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.We view the Moore-Tachikawa conjecture as a first step in systematically assigning new TQFTs to purely Lietheoretic 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 be the Lie algebra of 𝐺.Consider a conjugacy class C of parabolic subalgebras of 𝔤.This class determines partial Grothendieck-Springer resolutions 𝜇 C : 𝔤 C -𝔤 * = 𝔤 and 𝜈 C : 𝐺 C -𝐺.We construct a canonical symplectic groupoid (𝑇 * 𝐺) C - -𝔤 C and quasi-symplectic groupoid D(𝐺) C - -𝐺 C .By considering a Kostant slice Kos ⊆ 𝔤 and Steinberg slice Ste ⊆ 𝐺, we prove that the pairs ( ( (𝑇 * 𝐺) C ) reg - -(𝔤 C ) reg , 𝜇 -1 C (Kos) ) and ( (D(𝐺) C ) reg - -(𝐺 C ) reg , 𝜈 -1 C (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.