String transformation conjecture for quasimap and Gromov–Witten S-operators

Let $E\to E\sslash \textbf{G}$ be the GIT quotient in the paper, let $H^*(E\sslash \textbf{G},\Lambda)$ be its cohomology with coefficient ring $\Lambda$, and let $\gamma\in H^*(E\sslash \textbf{G},\Lambda)$ be of the form $1+O(q)$. Let $S^Q_{\mathbf{t}}(z)$ and $S_{\mathbf{t}}(z)$ denote the quasimap and Gromov–Witten S-operators, respectively, and let $\tau_\gamma(\mathbf{t})$ be the invertible change of variables defined from their first-order expansions. String transformation conjecture. For every such $\gamma$,

$$S^Q_{\tau_\gamma(\mathbf{t})}(z)(\gamma)=S_{\mathbf{t}}(z)(\gamma).$$

This conjecture asserts that the quasimap and Gromov–Witten theories are related by a string-type transformation of the input parameter. The preceding discussion establishes equality modulo $1/z^2$; the full equality is the conjectural strengthening.