The Sigma-Model Representation for the Duality-Symmetric D=11 Supergravity

In this paper which is based on the results obtained in collaboration with Igor Bandos and Dmitri Sorokin I present the extention of the "doubled fields" formalism by Cremmer, Julia, Lu and Pope to the supersymmetric case and lifting this construction onto the level of the proper duality-symmetric action for D = 11 supergravity. Further extension of the doubled field formulation to include dynamical sources is also discussed. This year we can truly commemorate the 25th anniversary from the day when eleven-dimensional supergravity (1) was born. The importance of this event is worth mentioning since supergravity in D = 11 space-time dimensions is a kind of special theory possessing a number of non-trivial features. First of all, this theory is the maximally-dimensional supersymmetric theory of gravity interacting with antisymmetric tensor fields for the standard signature of metric with one time- like direction. Second, the theory has relatively simple structure of the action in comparison to other low-dimensional maximal supergravities. The third point consists in the observation that the massless spectrum of a theory derived upon a compactification from D =1 1 down toD =4 does not contain particles with spin higher than two that is an indication of having a consistent four-dimensional interacting theory. Last but not least the eleven-dimensional supergravity is the low-energy limit of M-theory unifying non-perturbatively five different theories of superstrings. Since not much has been known about M-theory, D = 11 supergravity is very useful tool to investigate the M-theory structure. One of the ways to do so is to consider different routes of dimensional reduction to establish the connection between different low-energy effective actions for superstrings which are nothing but supergravities and to find a relevant scheme that would be in accordance with four-dimensional phenomenology. Following this way another notion, the duality, becomes important and allows us to establish the symmetry structure in reduced theories which is originally hidden. Moreover, the duality gives a fresh look on the symmetry structure in higher-dimensional supergravities that allows one to find new way for describing in a uniform duality invariant manner gauge and internal symmetries of these theories. It should be emphasized that the notion of duality in the context of searching new symme- tries in dimensionally reduced supergravities (2) appeared one year after discovering the eleven- dimensional supergravity. There discovered was a hidden symmetry structure of maximal low- dimensional supergravities that extends the naively expected global symmetry groups arising in the process of dimensional reduction on multidimensional tori. Some important observations were made that were crucial to develop the technique to derive later on the low-dimensional theories of supergravities by reduction on spheres and manifolds with non-trivial holonomy groups.