The matrix Chern–Simons one-form as a Universal Chern–Simons theory

Abstract

We consider different large N limits of the one-dimensional Chern–Simons action i ∫ d t Tr ( ∂ 0 + A 0 ) where A 0 is an N × N anti-Hermitian matrix. The Hilbert space on which A 0 acts as a linear transformation is taken as the quantization of a 2 k-dimensional phase space M with different gauge field backgrounds. For slowly varying fields, the large N limit of the one-dimensional CS action is equal to the ( 2 k + 1 ) -dimensional CS theory on M × R . Different large N limits are parametrized by the gauge fields and the dimension 2 k. The result is related to the bulk action for quantum Hall droplets in higher dimensions. Since the isometries of M are gauged, this has implications for gravity on fuzzy spaces. This is also briefly discussed.