On the Quantum K-Theory of the Quintic

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.