This article is cited in
2 papers
On the Quantum K-Theory of the Quintic
Stavros Garoufalidisa,
Emanuel Scheideggerb 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
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.
Keywords:
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.
MSC: 14N35,
53D45,
39A13,
19E20 Received: October 21, 2021; in final form
March 3, 2022; Published online
March 21, 2022
Language: English
DOI:
10.3842/SIGMA.2022.021
Fulltext:
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