Moving Frames Hierarchy and BF Theory

TL;DR

This paper establishes a connection between a projected Chern-Simons model and an Abelian gauge theory of spin chains, constructing an integrable hierarchy with geometric and gauge-theoretic insights.

Contribution

It introduces a novel hierarchy of integrable gauge theories derived from a projection of Chern-Simons models, linking geometric spin chain models with BF theory constraints.

Findings

Derived an equivalence between projected Chern-Simons models and Abelian gauge theories.

Constructed an infinite hierarchy of integrable gauge theories and magnetic models.

Interpreted the spectral parameter as a statistical gauge potential constrained by cocycle conditions.

Abstract

We show that the one-dimensional projection of Chern-Simons gauged Nonlinear Schrodinger model is equivalent to an Abelian gauge field theory of continuum Heisenberg spin chain. In such a theory, the matter field has geometrical meaning of coordinates in tangent plane to the spin phase space, while the U(1) gauge symmetry relates to rotation in the plane. This allows us to construct the infinite hierarchy of integrable gauge theories and corresponding magnetic models. To each of them a U(1) invariant gauge fixing constraint of non-Abelian BF theory is derived. The corresponding moving frames hierarchy is obtained and the spectral parameter is interpreted as a constant-valued statistical gauge potential constrained by the 1-cocycle condition.