40 bilinear relations of q-Painlevé VI from $$ \mathcal{N} $$ = 4 super Chern-Simons theory

Abstract

We investigate partition functions of the circular-quiver supersymmetric Chern-Simons theory which corresponds to the q-deformed Painlevé VI equation. From the partition functions with the lowest rank vanishing, where the circular quiver reduces to a linear one, we find 40 bilinear relations. The bilinear relations extend naturally to higher ranks if we regard these partition functions as those in the lowest order of the grand canonical partition functions in the fugacity. Furthermore, we show that these bilinear relations are a powerful tool to determine some unknown partition functions. We also elaborate the relation with some previous works on q-Painlevé equations.

Affine symmetries for ABJM partition function and its generalization

“…explained from the physical argument of duality cascades. The rules thus found around the fundamental domain are surprisingly simple (see [7] or (3.13) later), which we propose to remain valid even far away from the fundamental domain. Second, alternatively, we can "derive" the transformation rules directly from the matrix model.…”

“…However, such an approach is difficult for the (2, 2) model due to the convergence condition for the integrations in the matrix model. Indeed, as discussed in [7] and some earlier works [42,43], the convergence of the matrix model is guaranteed only in a range of relative ranks where the theory is "good" in the sense of [44]. For this reason, in this paper we explore the project of studying the transformation under duality cascades by overlooking these ambiguities and providing a few studies from slightly different approaches.…”

“…First, we trust the validity of analysis of the matrix model in the fundamental domain [7] and try to extend it outside using qP VI . With this method, we obtain the exact expressions of the matrix model slightly beyond the fundamental domain.…”

“…With the similarity in mind, it is natural that the grand partition functions of the ABJM theory and the above-mentioned four-node (2, 2) model were found to satisfy respectively the q-Painlevé equations qP III 3 [3] and qP VI [6,7], which enjoy respectively symmetries of the affine A 1 and D 5 Weyl groups. Here we would like to stress, however, that there is a big difference between the Weyl group and the affine Weyl group.…”

“…Later, it was found [3] that the grand partition function satisfies the q-Painlevé equation qP III 3 , which associates to symmetries of the affine A 1 Weyl group. Also, all of these relations are generalized to the theory with two NS5-branes and two (1, k)5-branes [4], where the spectral operator enjoys symmetries of the D 5 Weyl group [5] and the grand partition function satisfies the q-Painlevé equation qP VI associated to the affine D 5 Weyl group [6,7]. In this paper we would like to point out a big difference of symmetries between non-affine cases and affine ones.…”