Integrability and BRST invariance from BF topological theory

TL;DR

This paper demonstrates that various integrable equations derived from non-abelian BF topological theory in 1+1 dimensions possess BRST invariance and an infinite set of conserved quantities, linking topological field theory to integrable systems.

Contribution

It explicitly constructs BRST invariant systems for several well-known integrable equations from BF theory, revealing their underlying topological symmetry structure.

Findings

Integrable equations from BF theory are BRST invariant.

Explicit BRST transformations for KdV, mKdV, CKdV, and Harry Dym.

Infinite conserved quantities are associated with these systems.

Abstract

We consider the BRST invariant effective action of the non-abelian BF topological theory in $1+1$ dimensions with gauge group $Sl(2,\mathbb{R})$. By considering different gauge fixing conditions, the zero-curvature field equation give rise to several well known integrable equations. We prove that each integrable equation together with the associated ghost field evolution equation, obtained from the BF theory, is a BRST invariant system with an infinite sequence of BRST invariant conserved quantities. We construct explicitly the systems and the BRST transformation laws for the KdV sequence (including the KdV, mKdV and CKdV equations) and Harry Dym integrable equation.