Geodesic
On the Quantum K-Theory of the Quintic
Symmetry, integrability and geometry: methods and applications, Tome 18 (2022) Cet article a éte moissonné depuis la source Math-Net.Ru

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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, $q$-difference equations, $q$-Frobenius method, $J$-function, gauged linear $\sigma$ models, 3d-3d correspondence, Chern–Simons theory, $q$-holonomic functions.
Mots-clés : Gopakumar–Vafa invariants, reconstruction 
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Stavros Garoufalidis; Emanuel Scheidegger. On the Quantum K-Theory of the Quintic. Symmetry, integrability and geometry: methods and applications, Tome 18 (2022). http://geodesic.mathdoc.fr/item/SIGMA_2022_18_a20/

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