An SL(2, Z) Multiplet of Type IIB Super Five-Branes

It is well-known that the low energy string theory admits a non-singular solitonic super five-brane solution which is the magnetic dual to the fundamental string solution. By using the symmetry of the type IIB string theory, we construct an SL(2, Z) multiplet of magnetically charged super five-branes starting from this solitonic solution. These solutions are characterized by two integral three-form charges (q1, q2) and are stable when the integers are coprime. We obtain an expression for the tension of these (q1, q2) fivebranes as envisaged by Witten. The existence of similar five-brane solutions in type II string theory in D < 10 has also been discussed. ∗E-mail address: [email protected] String theories in the long wavelength limit are described by various kinds of supergravity theories in D = 10. The supergravity equations of motion are well-known [1–3] to admit various black p-brane solutions which are essentially the black hole solutions of the dimensionally reduced low energy string effective action spatially extended to the ten dimensional theory. These solutions are usually characterized by two parameters related to the charge and the mass of the black p-branes. In the extremal limit as usual in the Reissner-Nordstrom black hole, the charge and the ADM mass of the p-branes are related to each other and the solutions become supersymmetric saturating the BPS bound. The extremal solutions of string effective actions are particularly interesting since their masses and the charges can be calculated exactly due to certain non-renormalization theorems of the underlying supersymmetric theories. Thus although these solutions are obtained from the low energy effective theory, they are quite useful to identify certain non-perturbative symmetries of string theory. Low energy effective action of any string theory admits a fundamental string solution [4] and its magnetic dual the non-singular solitonic five-brane solution [5–8]. These are the extremal limit, as we have mentioned, of the black string and black five-brane solution of the corresponding supergravity equations of motion. Since it is known that the equations of motion of type IIB supergravity theory is invariant [9] under an SL(2, R) group one can construct a more general string like as well as five-brane solutions using this symmetry group. We should like to mention that the SL(2, R) symmetry of type IIB string theory is a non-perturbative symmetry. It transforms the string coupling constant in a nontrivial way and therefore mixes up the different orders of string perturbation theory. A discrete subgroup of this SL(2, R) group has been conjectured to be the exact symmetry group of the quantum type IIB string theory [10]. Using this symmetry, Schwarz [11] has constructed an SL(2, Z) multiplet of string like solutions in type IIB string theory starting from the fundamental string solution. Both the string tension and the charge were shown to be given by the SL(2, Z) covariant expressions. Since the tension and the charge of these extremal solutions remain unrenormalized, it provides a strong support in favor of the conjecture that SL(2, Z) is an exact symmetry group of the quantum theory. Given the symmetry of the type IIB theory, it is natural to expect that the solitonic five-brane solution should also form an SL(2, Z) multiplet as pointed out by Schwarz in ref.[11]. In this paper, we construct an infinite family of magnetically charged super five-branes, permuted by SL(2, Z) group, starting from the solitonic five-brane solution of