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# Advances in Theoretical and Mathematical Physics

## Volume 22 (2018)

### Number 6

### Cut-and-join description of generalized Brezin–Gross–Witten model

Pages: 1347 – 1399

DOI: http://dx.doi.org/10.4310/ATMP.2018.v22.n6.a1

#### Author

#### Abstract

We investigate the Brezin–Gross–Witten model, a tau-function of the KdV hierarchy, and its natural one-parameter deformation, the generalized Brezin–Gross–Witten tau-function. In particular, we derive the Virasoro constraints, which completely specify the partition function. We solve them in terms of the cut-and-join operator. The Virasoro constraints lead to the loop equations, which we solve in terms of the correlation functions. Explicit expressions for the coefficients of the tau-function and the free energy are derived, and a compact formula for the genus zero contribution is conjectured. A family of polynomial solutions of the KdV hierarchy, given by the Schur functions, is obtained for the half-integer values of the parameter. The quantum spectral curve and its classical limit are discussed.

Published 3 May 2019