Conformal internal symmetry of $2d$ $σ$-models coupled to gravity and a dilaton

Details Conformal internal symmetry of $2d$ $σ$-models coupled to gravity and a dilaton Conformal internal symmetry of $2d$ $σ$-models coupled to gravity and a dilaton

1996-08-13

arxiv.org/abs/hep-th/9608082v1

Nucl.Phys. B482 (1996) 431-465

Physics

High Energy Physics

High Energy Physics - Theory

44 pages, LATEX

Scientific paper

10.1016/S0550-3213(96)00551-2

General Relativity reduced to two dimensions possesses a large group of symmetries that exchange classical solutions. The associated Lie algebra is known to contain the affine Kac-Moody algebra $A_1^{(1)}$ and half of a real Witt algebra. In this paper we exhibit the full symmetry under the semi-direct product of $\Lie{A_1^{(1)}}$ by the Witt algebra $\Lie{\Wir}$. Furthermore we exhibit the corresponding hidden gauge symmetries. We show that the theory can be understood in terms of an infinite dimensional potential space involving all degrees of freedom: the dilaton as well as matter and gravitation. In the dilaton sector the linear system that extends the previously known Lax pair has the form of a twisted self-duality constraint that is the analog of the self-duality constraint arising in extended supergravities in higher spacetime dimensions. Our results furnish a group theoretical explanation for the simultaneous occurrence of two spectral parameters, a constant one ($=y$) and a variable one ($=t$). They hold for all $2d$ non-linear $\sigma$-models that are obtained by dimensional reduction of $G/H$ models in three dimensions coupled to pure gravity. In that case the Lie algebra is $\Lie{\Wir \semi G^{(1)}}$; this symmetry acts on a set of off shell fields (in a fixed gauge) and preserves the equations of motion.

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