Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String

TL;DR

This paper connects instanton partition functions with surface operators to irregular conformal blocks and open topological string amplitudes, providing a unified framework for understanding surface operators via CFT and string theory.

Contribution

It establishes a precise correspondence between instanton partition functions with surface operators, irregular conformal blocks, and open topological string amplitudes, extending geometric engineering insights.

Findings

Exact agreement between instanton partition functions and CFT correlation functions with degenerate operators.

Decoupling limit analysis from superconformal to asymptotically free theories.

Evidence of correspondence between open topological string amplitudes and gauge theory partition functions.

Abstract

Following a recent paper by Alday and Tachikawa, we compute the instanton partition function in the presence of the surface operator by the localization formula on the moduli space. For SU(2) theories we find an exact agreement with CFT correlation functions with a degenerate operator insertion, which enables us to work out the decoupling limit of the superconformal theory with four flavors to asymptotically free theories at the level of differential equations for CFT correlation functions (irregular conformal blocks). We also argue that the K theory (or five dimensional) lift of these computations gives open topological string amplitudes on local Hirzebruch surface and its blow ups, which is regarded as a geometric engineering of the surface operator. By computing the amplitudes in both A and B models we collect convincing evidences of the agreement of the instanton partition function with surface operator and the partition function of open topological string.