Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals

TL;DR

This paper explores the pure gauge limit of Selberg integrals within the AGT correspondence, revealing their connection to the Brezin-Gross-Witten model and proposing a new perspective on Nekrasov functions for pure SU(2) theories.

Contribution

It demonstrates that in the pure gauge limit, Selberg integrals become averages in the BGW model, linking Nekrasov functions to BGW models and suggesting a new approach to understanding these limits.

Findings

Selberg integrals turn into BGW model averages in the pure gauge limit.

Nekrasov functions for pure SU(2) theories relate to BGW models.

Proposes that the elliptic Selberg integral limit is also a BGW model.

Abstract

The AGT relation identifies the Nekrasov functions for various N=2 SUSY gauge theories with the 2d conformal blocks, which possess explicit Dotsenko-Fateev matrix model (beta-ensemble) representations the latter being polylinear combinations of Selberg integrals. The "pure gauge" limit of these matrix models is, however, a non-trivial multiscaling large-N limit, which requires a separate investigation. We show that in this pure gauge limit the Selberg integrals turn into averages in a Brezin-Gross-Witten (BGW) model. Thus, the Nekrasov function for pure SU(2) theory acquires a form very much reminiscent of the AMM decomposition formula for some model X into a pair of the BGW models. At the same time, X, which still has to be found, is the pure gauge limit of the elliptic Selberg integral. Presumably, it is again a BGW model, only in the Dijkgraaf-Vafa double cut phase.