SU(N) q-Toda equations from mass deformed ABJM theory

It is known that the partition functions of the U(N) x U(N+M) ABJM theory satisfy a set of bilinear relations, which, written in the grand partition function, was recently found to be the q-Painleve III_3 equation. In this paper we have suggested a similar bilinear relation holds for the ABJM theory with N=6 preserving mass deformation for an arbitrary complex value of mass parameter, to which we have provided several non-trivial checks by using the exact values of the partition functions for various N,k,M and the mass parameter. For particular choices of the mass parameters labeled by integers $\nu,a$ as $m_1=m_2=-\pi i(\nu-2a)/\nu$, the bilinear relation corresponds to the q-deformation of the affine SU($\nu$) Toda equation in $\tau$-form.


Introduction and Summary
To understand the M-theory, a postulated eleven dimensional theory which unifies various ten dimensional superstring theories non-perturbatively through an extra one dimension, is an important theme of research in theoretical particle physics.In the eleven dimensional supergravity there are BPS black brane solutions with dimension 2 + 1 (M2-brane) and 5 + 1 (M5-brane), each of which has the gravitational entropy scaling ∼ N 3/2 and ∼ N 3 [1] with respect to their charges N .These scalings of the entropy are in contrast to the scaling for the 3 + 1 dimensional black brane in the ten dimensional type IIB supergravity (D3-brane) which can be naturally understood as the degrees of freedom of the D3-brane worldvolume theory originating from the open strings ending on the stack of N D3-branes.The construction of the worldvolume theory of the M2-branes/M5branes which reproduce these scalings of the entropy, while preserving the required supersymmetry and being compatible with the string theory under the compactification of the M-theory direction, is an important step toward understanding the M-theory.
These theories have been mysterious for a long time.About the M2-branes, however, there has been a series of progress using Chern-Simons matter theories starting with [2], and finally the N = 6 U(N ) k × U(N ) −k superconformal Chern-Simons matter theory [3,4] was proposed as the theory of N M2-branes placed on the orbifold background C 4 /Z k .Later the S 3 partition function of this theory was calculated in the M-theory limit N → ∞ with k kept finite [5] by using the supersymmetric localization formula [6], and it was found that the free energy F = − log Z S 3 behaves as F ≈ (π √ 2k/3)N 3/2 , which perfectly agrees with the result obtained from the supergravity [7].
The supersymmetric localization formula [6] allows us to rewrite the S 3 partition function of the ABJM theory into an N dimensional matrix model, where the rewriting is exact even for finite N .Hence this matrix model have been studied intensively [8,9,10,7,11,12,13,14] to reveal the structure of the 1/N corrections which would be related to the quantum effects in the M-theory.One of the largest breakthroughs in this direction was [15], where it was found that the S 3 partition function of the ABJM theory has the same structure as the partition function of one dimensional quantum mechanical system of N non-interacting fermions with the one-particle density matrix given as1 ρ = (2 cosh x/2) −1 (2 cosh p/2) −1 .Here x and p are the canonical position/momentum operators satisfying [ x, p] = 2πik.
The Fermi gas formalism and various computational techniques developed based on it [16,17,18,19,20] enabled us to study the 1/N perturbative corrections as well as the non-perturbative effects of order O(e − √ N/k , e − √ kN ) [11] quantitatively at finite k.Finally in [21] the large N expansion of the S 3 partition function of the ABJM theory was completely determined including all of the non-perturbative effects.Defining the grand potential J(µ) as ∞ N =0 e µN Z(N ) =: n∈Z e J(µ+2πin) , they found that the large µ non-perturbative effects of order O(e −µ/k , e −µ ) in J(µ), which correspond to O(e − √ N/k , e − √ kN ) corrections in the partition function, are precisely given by the topological string free energy and the Nekrasov-Shatashvili limit of the refined topological string free energy both on local P 1 × P 1 .Later this result was also generalized to the case with non-equal ranks U(N ) k × U(N + M ) −k [22].
On the other hand, it was recently discovered that the partition function of the ABJM theory is also related to the Painlevé systems [23,24,25].They found [23] that the grand partition function of the ABJM theory coincides, under a particualr limit, to the Fredholm determinant type solution of the τ function of the Painlevé III 3 equation which was found in [26].They further found [24] that the exact relations between the partition functions of the rank deformed ABJM theory discovered in [27] can be reorganized into the τ -form of the q-deformed Painlevé III 3 equation.
