Lie Fu
email: fu@math.univ-lyon1.fr

Zhiyu Tian
email: zhiyu.tian@univ-grenoble-alpes.fr

Zhiyu Tian Motivic Multiplicative

Mckay Correspondence

MOTIVIC MULTIPLICATIVE MCKAY CORRESPONDENCE FOR SURFACES

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Finite subgroups of SL 2 (C) are classically studied by Klein [START_REF] Klein | Lectures on the icosahedron and the solution of equations of the fifth degree[END_REF] and Du Val [START_REF] Du | Homographies, quaternions and rotations[END_REF]. A complete classification (up to conjugacy) is available : cyclic, binary dihedral, binary tetrahedral, binary octahedral and binary icosahedral. The last three types correspond to the groups of symmetries of Platonic solids 1 as the names indicate. Let G ⊂ SL 2 (C) be such a (non-trivial) finite subgroup acting naturally on the vector space V := C 2 . The quotient X := V/G has a unique rational double point 2 . Let f : Y → X be the minimal resolution of singularities:

V π Y f / / X
which is a crepant resolution, that is, K Y = f * K X . The exceptional divisor, denoted by E, consists of a bunch of (-2)-curves 3 .

The classical McKay correspondence ( [START_REF] Mckay | Graphs, singularities, and finite groups[END_REF], cf. also [START_REF] Reid | La correspondance de McKay[END_REF]) establishes an bijection between the set Irr (G) of non-trivial irreducible representations of G on the one hand and the set Irr(E) of irreducible components of E on the other hand :

Irr (G)
Irr(E) ρ → E ρ .

Thus E = ρ∈Irr (G) E ρ . Moreover, this bijection respects the 'incidence relations' : precisely, for any ρ 1 ρ 2 ∈ Irr (G), the intersection number (E ρ 1 • E ρ 2 ), which is 0 or 1, is equal to the multiplicity of ρ 2 in ρ 1 ⊗ V (hence is also equal to the multiplicity of ρ 1 in ρ 2 ⊗ V), where V is the 2-dimensional natural representation via G ⊂ SL(V). All these informations can be encoded into Dynkin diagrams of A-D-E type, which is on the one hand the dual graph of the exceptional divisor E and on the other hand the McKay graph of the non-trivial irreducible representations of G, with respect to the prefered representation V. Apart from the original observation of McKay, there are many approaches to construct this correspondence geometrically and to extend it to higher dimensions : K-theory of sheaves [START_REF] Gonzalez-Sprinberg | Construction géométrique de la correspondance de McKay[END_REF], G-Hilbert schemes [25], [START_REF] Ito | McKay correspondence and Hilbert schemes[END_REF], [START_REF] Ito | McKay correspondence and Hilbert schemes in dimension three[END_REF], [START_REF] Ito | Hilbert schemes and simple singularities[END_REF], motivic integration [START_REF] Victor | Stringy Hodge numbers of varieties with Gorenstein canonical singularities, Integrable systems and algebraic geometry[END_REF], [START_REF]Birational Calabi-Yau n-folds have equal Betti numbers[END_REF], [START_REF] Denef | Germs of arcs on singular algebraic varieties and motivic integration[END_REF], [START_REF]Motivic integration, quotient singularities and the McKay correspondence[END_REF], [START_REF] Yasuda | Twisted jets, motivic measures and orbifold cohomology[END_REF], [START_REF] Lupercio | The global McKay-Ruan correspondence via motivic integration[END_REF] and derived categories [START_REF] Bridgeland | The McKay correspondence as an equivalence of derived categories[END_REF] etc. We refer the reader to Reid's note of his Bourbaki talk [START_REF] Reid | La correspondance de McKay[END_REF] for more details and history.

Following Reid [25], one can recast the above McKay correspondence (the bijection) as follows: the isomorphism classes of irreducible representations index a basis of the homology of the resolution Y. This is of course equivalent to say that the conjugacy classes of G index a basis of the cohomology of Y. We remark that, which is the starting point of this paper, the quotient X = V/G has a natural orbifold structure, meaning that X underlies the smooth Deligne-Mumford stack X := [V/G], and the (co)homology of the coarse moduli space |IX| of its inertia stack IX has a basis indexed by the conjugacy classes of G. Thus Reid's McKay correspondence can be stated as an isomorphism of vector spaces:

H * (Y) H * (|IX|). Chen and Ruan defined in [START_REF] Chen | A new cohomology theory of orbifold[END_REF] the orbifold cohomology H * orb ([V/G]), whose underlying vector space is exactly the cohomology of |IX| ; the supplementary ingredient is that they can put a highly nontrivial (associative and commutative) ring structure, the so-called orbifold product, on this orbifold cohomology. See Definition 2.1 for a down-to-earth construction. Therefore it is natural to ask whether there is a multiplicative isomorphism (of algebras)

H * (Y) H * orb ([V/G]
). However none of aforementioned beautiful theories takes care of the multiplicative structures. Nevertheless, the existence of such an isomorphism of algebras is known. For example, it is a baby case of the result of Ginzburg-Kaledin [START_REF] Ginzburg | Poisson deformations of symplectic quotient singularities[END_REF] on symplectic resolutions of symplectic quotient singularities. An explicit formula is proposed by Bryan-Graber-Pandharipande in [START_REF] Bryan | The orbifold quantum cohomology of C 2 /Z 3 and Hurwitz-Hodge integrals[END_REF], which is used there to prove (a stronger version of) the C 2 /Z 3 case. We will also use this formula to construct our multiplicative isomorphism. This isomorphism fits perfectly into Ruan's following more general Cohomological Crepant Resolution Conjecture (CCRC) : Conjecture 1.1 (CCRC [START_REF] Ruan | The cohomology ring of crepant resolutions of orbifolds, Gromov-Witten theory of spin curves and orbifolds[END_REF]). Let M be a smooth projective variety endowed with a faithful action of a finite group G. Assume that the quotient X := M/G is Gorenstein then for any crepant resolution Y → X, there is an isomorphism of graded C-algebras:

(1) C) . More generally, given a smooth proper orbifold X with underlying singular variety X being Gorenstein, then for any crepant resolution Y → X, we have an isomorphism of graded C-algebras:

H * qc (Y, C) H * orb ([M/G],
H * qc (Y, C) H * orb (X, C) .
Here the left hand side is the quantum corrected cohomology algebra, whose underlying graded vector space is just H * (Y, C), endowed with the cup product with quantum corrections related to Gromov-Witten invariants with curve classes contracted by the crepant resolution, as defined in [START_REF] Ruan | The cohomology ring of crepant resolutions of orbifolds, Gromov-Witten theory of spin curves and orbifolds[END_REF]. Since we only consider in this paper the 2-dimensional situation, the Gromov-Witten invariants always vanish hence there are no quantum corrections involved. See Lemma 2.3 for this vanishing. Conjecture 1.1 suggests that one should consider the existence of such multiplicative McKay correspondence in the global situation (instead of a quotient of a vector space by a finite group), that is, a Gorenstein quotient of a surface by a finite group action, or even more generally a 2-dimensional proper Gorenstein orbifold. Our following main result confirms this, which also pushes the (surface) McKay correspondence to the motivic level: Theorem 1.2 (Motivic multiplicative global McKay correspondence). Let X be a smooth proper 2dimensional Deligne-Mumford stack with isolated stacky points. Assume that X has projective coarse moduli space X with Gorenstein singularities. Let Y → X be the minimal resolution. Then we have an isomorphism of algebra objects in the category CHM C of Chow motives with complex coefficients:

(2) h(Y) C h orb (X) C .
In particular, one has an isomorphism of C-algebras:

CH * (Y) C CH * orb (X) C ; H * (Y, C) H * orb (X, C) ; K(Y) C K orb (X) C ; K top (Y) C K top orb (X) C .
This result also confirms the 2-dimensional case of the so-called Motivic HyperKähler Resolution Conjecture studied in [START_REF] Fu | Motivic Hyper-Kähler Resolution Conjecture : I. Generalized Kummer varieties[END_REF] and [START_REF] Fu | Motivic Hype-Kähler Resolution Conjecture : II. Hilbert schemes of K3 surfaces[END_REF].

As the definitions of the orbifold theories are particularly explicit and elementary for the global quotient stacks (cf. §2.1), we deliberately treat the global quotient case ( §3) and the general case ( §4) separately.

Convention : All Chow rings are with rational coefficients unless otherwise stated. CHM is the category of Chow motives with rational coefficients. h : SmProj op → CHM is the (contravariant) functor that associates a smooth projective variety its Chow motive and a morphism its graph as correspondence. An orbifold means a separated Deligne-Mumford stack of finite type with trivial stabilizer at the generic point.

