Superconformal Chern-Simons Partition Functions of Affine D-type Quiver from Fermi Gas

We consider the partition function of the superconformal Chern-Simons theories with the quiver diagram being the affine D-type Dynkin diagram. Rewriting the partition function into that of a Fermi gas system, we show that the perturbative expansions in 1/N are summed up to an Airy function, as in the ABJM theory or more generally the theories of the affine A-type quiver. As a corollary, this provides a proof for the previous proposal in the large N limit. For special values of the Chern-Simons levels, we further identify three species of the membrane instantons and also conjecture an exact expression of the overall constant, which corresponds to the constant map in the topological string theory.


Introduction
The ADE classification not only provides a beautiful structure in the mathematical science, but also plays important roles in M-theory. As an example related to the M5-brane, there is an ADE classification of six dimensional N = (2, 0) theories, coming from the classification of the singularities on which the M5-branes are placed [1]. Also, even for the six dimensional N = (1, 0) theories, the ADE classification continues to be crucial [2].
In the context of the M2-branes, the three dimensional U(N) quiver gauge theories can also be classified by the affine ADE-type, or A D E-type, Dynkin diagrams. A large class of three dimensional superconformal Chern-Simons theories can be constructed by quiver diagrams, by the rule where each vertex is assigned with a vector multiplet of a gauge group U(N) with N proportional to the comark and each edge connecting two vertices is assigned with a pair of hypermultiplets which are in the bifundamental representation under the gauge groups on these vertices. For example, the simplest quiver, A 1 , gives the ABJM theory [3]. For these theories, the localization technique allows us to express the partition function on S 3 as a finite dimensional matrix model [4]. Then, it was shown in [5] that, if we require that the long range force among the eigenvalues vanishes (also known as the balance condition in [6]), the quivers have to be of A D E-type. The behavior of these matrix models in the limit N → ∞ were studied for the A-type quivers in [7,8], for the D-type quivers in [5,9,10] and for the E-type quivers in [10]. Interestingly, they observed a universal scaling law in the partition function for all the A D E-type quivers with C some constant depending on the quivers. This scaling law is a characteristic property of the M2-branes in the context of the AdS/CFT correspondence [11].
Beyond this leading N 3 2 scaling behavior, the partition function of the ABJM theory was studied in full detail. After the scaling behavior above was obtained in [12], it was shown that the N 3 2 behavior of the partition function of the ABJM theory is completed by the perturbative 1/N corrections to be an Airy function [13], with A [14], B and C some constants, up to non-perturbative corrections in 1/N. Later it was pointed out that the partition function of this theory can be rewritten as that of an ideal Fermi gas system with N particles [15]. This formalism is so efficient that it not only provided a simple rederivation of the above result, but even allowed us the exact analysis of the non-perturbative corrections [16][17][18][19][20][21].
In [15] the authors also showed that the Fermi gas formalism works for theories of general A-type quivers, and that the completion by an Airy function is universal for these theories. The superconformal Chern-Simons theories of the A-type quivers were also studied in detail, including the perturbative coefficients and various non-perturbative corrections [22][23][24][25]. * In [15] they further conjectured that the completion by an Airy function is universal even for other theories of the M2-branes. However, in the case of the D E-type quivers, it was not trivial whether the universal behavior of the Airy function is valid because of the lack of the Fermi gas formalism due to its non-circular structure and the non-uniform comarks.
In this paper we consider the D-type quivers, and obtain a positive answer to this question. Here we shall explain our setup. We consider the quiver superconformal Chern-Simons theory whose quiver is the D r Dynkin diagram. We call the two vertices on the left end as µ and µ ′ , the ones on the right end as ν and ν ′ , and label the others as 1, 2, · · · , r − 3, as in figure 1. * The N f flavor matrix model also allows the Fermi gas formalism which turns out to be equivalent to that for some of the A-type quiver. These matrix models were studied in [26][27][28]. The equivalence between these matrix models is proven systematically in [29].
In addition to the number N m of the gauge group U(N m ), the Chern-Simons level is assigned on each vertex, which we take as (m = 1, 2, · · · , r − 3) with s m extended to arbitrary real numbers. This choice is known to be the most general one under the requirement of the superconformal symmetry of the theory [5]. Below, we shall first present a Fermi gas description for the partition function of this theory in section 2. In the Fermi gas formalism, it is rather convenient to introduce the chemical potential µ dual to N and study the grand potential. Other than s m , the grand potential is controlled by two parameters, the chemical potential µ and the overall Chern-Simons level k which plays the role of the Planck constant of this quantum statistical system, = 2πk. The grand potential turns out to be a cubic polynomial of µ if we neglect the non-perturbative effects. In this manner, all order perturbative corrections of the partition function in 1/N are taken into account. As a result, we obtain the expression of an Airy function (1.2) in section 3. The coefficient C relevant to the leading N 3 2 behavior is obtained as with the variables σ given as by the reordered s, 0 ≤ |s r | ≤ |s r−1 | ≤ · · · ≤ |s 1 |. (1.6) This coincides perfectly with the previous proposal in [10], where the authors further tried to give a Fermi surface interpretation to their proposal. In the limit of k → 0, we can also compute the coefficient B, and the result is Moreover, once we further restrict ourselves to the special values of s m s 1 = s 2 = · · · = s r = 1, (1.8) we are able to compute the coefficient A, and even the non-perturbative corrections which are the generalization of the membrane instantons in [30]. We analyse this model in section 4 and observe that there are three kinds of instantons with exponents e − 2µ r , e − 2µ r−2 and e − µ 2 respectively. This structure is reminiscent of the general instanton expansion in the N = 4 supersymmetric theory of the A-type quiver [23,24].

