Quantum Field Theory and the Space of All Lie Algebras

TL;DR

This paper introduces a novel class of topological invariants derived from quantum field theory to study the complex space of all n-dimensional Lie algebras, inspired by the Witten index generalizations.

Contribution

It proposes a new quantum field theory-based approach to analyze the topology of the space of Lie algebras, extending existing methods with invariants applicable in any dimension.

Findings

New invariants for Lie algebra classification

Applicable in any dimension

Inspired by Witten index generalizations

Abstract

The space M_n of all isomorphism classes of n-dimensional Lie algebras over a field k has a natural non-Hausdorff topology, induced from the Segal topology by the action of GL(n). One way of studying this complicated space is by topological invariants. In this article we propose a new class of invariants coming from quantum field theory, valid in any dimension, inspired by Jaffe's study of generalizations of the Witten index.