Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

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 of \({\mathcal{N}=2}\) SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture at genus 0, 1.

Liouville Correlation Functions from Four-dimensional Gauge Theories

Groupes et algébres de lie

We study four dimensional N=2 supersymmetric gauge theory in the Omega-background with the two dimensional N=2 super-Poincare invariance. We explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional N=2 theory. 
The epsilon-parameter of the Omega-background is identified with the Planck constant, the twisted chiral ring maps to quantum Hamiltonians, the supersymmetric vacua are identified with Bethe states of quantum integrable systems. This four dimensional gauge theory in its low energy description has two dimensional twisted superpotential which becomes the Yang-Yang function of the integrable system. We present the thermodynamic-Bethe-ansatz like formulae for these functions and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. The general construction is illustrated at the examples of the many-body systems, such as the periodic Toda chain, the elliptic Calogero-Moser system, and their relativistic versions, for which we present a complete characterization of the L^2-spectrum. We very briefly discuss the quantization of Hitchin system.

Quantization of Integrable Systems and Four Dimensional Gauge Theories

We propose a relation between correlation functions in the 2d A(N-1) conformal Toda theories and the Nekrasov instanton partition functions in certain conformal N = 2 SU(N) 4d quiver gauge theories. Our proposal generalises the recently uncovered relation between the Liouville theory and SU(2) quivers [1]. New features appear in the analysis that have no counterparts in the Liouville case.

https://iopscience.iop.org/article/10.1088/1126-6708/2009/11/002/pdf

A(N-1) conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories

We solve = 2 supersymmetric Yang-Mills theories for an arbitrary classical gauge group, i.e. SU(N), SO(N), Sp(N). In particular, we derive the prepotential of the low-energy effective theory, and the corresponding Seiberg-Witten curves. We manage to do this without resolving singularities of the compactified instanton moduli spaces.

ABCD of instantons

We solve N=2 supersymmetric Yang-Mills theories for arbitrary classical gauge group, i.e. SU(N), SO(N), Sp(N). In particular, we derive the prepotential of the low-energy effective theory, and the corresponding Seiberg-Witten curves. We manage to do this without resolving singularities of the compactified instanton moduli spaces.

We apply equivariant integration technique, developed in the context of instanton counting, to two dimensional = 2 supersymmetric Yang-Mills models. The expression for the twisted superpotential for U(N) model is conjectured. Connections to the four dimensional case are discussed.

On F-term contribution to effective action

N=2 supersymmetric Yang-Mills theories for all classical gauge groups, that is, for SU(N), SO(N), and Sp(N) is considered The formal expression for almost all models accepted by the asymptotic freedom are obtained The equations which define the Seiberg-Witten curve are proposed In some cases they are solved It is shown that for all considered the 1-instanton corrections which follows from these equations agree with the direct computations Also they agree with the computations based on Seiberg-Witten curves which come from the M-theory consideration It is shown that for a large class of models the M-theory predictions matches with the direct compuatations It is done for all considered models at the 1-instanton level For some models it is shown at the level of the Seiberg-Witten curves

On certain aspects of string theory/gauge theory correspondence

= 2 supersymmetric Yang-Mills theories for all classical gauge groups, that is, for SU(N), SO(N), and Sp(N) is considered. The equations which define the Seiberg-Witten curve are proposed. In some cases they are solved. It is shown that for (almost) all models allowed by the asymptotic freedom the 1-instanton corrections which follows from these equations agree with the direct computations and with known results.

Sergey Shadchin

Papers

ABCD of instantons

ABCD of instantons

On F-term contribution to effective action

On certain aspects of string theory/gauge theory correspondence

Saddle point equations in Seiberg-Witten theory