The Higgs-branch conjecture for Sicilian theories

Let $G$ be a simple group, let $g$ be the genus of a Riemann surface, and let $e^{(1)},\dots,e^{(p)}$ be nilpotent labels for its punctures. Write $\mathcal{M}(G^{\vee},V)$ for the Coulomb branch associated with the dual group $G^{\vee}$ and representation $V$, let $W$ be the Weyl group, and let $\mathcal{M}_e$ denote the Higgs branch associated with a puncture labeled by $e$. For an unpunctured surface, set

$$\mathcal{M}_{g-\operatorname{loop}}^{(G)}=\mathcal{M}(G^{\vee},(\mathfrak{g}^{\vee})^{\oplus g})\cong \mathcal{M}(T^{\vee},(\mathfrak{g}^{\vee})^{\oplus g-1})/W.$$

For a puncture labeled by $e$, let $\mathcal{S}_e$ be the Slodowy slice for the corresponding nilpotent element, and let $\mathcal{S}_{\mathsf{reg}}$ be the regular Kostant slice. The Higgs-branch conjecture for Sicilian theories. The Higgs branches of Sicilian theories are given by the following constructions: for an unpunctured surface, the Higgs branch is $\mathcal{M}_{g-\operatorname{loop}}^{(G)}$; for a genus-zero surface with one puncture labeled by $e$, it is

$$\mathcal{M}_e=\mu^{-1}(\mathcal{S}_{\mathsf{reg}}\times\mathcal{S}_e)\subseteq T^*G,$$

where $\mu\colon T^*G\to\mathfrak{g}^*\times\mathfrak{g}^*$ is the moment map for the left and right actions of $G$; and for a genus-$g$ surface with punctures labeled by $e^{(1)},\dots,e^{(p)}$, it is isomorphic to

$$\mathcal{M}_{g-\operatorname{loop}}^{(G)}\times_{\mathfrak{c}_{G^{\vee}}}\mathcal{M}_{e^{(1)}}\times_{\mathfrak{c}_{G^{\vee}}}\cdots\times_{\mathfrak{c}_{G^{\vee}}}\mathcal{M}_{e^{(p)}}/\mathcal{M}(G^{\vee})^{r+1}_0,$$

where

$$\mathcal{M}(G^{\vee})^{r+1}_0=\{(a_1,\dots,a_{r+1})\in\mathcal{M}(G^{\vee})\times_{\mathfrak{c}_{G^{\vee}}}\cdots\times_{\mathfrak{c}_{G^{\vee}}}\mathcal{M}(G^{\vee})\mid a_1\cdots a_{r+1}=1\}.$$

The action on $\mathcal{M}_{g-\operatorname{loop}}^{(G)}$ is induced from $\mathcal{M}(G^{\vee})_{\mathfrak{t}^{\vee}}\to\mathcal{M}(T^{\vee})\cong\mathfrak{t}^{\vee}\times T$, while the action on $\mathcal{M}_e$ is by left multiplication under the identification of $\mathcal{M}(G^{\vee})$ with the regular centralizers of $G$. These proposals describe how the Higgs branches assigned to the basic pieces of a Sicilian theory are assembled by connected sum; the individual constructions have appeared in earlier work, while the asserted general assembly is the conjectural part.