The Gromov-Witten potential associated to a TCFT

TL;DR

This paper constructs a Gromov-Witten potential for certain Calabi-Yau A-infinity categories using a novel approach that employs BV algebra and string vertices, extending the B model at all genera without relying on Deligne-Mumford spaces.

Contribution

It introduces a new method to define the Gromov-Witten potential via BV algebra and string vertices, bypassing traditional moduli space techniques.

Findings

Constructed a Gromov-Witten potential in Fock space for Calabi-Yau A-infinity categories.

Extended the B model potential to all genera.

Provided a new algebraic framework using BV algebra and quantum master equation.

Abstract

This is the sequel to my preprint "TCFTs and Calabi-Yau categories", math.QA/0412149. Here we extend the results of that paper to construct, for certain Calabi-Yau A-infinity categories, something playing the role of the Gromov-Witten potential. This is a state in the Fock space associated to periodic cyclic homology, which is a symplectic vector space. Applying this to a suitable A-infinity version of the derived category of sheaves on a Calabi-Yau yields the B model potential, at all genera. The construction doesn't go via the Deligne-Mumford spaces, but instead uses the Batalin-Vilkovisky algebra constructed from the uncompactified moduli spaces of curves by Sen and Zwiebach. The fundamental class of Deligne-Mumford space is replaced here by a certain solution of the quantum master equation, essentially the "string vertices" of Zwiebach. On the field theory side, the BV operator has an interpretation as the quantised differential on the Fock space for periodic cyclic chains. Passing to homology, something satisfying the master equation yields an element of the Fock space.