M2-branes and $\mathfrak{q}$-Painlevé equations

Abstract

In this paper we investigate a novel connection between the effective theory of M2-branes on (C 2 /Z 2 ×C 2 /Z 2 )/Z k and the q-deformed Painlevé equations, by proposing that the grand canonical partition function of the corresponding four-nodes circular quiver N = 4 Chern-Simons matter theory solves the q-Painlevé VI equation. We analyse how this describes the moduli space of the topological string on local dP 5 and, via geometric engineering, five dimensional N f = 4 SU(2) N = 1 gauge theory on a circle. The results we find extend the known relation between ABJM theory, q-Painlevé III 3 , and topological strings on local P 1 × P 1 . From the mathematical viewpoint the quiver Chern-Simons theory provides a conjectural Fredholm determinant realisation of the q-Painlevé VI τ -function. We provide evidence for this proposal by analytic and numerical checks and discuss in detail the successive decoupling limits down to N f = 0, corresponding to q-Painlevé III 3 . * bonelli(at)sissa.it † fgloblek(at)sissa.it ‡ naotaka.kubo(at)yukawa.kyoto-u.ac.jp § nosaka(at)yukawa.kyoto-u.ac.jp ¶ tanzini(at)sissa.it

Duality cascades and parallelotopes

“…It is important to elaborate the relations especially from integrabilities [58][59][60][61][62][63][64]. Sixthly, we have studied the relations between duality cascades and parallelotopes mainly having the setup of three-dimensional supersymmetric Chern-Simons theories in mind.…”

Duality Cascades and Parallelotopes

“…It is important to elaborate the relations especially from integrabilities [45][46][47][48][49][50][51].…”

tt∗ Toda equations for surface defects in SYM and instanton counting for classical Lie groups

“…It should be possible to apply the approach proposed here to general class-S theories [12] by studying the related isomonodromic deformation problem, for example for circular quivers, generalizing to other classical groups the results of [16,17]. It would be also interesting to extend the analysis to non-self-dual Ω-background, which should amount to the quantization of the τ -systems, and its lift to five-dimensional gauge theories on R 4 × S 1 , which should correspond to quantum q-difference τ -systems [18][19][20][21][22][23].…”