The Liu–Xu universal vanishing conjecture for Gromov–Witten invariants

Let $X$ be a smooth projective variety. Given a basis $\{\gamma_a\}$ for $H^*(X,\mathbb Q)$, let $x_i,y_i\in H^*(X)$ and let $k\geq 2g-3+r+s$. Using Gathmann's conventions

$$\langle\tau_{-2}(pt)\rangle_{0,0}^X=1$$

and

$$\langle\tau_m(\gamma_1)\tau_{-1-m}(\gamma_2)\rangle_{0,0}^X=(-1)^{\max(m,-1-m)}\int_X\gamma_1\cdot\gamma_2,\qquad m\in\mathbb Z,$$

with all other Gromov–Witten invariants containing a negative power of a cotangent line class defined to be zero, Liu–Xu's universal vanishing conjecture. The Gromov–Witten potential function satisfies

$$\sum_{g'=0}^g\sum_{j\in\mathbb Z}(-1)^j\langle\langle\tau_j(\gamma_a)\prod_{i=1}^r\tau_{p_i}(x_i)\rangle\rangle^X_{g'}\langle\langle\tau_{k-j}(\gamma^a)\prod_{i=1}^s\tau_{q_i}(y_i)\rangle\rangle^X_{g-g'}=0.$$

Here $j$ runs over all integers. This is a proposed universal system of equations for Gromov–Witten invariants, generalizing vanishing identities for $n$-point functions; the supplied source does not establish the assertion or indicate a resolution status.