The quintic quantum K-theory formula in terms of Gopakumar–Vafa invariants

Let $X$ be the quintic threefold, with rational K-theory

$$K(X)=\mathbb Q[x]/(x^4)$$

and basis $\Phi_\alpha=x^\alpha$ for $\alpha=0,1,2,3$. Let $Q$ and $q$ be the degree and loop variables, let $J(Q,q,0)$ be the small $J$-function of the quintic, and let $\operatorname{GV}_d$ denote its Gopakumar–Vafa invariants. Define $a(d,r,q)$ and $b(d,r,q)$ by

$$5a(d,r,q)=\frac{dr}{1-q}+\frac{dq}{(1-q)^2},$$ $$5b(d,r,q)=\frac{rd+r^2-d}{1-q}+\frac{d}{(1-q)^2}-\frac{q+q^2}{(1-q)^3}.$$

The quintic quantum K-theory conjecture. The small $J$-function of the quintic is expressed linearly in terms of the GV-invariants by

$$\frac{1}{1-q}J(Q,q,0)=1+x^2\sum_{d,r\geq1}a(d,r,q^r)\operatorname{GV}_dQ^{dr}+x^3\sum_{d,r\geq1}b(d,r,q^r)\operatorname{GV}_dQ^{dr}.$$

This gives an explicit genus-zero formula for the abstract expression of quantum K-theoretic Gromov–Witten invariants in terms of cohomological invariants established by Givental and Tonita. The conjecture is presented as a proposed reconstruction of the quintic’s quantum K-theory from its GV invariants.