Eynard-Orantin Topological Recursion, Moduli-Space Virasoro Constraints, and Semisimple CohFTs

We use Virasoro constraints to prove Eynard’s intersection-number formula for Eynard-Orantin topological recursion with several simple ramification points. The key construction couples the higher Weil-Petersson partition functions associated with the individual ramification points through a quadratic differential operator determined by the Bergman kernel. We identify this operator with the quadratic part of Givental’s quantized $R$-matrix, thereby obtaining a Virasoro-theoretic proof of the DOSS identification of topological recursion with Givental’s formula for semisimple cohomological field theories. Furthermore, while the DOSS identification fixes the distinguished $y$-function, we keep the Bergman kernel/$R$-matrix fixed and allow $y$ to vary, showing that this amounts to shifting the evaluation point of the corresponding ancestor potential.