A ug 2 00 2 On String Tensions in Supersymmetric Gauge Theory

In some models of N = 1 supersymmetric SU(M) gauge dynamics (hepth/9503163 and hep-th/9707244), the tension of a string ending on q external quarks is proportional to sin(πq/M), q = 1, . ,M − 1. In this paper we calculate the ratios of the q-string tensions using the recently derived type IIB gravity duals of N = 1 SUSY gauge theories. Far in the IR these gravity duals contain an S with M units of R-R 3-form flux which, upon S-duality, turns into NS-NS 3-form flux. The confining q-string is described by a D3-brane wrapping an S ⊂ S with q units of world volume flux. For one of the gravity dual backgrounds (Maldacena-Nuñez) a D3-brane probe calculation exactly reproduces the sin(πq/M) dependence, while for another (Klebanov-Strassler) we find approximate agreement. We speculate on the connection of the q-string tensions with D-brane tensions in the SU(2) WZW model. Consider a gauge theory with gauge group SU(M) and no dynamical fundamental matter. A Wilson loop in the fundamental representation corresponds to transporting a probe quark around a closed contour C. The classic criterion for confinement is that this Wilson loop obey the area law − ln〈W1(C)〉 = T1A(C) (1) in the limit of large area. An interesting generalization is to consider Wilson loops in antisymmetric tensor representations with q indices where q ranges from 1 to M − 1. q = 1 corresponds to the fundamental representation as denoted above, and there is a symmetry under q →M − q which corresponds to replacing quarks by anti-quarks. These Wilson loops can be thought of as confining strings which connect q probe quarks on one end to q corresponding probe anti-quarks on the other. For q = M the probe quarks combine into a colorless state (a baryon); hence the corresponding Wilson loop does not have an area law. It is interesting to ask how the tension of this class of confining strings depends on q. If it is a convex function, Tq+q′ < Tq + Tq′ , (2) then the q-string will not decay into strings with smaller q. This is precisely the situation found by Douglas and Shenker (DS) [1] in softly broken N = 2 gauge theory, and later by Hanany, Strassler and Zaffaroni (HSZ) [2] in the MQCD approach to confining N = 1 supersymmetric gauge theory: Tq = Λ 2 sin πq M , q = 1, 2, . ,M − 1 (3) where Λ is the overall IR scale. MQCD uses the 5-branes of 11-dimensional Mtheory to engineer theories which belong to the same universality class as conventional gauge theories [3, 4]. Using these tools it is possible to construct a model with N = 1 supersymmetry and gauge group SU(M) which far in the infrared exhibits confinement, chiral symmetry breaking and many other expected phenomena [4]. In the UV, however, this model has extra degrees of freedom compared to the usual 4-d gauge theory; hence there is no logarithmic running or dimensional transmutation. The formula (3) gives definite quantitative predictions for the ratios of the string tensions in MQCD, but it is not clear to what extent these predictions might also be true for supersymmetric QCD. In this paper we rederive this formula using a rather different set-up, gauge/string duality, suggesting that the formula may be quite robust.