Our motivation is to find other examples of the theories of M2-branes which enjoy the similar relation.On one hand, such new examples may provide us a new way to characterize a class of q-difference equations including q-Painlevé systems from the viewpoint of the M2-branes on various background geometries.Such generalization would be also useful in the other direction: solving the M-theory by using q-difference equations.As a generalization of the ABJM theory, we can consider its mass deformation [28,29] where the bifundamental chiral multiplets have supersymmetric mass term, whose partition function were also studied in the large N limit [30,31,32,33] together with the 1/N corrections [34,35].Recently the partition function of this theory was found to be useful to understand the correlation functions of the ABJM theory even for the supersymmetry nonprotected sectors by using the bootstrap technique [36,37,38,39], which were hoped to give a new information about the M-theory [40,41,42,43,44,45].The exact values of the partition functions for general N or an exact relation among them which holds beyond the 1/N perturbation may provide further hints toward understanding the M-theory.Indeed as a class of the generalization of the relation between the ABJM matrix model and q-PIII 3 , it was already proposed that the τ -function of the affine SU(ν) q-Toda equation also allows a Fredholm determinant type solution, together with the concrete expression for the matrix models [24], although it was not clear whether those matrix models can arise in the context of the worldvolume theories of the M2-branes.In this paper we have found that this matrix model with a special choice of the parameters, can be realized as the S 3 partition function of the U(N ) k × U(N + M ) −k ABJM theory with the N = 6 preserving mass deformation [3,28,29], where the mass parameter is pure imaginary and identified with the rank ν as m 1 = m 2 = −πi(ν − 2a)/ν (see (2.2) for the normalization).Here the rank difference M is related to the time variable of the Toda system.To this 3d partition function, various techniques which were used to solve the ABJM theory are applicable straighforwardly.In particular, we find that the partition function can be written in so called "open string formalism" [46], which allows us to calculate the partition function exactly at finite k, m 1 , m 2 , N, M .These exact values provide a non-trivial check that the matrix model proposed in [24] is actually a solution of the SU(ν) q-Toda equation.This paper is organized as follows.After fixing our notation for the partition function of the mass deformed ABJM theory in section 2, in section 3 we display a conjectural bilinear relation among the partition functions for m 1 = m 2 (3.1).We also compare our proposal with the known relation between the partition function of the ABJM theory without mass deformation and the q-Painlevé III 3 (or equivalently the affine SU(2) q-Toda) system [25] and its higher rank generalization [24,47].In section 4, we rewrite the partition function of the mass deformed ABJM theory in the open string formalism and calculate the exact values of the partition function.By using these exact values we have checked that our proposal (3.1) holds for a generic complex value of m 1 = m 2 up to O(z 5 ) or higher for some special values of k.In section 5 we summarize our results and discuss possible future works.In appendix A we review the proof of the Cauchy-Vandermonde determinant formula (3.4).In appendix B we display the explicit expressions for some of the exact values of the partition function obtained by using the methods in section 4.

Partition function of mass deformed ABJM theory
The ABJM theory is an N = 6 superconformal Chern-Simons matter theory which consists of the two vector multiplets of the gauge groups U(N ),U(N + M ) with the Chern-Simons actions with the Chern-Simons levels ±k, and the two pairs of the chiral multiplets in the bifundamental representation and the anti-bifundamental representation under the gauge group U(N ) × U(N + M ) [3,4,48].We can further add the mass term to the these chiral multiplets while preserving N = 2 supersymmetry by turning on the vevs of the background vector multiplets of the flavor symmetries [49].The partition function of the mass deformed ABJM theory can be calculated by the supersymmetric localization formula [6] where m 1 , m 2 are the vevs of the scalar components of the background vector multiplets [49,35], and λ i , λ m are the vevs of the scalar components of the vectormultiplets which label the localization locus.Here we have introduced the overall factor e N M (m1+m2)/2 for later convenience.For M = 0 it is known that the partition function can be rewritten as the partition function of N particle ideal Fermi gas [34,15] 2 The results for M = 0 in this paper overlap with those displayed in [34] where we have defined the partition function with the same normalization as (2.2) under the following parameter identifications: with the one particle density matrix ρ given as 3 (2.5) We also define the grand partition function Ξ k,M (z, m 1 , m 2 ) of the mass deformed ABJM theory as For M = 0 the Fermi gas formalism (2.3) implies that Ξ k,0 (z, m 1 , m 2 ) is given as a Fredholm determinant In this paper we conjecture the following bilinear relation among the partition functions of the mass deformed ABJM theory with m 1 = m 2 : where Ξ k,M (z; m 1 , m 2 ) is the grand partition function defined as (2.6) and Z k,M (0) is the partition function of This relation (3.1) can be checked order by order in z once we know the exact values of the partition functions Z k,M (N, m 1 , m 2 ), which we compute in section 4.
