Surface Operators and Separation of Variables

TL;DR

This paper explores the duality between 4d supersymmetric theories with surface operators and 2d conformal field theories, revealing a deep connection via brane constructions and integrable systems.

Contribution

It establishes an IR duality linking 4d theories with surface operators to 2d WZW and Liouville models, extending the AGT correspondence.

Findings

Demonstrates IR duality between 4d theories with surface operators and 2d conformal theories.

Connects the duality to brane creation in M-theory.

Expresses the duality through relations between Darboux coordinates on the Hitchin moduli space.

Abstract

Alday, Gaiotto, and Tachikawa conjectured relations between certain 4d N=2 supersymmetric field theories and 2d Liouville conformal field theory. We study generalizations of these relations to 4d theories with surface operators. For one type of surface operators the corresponding 2d theory is the WZW model, and for another type - the Liouville theory with insertions of extra degenerate fields. We show that these two 4d theories with surface operators exhibit an IR duality, which reflects the known relation (the so-called separation of variables) between the conformal blocks of the WZW model and the Liouville theory. Furthermore, we trace this IR duality to a brane creation construction relating systems of M5 and M2 branes in M-theory. Finally, we show that this duality may be expressed as an explicit relation between the generating functions for the changes of variables between natural sets of Darboux coordinates on the Hitchin moduli space.