡ t♦ ③❡r♦ ✇✐•• ❣✐✈❡ r✐s❡ t♦ ❛ s✐♥❣✉•❛r✐t② ❡q✉❛t✐♦♥✱ t❤❛t ✐s t❤❡ ❡q✉❛t✐♦♥ ❞❡s❝r✐❜✐♥❣ t❤❡ s✐♥❣✉•❛r ❤②♣❡rs✉r❢❛❝❡ ♦♥ M ❍▼ ✳ ❙✉❜st✐t✉t✐♥❣ ❡①♣r❡ss✐♦♥s ❢♦r U ❛♥❞ r ♦♥❡ ❛rr✐✈❡s ❛t K + 4c R 2 + 1 π γ Ω(γ) i 8π Z γ J (1,+) γ + Zγ J (1,-) γ -|Z γ | 2 J (2) γ ✭✹✳✺✼✮ + 1 8π 2 γ Ω(γ) Z γ J (2,+) γ + Zγ J (2,-) γ V γΛ M ΛΣ γ ′ Ω(γ ′ ) Z γ ′ J (2,+) γ ′ + Zγ ′ J (2,-) γ ′ Vγ ′ Σ = 0. ❚❤❡r❡ ✐s ❛❝t✉❛••② ❛♥♦t❤❡r ♠♦r❡ ❣❡♦♠❡tr✐❝ ✇❛② t♦ ❡①tr❛❝t t❤❡ ❡q✉❛t✐♦♥ ❢♦r s✐♥✲ ❣✉•❛r✐t②✳ ▲❡t ✉s ✐••✉str❛t❡ ✐t ✐♥ t❤❡ ❝❛s❡ ❢♦r ♦♥❡✲•♦♦♣ ♠❡tr✐❝ ✜rst✳ ❋r♦♠ ✭✹✳✺✮ ♦♥❡ ❝❛♥ ❡❛s✐•② ✜♥❞ t❤❡ ❡①♣r❡ss✐♦♥s ❢♦r ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ♥❡❛r t❤❡ ♥♦rt❤ ♣♦•❡✳ ❯s✐♥❣ t❤❡ ♣r♦❝❡❞✉r❡ t♦ ❝♦♥str✉❝t (1, 0)✲❢♦r♠s ✐♥ ✸✳✺ ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦••♦✇✐♥❣✱ Π a = dz a , Y Λ = d ζΛ -F ΛΣ dζ Σ , Σ = dr + 2cd log R + i 4 dσ + ζΛ dζ Λ -ζ Λ d ζΛ . ✭✹✳✺✽✮ ❚❤❡ ❝✉r✈❛t✉r❡ s✐♥❣✉•❛r✐t② ♦❝❝✉rs ✐❢ t❤❡ ❜❛s✐s ❢♦r♠s ❜❡❝♦♠❡ ❞❡❣❡♥❡r❛t❡✳ ❚❤❡ ♦♥•② ♣♦ss✐❜✐•✐t② ❢♦r t❤✐s t♦ ❤❛♣♣❡♥ ✐♥ ✭✹✳✺✽✮ ✐s Re Σ = de φ + 2cd log R = 0, ✭✹✳✺✾✮ ✇❤✐❝❤ ❣✐✈❡s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡q✉❛t✐♦♥ ❢♦r s✐♥❣✉•❛r✐t②✳ ❯s✐♥❣ t❤❡ ♦♥❡ •♦♦♣ r❡•❛t✐♦♥ ❜❡t✇❡❡♥ R ❛♥❞ t❤❡ ❞✐•❛t♦♥ ♦♥❡ ♦❜t❛✐♥s t❤❛t ✐t r❡❞✉❝❡s t♦ r + 2c = 0, ✭✹✳✻✵✮ ✇❤✐❝❤ t❤✉s r❡♣r❡s❡♥ts t❤❡ ❡q✉❛t✐♦♥ ❢♦r s✐♥❣✉•❛r✐t② ❢♦r t❤❡ ♦♥❡ •♦♦♣ ❝♦rr❡❝t❡❞ ❍▼ ♠♦❞✉•✐ s♣❛❝❡✳ ■♥ t❤❡ ♠♦r❡ ❝♦♠♣•✐❝❛t❡❞ ❝❛s❡ ✇❤❡♥ ✐♥st❛♥t♦♥ ❝♦rr❡❝t✐♦♥s ❛r❡ ♣r❡s❡♥t ♦♥❡ ❝❛♥ ❞♦ t❤❡ s✐♠✐•❛r ❡①❡r❝✐s❡ ❛s ❛❜♦✈❡ ✇✐t❤ t❤❡ (1, 0)✲❢♦r♠s ❞❡✜♥❡❞ ✐♥ ✭✹✳✸✹✮ t♦ r❡❝♦✈❡r ✭✹✳✺✼✮✳ ❲❡ ✇r✐t❡ t❤❡ ❡q✉❛t✐♦♥ ✭✹✳✺✼✮ ✉s✐♥❣ R ✐♥st❡❛❞ ♦❢ t❤❡ ❞✐•❛t♦♥ ❜❡❝❛✉s❡ ✐♥ ❛ s❡♥s❡ ✐t ✐s ❛ ♠♦r❡ ❢✉♥❞❛♠❡♥t❛• q✉❛♥t✐t②✱ ❛s ✐t ✐s r❡•❛t❡❞ t♦ t❤❡ ✶✵✲❞✐♠❡♥s✐♦♥❛• str✐♥❣ ✺✷ ❈✉r✈❛t✉r❡ s✐♥❣✉•❛r✐t② ✐♥ ♣r❡s❡♥❝❡ ♦❢ ❉✲✐♥st❛♥t♦♥s✳ ✹✳✺ ❝♦✉♣•✐♥❣✳ ■♥ t❤❡ ✇❡❛❦ ❝♦✉♣•✐♥❣ r❡❣✐♠❡ R → ∞ t❤❡ ✜rst t❡r♠ ✐♥ t❤✐s ❡q✉❛t✐♦♥ ❞♦♠✐♥❛t❡s t♦ ❛•• ♦t❤❡rs✳ ▲❡t ✉s ❛•s♦ ♥♦t❡ t❤❛t t❤✐s t❡r♠ ✐s ♣♦s✐t✐✈❡✳ ■t ♠❡❛♥s t❤❛t ✭✹✳✺✼✮ ❝❛♥ ♦♥•② ❤❛✈❡ s♦•✉t✐♦♥s ✐❢ t❤❡ •❡❢t ❤❛♥❞ s✐❞❡ ❜❡❝♦♠❡s ♥❡❣❛t✐✈❡ s♦♠❡✇❤❡r❡ ✐♥ t❤❡ ♠♦❞✉•✐ s♣❛❝❡✳ ❋♦r ✉♥❞❡rst❛♥❞✐♥❣ ✇❤❡t❤❡r t❤✐s ❝❛♥ ❤❛♣♣❡♥✱ ✇❡ ♥❡❡❞ t♦ ❛♥❛•②③❡ t❤❡ ❡q✉❛t✐♦♥ ✐♥ str♦♥❣ ❝♦✉♣•✐♥❣ r❡❣✐♠❡ R → 0✳ ❚❤❡ ♦♥•② ✇❛② t♦ ❞♦ t❤✐s ✐s t♦ ❡①♣•♦✐t ❙✲❞✉❛•✐t② s②♠♠❡tr②✳ ❇✉t t♦ ❞♦ t❤✐s✱ ✐t ✐s r❡q✉✐r❡❞ t♦ ♠❛❦❡ t❤✐s s②♠♠❡tr② ♠❛♥✐❢❡st✳ ❚♦ t❤✐s ❡♥❞✱ ✇❡ ✜rst ♥❡❡❞ t♦ ❣♦ t♦ t❤❡ t②♣❡ ■■❇ ❢♦r♠✉•❛t✐♦♥✳ ❚❤❡ ❡•❡❝tr✐❝❛••② ❝❤❛r❣❡❞ ❉✲✐♥st❛♥t♦♥s ♦♥ t❤❡ t②♣❡ ■■❆ s✐❞❡ ❛r❡ t❤❡♥ ♠❛♣♣❡❞ t♦ ❝♦♥tr✐❜✉t✐♦♥s ❝♦♠✲ ✐♥❣ ❢r♦♠ t❤❡ ❉✭✲✶✮ ❛♥❞ ❉✶✲❜r❛♥❡ ✐♥st❛♥t♦♥s✳ ❚❤❡ ❉✭✲✶✮✲✐♥st❛♥t♦♥s ❛r❡ ♣♦✐♥t •✐❦❡ ♦❜•❡❝ts ❛♥❞ ❛r❡ ♦♥•② •❛❜❡•❡❞ ❜② ❛ s✐♥❣•❡ ❝❤❛r❣❡ q 0 ✳ ❚❤❡ •❛tt❡r✱ ❉✶✲✐♥st❛♥t♦♥s ❝❛♥ ✇r❛♣ t✇♦ ❞✐♠❡♥s✐♦♥❛• ❝②❝•❡s ✐♥s✐❞❡ t❤❡ ❈❛•❛❜✐✲(❛✉ ❛♥❞ ❛r❡ •❛❜❡•❡❞ ❜② ❝❤❛r❣❡s q a ✳ ❆♣❛rt ❢r♦♠ t❤❡ ❡①♣♦♥❡♥t✐❛• ❝♦rr❡❝t✐♦♥s ❝♦♠✐♥❣ ❢r♦♠ g s ✱ t❤❡r❡ ❛r❡ ♣❡rt✉r❜❛t✐✈❡ ❛♥❞ ♥♦♥✲♣❡rt✉r❜❛t✐✈❡ α ′ ✲❝♦rr❡❝t✐♦♥s✳ ❚❤❡ ♥♦♥✲♣❡rt✉r❜❛t✐✈❡ ❝♦rr❡❝t✐♦♥s ✐♥ α ′ ❝♦♠❡ ❢r♦♠ t❤❡ ✇♦r•❞s❤❡❡t ✐♥st❛♥t♦♥s✳ ❚❤❡ r❡s✉•t✐♥❣ ♠♦❞✉•✐ s♣❛❝❡ t❤❡♥ ❛❞♠✐ts ❛♥ ✐s♦♠❡tr✐❝ ❛❝t✐♦♥ ♦❢ SL(2, Z) ❣r♦✉♣ ✇❤✐❝❤ ✐s t❤❡ ❙✲❞✉❛•✐t② s②♠♠❡tr② ♦♥ t❤❡ t②♣❡ ■■❇ s✐❞❡✳ ❯♥❞❡r t❤✐s s②♠♠❡tr② t❤❡ ♣❡rt✉r❜❛t✐✈❡ α ′ ✲❝♦rr❡❝t✐♦♥s ♠✐① ✇✐t❤ ❉✭✲✶✮✲✐♥st❛♥t♦♥s ❛♥❞ t❤❡ ✇♦r•❞s❤❡❡t ✐♥st❛♥t♦♥ ❝♦rr❡❝t✐♦♥s ♠✐① ✇✐t❤ ❉✶✲✐♥st❛♥t♦♥s✳ ❆t t❤✐s st❛❣❡ ✇❡ ✉s❡ t❤❡ r❡•❡✈❛♥t ♠✐rr♦r ♠❛♣s t♦ r❡✇r✐t❡ ✭✹✳✺✼✮ ✐♥ ♠❛♥✐❢❡st•② SL(2, Z)✲✐♥✈❛r✐❛♥t ❢♦r♠✳ ❲❡ ✇✐•• ✉s❡ t❤❡♠ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ t♦ r❡✇r✐t❡ ✭✹✳✺✼✮ ✐♥ ♠❛♥✐❢❡st•② ❙✲❞✉❛•✐t② ✐♥✈❛r✐❛♥t ❢♦r♠ ❜② ♣❡r❢♦r♠✐♥❣ P♦✐ss♦♥ r❡s✉♠♠❛t✐♦♥ ♦✈❡r q 0 ✳ R = τ 2 2 , z a = b a + it a , ζ 0 = τ 1 , ζ a = -(c a -τ 1 b a ). ✭✹✳✻✶✮ ❲❡ r❡♠✐♥❞ t❤❡ r❡❛❞❡r ❢r♦♠ ❝❤❛♣t❡r ✷ t❤❛t ❛❝t✐♦♥ ♦❢ t❤❡ SL(2, Z) tr❛♥s❢♦r♠❛t✐♦♥ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ♠❛tr✐① a b c d ✐s r❡❛•✐③❡❞ ♦♥ t❤❡ t②♣❡ ■■❇ ❝♦♦r❞✐♥❛t❡s ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ✇❛②✱ τ → aτ + b cτ + d , t a → |cτ + d| t a , c a b a → a b c d c a b a , ✭✹✳✻✷✮ ✇❤❡r❡ ✇❡ ❝♦♠❜✐♥❡❞ t❤❡ ✐♥✈❡rs❡ ✶✵✲❞✐♠❡♥s✐♦♥❛• str✐♥❣ ❝♦✉♣•✐♥❣ τ 2 = 1/g s ✇✐t❤ t❤❡ ||✲✜❡•❞ τ 1 ✐♥t♦ ❛♥ ❛①✐♦✲❞✐•❛t♦♥ τ = τ 1 + iτ 2 ✳ ❲❡ ✇✐•• r❡str✐❝t t♦ t❤❡ ❝❛s❡ ✇✐t❤ ❉✭✲✶✮✲✐♥st❛♥t♦♥s ♦♥•②✳ ❙✐♥❝❡ t❤❡s❡ ❉✭✲✶✮✲ ✐♥st❛♥t♦♥s ♠✐① ✇✐t❤ ♦♥❡✲•♦♦♣ ❝♦rr❡❝t✐♦♥s ✐♥ t❤❡ s❛♠❡ SL(2, Z) ♠✉•t✐♣•❡t✱ ✇♦r❦✐♥❣ ✐♥ t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ ✐s •✉st✐✜❡❞✳ ❖♥❡ ❝♦✉•❞ ❛•r❡❛❞② ❤♦♣❡ ❢♦r t❤❡ ♣♦ss✐❜✐•✐t② ♦❢ r❡♠♦✈✐♥❣ t❤❡ ♦♥❡✲•♦♦♣ s✐♥❣✉•❛r✐t② ❜② t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❉✭✲✶✮✲✐♥st❛♥t♦♥s ♦♥•②✳ ❙✐♥❝❡ ✇❡ ❞r♦♣ t❤❡ ✇♦r•❞s❤❡❡t ✐♥st❛♥t♦♥s t❤❡ ♣r❡♣♦t❡♥t✐❛• ✐♥ t❤✐s ❝❛s❡ ✐s ❣✐✈❡♥ ❜② F (X) = -κ abc X a X b X c 6X 0 + iζ(3)χ Ŷ 16π 3 (X 0 ) 2 , ✭✹✳✻✸✮ ✇❤❡r❡ t❤❡ ✜rst t❡r♠ ✐s t❤❡ ✉s✉❛• ❝✉❜✐❝ t❡r♠ ❛♥❞ t❤❡ s❡❝♦♥❞ ♦♥❡ ❡♥❝♦❞❡s t❤❡ ♣❡r✲ t✉r❜❛t✐✈❡ α ′ ✲❝♦rr❡❝t✐♦♥✳ ❙✐♥❝❡ ✇❡ ❝♦♥s✐❞❡r ♦♥•② D(-1)✲✐♥st❛♥t♦♥s t❤❡ ❉✲❜r❛♥❡ ✺✸ ✹✳ ◗✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s t♦ t❤❡ ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡ ❛♥❞ ❉✲✐♥st❛♥t♦♥s ❝❤❛r❣❡s ❛r❡ •❛❜❡•❡❞ ❜② ♦♥•② ♦♥❡ ♥✉♠❜❡r q 0 ✳ ■♥ t❤✐s ❝❛s❡ t❤❡ ❉❚ ✐♥✈❛r✐❛♥t ✐s ✐♥❞❡✲ ♣❡♥❞❡♥t ♦❢ t❤❡ ❝❤❛r❣❡ ❛♥❞ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ❊✉•❡r ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ t❤❡ ❈❛•❛❜✐✲(❛✉✱ Ω(q 0 ) = χ Y = -χ Ŷ✱ ✇❤❡r❡ Ŷ ✐s t❤❡ ♠✐rr♦r ❈❛•❛❜✐✲(❛✉✳ ❚❤❡ ♣r❡♣♦t❡♥t✐❛• ✭✹✳✻✸✮ •❡❛❞s t♦ t❤❡ ❢♦••♦✇✐♥❣ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ❡①♣♦♥❡♥t✐❛• ♦❢ ❑ä❤•❡r ♣♦t❡♥t✐❛•✱ K = 8V - ζ(3)χ Ŷ 4π 3 , ✭✹✳✻✹✮ ✇❤❡r❡ V = 1 6 κ abc t a t b t c ✐s t❤❡ ❈❛•❛❜✐✲(❛✉ ✈♦•✉♠❡ ❛♥❞ ✇❡ ✉s❡❞ t❤❡ r❡•❛t✐♦♥ z a = b a + it a ❢r♦♠ t❤❡ ♠✐rr♦r ♠❛♣ ✭✹✳✻✶✮✳ ❲❡ ✇✐•• ♥♦t ❣✐✈❡ ❞❡t❛✐•s ♦❢ t❤❡ r❡s✉♠♠❛t✐♦♥ ♣r♦❝❡❞✉r❡ ❤❡r❡✳ ■t ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✶✶(✳ ❲❡ ✇r✐t❡ t❤❡ r❡s✉•t ♦♥•② ❛♥❞ ❛♥❛•②③❡ ✐ts ❝♦♥s❡q✉❡♥❝❡s✳ ❚❤❡ ❡q✉❛t✐♦♥ ❢♦r s✐♥❣✉•❛r✐t② ✐♥ ♠❛♥✐❢❡st•② ❙✲❞✉❛•✐t② ✐♥✈❛r✐❛♥t ❢♦r♠ t✉r♥s ♦✉t t♦ ❜❡ 1 + χ Ŷ E 3/2 (τ ) 64π 3 V τ 3/2 2 - 2χ 2 Ŷ 64π 3 V τ 3/2 2 2 E 2 3/2 (τ ) - 9 4 ′ m,n∈Z ′ m ′ ,n ′ ∈Z τ 5 2 (mn ′ -nm ′ ) 2 |mτ + n| 5 |m ′ τ + n ′ | 5 = 0, ✭✹✳✻✺✮ ✇❤❡r❡ t❤❡ E 3/2 ✐s t❤❡ ♥♦♥✲❤♦•♦♠♦r♣❤✐❝ ❊✐s❡♥st❡✐♥ s❡r✐❡s ❛♥❞ ✐s ♠♦❞✉•❛r ✐♥✈❛r✐❛♥t✱ E 3/2 (τ ) = ′ m,n∈Z τ 3/2 2 |mτ + n| 3 . ✭✹✳✻✻✮ ◆♦✇ t❤❛t ✇❡ ❤❛✈❡ ♦❜t❛✐♥❡❞ ✭✹✳✻✺✮✱ ✇❡ ❝❛♥ ❡①♣•♦✐t t❤❡ ♣♦✇❡r ♦❢ ❙✲❞✉❛•✐t② ❢♦r ❣♦✐♥❣ t♦ str♦♥❣ ❝♦✉♣•✐♥❣ r❡❣✐♠❡ t♦ ✇❡❛❦ ❝♦✉♣•✐♥❣✳ ❚♦ ❞♦ t❤✐s✱ •❡t ✉s ✜rst ♣✉t τ 1 = 0 ❢♦r ❝♦♥✈❡♥✐❡♥❝❡✳ ❚❤❡♥ ❙✲❞✉❛•✐t② tr❛♥s❢♦r♠❛t✐♦♥ ❛❝ts ❜② τ 2 → τ -1 2 , V → V τ 3 2 . ✭✹✳✻✼✮ ✇✐t❤ t❤✐s ✇❡ ❝❛♥ ❡①tr❛❝t ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ str♦♥❣ ❝♦✉♣•✐♥❣ •✐♠✐t✳ ❙✐♥❝❡ ✐♥ ❤✐♥❞s✐❣❤t ✇❡ ❦♥♦✇ t❤❛t t❤❡ ❡q✉❛t✐♦♥ ✭✹✳✺✼✮ ✐s ❙✲❞✉❛•✐t② ✐♥✈❛r✐❛♥t✱ ✇❡ ❝❛♥ ✇♦r❦ ❞✐r❡❝t•② ✇✐t❤ ✐t✳ ❆❢t❡r ✇❡ ♣❡r❢♦r♠ t❤❡ ❙✲❞✉❛•✐t② tr❛♥s❢♦r♠❛t✐♦♥ ✭✹✳✻✼✮ ❛s ❛❜♦✈❡ ✇❡ ❝❛♥ t❛❦✐♥❣ t❤❡ •✐♠✐t t❤❡ •✐♠✐t τ 2 → ∞✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ str♦♥❣ ❝♦✉♣•✐♥❣✱ ❛s ✐t ✇❛s t❤❡ ✉s✉❛• ✇❡❛❦ ❝♦✉♣•✐♥❣ •✐♠✐t✳ ❚❤❡ ✐♥st❛♥t♦♥ ❝♦♥tr✐❜✉t✐♦♥s ❜❡❝♦♠❡ ❡①♣♦♥❡♥t✐❛••② s✉♣♣r❡ss❡❞ ❛♥❞ t❤❡ •❡❢t ❤❛♥❞ s✐❞❡ ♦❢ t❤❡ s✐♥❣✉•❛r✐t② ❡q✉❛t✐♦♥ ❤❛s t❤❡ ❡①♣❛♥s✐♦♥✱ - ζ(3) 2 χ 2 Ŷ 2(16π 3 V ) 2 τ -6 2 + ζ(3)χ 2 Ŷ 6(16π 2 V ) 2 τ -4 2 + ζ(3)χ Ŷ 32π 3 V τ -3 2 + χ Ŷ 96πV τ -1 2 + 1 + O e -2πτ -1 2 . ✭✹✳✻✽✮ ❯♥•✐❦❡ t❤❡ ❝❛s❡ ❜❡❢♦r❡ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥✱ ✇❤❡r❡ t❤❡ •❡❢t ❤❛♥❞ s✐❞❡ ✇❛s ❛•✇❛②s ♣♦s✐t✐✈❡✱ t❤❡ ❞♦♠✐♥❛♥t ❝♦♥tr✐❜✉t✐♦♥ ♥♦✇ ❝♦♠❡s ❢r♦♠ t❤❡ ✜rst t❡r♠ ❣♦✐♥❣ -τ -6 2 ✳ ✺✹ ❈✉r✈❛t✉r❡ s✐♥❣✉•❛r✐t② ✐♥ ♣r❡s❡♥❝❡ ♦❢ ❉✲✐♥st❛♥t♦♥s✳ ✹✳✺ ❚❤✉s ✇❡ ❝♦♥❝•✉❞❡ t❤❛t t❤❡ s✐♥❣✉•❛r✐t② r❡♠❛✐♥s ❡✈❡♥ ✐❢ ♦♥❡ ✐♥❝♦r♣♦r❛t❡s ❉✭✲✶✮✲ ✐♥st❛♥t♦♥s✳ ❚❤❡ s✐t✉❛t✐♦♥ ❛❝t✉❛••② ❜❡❝♦♠❡s ✇♦rs❡ ❛s ❝♦♠♣❛r❡❞ t♦ t❤❡ ♦♥❡✲•♦♦♣ ❝♦rr❡❝t❡❞ ❝❛s❡✳ ❚❤❡r❡ t❤❡ s✐♥❣✉•❛r✐t② ❡①✐st❡❞ ❢♦r ♦♥•② χ Y > 0✱ ✇❤❡r❡❛s ♥♦✇ ✐t ❡①✐sts ❢♦r ❛♥② s✐❣♥ ♦❢ t❤❡ ❊✉•❡r ❝❤❛r❛❝t❡r✐st✐❝✳ ❋✐♥❛••② t♦ r❡♠❛r❦✱ ✇❡ ❞♦ ♥♦t ❡①♣❡❝t t❤❛t ❛♥② ♦❢ t❤❡ ❉✲✐♥st❛♥t♦♥s ❝❛♥ ❝✉r❡ t❤✐s s✐t✉❛t✐♦♥✳ ❚❤❡ ♦♥•② ♣♦ss✐❜✐•✐t② t♦ ❣❡t r✐❞ ♦❢ t❤❡ s✐♥❣✉•❛r✐t② ❛♣♣❡❛rs t♦ r❡q✉✐r❡ t❤❡ ✐♥❝•✉s✐♦♥ ♦❢ ◆❙✺✲✐♥st❛♥t♦♥s✳ ❚❤❡✐r ❡①♣♦♥❡♥t✐❛••② s✉♣♣r❡ss❡❞ ❝♦♥tr✐❜✉t✐♦♥ ❛t t❤❡ ✇❡❛❦ ❝♦✉♣•✐♥❣ r❡❣✐♠❡ ❝❛♥ ❞r❛st✐❝❛••② ❛•t❡r t❤❡ s✐t✉❛t✐♦♥ ❛t t❤❡ str♦♥❣ ❝♦✉♣•✐♥❣ t❤♦✉❣❤ ❙✲❞✉❛•✐t②✳ ■t ✇✐•• ❜❡ ✈❡r② ✐♥t❡r❡st✐♥❣ t♦ ✐♥✈❡st✐❣❛t❡ t❤❡✐r ❡✛❡❝t ❛♥❞ ✐s ♦♥❡ ♦❢ t❤❡ ✇♦r❦s ❢♦r ❢✉t✉r❡✳ ✺✺ ❈❍❆P❚❊| ✺ ▼✐rr♦r s②♠♠❡tr② ❛t t❤❡ q✉❛♥t✉♠ •❡✈❡• ■♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡r✱ ✇❡ ✇♦r❦❡❞ ✇✐t❤ t❤❡ ❤②♣❡r♠✉•t✐♣•❡t ♠♦❞✉•✐ s♣❛❝❡ ✐♥ t❤❡ t②♣❡ ■■❆ ♣✐❝t✉r❡✳ ❆s ✇❛s ♠❡♥t✐♦♥❡❞✱ ♠✐rr♦r s②♠♠❡tr② r❡•❛t❡s t❤❡ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ♦❢ t②♣❡ ■■❆ str✐♥❣ t❤❡♦r② ❝♦♠♣❛❝t✐✜❡❞ ♦♥ ❛ ❈❛•❛❜✐✲(❛✉ Y ❛♥❞ t❤❛t ♦❢ t②♣❡ ■■❇ str✐♥❣ t❤❡♦r② ❝♦♠♣❛❝t✐✜❡❞ ♦♥ ❛ ♠✐rr♦r ❈❛•❛❜✐✲(❛✉ Ŷ✳ ▲❡t ✉s ♥♦✇ t✉r♥ t♦ t❤❡ •❛tt❡r ❢♦r♠✉•❛t✐♦♥✳ ❊✈❡♥ t❤♦✉❣❤ t❤❡s❡ t✇♦ ❞❡s❝r✐♣t✐♦♥s s❤♦✉•❞ ❜❡ ✐❞❡♥t✐❝❛• ❞✉❡ t♦ ♠✐rr♦r s②♠♠❡tr②✱ ✐t ✐s ❝♦♥✈❡♥✐❡♥t t♦ ❤❛✈❡ ❜♦t❤ ♦❢ t❤❡♠ ❛t ♦✉r ❞✐s♣♦s❛•✱ ❜❡❝❛✉s❡ t❤❡r❡ ❛r❡ s❡✈❡r❛• s✐t✉❛t✐♦♥s ✇❤✐❝❤ s✐♠♣•✐❢② ✐♥ ❡✐t❤❡r ♦❢ t❤❡s❡ ❢r❛♠❡✇♦r❦s✳ ■♥ ♣❛rt✐❝✉•❛r✱ ❢♦r t②♣❡ ■■❇ t❤❡♦r② t❤❡r❡ ✐s ❛ ✈❡r② ♣♦✇❡r❢✉• ❙✲❞✉❛•✐t②✳ ▼❛♥② ♦❢ t❤❡ r❡s✉•ts ♣r❡s❡♥t❡❞ ✐♥ t❤❡ t❤❡s✐s ❞❡♣❡♥❞ ❝r✉❝✐❛••② ♦♥ ❝•❡✈❡r ✉s❡ ♦❢ t❤✐s s②♠♠❡tr②✱ ❢♦r ❡①❛♠♣•❡✱ ✜♥❞✐♥❣ ✜✈❡❜r❛♥❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ♦♥ t❤❡ t②♣❡ ■■❇ s✐❞❡✱ ✇❤✐❝❤ ✇❡ ✇✐•• ♣r❡s❡♥t ✐♥ ❝❤❛♣t❡r ✼✳ ❋✐♥❛••②✱ t❤❡ ♠✐rr♦r s②♠♠❡tr② ❛t t❤❡ q✉❛♥t✉♠ •❡✈❡• ✐s st✐•• ❛ ❝♦♥•❡❝t✉r❡✱ ❛•t❤♦✉❣❤ ✐♥ t❤❡ •✐❣❤t ♦❢ t❤❡ r❡❝❡♥t r❡s✉•ts✱ ✐t ❞♦❡s ♥♦t s❡❡♠ t♦ ❜❡ ❛♥ ✉♥❛tt❛✐♥❛❜•❡ ❣♦❛• t♦ ❡st❛❜•✐s❤ ✐t ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛•• ♣♦ss✐❜•❡ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s✳ ▲❡t ✉s ❜r✐❡✢② s✉♠♠❛r✐③❡ t❤❡ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ t❤❡ ❍▼ ♠♦❞✉•✐ s♣❛❝❡s ✐♥ t❤❡ t✇♦ ♠✐rr♦r ❢♦r♠✉•❛t✐♦♥s✳ ❋✐rst✱ ✐♥st❡❛❞ ♦❢ ❝♦♠♣•❡① str✉❝t✉r❡ ❞❡❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ❈❛•❛❜✐✲(❛✉ z a ✱ ♦♥❡ ❝♦♥s✐❞❡rs t❤❡ ❝♦♠♣•❡①✜❡❞ ❑ä❤•❡r ♠♦❞✉•✐ v a ✱ ✇❤❡r❡ ✇❡ ❛r❡ ✉s✐♥❣ t❤❡ s❛♠❡ ✐♥❞❡① ❜❡❝❛✉s❡ ❢♦r ♠✐rr♦r ♣❛✐r ♦❢ ❈❛•❛❜✐✲(❛✉s h 2,1 (Y) = h 1,1 ( Ŷ)✳ ❚❤❡ ♣❡r✐♦❞s ♦❢ || ✸✲❢♦r♠ ❛r❡ r❡♣•❛❝❡❞ ❜② t❤❡ ♣❡r✐♦❞s ♦❢ t❤❡ ❡✈❡♥ ❞✐♠❡♥s✐♦♥❛• || ❢♦r♠s ✭c 0 , c a , c a , c 0 ✮✳ ❋✉rt❤❡r♠♦r❡✱ ♦♥❡ ❝❛♥ ❝♦♠❜✐♥❡ τ 1 = c 0 ✇✐t❤ t❤❡ t❡♥ ❞✐♠❡♥s✐♦♥❛• str✐♥❣ ❝♦✉♣•✐♥❣ ❝♦♥st❛♥t τ 2 = 1/g s t♦ ❢♦r♠ ❛♥ ❛①✐♦✲❞✐•❛t♦♥✐❝ ✜❡•❞ τ = τ 1 + iτ 2 ✳ ◆❡①t✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ t②♣❡ ♦❢ s②♠♠❡tr✐❡s t❤❛t ❝❤❛r❛❝t❡r✐③❡ ❡❛❝❤ ♦❢ t❤❡ ♣❛r✲ t✐❝✉•❛r ❢♦r♠✉•❛t✐♦♥s✳ ❚❤❡ t②♣❡ ■■❆ t❤❡♦r② ✐s ♠❛♥✐❢❡st•② s②♠♣•❡❝t✐❝❛••② ❝♦✈❛r✐❛♥t✱ ✇❤❡r❡❛s t❤❡ ♣❡rt✐♥❡♥t s②♠♠❡tr② ❢♦r t②♣❡ ■■❇ ✐s SL(2, Z) ❞✉❛•✐t②✳ ❚❤❡ ♣❤②s✐❝❛• ✜❡•❞s ✐♥ t②♣❡ ■■❇ ❢♦r♠ ❛ r❡♣r❡s❡♥t❛t✐♦♥ ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ t❤✐s s②♠♠❡tr② ❣r♦✉♣✳ ✺✻ ❉✲❜r❛♥❡s ❛♥❞ ❝❤❛r❣❡ q✉❛♥t✐③❛t✐♦♥✳ ✺✳✶ ■♥ ♣❛rt✐❝✉•❛r✱ t❤✐s ✐♠♣•✐❡s t❤❛t✱ ❛t t❤❡ q✉❛♥t✉♠ •❡✈❡•✱ ♦✉r ❝♦♥str✉❝t✐♦♥ ✐♥ t②♣❡ ■■❆ ❢r❛♠❡✇♦r❦ ♠✉st ❤❛✈❡ ❛ ❤✐❞❞❡♥ SL(2, Z) s②♠♠❡tr②✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ♠❛❞❡ ♠❛♥✐❢❡st ❜② ♣❛ss✐♥❣ t♦ t❤❡ t②♣❡ ■■❇ s✐❞❡✳ ■♥ t❤✐s ❝❤❛♣t❡r ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ♣r♦✈✐❞❡ ❛ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ t✇✐st♦r s♣❛❝❡ ♦❢ ❛ ❝•❛ss ♦❢ ◗❑ ♠❛♥✐❢♦•❞s ❛❞♠✐tt✐♥❣ ❛♥ ✐s♦♠❡tr✐❝ ❛❝t✐♦♥ ♦❢ SL(2, Z)✳ ❲❡ ❛•s♦ s❤♦✇ ❤♦✇ ✐t ❝❛♥ ❜❡ s✐❣♥✐✜❝❛♥t•② s✐♠♣•✐✜❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s✳ ❚❤✐s s✐♠♣•✐✜❝❛t✐♦♥ ✇✐•• ❜❡ ❝r✉❝✐❛• ✐♥ •❛t❡r ❝❤❛♣t❡rs ✇❤❡r❡ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ♣r❡s❡♥t ♦✉r r❡s✉•ts ❢♦r ✜✈❡❜r❛♥❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s✳ ✺✳✶ ❉✲❜r❛♥❡s ❛♥❞ ❝❤❛r❣❡ q✉❛♥t✐③❛t✐♦♥ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ❛❞❞r❡ss t❤❡ ♣r♦❜•❡♠ ♦❢ ❝•❛ss✐❢②✐♥❣ ❉✲✐♥st❛♥t♦♥s ♦♥ t❤❡ t②♣❡ ■■❇ s✐❞❡✳ ■t t✉r♥s ♦✉t t❤❛t t❤❡ s✐♠♣•✐st✐❝ ❞❡s❝r✐♣t✐♦♥ ♦❢ ✐♥st❛♥t♦♥s ❛s s♦♠❡ n✲ ❞✐♠❡♥s✐♦♥❛• ♦❜•❡❝ts ✇r❛♣♣✐♥❣ n✲❞✐♠❡♥s✐♦♥❛• ❝②❝•❡s ❛♥❞ ❝•❛ss✐✜❡❞ ❜② ✐♥t❡❣❡rs ❡♥✉✲ ♠❡r❛t✐♥❣ t❤❡s❡ ❝②❝•❡s ✐s ♥♦t ❡♥♦✉❣❤✳ ❚❤❡ ❝♦rr❡❝t ✇❛② t♦ ❞❡s❝r✐❜❡ t❤❡♠ ✐s t❤r♦✉❣❤ t❤❡ ❞❡r✐✈❡❞ ❝❛t❡❣♦r② ♦❢ ❝♦❤❡r❡♥t s❤❡❛✈❡s ❬✺✶(✳ ■♥ ♣❛rt✐❝✉•❛r✱ ❢♦r ❛ ♥♦♥✲✈❛♥✐s❤✐♥❣ ❉✺✲❜r❛♥❡ ❝❤❛r❣❡ p 0 ✱ ❛ ❜♦✉♥❞ st❛t❡ ♦❢ ❉✺✲❉✸✲❉✶✲❉✭✲✶✮ ❜r❛♥❡s ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❛ ❝♦❤❡r❡♥t s❤❡❛❢ E ♦♥ Ŷ✱ ♦❢ r❛♥❦ rk(E) = p 0 ✳ ❚❤❡ ❝❤❛r❣❡ ✈❡❝t♦r ✐s ❣✐✈❡♥ ❜② t❤❡ s♦ ❝❛••❡❞ ❣❡♥❡r❛•✐③❡❞ ▼✉❦❛✐ ✈❡❝t♦r ✐♥ t❤❡ ❡✈❡♥✲❞✐♠❡♥s✐♦♥❛• ❝♦❤♦♠♦•♦❣② ❜❛s✐s ❬✺✷(✱ γ ′ = ch(E) Td( Ŷ) = p 0 + p a ω a -q ′ a ω a + q ′ 0 ω Ŷ, ✭✺✳✶✮ ✇❤❡r❡ ch(E) ❛♥❞ Td( Ŷ) ❛r❡ t❤❡ ❈❤❡r♥ ❝❤❛r❛❝t❡r ♦❢ t❤❡ s❤❡❛❢ ❛♥❞ t❤❡ ❚♦❞❞ ❝•❛ss ♦❢ t❤❡ ❈❛•❛❜✐✲(❛✉ Ŷ r❡s♣❡❝t✐✈❡•②✳ ❋r♦♠ t❤✐s ♦♥❡ ❝❛♥ ♦❜t❛✐♥ ❡①♣r❡ss✐♦♥s ❢♦r ❝❤❛r❣❡s ❡①♣•✐❝✐t•②✱ p 0 = rk(E), p a = γ a c 1 (E), q ′ a = - γa ch 2 (E) + p 0 24 c 2,a , q ′ 0 = Ŷ ch 3 (E) + 1 24 c 1 (E)c 2 ( Ŷ) , ✭✺✳✷✮ s❤♦✇✐♥❣ t❤❛t ❝❤❛r❣❡s q ′ Λ ❛r❡ ♥♦t ✐♥t❡❣❡rs✳ ❚❤❡r❡ s❡❡♠s t♦ ❜❡ ❛♥ ❛♣♣❛r❡♥t ❝•❛s❤ ✇✐t❤ t❤❡ ♠✐rr♦r s②♠♠❡tr② ❜❡❝❛✉s❡ t❤❡ ❝❤❛r❣❡ ✈❡❝t♦r ✐♥ t❤❡ t②♣❡ ■■❆ ❢♦r♠✉•❛t✐♦♥ ✐s ✐❞❡♥t✐✜❡❞ ✇✐t❤ t❤❡ ✐♥t❡❣❡r ❤♦♠♦•♦❣② ❝•❛ss H 3 (Y, Z)✳ ❚❤✐s ♣✉③③•❡ ❝❛♥ ❜❡ r❡s♦•✈❡❞ ❛s ❢♦••♦✇s ❬✷✶(✳ ▲❡t ✉s ❝♦♠♣❛r❡ t❤❡ ❝❡♥tr❛• ❝❤❛r❣❡ ❛ss♦❝✐❛t❡❞ t♦ ❉✲✐♥st❛♥t♦♥s ✐♥ t②♣❡ ■■❇ Z γ ′ = Ŷ e -v a ωa γ ′ , ✭✺✳✸✮ ✺✼ ✺✳ ▼✐rr♦r s②♠♠❡tr② ❛t t❤❡ q✉❛♥t✉♠ •❡✈❡• ❛♥❞ t❤❡ ❝❡♥tr❛• ❝❤❛r❣❡ ✐♥ t②♣❡ ■■❆ ❣✐✈❡♥ ❜② t❤❡ ♣r❡♣♦t❡♥t✐❛• r❡str✐❝t❡❞ t♦ t❤❡ •❛r❣❡ ✈♦•✉♠❡ •✐♠✐t F (X) = - κ abc 6 X a X b X c X 0 + 1 2 A ΛΣ X λ X Σ . ✭✺✳✹✮ ■t ✐s ✐♠♣♦rt❛♥t t♦ ❦❡❡♣ t❤❡ q✉❛❞r❛t✐❝ ♣✐❡❝❡ ❤❡r❡✱ ❡✈❡♥ t❤♦✉❣❤ ✐t ❞♦❡s ♥♦t ❛✛❡❝t ❑ä❤•❡r ♣♦t❡♥t✐❛•✳ ❚❤❡♥ ✐❞❡♥t✐❢②✐♥❣ z a ✇✐t❤ v a ✱ t❤❡ t✇♦ ❡①♣r❡ss✐♦♥s ❝♦✐♥❝✐❞❡ ✐❢ q ′ Λ = q Λ -A ΛΣ p Σ . ✭✺✳✺✮ ❋♦r t❤✐s ✐❞❡♥t✐✜❝❛t✐♦♥✱ ♦♥❡ s❤♦✉•❞ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ❝♦♥❞✐t✐♦♥s s❛t✐s✜❡❞ ❜② t❤❡ ♠❛tr✐① A ΛΣ ✐s ❝♦♥str❛✐♥❡❞ ❜② ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s✱ (a) A 00 ∈ Z, (b) A 0a ∈ c 2,a 24 + Z, (c) A ab p b - 1 2 κ abc p b p c ∈ Z, (d) 1 6 κ abc p a p b p c + 1 12 c 2,a p a ∈ Z, ✭✺✳✻✮ ❢♦r ❛♥② ❛r❜✐tr❛r② p a ∈ Z✳ ❚❤❡ ❛❜♦✈❡ ❝♦♥❞✐t✐♦♥s ❡♥s✉r❡ t❤❛t t❤❡ s②♠♣•❡❝t✐❝ tr❛♥s✲ ❢♦r♠❛t✐♦♥ ✭✺✳✺✮ ♠❛♣s t❤❡ r❛t✐♦♥❛• ❝❤❛r❣❡s ♦❢ t❤❡ t②♣❡ ■■❇ ❢♦r♠✉•❛t✐♦♥ ✭✺✳✶✮ t♦ t❤❡ ✐♥t❡❣❡r ❝❤❛r❣❡s ♦❢ t❤❡ t②♣❡ ■■❆ ❢♦r♠✉•❛t✐♦♥✳ ❚❤❡ q✉❛♥t✐③❛t✐♦♥ ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ♣r✐♠❡❞ ❝❤❛r❣❡s ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦••♦✇✐♥❣✱ q ′ 0 ∈ Z - p 0 24 c 2,a - 1 2 κ abc p b p c , q ′ a ∈ Z - 1 24 p a c 2,a .
