Brezin-Gross-Witten tau function and isomonodromic deformations

Understand

The Brezin-Gross-Witten tau function is a tau function of the KdV hierarchy which arises in the weak coupling phase of the Brezin-Gross-Witten model.

• It falls within the family of generalized Kontsevich matrix integrals, and its algebro--geometric interpretation has been unveiled in recent works of Norbury.
• We prove that a suitably generalized Brezin-Gross-Witten tau function is the isomonodromic tau function of a $2\times 2$ isomonodromic system and consequently present a study of this tau function purely by means of this isomonodromic interpretation.
• Within this approach we derive effective formul\ae\ for the generating functions of the correlators in terms of simple generating series, the Virasoro constraints, and discuss the relation with the Painlev\'{e} XXXIV hierarchy.