BCOV Feynman rule of the quintic threefold

We provide a proof for the Bershadsky–Cecotti–Ooguri–Vafa (BCOV) Feynman rule conjecture, which applies to all genus Gromov–Witten potentials of the quintic threefold. This is achieved by extracting an equivalent A-model Feynman rule from the N-Mixed-Spin-P field (NMSP) theory. For genus greater than 1, the Feynman rule effectively computes the genus g Gromov–Witten potential using 3g−3 initial data. As a result, the Yamaguchi–Yau equations and the genus 1 and 2 mirror formulas are confirmed. A notable discovery is that the propagators in the BCOV theory can be interpreted in the A-model as enumerative counts of chains of rational curves in the NMSP master space outside the quintic threefold. In this way these propagators afford direct visibility of BCOV Feynman structures in the moduli of NMSP fields. Furthermore, we demonstrate that the set of all propagators capable of yielding Feynman rules forms a torsor under a “gauge” group action, which includes all BCOV’s “gauge” freedom of propagators.