On quantum determinants in integrable quantum gravity

Abstract

Einstein–Rosen waves with two polarizations are cylindrically symmetric solutions to vacuum Einstein equations. Einstein equations in this case reduce to an integrable system. In 1971, Geroch has shown that this system admits an infinite-dimensional group of symmetry transformations known as the Geroch group. The phase space of this system can be parametrized by a matrix-valued function of spectral parameter, called monodromy matrix. The latter admits the Riemann–Hilbert factorization into a pair of transition matrices, i.e. matrix-valued functions of spectral parameter such that one of them is holomorphic in the upper half-plane, and the other is holomorphic in the lower half-plane. The classical Geroch group preserves the determinants of transition and monodromy matrices by construction. The algebraic quantization of the quadratic Poisson algebra generated by transition matrices of Einstein–Rosen system was proposed by Korotkin and Samtleben in 1997. Paternain, Peraza and Reisenberger have recently suggested a quantization of the Geroch group, which can be considered as a symmetry of the quantum algebra of observables. They have shown that commutation relations involving quantum monodromy matrices are preserved by the action of the quantum Geroch group. In present paper we introduce the notion of the determinant of the quantum monodromy matrix. We derive a factorization formula expressing the quantum determinant of the monodromy matrix as a product of the quantum determinants of the transition matrices. The action of the quantum Geroch group is extended from the subalgebra generated by the monodromy matrixonto the full algebra of observables. This extension is used to prove that the quantum determinant of the quantum monodromy matrix is invariant under the action of quantum Geroch group.