A real analogue of the Moore--Tachikawa category

TL;DR

This paper constructs a real analogue of the Moore--Tachikawa category, providing a rigorous proof that it forms a well-defined category, inspired by string theory and supersymmetric quantum field theories.

Contribution

It defines a real version of the target category for the Moore--Tachikawa conjecture and proves its categorical structure rigorously.

Findings

Successfully constructed the real analogue category.

Proved the categorical properties of the new construction.

Provides a foundation for further mathematical and physical applications.

Abstract

For each complex semisimple group $G_{\mathbb{C}}$, Moore and Tachikawa conjectured the existence of a certain two-dimensional topological quantum field theory $\eta_{G_{\mathbb{C}}} : \mathrm{Cob}_2 \to \mathrm{MT}$ whose target category has complex Lie groups as objects and holomorphic symplectic varieties with Hamiltonian actions of the groups as morphisms. The conjecture is motivated by string theory, where $\eta_{G_{\mathbb{C}}}$ is obtained by taking the Higgs branch of some supersymmetric quantum field theories (called theories of class $S$) depending on $G_{\mathbb{C}}$ and a Riemann surface. The goal of this paper is to define a real analogue $\mathrm{MT}_{\mathbb{R}}$ of the target category $\mathrm{MT}$ and to rigorously prove that it is a category.