S D-IIB = 2πg -1 s |Z γ | + 2πi Ŷ γ ′ ∧ Âeven e -B2 , ✭✺✳✽✮ ✇❤❡r❡ t❤❡ || ♣♦t❡♥t✐❛• Âeven ✐s Âeven e -B2 = ζ 0 -ζ a ω a -ζ′ a ω a -ζ′ 0 ω Ŷ, ✭✺✳✾✮ ❛♥❞ t❤❡ s❤✐❢t❡❞ || ✜❡•❞s ❛r❡ r❡•❛t❡❞ t♦ t❤❡ ✉♥s❤✐❢t❡❞ ♦♥❡s ❜② ζ′ Λ = ζΛ -A ΛΣ ζ Σ . ✭✺✳✶✵✮ ❚❤✐s s❤✐❢t ✐s t❤❡ s❛♠❡ ❛s ✐♥ ✭✺✳✺✮ ❛♥❞ ❛❜s♦r❜s t❤❡ q✉❛❞r❛t✐❝ t❡r♠ ✐♥ t❤❡ ♣r❡♣♦t❡♥t✐❛•✳ ✺✽ ❙✲❞✉❛•✐t② ✐♥ t❤❡ ❍▼ ♠♦❞✉•✐ s♣❛❝❡✳ ✺✳✷ ✺✳✷ ❙✲❞✉❛•✐t② ✐♥ t❤❡ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ■♥ ❝❤❛♣t❡r ✷ ✇❡ ❤❛✈❡ ❛•r❡❛❞② ❞✐s❝✉ss❡❞ ❛❜♦✉t t❤❡ ❛❝t✐♦♥ ♦❢ ❙✲❞✉❛•✐t② ♦♥ t❤❡ ❝•❛ss✐❝❛• ❍▼ ♠♦❞✉•✐ s♣❛❝❡✳ ■♥ t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥✱ t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ SL(2, R) ❣r♦✉♣ ♦♥ t❤❡ t②♣❡ ■■❇ ✜❡•❞s ✐s ❣✐✈❡♥ ❜② ✭✷✳✹✶✮ ❛♥❞ t❤❡ ❝•❛ss✐❝❛• ♠✐rr♦r ♠❛♣ ✐s ❣✐✈❡♥ ❜② ✭✷✳✹✷✮✳ ■♥ t❤✐s ❝❤❛♣t❡r ♦✉r ❣♦❛• ✐s t♦ ❣✐✈❡ ❛ ❞❡s❝r✐♣t✐♦♥ ♦❢ ❛ ❣❡♥❡r✐❝ ◗❑ ♠❛♥✐❢♦•❞ ❛❞♠✐tt✐♥❣ ♦♥•② ❛♥ ✐s♦♠❡tr✐❝ ❛❝t✐♦♥ ♦❢ SL(2, Z)✳ ❲❡ ❛r❡ ❣♦✐♥❣ t♦ ♣r♦✈✐❞❡ ❡①♣•✐❝✐t ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♠✐rr♦r ♠❛♣ ✐♥ ♣r❡s❡♥❝❡ ♦❢ ❛•• ♣♦ss✐❜•❡ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s✳ ❆•t❤♦✉❣❤✱ ✇❡ ✇✐•• ♥♦t ❜❡ ♣r♦✈✐❞✐♥❣ ❛♥② ❝♦♥❝r❡t❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ❤❡r❡✱ ✇❡ ✇✐•• ❞♦ t❤❡ ♥❡❝❡ss❛r② ❣r♦✉♥❞✇♦r❦ t♦ ❡st❛❜•✐s❤ t❤❛t ✐t ✐s ✐♥❞❡❡❞ ♣♦ss✐❜•❡ t♦ ♣r❡s❡r✈❡ q✉❛t❡r♥✐♦♥✲❑ä❤•❡r str✉❝t✉r❡ ✇❤✐•❡ ❜r❡❛❦✐♥❣ ❛•• t❤❡ ❝♦♥t✐♥✉♦✉s ✐s♦♠❡tr✐❡s✳ ❇✉t ❜❡❢♦r❡ ♣r♦❝❡❡❞✐♥❣ ❢✉rt❤❡r t❤❡r❡ ❛r❡ ❛ ❢❡✇ ❝♦♠♠❡♥ts t♦ ♠❛❦❡✳ ❼ ❚❤❡ ❝♦♥t✐♥✉♦✉s SL(2, R) s②♠♠❡tr② ✐s ♣r❡s❡♥t ♦♥•② ❛t t❤❡ ❝•❛ss✐❝❛• •❡✈❡•✳ ❆•r❡❛❞②✱ α ′ ❝♦rr❡❝t✐♦♥s ❜r❡❛❦ ✐t✳ ❍♦✇❡✈❡r ✐t ✇❛s s❤♦✇♥ ✐♥ ❬✺✺(✱ ❬✷✵( t❤❛t ✐❢ ❉✭✲✶✮ ❛♥❞ ❉✶ ✐♥st❛♥t♦♥ ❝♦rr❡❝t✐♦♥s ❛r❡ ✐♥❝♦r♣♦r❛t❡❞✱ ♦♥❡ ❝❛♥ r❡st♦r❡ ❛ s✉❜❣r♦✉♣ ♦❢ ✐t✱ ♥❛♠❡•② SL(2, Z)✳ ❋✉rt❤❡r♠♦r❡✱ ✐s♦♠❡tr✐❝ ❛❝t✐♦♥ ♦❢ t❤❡ •❛tt❡r ✐s ❛•s♦ ❡①♣❡❝t❡❞ t♦ ❤♦•❞ ❡✈❡♥ ✇❤❡♥ ❛•• q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ❛r❡ ✐♥❝♦r♣♦r❛t❡❞✳ ❼ ❚❤❡ || ✜❡•❞s ✐♥ t❡r♠s ♦❢ ♣❡r✐♦❞ ✐♥t❡❣r❛•s ♦❢ ✶✵✲❞✐♠❡♥s✐♦♥❛• ❣❛✉❣❡ ♣♦t❡♥t✐❛•s ❛r❡ ❬✺✱ ✶✽(✱ c 0 = Â0 , c a = γ a Â2 , c a = - γa Â4 - 1 2 B2 ∧ Â2 , c 0 = - Ŷ Â6 -B2 ∧ Â4 + 1 3 B2 ∧ Â2 ∧ Â2 . ✭✺✳✶✶✮ ❚❤❡ ❛❜♦✈❡ ❡①♣r❡ss✐♦♥s ❛r❡ ❡①❛❝t ♦♥•② ❛t t❤❡ ❝•❛ss✐❝❛• •❡✈❡•✳ ■♥ ❣❡♥❡r❛• ✇❡ ❞❡✜♥❡ t❤❡ t②♣❡ ■■❇ ✜❡•❞s ❛s t❤♦s❡ ✇❤✐❝❤ tr❛♥s❢♦r♠ ❛❝❝r❞✐♥❣ t♦ ✭✷✳✹✶✮✳ ❖❜✈✐✲ ♦✉s•②✱ t❤✐s ♠❡❛♥s t❤❛t t❤❡ ♠✐rr♦r ♠❛♣ ✭✷✳✹✷✮ r❡❝❡✐✈❡s q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s✳ ❆s ❤❛s ❜❡❡♥ ♠❡♥t✐♦♥❡❞✱ ♦♥❡ ♦❢ ♦✉r ❣♦❛•s ✐♥ t❤✐s ❝❤❛♣t❡r ✐s t♦ ♦❜t❛✐♥ ❡①♣•✐❝✐t ❡①♣r❡ss✐♦♥s ❢♦r t❤❡♠✳ ❼ ❆t t❤❡ q✉❛♥t✉♠ •❡✈❡•✱ SL(2, Z) tr❛♥s❢♦r♠❛t✐♦♥s ❛r❡ ❣✐✈❡♥ ❜② ✭✷✳✹✶✮ ✇✐t❤ a, b, c, d ∈ Z✱ ❡①❝❡♣t t❤❛t t❤❡ ❉✸✲❛①✐♦♥ c a ✇❤✐❝❤ ✇❛s ✐♥✈❛r✐❛♥t✱ ♥♦✇ ❛❝q✉✐r❡s ❛ ❝♦♥st❛♥t s❤✐❢t c a → c a -c 2,a ε(g) ❬✷✶(✳ ❍❡r❡✱ ε(g) ✐s t❤❡ ♠✉•t✐♣•✐❡r s②st❡♠ ♦❢ t❤❡ ❉❡❞❡❦✐♥❞ ❢✉♥❝t✐♦♥ η(τ ) ❣✐✈❡♥ ❜② e 2πi ε(g) = η aτ +b cτ +d (cτ + d) 1/2 η(τ ) . ✭✺✳✶✷✮ ■♥ ♣❛rt✐❝✉•❛r✱ e 2πi ε(g) ✐s t❤❡ 24 th r♦♦t ♦❢ ✉♥✐t② ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❢r❛❝t✐♦♥❛• t❡r♠s ❛♣♣❡❛r✐♥❣ ✐♥ t❤❡ q✉❛♥t✐③❛t✐♦♥ ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ ❝❤❛r❣❡s q ′ Λ ✳ ■ts ❛♣♣❡❛r✲ ❛♥❝❡ ❝❛♥ ❜❡ tr❛❝❡❞ ❜❛❝❦ t♦ t❤❡ ❢❛❝t t❤❛t t❤❡ ❙✲❞✉❛•✐t② tr❛♥s❢♦r♠❛t✐♦♥s ❛♥❞ ✺✾ ✺✳ ▼✐rr♦r s②♠♠❡tr② ❛t t❤❡ q✉❛♥t✉♠ •❡✈❡• t❤❡ ❍❡✐s❡♥❜❡r❣ s❤✐❢ts ✭✷✳✹✸✮ ❛r❡ ♥♦t ❝♦♠♣•❡t❡•② ✐♥❞❡♣❡♥❞❡♥t✳ ❆❝t✉❛••② ♦♥❡ s❤♦✉•❞ ✐❞❡♥t✐❢② t❤❡ ❍❡✐s❡♥❜❡r❣ tr❛♥s❢♦r♠❛t✐♦♥ ✇✐t❤ t❤❡ ♣❛r❛♠❡t❡r η 0 = b ✇✐t❤ t❤❡ SL(2, Z) tr❛♥s❢♦r♠❛t✐♦♥ ✐♥❞✉❝❡❞ ❜② t❤❡ ♠❛tr✐① 1 b 0 1 ✳ ❚❤✐s ✐❞❡♥✲ t✐✜❝❛t✐♦♥ ✐♥❞❡❡❞ ✇♦r❦s ✐❢ ♦♥❡ t❛❦❡s ✐♥t♦ ❛❝❝♦✉♥t t❤✐s ❝♦♥st❛♥t s❤✐❢t ♦❢ t❤❡ ❛①✐♦♥ c a ✳ ❊✈❡♥ t❤♦✉❣❤ t❤✐s ✐ss✉❡ ❞♦❡s ♥♦t s❡❡♠ t♦ ❜❡ ❝r✉❝✐❛•✱ t❤❡r❡ ✐s ❛♥✲ ♦t❤❡r r❡❛s♦♥ ✇❤✐❝❤ s❤♦✇s t❤❛t t❤✐s ♠♦❞✐✜❝❛t✐♦♥ ✐s r❡❛••② ♥❡❡❞❡❞✳ ▲❛t❡r✱ ✐♥ ❝❤❛♣t❡r ✼ ✇❡ s❤❛•• s❡❡ t❤❛t t❤✐s ♠♦❞✐✜❝❛t✐♦♥ ✐s ✐♠♣♦rt❛♥t ❛•s♦ ❢♦r ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❛❝t✐♦♥s ♦❢ t❤❡ ✐s♦♠❡tr② ❣r♦✉♣s ❣❡♥❡r❛t❡❞ ❜② SL(2, Z)✱ ♠♦♥♦❞r♦♠② ❛♥❞ ❍❡✐s❡♥❜❡r❣ s②♠♠❡tr②✱ ♦♥ t❤❡ ✜✈❡❜r❛♥❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s✳ ✺✳✸ ▼♦♥♦❞r♦♠② ✐♥✈❛r✐❛♥❝❡ ❛♥❞ s♣❡❝tr❛• ✢♦✇ ❆♥♦t❤❡r ✐♠♣♦rt❛♥t s②♠♠❡tr② t❤❛t ❤❛s t♦ ❜❡ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ❛t t❤❡ ♥♦♥✲ ♣❡rt✉r❜❛t✐✈❡ •❡✈❡• ✐s t❤❡ ♠♦♥♦❞r♦♠② ✇❤✐❝❤ ❛r✐s❡s ❢r♦♠ t❤❡ ♣♦ss✐❜✐•✐t② ♦❢ ❣♦✐♥❣ ❛r♦✉♥❞ t❤❡ s✐♥❣✉•❛r✐t✐❡s ✐♥ t❤❡ ♠♦❞✉•✐ s♣❛❝❡ ♦❢ ❝♦♠♣•❡①✐✜❡❞ ❑ä❤•❡r ❝•❛ss ❞❡❢♦r✲ ♠❛t✐♦♥s M K ( Ŷ)✳ ❚❤✐s s❤♦✉•❞ ❜❡ ❛ s②♠♠❡tr② ❜❡❝❛✉s❡ t❤❡ ✜♥❛• ♠❡tr✐❝ ♠✉st ❜❡ s♠♦♦t❤ ❡✈❡r②✇❤❡r❡✳ ❍❡r❡ ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ♠♦♥♦❞r♦♠② ❛r♦✉♥❞ t❤❡ •❛r❣❡ ✈♦•✉♠❡ ♣♦✐♥t✳ ❚❤✐s s②♠♠❡tr② ✐s ❣❡♥❡r❛t❡❞ ❜② b a → b a + ǫ a , ǫ a ∈ Z. ✭✺✳✶✸✮ ❙✐♥❝❡ ✐♥ ✭✺✳✾✮✱ t❤❡ B✲✜❡•❞ ✐s ♣r❡s❡♥t ❛•♦♥❣✇✐t❤ t❤❡ || ♣♦t❡♥t✐❛•✱ t♦ r❡❛•✐③❡ s✉❝❤ s②♠♠❡tr② ♦♥❡ s❤♦✉•❞ ❛•s♦ ♣❡r❢♦r♠ ❛ s②♠♣•❡❝t✐❝ r♦t❛t✐♦♥ ♦❢ ||✲✜❡•❞s ❬✷✶(✱ M ǫ a : ζ 0 → ζ 0 , ζ a → ζ a + ǫ a , ζ′ a → ζ′ a -κ abc ζ b ǫ c - 1 2 κ abc ǫ b ǫ c ζ 0 , ζ′ 0 → ζ′ 0 -ζ′ a ǫ a + 1 2 κ abc ǫ a ǫ b ζ c + 1 6 κ abc ǫ a ǫ b ǫ c ζ 0 . ✭✺✳✶✹✮ ❚❤❡ ❛❜♦✈❡ tr❛♥s❢♦r♠❛t✐♦♥ ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ✐♥ t❡r♠s ♦❢ t❤❡ ✉♥♣r✐♠❡❞ ✜❡•❞s✳ ❚❤❡♥ t❤❡ ♠❛tr✐① ✐♥❞✉❝✐♥❣ t❤❡ s②♠♣•❡❝t✐❝ tr❛♥s❢♦r♠❛t✐♦♥ ❜❡❝♦♠❡s ρ(M ǫ a ) =     1 0 0 0 ǫ a δ a b 0 0 -L a (ǫ) -κ abc ǫ c 0 δ a b L 0 (ǫ) L b (ǫ) + 2A bc ǫ c 1 -ǫ b     , ✭✺✳✶✺✮ ✇❤❡r❡ ✇❡ ✐♥tr♦❞✉❝❡❞ t✇♦ ❢✉♥❝t✐♦♥s L a (ǫ) ≡ 1 2 κ abc ǫ b ǫ c -A ab ǫ b , L 0 (ǫ) ≡ 1 6 κ abc ǫ a ǫ b ǫ c + 1 12 c 2,a ǫ a , ✭✺✳✶✻✮ ✇❤✐❝❤ ❛r❡ ✐♥t❡❣❡r ✈❛•✉❡❞ ❜❡❝❛✉s❡ ♦❢ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♠❛tr✐① A ΛΣ ✭✺✳✻✮✳ ✻✵ ❙✲❞✉❛•✐t② ✐♥ t❤❡ t✇✐st♦r s♣❛❝❡✳ ✺✳✹ ◆♦t❡ t❤❛t✱ t❤❡ ❉✲❜r❛♥❡ ❛❝t✐♦♥ S IIB r❡♠❛✐♥s ✐♥✈❛r✐❛♥t ✐❢ t❤❡ ♠♦♥♦❞r♦♠② tr❛♥s✲ ❢♦r♠❛t✐♦♥ ✐s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ s✐♠✐•❛r tr❛♥s❢♦r♠❛t✐♦♥ ♦♥ t❤❡ ❝❤❛r❣❡ •❛tt✐❝❡✱ t❤❡ s♦ ❝❛••❡❞ s♣❡❝tr❛• ✢♦✇ ❞❡✜♥❡❞ ❜② γ → γ[ǫ] = ρ(M ǫ a ).γ. ✭✺✳✶✼✮ ❚❤❡ s②♠♣•❡❝t✐❝ ♠❛tr✐① ✐s ✐♥t❡❣❡r ✈❛•✉❡❞✱ ❛♥❞ ❤❡♥❝❡ ✐t ♣r❡s❡r✈❡s t❤❡ ❝❤❛r❣❡ •❛tt✐❝❡ ❛♥❞ ❤❡♥❝❡ ✐s ❛♥ ❛❞♠✐ss✐❜•❡ tr❛♥s❢♦r♠❛t✐♦♥✳ ❙✐♥❝❡ t❤❡ ❉✲✐♥st❛♥t♦♥ ❛❝t✐♦♥ r❡♠❛✐♥s ✐♥✈❛r✐❛♥t ✉♥❞❡r ❝♦♠❜✐♥❡❞ ❛❝t✐♦♥s ♦❢ ♠♦♥✲ ♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ❛♥❞ t❤❡ s♣❡❝tr❛• ✢♦✇ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ✐t✱ t❤❡ ❇P❙ ✐♥❞✐❝❡s Ω(γ; v) s❤♦✉•❞ ❛•s♦ ❜❡ ✐♥✈❛r✐❛♥t✳ ❆•t❤♦✉❣❤ t❤❡ ♠♦❞✉•✐ ❞❡♣❡♥❞❡♥❝❡ ❢♦r t❤❡♠ ✐s ♠✐•❞✱ ✐t ✐s ❝r✉❝✐❛• ❢♦r t❤❡ ✐♥✈❛r✐❛♥❝❡ t♦ t❛❦❡ ✐t ✐♥t♦ ❛❝❝♦✉♥t✳ ✺✳✹ ❙✲❞✉❛•✐t② ✐♥ t❤❡ t✇✐st♦r s♣❛❝❡ ❚❤❡ ♠♦st ♣r♦♠✐♥❡♥t ❢❡❛t✉r❡ ♦❢ t②♣❡ ■■❇ ❢♦r♠✉•❛t✐♦♥ ✐s ✐ts ❙✲❞✉❛•✐t② s②♠♠❡tr②✳ ❋♦r ♦✉r ❝❛s❡ ✐t ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❛❝t✐♦♥ ♦❢ SL(2, Z) ❣r♦✉♣ ♦♥ t❤❡ t②♣❡ ■■❇ ✜❡•❞s ❛❝❝♦r❞✐♥❣ t♦ ✭✷✳✹✶✮✳ ❲❡ ❤❛✈❡ ❛•r❡❛❞② s❡❡♥ t❤❛t t❤❡ ♣r♦❜•❡♠ ♦❢ ✐♥❝♦r♣♦r❛t✐♥❣ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ❝❛♥ ❜❡ ♠♦st ❝♦♥✈❡♥✐❡♥t•② t❛❝❦•❡❞ ❛t t❤❡ •❡✈❡• ♦❢ t❤❡ t✇✐st♦r s♣❛❝❡✳ ❚❤❡r❡❢♦r❡✱ ✐t ✐s ✐♠♣♦rt❛♥t t♦ ✉♥❞❡rst❛♥❞ ❤♦✇ ❙✲❞✉❛•✐t② ❛❝ts ♦♥ t❤❡ t✇✐st♦r s♣❛❝❡✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ ✇❡ ✇❛♥t t♦ ♦❜t❛✐♥ t❤❡ t✇✐st♦r ❞❡s❝r✐♣t✐♦♥ ✇❤❡♥ t❤❡r❡ ✐s ♦♥•② ❛♥ ✐s♦♠❡tr✐❝ ❛❝t✐♦♥ SL(2, Z) ❣r♦✉♣ ♣r❡s❡♥t✳ ❖✉r ❣♦❛• ✐s t✇♦❢♦•❞ ❤❡r❡✳ ❋✐rst✱ ✇❡ ✇❛♥t t♦ ✉♥❞❡rst❛♥❞ ✇❤❛t ❦✐♥❞ ♦❢ ❝♦♥str❛✐♥ts ❛r❡ ✐♠♣♦s❡❞ ❛t t❤❡ •❡✈❡• ♦❢ t❤❡ t✇✐st♦r s♣❛❝❡ ✐♥ t❤✐s ❝❛s❡✳ ❙❡❝♦♥❞•②✱ ✇❡ ✇❛♥t t♦ ♦❜t❛✐♥ t❤❡ ♠✐rr♦r ♠❛♣ r❡•❛t✐♥❣ t❤❡ t②♣❡ ■■❆ ❛♥❞ t②♣❡ ■■❇ ✜❡•❞s✳ ❖❢ ❝♦✉rs❡✱ s✐♥❝❡ ✇❡ ❞♦ ♥♦t s♣❡❝✐❢② ❝♦♥❝r❡t❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s✱ t❤❡ •❛tt❡r r❡s✉•ts ✇✐•• ❜❡ ♦♥•② ❢♦r♠❛•✳ ❚❤✐s q✉❡st✐♦♥ ♦❢ ✜♥❞✐♥❣ ❛♣♣r♦♣r✐❛t❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ✇✐•• ❜❡ ❛❞❞r❡ss❡❞ ✐♥ ❛ •❛t❡r ❝❤❛♣t❡r✳ ✺✳✹✳✶ ▲✐❢t ♦❢ t❤❡ SL(2, R) ❢♦r tr❡❡ •❡✈❡• ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ❲❡ st❛rt ❢r♦♠ t❤❡ ❝•❛ss✐❝❛• t✇✐st♦r s♣❛❝❡ ♦❢ t❤❡ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ✇❤✐❝❤ ❝❛rr✐❡s ❛♥ ✐s♦♠❡tr✐❝ ❛❝t✐♦♥ ♦❢ t❤❡ ❝♦♥t✐♥✉♦✉s ❣r♦✉♣ SL(2, R)✳ ■♥ t❤❡ ❝•❛ss✐❝❛• ❛♣♣r♦①✐♠❛t✐♦♥✱ t❤❡ •❛st t✇♦ t❡r♠s ✐♥ t❤❡ ♣r❡♣♦t❡♥t✐❛• ✐♥ ✭✷✳✸✷✮ ❛r❡ ❛❜s❡♥t✳ ❋✉rt❤❡r♠♦r❡ ✐❢ ✇❡ r❡❞❡✜♥❡ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ❛s ξ′ Λ = ξΛ -A ΛΣ ξ Λ , α ′ = α + 1 2 A ΛΣ ξ Λ ξ Σ , ✭✺✳✶✽✮ ✻✶ ✺✳ ▼✐rr♦r s②♠♠❡tr② ❛t t❤❡ q✉❛♥t✉♠ •❡✈❡• t❤❡ q✉❛❞r❛t✐❝ ♣✐❡❝❡ ✐♥ t❤❡ ♣r❡♣♦t❡♥t✐❛• ✐s ❛•s♦ r❡♠♦✈❡❞ ❛♥❞ ♦♥❡ r❡♠❛✐♥s •✉st ✇✐t❤ t❤❡ ✜rst ❝✉❜✐❝ t❡r♠✳ ❚❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ❛r❡ ✶ ξ Λ [0] = ζ Λ + R(t -1 z Λ -tz Λ ), ξ[0] Λ = ζΛ + R(t -1 F cl Λ (z) -t F cl Λ (z)), ✭✺✳✶✾✮ α [0] = - 1 2 (σ + ζ Λ ζΛ ) -Rζ Λ (t -1 F cl Λ -t F cl Λ ) -R 2 (t -2 F cl + t 2 F cl ) + R 2 2 (z Λ F cl Λ + zΛ F cl Λ ). ■♥ ❛♥② ❝❛s❡ t❤❡ ✐s♦♠❡tr✐❝ ❛❝t✐♦♥ ♦❢ SL(2, R) ♠✉st ❤❛✈❡ ❛ ❤♦•♦♠♦r♣❤✐❝ r❡♣r❡s❡♥t❛✲ t✐♦♥ ♦♥ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s✳ ❚♦ ❛❝❤✐❡✈❡ t❤✐s ♦♥❡ ❤❛s t♦ ❛•s♦ ♣r❡s❝r✐❜❡ ❤♦✇ t❤❡ ✜❜❡r ❝♦♦r❞✐♥❛t❡ t tr❛♥s❢♦r♠s✳ ❚❤✐s ✇❛s ❢♦✉♥❞ ✐♥ ❬✷✾( ❛♥❞ t❤❡ r❡s✉•t r❡❛❞s ❛s t → cτ 2 + t(cτ 1 + d) + t|cτ + d| (cτ 1 + d) + |cτ + d| -tcτ 2 = 1 + t c,d -t t c,d --t = - t c,d + -t 1 + t c,d + t , ✭✺✳✷✵✮ ✇❤❡r❡ t c,d ± ❛r❡ t❤❡ t✇♦ r♦♦ts ♦❢ t❤❡ ❡q✉❛t✐♦♥ cξ 0 + d = 0 ❛♥❞ ❛r❡ ❣✐✈❡♥ ❜② t c,d ± = cτ 1 + d ∓ |cτ + d| cτ 2 , t c,d + t c,d -= -1. ✭✺✳✷✶✮ ❚❤❡♥ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ SL(2, R) ❛r❡ ❢♦✉♥❞ t♦ ❜❡ ❬✶✽(✱ ξ 0 → aξ 0 + b cξ 0 + d , ξ a → ξ a cξ 0 + d , ξa → ξa + cκ abc 2(cξ 0 + d) ξ a ξ b , ξ0 α → d -c -b a ξ0 α + 1 6 κ abc ξ a ξ b ξ c c 2 /(cξ 0 + d) -[c 2 (aξ 0 + b) + 2c]/(cξ 0 + d) 2 . ✭✺✳✷✷✮ ▼♦r❡♦✈❡r✱ ♦♥❡ ♦❜t❛✐♥s t❤❡ ❢♦••♦✇✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥s ❢♦r t❤❡ ❝♦♥t❛❝t ♣♦t❡♥t✐❛•✱t❤❡ ❑ä❤•❡r ♣♦t❡♥t✐❛• ♦♥ t❤❡ t✇✐st♦r s♣❛❝❡ ❛♥❞ t❤❡ ❝♦♥t❛❝t ♦♥❡✲❢♦r♠✱ e Φ → e Φ |cτ + d| , K Z → K Z -log(|cξ 0 + d|), X [i] → X [i] cξ 0 + d . ✭✺✳✷✸✮ ❙♦✱ ✐t t✉r♥s ♦✉t t❤❛t t❤❡ ❑ä❤•❡r ♣♦t❡♥t✐❛• ❝❤❛♥❣❡s ❜② ❛ ❑ä❤•❡r tr❛♥s❢♦r♠❛t✐♦♥ ❝♦♥s✐st❡♥t•②✱ •❡❛✈✐♥❣ t❤❡ ♠❡tr✐❝ ♦♥ t❤❡ t✇✐st♦r s♣❛❝❡ Z ♦r t❤❡ ♠❡tr✐❝ ♦♥ t❤❡ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ M HM ✐♥✈❛r✐❛♥t✳ ❋✉rt❤❡r♠♦r❡ t❤❡ ❝♦♥t❛❝t str✉❝t✉r❡ r❡♠❛✐♥s ✐♥✈❛r✐✲ ❛♥t✱ ❛s t❤❡ ❝♦♥t❛❝t ♦♥❡ ❢♦r♠ r❡s❝❛•❡s ❜② ❛ ❤♦•♦♠♦r♣❤✐❝ ❢❛❝t♦r ♦♥•②✱ ❡♥s✉r✐♥❣ t❤❛t t❤❡ ❙✲❞✉❛•✐t② ❛❝ts ❜② ✐♥❞✉❝✐♥❣ ❛ ❝♦♥t❛❝t tr❛♥s❢♦r♠❛t✐♦♥✳ ✶ ❋♦r ❝•❛r✐t② ♦❢ ♥♦t❛t✐♦♥s✱ ✇❡ ❞r♦♣ t❤❡ ♣r✐♠❡s ✐♥ t❤❡ r❡st ♦❢ t❤✐s ❝❤❛♣t❡r ❛♥❞ ❜② F cl ✱ ✇❡ ❞❡♥♦t❡ t❤❡ ❝•❛ss✐❝❛• ❝✉❜✐❝ ♣❛rt ♦❢ t❤❡ ♣r❡♣♦t❡♥t✐❛•✳ ✻✷ ❙✲❞✉❛•✐t② ✐♥ t❤❡ t✇✐st♦r s♣❛❝❡✳ ✺✳✹ ✺✳✹✳✷ ▼♦❞✉•❛r ❝♦♥str❛✐♥t ♦♥ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ◆♦✇ ✇❡ ✇❛♥t t♦ ❞❡❢♦r♠ t❤❡ ◗❑ s♣❛❝❡ ♦❢ t❤❡ ♣r❡✈✐♦✉s s✉❜s❡❝t✐♦♥ ✐♥ ❛ ♠♦❞✉•❛r ✐♥✈❛r✐❛♥t ❢❛s❤✐♦♥✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ ✇❡ ✇✐•• ❛ss✉♠❡ t❤❛t ❛•• t❤❡ ❝♦♥t✐♥✉♦✉s ✐s♦♠❡tr✐❡s ❛r❡ ❜r♦❦❡♥ ❛♥❞ ✜♥❞ ❛ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ◗❑ s♣❛❝❡s ❛❞♠✐tt✐♥❣ ✐s♦♠❡tr✐❝ ❛❝t✐♦♥ ♦❢ SL(2, Z)✳ ■♥ s✉❝❤ s✐t✉❛t✐♦♥ t❤❡ t✇✐st♦r s♣❛❝❡ ❞❡s❝r✐♣t✐♦♥ ✐s t❤❡ ♠♦st ❡✣❝✐❡♥t ♦♥❡✳ ❚❤❡ ✐♥❝•✉s✐♦♥ ♦❢ ✐♥st❛♥t♦♥s t❤❛t ❛r❡ r❡s♣♦♥s✐❜•❡ ❢♦r ❜r❡❛❦✐♥❣ ♦❢ t❤❡ ❝♦♥t✐♥✉♦✉s SL(2, R) s②♠♠❡tr② t♦ ✐ts ❞✐s❝r❡t❡ s✉❜❣r♦✉♣ SL(2, Z)✱ r❡✜♥❡s t❤❡ ❝♦✈❡r✐♥❣ ♦♥ t❤❡ t✇✐st♦r s♣❛❝❡ ❛♥❞ t❤❡ s❛♠❡ ❤❛♣♣❡♥s ❢♦r t❤❡ ❛ss♦❝✐❛t❡❞ s❡t ♦❢ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s✳ ❚❤❡ ✜rst ✐ss✉❡ t❤❛t ♦♥❡ ♥❡❡❞s t♦ ❛❞❞r❡ss ❤❡r❡ ✐s ❤♦✇ ❙✲❞✉❛•✐t② ✐s r❡❛•✐③❡❞ ✐♥ t❤✐s ❝♦♥t❡①t✳ ❋♦r t❤✐s ♣✉r♣♦s❡✱ t♦ ❛✈♦✐❞ ✉♥♥❡❝❡ss❛r② ❝♦♠♣•✐❝❛t✐♦♥s✱ ✇❡ ✜rst ♣r♦✈✐❞❡ t❤❡ s✐♠✲ ♣•❡st ❝♦♥str✉❝t✐♦♥ ✐♥ t❡r♠s ♦❢ ♦♣❡♥ ♣❛t❝❤❡s ❛♥❞ ❝•♦s❡❞ ❝♦♥t♦✉rs ❬✷✾✱ ✸✵( ❛♥❞ •❛t❡r ❝♦♠♠❡♥t ♦♥ ❣❡♥❡r❛•✐③❛t✐♦♥s✳ ▲❡t t❤❡ t✇✐st♦r s♣❛❝❡ Z ❜❡ ❞❡✜♥❡❞ ❜② t❤❡ ❝♦✈❡r✐♥❣ Z = U + ∪ U -∪ (∪ m,n U m,n ) ✭✺✳✷✹✮ ✇❤❡r❡ U ± ❝♦✈❡r t❤❡ ♥♦rt❤ ❛♥❞ s♦✉t❤ ♣♦•❡s ♦❢ CP 1 ❛s ❛❜♦✈❡ ❛♥❞ U m,n ❛r❡ ♠❛♣♣❡❞ t♦ ❡❛❝❤ ♦t❤❡r ✉♥❞❡r t❤❡ ❛♥t✐♣♦❞❛• ♠❛♣ ❛♥❞ SL(2, Z) tr❛♥s❢♦r♠❛t✐♦♥s ς [U m,n ] = U -m,-n , ✭✺✳✷✺✮ U m,n → U m,n , m n = d -c -b a m n , ✭✺✳✷✻✮ ✐♥❝•✉❞✐♥❣ ❛♥ ✐♥✈❛r✐❛♥t ♣❛t❝❤ U 0 ≡ U 0,0 ✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❛ss✉♠❡ ✐♥ ❛❞❞✐t✐♦♥ t❤❛t cξ 0 [m,n] + d ✐s ♥♦♥✲✈❛♥✐s❤✐♥❣ ✐♥ U m,n ❢♦r ❛•• (c, d) = (±m, ±n)✱ ✐♥❝•✉❞✐♥❣ (m, n) = (0, 0)✱ ❛♥❞ ❤❛s ❛ s✐♠♣•❡ ③❡r♦ ❢♦r (c, d) = (m, n)✳ ❉❡♥♦t✐♥❣ t❤✐s ③❡r♦ ❜② t c,d + ✱ t❤❡ r❡❛•✐t② ❝♦♥❞✐t✐♦♥ ❢♦r ξ 0 ❛♥❞ ✭✺✳✷✺✮ ✐♠♣•② t❤❛t ς t c,d + = t -c,-d + ≡ t c,d -. ✭✺✳✷✼✮ ❲✐t❤ t❤❡ ❝♦✈❡r✐♥❣ ✭✺✳✷✹✮ ✇❡ ❛ss♦❝✐❛t❡ t❤❡ ❢♦••♦✇✐♥❣ s❡t ♦❢ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s H [+0] = F cl (ξ [+] ), H [-0] = F cl (ξ [-] ), H [(m,n)0] = G m,n (ξ [m,n] , ξ[0] , α [0] ), ✭✺✳✷✽✮ ✇❤❡r❡ F cl (X) ✐s t❤❡ ❝✉❜✐❝ ♣r❡♣♦t❡♥t✐❛•✳ ✷ ❚❤❡ ❢✉♥❝t✐♦♥s G m,n ❛r❡ ♥♦t ❛r❜✐tr❛r②✳ ❇❡s✐❞❡s t❤❡ r❡❛•✐t② ❝♦♥❞✐t✐♦♥s ς G m,n = G -m,-n , ✭✺✳✷✾✮ ✷ ◆♦t❡ t❤❛t t❤✐s ❛ss✉♠♣t✐♦♥ ✐♠♣•✐❡s t❤❛t U km,kn ❢♦r k ≥ 1 ❛r❡ ❛•• ✐❞❡♥t✐❝❛•✱ ✐♥ ♣❛rt✐❝✉•❛r U 0,±k = U ± ✳ ❍♦✇❡✈❡r✱ ✇❡ ❝♦♥s✐❞❡r ❛•• s✉❝❤ ♣❛t❝❤❡s t♦ ❜❡ ❞✐✛❡r❡♥t✳ ❚❤✐s s❛❝r✐✜❝❡ ✐♥ r✐❣♦r r❡♥❞❡rs t❤❡ ♣r❡s❡♥t❛t✐♦♥ t♦ ❜❡ •❡ss ❝✉♠❜❡rs♦♠❡✳ ❆ ♠♦r❡ ♣r❡❝✐s❡ ❛♥❛•②s✐s ✐s ❛❝t✉❛••② ♣♦ss✐❜•❡✱ ❜✉t ✐t ❞♦❡s ♥♦t ❝❤❛♥❣❡ ❛♥② r❡s✉•t✳ ✻✸ ✺✳ ▼✐rr♦r s②♠♠❡tr② ❛t t❤❡ q✉❛♥t✉♠ •❡✈❡• t❤❡② ♠✉st tr❛♥s❢♦r♠ ✐♥ s✉❝❤ ❛ ✇❛② s♦ t❤❛t t❤❡ ♠♦❞✉•❛r ✐♥✈❛r✐❛♥❝❡ ✐s ❡♥s✉r❡❞✳ ❚❤❡ ❛❝t✐♦♥ ♦❢ ❙✲❞✉❛•✐t② tr❛♥s❢♦r♠❛t✐♦♥ ♠✉st ♣r❡s❡r✈❡ t❤❡ ❝♦♥t❛❝t str✉❝t✉r❡✳ ❆ s✐♠♣•❡ ✇❛② t♦ r❡❛•✐③❡ t❤✐s ✐s t❤r♦✉❣❤ ❞❡♠❛♥❞✐♥❣ t❤❛t t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s tr❛♥s❢♦r♠ ❛❝❝♦r❞✐♥❣ t♦ ✭✺✳✷✷✮✱ ✇❤❡r❡ t❤❡ ♦♥•② ♥❡✇ ❢❡❛t✉r❡ ✐s t❤❛t t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ❛r❡ •❛❜❡•❡❞ ❜② ♣❛t❝❤ ✐♥❞✐❝❡s [m, n]✳ ❯♥❞❡r ❙✲❞✉❛•✐t②✱ t❤❡ ♣❛t❝❤❡s t❤❛t ❝♦✈❡r t❤❡ t✇✐st♦r s♣❛❝❡ s❤♦✉•❞ r❡s❤✉✤❡ t♦ ❡❛❝❤ ♦t❤❡r✳ ■♥ ♣❛rt✐❝✉•❛r ✐t ♠❡❛♥s t❤❛t ❛ ❞✐✛❡r❡♥t ♣❛t❝❤ ✇✐t❤ ✐♥❞✐❝❡s [m ′ , n ′ ] s❤♦✉•❞ ❜❡ r❡•❛t❡❞ t♦ t❤❡ ♣❛t❝❤ ✐♥❞❡①❡❞ ❜② [m, n] ❜② ❙✲❞✉❛•✐t② tr❛♥s❢♦r♠❛t✐♦♥ ❛s t❤❡ ❢♦••♦✇✐♥❣✱ m ′ n ′ = a c b d m n . ✭✺✳✸✵✮ P❡r❢♦r♠✐♥❣ SL(2, Z) tr❛♥s❢♦r♠❛t✐♦♥s ♦♥ t❤❡ ❣•✉✐♥❣ ❝♦♥❞✐t✐♦♥s ✭✸✳✼✮ ♦♥❡ t❤❡♥ ✜♥❞s t❤❛t t❤❡ ❤♦•♦♠♦r♣❤✐❝ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ❛r❡ r❡q✉✐r❡❞ t♦ s❛t✐s❢② t❤❡ ❝♦♥❞✐✲ t✐♦♥✱ G m,n → G m ′ ,n ′ cξ 0 [m ′ ,n ′ ] + d - c 6 κ abc 3ξ a [m ′ ,n ′ ] -2T a m ′ ,n ′ (cξ 0 [m ′ ,n ′ ] + d)(cξ 0 [0] + d) T b m ′ ,n ′ T c m ′ ,n ′ + c 2 6 κ abc 3ξ a [0] ξ b [m ′ ,n ′ ] + T a m ′ ,n ′ T b m ′ ,n ′ (cξ 0 [m ′ ,n ′ ] + d)(cξ 0 [0] + d) 2 T c m ′ ,n ′ T 0 m ′ ,n ′ - c 3 6 κ abc ξ a [m ′ ,n ′ ] ξ b [m ′ ,n ′ ] ξ c [m ′ ,n ′ ] (cξ 0 [m ′ ,n ′ ] + d) 2 (cξ 0 [0] + d) 2 T 0 m ′ ,n ′ 2 , ✭✺✳✸✶✮ ✇❤❡r❡ T Λ m,n = ∂ ξ[0] Λ G m,n -ξ Λ [0] ∂ α [0] G m,n . ✭✺✳✸✷✮ ❚❤✐s ❝♦♥str❛✐♥t ✐s ♦♥❡ ♦❢ t❤❡ ♠❛✐♥ ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ SL(2, Z) ✐♥✈❛r✐❛♥t ❝♦♥str✉❝✲ t✐♦♥✳ ❆s ✇❡ ✇✐•• s❡❡ ❜❡•♦✇✱ ❡✈❡♥ t❤♦✉❣❤ t❤✐s ❝♦♥str❛✐♥t •♦♦❦s ❝♦♠♣•✐❝❛t❡❞ ❞✉❡ t♦ ♥♦♥✲•✐♥❡❛r✐t✐❡s✱ t❤❡② ❝❛♥ ❜❡ r❡♠♦✈❡❞ ✐❢ ♦♥❡ ❝❤♦♦s❡s t♦ ✇♦r❦ ✉s✐♥❣ t❤❡ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s ✇❤✐❝❤ ✇❡r❡ ✐♥tr♦❞✉❝❡❞ ✐♥ ❝❤❛♣t❡r ✸✳ ✺✳✹✳✸ ▼♦❞✉•❛r ❛❝t✐♦♥ ♦♥ t❤❡ t✇✐st♦r ✜❜❡r ❆•t❤♦✉❣❤ ✐t ✇❛s ♥♦t ♠❡♥t✐♦♥❡❞ ❡①♣•✐❝✐t•②✱ ✐♥ t❤❡ ♣r❡✈✐♦✉s s✉❜s❡❝t✐♦♥ ✇❤✐•❡ ❞❡s❝r✐❜✲ ✐♥❣ t❤❡ ♠♦❞✉•❛r ✐♥✈❛r✐❛♥t ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ t✇✐st♦r s♣❛❝❡✱ ✇❡ t❛❝✐t•② ❛ss✉♠❡❞ t❤❛t t❤❡ CP 1 ❝♦♦r❞✐♥❛t❡ tr❛♥s❢♦r♠s ❛♣♣r♦♣r✐❛t❡•②✳ ❇② t❤✐s✱ ✇❡ ♠❡❛♥ t❤❛t t tr❛♥s✲ ❢♦r♠s ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ❛❞♠✐t t❤❡ ❛❝t✐♦♥ ♦❢ SL(2, Z) ❣r♦✉♣ ❛s ✐♥ ✭✺✳✷✷✮✳ ■♥ ♦✉r ❝♦♥s✐❞❡r❛t✐♦♥s✱ t❤❡ ♠❛✐♥ ❝♦♠♣•✐❝❛t✐♦♥ ❝♦♠❡s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t ξ 0 ✐s ♥♦t ❣•♦❜❛••② ❞❡✜♥❡❞ ❛♥②♠♦r❡✱ ❛♥❞ ❛s ❛ r❡s✉•t t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ✜❜❡r ❝♦♦r❞✐♥❛t❡ ❛•s♦ r❡❝❡✐✈❡s ❝♦rr❡❝t✐♦♥s✳ ❍♦✇❡✈❡r✱ q✉✐t❡ r❡♠❛r❦❛❜•②✱ ✐t ✐s st✐•• ♣♦ss✐❜•❡ t♦ ✜♥❞ t❤❡ ✏❛♣♣r♦♣r✐❛t❡✧ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t ❜② ❝♦♥s✐❞❡r✐♥❣ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ❑ä❤•❡r ♣♦t❡♥t✐❛• K Z ♦♥ t❤❡ t✇✐st♦r s♣❛❝❡ ❛♥❞ r❡q✉✐r✐♥❣ t❤❛t ✐t tr❛♥s❢♦r♠s ❜② ❑ä❤•❡r tr❛♥s❢♦r♠❛t✐♦♥ ✭✺✳✷✸✮ ❬✸✵(✳ ■♥ t❤❡ ♣r♦❝❡ss ♦❢ ✜♥❞✐♥❣ ❤♦✇ t ✻✹ ❙✲❞✉❛•✐t② ✐♥ t❤❡ t✇✐st♦r s♣❛❝❡✳ ✺✳✹ tr❛♥s❢♦r♠s✱ ✇❡ ❤❛✈❡ ❝♦♠❡ ❛❝r♦ss ❛ tr✉❡•② ❜❡❛✉t✐❢✉• ♠❛t❤❡♠❛t✐❝❛• ❝♦♥str✉❝t ✇❤✐❝❤ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ❝❛•• ❛s ✐♥✈❛r✐❛♥t ♣♦✐♥ts✳ ❚❤✐s ♦❜ •❡❝ts ❛r❡ ❣❡♥❡r❛•✐③❛t✐♦♥s ♦❢ ∓i ✐♥ t❤❡ ❝•❛ss✐❝❛• ❝❛s❡✱ ✇❤✐❝❤ r❡♠❛✐♥s ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ SL(2, R) ✭✺✳✷✵✮✳ ❲❡ ❛r❡ ❣♦✐♥❣ t♦ ❡①♣•❛✐♥ ✇❤❛t t❤❡s❡ ♦❜•❡❝ts ❛r❡ ✐♥ ♠♦r❡ ❞❡t❛✐•s ✐♥ t❤❡ ❢♦••♦✇✐♥❣✳ ❼ ❚❤❡ ✜rst ❛♥❞ ♠♦st ✐♠♣♦rt❛♥t t❤✐♥❣ t♦ ❝♦♥s✐❞❡r ✐s t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t✳ ■t ✇❛s ❢♦✉♥❞ ✐♥ ❬✸✵( t❤❛t t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ ✐s r❛t✐♦♥❛• ❛♥❞ ✐s ❣✐✈❡♥ ❜②✱ g c,d [t] = C c,d - t -t c,d + t -t c,d - , ✭✺✳✸✸✮ ✇❤❡r❡ t❤❡ ❝♦♥st❛♥t ♣r❡❢❛❝t♦r s❤♦✉•❞ s❛t✐s❢② |C c,d -| = |t c,d -| ❛♥❞ ✇❡ ✉s❡❞ t❤❡ ♥♦t❛t✐♦♥ g c,d ❢♦r t❤❡ SL(2, Z) ❛❝t✐♦♥✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ✭✺✳✸✸✮ s❤♦✉•❞ s❛t✐s❢② ❛ ❝❡rt❛✐♥ ❣r♦✉♣ •❛✇✳ ■♥ ♣❛rt✐❝✉•❛r✱ t❤✐s ✐♠♣•✐❡s t❤❛t g c,d [g -c,a [t]] = t s✐♥❝❡ t❤❡ t✇♦ SL(2, Z) tr❛♥s✲ ❢♦r♠❛t✐♦♥s ❛r❡ ✐♥✈❡rs❡ ♦❢ ❡❛❝❤ ♦t❤❡r✳ ❚❤✐s ❛••♦✇s t♦ ✜① t❤❡ ♣r❡❢❛❝t♦r ✐♥ ✭✺✳✸✸✮ ❛s C c,d ± = g c,d t -c,a ± , t c,d + C c,d -= t c,d -C c,d + . ✭✺✳✸✹✮ ■♥ ❢❛❝t✱ ❢r♦♠ t❤❡ r❡❛•✐t② ♣r♦♣❡rt✐❡s ♦❢ t c,d ± ✱ ♦♥❡ ❝❛♥ ❡❛s✐•② s❡❡ t❤❛t t❤❡ s❡❝♦♥❞ ♦❢ t❤❡ ❛❜♦✈❡ ❝♦♥❞✐t✐♦♥s ✐s t❤❡ s❛♠❡ ❛s t❤❡ ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ ♠♦❞✉•✉s ♦❢ C c,d -✳ ■t ✐s ❡❛s② t♦ s❡❡ t❤❛t t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❛ ♥❛t✉r❛• ❣❡♥❡r❛•✐③❛t✐♦♥ ♦❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ✭✺✳✷✵✮ ✈❛•✐❞ ✐♥ t❤❡ ❝•❛ss✐❝❛• ❝❛s❡✳ ❼ ◆♦✇ ✇❡ t✉r♥ t♦ ♦♥❡ ♦❢ t❤❡ ♦t❤❡r ❦❡② ❢❡❛t✉r❡s✳ ■t t✉r♥s ♦✉t t❤❛t ❛•♦♥❣ ✇✐t❤ t❤❡ r♦♦ts ♦❢ cξ 0 + d = 0✱ t❤❡ ❝r✐t✐❝❛• ♣♦✐♥ts ♦❢ ξ 0 ❛•s♦ ♣•❛② ❛♥ ✐♠♣♦rt❛♥t r♦•❡✳ ❚❤❡② ❛r❡ ❞❡✜♥❡❞ ❜② ∂ t ξ 0 [0] (s ± ) = 0, ✭✺✳✸✺✮ ❛♥❞ ❛r❡ r❡•❛t❡❞ t♦ ❡❛❝❤ ♦t❤❡r ❜② t❤❡ ❛♥t✐♣♦❞❛• ♠❛♣ ς[s + ] = s -✳ ❚❤❡② ❛r❡ ✐♥st❛♥t♦♥ ❝♦rr❡❝t❡❞ ✈❡rs✐♦♥s ♦❢ t❤❡ ♣♦✐♥ts s ± = ∓i✳ ❚❤❡s❡ ♦❜•❡❝ts ❛r❡ ❝❛••❡❞ ✐♥✈❛r✐❛♥t ♣♦✐♥ts ❜❡❝❛✉s❡ t❤❡② ❛r❡ ✜①❡❞ ♣♦✐♥ts ♦❢ SL(2, Z) tr❛♥s❢♦r♠❛t✐♦♥✳ ❚❤✐s ❝❛♥ ❜❡ s❡❡♥ ❢r♦♠ t❤❡ ❢❛❝t t❤❛t g c,d [∂ t ξ 0 [0] ] ∼ ∂ t ξ 0 [0] ✸ ✱ ✇❤✐❝❤ ✇✐•• ❜❡❝♦♠❡ ❝•❡❛r ❢r♦♠ t❤❡ ❡①♣r❡ss✐♦♥s ♦❢ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s t❤❛t ✇❡ ✇✐•• ♣r❡s❡♥t ✐♥ ❛ s❤♦rt ✇❤✐•❡✳ ■t ✐s ✐♠♣♦rt❛♥t t♦ r❡❛•✐③❡ t❤❛t s ± s❤♦✉•❞ ❜❡ ✈✐❡✇❡❞ ♥♦t •✉st ❛s ❢✉♥❝t✐♦♥s ♦♥ t❤❡ ◗❑ s♣❛❝❡ M✱ ❜✉t ❛•s♦ ❛s s❡❝t✐♦♥s ♦❢ t❤❡ ❜✉♥❞•❡ ✸ ❋r♦♠ ❣❡♥❡r❛• ❡①♣❡❝t❛t✐♦♥s✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡s❡ ♣♦✐♥ts ❜❡•♦♥❣ t♦ t❤❡ ♣❛t❝❤ U 0 ✳ ❚❤✐s ♠❛❦❡s t❤❡ ❝♦♥str✉❝t✐♦♥ •❡ss ❝✉♠❜❡rs♦♠❡✳ ❋✉rt❤❡r♠♦r❡✱ ✐♥ t❤❡ ❝•❛ss✐❝❛• ❝❛s❡ ♦r ✐♥ t❤❡ ❝❛s❡ ✇❤❡♥ t❤❡r❡ ❛r❡ t✇♦ ❝♦♥t✐♥✉♦✉s ✐s♦♠❡tr✐❡s✱ t❤❡s❡ ♣♦✐♥ts ✐♥❞❡❡❞ ❛r❡ ✐♥ U 0 ✳ ❚❤❡ ✐♥st❛♥t♦♥s ❝♦rr❡❝t✐♦♥s✱ ✇❤❡♥ s♣❡❝✐✜❝ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♥s✐❞❡r❡❞✱ ❛r❡ s✉♣♣♦s❡❞ t♦ ❜❡ ♥♦t ❜❡❤❛✈✐♥❣ t♦♦ ✏✇✐•❞•②✧ ✳ ▼♦r❡ ♣r❡❝✐s❡•②✱s ± s❤♦✉•❞ ❜❡ ♣❡rt✉r❜❛t✐♦♥s ❛r♦✉♥❞ t❤❡✐r ❝•❛ss✐❝❛• ✈❛•✉❡s✳ ❖❢ ❝♦✉rs❡✱ ✇❡ ❝❛♥ r❡•❛① t❤✐s ❛ss✉♠♣t✐♦♥✱ ❜✉t ✇❡ ❞♦ ♥♦t ❡①♣❡❝t t❤❡ r❡s✉•ts t♦ ❝❤❛♥❣❡ s✐❣♥✐✜❝❛♥t•② ❜② t❛❦✐♥❣ t❤✐s ✐♥t♦ ❛❝❝♦✉♥t✳ ✻✺ ✺✳ ▼✐rr♦r s②♠♠❡tr② ❛t t❤❡ q✉❛♥t✉♠ •❡✈❡• Z → M✳ ❚❤❡r❡❢♦r❡✱ t❤❡✐r ✐♥✈❛r✐❛♥❝❡ ✐s ♣r♦♣❡r•② ✐♥t❡r♣r❡t❡❞ ❛s ✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ s❡❝t✐♦♥s✳ ❚❤✉s t❤❡② s❛t✐s❢② t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣❡rt②✱ g c,d [s ± ] = g c,d [t] t=s ± = C c,d - s ± -t c,d + s ± -t c,d - . ✭✺✳✸✻✮ ❼ ❍❛✈✐♥❣ ✐♥tr♦❞✉❝❡❞ t❤❡ ❝r✐t✐❝❛• ♣♦✐♥ts✱ ✇❡ ❛r❡ ♥♦✇ ✐♥ ❛ ♣♦s✐t✐♦♥ t♦ ✐♥tr♦❞✉❝❡ t❤❡ ✐♥✈❛r✐❛♥t ❦❡r♥❡• ✇❤✐❝❤ ♣•❛② ❛ ❝r✉❝✐❛• r♦•❡ ❢♦r ♠❛❦✐♥❣ t❤❡ ♠♦❞✉•❛r s②♠✲ ♠❡tr② ❡①♣•✐❝✐t✳ ❲❡ ❞❡✜♥❡ ✐t ❛s ❛ t✲✐♥❞❡♣❡♥❞❡♥t s❤✐❢t ♦❢ t❤❡ ❦❡r♥❡• ❛♣♣❡❛r✐♥❣ ✐♥ ✭✸✳✾✮ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ✇❛② s✉❝❤ t❤❛t t❤❡ ✐♥t❡❣r❛t✐♦♥ ♠❡❛s✉r❡ r❡♠❛✐♥s ❙✲ ❞✉❛•✐t② ✐♥✈❛r✐❛♥t✱ K(t, t ′ ) = 1 2 t ′ + t t ′ -t + k(t ′ ) , dt ′ t ′ K(t, t ′ ) → dt ′ t ′ K(t, t ′ ) ✭✺✳✸✼✮ ✇❤❡r❡ k(t) = s + s --t 2 (t -s + )(t -s -) . ✭✺✳✸✽✮ ❍❡♥❝❡ ✇❡ ❤❛✈❡ t❤❡ ✐♥✈❛r✐❛♥t ❦❡r♥❡• ❛s dt ′ t ′ K(t, t ′ ) = dt ′ 2 (t -s + )(t ′ -s -) + (t -s -)(t ′ -s + ) (t ′ -t)(t ′ -s + )(t ′ -s -) . ✭✺✳✸✾✮ ◆♦✇✱ ❛r♠❡❞ ✇✐t❤ t❤❡ ❙✲❞✉❛•✐t② ✐♥✈❛r✐❛♥t ❦❡r♥❡•✱ ✐♥✈❛r✐❛♥t ♣♦✐♥ts ❛♥❞ t❤❡ ♣♦✐♥ts t m,n ± ✱ ✇❡ ❝❛♥ ♣r♦❝❡❡❞ t♦ ✜♥❞ ❡①♣r❡ss✐♦♥s ❢♦r ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s s❛t✐s❢②✐♥❣ ✭✺✳✷✷✮ ✹ ✳ ❚❤❡ ♥♦♥✲♣❡rt✉r❜❛t✐✈❡ ♠✐rr♦r ♠❛♣ ❋r♦♠ ❡q✉❛t✐♦♥ ✭✸✳✾✮✱ ✇❡ s❡❡ t❤❛t t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ❞❡♣❡♥❞ ♦♥ ❜♦t❤ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦♥ t❤❡ ❜❛s❡ A Λ , B Λ , B α , Y Λ ❛♥❞ t❤❡ ✜❜❡r ❝♦♦r❞✐♥❛t❡ t✳ ❚②♣✐❝❛••② t❤❡s❡ ❝♦♦r❞✐♥❛t❡s ♦♥ M ❤❛s ✈❡r② s✐♠♣•❡ r❡•❛t✐♦♥s ✇✐t❤ t②♣❡ ■■❆ ❝♦♦r❞✐♥❛t❡s ❛♥❞ ❛r❡ ✜①❡❞ ❜② s②♠♣•❡❝t✐❝ ✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ t②♣❡ ■■❆ ❝♦♥str✉❝t✐♦♥✳ ❍♦✇❡✈❡r✱ ♦✉r ❣♦❛• ❤❡r❡ ✐s t♦ r❡✇r✐t❡ t❤❡ ✐♥t❡❣r❛• ❡q✉❛t✐♦♥s ❢♦r t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ❛❝t✐♦♥ ♦❢ SL(2, Z) ♦♥ t❤❡♠ ✐s r❡❛•✐③❡❞ ❛s ✭✺✳✷✷✮✱ ♣r♦✈✐❞❡❞ t❤❡ r✐❣❤t tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t ❣✐✈❡♥ ❜② ✭✺✳✸✸✮✳ ❋♦r t❤✐s ♦♥❡ ❤❛s t♦ ❡①♣r❡ss t❤❡ ❝♦♦r❞✐♥❛t❡s A Λ , B Λ , B α , Y Λ ❜② t②♣❡ ■■❇ ♣❤②s✐❝❛• ✜❡•❞s τ, b a , t a , c a , c a , ψ✱ ✇❤✐❝❤ ❛❞♠✐t ❛ s✐♠♣•❡ ❛♥❞ ♠♦r❡ ♥❛t✉r❛• ❛❝t✐♦♥ ♦❢ t❤❡ ♠♦❞✉•❛r ❣r♦✉♣ ✭✷✳✹✶✮✳ ❍❡r❡✱ ✇❡ ❝❛•• t❤✐s r❡•❛t✐♦♥ ❛s t❤❡ ✏♠✐rr♦r ♠❛♣✧✳ ❊ss❡♥t✐❛••②✱ s✉❝❤ ❝♦♦r❞✐♥❛t❡ tr❛♥s❢♦r♠❛t✐♦♥s ♣r♦✈✐❞❡ ❣❡♥❡r❛•✐③❛t✐♦♥s ♦❢ t❤❡ ❝•❛ss✐❝❛• ♠✐rr♦r ♠❛♣ ✭✷✳✹✷✮✱ ❜❡❝❛✉s❡ t❤❡ r❡•❛t✐♦♥s ♦❢ A Λ , B Λ , B α , Y Λ ✇✐t❤ t②♣❡ ■■❆ ❝♦♦r❞✐♥❛t❡s ❛r❡ s✐♠♣•❡✱ ❛s ✇❡ ❤❛✈❡ ❛•r❡❛❞② ♠❡♥t✐♦♥❡❞✳ ✹ ❋♦r ❝♦♥✈❡♥✐❡♥❝❡✱ ✇❡ ❞r♦♣ t❤❡ ❝♦♥st❛♥t s❤✐❢t ✐♥ c a ✇❤✐❝❤ ❝❛♥ ❜❡ ❛•✇❛②s r❡st♦r❡❞ ✻✻ ❙✲❞✉❛•✐t② ✐♥ t❤❡ t✇✐st♦r s♣❛❝❡✳ ✺✳✹ ❚♦ t❤✐s ❡♥❞✱ •❡t ✉s ✜rst ❣✐✈❡ t❤❡ ♠✐rr♦r ♠❛♣ r❡•❛t✐♦♥✳ ❚❤❡ ❝♦♥st❛♥t t❡r♠s ✐♥ t❤❡ ✐♥t❡❣r❛• ❡q✉❛t✐♦♥s ❢♦r t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ❛r❡ ❣✐✈❡♥ ❜② ✺ ❬✸✵(✱ A Λ = ζ Λ cl - 1 2 (s + + s -) Y Λ s + s - -Ȳ Λ + 1 2 j C j dt 2πit k(t) ∂ ξ[0] Λ H [j0] -ξ Λ [0] ∂ α [0] H [j0] , B Λ = ζcl Λ - 1 2 j C j dt 2πit k(t)∂ ξ Λ [j] H [j0] , ✭✺✳✹✵✮ B α = (σ + ζ Λ ζΛ ) cl - 1 2 j C j dt 2πit k(t) H [j0] -ξ Λ [j] ∂ ξ Λ [j] H [j0] , ✇❤❡r❡ t❤❡ ✐♥❞❡① ✏❝•✧ r❡❢❡rs t♦ t❤❡ q✉❛♥t✐t✐❡s ❣✐✈❡♥ ❜② t❤❡ ❝•❛ss✐❝❛• ♠✐rr♦r ♠❛♣✳ ◆♦✇ ✇✐t❤ t❤✐s ❛♥❞ t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t ❢♦r ♠✐rr♦r ♠❛♣ ♦❢ Y Λ ✱ ♦♥❡ ❝❛♥ ❝❤❡❝❦ t❤❛t t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ✐♥❞❡❡❞ s❛t✐s❢② ✭✷✳✹✶✮✳ ▲❡t ✉s ❞❡✜♥❡ ✻ Y Λ = iτ 2 s + s - s --s + Y Λ + s + s - 2 m,n ′ Cm,n dt 2πi s + (s --t) + s -(s + -t) (t -s + ) 2 (t -s -) 2 T Λ m,n , ✭✺✳✹✶✮ ✇❤❡r❡ Y 0 = 1, Y a = b a s -e iU -s + e -iU -2 √ s + s -t a (s --s + ) cos U + iτ -1 2 tan U I a ✭✺✳✹✷✮ ❛♥❞ I a = (s --s + ) 2 4π m,n ′ Cm,n dt t T a m,n (t -s + ) 2 (t -s -) 2 , ✭✺✳✹✸✮ U = s --s + 4π m,n ′ Cm,n dt log (1 -∂ α [0] G m,n ) (t -s + )(t -s -) . ✭✺✳✹✹✮ ❚❤❡♥✱ ♣r♦✈✐❞❡❞ ❛t t❤❡ ❝r✐t✐❝❛• ♣♦✐♥ts ♦♥❡ ❤❛s ξ 0 [0] (s + ) = τ, ξ 0 [0] (s -) = τ , ξ a [0] (s + ) = -c a + τ b a , ξ a [0] (s -) = -c a + τ b a , ✭✺✳✹✺✮ s ± ∂ t ξ a [0] (s ± ) = ± 2iτ 2 t a cos U √ s + s - s --s + -ie ±iU L a , ✭✺✳✹✻✮ ✇❤❡r❡ L a ✐s ❛♥ ❛r❜✐tr❛r② r❡❛• ✈❡❝t♦r✳ ❚❤❡♥ Y Λ s❤♦✉•❞ tr❛♥s❢♦r♠ ❛s ✼ Y 0 → C c,d -t c,d + c(t c,d + -t c,d -) 2 ρ [c,d] (t c,d + ) , Y a → - C c,d -t c,d + ξ a [c,d] (t c,d + ) (t c,d + -t c,d -) 2 ρ [c,d] (t c,d + ) . ✭✺✳✹✼✮ ✺ ❚❤❡ ✐♥❞❡① j st❛♥❞s ❢♦r ❛•• ♣♦ss✐❜•❡ ♣❛t❝❤❡s ❛♥❞ ❝♦♥t♦✉rs✳ ✻ ❚❤❡ ♣r✐♠❡ ♦♥ t❤❡ s✉♠ ✐♥❞✐❝❛t❡s t❤❛t ✐t r✉♥s ♦✈❡r (m, n) ∈ Z 2 /(0, 0)✳ ✼ ρ [m,n] ❛r❡ ❞❡✜♥❡❞ ❛s cξ 0 [m,n] + d ≡ -t -1 (t -t m,n + )(t -t m,n -)ρ [m,n] (t)✳ ✻✼ ✺✳ ▼✐rr♦r s②♠♠❡tr② ❛t t❤❡ q✉❛♥t✉♠ •❡✈❡• ❆•• t❤❡s❡ ❛❜♦✈❡ r❡s✉•ts ❛r❡ ♥♦t t♦♦ ❞✐✣❝✉•t t♦ ♣r♦✈❡ ❛♥❞ t❤❡ ♣r♦♦❢s ❛r❡ ❣✐✈❡♥ ✐♥ ❛♣♣❡♥❞✐① ❈ ♦❢ ❬✸✵(✳ ◆♦✇✱ ✇❡ ❝❛♥ ✜♥❞ ❛♥ ❡①♣•✐❝✐t ❡q✉❛t✐♦♥ ❞❡t❡r♠✐♥✐♥❣ t❤❡ ✐♥✈❛r✐❛♥t ♣♦✐♥ts✳ ❯s✐♥❣ ✭✺✳✹✵✮ ❛♥❞ ✭✺✳✹✶✮✱ t❤❡ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡ ξ 0 ✐♥ t❤❡ ♣❛t❝❤ U 0 ❜❡❝♦♠❡s✱ ξ 0 [0] (t) = τ 1 + iτ 2 s --s + s + s - t -t + (t -s + )(t -s -) 2t m,n ′ Cm,n dt ′ 2πi t ′ (t -s + )(t ′ -s -) + (t -s -)(t ′ -s + ) (t ′ -t)(t ′ -s + ) 2 (t ′ -s -) 2 T 0 m,n . ✭✺✳✹✽✮ ❚❤✉s✱ ❢r♦♠ ∂ t ξ 0 [0] (s ± ) = 0✱ ♦♥❡ ❝❛♥ ♦❜t❛✐♥ t❤❡ ❡q✉❛t✐♦♥ s + + s - (s + -s -) 3 = 1 4πτ 2 m,n ′ Cm,n dt t T 0 m,n (t -s + ) 2 (t -s -) 2 . ✭✺✳✹✾✮ ❙✉♣♣•❡♠❡♥t❡❞ ❜② t❤❡ ❝♦♥❞✐t✐♦♥ ς [s + ] = s -✱ ✐t ❝❛♥ ❜❡ s♦•✈❡❞ ❜② ♣❡rt✉r❜❛t✐✈❡ ❡①✲ ♣❛♥s✐♦♥ ✐♥ ♣♦✇❡rs ♦❢ ✭✐♥t❡❣r❛•s ♦❢ ✮ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s T 0 m,n ❣❡♥❡r❛t✐♥❣ ❞❡❢♦r✲ ♠❛t✐♦♥s✳ ❋✐♥❛••②✱ •❡t ✉s ♠❡♥t✐♦♥ t❤❛t ♦✉r ❝♦♥str✉❝t✐♦♥ ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ ❛ U (1) s②♠♠❡tr② t❤❛t r♦t❛t❡s t❤❡ ♣❤❛s❡ ♦❢ t❤❡ ✜❜❡r ❝♦♦r❞✐♥❛t❡ ❛s t → e iφ t✳ ❆s ❛ ♣❛rt✐❝✉•❛r ❝❛s❡✱ t❤✐s ❡♥t❛✐•s t❤❛t t❤❡ ✐♥✈❛r✐❛♥t ♣♦✐♥ts ❛r❡ ❞❡✜♥❡❞ ✉♣t♦ t❤✐s r❡s❝❛•✐♥❣ s ± → e iφ s ± ✳ ❚❤✐s ✐♥❞✉❝❡s ❛♥ ❡①tr❛ s②♠♠❡tr② ✐♥ t❤❡ ❝♦♠♣•❡① ❝♦♦r❞✐♥❛t❡s Y Λ → e iφ Y Λ ❛♥❞ ✐s r❡s♣♦♥s✐❜•❡ ❢♦r t❤❡ ❛✉①✐•✐❛r② ❝♦♦r❞✐♥❛t❡ ❞✐s❝✉ss❡❞ ❜❡•♦✇ ✭✸✳✾✮✳ ❆ ✇❛② t♦ ✜① t❤✐s ✐s t♦ ❞❡♠❛♥❞ s ± t♦ ❜❡ ♣✉r❡•② ✐♠❛❣✐♥❛r② ❬✸✵(✳ ❚❤✐s tr✉•② ✐♠♣♦s❡s ❛♥ ❛❞❞✐t✐♦♥❛• ❝♦♥❞✐t✐♦♥✱ ❛s t❤❡ ❞❡✜♥✐♥❣ ❡q✉❛t✐♦♥ ❢♦r s ± ✭✺✳✹✾✮ ❞♦❡s ♥♦t ✜① ✐t✳ ❚❤❛t t❤❡ ✐♥✈❛r✐❛♥t ♣♦✐♥ts ❛r❡ ✐♠❛❣✐♥❛r② ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s ✏❣❛✉❣❡ ✜①✐♥❣✧ ♦❢ t❤❡ U (1) s②♠♠❡tr② ❛♥❞ ✐t ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❛ ♣❛rt ♦❢ ♦✉r ❝♦♥str✉❝t✐♦♥✳ ✽ ▼♦❞✉•❛r ✐♥✈❛r✐❛♥t ♣♦t❡♥t✐❛• ❆s ❛ ❜②♣r♦❞✉❝t ✐♥ ❬✸✵( ❛ ♠♦❞✉•❛r ✐♥✈❛r✐❛♥t ♣♦t❡♥t✐❛• ✇❛s ❛•s♦ ❝♦♥str✉❝t❡❞ ✉s✐♥❣ t❤❡ ❝♦♥st❛♥t ♣❛rt ♦❢ t❤❡ ❝♦♥t❛❝t ♣♦t❡♥t✐❛•✳ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s s✉r♣r✐s✐♥❣ ❛s ✐t ❡♥❝♦❞❡s ❛•• ❞❡❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ◗❑ s♣❛❝❡ r❡s♣❡❝t✐♥❣ ♠♦❞✉•❛r s②♠♠❡tr② ✐♥ ❛ ♥♦♥✲tr✐✈✐❛• ✇❛②✳ ■t ✐s ❣✐✈❡♥ ❜② R = |s + -s -| e -1 2 m,n ′ Cm,n dt 2πit s + s --t 2 (t-s + )(t-s -) log(1-∂ α [0] Gm,n) τ 1/2 2 cos 1 4π m,n ′ Cm,n dt t log (1 -∂ α [0] G m,n ) 1 2 Im Y Λ FΛ ( Ȳ ) - 1 16π m,n ′ Cm,n dt t t -1 Y Λ -t Ȳ Λ ∂ ξ Λ [m,n] G m,n + t -1 F Λ (Y ) -t FΛ ( Ȳ ) T Λ m,n . ✭✺✳✺✵✮ ✽ ◆♦t✐❝❡ t❤❛t t❤❡r❡ ❡①✐sts ♦t❤❡r ❝❤♦✐❝❡s ❢♦r ✜①✐♥❣ t❤❡ ❣❛✉❣❡ ❛s ✇❡••✳ ❖♥❡ ❝❛♥ ❝♦♥s✐❞❡r Im Y 0 = 0✱ ❢♦r ❡①❛♠♣•❡✳ ❍♦✇❡✈❡r✱ ✇❡ ❝❤♦♦s❡ Re s ± = 0✱ ❛s t❤✐s ✐s t❤❡ ♠♦st ❝♦♥✈❡♥✐❡♥t ♦♥❡✳ ✻✽ ❙✲❞✉❛•✐t② ✐♥ t❤❡ t✇✐st♦r s♣❛❝❡✳ ✺✳✹ ■♥ ❬✺✻(✱ ❬✺✼(✱ ❛♥ ❛tt❡♠♣t t♦ ❝♦♥str✉❝t s✉❝❤ ❛ ❢✉♥❝t✐♦♥ ✇❛s ♠❛❞❡ ❛•t❤♦✉❣❤ ✐ts ❣❡♦✲ ♠❡tr✐❝ ♦r✐❣✐♥ ✇❛s ✉♥❝•❡❛r✳ ❚❤❡ ✇♦r❦ ✐♥ ❬✸✵( s✉❣❣❡sts t❤❛t t❤❡ ♠♦❞✉•❛r ✐♥✈❛r✐❛♥t ❢✉♥❝t✐♦♥ R ❤❛s ✐ts ♦r✐❣✐♥ ✐♥ t❤❡ ❝♦♥st❛♥t ♣❛rt ♦❢ t❤❡ ❝♦♥t❛❝t ♣♦t❡♥t✐❛•✳ ❋r♦♠ ♣❤②s✐❝s ♣♦✐♥t ♦❢ ✈✐❡✇✱ ♣❡r❤❛♣s✱ R ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❛♥ ❙✲❞✉❛•✐t② ✐♥✈❛r✐❛♥t ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥✳ ✺✳✹✳✹ ❱✐rt✉❡ ♦❢ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s ✿ ❛♥ ❡①❛♠♣•❡ ■t ✐s ❤❛r❞ t♦ ♦✈❡rst❛t❡ t❤❡ s✐♠♣•✐❝✐t② ♦❢ t❤❡ ❞❡s❝r✐♣t✐♦♥ ✉s✐♥❣ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s✳ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ♣r♦✈✐❞❡ ❛♥♦t❤❡r ✐♥st❛♥❝❡ ♦❢ ✉s❡❢✉•♥❡ss ♦❢ t❤✐s✳ ❖❢ ❝♦✉rs❡✱ ❛s ❤❛❞ ❜❡❡♥ ♠❡♥t✐♦♥❡❞ t❤✐s ❝♦♥str✉❝t✐♦♥ ✐s ❝♦♠♣•❡t❡•② ❡q✉✐✈❛•❡♥t t♦ t❤❡ ♦♥❡ ♣r❡s❡♥t❡❞ ❜❡❢♦r❡ ✉s✐♥❣ ♥♦r♠❛• tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ ❞❡♣❡♥❞✐♥❣ ♦♥ ❝♦♦r❞✐♥❛t❡s ♦❢ t✇♦ ❞✐✛❡r❡♥t ♣❛t❝❤❡s✳ ❍♦✇❡✈❡r✱ t❤❡ ❛❞✈❛♥t❛❣❡ ✐s t❤❛t✱ t❤❡ •❛tt❡r ✐s ♠♦r❡ ❡✣❝✐❡♥t t❤❛♥ t❤❡ ❢♦r✲ ♠❡r ❛♥❞ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s ❝♦♥t❛✐♥ s✉✣❝✐❡♥t ✐♥❢♦r♠❛t✐♦♥ s♦ t❤❛t t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ❝❛♥ ❜❡ ❝♦♥str✉❝t❡❞ ❢r♦♠ t❤❡♠ ✈❛•✐❞ t♦ ❛•• ♦r❞❡rs✳ ❋r♦♠ ✭✸✳✸✻✮✱ λ = cξ 0 [m,n] + d ❢♦r ♦✉r ❝❛s❡ ❜❡❝❛✉s❡ ♦❢ ✭✺✳✷✸✮✱ ✇❤✐❝❤ ✐♠♣•✐❡s t❤❛t t❤❡ ❝♦♥str❛✐♥t ♦♥ t❤❡ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥ ✇✐•• ❜❡ h [i] m,n → h [i] m ′ ,n ′ cξ 0 [m ′ ,n ′ ] + d , m ′ n ′ = a c b d m n , ✭✺✳✺✶✮ ✇❤✐❝❤ ✐s ❛ s✐♠♣•❡ ♠♦❞✉•❛r ❝♦♥str❛✐♥t✱ ❝♦♠♣❛r❡❞ t♦ t❤❡ ✈❡r② ❝♦♠♣•✐❝❛t❡❞ ♥♦♥✲•✐♥❡❛r ♦♥❡ ❣✐✈❡♥ ✐♥ ✭✺✳✸✶✮✳ ■♥st❡❛❞ ♦❢ ✉s✐♥❣ ✭✸✳✸✻✮✱ ♦♥❡ ❝❛♥ ❝❤♦♦s❡ ❛ ♠♦r❡ ♣❡❞❡str✐❛♥ ✇❛② ✾ ❖♥❡ ❝❛♥ ❡①✲ ♣•✐❝✐t•② ✜♥❞ ❢r♦♠ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ✭✺✳✷✷✮ t❤❛t {g c,d • h, g c,d • f } = g c,d • {h, f } cξ 0 + d -c g c,d • h ∂ ξ0 f -ξ 0 ∂ α f , ✭✺✳✺✷✮ ✇❤✐❝❤ ❝❛♥ ❛•s♦ ❜❡ ✐♠♠❡❞✐❛t❡•② s❡❡♥ ❢r♦♠ t❤❡ ▲❡✐❜♥✐t③ ♣r♦♣❡rt② ♦❢ t❤❡ ❝♦♥t❛❝t ❜r❛❝❦❡ts✳ ❚❤✐s ❝♦♠❡s ✐♥ ❤❛♥❞② ❜❡❝❛✉s❡ t❤❡ ❛❝t✐♦♥ ♦❢ ❙✲❞✉❛•✐t② ❝❛♥ ❜❡ ❝♦♥✈❡♥✐❡♥t•② ❡♥✲ ❝♦❞❡❞ ✐♥ t❡r♠s ♦❢ ✐ts ❛❝t✐♦♥ ♦♥ t❤❡ ❝♦♥t❛❝t ❜r❛❝❦❡t ❛s✱ g c,d • e {h [i] , • } • g -1 c,d = e g c,d •{h [i] , • }•g -1 c,d = e g c,d •{h [i] , g -1 c,d • } = e {(cξ 0 +d)g c,d •h [i] , • } , ✭✺✳✺✸✮ ♣r♦✈✐♥❣ ✭✺✳✺✶✮✳ ■♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✱ ✇❡ ✇✐•• ✜rst ❞✐s❝✉ss ❤♦✇ t♦ r❡❢♦r♠✉•❛t❡ t❤❡ t②♣❡ ■■❆ ❝♦♥✲ str✉❝t✐♦♥ ❢♦r ❉✲✐♥st❛♥t♦♥s ✐♥ ❝❤❛♣t❡r ✹ ✐♥ t❤❡ t②♣❡ ■■❇ s✐❞❡✱ ✇✐t❤ ❡①♣•✐❝✐t ✐♥✈❛r✐❛♥❝❡ ✉♥❞❡r ❙✲❞✉❛•✐t②✳ ❚❤❡♥ ✇❡ ✇✐•• ♣r♦❝❡❡❞ ♦♥ t♦ ♣r♦✈✐❞✐♥❣ t❤❡ ✜✈❡❜r❛♥❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ✈❛•✐❞ t♦ ❛•• ♦r❞❡rs ✐♥ ✐♥st❛♥t♦♥ ❡①♣❛♥s✐♦♥✳ ✾ ❖❢ ❝♦✉rs❡✱ ❜♦t❤ ②✐❡•❞ t❤❡ s❛♠❡ r❡s✉•t✳ ❲❡ ❣✐✈❡ t❤✐s •✉st t♦ ♣r❡s❡♥t ②❡t ❛♥♦t❤❡r ♠❡t❤♦❞ t♦ ✜♥❞ t❤❡ s❛♠❡ r❡s✉•t ✐♥ t❤✐s s♣❡❝✐✜❝ ❝❛s❡✳ ✻✾ ❈❍❆P❚❊| ✻ ❚❤❡ s②♠♠❡tr② ❣r♦✉♣ ♦❢ M HM ■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ❞✐s❝✉ss ❛❜♦✉t t❤❡ ❢❛t❡ ♦❢ t❤❡ s②♠♠❡tr② ❣r♦✉♣ ♦❢ t❤❡ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ M HM ✳ ❲❡ ❤❛✈❡ ❛•r❡❛❞② •❡❛r♥❡❞ t❤❛t ✐♥st❛♥t♦♥ ❝♦rr❡❝t✐♦♥s ❜r❡❛❦ t❤❡ ❝•❛ss✐❝❛• ❝♦♥t✐♥✉♦✉s ✐s♦♠❡tr✐❡s t♦ t❤❡✐r ❞✐s❝r❡t❡ s✉❜❣r♦✉♣s ✐♥ ❝❤❛♣t❡r ✷✳ ❲❡ ❤❛✈❡ ❛•s♦ s❡❡♥ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡r t❤❛t✱ ♠♦♥♦❞r♦♠② ✐♥✈❛r✐❛♥❝❡ ❛r♦✉♥❞ t❤❡ •❛r❣❡ ✈♦•✉♠❡ ♣♦✐♥t ✐s ❣❡♥❡r❛t❡❞ ❞✉❡ t♦ t❤❡ s❤✐❢t ♦❢ t❤❡ B✲✜❡•❞ ✭✷✳✹✺✮✳ ❋✉rt❤❡r♠♦r❡✱ s②♠♠❡tr✐❡s ✉♥❞❡r t❤❡ ❝♦♥t✐♥✉♦✉s tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ || ❛♥❞ ◆❙ s❝❛•❛rs ✐♥ ❝❤❛♣✲ t❡r ✷✱ ❣❡t ❜r♦❦❡♥ ❞♦✇♥ t♦ ❞✐s❝r❡t❡ tr❛♥s❢♦r♠❛t✐♦♥s ❛s ✇❡••✳ ❋✐♥❛••②✱ t❤❡r❡ ✐s t❤❡ SL(2, Z) s②♠♠❡tr② ♦❢ t❤❡ ❍▼ ♠♦❞✉•✐ s♣❛❝❡✳ ❚❤❡ ❣♦❛• ♦❢ t❤✐s ❝❤❛♣t❡r ✐s t♦ ♦❜t❛✐♥ ❝•♦s✉r❡ ♦❢ t❤❡ ❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② ♠♦♥✲ ♦❞r♦♠② ❛r♦✉♥❞ t❤❡ •❛r❣❡ ✈♦•✉♠❡ ♣♦✐♥t✱ ❍❡✐s❡♥❜❡r❣ s❤✐❢ts ❛♥❞ SL(2, Z) t♦❣❡t❤❡r✳ ■t t✉r♥s ♦✉t✱ ❞❡♠❛♥❞✐♥❣ t❤❛t t❤❡ ❢✉•• q✉❛♥t✉♠ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ r❡♠❛✐♥s ✐♥✈❛r✐✲ ❛♥t ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡s❡ s②♠♠❡tr② ❣r♦✉♣s ✐s s✉✣❝✐❡♥t t♦ ❛❝❤✐❡✈❡ t❤✐s t❛s❦✳ ■♥ t❤❡ ♣r♦❝❡ss ✇❡ ✇✐•• ✜♥❞ t❤❛t✱ ❛ r❛t❤❡r ♥♦♥✲tr✐✈✐❛• ♠♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣❡r ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥s ❣✐✈❡♥ ✐♥ ❬✷✶( ✐s r❡q✉✐r❡❞✳ ❙✉❝❤ ♠♦❞✐✜❝❛t✐♦♥s ❛r❡ ❛❝✲ t✉❛••② ♥♦t ❛❞ ❤♦❝✱ ❜✉t r❛t❤❡r ♦r✐❣✐♥❛t❡s ❢r♦♠ s✉❜t•❡t✐❡s ✐♥ t❤❡ q✉❛❞r❛t✐❝ r❡✜♥❡♠❡♥t ✭✹✳✶✷✮✱ ✐♥tr♦❞✉❝❡❞ ✐♥ ❝❤❛♣t❡r ✹✳ ❋✐♥❛••②✱ ✇❡ ✇✐•• ❛•s♦ tr② t♦ ✉♥r❛✈❡• t❤❡ ♣♦ss✐❜•❡ ❝♦♥♥❡❝t✐♦♥s ❛♥❞ t❤❡r❡❜② ♣❡r❤❛♣s s✐♠✐•❛r ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤✐s ♠♦❞✐✜❝❛t✐♦♥ t♦ t❤❡ ❋r❡❡❞✲❲✐tt❡♥ ❛♥♦♠❛•② ❬✺✽(✳ ✻✳✶ |❡✈✐s✐t✐♥❣ ❝•❛ss✐❝❛• ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ❆t t❤❡ ❝•❛ss✐❝❛• •❡✈❡•✱ ♦♥❡ ❤❛s t♦ ❝♦♥s✐❞❡r t❤❡ ❛•❣❡❜r❛ ♦❢ ❛ s❡♠✐✲❞✐r❡❝t ♣r♦❞✉❝t SL(2, R) ⋉ N ✱ ✇❤❡r❡ N = N (1) ⊕ N (2) ⊕ N (3) ✐s ❛ ♥✐•♣♦t❡♥t ❛•❣❡❜r❛ ♦❢ ❞✐♠❡♥s✐♦♥ ✼✵ ◗✉❛♥t✉♠ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ❛♥❞ ✐ts s②♠♠❡tr✐❡s✳ ✻✳✷ 3b 2 + 2✱ s❛t✐s❢②✐♥❣ ✶ N (1) , N (1) ⊂ N (2) , N (p) , N (q) = 0, ✐❢ p + q ≥ 3, ✭✻✳✶✮ ✇❤❡r❡ t❤❡ ❣❡♥❡r❛t♦rs N (1) , N (2) , N (3) tr❛♥s❢♦r♠ ❛s b 2 ❞♦✉❜•❡ts✱ b 2 s✐♥❣•❡ts ❛♥❞ ♦♥❡ ❞♦✉❜•❡t ✉♥❞❡r SL(2, R) r❡s♣❡❝t✐✈❡•② ❬✷✷(✳ ❚❤❡ r❡•❛t✐♦♥ ♦❢ t❤✐s ❛•❣❡❜r❛ ❞❡❝♦♠♣♦s✐✲ t✐♦♥ ✇✐t❤ t❤❡ s②♠♠❡tr✐❡s ❞✉❡ t♦ SL(2, R)✱ ✭✷✳✹✺✮ ❛♥❞ ✭✷✳✹✸✮ ❛r❡ ❛s t❤❡ ❢♦••♦✇✐♥❣✳ ❼ ❚❤❡ ❣r♦✉♣ ❡•❡♠❡♥ts T (1) (ǫ a ,η a ) ❛r❡ ♦❜t❛✐♥❡❞ ❜② ❡①♣♦♥❡♥t✐❛t✐♥❣ N (1) ✭✇✐t❤ s✉✐t✲ ❛❜•❡ ❛❞♠✐①t✉r❡ ♦❢ ❤✐❣❤❡r ♦r❞❡r ❣❡♥❡r❛t♦rs✮ ❝♦♥s✐st✐♥❣ ♦❢ ✭✷✳✹✺✮ ❛♥❞ ✭✷✳✹✸✮✱ ✇✐t❤ ♥♦♥✲✈❛♥✐s❤✐♥❣ η a ✳ ❼ ❚❤❡ ❣r♦✉♣ ❡•❡♠❡♥ts T (2) ηa ❛r❡ ♦❜t❛✐♥❡❞ ❜② ❡①♣♦♥❡♥t✐❛t✐♥❣ N (2) ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❍❡✐s❡♥❜❡r❣ s❤✐❢ts ✇✐t❤ ♥♦♥✲✈❛♥✐s❤✐♥❣ ηa ✳ ❼ ❚❤❡ ❣r♦✉♣ ❡•❡♠❡♥ts T (3) (η 0 ,κ) ✐s ♦❜t❛✐♥❡❞ ❜② ❡①♣♦♥❡♥t✐❛t✐♥❣ N (3) ❛♥❞ ❝♦✐♥❝✐❞❡ ✇✐t❤ ❍❡✐s❡♥❜❡r❣ s❤✐❢ts ✇✐t❤ ♥♦♥✲✈❛♥✐s❤✐♥❣ η0 , κ✳ ❼ ❯♥•✐❦❡ t❤❡ ❍❡✐s❡♥❜❡r❣ s❤✐❢ts ❜❡•♦♥❣✐♥❣ t♦ N ✱ t❤❡ ❍❡✐s❡♥❜❡r❣ s❤✐❢t ζ 0 → ζ 0 +η 0 ❜❡•♦♥❣s t♦ t♦ SL(2, R) ❛♥❞ ✐♥ ❢❛❝t ❝♦✐♥❝✐❞❡s ✇✐t❤ SL(2, R) tr❛♥s❢♦r♠❛t✐♦♥ τ → τ + η 0 ✳ ✷ ✻✳✷ ◗✉❛♥t✉♠ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ❛♥❞ ✐ts s②♠♠❡✲ tr✐❡s ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ❞❡s❝r✐❜❡ t❤❡ ❡✛❡❝ts ♦❢ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s t♦ t❤❡ ❝•❛ss✐❝❛• s②♠♠❡tr✐❡s ♦❢ M HM ✳ ❲❡ ❤❛✈❡ ❛•r❡❛❞② s❡❡♥ ❤♦✇ t♦ r❡❛•✐③❡ q✉❛♥t✉♠ ❙✲ ❞✉❛•✐t② s②♠♠❡tr② ❢♦r t❤❡ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ✐♥ ❝❤❛♣t❡r ✺✳ ❋✉rt❤❡r♠♦r❡✱ ✐♥ ♣r❡s❡♥❝❡ ♦❢ ✐♥st❛♥t♦♥s✱ ✇❡ ❤❛✈❡ ❛•s♦ s❡❡♥ t❤❛t t♦ ❜❡ ❛❜•❡ t♦ ✐❞❡♥t✐❢② SL(2, Z) tr❛♥s❢♦r♠❛t✐♦♥ ❣❡♥❡r❛t❡❞ ❜② 1 b 0 1 ❛♥❞ ❍❡✐s❡♥❜❡r❣ s❤✐❢ts ζ 0 → ζ 0 + η 0 ✱ ✇❡ ♥❡❡❞ t♦ ♠♦❞✐❢② t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ❛①✐♦♥ c a ❜② ❛ ❝♦♥st❛♥t s❤✐❢t ♣r♦♣♦rt✐♦♥❛• t♦ ε(g)✳ ◆♦✇ ✇❤❛t ✇❡ ❤❛✈❡ t♦ ❞♦ ✐s t♦ ♠❛❦❡ t❤❡ ♦t❤❡r ♣✐❡❝❡s ✜t t♦❣❡t❤❡r✳ ❋♦r ✇❤❛t ❝♦♥❝❡r♥s ✉s✱ ✇❡ ✇✐•• ❤❛✈❡ t♦ ❝♦♥s✐❞❡r t❤❡ s❡♠✐❞✐r❡❝t ♣r♦❞✉❝t ❣r♦✉♣ SL(2, Z) ⋉ N (Z)✱ ✇❤❡♥ q✉❛♥✲ t✉♠ ❝♦rr❡❝t✐♦♥s ❛r❡ ♣r❡s❡♥t✳ ■t ✐s ♥❛t✉r❛• t♦ ❛ss✉♠❡ t❤❛t ✉♥❞❡r t❤❡ ❣r♦✉♣ N (Z) ❝♦♥s✐st✐♥❣ ♦❢ ✐♥t❡❣❡r ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ❛♥❞ •❛r❣❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ t❤❡ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ r❡♠❛✐♥s ✐♥✈❛r✐❛♥t✳ ❍♦✇❡✈❡r✱ t♦ ❛❝❤✐❡✈❡ t❤❛t✱ •✉st ❛ ♥❛✐✈❡ ❞✐s❝r❡t✐③❛t✐♦♥ ♦❢ t❤❡ ✭✷✳✹✺✮ ❛♥❞ ✭✷✳✹✸✮ ❞♦❡s ♥♦t ✇♦r❦✳ ❍❡r❡ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ♣r♦✈✐❞❡ t❤❡ ❝♦♥s✐st❡♥t s❡t ♦❢ ❞✐s❝r❡t❡ tr❛♥s❢♦r♠❛t✐♦♥s ❬✷✶(✳ ❇❡❢♦r❡ ❣♦✐♥❣ ❢✉rt❤❡r✱ •❡t ✉s r❡❝❛•• s♦♠❡ ❢❛❝ts ❛❜♦✉t t❤❡ q✉❛❞r❛t✐❝ r❡✜♥❡♠❡♥ts ❛♥❞ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝s ❛♣♣❡❛r t❤❡r❡✐♥✱ ❛s t❤❡s❡ ❝❤❛r❛❝t❡r✐st✐❝s ✇✐•• ❜❡ ♣r❡s❡♥t ✐♥ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ❛①✐♦♥s ❞✉❡ t♦ t❤❡ ❛❝t✐♦♥ ♦❢ •❛r❣❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥s✳ ✶ ❲❡ ✉s❡ t❤❡ ❛❜❜r❡✈✐❛t✐♦♥ [A, B] = A -1 B -1 AB t♦ ✇r✐t❡ t❤❡ ❝♦♠♠✉t❛t♦rs✳ ✷ ■♥ ♣r❡s❡♥❝❡ ♦❢ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ❤♦✇ t♦ ♠❛❦❡ t❤✐s ✐❞❡♥t✐✜❝❛t✐♦♥ ✇♦r❦ ✇❛s ❛•r❡❛❞② ❞✐s❝✉ss❡❞ ✐♥ ❝❤❛♣t❡r ✺✳ ✼✶ ✻✳ ❚❤❡ s②♠♠❡tr② ❣r♦✉♣ ♦❢ M HM ◗✉❛❞r❛t✐❝ r❡✜♥❡♠❡♥ts ❛♥❞ ❝❤❛r❛❝t❡r✐st✐❝s ❚❤❡ q✉❛❞r❛t✐❝ r❡✜♥❡♠❡♥t σ D (γ) t②♣✐❝❛••② ❛♣♣❡❛rs ✐♥ ❝❤✐r❛• ❜♦s♦♥✐❝ ♣❛rt✐t✐♦♥ ❢✉♥❝✲ t✐♦♥ ❬✻✶✕✻✹(✳ ■♥ ♦✉r ❝♦♥t❡①t✱ ❛s ✇❡ ❤❛✈❡ ❛•r❡❛❞② ❞✐s❝✉ss❡❞ ✐t ✐s r❡q✉✐r❡❞ ❢♦r t❤❡ ❝♦♥s✐st❡♥❝② ✇✐t❤ t❤❡ ✇❛••✲❝r♦ss✐♥❣ ♣❤❡♥♦♠❡♥❛✳ ❖♥❡ ❝❛♥ ✇r✐t❡ ❛ ❣❡♥❡r❛• s♦•✉t✐♦♥ t♦ ✭✹✳✶✶✮ ❛s σ D (γ) = exp 2πi - 1 2 p Λ q Λ + A ΛΣ p Σ + q Λ + A ΛΣ p Σ θ Λ ❉ -p Λ φ ❉,Λ , ✭✻✳✷✮ ✇❤r❡ θ Λ ❉ ❛♥❞ φ ❉,Λ ❛r❡ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝s ✇❤✐❝❤ ❛r❡ ❛•s♦ ❦♥♦✇♥ ❛s ❣❡♥❡r❛•✐③❡❞ s♣✐♥ str✉❝t✉r❡s ♦♥ Y✳ ❚❤❡s❡ ❛r❡ ❞❡✜♥❡❞ ♠♦❞✉•♦ ✐♥t❡❣❡rs✳ ❚❤❡ t❡r♠s ♣r♦♣♦rt✐♦♥❛• t♦ t❤❡ ♠❛tr✐① A ❛r✐s❡ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ✇❡ ❤❛✈❡ ❝❤❛♥❣❡❞ t❤❡ ❜❛s✐s ❛♥❞ ♥♦♥✲ ✐♥t❡❣r❛•✐t② ♦❢ t❤❡ ❝❤❛r❣❡ γ ✐♥ t❤❡ t②♣❡ ■■❇ s✐❞❡ ✭✺✳✺✮✱ ❛s ✇❡ ❤❛✈❡ ❛•r❡❛❞② s❡❡♥ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡r✳ ❆•t❤♦✉❣❤✱ ♦♥❡ ❝❛♥ t❤✐♥❦ ❛❜♦✉t t❤❡s❡ ❝❤❛r❛❝t❡r✐st✐❝s ❛s •✉st ❤❛•❢✲✐♥t❡❣❡r ♥✉♠✲ ❜❡rs✱ t❤❡ s②♠♣•❡❝t✐❝ ✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ ❉✲✐♥st❛♥t♦♥ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ r❡q✉✐r❡s t❤❡ q✉❛❞r❛t✐❝ r❡✜♥❡♠❡♥t σ D (γ) t♦ tr❛♥s❢♦r♠ ✉♥❞❡r s②♠♣•❡❝t✐❝ r♦t❛t✐♦♥s ❛s t❤❡ ❢♦••♦✇✲ ✐♥❣✱ Sp(2h 1,1 + 2, Z) ∋ ρ = D C B A : θ Λ ❉ φ ❉,Λ → ρ • θ Λ ❉ φ ❉,Λ - 1 2 (A T C) d (D T B) d , ✭✻✳✸✮ ✇❤❡r❡ (A) d ❞❡♥♦t❡s t❤❡ ❞✐❛❣♦♥❛• ♦❢ ❛ ♠❛tr✐① A✳ ■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ❉✲✐♥st❛♥t♦♥ ❝❤❛r❛❝t❡r✐st✐❝s✱ t❤❡r❡ ❡①✐st ❛ s✐♠✐•❛r s❡t ♦❢ ♦❜•❡❝ts Θ = (θ Λ , φ Λ )✱ ❝❤❛r❛❝t❡r✐③✐♥❣ t❤❡ ✜❜r❛t✐♦♥ ♦❢ t❤❡ ◆❙✲❛①✐♦♥ ❝✐r❝•❡ ❜✉♥❞•❡ ♦✈❡r t❤❡ t♦r✉s ♣❛r❛♠❡tr✐③❡❞ ❜② t❤❡ || s❝❛•❛rs✳ ❯♥❞❡r ❛ s②♠♣•❡❝t✐❝ tr❛♥s❢♦r♠❛t✐♦♥✱ t❤❡② ❜❡❤❛✈❡ ✐♥ t❤❡ s❛♠❡ ✇❛② ❛s t❤❡ ❉✲✐♥st❛♥t♦♥ ❝❤❛r❛❝t❡r✐st✐❝s t❤r♦✉❣❤ ✭✻✳✸✮✳ ◆♦✇ ✇❡ ❛r❡ ✐♥ ❛ ♣♦s✐t✐♦♥ t♦ ❣✐✈❡ t❤❡ q✉❛♥t✉♠ ❝♦rr❡❝t❡❞ tr❛♥s❢♦r♠❛t✐♦♥s ❢♦r ❍❡✐s❡♥❜❡r❣ s❤✐❢ts ❛♥❞ ♠♦♥♦❞r♦♠②✳ ❼ ❋✐rst✱ •❡t ✉s ❝♦♥s✐❞❡r t❤❡ ❍❡✐s❡♥❜❡r❣ s❤✐❢ts✳ ❚❤❡② ✇❡r❡ ❣✐✈❡♥ ✐♥ ❬✷✶(✱ ❛♥❞ ❛r❡ ❛s t❤❡ ❢♦••♦✇✐♥❣✱ T η Λ ,η Λ ,κ : ζ Λ → ζ Λ + η Λ , ζΛ → ζΛ + ηΛ , ✭✻✳✹✮ σ → σ + 2κ -ηΛ (ζ Λ -2θ Λ ) + η Λ ( ζΛ + A ΛΣ -2φ Λ ) -η Λ ηΛ . ❍❡r❡ (η Λ , ηΛ , κ) ∈ Z 2h 1,1 +3 ✳ ❼ ◆❡①t✱ ❛s ✇❛s ❞✐s❝✉ss❡❞ ✐♥ ❬✷✶✱ ✻✾(✱ t❤❡ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ✐♥ ✭✺✳✶✹✮ s❤♦✉•❞ ❜❡ ❛♣♣❡♥❞❡❞ ❜② ❛ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ◆❙✲❛①✐♦♥ ❛s σ → σ + 2κ(M ǫ a ), ✭✻✳✺✮ ✇❤❡r❡ κ(M ) ✐s t❤❡ ❝❤❛r❛❝t❡r ♦❢ t❤❡ ♠♦♥♦❞r♦♠② ❣r♦✉♣✱ ✇❤✐•❡ ❝♦♥s✐❞❡r✐♥❣ ❛•• ✐♥st❛♥t♦♥ ❝♦rr❡❝t✐♦♥s✳ ❙✐♥❝❡ t❤❡ ♠♦♥♦❞r♦♠② ❣r♦✉♣ ✐s ❛❜❡•✐❛♥ ✐♥ ♥❛t✉r❡✱ ✼✷ ❋❛✐•✉r❡ ♦❢ t❤❡ ❣r♦✉♣ •❛✇✳ ✻✳✹ t❤✐s ❝❤❛r❛❝t❡r ❜❡❝♦♠❡s κ(M ǫ a ) = κ a ǫ a ✳ ❚❤❡ ❝♦♠♣❡♥s❛t✐♥❣ s♣❡❝tr❛• ✢♦✇ tr❛♥s❢♦r♠❛t✐♦♥ ✭✺✳✶✼✮ t❛❦❡s t❤❡ ❢♦••♦✇✐♥❣ ❢♦r♠ ❡①♣•✐❝✐t•②✱ p a [ǫ] = p a + ǫ a p 0 , q a [ǫ] = q a -κ abc p b ǫ c - p 0 2 κ abc ǫ b ǫ c , q 0 [ǫ] = q 0 -q a ǫ a + 1 2 κ abc p a ǫ b ǫ c + p 0 6 κ abc ǫ a ǫ b ǫ c . ✭✻✳✻✮ ✻✳✸ ❋❛✐•✉r❡ ♦❢ t❤❡ ❣r♦✉♣ •❛✇ ❊✈❡♥ t❤❡s❡ ❛❜♦✈❡ ♥♦♥✲tr✐✈✐❛• ❛❞•✉st♠❡♥ts t✉r♥ ♦✉t t♦ ❜❡ ♥♦t s❛t✐s❢❛❝t♦r② ❢♦r ♦❜✲ t❛✐♥✐♥❣ ❛ ❢❛✐t❤❢✉• r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❞✉❛•✐t② ❣r♦✉♣s✳ ▲❡t ✉s ✜rst s❡❡ ✇❤❡r❡ ✐t ❢❛✐•s ♣r❡❝✐s❡•②✳ ❚❤❡ r❡•❡✈❛♥t ❛❝t✐♦♥s ♦❢ ✈❛r✐♦✉s s②♠♠❡tr✐❡s ♦♥ t❤❡ t②♣❡ ■■❇ ✜❡•❞s ❛r❡ ❣✐✈❡♥ ✐♥ t❤❡ ❛♣♣❡♥❞✐① ❆✳✷ ✐♥ ❛ t❛❜✉•❛r ❢♦r♠ ✐♥ ❬✷✹(✳ ❈♦♠♣✉t✐♥❣ t❤❡ ❝♦♠♠✉✲ t❛t♦r ❜❡t✇❡❡♥ ❙✲❞✉❛•✐t② ❛♥❞ ❍❡✐s❡♥❜❡r❣ tr❛♥s❢♦r♠❛t✐♦♥ ✱ ✇❡ ✜♥❞ t❤❛t ❛♣❛rt ❢r♦♠ ♠♦♥♦❞r♦♠② ✇❡ ❣❡t s♦♠❡ ❛♥♦♠❛•♦✉s t❡r♠s✱ ✸ S -1 T (1) 0,η a S •       b a c a c a c 0 ψ       = M η a •       b a c a c a c 0 ψ       +       0 0 A ab η b η a φ a + c 2,a 8 -1 2 A ab η b η a κ a + c 2,a 24       ,✭✻✳✼✮ ✇❤❡r❡ ✇❡ r❡♠❡♠❜❡r t♦ ❤❛✈❡ s❡t t❤❡ ◆❙✺✲✐♥st❛♥t♦♥ ❝❤❛r❛❝t❡r✐st✐❝s θ Λ = 0✳ ❚❤✐s r❡q✉✐r❡♠❡♥t ❝♦♠❡s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t ❢♦r ❍❡✐s❡♥❜❡r❣ s❤✐❢t η Λ = 0✱ t❤❡ ❣r♦✉♣ •❛✇ ✭s❡❡ ❜❡•♦✇ ✭✻✳✶✶✮✮ s❤♦✉•❞ ❤♦•❞✳ ❋✉rt❤❡r♠♦r❡✱ ✐❢ ♦♥❡ ❝♦♥s✐❞❡rs t❤❡ ✜✈❡❜r❛♥❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ✇❤✐❝❤ ✇❡ ♣r❡s❡♥t ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✱ ✉♥❞❡r ❍❡✐s❡♥❜❡r❣ s❤✐❢t ✇✐t❤ η Λ = 0✱ s❤♦✉•❞ ❜❡ ✐♥✈❛r✐❛♥t ❬✷✶(✳ ❚❤✐s ✐s ❡♥s✉r❡❞ ♦♥•② ✐❢ θ Λ = 0✳ ❚❤✐s s♦♠❡✇❤❛t s✐♠♣•✐✜❡s t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝s ❜② ❡•✐♠✐♥❛t✐♥❣ s♦♠❡ t❡r♠s ✐♥ ✭✻✳✸✮✱ φ a → φ a + 1 2 κ aac ǫ c , φ 0 → φ 0 -ǫ a φ a - 1 2 (L 0 (ǫ) -ǫ a L a (ǫ) + κ aac ǫ a ǫ c ) . ✭✻✳✽✮ ❍♦✇❡✈❡r✱ t❤❡ s✐t✉❛t✐♦♥ ✐s st✐•• ♥♦t s❛t✐s❢❛❝t♦r②✱ ❜❡❝❛✉s❡ ✐t ✐s ❝•❡❛r ❢r♦♠ ✭✻✳✼✮ ❛ ❝♦♥s✐st❡♥t ❛❝t✐♦♥ ♦❢ ❛•• s②♠♠❡tr✐❡s ✐s ♥♦t ❣❡♥❡r❛t❡❞ ❡♥s✉r✐♥❣ ❝•♦s✉r❡ ♦❢ t❤❡ ❣r♦✉♣✳ ❆•t❤♦✉❣❤ ✐♥ ✭✻✳✼✮ t❤❡ ❛♥♦♠❛•♦✉s t❡r♠s ✭✐✳❡✳ t❤❡ t❡r♠s ❞✉❡ t♦ ✇❤✐❝❤ ✭✻✳✶✶✮ ❢❛✐•s t♦ ❤♦•❞✮✱ ✐♥ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ c 0 ❛♥❞ ψ ❝❛♥ ❜❡ r❡♠♦✈❡❞ ❜② ❛♣♣r♦♣r✐❛t❡•② ❝❤♦♦s✐♥❣ φ a ❛♥❞ t❤❡ ❝❤❛r❛❝t❡r κ a ✱ t❤❡ ♦♥❡ ✐♥ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ c a ❝❛♥ ♥♦t ❜❡ ❝❛♥❝❡•❡❞✳ ❚❤✉s ✇❡ ❤❛✈❡ t♦ ❝♦♥❝•✉❞❡ t❤❛t t❤❡ ♠♦♥♦❞r♦♠②✱ ❍❡✐s❡♥❜❡r❣ ❛♥❞ ❙✲❞✉❛•✐t② tr❛♥s❢♦r♠❛t✐♦♥s ❢❛✐• t♦ ❢♦r♠ ❛ ❝♦♥s✐st❡♥t ❣r♦✉♣ r❡♣r❡s❡♥t❛t✐♦♥✳ ✸ S = 0 -1 1 0 ✐s t❤❡ ♥♦♥✲tr✐✈✐❛• ❣❡♥❡r❛t♦r ♦❢ SL(2, Z)✳ ✼✸ ✻✳ ❚❤❡ s②♠♠❡tr② ❣r♦✉♣ ♦❢ M HM ✻✳✹ ❈♦♥str✉❝t✐♥❣ ❣r♦✉♣ r❡♣r❡s❡♥t❛t✐♦♥s ❚♦ r❡s♦•✈❡ t❤❡ ✐♥❝♦♥s✐st❡♥❝✐❡s✱ ✜rst ✇❡ ♥♦t✐❝❡ t❤❛t ✇❡ ❝❛♥ ❛❜s♦r❜ t❤❡ ❉✲✐♥st❛♥t♦♥ ❝❤❛r❛❝t❡r✐st✐❝s t♦ || ❛♥❞ ◆❙ s❝❛•❛rs✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ t❤❡s❡ ❝❤❛r❛❝t❡r✐st✐❝s ❛♣♣❡❛r ✇✐t❤ t❤❡ || s❝❛•❛rs ✐♥ ♣❛rt✐❝✉•❛r ❝♦♠❜✐♥❛t✐♦♥s✱ ❛s ✐♥ ✭✻✳✾✮✱ ❛♥❞ s✉❜s❡q✉❡♥t•②✱ t♦ ❡♥s✉r❡ t❤❡ ✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ♠❡tr✐❝ ♦♥❡ ❤❛s t♦ ❛❜s♦r❜ t❤❡♠ ✐♥ t❤❡ ◆❙ ❛①✐♦♥ ❛s ✇❡••✳ ζ Λ -θ Λ D → ζ Λ , ζΛ -φ ❉,Λ + A ΛΣ θ Σ D → ζΛ , σ + φ ❉,Λ ζ Λ -θ Λ D ζΛ + A ΛΣ ζ Σ → σ. ✭✻✳✾✮ ❚❤✐s ♥♦✇ ❛♠❡♥❞s t❤❡ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥✱ M ǫ a : b a → b a + ǫ a , ζ a → ζ a + ǫ a ζ 0 , ζa → ζa -κ abc ζ b ǫ c - 1 2 κ abc ǫ b ǫ c ζ 0 + A ab ǫ b , ζ0 → ζ0 -ζa ǫ a + 1 2 κ abc ζ a ǫ b ǫ c + 1 6 κ abc ǫ a ǫ b ǫ c ζ 0 - 1 2 A ab ǫ a ǫ b + c 2,a 8 ǫ a , σ → σ -A ab ǫ a ζ b - 1 2 A ab ǫ a ǫ b + 1 4 c 2,a ǫ a ζ 0 + 2κ a ǫ a . ✭✻✳✶✵✮ ❚❤❡ ♠♦❞✐✜❝❛t✐♦♥ ✐♥ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ❝♦♠♣❛r❡❞ t♦ ✭✺✳✶✹✮ ✐s ❝•❡❛r•② ❞✉❡ t♦ t❤❡ ❛♥♦♠❛•♦✉s t❡r♠s t❤❛t ❛♣♣❡❛r❡❞ ✐♥ ✭✻✳✼✮✳ ❲✐t❤ t❤✐s✱ ✇❡ ♣r♦❝❡❡❞ t♦ s❤♦✇ ❤♦✇ t♦ ❝♦♥str✉❝t ❛ ❣r♦✉♣ r❡♣r❡s❡♥t❛t✐♦♥ ✐♥✈♦•✈✐♥❣ t❤❡ ❞✐s❝r❡t❡ s②♠♠❡tr② ❣r♦✉♣s ♦❢ ❍❡✐s❡♥❜❡r❣ s❤✐❢t✱ ▼♦♥♦❞r♦♠② ❛♥❞ ❙✲❞✉❛•✐t②✳ ❇❡•♦✇ ✐♥ t❛❜•❡ ✻✳✶ ✇❡ ❞✐s♣•❛② t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ ❝♦rr❡❝t❡❞ s②♠♠❡tr② tr❛♥s❢♦r✲ ♠❛t✐♦♥s ♦♥ t❤❡ t②♣❡ ■■❇ ❝♦♦r❞✐♥❛t❡s✳ Pr♦✈✐❞❡❞ t❤❛t ♦♥❡ ✜①❡s t❤❡ ❝❤❛r❛❝t❡r ♦❢ t❤❡ ♠♦♥♦❞r♦♠② ❣r♦✉♣ ❛s κ(M ǫ a ) = -c 2,a 24 ❬✷✹( ✐♥❞❡❡❞ ✇❡ ✜♥❞ t❤❡ ❣r♦✉♣ r❡♣r❡s❡♥t❛t✐♦♥✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✇❡ ♦❜t❛✐♥ S -1 T (1) 0,η a S = M η a , ✭✻✳✶✶✮ ✇❤❡r❡ S = 0 -1 1 0 ✐s ♦♥❡ ♦❢ t❤❡ ❣❡♥❡r❛t♦rs ♦❢ t❤❡ SL(2, Z) ❣r♦✉♣✱ t❤❡ s♦ ❝❛••❡❞ ❙✲❣❡♥❡r❛t♦r✳ ❚❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ ♦t❤❡r ❣❡♥❡r❛t♦r T = 1 1 1 0 ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ❍❡✐s❡♥❜❡r❣ s❤✐❢t τ → τ + 1✳ ❙✐♥❝❡ SL(2, Z) ❣r♦✉♣ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ✏✇♦r❞✧ ♦❢ t❤❡s❡ ❙ ❛♥❞ ❚✲tr❛♥s❢♦r♠❛t✐♦♥s g = S m 1 T n 1 S m 2 T m 2 ...✱ ✐t s✉✣❝❡s t♦ ✈❡r✐❢② t❤❡ ❣r♦✉♣ •❛✇ s❡♣❛r❛t❡•② ❢♦r t❤❡♠✳ ❆s ❚✲tr❛♥s❢♦r♠❛t✐♦♥ ✐s ♦♥❡ ♦❢ t❤❡ ❍❡✐s❡❜❡r❣ s❤✐❢ts ✐t ❝♦♠♠✉t❡s ✇✐t❤ T (1) 0,η a ❛♥❞ t❤❡ ♦♥•② ♥♦♥✲tr✐✈✐❛• ❝♦♠♠✉t❛t♦r t❤❛t ✇❡ ❤❛✈❡ t♦ ❝❤❡❝❦ ✐s ✭✻✳✶✶✮✳ ✼✹ ❈♦♥str✉❝t✐♥❣ ❣r♦✉♣ r❡♣r❡s❡♥t❛t✐♦♥s✳ ✻✳✹ b a c a c a c 0 ψ S c a -b a c a + c 2,a 8 -ψ c 0 T b a c a + b a c a -c 2,a 24 c 0 ψ -c 0 M ǫ a b a + ǫ a c a c a + 1 2 κ abc ǫ b c c + A ab ǫ b c 0 -ǫ a c a -1 6 κ abc ǫ a (b b + 2ǫ b )c c -1 2 A ab ǫ a ǫ b + c 2,a 8 ǫ a ψ + 1 6 κ abc ǫ a c b c c -κ a ǫ a T (1) 0,η a b a c a + η a c a -1 2 κ abc η b b c + A ab η b c 0 + 1 6 κ abc η a b b b c + c 2,a 24 η a ψ + η a c a + 1 2 A ab η a η b -1 6 κ abc η a b b (c c + 2η c ) T (2) ηa b a c a c a + ηa c 0 ψ T (3) η0 ,κ b a c a c a c 0 + η0 ψ + κ ❚❛❜•❡ ✻✳✶✿ ❚❤❡ ❛❝t✐♦♥ ♦❢ ❣❡♥❡r❛t♦rs ♦❢ t❤❡ ❞✐s❝r❡t❡ s②♠♠❡tr② tr❛♥s❢♦r♠❛t✐♦♥s ✐♥ t❤❡ t②♣❡ ■■❇ ❝♦♦r❞✐♥❛t❡ ❜❛s✐s✳ ✻✳✹✳✶ ❈♦♠♠✉t❛t♦rs ♦❢ ❞✐✛❡r❡♥t ❡•❡♠❡♥ts ❍❡r❡ ✇❡ ♣r❡s❡♥t t❤❡ r❡s✉•ts ♦❢ ❞✐✛❡r❡♥t ❝♦♠♠✉t❛t♦rs ❛♠♦♥❣ ✈❛r✐♦✉s ❡•❡♠❡♥ts ❣❡♥✲ ❡r❛t✐♥❣ t❤❡ ❣r♦✉♣ ✉s✐♥❣ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ r✉•❡s ❢r♦♠ t❛❜•❡ ✻✳✶ ✭ T (1) ǫ a ,0 ≡ M ǫ a ✮✳ ❼ ❈♦♠♠✉t❛t♦rs ✇✐t❤ S ✿ S -1 M ǫ a S = T (1) 0,-ǫ a T (2) 2A ab ǫ b T (3) 0, 3L 0 (ǫ)-ǫ a La(-ǫ) , S -1 T (1) 0,η a S = M η a , S -1 T (2) ηa S = T (2) ηa , S -1 T (3) η0 ,κ S = T (3) -κ,η 0 ; ✭✻✳✶✷✮ ❼ ❈♦♠♠✉t❛t♦rs ✇✐t❤ T ✿ [M ǫ a , T ] = T (1) 0,ǫ a T (2) La(ǫ) T (3) -L 0 (ǫ a ), ǫ a La(ǫ)-L 0 (ǫ) , [T (1) 0,η a , T ] = [T (2) ηa , T ] = 1, [T (3) η0 ,κ , T ] = T (3) 0,-η 0 ; ✭✻✳✶✸✮ ❼ ❚❤❡ ♥✐•♣♦t❡♥t s✉❜❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② ❍❡✐s❡♥❜❡r❣ ❛♥❞ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r✲ ♠❛t✐♦♥s ✿ ✼✺ ✻✳ ❚❤❡ s②♠♠❡tr② ❣r♦✉♣ ♦❢ M HM M ǫ a , T (1) 0,η a = T (2) -κ abc ǫ b η c T (3) η a La(-ǫ),-ǫ a La(η) , M ǫ a , T (2) ηa = T (3) ǫ a ηa,0 . T (1) 0,η a , T (2) ηa = T (2) 0,-η a ηa , T (1) 0,η a , T (3) η0 ,κ = M ǫ a , T (3) η0 ,κ = T (2) ηa , T (3) η0 ,κ = 1. ✭✻✳✶✹✮ ❋r♦♠ t❤❡ ❛❜♦✈❡ ✇❡ s❡❡ t❤❛t t❤❡ ♠♦❞✐✜❝❛t✐♦♥s ♣r♦♣♦s❡❞ ✐♥ ❬✷✹( ✐♥❞❡❡❞ ❣✐✈❡s r✐s❡ t♦ ❛ ♥✐❝❡ ❣r♦✉♣ r❡♣r❡s❡♥t❛t✐♦♥✳ ✻✳✺ ❙♦♠❡ r❡♠❛r❦s ♦♥ ♠♦❞✐✜❝❛t✐♦♥s ♦❢ t❤❡ ♠♦♥✲ ♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ❚❤❡ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ❛①✐♦♥s ❡♥s✉r✐♥❣ ❝♦♥s✐st❡♥t ❝♦♥s✐st❡♥t ❣r♦✉♣ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❞✐s❝r❡t❡ s②♠♠❡tr✐❡s ✇❛s ❣✐✈❡♥ ✐♥ ✭✻✳✶✵✮✳ ❲❡ ❤❛✈❡ s❡❡♥ ❤♦✇ t❤❡ ❛♠❡♥❞♠❡♥ts ❝❛♠❡ ✐♥t♦ ❜❡✐♥❣ ❝♦♠♣❛r❡❞ t♦ ✭✺✳✶✹✮ ❬✷✶(✳ ❚❤❡ ❝r✉❝✐❛• ❢❡❛t✉r❡ ♦❢ t❤❡ ♥❡✇ t❡r♠s ❛♣♣❡❛r✐♥❣ ✐♥ ✭✻✳✶✵✮ ✐s t❤❛t t❤❡② ❤❛✈❡ ♦r✐❣✐♥s ✐♥ t❤❡ q✉❛❞r❛t✐❝ r❡✜♥❡♠❡♥t ❢♦r ❉✲✐♥st❛♥t♦♥s✱ ♦r ♠♦r❡ ♣r❡❝✐s❡•② t❤❡ ❉✲✐♥st❛♥t♦♥ ❝❤❛r❛❝t❡r✐st✐❝s ❛♣♣❡❛r✐♥❣ ✐♥ ✭✻✳✷✮✳ ❆❞❞✐t✐♦♥❛••②✱ t❤❡s❡ ♠♦❞✐✜❝❛t✐♦♥s ✐♥ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ❡♥s✉r❡ t❤❛t t❤❡ ◆❙✺✲❜r❛♥❡ ❝❤❛r❛❝t❡r✐st✐❝s tr❛♥s❢♦r♠ ❤♦♠♦❣❡♥❡♦✉s•②✱ ❞✉❡ t♦ ✇❤✐❝❤ t❤❡② ❝❛♥ ♥♦✇ ❜❡ ❞r♦♣♣❡❞✳ ❙✐♥❝❡ t❤❡ ♦t❤❡r ❝❤❛r❛❝t❡r✐st✐❝s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❉✲✐♥st❛♥t♦♥s ❛r❡ ❛•r❡❛❞② ❛❜s♦r❜❡❞ ✐♥ t❤❡ ❛①✐♦♥s✱ ✐♥ ♦✉r r❡s✉•ts t❤❡② ❞♦ ♥♦t ❛♣♣❡❛r ✭s❡❡ ❢♦r ❡①❛♠♣•❡ ✭✼✳✷✹✮ ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✮✳ ❆s ✇❛s ♠❡♥t✐♦♥❡❞ ✐♥ ❝❤❛♣t❡r ✹✱ t❤❡ q✉❛❞r❛t✐❝ r❡✜♥❡♠❡♥t ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ❢♦r♠ ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❤♦♠♦♠♦r♣❤✐s♠ σ : Γ → U (1)✳ ❋♦r t❤❡ s❡•❢✲❞✉❛• t❤r❡❡ ❢♦r♠ ✢✉① H = dB✱ t❤❡ s♦•✉t✐♦♥ ❢♦r t❤❡♠ ✐s ❣✐✈❡♥ ❜② ❬✷✶(✱ σ Θ (H) = e 2πi(-1 2 m Λ n Λ +m Λ θ Λ -n Λ φ Λ ) , ✭✻✳✶✺✮ ✇❤❡r❡ Θ = (θ Λ , φ Λ ) ❛r❡ ◆❙✺✲❜r❛♥❡ ❝❤❛r❛❝t❡r✐st✐❝s✱ ❛s ✐♥tr♦❞✉❝❡❞ ❜❡❢♦r❡ ❛♥❞ (m Λ , n Λ ) ❛r❡ ✐♥t❡❣❡rs✳ ✹ ❆❢t❡r ❛❜s♦r❜✐♥❣ t❤❡ ❉✲✐♥st❛♥t♦♥ ❝❤❛r❛❝t❡r✐st✐❝s t♦ t❤❡ || ❛♥❞ ◆❙✲❛①✐♦♥s✱ t❤❡✐r tr❛♥s❢♦r♠❛t✐♦♥ ✐s ♥♦✇ ♠♦❞✐✜❡❞ ❜② ❤❛•❢✲✐♥t❡❣❡r s❤✐❢ts✳ ❆s ✇❡ ❛r❡ ❣♦✐♥❣ t♦ s❡❡ ❜❡•♦✇✱ s✐♠✐•❛r ❤❛•❢✲✐♥t❡❣❡r s❤✐❢ts ❛r✐s✐♥❣ ❢r♦♠ ✏♠♦♥♦❞r♦♠②✲•✐❦❡✧ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❝❡rt❛✐♥ U (1) ❣❛✉❣❡ ✜❡•❞s ❛♣♣❡❛r ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ t❤❡ s♦ ❝❛••❡❞ ❋r❡❡❞✲❲✐tt❡♥ ❛♥♦♠❛•② ❬✺✽(✳ ✹ ■t ♠✐❣❤t s❡❡♠ t♦ ❜❡ ❛ ❜✐t ♦❢ ♦✈❡r❦✐••✐♥❣ t♦ ❞✐s❝✉ss t❤❡s❡ ❝❤❛r❛❝t❡r✐st✐❝s ❛❣❛✐♥✱ ❛s ✇❡ ❤❛✈❡ ❛•r❡❛❞② s❡❡♥ ✐♥ ❝❛s❡ ♦❢ ❛♠❡♥❞❡❞ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ❛•• ♦❢ t❤❡♠ ❞r♦♣✳ ❇✉t ✇❡ ❞✐s❝✉ss t❤❡♠ ❤❡r❡ ❛s ✇❡ ✇❛♥t t♦ ❝♦♥tr❛st ❤♦✇ t❤❡ s✉❜t•❡t✐❡s r❡•❛t❡❞ t♦ ❝❤❛r❛❝t❡r✐st✐❝s ❛r❡ tr❛❞❡❞ ❛❣❛✐♥st t❤❡ s✉❜t•❡t✐❡s ✐♥ t❤❡ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ✭✻✳✶✵✮✳ ❲❡ r❡✐t❡r❛t❡ ❛❣❛✐♥ t❤❛t ❞♦✐♥❣ s♦ ✇❛s ❝r✉❝✐❛• ❬✷✹( ❛s ✐t ♠❛❞❡ s✉r❡ t❤❡ ❞✐s❝r❡t❡ s②♠♠❡tr✐❡s ❢♦r♠ ❛ ❣r♦✉♣ r❡♣r❡s❡♥t❛t✐♦♥✳ ✼✻ ❙♦♠❡ r❡♠❛r❦s ♦♥ ♠♦❞✐✜❝❛t✐♦♥s ♦❢ t❤❡ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥✳ ✻✳✺ ❊✈❡♥ t❤♦✉❣❤✱ t❤❡ ❛♥♦♠❛•✐❡s ❝♦♥s✐❞❡r❡❞ ❜② ❋r❡❡❞ ❛♥❞ ❲✐tt❡♥ ❛r❡ ♦♥ t❤❡ ✇♦r•❞s❤❡❡t ✐♥ ♣r❡s❡♥❝❡ ♦❢ ❉✲❜r❛♥❡s✱ ✐t s❡❡♠s t❤❛t t❤❡② ❛r❡ ❝•♦s❡•② ❛♥❛•♦❣♦✉s t♦ t❤❡ ❛♥♦♠❛•♦✉s t❡r♠s ❛♣♣❡❛r✐♥❣ ✐♥ t❤❡ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ✭✻✳✶✵✮✳ ❇❡❢♦r❡ ♣r♦❝❡❡❞✐♥❣✱ •❡t ✉s ❞❡s❝r✐❜❡ ❋r❡❡❞✲❲✐tt❡♥ ❛♥♦♠❛•② ❜r✐❡✢② ✭❛♥❞ r❛t❤❡r ❝r✉❞❡•②✮✳ ❚❤❡♥ ✇❡ ✇✐•• ✜♥✐s❤ ❜② ❡①♣•♦r✐♥❣ ♣♦ss✐❜•❡ ✭❛♥❞ ❝♦♥•❡❝t✉r❛•✮ ❝♦♥♥❡❝t✐♦♥s ♦❢ ✭✻✳✶✵✮ ✇✐t❤ t❤❡♠✳ ✻✳✺✳✶ ❋r❡❡❞✲❲✐tt❡♥ ❛♥♦♠❛•② ❍❡r❡ ✇❡ r❡❝❛•• s♦♠❡ ❢❛❝ts ❛❜♦✉t ❣•♦❜❛• ❛♥♦♠❛•✐❡s ✐♥ t❤❡ ✇♦r•❞s❤❡❡t ♣❛t❤ ✐♥t❡❣r❛• ♦❢ t②♣❡ ■■ str✐♥❣ t❤❡♦r✐❡s ✐♥ ♣r❡s❡♥❝❡ ♦❢ ❉✲❜r❛♥❡s ❬✺✽(✳ ❚❤❡ ✇♦r•❞s❤❡❡ts t❤❛t ✇❡ ❝♦♥s✐❞❡r ❛r❡ ♦r✐❡♥t❡❞ |✐❡♠❛♥♥ s✉r❢❛❝❡s✱ ♠❛♣♣❡❞ t♦ s♣❛❝❡t✐♠❡ Y ✇❤✐❝❤ ❛❞♠✐ts s♣✐♥ str✉❝t✉r❡ ❛s ♦✉r ♠♦❞❡• ❝♦♥t❛✐♥s ❢❡r♠✐♦♥s✳ ❋♦r t❤❡ s✐♠♣•❡st ❝❛s❡ ♦❢ ❝•♦s❡❞ ✇♦r•❞s❤❡❡ts ✇✐t❤♦✉t ❜♦✉♥❞❛r②✱ t❤❡ ✇♦r•❞s❤❡❡t ♠❡❛s✉r❡ ✐s ✇❡•• ❞❡✜♥❡❞✱ ♣r♦✈✐❞❡❞ Y ✐s s♣✐♥✳ ❚❤❡ s✐t✉❛t✐♦♥ ❣❡ts ❝♦♠♣•✐❝❛t❡❞ ✐❢ ♦♥❡ st❛rts ❝♦♥s✐❞❡r✐♥❣ ❉✲❜r❛♥❡s✳ ❚❤❡② ❛r❡ s✉♣♣♦rt❡❞ ♦♥ ❛♥ ♦r✐❡♥t❡❞ s✉❜♠❛♥✐❢♦•❞ Q ♦♥ ✇❤✐❝❤ str✐♥❣s ❝❛♥ ❡♥❞✳ ❚❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ✇♦r•❞s❤❡❡t ∂Σ ♠❛♣s t♦ t❤✐s Q✳ ❖♥ Q t❤❡r❡ ✐s ❛ ✜❡•❞ A ✇❤✐❝❤ ✐s ❝♦♥s✐❞❡r❡❞ t♦ ❜❡ ❛ U (1) ❣❛✉❣❡ ✜❡•❞ ❝♦♥✈❡♥t✐♦♥❛••②✱ t✉r♥s ♦✉t t♦ ❤❛✈❡ ❝♦rr❡❝t ✐♥t❡r♣r❡t❛t✐♦♥ ❛s ❛ ❙♣✐♥ c ❝♦♥♥❡❝t✐♦♥ ❬✺✽(✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ B ✜❡•❞ ❝♦♥s✐❞❡r❡❞ ✐♥ t❤✐s ❞✐s❝✉ss✐♦♥ ✐s t♦♣♦•♦❣✐❝❛••② tr✐✈✐❛• ❛♥❞ ❢♦r ❝♦♥✈❡♥✐❡♥❝❡✱ ❛❞❞✐t✐♦♥❛••② ✇❡ s❡t ✐t t♦ ③❡r♦✳ ❚❤❡ r❡•❡✈❛♥t ❝♦♥tr✐❜✉t✐♦♥ t♦ t❤❡ ✇♦r•❞s❤❡❡t ♣❛t❤ ✐♥t❡❣r❛• t❤❛t ♦♥❡ ❤❛s t♦ ❝♦♥✲ s✐❞❡r ✐s t❤❡♥ pfaff (D) × e 2πi ∂Σ A , ✭✻✳✶✻✮ ✇❤❡r❡ pfaff (D) ✐s t❤❡ ♣❢❛✣❛♥ ♦❢ t❤❡ ✇♦r•❞s❤❡❡t ❉✐r❛❝ ♦♣❡r❛t♦r ❛♥❞ t❤❡ s❡❝♦♥❞ ❢❛❝t♦r ✐s t❤❡ ❤♦•♦♥♦♠② ♦❢ A ❛r♦✉♥❞ t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ❜♦✉♥❞❛r② ♦❢ Σ✳ ❚❤❡ ♠❛✐♥ r❡s✉•t ♦❢ ❬✺✽( ❝♦♥❝❡r♥s t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❛♥♦♠❛•② ✐♥ pfaff (D)✱ ✇❤✐❝❤ ✐s ❛ s❡❝t✐♦♥ ♦❢ ❛ •✐♥❡ ❜✉♥❞•❡ ♦✈❡r t❤❡ s♣❛❝❡ ♦❢ ♣❛r❛♠❡t❡rs ♠❛♣♣✐♥❣ t❤❡ ✇♦r•❞✲ s❤❡❡ts Σ t♦ t❤❡ s♣❛❝❡t✐♠❡ Y ✳ ❚❤✐s ❜✉♥❞•❡ ❝❛rr✐❡s ❛ ♥❛t✉r❛• ♠❡tr✐❝ ❛♥❞ ❝♦♥♥❡❝t✐♦♥✳ ❚❤❡ ❛♥♦♠❛•② ❝❛♥ t❤❡♥ ❜❡ ✉♥❞❡rst♦♦❞ ❛s ❛♥ ♦❜str✉❝t✐♦♥ t♦ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ❣•♦❜❛• ✢❛t s❡❝t✐♦♥ ❤❛✈✐♥❣ ✉♥✐t ♥♦r♠✳ ■t ✇❛s s❤♦✇♥ ❜② ❋r❡❡❞ ❛♥❞ ❲✐tt❡♥ t❤❛t t❤✐s ❝♦♥✲ ♥❡❝t✐♦♥ ✐s ✢❛t✱ ❜✉t t❤❡r❡ ✐s ❛ ❤♦•♦♥♦♠② ±1 ❞❡t❡r♠✐♥❡❞ ❜② ❛ ❝❡rt❛✐♥ ♠♦❞ ✷ ❝•❛ss ✭✇❤✐❝❤ ✐♥ t❤✐s ❝❛s❡ ✐s t❤❡ s❡❝♦♥❞ ❙t❡✐❢❡•✲❲❤✐t♥❡② ❝•❛ss✮ ♦❢ t❤❡ ♥♦r♠❛• ❜✉♥❞•❡ N t♦ t❤❡ ❉✲❜r❛♥❡ ✇♦r•❞✈♦•✉♠❡ Q✳ ❋♦r t❤❡ ♣❛t❤ ✐♥t❡❣r❛• t♦ ❜❡ ✇❡••✲❞❡✜♥❡❞✱ t❤❡r❡ ♠✉st ❜❡ ❛ ❝♦♠♣❡♥s❛t✐♥❣ ❛♥♦♠❛•② ❝♦♠✐♥❣ ❢r♦♠ t❤❡ A ✜❡•❞✳ ❋♦r t❤❡ ❝❛s❡ ✇❤❡♥ B = 0✱ t❤✐s ❝❛♥ ❜❡ ✉♥❞❡rst♦♦❞ ❛s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ A ❜❡✐♥❣ ❛ ❙♣✐♥ c ❝♦♥♥❡❝t✐♦♥✱ ❢r♦♠ ✇❤✐❝❤ ✐t ❝❛♥ ❜❡ ❞❡❞✉❝❡❞ t❤❛t t❤❡ W 3 (N ) = 0✳ ✺ ✺ W 3 ✐s t❤❡ ❝❛♥♦♥✐❝❛• ✐♥t❡❣r❛• •✐❢t ♦❢ t❤❡ t❤✐r❞ ❙t✐❡❢❡•✲❲❤✐t♥❡② ❝•❛ss w 3 ❛♥❞ ✐s t❤❡ ♦❜str✉❝t✐♦♥ t♦ Spin c str✉❝t✉r❡✳ ■t ✐s r❡•❛t❡❞ ❜② t❤❡ ❝♦♥♥❡❝t✐♥❣ ❤♦♠♦♠♦r♣❤✐s♠✱ t❤❡ s♦ ❝❛••❡❞ ❇♦❝❦st❡✐♥ ♠❛♣ t♦ w 2 ✱ ❛s W 3 (N ) = β(w 2 (N ))✳ ❋r♦♠ t❤❡ •♦♥❣ ❡①❛❝t s❡q✉❡♥❝❡ ✐♥ ❝♦❤♦♠♦•♦❣② ♦❢ Q ✐♥❞✉❝❡❞ ❢r♦♠ t❤❡ s❤♦rt ❡①❛❝t ❝♦❡✣❝✐❡♥t s❡q✉❡♥❝❡ ❬✺✽(✱ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t 2W 3 (N ) = 0 ❛♥❞ W 3 (N ) = 0 ♣r❡❝✐s❡•② ✇❤❡♥ w 2 (N ) ❝❛♥ ❜❡ •✐❢t❡❞ t♦ ❛ ❝•❛ss ✐♥ H 2 (Q, Z)✳ ■♥ ❢❛❝t✱ W 3 r❡❞✉❝❡s t♦ mod 2 t♦ t❤❡ ❙t✐❡❢❡•✲❲❤✐t♥❡② ❝•❛ss w 3 ✱ ❛♥❞ t❤✐s ✐s ❛❝t✉❛••② tr✉❡ ❢♦r ❛•• ♦❞❞ ❙t✐❡❢❡•✲❲❤✐t♥❡② ❝•❛ss❡s ✼✼ ✻✳ ❚❤❡ s②♠♠❡tr② ❣r♦✉♣ ♦❢ M HM ▲❡t ✉s ❛♥❛•②③❡ t❤✐s ❝❛s❡ ❛ ❜✐t ❢✉rt❤❡r✳ ❚❤❡ ❛♥♦♠❛•② ❝❛♥❝❡••❛t✐♦♥ ❢♦r ❝•♦s❡❞ s✉r❢❛❝❡s ✐♠♣•② t❤❛t t❤❡ r❡s✉•t ❢♦r ✭✻✳✶✻✮ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ s♣✐♥ str✉❝t✉r❡s ❛♥❞ ♦♥❡ ❝❛♥ s❡t t❤❡ ♥✉♠❜❡r ♦❢ r✐❣❤t ❛♥❞ •❡❢t ♠♦✈❡rs t♦ ❜❡ ❡q✉❛•✳ ❚❤✐s ♠❛❦❡s t❤❡ ❉✐r❛❝ ♦♣❡r❛t♦r r❡❛• ❛♥❞ ❤❡♥❝❡ pfaff (D) ✐s r❡❛•✳ ❚❤✐s pfaff (D) ✐s ♥♦t ❛ ♥✉♠❜❡r ❛♥❞ ♦♥❡ ❝❛♥ ♥♦t ❝❤♦♦s❡ ❛ ♣❛rt✐❝✉•❛r s✐❣♥ ❢♦r ✐t✳ ❈♦♥s✐❞❡r ❛ ❢❛♠✐•② ♦❢ Σ ♣❛r❛♠❡tr✐③❡❞ ❜② ❛ ❝✐r❝•❡ C✳ ❚❤❡♥ ✇❡ ❤❛✈❡ ❛ ♠❛♣ φ : Σ × C → Y ❛♥❞ φ(∂Σ × C) ⊂ Q✳ ■t ✇❛s s❤♦✇♥ ✐♥ ❬✺✽( t❤❛t ❣♦✐♥❣ ❛r♦✉♥❞ t❤❡ •♦♦♣ ❝❤❛♥❣❡s t❤❡ s✐❣♥ ❛s pfaff (D) → (-1) α pfaff (D), ✭✻✳✶✼✮ ✇❤❡r❡ α = (∂Σ × C, φ * (w 2 (Q)))✳ ■♥ ♣❛rt✐❝✉•❛r✱ t❤❡ ❛♠❜✐❣✉✐t② ✐♥ ❞❡✜♥✐t✐♦♥ ❡①✐sts ✐❢ w 2 (Q) ✐s ♥♦♥✲③❡r♦ ✭s❡❝♦♥❞ ❙t✐❡❢❡•✲❲❤✐t♥❡② ❝•❛ss✮✳ •✐✈❡♥ t❤❡ ♥♦r♠❛• ❜✉♥❞•❡ N t♦ Q ✐♥ Y ♦♥❡ ❤❛s t❤❡ ❢♦••♦✇✐♥❣ (1 + w 1 (Y ) + w 2 (Y ) + ...) = (1 + w 1 (Q) + w 2 (Q) + ...)(1 + w 1 (Q) + w 2 (Q) + ...). ✭✻✳✶✽✮ ❙✐♥❝❡ Y ✐s s♣✐♥✱ w 1 (Y ) = w 2 (Y ) = 0 ❛♥❞ ❤❡♥❝❡ α = (∂Σ × C, φ * (w 2 (N ))) ❛s w 2 (Q) = w 2 (N ), ✭✻✳✶✾✮ ✇❤✐❝❤ ❤❛✈❡ ❛ ♠♦r❡ ♥❛t✉r❛• ❢♦r♠✉•❛t✐♦♥ ✐♥ ❑✲t❤❡♦r② ❬✺✽(✳ ◆♦✇ ❢♦r t❤❡ ♣❛t❤ ✐♥t❡❣r❛• t♦ ❜❡ ✇❡••✲❞❡✜♥❡❞ t❤❡ s❡❝♦♥❞ ❢❛❝t♦r s❤♦✉•❞ ❤❛✈❡ t❤❡ ❡①❛❝t s❛♠❡ ❛♠❜✐❣✉✐t②✱ s♦ ❛s t♦ ❝❛♥❝❡• t❤❛t ♦❢ t❤❡ ✜rst✳ ❚♦ ✉♥❞❡rst❛♥❞ ✐ts ❣❡♦♠❡tr✐❝ ♦r✐❣✐♥✱ •❡t ✉s ❝♦♥s✐❞❡r t❤❡ ▲❡✈✐✲❈✐✈✐t❛ ❝♦♥♥❡❝t✐♦♥ ω ♦♥ Q✳ ❚❤❡ str✉❝t✉r❡ ❣r♦✉♣ SO(n) ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❛ ❞♦✉❜•❡ ❝♦✈❡r ❙♣✐♥ (n)✳ ❚❤❡ tr❛❝❡ ♦❢ t❤❡ ❤♦•♦♥♦♠② ❝♦♥♥❡❝t✐♦♥
SO(n) t♦ Spin(n)✳ ❚❤❡♥ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ s❡❝♦♥❞ ❢❛❝t♦r ❛♥❞ t❤❡ ❛❜♦✈❡ ✐s ✇❡••✲ ❞❡✜♥❡❞ ✐❢ t❤❡ ✇♦r•❞s❤❡❡t ✐♥t❡❣r❛t✐♦♥ ♠❡❛s✉r❡ ✐s ✇❡••✲❞❡✜♥❡❞ ❛s ❛r❣✉❡❞ ❜② ❋r❡❡❞ ❛♥❞ ❲✐tt❡♥✳ ❚❤❡ ✐♠♣♦rt❛♥t ♣♦✐♥t ✐s t❤❛t ✐♥ t❤❡ tr❛❝❡ ♦❢ t❤❡ ❤♦•♦♥♦♠② ❛r♦✉♥❞ ∂Σ t❤❡ ♦♥•② s♣✐♥♦rs t❤❛t ❛♣♣❡❛r ❤❛✈❡ ❝❤❛r❣❡ 1 ✇✐t❤ r❡s♣❡❝t t♦ A✳ ❙✉❝❤ s♣✐♥♦rs ❛r❡ ❛❝t✉❛••② s❡❝t✐♦♥s ♦❢ S(Q) ⊗ L✱ ✇❤❡r❡ L ✐s t❤❡ •✐♥❡ ❜✉♥❞•❡ ✇❤♦s❡ ❝♦♥♥❡❝t✐♦♥ ✐s A ❛♥❞ S ✐s t❤❡ s♣✐♥ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ SO(n)✳ ❙✉❝❤ t❡♥s♦r ♣r♦❞✉❝t ✐s ❝❛••❡❞ t♦ ❜❡ ❛ ❙♣✐♥ c str✉❝t✉r❡ ♦❢ Q✳ ❚❤✐s s❤♦✇s ✇❤② t❤❡ ❣❡♦♠❡tr✐❝ ♥❛t✉r❡ ♦❢ t❤❡ ✏❣❛✉❣❡ ✜❡•❞✧ A s❤♦✉•❞ ❜❡ ✐♥❞❡❡❞ t❤❛t ♦❢ ❛ ❙♣✐♥ c ❝♦♥♥❡❝t✐♦♥✳ ❲❤❡♥ ✇❡ s❛② ✏❋r❡❡❞✲❲✐tt❡♥✧ ❛♥♦♠❛•② ✐♥ t❤❡ ❢♦••♦✇✐♥❣✱ ✇❡ ❞♦ ♥♦t ❛s❝r✐❜❡ t❤❡ s❛♠❡ ♠❡❛♥✐♥❣ t♦ ✐t ❛s ✐♥ t❤❡ •✐t❡r❛t✉r❡✳ ■♥st❡❛❞ ❝❛✈❛•✐❡r•② ✇❡ ♠❡❛♥ t❤✐s ♣❛rt✐❝✉•❛r s✐❣♥ ❛♠❜✐❣✉✐t②✱ t❤❡ ❝♦♥s✐❞❡r❛t✐♦♥ ♦❢ ✇❤✐❝❤ ❛s ✇❡ ❤❛✈❡ s❡❡♥✱ ✐s ❡①tr❡♠❡•② ❝r✉❝✐❛• ❢♦r t❤❡ ✇♦r•❞s❤❡❡t ♣❛t❤ ✐♥t❡❣r❛• t♦ ❜❡ ✇❡••✲❞❡✜♥❡❞✳ ■♥ t❤❡ ♥❡①t s❡❝t✐♦♥ ✇❡ s♣❡❝✉•❛t❡ ✐❢ t❤❡ ❤❛•❢✲✐♥t❡❣❡r t❡r♠s ✐♥ ♦✉r ♠♦❞✐✜❡❞ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ❝❛♥ ❜❡ ❣✐✈❡♥ ❛ s✐♠✐•❛r ❣❡♦♠❡tr✐❝ ✐♥t❡r♣r❡t❛t✐♦♥ t♦ t❤❛t ♦❢ t❤❡ s✐❣♥ ❛♠❜✐❣✉✐t② ❝♦♥s✐❞❡r❡❞ ❜② ❋r❡❡❞ ❛♥❞ ❲✐tt❡♥✳ ✼✽ ❙♦♠❡ r❡♠❛r❦s ♦♥ ♠♦❞✐✜❝❛t✐♦♥s ♦❢ t❤❡ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥✳ ✻✳✺ ✻✳✺✳✷ ❈♦♥♥❡❝t✐♦♥ ♦❢ t❤❡ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ✭✻✳✶✵✮ ✇✐t❤ ❋r❡❡❞✲❲✐tt❡♥ ❛♥♦♠❛•② ❖♥❡ ♦❢ t❤❡ ✐♠♣♦rt❛♥t ❝♦♥t❡♥ts ♦❢ t❤❡ ❣•♦❜❛• ✇♦r•❞s❤❡❡t ❛♥♦♠❛•✐❡s ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s✉❜s❡❝t✐♦♥ ✐s t❤❛t ✐t ♦r✐❣✐♥❛t❡s ❢♦r♠ ♠♦♥♦❞r♦♠② ❛r♦✉♥❞ ❛ ❝•♦s❡❞ •♦♦♣ ♦❢ t❤❡ ✏ U (
a → b a + ǫ a ✱ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ r❡•❛t✐♦♥✱ A even e -B = ζ 0 -ζ a ω a -ζa ω a -ζ0 ω Ŷ, ✭✻✳✷✶✮ ✇❤❡r❡ t❤❡ ||✲♣♦t❡♥t✐❛• ✐♥ A even ∈ H even ( Ŷ, R)✳ ❲❤✐•❡ ❞♦✐♥❣ t❤❛t t❤❡ ||✲♣♦t❡♥t✐❛• ❤❛s t♦ ❜❡ ❦❡♣t ✜①❡❞✳ ❆s ✇❡ ❤❛✈❡ ❛•r❡❛❞② s❡❡♥✱ ❢♦r t❤❡ ❝•❛ss✐❝❛• ❝❛s❡ ✇❤❡r❡❛s t❤❡ ◆❙✲❛①✐♦♥ r❡♠❛✐♥s ✐♥✈❛r✐❛♥t✱ ❢♦r t❤❡ q✉❛♥t✉♠ ❝❛s❡ ✐t tr❛♥s❢♦r♠❡❞ ❜② t❤❡ ❝❤❛r❛❝t❡r ♦❢ t❤❡ ♠♦♥♦❞r♦♠② ❣r♦✉♣ ❢♦r ✭✺✳✶✹✮✳ ❆s ❛♥ ❛tt❡♠♣t t♦ ❡①♣•❛✐♥ t❤❡ ❛♥♦♠❛•♦✉s t❡r♠s ✐♥ ✭✻✳✶✵✮✱ ♦♥❡ ❝❛♥ t❤✐♥❦ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ s✐♠♣•❡ r❡s♦•✉t✐♦♥✳ ■❢ ♦♥❡ ❞❡✜♥❡s t❤❡ ||✲♣♦t❡♥t✐❛•s ❛s A even + 1 8 c 2 ( Ŷ) + A e -B = ζ 0 -ζ a ω a -ζa ω a -ζ0 ω Ŷ, ✭✻✳✷✷✮ s❤✐❢t✐♥❣ t❤❡ B ✜❡•❞ ❛❝t✉❛••② ❣❡♥❡r❛t❡s t❤❡ ❛♥♦♠❛•♦✉s t❡r♠s ✐♥ ✭✻✳✶✵✮✱ ♣r♦✈✐❞❡❞ ♦♥❡ ❞♦❡s ♥♦t tr❛♥s❢♦r♠ t❤❡ ||✲♣♦t❡♥t✐❛• ❛♥❞ A ✐s ❛ t✇♦✲❢♦r♠ s✉❝❤ t❤❛t A ∧ B = A ab ω a b b ❛♥❞ A ∧ B ∧ B = A ab b a b b ω Ŷ✳ ❇✉t t❤❡♥ ✐t r❡q✉✐r❡s ✉s t♦ ❤❛✈❡ A ab = κ abc A c . ✭✻✳✷✸✮ ❲❤❡♥ s✉❝❤ ♠♦❞✐✜❡❞ ❞❡✜♥✐t✐♦♥ ❢♦r t❤❡ ||✲♣♦t❡♥t✐❛• ✐s ❝♦♥s✐❞❡r❡❞✱ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rt ♦❢ t❤❡ ❉✲✐♥st❛♥t♦♥ ❛❝t✐♦♥ ✐s ❛•s♦ ❛✛❡❝t❡❞✱ S γ = 8πe (φ+K)/2 X e -J γ ′ + 2πi X γ ′ ∧ A even + 1 8 c 2 ( X ) + A e -B . ✭✻✳✷✹✮ ❇✉t t❤❡ ♣r♦❜•❡♠ ✐s t❤❛t ✭✻✳✷✸✮ ❢❛✐•s t♦ ❤♦•❞✳ ❲❡ ❛❝t✉❛••② ❤❛✈❡ ❝❤❡❝❦❡❞ ✇✐t❤ ✈❛r✐♦✉s ❈❛•❛❜✐✲(❛✉ ❞❛t❛✳ ❙✉r♣r✐s✐♥❣•② ❢♦r ♠❛♥② ♦❢ t❤❡♠ ✭✻✳✷✸✮ ❛❝t✉❛••② ✇♦r❦s✳ ❍♦✇❡✈❡r ❢♦r ❛ ❢❡✇ ✐t ❢❛✐•s t♦ ❤♦•❞ ❛♥❞ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ♣r♦✈✐❞❡ ❛♥ ❡①❛♠♣•❡ ♦❢ s✉❝❤✳ ▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ ❈❛•❛❜✐✲(❛✉ ❡♠❜❡❞❞❡❞ ✐♥ t❤❡ ♣r♦❞✉❝t ♦❢ ♣r♦•❡❝t✐✈❡ s♣❛❝❡s P 2 × P 3 ✱ 3| 3 1 2| 2 1 . ✭✻✳✷✺✮ ✼✾ ✻✳ ❚❤❡ s②♠♠❡tr② ❣r♦✉♣ ♦❢ M HM ❚❤❡ ❛❜♦✈❡ ✐s t❤❡ s♦ ❝❛••❡❞ ❝♦♥✜❣✉r❛t✐♦♥ ♠❛tr✐① ❞❡✜♥❡❞ ❛s ❛ ♣❛✐r [n|q]✳ ■t s♣❡❝✐✜❡s t❤❡ ❡♠❜❡❞❞✐♥❣ s♣❛❝❡ ❛♥❞ ❞❡❣r❡❡s ♦❢ t❤❡ ❤♦♠♦❣❡♥❡✐t② ♦❢ t❤❡ ❞❡✜♥✐♥❣ ♣♦•②♥♦♠✐❛•s✳ ❋♦r t❤✐s ❡①❛♠♣•❡✱ ✇❡ ❤❛✈❡ f abcαβ x a x b x c y α y β = 0, g aα x a y α = 0. ✭✻✳✷✻✮ ❍♦✇❡✈❡r✱ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ♠❛tr✐① ❞♦❡s ♥♦t s♣❡❝✐❢② t❤❡ ❝♦❡✣❝✐❡♥ts f abcαβ ❛♥❞ g aα ♦t❤❡r✇✐s❡ ❬✻✵(✳ ❚❤❡♥ ❢r♦♠ t❤❡ ❛❞•✉♥❝t✐♦♥ ❢♦r♠✉•❛✱ ♦♥❡ ❝❛♥ ✜♥❞ t❤❛t c 2,1 = 44, c 2,2 = 36✳ ❖♥❡ ❝❛♥ ❛•s♦ ❝♦♠♣✉t❡ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♥✉♠❜❡rs✱ κ 111 = 2, κ 112 = 5, κ 122 = 3, κ 222 = 0✳ ❋✉rt❤❡r♠♦r❡✱ ♦♥❡ ❝❛♥ ❝❛•❝✉•❛t❡ t❤❡ ❊✉•❡r ❝❤❛r❛❝t❡r✐st✐❝ ❛♥❞ ❍♦❞❣❡ ♥✉♠❜❡rs ❢♦••♦✇✐♥❣ ❬✻✵(✳ ❚❤❡② ❝♦♠❡ ♦✉t t♦ ❜❡ -150 ❛♥❞ h 11 = 2, h 12 = 77 r❡s♣❡❝t✐✈❡•②✳ ❙✐♥❝❡ ✇❡ ❤❛✈❡ 1 2 κ aab ǫ b -κ abc ǫ b ǫ c = 0 mod Z✱ ♦♥❡ ✜♥❞s t❤❛t A ab = 1 2 κ aab mod Z, ✭✻✳✷✼✮ ✇❤❡r❡ ✇❡ ❤❛✈❡ ✉s❡❞ t❤❡ ❢❛❝t t❤❛t L a (ǫ) ✐s ✐♥t❡❣r❛• ❢r♦♠ ✭✺✳✻✮✳ ❚❤❡♥ ✇❡ ♦❜t❛✐♥ A 11 = 0, A 22 = 0, A 12 = 1 2 . ✭✻✳✷✽✮ ❚❤❡ ❛❜♦✈❡ ✈❛•✉❡s ♦❢ ❢♦r t❤❡ ❡•❡♠❡♥ts ♦❢ t❤❡ ♠❛tr✐① A ✐s ✐♥❝♦♥s✐st❡♥t ✇✐t❤ ✭✻✳✷✸✮✳ ❋r♦♠ ✭✻✳✷✸✮✱ ✇❡ ❤❛✈❡ A 11 = κ 111 A 1 + κ 112 A 2 = 2A 1 + 5A 2 = 0 mod Z, A 12 = κ 121 A 1 + κ 122 A 2 = 5A 1 + 3A 2 = 1 2 mod Z A 22 = κ 221 A 1 + κ 222 A 2 = 3A 1 = 0 mod Z. ✭✻✳✷✾✮ ❚❤❡♥ ✐t ♠❡❛♥s t❤❛t✱ ❢♦r s♦♠❡ ✐♥t❡❣❡r m✱ ✇❡ ❤❛✈❡ A 1 = m/3✳ ❍❡♥❝❡✱ ❢r♦♠ t❤❡ ✜rst ♦❢ t❤❡ ❡q✉❛t✐♦♥s✱ ❢♦r s♦♠❡ ✐♥t❡❣❡r n✱ ♦♥❡ ♦❜t❛✐♥s A 2 = 1 5 n -2m 3 ✳ P•✉❣❣✐♥❣ ✐♥t♦ t❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥✱ ✇❡ ❣❡t A 12 = 4m+9n 15 mod Z✱ ✇❤✐❝❤ ❝•❡❛r•② ✐s ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❚❤✉s✱ ✇❡ ❝♦♥❝•✉❞❡ t❤❛t t❤❡ ✇❛② ♦❢ ♠♦❞✐❢②✐♥❣ ✭✻✳✷✷✮ ✐s t♦♦ ♥❛✐✈❡✳ ❚❤❡ ♦t❤❡r ♥❛t✉r❛• ♣♦ss✐❜✐•✐t② t❤❛t ♦♥❡ ❝❛♥ ❝♦♥s✐❞❡r ✐s t❤❛t t❤❡ ||✲♣♦t❡♥t✐❛• tr❛♥s❢♦r♠s ✐ts❡•❢✱ ❛•t❤♦✉❣❤✱ t❤✐s ♣r♦♣♦s❛• ✐s st✐•• ❝♦♥•❡❝t✉r❛•✳ ■❢ t❤❛t ✐s t❤❡ ❝❛s❡ ✐♥✲ ❞❡❡❞✱ ♦♥❡ ❝❛♥ ✐♥t❡r♣r❡t t❤❡ ❛♥♦♠❛•♦✉s t❡r♠s ✐♥ ✭✻✳✶✵✮ ❤❛✈❡ s✐♠✐•❛r ♦r✐❣✐♥s t♦ t❤❛t ♦❢ ❋r❡❡❞✲❲✐tt❡♥ ❛♥♦♠❛•②✱ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ r❡❛s♦♥s✳ ❚❤❡r❡ ✐s ❛ str✐❦✐♥❣ s✐♠✐•❛r✐t② ❜❡t✇❡❡♥ t❤❡ ♦r✐❣✐♥ ♦❢ t❤❡ ❛♥♦♠❛•♦✉s t❡r♠s ❛❝q✉✐r❡❞ ❜② t❤❡ tr❛♥s❢♦r✲ ♠❛t✐♦♥ ♦❢ t❤❡ ||✲♣♦t❡♥t✐❛• A even ✉♥❞❡r ✐♥t❡❣❡r ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ❛♥❞ t❤❡ ❤❛•❢✲✐♥t❡❣❡r t❡r♠s t❤❛t ❛♣♣❡❛r❡❞ ❢r♦♠ t❤❡ U (1) ❣❛✉❣❡ ✜❡•❞ A ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❋r❡❡❞✲❲✐tt❡♥ ❛♥♦♠❛•②✳ ❙✐♥❝❡ ✐♥ ♦✉r ❝❛s❡ t❤❡ ||✲s❝❛•❛rs ❛♣♣❡❛r ❛s ♣❡r✐♦❞s ♦❢ t❤❡ ||✲♣♦t❡♥t✐❛•✱ ✐t ✐s r❡❛s♦♥❛❜•❡ t♦ ❡①♣❡❝t t❤❛t t❤❡ ❛♥♦♠❛•♦✉s t❡r♠s ✐♥ ✭✻✳✶✵✮ ❤❛✈❡ ❛ s✐♠✐•❛r ❣❡♦♠❡tr✐❝ ♦r✐❣✐♥✳ ❋✉rt❤❡r♠♦r❡✱ ♦♥❡ ❡①♣❡❝ts t❤❛t t❤❛t t❤❡s❡ ❤❛•❢✲ ✐♥t❡❣❡r t❡r♠s ❛♣♣❡❛r ❢r♦♠ t❤❡ s✉❜t•❡t✐❡s ♦❢ t❤❡ ♦♥❡✲•♦♦♣ ❞❡t❡r♠✐♥❛♥t ❛r♦✉♥❞ t❤❡ ❉✲✐♥st❛♥t♦♥ ❜❛❝❦❣r♦✉♥❞✱ ✐♥ ❛ s✐♠✐•❛r ✈❡✐♥ ✇✐t❤ ❬✺✽(✳ ✽✵ ❈❍❆P❚❊| ✼ ▼✐rr♦r s②♠♠❡tr② ✐♥ ❛❝t✐♦♥ ✿ ✐♥st❛♥t♦♥s ✐♥ t②♣❡ ■■❇ ❢♦r♠✉•❛t✐♦♥ ■♥ t❤✐s ❝❤❛♣t❡r ♦✉r ❣♦❛• ✐s t♦ ❞❡s❝r✐❜❡ M HM ✐♥❝•✉❞✐♥❣ ❛•• ♣♦ss✐❜•❡ ✐♥st❛♥t♦♥ ❝♦r✲ r❡❝t✐♦♥s✳ ■♥ ❝♦♥tr❛st t♦ t❤❡ s✐t✉❛t✐♦♥ ❛ ❢❡✇ ②❡❛rs ❛❣♦✱ ♥♦✇ ✇❡ ❤❛✈❡ ❛ ❜❡tt❡r ❤❛♥❞•❡ ♦♥ t❤❡ ✐♥st❛♥t♦♥ ❝♦rr❡❝t✐♦♥s✱ t❤❛♥❦s t♦ t❤❡ ✇♦r❦s ✐♥ ❬✶✽✕✷✹(✱ ✇❤✐❝❤ ❝r✉❝✐❛••② ❞❡✲ ♣❡♥❞❡❞ ♦♥ t❤❡ ❛❞✈❡♥t ♦❢ t❤❡ t✇✐st♦r s♣❛❝❡ ❢♦r♠❛•✐s♠ t♦ ❞❡s❝r✐❜❡ q✉❛t❡r♥✐♦♥ ❑ä❤•❡r ♠❛♥✐❢♦•❞s✱ ✇❤✐❝❤ ❞❡✈❡•♦♣❡❞ ✐♥ ❬✷✺(✳ ❚❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ ✇♦r❦✐♥❣ ❛t t❤❡ •❡✈❡• ♦❢ t❤❡ t✇✐st♦r s♣❛❝❡ ✇❛s ♠❡♥t✐♦♥❡❞ ✐♥ ❝❤❛♣t❡r ✸✳ ❚❤❡ ❤♦•♦♠♦r♣❤✐❝ str✉❝t✉r❡ ♦❢ t❤❡ t✇✐st♦r s♣❛❝❡ Z ❛♥❞ ✈❛r✐♦✉s s②♠♠❡tr✐❡s ♦❢ t❤❡ ◗❑ s♣❛❝❡ •✐❢t❡❞ t♦ t❤❡ t✇✐st♦r s♣❛❝❡✱ ❝❛♥ ❜❡ ❡①♣•♦✐t❡❞ t♦ ❝♦♠♣✉t❡ t❤❡s❡ ✐♥st❛♥t♦♥ ❝♦rr❡❝t✐♦♥s✳ ■♥ ❝❤❛♣t❡r ✹✱ ✇❡ ❤❛✈❡ ❛•r❡❛❞② s❡❡♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❉✲✐♥st❛♥t♦♥ ❝♦rr❡❝t❡❞ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ♦♥ t❤❡ t②♣❡ ■■❆ s✐❞❡ t♦ ❛•• ♦r❞❡rs ✐♥ ✐♥st❛♥t♦♥ ❡①♣❛♥s✐♦♥✳ ❚❤❡ ♣r♦❜•❡♠ ♦❢ ✐♥❝♦r♣♦r❛t✐♥❣ ◆❙✺✲✐♥st❛♥t♦♥ ❝♦rr❡❝t✐♦♥ ❝❛♥ ❜❡ ❛❞❞r❡ss❡❞ ♠♦st ❡❝♦✲ ♥♦♠✐❝❛••② ♦♥ t❤❡ t②♣❡ ■■❇ s✐❞❡ ❬✷✸(✳ ❙✐♥❝❡ ❉✺ ❛♥❞ ◆❙✺✲✐♥st❛♥t♦♥s tr❛♥s❢♦r♠ ❛s ❛ ❞♦✉❜•❡t ✉♥❞❡r SL(2, Z)✱ ♦♥❝❡ ✇❡ ❦♥♦✇ t❤❡ ❉✺✲✐♥st❛♥t♦♥s ✐♥ t②♣❡ ■■❇✱ ✇❡ ❝❛♥ ✐♥✈♦❦❡ t❤❡ ❙✲❞✉❛•✐t② ❝♦♥str❛✐♥t ✭✺✳✸✶✮✱ ♦r ❡q✉✐✈❛•❡♥t•② ✭✺✳✺✶✮ t♦ ✜♥❞ ✜✈❡❜r❛♥❡ ✐♥st❛♥t♦♥s✳ ❙t❛rt✐♥❣ ❢r♦♠ t❤❡ ❉✲✐♥st❛♥t♦♥s ♦♥ t❤❡ t②♣❡ ■■❆ s✐❞❡✱ ♦♥❡ •✉st ❛♣♣•✐❡s ♠✐rr♦r ♠❛♣ r❡•❛t✐♦♥s t♦ ✜♥❞ t❤❡ ❜♦✉♥❞ st❛t❡ ♦❢ ❉✺✲❉✸✲❉✶✲❉✭✲✶✮ ✐♥st❛♥t♦♥s ♦♥ t❤❡ t②♣❡ ■■❇ ❢♦r♠✉•❛t✐♦♥✳ ❚❤❡♥ ♦♥❡ ❣❡♥❡r❛t❡s ✜✈❡❜r❛♥❡ ✐♥st❛♥t♦♥s ❜② ❛♣♣•✐❝❛t✐♦♥ ♦❢ ❙✲❞✉❛•✐t② ❝♦♥str❛✐♥t✳ ❲❤❛t ✐s st✐•• ♠✐ss✐♥❣✱ ✐s t❤❡ ♠♦❞✉•❛r ✐♥✈❛r✐❛♥t ❝♦♥str✉❝t✐♦♥ ✇❤❡♥ ♦♥❡ ❝♦♥s✐❞❡rs t❤❡ ❜♦✉♥❞ st❛t❡ ♦❢ ❉✸✲❉✶✲❉✭✲✶✮ ✐♥st❛♥t♦♥s✳ ❆♥ ❛tt❡♠♣t t♦ ✜♥❞ ✐t ✇❛s ♠❛❞❡ ✐♥ ❬✷✷( ❛t t❤❡ •❛r❣❡ ✈♦•✉♠❡ •✐♠✐t ❛♥❞ ♦♥❡✲✐♥st❛♥t♦♥ ❛♣♣r♦①✐♠❛t✐♦♥✳ ❊✈❡♥ ❛t t❤❛t •❡✈❡•✱ t❤❡ ♣✐❝t✉r❡ ✇❛s ♥♦t s❛t✐s❢❛❝t♦r②✳ ❚❤❡ ♠❛✐♥ ♦❜st❛❝•❡ ✐♥ t❤✐s r❡❣❛r❞ st❡♠s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t ❉✸✲❜r❛♥❡ ✐♥st❛♥t♦♥s ❛r❡ ♠♦❞✉•❛r ✐♥✈❛r✐❛♥t t❤❡♠s❡•✈❡s✳ ❯♥•✐❦❡ t❤❡ ❝❛s❡ ❢♦r ✜✈❡❜r❛♥❡ ✐♥st❛♥t♦♥s ✇❤✐❝❤ ✇❡r❡ ❞❡r✐✈❡❞✱ ❢♦r ❉✸✲✐♥st❛♥t♦♥s t❤❡ ❝♦♥str✉❝t✐♦♥ s❤♦✉•❞ ❜❡ ❝♦♥s✐st❡♥t ✇✐t❤ ♠♦❞✉•❛r✐t②✳ ❚♦ ❡♥s✉r❡ t❤✐s ❝♦♥s✐st❡♥❝② ✐s ❛ r❛t❤❡r ♥♦♥✲tr✐✈✐❛• ♣r♦❜•❡♠✳ ❍♦✇❡✈❡r✱ t❤❡ ♠❛t❤❡♠❛t✐❝❛• str✉❝t✉r❡ ♦❢ ❉✸✲✐♥st❛♥t♦♥s ❢♦✉♥❞ ✐♥ ❬✷✷( ✐s ♥♦ •❡ss t❤❛♥ ❢❛s❝✐♥❛t✐♥❣✳ ■t ❞✐s❝❡r♥s ❛ r✐❝❤ ✐♥t❡r♣•❛② ❜❡t✇❡❡♥ ✽✶ ✼✳ ▼✐rr♦r s②♠♠❡tr② ✐♥ ❛❝t✐♦♥ ✿ ✐♥st❛♥t♦♥s ✐♥ t②♣❡ ■■❇ ❢♦r♠✉•❛t✐♦♥ ♠♦❝❦ ♠♦❞✉•❛r ❢✉♥❝t✐♦♥s ❛♥❞ ❝♦♥t❛❝t ❣❡♦♠❡tr②✱ ❛ ♣r✐♦r✐ t✇♦ ❝♦♠♣•❡t❡•② ❞✐✛❡r❡♥t ❜r❛♥❝❤❡s ♦❢ ♠❛t❤❡♠❛t✐❝s✱✳ ■t ✐s ♦♥❡ ♦❢ ♦✉r ❣♦❛•s t♦ ✐♥✈❡st✐❣❛t❡ t❤✐s ♣r♦❜•❡♠ ❢✉rt❤❡r t♦ ❤❛✈❡ t❤❡ ✜♥❛• ✇♦r❞ ❢♦r ❉✸✲✐♥st❛♥t♦♥s✳ ✼✳✶ ❉✭✲✶✮✲❉✶ ✐♥st❛♥t♦♥s ▲❡t ✉s ✜rst r❡❝❛•• t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❉✭✲✶✮✲❉✶ ✐♥st❛♥t♦♥s ❛•♦♥❣ t❤❡ •✐♥❡s ♦❢ ❬✷✵(✳ ❚❤❡② ❝♦rr❡s♣♦♥❞ t♦ ✐♥st❛♥t♦♥s ✇✐t❤ ✈❛♥✐s❤✐♥❣ ♠❛❣♥❡t✐❝ ❝❤❛r❣❡ p Λ ✳ ❚❤r♦✉❣❤ ♠✐rr♦r s②♠♠❡tr②✱ t❤❡② ❛r❡ r❡•❛t❡❞ t♦ t❤❡ ❆✲❉✷ ✐♥st❛♥t♦♥s ✐♥ t❤❡ t②♣❡ ■■❆ s✐❞❡✳ ❚❤❡ ♠❡tr✐❝ ✇❤❡♥ t❤❡s❡ ✐♥st❛♥t♦♥s ❛r❡ t❤❡ ♦♥•② ♦♥❡s ❝♦♥s✐❞❡r❡❞ ✇❛s ❣✐✈❡♥ ✐♥ ❝❤❛♣t❡r ✹ ❡q✉❛t✐♦♥ ✭✹✳✹✻✮ ✇❤❡r❡ t❤❡ ❡①♣r❡ss✐♦♥ ✐s ✇r✐tt❡♥ ✐♥ t②♣❡ ■■❆ ❝♦♦r❞✐♥❛t❡s✳ ■♥ t❤✐s ❝❛s❡ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ξ Λ ❛r❡ ❣•♦❜❛••② ❞❡✜♥❡❞ ❛♥❞ t❤❡ ✐♥t❡❣r❛•s ❛♣♣❡❛r✐♥❣ ✐♥ t❤❡ t✇✐st♦r •✐♥❡s ❢♦r t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ❝❛♥ ❜❡ ❡①♣•✐❝✐t•② ❡✈❛•✉❛t❡❞✳ ❋✉rt❤❡r♠♦r❡✱ s✐♥❝❡ t❤❡r❡ ✐s ♥♦ ξΛ ❛♥❞ α ❞❡♣❡♥❞❡♥❝❡ ✐♥ t❤✐s ❝❛s❡✱ t❤❡ ♥♦♥✲•✐♥❡❛r✐t✐❡s ✐♥ t❤❡ ❙✲ ❞✉❛•✐t② ❝♦♥str❛✐♥t ✭✺✳✸✶✮ ❞✐s❛♣♣❡❛r ❛♥❞ ✐t ❝♦✐♥❝✐❞❡s ✇✐t❤ ✭✺✳✺✶✮✱ s❛t✐s❢②✐♥❣ ❛ s✐♠♣•❡r ♠♦❞✉•❛r ❝♦♥str❛✐♥t G m,n → G m,n cξ 0 + d + reg. ✭✼✳✶✮ ❋✉rt❤❡r♠♦r❡✱ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ♦r✐❣✐♥❛t❡ ❢r♦♠ s✐♠♣•❡ ♠♦❞✉•❛r ❢♦r♠s✱ ❞✉❡ t♦ ✇❤✐❝❤ ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s✱ ❛s ✇❛s r❡♠❛r❦❡❞ ✐♥ ✸✳ ▲❡t ✉s ❜r✐❡✢② ❞❡s❝r✐❜❡ ❤♦✇ SL(2, Z) ✐♥✈❛r✐❛♥t ❝♦♥str✉❝t✐♦♥ ♦❢ ❉✭✲✶✮✲❉✶ ✐♥st❛♥✲ t♦♥s ✇❛s ♦❜t❛✐♥❡❞ st❛rt✐♥❣ ❢r♦♠ ❡•❡❝tr✐❝❛••② ❝❤❛r❣❡❞ ❉✷ ✐♥st❛♥t♦♥s ✐♥ t❤❡ t②♣❡ ■■❆ ❢♦r♠✉•❛t✐♦♥ ❛s ✇❛s ❞♦♥❡ ✐♥ ❬✷✵(✳ ❖♥❡ st❛rts ✇✐t❤ t❤❡ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ❉✲✐♥st❛♥t♦♥ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ ✭✹✳✶✵✮ ✇❤❡r❡ t❤❡ ❝❤❛r❣❡ •❛tt✐❝❡ ✐s r❡str✐❝t❡❞ t♦ p Λ = 0✳ ❚❤❡♥ ❛ P♦✐ss♦♥ r❡s✉♠♠❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ q 0 ✐s ♣❡r❢♦r♠❡❞✳ ❚❤❡r❡❜②✱ ♦♥❡ ✜♥❞s ❛ s✉♠ ♦✈❡r t✇♦ ✐♥t❡❣❡rs (m, n) ✇❤❡r❡ t❤❡ ✜rst ♦♥❡ ❛♣♣❡❛rs ❞✉❡ t♦ ❡①♣❛♥s✐♦♥ ♦❢ t❤❡ ❞✐•♦❣✲ ❛r✐t❤♠ ❝♦✉♥t✐♥❣ ♠✉•t✐✲❝♦✈❡r✐♥❣ ❡✛❡❝ts ❛♥❞ t❤❡ s❡❝♦♥❞ ♦♥❡ ❛♣♣❡❛rs ❛s ❛ r❡s✉•t ♦❢ r❡s✉♠♠❛t✐♦♥✱ ✐✳❡✳ q 0 ✐s tr❛❞❡❞ ❢♦r n✳ ❙✐♠✐•❛r s✐t✉❛t✐♦♥ ❛r✐s❡s ❢♦r t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ❛s ✇❡••✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✐t t✉r♥s ♦✉t t❤❛t ❢♦r ❡❛❝❤ ♦❢ t❤❡ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s✱ t❤❡r❡ ✐s ❛ t❡r♠ t❤❛t tr❛♥s❢♦r♠s ❛s ✭✺✳✷✷✮ ❛♥❞ ❛♥♦t❤❡r ❛♥♦♠❛•♦✉s ♣✐❡❝❡✳ ❍♦✇❡✈❡r✱ s✉❝❤ ❛♥♦♠❛•♦✉s t❡r♠s ❝❛♥ ❜❡ r❡♠♦✈❡❞ ✐❢ ♦♥❡ ♣❡r❢♦r♠s •♦❝❛• ❝♦♥t❛❝t tr❛♥s❢♦r♠❛t✐♦♥s ❛♣♣r♦♣r✐❛t❡•② ✐♥ t❤❡ ♣❛t❝❤❡s ♦❢ ❞❡✜♥✐t✐♦♥ ♦❢ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s✳ ❆s ❛ r❡s✉•t ♦♥❡ ✐♥❞❡❡❞ ✜♥❞s ❛♥ SL(2, Z) ✐♥✈❛r✐❛♥t ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ t✇✐st♦r s♣❛❝❡✳ ❚❤❡ r❡s✉•t✐♥❣ t✇✐st♦r s♣❛❝❡ ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❜② ✭✺✳✷✹✮ ✇❤❡r❡ t❤❡ ♣❛t❝❤❡s U m,n s✉rr♦✉♥❞ t❤❡ r♦♦ts t m,n + ♦❢ t❤❡ ❡q✉❛t✐♦♥ mξ 0 + n = 0✳ ❚❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ❛r❡ ❣✐✈❡♥ ❜② G D1 m,n (ξ) = - i (2π) 3 qa∈H + 2 ∪{0} n (0) qa        e -2πimqaξ a m 2 (mξ 0 + n) , m = 0, (ξ 0 ) 2 e 2πinqaξ a /ξ 0 n 3 , m = 0. ✭✼✳✷✮ ✽✷ ❙✉❜t•❡t✐❡s ♦❢ ◆❙✺✲❜r❛♥❡ ❝♦♥tr✐❜✉t✐♦♥s✳ ✼✳✷ ❍❡r❡ ✇❡ s❡t n (0) 0 = -χ Ŷ/2 ❛♥❞ ✉s❡❞ t❤❛t Ω(γ D1 ) = n (0) qa ❢♦r γ D1 = (0, 0, ±q a , q 0 ), {q a } = 0, Ω(γ D(-1) ) = 2n (0) 0 ❢♦r γ D(-1) = (0, 0, 0, q 0 ). ✭✼✳✸✮ ■t ✐s ✉s❡❢✉• t♦ ♥♦t❡ ❛•s♦ t❤❛t t❤❡ ❝♦♥tr✐❜✉t✐♦♥s t♦ ✭✼✳✷✮ ✇✐t❤ m = 0 ❛r❡ ♥♦t❤✐♥❣ ❜✉t t❤❡ α ′ ✲❝♦rr❡❝t❡❞ ♣❛rt ♦❢ t❤❡ ♣r❡♣♦t❡♥t✐❛•✱ n>0 G D1 0,n = F α ′ ✲•♦♦♣ + F ws ✳ ❙✐♥❝❡ U 0,n = U + ✱ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s t♦ t❤❡ ♣♦•❡s ❛r❡ ❞❡t❡r♠✐♥❡❞ ❜② ❛❞❞✐♥❣ t❤❡ r❡s♣❡❝t✐✈❡ ❝•❛ss✐❝❛• ♣✐❡❝❡s✳ ❚❤❡ r❡s✉•t✐♥❣ ❝♦✈❡r✐♥❣ ♦❢ t❤❡ t✇✐st♦r s♣❛❝❡ •♦♦❦s ❝♦♠♣•❡t❡•② ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ♠❡•♦♥✲s❤❛♣❡❞ ❝♦✈❡r✐♥❣ ♦♥ t❤❡ t②♣❡ ■■❆ s✐❞❡ ❢r♦♠ ✇❤✐❝❤ ✇❡ st❛rt❡❞✳ ❚❤✐s ❢❛❝t ❝❛♥ ❜❡ r❡❝♦♥❝✐•❡❞ ❜② r❡♠❡♠❜❡r✐♥❣ t❤❛t ✐♥ ♦r❞❡r t♦ ✜♥❞ ♠♦❞✉•❛r ✐♥✈❛r✐❛♥t ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ t✇✐st♦r s♣❛❝❡ ✇❡ ❤❛❞ t♦ ♣❡r❢♦r♠ ❝❡rt❛✐♥ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥s✳ ❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ❝❛♥ ❜❡ s✐♥❣✉•❛r ♦✉ts✐❞❡ t❤❡✐r ♣❛t❝❤ ♦❢ ❞❡✜♥✐t✐♦♥✳ ❆s ❛ r❡s✉•t✱ t❤❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❡❞ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s ❝❛♥ ❤❛✈❡ ❞✐✛❡r❡♥t s✐♥❣✉•❛r✐t② str✉❝t✉r❡ t❤❛♥ t❤❡ ♦♥❡s ✇❡ st❛rt❡❞ ✇✐t❤✳ ■♥ ❬✷✵( t❤r♦✉❣❤ ❛ ❝❛r❡❢✉• ❛♥❛•②s✐s✱ ✐t ✇❛s s❤♦✇♥ t❤❛t t❤❡ ❝♦♥str✉❝t✐♦♥s ✐♥ t❤❡ t②♣❡ ■■❆ ❛♥❞ t②♣❡ ■■❇ ❢♦r♠✉•❛t✐♦♥s ❛r❡ ✐♥❞❡❡❞ t❤❡ s❛♠❡✳ ❍❛✈✐♥❣ ❢♦✉♥❞ t❤❡ SL(2, Z) ✐♥✈❛r✐❛♥t ❞❡s❝r✐♣t✐♦♥ ❢♦r ❉✭✶✮✲❉✶ ✐♥st❛♥t♦♥s✱ ✇❡ ✜rst ♣r♦❝❡❡❞ t♦ ♦❜t❛✐♥ t❤❡ ✜✈❡❜r❛♥❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s✱ ❛s ✇❛s ❞♦♥❡ ✐♥ ❬✷✶✱ ✷✹(✳ ❲❡ ❛r❡ ❣♦✐♥❣ t♦ ❛ss✉♠❡ t❤❛t t❤❡r❡ ✐s ♥♦ ❜♦✉♥❞ st❛t❡ ♦❢ t❤❡ ❢♦r♠ ❉✸✲❉✶✲❉✭✲✶✮ ♣r❡s❡♥t✱ ✇❤✐❝❤ ❝❛♥ ❜❡ r❡st❛t❡❞ ❛s s❡tt✐♥❣ Ω(γ) = 0 ❢♦r t❤❡ ❝❤❛r❣❡s γ = (0, p a , q a , q 0 ), p a = 0✳ ❲❡ ✇✐•• ✜♥✐s❤ t❤✐s ❝❤❛♣t❡r ❜② s❤♦✇✐♥❣ t❤❛t t❤❡ ♠♦❞✉•✐ s♣❛❝❡ M HM ✐♥ ♣r❡s❡♥❝❡ ♦❢ ✜✈❡❜r❛♥❡ ✐♥st❛♥t♦♥s ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ ❞✐s❝r❡t❡ ✐s♦♠❡tr✐❡s ❞✐s❝✉ss❡❞ ✐♥ ❝❤❛♣t❡r ✻✳ ✼✳✷ ❙✉❜t•❡t✐❡s ♦❢ ◆❙✺✲❜r❛♥❡ ❝♦♥tr✐❜✉t✐♦♥s ❇❡❢♦r❡ ❣✐✈✐♥❣ t❤❡ ♣r♦❝❡❞✉r❡ ♦❢ ✜♥❞✐♥❣ t❤❡ ✜✈❡❜r❛♥❡ ❝♦♥tr✐❜✉t✐♦♥s✱ •❡t ✐s ♠❛❦❡ s♦♠❡ r❡♠❛r❦s ♦♥ ✇❤② ◆❙✺✲❜r❛♥❡ ❝♦♥tr✐❜✉t✐♦♥s ❛r❡ r❛t❤❡r ❞✐✣❝✉•t t♦ ✜♥❞✳ ❊✈❡♥ t❤♦✉❣❤ ❛ r❡s✉•t t♦ ❛•• ♦r❞❡rs ✐♥ t❤❡ ✐♥st❛♥t♦♥ ❡①♣❛♥s✐♦♥ ✇❛s ❢♦✉♥❞ ✐♥ ❬✷✸✱ ✷✹(✱ t❤❡r❡ ❛r❡ s❡✈❡r❛• ✐ss✉❡s t❤❛t r❡q✉✐r❡ ❜❡tt❡r ✉♥❞❡rst❛♥❞✐♥❣✳ ❼ ❋✐rst ♦❢ ❛•• t❤❡ ❞②♥❛♠✐❝s ♦❢ ◆❙✺✲❜r❛♥❡s ✐s ♥♦t ❛❝t✉❛••② ✇❡•• ✉♥❞❡rst♦♦❞✳ ❚❤❡ ✇♦r•❞✈♦•✉♠❡ t❤❡♦r②✱ ✐♥ ♣❛rt✐❝✉•❛r ❢♦r t❤❡ ❝❛s❡ ♦❢ ♠✉•t✐♣•❡ ✜✈❡❜r❛♥❡s ✐s ♥♦t ❦♥♦✇♥ t♦ t❤❡ ❢✉•• ❡①t❡♥t✳ ▲❡t ✉s ❝♦♥s✐❞❡r t②♣❡ ■■❆ ❢♦r♠✉•❛t✐♦♥ ❝♦♠♣❛❝t✐✜❡❞ ♦♥ ❛ ❈❛•❛❜✐✲(❛✉ Y✱ ✇✐t❤ ❛ st❛❝❦ ♦❢ k ❊✉❝•✐❞❡❛♥ ◆❙✺✲❜r❛♥❡s ✇❤♦s❡ ✇♦r•❞✈♦•✉♠❡ W ✐s Y ✐ts❡•❢✳ ❋♦r k = 1 t❤❡ ✇♦r•❞✈♦•✉♠❡ s✉♣♣♦rts ✜✈❡ s❝❛•❛r ✜❡•❞s ✭❞❡s❝r✐❜✐♥❣ tr❛♥s✈❡rs❡ ✢✉❝t✉❛t✐♦♥s ♦❢ W ✐♥ R 10 ×S 1 ✱ ✇❤❡r❡ S 1 ✐s t❤❡ ▼✲t❤❡♦r② ❝✐r❝•❡✮✱ t✇♦ s②♠♣•❡❝t✐❝ ▼❛•♦r❛♥❛✲ ❲❡②• ❢❡r♠✐♦♥s ❛♥❞ t❤❡ t✇♦✲❢♦r♠ ♣♦t❡♥t✐❛• B✱ ✇❤♦s❡ ✜❡•❞ str❡♥❣t❤ H ✐s s❡•❢✲ ❞✉❛•✱ * W H = iH. ✭✼✳✹✮ ✽✸ ❚❤❡ ✇♦r•❞✈♦•✉♠❡ W ❜❡❤❛✈❡s ❛s ❛ ♠❛❣♥❡t✐❝ s♦✉r❝❡ ❢♦r t❤❡ t✇♦ ❢♦r♠ ♣♦t❡♥t✐❛• B ✇❤✐❝❤ ♣r♦♣❛❣❛t❡s ✐♥ t❤❡ ✶✵ ❞✐♠❡♥s✐♦♥❛• ❜✉•❦ ♦❢ s♣❛❝❡t✐♠❡✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t❤❡ ✜❡•❞ str❡♥❣t❤ H ❢♦r t❤❡ ◆❙ t✇♦ ❢♦r♠ ✜❡•❞ ❇ ❛❝ts ❛ s♦✉r❝❡ ♦❢ t❤❡ || ✸✲❢♦r♠ ♣♦t❡♥t✐❛• Â3 ✳ ❚❤✐s ✢✉① ❜❡❤❛✈❡s ❛s ✐❢ ❉✲❜r❛♥❡s ❛r❡ ❜♦✉♥❞ t♦ ❛ s✐♥❣•❡ ◆❙✺✲❜r❛♥❡✳ ❚❤❡ ♣r♦❜•❡♠s ❛r❡ t✇♦ ❢♦•❞✳ ❋✐rst✱ ❞✉❡ t♦ t❤❡ s❡•❢✲❞✉❛•✐t② ❝♦♥❞✐t✐♦♥ ✭✼✳✹✮ t❤❡ ✸✲❢♦r♠ ✢✉① ❝❛♥ ♥♦t ❜❡ ❞❡✜♥❡❞ ♣r♦♣❡r•② ❢♦r t❤❡ ❜♦✉♥❞ st❛t❡ ♦❢ ❉✲❜r❛♥❡s ❛♥❞ ◆❙✺✲❜r❛♥❡s ✐❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♣r♦❞✉❝t ♦❢ ✸✲❝②❝•❡s s✉♣♣rt✐♥❣ ❉✲❜r❛♥❡s ✐s ♥♦♥✲ ✈❛♥✐s❤✐♥❣✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥ ❞❡s❝r✐❜✐♥❣ t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ ◆❙✺✲❜r❛♥❡ s❤♦✉•❞ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❛ s❡❝t✐♦♥ ♦❢ s♦♠❡ ❜✉♥❞•❡✳ ❖♥❡ ❤❛s t♦ ❜❡ r❛t❤❡r ❝❛r❡❢✉• ✇❤✐•❡ ❝♦♥s✐❞❡r✐♥❣ t❤❡ s✐t✉❛t✐♦♥ k > 1✳ ❆ ✇❛② t♦ ❞♦ t❤❛t ✐s ❜② ❢♦••♦✇✐♥❣ ❬✻✼(✱ ❛•t❤♦✉❣❤ ❛ ❞❡❡♣❡r ✉♥❞❡rst❛♥❞✐♥❣ ✐s r❡q✉✐r❡❞ ❆❝❝♦r❞✐♥❣ t♦ ✐t✱ k✲✜✈❡❜r❛♥❡s s♣•✐t ✐♥t♦ t✇✐st❡❞ s❡❝t♦rs •❛❜❡•❡❞ ❜② t❤❡ ♣❛✐rs ♦❢ ✐♥t❡❣❡rs (p, q) s✉❝❤ t❤❛t pq = k✳ ■t ✐s ✐♥t❡r♣r❡t❡❞ ❛s k ✜✈❡❜r❛♥❡s ✇r❛♣♣✐♥❣ Y r❡❝♦♠❜✐♥❡ ✐♥t♦ q ✜✈❡❜r❛♥❡s ✇r❛♣♣✐♥❣ p ❝♦♣✐❡s ♦❢ Y✳ ❚❤❡ ❞②♥❛♠✐❝s ♦❢ ❛ (p, q) t✇✐st❡❞ s❡❝t♦r ✐s ❞❡s❝r✐❜❡❞ ❜② q 3 ✐♥t❡r❛❝t✐♥❣ s❡•❢✲❞✉❛• t✇♦ ❢♦r♠s B ijk ✳ q ❛♠♦♥❣ t❤❡♠ B iii st❛② ♠❛ss•❡ss ❛♥❞ t❤❡ ❛❜❡•✐❛♥ t✇♦ ❢♦r♠ B = q i=1 B iii ❞❡s❝r✐❜❡s ❝❡♥t❡r ♦❢ ♠❛ss ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✳ ❼ ❚❤❡ ◆❙✺✲✐♥st❛♥t♦♥s ✐♠♣❛rt tr✉❡ str✐♥❣② ❝♦rr❡❝t✐♦♥s t♦ t❤❡ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ M HM ✳ ■♥ ❛❜s❡♥❝❡ ♦❢ α ❞❡♣❡♥❞❡♥❝❡ ✐♥ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s✱ ♦♥❡ ❝❛♥ ❡①♣•♦✐t t❤❡ ◗❑✴❍❑ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❬✸✾(✳ ■t ✐s ❛ ♣r♦❝❡❞✉r❡ t♦ ♠❛♣ t❤❡ ◗❑ s♣❛❝❡ ❤❛✈✐♥❣ ❛♥ ✐s♦♠❡tr② t♦ ❛♥ ❍❑ s♣❛❝❡ ✇✐t❤ ❛ ❤②♣❡r❤♦•♦♠♦r♣❤✐❝ •✐♥❡ ❜✉♥❞•❡ ✭✇❤♦s❡ ❝✉r✈❛t✉r❡ ✐s ❛ (1, 1)✲❢♦r♠ ✇✐t❤ r❡s♣❡❝t t♦ ❛•• ♦❢ t❤r❡❡ ❝♦♠♣•❡① str✉❝t✉r❡s✮ ❛♥❞ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ♦♥ t✇✐st♦r s♣❛❝❡ ♦❢ t❤❡ ❍❑ s♣❛❝❡ ❞♦❡s ♥♦t ❤❛✈❡ ❡①♣•✐❝✐t t ❞❡♣❡♥❞❡♥❝❡✳ ❆•t❤♦✉❣❤✱ ✇❤✐•❡ ❣✐✈✐♥❣ t❤❡ t✇✐st♦r ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❉✲✐♥st❛♥t♦♥s ✐♥ ❝❤❛♣t❡r ✹✱ ✇❡ ❞✐❞ ♥♦t r❡s♦rt t♦ ✐t✱ ✐t ✇❛s q✉✐t❡ ❝•❡❛r t❤❛t ✇❡ r❡str✐❝t❡❞ t♦ t❤❡ s②♠♣•❡❝t✐❝ s✉❜s♣❛❝❡ ♦❢ (ξ, ξ) ♦♥ t❤❡ t✇✐st♦r s♣❛❝❡✳ ❲❤❡♥ ◆❙✺✲✐♥st❛♥t♦♥s ❛r❡ ♣r❡s❡♥t ❛•• ❝♦♥t✐♥✉♦✉s ✐s♦♠❡tr✐❡s ❛r❡ ❜r♦❦❡♥ ❛s ✇❡ ❤❛✈❡ s❡❡♥ ✐♥ ❝❤❛♣t❡r ✷ ❛♥❞ ♦♥❡ ❤❛s t♦ ✇♦r❦ ✉s✐♥❣ t❤❡ ❢✉••✲✢❡❞❣❡❞ ❝♦♥t❛❝t ❣❡♦♠❡tr②✳ ✼✳✸ ❋✐✈❡❜r❛♥❡ ❝♦rr❡❝t✐♦♥s ❇❡❢♦r❡ ♣r♦❝❡❡❞✐♥❣✱ •❡t ✉s s✉♠♠❛r✐③❡ t❤❡ str❛t❡❣② ✇❡ ❛r❡ ❣♦✐♥❣ t♦ t❛❦❡ t♦ ❞❡r✐✈❡ ✜✈❡❜r❛♥❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ♦♥ t❤❡ t②♣❡ ■■❇ s✐❞❡✳ ❲❡ ✇✐•• ✐♥❝•✉❞❡ ❉✭✲✶✮✲❉✶ ✐♥✲ st❛♥t♦♥s ❢r♦♠ t❤❡ ✈❡r② ❜❡❣✐♥♥✐♥❣✳ ❚❤❡ ❇P❙ r❛②s ❢♦r ❡•❡❝tr✐❝ ❉✲✐♥st❛♥t♦♥s ✇✐t❤ ❝❤❛r❣❡ γ = (0, 0, 0, q0 ) ❛r❡ ❛•✐❣♥❡❞ ❛•♦♥❣ t❤❡ ✐♠❛❣✐♥❛r② ❛①✐s ❛♥❞ ❇P❙ r❛②s ❝♦rr❡✲ s♣♦♥❞✐♥❣ t♦ ❉✲✐♥st❛♥t♦♥s ✇✐t❤ ❝❤❛r❣❡ ✈❡❝t♦r γ = (0, 0, qa , q0 ) ❛r❡ ❝♦•♦r❡❞ ❜② ❣r❡❡♥ ✐♥ ✜❣✉r❡ ✼✳✶✳ ■t ✇❛s s❤♦✇♥ ✐♥ ❬✷✵( ❤♦✇ t♦ r♦t❛t❡ t❤❡s❡ ❇P❙ r❛②s ❜② ✜rst ♣❡r❢♦r♠✐♥❣ ❛ P♦✐ss♦♥ r❡s✉♠♠❛t✐♦♥ ♦♥ t❤❡ q0 ❛♥❞ t❤❡♥ ❞♦✐♥❣ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ❛s s❤♦✇♥ ✐♥ ✜❣✉r❡ ✼✳✶✳ ■t ✇❛s ❢♦✉♥❞ t❤❛t✱ st❛rt✐♥❣ ❢r♦♠ t❤❡ ❆✲❉✷ ✐♥st❛♥t♦♥s ♦♥ t❤❡ t②♣❡ ✽✹ ❋✐✈❡❜r❛♥❡ ❝♦rr❡❝t✐♦♥s✳ ✼✳✸ ❋✐❣✉r❡ ✼✳✶✿ ❊①❛♠♣•❡ ♦❢ t❤❡ ❇P❙ r❛②s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❉✺ ✭r❡❞✮✱ ❉✶ ✭❣r❡❡♥✮ ❛♥❞ ❉✭✲✶✮ ✭❜r♦✇♥✮ ❜r❛♥❡s ❛♥❞ t❤❡ ❡✛❡❝t ♦❢ t❤❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ✇❤✐❝❤ r♦t❛t❡s t❤❡ t✇♦ •❛tt❡r t②♣❡s ♦❢ r❛②s t♦ t❤❡ r❡❛• •✐♥❡✳ U γ ❞❡♥♦t❡s t❤❡ ♣❛t❝❤ •②✐♥❣ ✐♥ t❤❡ ❝♦✉♥t❡r❝•♦❝❦✇✐s❡ ❞✐r❡❝t✐♦♥ ❢r♦♠ t❤❡ ❇P❙ r❛② ℓ γ ✳ ■■❆ s✐❞❡✱ ♦♥❡ ✜♥❞s ❛ ♠♦❞✉•❛r ✐♥✈❛r✐❛♥t ❝♦♥str✉❝t✐♦♥ ❢♦r ❉✭✲✶✮✲❉✶ ✐♥st❛♥t♦♥s ♦♥ t❤❡ t②♣❡ ■■❇ s✐❞❡✳ ❍♦✇❡✈❡r✱ t❤❡ ❉✺✲✐♥st❛♥t♦♥ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ✭❛❝r♦ss t❤❡ r❡❞ ❇P❙ r❛②s ✐♥ ✜❣✉r❡ ✼✳✶✮ ❞♦ ♥♦t r❡♠❛✐♥ ✉♥❛✛❡❝t❡❞ ❞✉❡ t♦ t❤✐s ♣r♦❝❡❞✉r❡✳ ❚❤❡② ❜❡✲ ❝♦♠❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❡❞ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s ❛r❡ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✼✳✷✳ ❲❡ ❛r❡ ❛•♠♦st ❞♦♥❡ ♥♦✇✳ ❚❤❡ ❛♣♣•✐❝❛t✐♦♥ ♦❢ ❙✲❞✉❛•✐t② ❝♦♥str❛✐♥t ✭✺✳✺✶✮ s❤♦✉•❞ ❣✐✈❡ ✉s ✜✈❡❜r❛♥❡ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥ ✐♥ ♣r❡s❡♥❝❡ ♦❢ ❉✭✲✶✮✲❉✶ ✐♥st❛♥t♦♥s✳ ■♥ t❤❡ ❢♦••♦✇✐♥❣✱ ✐♥ ❛ st❡♣ ❜② st❡♣ ❢❛s❤✐♦♥✱ ✇❡ s❤♦✇ ❤♦✇ t♦ ✐♠♣•❡♠❡♥t t❤✐s str❛t❡❣② ❝♦♥❝r❡t❡•②✳ ✼✳✸✳✶ •❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ❞✉❡ t♦ ❉✭✲✶✮✲❉✶ ✐♥st❛♥t♦♥s ❚♦ ❞✐s♣•❛② t❤❡ ❡✛❡❝t ♦❢ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥✱ ✇❡ ♥❡❡❞ t♦ ✜rst ✐♥tr♦❞✉❝❡ s♦♠❡ ❝♦♥✈❡♥t✐♦♥s✳ ❲❡ ❞❡✜♥❡ ❛♥ ♦r❞❡r✐♥❣ ♦❢ t❤❡ ❝❤❛r❣❡s ❛s γ > γ ′ ✐✛ 0 < arg(Z γ Z -1 γ ′ ) < π✳ ❚❤❡♥ ❢♦r ❡❛❝❤ ❝❤❛r❣❡ γ ✇❡ ❞❡✜♥❡ ❛♥ ❛ss♦❝✐❛t❡❞ s❡t ♦❢ ❉✭✲✶✮✲❜r❛♥❡ ❝❤❛r❣❡s ✇❤♦s❡ ❇P❙ r❛②s •✐❡ ✐♥ t❤❡ s❛♠❡ ❤❛•❢✲♣•❛♥❡ ❛s ℓ γ Γ (-1) γ = {γ = (0, 0, 0, q0 ) : q0 Re Z γ > 0} , ✭✼✳✺✮ ❛♥❞ ❛♥♦t❤❡r s❡t ♦❢ ❉✶✲❜r❛♥❡ ❝❤❛r❣❡s ❢♦r ✇❤✐❝❤ t❤❡ ❇P❙ r❛②s ❛r❡ ❜❡t✇❡❡♥ ℓ γ ❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ❛①✐s Γ (1) γ = γ = (0, 0, qa , q0 ) ∈ H + 2 ∪ H - 2 : N (γ) = N (γ) and γ > γ ❢♦r N (γ) odd γ ≤ γ ❢♦r N (γ) even , ✭✼✳✻✮ ✇❤❡r❡ H + 2 ✐s t❤❡ s❡t ♦❢ ❝❤❛r❣❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❡✛❡❝t✐✈❡ ❤♦♠♦•♦❣② ❝•❛ss❡s ♦♥ Ŷ✱ H - 2 ✐s t❤❡ s❡t ♦❢ ♦♣♣♦s✐t❡ ❝❤❛r❣❡s✱ ❛♥❞ N (γ) ❞❡♥♦t❡s t❤❡ q✉❛❞r❛♥t ✇❤✐❝❤ ℓ γ ✽✺ ✼✳ ▼✐rr♦r s②♠♠❡tr② ✐♥ ❛❝t✐♦♥ ✿ ✐♥st❛♥t♦♥s ✐♥ t②♣❡ ■■❇ ❢♦r♠✉•❛t✐♦♥ ❜❡•♦♥❣s t♦✳ ✶ ◆♦t❡ t❤❛t ❜♦t❤ t❤❡ ♦r❞❡r✐♥❣ ❛♥❞ t❤❡ t✇♦ ❝❤❛r❣❡ s❡ts Γ (±1) γ ♠❛② ❝❤❛♥❣❡ ❛❢t❡r ❝r♦ss✐♥❣ ❛ ✇❛•• ♦❢ ♠❛r❣✐♥❛• st❛❜✐•✐t②✳ •✐✈❡♥ t❤✐s✱ ✇❡ t❛❦❡ t❤❡ ❇P❙ r❛②s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ❣❡♥❡r❛t✐♥❣ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ ♦♥ U γ t♦ •✐❡ ♦♥ t❤❡ ❛♥t✐✲❝•♦❝❦✇✐s❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❢r♦♠ t❤❡ ❇P❙ r❛② ℓ γ ✭s❡❡ t❤❡ ✜❣✉r❡ ✼✳✶✮✳ ❚❤❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥ ✐s t❤❡♥ ❣✐✈❡♥ ❜② g [γ] = (-1) N (γ)    1 2 γ∈Γ (-1) γ h [γ] + γ∈Γ (1) γ h [γ]    . ✭✼✳✼✮ ❚❤✐s ❤❛s ❛ ✈❡r② s✐♠♣•❡ ♠❡❛♥✐♥❣✳ ❚❤❡ ❉✶✲✐♥st❛♥t♦♥ ❇P❙ r❛②s ❛♥❞ ❝♦rr❡s♣♦♥❞✲ ✐♥❣•② t❤❡ ❞✐s❝♦♥t✐♥✉✐t✐❡s ❛r❡ r♦t❛t❡❞ t♦ ❡✐t❤❡r ♦❢ t❤❡ ♣♦s✐t✐✈❡ ♦r ♥❡❣❛t✐✈❡ r❡❛• ❛①✐s✱ ✇❤✐❝❤❡✈❡r ✐s t❤❡ ❝•♦s❡st✳ ❚❤❡ ❉✭✲✶✮✲✐♥st❛♥t♦♥s ❜❡❢♦r❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ✇❡r❡ ❛•✐❣♥❡❞ ❛•♦♥❣ t❤❡ ✐♠❛❣✐♥❛r② ❛①✐s✳ ❆❢t❡r t❤❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥✱ t❤❡② ❛r❡ s♣•✐t ✐♥t♦ t✇♦ ❤❛•✈❡s ❛♥❞ ❡❛❝❤ ♦❢ t❤❡♠ ❛r❡ ❛•s♦ r♦t❛t❡❞ t♦ t❤❡ r❡❛• ❛①❡s✳ ❆s ❛ r❡s✉•t t❤❡ ❝♦♥t♦✉rs ♦❢ ❉✭✲✶✮ ❛♥❞ ❉✶ ✐♥st❛♥t♦♥s ❛r❡ ♥♦✇ ❛•♦♥❣ t❤❡ ♣♦s✐t✐✈❡ ♦r ♥❡❣❛t✐✈❡ r❡❛• ❛①❡s ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s ❝❛♥ ❜❡ s✉♠♠❡❞ ✉♣ ❛s t❤❡② ❞❡♣❡♥❞ ♦♥•② ♦♥ ξ Λ ✳ P♦✐ss♦♥ r❡s✉♠♠✐♥❣ ♦✈❡r q0 •❡❛❞s t♦ t❤❡ ❛•t❡r♥❛t✐✈❡ t✇✐st♦r ❞❡s❝r✐♣t✐♦♥✱ ❝♦♠♣•②✐♥❣ ✇✐t❤ t❤❡ ❙✲❞✉❛•✐t② ❝♦♥str❛✐♥ts✳ ■♥st❡❛❞ ♦❢ ❇P❙ r❛②s ♥♦✇ ✇❡ ❝♦♥s✐❞❡r t❤❡ ✉s✉❛• ❙✲❞✉❛•✐t② ✐♥✈❛r✐❛♥t ❝♦♥t♦✉rs ❢♦r ❉✭✲✶✮✲❉✶ ❝❛s❡✱ ✇❤✐❝❤ ❛r❡ ❝✐r❝•❡s s✉rr♦✉♥❞✐♥❣ t❤❡ ♣♦✐♥ts t m,n ± ✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s ❝♦✐♥❝✐❞❡ ✇✐t❤ ✭✼✳✷✮ ❛s h D1 m,n = G m,n ✱✳ ✼✳✸✳✷ ❉✺✲✐♥st❛♥t♦♥s ❛❢t❡r ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ❚❤❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ✐♠♣❛rts ♥♦♥✲tr✐✈✐❛• ❡✛❡❝ts ♦♥ t❤❡ ❉✺✲✐♥st❛♥t♦♥ tr❛♥s✐✲ t✐♦♥ ❢✉♥❝t✐♦♥s✳ ❚❤✐s ✐s ❞✉❡ t♦ t❤❡ ❢❛❝t t❤❛t ✇❤✐•❡ r♦t❛t✐♥❣ t❤❡ ❇P❙ r❛②s ❢♦r ❉✭✲✶✮ ❛♥❞ ❉✶✲✐♥st❛♥t♦♥s✱ ♦♥❡ ♥❡❝❡ss❛r✐•② ❝r♦ss❡s t❤❡ ❇P❙ r❛②s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❉✸ ♦r ❉✺ ✐♥st❛♥t♦♥s ❢♦r ❣❡♥❡r✐❝ ❝❛s❡s ✇❤❡r❡ t❤❡ ❝❤❛r❣❡s ❛r❡ ♠✉t✉❛••② ♥♦♥✲•♦❝❛•✳ ❍♦✇❡✈❡r✱ s✐♥❝❡ t❤❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s ❛r❡ ❞❡♣❡♥❞❡♥t ♦❢ ξ ♦♥•②✱ t❤❡ r❡s✉•t✐♥❣ s❤✐❢t ✐♥ t❤❡ ❉✺ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s ❛r❡ •✐♥❡❛r s❤✐❢ts ✭✸✳✷✾✮✱ h [γ] g (ξ, ξ) = H γ (Ξ (g) γ ), Ξ (g) γ = Ξ γ + p Λ ∂ ξ Λ g [γ] (ξ). ✭✼✳✽✮ ❖♥❡ ❝❛♥ ❛•s♦ ❝♦♠♣✉t❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ✇❤✐❝❤ t❛❦❡ t❤❡ ❢♦•✲ •♦✇✐♥❣ ❢♦r♠✱ H [γ] g = h [γ] g + 2π 2 q Λ p Λ (h [γ] g ) 2 - (-1) N (γ) 4π 2 γ∈Γ (-1) γ ∪Γ (1) γ nq e -2πiq Λ ξ Λ E 4π 2 qΛ p Λ h [γ] g , ✭✼✳✾✮ ✶ ❖♥❡ ❝❛♥ ✇r✐t❡ N (γ) = 2 π arg (iZ γ ) ✳ ✽✻ ❋✐✈❡❜r❛♥❡ ❝♦rr❡❝t✐♦♥s✳ ✼✳✸ ❋✐❣✉r❡ ✼✳✷✿ ❙❝❤❡♠❛t✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ t✇✐st♦r ❞❛t❛ ❣❡♥❡r❛t✐♥❣ ❉✭✲✶✮✲❉✶ ❛♥❞ ❉✺✲✐♥st❛♥t♦♥s ✐♥ t❤❡ t②♣❡ ■■❇ ♣✐❝t✉r❡ ✇❤❡r❡ E(x) = 1 -(1 + x) e -x . ✭✼✳✶✵✮ ❚❤❡ t✇✐st♦r s♣❛❝❡ t❛❦❡s t❤❡ ❢♦r♠ ♦❢ ✜❣✉r❡ ✼✳✷✳ ✼✳✸✳✸ ❲❛••❝r♦ss✐♥❣ ❛♥❞ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ▲❡t ✉s ❞❡♠♦♥str❛t❡ t❤❛t t❤❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ✭✼✳✽✮ ✐s ❝♦♥s✐st❡♥t ✇✐t❤ ✇❛••✲ ❝r♦ss✐♥❣ ♣❤❡♥♦♠❡♥❛✳ ❋♦r ❜r❡✈✐t②✱ •❡t ✉s ✐♥tr♦❞✉❝❡ t❤❡ ♥♦t❛t✐♦♥ E(x) = e 2πix ✳ ❋✐rst✱ ❝♦♥s✐❞❡r t❤❡ ♦r✐❣✐♥❛• ❑♦♥ts❡✈✐❝❤✲❙♦✐❜❡•♠❛♥ ✇❛••❝r♦ss✐♥❣ ❢♦r♠✉•❛ ✭✹✳✷✵✮ ✇❤❡♥ ❛♣❛rt ❢r♦♠ ❉✺✲✐♥st❛♥t♦♥s✱ t❤❡r❡ ❛r❡ ♦♥•② ❡•❡❝tr✐❝ ❉✲✐♥st❛♥t♦♥s ❤❛✈✐♥❣ ❝❤❛r❣❡ ✈❡❝t♦r γ = (0, 0, 0, .q 0 ) ❛•♦♥❣ ✇✐t❤ t❤❡ ❜♦✉♥❞ st❛t❡s ♦❢ ❉✲✐♥st❛♥t♦♥s ✇✐t❤ ❝❤❛r❣❡s γ = (p 0 , p a , q a , q 0 ) p 0 = 0✳ ❚❤❡ ❉✲✐♥st❛♥t♦♥s ✇✐t❤ ❝❤❛r❣❡ γ ❛r❡ ❛•✐❣♥❡❞ ❛•♦♥❣ t❤❡ ✐♠❛❣✐♥❛r② ❛①✐s✳ ❚❤❡♥ t❤❡ ❑❙ ❢♦r♠✉•❛ ✐s γ j ∈Γ (5) 1 U -Ω + (γ j ) γ j q0 >0 U -χ γ γ i ∈Γ (5) 2 U -Ω + (γ i ) γ i γ i ∈Γ (5) 1 U Ω -(γ i ) γ i q0 >0 U χ γ γ j ∈Γ (5) 2 U Ω -(γ j ) γ j = 1, ✭✼✳✶✶✮ ✇❤❡r❡ ✇❡ •♦♦❦ ❛t t❤❡ ✉♣♣❡r ❤❛•❢ ♣•❛♥❡ ♦♥•② ✐♥ ✜❣✉r❡ ✼✳✶✳ ✰ ❛♥❞ ✲ ✐♥❞✐❝❡s ♦✈❡r t❤❡ ❉❚ ✐♥✈❛r✐❛♥ts ❞❡♥♦t❡ ❝❤❛♠❜❡rs ❛❢t❡r ❛♥❞ ❜❡❢♦r❡ ✇❛••❝r♦ss✐♥❣ r❡s♣❡❝t✐✈❡•②✳ ❆ s✐♠✐•❛r ❛♥❛•②s✐s ❝❛♥ ♦❢ ❝♦✉rs❡ ❜❡ ♣❡r❢♦r♠❡❞ ❢♦r t❤❡ •♦✇❡r ❤❛•❢ ♣•❛♥❡✳ Γ (5) 1 ❛♥❞ Γ (5) 2 ❞❡♥♦t❡ ❇P❙ r❛②s ✐♥ t❤❡ ✜rst ❛♥❞ s❡❝♦♥❞ q✉❛❞r❛♥ts r❡s♣❡❝t✐✈❡•②✳ ❍♦✇❡✈❡r✱ ✇❤❡♥ ♦♥❡ ♣❡r❢♦r♠s ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥✱ t❤❡ ❝♦♥t♦✉rs ❝❤❛♥❣❡✳ ❚❤❡ ❇P❙ r❛② ❝♦rr❡s♣♦♥❞✐♥❣ t♦ γ ♥♦✇ ❣❡ts r♦t❛t❡❞ t♦ t❤❡ r❡❛• ❛①✐s ❛♥❞ t❤❡ t✇✐st♦r s♣❛❝❡ •♦♦❦s •♦❦❡ ✜❣✉r❡ ✼✳✷✳ ❖♥❡ ❝❛♥ ♥♦✇ r❛✐s❡ t❤❡ q✉❡st✐♦♥ ✿ ✇❤❛t ❞♦❡s ❡♥s✉r❡ ❝♦♥s✐st❡♥❝② ✇✐t❤ ✇❛••❝r♦ss✐♥❣✱ ❣✐✈❡♥ s✉❝❤ ❞r❛st✐❝❛••② ❞✐✛❡r❡♥t t✇✐st♦r s♣❛❝❡ ❄ ❚❤❡ ♣♦✐♥t ✐s t❤❛t t❤❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ❛✛❡❝ts t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❝❤❛r❣❡ γ✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❑❙ ♦♣❡r❛t♦rs ❛r❡ U γ = W -1 N (γ) U γ W N (γ) , ✭✼✳✶✷✮ ✽✼ ✼✳ ▼✐rr♦r s②♠♠❡tr② ✐♥ ❛❝t✐♦♥ ✿ ✐♥st❛♥t♦♥s ✐♥ t②♣❡ ■■❇ ❢♦r♠✉•❛t✐♦♥ ✇❤❡r❡ U γ ✇❛s t❤❡ ❑❙ ♦♣❡r❛t♦r ❢♦r ❝❤❛r❣❡ γ ❜❡❢♦r❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ❛♥❞ W n ✐s t❤❡ ♦♣❡r❛t♦r ❣❡♥❡r❛t✐♥❣ t❤❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ✐♥ t❤❡ nt❤ q✉❛❞r❛♥t✳ ■♥ ✜rst ❛♥❞ s❡❝♦♥❞ q✉❛❞r❛♥ts t❤❡② ❜❡❝♦♠❡ W 1 = q0 >0 U -χ 2 γ = exp - χ 8π 2 q0 >0 E -q 0 ξ 0 , W 2 = q0 >0 U χ 2 γ = exp χ 8π 2 q0 >0 E -q 0 ξ 0 , ✭✼✳✶✸✮ ✇❤❡r❡❛s ❢♦r t❤❡ ♦t❤❡r t✇♦ q✉❛❞r❛♥ts t❤❡ ♣r♦❞✉❝ts r✉♥ ♦✈❡r ♥❡❣❛t✐✈❡ ❝❤❛r❣❡s ✷ ✳ ❚❤❡♥ ✐t ✐s ❡❛s② t♦ ❝❤❡❝❦ t❤❛t γ j ∈Γ (5) 1 U -Ω + (γ j ) γ j γ i ∈Γ (5) 2 U -Ω + (γ i ) γ i γ i ∈Γ (5) 1 U Ω -(γ i ) γ i γ j ∈Γ (5) 2 U Ω -(γ j ) γ j = γ j ∈Γ (5) 1 W -1 2 U -Ω + (γ j ) γ j W 2 γ i ∈Γ (5) 2 W -1 1 U -Ω + (γ i ) γ i W 1 γ i ∈Γ (5) 1 W -1 1 U Ω -(γ i ) γ i W 1 γ j ∈Γ (5) 2 W -1 2 U Ω -(γ j ) γ j W 2 = W -1 2    γ j ∈Γ (5) 1 U -Ω + (γ j ) γ j q0 >0 U -χ γ γ i ∈Γ (5) 2 U -Ω + (γ i ) γ i γ i ∈Γ (5) 1 U Ω -(γ i ) γ i q0 >0 U χ γ γ j ∈Γ (5) 2 U Ω -(γ j ) γ j    W 2 = 1, ✇❤❡r❡ ✇❡ ❤❛✈❡ ✉s❡❞ ✭✼✳✶✶✮ ❛♥❞ t❤❡ ❝♦♥t❛❝t str✉❝t✉r❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❡❞ ♦♣❡r❛t♦rs ✐s s♠♦♦t❤ ❛❝r♦ss t❤❡ ✇❛••✳ ◆♦✇ ♦♥❡ ❝❛♥ ❝♦♥s✐❞❡r ❝❤❛r❣❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❉✶✲❉✭✲✶✮ ❜♦✉♥❞ st❛t❡ ❤❛✈✐♥❣ ❝❤❛r❣❡ γ = (0, 0, qa , q0 ) ❛♥❞ ❝♦rr❡s♣♦♥❞✐♥❣ ❇P❙ r❛②s ✐♥ t②♣❡ ■■❆ ❛•♦♥❣✇✐t❤ ❇P❙ r❛②s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❝❤❛r❣❡s γ✱ ❛s ✐♥ ✜❣✉r❡ ✼✳✶✳ ✸ ❋♦r ❜❡✐♥❣ ❛❜•❡ t♦ ❞♦ t❤❡ ❛♥❛•②s✐s ❛s ❛❜♦✈❡✱ ♦♥❡ ♥❡❡❞s t❤❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦♣❡r❛t♦rs W γ = εγ q0 >0 U (-1) N (γ) χ 2 γ γ∈Γ (1) γ U (-1) N (γ) n (0) qa γ , ✭✼✳✶✹✮ ✷ ❍❡r❡ χ = χ d|q0 1 d 2 ✸ ❲❡ ❛ss✉♠❡ t❤❛t ✇❡ ❛r❡ ❛t t❤❡ ♣♦✐♥t ✐♥ t❤❡ ♠♦❞✉•✐ s♣❛❝❡ ✇❤✐❝❤ ❞♦❡s ♥♦t ❜❡•♦♥❣ t♦ ❛♥② •✐♥❡ ♦❢ ♠❛r❣✐♥❛• st❛❜✐•✐t②✳ ✽✽ ❋✐✈❡❜r❛♥❡ ❝♦rr❡❝t✐♦♥s✳ ✼✳✸ ✇❤❡r❡ ε γ = sign ( Re Z γ )✳ ❚❤❡ tr❛♥s❢♦r♠❡❞ ♦♣❡r❛t♦rs r❡❛❞ ❛s U γ = W -1 γ U γ W γ , ✭✼✳✶✺✮ ❲✐t❤ t❤✐s ♦♥❡ s❤♦✉•❞ ❝♦♥s✐❞❡r ❣❛✉❣❡ tr❛♥s❢♦r♠✐♥❣ U γ ❛s ✇❡ ❞✐❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❛s❡✳ ❉♦✐♥❣ ❡①❛❝t•② s✐♠✐•❛r ❛♥❛•②s✐s ♦♥❡ ❝❛♥ ❡①♣•✐❝✐t•② ♣r♦✈❡ t❤❛t t❤❡ ❝♦♥t❛❝t str✉❝t✉r❡ ❛♥❞ ❤❡♥❝❡ t❤❡ t✇✐st♦r s♣❛❝❡ ❞♦❡s r❡♠❛✐♥ s♠♦♦t❤ ❛❢t❡r t❤❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥✳ ✼✳✸✳✹ ❋✐✈❡❜r❛♥❡ ✐♥st❛♥t♦♥s ❢r♦♠ ❙✲❞✉❛•✐t② ◆♦✇ ✇❡ ❤❛✈❡ ❛•• ✐♥❣r❡❞✐❡♥ts t♦ r❡❛❝❤ ♦✉r ♠❛✐♥ ❣♦❛• ✖ t❤❡ t✇✐st♦r✐❛• ❞❡s❝r✐♣t✐♦♥ ♦❢ ✜✈❡❜r❛♥❡ ✐♥st❛♥t♦♥s ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❉✶✲❉✭✲✶✮✲✐♥st❛♥t♦♥ ❝♦rr❡❝t✐♦♥s✳ ✹ ❚♦ t❤✐s ❡♥❞✱ ✇❡ s✐♠♣•② ❛♣♣•② t❤❡ ♠♦❞✉•❛r ❝♦♥str❛✐♥t ✭✺✳✺✶✮ t♦ t❤❡ ❣❛✉❣❡ tr❛♥s❢♦r♠❡❞ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥ ✐♥ ✭✼✳✽✮✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ ✇❡ s♣•✐t t❤❡ ❉✺✲❉✸✲❉✶✲❉✭✲✶✮ ❜r❛♥❡ ❜♦✉♥❞ st❛t❡ ❝❤❛r❣❡ γ ✐♥t♦ t✇♦ ❝♦♠♣♦♥❡♥ts✱ p 0 ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❉✺✲❜r❛♥❡ ❝❤❛r❣❡ ❛♥❞ γ = (p a , q a , q 0 ) ❛s t❤❡ r❡st✳ ❚❤❡♥ ✇❡ ✐❞❡♥t✐❢② h [γ] 0,p 0 = h [γ] g ❛♥❞ ✇❡ ❛♣♣•② t❤❡ SL(2, Z) ❝♦♥str❛✐♥t ✇✐t❤ t❤❡ ♠❛tr✐① g = a b k/p 0 p/p 0 ∈ SL(2, Z) ,
[γ] k,p ) 2 q0 (p 0 ) 2 k(ξ 0 + n 0 ) + 2k 2 F cl (ξ + n) (1 + 2πikh [γ]
k,p ) 2 -(-1) N k,p (γ) k(ξ 0 + n 0 ) 4π 2 p 0 γ∈Γ k,p;γ nq E Sk,p;γ E 4π 2 p 0 (q Λ p Λ ) k(ξ 0 + n 0 ) h

∂ ξ a [i] (M ǫ a • H [ij] ) = ∂ ξ a [i] H [ij] -κ abc ǫ b T c [ij] - 1 2 ǫ b ǫ c T 0 [ij] , ∂ ξ 0 [i] (M ǫ a • H [ij] ) = ∂ ξ 0 [i] H [ij] -ǫ a ∂ ξ 0 [i] H + 1 2 κ abc ǫ a ǫ b T c [ij] + 1 6 κ abc ǫ a ǫ b ǫ c T 0 [ij] ,

  ✭✶✳✸✮ ✇❤❡r❡ t❤❡ ✇♦r•❞s❤❡❡t ♠❡tr✐❝ h ab✐s ♥♦✇ ❝♦♥s✐❞❡r❡❞ ❛s ❞②♥❛♠✐❝❛• ✈❛r✐❛❜•❡ ❛♥❞ ✭✶✳✷✮ ❛♣♣❡❛rs ❛s ♦♥❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥s ♦❢ ♠♦t✐♦♥✳ ❚❤✐s ❝❛♥ ❜❡ ❢✉rt❤❡r ❡①t❡♥❞❡❞ ❜② ❛❞❞✐♥❣ t❤❡ st❛♥❞❛r❞ ❊✐♥st❡✐♥ t❡r♠ ✇❤✐❝❤ ✐s ❛ t♦t❛• ❞❡r✐✈❛t✐✈❡ ✐♥ t✇♦ ❞✐♠❡♥s✐♦♥s✱ 1 4π Σ dσdτ √ -hR = χ, ✭✶✳✹✮ ✇❤✐❝❤ ✐s ❛ t♦t❛• ❞❡r✐✈❛t✐✈❡ ✐♥ t✇♦ ❞✐♠❡♥s✐♦♥s ❣✐✈❡♥ ❜② t❤❡ ❊✉•❡r ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ t❤❡ s✉r❢❛❝❡✳ ❋♦r ❛ |✐❡♠❛♥♥ s✉r❢❛❝❡ ✇✐t❤ g ❤❛♥❞•❡s ❛♥❞ b ❜♦✉♥❞❛r✐❡s ✐t ✐s ❣✐✈❡♥ ❜②✱ χ = 2 -2gb. ✭✶✳✺✮ ❚❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤✐s ❝♦♥tr✐❜✉t✐♦♥✱ t❤❡ P♦•②❛❦♦✈ ❛❝t✐♦♥ ❜❡❝♦♠❡s✱

  r•❞✲s❤❡❡t ∂Σ✳ ▼♦r❡♦✈❡r✱ ♦♥❡ ✐❞❡♥t✐✜❡s ❝•♦s❡❞ str✐♥❣s ❛s ❤❛✈✐♥❣ ♣❡r✐♦❞✐❝ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ✇♦r•❞✲s❤❡❡t✳ ✸ ✶✳ ❙♦♠❡ ❜❛s✐❝ ❢❛❝ts ❛❜♦✉t str✐♥❣ t❤❡♦r② ✶✳✷✳✶ ❲♦r•❞ s❤❡❡t ♣♦✐♥t ♦❢ ✈✐❡✇ ❙✐♥❝❡ ✇❡ ❝♦♥s✐❞❡r t❤❡ ✇♦r•❞ s❤❡❡t ♠❡tr✐❝ ❛s ❛ ❞②♥❛♠✐❝❛• ✈❛r✐❛❜•❡✱ t❤❡ P♦•②❛❦♦✈ ❛❝t✐♦♥ ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s ❛ t❤❡♦r② ♦❢ t✇♦ ❞✐♠❡♥s✐♦♥❛• ❣r❛✈✐t② ❝♦✉♣•❡❞ t♦ t❤❡ ♠❛tt❡r ✜❡•❞s X µ ✳ ❲❡ ✇♦✉•❞ •✐❦❡ t♦ q✉❛♥t✐③❡ t❤❡ ✇♦r•❞✲s❤❡❡t t❤❡♦r② t❤❛t ✇❡ ❢♦✉♥❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✳ ❲❡ ✇✐•• ❞♦ t❤✐s ✉s✐♥❣ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ♣❛t❤ ✐♥t❡❣r❛• q✉❛♥t✐③❛t✐♦♥✳ ❚♦ ❞❡✜♥❡ t❤❡ ♣❛t❤ ✐♥t❡❣r❛•✱ ✇❡ ♣❡r❢♦r♠ t❤❡ ❲✐❝❦ r♦t❛t✐♦♥ t♦ t❤❡ ❊✉❝•✐❞❡❛♥ s✐❣♥❛t✉r❡ ♦♥ t❤❡ ✇♦r•❞s❤❡❡t Z = s✉r❢❛❝❡s Σ DX µ e -S P (X,Σ) . ✭✶✳✼✮ ❚❤❡ s✉♠ ♦✈❡r t❤❡ s✉r❢❛❝❡s s❤♦✉•❞ ❜❡ ✉♥❞❡rst♦♦❞ ❛s ❛ s✉♠ ♦✈❡r ❛•• ♣♦ss✐❜•❡ t♦♣♦•♦✲ ❣✐❡s ✇❤✐❝❤ ✐s ❡q✉✐✈❛•❡♥t t♦ t❤❡ s✉♠ ♦✈❡r t❤❡ ❣❡♥✉s ♦❢ t❤❡ |✐❡♠❛♥♥ s✉r❢❛❝❡✱ s✉♣♣•❡✲ ♠❡♥t❡❞ ❜② t❤❡ ✐♥t❡❣r❛• ♦✈❡r ❞✐✛❡♦♠♦r♣❤✐s♠ ✐♥✈❛r✐❛♥t ❝•❛ss❡s ♦❢ t❤❡ ♠❡tr✐❝✱ s✉r❢❛❝❡s Σ → g Dρ(h ab ). ✭✶✳✽✮ ◆♦✇ t♦ ♣❡r❢♦r♠ t❤❡ ✐♥t❡❣r❛• ♦♥❡ ✉s❡s t❤❡ st❛♥❞❛r❞ tr✐❝❦ ♦❢ ❋❛❞❞❡❡✈✲P♦♣♦✈ ❣❛✉❣❡ ✜①✐♥❣ ✇❤❡r❡ t❤❡ s②♠♠❡tr② ✐s t❤❡ ✇♦r•❞ s❤❡❡t ❞✐✛❡♦♠♦r♣❤✐s♠s✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ✐s ❛♥♦t❤❡r s②♠♠❡tr② t❤❛t ♦♥❡ ♥❡❡❞s t♦ ❝♦♥s✐❞❡r✱ t❤❡ ❲❡②• ✐♥✈❛r✐❛♥❝❡✳ ✶✳✷✳✷ ❲❡②• ✐♥✈❛r✐❛♥❝❡ ❆♣❛rt ❢r♦♠ P♦✐♥❝❛r❡ ❛♥❞ ❞✐✛❡♦♠♦r♣❤✐s♠ s②♠♠❡tr✐❡s✱ t❤❡ P♦•②❛❦♦✈ ❛❝t✐♦♥ ❞♦❡s ♣♦ss❡ss ❛♥♦t❤❡r s②♠♠❡tr② ❛t t❤❡ ❝•❛ss✐❝❛• •❡✈❡•✱ t❤❡ r❡s❝❛•✐♥❣ s②♠♠❡tr② ♦❢ t❤❡ ✇♦r•❞ s❤❡❡t ♠❡tr✐❝ ♦r t❤❡ ❲❡②• ✐♥✈❛r✐❛♥❝❡✱ h ab → e φ h ab . ✭✶✳✾✮ ❚❤✐s s②♠♠❡tr② t♦❣❡t❤❡r ✇✐t❤ t❤❡ ❞✐✛❡♦♠♦r♣❤✐s♠s ✐s r❡s♣♦♥s✐❜•❡ ❢♦r ❡q✉✐✈❛•❡♥❝❡ ♦❢ t❤❡ P♦•②❛❦♦✈ ❛♥❞ ◆❛♠❜✉✲•♦t♦ ❛❝t✐♦♥s✳ ❍♦✇❡✈❡r✱ ❛t t❤❡ q✉❛♥t✉♠ •❡✈❡• t❤❡ ❛❜♦✈❡ s②♠♠❡tr② ✭❲❡②• ✐♥✈❛r✐❛♥❝❡✮ ❝❛♥ ❜❡ ❜r♦❦❡♥✱ ❛s t❤❡ ✐♥t❡❣r❛t✐♦♥ ♠❡❛s✉r❡ ♦❢ t❤❡ ♣❛t❤ ✐♥t❡❣r❛• ♦✈❡r t❤❡ ♠❡tr✐❝s ❤❛s t♦ ❜❡ r❡❣✉•❛r✐③❡❞ ❛♥❞ ✐♥ ❣❡♥❡r❛•t❤❡r❡ ✐s ♥♦ r❡❣✉•❛r✐③❛t✐♦♥ s❝❤❡♠❡ t❤❛t ❦❡❡♣s ❛•• t❤❡ s②♠♠❡tr✐❡s ✐♥t❛❝t✳ ❚❤✐s ❛♥♦♠❛•② ❝❛♥ ❜❡ ❜❡st ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ str❡ss✲❡♥❡r❣② t❡♥s♦r ✇❤✐❝❤ ✐s

  ✭✷✳✶✮ ❲✐t❤♦✉t ❣♦✐♥❣ ✐♥t♦ t❤❡ ❞❡t❛✐•s ♦❢ ✐♥tr✐♥s✐❝ ❞❡✜♥✐t✐♦♥✱ •❡t ✉s r❡♠✐♥❞ t❤❡ r❡❛❞❡r t❤❛t ❛ s♣❡❝✐❛• ❑ä❤•❡r ❣❡♦♠❡tr② ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s ❛ ❑ä❤•❡r ❣❡♦♠❡tr② ✇✐t❤ t❤❡ s♣❡❝✐❛• str✉❝t✉r❡✱ t❤❛t ❛•• ❣❡♦♠❡tr✐❝ q✉❛♥t✐t✐❡s ❝❛♥ ❜❡ ❞❡t❡r♠✐♥❡❞ ✐♥ t❡r♠s ♦❢ ❛ s✐♥❣•❡ ❤♦•♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥ ❝❛••❡❞ t❤❡ ♣r❡♣♦t❡♥t✐❛•✳ ❇♦t❤ t❤❡ ❛❜♦✈❡ s❡❝t♦rs ❛r❡ ❡q✉✐♣♣❡❞ ✇✐t❤ s✉❝❤ ❣❡♦♠❡tr✐❡s✱ ♠♦r❡ ♣r❡❝✐s❡•②✱ •♦❝❛• s♣❡❝✐❛• ❑ä❤•❡r ❣❡♦♠❡tr✐❡s✳ ❋♦r t❤✐s ❝❛s❡✱ t❤❡ s♣❡❝✐❛• ❑ä❤•❡r ♠❛♥✐❢♦•❞s ❛r❡ ❝♦♠♣•❡t❡•② ❞❡t❡r♠✐♥❡❞ ❜② ❛ ❞❡❣r❡❡ t✇♦ ❤♦♠♦❣❡♥❡♦✉s ❢✉♥❝t✐♦♥ F (X)✱ ✇❤❡r❡ X Λ ❛r❡ t❤❡ ❝♦♦r❞✐♥❛t❡s ✇✐t❤ ✐♥❞✐❝❡s Λ = 0, ..., n ❛♥❞ n ✐s t❤❡ ❝♦♠♣•❡① ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ♠❛♥✐❢♦•❞✳ ❚❤❡♥ t❤❡ ❑ä❤•❡r ♣♦t❡♥t✐❛• ✐s ❣✐✈❡♥ ❜②

  ✭✷✳✶✺✮ ✇❤❡r❡ ✭✷✳✷✮ ✐♥❞❡❡❞ •❡❛❞s t♦ t❤❡ ❑ä❤•❡r ♣♦t❡♥t✐❛• ❢r♦♠ ✭✷✳✶✸✮✳ ❲❡ ✇✐•• s❡❡ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ t❤❛t t❤✐s ❝✉❜✐❝ ♣r❡♣♦t❡♥t✐❛• ✐s ♦♥•② ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ t❤❡ •❛r❣❡ ✈♦•✉♠❡ •✐♠✐t t i → ∞ ♦❢ t❤❡ ♣r❡♣♦t❡♥t✐❛• t❤❛t ❛♣♣❡❛rs ✐♥ str✐♥❣ ❝♦♠♣❛❝t✐✜❝❛t✐♦♥✳ ✷✳✷ ❋✐❡•❞ ❝♦♥t❡♥t ♦❢ t②♣❡ ■■ s✉♣❡r❣r❛✈✐t✐❡s ✐♥ ❢♦✉r ❞✐♠❡♥s✐♦♥s ■♥ ♦r❞❡r t♦ ♦❜t❛✐♥ t❤❡ •♦✇ ❡♥❡r❣② ❡✛❡❝t✐✈❡ ❛❝t✐♦♥ ✐♥ ❢♦✉r ❞✐♠❡♥s✐♦♥s✱ ♦♥❡ st❛rts ❢r♦♠ t❡♥ ❞✐♠❡♥s✐♦♥❛• s✉♣❡r❣r❛✈✐t② t❤❡♦r✐❡s✳ ❲❡ ✇✐•• ❢♦❝✉s ♣❛rt✐❝✉•❛r•② ♦♥ t②♣❡ ■■❆ ❛♥❞ t②♣❡ ■■❇ s✉♣❡r❣r❛✈✐t② t❤❡♦r✐❡s✳ ❲❡ ✇✐•• ❜❡ ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ❜♦s♦♥✐❝ s❡❝t♦rs ♦♥•②✱ ❜❡❝❛✉s❡ t❤❡ ❢❡r♠✐♦♥✐❝ s❡❝t♦rs ❝❛♥ ✐♥ ♣r✐♥❝✐♣•❡ ❜❡ r❡st♦r❡❞ ❜② s✉♣❡rs②♠♠❡t✐❝ tr❛♥s❢♦r♠❛t✐♦♥s✳ ❚❤❡ ❜♦s♦♥✐❝ s❡❝t♦rs ✐♥ ✶✵ ❞✐♠❡♥s✐♦♥s ❛r❡ s✉❜❞✐✈✐❞❡❞ ✐♥t♦ t✇♦ ♣❛rts✱ t❤❡ ◆❙ ❛♥❞ t❤❡ || s❡❝t♦rs✳ ❚❤❡ ❢♦r♠❡r ❝♦♠♣r✐s❡s t❤❡ t❡♥✲❞✐♠❡♥s✐♦♥❛• ♠❡tr✐❝ ĝXY ✱ t❤❡ t✇♦ ❢♦r♠ ✜❡•❞ B2 ❛♥❞ t❤❡ t❡♥ ❞✐♠❡♥s✐♦♥❛• ❞✐•❛t♦♥ φ✱ ✇❤❡r❡❛s t❤❡ •❛tt❡r s❡❝t♦r ❝♦♥t❛✐♥s ♣✲❢♦r♠ ♣♦t❡♥t✐❛•s Âp ✳ ❋♦r t❤❡ t②♣❡ ■■❆ t❤❡♦r② p = 1, 3 ❛♥❞ ❢♦r t②♣❡ ■■❇✱ p = 0, 2, 4✱ ✇✐t❤ ❛♥ ❛❞❞✐t✐♦♥❛• ❝♦♥str❛✐♥t ♦♥ Â4 ✱ s✉❝❤ t❤❛t ✐ts ✜❡•❞ str❡♥❣t❤ ✐s s❡•❢✲❞✉❛•✱ F 5 = * F 5 ✳ ❚♦ ♣❡r❢♦r♠ t❤❡ ❝♦♠♣❛❝t✐✜❝❛t✐♦♥ t❤❡s❡ ❞✐✛❡r❡♥t✐❛• ❢♦r♠s s❤♦✉•❞ ❜❡ ❡①♣❛♥❞❡❞ ✐♥ t❤❡ ❜❛s✐s ♦❢ ❤❛r♠♦♥✐❝ ❢♦r♠s✳ ❋♦r t②♣❡ ■■❇ s✐❞❡ s✉❝❤ ❢♦r♠s ❤❛✈❡ ❛•r❡❛❞② ❜❡❡♥ ✶✺ ✷✳ ❈♦♠♣❛❝t✐✜❝❛t✐♦♥ ♦❢ t②♣❡ ■■ str✐♥❣ t❤❡♦r✐❡s ♦♥ ❈❛•❛❜✐✲(❛✉ t❤r❡❡❢♦•❞s ❛♥❞ t❤❡✐r ♠♦❞✉•✐ s♣❛❝❡s ✐♥tr♦❞✉❝❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s✉❜s❡❝t✐♦♥✳ ❋♦r t②♣❡ ■■❆ ♦♥❡ ❤❛s t♦ r❡♠❡♠❜❡r t❤❛t t❤❡r❡ ✐s ♥♦ ♥♦♥tr✐✈✐❛• ✶ ♦r ✺✲❝②❝•❡s s♦ t❤❛t ♦♥❡ ❤❛s t♦ ❝♦♥s✐❞❡r ♦♥•② ✸✲❢♦r♠s✳ ❖♥❡ ❝❛

  ✭✷✳✶✼✮ ✇❤❡r❡ ❤❛ts st❛♥❞ ❢♦r ✜❡•❞s ✐♥ 10✲❞✐♠❡♥s✐♦♥s ❛♥❞ Ĥ3 = d B2 , F2 = d Â1 , F4 = d Â3 -Â1 ∧ Ĥ3 . ✭✷✳✶✽✮ ▲❡t ✉s ♥♦✇ ❛♥❛•②③❡ ✇❤❛t ❤❛♣♣❡♥s ❞✉❡ t♦ ❝♦♠♣❛❝t✐✜❝❛t✐♦♥✳ ❼ ❋✐rst •❡t ✉s tr② t♦ ✉♥❞❡rst❛♥❞ t❤❡ ◆❙✲◆❙ s❡❝t♦r t♦ ✇❤✐❝❤ ĝXY , B2 , φ ❜❡•♦♥❣✳ ❚❤❡♥

  ✭✺✳✼✮ ■t t✉r♥s ♦✉t t❤❛t t❤❡ ❛❜♦✈❡ q✉❛♥t✐③❛t✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥ ❉✲❜r❛♥❡ ❝❤❛r❣❡s ❣❡ts r❡✢❡❝t❡❞ ✐♥ t❤❡ ♠✐rr♦r s②♠♠❡tr② ❛♥❞ ❙✲❞✉❛•✐t② tr❛♥s❢♦r♠❛t✐♦♥s ❛s ✇❡••✳ ❚♦ ✐♥✲ ✈❡st✐❣❛t❡ t❤✐s✱ ✜rst •❡t ✉s s❡❡ ❤♦✇ ♠✐rr♦r s②♠♠❡tr② r❡•❛t❡s t❤❡ t②♣❡ ■■❆ ❛♥❞ t②♣❡ ■■❇ || ✜❡•❞s✳ ❚❤❡② ❛♣♣❡❛r ✐♥ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rt ♦❢ t❤❡ ❉✲❜r❛♥❡ ❛❝t✐♦♥✱ ✇❤❡r❡❛s t❤❡ r❡❛• ♣❛rt ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❝❡♥tr❛• ❝❤❛r❣❡ ✇❤✐❝❤ ❣✐✈❡s t❤❡ str❡♥❣t❤ ♦❢ t❤❡ ❉✲✐♥st❛♥t♦♥s✳ ❲❡ ❤❛✈❡ ❛•r❡❛❞② ❞✐s❝✉ss❡❞ t❤❡ ❉✲❜r❛♥❡ ❛❝t✐♦♥ ✐♥ t❤❡ t②♣❡ ■■❆ ❢♦r✲ ♠✉•❛t✐♦♥ ✐♥ ❝❤❛♣t❡r ✷✳ ■♥ t②♣❡ ■■❇✱ ✐♥st❡❛❞ ✐t ✐s ❣✐✈❡♥ ❜② ❬✺✸✱ ✺✹(

  1) ❣❛✉❣❡ ✜❡•❞✧ A✳ ❆s ✇❡ ❤❛✈❡ ❛•r❡❛❞② ❢♦✉♥❞✱ ✐♥ ♦✉r ❝❛s❡ t❤❡ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ❛❝q✉✐r❡s s✐♠✐•❛r ❤❛•❢✲✐♥t❡❣❡r t❡r♠s ❛s ✐♥ ✭✻✳✶✵✮✳ ■♥ ✭✷✳✹✺✮ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ❤❛s ❛ s✐♠♣•❡ ✐♥t❡r♣r❡t❛t✐♦♥✳ ❆s ❢❛r ❛s t❤❡② ❛r❡ ❝♦♥❝❡r♥❡❞✱ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜② s❤✐❢t✐♥❣ t❤❡ ✜❡•❞s b

