Solving four-dimensional field theories with the Dirichlet fivebrane (original) (raw)

The realization of N = 2 four dimensional super Yang-Mills theories in terms of a single Dirichlet fivebrane in type IIB string theory is considered. A classical brane computation reproduces the full quantum low energy effective action. This result has a simple explanation in terms of mirror symmetry. A particularly fruitful approach to the study of supersymmetric quantum field theories has been to realize these theories as a limit of string or M theory where gravitational effects decouple. There are two complementary approaches to this problem-the geometric engineering [1] approach and the Hanany-Witten brane set up [2]. To study N = 2 super Yang-Mills theories in 3 + 1 dimensions within the geometric engineering approach, one typically compactifies type IIA/B string theory on a Calabi-Yau threefold. The full non-perturbative solution of the N = 2 super Yang-Mills theory is then obtained by invoking mirror symmetry. In the Hanany-Witten approach, one typically considers a web of branes in a flat space. In order to study N = 2 super Yang-Mills theory in 3 + 1 dimensions, one considers two parallel solitonic fivebranes with a number of Dirichlet fourbranes stretched between them [3]. In this approach, all perturbative and non-perturbative corrections to the field theory are coded into the shape of the branes. The solution of these theories is performed by lifting to M theory. After the lift, the original type IIA brane set up is reinterpreted as a single fivebrane in M theory, wrapping the Seiberg-Witten curve Σ. The relationship between these two approaches has been explained in [4]. In this report, we will study N = 2 super Yang-Mills theory using the Hanany-Witten approach. Up to now, the description of the M theory fivebrane relevant for N = 2 super Yang-Mills theory has been in terms of eleven dimensional supergravity, which is a valid description of M theory at low energy [5]. A number of holomorphic quantities [6] including the exact low energy effective action [5] can be recovered using the supergravity description. The supergravity description corresponds to a strong coupling description of the original type IIA setup. However, one expects the field theory to emerge in the opposite limit, where the string theory is weakly coupled [7]. This limit is not captured by the supergravity approximation, so that one expects that the supergravity approach will only be capable of reproducing field theory quantities which are protected by supersymmetry. In this note, we will provide a direct construction in string theory which realizes the N = 2 super Yang-Mills theory in terms of a single Dirichlet fivebrane wrapping the Seiberg-Witten curve. We will be mainly concerned with two important issues: how a matrix description is obtained and how the string theory configurations described in this article are related to the original type IIA brane set up [3]. In particular the single D5 in Type IIB string theory will be seen to be related by T-duality to what has been described in the literature as the "magnetic" IIA brane configuration. We will then show how the D5 provides a strongly coupled, low energy description of weakly coupled IIA string theory in the original brane set up. We will start with a brane construction consisting of a number of Dirichlet fourbranes suspended between Dirichlet sixbranes in type IIA string theory on R 9 × S 1. The coordinate x 7 is compact, with radius R 7. In the classical approximation, the sixbranes are located at x 8 = x 9 = 0 and at some fixed x 6. The world volume coordinates for the sixbranes are x 0 , x 1 , x 2 , x 3 , x 4 , x 5 , x 7. The fourbranes are located at x 8 = x 9 = 0 and at some fixed values of x 4 , x 5 , x 7. The fourbranes have world volume coordinates x 0 , x 1 , x 2 , x 3 , x 6. Since the fourbranes stretch between the two sixbranes, the x 6 coordinate is restricted to a finite interval. This brane configuration is related to the configuration studied in [8] by T duality along x 1 , x 2 , x 3. The supersymmetries preserved by the fourbranes are of the form [9] ǫ L Q L + ǫ R Q R where ǫ L = Γ 0 Γ 1 Γ 2 Γ 3 Γ 6 ǫ R. Thus the fourbrane breaks one half of the supersymmetry. The sixbranes preserve supersymmetries of the form ǫ