The adiabatic-limit correspondence for gauged Witten invariants

Let $\bar{X}_W$ be the symplectic reduction in the geometric phase, let $H_{\rm CR}^*(\bar{X}_W;\Lambda)$ denote its Chen–Ruan cohomology, and let $\langle \alpha_1 \otimes \cdots \otimes \alpha_k; \beta \rangle_{g,k}^{\mathrm{GLSM};\epsilon}$ be the GLSM correlation function with parameter $\epsilon$. Write

$$\pi:\overline{\mathcal M}_{g,k+l}\longrightarrow\overline{\mathcal M}_k$$

for the forgetful map, and let $\langle \alpha_1 \otimes \cdots \otimes \alpha_k \otimes a^{\otimes l};\pi^*\beta\rangle_{g,k+l}^{\mathrm{GW}}$ denote the orbifold Gromov–Witten invariant of $\bar{X}_W$. Adiabatic-limit conjecture. There is a class $a \in H_{\rm CR}^*(\bar{X}_W;\Lambda)$ satisfying

$$\lim_{\epsilon \to 0} \langle \alpha_1 \otimes \cdots \otimes \alpha_k; \beta \rangle_{g,k}^{\mathrm{GLSM};\epsilon} = \sum_{l \geq 0} \langle \alpha_1 \otimes \cdots \otimes \alpha_k \otimes \underbrace{a \otimes \cdots \otimes a}_{l}; \pi^* \beta\rangle_{g,k+l}^{\mathrm{GW}}.$$

The class $a$ is defined by counting affine vortices whose images are contained in $X_W$, analogously to the quantum Kirwan map. The conjecture predicts that the adiabatic limit of gauged Witten solutions yields orbifold Gromov–Witten theory of the symplectic reduction, with affine-vortex contributions encoded by $a$; it is proposed as a correspondence to be studied further, and no resolution is supplied here.