Acknowledgement :

The authors want to thank Cédric Bonnafé, Philippe Caldero, Jér ôme Germoni and Dmitry Kaledin for interesting discussions. The most part of the paper is prepared when L. Fu is staying with his family at the Hausdorff Institute of Mathematics for the trimester program on K-theory. He thanks Bonn University for providing the perfect working condition in HIM and the relaxing living style in such a beautiful city.

Crepant resolution conjecture

Let us briefly recall the construction of the orbifold Chow motive (as an algebra object) and the orbifold Chow ring. We will first give the down-to-earth definition for an orbifold which is a global Gorenstein quotient by a finite group ; then we invoke the techniques in [START_REF] Abramovich | Gromov-Witten theory of Deligne-Mumford stacks[END_REF] to give the construction in the general setting of Deligne-Mumford stacks. We refer to our previous work [START_REF] Fu | Motivic Hyper-Kähler Resolution Conjecture : I. Generalized Kummer varieties[END_REF] (joint with Charles Vial), [START_REF] Fu | Motivic Hype-Kähler Resolution Conjecture : II. Hilbert schemes of K3 surfaces[END_REF] as well as the original sources (for cohomology and Chow rings) [START_REF] Chen | A new cohomology theory of orbifold[END_REF], [START_REF] Fantechi | Orbifold cohomology for global quotients[END_REF], [START_REF] Abramovich | Gromov-Witten theory of Deligne-Mumford stacks[END_REF], [START_REF] Jarvis | Stringy K-theory and the Chern character[END_REF] for the history and more details.

2.1. Orbifold theory : global quotient case. Let M be a smooth projective variety and G be a finite group acting faithfully on M. Assume that G preserves locally the canonical bundle : for any x ∈ M fixed by ∈ G, the differential D ∈ SL(T x M). This is equivalent to require that the quotient X := M/G has only Gorenstein singularities. Denote by M = x ∈ M | x = x the fixed locus of ∈ G, M ,h = M ∩ M h (with the reduced scheme structure) and X := [M/G] the quotient smooth Deligne-Mumford stack. Definition 2.1 (Orbifold theories). We define an auxiliary algebra object h(M, G) in CHM with G-action, and the orbifold motive h([M/G]) will be its subalgebra of invariants. The definitions for Chow rings and cohomology are similar.

(1

( )| Z := r-1 j=0 j r rank(W j ),
where r is the order of , W j is the eigen-sub-bundle of the restricted tangent bundle TM| Z , for the natural automorphism induced by , with eigenvalue e 2πi r j . The age function is invariant under conjugacy.

(2 • ) We endow the direct sums

h(M, G) := ∈G h(M )(-age( )) CH * (M, G) := ∈G CH * -age( ) (M ) H * (M, G) := ∈G H * -2 age( ) (M ) K(M, G) := ∈G K(M )
with the natural G-action induced by the following action: for any , h ∈ G,

h : M -→ M h h -1
x → hx.

(3 • ) For any ∈ G, define V := r-1 j=0 j r [W j ] ∈ K 0 (M ) Q ,
whose virtual rank is age( ), where r and W j 's are as in (1 • ).

(

• ) For any 1 , 2 ∈ G, let 3 = -1 2 -1 4
1 , we define the (virtual class of ) the obstruction bundle on the fixed locus M 1 , 2 by

(3) F 1 , 2 := V 1 M < 1 , 2 > + V 2 M < 1 , 2 > + V 3 M < 1 , 2 > + TM < 1 , 2 > -TM| M < 1 , 2 > ∈ K 0 M < 1 , 2 > Q .
(5 • ) The orbifold product orb is defined as follows: given , h ∈ G, let ι : M < ,h> → M be the natural inclusion.

• For cohomology:

orb : H i-2 age( ) (M ) × H j-2 age(h) (M h ) → H i+j-2 age( h) (M h ) (α, β) → ι * α| M < ,h> β| M < ,h> c top (F ,h )
• For Chow groups:

orb : CH i-age( ) (M ) × CH j-age(h) (M h ) → CH i+j-age( h) (M h ) (α, β) → ι * α| M < ,h> • β| M < ,h> • c top (F ,h )
• For K-theory:

orb : K(M ) × K(M h ) → K(M h ) (α, β) → ι ! α| M < ,h> • β| M < ,h> • λ -1 (F ∨ ,h ) • For motives: orb : h(M )(-age( )) ⊗ h(M h )(-age(h)) → h(M h )(-age( h)) is determined by the correspondence δ * (c top (F ,h )) ∈ CH dim M +dim M h +age( )+age(h)-age( h) (M × M h × M h ),
where δ : M < ,h> → M × M h × M h is the natural morphism sending x to (x, x, x). (6 • ) Finally, we take the subalgebra of invariants whose existence is guaranteed by the idempotent completeness of CHM :

h orb ([M/G]) := h (M, G) G ; CH * orb ([M/G]) := (CH * (M, G), orb ) G ; and similarly H * orb ([M/G]) := (H * (M, G), orb ) G ; K orb ([M/G]) := (K(M, G), orb ) G .
These are commutative Q-algebras and depend only on the stack [M/G] (not the presentation).

2.2. Orbifold theory : general case. Let X be a smooth proper orbifold with projective coarse moduli space X with Gorenstein singularities. Recall that under the Gorenstein assumption, the age function takes values in integers. Define the orbifold Chow motive and orbifold Chow group as follows:

h orb (X) := h(IX)(-age) := ⊕ i h(IX i )(-age i ), CH * orb (X) := CH * -age (IX) := ⊕ i CH * -age i (IX i ) ;
where the theory of Chow ring (with rational coefficients) as well as the intersection theory of a stack is the one developed by Vistoli in [START_REF] Vistoli | Intersection theory on algebraic stacks and on their moduli spaces[END_REF] ; the theory of Chow motives for smooth proper Deligne-Mumford stacks is the so-called DMC motives4 developed by Behrend-Manin in [START_REF] Behrend | Stacks of stable maps and Gromov-Witten invariants[END_REF] and reviewed in Toën [28, §2. First construction], which is proven in [START_REF] Toën | On motives for Deligne-Mumford stacks[END_REF]Theorem 2.1] to be equivalent to the usual category of Chow motives ; IX = i IX i is the decomposition into connected components while the age function age is the locally constant function whose value on IX i is age i which is Chen-Ruan's degree shifting number defined in [8, §3.2]. Let us also point out that Toën's second construction in [28, §3] of Chow motives of Delign-Mumford stacks is very close to the orbifold Chow motive defined above with the only difference being the age-shifting. Now the key point is to put a product structure on h orb (X) and CH * orb (X). Consider the moduli space K 0,3 (X, 0), constructed by Abramovich-Vistoli [START_REF] Abramovich | Compactifying the space of stable maps[END_REF], of 3-pointed twisted stable maps of genus zero with trivial curve class. It comes equipped with a virtual fundamental class [K 0,3 (X, 0)] vir ∈ CH dim X K 0,3 (X, 0) together with three (proper) evaluation maps: e i : K 0,3 (X, 0) → IX with target being the inertia stack ( [START_REF] Abramovich | Gromov-Witten theory of Deligne-Mumford stacks[END_REF]). Note that in general, the evaluation morphism has target in a different stack, the rigidified cyclotomic inertial stack ([1, Section 3.4]). However, in the smooth orbifold case, one can prove that the evaluation morphisms of the degree 0 twisted stable maps land in the inertial stack [START_REF] Edidin | Logarithmic trace and orbifold products[END_REF]Section 1.3.1].

Pushing forward the virtual fundamental class gives the class

γ := (e 1 , e 2 , ě3 ) * [K 0,3 (X, 0)] vir ∈ CH dim X (IX 3 ),
where ě3 is the composition of the evaluation map e 3 and the involution IX → IX inverting the group element (cf. [START_REF] Abramovich | Gromov-Witten theory of Deligne-Mumford stacks[END_REF]); and we are using again Visotli's Chow groups ( [START_REF] Vistoli | Intersection theory on algebraic stacks and on their moduli spaces[END_REF]). The orbifold product for the orbifold Chow ring is defined as the action of the correspondence γ:

CH * orb (X)× CH * orb (X) → CH * orb (X) CH * -age (IX)× CH * -age (IX) → CH * -age (IX) (α, β) → pr 3, * pr * 1 (α) • pr * 2 (β) • γ It can be checked (cf. [1, Theorem 7.4.1]
) that the age shifting makes the above orbifold product additive with respect to the degrees (otherwise, it is not!). Similarly, we can define the multiplicative structure on h orb (X) to be

γ ∈ CH dim X (IX 3 ) = Hom CHM h(IX)(-age) ⊗ h(IX)(-age), h(IX)(-age) = Hom CHM (h orb (X) ⊗ h orb (X), h orb (X)) .
Thanks to [1, Theorem 7.4.1], this product structure is associative. On the other hand, when X is a finite group global quotient stack, the main result of [21, §8] implies that the elementary construction in §2.1 actually recovers the above abstract construction.

