Branes and N=1 Duality in String Theory

We propose a construction of dual pairs in four dimensional N=1 supersymmetric Yang-Mills theory using branes in type IIA string theory. Supersymmetric gauge theories invariant under eight or sixteen supercharges (e.g. N=2 and N=4 supersymmetric Yang-Mills theories in four dimensions, respectively) have been intensively studied in the last few years. Many of the diverse strong coupling phenomena that they exhibit were found to have a natural explanation in string theory (some were even discovered this way). In gauge theory, a lot is known about systems with four supercharges (in particular, N=1 supersymmetric gauge theories in four dimensions [1,2]), and it is natural to ask what string theory can teach us about such systems. The purpose of this note is to construct, in analogy to [3], a configuration of branes in type II string theory, which appears to exhibit one of the most remarkable field theory phenomena – Seiberg's duality in four dimensional N=1 supersymmetric gauge theory [4]. This duality is the statement that two different gauge theories (" electric " and " magnetic ") may sometimes give rise to the same long distance physics, i.e. flow to the same infrared conformal field theory. Its interpretation in string theory was recently studied in [5] from a different point of view. We will also discuss certain extensions of Seiberg's work [6]. Below, we consider a configuration of branes whose low energy worldvolume dynamics is that of an N=1 four dimensional supersymmetric Yang-Mills theory with gauge group U (N c) and N f flavors of quarks in the fundamental representation. We focus on two microscopic coupling constants, the gauge coupling and the Fayet-Iliopoulos (FI) D-term. We show that by varying the gauge coupling (using the FI D-term to avoid a strong coupling singularity), we recover in different limits the electric and magnetic descriptions of the theory, given by Seiberg. This shows that the two models have the same moduli space of vacua and, therefore, the same chiral ring. It is natural to expect that the full infrared conformal field theories agree as well. 2. The brane configuration. The configurations we will study involve four kinds of branes in type IIA string theory: a Neveu-Schwarz (NS) fivebrane, Dirichlet (D) sixbrane, Dirichlet fourbrane and a differently oriented NS fivebrane, which we will refer to as the NS ′ fivebrane. Specifically, the four kinds of branes are: