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On the Quantum K-Theory of the Quintic
Algebraic Geometry · Submitted 19 Jan 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 $Q$. | background 1.00 | 0.06 | 0.04 | asserted 1.00 |
| s1 | 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. | background 1.00 | 0.05 | 0.04 | asserted 0.99 |
| s2 | 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. | contribution 0.94 | 0.86 | 0.66 | speculative 0.91 |
| s3 | 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. | result 0.79 | 0.66 | 0.19 | asserted 1.00 |
| s4 | Our conjecture for the small $J$-function agrees with a proposal of Jockers-Mayr. | result 0.87 | 0.74 | 0.25 | asserted 0.97 |
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The paper
Original 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.
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How the literature uses it
4 arXiv papers cite this in 4 sentences
Papers citing it · 2 sentences without a confident label
Cites it as background 2 papers
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See [ref] for additional discussion on $q$-deformed Picard–Fuchs equation.
Difference Equation for Quintic 3-Fold · Solutions at $Q=0$ · 0.98
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And see [ref] for additional discussion on q-deformed Picard-Fuchs equation and Frobenius method for quintic 3-fold.
Quantum $K$-theory and $q$-Difference equations · Difference equation for quintic 3-fold · 0.93
What it cites · 39 sentences without a confident label
Uses the method or tool of 2 papers
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Having computed the fundamental matrix $T$ [ref], we use [ref] , with $P^-1 q^Q ∂_Q$ replaced by $1-(1-x)E$ to deduce the small A-matrix [ref].
Reconstruction and Convergence in Quantum $K$-Theory via Difference Equations · The small $q$-difference equation of the quintic · 0.96
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In that case, we need to apply the Birkhoff factorization [ref] to the matrix [equation] where the $q$-coefficients of the entries of $T$ are in $_-(q)$ and of $U$ are in $_+(q)$ (compare also with Lemma 3.3 of [ref] ).
Explicit reconstruction in quantum cohomology and K-theory · The small $q$-difference equation of the quintic · 0.98
Uses data or values from 2 papers
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Such a value was given by Givental in [ref] and by Tonita in [ref] who proved that if $J_d$ denotes the coefficient of $Q^d$ in $J_^4(Q,q,0)$ given in [ref], then [equation] lies on the K-theoretic Lagrangian cone of the quintic $X$.
Permutation-equivariant quantum K-theory V. Toric $q$-hypergeometric functions · A special value for the $J$-function of the quintic · 0.89
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Such a value was given by Givental in [ref] and by Tonita in [ref] who proved that if $J_d$ denotes the coefficient of $Q^d$ in $J_^4(Q,q,0)$ given in [ref], then [equation] lies on the K-theoretic Lagrangian cone of the quintic $X$.
Twisted K-theoretic Gromov-Witten invariants · A special value for the $J$-function of the quintic · 0.89
Compares with 2 papers
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Their conjecture [ref] can now be stated (in the case of the quintic) as follows [ref] : [equation] where $p_2$, $p_1,1$, $ℱ$, $ℱ_1$ are certain explicitly given functions of $t$ and the Gopakumar–Vafa invariants $_d$, [ref] .
Quantum K-Theory of Calabi-Yau Manifolds · A comparison with Jockers–Mayr · 0.96
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Our conjecture also agrees with the results of Jockers–Mayr presented in [ref] .
A 3d Gauge Theory/Quantum K-Theory Correspondence · The small $J$-function for the quintic · 0.88
Cites as support 1 paper
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In [ref] this “coincidence” is proven in abstractly.
The Hirzebruch--Riemann--Roch theorem in true genus-0 quantum K-theory · The small $J$-function for the quintic · 0.99
Cites as background 18 papers
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Among them is the primary potential [equation] (where the summation is over $d ∈Eff(X)$ and $n ≥0$), the $J$-function [equation] (where ${Φ_}$ is a basis for $K(X)$ for $=0,…,N$ with $Φ_0=1$), and the $T$ matrix $T_,(Q,q,t) ∈(K(X)) ⊗(q)[[Q,t]]$ and its inverse, whose definition we omit but may be found in [ref] .
Reconstruction and Convergence in Quantum $K$-Theory via Difference Equations · Reconstruction theorems for quantum K-theory · 0.97
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Our choice of generating series will be the so-called small $J$-function [equation] (with the notation of Section [ref]), which determines the genus 0 quantum K-theory $X$, i.e., the integers [ref] [ref] with $g=0$, as well as the genus 0 permutation-equivariant quantum K-theory $X$ [ref] (when $K(X)$ is generated by line bundles).
Permutation-equivariant quantum K-theory VIII. Explicit reconstruction · Quantum K-theory, the small $J$-function and its $q$-difference equation · 0.87
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The K-theoretic Gromov–Witten invariants of a compact Kähler manifold $X$ (often omitted from the notation) is a collection of integers (see [ref] ) [equation] defined for vector bundles $E_1,…,E_n$ on $X$ and nonnegative integers $k_1,…,k_n$ as the holomorphic Euler characteristic of $^vir ⊗( ⊗_i=1^n ev_i^*(E_i) ⊗L_i^k_1)$ over the moduli space $^X,d_g,n$ of genus $g$ degree $d$ stable maps to $X$ with $n$ marked points.
The Hirzebruch--Riemann--Roch theorem in true genus-0 quantum K-theory · Quantum K-theory, the small $J$-function and its $q$-difference equation · 0.94
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