Circular quiver gauge theories, isomonodromic deformations and $W_N$ fermions on the torus

Abstract

We study the relation between class S theories on punctured tori and isomonodromic deformations of flat SL(N) connections on the two dimensional torus with punctures. Turning on the self dual $\Omega$-background corresponds to a deautonomization of the Seiberg-Witten integrable system which implies a specific time dependence in its Hamiltonians. We show that the corresponding $\tau$-function is proportional to the dual gauge theory partition function, the proportionality factor being a non trivial function of the solution of the deautonomized Seiberg-Witten integrable system. This is obtained by mapping the isomonodromic deformation problem to $W_N$ free fermion correlators on the torus.

35 pages, 7 figures

References in corpus (6)

• Cluster integrable systems, q-Painleve equations and their quantization
• Irregular conformal blocks and connection formulae for Painlevé V functions
• Cluster Toda chains and Nekrasov functions
• CFT approach to the $q$-Painlevé VI equation
• q-Painleve equation from Virasoro constraints
• Higher rank isomonodromic deformations and W-algebras

Cited by in corpus (1)

• Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions