Pure 𝑆𝑈(2) gauge theory partition function and generalized Bessel kernel

@article{Gavrylenko2017PureG,
  title={Pure 𝑆𝑈(2) gauge theory partition function
 and generalized Bessel kernel},
  author={Pavlo Gavrylenko and O Lisovyy},
  journal={Proceedings of Symposia in Pure
                        Mathematics},
  year={2017}
}
We show that the dual partition function of the pure $\mathcal N=2$ $SU(2)$ gauge theory in the self-dual $\Omega$-background (a) is given by Fredholm determinant of a generalized Bessel kernel and (b) coincides with the tau function associated to the general solution of the Painleve III equation of type $D_8$ (radial sine-Gordon equation). In particular, the principal minor expansion of the Fredholm determinant yields Nekrasov combinatorial sums over pairs of Young diagrams. 

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References

SHOWING 1-10 OF 44 REFERENCES

Wild quiver gauge theories

A bstractWe study $ \mathcal{N} = {2} $ supersymmetric SU(2) gauge theories coupled to non-Lagrangian superconformal field theories induced by compactifying the six dimensional A1 (2,0) theory on

Seiberg-Witten theory and random partitions

We study \( \mathcal{N} = 2 \) supersymmetric four-dimensional gauge theories, in a certain 525-02 = 2 supergravity background, called theΩ-background. The partition function of the theory in the

Liouville Correlation Functions from Four-Dimensional Gauge Theories

We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class

Connection Problem for the Tau-Function of the Sine-Gordon Reduction of Painlevé-III Equation via the Riemann-Hilbert Approach

We evaluate explicitly, in terms of the Cauchy data, the constant pre-factor in the large $x$ asymptotics of the Painleve III tau-function. Our result proves the conjectural formula for this

Isomonodromic Tau-Functions from Liouville Conformal Blocks

The goal of this note is to show that the Riemann–Hilbert problem to find multivalued analytic functions with $${{\rm SL}(2,\mathbb{C})}$$SL(2,C)-valued monodromy on Riemann surfaces of genus zero

Seiberg–Witten theory as a Fermi gas

We explore a new connection between Seiberg–Witten theory and quantum statistical systems by relating the dual partition function of SU(2) Super Yang–Mills theory in a self-dual $$\Omega $$Ω

Fredholm Determinant and Nekrasov Sum Representations of Isomonodromic Tau Functions

We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with n regular singular points on the Riemann sphere and generic monodromy in GL

How instanton combinatorics solves Painlevé VI, V and IIIs

We elaborate on a recently conjectured relation of Painlevé transcendents and 2D conformal field theory. General solutions of Painlevé VI, V and III are expressed in terms of c = 1 conformal blocks

Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I

We discuss the relation between Liouville theory and the Hitchin integrable system, which can be seen in two ways as a two step process involving quantization and hyperkaehler rotation. The modular