Fermi gas formalism
In this section, we shall present a Fermi gas description for the superconformal Chern-Simons theories of the D-type quiver.
The partition function of this theory is given by with the integration measure Here the numerator V , coming from the vector multiplets in the adjoint representation, is given by and the denominator H, coming from the hypermultiplets in the bifundamental representation, is

Density matrix from matrix model
To express the partition function (2.1) of the superconformal Chern-Simons matrix model of the D-type quiver in terms of that of a Fermi gas system, in this subsection let us first rewrite the generating function of the matrix model into a Fredholm determinant.
First, we rewrite the integrand of the matrix model Here we have introduced the combined variables (µ) a=1,··· ,2N = (µ) a=1,··· ,N , (µ ′ ) a−N =1,··· ,N and (ν) a=1,··· ,2N = (ν) a=1,··· ,N , (ν ′ ) a−N =1,··· ,N . Namely, the second factor in (2.5) is the determinant of a matrix which is a vertical array of two N × 2N rectangular matrices, one with components 2 cosh 2 −1 and the other with components 2 cosh Similarly, the second last factor is the determinant of a matrix which is a horizontal array of two 2N × N rectangular matrices, 2 cosh λ (r−3) 2N ×N . Note that in the above rewriting we have used the formulae (2.6) † We knew of a related work [31] from the reference list of [29]. In a seminar by Nadav Drukker at Nagoya university, we learned that actually they had a similar idea in rewriting the integration measure into a determinant of hyperbolic cosecant functions as in (2.5). We are grateful to Nadav Drukker for valuable discussions. ‡ Since the singularities appearing in the first and last determinants are originally absent in (2.4), we expect them to be harmless.
which follow from the Cauchy identity by the substitutions x i = e µ i and y j = e ν j or y j = −e ν j .
Then, using the formula proved in appendix A of [32] with the same ranks, we can combine the series of 2N × 2N determinants into Here the functions M, N and L denote § with the matrix in the second determinant in (2.8) given explicitly by . (2.10) Furthermore, if we use the formula in [32] for different ranks, we can perform the µ ′ and ν ′ integrations to find where • stands for either the µ ′ integration or the ν ′ integration in (2.2). It should be clear from the context which integration it stands for. For example, Now we can apply the formula (A.1) in our appendix A and find with the four N × N blocks given by (2.14) Here • denotes the ν integration. If we further use the formula presented in appendix B, the grand potential J(µ), is given by where the density matrix ρ is and other matrices are given in (B.2). Then it is not difficult to observe that the density matrix can be put into Here we have regarded the functions M, N and L as matrices, and contracted the adjacent indices by using the integrations in (2.2), without displaying • or • explicitly. After suitable similarity transformations and rearrangements, we can further put the density matrix into

