On the Quantum K-Theory of the Quintic - Garoufalidis, Stavros; Scheidegger, Emanuel (Symmetry, Integrability and Geometry: Methods and Applications, 2022)
Quantum K-theory of a smooth projective variety at genus zero is a collection of integers that can be assembled into a generating series (, , ) that satisfies a system of linear differential equations with respect to and -difference equations with respect to . With some mild assumptions on the variety, it is known that the full theory can be reconstructed from its small -function (, , 0), which, in the case of Fano manifolds, is a vector-valued -hypergeometric function. On the other hand, for the quintic 3-fold, we formulate an explicit conjecture for the small -function and its small linear -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 -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 -function agrees with a proposal of Jockers-Mayr.