Atsuo Shitade

Taro Kimura

Bulk angular momentum and Hall viscosity in chiral superconductors

Keywords: number(s): 04.62.+v, 66.20.Cy, 74.25.Ld

We establish the Berry-phase formulas for the angular momentum (AM) and the Hall viscosity (HV) to investigate chiral superconductors (SCs) in two and three dimensions. The AM is defined by the temporal integral of the antisymmetric momentum current induced by an adiabatic deformation, while the HV is defined by the symmetric momentum current induced by the symmetric torsional electric field. Without suffering from the system size or geometry, we obtain the macroscopic AM L z = mN 0 /2 at zero temperature in full-gap chiral SCs, where m is the magnetic quantum number and N 0 is the total number of electrons. We also find that the HV is equal to half the AM at zero temperature not only in full-gap chiral SCs as is well known but also in nodal ones, but its behavior at finite temperature is different in the two cases.

I. INTRODUCTION

Chiral superfluids and superconductors (SCs) are exotic states whose time-reversal symmetry is spontaneously broken and Cooper pairs carry nonzero angular momentum (AM). One well-known example is 3 He-A whose pairing symmetry is p x + ip y [1]. Among electron systems, there are a few candidates for chiral SCs such as Sr 2 RuO 4 with p x + ip y [2,3] and URu 2 Si 2 with d zx + id yz [4][5][6][7]. Recently the d x 2 -y 2 + id xy pairing symmetry was theoretically proposed in SrPtAs [8][9][10].

There is a long-standing problem on the AM in chiral -wave SCs, the so-called intrinsic AM paradox. This paradox is summarized as L z = mN 0 /2 × ( 0 /E F ) γ , where |m| , N 0 , 0 , and E F are the magnetic quantum number, the total number of electrons, the gap strength, and the Fermi energy, respectively. γ = 0 [11][12][13][14][15][16][17][18][19] is the most natural if all electrons form Cooper pairs with the AM z = m. On the other hand, γ = 1 [1,20] is intuitively plausible if a few electrons near the Fermi surface form Cooper pairs. γ = 2 was also proposed by using the Ginzburg-Landau theory [21,22]. Recent microscopic studies have settled this paradox, at least theoretically, to γ = 0 [15,[17][18][19] but have been employed only in finite systems such as a cylinder [15] and a disk [17][18][19].

Generally, the physical quantities involving the position operator are ill defined in periodic systems. The most famous examples are the charge polarization and the orbital magnetization. The former is classically coupled to an electric field but is quantum-mechanically defined by the temporal integral of the charge current under an adiabatic deformation [23][24][25][26]. On the other hand, the latter is coupled to a magnetic field and can be defined by the magnetic field derivative of the free energy [27][28][29]. Consequently, the charge polarization is associated with the Berry connection in the Bloch basis, while the orbital magnetization with the Berry curvature and the magnetic moment. These formalisms were extended to heat analogs such as the heat polarization [26] and magnetization [30]. However, the AM has not been formulated yet.

Another interesting clue to the AM is the Hall viscosity (HV), which has been intensively discussed in the context of the quantum Hall effect [31,32]. Similar to the Hall conductivity, the HV is nonzero only when the time-reversal symmetry is broken. The important relation η H = N 0 s/2 holds in general gapped systems at zero temperature [33][34][35], in which the orbital spin s is equal to /2 in chiral -wave SCs. In addition, it was also related to the momentum-dependent Hall conductivity [36,37]. The torsional Chern-Simons term was proposed to describe the quantum HV in the twodimensional massive Dirac system [38][39][40].

In this paper, we derive the Berry-phase formulas for the AM and the HV to apply to chiral SCs in two and three dimensions. The antisymmetric and symmetric components of a torsional electric field describe an angular velocity and a strain-rate tensor, respectively. Since the AM is conjugate to the former, it can be formulated in the same way as in the charge polarization, namely, by the temporal integral of the antisymmetric momentum current induced by an adiabatic deformation. Viscosity is defined by the symmetric momentum current, i.e., the stress tensor, induced by the latter. In contrast to the previous works regarding the intrinsic AM paradox, we obtain L z = mN 0 /2 without suffering from the finite-size effects. We also investigate the temperature dependence of the HV for two-dimensional gapped chiral SCs and threedimensional nodal ones.

Hereafter we assign the Latin (a,b, . . . = 0, 1, . . . , d) and Greek (μ,ν, . . . = 0,1, . . . ,d) alphabets to locally flat and global coordinates, respectively. We follow the Einstein convention, which implies summation over the spacetime dimension D = d + 1 when an index appears twice in a single term. The Minkowski metric is taken as η ab = diag(-1,+1, . . . ,+1). The Planck constant and the charge are denoted by and q, while the speed of light and the Boltzmann constant are put to c = k B = 1. The upper or lower signs in equations correspond to boson or fermion.

II. CARTAN FORMALISM

To begin with, we examine an angular velocity from the gauge-theoretical viewpoint. Now that the system is rotated, we have to deal with a theory in a curved spacetime. Here we use the Cartan formalism, which consists of two gauge potentials such as a vielbein and a spin connection [START_REF] Kleinert | Multivalued Fields in Condensed Matter, Electromagnetism, and Gravitation[END_REF]. A vielbein h a μ is a gauge potential corresponding to local spacetime translations, while a spin connection ω ab μ is that corresponding to local Lorentz transformations. In these gauge potentials as well as a vector potential A μ , the partial derivative is replaced by the covariant derivative ∂ a → D a ≡ h μ a (∂ μ -iqA μ /iω ab S ab /2 ). Here S ab is the generator of local Lorentz transformations. The spatial component of a vielbein is related to a displacement vector [38][39][40] as

h ki = δ ki + ∂ i u k .
Moreover, h k0 describes rotation as illustrated below.

For simplicity, let us consider a Dirac fermion in a curved spacetime. The Dirac Lagrangian density is given by

hL = h ψ( γ a D a -m)ψ, ( 1
)
in which h ≡ det h a μ is the determinant of a vielbein, ψ ≡ i -1 ψ † γ 0 is the Dirac conjugate, and the γ matrices satisfy the Clifford algebra {γ a ,γ b } = 2η ab . When we introduce the nonzero off-diagonal component h k0 = φ k r in addition to the identity, the square line element is given by

ds 2 = -dt 2 + (d x + φ r dt) 2 , ( 2
)
and the Dirac Lagrangian is reduced to

hL = ψ † [i D 0 + i α • D -mβ -i φ r • D]ψ. (3)
Here α and β are the Dirac matrices. If we put φ r = × x, is assigned to an angular velocity, and the fourth term represents the coupling between an angular velocity and the AM. Such a discussion relies on the translational symmetry and gauge principle of gravity and hence is not restricted to relativistic systems. Since a vielbein is a gauge potential, it induces a field strength called torsion. Here a spin connection is neglected, and hence torsion is written by

T l j 0 = ∂ j h l0 -∂ 0 h lj , ( 4a
)
T l ij = ∂ i h lj -∂ j h li . ( 4b
)
The former is "electric." The first term describes an angular velocity if l and j are antisymmetric, while the second term describes a strain-rate tensor if symmetric. On the other hand, the latter is "magnetic" and characterizes edge and screw dislocations in crystals. Especially its flux is identified as the Burgers vector.

It is a natural question why rotation is described by a vielbein, i.e., a gauge potential corresponding to space translations. In fact, global space translations do not include rotations. However, a gauge potential is associated with a local symmetry, and local space translations do include rotations. Note that an angular velocity is coupled not only to the AM but to the spin, which is implemented by a spin connection. Such spin responses are out of scope here.

III. ANGULAR MOMENTUM

Since an angular velocity is "electric," the AM is similar to the charge polarization rather than to the orbital magnetization. Below we define the momentum polarization by the temporal integral of the nonsymmetric momentum current induced by an adiabatic deformation. The momentum current is conjugate to a "vector potential" h ki and is given by the product of the momentum and the velocity. In the Wigner representation of the Keldysh formalism, it is written by

T î k = ± i 2 d D π (2π ) D tr -∂ π î Ĝ-1 Ĝ π k < + c.c. ( 5
)
Here Ĝ is the Keldysh Green's function, (-∂ π î Ĝ-1 ) is the renormalized velocity to satisfy the local conservation law, and π k is the Wigner representation of the covariant derivative corresponding to the momentum. Note that this momentum current is Hermitian but not symmetric over k and î. An adiabatic deformation is implemented by the gradient expansion up to the first order [26]. Since both the Keldysh Green's function Ĝ and the renormalized velocity (-∂ π î Ĝ-1 ) are perturbed, the gradient expansion of the momentum current Eq. ( 5) is given by

T î k = ± i 2 4 d D π (2π ) D π k tr ∂ π î ˆ 1 Ĝ0 -∂ π î Ĝ-1 0 Ĝ1 < + c.c., (6)
in which the gradient expansion of the Keldysh Green's function Ĝ1 is given by Ĝ1

= Ĝ0 ˆ 1 Ĝ0 + i Ĝ0 ∂ X 0 Ĝ-1 0 Ĝ0 ∂ πˆ0 Ĝ-1 0 Ĝ0 -(X 0 ↔ π 0) , (7)
and the corresponding self-energy ˆ 1 is determined selfconsistently. Now the momentum current is transformed into

T î k =± i 2 4 d D π (2π ) D π k tr ∂ π î ˆ 1 Ĝ0 -∂ π î Ĝ-1 0 Ĝ0 ˆ 1 Ĝ0 < + c.c. ± 2 4 d D π (2π ) D π k tr ∂ π î Ĝ-1 0 Ĝ0 ∂ X 0 Ĝ-1 0 Ĝ0 ∂ πˆ0 Ĝ-1 0 Ĝ0 -(X 0 ↔ π 0) < + c.c. ( 8
)
By extracting the lesser component and employing the temporal integral, we obtain the change in the momentum polarization, but not the momentum polarization itself,

P î k ≡ dX 0 T î k = 2 6 ABC d D π (2π ) D dX 0 f (-π 0)π k tr G R 0 ∂ A G R-1 0 G R 0 ∂ B G R-1 0 G R 0 ∂ C G R-1 0 + c.c. (9a) + 2 4 d D π (2π ) D dX 0 f (-π 0)π k tr G R 0 -G A 0 ∂ π î G R-1 0 + G A-1 0 G R 0 ∂ X 0 G R-1 0 + c.c. (9b) ∓ i 2 4 δ î k d D π (2π ) D dX 0 tr[ ˆ 1 Ĝ0 ] < + c.c. (9c) ± i 2 4 d D π (2π ) D dX 0 f (-π 0)π k tr <(1) 1 G R 0 -G A 0 ∂ π î G R-1 0 G R 0 + c.c., (9d)
where ABC in Eq. ( 9a) is the antisymmetric tensor with

πˆ0π î X 0 = 1. f (ω) = (e βω ∓ 1) -1
is the distribution function for boson or fermion.