  ✭✼✳✶✻✮✇❤❡r❡ t❤❡ t✇♦ ✐♥t❡❣❡rs (p, k) = (0, 0) ❤❛✈❡ p 0 ❛s t❤❡ ❣r❡❛t❡st ❝♦♠♠♦♥ ❞✐✈✐s♦r✱ ✇❤❡r❡❛s a ❛♥❞ b ♠✉st s❛t✐s❢② apbk = p 0 ✳ ❚❤❡ ✐♥t❡❣❡r k ✇✐•• ❛♣♣❡❛r ❛s ◆❙✺✲❜r❛♥❡ ❝❤❛r❣❡✳ ❆s ❢♦r t❤❡ ♦t❤❡r ❝❤❛r❣❡s✱ ✐t ✐s ❝♦♥✈❡♥✐❡♥t t♦ ♣❛❝❦ t❤❡♠ ✐♥t♦ r❛t✐♦♥❛• ❝❤❛r❣❡sn a = p a /k✱ n 0 = p/k ❛♥❞ t❤❡ s♦✲❝❛••❡❞ ✐♥✈❛r✐❛♥t ❝❤❛r❣❡s ❬✷✶( p a p b p c (p 0 ) 2 , ✭✼✳✶✼✮ ✇❤✐❝❤ ❛r❡ ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ s♣❡❝tr❛• ✢♦✇ tr❛♥s❢♦r♠❛t✐♦♥✳ ◆♦✇ ❛ s✐♠♣•❡ ❛♣♣•✐❝❛t✐♦♥ ♦❢ t❤❡ ❙✲❞✉❛•✐t② ❝♦♥str❛✐♥t ✭✺✳✺✶✮ ❣✐✈❡s t❤❡ ✜✈❡❜r❛♥❡ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥ h [γ] k,p = (p 0 ) -1 (kξ 0 + p) g • h [γ] ξ 0 + n 0 )σ D (γ) E(S k,p;γ ) , ✭✼✳✶✽✮ ✇❤❡r❡ ✇❡ ❤❛✈❡ ✉s❡❞ t❤❡ ❢♦••♦✇✐♥❣ ♥♦t❛t✐♦♥s ✹ ❲❡ r❡♠✐♥❞ t❤❛t ♦✉r ❝♦♥str✉❝t✐♦♥ ✐❣♥♦r❡s t❤❡ ❡✛❡❝t ♦❢ ❉✸✲✐♥st❛♥t♦♥s✳ ❆•t❤♦✉❣❤ s✉❝❤ ❛♣✲ ♣r♦①✐♠❛t✐♦♥ ✐s ♣❤②s✐❝❛••② ✉♥•✉st✐✜❡❞✱ ❛t ❛ ❢♦r♠❛• •❡✈❡• ✐t ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ❜② s❡tt✐♥❣ t♦ ③❡r♦ ❛•• ❉❚✲✐♥✈❛r✐❛♥ts Ω(γ) ❢♦r ❝❤❛r❣❡s ✇✐t❤ p 0 = 0, p a = 0✳ ◆♦t❡ ❤♦✇❡✈❡r t❤❛t ✇❡ ❞♦ ✐♥❝•✉❞❡ t❤❡ ❡✛❡❝t ♦❢ ❉✸✲❜r❛♥❡s ❜♦✉♥❞ t♦ ❉✺✲❜r❛♥❡s✱ ❛s r❡q✉✐r❡❞ ❜② ✐♥✈❛r✐❛♥❝❡ ✉♥❞❡r ♠♦♥♦❞r♦♠✐❡s✳ ✽✾ ✼✳ ▼✐rr♦r s②♠♠❡tr② ✐♥ ❛❝t✐♦♥ ✿ ✐♥st❛♥t♦♥s ✐♥ t②♣❡ ■■❇ ❢♦r♠✉•❛t✐♦♥ ✭✶✮ S k,p;γ = -kS n Λ + p 0 (p 0 q0k qa (ξ a + n a )) k 2 (ξ 0 + n 0 ) -a k p 0 q 0c 2,a p a ε(g) -(-1) N k,p (γ) 2πi γ∈Γ k,p;γ nq p Λ qΛ E Sk,p;γ ✭✼✳✶✾✮ ✇✐t❤ S n Λ = αn Λ ξΛ + F cl (ξ + n)✳ ✭✷✮ ❙✲❞✉❛•✐t② tr❛♥s❢♦r♠❡❞ ❉✶✲❜r❛♥❡ t✇✐st♦r✐❛• ❛❝t✐♦♥ ✭♥♦t❡ t❤❛t ❜♦t❤ ❛❝t✐♦♥s ✭✼✳✶✾✮ ❛♥❞ ✭✼✳✷✵✮ ❛r❡ r❡❣✉•❛r ❛t k = 0 ❛♥❞ r❡❞✉❝❡ ✐♥ t❤✐s •✐♠✐t t♦ t❤❡ ✭❣❛✉❣❡ tr❛♥s❢♦r♠❡❞✮ ❉✲✐♥st❛♥t♦♥ t✇✐st♦r✐❛• ❛❝t✐♦♥s -Ξ (g) γ ❛♥❞ -Ξ γ ✱ r❡s♣❡❝t✐✈❡•②✳✮ Sk,p;γ = q0 (p 0 ) 2 k 2 (ξ 0 + n 0 ) -p 0 qa ξ a k(ξ 0 + n 0 ) -a k p 0 q0 . ✭✼✳✷✵✮ ✭✸✮ |❛t✐♦♥❛• •♦♣❛❦✉♠❛r✲❱❛❢❛ ✐♥✈❛r✐❛♥ts n (0) qa ❝♦♥str✉❝t❡❞ ❢r♦♠ st❛♥❞❛r❞ ✐♥✈❛r✐✲ ❛♥ts ❜② ✉s✐♥❣ s❢♦r♠❡❞ ❇P❙ ✐♥❞✐❝❡s t❤❛t Ωk,p (γ) = Ω(γ, g • z)✱ t❤❛t t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❛t t❤❡ ❉❚ ✐♥✈❛r✐❛♥ts ❛r❡ ♦♥•② ♣✐❡❝❡✇✐s❡ ❝♦♥st❛♥t ❛♥❞ t❤❡ ♠♦❞✉•✐ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡♠ ✐s ❛✛❡❝t❡❞ ❜② ❙✲❞✉❛•✐t② tr❛♥s❢♦r♠❛t✐♦♥✳ ✭✺✮ ❚❤❡ tr❛♥s❢♦r♠❡❞ ❝❤❛r❣❡ •❛tt✐❝❡ Γ k,p;γ = Γ (1) γ (g • z) ∪ Γ (-1) γ (g • z). ✭✼✳✷✷✮ ❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦♥ z a ❝♦♠❡s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t t❤❡ ❞❡✜♥✐t✐♦♥s ✭✼✳✺✮ ❛♥❞ ✭✼✳✻✮ ❞❡♣❡♥❞ ♦♥ t❤❡ ❝❡♥tr❛• ❝❤❛r❣❡ ❢✉♥❝t✐♦♥✳ ✭✼✮ ❚❛r❣❡t q✉❛❞r❛♥t ✐♥ t❤❡ ❝♦♠♣•❡① ♣•❛♥❡ N k,p (γ) = 2 π arg (ig • Z γ ) ✳ ❚❤❡ ❝♦♥t♦✉rs ♦♥ t❤❡ CP 1 ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s ✭✼✳✶✽✮ ❛r❡ ❣✐✈❡♥ ❜② t❤❡ ✐♠❛❣❡ ✉♥❞❡r ❙✲❞✉❛•✐t② ♦❢ t❤❡ ♦r✐❣✐♥❛• ❇P❙ r❛② ℓ γ ✳ ℓ k,p;γ = {t : Z γ (g • z)/(g • t) ∈ iR -}. ✭✼✳✷✸✮ ■t ❝❛♥ ❜❡ ❢♦✉♥❞ t❤❛t t❤❡② •♦✐♥ t❤❡ ♣♦✐♥ts t k,p ± ✳ ✭ ❛s ✐♥ ✜❣✉r❡ ✼✳✸✮✳ ❚❤❡ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❢r♦♠ t❤❡ ❝♦♥t❛❝t ✾✵ ▼♦♥♦❞r♦♠② ❛♥❞ ❍❡✐s❡♥❜❡r❣ ✐♥✈❛r✐❛♥❝❡ ♦❢ ✜✈❡❜r❛♥❡ ✐♥st❛♥t♦♥ ❝♦rr❡❝t❡❞ ❍▼ ♠♦❞✉•✐ s♣❛❝❡✳ ✼✳✹ ❋✐❣✉r❡ ✼✳✸✿ ❙❝❤❡♠❛t✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ t✇✐st♦r ❞❛t❛ ❣❡♥❡r❛t✐♥❣ ❉✭✲✶✮✲❉✶ ❛♥❞ ❛•• ✜✈❡❜r❛♥❡ ✐♥st❛♥t♦♥s✳ ❇P❙ r❛②s •♦✐♥✐♥❣ ❞✐✛❡r❡♥t ♣♦✐♥ts t k,p ± ❝♦rr❡s♣♦♥❞ t♦ ❞✐❢✲ ❢❡r❡♥t ✜✈❡❜r❛♥❡ ❝❤❛r❣❡s k ❛♥❞ p✳ ❉✐✛❡r❡♥t ❇P❙ r❛②s •♦✐♥✐♥❣ t❤❡ s❛♠❡ ♣♦✐♥ts ❝♦rr❡s♣♦♥❞ t♦ ❞✐✛❡r❡♥t r❡❞✉❝❡❞ ❝❤❛r❣❡s γ✳ ❤❛♠✐•t♦♥✐❛♥ ✭✼✳✶✽✮✳ ❚❤❡ ❡①♣r❡ss✐♦♥ ✐s ✈❛•✐❞ t♦ ❛•• ♦r❞❡rs ❛♥❞ ✐s ❣✐✈❡♥ ❜② H

  ✐♥ ❬✷✹( t❤❛t t❤❡ ❛❜♦✈❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ ✐♥❞❡❡❞ s❛t✐s✜❡s ✭✺✳✸✶✮✳ ■t ❡①❡♠♣•✐✜❡s t❤❡ ♣♦✇❡r ♦❢ t❤❡ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥ ❢♦r♠✉•❛t✐♦♥✳ ❙♦•✈✐♥❣ t❤❡ ❝♦♥✲ str❛✐♥t ♦♥ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ✐s ❛❝t✉❛••② ✈❡r② ❝✉♠❜❡rs♦♠❡✳ ■♥st❡❛❞✱ t❤❡ s♦•✉✲ t✐♦♥ t❤r♦✉❣❤ t❤❡ ♥❡✇ ♣❛r❛♠❡tr✐③❛t✐♦♥ ✐s ♠✉❝❤ ♠♦r❡ ❡✣❝✐❡♥t✳ ✼✳✹ ▼♦♥♦❞r♦♠② ❛♥❞ ❍❡✐s❡♥❜❡r❣ ✐♥✈❛r✐❛♥❝❡ ♦❢ ✜✈❡✲ ❜r❛♥❡ ✐♥st❛♥t♦♥ ❝♦rr❡❝t❡❞ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ■♥ ❝❤❛♣t❡r ✻✱ ✇❡ ❤❛✈❡ s❡❡♥ t❤❛t r❡q✉✐r✐♥❣ t❤❛t t❤❡ ❍▼ ♠♦❞✉•✐ s♣❛❝❡ ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ ❛•❣❡❜r❛ ❣❡♥❡r❛t❡❞ ❜② ❞✐s❝r❡t❡ s②♠♠❡tr✐❡s✱ ♦♥❡ ❝❛♥ ✜♥❞ ❛ ♥✐❝❡ ❣r♦✉♣ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r SL(2, Z) ⋉ N (Z)✳ ❚❤✐s •❡❞ t♦ t❤❡ ♠♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ✭✻✳✶✵✮ ❛♥❞ ❛s ❛ ❝♦♥s❡q✉❡♥❝❡✱ ✉♥•✐❦❡ ✭✻✳✼✮ t❤❡ ❛♥♦♠❛✲ •♦✉s t❡r♠s ❞✐s❛♣♣❡❛r ✐♥ ✭✻✳✶✵✮✱ s♦ t❤❛t ♦♥❡ ❤❛s T (1) 0,η a S = SM ηa ✳ ❚❤❡ ✜✈❡❜r❛♥❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s t❤❛t ✇❡ ❤❛✈❡ ❞❡r✐✈❡❞ ✐♥ ✭✼✳✷✹✮✱ ✐s ♦❜t❛✐♥❡❞ ❜② t❤❡ ❛♣♣•✐❝❛t✐♦♥ ♦❢ t❤❡ ❙✲❞✉❛•✐t② ❝♦♥str❛✐♥t ✭✺✳✸✶✮ ♦♥ t❤❡ ❉✺✲✐♥st❛♥t♦♥ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ✾✶ ✼✳ ▼✐rr♦r s②♠♠❡tr② ✐♥ ❛❝t✐♦♥ ✿ ✐♥st❛♥t♦♥s ✐♥ t②♣❡ ■■❇ ❢♦r♠✉•❛t✐♦♥ t②♣❡ ■■❇ s✐❞❡✳ ❚❤✉s ✇✐t❤ t❤❡ ❛♠❡♥❞❡❞ tr❛♥s❢♦r♠❛t✐♦♥s✱ ✐t s❤♦✉•❞ ❜❡ ♣♦ss✐❜•❡ t♦ ❝❤❡❝❦ ✭✻✳✶✶✮ ❡①♣•✐❝✐t•② ❜② ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❛❝t✐♦♥ ♦❢ s②♠♠❡tr✐❡s ♦♥ t❤❡ ✜✈❡❜r❛♥❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ ✭✼✳✷✹✮✳ ❚❤✐s s❡r✈❡s ❛s ❛ ❝r♦ss❝❤❡❝❦ t❤❛t t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ ❞✐s✲ ❝r❡t❡ s②♠♠❡tr✐❡s ♦♥ t❤❡ ♣❤②s✐❝❛• ✜❡•❞s ❣✐✈❡♥ ✐♥ ❝❤❛♣t❡r ✻ ✐s ✐♥❞❡❡❞ ❝♦♥s✐st❡♥t✳ ■♥ t❤❡ ❢♦••♦✇✐♥❣✱ ✇❡ ❣✐✈❡ t❤❡ ✈❡r✐✜❝❛t✐♦♥ t❤❛t t❤❡ ❢✉•• ❝♦♥t❛❝t str✉❝t✉r❡ ♦❢ t❤❡ t✇✐st♦r s♣❛❝❡ r❡♠❛✐♥s ✐♥✈❛r✐❛♥t✱ ♦r ❡q✉✐✈❛•❡♥t•② M HM r❡♠❛✐♥s ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ ✐♥t❡❣❡r ♠♦♥♦❞r♦♠②✱ ❍❡✐s❡♥❜❡r❣ ❛♥❞ ❙✲❞✉❛•✐t② s②♠♠❡tr✐❡s✳ ❇❡❢♦r❡ ❣♦✐♥❣ ❢✉rt❤❡r✱ •❡t ✉s r❡❝♦••❡❝t s♦♠❡ ❢❛❝ts ❛❜♦✉t t❤❡ ❝❤❛r❛❝t❡rs ε(g) ❛♣♣❡❛r✐♥❣ ✐♥ t❤❡ ❙✲❞✉❛•✐t② tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ||✲❛①✐♦♥ c a ✳ ✼✳✹✳✶ ❚❤❡ ❝❤❛r❛❝t❡r ε(g) ❛♥❞ ❉❡❞❡❦✐♥❞ s✉♠ ❚❤❡ ❝❤❛r❛❝t❡r ε(g)✱ ❛s ✇❡ ❤❛✈❡ ❛•r❡❛❞② s❡❡♥ ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❢♦••♦✇✐♥❣✱ e 2πi ε(g) = η aτ +b cτ +d (cτ + d) 1/2 η(τ ) . ✭✼✳✷✺✮ ■♥ ♣❛rt✐❝✉•❛r✱ 24ε(g) ✐s ❛♥ ✐♥t❡❣❡r ❛♥❞ ✐t ❤❛s t❤❡ ❡①♣•✐❝✐t r❡♣r❡s❡♥t❛t✐♦♥✱ d, c) ✐s t❤❡ ❉❡❞❡❦✐♥❞ s✉♠✳ ■t ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ t❡r♠s ♦❢ t❤❡ ❢♦••♦✇✐♥c > 0✳ ❚❤✉s✱ ❢♦r t❤❡ ❣❡♥❡r❛t♦rs S ❛♥❞ T ♦❢ t❤❡ SL(2, Z) ❣r♦✉♣✱ s♦ s❛t✐s✜❡s ❛ r❡❝✐♣r♦❝✐t② r❡•❛t✐♦♥ ✇❤✐❝❤ ✇❡ ✇✐•• r❡q✉✐r❡ ❢♦r t❤❡ ♣r♦♦❢✱ s(d, c) + s(c, d) ♦❞r♦♠② ❛♥❞ ❍❡✐s❡♥❜❡r❣ ✐♥✈❛r✐❛♥❝❡ ♦❢ ✜✈❡❜r❛♥❡ ✐♥st❛♥t♦♥ ❝♦rr❡❝t❡❞ ❍▼ ♠♦❞✉•✐ s♣❛❝❡✳ ✼✳✹ ✼✳✹✳✷ ■♥✈❛r✐❛♥❝❡ ✉♥❞❡r ♠♦♥♦❞r♦♠② ❲❡ ✜rst ❞✐s❝✉ss t❤❡ ✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ ✜✈❡❜r❛♥❡ ✐♥st❛♥t♦♥ ❝♦rr❡❝t❡❞ ❝♦♥t❛❝t str✉❝✲ t✉r❡ ✉♥❞❡r ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥s✳ ❚❤❡ •✐❢t ♦❢ ✭✻✳✶✵✮ t♦ t❤❡ t✇✐st♦r s♣❛❝❡ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦••♦✇✐♥❣ ❛❝t✐♦♥ ♦♥ t❤❡ ❉❛r❜♦✉① ❝♦♦r❞✐♥❛t❡s M ǫ a : ξ 0 → ξ 0 , ξ a → ξ a + ǫ a ξ 0 , ξa → ξaκ abc ǫ b ξ c -1 2 κ abc ǫ b ǫ c ξ 0 + A ab ǫ b , ξ0 → ξ0ǫ a ξa + 1 2 κ abc ǫ a ǫ b ξ c + 1 6 κ abc ǫ a ǫ b ǫ c ξ 0ǫ a ξ b ξ c + κ abc ǫ a ǫ b ξ c ξ 0 + 1 3 κ abc ǫ a ǫ b ǫ c (ξ 0 ) 2 + c 2,a24ǫ a , ✭✼✳✸✷✮ ❛♥❞ t❤❡ ✜❜❡r ❝♦♦r❞✐♥❛t❡ t ❞♦❡s ♥♦t tr❛♥s❢♦r♠✳ ▲❡t ✉s ✜rst ✜♥❞ ✇❤❛t ❝♦♥str❛✐♥t ✐t ❡♥t❛✐•s ♦♥ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✳ ❖♥❡ ❝❛♥ ❡❛s✐•② ✜♥❞ t❤❛t

  ✐s ✐♥tr♦❞✉❝❡❞ ✐♥ ✭✺✳✸✶✮✳ ❋r♦♠ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ α ♦♥❡ ❝❛♥ ♥♦✇ ♦❜t❛✐♥ t❤❡ ❝♦♥str❛✐♥t ♦♥ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥sM ǫ a • H [ij] = H [ij] + 1 2 κ abc ǫ a T b [ij] T c [ij] + 1 2 κ abc ǫ a ǫ b T c [ij] T 0 [ij] + 1 6 κ abc ǫ a ǫ b ǫ c (T 0 [ij] ) 2 . ✭✼✳✸✹✮ ❚❤❡ ❉✲✐♥st❛♥t♦♥ ❛❝t✐♦♥ r❡♠❛✐♥s ✐♥✈❛r✐❛♥t ✉♥❞❡r ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥✱ ✐❢ ♦♥❡ s✉♣♣•❡♠❡♥ts ✐t ✇✐t❤ t❤❡ s♣❡❝tr❛• ✢♦✇ tr❛♥s❢♦r♠❛t✐♦♥ ♦♥ t❤❡ •❛tt✐❝❡ ♦❢ ❝❤❛r❣❡s ✭✻✳✻✮ ✇✐t❤ t❤❡ ♣❛r❛♠❡t❡r ǫ ′ a = p p 0 ǫ a ✳ ❚♦ ♣r♦❝❡❡❞ ✇❡ ❤❛✈❡ t♦ ✜rst ❝❤❡❝❦ t❤❡ ✐♥✈❛r✐❛♥❝❡ ♦❢ ✭✼✳✶✾✮✳ ❋♦r t❤❡ •❛st t❡r♠✱ ♦♥❡ s❤♦✉•❞ ❝❤❛♥❣❡ t❤❡ s✉♠♠❛t✐♦♥ ✈❛r✐❛❜•❡ ❛s q0 → qaǫ ′ a qa , qa → qa . ✭✼✳✸✺✮ ❚❤❡♥ ❜♦t❤ t❤❡ ❢❛❝t♦rs p Λ qΛ ❛♥❞ t❤❡ ❡①♣♦♥❡♥t✐❛• ✐♥ Sk,p;γ r❡♠❛✐♥ ✐♥✈❛r✐❛♥t✳ ❖♥❡ ❝♦✉•❞ ✇♦rr② ❛❜♦✉t ❤❛✈✐♥❣ ❝❤❛♥❣❡❞ t❤❡ s✉♠♠❛t✐♦♥ r❛♥❣❡ ✇❤✐•❡ ❞♦✐♥❣ t❤✐s✳ ❇✉t ✐♥ ❢❛❝t ✐t st❛②s t❤❡ s❛♠❡✳ ❇❡❝❛✉s❡ ❜♦t❤ g • Z γ ❛♥❞ g • Z γ r❡♠❛✐♥ ✐♥✈❛r✐❛♥t ✐❢ ♦♥❡ ♣❡r❢♦r♠s t❤❡ s♣❡❝tr❛• ✢♦✇ tr❛♥s❢♦r♠❛t✐♦♥ ♦♥ t❤❡ ❝❤❛r❣❡ •❛tt✐❝❡✳ ❚❤✐s ♣r♦✈❡s t❤❛t t❤❡ •❛st t❡r♠ ✐s ✐♥✈❛r✐❛♥t✳ ✾✸ ✼✳ ▼✐rr♦r s②♠♠❡tr② ✐♥ ❛❝t✐♦♥ ✿ ✐♥st❛♥t♦♥s ✐♥ t②♣❡ ■■❇ ❢♦r♠✉•❛t✐♦♥ ▲❡t ✉s ❝❛•❝✉•❛t❡ t❤❡ ♣❤❛s❡ ν(ǫ) ❛♣♣❡❛r✐♥❣ ✐♥ M ǫ a h [γ] k,p = ν(ǫ) h [γ] k,p ✳ ❚❤❡♥ ν(ǫ) = E -kǫ a κ apc 2,a ǫ a abc ǫ a p b p c + p 2p 0 (pb)κ abc ǫ a ǫ b p cp 2 6p 0 (pb)κ abc ǫ a ǫ b ǫ c = Ec 2,a ǫ a bp 0 12 1p/p 0 ) 2p 0 ) + p/p 0 + 1 (1 + b) A ab ǫ a p b , ✭✼✳✸✻✮ ✇❤❡r❡ ✇❡ ❤❛✈❡ ✉s❡❞ t❤❡ ❡①♣r❡ss✐♦♥ ❢♦r ✭✼✳✷✻✮ ❛♥❞ ❡①❝❤❛♥❣❡❞ t❤❡ ❛r❣✉♠❡♥ts ✐♥ t❤❡ ❉❡❞❡❦✐♥❞ s✉♠ ❜② ✭✼✳✸✶✮✱ ❛♥❞ ✭✺✳✻✮✳ ❚❤❡ •❛st t❡r♠ ✐♥ t❤❡ ❛❜♦✈❡ ❡①♣r❡ss✐♦♥ ✐s ❝•❡❛r•② ✐♥t❡❣❡r✱ ❜❡❝❛✉s❡ ❞✉❡ t♦ t❤❡ SL(2, Z) r❡•❛t✐♦♥ ap/p 0bk/p 0 = 1✱ ❜♦t❤ ♦❢ b ❛♥❞ p/p 0 ❝❛♥ ♥♦t ❜❡ ❡✈❡♥✳ ❋♦r t❤❡ ♦t❤❡r t❡r♠✱ ✇❡ ✜rst ✉s❡ t❤❡ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ❉❡❞❡❦✐♥❞ s✉♠ ✭✼✳✷✾✮✳ ❲❡ ❛•s♦ r❡♠❡♠❜❡r t❤❛t c 2,a ✐s ❡✈❡♥ ❛♥❞ ❤❡♥❝❡ t❤❡ •❛st t❡r♠ ✐♥ ✭✼✳✷✾✮ ❞♦❡s ♥♦t ❝♦♥tr✐❜✉t❡✳ ❚❤❡♥ ❛❢t❡r ❛ ❜✐t ♦❢ ❛•❣❡❜r❛ ♦♥❡ ❛rr✐✈❡s ❛t t❤❡ ❢♦••♦✇✐♥❣ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ♣❤❛s❡✱ ν(ǫ) = E -c 2,a ǫ a 12 (d -1) (b(d + 1)c(2d -1)) , ✭✼✳✸✼✮ ✇❤❡r❡ c = p/p 0 ❛♥❞ d = k/p 0 ✳ ■t ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s ν(ǫ) = E -c 2,a ǫ a 12 (d -1) (b(adbc)(d + 1)c(2d -1)) = E -c 2,a ǫ a 12 (d -1) -b 2 c(d + 1) + c(d + 1) -3cd = E c 2,a ǫ a 12 (d -1)(d + 1)(b -1)(b + 1) . ✭✼✳✸✽✮ ■♥ t❤❡ s❡❝♦♥❞ ❛♥❞ t❤✐r❞ st❡♣s ✇❡ ❤❛✈❡ ✉s❡❞ t❤❡ ❢❛❝t t❤❛t 1 6 d(d -1)(d + 1) ❛♥❞ 1 2 d(d -1) ❛r❡ ✐♥t❡❣❡rs r❡s♣❡❝t✐✈❡•②✳ ◆♦✇ ✐t ✐s ♥♦t ❞✐✣❝✉•t t♦ s❡❡ t❤❛t ν(ǫ) = 1✳ ❋✐rst✱b ❛♥❞ d ❝❛♥ ♥♦t ❜❡ s✐♠✉•t❛♥❡♦✉s•② ❡✈❡♥✳ ❙❡❝♦♥❞•②✱ (b 2 -1)(d 2 -1) ✐s ♥♦t ❞✐✈✐s✐❜•❡ ❜② t❤r❡❡ ♦♥•② ✐❢ b = 3k ❛♥❞ d = 3l✱ ✇❤❡r❡ k, l ∈ Z✳ ❇✉t t❤✐s ✐s ✐♠♣♦ss✐❜•❡✱ ❛s ✐t ❝♦♥tr❛❞✐❝ts adbc = 1✳ ❍❡♥❝❡✱ ✇❡ ❤❛✈❡ ♣r♦✈❡♥ t❤❛t t❤❡ ❝♦♥t❛❝t str✉❝t✉r❡ r❡♠❛✐♥s ✐♥✈❛r✐❛♥t ✉♥❞❡r ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ♠❛♣♣✐♥❣ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s t♦ ❡❛❝❤ ♦t❤❡r ❛s M ǫ a • h [γ] k,p = h [γ[-pǫ a /p 0 ]] k,p , M ǫ a • ℓ k,p;γ = ℓ k,p;γ[-pǫ a /p 0 ] , ✭✼✳✸✾✮ ❚❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ✭✼✳✷✹✮ ❛r❡ ❛•s♦ ♠❛♣♣❡❞ t♦ ❡❛❝❤ ♦t❤❡r ❛s ❛ ❝♦♥s❡q✉❡♥❝❡✳ ❍♦✇❡✈❡r✱ t❤❡② s❛t✐s❢② t❤❡ ♥♦♥✲•✐♥❡❛r ❝♦♥str❛✐♥t ✭✼✳✸✹✮ ❛s t❤❡ ❢♦••♦✇✐♥ǫ a T b T c + 1 2 κ abc ǫ a ǫ b T c T 0 + 1 6 κ abc ǫ a ǫ b ǫ c (T 0 ) 2 . ✭✼✳✹✵✮ ✺ ❚❤✐s ✐s ②❡t ❛♥♦t❤❡r ✐♥st❛♥❝❡ ♦❢ t❤❡ ✈✐rt✉❡ ♦❢ t❤❡ ❝♦♥t❛❝t ❜r❛❝❦❡t ❢♦r♠✉•❛t✐♦♥✳ ■♥st❡❛❞ ♦❢ ❤❛✈✐♥❣ t♦ ❞❡❛• ✇✐t❤ ♥♦♥✲•✐♥❡❛r ✭✼✳✸✹✮ ♦♥❡ ❝❛♥ ✇♦r❦ ✇✐t❤ s✐♠♣•❡r ✭✼✳✸✾✮✳ ✾✹ ▼♦♥♦❞r♦♠② ❛♥❞ ❍❡✐s❡♥❜❡r❣ ✐♥✈❛r✐❛♥❝❡ ♦❢ ✜✈❡❜r❛♥❡ ✐♥st❛♥t♦♥ ❝♦rr❡❝t❡❞ ❍▼ ♠♦❞✉•✐ s♣❛❝❡✳ ✼✳✹ ❚♦ ❡♥❞ t❤❡ ♣r♦♦❢ t❤❛t M HM r❡♠❛✐♥s ✐♥✈❛r✐❛♥t ✉♥❞❡r ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥ ✭✻✳✶✵✮ ❛❝❝♦♠♣❛♥✐❡❞ ❜② s♣❡❝tr❛• ✢♦✇ tr❛♥s❢♦r♠❛t✐♦♥✱ ✇❡ ♥❡❡❞ t♦ ❡♥s✉r❡ t❤❛t t❤❡ r❛t✐♦♥❛• ❉❚ ✐♥✈❛r✐❛♥ts Ωk,p (γ) r❡♠❛✐♥s ✐♥✈❛r✐❛♥t✳ ■t ✇❛s s❤♦✇♥ ✐♥ ❬✻✽( t❤❛t t❤❡ ❇P❙ ✐♥❞✐❝❡s ✐♥❞❡❡❞ r❡♠❛✐♥ ✐♥✈❛r✐❛♥t ✉♥❞❡r ❝♦♠❜✐♥❡❞ ❛❝t✐♦♥ ♦❢ ♠♦♥♦❞r♦♠② tr❛♥s✲ ❢♦r♠❛t✐♦♥s ❛♥❞ s♣❡❝tr❛• ✢♦✇ tr❛♥s❢♦r♠❛t✐♦♥ ♦♥ t❤❡ ❝❤❛r❣❡ •❛tt✐❝❡ ✇✐t❤ t❤❡ s❛♠❡ ♣❛r❛♠❡t❡r✳ ❍❡♥❝❡ t❤❡ ♣r♦♦❢ ✐s ❝♦♠♣•❡t❡✳ ✼✳✹✳✸ ■♥✈❛r✐❛♥❝❡ ✉♥❞❡r ❍❡✐s❡♥❜❡r❣ tr❛♥s❢♦r♠❛t✐♦♥ ❋♦r ❛ ❞✐r❡❝t ❝❤❡❝❦ ♦❢ t❤❡ ❣r♦✉♣ •❛✇ ♣r❡s❡♥t❡❞ ✐♥ ❝❤❛♣t❡r ✻✱ ♦♥❡ ❤❛s t♦ ❛•s♦ ✈❡r✐❢② t❤❛t t❤❡ ❛❝t✐♦♥ ♦❢ ❍❡✐s❡♥❜❡r❣ s❤✐❢ts ❦❡❡♣ t❤❡ ❝♦♥t❛❝t str✉❝t✉r❡ ✐♥✈❛r✐❛♥t✳ ❋r♦♠ t❤❡ ❝♦♠♠✉t❛t♦rs ❣✐✈❡♥ ✐♥ ❝❤❛♣t❡r ✻ ❝•❡❛r•②✱ t❤❡ ♦♥•② ❍❡✐s❡♥❜❡r❣ tr❛♥s❢♦r♠❛t✐♦♥ ❣❡♥❡r❛t♦r t❤❛t ❝❛♥ ♣♦ss✐❜•② ❤❛✈❡ ♥♦♥✲tr✐✈✐❛• ❛❝t✐♦♥ ♦♥ t❤❡ ✜✈❡❜r❛♥❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ ✭✼✳✷✹✮♦r t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥ ✭✼✳✶✽✮ ✐s T (1) 0,η a ✳ ■ts ❛❝t✐♦♥ ❛t t❤❡ •❡✈❡• ♦❢ t❤❡ t✇✐st♦r s♣❛❝❡ ✐s ❣✐✈❡♥ ❜②✱ T (1) 0,η a : ξ a → ξ aη a , ξa → ξa + A ab η b , ξ0 → ξ0 + c 2,a 24 η a , α → α + η a ξa + 1 2 A ab η a η b . ✭✼✳✹✶✮ ❆s ✇❛s t❤❡ ❝❛s❡ ✇✐t❤ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r♠❛t✐♦♥✱ t♦ ❡♥s✉r❡ ✐♥✈❛r✐❛♥❝❡ ♦❢ M HM ❛❢t❡r ✜✈❡❜r❛♥❡ ✐♥st❛♥t♦♥ ❝♦rr❡❝t✐♦♥s✱ ♦♥❡ ❤❛s t♦ ♣❡r❢♦r♠ ❛ s♣❡❝tr❛• ✢♦✇ tr❛♥s❢♦r✲ ♠❛t✐♦♥ ♦♥ t❤❡ ❝❤❛r❣❡ •❛tt✐❝❡ γ → γ[ǫ]✱ ❜✉t ♥♦✇ ✇✐t❤ t❤❡ ♣❛r❛♠❡t❡r ǫ a = k η a p 0 ✳ ❚❤✉s✱ ✇❡ ❤❛✈❡ t♦ ❡st❛❜•✐s❤ t❤❛t T 0,η a • h η a • ℓ k,p;γ = ℓ k,p;γ[-kη a /p 0 ] . ✭✼✳✹✷✮ ❋♦r t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s✱ t❤❡ ✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ ❝♦♥t❛❝t str✉❝t✉r❡ ✐s ❡♥❝♦❞❡❞ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ✐♥t♦ ❡❛❝❤ ♦t❤❡r✱ T 0,η a • H [γ[kη a /p 0 ]] t❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥ ✐♥ ✭✼✳✹✷✮ ✐s ❛ r❡q✉✐r❡♠❡♥t ❢♦r t❤❡ ❝♦♥t❛❝t str✉❝t✉r❡ t♦ ❜❡ ✐♥✈❛r✐❛♥t✱ ✇❡ ❤❛✈❡ t♦ ❝❤❡❝❦ t❤❡ ✜rst ♦♥❡ t❤❛t ♠❛♣s ❝♦♥t❛❝t ❤❛♠✐•t♦♥✐❛♥s t♦ ❡❛❝❤ ♦t❤❡r ❢♦r ❝♦♥s✐st❡♥❝②✳ ▲❡t ✉s ❝♦♠♣✉t❡ t❤❡ ♣❤❛s❡ ν(η) ✐♥ T 0,η a h ❞✐r❡❝t ❝❛•❝✉•❛t✐♦♥ •❡❛❞s t♦ ν(η) = E k 2p 0 κ abc p a p b η c + η a η b + η a A ab p bap 0 k p Λ L Λ (ǫ) . ✭✼✳✹✹✮ ❯s✐♥❣ ✭✺✳✻✮ t❤❡ ❛❜♦✈❡ ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s ν(η) = E p 0 c 2,a η a c 2 s (d, c) -cp 0 8 (1c) + a 12 c 2 -1 -(a -1) (c + 1) A ab η a p b , ✭✼✳✹✺✮ ✾✺ ✼✳ ▼✐rr♦r s②♠♠❡tr② ✐♥ ❛❝t✐♦♥ ✿ ✐♥st❛♥t♦♥s ✐♥ t②♣❡ ■■❇ ❢♦r♠✉•❛t✐♦♥ ✇❤❡r❡ ✇❡ ♣r❡❢❡rr❡❞ t♦ ✇r✐t❡ t❤❡ r❡s✉•t ✐♥ t❡r♠s ♦❢ c = k/p 0 ❛♥❞ d = p/p 0 ✳ ◆♦✇ ♥♦t❡ t❤❛t t❤❡ r❡•❛t✐♦♥ adbc = 1 ❡♥s✉r❡s t❤❛t a ❛♥❞ c ❝❛♥ ♥♦t ❜❡ s✐♠✉•t❛♥❡♦✉s•② ❡✈❡♥✳ ❚❤❡r❡❢♦r❡✱ t❤❡ •❛st t❡r♠ ✐♥ ✭✼✳✹✺✮ ✐s ❛♥ ✐♥t❡❣❡r ❛♥❞ t❤✉s ❞✐s❛♣♣❡❛rs✳ ❋✉rt❤❡r♠♦r❡✱ ✉s✐♥❣ t❤❡ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ❉❡❞❡❦✐♥❞ s✉♠ ✭✼✳✷✾✮ ❛♥❞ t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤❛t c 2,a η a ✐s ❡✈❡♥✱ ♦♥❡ ✜♥❞s ν(η) = E p 0 2,a η a (c -t❤❡ s✐t✉❛t✐♦♥ ✐s s✐♠✐•❛r t♦ t❤❡ ❝❛s❡ ♦❢ ♠♦♥♦❞r♦♠②✳ ❚❤❡ ✜rst t❡r♠ ❝❛♥ ❜❡ ❞r♦♣♣❡❞ ❜❡❝❛✉s❡ 1 4 c(c -1)p 0 (p 0 -1) ∈ Z✳ ❚❤❡ r❡st ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s ν(η) = E d 12 c 2,a η a (c + 1)(c -1)(a -1)(a + 1) , ✭✼✳✹✼✮ ✇❤✐❝❤ ✐s ❡q✉❛• t♦ ♦♥❡✱ ❛s 6|(c + 1)(c -1)(a -1)(a + 1)✳ ❚❤✐s ❡♥❞s t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ ✜✈❡❜r❛♥❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ ✉♥❞❡r ❍❡✐s❡♥❜❡r❣ tr❛♥s❢♦r♠❛t✐♦♥s✳ ❚❤✉s ✇❡ ❤❛✈❡ ♣r♦✈❡♥ ❜② ❞✐r❡❝t ❛❝t✐♦♥ ♦❢ t❤❡ s②♠♠❡tr✐❡s ♦♥ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ t❤❛t t❤❡ ❣r♦✉♣ •❛✇ ✭✻✳✶✶✮ ❤♦•❞s ✇✐t❤ t❤❡ ♠♦❞✐✜❡❞ ♠♦♥♦❞r♦♠② tr❛♥s❢♦r✲ ♠❛t✐♦♥s✱ ✇❤✐❝❤ ✇❛s ♦♥❡ ♦❢ t❤❡ ♠❛•♦r r❡s✉•ts ✐♥ ❬✷✹(✳ ❇❡❢♦r❡ ✇❡ ✜♥✐s❤✱ ✇❡ ✇♦✉•❞ •✐❦❡ t♦ ♣♦✐♥t ♦✉t s♦♠❡ s❤♦rt❝♦♠✐♥❣s ♦❢ ♦✉r ❝♦♥str✉❝✲ t✐♦♥s✳ ❚❤❡ t✇✐st♦r s♣❛❝❡ ♣r❡s❡♥t❡❞ ✐♥ ✜❣✉r❡ ✼✳✸ ❤❛s ❛ r❛t❤❡r ❝♦♠♣•✐❝❛t❡❞ str✉❝t✉r❡✳ ❚❤❡r❡ ❛r❡ ♦♣❡♥ ♣❛t❝❤❡s s✉rr♦✉♥❞✐♥❣ t m,n ± ✇❤✐❝❤ •✐❡ ❛•♦♥❣ t❤❡ r❡❛• •✐♥❡✳ ❖♥ t♦♣ ♦❢ t❤❛t✱ t❤❡r❡ ❛r❡ ✜✈❡❜r❛♥❡ ❝♦♥t♦✉rs t❤❛t ❞✐✈✐❞❡ t❤❡ t✇✐st♦r s♣❛❝❡ ✐♥t♦ s❡✈❡r❛• ♦t❤❡r ♣❛t❝❤❡s ❛s t❤❡② ❝♦♥♥❡❝t t❤❡ ♣♦✐♥ts t m,n ± ✳ ❚❤❡ ✜rst ♣r♦❜•❡♠ ❝♦♠❡s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t ✇❡ ❤❛✈❡ ❛ss✉♠❡❞ t❤❡ ♣♦✐♥ts t m,n ± ❞♦ ♥♦t ❤❛✈❡ ❛♥② ❛❝❝✉♠✉•❛t✐♦♥ ♣♦✐♥t ♦♥t❤❡ r❡❛• •✐♥❡✱ t❤❛t ✐s t❤❡② ❛r❡ s❡♣❛r❛❜•❡✳ ❆•t❤♦✉❣❤✱ ✐t s❡❡♠s t♦ ❜❡ ❛ ♠✐•❞ ❛ss✉♠♣t✐♦♥✱ ✐t ✐s ❞❡s✐r❛❜•❡ t♦ ❣❡t r✐❞ ♦❢ t❤✐s✳ ❆♥ ❡✈❡♥ ♠♦r❡ s❡r✐♦✉s ♣r♦❜•❡♠ ❛r✐s❡s ✇✐t❤ t❤❡ ✜✈❡❜r❛♥❡ ❝♦♥t♦✉rs✳ ❚❤❡② ✐♥t❡rs❡❝t ✐♥ ✐♥✜♥✐t❡•② ♠❛♥② ♣♦✐♥ts ♦♥ t❤❡ t✇✐st♦r s♣❛❝❡✳ ❯♥❢♦rt✉♥❛t❡•②✱ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ t❤❛t ✇❡ ❤❛✈❡ ❞❡r✐✈❡❞ ✭✼✳✷✹✮ ✐s ♥♦t ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ ❝♦❝②❝•❡ ❝♦♥❞✐t✐♦♥ ❛r♦✉♥❞ t❤❡♠✳ ❚❤❡ s✐t✉❛t✐♦♥ ❝❛♥ ❣❡t ❡✈❡♥ ✇♦rs❡ ✐❢ t❤❡ ♣♦✐♥ts ❢♦r♠ ❛ ❞❡♥s❡ s❡t ♦♥ t❤❡ t✇✐st♦r s♣❛❝❡✳ ❯♥❢♦rt✉♥❛t❡•②✱ t❤❡ ❛✉t❤♦r ❞♦❡s ♥♦t ❦♥♦✇ ❤♦✇ t♦ r❡❝♦♥❝✐•❡ SL(2, Z) ✐♥✈❛r✐❛♥❝❡ ✇✐t❤ t❤❡ ❝♦❝②❝•✐❝✐t② ❛t t❤❡s❡ ♣♦✐♥ts ❛♥❞ t❤✐s r❡♥❞❡rs t❤❡ ♣r♦❜•❡♠ ❢♦r ❢✉t✉r❡ ✐♥✈❡st✐❣❛t✐♦♥s✳ ✾✻

  ✐s

	Tr P exp 2π	∂Σ	ω .	✭✻✳✷✵✮
	❚❤❡ s✐❣♥ ❛♠❜✐❣✉✐t② ❤❡r❡ ❝♦♠❡s ❞✉❡ t♦ t❤❡ ❢❛❝t t❤❛t t❤❡r❡ ❛r❡ t✇♦ ✇❛②s t♦ •✐❢t