On the Hodge-BGW correspondence

Abstract

We establish an explicit relationship between the partition function of certain special cubic Hodge integrals and the generalized Brézin-Gross-Witten (BGW) partition function, which we refer to as the Hodge-BGW correspondence. As an application, we obtain an ELSV-like formula for generalized BGW correlators.

GUE via Frobenius Manifolds. I. From Matrix Gravity to Topological Gravity and Back

“…Usually, to come back to the matrix gravity, one needs a deformation theory [25,26,38,46]. But, the recent studies [26,28,30,68] all together show that one can start with Witten's topological gravity and come back to the matrix gravity without a deformation theory, again at least in the higher genera. More precisely, we first go to the special cubic Hodge partition function by a space/time duality [18,67,68] (see also [2,3]) (in [68] this is revealed by the Hodge-BGW correspondence), and then go to the so-called modified even GUE partition function by the Hodge-GUE correspondence [26,28], and finally back to the even GUE partition function via a product formula [30], again at least to the higher genera in jets.…”

“…where the righthand side of (64) should be viewed as res L j . By using (64) and the compatibility between (40), (41) we have (68) i,j≥1…”

“…The proof of Proposition 2 using the Hodge-BGW correspondence is given in [68]. An equivalent version of this proposition and the proof are given in [67].…”

Higher Brézin-Gross-Witten tau-functions and intersection theory of Witten's and Norbury's classes

“…Geometric interpretation of the generalized higher BGW models, considered in Section 5, is not clear yet. We expect that they can be related to certain Hodge integrals, see [YZ21].…”

Gelfand--Dickey hierarchy, generalized BGW tau-function, and $W$-constraints

“…Again, let us look at the KdV case first (i.e., the case with r = 2). For this case, it is known that there exists a solution to the KdV hierarchy, called the generalized BGW solution, depending non-trivially on one arbitrary parameter [3,18], having bispectral properties [18,20], and possessing enumerative meanings [33,38,42]. This motivates us to generalize the generalized BGW solution to an arbitrary r ≥ 2.…”