Crepant resolution conjectures.

With orbifold theories being defined, we can speculate that a motivic or K-theoretic version of the Crepant Resolution Conjecture 1.1 should hold. But the problem is that in the definition of the quantum corrections, there is the subtle convergence property which is difficult to make sense in general for Chow groups / motives or for K-theory. Therefore, we will look at some cases that these quantum corrections actually vanish a priori : Case 1: Hyper-Kähler resolution. The first one is when the resolution Y is holomorphic symplectic, which implies that all (Chow-theoretic, K-theoretic or cohomological) Gromov-Witten invariants vanish (see the proof of [START_REF] Fu | Motivic Hype-Kähler Resolution Conjecture : II. Hilbert schemes of K3 surfaces[END_REF]Lemma 8.1]). In this case, we indeed have the following Motivic HyperKähler Resolution Conjecture (MHRC), proposed in [START_REF] Fu | Motivic Hyper-Kähler Resolution Conjecture : I. Generalized Kummer varieties[END_REF]: Conjecture 2.2 (MHRC [START_REF] Fu | Motivic Hyper-Kähler Resolution Conjecture : I. Generalized Kummer varieties[END_REF], [START_REF] Fu | Motivic Hype-Kähler Resolution Conjecture : II. Hilbert schemes of K3 surfaces[END_REF]). Let M be a smooth projective holomorphic symplectic variety endowed with a faithful symplectic action of a finite group G. If quotient X := M/G has a crepant resolution Y → X, then there is an isomorphism of algebra object in CHM C :

h(Y) h orb ([M/G]).
In particular, we have an isomorphism of graded C-algebras:

CH * (Y) C CH * orb ([M/G]) C .
Thanks to the orbifold Chern character isomorphism constructed by Jarvis-Kaufmann-Kimura in [START_REF] Jarvis | Stringy K-theory and the Chern character[END_REF], MHRC also implies the K-theoretic HyperKähler Resolution Conjecture of loc.cit. . Conjecture 2.2 is proven in our joint work with Charles Vial [START_REF] Fu | Motivic Hyper-Kähler Resolution Conjecture : I. Generalized Kummer varieties[END_REF] for Hilbert schemes of abelian varieties and generalized Kummer varieties and in [START_REF] Fu | Motivic Hype-Kähler Resolution Conjecture : II. Hilbert schemes of K3 surfaces[END_REF] for Hilbert schemes of K3 surfaces.

Case 2: Surface minimal resolution. The second one is the main purpose of the article, namely the surface case, i.e. dim(Y) = 2. In this case, the vanishing of quantum corrections is explained in the following lemma. Lemma 2.3. Let X be a surface with Du Val singularities and π : Y → X be the minimal resolution. Then the virtual fundamental class of M 0,3 Y, β is rationally equivalent to zero for any curve class β which is contracted by π.

Proof. Consider the forgetful-stabilization morphism

f : M 0,3 Y, β → M 0,0 Y, β .
By the general theory, the virtual fundamental class of M 0,3 Y, β is the pull-back of the virtual fundamental class of M 0,0 Y, β . However, the virtual dimension of M 0,0 Y, β is (β•K Y )+(dim Y-3) = -1 since π is crepant. Therefore, both moduli spaces have zero virtual fundamental class in Chow group, cohomology or K-theory.

Thanks to the vanishing of quantum corrections, the motivic version of the Crepant Resolution Conjecture 1.1 for surfaces is exactly the content of our main Theorem 1.2. See the precise statement in Introduction. We will first give the proof for stacks which are finite group quotients in §3, then the proof in the general case in §4.

Proof of Theorem 1.2 : global quotient case

In this section, we show Theorem 1.2 in the following setting: S is a smooth projective surface, G is a finite group acting faithfully on S such that the canonical bundle is locally preserved (Gorenstein condition), X := S/G is the quotient surface (with Du Val singularities) and Y → X is the minimal (crepant) resolution. We denote by L := 1(-1) the Lefschetz motive in CHM.

For any x ∈ S, let G x := { ∈ G | x = x} be the stabilizer. Let Irr(G x ) be the set of isomorphism classes of irreducible representations of G x and Irr (G x ) be that of non-trivial ones. We remark that by assumption, there are only finitely many points of S with non-trivial stabilizer.

Resolution side. We first compute the Chow motive algebra (or Chow ring) of the minimal resolution Y.

For any x ∈ S, we denote by x its image in S/G. The Chow motive of Y has the decomposition in CHM ( 4)

h (Y) h(S) G ⊕ x∈S/G ρ∈Irr (G x ) L x,ρ         h(S) ⊕ x∈S ρ∈Irr (G x ) Lx,ρ         G ,
where Lx,ρ is the Lefschetz motive corresponding to the irreducible component of the exceptional divisor over x, indexed by the non-trivial irreducible representation ρ of G x via the classical McKay correspondence. The product structure is determined as follows via the above decomposition, which is also part of the classical McKay correspondence. Let i x : {x} → S be the natural inclusion.

• h(S) ⊗ h(S)

δ S -→ h(S) is the usual product induced by the small diagonal of S 3 . • For any x with nontrivial stabilizer G x and any ρ ∈ Irr (G x ),

h(S) ⊗ Lx,ρ i * x -→ Lx,ρ
is determined by the class x ∈ CH 2 (S) = Hom(h(S) ⊗ L, L).

• For any ρ ∈ Irr (G x ) as above,

Lx,ρ ⊗ Lx,ρ -2i x, * ----→ h(S), is determined by -2x ∈ CH 2 (S). • For any ρ 1 ρ 2 ∈ Irr (G x ),
-If they are adjacent, that is, ρ 1 appears (with multiplicity 1) in the G x -module ρ 2 ⊗ T x S, then First of all, it is easy to see that age( ) = 1 for any element id of G, and age(id) = 0. By Definition 2.1,

Lx,ρ 1 ⊗ Lx,ρ 2 i x, * --→ h(S),
(6) h(S, G) = h(S) ⊕ ∈G id x∈S Lx, = h(S) ⊕ x∈S ∈G x id Lx, ,
where Lx, is the Lefschetz motive 1(-1) indexed by the fixed point x of .

Lemma 3.1 (Obstruction class).

For any , h ∈ G different from id, the obstruction class is

c ,h =      1 if = h -1 0 if h -1
Proof. For any id and any x ∈ S , the action of on T x S is diagonalizable with a pair of conjugate eigenvalues, therefore V in Definition 2.1 is a trivial vector bundle of rank one on S . Hence for any , h ∈ G different from id and x ∈ S fixed by and h, the dimension of the fiber of the obstruction bundle

F ,h at x is dim F ,h (x) = dim V (x) + dim V h (x) + dim V ( h) -1 (x) -dim T x S, which is 1 if h -1 and is 0 if = h -1 .
The computation of c ,h follows.

Once the obstruction classes are computed, we can write down explicitly the orbifold product from Definition 2.1, which is summarized in the following proposition.

Proposition 3.2. The orbifold product on h(S, G) is given as follows via the decomposition (6):

h(S) ⊗ h(S) δ S -→ h(S); h(S) ⊗ Lx, i * x -→ Lx, ∀ x = x; Lx, ⊗ L x, -1 i x, * --→ h(S).
where the first morphism is the usual product given by small diagonal; the second and the third morphisms are given by the class x ∈ CH 2 (S) and i x : {x} → S is the natural inclusion; all the other possible maps are zero.

The G-action on ( 6) is as follows by Definition 2.1:

• The G-action on h(S) is the original action.

• For any h ∈ G, it maps for any x ∈ S and id ∈ G x , the Lefschetz motive Lx, isomorphically to L hx,h h -1 .

The multiplicative correspondence.

With both sides of the correspondence computed, we can give the multiplicative McKay correspondence morphism, which is in the category CHM C of complex Chow motives. [START_REF] Bryan | The orbifold quantum cohomology of C 2 /Z 3 and Hurwitz-Hodge integrals[END_REF] Φ : h(S)

⊕ x∈S ρ∈Irr (G x ) Lx,ρ → h(S) ⊕ x∈S ∈G x id
Lx, , which is given by the following 'matrix by blocs':

• id : h(S) → h(S);

• For each x ∈ S (with nontrivial stabilizer G x ), the morphism

ρ∈Irr (G x ) Lx,ρ → ∈G x id
Lx, is the 'matrix' with coefficient 1

√ |G x | χ ρ 0 ( ) -2 • χ ρ ( ) at place (ρ, ) ∈ Irr (G x ) × (G x \{id})
, where χ denotes the character, ρ 0 is the natural 2-dimensional representation T x S of G x . Note that ρ 0 ( ) has determinant 1, hence its trace χ ρ 0 ( ) is a real number.

