Supersymmetry and Dualities ∗

Duality transformations with respect to rotational isometries relate supersymmetric with non-supersymmetric backgrounds in string theory. We find that non-local world-sheet effects have to be taken into account in order to restore supersymmetry at the string level. The underlying superconformal algebra remains the same, but in this case T-duality relates local with non-local realizations of the algebra in terms of parafermions. This is another example where stringy effects resolve paradoxes of the effective field theory. CERN-TH/95-258 FTUAM/95-34 October 1995 Contribution to the proceedings of the Trieste conference on S-Duality and Mirror Symmetry, 5–9 June 1995; to be published in Nuclear Physics B, Proceedings Supplement Section, edited by E. Gava, K. Narain and C. Vafa Permanent address: Departamento de Fisica Teorica, Universidad Autonoma, 28049 Madrid, Spain Permanent address: Department of Physics, University of Patras, 26110 Patras, Greece 1 Outline and summary These notes, based on two lectures given at the Trieste conference on “S-Duality and Mirror Symmetry”, summarize some of our recent results on supersymmetry and duality [1, 2, 3]. The problem that arises in this context concerns the supersymmetric properties of the lowest order effective theory and their behaviour under T-duality (and more generally U-duality) transformations (see also [4]). We will describe the geometric conditions for the Killing vector fields that lead to the preservation or the loss of manifest spacetime supersymmetry under duality, and classify them as translational versus rotational respectively. Analyzing this problem in detail we find that there is no contradiction at the level of string theory. This is part of the standard lore that stringy effects manifest as paradoxes of the effective field theory, and supersymmetry is no exception to it. In toroidal compactifications of superstring theory T-duality is a symmetry that interchanges momentum with winding modes, and it manifests geometrically as a smalllarge radius equivalence in the effective theory of the background massless fields. This equivalence has been generalized to arbitrary string backgrounds with Abelian isometries following Buscher’s formula that was originally derived for the β-function equations of the metric Gμν , antisymmetric tensor field Bμν and dilaton Φ to lowest order in the σmodel coupling constant (inverse string tension) α (see for instance [5], and references therein). One might naively expect that if a background is supersymmetric (eg a bosonic solution of effective supergravity), the dual background will also share the same number of supersymmetries. This is a reasonable expectation provided that there are no nonlocal world-sheet effects, so that the dual faces of the theory provide a trustworthy low energy approximation to string dynamics in the α-expansion. There are circumstances, however, when the dual geometry has strong curvature singularities and the effective field theory description breaks down in their vicinity. This is a typical property of gravitational backgrounds having Killing isometries with fixed points, in which case the T-dual background exhibits curvature singularities. Then, T-duality appears to be incompatible with space-time supersymmetry, and non-local world-sheet effects have to be taken into account for resolving this issue consistently. Afterall, it is a well-known fact that symmetries not commuting with a given Killing vector field are not manifest in the dual background of the effective field theory description. In the first part most of our discussion will concentrate on 4-dim space-time backgrounds, thinking of superstring vacua as arising from ten dimensions by compactification on a 6-dim internal space K. It is much simpler to expose the main ideas in this case and employ the notion of self-duality (when it is appropriate) to refine the distinction we want to make. In four dimensions, and in the presence of some Killing isometries, it is also possible to intertwine T with S-duality, thus forming a much larger symmetry group known as U-duality. We will present the essential features of this idea at the level of the effective theory by considering 4-dim backgrounds with (at least) one isometry, in which case the U-duality corresponds to O(2, 2) transformations. The supersymmetric properties of the effective theory will also be considered under the action of such generalized