Below we focus on the clean and noninteracting limit ˆ = 0 to derive the Berry-phase formula in the Bloch basis. The retarded Green's function is given by

G R 0 = [(-π 0) - H(X 0 ) + μ + iη] -1 with η → +0, leading to ∂ πˆ0 G R-1 0 = -1, ∂ π î G R-1 0 = -v î , and ∂ X 0 G R-1 0 = -Ḣ.
In an adiabatic deformation, the trace can be expanded with respect to the eigenstates satisfying H(X 0 )|u n πX 0 = n πX 0 |u n πX 0 . We evaluate the integral over (-π 0) by the residue theorem to obtain

P î k = - n d d π (2π ) d dX 0 π k î n πX 0 f n πX 0 , ( 10
)
with f n πX 0 ≡ f ( n πX 0μ) and the Berry curvature in the (π î / ,X 0 ) space being defined by î

n πX 0 ≡ i ∂ π î u n πX 0 ∂ X 0 u n πX 0 -(X 0 ↔ π î ). (11)
Similar to the charge polarization, Eq. ( 10) depends on the choice of an adiabatic deformation at finite temperature and hence is not well defined. At zero temperature in a gapped fermion system, its integrand becomes the total derivative with respect to X 0 when k = î, and the momentum polarization itself is given by

P î k = occ n d d π (2π ) d π k A î n π , ( 12
)
where we introduce the Berry connection,

A î n π ≡ i u n π ∂ π î u n π . ( 13
)
Indeed this expression is quite similar to that for the charge polarization, which is given by the integral of the Berry connection itself. Note that at the initial time, the reference system has time-reversal symmetry, and the momentum polarization should be zero. After all, the AM is obtained by the antisymmetric part of the momentum polarization,

L k ≡ î ĵ k P ĵ î = occ n d d π (2π ) d î ĵ k A î n π π ĵ . ( 14
)
Since the Berry connection is regarded as the expectation value of the position operator in the Wannier basis, this Berry-phase formula really indicates = x × p in the momentum space.

In the above derivation, it is not obvious why the system should be gapped. Wave functions have the phase degree of freedom; namely, physical quantities should be invariant under a unitary transformation |u n π = e -iθ n π |u n π . Correspondingly, the Berry connection Eq. ( 13) is transformed as A n π = A n π + ∂ π θ n π , and the integrand in Eq. ( 14) is transformed as

A n π × π = A n π × π + ∂ π × (θ n π π
). In a gapless system where the momentum space is restricted, the AM is not invariant. On the other hand, in a gapped system, the AM is found to be invariant by using the Stokes theorem and the single-valued property of wave functions.

IV. HALL VISCOSITY

Here we begin with a brief introduction of elasticity and viscosity. These are the mechanical properties of a system and are characterized by

T (ki) = λ (ki)(lj ) u (lj ) + η (ki)(lj ) u(lj) . ( 15
)
Here T (ki) , u (lj ) , and u(lj) are stress, strain, and strainrate tensors, respectively, and round brackets indicate the symmetry over their indexes. The linear coefficients λ (ki)(lj ) and η (ki)(lj ) are dubbed the elastic modulus and the viscosity, although their sign convention is not fixed. We can decompose the viscosity into the symmetric and antisymmetric parts:

η (ki)(lj ) = η (ki)(lj ) S + η (ki)(lj ) A
, where η

(ki)(lj ) S = η (lj )(ki) S
and η

(ki)(lj ) A = -η (lj )(ki) A
. The latter is dubbed the HV and is formulated below.

As discussed above, a strain-rate tensor is the symmetric torsional electric field, and the stress tensor is the symmetric part of the momentum current. Therefore the viscosity can be formulated by the perturbation theory of the momentum current Eq. ( 5) with respect to torsion [30]. Here we define the nonsymmetric viscosity by

η î ĵ k l ≡ ∂T î k ∂ -T lĵ 0 = ∓ i 2 2 d D π (2π ) D π k tr ∂ π î ˆ T lĵ 0 Ĝ0 -∂ π î Ĝ-1 0 ĜT lĵ 0 < + c.c. ( 16
)
Note that the negative sign in the definition is necessary because ∂ 0 h lj in Eq. (4a) gives rise to a strain-rate tensor.

The Keldysh Green's function in the presence of torsion is given by ĜT

a cd = Ĝ0 ˆ T a cd Ĝ0 -π a Ĝ0 ∂ π c Ĝ-1 0 Ĝ0 ∂ π d Ĝ-1 0 Ĝ0 -(c ↔ d) 2i, (17)
and the corresponding self-energy T a cd is determined self-consistently. Equation ( 16) is rewritten by

η î ĵ k l = ∓ i 2 2 d D π (2π ) D π k tr ∂ π î ˆ T lĵ 0 Ĝ0 -∂ π î Ĝ-1 0 Ĝ0 ˆ T lĵ 0 Ĝ0 < + c.c. ∓ 2 4 d D π (2π ) D π k π l tr ∂ π î Ĝ-1 0 Ĝ0 ∂ π ĵ Ĝ-1 0 Ĝ0 ∂ πˆ0 Ĝ-1 0 Ĝ0 -(j ↔ 0) < + c.c., (18)
and then

η î ĵ k l = - 2 6 î ĵ k abc k d D π (2π ) D f (-π 0)π k π l tr G R 0 ∂ π a G R-1 0 G R 0 ∂ π b G R-1 0 G R 0 ∂ π c G R-1 0 + c.c. ( 19a
) - 2 4 d D π (2π ) D f (-π 0)π k π l tr G R 0 -G A 0 ∂ π î G R-1 0 + G A-1 0 G R 0 ∂ π ĵ G R-1 0 + c.c. ( 19b
) ± i 2 2 δ î k d D π (2π ) D tr ˆ T lĵ 0 Ĝ0 < + c.c. ( 19c
) ∓ i 2 2 d D π (2π ) D f (-π 0)π k tr <(1) T lĵ 0 G R 0 -G A 0 ∂ π î G R-1 0 G R 0 + c.c. ( 19d
)
Again we focus on the clean and noninteracting limit ˆ = 0. The retarded Green's function is given by G R 0 = [(-π 0) -H + μ + iη] -1 . We expand the trace with respect to the Bloch basis and evaluate the integral over (-π 0) by the residue theorem. As a result, we obtain the Berry-phase formula for the nonsymmetric HV,

η î ĵ k l = 1 î ĵ m n d d π (2π ) d π k π l n π mf n π , (20)
with f n π ≡ f ( n πμ) and the Berry curvature being defined by

n π k = i 2 î ĵ k ∂ π î u n π ∂ π ĵ u n π . ( 21
)
The proper HV should be symmetric as shown in Eq. ( 15). In the conventional approach involving a metric, the stress tensor is defined by T μν ≡ 2δS/δg μν . This is manifestly symmetric because a metric is symmetric. By using g μν = η ab h a μ h b ν , it can be related to the momentum current T μ a ≡ δS/δh a μ by T μν = (h aμ T ν a + h aν T μ a )/2. Therefore the symmetric HV is given by

η ( kî)( l ĵ ) ≡ (η kî l ĵ + η î k l ĵ + η kî ĵ l + η î k ĵ l )/4. ( 22
)
Although a strain-rate tensor is described by torsion, Eq. ( 22) is different from the torsional HV discussed in Refs. [38][39][40].

In two dimensions, only three components may be nonzero,

η (xx)(xy) = 1 2 n d 2 π (2π ) 2 π x2 n πz f n π , (23a) η (xx)(yy) = 1 n d 2 π (2π ) 2 π x π y n πz f n π , ( 23b
)
η (xy)(yy) = 1 2 n d 2 π (2π ) 2 π y2 n πz f n π . (23c)
Furthermore, if a system is rotationally invariant, there is only

one nonzero component η H = η (xx)(xy) = η (xy)(yy) , η H = 1 4 n d 2 π (2π ) 2 π 2 n πz f n π . ( 24
)
In three dimensions, all the components of the HV should vanish if a system is rotationally invariant. On the other hand, if a system is axially invariant along the z axis, two components are independent,

η (xx)(xy) = 1 2 n d 3 π (2π ) 3 π x2 n πz f n π , ( 25a
)
η (zx)(yz) = 1 4 n d 3 π (2π ) 3 × (π z2 n πz -π z π x n πx -π z π y n πy )f n π . (25b)
As in the case of the momentum polarization Eq. ( 12), these expressions are quite analogous to that for the Hall conductivity, corresponding to the charge transport. The integrand just differs in the factor of π .

Finally, let us comment on the interacting cases. In the conventional metric approach, the effects of interactions are fully taken into account by using the many-body ground-state wave function. On the other hand, in our approach, such effects are compiled into the self-energy and can be taken into account by using the Feynman diagrams or the dynamical mean-field theory. Therefore these two approaches are complementary and coincide in the noninteracting limit, which reminds us that the charge polarization can be calculated by averaging over boundary conditions [24] or by using the Green's function formula [25,26].

V. APPLICATIONS TO CHIRAL SUPERCONDUCTORS

Now we apply our results to chiral SCs. For simplicity, we restrict ourselves to the single-band model with a singlet or 134510-4 unitary triplet pairing described by

H -μN = k [c † k↑ c -k↓ ] ξ k k * k -ξ k c k↑ c -k↓ , ( 26
)
in which ξ k ≡ kμ is even, and the gap k is even or odd for singlet or triplet with d k z, respectively. This Hamiltonian has the positive and negative dispersions

±E k = ± √ ξ 2 k + | k | 2
, whose wave functions are given by

[|u k+ |u k-] = u k -v k v * k u k . ( 27
)
Here

u k = (1 + ξ k /E k )/2 and v k = (1 -ξ k /E k )/2 k /| k | satisfy the normalization condition u 2 k + |v k | 2 = 1.
The temperature dependence of the gap strength is obtained by solving the gap equation. Now that we are not interested in the competition between several phases but in the AM and the HV for chiral SCs, we assume the attractive interaction favoring a chiral pairing symmetry k = w k with w k being normalized by d k |w k | 2 /4π = 1 in three dimensions. By using the standard approximations, the gap equation is given by

ln | 0 / | = ∞ 0 dξ k d k 4π 2|w k | 2 f (E k )/E k . ( 28
)
Next, we calculate the Berry connection and curvature. The Berry connections for the positive-and negative-energy states have opposite signs; namely,

A î + k = -A î -k ≡ A î k , A î k = i v k ∂ k î v * k -∂ k î v k v * k 2. ( 29
)
Correspondingly, the Berry curvatures change their signs; i.e.,

+ k m = --k m ≡ k m, k m = i î ĵ m∂ k î v k ∂ k ĵ v * k . ( 30
)
For chiral SCs with w k ∝ e imφ in the polar coordinate k = (k sin θ cos φ,k sin θ sin φ,k cos θ ), the Berry connection and curvature are given by

A x k = -m|v k | 2 sin φ/k sin θ, ( 31a
) A y k = m|v k | 2 cos φ/k sin θ, ( 31b
)
A z k = 0, ( 31c
)
and respectively. Consequently, the AM for gapped chiral SCs at zero temperature is given by

kz = m(k∂ k + cot θ∂ θ )|v k | 2 /k 2 , ( 32a
) kx = m cos φ(-k cot θ∂ k + ∂ θ )|v k | 2 /k 2 , ( 32b
) ky = m sin φ(-k cot θ∂ k + ∂ θ )|v k | 2 /k 2 , ( 32c)
L z = - k ( A k × k) z = m k |v k | 2 = mN 0 /2, ( 33
)
where we introduce the number of electrons at zero temperature N 0 = k 2|v k | 2 . This result is consistent with the recent microscopic studies suggesting γ = 0 [11][12][13][14][15][16][17][18][19]. However, the bulk AM is well defined only in a gapped system at zero temperature. Therefore, we cannot apply this result to threedimensional chiral SCs with point or line nodes. Below we calculate the HV, which is well defined in a gapless system or at finite temperature.

