Moore-Tachikawa Varieties: Beyond Duality

TL;DR

This paper generalizes Moore-Tachikawa varieties to hyperkähler quotients in 2D TFTs, addressing the challenge of defining a Drinfeld center for composite class S theories and extending duality concepts.

Contribution

It introduces a new framework for Moore-Tachikawa varieties involving hyperkähler quotients, expanding the mathematical structures used in 2D topological field theories.

Findings

Extended Moore-Tachikawa varieties to hyperkähler quotients.

Linked bordism operators to the Drinfeld center in complex theories.

Provided a new perspective on duality beyond traditional cases.

Abstract

We propose a generalisation of the Moore-Tachikawa varieties for the case in which the target category of the 2D TFT is a hyperk$\ddot{\text{a}}$hler quotient. The setup requires generalising the bordism operators of Moore and Segal to the case involving lack of reparametrisation-invariance on the Riemann surface, ultimately enabling to relate this to the issue of defining a Drinfeld center for composite class ${\cal S}$ theories.