Chern-Simons-Witten Theory as a Topological Fermi Liquid

TL;DR

This paper reinterprets Chern-Simons-Witten theory on a torus as a free fermion system, exploring its large N limit, bosonization, and potential string interpretations, highlighting differences from Yang-Mills theory.

Contribution

It provides a novel free fermion representation of Chern-Simons-Witten theory and discusses its large N limit, bosonization, and implications for string theory interpretations.

Findings

Hilbert space relates to group quantum mechanics

Large N limit described by matrix quantum mechanics techniques

Differences from YM_2 identified for subleading orders

Abstract

We reinterpret U(N) Chern-Simons-Witten theory quantized on a torus as a free fermion system. Its Hilbert space and some observables are simply related to those of group quantum mechanics, even at finite N and k. Its large N limit can be described using techniques developed for matrix quantum mechanics and two-dimensional Yang-Mills theory. We discuss the bosonization of this theory, which for YM_2 gave a precise interpretation of Wilson loop operators in terms of string creation and annihilation operators, and examine its consequences for a string interpretation here. The formalism seems entirely adequate for the leading large N results and in a sense can be thought of as a `classical string field theory'. In considering subleading orders in 1/N, we identify some major differences between CSW and YM_2, which must be dealt with to find a CSW gauge string interpretation. Although these particular differences are probably not relevant for `QCD string,' they do illustrate some of the issues there, and we comment on this. We also propose an approach to dealing with large N transitions.