Categorical Enumerative Invariants, II: Givental formula

Abstract: To a pair $(A,s)$ consisting of a smooth, cyclic $A_\infty$-algebra $A$ and a splitting $s$ of the Hodge filtration on its Hochschild homology Costello (2005) associates an invariant which conjecturally generalizes the total descendant Gromov-Witten potential of a symplectic manifold. In this paper we give explicit, computable formulas for Costello's invariants, as Feynman sums over partially directed stable graphs. The formulas use in a crucial way the combinatorial string vertices defined earlier by Costello and the authors. Explicit computations elsewhere confirm in many cases the equality of categorical invariants with known Gromov-Witten, Fan-Jarvis-Ruan-Witten, and Bershadsky-Cecotti-Ooguri-Vafa invariants.