Grothendieck–Springer resolutions and TQFTs

Abstract The Moore–Tachikawa conjecture is that each connected complex semisimple group G $G$ upper G determines a two-dimensional topological quantum field theory (TQFT) in a category of Hamiltonian symplectic varieties. We 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. In more detail, let g $\mathfrak {g}$ German g be the Lie algebra of G $G$ upper G . Consider a conjugacy class C $\mathcal {C}$ script upper C of parabolic subalgebras of g $\mathfrak {g}$ German g . This class determines partial Grothendieck–Springer resolutions μ C : g C ⟶ g ∗ = g $\mu _{\mathcal {C}}:\mathfrak {g}_{\mathcal {C}}\longrightarrow \mathfrak {g}^*=\mathfrak {g}$ mu Subscript script upper C Baseline colon German g Subscript script upper C Baseline long right arrow German g Superscript asterisk Baseline equals German g and ν C : G C