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

SIGMA 18 (2022), 021, 20 pages      arXiv:2101.07490      https://doi.org/10.3842/SIGMA.2022.021

### On the Quantum K-Theory of the Quintic

Stavros Garoufalidis a and Emanuel Scheidegger b
a) International Center for Mathematics, Department of Mathematics, Southern University of Science and Technology, Shenzhen, China
b) Beijing International Center for Mathematical Research, Peking University, Beijing, China

Received October 21, 2021, in final form March 03, 2022; Published online March 21, 2022

Abstract
Quantum K-theory of a smooth projective variety at genus zero is a collection of integers that can be assembled into a generating series $J(Q,q,t)$ that satisfies a system of linear differential equations with respect to $t$ and $q$-difference equations with respect to $Q$. With some mild assumptions on the variety, it is known that the full theory can be reconstructed from its small $J$-function $J(Q,q,0)$ which, in the case of Fano manifolds, is a vector-valued $q$-hypergeometric function. On the other hand, for the quintic 3-fold we formulate an explicit conjecture for the small $J$-function and its small linear $q$-difference equation expressed linearly in terms of the Gopakumar-Vafa invariants. Unlike the case of quantum knot invariants, and the case of Fano manifolds, the coefficients of the small linear $q$-difference equations are not Laurent polynomials, but rather analytic functions in two variables determined linearly by the Gopakumar-Vafa invariants of the quintic. Our conjecture for the small $J$-function agrees with a proposal of Jockers-Mayr.

Key words: quantum K-theory; quantum cohomology; quintic; Calabi-Yau manifolds; Gromov-Witten invariants; Gopakumar-Vafa invariants; $q$-difference equations; $q$-Frobenius method; $J$-function; reconstruction; gauged linear $\sigma$ models; 3d-3d correspondence; Chern-Simons theory; $q$-holonomic functions.

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