Enumerative mirror symmetry formulas for rank-two quasimap invariants

Let $g\geq 2$ be the genus, let ${\mathsf{r}}=2$ be the rank, and let ${\mathsf{d}}$ be the degree. Define the generating series $\check{\mathsf{QM}}^i_{\mathsf{d}}(q)$ and $\hat{\mathsf{QM}}^i_{\mathsf{d}}(q)$, for $i\in\{0,1\}$, and the functions ${\mathsf{U}}_1(q)$, ${\mathsf{U}}_2(q)$, and ${\mathsf{U}}_3(q)$ as above. The rank-two enumerative mirror symmetry conjecture. If $g\geq 2$ and ${\mathsf{r}}=2$, then

$$\begin{aligned} \check{\mathsf{QM}}^0_{\mathsf{d}}(q)&=(-1)^{\mathsf{d}}(2-2g)2^{4g-1}{\mathsf{U}}_1(q) \\ \check{\mathsf{QM}}^1_{\mathsf{d}}(q)&=(2-2g)2^{2g-1}\bigl({\mathsf{U}}_2(q)+(-1)^{\mathsf{d}}{\mathsf{U}}_3(q)\bigr) \\ \hat{\mathsf{QM}}^0_{\mathsf{d}}(q)&=(-1)^{{\mathsf{d}}}(2-2g)2^{2g-1}{\mathsf{U}}_1(q) \\ \hat{\mathsf{QM}}^1_{\mathsf{d}}(q)&=(2-2g)\bigl(2^{4g-1}{\mathsf{U}}_2(q)+(-1)^{{\mathsf{d}}}2^{2g-1}{\mathsf{U}}_3(q)\bigr). \end{aligned}$$

These formulas are intended to completely determine the genus-one theory of the relevant moduli spaces and are obtained by translating physical calculations into mathematical predictions. The source presents them as conjectural formulas for rank two; analogous conjectures are referenced for arbitrary prime rank.