Grothendieck-Springer resolutions and TQFTs (2504.10285v1)

Abstract: Let $G$ be a connected complex semisimple group with Lie algebra $\mathfrak{g}$ and fixed Kostant slice $\mathrm{Kos}\subseteq\mathfrak{g}^*$. In a previous work, we show that $((T^G)_{\text{reg}}\rightrightarrows\mathfrak{g}^_{\text{reg}},\mathrm{Kos})$ yields the open Moore-Tachikawa TQFT. Morphisms in the image of this TQFT are called open Moore-Tachikawa varieties. By replacing $T^G\rightrightarrows\mathfrak{g}^$ and $\mathrm{Kos}\subseteq\mathfrak{g}^*$ with the double $\mathrm{D}(G)\rightrightarrows G$ and a Steinberg slice $\mathrm{Ste}\subseteq G$, respectively, one obtains quasi-Hamiltonian analogues of the open Moore-Tachikawa TQFT and varieties. We consider a conjugacy class $\mathcal{C}$ of parabolic subalgebras of $\mathfrak{g}$. This class determines partial Grothendieck-Springer resolutions $\mu_{\mathcal{C}}:\mathfrak{g}{\mathcal{C}}\longrightarrow\mathfrak{g}^*=\mathfrak{g}$ and $\nu{\mathcal{C}}:G_{\mathcal{C}}\longrightarrow G$. We construct a canonical symplectic groupoid $(T^*G){\mathcal{C}}\rightrightarrows\mathfrak{g}{\mathcal{C}}$ and quasi-symplectic groupoid $\mathrm{D}(G){\mathcal{C}}\rightrightarrows G{\mathcal{C}}$. In addition, we prove that the pairs $(((T^*G){\mathcal{C}}){\text{reg}}\rightrightarrows(\mathfrak{g}{\mathcal{C}}){\text{reg}},\mu_{\mathcal{C}}^{-1}(\mathrm{Kos}))$ and $((\mathrm{D}(G){\mathcal{C}}){\text{reg}}\rightrightarrows(G_{\mathcal{C}}){\text{reg}},\nu{\mathcal{C}}^{-1}(\mathrm{Ste}))$ determine TQFTs in a $1$-shifted Weinstein symplectic category. Our main result is about the Hamiltonian symplectic varieties arising from the former TQFT; we show that these have canonical Lagrangian relations to the open Moore-Tachikawa varieties. Pertinent specializations of our results to the full Grothendieck-Springer resolution are discussed throughout this manuscript.