Note that when m 1 = m 2 = −πi(ν − 2a)/ν with ν, a ∈ N (1 ≤ a ≤ ν − 1, ν, a: coprime), (3.1) is a higher rank generalization of the relation between the grand partition function of the ABJM theory and the affine SU(2) q-Toda equation obtained in [25].Indeed as we see below the partition function Z k,M (N, m 1 , m 2 ) for m 1 = m 2 coincides with the fermionic spectral trace of the inverse quantum mirror curve of Y ν,0 geometry [24] whose relation to the affine SU(ν) q-Toda system was also argued in [47].

Closed string formalism and relation to q-Toda system
Our goal here is to rewrite the partition function 3 Here we adopt the following notation and normalization: • x, p: canonical position/momentum operators with [ x, p] = 2πik, • |x : position eigenstate, |p : momentum eigenstate, • x|x = 2πδ(x − x ), p|p = 2πδ(p − p ), and correspondingly, dx 2π |x x| = dp 2π |p p| = 1, We also use the following formulas several times where in the second formula |p is a momentum eigenstate, while |p is a position eigenstate with the eigenvalue p.
with some one dimensional quantum mechanical operator ρ M . 4Here Z k,M (0) is the partition function (2.2) for N = 0, which is simply given as (3.2).For this purpose, first of all we rewrite the one-loop determinant factors in the integrand of the partition function (2.2) by using the following Cauchy-Vandermonde determinant formula [46] (see appendix A for the proof) where Here we have also rescaled the integration variables λ i , λ m as (λ i , λ m ) → (λ i /k, λ m /k) for later convenience.If we introduce the 1d quantum mechanical bra-ket notations (see footnote 3), we can further rewrite the partition function as Second we simplify the matrix elements of the second determinant by replacing which results in Here to obtain the last two expressions we have used the formulas (2.4).Third, after these replacements we further "trivialize" the second determinant in (3.6) as follows to obtain Now we can perform the λ m -integrations trivially just by replacing them according to the δ-functions.If we use the Cauchy-Vandermonde determinant formula (3.4) inversely we can rewrite the determinant factor in the first line of (3.10) back into the product of hyperbolic functions.This product can be rearranged into those which we can combine again into the Cauchy determinant (3.4) with M = 0 and the rest.Thus we finally obtain the following with Z k,M (0) given as (3.2) and (3.12) which satisfy the following recursive relation , the partition function (3.11) can be rewritten as Comparing this result for m 1 = m 2 = −πi(ν − 2a)/ν with a, ν ∈ N with the fermionic spectral trace of the inverse quantum mirror curve of Y ν,0 geometry at the one period phases obtained in [24] Z ν (0, 0, we find that the two results coincide with the following parameter identifications Hence our conjecture (3.1) can be rewritten as with This is indeed a natural higher rank generalization of the relation between the spectral determinant of quantum mirror curve of local P 1 × P 1 and the affine SU(2) q-Toda equation found in [25] More explicitly, by defining the τ -function τ j in the following way the bilinear relation (3.1) for m 1 = m 2 = −πi(ν − 2a)/ν reduces to the bilinear form of the affine SU(ν) q-Toda equation written in [47]: 4 Check of (3.1) for small z using open string formalism In this section we consider another way to rewrite the partition function of the mass deformed ABJM theory (2.2), the variation of so called open string formalism [46] which is useful for the computation of the exact values of the partition function.Starting from (3.6) we use the following formula for the integration variables { λ m } N +M m=1 to rewrite the partition function With this result, the grand partition function Ξ k,M (z; m 1 , m 2 ) (2.6) can be written as where By performing similarity transformations of the 1d quantum mechanical operators we can further rewrite Ξ k,0 (z, m 1 , m 2 ) and det H r,s (z) as with ρ 0 given as (2.5).
