Brezin–Gross–Witten tau function and isomonodromic deformations

Abstract

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. This tau function admits a natural extension, called generalized Brezin-Gross-Witten tau function. We prove that the latter is the isomonodromic tau function of a 2 × 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 formulae for the generating functions of the correlators in terms of simple generating series, the Virasoro constraints, and discuss the relation with the Painlevé XXXIV hierarchy.

Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

“…They are directly related to the theory of tau functions (formal [21] and isomonodromic [9,31]) and to topological recursion theory [5,6,15,28]. Incidentally, similar formulae also appear for matrix models with external source [7,8,[11][12][13]41]. In Sect.…”

On tau-functions for the KdV hierarchy

“…Note added. After this paper was finished we learned that the identities ( 36)-( 37), ( 39)-( 41), which had been presented together with their proofs by one of the authors [67], had also been found by M. Bertola and G. Ruzza (see [17], whose arXiv version appeared slightly before the arXiv version of this paper), but with a quite different method using matrix models together with the construction of appropriate Riemann-Hilbert problems.…”

“…], one could use alternatively the Sato Grassmannian approach [13,26,63,72] to prove the identity (19). It would be interesting to investigate if the identity (19) could also be proved, say for V = C((x)), by using the approach of Bergère and Eynard with the employment of topological recursion (loop equation) [1,8,10,24,29,45,59,69], or by using matrix models together with appropriate Riemann-Hilbert problems (isomonodromic deformations) [12,16,17,27,29,37,39,49,57,60,66], or by using OPE from appropriate vertex algebras [5,26,47,72]; we expect that at least for the initial data f (x) ∈ V having the bispectral property defined by Duistermaat and Grünbaum [43] (see also the M -bispectrality given in Section 6.2) this might be possible. We also note that the matrix resolvent method and some of the above-mentioned methods extend to new situations (see e.g.…”

“…We also note that the matrix resolvent method and some of the above-mentioned methods extend to new situations (see e.g. [14,15,37,38,39,40], [3], [8,9,11,59] and [16,17,18,51]).…”

JT supergravity and Brezin-Gross-Witten tau-function

“…Nevertheless, the exact result in the first line of (4.15) is a positive function for β > 0. A non-perturbative resummation of c 0 (g) is proposed in [46]…”