Isomonodromic τ-functions and W N conformal blocks

et al. 2020

Commun. Math. Phys.

In this paper we study the extension of Painlevé/gauge theory correspondence to circular quivers by focusing on the special case of SU (2) 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) N = 2 * theory on self-dual Ωbackground and, in the Seiberg-Witten limit, an elegant relation between the IR and UV gauge couplings.

Section: Review Of Painlevé/cft Correspondence On the Spherementioning

confidence: 99%

Section: Review Of Painlevé/cft Correspondence On the Spherementioning

confidence: 99%

$${\mathcal {N}}$$ = $$2^*$$ Gauge Theory, Free Fermions on the Torus and Painlevé VI

Bonelli

Monte

et al. 2020

Commun. Math. Phys.

“…Second, the Nekrasov partition functions for these theories are equal to the conformal blocks in 2d CFT [1]. Finally, the isomonodromic tau-functions are expressed as the series of conformal blocks (or Nekrasov functions) with special, integer central charge, see [25,27,35].…”

Section: Introductionmentioning

confidence: 99%

Cluster integrable systems, q-Painlevé equations and their quantization

Bershtein

Marshakov

2018

J. High Energ. Phys.

Self Cite

We discuss the relation between the cluster integrable systems and q-difference Painlevé equations. The Newton polygons corresponding to these integrable systems are all 16 convex polygons with a single interior point. The Painlevé dynamics is interpreted as deautonomization of the discrete flows, generated by a sequence of the cluster quiver mutations, supplemented by permutations of quiver vertices.We also define quantum q-Painlevé systems by quantization of the corresponding cluster variety. We present formal solution of these equations for the case of pure gauge theory using q-deformed conformal blocks or 5-dimensional Nekrasov functions. We propose, that quantum cluster structure of the Painlevé system provides generalization of the isomonodromy/CFT correspondence for arbitrary central charge.

“…As in the Virasoro case, recent efforts have been focused in finding new boostrap solutions in Toda theories [44]. Definitely, our construction should work for the N × N case related to W N algebras with c = N − 1, since relation between higherrank isomonodromic deformations and W-algebras is also known [45]. We conjecture that in this case there should be the analog of (5.20), stating that the product of two connection constants equals to the ratio of 3-point functions.…”

Section: Discussionmentioning

confidence: 99%

Crossing invariant correlation functions at c = 1 from isomonodromic τ functions

Santachiara

2019

J. High Energ. Phys.

We present an approach that gives rigorous construction of a class of crossing invariant functions in c = 1 CFTs from the weakly invariant distributions on the moduli space M SL(2,C) 0,4 of SL(2, C) flat connections on the sphere with four punctures. By using this approach we show how to obtain correlation functions in the Ashkin-Teller and the Runkel-Watts theory. Among the possible crossing-invariant theories, we obtain also the analytic Liouville theory, whose consistence was assumed only on the basis of numerical tests.