Linear Systems for 2D Poincaré Supergravities

Abstract

This contribution contains a summary of [1], which generalizes the linear systems that were derived already some time ago for the dimensionally reduced field equations of Einstein Yang-Mills theories [2, 3] and their locally supersymmetric extensions [4, 5]. These reductions correspond to solutions of the field equations, which depend on two coordinates only and thus possess at least two commuting Killing vectors. The construction of [1] differs from earlier treatments, which were all based on the (super) conformai gauge, in that it allows for non-trivial topologies of the two dimensional world sheets by taking into account the topological degrees of freedom of the world sheet, i.e. its moduli and supermoduli. These constitute extra physical (but non-propagating) degrees of freedom not present in the corresponding flat space integrable sigma models, and affect the dynamics in a non-trivial fashion. In particular, there is a “back reaction” of the matter fields on the topological degrees of freedom, in contrast to conformai field theories, where the moduli determining the background can be freely chosen. The spectral parameter t entering the linear system is now not only a function of the “dilaton” field as in [3, 4], but also depends on the moduli and super-moduli of the world sheet. It is subject to a pair of differential equations, whose integrability condition yields one of the equations of motion obtained by dimensional reduction of Einstein’s equations. Apart from these intriguing new structures, an important motivation for investigating the 2d supergravity models is the search for new symmetries generalizing the Geroch group [6] and the “hidden symmetries” of dimensionally reduced supergravities [7, 8, 9]. The results obtained in [1] indicate that, if such extensions of the Geroch group exist, they are likely to involve the topological degrees of freedom. It should be stressed, however, that even for the known classes of solutions, the global structure of the Geroch group is not fully understood (see [3] for a discussion).