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$\mathcal{N}=2^*$ gauge theory, free fermions on the torus and Painlev\'e VI

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arxiv 1901.10497 v2 pith:G6J3UKJ4 submitted 2019-01-29 hep-th math-phmath.MPnlin.SI

classification hep-thmath-phmath.MPnlin.SI
keywords theorygaugemathcalpainlevprovidestorusachievedalgebra
verification ladder T0 review T1 audit T2 compute T3 formal
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abstract

In this paper we study the extension of Painlev\'e/gauge theory correspondence to circular quivers by focusing on the special case of $SU(2)$ $\mathcal{N}=2^*$ theory. We show that the Nekrasov-Okounkov partition function of this gauge theory provides an explicit combinatorial expression and a Fredholm determinant formula for the tau-function describing isomonodromic deformations of $SL_2$ flat connections on the one-punctured torus. This is achieved by reformulating the Riemann-Hilbert problem associated to the latter in terms of chiral conformal blocks of a free-fermionic algebra. This viewpoint provides the exact solution of the renormalization group flow of the $SU(2)$ $\mathcal{N}=2^*$ theory on self-dual $\Omega$-background and, in the Seiberg-Witten limit, an elegant relation between the IR and UV gauge couplings.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Modular transformations of tau functions and conformal blocks on the torus

    math-ph 2025-08 conditional novelty 8.0 of 10

    The paper derives the modular connection constant for tau functions on the one-punctured torus and obtains an exact closed formula for the c=1 Virasoro modular kernel.

  2. Blowing-up the edge: connection formulae and stability chart of the Lam\'e equation

    hep-th 2025-07 conditional novelty 7.0 of 10

    The paper derives the resummed Nekrasov-Shatashvili free energy from blow-up equations and uses it to compute the band-gap structure, connection formulas, and stability chart of the Lamé equation.

  3. Many-faced Painlev\'e I: irregular conformal blocks, topological recursion, and holomorphic anomaly approaches

    math-ph 2025-05 conditional novelty 7.0 of 10

    The paper proves the conifold gap property for Weierstrass elliptic topological recursion and gives an algebraic construction of the rank 5/2 Virasoro Whittaker state, linking the Painlevé I tau function to CFT and ho...

  4. (1,k) CFT and RH problem with the c=-2 case

    math-ph 2026-07 conditional novelty 6.0 of 10

    For (1,k) Virasoro models, periodic vertex operators plus two degenerate fields solve a modified Riemann–Hilbert problem; in the k=2, c=-2 case the solution is explicit and satisfies new bilinear identities.

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