• The other morphisms are zero.

To conclude the main theorem, one has to show three things: (i) Φ is compatible with the G-action; (ii) Φ is multiplicative and (iii) Φ induces an isomorphism Φ G of complex Chow motives on G-invariants.

Lemma 3.3. Φ is G-equivariant.
Proof. The G-action on the first direct summand h(S) is by definition the same, hence is preserved by Φ| h(S) = id. For the other direct summands, since it is a matrix computation, we can treat the Lefschtez motives as 1-dimensional vector spaces: let E x,ρ be the 'generator' of Lx,ρ and e x, be the 'generator' of Lx, . Then the G-actions computed in the previous subsections say that for any x and any h ∈ G x , h.E x,ρ = E hx,hρ and h.e x, = e hx,h h -1

,
where hρ is defined in [START_REF] Behrend | Stacks of stable maps and Gromov-Witten invariants[END_REF].

Therefore

Φ(h.E x,ρ ) = Φ(E hx,hρ ) = 1 √ |G hx | ∈G hx χ ρ 0 ( ) -2 χ hρ ( ) e hx, = 1 √ |G x | ∈G x χ ρ 0 ( ) -2 χ hρ (h h -1 ) e hx,h h -1 = 1 √ |G x | ∈G x χ ρ 0 ( ) -2 χ ρ ( ) e hx,h h -1 = 1 √ |G x | ∈G x χ ρ 0 ( ) -2 χ ρ ( ) h.e x, = h.Φ(E x,ρ ),
where the third equality is a change of variable: replace by h h -1 , the fourth equality follows from the definition of hρ in (5) Proposition 3.4 (Multiplicativity). Φ preserves the multiplication, i.e. Φ is a morphism of algebra objects in CHM C .

Proof. The cases of multiplying h(S) with itself or with a Lefschetz motive Lx,ρ are all obviously preserved by Φ. We only need to show that for any x ∈ S with non-trivial stabilizer G x , the morphism

ρ∈Irr (G x ) Lx,ρ → ∈G x id Lx,
given by the matrix with coefficient 1

√ |G x | χ ρ 0 ( ) -2 • χ ρ ( ) at place (ρ,
) is multiplicative (note that the result of the multiplication could go outside of these direct sums to h(S)). Since this is just a matrix computation, let us treat Lefschetz motives as 1-dimensional vector spaces (or equivalently, we are looking at the corresponding multiplicativity of the realization of Φ for Chow rings): let E x,ρ be the 'generator' of Lx,ρ and e x, be the 'generator' of Lx, . Then the computations of the products in the previous two subsections say that:

E x,ρ 1 • E x,ρ 2 =            -2x if ρ 1 = ρ 2 ; x if ρ 1 , ρ 2 are adjacent; 0 if ρ 1 , ρ 2 are not adjacent; (8) e x, • e x,h =      x if = h -1 ; 0 if h -1 ; (9)
Therefore for any ρ 1 , ρ 2 ∈ Irr (G x ), we have

Φ(E x,ρ 1 ) • Φ(E x,ρ 2 ) = 1 |G x | ∈G x h∈G x χ ρ 0 ( ) -2 χ ρ 0 (h) -2 χ ρ 1 ( )χ ρ 2 (h) e x, • e x,h = 1 |G x | ∈G x χ ρ 0 ( ) -2 χ ρ 0 ( -1 ) -2 χ ρ 1 ( )χ ρ 2 ( -1 ) • x = 1 |G x | ∈G x (χ ρ 0 ( ) -2) χ ρ 1 ( )χ ρ 2 ( ) • x = 1 |G x |         ∈G x χ ρ 0 ⊗ρ 1 ( )χ ρ 2 ( ) -2 ∈G x χ ρ 1 ( )χ ρ 2 ( )         • x = ρ 0 ⊗ ρ 1 , ρ 2 -2 ρ 1 , ρ 2 • x = Φ E x,ρ 1 • E x,ρ 2
where the first equality is the definition of Φ (and we add the non-existent e x,1 with coefficient 0), the second equality uses (9) the orthogonality among e x, 's (i.e. Lx, 's), the third equality uses the fact that χ ρ 0 takes real value; the last equality uses all three cases of [START_REF] Chen | A new cohomology theory of orbifold[END_REF].

where ρ runs over the set Irr (G x ) of isomorphism classes of non-trivial irreducible representations and [ ] runs over the set of conjugacy classes of G x different from id.

As this is about a matrix, it is enough to look at the realization of [START_REF]Motivic integration, quotient singularities and the McKay correspondence[END_REF]:

ρ∈Irr (G x ) E x,ρ →               ∈G x id e x,               G x
, where both sides come equipped with non-degenerate quadratic forms given by intersection numbers and degrees of the orbifold product respectively. More precisely, by ( 8) and ( 9):

(E x,ρ 1 • E x,ρ 2 ) =            -2 if ρ 1 = ρ 2 ; 1 if ρ 1 , ρ 2 are adjacent; 0 if ρ 1 , ρ 2 are not adjacent; (e x, • e x,h ) =      1 if = h -1 ; 0 if h -1 ;
which are both clearly non-degenerate. Now Proposition 3.4 shows that our matrix [START_REF] Du | Homographies, quaternions and rotations[END_REF] respects the non-degenerate quadratic forms on both sides, therefore it is non-degenerate.

Let us note here also an elemetary proof which does not use the orbifold product. We first remark that for any id, ρ 0 ( ) ∈ SL 2 (C) which is of finite order and different from the identity, hence its trace χ 0 ( ) 2. Therefore the non-degeneratedness of the matrix ( 11) is equivalent to the non-degeneratedness of the matrix χ ρ ( )

(ρ,[ ])
, which is obtained from the character table of the finite group G x by removing the first row (corresponding to the trivial representation) and the first column (corresponding to id ∈ G x ). The non-degeneratedness of this matrix is a completely general fact, which holds for all finite groups. We will give a proof in Lemma 3.6 at the end of this section.

The combination of Lemma 3.3, Proposition 3.4 and Proposition 3.5 proves the isomorphism of algebra objects (2) in the main Theorem 1.2 in the global quotient case. For the isomorphisms for the Chow rings and cohomology rings, it is enough to apply realization functors. For the isomorphisms for the K-theory and topological K-theory, it suffices to invoke the construction of orbifold Chern characters in [START_REF] Jarvis | Stringy K-theory and the Chern character[END_REF] which induce isomorphisms of algebras from (orbifold) K-theory to (orbifold) Chow ring as well as from (orbifold) topological K-theory to (orbifold) cohomology ring. The proof of Theorem 1.2 in the global quotient case is now complete.

The following lemma is used in the second proof of Proposition 3.5. The elegant proof below is due to Cédric Bonnafé. We thank him for allowing us to use it. Recall that for a finite group G, its character table is a square matrix whose rows are indexed by isomorphism classes of irreducible complex representations of G and columns are indexed by conjugacy classes of G. Lemma 3.6. Let G be any finite group. Then the matrix obtained from the character table by removing the first row corresponding to the trivial representation and the first column corresponding to the identity element, is non-degenerate.

Proof. Denote by 1 the trivial representation and by ρ 1 , • • • , ρ n the set of isomorphism classes of non-trivial representations of G. Suppose we have a linear combination n i=1 c i χ ρ i , with c i ∈ C, which vanishes for all non-identity conjugacy class, hence for all non-identity elements of G:

(12) n i=1 c i χ ρ i ( ) = 0, ∀ id ∈ G. Set c 0 := - 1 |G| n i=1 c i dim(ρ i ),
and denote by χ re be the character of the regular representation, then [START_REF] Edidin | Logarithmic trace and orbifold products[END_REF] implies that the following linear combination vanishes for all ∈ G:

c 0 χ re + n i=1 c i χ ρ i = 0.
If c 0 0, it contradicts to the fact that the trivial representation should appear (with multiplicity 1) in the regular representation. Hence we have c 0 = 0. Then by the linear independency among the characters of irreducible representations, we must have

c 1 = • • • = c n = 0.
4. Proof of Theorem 1.2 : general orbifold case

In this section, we give the proof of Theorem 1.2 in the full generality. As the proof goes essentially in the same way as the global quotient case in §3, we will focus on the different aspects of the proof and refer to the arguments in §3 whenever possible.

