T-duality, fiber bundles and matrices

Abstract

We extend the T-duality for gauge theory to that on curved space described as a nontrivial fiber bundle. We also present a new viewpoint concerning the consistent truncation and the T-duality for gauge theory and discuss the relation between the vacua on the total space and on the base space. As examples, we consider S 3 (/Z k ), S 5 (/Z k ) and the Heisenberg Emergence of space-time is one of the key concepts in nonperturbative definition of superstring or M-theory by matrix models [1,2]. This phenomenon in field theory was found over two decades ago in the large N reduction of gauge theories [3][4][5][6][7], which states equivalence under some conditions between a large N gauge theory and the matrix model that is its dimensional reduction to a point. This equivalence originates from the fact that the eigenvalues of matrices can be interpreted as momenta. This interpretation reappeared in the T-duality between the low-energy effective theories for Dp-branes and for D(p − 1)-branes [8,9]. More concretely, this T-duality tells that a U(N) gauge theory on R p × S 1 is equivalent to the U(N × ∞) gauge theory that is its dimensional reduction to R p if a periodicity condition is imposed to the theory on R p .The main purpose of this paper is to extend the T-duality for gauge theory to that on curved space described as a nontrivial fiber bundle. The above mentioned T-duality is concerning a trivial S 1 bundle, R p × S 1 . We restrict ourselves to principal S 1 bundles and show the T-duality between the gauge theories on the total space and on the base space.We also present a new viewpoint concerning the consistent truncation and the T-duality for gauge theory. Furthermore, we discuss the properties of the vacua 1 on the total space and the base space. In our previous publication [10] on the gauge/gravity correspondence for the SU(2|4) symmetric theories [11] (see also [12][13][14][15][16][17]), we showed the T-duality between N = 4 super Yang Mills (SYM) on R × S 3 (/Z k ) and 2 + 1 SYM on R × S 2 , which is suggested from the gravity side. This is regarded as the T-duality on S 3 (/Z k ), which is a nontrivial S 1 fibration over S 2 . In this paper, we generalize this result. Our findings would be useful for the study of describing curved space-time in matrix models [18][19][20] as well as the study of curved D-branes. This paper is organized as follows. In section 2, we review the T-duality for gauge theory in a standard way. In section 3, we present a new viewpoint concerning the consistent truncation and the T-duality for gauge theory. Although this viewpoint is not necessarily needed for the proof of the T-duality on fiber bundle, it is interesting itself and indeed 1 Throughout this paper, we consider gauge theories on Riemannian manifolds with a positive-definite metric. In the following arguments, we can easily add the time direction as direct product. To be precise, the 'vacua' in this paper mean the classical vacua of the corresponding gauge theories on this direct product space.

Fiber bundles and matrix models

“…This also enables us to dimensionally reduce YM on a group manifold to a matrix model. Second, as an extension of the work [29], in the case in which the fiber is SU(2), we show that YM on the total space is equivalent to a certain vacuum 1 of YM-higgs on the base space with the periodicity imposed. This enables us to realize YM on an SU(2) k × U(1) l bundle in YM-higgs on its base space.…”

“…This vacuum is given by multimonopole configuration on the base space. We already found the U(1) case of this equivalence in [29]. In the SU(2) case, we also use the result in section 2 that YM on S 3 is realized in the matrix model.…”

“…The case of principal U(1) bundles was already given in [29]. Here we consider the case where G is nonabelian.…”

“…The relation between N = 4 SYM on R × S 3 /Z k and 2+1 SYM on R × S 2 can be regarded as an extension of the matrix T-duality to that on a nontrivial U(1) bundle, S 3 /Z k , whose base space is S 2 . Furthermore, in [29], we generalized the matrix T-duality to that on an arbitrary U(1) bundle. As an application of these results, in [30], Ohta and the present authors investigated relationships among Chern-Simons theory on a U(1) bundle over a Riemann surface, BF theory with a mass term on the Riemann surface, which is equivalent to two-dimensional Yang-Mills on the Riemann surface, and a matrix model.…”

“…In section 6, we discuss how the extended matrix T-dulaity found in [29] and reviewed in section 4.2 is interpreted as Buscher's T-duality. In appendices A-D, we describe some details.…”