Graph integrals arise in quantum field theory and encode particle interactions. They also play important roles in mathematics, including knot theory, mirror symmetry, and enumerative algebraic geometry. However, their rigorous definition is subtle, since the corresponding integrands are often singular and not Lebesgue integrable.

In joint work with Minghao Wang, we prove a convergence result for Feynman graph integrals on closed real-analytic Kähler manifolds. Using Getzler's rescaling technique, we show that the graph integrands extend to the Fulton--MacPherson compactification as differential forms with mild divisorial-type singularities, which allows us to define the integrals rigorously as Cauchy principal value integrals.

As an application, we construct the higher-genus B-model invariants on Calabi--Yau threefolds predicted by Bershadsky--Cecotti--Ooguri--Vafa. Through mirror symmetry, these invariants are expected to correspond to higher-genus Gromov--Witten invariants, which are notoriously difficult to compute directly.