On the Quantum K-Theory of the Quintic

Symmetry Integrability and Geometry-Methods and Applications

article

Stavros Garoufalidis¹  Emanuel Scheidegger² 

[1] International Center for Mathematics, Department of Mathematics, Southern University of Science and Technology;Beijing International Center for Mathematical Research, Peking University

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.;   

DOI  :  10.3842/SIGMA.2022.021

来源: National Academy of Science of Ukraine

【 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.

【 授权许可】

Unknown