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

Abstract

We consider different large $\mathcal{N}$ limits of the one-dimensional Chern-Simons action $i\int dt~ \Tr (\del_0 +A_0)$ where $A_0$ is an $\mathcal{N}\times \mathcal{N}$ 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 $\mathcal{M}$ with different gauge field backgrounds. For slowly varying fields, the large $\mathcal{N}$ limit of the one-dimensional CS action is equal to the $(2k+1)$-dimensional CS theory on $\mathcal{M}\times \mathbf{R}$. Different large $\mathcal{N}$ 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 $\mathcal{M}$ are gauged, this has implications for gravity on fuzzy spaces. This is also briefly discussed.

37 pages, references and a comment added

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• The role of the spin connection in quantum Hall effect: A perspective from geometric quantization
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