Nuclear Physics B243 (1984) 335-349 © North-Holland Publishing Company INFINITE SYMMETRY ALGEBRAS OF EXTENDED SUPERGRAVITY THEORIES* Neil MARCUS, Augusto SAGNOTTI and John H. SCHWARZ California Institute of Technology, Pasadena, California 91125, USA Received 26 October 1983 (Revised 24 February 1984) When extended supergravity theories with noncompact symmetry groups are written in a physical gauge, the noncompact symmetries join with the supersymmetries to generate an infinite- dimensional algebra. The details are worked out explicitly for a two-dimensional theory with an SU(1,1) internal symmetry. Our analysis confirms the observation of Elfis et al. that the infinite rigid superalgebra should be obtained from the finite-dimensional local superalgebra by replacing scalar fields with their asymptotic values at infinity. The infinite algebra is described by extending the super-Poincar6 generators to functions on the coset space defined by the scalar fields at infinity. While mathematically nontrivial, this result is, in a certain sense, trivial from a physical point of view. I. Introduction One of the remarkable facts about extended supergravity theories is that the scalar fields parametrize a coset space defined by the quotient of a noncompact symmetry group G and its maximal compact subgroup H. This was first recognized for a form of N = 4, D = 4 supergravity [1] with G/H = SU(1,1)/U(1), and was later utilized in the construction of N = 8, D = 4 supergravity [2], where the coset space was found to be Ev,7//SU(8). The supersymmetry charges typically transform as a representation of the group H, but not of its extension to G. In the usual covariant formulation with local gauge invariances this is a complete description of the symmetry, because the closure of the algebra involves field-dependent coefficients (structure functions). When one chooses a physical gauge in which all auxiliary and gauge degrees of freedom are eliminated, however, only a rigid algebra with fixed numerical structure constants can occur. The way in which this happens and the determination of the specific physical algebra that emerges is quite nontrivial. The algebra is always infinite. The simplest way to convince oneself of this is to note, for * Work supported in part by the US Department of Energy under contract no. DE-AC-03-81-ER40050 and by the Fleischmann Foundation. 335