The combinatorial formula for open gravitational descendents

Pandharipande, Solomon and Tessler (2014) defined descendent integrals on the moduli space of Riemann surfaces with boundary, and conjectured that the generating function of these integrals satisfies the open KdV equations. We prove a formula for these integrals in terms of sums of Feynman diagrams. This formula is a generalization of the combinatorial formula of Kontsevich (1992) to the open setting. In order to overcome the main challenges of the open setting, which are orientation questions and the existence of boundary and boundary conditions, new techniques are developed. These techniques, which are interesting in their own right, include a characterization of graded spin structure in terms of open and nodal Kasteleyn orientations, and a new formula for the angular form of S2n-1-bundles. Buryak and Tessler (2017) proved the conjecture of Pandharipande, Solomon and Tessler based on the work presented here.