Abstract
In their recent inspiring paper, Mironov and Morozov claim a surprisingly simple expansion formula for the Kontsevich-Witten tau-function in terms of the Schur Q-functions. Here we provide a similar description for the Brézin-Gross-Witten tau-function. Moreover, we identify both tau-functions of the KdV hierarchy, which describe intersection numbers on the moduli spaces of punctured Riemann surfaces, with the hypergeometric solutions of the BKP hierarchy.
Highlights
In this paper we consider1 a Schur Q-function expansion of the BGW tau-function
We provide a similar description for the Brézin-Gross-Witten tau-function. We identify both tau-functions of the KdV hierarchy, which describe intersection numbers on the moduli spaces of punctured Riemann surfaces, with the hypergeometric solutions of the BKP hierarchy
After the first version of this paper was posted on the arXiv, the author has proven [4] that any solution of the KdV hierarchy solves the BKP hierarchy, or symbolically
Summary
Denote by Mg,n the Deligne-Mumford compactification of the moduli space of all compact Riemann surfaces of genus g with n distinct marked points It is a non-singular complex orbifold of dimension 3g − 3 + n. An interesting family of such intersection numbers was recently considered by Norbury [31] He introduced Θ-classes, Θg,n ∈ H4g−4+2n(Mg,n), see [32, section 2] for the definition. We refer the reader to [31, 32] for a detailed description Both KW and BGW tau-functions can be described by matrix models. The BGW tau-function in the Miwa parametrization is given by the unitary matrix integral. It has another integral description [27], similar to the Kontsevich integral (2.12), see section 2.3 below. KdV integrability of the BGW model follows from this description
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
