Virtual dimension formula for the moduli of stable supermaps

Let ${\mathcal Y}$ be a projective smooth superscheme of dimension $r|s$, with bosonic reduction $Y$, and let $\beta_0\in A_1(Y)$. Let ${\mathfrak S}{\mathfrak M}_{g,{\mathfrak n}_{NS},{\mathfrak n}_{RR}}({\spacecal Y},\beta)$ be the moduli superstack of stable supermaps into ${\spacecal Y}$ of degree $\beta=(1-\Pi)\beta_0$, with ${\mathfrak n}_{NS}$ Neveu–Schwarz punctures and ${\mathfrak n}_{RR}$ Ramond–Ramond punctures. Write ${\mathcal F}_{\spacecal Y}=\Pi{\mathcal J}_{\spacecal Y}/{\mathcal J}_{\spacecal Y}^2$ on $Y$. Virtual dimension formula. The virtual dimension is

$$\begin{aligned} \operatorname{vdim}\, {\mathfrak S}{\mathfrak M}_{g,{\mathfrak n}_{NS},{\mathfrak n}_{RR}}({\spacecal Y},\beta) ={}&(r-3)(1-g)+{\mathfrak n}_{NS}+{\mathfrak n}_{RR}(1+s/2)\\ &-\Pi\left[(1-g)(s-2)+{\mathfrak n}_{NS}+({\mathfrak n}_{RR}/2)(r+1)\right]\\ &+(1-\Pi)\int_{\beta_0}\left[\operatorname{ch}_1(T_Y)-\operatorname{ch}_1({\mathcal F}_{\spacecal Y})\right]. \end{aligned}$$

The formula is obtained by a formal application of super Grothendieck–Riemann–Roch. A rigorous justification via a perfect obstruction theory and a deformation theory for the moduli superstack is deferred to future work, so the assertion is presently open.