The abelian/nonabelian correspondence for equivariant Frobenius structures

Let $P$ be the formal ${\bf S}$-equivariant Frobenius manifold over $N(X//{\bf G})\otimes{\mathbb C}[\lambda]$ defined by the genus-zero ${\bf S}$-equivariant Gromov–Witten theory of $X//{\bf G}$, with flat coordinates $t_i$. Let $N$ be the formal scheme carrying the induced Frobenius structure from the abelian quotient, with flat coordinates $s_i$, and let $\xi_i$ and $F'$ denote its corresponding basis fields and potential.

Equivariant abelian/nonabelian correspondence. Let $\varphi:P\longrightarrow N$ be the isomorphism of formal schemes over $N(X//{\bf G})\otimes{\mathbb C}[\lambda]$ defined by $\varphi^*(s_i)=t_i$. Then $\varphi$ induces an isomorphism of formal ${\bf S}$-equivariant Frobenius structures such that $\varphi^*(\xi_i)=\sigma_i$ and $\varphi^*F'=F^{X//{\bf G},{\bf S}}$ up to quadratic terms.

This is the equivariant formulation of the abelian/nonabelian correspondence: the Frobenius structure induced from the torus quotient is identified with that arising from the equivariant Gromov–Witten theory of the nonabelian quotient.