The Gromov–Witten invariant energy-bound conjecture

Let $(X,\omega)$ be a compact symplectic manifold and let $g\in\mathbb Z$. For fixed classes $H_1,\ldots,H_k\in H^*(X)$, let $\langle\tau_{b_1}H_{c_1},\ldots,\tau_{b_N}H_{c_N}\rangle^X_{g,\beta}$ denote the descendant Gromov–Witten invariant, where $b_s\in\mathbb Z^{\ge0}$ and $c_s\in\{1,\ldots,k\}$. The Gromov–Witten invariant energy-bound conjecture. There exists $C_{X,g}\in\mathbb R^+$ such that

$$\left|\frac{b_1!\,\langle\tau_{b_1}H_{c_1},\ldots,b_N!\,\tau_{b_N}H_{c_N}\rangle^X_{g,\beta}}{N!}\right|\le C_{X,g}^{\langle\omega,\beta\rangle+N}$$

for all $\beta\in H_2(X)$, $N,b_s\in\mathbb Z^{\ge0}$, and $c_s\in\{1,\ldots,k\}$. Such bounds would control descendant invariants uniformly in the curve energy and number of insertions. The source notes that the conjecture is known in genus zero for all projective spaces and in arbitrary genus in substantial settings, but remains open in the stated generality.