A. Hall viscosity in two dimensions

First, we consider w k = e i φ in two dimensions. For odd , the system is triplet and is classified into a class-D topological SC in terms of the topological periodic table [START_REF] Schnyder | [END_REF][START_REF] Kitaev | AIP Conf. Proc[END_REF]. On the other hand, for even , it is singlet and is classified into a class-C topological SC. Anyway it is gapped and the AM Eq. ( 33) is well defined at zero temperature. Since there is the rotational symmetry, only Eq. ( 24) is nonzero,

2η H = - 2 k k 2 kz [1 -2f (E k )] = - 2 k k∂ k |v k | 2 [1 -2f (E k )]. (34)
At zero temperature, we employ the partial integral to obtain 2η H = N 0 /2, which is consistent with the previous result for gapped chiral SCs [33,34,37].

Let us turn to finite temperature. By solving the gap equation Eq. ( 28) and using the temperature dependence of the gap strength, we evaluate the HV normalized by that at zero temperature, i.e., half the AM,

2η H N 0 /2 = ∞ 0 dξ ∂ ξ (ξ/E)[1 -2f (E)]. ( 35
)
As shown in Fig. 1, both the gap strength and the HV exponentially converge since the system is gapped.

B. Hall viscosity in three dimensions

Next, we consider w k = √ 4πY m (θ,φ) in three dimensions, where Y m is a spherical harmonic function. As emphasized above, three-dimensional chiral SCs have nodes generally, and the bulk AM is not well defined. For example, p x ± ip y with ( ,m) = (1,±1) and d x 2 -y 2 ± id xy with ( ,m) = (2,±2) have point nodes at poles, while d zx ± id yz with ( ,m) = (2,±1) has both point and line nodes. Since there is the axial symmetry along the z axis, two components Eq. ( 25) are nonzero,

2η (xx)(xy) = - k k x2 kz [1 -2f (E k )] = -m k sin 2 θ cos 2 φ(k∂ k + cot θ∂ θ )|v k | 2 × [1 -2f (E k )], (36a) 2η
(zx)(yz) = - 2 k (k z2 kz -k z k x kx -k z k y ky ) × [1 -2f (E k )] = - m 2 k (2 cos 2 θk∂ k + cos 2θ cot θ∂ θ )|v k | 2 × [1 -2f (E k )], (36b)
both of which are reduced to mN 0 /2 at zero temperature.

We numerically calculate the normalized HV at finite temperature,

2η (xx)(xy) mN 0 /2 = 3 4 ∞ 0 dξ π 0 sin θdθ sin 2 θ∂ ξ (ξ/E)[1 -2f (E)], (37a) 2η
(zx)(yz) mN 0 /2 = 3 2 ∞ 0 dξ π 0 sin θdθ cos 2 θ∂ ξ (ξ/E)[1 -2f (E)], (37b)
instead of Eq. (36). In Figs. 2,3, and 4, we show the temperature dependences of the gap strength and the HV for p x + ip y , d zx + id yz , and d x 2 -y 2 + id xy , respectively. All of them converge by the power laws since the systems have nodes. These powers can be analytically obtained by expanding the integrands in Eqs. ( 28) and ( 37) around nodes as in the Appendix. Especially for d x 2 -y 2 + id xy , we find that the temperature dependence of | / 0 | and 2η (xx)(xy) /( mN 0 /2) is almost determined by the line node, while that of 2η (zx)(yz) /( mN 0 /2) is by the point nodes. We also summarize their powers in Table I. The temperature dependence of the AM for p x + ip y was calculated in finite systems such as a mesoscopic cylinder [15] and a macroscopic disk [18,19] compared to the coherence length. In three dimensions, the AM decreases from N 0 /2 by T 2 owing to the presence of the point nodes [15]. In two dimensions, the AM also decreases from N 0 /2 by T 2 , which is attributed to the presence of the Majorana edge modes [18,19]. Although the HV is equal to half the AM at zero temperature not only in two dimensions but also in three dimensions, the temperature dependence of the AM and the HV is generally different. This discrepancy does not conflict with the relation between the AM and the HV [33,34], because it is available only for gapped systems at zero temperature. Indeed such a difference was recently found in two-dimensional gapless systems by using the holographic approach [START_REF] Liu | [END_REF], too. As pointed out in this paper, the AM is given by the polarization of the momentum. In this sense, it is seen as a thermodynamic quantity, while the HV is just a transport coefficient but not a bulk quantity. Therefore it is natural to think that physical origins of those quantities and also their temperature dependences are different in general, even if they have a special relation at zero temperature. It may be interesting to study other models, e.g., quantum Hall systems and chiral SCs with higher , at finite temperature. Let us note that such a relation between a thermodynamic quantity and a transport coefficient was also proposed for the entropy density and the shear viscosity [START_REF] Kovtun | [END_REF]. Nevertheless it is expected that they would have a universal relation only in the extremely strong coupling situation.

To summarize, we derive the Berry-phase formulas for the AM and the HV by using the Keldysh formalism in a curved spacetime. First, we examine the physical quantity conjugate to the AM, namely, an angular velocity of rotation, from the gauge-theoretical viewpoint of gravity. Since an angular velocity is the antisymmetric torsional electric field, we define the AM by the temporal integral of the antisymmetric momentum current induced by an adiabatic deformation, which is implemented by the gradient expansion. Viscosity is the response of the stress tensor, i.e., the symmetric momentum current, to a strain-rate tensor. It can be derived by the perturbation theory with respect to torsion because a strain-rate tensor is described by the symmetric torsional electric field. We also apply these results to chiral SCs in two and three dimensions. In two dimensions at zero temperature, we reproduce L z = 2η H = N 0 /2 without any finite-size effects. In three dimensions at zero temperature, where the AM is not well defined owing to the presence of nodes, we find that the HV is equal to half the AM calculated in finite systems previously. Although it is not related to the AM at finite temperature, it is useful to determine the gap structure for chiral SCs.

Thus, all the power-law behaviors can be explained by nodal excitations.

18 FIG. 1 .

 181 FIG. 1. (Color online) The gap strength (red cross) and the HV (green filled square) for two-dimensional chiral SCs as functions of (a) temperature T /| 0 | and (b) inverse temperature | 0 |/T . The green broken line indicates e -| 0 |/T .

6 FIG. 2 . 9 FIG. 3 .

 6293 FIG. 2. (Color online) Temperature dependences of | / 0 | (red cross), 2η (xx)(xy) /( mN 0 /2) (green filled square), and 2η (zx)(yz) /( mN 0 /2) (blue open circle) for a three-dimensional p x + ip y SC with ( ,m) = (1,1). The green broken and blue dotted lines indicate (T /| 0 |) 4 and (T /| 0 |) 2 , respectively.

5 FIG. 4 .

 54 FIG. 4. (Color online) Temperature dependences of | / 0 | (red cross), 2η (xx)(xy) /( mN 0 /2) (green filled square), and 2η (zx)(yz) /( mN 0 /2) (blue open circle) for a three-dimensional d x 2 -y 2 + id xy SC with ( ,m) = (2,2). The red solid, green broken, and blue dotted lines indicate (T /| 0 |) 3 , (T /| 0 |) 2 , and T /| 0 |, respectively.

2 15| | 2 θ 2 / 2 (ξ 2 +√ ξ 2 +15| | 2 θ 2 / 2 + 1 8 ξ 2 +√ ξ 2 +15| | 2 θ 4 / 8 + 1 2 15| | 2 θ 4 √ ξ 2 +15| | 2 θ 4 / 8 + 1

 2222221822481242481 15| | 2 θ 2 /2) 3/2 (e β power, the gap strength | / 0 | and one component of the HV 2η (xx)(xy) /( mN 0 /2) are mainly contributed from the line node, while the other component 2η (zx)(yz) /( mN 0 /2) is from the point nodes.Fordx 2 -y 2 + id xy , where |w k | 2 = 15 sin 4 θ/8, we expand Eqs. (28) and (37) around the point node θ = 0 and introduce y = 15/8β| |θ 2 . Then we obtain ln | 0 / | 2 15| | 2 θ 4 /8 (e β /8 (ξ 2 + 15| | 2 θ 4 /8) 3/2 (e β θ 4 /8 (ξ 2 + 15| | 2 θ 4 /8) 3/2 (e β √ ξ 2 +15| | 2 θ 4 /8 + 1)

TABLE I .

 I Power-law behaviors of the gap strength and the HV at low temperature for several pairing symmetries in three dimensions. See also Figs.2, 3, and 4.

						VI. DISCUSSION AND SUMMARY
	Symmetry	( ,m)	Nodes	| | 2η (xx)(xy) 2η (zx)(yz)
	p x + ip y	(1,1)	point	T 4	T 4	T 2
	d zx + id yz	(2,1) point and line T 3	T	T 2
	d x 2 -y 2 + id xy (2,2)	point	T 3	T 2	T

ACKNOWLEDGMENTS

We thank H. Sumiyoshi and Y. Tada for fruitful discussions. This work was supported by Grants-in-Aid for the Japan Society for the Promotion of Science, Fellows No. 24-600 and No. 25-4302.

APPENDIX: TEMPERATURE DEPENDENCE OF HALL VISCOSITY

In this appendix, we approximately but analytically calculate the gap strength and the HV to explain their power-law behaviors at low temperature.

