2021/01/31 by Stavros Garoufalidis, Emanuel Scheidegger
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Cohomology #Conjecture #Differential equation #Generating function #Linear differential equation #Mathematical analysis #Mathematics #Nonlinear Waves and Solitons #Nonlinear system #Physics #Pure mathematics #Quantum cohomology #Quantum mechanics #Quintic function #hep-th #math.AG #math.GT
paper · pdf · doi:10.3842/sigma.2022.021
published as SIGMA 18 (2022), 021, 20 pages
arxiv created 2022/03/21 · openalex publication_date 2022/03/21 · arxiv updated 2022/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/22
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.