On Target-space Duality in P–branes

We study the target-space duality transformations in p–branes as transformations which mix the worldvolume field equations with Bianchi identities. We consider an (m + p + 1)-dimensional spacetime with p + 1 dimensions com-pactified, and a particular form of the background fields. We find that while a GL(2) = SL(2) × R group is realized when m = 0, only a two parameter group is realized when m > 0. 1 Ten dimensional string theory compactified on a six–torus has a symmetry group O(6, 6; Z), called " T duality " , which mixes momentum modes with winding modes [1-4] (in the heterotic case the group is O(6, 22; Z)). This is also referred to as target space duality. (For an extensive review and further references, see [5]). It has been conjectured that string theory possesses also a symmetry group SL(2, Z), called " S duality " , which transforms Fock states into soliton states [6-8]. This is also known as strong–weak coupling duality. (For a recent review, see [9]). Such a symmetry could only be seen in a nonperturbative approach and its existence has not yet been firmly established. One possible way of studying the S duality of strings is via fivebranes. A fivebrane in ten dimensions has been conjectured to be " dual " to a string in a certain well defined sense [10], and it has been further conjectured that under this transformation the role of the S and T dualities would interchange [11], in the sense that the SL(2, Z) S duality of the string would play the role of a T duality for the fivebrane, and similarly the O(6, 6; Z) T duality of the string would play the role of a nonperturbative S duality for the fivebrane. Since a T duality symmetry can be seen perturbatively, it would seem convenient to study SL(2, Z) symmetry in the context of five–branes. All these groups have been found either directly in four dimensional N = 4 super-gravity [3] or in the toroidal compactification of ten dimensional supergravity theories [4] which one obtains as low energy limits of strings and fivebranes. It was conjectured by Duff and proven by Cecotti et al. that the T duality group O(6, 6; Z) is a symmetry of string theory, transforming the worldsheet field equations and Bianchi identities into each other [2]. Although it has been shown that an SL(2, Z) …