For p x + ip y , where |w k | 2 = 3 sin 2 θ/2, we expand Eqs. ( 28) and (37) around the point node θ = 0 to obtain

In the second lines, we change the variables by x = βξ and y = √ 3/2β| |θ , and in the third lines, we take the low-temperature limit √ 3/2β| |π/2 → ∞. These integrals can be analytically estimated in the polar coordinate. For d zx + id yz , where |w k | 2 = 15 sin 2 θ cos 2 θ/2, there are both point and line nodes. First, we expand Eqs. ( 28) and (37) around the point node θ = 0 and change the variables by x = βξ and y = 15/2β| |θ , which results in

On the other hand, for the line node, we redefine θ → π/2θ and expand Eqs. ( 28) and ( 37) around θ = 0, leading to


Tawsif Gokhool
email: tawsif.gokhool@inria.fr

Maxime Meilland
email: meilland@i3s.unice.fr

Patrick Rives
email: patrick.rives@inria.fr

Eduardo Fernández-Moral

A Dense Map Building Approach from Spherical RGBD Images

Keywords: Visual Odometry, RGBD Dense Tracking, Spherical/Omnidirectional Vision, Optimisation, Pose Graph

Visual mapping is a required capability for practical autonomous mobile robots where there exists a growing industry with applications ranging from the service to industrial sectors. Prior to map building, Visual Odometry(VO) is an essential step required in the process of pose graph construction. In this work, we first propose to tackle the pose estimation problem by using both photometric and geometric information in a direct RGBD image registration method. Secondly, the mapping problem is tackled with a pose graph representation, whereby, given a database of augmented visual spheres, a travelled trajectory with redundant information is pruned out to a skeletal pose graph. Both methods are evaluated with data acquired with a recently proposed omnidirectional RGBD sensor for indoor environments.

INTRODUCTION

This paper is focussed in the context of Visual Odometry (VO), defined as the process of estimating the relative motion of a mobile agent using vision sensors. This incremental technique computes the pose of a vehicle based on the movements induced by onboard cameras. Over the years, VO has been useful to compensate other similar techniques such as wheel odometry which is highly affected by dead reckoning in uneven terrains. On the other hand, global positioning system (GPS) has shown its limitation in aerial and underwater applications [START_REF] Fraundorfer | Visual Odometry: PartII: Matching, Robustness, Optimisation and Applications[END_REF].

Odometry techniques in general require accurate relative motion estimation to reduce trajectory drift. Over the last decade, VO coupled with SLAM approaches have evolved in two main categories; feature based and dense techniques. Feature based methods rely on a preceeding identification and extraction phase. Registration allows images which are further apart , but are affected by outliers . Dense technique, which uses the entire information content has become increasingly popular recently as registration is performed using a direct image alignment [START_REF] Meilland | Simultaneous superresolution tracking and mapping[END_REF]. The latter is generally more accurate but is restricted by smaller interframe displacements.

An important attribute of a reliable mapping system is scalability. Accumulation of frames over ex-tended trajectories causes the issue of memory space over time. In order to reduce data redundancy, maintaining a sparse pose graph in a skeletal representation is often desirable.Pose Graph pruning is therefore a central issue as discussed in [START_REF] Konolige | FrameSLAM: From Bundle Adjustment to Real-Time Visual Mapping[END_REF].

This paper extends the results presented in [START_REF] Meilland | Dense visual mapping of large scale environments for realtime localisation[END_REF] in several ways. We introduce a cost function combining both geometric and photometric constraints and use it in a direct image registration method for pose estimation. Additionally, the problem of pose graph optimization is addressed, whereby, given a database of augmented visual spheres, an explored trajectory with redundant information is pruned out to a skeletal pose graph.

RELATED WORKS

Visual Odometry

After the well established theory of optical flow for motion estimation using Intensity based images, the community has now turned to the fusion of both intensity and depth information for registration and tracking. This trend is recurrent especially due to the advent of consumer based RGB-D sensors such as Microsoft's Kinect or Asus's Xtion Pro. A break-through of this technique was devised by [START_REF] Harville | 3D Pose Tracking with Linear Depth and Brightness Constraints[END_REF], who were among the first to formalize registration as a Brightness Change Constraint Equation (BCCE) and a Depth Change Constraint Equation (DCCE). They argued that tracking is best achieved with intelligent fusion of the two constraints mentioned above in order to reduce the effect of drift, occlusions or illumination changes.

Their work was further extended in [START_REF] Rahimi | Reducing Drift in Parametric Motion Tracking[END_REF], where a Maximum Likelihood (ML) based differential tracker was developed and the problem of drift and loop closure were also addressed. To improve the tracking performance, the measurement model incorporated the fusion of multiple frames. A similar formulation was proposed in [START_REF] Wang | Real Time Bayesian 3-D Pose Tracking[END_REF], but encapsulated in a Bayesian framework. A 2D-3D pose estimation along with an intensity cost function helped to improve feature correspondence as well as drift reduction.

Recent works of the same domain includes that of [START_REF] Newcombe | KinectFusion: Real-Time Dense Surface Mapping and Tracking[END_REF] whereby a preliminary frame to frame registration is fed to a surface reconstruction module which improves the perceived model over time together with the estimated pose. The RGB-D slam framework of [START_REF] Henry | RGB-D Mapping: Using Kinect-Style Depth Cameras for Dense 3D Modeling of Indoor Environments[END_REF] used a variant of the Iterative Closest Point algorithm (ICP) -direct alignment of 3D point clouds [START_REF] Besl | A method for registration of 3-D shapes[END_REF], [START_REF] Zhang | On local matching of free-form curves[END_REF], together with photometry. Their work also included a surfel representation of the environment to have a more compact representation of the information available. A robust ICP flavour was experimented in [START_REF] Lui | Robust Egomotion Estimation using ICP in Inverse Depth Coordinates[END_REF] using an inverse depth parameterisation. Other related works merging photometric and geometric information for Visual Odometry (VO) can be found in [START_REF] Tykkälä | Direct Iterative Closest Point for Real-time Visual Odometry[END_REF], [START_REF] Tykkälä | RGB-D tracking and reconstruction for TV broadcasts[END_REF], [START_REF] Kerl | Dense Visual SLAM for RGB-D Cameras[END_REF].

Keyframe Selection

When exploring vast scale environments, many frames sharing redundant information clutter the memory space considerably. The idea to select keyframes based on a predefined criteria happens to be very useful in the conception of a sparse skeletal pose graph. Furthermore, performing frame to frame registration introduces drift in the trajectory due to uncertainty in the estimated pose as pointed out in [START_REF] Kerl | Dense Visual SLAM for RGB-D Cameras[END_REF].

Therefore, in order to overcome this issue, frame to keyframe odometry is rather desirable. Common techniques applied constitute of introducing keyframes when two such frames share very few features between them as defined by the view cluster-ing criteria of [START_REF] Konolige | Towards lifelong Visual Maps[END_REF], or a threshold on the number of features shared between a central frame and its corresponding adjacent frames [START_REF] Royer | Monocular vision for mobile robot localization and autonomous navigation[END_REF]. [START_REF] Strasdat | Scale Drift-Aware Large Scale Monocular SLAM[END_REF], introduce a new frame whenever a certain distance threshold between camera poses is exceeded. [START_REF] Wang | Real Time Bayesian 3-D Pose Tracking[END_REF] modeled a temporal criteria to take into account the interpose frame difference as well as feature overlap among them. On the other hand Meilland et al. [START_REF] Meilland | Dense visual mapping of large scale environments for realtime localisation[END_REF] used a selection criteria based on the Median Absolute Deviation (MAD) in intensity error between a reference and a current frame to reinitialize on a predefined threshold.

Recently, information theory [START_REF] Kretzschmar | Lifelong map learning for graph-based slam in static environments[END_REF] was introduced to prune out nodes with a minimal expected information gain. On a similar note, [START_REF] Kim | Real-time visual SLAM for autonomous underwater hull inspection using visual saliency[END_REF] set up a salient keyframe selection criteria based on the ratio between the covariance of the measurement and that of the innovation to encode the entropy between two corresponding nodes. However, this criteria is modeled based on the probabilistic framework of iSAM where the covariances are easily extracted. On the other hand, we found that the criteria based on a differential entropy approach introduced by [START_REF] Kerl | Dense Visual SLAM for RGB-D Cameras[END_REF] is more suitable for our system of geo-referenced spheres which will be discussed in section 4.3.

SPHERICAL SYSTEM REPRESENTATION

A new sensor for a large field of view RGBD image acquisition has been used in this paper. This device integrates a set of Asus Xtion Pro Live (Asus XPL) sensors and allows to build a spherical representation as sketched in figure 1. 1The chosen configuration offers the advantage of creating full 360 • RGBD images of the scene isometrically, i.e. the same solid angle is assigned to each pixel. This permits to apply directly some operations, like point cloud reconstruction, photo consistency alignment or image subsampling. To build the images, the sphere S 2 is sampled according to the resolution of our device, so that the equator (θ direction) contains 1920 samples in the range [0, 2π], while the meridian (φ direction) is sampled with the same spacing, containing 960 samples in the range [-π/2, π/2].

For spherical warping, a virtual sphere with the above sampling and radius ρ = 1 is used to project the sample points into image coordinates (u, v), (see figure 2). For that, the extrinsic calibration of each sensor is taken into account. Thus, a point p in S 2 is parameterized in

C 1 C 2 C 3 C 4 C 5 C 6
R 3 with   x y z   =   ρ cos(φ) sin(θ) ρ sin(φ) ρ cos(φ) cos(θ)   (1)
where ρ in this case equals 1 (unit sphere). This point q = (u, v) on image coordinates is found by applying perspective projection to p, through the homogeneous matrix M = K[R|t] where K ∈ R 3×3 is the camera projection model and [R|t] ∈ SE(3) is the relative position of the camera (extrinsic calibration). Nearest neighbor interpolation is used to assign the intensity and depth values to the respective spherical images.

The point cloud from the spherical representation (figure 3) is obtained by applying equation (1), substituting ρ by the measured depth and the values of θ and φ by its corresponding image location.

SPHERICAL RGBD REGISTRATION

To accurately recover the position of the RGBD spheres with respect to one-another, a 6 dof pose estimation method is proposed based on accurate dense localization. Considering an RGBD sphere S(I, ρ) as defined in Section 3, the objective is now to compute a pose between a reference sphere and the next one.

The localisation problem is then to estimate the relative pose T between the two consecutive spheres.

Inter-frame incremental displacement is defined as an element of the Lie groups applied on the smooth differential manifold of SE(3) [START_REF] Blanco | A tutorial on se(3) transformation parameterizations and on-manifold optimization[END_REF], also known as the group of direct affine isometries. Motion is parametrized as a twist (a velocity screw motion around an axis in space), denoted as

x = {[ω, υ]|υ ∈ R 3 , ω ∈ so(3)} ∈ se(3): ω = ω x ω y ω z , υ = υ x υ y υ z , with so(3) = { ω ∈ R 3×3 | ω = -ωT ),
where ω and υ are the angular and linear velocities respectively . The reconstruction of a group action T ∈ SE(3) from the twist consists of applying the exponential map using Rodriguez formula [START_REF] Ma | An Invitation to 3-D Vision[END_REF]. Thereon, T is denoted as the transformation (pose) recovered between the current frame I c ∈ R m×n observed at time t and the reference frame I * ∈ R m×n .

