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The partition function of a topological field theory

This is the sequel to my paper ‘TCFTs and Calabi–Yau categories’, Advances in Mathematics 210 (2007) no. 1, 165–214. Here we extend the results of that paper to construct, for certain Calabi–Yau A∞ 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∞ version of the derived category of sheaves on a Calabi–Yau yields the B model potential, at all genera. The construction does not 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 to the quantum master equation, essentially the ‘string vertices’ of Zwiebach. On the field theory side, the BV operator has an interpretation as the quantized differential on the Fock space for periodic cyclic chains. Passing to homology, something satisfying the master equation yields an element of the Fock space.