4.1 Recursive calculation of x| ρ n 0 |y and 2πi r | ρ n 0 |x In the previous section we have rewritten the grand partition function of the mass deformed ABJM theory Ξ k,M (z, m 1 , m 2 ) as (4.3), which implies that we can calculate the partition function Z k,M (N, m 1 , m 2 ), the expansion coefficients of Ξ k,M (z, m 1 , m 2 ) in z, by calculating Tr ρ n 0 and 2πi r | ρ n 0 |x . 6Though the first terms of these sequences can be obtained easily the calculation is non-trivial for higher n.Below we show two ways to calculate these quantities recursively in n.The results are listed in appendix B, with which we can check that the conjectured bilinear relation (3.1) indeed holds for various values of k, M and m 1 = m 2 .

Algorithm with fixing im2
2π ∈ Q To explain the first method, which was used in [34,35] by generalizing the method used for the massless case of [55,17], first we write the matrix element x| ρ 0 |y as (e which implies Using this relation repeatedly we obtain This implies that the matrix element of ρ n 0 is written as < l a t e x i t s h a 1 _ b a s e 6 4 = " / z / k 6 B u m r N r V p q d D + L w W 9 u h T j with The vector φ (x) obeys the following recursion relation and the initial condition When k ∈ Q and im 2 /(2π) ∈ Q we can further rewrite the recursion relation (4.12) as follows.By choosing R ∈ N such that the following S, T are integers the integration can be rewritten as (u = e x/R , v = e y/R ) If we further assume that φ (u) has the following structure7 the integration (4.14) can be rewritten as [17] φ where log (+) is the logarithm with branch cut R ≥0 (i.e.log (+) (re iθ ) = log r + iθ with 0 < θ ≤ 2π), γ is the contour depicted in figure 1 (left) and B j+1 (z) is the Bernoulli polynomial.The poles to be collected when we calculate φ +1 from φ are the union of {v = e −m1/S ue (2πi/S)(a−1/2) } S a=1 , {v = e (2πi/R)(b−1/2) } R b=1 and the poles of φ (v).The poles of φ (u) can be listed exhaustively in the following way [34].As the locations of the first set of the poles {v = e −m1/S ue (2πi/S)(a−1/2) } S a=1 are u-dependent, the recursion relation (4.16) generates new poles through the resudies at these u-dependent poles.Therefore from {v = e (2πi/R)(b−1/2) } R b=1 we find φ ≥1 (u) has poles at R b=1 {u = e m1/S e −(2πi/S)(a−1/2) e (2πi/R)(b−1/2) } S a=1 .Also, if φ (u) has a pole at u = u 0 , it follows through the same process that φ +1 (u) has poles at {u = e m1/S e −(2πi/S)(a−1/2) u 0 } S a=1 .Taking into account that the initial condition φ 0 (u) has no poles, we conclude that the poles of φ ≥1 (u) are (4.17) Once we obtain φ (u) with = 0, 1, • • • , n − 1 we can calculate Tr ρ n 0 (4.10) by the same change of integration variable and the contour deformation: 2 ) where n (u) are the rational functions given by The poles to be collected for the calculation of Tr ρ n 0 in (4.18) are where the poles of φ (u) are given in (4.17).
The calculation of 2πi r | ρ n 0 |x can be performed in the same way as φ (u).Writing 2πi χ r,n (x) obeys the following recursion relation and the initial condition The recursion relation is nothing but the one for ψ (x) (except an extra overall factor e m1/2 ).When k ∈ Q and im 2 /(2π) ∈ Q it is rewritten as < l a t e x i t s h a 1 _ b a s e 6 4 = " r 0 i c 2 Z W P f V 6 5 k n T i e B 7 w h l 8 t W J  The poles to be collected for the calculation of χ r,n+1 (u) are with (poles of χ r,n (v)) = (poles of φ n (v))| m1→−m1 .

