Integrable Classical and Quantum Gravity

Abstract

In these lectures we report recent work on the exact quantization of dimensionally reduced gravity [1, 2, 3, 4]. Assuming the presence of commuting Killing symmetries which effectively eliminate the dependence on all but two space-time coordinates allows us to cast the models into the form of 2d non-linear (G/H)-coset space σ-models coupled to gravity and a dilaton. This construction includes a variety of models described by different coset spaces — ranging from pure 4d Einstein gravity with two commuting Killing vector fields to dimensionally reduced maximal supergravity with the coset space E 8(+8)/SO(16) [5]. Although these models superficially resemble the 2d dilaton-gravity models considered more recently in the context of string theory and Liouville theory (see [6] and references therein), the latter generically admit more general couplings of the dilaton and the Liouville field while restricting the matter sector essentially to a set of free fields. By contrast, the matter sector of the models considered here is governed by highly non-linear interactions; its underlying group theoretical structure is a crucial ingredient in our analysis.KeywordsQuantum GravitySpectral ParameterPoisson StructureConformal FactorCoset SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.