Deformed Prepotential, Quantum Integrable System and Liouville Field Theory

TL;DR

This paper explores the duality between Seiberg-Witten theory with surface operators, integrable systems, and Liouville field theory, verifying conjectures through null state conditions and deriving instanton partition functions.

Contribution

It confirms the AGT relation's predictions by linking null state conditions to Schrödinger equations and relates deformed prepotentials to monodromy operations in conformal blocks.

Findings

Null state conditions lead to Schrödinger equations of integrable systems.

Deformed prepotential corresponds to monodromy operations in conformal blocks.

Instanton partition functions involve counting of 2D and 4D instantons.

Abstract

We study the dual descriptions recently discovered for the Seiberg-Witten theory in the presence of surface operators. The Nekrasov partition function for a four-dimensional N=2 gauge theory with a surface operator is believed equal to the wave-function of the corresponding integrable system, or the Hitchin system, and is identified with the conformal block with a degenerate field via the AGT relation. We verify the conjecture by showing that the null state condition leads to the Schrodinger equations of the integrable systems. Furthermore, we show that the deformed prepotential emerging from the period integrals of the principal function corresponds to monodromy operation of the conformal block. We also give the instanton partition functions for the asymptotically free SU(2) gauge theories in the presence of the surface operator via the AGT relation. We find that these partition functions involve the counting of two- and four-dimensional instantons.