The multiple-cover formula for quotient invariants of abelian threefolds

Let $X$ be an abelian threefold, and let $\mathsf{N}_{g,(d_1,d_2,d_3)}$ denote its reduced quotient Gromov–Witten invariant for a curve class of type $(d_1,d_2,d_3)$. Define

$$\mathsf{n}(d_1,d_2,d_3,k)=\sum_{\delta\mid G(d_1,d_2,d_3,k)}\delta^2,$$

where

$$G(d_1,d_2,d_3,k)=\gcd\left(k,d_1,d_2,d_3,\frac{d_1d_2}{k},\frac{d_1d_3}{k},\frac{d_2d_3}{k},\frac{d_1d_2d_3}{k^2}\right).$$

Multiple-cover formula. For all $g\geq2$, $d_1,d_2>0$, and $d_3\geq0$,

$$\mathsf{N}_{g,(d_1,d_2,d_3)}=\sum_k\mathsf{n}(d_1,d_2,d_3,k)k^{2g-3}\mathsf{N}_{g,\left(1,1,\frac{d_1d_2d_3}{k^2}\right)},$$

where $k$ runs over all divisors of $\gcd(d_1d_2,d_1d_3,d_2d_3)$ such that $k^2$ divides $d_1d_2d_3$.

This conjecture gives a uniform imprimitive-class formula, reducing all quotient invariants to classes of type $(1,1,*)$. Its shape is motivated by earlier physics predictions, but the source notes that those predictions do not match these invariants.