Quantization of algebraic invariants through Topological Quantum Field Theories

TL;DR

This paper explores how algebraic invariants can be quantized using Topological Quantum Field Theories, establishing conditions for quantizability and demonstrating limitations with specific invariants.

Contribution

It introduces necessary and partial sufficient conditions for quantizing algebraic invariants via TQFTs, and shows certain invariants are not quantizable.

Findings

Necessary conditions for quantizability based on Euler characteristics.

Partial sufficient conditions using almost-TQFTs and almost-Frobenius algebras.

The Poincaré polynomial of G-representation varieties is not quantizable by monoidal TQFTs.

Abstract

In this paper we investigate the problem of constructing Topological Quantum Field Theories (TQFTs) to quantize algebraic invariants. We exhibit necessary conditions for quantizability based on Euler characteristics. In the case of surfaces, also provide a partial answer in terms of sufficient conditions by means of almost-TQFTs and almost-Frobenius algebras for wide TQFTs. As an application, we show that the Poincar\'e polynomial of $G$-representation varieties is not a quantizable invariant by means of a monoidal TQFTs for any algebraic group $G$ of positive dimension.