Operator formalism for density matrix
In the previous subsection, we have reduced the study of the superconformal Chern-Simons matrix model of the D-type quiver into an integration kernel ρ. To express the superconformal Chern-Simons matrix model in terms of a Fermi gas system with N particles, we need to further rewrite the integration kernel in the operator formalism. For this purpose, we introduce the canonical coordinate/momentum operators q and p obeying the canonical commutation relation with the Planck constant given by = 2πk and normalize the coordinate eigenstate as It is not difficult to spell out each block in ρ explicitly in terms of the hyperbolic functions and the integrations. Then, after rescaling the integration variables by 1/k, we find that the density matrix is given by where the operators ρ ± are In the derivation we have used the formulae (2.24) Using this result we can simplify the grand potential (2.16). Though in (2.16) the determinant is taken simultaneously over the functional space (or, in the operator formalism, the phase space) and over the two dimensional space, since the left two components and the right two components are commutative respectively, after the multiplication by the elementary matrix, we note that the determinant over the two dimensional space can be taken trivially e J(µ) = det(I + e µ ρ), (2.25) with the density matrix purely in the phase space given by ρ = ρ + + ρ − . After the similarity transformation to move e

Large N behavior from Fermi surface analysis
In the previous section, after switching from the partition function Z(N) to the grand potential J(µ) (2.15), we find that the grand potential J(µ) is expressed in terms of the Fredholm determinant of the density matrix (2.25). Since the relation is very similar to the case of the A-type quiver [15], we expect that the perturbative corrections to the partition function sum up to an Airy function as (1.2) with some constants C, B and A. Indeed, the expression of the Airy function follows from the fact that the number of states with energy smaller than E behaves in large E as [15] n(E) Here the overall factor 2 compared with the case of the A-type quivers is due to the square-root in (2.25). In this section we shall show this relation, with explicit expressions of C and B up to O(k), by the technique used in [15].
For this purpose, let us consider the classical limit ( → 0). Classically, the number of states n(E) is given by the phase space volume where ρ 0 is the classical density matrix obtained from ρ (2.27) by neglecting the commutators log(ρ 0 (q, p)) −1 = quadrant R 2 ≥0 . We can further divide the volume vol{(q, p) ∈ R 2 ≥0 | log ρ −1 0 ≤ E} into the leading contribution in the limit of large E, V pol , and the deviation from it, δV , as The main contribution V pol can be computed by approximating the hyperbolic functions by rational functions Since the above subregion on the first quadrant is a polygon (see figure 2), whose vertices are located at with σ m given by s m as in (1.5), the volume of this subregion is Now we consider the deviation from the volume of the limit polygon, δV . First we divide the region between the classical Fermi surface and the polygon into the pieces around each line of p = s m q and p = 0, as in figure 3, and call the volume of each piece v m and v r+1 respectively, where q ′ ( p) and q( p) are the q-coordinate of a point ( q, p) on the Fermi surface and that on the approximant (3.10) and p ± correspond to the midpoints on the edges of the polygon. Noting p − s ℓ q < 0 for ℓ < m and p − s ℓ q > 0 for ℓ > m in this piece, we can compute v m as Here, although originally the integral interval, [ p − , p + ] is finite, we can replace it by the whole real axis (−∞, ∞) without affecting the perturbative behavior in (3.2), since the integrand is exponentially small for large p. The contribution from the piece around p = 0 can be calculated similarly to obtain v r+1 = π 2 3σ r+1 . (3.13) Substituting these results we obtain the large E expression of n(E) (3.2) with C and B given by (1 .4) and (1.7).
So far we have been neglecting the quantum corrections. Though it is difficult to take care of them, especially due to the variety of arguments of the hyperbolic functions in the density matrix (2.27), we can make the following estimation. There are two kinds ofcorrections, the Wigner transformation of each operator and the commutators of operators coming from the Baker-Campbell-Hausdorff formula. According to the Wigner transformation formula, the former always starts with the second derivatives of each term in (3.4). Also, if the Hamiltonian is hermitian, there are only nested commutators, which again start with the second derivatives. ¶ Therefore, since the second derivatives of hyperbolic functions are always exponentially suppressed, the quantum corrections never change the asymptotic polygon of the Fermi surface in the limit of E → ∞. This ensures the behavior of n(E) (3.2), with C uncorrected, and therefore that the perturbative partition function is given as an Airy function even with all order quantum corrections. On the other hand, B is possibly corrected due to the quantum effect.