Cost Function Formulation

In [START_REF] Meilland | Dense visual mapping of large scale environments for realtime localisation[END_REF], the pose estimation was tackled as an image registration problem whose ob-jective was to perform a direct image alignment by minimizing a cost based on the image intensity only:

F I = 1 2 k ∑ i η HUB I w( TT(x); P * i ) -I * w(I; P * i ) 2 ,
(2) where w(.) is the warping function that projects a 3-D point P i , given the pose T , onto the sphere and η HUB is a robust weighting function on the error given by Huber's M-estimator [START_REF] Huber | Robust Statistics[END_REF]. The latter plays an important role in reducing the effect of outliers by measuring the similarity between two corresponding pixels. Hence the weight computed accommodates partially the uncertainty associated to each pairing between a reference and a current frame.

With the aim of robustifying the above cost function, a geometric point to plane constraint [START_REF] Chen | Object modelling by registration of multiple range images[END_REF]) is added to the equation (2) as follows:

F S = 1 2 e I 2 D I + λ 2 2 e ρ 2 D D , (3)
which can be written in its explicit form:

F S = 1 2 k ∑ i η HU B I w( TT(x); P * i ) -I * w(I; P * i ) 2 + λ 2 2 k ∑ i η HU B n T P i -TT(x)P * i 2 , (4)
such that P ∈ (X,Y, Z) -→ (θ, φ, ρ) and λ is a tuning parameter to effectively balance the two cost functions. n T is the normal map computed from the cross product of adjacent points on the grid structured depth map.

Since the unknown x is common in both parts of equation ( 4), the error function is stacked in a single vector computed simultaneously as shown: (5) The Jacobian matrix J S is the total Jacobian relative to the augmented cost function defined above and is given as:

J S = J I * J w J T λn T J D , (6)
Where, respectively, J I * is the jacobian w.r.t. the intensity, and J w is the jacobian w.r.t. the warping function, J T is the jacobian w.r.t. the pose and J D is the jacobian w.r.t. the depth.

Similarly, the weighting function for each part of cost function is stacked in a block diagonal matrix where D I , D D ∈ R mn×mn are the confidence level in illumination and depth respectively for each corresponding feature pair:

D S = D I 0 0 D D (7)
Linearization of the above cost function leads to a classic closed form solution given by an Iterative Least Mean Squares (ILMS) and the incremental motion x is given by the following expression:

x = -(J T S D S J S ) -1 J T S D S e(x) S (8)
Using an iterative optimization scheme, the estimate is updated at each step by an homogeneous transformation:

T ←-TT(x), (9)
where T = [R t] is the current pose estimate with respect to the reference available from the previous iteration. This method, requires an initial guess of the pose and it happens that an initial solution does not give an optimum of x depending on how far the initialization has been made with respect to the reference. To be able to recover a pose close to the solution, equations (5) to ( 9) are evaluated iteratively until a tolerated threshold in x < ε is reached. This leads us to the term Iterative Re-weighted Least Squares (IRLS) where the Gauss-Newton algorithm is well adapted for these genres of non linear unconstrained optimization problem.

Information Selection

In order to estimate displacement between two frames, a set of correspondences between them has to be found to constrain the motion model (R,t) efficiently. This is a vital step in visual odometry as bad feature matches lead to pronounced deviation from the real motion. In literature, two mainstreams are identified: the first one based on feature extraction while the second one uses dense (correspondencefree) methods [START_REF] Fraundorfer | Visual Odometry: PartII: Matching, Robustness, Optimisation and Applications[END_REF].

Both approaches exhibit their advantages and inconveniences. The former, based on point feature detection needs to undergo an identification phase where primitives such as blobs, corners, lines or edges are usual candidates. Good features are characterized in terms of several properties such as stability, computational efficiency, distinctiveness or invariance to geometric and photometric changes.

On the other hand, dense methods make use of the entire photometric and geometric information content for tracking. Along that streamline, [START_REF] Dellaert | Fast Image-based tracking by selective pixel integration[END_REF] argued that instead of using all the information content which is computation intensive, selection of a subset of good pixels that yield enough information about the 6 degrees of freedom (DOF) state vector could considerably reduce computational cost as well as data redundancy without compromising on the accuracy of the estimation.

Recently, a similar approach was independently devised in [START_REF] Meilland | A spherical robot-centered representation for urban navigation[END_REF] which relies on finding the most informative pixel subset based on the decomposition of the Jacobian matrix obtained from equation ( 2). We hereby outline the backbone of the algorithm which has been implemented in the system and is also illustrated in figure 4: • Decomposition of the Jacobian J gives in itself a saliency map pertaining to each DOF of x • Columnwise, J is sorted out for the best pixel which is indexed in decreasing order of magnitude • Each DOF is then looked-up for the best ranked pixel i in ascending order and the i th line is lifted to a new table as shown in figure 4 • In case a particular pixel gives the best indexing in more than one DOF and given that it has already been selected, we proceed to the second best pixel and so on. • The selection process is performed iteratively until all the pixels have been successfully sorted out • The final result is a table of best ranked pixels with respect to their response to a particular DOF Therefore, instead of using all the pixels from the intensity and depth map, a wise selection of the top 10-20% of the pixels are used for registration.

Salient Frames Criterion

We define the differential entropy of a random variable x with dimensions n such that x ∼ N (µ, Σ) as:

H(x) = n 2 1 + ln(2π) + 1 2 ln(|Σ|), (10)
where, Σ is the covariance matrix of the estimate x which is obtained by the inverse of the Fisher Information matrix computed from the normal equations ( 8):

Σ = J T S D S J S -1 (11)
The entropy ratio between a motion estimate x k:k+ j from a reference frame k to a current frame k+ j is obtained by the following deduction:

α = H(x k:k+ j ) H(x k:k+1 ) , ( 12
)
where the denominator is just the entropy relative to the consecutive of the k th frame in question. The greater the gap between the reference and the current frame, the smaller the value of α. Hence a preset on the value of α is used to decide whenever a keyframe needs to be inserted or not.

RESULTS

Simulated Data

Our first fold of experiments have been performed on a synthetic dataset available along with the provided ground truth. Our algorithms have been thoroughly tested on this dataset of around 600 images in order to validate the convergence of the various cost functions discussed in the paper before moving on to real data. Figure 6 shows the trajectory plot obtained from the augmented cost function (4). As illustrated, we observe a small but noticeable offset between ground truth and the computed trajectory using the proposed approach. This is accountable for the fact that VO is prone to drift phenomena.

Real Data

Our experimentations are performed on a dataset of around 170 intensity and depth images within an office environment constituted of several rooms and corridors. Figure 7 illustrates the different types of places observed along the trajectory. 2) and (4) respectively. We observe that the algorithm using vision-only performs poorly in low-textured regions such as corridors or in the presence of reflections from window panes. Such circumstances lead to erroneous estimated poses coming from poor convergence of the algorithm. On the other hand, the photometry + geometry cost function takes care of these discrepancies by relying more on the depth information available. This is justified by the overall faster convergence of the algorithm as profiled in figure 8(b). The high spikes of the figure capped at 200 iterations are due to the nonconvergence of the intensity cost function while the new approach still manage to converge at lower iterations. Finally, figures 8(c) and 8(d), depicts the error norm of each frame at convergence.

Figure 9 focusses on the keyframe criteria discussed in section 4.3 for the same dataset: MAD (method 1) and Entropy ratio (method 2). Over here, we fix the photometry + geometry approach with the keyframe criteria as the only variants. While the MAD acts on the residual warping error after convergence, the entropy ratio α abstracts the uncertainty in the estimated pose along the trajectory. The number of spheres initialized for method 1 is around 50 while method 2 revealed 27 re-initializations. However, we believe that greater reduction is achievable with lesser inter frame acquisition so that the pose estimation algorithm is better initialized leading to a faster and more accurate convergence. A heuristic threshold of 15 was chosen for the case of the MAD and 0.96 for that of α.

CONCLUSIONS

In this paper, a spherical RGBD registration method has been proposed and we have demonstrated that both photometry and geometry can co-exist within the same framework. The idea is to implement a robust registration process that compensates mutually for poorly conditioned photometric images as well as depth maps. Furthermore, we have compared two different criteria for keyframe selection and have shown an improvement by using a criterion based on differential entropy. Results have been presented with synthetic and real data. For the latter, a novel sensor was used to acquire an indoor dataset at the INRIA Sophia Antipolis offices.

Our future works will be directed towards the exploitation of our new spherical mapping system on board of a mobile robot platform for simultaneous localisation and mapping (SLAM) as well as autonomous navigation. Frame Nos.

(d) Error norm vs Frame Nos.

Int +D Int

Int + D Int

Figure 1 :

 1 Figure 1: Multi RGBD acquisition system.

Figure 2 :

 2 Figure 2: Spherical image construction.

Figure 3 :

 3 Figure 3: Point cloud obtained from the spherical RGBD acquisition system.

η

  HUB I w( TT(x); P * ) -I * (P * ) λη HUB n T P -TT(x

Figure 4 :

 4 Figure 4: Saliency map.

Figure 5 :

 5 Figure 5: Example of spherical views obtained from the Synthetic dataset.

Figure 6 :

 6 Figure 6: Trajectory comparison between ground truth and that of the Intensity with Geometric cost function with the keyframe criteria based on the differential entropy approach

Figure 7 :

 7 Figure 7: Example of spherical views from the Office dataset.

Figure 8

 8 Figure 8(a) shows the trajectory obtained using the cost functions of equations (2) and (4) respectively. We observe that the algorithm using vision-only performs poorly in low-textured regions such as corridors or in the presence of reflections from window

Figure 8 :

 8 Figure 8: Performance comparison between Intensity only and Intensity and Depth cost functions.

Figure 9 :

 9 Figure 9: Comparison between MAD vs entropy ratio α using the same augmented photometry + geometry cost function.

Figure 10 :

 10 Figure 10: Top view of real trajectory with fusion of intensity and depth with entropy ratio α.

The conception of this sensor system is out of scope of this paper and details are omitted due to ongoing patent consideration.

ACKNOWLEDGEMENTS

This work has been developed at INRIA Sophia-Antipolis and extends a previous collaboration with the I3S/CNRS team. It is funded by EADS/Astrium under the contract N. 7128 and by the "European Re-gional Development Fund ERDF" through the Spanish Government under the contract DPI2011-25483.


Taro Kimura
email: taro.kimura@cea.fr

Note on a duality of topological branes

We show a duality of branes in the topological B-model by inserting two kinds of non-compact branes simultaneously. We explicitly derive the integral formula for the matrix model partition function describing this situation, which correspondingly includes both the characteristic polynomial and the external source. We show that these two descriptions are dual to each other through the Fourier transformation, and the brane partition function satisfies integrable equations in one and two dimensions. . . . . . .

Introduction

The D-brane is the most fundamental, but also non-perturbative object in superstring theory. It gives a nontrivial boundary condition for open strings, and consequently a lot of gauge theories can be constructed as effective theories for stringy modes on them. When we implement various gauge groups and matter contents for the effective field theory, branes often have to be arranged in a complicated way. In such a case, we have to deal with the dynamics of brane intersection appropriately, and discuss what kind of stringy excitation can be allowed there. In this sense it is important to study the properties of a brane complex in detail.

