constants, see Remark \[degree one\].

-   When $g=1$ a correction term is needed, to account for the failure of the dilaton equation on $\overline{M}_{1,1}$. Namely, we redefine $$F^\ke_1({\bf t}):= \frac{1}{24}\chi_{\mathrm{top}}({W/\!\!/{{\bf G}}}) \log J_0^\ke+\sum_{(\beta, m)\neq (0,0)} \frac{q^\beta}{m!}{\langle}{\bf t}(\psi_1),\dots, {\bf t}(\psi_m) {\rangle}^\ke_{1,m,\beta},$$ where $\chi_{\mathrm{top}}$ denotes the topological Euler characteristic.

By matching Taylor coefficients in the $t_{ij}$’s in Conjecture \[main conj intro\] we get the equivalent formulation:

\[equivalent\] Let ${W/\!\!/{{\bf G}}}$ be a GIT quotient corresponding to a semi-positive $(W,{{\bf G}},\theta)$. Fix $(g, n)\neq (1,0)$, with $2g-2+n\geq 0$, and a stability parameter $\ke\geq 0+$. Then for arbitrary integers $a_1,\dots a_n\geq 0$ and arbitrary even cohomology classes $\delta_1,\dots \delta_n \in H^*({W/\!\!/{{\bf G}}},\QQ)$ $$\begin{aligned}
\label{general}
&(J_0^\ke(q))^{2g-2+n}\sum_\beta q^\beta \langle \delta_1\psi_1^{a_1},\dots,\delta_n\psi_n^{a_n}
\rangle^\ke_{g,n,\beta}=\\
&\nonumber \sum_\beta q^\beta\sum_{m=0}^\infty \frac{1}{m!}\left{\langle}\delta_1\psi_1^{a_1},\dots,\delta_n\psi_n^{a_n}, \frac{J^\ke_1(q)}{J^\ke_0(q)},\dots, \frac{J^\ke_1(q)}{J^\ke_0(q)}
\right{\rangle}_{g,n+m,\beta}^\infty.\end{aligned}$$ If $(g,n)=(1,0)$ and $\ke$ is arbitrary,

$$\label{unpointed genus 1}
\frac{1}{24}\chi_{\mathrm{top}}({W/\!\!/{{\bf G}}}) \log J_0^\ke+\sum_{\beta\neq 0}{\langle}\;{\rangle}^\ke_{1,0,\beta}=
\sum_{(\beta, m)\neq (0,0)} \frac{q^\beta}{m!}\left{\langle}\frac{J^\ke_1(q)}{J^\ke_0(q)},\dots, \frac{J^\ke_1(q)}{J^\ke_0(q)} \right{\rangle}^\infty_{1,m,\beta}.$$

Note that the string and divisor equations in Gromov-Witten theory allows one to rewrite the right-hand sides of and in terms of invariants with the same insertions as in the left-hand sides.

Results
-------

The first evidence we give for the conjectures is that they hold in genus zero for a large class of targets.

\[genus zero intro\] 