Recall the setting: X is a two-dimensional Deligne-Mumford stack with only finitely many points with non-trivial stabilizers ; X is the underlying (projective) singular surface with only Du Val singularities and Y → X is the minimal resolution. For each x ∈ X, denote by G x its stabilizer, which is contained in SL 2 .

Throughout this section, Chow groups of stacks are as in [START_REF] Vistoli | Intersection theory on algebraic stacks and on their moduli spaces[END_REF] and Chow motives of stacks or singular Q-varieties are as in [28, §2] Lx,ρ and the multiplication is the following:

• h(X) ⊗ h(X) δ X --→ h(X)
is the usual intersection product.

• For any ρ ∈ Irr (G x ), h(X) ⊗ Lx,ρ i *

x -→ Lx,ρ is given by the class x ∈ CH 2 (X) = Hom(h(X) ⊗ L, L).

• For any ρ ∈ Irr (G x ), Lx,ρ ⊗ Lx,ρ -2i x, * ----→ h(X), is determined by -2x ∈ CH 2 (X). • For any ρ 1 ρ 2 ∈ Irr (G x ), -If they are adjacent, that is, ρ 1 appears (with multiplicity 1) in the G x -module ρ 2 ⊗ C 2 , where C 2 is such that C 2 /G x is the singularity type of x, then Lx,ρ 1 ⊗ Lx,ρ 2 i x, * --→ h(X), is determined by x ∈ CH 2 (X).

-If they are not adjacent, then Lx,ρ 1 ⊗ Lx,ρ 2 0 -→ h(X) is the zero map. • The other multiplication maps are zero. 4.2. Orbifold side. Similar to (6), we have [START_REF] Fu | Motivic Hype-Kähler Resolution Conjecture : II. Hilbert schemes of K3 surfaces[END_REF] h(X) = h(X)

⊕ x∈X               ∈G x id Lx,               G x
, where the action of G x is by conjugacy.

Note that degree 0 twisted stable maps with 3 marked points to X are either untwisted stable maps to X or a twisted map to one of the stacky points of X. In the latter case, the irreducible components of the moduli space around these twisted stable maps and the obstruction bundle are the same as those of the twisted stable maps to the orbifold [C 2 /G]. It is then clear that the orbifold product can be described as if X is a global quotient. Therefore the orbifold product on h(X) is given by the following, via (14):

• h(X) ⊗ h(X) δ S -→ h(X) is the usual intersection product.

• For all ∈ G x , h(X) ⊗ Lx, • For all ∈ G x , Lx, ⊗ L x, -1 i x, * --→ h(X) determined by the class of x ∈ X.

• The other multiplication maps are zero. 4.3. The multiplicative isomorphism. Similar to [START_REF] Bryan | The orbifold quantum cohomology of C 2 /Z 3 and Hurwitz-Hodge integrals[END_REF], we define [START_REF] Fu | Motivic Hyper-Kähler Resolution Conjecture : I. Generalized Kummer varieties[END_REF] φ : h(X)

⊕ x∈X ρ∈Irr (G x ) Lx,ρ → h(X) ⊕ x∈X ∈G x id
Lx, , which is given by the following 'matrix by blocs':

• id : h(X) → h(X);

• For each x ∈ X (with nontrivial stabilizer G x ), the morphism

ρ∈Irr (G x ) Lx,ρ → ∈G x id
Lx, is the 'matrix' with coefficient 1 √ |G x | χ ρ 0 ( ) -2 • χ ρ ( ) at place (ρ, ) ∈ Irr (G x ) × (G x \{id}), where χ denotes the character, ρ 0 is the natural 2-dimensional representation C 2 of G x such that C 2 /G x is the singularity type of x. Note that ρ 0 ( ) has determinant 1, hence its trace χ ρ 0 ( ) is a real number.

• The other morphisms are zero.

- 3 . 2 .

 32 is determined by x ∈ CH 2 (S). If they are not adjacent, then Lx,ρ 1 ⊗ Lx,ρ 2 0 -→ h(S) is the zero map. • The other multiplication maps are zero. The G-action on (4) is as follows: • The G-action of h(S) is induced by the original action on S. • For any h ∈ G, it maps for any x ∈ S and ρ ∈ Irr (G x ), the Lefschetz motive Lx,ρ isomorphically to L hx,hρ , where hρ ∈ Irr (G hx ) is the representation which makes the following diagram commutes: Orbifold side. Now we compute the orbifold Chow motive algebra (or Chow ring) of the quotient stack [S/G]. The compute is quite straight-forward. Here L := 1(-1) is the Lefschetz motive.

  .

4. 1 .

 1 Resolution side. Similar to (4), we have the following decomposition given by the classical McKay correspondence (see Introduction): (13) h(Y) h(X) ⊕ x∈X ρ∈Irr (G x )

  Lx, determined by the class of x ∈ X.

DMC stands for Deligne-Mumford Chow.

Lie Fu is supported by the Agence Nationale de la Recherche (ANR) through ECOVA (ANR-15-CE40-0002) within the program Défi de tous les savoirs and LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program Investissements d'Avenir (ANR-11-IDEX-0007). Zhiyu Tian is partially supported by the funding "Accueil des Nouveaux Arrivants" of IDEX, Université Grenoble Alpes. Lie Fu and Zhiyu Tian are supported by the Agence Nationale de la Recherche (ANR) through HodgeFun (ANR-16-CE40-0011), and by Projet Exploratoire Premier Soutien (PEPS) Jeunes chercheur-e-s 2016 operated by Insmi and Projet Inter-Laboratoire 2016 and 2017 by Fédération de Recherche en Mathématiques Rh ône-Alpes/Auvergne CNRS 3490. 1 i.e. regular polyhedrons in R 3 .


Martina Hentschel

Henning Schomerus

Roman Schubert

Husimi functions at dielectric interfaces: Inside-outside duality for optical systems and beyond

Keywords: numbers: 03.65.Sq, 03.65.Nk, 42.25.-p, 42.60.Da

We introduce generalized Husimi functions at the interfaces of dielectric systems. Four different functions can be defined, corresponding to the incident and departing wave on both sides of the interface. These functions allow to identify mechanisms of wave confinement and escape directions in optical microresonators, and give insight into the structure of resonance wave functions. Off resonance, where systematic interference can be neglected, the Husimi functions are related by Snell's law and Fresnel's coefficients.

Optical microresonators receive growing interest over the last years, because of the intricate interplay of shape (leading to irregular classical ray dynamics), openness of the system (offering means of excitation and escape), and the wave nature of the field. This interplay, together with the promising prospect of applications in future communication devices, has stimulated experiments [1-3] as well as theoretical investigations [4,5] that were based on concepts well-known from scattering theory, classical ray dynamics, semiclassics, and quantum chaos [START_REF] Stöckmann | Quantum Chaos[END_REF]. A particularly useful tool to study waves in dynamical systems is the Husimi representation of the wave function in classical phase space. So far, the Husimi representation was mostly used to study the closed analogues of optical microsystems, with the dielectric interface being replaced by hard walls, and the principal confinement and radiation directions have been inferred by adding the laws of reflection and refraction by hand. Most notably, efforts in this direction suffer from the fact that the incident and emerging wave components cannot be discriminated by the conventional Husimi representation. The reasons for these fundamental shortcomings arise from the facts that the wave function of the dielectric system is only partially confined by the internal reflection at the refractive index boundary, and that it is affected by the different nature of the boundary conditions, which are neither of Dirichlet nor of von-Neumann type but of a mixed type that follows from Maxwell's equations (both the wave function and its derivative are non-vanishing at the interface).

In this paper we introduce four Husimi functions appropriate for dielectric interfaces, corresponding to the intensity of incident and emerging waves at both sides of the interface. In the regime of ray optics it will be demonstrated that these Husimi functions are related across the interface via Fresnel's formulas, with phase space being deformed according to Snell's law. This connection can be seen as a new variant of the inside-outside duality [START_REF] Doron | [END_REF]. However, ray optics only applies when the wave length is short and when systematic interference effects can be neglected. The Husimi functions do

χ 1 χ 1 χ 0 χ 0 I 0 em I 0 inc I 1 inc I 1 em r c n 0 = 1 n 1 = n φ FIG. 1: Refractive-index boundary of a dielectric system.
not require these limitations and develop their full predictive power especially when systematic interference effects lead to strong deviations from Fresnel's and Snell's laws. In particular, ray optics breaks down close to resonances, where the internal part of the scattering wave function is known to be almost independent of the incoming wave that excites the system. The Husimi functions still provide an accurate representation of the wave function (in particular, they nicely display the radiation directions of the field). We illustrate these features using the dielectric circular disk and an annular system as examples [8].

