Topological conformal field theories and Calabi–Yau categories

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∞ category, which plays the role of the target in ordinary Gromov–Witten theory. When we use an appropriate A∞ 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 open-closed Gromov–Witten theory can be constructed 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.