Intersection numbers on $$ {\overline{M}}_{g,n} $$ and BKP hierarchy

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.

KdV solves BKP

“…This theorem partially explains the results of [Ale20,MM20]. Indeed, in [MM20] it was noted that the Kontsevich-Witten tau-function -one of the most important solutions of the KdV hierarchy -has a simple expansion in terms of the Schur Q-functions.…”

“…In this paper we answer the question about the relation between the KdV and BKP hierarchies, raised in [Ale20]. Namely, in [Ale20] it was observed that several families of the KdV tau-functions important in enumerative geometry and theoretical physics also solve the BKP hierarchy after a simple rescaling of times. Here we prove that any tau-function of KdV solves the BKP hierarchy.…”

A new kind of anomaly: on W-constraints for GKM

“…The observation of "anomaly" SE = ME even for r-reduced τ -functions leaves the puzzle of W -constraints and the origin of W -representation for GKM with r > 2 [25] unsolved. This adds to the equally puzzling complication of superintegrability formulas and character calculus for r > 2: at least the appropriate basis of Q-functions [16][17][18][19][20][21][22] remains unknown. It is unclear if there is a direct connection between these two complications -anyhow, the story of GKM is still incomplete and at least one additional idea is still lacking.…”

“…and, as established recently, 6) Z r has a peculiar non-Abelian W -representation [25,[29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44], i.e. can be unambiguously described by a single combination of W -constraints, ) Z r possesses character expansion in terms of Hall-Littlewood polynomials [16][17][18][19][20][21][22].…”

“…No doubt, at technical level these observations are well known to people who worked with GKM, but we make an attempt to summarize them and promote to a more conceptual level. This is needed because of the new accents introduced into the GKM theory quite recently, in [16][17][18][19][20][21][22] and [23]- [25], and the need to explain the origins and the form of the "single equation" [24] and the character expansion in terms of Hall-Littlewood polynomials [16][17][18][19][20][21][22]. We do not achieve these goals in the present paper, but we try to better explain the difficulties of one particularly promising suggestion from [25] -with the hope that it gains attention and will be somehow resolved in the near future.…”