Microresonators and ray optics. Consider the light that illuminates and permeates a dielectric system confined by a refractive-index boundary, as shown in Fig. 1. The disk (and also the annulus) are confined by a circular dielectric interface of radius ¡ £¢ ¥¤ §¦ . Polar coordinates ¡ , ¨will be used to parameterize position space. We distinguish four wave components: The incident (inc) wave and the emerging (em) wave on both sides of the interface (region 0 with refractive index emerging rays are related by the laws of reflection and Snell's law, © ! #" $ ! %" . For our circular interfaces Snell's law is equivalent to conservation of the angular-momentum variable & (' 0) 1 2 3 4" 5) , where ' and ' (© ' are the wave number in each region. The ray intensities on either side follow from Fresnel's laws, 6 87 @9 BA C ED F" G 0H 6 EI P RQ TS VU ED W" X YH 6 8I P RQ §`( 1) 6 7 @9 BA D F" H 6 I P RQ BS VU D W" H 6 I P RQ ba

(2)

The reflection and transmission coefficients A c and U c are related by

A C ED W" d eH f gA h ED W" X iH ¤ A pU qD F" G RH X U rD F" f RH d ¦ #s A `(3) with [8] A t g ! vu wD W" X s " d eH 2x d ! vu 8D W" f S " d RH .
Husimi functions at a dielectric interface. The Husimi function at the system boundary of closed systems was introduced in Refs. [9,10] by projection of the conventional Husimi function from full phase space [coordinates D ¡ `¨H , momentum D y' ) ! 4" ) `' ) 1 £ " ) H ] onto the reduced phase space at the boundary ¡ ¡ £¢ with coordinates ¨and ! 4" ) [11]. The four different Husimi functions (corresponding to the incident and emerging wave at both sides of a dielectric interface) can be constructed by the same procedure when the appropriate boundary conditions are employed. The intensities 6 D ` ! 4" GH D ` 2 ! %" dH X ! f" will turn out to be related by the laws (1), (2) when ray optics applies, but also accurately describe the wave function when ray-optical relations across the interface break down, as expected, e.g., for resonances. The conventional Husimi function for a given wave function D ¡ `¨H of the dielectric system is obtained as the overlap with a wave packet with minimal uncertainty in the variables D ¡ `¨H for real space and D y' ) ! 4" ) `' ) 1 £ " ) H for momentum space. The projection onto the boundary can be formulated rigorously [10]: The wave function is expressed by means of advanced and retarded Green's functions, which in turn allow to distinguish between incident and emerging waves. Green's formula is then used to express the solution ) of the Helmholtz equation in region , ¦ as an integral over the boundary, involving both ) and its normal (radial) derivative ) . A semiclassical (saddle-point) approximation then allows to identify in these expressions the following four different Husimi functions on the interface,

I P RQ 7 @9 f ) D ` ! #" GH ' 0) q d d d d D s h¦ H ) Ee ) Ef w) D ` ! #" GH S D s H pg ' ) e ) f ) D ` ! %" dH d d d d u ` (4)
with the angular-momentum dependent weighting factor e ) C ih © j) R k" v) [START_REF]The unphysical singularity[END_REF]. Here the functions

f w) pl nm ¡ ¡ l u o p ) D ¡ `¨H yq vD ¡ `Cr ` ! d" GH `(5) f ) l nm ¡ ¡ l u o p ) D ¡ `¨H yq vD ¡ `Cr ` ! d" GH (6)
are overlaps with the minimal-uncertainty wave packet q vD ¡ `Cr ` 2 3 X" GH n ps 5t tu 8v wx yy 3z 5{ u io

t v | x v c !} 5~I P 5 !z v{ u o t D H Yv w E D ¡ s n¡ R¢ H (7)
(a periodic function in ¨), which is restricted to the interface and centered around D ` ! %" dH . The parameter controls its extension in -direction, thereby also fixing the uncertainty in 2 ! 4" . We set h x w' k . The scaling with ' k results in the same resolution in for all four Husimi functions.

Inside-outside duality. As a consequence of the boundary conditions derived from Maxwell's equations we find the identities f qD ` 2 ! %" d eH f eD ` 2 3 4" X iH , f D ` ! #" G 0H f D ` 2 3 4" X iH , where the angles " c are related by Snell's law.

From these relations it follows that our Husimi functions strictly fulfill the condition of intensity conservation,

© 7 @9 D ` 2 ! %" d eH S 7 @9 D ` 2 ! 4" f iH p© I P £Q D ` 2 ! %" d eH S I P RQ D ` ! 4" f H `(8)
where the factor © $ f 2 3 #" x q X ! #" accounts for the phasespace deformation by Snell's law. Additional relations between the Husimi functions can be anticipated in the regime of ray optics: The intensities on one side of the interface should be related to the intensities on the other side by Eqs. (1), (2).

We then expect validity of the resulting inside-outside duality relations which express the Husimi functions in region by the Husimi functions in region ¦ . The notation D H indicates that the approximation is of semiclassical (short wave length) nature; most noticeably, the Husimi functions (intensities) are added incoherently. The duality relations can also be inverted,

7 @9 p D 7 @9 H f $© ¦ Xs A ¦ #s A I P RQ S © A ¦ #s A 7 @9 §`( 11
) I P RQ p D I P RQ H G s © A ¦ #s A I P RQ S © ¦ ¦ #s A 7 @9 §a (12)
However, the Husimi functions in region 1 can only be reconstructed in the strip 2 ! %" X w ¦ x q© , because the rest of phase space is isolated from region 0 by total internal reflection. The duality relations are exactly fulfilled in two simple cases, namely, if one incident or emerging wave vanishes or when the two incident waves have the same intensity (the two emerging waves then have the same intensity, as well). We now test the duality relations in more general situations.

The panels on the left in Fig. 2 show the exact Husimi functions from Eq. ( 4) for the case of the circular disk which is excited by a plane wave at an off-resonant excitation frequency. The illuminating plane wave is clearly visible in the Husimi 4). The reconstructed Husimi functions are obtained by Eqs. ( 9)- [START_REF]The unphysical singularity[END_REF]. Negative Husimi densities are shown in black. The dashed lines in the panels for region mark the critical angle of incidence for total internal reflection.

function I P RQ

, around the polar angle I , while the focal point of the dielectric disk results in a bright spot in 7 @9 that is located around $ . There is a close correspondence between the Husimi functions of the incident and emerging waves, and the deformation of phase space by the stretching factor © of Snell's law is clearly visible. The right panels of Fig. 2 show for comparison the predictions of Eqs. ( 9)- [START_REF]The unphysical singularity[END_REF]. In the reconstruction we used the slightly modified semiclassical versions of Fresnel's coefficients and Snell's laws given in Ref. [13], which are appropriate for the present case of a curved interface (this results in a slight, but still noticeable quantitative improvement of the reconstruction). We observe a good qualitative and quantitative agreement with the exact Husimi functions. Regions with unphysical negative intensities are small. The most interesting deviations between the exact and the reconstructed Husimi functions occur around the central spot at ` 2 ! %" 5) , where the incoherent predictions of Eqs. ( 9)-( 12) underestimate the exact Husimi densities I P RQ , 7 @9 , while they overestimate the intensities I P £Q , 7 @9 in the same area of phase space. These deviations arise from a Fabry-Perot like systematic interference which is constructive in backward direction at the presently chosen frequency. At other frequencies the interference is destructive, and both cases alternate periodically. Resonances. Figure 3 displays the situation for excitation at a frequency which is close to a narrow resonance, a whispering gallery (WG) mode located around 2 3 4" X a w q . The top panel shows that the wave function is now well confined inside the disk (region 1). Correspondingly, the Husimi functions I P £Q and 7 @9 noticeably exceed the Husimi functions I P RQ and 7 @9 . Moreover, the Husimi functions I P RQ and 7 @9 are dominated by the characteristics of the resonance wave function and consequentially are almost independent of the choice of the exciting wave. (The remnants of the exciting plane can be identified when comparing Fig. 3 with Fig. 2.) Hence the reconstructed Husimi functions deviate noticeably from the exact Husimi functions around ! #" f E ¦ x q© . This is no surprise since resonances are formed by systematic constructive interference, and incoherent ray optics cannot be expected to apply under these circumstances. Most importantly, by principle, the confined wave intensity in the region ! #" f w ¦ x q© cannot be reconstructed because classically no refracted rays ever reach this region (which is dark off resonance). On the other hand, the exact Husimi functions display nicely all the features of the resonance wave function in phase space.

