The Matrix Chern-Simons One-form as a Universal Chern-Simons Theory

TL;DR

This paper shows that the large N limit of a one-dimensional Chern-Simons action can reproduce higher-dimensional Chern-Simons theories, with implications for quantum Hall effects and gravity on fuzzy spaces.

Contribution

It establishes a universal connection between one-dimensional and higher-dimensional Chern-Simons theories through large N limits, incorporating various gauge backgrounds.

Findings

Large N limit yields (2k+1)-dimensional CS theory on phase space times real line.

Different limits depend on gauge fields and phase space dimension.

Results relate to quantum Hall bulk actions in higher dimensions.

Abstract

We consider different large ${\cal N}$ limits of the one-dimensional Chern-Simons action $i\int dt~ \Tr (\del_0 +A_0)$ where $A_0$ is an ${\cal N}\times{\cal N}$ antihermitian matrix. The Hilbert space on which $A_0$ acts as a linear transformation is taken as the quantization of a $2k$-dimensional phase space ${\cal M}$ with different gauge field backgrounds. For slowly varying fields, the large ${\cal N}$ limit of the one-dimensional CS action is equal to the $(2k+1)$-dimensional CS theory on ${\cal M}\times {\bf R}$. Different large ${\cal N}$ limits are parametrized by the gauge fields and the dimension $2k$. The result is related to the bulk action for quantum Hall droplets in higher dimensions. Since the isometries of ${\cal M}$ are gauged, this has implications for gravity on fuzzy spaces. This is also briefly discussed.