Fiber bundles and matrix models

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 YangMills (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 [29], 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) ≃ S 2n+1 and on SU (n + 1)/(SU (n) × U (1)) ≃ 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 combining this with the relationship concerning a U (1) bundle, we realize YM-higgs on SU (n + 1)/SU (n) ≃ S 2n+1 in the matrix model. We see that the relationship concerning a U (1) bundle can be interpreted as Buscher's

Perturbative tests for a large-N reduced model of $ \mathcal{N} = {4} $ super Yang-Mills theory

“…To check the validity of the formulation, some one-loop perturbative calculations were performed in [12]. It was shown that the tadpole vanishes and that the fermion one-loop self-energy agrees with the one in the continuum theory.…”

“…In [12], a non-perturbative formulation of N = 4 SYM on R × S 3 in the planar ('t Hooft) limit was proposed by using the plane wave matrix model (PWMM) [13] (for earlier discussions, see [14,15]). Note that N = 4 SYM on R 4 at a conformal point has the P SU(2, 2|4) symmetry with thirty-two supercharges and is equivalent to N = 4 SYM on R × S 3 through the conformal mapping.…”

“…In this section, we review the novel large-N reduction for N = 4 SYM on R × S 3 proposed in [12]. In section 2.1, as a warm-up, we review the large-N reduction on S 1 developed in [12], by taking the φ 3 theory on S 1 as an example.…”

“…In section 2, we review the non-perturbative formulation of the planar N = 4 SYM on R×S 3 proposed in [12]. In section 3, after reviewing Wilson loops in the AdS/CFT correspondence, we construct operators in the non-perturbative formulation which correspond to the Wilson loops in N = 4 SYM on R × S 3 in the continuum limit.…”

“…In section 2.1, as a warm-up, we review the large-N reduction on S 1 developed in [12], by taking the φ 3 theory on S 1 as an example. In section 2.2, to explain the mechanism of the large-N reduction on S 3 , we consider the large N reduction for the φ 3 theory on S 3 .…”