A and instantons for special quivers
In this section, we restrict ourselves to the cases where the Chern-Simons levels are given by (1.3) with a uniform value of s (1.8), where we set this value to be 1, which is always possible by the redefinition of k. This restriction allows us to obtain the constant part A and the nonperturbative corrections (O(e −µ )) in the classical grand potential by performing the phase space integral explicitly, as in the theory of the A-type quiver with N = 4 supersymmetry enhancement [15,23,24].
First we rewrite the classical grand potential as Here, ρ 0 is the product of the hyperbolic functions (3.4). Under the restriction (1.8) there only appear three kinds of arguments, P ≡ p − q and Q ≡ p + q of the hyperbolic cosine function ¶ The requirement of hermiticity is also essential in the discussion in the A-type quiver [15,23]. and p = (P + Q)/2 of the hyperbolic sine function. Using the sum-to-product identity of the hyperbolic sine function, we can factorize the integration as which can be derived recursively by integration by parts, starting with , (4.4) and the formula a,b≥0,a+b=n which can be shown by considering the generating function with respect to n, we finally obtain the following expression for J 0 (µ) with Z(n) = 1 8π Γ(2n + 1) Γ(n + 1) To rewrite this J 0 (µ) into the expansion around µ → ∞ by analytic continuation, it is useful to consider the following Mellin-Barnes representation for (4.6) [33] with 0 < ǫ < 1, which is a natural generalization of the technique used in [24]. Indeed, assuming µ < 0, we can evaluate the right-hand side by collecting the residues around the poles in Re(t) > ǫ to reproduce the series expansion (4.6). Assuming µ > 0, on the other hand, we can evaluate the integration by pinching the contour so that it encloses the region Re(t) < ǫ. As a result, we can obtain the large µ expansion of the grand potential just by looking at the poles of the integrand in Re(t) < ǫ, as Here the first three perturbative terms come from the residue around t = 0. The constants C and B are consistent with the classical Fermi surface analysis in section 3, and A is (4.10) The non-perturbative part J np 0 (µ) consists of three kinds of instantons where ψ(x) is the di-gamma function ψ(x) = ∂ x log Γ(x).

Discussion
In this paper we have studied the partition function of the superconformal Chern-Simons theories of the D-type quiver, and have shown that we can rewrite the partition function into that of the Fermi gas system as in the case of the A-type quiver. We find that, again, the perturbative corrections of the partition function are summed up to the Airy function, if the Hamiltonian of the Fermi gas system is hermitian. Though for the general D-type quiver, in section 3 we only consider the perturbative coefficients in the classical limit k → 0, the Fermi gas formalism is very powerful and allows us in principle to determine the quantum corrections and the non-perturbative instanton corrections.
To further proceed to studying the membrane instanton of the general D-type quivers quantum-mechanically by the WKB expansion, it is, however, difficult to handle the noncommutative operators in the density matrix, or the exact integration over the phase space without taking the large µ limit. In the theories of the general A-type quivers, we have overcome the difficulties [23][24][25] by restricting ourselves to those with N = 4 supersymmetry [34]. Similarly, here in the theories of the D-type quivers, the difficulty is moderated by choosing the quivers with uniform s m , as in section 4. For these special cases, we have found that the non-perturbative corrections consist of three kinds of instantons, which is reminiscent of the results for the theories of the A-type quivers with the N = 4 supersymmetry [24].
It is interesting to see whether the symmetry is enhanced for these cases with uniform s m . Also, we hope to interpret these instanton exponents from the dual supergravity picture, as membranes wrapping on the tri-Sasaki Einstein manifold, though the geometry is more complicated than that for the A-type quivers. Furthermore, we hope to proceed to all the non-perturbative corrections including both the membrane instantons and the worldsheet instantons which have not been discussed at all in this work.
After seeing the Fermi gas formalism for the theories of the A-type and D-type quivers, it should be interesting to ask whether a Fermi gas formalism exists also for the E-type quivers. Also, it is interesting to study other quivers with orthosymplectic groups in [9] from the Fermi gas formalism. See e.g. [28].
We hope to study these directions in the future. with the skew-symmetric matrix P Remark. The definition of the pfaffian for any skew-symmetric matrix P is given by pf P = (−1) Proof. We can prove it by skew-symmetrizing the matrix elements,

B Another pfaffian formula
Proposition. Let P ab (x, y) with a, b = 1, 2 be functions of two variables satisfying P ba (y, x) = −P ab (x, y). Let P be a 2N × 2N skew-symmetric matrix consisting of four N × N blocks P ab whose (i, j)-component is P ab (x i , x j ). Then, we have Here the pfaffian on the left-hand side is the finite dimensional one, while on the right-hand side the determinant denotes simultaneously the 2 × 2 determinant and the Fredholm determinant.
Remark. This is the continuum limit N ∞ → ∞ of the following proposition (See e.g. Proposition 2.1 in [35]). Note that, in taking the limit, we use pf(Ω + zP ) 2 = det(Ω + zP ) = det(I − z Ω P ), which follows from Ω −1 = −Ω and det Ω = 1, and fix the overall signs by setting P to be zero.