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The combinatorial formula for open gravitational descendents

Ran J Tessler

Geometry & Topology 27 (2023) 2497–2648
Abstract

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 S2n1–bundles.

Buryak and Tessler (2017) proved the conjecture of Pandharipande, Solomon and Tessler based on the work presented here.

Keywords
open Gromov Witten, open gravitational descendents, Kontsevich's matrix model, open KdV, open Witten conjecture
Mathematical Subject Classification 2010
Primary: 53D45
References
Publication
Received: 3 December 2018
Revised: 25 December 2021
Accepted: 15 January 2022
Published: 19 September 2023
Proposed: Jim Bryan
Seconded: Richard P Thomas, Mark Gross
Authors
Ran J Tessler
Department of Mathematics
Weizmann Institute of Science
Rehovot
Israel

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