Dressing operators in equivariant Gromov–Witten theory of CP1

Abstract

Okounkov and Pandharipande proved that the equivariant Toda hierarchy governs the equivariant Gromov–Witten theory of C P 1 . A technical clue of their method is a pair of dressing operators on the Fock space of 2D charged free fermion fields. We reformulate these operators as difference operators in the Lax formalism of the 2D Toda hierarchy. This leads to a new explanation to the question of why the equivariant Toda hierarchy emerges in the equivariant Gromov–Witten theory of C P 1 . Moreover, the non-equivariant limit of these operators turns out to capture the integrable structure of the non-equivariant Gromov–Witten theory correctly.

Generalized ILW hierarchy: solutions and limit to extended lattice GD hierarchy

“…The factorization problem itself can generate all solutions of the generalized ILW hierarchy. We find a special solution of this type in the context of our previous work [22] on the equivariant Toda hierarchy. We constructed therein a pair of difference operators that play the role of Okounkov and Pandharipande's 'dressing operators' [23,24] in their fermionic description of the equivariant Gromov-Witten theory of CP 1 .…”

“…More precisely, the work of Okounkov and Pandharipande [23] amounts to the case of N = 1; the case of N > 1 is an orbifold generalization [24]. 1 The definition of H is slightly different from our previous work [22] in two aspects. First, s is rescaled by ℏ so as to preserve the commutation relation [log Λ, s] = 1.…”

“…They use what they call 'dressing operators' to convert a fermionic formula of the generating function of equivariant Gromov-Witten invariants to the standard fermionic formalism of 2D Toda tau functions [12,25]. In our previous work [22], those operators (denoted by V and V therein) are reformulated as difference operators in the spatial variable s, and used to explain how the equivariant Toda hierarchy emerges in the equivariant Gromov-Witten theory of CP 1 .…”

“…Let us briefly recall the construction of V [22]. V is constructed order-by-order in the ν-expansion:…”

“…The factorization problem for the 2D Toda hierarchy thereby generates a solution of the equivariant bigraded Toda hierarchy of type (N, N). In order to accommodate the equivariant Gromov-Witten theory of CP 1 , however, one has to modify the intertwining relation ( 46) by a constant term [13,22] (see appendix, remark 11).…”