Surface Operator, Bubbling Calabi-Yau and AGT Relation

TL;DR

This paper explores the connection between surface operators in N=2 gauge theories, topological string theory, and conformal field theory, revealing how geometric transitions in string theory correspond to surface operator insertions.

Contribution

It proposes that topological open string amplitudes geometrically engineer surface operator partition functions and Gaiotto curves, linking geometric transitions to degenerate field insertions in CFT.

Findings

Surface operators correspond to degenerate field insertions in CFT.

Geometric transition in topological strings models surface operator effects.

Bubbling Calabi-Yau geometries represent surface operator insertions.

Abstract

Surface operators in N=2 four-dimensional gauge theories are interesting half-BPS objects. These operators inherit the connection of gauge theory with the Liouville conformal field theory, which was discovered by Alday, Gaiotto and Tachikawa. Moreover it has been proposed that toric branes in the A-model topological strings lead to surface operators via the geometric engineering. We analyze the surface operators by making good use of topological string theory. Starting from this point of view, we propose that the wave-function behavior of the topological open string amplitudes geometrically engineers the surface operator partition functions and the Gaiotto curves of corresponding gauge theories. We then study a peculiar feature that the surface operator corresponds to the insertion of the degenerate fields in the conformal field theory side. We show that this aspect can be realized as the geometric transition in topological string theory, and the insertion of a surface operator leads to the bubbling of the toric Calabi-Yau geometry.