PROCEEDINGS SUPPLEMENTS www.elsevier.nl/locate/npe zyxwvu ELSEVIER Nuclear Physics B (Proc. Suppl.) 88 (2000) 175-183 A supergravity view of string dualities K.S. Stelle * The Blackett Laboratory, Imperial College Prince Consort Road, London SW7 2BW, UK We give an overview of the way in which the dualities of string theory are realized explicitly in a supergravity context. An illustrative example is the relation between the 8-brane of massive Romans type II string theory and M-theory. 1. INTRODUCTION String theories and supergravity theories lock together in an overall picture that is now called M-theory. Many of the characteristic string relations have direct analogues in supergravity. For example, both the perturbative and non- perturbative string dualities have direct super- gravity analogues. Supergravity also displays the characteristic worldvolume/spacetime com- plementarity of string theory, exemplified in the soliton viewpoint for pbranes. In the following, we shall consider specifically the relationship be- tween string T-duality and the Kaluza-Klein re- duction of supergravity theories. In Kaluza-Klein dimensional reduction. a field theory is considered in which some spacetime di- mensions are taken to be compact and small, so that an expansion in powers of the small dimen- sion lengths is sensible. In the limit where such lengths tend to zero, only the field excitations in directions orthogonal to the compactification di- mensions are retained. The field-theoretic context of Kaluza-Klein re- duction should be contrasted to that of string the- oretic T-duality. String theory is currently for- mulated about a chosen spacetime background (liberation from such a background-dependent formalism remains still an important challenge for the subject). In order to derive T-duality relationships, this background must be taken to have zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA isometrics. Such isometric backgrounds are identical in structure to those considered as a ‘Research supported in part by PPARC under SPG grant No. 613 prelude to Kaluza-Klein reduction in field the- ory, where the background fields are taken to be isometric with respect to translations around the small compact dimensions. However, there is a principal difference between the two situations: in string theory, although the background fields are restricted to have excitations only in direc- tions orthogonal to the compact dimensions, the step of shrinking the lengths of such dimensions down to zero is not taken. One may think of the prelude to a T-duality transformation as setting up a cylindrical tube on which to let strings freely propagate: the background is nicely symmetric in a compact dimension y, but the strings them- selves (open and closed alike) are not required to share in this cylindrical symmetry, i.e. they may have extent in the y direction. One may also exploit the finite size of a background, compact dimension y by allowing closed strings to wrap around such a dimension (giving winding states). The essential difference, then, between the field-theoretic Kaluza-Klein context and the string-theoretic T-duality context is that in field theory the requirement of field isometries amounts to a truncation of the theory down to a lower dimensionality, while in string theory the isometry is declared only for the background, while the dynamical strings themselves are still allowed to propagate in all dimensions of the background spacetime, compact and noncompact alike. The isometry in string theory thus affects only the arena for string propagation, not the dy- namics of the strings themselves. In both the Kaluza-Klein field-theoretic con- text and the string theory T-duality context, 0920-5632/00/$ - see. front matter 0 2000 Elsevier Science B.V. All rights reserved. PI1 SO920-5632(00)00766-O