关于五次曲线的量子K理论
On the Quantum K-Theory of the Quintic
摘要：
亏格为零的光滑射影簇的量子K-理论是一个整数集合，这些整数可以被组装成一个生成序列$J(Q,q,t)$，该生成序列满足关于$t$的线性微分方程组和关于$Q$的$q$ -差分方程组。通过对簇的一些温和的假设，已知完整的理论可以从它的小$J$ -函数$J(Q,q,0)$重建，在Fano流形的情况下，它是向量值$q$ -超几何函数。另一方面，对于五次3重，我们制定了一个明确的猜想，小$J$ -函数和它的小线性$q$ -差分方程的Gopakumar-Vafa不变量线性表示。与量子纽结不变量和法诺流形的情况不同，小线性$q$ -差分方程的系数不是劳伦多项式，而是由五次函数的Gopakumar-Vafa不变量线性确定的二元解析函数。我们对小$J$ -函数的猜想与Jockers-Mayr的建议一致。
英文摘要：
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.
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