Fiber Bundles and Matrix Models

TL;DR

This paper explores the relationship between gauge theories on principal bundles and their base spaces, demonstrating dimensional reduction to matrix models and relating monopole vacua to matrix model vacua, extending previous work to non-Abelian bundles.

Contribution

It extends the relationship between gauge theories on principal bundles and matrix models from U(1) to SU(n+1) bundles, including monopole vacua and T-duality interpretations.

Findings

Dimensional reduction of YM on principal bundles to YM-Higgs on base spaces.

Equivalence of YM on SU(2) bundles to YM-Higgs around specific backgrounds.

Connection between monopole vacua on CP^n and matrix model vacua.

Abstract

We investigate relationship between a gauge theory on a principal bundle and that on its base space. In the case where the principal bundle is itself a group manifold, we also study relations of those gauge theories with a matrix model obtained by dimensionally reducing them to zero dimensions. First, we develop the dimensional reduction of Yang-Mills (YM) on the total space to YM-higgs on the base space for a general principal bundle. Second, we show a relationship that YM on an SU(2) bundle is equivalent to the theory around a certain background of YM-higgs on its base space. This is an extension of our previous work (hep-th/0703021), in which the same relationship concerning a U(1) bundle is shown. We apply these results to the case of $SU(n+1)$ as the total space. By dimensionally reducing YM on $SU(n+1)$, we obtain YM-higgs on $SU(n+1)/SU(n)\simeq S^{2n+1}$ and on $SU(n+1)/(SU(n)\times U(1))\simeq CP^n$ and a matrix model. We show that the theory around each monopole vacuum of YM-higgs on $CP^n$ is equivalent to the theory around a certain vacuum of the matrix model in the commutative limit. By combing this with the relationship concerning a U(1) bundle, we realize YM-higgs on $SU(n+1)/SU(n)\simeq S^{2n+1}$ in the matrix model. We see that the relationship concerning a U(1) bundle can be interpreted as Buscher's T-duality.