Topological conformal field theories and Calabi-Yau categories
Abstract
Description
This is the first of two papers which construct a purely algebraic counterpart to the theory of Gromov-Witten invariants (at all genera). These Gromov-Witten type invariants depend on a Calabi-Yau A-infinity category, which plays the role of the target in ordinary Gromov-Witten theory. When we use an appropriate A-infinity version of the derived category of coherent sheaves on a Calabi-Yau variety, this constructs the B model at all genera. When the Fukaya category of a compact symplectic manifold X is used, it is shown, under certain assumptions, that the usual Gromov-Witten invariants are recovered. The assumptions are that a good theory of open-closed Gromov-Witten invariants exists for X, and that the natural map from the Hochschild homology of the Fukaya category of X to the ordinary homology of X is an isomorphism.
50 pages, 13 figures. Cosmetic and organisational changes in this version. Some references have been added, as has discussion of the non-unital case. Results about topology of moduli space now appear in math.GT/0601130. Discussion of Deligne-Mumford spaces and the Gromov-Witten potential is removed from the introduction of this paper; this now appears in math.QA/0509264
50 pages, 13 figures. Cosmetic and organisational changes in this version. Some references have been added, as has discussion of the non-unital case. Results about topology of moduli space now appear in math.GT/0601130. Discussion of Deligne-Mumford spaces and the Gromov-Witten potential is removed from the introduction of this paper; this now appears in math.QA/0509264