Proof of Two Multivariate $q$-Binomial Sums Arising in Gromov-Witten Theory

TL;DR

This paper proves two complex multivariate $q$-binomial identities related to Gromov-Witten invariants of log Calabi-Yau surfaces, using Jackson's $q$-Pfaff-Saalschütz summation formula.

Contribution

It establishes two previously conjectured identities connecting $q$-binomial sums with Gromov-Witten invariants, expanding mathematical understanding in algebraic geometry and hypergeometric series.

Findings

Proved two conjectured multivariate $q$-binomial identities.

Connected identities to Gromov-Witten invariants of specific surfaces.

Utilized Jackson's $q$-Pfaff-Saalschütz summation formula.

Abstract

We prove two multivariate $q$-binomial identities conjectured by Bousseau, Brini and van Garrel [Geom. Topol. 28 (2024), 393-496, arXiv:2011.08830] which give generating series for Gromov-Witten invariants of two specific log Calabi-Yau surfaces. The key identity in all the proofs is Jackson's $q$-analogue of the Pfaff-Saalsch\"utz summation formula from the theory of basic hypergeometric series.