Darboux transformations and Fay identities for the extended bigraded Toda hierarchy

Abstract

The extended bigraded Toda hierarchy (EBTH) is an integrable system satisfied by the Gromov-Witten total descendant potential of CP 1 with two orbifold points. We write a bilinear equation for the tau-function of the EBTH and derive Fay identities from it. We show that the action of Darboux transformations on the tau-function is given by vertex operators. As a consequence, we obtain generalized Fay identities.

The Lax operator fixed under the additional symmetries of the extended Toda hierarchy

“…We now summarize the reviews of the Toda hierarchy and its extensions found in [1,2], and [19]. First, define the shift operator Λ to act on functions f : R → R by (Λf )(s) = f(s + 1) for all s ∈ R. A difference operator is a formal power series in the shift operator of the form…”

“…For a more comprehensive review of the Darboux transformations we refer the reader to [2,14] (section 4), and the references therein.…”

“…To better understand these additional symmetries, we find necessary properties of Lax operators fixed under these symmetries at the initial condition of t n+1 = 0, tn+1 = 0, and x n = δ n,1 for n ⩾ 0. Applying the Darboux transformations, found in [14] and examined further in [2], to such solutions then produces more interesting solutions of the EBTH. In the special case of the ETH, we determine the unique such Lax operator at the aforementioned values of t n+1 , tn+1 , and x n .…”