Algorithm without fixing m 2
We can also calculate x| ρ n 0 |y and 2πi r | ρ n 0 |x without fixing m 1 , m 2 (and r ) in the following way [14,57].Since these quantities are meromorphic functions in m 1 , m 2 , we assume m 1 , m 2 ∈ R in the calculation and the results for m 1 , m 2 ∈ C are obtained by the analytic continuation.The algorithm works for general k ∈ Q, which we denote k = p/q with p, q coprime.To perform the integration (4.27), let us consider the following integration ), then we find that the integration (4.28) is invariant, up to an overall factor, under the shift of the integration contour Hence the integration I[f ] can be evaluated as where the integration contour γ is as displayed in figure 2. For example, applying this calculation to 2πi r | ρ 0 |x (4.26) we obtain Note that the integrand does not enjoy the quasi-periodicity in y any more.Nevertheless, if we look at the contributions from each pole (4.31), they separately enjoy the quasi-periodicity, hence we can repeat the same calculation as above.In principle we can repeat this algorithm indefinitely for higher n by applying the formula (4.30) to each term in 2πi r | ρ n−1 0 |y y| ρ 0 |x which takes the following form b ∈∆ (2)   1 cosh with α, β being some constants and ∆ (1) = {δ (1) b } b being some sets of constants.Explicitly, the multiplication of ρ 0 to 2πi r | ρ n−1 0 from the right transforms each term of the form (4.33) as follows b ∈∆ (2)   1 cosh ) a ∈∆ (1) b ∈∆ (2)   1 cosh where (α , β , ∆ (1) , ∆ (2) ) runs over all of the following (i),( , ∆ (2) ) for each δ (1) a ) given as (ii) (α , β , ∆ (1) , ∆ (2) ) for each δ b ∈ ∆ (2) and j ∈ J(δ b ) given as (iii) and (α , β , ∆ (1) , ∆ (2) ) for each j = 1, 2, • • • , q given as with b ] + 2πk j − where |∆ (1) |, |∆ (2) | are the length of ∆ (1) , ∆ (2) .Here the elements of I, Ī label the poles of the first cosine factor in (4.33) and the elements of J, J label the poles of the second cosine factors in (4.33).Note that the integrations are well defined by the principal value prescription when the poles are on top of the integration contour γ depicted in figure 2. As a result, these poles contributes in the integration (4.30) with the residue weighted by 1/2.We have counted these poles separately from I(δ The same integration technique is also applicable to the calculation of Tr ρ n through the decomposition (4.10), which we can also rewrite as The integrand in the last line is a linear combination of the terms of the form (4.33), hence the integration can be evaluated with the following formula b ∈∆ (2)   1 cosh 2i(−1) j e β(δ (1) b ∈∆ (2) j∈J(δ with I, Ī, J, J defined as (4.38).
We have displayed the results for n ≤ 2 in appendix B.1.Though the exact expressions for 2πi r | ρ n 0 |x and Tr ρ n 0 with n > 2 are too lengthy to display, we can proceed by using Mathematica for higher n.We have performed the computation for n ≤ n max with n max = 7 for k = 2, n max = 6 for k = 3, n max = 5 for k = 2/3, 3/2, 4, 5, 6 and n max = 4 for several other k ∈ Q. Plugging them into the grand partition function Ξ k,M (z, m 1 , m 2 ) and substituting a generic complex irrational value to m 1 = m 2 (see table 1 for an example) we have checked that the conjectured difference identity (3.1) indeed holds up to O(z nmax+1 ). 8

Discussion
In this paper we have studied the S 3 partition function of the mass deformed U(N ) k × U(N + M ) −k ABJM theory for finite k, N, M .For M = 0 this model was studied by the Fermi gas formalism in [34].In this paper we have extended the Fermi gas formalism to the case with non-equal ranks of the gauge group, M > 0. As a result we have found non-trivial evidences that the partition functions for m 1 = m 2 satisfies an infinite set of exact bilinear relations (3.1).For m 1 = m 2 = −πi(ν − 2a)/ν with ν, a ∈ N, 1 ≤ a ≤ ν − 1 our proposal (3.1) coincides with the higher rank generalization of the result in [25] which relates the partition function of the ABJM theory without mass deformation to the τ -function of the q-Painlevé III 3 (or affine SU(2) q-Toda) equation.Remarkably, we have observed that the bilinear relation (3.1) is satisfied not only for m 1 = m 2 = −πi(ν − 2a)/ν but also for general m 1 = m 2 ∈ C. Hence (3.1) also provides relations between the derivatives of the partition functions with respect to the mass parameter which gives some of the OPE coefficients for the stress tensor multiplet [40].