Finally, let us illustrate the usefulness of the Husimi functions (4) also for a more complex system than the circular disk, the annular system formed by regions of different refractive indices that are confined by two eccentric disks. The ray optics in this system corresponds to nonintegrable dynamics in phase space, which allows for a multifaceted set of resonance wave functions [5]. Off resonance (Fig. 4) the scattering wave function enters the dielectric system only barely, and the sit-FIG. 4: Same as Fig. 2, but for a dielectric annulus (refractive indices j¡ % ¢ outside, 5£ d ¤ in the annulus, j¥ # ¦ in the inner disk, radii § ¨ © 0ª § £« ¨ ¬ w ¦ , displaced by ® 4 ¬ q e ). The system is illuminated by a plane wave at an off-resonant frequency (illumination direction j°± ± ² ).

uation is similar to the circular disk because the internal disk is not explored extensively. At resonance the situation is very different. Figure 5 shows a typical resonance wave function in real space and its Husimi representation in phase space. The intensity of the resonance wave function is concentrated on straight segments which can be identified as a short stable periodic trajectory of the corresponding classical ray dynamics. The Husimi functions display a strong intensity exactly in the vicinity of this trajectory in classical phase space.

In conclusion, we introduced four Husimi representations of the scattering wave function at the interfaces of dielectric microresonators, corresponding to the incident and emerging waves at both sides of the interface. These Husimi functions are easily computed from the wave function and have many desirable properties: They are related by the laws of Fresnel and Snell in the ray-optics regime (i.e., short and offresonant wavelength) and generally provide valuable detailed insight into the wave dynamics in complex dielectric systems, most notably even close to resonances where ray optics breaks down.

We gratefully acknowledge helpful discussions with Jan Wiersig, Arnd Bäcker, and Christian Miniatura.

[1] V. for " ´ x is regularized by the corrections to the semiclassical derivation of Eq. ( 4). In our calculations we regularized

FIG. 2 :

 2 FIG. 2: Exact (left) and reconstructed (right) Husimi functions for the circular dielectric disk ( ) illuminated by a plane wave at an off-resonant frequency. The top panel shows the scattering wave function in real space. The exact Husimi functions are obtained from Eq. (4). The reconstructed Husimi functions are obtained by Eqs. (9)-[START_REF]The unphysical singularity[END_REF]. Negative Husimi densities are shown in black. The dashed lines in the panels for region mark the critical angle of incidence for total internal reflection.

FIG. 3 :

 3 FIG.3: Same as Fig.2, but for illumination at a resonance frequency.

  Doya, O. Legrand, F. Mortessagne, and C. Miniatura, Phys. Rev. Lett. 88, 014102 (2002). [2] S.-B. Lee et al., Phys. Rev. Lett. 88, 033903 (2002). [3] N. B. Rex et al., Phys. Rev. Lett. 88, 094102 (2002).

FIG. 5 :

 5 FIG.5: Same as Fig.4, but for illumination at a resonance frequency.

  these corrections.[13] M. Hentschel and H. Schomerus, Phys. Rev. E 65, 045603(R) (2002).


Yosra Zguira

Hervé Rivano

Evaluation of IoB-DTN protocol for mobile IoT Application

Keywords: Delay Tolerant Network, Internet of Things, Public bike sharing system

Communication is essential to the coordination of public transport systems. Nowadays, cities are facing an increasing number of bikes used by citizens therefore the need of monitoring and managing their traffic becomes crucial. Public bike sharing system has been introduced as an urban transportation system that can collect data from mobile devices. In this context, we introduce "IoB-DTN", a protocol based on the Delay/Disruption Tolerant Network (DTN) paradigm adapted for an IoT-like applications running on bike sharing system based sensor network. This document presents the simulation results obtained by evaluating the Binary Spray and Wait inspired variant of IoB-DTN with four buffer management policies and by comparing three variants of IoB-DTN by varying the number of packet copies sprayed in the network.

Evaluation du protocole IoB-DTN pour l'application IoT mobile

Résumé : La communication est essentielle à la coordination des systèmes de transport public. De nos jours, les villes font face à un nombre élevé de vélos utilisés par les citoyens, d'où la nécessité de surveiller et de gérer leur trafic. Le système de partage de vélo public a été introduit en tant que un système de transport urbain capable de collecter des donnés à partir des appareils mobiles. Dans ce contexte, nous introduisons « IoB-DTN » un protocole basé sur le paradigme DTN (Delay / Disruption Tolerant Network), adapté aux applications de type IoT et fonctionnant sur un réseau de capteurs basé sur un système de partage de vélos. Ce document présente les résultats de simulation obtenus en évaluant la variante Binary Spray and Wait inspirée de IoB-DTN avec quatre politiques de gestion de tampons et en comparant trois variantes de IoB-DTN en faisant varier le nombre de copies de paquets diffusés dans le réseau.

Mots-clés : Réseaux tolérants aux délais, Internet des objets, Système de partage de vélo public

Introduction

The United Nations Population Division shows that from 1950 to 2050, the part of world's population living in urban areas will grow from 30% to 66% [START_REF]World Urbanization Prospects: The 2014 Revision[END_REF]. This generates multiple undesirable effects such as air pollution, dirty streets, noise, dangers to our health, a lot of traffic jamming and crowded neighborhoods. Therefore, there is a need to a mechanism to monitor the air pollution, highlighting the environment and transportation issues. In our work, we are interested to the last problem. One of the major solutions of urban transportation issues is public transportation system. They have been designed to connect smart devices to the Internet and they have been used to collect data from mobile devices. Public bike sharing systems have been introduced as part of the public transportation system and could be used as a mobile sensor network. In the present document, we give a description of IoB-DTN protocol applied to mobile network IoT (Internet of Things) devices running a data collection application and a performance evaluation of the protocol on real networks and in particular on connected bicycles.

Related Work

Many works focused on communication based public transport network have been proposed. Latora and Marichiori present the first network-based studies of urban transportation systems [START_REF] Latora | Is the boston subway a small-world network?[END_REF]. The DakNet [START_REF] Pentland | Daknet: Rethinking connectivity in developing nations[END_REF] provides low-cost digital communication to remote villages in India and Cambodia. Buses are used in DakNet to transfer data between Internet access points and Internet kiosks in villages. The KioskNet [START_REF] Seth | Lowcost communication for rural internet kiosks using mechanical backhaul[END_REF] uses buses and cars as "mechanical backhaul" devices to transfer data between remote villages and Internet gateway. Authors rest on the pioneering lead of Daknet and extend the Delay Tolerant Networking Research Group architecture [START_REF]Dtn research group[END_REF] to present a detailed architecture that addresses the problem of low-cost and reliable connectivity for rural kiosks. Zhao et al. [29] present a vehicle assisted data delivery (VADD) for vehicular ad hoc networks. The VADD is based on the carry and forward paradigm [START_REF] Vahdat | Epidemic routing for partially connected ad hoc networks[END_REF] where vehicles are used as data carriers and the path to the destination is determined based on the ad hoc connectivity of the vehicles. The DieselNet [START_REF] Burgess | Maxprop: Routing for vehicle-based disruption-tolerant networks[END_REF] is equipped up to 40 buses with access points in Amherst. Each bus has two 802.11 radios, a GPS and 40GB hard drive. Data is transmitted via bus-to-bus communications enabling their intermittent connectivity. DieselNet has led to the design of the MaxProp routing protocol. The authors in [START_REF] De Zoysa | A public transport system based sensor network for road surface condition monitoring[END_REF] propose a public transport based sensor network, called BusNet, to monitor road surface condition. BusNet is a sensor network designed to monitor environmental pollution using sensors embedded on public transport buses. This network generates and delivers data by using the stable transport infrastructure and it does not rely on the ad hoc connectivity between vehicles. The BikeNet project [START_REF] Shane B Eisenman | Bikenet: A mobile sensing system for cyclist experience mapping[END_REF] defined a mobile networked sensing system embedded into a cyclist's bike. Nakamura et al. propose a web framework in Tokyo involving bicycles with sensors communication in a Wide Area Ubiquitous Network (WAUN) [START_REF] Nakamura | Proposal of web framework for ubiquitous sensor network and its trial application using no2 sensor mounted on bicycle[END_REF]. There are many public bicycle schemes that collect real-time data. Most of hire schemes collect data when bikes leave their bicycle stations include those in Germany [START_REF] Bracher | Innovative öffentliche fahrradverleihsysteme: Modellprojekte am start[END_REF] and Netherlands [START_REF] Gopedelec | electric rental bike now inaugurated in the netherlands[END_REF]. Several studies focused on analyzing the movement of bikes in public hire systems such as the public cycle hire scheme in London [START_REF] Wood | Visualizing the dynamics of london's bicyclehire scheme[END_REF], Lyon [START_REF] Borgnat | Shared bicycles in a city: A signal processing and data analysis perspective[END_REF] and Barcelona [START_REF] Froehlich | Sensing and predicting the pulse of the city through shared bicycling[END_REF]. In [START_REF] Wirtz | Opportunistic interaction in the challenged internet of things[END_REF], Wirtz et al. propose Direct Interaction with Smart Challenged Objects (DISCO) enabling objects to define and provide their interaction patterns and interface immediately to users. The authors, in [START_REF] Fadi | A delay-tolerant framework for integrated rsns in iot[END_REF], introduce DIRSN, an optimized delay-tolerant approach for integrated RFID-sensor networks (RSNs) in the IoT. Their framework provides an optimized architecture for integrated RSNs in addition to a delay-tolerant routing scheme. Elmangoush et al. [START_REF] Elmangoush | Interconnecting standard m2m platforms to delay tolerant networks[END_REF] propose an architecture enabling the interconnection of standard-based machine-to-machine (M2M) platforms to DTNs in order to collect data from sensor devices.

