Non-perturbative quantization of Lagrangian quantum field theories for strongly coupled topological systems

Develop non-perturbative quantization methods for Lagrangian quantum field theories that apply to strongly coupled topological quantum systems, providing a consistent treatment of global topological degrees of freedom and flux quantization beyond perturbation theory.

Background

The paper argues that Lagrangian descriptions of gauge theories, such as abelian Chern–Simons models used for fractional quantum Hall (FQH) systems, typically capture only local gauge potentials and miss the global topological data crucial for topological phases. This leads to conflicts with flux quantization when effective fractional charges are modeled.

The authors emphasize that addressing these issues requires a non-perturbative framework, since perturbative quantization cannot encode the necessary global structure. They point out that this gap is part of a long-standing challenge in mathematical physics, akin to the general mass gap problem.

Their work proposes a non-Lagrangian, homotopy-theoretic approach based on classifying spaces and exotic (non-abelian) cohomology, but the broader open challenge of developing general non-perturbative quantization methods for Lagrangian theories remains.

References

This issue is an example of the notorious open problem of finding non-perturbative quantizations of Lagrangian theories as needed for strongly coupled topological quantum systems (the analog in solid state physics of what in mathematical high energy physics is known as the mass gap problem which has famously been pronounced a "Millennium Problem" ).

— Fractional Quantum Hall Anyons via the Algebraic Topology of Exotic Flux Quanta  (2505.22144 - Sati et al., 28 May 2025) in Section 1, Introduction and Survey – The problem of flux quantization in FQH systems

We should be careful however, since the perturbative prescription does not necessarily determine the nonperturbative infrared state space, the treatment of global zero modes or the vacuum central charge. These remain questions about the infrared completion of the induced description.

— BRST quantization of the induced gauge theory for the Higgs-branch CFT${}_2$  (2609.26571 - Stefański, 22 Sep 2026) in Section 6, “The two-dimensional limit and evanescence”