In this article we investigate some aspects of brane-on-brane structure in topological string theory, especially through its matrix model description, by introducing two kinds of branes simultaneously. It is well known that the Riemann surface plays a crucial role in the topological B-model on Calabi-Yau threefolds, and this Riemann surface can be identified with the spectral curve appearing in the large N limit of the matrix model [1]. A brane is also introduced into the B-model, and its realization in terms of the matrix model has been extensively investigated [2]: There are seemingly two kinds of non-compact branes, which correspond to the characteristic polynomial and the external source in the matrix model. While these two descriptions are apparently different, we will see that they are essentially the same, as a consequence of the symplectic invariance of the topological B-model. From the viewpoint of the matrix model, this situation just corresponds to insertion of both the characteristic polynomial and the external source. We will explicitly show that two descriptions are dual to each other by seeing a newly derived matrix integral formula.

It is known that the correlation function of the characteristic polynomial and the matrix integral with the external source are deeply related to the integrable hierarchy [3]. Thus it is naively expected that the hybrid of them also possesses a similar connection to the integrability. Since in this case we have two kinds of variables corresponding to the characteristic polynomials and the external PTEP 2014, 103B04 T. Kimura sources, we obtain the 2D integrable equation as a consequence, and identify the partition function as the τ -function of the corresponding integrable hierarchy with the Miwa coordinates.

This article is organized as follows. In Sect. 2 we start with a review on how to consider branes in the topological B-model in terms of the matrix model. We introduce two descriptions of branes using the characteristic polynomial and the external source of the matrix integral with emphasis on the similarities and differences between them. In Sect. 3 we show a duality between two kinds of non-compact branes in the B-model by introducing both of them at the same time. We derive a new matrix integral formula corresponding to this situation, and show that the characteristic polynomial and the external source are dual to each other in the sense of Fourier transformation. We also show that the brane partition function satisfies the integrable equation in one and two dimensions, which is well known as the Toda lattice equation. In Sect. 4 we consider the Gaussian matrix model as an example. In this case we can show the duality using the fermionic variables, which is useful to discuss the effective degrees of freedom on the branes. We close this article in Sect. 5 with some discussions and concluding remarks.

Branes in the topological B-model

We first review the large N matrix model description of the topological B-model, mainly following Ref. [2], and how to introduce the brane, which is also called the B-brane. The non-compact Calabi-Yau threefold that we discuss in this article is obtained as a hypersurface in (u, v, p, x) ∈ C 4 :

u v -H ( p, x) = 0. (1)
Here H ( p, x) determines the Riemann surface

: H ( p, x) = 0 (2)
with

H ( p, x) = p 2 -W (x) 2 -f (x), (3)
where W (x) and f (x) are polynomials of degree n + 1 and n, respectively. This smooth Calabi-Yau threefold is given by deformation of the singular one:

H ( p, x) = p 2 -W (x) 2 . ( 4
)
The branes, which are compact, are wrapping n S 3 at critical points W (x) = 0, and the sizes of the S 3 s are parametrized by f (x), which describes the quantum correction around them. The holomorphic (3, 0)-form in the Calabi-Yau threefold ( 1) is chosen to be

= du ∧ dp ∧ dx u . ( 5
)
Then the periods of this (3, 0)-form over three-cycles reduce to the integral of the symplectic 2-form

D dp ∧ dx, ( 6
)
where D stands for domains in the complex 2D ( p, x)-plane. The boundary of this domain denoted by γ is related to the Riemann surface (2) as γ = ∂ D ⊂ . Thus this integral further reduces to a one-cycle on the Riemann surface :

γ p dx. ( 7
)
In this way we can focus only on the complex 1D subspace by keeping the dependence on u and v fixed.

PTEP 2014, 103B04

T. Kimura

Characteristic polynomial

The algebraic curve (2) can be identified with the spectral curve of the matrix model: It is just given by the loop equation in the large N limit of the matrix integral:

Z N = dX e -1 gs Tr W (X ) . ( 8
)
The polynomial W (x) is the matrix potential. This is the reason why we can discuss the topological B-model using the matrix model. In this description the genus expansion with 1/N corresponds to the quantum correction, and the 't Hooft parameter t = g s N gives the size of S 3 . The other canonical variable in (3) is related to the resolvent of the matrix model:

p(x) = W (x) -2g s Tr 1 x -X . ( 9
)
The saddle point equation of the matrix model is equivalent to the condition p(x) = 0. Since the one-form on the spectral curve is given by these two canonical variables,

λ = p(x) dx = dφ, ( 10
)
we can naturally introduce the chiral boson φ(x) on the Riemann surface , which is interpreted as the Kodaira-Spencer field describing deformation of the complex structure at infinity:

φ(x) = W (x) -2g s Tr log(x -X ). ( 11
)
Here this φ(x) also has the meaning of the effective potential for the matrix model, and thus the eigenvalues are degenerate at the critical point, ∂φ = p(x) = 0. The vertex operator, which creates a non-compact brane at a position x, is constructed by the standard bosonization scheme:

V (x) = e -1 2gs φ(x) = e -1 2gs W (x) det(x -X ). ( 12
)
The prefactor e -1 2gs W (x) corresponds to the classical part of the operator, while the determinant part, namely the characteristic polynomial, gives the quantum fluctuation as a gravitational back reaction. This brane creation operator gives a pole at x on the Riemann surface, and its residue is given by λ = g s . (13) This is just interpreted as one brane contribution, and also shows that g s plays the role of the Planck constant for the canonical pair ( p, x).

The partition function of the branes is represented as a correlation function of the characteristic polynomials, e.g., the k-point function given by

k α=1 det(x α -X ) . ( 14
)
This expectation value is taken with respect to the matrix measure

O(X ) = 1 Z N dX O(X ) e -1
gs Tr W (X ) (15) with the standard normalization 1 = 1. Including the classical part, the brane partition function is then given by

k (x 1 , . . . , x k ) = k α=1 e -1 2gs W (x α ) k α=1 det(x α -X ) . ( 16
)
This correlation function can be exactly evaluated using the orthogonal polynomial method. We will come back to this formula in Sect. 3.2 (see (30)).

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External source

Let us then consider another kind of non-compact brane in the B-model, which is described by the external source in the matrix model. We consider the matrix action written in the form of

S(P) = 1 g s Tr[W (P) -AP], (17)
where the potential is regarded as an integral of the one-form along an open path to a certain point p on the Riemann surface:

W ( p) = p x( p ) dp . ( 18
)
We can assume that the matrix A is diagonal A = diag(a 1 , . . . , a N ) without loss of generality. The action (17) corresponds to the matrix model with the external source

Z N (A) = dP e -1 gs Tr[W (P)-AP] , (19)
which is analogous to the Kontsevich model [4]. The external source implies the positions of N branes, at least at the classical level, because the extremum of the action W (P) -A = 0 gives the classical solution, X = diag(a 1 , . . . , a N ), since we have W (P) = X according to (18). The one-form used in ( 18) is apparently different from ( 10), but they are equivalent up to the symplectic invariance for a pair of the canonical variables ( p, x). This symmetry is manifest by construction of the topological B-model as seen in ( 5) and [START_REF] Kharchev | [END_REF]. Therefore, the two descriptions of the non-compact branes based on the characteristic polynomial and the external source in the matrix model are dual to each other in this sense. We will show in Sect. 3 that they are converted through the Fourier transformation by deriving the explicit matrix integral representation.

A duality of branes

As seen in the previous section, there are two kinds of non-compact branes in the topological B-model, which are related through the symplectic transformation. We now study the situation such that both kinds of branes are applied at once. This is realized by inserting the characteristic polynomial to the matrix model in addition to the external source. Let us consider the corresponding partition function denoted by

N , M {a j } N j=1 ; {λ α } M α=1 = dX e -1 gs Tr W (X )+Tr AX M α=1 det(λ α -X ). ( 20
)
We now have to be careful of the meaning of the matrix potential and the external source. As discussed in Sect. 2.2, if the matrix potential is given by the integral of the one-form in the form of ( 18), the external source gives the classical positions of branes in the x-coordinate. In this case, however, the roles of x and p are exchanged. Instead of (18), the potential should be written as

W (x) = x p(x) dx. ( 21
)
With this choice, the corresponding external source determines the positions of branes in the p-coordinate. We note that, in this case, the extremum of the action does not simply imply the classical positions of branes as W (X ) = A, because of the potential shift due to the characteristic polynomial. Although there exists this kind of back reaction, the interpretation of the external source as the p-coordinate still holds, as shown in the following.

PTEP 2014, 103B04

T. Kimura The characteristic polynomial and the external source put punctures on the Riemann surface for x and p coordinates because they correspond to the vertex operators creating the non-compact branes. Since ( p, x) is a pair of the canonical variables, if once its x-coordinate is fixed, one cannot determine the other p-coordinate, and vice versa. This means that the positions of the branes created by the characteristic polynomial and the external source are fixed by the x and p coordinates. For example, in the case with A = 0 for simplicity, all the N eigenvalues are distributed on the extended line, namely the cut in the complex x-plane described by p(x) = 0, which corresponds to the oneform p(x)dx. On the other hand, if we apply the other representation based on the p-plane, the one-form x( p)dp has a pole at p = 0 with degree N . Then, when one turns on the external source A = diag(a 1 , . . . , a N ), the pole at p = 0 is split and located at p = a i for i = 1, . . . , N . In this way the external source and also the characteristic polynomial characterize a pole with the p and x coordinates, respectively. In Fig. 1 we illustrate the situation such that both kinds of branes are inserted to the Riemann surface . We will see in the following that these coordinates can be exchanged through the Fourier transformation.

Matrix integral formula

We now provide an explicit formula for the partition function (20). In order to derive the formula, we apply a method to compute the matrix model partition function with the external source [START_REF] Dijkgraaf | New Symmetry Principles in Quantum Field Theory[END_REF][START_REF] Kharchev | [END_REF], which is also applicable to this situation.