Simulated scenario

In our work, we are interested to the use of IoT on connected bikes. We propose a simplified mechanism enabling bicycles to collect and send data to different sinks. Each bike embeds a sensor and a 802.11p communication device. Bicycles generate data every second, store it in their buffers and then transfer it to its neighbors depending on the duty cycle used. They exchange data until they reach a sink. Each neighbor is in communication range with a base station, it forwards all data stored in its buffer. In the next section, we details more the description of the protocol.

IoB-DTN: Internet Of Bikes-DTN protocol

The IoB-DTN proposed protocol is inspired by DTN flooding protocols [START_REF] Evan | Routing strategies for delay-tolerant networks[END_REF]. Flooding strategies are based on the principal of replicating messages to enough nodes so that destination nodes must receive it. These protocols can not be applicable for the context of IoT because of their need for high memory storage and energy consumption. In this context, we introduce "IoB-DTN" protocol inspired, more precisely, by Binary Spray and Wait protocol [START_REF] Maitreyi | Design of binary spray and wait protocol for intermittently connected mobile networks[END_REF] which can be applied to a mobile network IoT devices for data collection application.

The number of packet copies (N 0 ) sprayed in the network is an interesting technique to control the level of flooding. For our study, we evaluate three variants of IoB-DTN:

N 0 = 2: Two-Hop Relay protocol [START_REF] Al Hanbali | Performance of ad hoc networks with two-hop relay routing and limited packet lifetime[END_REF] N 0 = 8: Binary Spray and Wait protocol [START_REF] Spyropoulos | Spray and wait: an efficient routing scheme for intermittently connected mobile networks[END_REF] N 0 = ∞: Epidemic Routing protocol [START_REF] Vahdat | Epidemic routing for partially connected ad hoc networks[END_REF] The buffer management policies is an important parameter of IoB-DTN which leads to find a slot in the buffer node when generating or receiving a packet. When the buffer is not full, it gives the next free slot else it decides which packet will be rejected and which should be kept.

We propose four buffer management policies:

1. KONP: Keep Oldest No Priority: The packet generated or received is discarded if the buffer is full.

NP: No Priority:

The new packet is replaced by the oldest one which has a high probability to be sent to neighbors and reached a sink.

3. GPP: Generated Packet Priority: If the new packet is generated, it replaces the oldest received packet. If it is a received packet, it is discarded. If the full buffer contains only generated packets, the oldest one is then rejected.

LC: Lesser Copy:

The new packet replaces the packet having the smallest number of copies.

As indicated in the previous section, each bicycle generates periodically a packet and stores N 0 copies in its buffer. When the duty cycle is over and the base station is in the communication range of a bike, this latter sends all data stored to the sink. If not, the bicycle verifies if it has at least two copies of each packet, then it transfers one copy and keeps the other one. If it has one copy of the packet, it keeps it until it reaches the destination.

When receiving a packet, the bike determines its position number in the neighbors list sent with the corresponding packet and calculates the number of copies that it should take as shown by this equation:

Inria N ← N 2 pos+1
The variable "pos" indicates the position of the bicycle in the neighbors list, "N" corresponds to the initial number of copies and "N " is the new number of copies calculated. If the buffer management policy provides a slot in the buffer and the receiver bike has at least one copy , it stores the packet with the corresponding number of copies and sends an ACK to the transmitter. If not, the packet is rejected.

At the reception of an ACK, the bike verifies the sender node. If it is a base station, it deletes the packet. If it is another bicycle, it updates the number of copies of the packet.

Simulation dataset

The urban environment we used to evaluate the IoB-DTN proposed protocol is the city of Lyon, France. Lyon is situated in east-central France, in the Auvergne-Rhône-Alpes region and it is the third-largest city after Paris and Marseille. Lyon is one of the first cities in the world that tested the self-service bicycles called Vélo'v [2]. The system, launched in May 2005, provides more than 4000 bikes available 24/7 from over 350 stations situated around the cities of Lyon and Villeurbanne. Each citizen takes his bike as close to his starting point and returns it as close to his point of arrival. Figure 1 depicts one of the bicycle station in the city center. Table 1 summarizes the simulation configuration used for our scenario. We simulate four cases as shown in Table 2 In this part, we present our simulation results. We compare the performances by evaluating two metrics: the loss rate and the delivery delay which is the time between the generation of a packet by a node and its reception by a base station.

First scenario

We compare the performances of the IoB-DTN protocol by fixing the number of copies to 8.

From Figure 3 we notice that GPP and LC behave better than NP and KONP policies by evaluating the loss rate in all cases. When the duty cycle is lower, GPP has better performances since it prioritizes the generated packets and therefore the redundancy provides robustness. LC has almost same performances than GPP policy since it rejects packets having the smallest number of copies and a high probability to be arrived to a base station. NP outperforms slightly KONP. It provides bad performances comparing to GPP and LC since it discards the oldest of generated and received packets. KONP is the worst, more precisely when the buffer is small. In this case, the buffer is saturated by the first generated packets quickly and then all others are discarded.

Figure 4 presents the delivery delay of received packets. NP offers the best performances since it drops both the oldest of generated and received packets. GPP and LC behave in the same way and have almost similar performances than NP. KONP gives a bad transmission delay since it forwards only the oldest packets. By evaluating the both metrics in the simulated scenario, we can classify two categories of policies: KONP and NP; GPP and LC. From the first one, KONP has bad performances on the loss rate and the delivery delay. GPP provides softly best performances than LC according to the second category.

In order to confirm our results obtained for the previous scenario, we simulated ten scenarios with different paths of bicycles. We considered only the best policy for each category: GPP and NP policies.

Figure 5 depicts the average loss rate obtained for the ten scenarios by varying the paths of bikes. From figures 5 and 6 we notice that the results obtained for the ten scenarios are similar to those presented for the previous scenario.

In our second scenario, we evaluate three variants of IoB-DTN by varying the number of copies sprayed in the network and comparing the two buffer management policies NP and GPP.

Second scenario

As previously mentioned, in this section, we evaluate three cases by varying the number of copies.

From Figures 7 and8, Epidemic Routing protocol has better performances on loss rate for GPP and NP policies since it diffuses many copies in the network. GPP outperforms NP in all cases thanks to its priority to generated packets. Binary Spray and Wait has almost similar results as Epidemic Routing protocol while Two Hop relay is the worst. This phenomenon is due to the bed choice of the neighbor node chosen as relay or to the need of more than two hops to transmit the redundant packet.

Conclusion

In this report, we have presented IoB-DTN protocol inspired by Binary Spray and Wait applied to an IoT-like applications running on bicycle sharing system based sensor network. Our simulation results shows that by giving priority to self generated packets as buffer management and by limiting the number of copies sprayed in the network gives the best performances on loss rate and delivery delay.

[

Figure 1 :

 1 Figure 1: Vélo'v station

Figure 2 :

 2 Figure 2: City area of Lyon

  Figure 3: Loss rate

Figure 5 :Figure 6 :

 56 Figure 5: Average loss rate for ten scenarios by varying the paths of bicycles

Figure 7 :Figure 8 :

 78 Figure 7: Loss rate for GPP by varying the number of copies

Figure 9 :Figure 10 :

 910 Figure 9: Delivery delay for GPP by varying the number of copies

  by varying the buffer size and the duty cycle.

	Number of bike stations 49
	Number of bikes	47
	Travel time	Min = 550s / Max = 1418s
	Packet generation time	Every second
	Communication model	802.11p
	Table 1: Simulation parameters
	Buffer size Duty cycle
	Case 1 250	50
	Case 2 250	150
	Case 3 500	50
	Case 4 500	150
	Table 2: simulated cases

Inria

6 Simulation results

  29] Jing Zhao and Guohong Cao. Vadd: Vehicle-assisted data delivery in vehicular ad hoc networks. IEEE transactions on vehicular technology, 57(3):1910-1922, 2008.

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