Poisson Realization and Quantization of the Geroch Group

Abstract

The conserved nonlocal charges generating the Geroch group with respect to the canonical Poisson structure of the Ernst equation are found. They are shown to build a quadratic Poisson algebra, which suggests to identify the quantum Geroch algebra with Yangian structures. Geroch’s discovery of an infinite-dimensional group of symmetries [1] was one of the essential steps in the study of Einstein’s equations with two Killing vector fields. The understanding of the algebraic structure underlying the Geroch group essentially improved with the construction of the corresponding linear system [2, 3]. Closer analysis of this system further provided different insightful realizations [4, 5, 6] of the symmetry algebra — meanwhile identified as half of an affine Kac-Moody algebra. In this letter, the Poisson realization of the Geroch group that has been missed so far is revealed. We identify the infinite set of conserved charges that generate the infinitesimal symmetry transformations with respect to the canonical Poisson structure. For the quantum theory, where the classical algebra of symmetries turns into a spectrum-generating algebra, the Poisson realization is of essential importance, since it is exactly the quantum pendant of this algebra according to the representations of which the spectrum of physical states is classified. ∗On leave of absence from Steklov Mathematical Institute, Fontanka, 27, St.Petersburg 191011 Russia

Cite this paper

@inproceedings{Korotkin2008PoissonRA, title={Poisson Realization and Quantization of the Geroch Group}, author={Dmitry Korotkin and Henning Samtleben}, year={2008} }