First of all, we move to "eigenvalue representation" from the N × N Hermitian matrix integral (20) by integrating out the angular part of X . This can be done by using the Harish-Chandra-Itzykson-Zuber formula [7,8]

dU e Tr U XU † Y = det e x i y j (x) (y) , ( 22
)
where the integral is taken over the U(N ) group with the Haar measure normalized by the volume factor Vol(U(N )), and (x) is the Vandermonde determinant. Thus we have the eigenvalue representation of (20):

d N x (x) (a) N j=1 e -1 gs W (x j )+a j x j N j=1 M α=1 (λ α -x j ). ( 23
)
We here apply the formula

(x) (λ) M α=1 N j=1 (λ α -x j ) = (x, λ). ( 24
)
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T. Kimura

The RHS of this equation (x, λ) is the Vandermonde determinant for N + M variables (x 1 , . . . , x N , λ 1 , . . . , λ M ), which is represented as an (N + M) × (N + M) determinant:

(x, λ) = det x j-1 k x N +α-1 k λ j-1 β λ N +α-1 β = (-1) M N det λ α-1 β λ M+ j-1 β x α-1 k x M+ j-1 k . ( 25
)
We recall that j, k = 1, . . . , N and α, β = 1, . . . , M. Since each matrix element can be replaced with any monic polynomials, 25) is written in a more general form:

P k (x) = x k + • • • , this determinant (
(x, λ) = det P j-1 (x k ) P N +α-1 (x k ) P j-1 (λ β ) P N +α-1 (λ β ) = (-1) M N det P α-1 (λ β ) P M+ j-1 (λ β ) P α-1 (x k ) P M+ j-1 (x k ) . ( 26
)
Thus the integral is given by

1 (a) (λ) d N x (x, λ) N j=1 e -1 gs W (x j )+a j x j . ( 27
)
To perform this integral, we then introduce an auxiliary function:

Q k (a) = dx P k (x) e -1 gs W (x)+ax . ( 28
)
Using this function, we arrive at the final expression of the partition function (20):

N , M {a j } N j=1 ; {λ α } M α=1 = 1 (a) (λ) det Q j-1 (a k ) Q N +α-1 (a k ) P j-1 (λ β ) P N +α-1 (λ β ) = (-1) M N (a) (λ) det P α-1 (λ β ) P M+ j-1 (λ β ) Q α-1 (a k ) Q M+ j-1 (a k ) . ( 29
)
This expression is manifestly symmetric under the exchange of (a 1 , . . . , a N ) and (λ 1 , . . . , λ M ) with the transformation P k (λ) ↔ Q k (a). As seen in (28), this is nothing but a Fourier (Laplace) transform

of x k e -1
gs W (x) . Thus we can say that the characteristic polynomial and the external source in the matrix model are dual to each other in the sense of Fourier transformation.

In terms of the topological strings, this duality reflects the symplectic invariance of the canonical pair ( p, x) in the B-model, as seen in ( 6). Thus such a symmetry, which allows the exchange of the canonical variables x and p, is directly related to the duality for the (r, s) minimal model described by H ( p, x) = p r + x s + • • • , and also the open/closed string duality [9][10][11]. It should also be mentioned that the symplectic invariance, or SL(2, Z) symmetry exchanging the canonical pair, realizes the S-duality of the topological string or topological M-theory [12]. Since the two descriptions of the non-compact branes are just converted into each other through the Fourier transformation, they are essentially equivalent. Although this equivalence has already been pointed out for the Gaussian matrix model, as in Ref. [2], we can see that this symmetry even holds with a generic matrix potential W (x). This matrix potential determines the geometry of the Calabi-Yau threefold in the form of (3). In addition, the positions of the branes are in a relationship satisfying the uncertainty principle: one cannot determine both the x and p coordinates at the same time, but only one or the other. We also note that this kind of symplectic invariance appears quite generally in the topological expansion of the spectral curve [13][14][15].

PTEP 2014, 103B04 T. Kimura

Integrability

The formula ( 29) is a quite natural generalization of the well known formulae for the expectation value of characteristic polynomial product

M α=1 det(λ α -X ) = 1 (λ) det 1≤α,β≤M P N +α-1 (λ β ), ( 30
)
where P k (x) is the kth monic orthogonal polynomial with respect to the weight function w(x) = e -1 gs W (x) , and also the matrix integral with the external source dX e

-1 gs Tr W (X )+Tr AX = 1 (a) det 1≤ j,k≤N Q j-1 (a k ). ( 31
)
It is convenient to apply the simplest choice of the polynomial P k (x) = x k to this formula (31). In this case the function Q k (a) is given by

Q k (a) = dx x k e -1 gs W (x)+ax = d da k Q(a), (32)
with an Airy-like function

Q(a) = dx e -1 gs W (x)+ax ; (33)
see Ref. [3] and references therein for details.

It is known that this kind of determinantal formula generically plays a role as the τ -function [START_REF] Kharchev | [END_REF], and satisfies the Toda lattice equation by taking the equal parameter limit; see, e.g., Ref. [16]. We show that the formula (29) indeed satisfies a similar integrable equation in the following.

Let us parametrize the positions of branes by "center of mass" and deviations from it as a j = a + δa j and λ α = λ + δλ α . We rewrite the numerator in terms of the deviations {δa j } and {δλ α } by considering the Taylor expansion around the centers of mass:

det Q j-1 (a k ) Q N +α-1 (a k ) P j-1 (λ β ) P N +α-1 (λ β ) = det ⎛ ⎜ ⎜ ⎝ (δa k ) l-1 (l -1)! (δλ β ) γ -1 (γ -1)! ⎞ ⎟ ⎟ ⎠ det ⎛ ⎝ Q (l-1) j-1 (a) Q (l-1) N +α-1 (a) P (γ -1) j-1 (λ) P (γ -1) N +α-1 (λ) ⎞ ⎠ , ( 34
)
where

P (γ -1) j-1 (λ) = d dλ γ -1 P j-1 (λ), Q (l-1) j-1 (a) = d da l-1 Q j-1 (a) = Q j+l-2 (a)
and so on.

The first determinant in the RHS is almost canceled by the Vandermonde determinants in the denominator of (29), since they are invariant under the constant shift as (a) = (δa) = det(δa k ) j-1 and (λ) = (δλ) = det(δλ β ) α-1 , respectively. Therefore the partition function in the equal position limit becomes

N , M {a j = a} N j=1 ; {λ α = λ} M α=1 = c N , M det ⎛ ⎝ Q (k-1) j-1 (a) Q (k-1) N +α-1 (a) P (β-1) j-1 (λ) P (β-1) N +α-1 (λ) ⎞ ⎠ (35) = (-1) M N c N , M det ⎛ ⎝ P (β-1) α-1 (λ) P (β-1) M+ j-1 (λ) Q (k-1) α-1 (a) Q (k-1) M+ j-1 (a) ⎞ ⎠ (36) ≡ c N , M det R N , M (a, λ), 7/15 PTEP 2014, 103B04 T. Kimura with c N , M = N j=1 1 ( j -1)! M α=1 1 (α -1)! , ( 37
)
which corresponds to the volume factor for the U(N ) and U(M) groups. This expression is seen as a hybridized version of the Wronskian. In the following we apply P k (x) = x k and (32) as in the case of (31) for simplicity.

In order to derive the integrable equation, we now use the Jacobi identity for determinants, which is given by

D • D i j k l = D i k • D j l -D i l • D j k , ( 38
)
where D is a determinant, and the minor determinant D i j is obtained by removing the ith row and the jth column from the matrix. Similarly. D i j k l is obtained by eliminating the i, jth row and the k, lth column. Putting i = k = N + M and j = l = N + M -1 for the determinant in (36), we have

det R N , M • det R N -2, M = det R N -1, M • Mλ∂ 2 a det R N -1, M -∂ a det R N -1, M • Mλ∂ a det R N -1, M . ( 39
)
Here we have used a formula (A1) discussed in Appendix A. This provides the following relation for the equal position partition function:

N +1, M • N -1, M N , M 2 = M N λ ∂ 2 ∂a 2 log N , M . ( 40
)
This is just the Toda lattice equation along the a-direction, but with a trivial factor that can be removed by rescaling the function.

We can assign another relation to the partition function by the identity (38) for the expression (35)

with i = k = N + M and j = l = N + M -1, which reads det R N , M • det R N , M-2 = det R N , M-1 • (M -1)∂ a ∂ λ λ det R N , M-1 -∂ λ det R N , M-1 • (M -1)λ∂ a det R N , M-1 . ( 41
)
We have again used the relation (A1). Rewriting this relation in terms of the partition function (35), we have

N , M+1 • N , M-1 N , M 2 = ∂ 2 ∂a∂λ λ log N , M . ( 42
)
We then obtain the 2D Toda lattice equation [17] with an extra factor. In order to remove this irrelevant factor, we rescale the partition function:

˜ N , M (a, λ) = e -λ N , M (a, λ). ( 43
)
Thus the Toda lattice equations ( 40) and ( 42) are now written in the well known form1

˜ N +1, M • ˜ N -1, M ˜ N , M 2 = M N ∂ 2 ∂a 2 log ˜ N , M , ˜ N , M+1 • ˜ N , M-1 ˜ N , M 2 = ∂ 2 ∂a∂λ log ˜ N , M . ( 44
)
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T. Kimura

This means that the brane partition function (43) plays the role of the τ -function for the 1D and 2D Toda lattice equations simultaneously. We note that this is an exact result for finite N (and also M). If one takes the large N limit, corresponding to the continuum limit for the Toda lattice equations, it reduces to the Korteweg-de Vries/Kadomtsev-Petviashvili (KdV/KP) equations. We also comment that the τ -function of the 2D Toda lattice hierarchy can be realized as the two-matrix model integral [19,20].

Although in this section we have focused only on the equal position limit of the partition function (29), it can be regarded as the τ -function for the corresponding integrable hierarchy. In this case we can introduce two kinds of Miwa coordinates:

t n = 1 n Tr A -n , tn = 1 n tr -n . ( 45
)
It is shown that all the time variables, t n and tn , are trivially related to each other in the equal parameter limit. After taking the continuum limit, namely the large N limit of the matrix model, it shall behave as the τ -function for the KdV/KP hierarchies.

Gaussian matrix model

We now study a specific example of the matrix model with the harmonic potential W (x) = 1 2 x 2 , namely the Gaussian matrix model. In this case we can check the duality formula more explicitly [21][22][23]:

1 Z N dX e -1 2gs Tr(X -A) 2 M α=1 det(λ α -X ) = (-1) M N 1 Z M dY e -1 2gs tr(Y -i ) 2 N j=1 det(a j + iY ), (46)
where X and Y are Hermitian matrices with matrix sizes N × N and M × M, and "Tr" and "tr" stand for the trace for them, respectively. This equality is just rephrased as From the generic formula (29), we can understand this duality as a consequence of the self-duality of Hermite polynomials with respect to Fourier transformation. Actually, the Hermite polynomial, which is an orthogonal polynomial with the Gaussian weight function w(x) = e -1 2 x 2 , has an integral representation:

H k (x) = ∞ -∞ dt √ 2π (it) k e -1 2 (t+i x) 2 . ( 48
)
This expression is essentially the same as (32) in the case of the harmonic potential. This is a specific property for the Gaussian model.

Fermionic formula

For the Gaussian matrix model, there is another interesting derivation of the duality (46) using fermionic variables, instead of the method used in Sect. 3.1. Following the approach applied in Refs. [21,22], basically, we discuss it from the viewpoint of the topological strings. We will actually show that the effective fermionic action for this partition function gives a quite natural perspective on the topological brane.