There are several possible future directions related to this work.First, although in this paper we have found the relation only for m 1 = m 2 , it would be nice if we can generalize the result to m 1 = m 2 .Second, we can consider 3d superconformal Chern-Simons theories with more general quivers.It is known that the S 3 partition function of a quiver superconformal Chern-Simons theory with the quiver given by the affine Dynkin diagram of A n , D n , E n scales in the large N limit as − log Z ∼ N 3/2 in the limit of N → ∞ if the ranks of the gauge groups are proportional to the comarks of each node and the total Chern-Simons level weighted by the comarks vanishes [56].For the affine A n quivers and the affine D n quivers the fermi gas formalism was already found [15,58,59], and for the affine A n quivers it was further extended to the case with mass deformations [34] as well as to the cases with non-equal ranks [60,61,62,63] separately.We expect that for the latter cases it is also possible to turn on both of the mass deformation and the rank deformation at the same time, and if so, it would be interesting to identify the corresponding q-difference equations and check them as the identities among the partition functions for different ranks.It would also be interesting to consider the generalization which are natural from the viewpoint of the q-difference equations, such as the q-deformed Painlevé of the other types or the generalization of the affine SU(ν) Toda system to the Toda systems for other Lie groups.

2 i
,n:N ×(N +M ) [e r yn ] r,n:M ×(N +M ) , :(N +M )×N [e − sym ] m,s:(N +M )×M , y p X 6 t e l 6 l 0 W R x 7 O 4 B w u w Y M b q M I D 1 K A B D B C e 4 R X e n C f n x X l 3 P p a t O S e b O Y U / c D 5 / A O R / j P 4 = < / l a t e x i t > = X w2 2⇡iRes[•, v !w]

Figure 1 :
Figure 1: Left: The integration contour γ used in the first line of (4.16); Center and Right: deformation of the contour to derive the second line of (4.16).
t e x i t s h a 1 _ b a s e 6 4 = " + X K F c 0 p p l H p P S 9 V 3 r y R E t F F C A G 4 = " > A A A B 7 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b R U 0 m q o M e i F 4 8 V 7 A e 0 o U y 2 m 3 b p b h J 2 N 0 I J / R F e P C j i 1 d / j z X / j t s 1 B W x 8 M P N 6 b Y W Z e k A i u j e t + O 4 W 1 9 Y 3 N r e J 2 a W d 3 b / 7 H c y 4 I p R I y a W I F X c 3 k r o C B V S Y x M q 2 R C 8 5 Z d X S a t W 9 S 6 r t Y e r S v 0 2 j 6 M I J 3 A K F + D B N d T h H h r Q B A p j e I Z X e H M S 5 8 V 5 d z 4 W r Q U n n z m G P 3 A + f w D q o o 9 K < / l a t e x i t > 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " + X K F c 0 p p l H p P S 9 V 3 r y R E t F F C A G 4 = " > A A A B 7 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b R U 0 m q o M e i F 4 8 V 7 A e 0 o U y 2 m 3 b p b h J 2 N 0 I J / R F e P C j i 1 d / j z X / j t s 1 B W x 8 M P N 6 b Y W Z e k A i u j e t + O 4 W 1 9 Y 3 N r e J 2 a W d 3 b / 7 H c y 4 I p R I y a W I F X c 3 k r o C B V S Y x M q 2 R C 8 5 Z d X S a t W 9 S 6 r t Y e r S v 0 2 j 6 M I J3 A K F + D B N d T h H h r Q B A p j e I Z X e H M S 5 8 V 5 d z 4 W r Q U n n z m G P 3 A + f w D q o o 9 K < / l a t e x i t > z < l a t e x i t s h a 1 _ b a s e 6 4 = " V L E o 6 V g U n u 2 T n O x o O k q s M P X v y T o = " > A A A B 6 H i c b V D L T g J B E O z F F + I