Identification of the Givental Formula with the Spectral Curve Topological Recursion Procedure

Abstract

We identify the Givental formula for the ancestor formal Gromov-Witten potential with a version of the topological recursion procedure for a collection of isolated local germs of the spectral curve. As an application we prove a conjecture of Norbury and Scott on the reconstruction of the stationary sector of the Gromov-Witten potential of CP 1 via a particular spectral curve.

Loop Equations for Gromov-Witten Invariant of P1

“…The Virasoro operators can be obtained from conjugation of local Virasoro operators by operators that reconstruct the partition function of the Gromov-Witten invariants from the partition function of Gromov-Witten invariants of a point. The work of [5] showed that topological recursion is equivalent to this reconstruction of Givental and Teleman. Hence one would expect that the global Virasoro constraints of Theorem 2 can be derived directly from the local Virasoro constraints.…”

“…In Section 2 we recall the definition of descendant and ancestor Gromov-Witten invariants which are needed in the proof of Theorem 1. In Section 3 we review the topological recursion and its application to Gromov-Witten invariants of P 1 proven in [5], and derive a preliminary form of the global constraints from the local ones. Section 5 develops the regularised integral and its properties which is the main technical tool of this paper.…”

“…. , x n ) analytically continues to a rational function is a consequence of topological recursion proven in [5] -see Remark 3or topological recursion relations obtained by pulling back relations in H * M g,n to relations in H * M g,n P 1 , d [12] which are satisfied quite generally by Gromov-Witten invariants. We expect that it can also be derived from the semi-infinite wedge formalism of Okounkov and Pandharipande [14].…”

“…The Gromov-Witten invariants of P 1 were expressed as expectation values of Plancherel measure by Okounkov and Pandharipande in [14]. This viewpoint naturally led to the conjecture that the Gromov-Witten invariants of P 1 satisfy topological recursion applied to the complex curve defined by a dual Landau-Ginzburg model [13], proven in [5] using a completely different point of view. Topological recursion, defined in Section 3, produces a collection of meromorphic multidifferentials ω g,n on a complex curve via a recursive relation between the expansion of ω g,n at its poles and the expansion of ω g ,n for 2g − 2 + n < 2g − 2 + n at their poles.…”

A Givental-like formula and bilinear identities for tensor models

“…An other important development which aroses from matrix models around the same time is the topological recursion [16], which has grown into a polyvalent technique allowing to solve many problems of algebraic and enumerative geometry [17]. The relation between these two techniques has been shown in [19].…”