Gerbe duality conjecture for Gromov–Witten theory

Let $G$ be a finite group, let $Q$ act on the finite set of isomorphism classes of irreducible $G$-representations, and let $\mathcal{Y}$ be a gerbe with dual $\widehat{\mathcal{Y}}$ equipped with its canonical $\mathbb{C}^*$-gerbe $c$. Write $GW_g(\mathcal{Y})$ for the genus $g$ Gromov–Witten generating function, and $GW_g(\widehat{\mathcal{Y}},c)$ for the corresponding $c$-twisted theory. Gerbe duality conjecture. As generating functions, the genus $g$ Gromov–Witten theory of $\mathcal{Y}$ is equal to the genus $g$ Gromov–Witten theory of $(\widehat{\mathcal{Y}},c)$:

$$GW_g(\mathcal{Y})=GW_g(\widehat{\mathcal{Y}},c).$$

This conjecture expresses the proposed equivalence between the Gromov–Witten theory of a gerbe and the $c$-twisted theory of its dual. It has been proved in increasing generality, including for banded $G$-gerbes, so the conjecture is solved.