Towards Super Teichmüller Spin TQFT

Abstract

The quantization of the Teichmüller theory has led to the formulation of the so-called Teichmüller TQFT for 3-manifolds. In this paper we initiate the study of "supersymmetrization" of the Teichmüller TQFT, which we call the super Teichmüller spin TQFT. We obtain concrete expressions for the partition functions of the super Teichmüller spin TQFT for a class of spin 3-manifold geometries, by taking advantage of the recent results on the quantization of the super Teichmüller theory. We then compute the perturbative expansions of the partition functions, to obtain perturbative invariants of spin 3-manifolds. We also comment on the relations of the super Teichmüller spin TQFT to 3dimensional Chern-Simons theories with complex gauge groups, and to a class of 3d N = 2 theories as arising from the compactifications of the M5-branes.

Solving 3d gravity with Virasoro TQFT

“…There is a conceptual difficulty in the definition of W N conformal blocks since not all descendant correlation functions can be reduced to primary correlation functions [140]. It stands to reason that at least in the case without higher spin fields one can similarly define a (spin) TQFT based on the crossing symmetry of super Virasoro blocks, see [141][142][143] for progress in this direction. Except for the N = 1 Virasoro case, the existence of this TQFT has not been established.…”