The combinatorial formula for open gravitational descendents

Tessler, Ran J

doi:10.2140/gt.2023.27.2497

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The combinatorial formula for open gravitational descendents

Ran J Tessler

Geometry & Topology 27 (2023) 2497–2648

DOI: 10.2140/gt.2023.27.2497

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 $S^{2 n - 1}$ –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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Volume 27, issue 7 (2023)