B-model Categorical Enumerative Invariants and holomorphic anomaly equations

In this paper, we study the B-model categorical enumerative invariants (CEI) associated with derived categories of coherent sheaves on smooth projective Calabi-Yau $3$-folds. We first prove the analogs of the dilaton, string, and divisor equations of CEI in a general context. Then we use these equations and the Givental quantization formula to prove that the B-model CEI for any miniversal family of smooth projective Calabi-Yau $3$-folds satisfies the holomorphic anomaly equations introduced by Bershadsky-Cecotti-Ooguri-Vafa. This provides strong evidence that CEI may be taken as a rigorous mathematical definition of the B-model topological string partition function.