Fiber bundles and matrix models

We investigate the 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) theory on the total space to YM-Higgs theory 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 [T. Ishii, G. Ishiki, S. Shimasaki, and A. Tsuchiya, J. High Energy Phys. 05 (2007) 014.], 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)S²ⁿ⁺¹ and on SU(n+1)/(SU(n)×U(1))CPⁿ and a matrix model. We show that the theory around each monopole vacuum of YM-Higgs on CPⁿ is equivalent to the theory around a certain vacuum of the matrix model in the commutative limit. By combining this with the relationship concerning a $U(1)$ bundle, we realize YM-Higgs on SU(n+1)/SU(n)S²ⁿ⁺¹ in the matrix model. We see that the relationship concerning a $U(1)$ bundle can be interpreted as Buscher's T-duality.