L 9 e h l I j H x R H b R R I 9 E L x 4 h k U c C G z I 7 9 M L I 7 O x m Z t Y E C V /g x Y P G e P W T v P k 3 D r A H B S v p p F L V n e 6 u I B F c G 9 f 9 d n J r 6 x u b W / n t w s 7 u 3 v 5 B 8 f C o q e N U M W y w W r H h 7 7 J 4 k b e 9 Y X y d 3 5 W v + q n g g o f 4 R 7 z S z D 9 e r T 1 sL 3 z 4 G v R a i P T O u n S b t A o l d 2 K O w L 9 S 7 w x K Z M x z h u l g d 9 M e B a D Q i 6 Z M X X P T T H I m U b B J f S L f m Y g Z b z D W l C 3 V L E Y T J C P T u 3 T T a s 0 a Z R o 2 w r p S P 2 + k b P Y m F 4 c 2 s m Y Y d v 8 9 o b i f 1 4 9 w + g g y I V K M w T F v x 6 K M k k x o c P c a F N o 4 C h 7 l j C u h f 0 r 5 W 2 m G Ue b b t G G 4 P 0 + + S + 5 r l a 8 3 U r 1 Y q 9 8 d D K O o 0 D W y Q b Z I h 7 Z J 0 f k j J y T G u H k k T y T F / L q P D k D 5 8 1 5 / x q d c M Y 7 a + Q H n I 9 P 7 E + m Z g = = < / l a t e x i t > 2⇡ip < l a t e x i t s h a 1 _ b a s e 6 4 = " u y K l / f / M 3 B v 7 e S e E g h M h a x C C 2 s M = " > A A A B 7 n i c b V D L S g N B E O z 1 G e M r 6 t H L Y B A 8 h d 0 o 6 D H o x W M E 8 4 B k C b O T S T J k d n a Y 6 R X C k o / w 4 k E R r 3 6 P N / / G S b I H T

2 .
Together with the exact values of Z k,M (N, m 1 , m 2 ) for N = 1, 2 in appendix B.1, the displayed digits of the numerical values of Z k,M (N, m 1 , m 2 ) for N ≥ 3 are sufficient for checking the bilinear relation (3.1) at order ∼ 10 −35 which is smaller than the values of Z k,M (N, m 1 , m 2 ) themselves.

2 .
Together with the exact values of Z k,M (N, m 1 , m 2 ) for N = 1, 2 in appendix B.1, the displayed digits of the numerical values of Z k,M (N, m 1 , m 2 ) for N ≥ 3 are sufficient for checking the bilinear relation (3.1) at order ∼ 10 −35 which is smaller than the values of Z k,M (N, m 1 , m 2 ) themselves.

2 . 2 N 2 =1( 7 ) 8 ) 2 B
Together with the exact values of Z k,M (N, m 1 , m 2 ) for N = 1, 2 in appendix B.1, the displayed digits of the numerical values of Z k,M (N, m 1 , m 2 ) for N ≥ 3 are sufficient for checking the bilinear relation (3.1) at order ∼ 10 −35 which is smaller than the values of Z k,M (N, m 1 , m 2 ) themselves.where in the expansion of the (N + M − 1)-th row we picked up the sub-leading terms − bn a +M −1 instead of the leading terms 1 a N +M −1 since the leading terms are proportional to the N + M -th row and can be subtracted by the elementary row operation.Hence from the leading part of (A.4) in the limit a N +M −1 → ∞ we obtainN +M −2 m<n (a m − a n ) N +M m<n (b m − b n ) N +M −2 m=1 N +M n=1 (a m + b n ) = −1) N +M − (−1) n:(N +M −2)×(N +M ) [b n ] n:N +M By repeating the same procedures for a N +M −2 , a N +M −3 , • • • , a N +1 we obtain N i<j (a i − a j ) N +M m<n (b m − b n ) If we substitute a i =e xi , b m = e ym in (A.8) we finally obtain the first line of the formulas (3.4).The second line of (3.4) is obtained if we instead substitute a i = e −xi , b m = e −ym to (A.8).B List of exact values of tr ρ n and 2πi r | ρ n |x As we explained in section 4 the grand partition functions Ξ k,M (N, m 1 , m 2 ), or its expansion coefficients Z k,M (N, m 1 , m 2 ), can be obtained from Tr ρ n 0 (n = 1, 2, • • • , N ) and 2πi r | ρ n 0 |x (n = 0, 1, • • • , N , r = 1, 2, • • • , M ) through (4.3) and (4.5).Below we display the exact values of these quantities.B.1 n ≤ 2 without fixing m 1 , m