Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 089, 6 pages      arXiv:2102.02360      https://doi.org/10.3842/SIGMA.2024.089
Contribution to the Special Issue on Basic Hypergeometric Series Associated with Root Systems and Applications in honor of Stephen C. Milne

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

Christian Krattenthaler
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria

Received February 03, 2024, in final form October 07, 2024; Published online October 10, 2024

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-Saalschütz summation formula from the theory of basic hypergeometric series.

Key words: Looijenga pairs; log Calabi-Yau surfaces; Gromov-Witten invariants; $q$-binomial coefficients; basic hypergeometric series; Pfaff-Saalschütz summation formula.

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References

1. Bousseau P., Brini A., van Garrel M., Stable maps to Looijenga pairs, Geom. Topol. 28 (2024), 393-496, arXiv:2011.08830.
2. Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Encyclopedia Math. Appl., Vol. 96, Cambridge University Press, Cambridge, 2004.