Topological Conformal Field Theories and Calabi-yau Categories

This paper concerns open, closed, and open-closed topological conformal field theories. We show that the category of open topological conformal field theories, at all genera, is homotopy equivalent to a category of Calabi-Yau A∞ categories. For each such, we show that there is a universal closed TCFT, which is the initial element in the category of compatible open-closed TCFTs. The homology of this universal closed TCFT is calculated to be the Hochschild homology of the CalabiYau A∞ category. As a corollary, the TCFT part of the higher-genus B model is constructed. This result is also used to relate the Gromov-Witten invariants and the Fukaya category of a symplectic manifold, assuming the existence of open-closed Gromov-Witten invariants.