Deformed Prepotential, Quantum Integrable System and Liouville Field Theory

TL;DR

The paper develops a cohesive framework linking ${\mathcal{N}}=2$ gauge theories with surface operators to quantum integrable systems and Liouville CFT via AGT. By tracing the null-state conditions to Schrödinger equations, it shows that the quantum-corrected prepotential $\mathcal{F}(\epsilon_1)$ emerges from monodromies of conformal blocks with a degenerate field, unifying Hitchin-system quantization with gauge-theory observables. It further computes Nekrasov-like ramified instanton partition functions through irregular conformal blocks, explicitly counting 2d and 4d instantons and extending to theories with fundamental flavors. The results strengthen the AGT bridge, provide practical tools for ramified instanton counting, and point to rich extensions to higher-rank theories and matrix-model approaches.

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.