Gauge fixing and classical dynamical r-matrices in ISO(2,1)-Chern-Simons theory

TL;DR

This paper applies gauge fixing to ISO(2,1) Chern-Simons theory on punctured surfaces, explicitly describing the resulting Poisson structures via classical dynamical r-matrices and classifying them through dynamical transformations.

Contribution

It provides an explicit description of Poisson structures in ISO(2,1) Chern-Simons theory using gauge fixing and classifies the resulting dynamical r-matrices via transformations.

Findings

Poisson structures are described by classical dynamical r-matrices.

Different gauge fixings relate Poisson structures through dynamical transformations.

Poisson structures often combine r-matrices for non-conjugate Cartan subalgebras.

Abstract

We apply Dirac's gauge fixing procedure to Chern-Simons theory with gauge group ISO(2,1) on manifolds RxS, where S is a punctured oriented surface of general genus. For all gauge fixing conditions that satisfy certain structural requirements, this yields an explicit description of the Poisson structure on the moduli space of flat ISO(2,1)-connections on S via the resulting Dirac bracket. The Dirac bracket is determined by classical dynamical r-matrices for ISO(2,1). We show that the Poisson structures and classical dynamical r-matrices arising from different gauge fixing conditions are related by dynamical ISO(2,1)-valued transformations that generalise the usual gauge transformations of classical dynamical r-matrices. By means of these transformations, it is possible to classify all Poisson structures and classical dynamical r-matrices obtained from such gauge fixings. Generically these Poisson structures combine classical dynamical r-matrices for non-conjugate Cartan subalgebras of iso(2,1).