On relationships among Chern-Simons theory, BF theory and matrix model

TL;DR

This paper explores the deep connections between Chern-Simons theory, BF theory, and matrix models, revealing how they relate through dimensional reduction, extended T-duality, and matrix representations, especially on S^3 and lens spaces.

Contribution

It demonstrates the inverse relationship between Chern-Simons and BF theories via extended matrix T-duality and constructs matrix model representations of Wilson loops on S^3.

Findings

Chern-Simons theory reduces to BF theory with a mass term on a Riemann surface.

Both theories can be realized within a matrix model framework on S^3.

Wilson loops in Chern-Simons theory are represented in the matrix model.

Abstract

Chern-Simons theory on a U(1) bundle over a Riemann surface \Sigma_g of genus g is dimensionally reduced to BF theory with a mass term, which is equivalent to the two-dimensional Yang-Mills on \Sigma_g. We show that the former is inversely obtained from the latter by the extended matrix T-duality developed in hep-th/0703021. For the case of g=0 (i.e. S^2), the U(1) bundle represents the lens space S^3/Z_p. We find that in this case both the Chern-Simons theory and the BF theory with the mass term are realized in a matrix model. We also construct Wilson loops in the matrix model that correspond to those in the Chern-Simons theory on S^3.