This is joint work with Minghao Wang. In this talk, I will explain the convergence of Feynman graph integrals on closed real-analytic K\"ahler manifolds: Using Getzler’s rescaling technique, the graph integrands extend naturally to the Fulton–MacPherson compactification as forms with divisorial singularities, allowing a rigorous definition as Cauchy principal value integrals. As an application, this yields a mathematical construction of the higher-genus B-model invariants on Calabi–Yau threefolds in the sense of Bershadsky–Cecotti–Ooguri–Vafa (BCOV).

Abstract: The classical Yamabe problem—solved through the work of Yamabe, Trudinger, Aubin, and Schoen—asserts that on any closed smooth connected Riemannian manifold $(M^n,g)$, $n\geq 3$, one can find a metric conformal to $g$ with constant scalar curvature. A fully nonlinear analogue replaces the scalar curvature by symmetric functions of the Schouten tensor. Traditionally, the existing theory has required the scalar curvature to have a fixed sign. In a recent work, we broaden the scope of fully nonlinear Yamabe problem by establishing optimal Liouville-type theorems, local gradient estimates, and new existence and compactness results. Our results allow conformal metrics with scalar curvature of varying signs. A crucial new ingredient in our proofs is our enhanced understanding of solution behavior near isolated singularities. I will also discuss extensions to manifolds with boundary, treating prescribed boundary mean curvature and the boundary curvature arising from the Chern–Gauss–Bonnet formula.

Joint with Analysis seminar at 3pm.