The dynamical structure of higher dimensional Chern-Simons theory

TL;DR

This work investigates the dynamical content of higher-dimensional Chern-Simons theories, showing that when the invariant tensor is generic, such theories possess local degrees of freedom in contrast to 2+1 dimensions. Using a complete Dirac constraint analysis for $G\times U(1)$ and $U(1)$ cases, the authors derive the full Dirac brackets, separate first- and second-class constraints, and count local DOF via ${\cal N}=nN-n-N$, with $N$ the dimension of the gauge group and $n$ related to the spacetime dimension. They demonstrate that boundary degrees of freedom form a $WZW_4$-type algebra in five dimensions, extendable to $WZW_{2n}$ in general, and provide explicit reduced actions and algebras for several cases, including $N=1$ Abelian and Lovelock-Chern-Simons gravity. The results show that higher-dimensional CS theories can be generally covariant without a dynamical metric yet possess propagating torsion, with significant implications for higher-dimensional gravity and quantum constraint quantization.

Abstract

Higher dimensional Chern-Simons theories, even though constructed along the same topological pattern as in 2+1 dimensions, have been shown recently to have generically a non-vanishing number of degrees of freedom. In this paper, we carry out the complete Dirac Hamiltonian analysis (separation of first and second class constraints and calculation of the Dirac bracket) for a group GxU(1). We also study the algebra of surface charges that arise in the presence of boundaries and show that it is isomorphic to the WZW4 discussed in the literature. Some applications are then considered. It is shown, in particular, that Chern-Simons gravity in dimensions greater than or equal to five has a propagating torsion.