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T. Kimura

Introducing fermionic variables in bifundamental representations (N , M) and ( N , M) of U(N ) × U(M), the characteristic polynomial is involved in an exponential form:

M α=1 det(λ α -X ) = d[ ψ, ψ] e M α=1 ψα i (λ α -X ) i j ψ α j . ( 49
)
Using this formula, the LHS of (47) becomes

1 Z N d[X, ψ, ψ] e -1 2gs Tr(X -A) 2 + ψα i (λ α -M) i j ψ α j . ( 50
)
Then the effective action yields

S eff (X, ψ, ψ) = - 1 2g s Tr(X -A) 2 + ψα i (λ α 1 -M) i j ψ α j = - 1 2g s Tr(X -A -g s ψ α ψα ) 2 + Tr A ψ α ψα + g s 2 ψ α i ψα j ψ β j ψβ i -tr ψ j ψ j , (51)
where (ψ α ψα ) i j ≡ ψ α i ψα j and (ψ j ψ j ) αβ ≡ ψ α j ψβ j are N × N and M × M matrices, respectively. Integrating out the matrix X , we obtain the intermediate form of the formula, which can be represented only in terms of the fermionic variables

e -1 2gs Tr A 2 e 1 gs Tr X A M α=1 det(λ α -X ) = d[ ψ, ψ] e gs 2 ψ α i ψα j ψ β j ψβ i +Tr A ψ α ψα -tr ψ j ψ j . ( 52
)
Since the four-point interaction is also represented in terms of the M × M matrix as Tr ψ α ψα 2 = tr ψ i ψi 2 , this term can be removed by inserting an M × M auxiliary matrix Y :

1 Z M d[Y, ψ, ψ] e -1 2gs tr(Y -i -ig s ψ j ψ j ) 2 -tr ψ j ψ j -gs 2 tr(ψ i ψi ) 2 +Tr A ψ α ψα = (-1) M N Z M dY N j=1 det(a j + iY ) e -1 2gs tr(Y -i ) 2 . ( 53
)
This is just the RHS of the duality formula (47).

Let us comment on the meaning of this formula in terms of the topological strings. When we apply m distinct values to A as A = diag(a (1) , . . . , a (1) N 1 , a (2) , . . . , a (2)

N 2 , . . . . . . , a (m) , . . . , a (m) N m ), ( 54
)
stacked N branes are decoupled into N 1 + • • • + N m , as shown in Fig. 2. This means that the U(N ) symmetry of the original matrix model is broken into its subsector:

U(N ) -→ U(N 1 ) × • • • × U(N m ). ( 55
)
PTEP 2014, 103B04

T. Kimura We find a similar symmetry breaking in the dual representation. In particular, when we put as = diag(λ (1) , . . . , λ (1) M 1

, λ (2) , . . . , λ (2) M 2 , . . . . . . , λ (l) , . . . , λ (l)

M l ), ( 56
)
the U(M) symmetry is broken as

U(M) -→ U(M 1 ) × • • • × U(M l ). ( 57
)
We can also discuss the symmetry breaking of the fermions by seeing the fermionic effective action in (52),

S eff (ψ, ψ) = g s 2 ψ α i ψα j ψ β j ψβ i + Tr A ψ α ψα -tr ψ j ψ j . ( 58
)
Although the four-point interaction term is invariant under the full symmetry of U(N ) × U(M), this symmetry is partially broken, because the source term plays the role of the non-singlet mass term. The remaining symmetry is U(N i ) × U(M j ) for i = 1, . . . , m and j = 1, . . . , l, as a subset of U(N ) × U(M). This fermionic excitation should be seen as a remnant of the chiral fermion associated with the anomaly on the intersecting branes [24][25][26][27][28].

Bosonic formula

We can extend the duality formula (47) for the inverse characteristic polynomial [23]: In this case a bifundamental bosonic field plays a similar role to the fermionic field, which is used to represent the characteristic polynomial in the numerator. Actually, we can derive this duality formula in almost the same manner as that discussed in Sect. 4.1.

The average shown in the LHS of ( 59) is explicitly written as

e -1 2gs Tr A 2 e 1 gs Tr X A M α=1 det(λ α -X ) -1 = 1 Z N dX M α=1 det(λ α -X ) -1 e -1 2gs Tr(X -A) 2 . ( 60
)
Since the inverse of a determinant is written as a Gaussian integral with a bosonic variable in the bifundamental representations,

M α=1 det(λ α -X ) -1 = d[φ, φ † ] e -M α=1 φ * α i (λ α -X ) i j φ α j , (61)
the intermediate form, corresponding to (52), is given by Then, inserting an auxiliary M × M Hermitian matrix Y in order to eliminate the four-point interaction, it is written as

d[φ, φ † ] e gs 2 φ α i φ * α j φ β j φ * β i +Tr A φ α φ * α -tr φ j φ * j . ( 62
1 Z M d[Y, φ, φ † ] e -1 2gs tr(Y -+g s φ j φ * j ) 2 -tr φ j φ * j + gs 2 tr(φ i φ * i ) 2 +Tr A φ α φ * α = e -1 2gs tr 2 e 1 gs tr Y N j=1 det(a j -Y ) -1 . ( 63
)
This is the RHS of the duality formula (59). The symmetry of the bifundamental bosons is partially broken, as seen in the bosonic effective action (62), as well as the fermionic case. The difference from the previous case is the role of branes. From the topological string point of view, the inverse of the characteristic polynomial is regarded as the partition function of the non-compact anti-brane, which is created by the vertex operator with the opposite charge V (x) = e 1 2gs φ(x) , instead of (12). In this case, the correlation function is written in terms of the Cauchy transform of the corresponding orthogonal polynomial [29].

Discussion

In this article we have investigated the symplectic invariance of branes in the topological B-model using its matrix model description. In particular, since two different descriptions of the non-compact brane correspond to the characteristic polynomial and the external source in the matrix model, we have considered the brane partition function given by inserting both of them simultaneously. We have derived the determinantal formula for this partition function, and shown that the two descriptions of the branes are dual to each other in the sense of the Fourier transformation. We have also shown that the brane partition function plays the role of the τ -function, and satisfies the Toda lattice equations in one and two dimensions. We have investigated the Gaussian matrix model as an example, and discussed the effective action of the topological branes in terms of the bifundamental fermion/boson.

Although we have focused on the U(N ) symmetric matrix model throughout this article, we can apply essentially the same argument to O(N ) and Sp(2N ) symmetric matrix models. In such a case the Hermitian matrix is replaced with real symmetric and self-dual quaternion matrices, respectively. Actually, when the Gaussian potential is assigned, one can obtain a similar duality formula [23,30,31], which claims that the dual of the O(N ) model is the Sp(2N ) model and vice versa. This relation is extended to an arbitrary β-ensemble, and a generic duality between β and 1/β is found. From the string theoretical point of view, this property is naturally understood as insertion of an orientifold plane. Correspondingly, the U(N ) × U(M) bifundamental fermion/boson used in Sect. 4 is replaced with the O(N ) × Sp(2M) bifundamental variables.

Let us comment on some possible applications of the duality discussed in this article. The exact low energy dynamics of N = 2 gauge theory, which is described by Seiberg-Witten theory, is solved by the world volume theory of D4-branes suspended between NS5-branes, and also an M5-brane, appearing in its M-theory lift [32]. In this case, since the geometry of this M5-brane indicates the Seiberg-Witten curve of the corresponding N = 2 theory, the positions of branes are directly related to the gauge theory dynamics. Thus it is expected that a nontrivial gauge theory duality is derived from the duality between the two coordinates of branes. Actually, a similar duality is discussed along this direction [33].

From the matrix model perspective, it is interesting to consider the ratio of the characteristic polynomials in the presence of the external source [29], and its interpretation in terms of topological PTEP 2014, 103B04 T. Kimura strings. The characteristic polynomial in the numerator and the denominator plays the role of the creation operator for the brane and anti-brane. Thus the ratio should describe the pair creation and annihilation of branes. In particular, it is expected that the scaling limit of the ratio extracts some interesting features of the tachyon condensation in topological strings. This kind of problem is also interesting in the context of the matrix model itself, because one can often find an universal property of the matrix model in such a scaling limit.

where we have introduced two M × M matrices:

A M = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ x N (0) • • • x N +M-2 (0)
x N +M-1 (0)

x N (1) • • • x N +M-2 (1) x N +M-1 (1) . . . . . . . . . . . . . . . . . .

x N (M-1) • • • x N +M-2 (M-1) x N +M-1 (M-1)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , ( A2
)
and

B M = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ x N (0) • • • x N +M-2 (0)
x N +M (0)

x N (1) • • • x N +M-2 (1) x N +M (1) . . . . . . . . . . . . . . . . . .

x N (M-1) • • • x N +M-2 (M-1) x N +M (M-1) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . ( A3
)
Here we denote (x j ) (l) = (d/dx) l x j and so on. Using the Jacobi identity (38) for det A M with i = k = M and j = l = M -1, we have

det A M • det A M-2 = det A M-1 • ∂ x det B M-1 -∂ x det A M-1 • det B M-1 . (A4)
It is convenient to rewrite this relation as

det A M • det A M-2 (det A M-1 ) 2 = ∂ ∂ x det B M-1 det A M-1 . ( A5
)
This is interpreted as a remnant of the Toda lattice equation [START_REF] Kharchev | [END_REF].

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Fig. 1 .

 1 Fig. 1. (Left) In the case with A = 0, the one-cut matrix model has a single cut in the complex x-plane, which is seen as a pole at p = 0 with degree N in the p-plane. There are also simple poles created by the characteristic polynomials. (Right) When one applies the external source A = diag( p 1 , . . . , p 1 , . . . , p m , . . . , p m ), it is split into m distinct poles with the corresponding multiplicity.

  α -X ) = (-1) j + iY ) . (47)

Fig. 2 .

 2 Fig. 2. When we turn on the external source A, the degenerate positions of N branes are lifted. Correspondingly, the U(N ) symmetry is broken into U(N 1 ) × • • • × U(N m ) by applying m distinct values to A.

  j -Y ) -1 . (59)

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Acknowledgements

I would like to thank Bertrand Eynard for his insightful comment on the symplectic invariance of the spectral curve. I am also grateful to Masato Taki for valuable discussions on topological string theory. This work is supported in part by a Grant-in-Aid for JSPS Fellows (#25-4302). Finally, it is my pleasure to submit the first version of this article on my 30th birthday. I would like to express my sincere gratitude to my parents on this occasion, and dedicate this article to the memory of them.

Funding

Open Access funding: SCOAP 3 .

Appendix A. A formula for determinants

In order to obtain (39) and (41), it is convenient to use the following relation:

PTEP 2014, 103B04

T. Kimura

What we have to do next is to evaluate det A M . To obtain this, we consider a ratio of determinants, and then take the equal parameter limit, x α → x for all α = 1, . . . , M,

On the other hand, it is written in another form:

Comparing these two expressions, we obtain the following result:

Substituting this expression into the relation (A5), we arrive at the relation (A1).