Effective Categorical Enumerative Invariants

We introduce enumerative invariants $F_{g,n}$ $(g\geq0$, $n \geq 1)$ associated to a cyclic $A_\infty$ algebra and a splitting of its non-commutative Hodge filtration. These invariants are defined by explicitly computable Feynman sums, and encode the same information as Costello's partition function of the corresponding field theory. Our invariants are stable under Morita equivalence, and therefore can be associated to a Calabi-Yau category with splitting data. This justifies the name categorical enumerative invariants (CEI) that we use for them. CEI conjecturally generalize all known enumerative invariants in symplectic geometry, complex geometry, and singularity theory. They also provide a framework for stating enumerative mirror symmetry predictions in arbitrary genus, whenever homological mirror symmetry holds.