Hirota equations for the extended bigraded Toda hierarchy and the total descendent potential of $\mathbb {C}P^1$ orbifolds

Abstract

We prove that the Hirota quadratic equations of Milanov and Tseng define an integrable hierarchy which is equivalent to the extended bigraded Toda hierarchy. In particular this proves a conjecture of Milanov-Tseng that relates the total descendent potential of the orbifold C k,m with a tau function of the bigraded Toda hierarchy.Date: 4/4/2013.

Multifold Darboux Transformations of the Extended Bigraded Toda Hierarchy

“…The Hirota bilinear equation of EBTH were equivalently constructed in our early paper [14] and a very recent paper [15], because of the equivalence of t 1,N flow and t 0,N flow of the EBTH in [14]. Meanwhile it was proved to govern Gromov-Witten invariant of the total descendent potential of P 1 orbifolds [15]. This hierarchy also lead to a series of results from analytical and algebraic considerations [16,17,18,19].…”

“…In the paper of Todor E. Milanov and Hsian-Hua Tseng ( [13]), they described conjecturally a kind of Hirota bilinear equations (HBEs) which was similar to the Lax operators of the EBTH and proved that it governed the Gromov-Witten theory of orbiford c N M . The Hirota bilinear equation of EBTH were equivalently constructed in our early paper [14] and a very recent paper [15], because of the equivalence of t 1,N flow and t 0,N flow of the EBTH in [14]. Meanwhile it was proved to govern Gromov-Witten invariant of the total descendent potential of P 1 orbifolds [15].…”

Darboux transformations and Fay identities for the extended bigraded Toda hierarchy

“…The EBTH is defined for every pair (k, m) of positive integers, and it coincides with the ETH for k = m = 1. The total descendant potential of CP 1 with two orbifold points of orders k and m is a tau-function of the EBTH (see [30,14]). The EBTH contains the bigraded Toda hierarchy, which is a reduction of the 2D Toda hierarchy (see [34,37]).…”

Additional symmetries of the extended bigraded Toda hierarchy

“…There is an EBTH for every pair (k, m) of positive integers, with k = m = 1 corresponding to the extended Toda hierarchy. The total descendant potential of CP 1 with two orbifold points of orders k and m is a tau-function of the EBTH (see [23,8]). Note that a part of the EBTH is the bigraded Toda hierarchy, which is a reduction of the 2D Toda hierarchy (see [26,27]).…”

“…In Section 2, we review the extended Toda hierarchy and the extended bigraded Toda hierarchy (EBTH), following the approach of K. Takasaki [26]. Our version of the EBTH is related to the original definition of G. Carlet [6] (or to [8]) by an explicit change of variables, and we believe it is more convenient. We discuss the Lax operator L, the wave operators, and wave functions of the EBTH.…”