Grothendieck-Springer resolutions and TQFTs

Abstract

The Moore-Tachikawa conjecture is that each connected complex semisimple group $G$ determines a two-dimensional TQFT in a category of Hamiltonian symplectic varieties. While it would be worthwhile to prove this conjecture outright, our objectives are drastically different. We instead view the Moore--Tachikawa conjecture as a first step in systematically assigning new TQFTs to purely Lie-theoretic data. At the same time, one should expect these new TQFTs to bear a close relation to those conjectured by Moore and Tachikawa. Our manuscript aims to integrate these two points of view. Let $\mathfrak{g}$ be the Lie algebra of $G$. Consider a conjugacy class $\mathcal{C}$ of parabolic subalgebras of $\mathfrak{g}$. This class determines partial Grothendieck--Springer resolutions $\hat{I}{¼}_{\mathcal{C}}:{\mathfrak{g}}_{\mathcal{C}}⟶{\mathfrak{g}}^{\ast }=\mathfrak{g}$ and $\hat{I}{½}_{\mathcal{C}}:G_{\mathcal{C}}⟶G$. We construct a canonical symplectic groupoid $(T^{\ast }G{)}_{\mathcal{C}}\begin{array}{c}⟶\\ ⟶\end{array}{\mathfrak{g}}_{\mathcal{C}}$ and quasi-symplectic groupoid $\mathrm{D}(G{)}_{\mathcal{C}}\begin{array}{c}⟶\\ ⟶\end{array}{G}_{\mathcal{C}}$. By considering a Kostant slice $\mathrm{Kos}\subseteq \mathfrak{g}$ and Steinberg slice $\mathrm{Ste}\subseteq G$, we prove that the pairs $(((T^{\ast }G{)}_{\mathcal{C}}{)}_{\text{reg}}\begin{array}{c}⟶\\ ⟶\end{array}({\mathfrak{g}}_{\mathcal{C}}{)}_{\text{reg}},\hat{I}{¼}_{\mathcal{C}}^{-1}(\mathrm{Kos}))$ and $((\mathrm{D}(G{)}_{\mathcal{C}}{)}_{\text{reg}}\begin{array}{c}⟶\\ ⟶\end{array}(G_{\mathcal{C}}{)}_{\text{reg}},\hat{I}{½}_{\mathcal{C}}^{-1}(\mathrm{Ste}))$ determine new and explicit TQFTs in a $1$-shifted Weinstein symplectic category. We then show that certain symplectic varieties arising from our new TQFTs have canonical Lagrangian relations to the open Moore-Tachikawa varieties.