On the Quantum K-Theory of the Quintic

[摘要] 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.

[关键词] 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.