Infinite-dimensional algebras in dimensionally reduced string theory

We examine 4-dimensional string backgrounds compactified over a two torus. There exist two alternative effective Lagrangians containing each two $SL(2)/U(1)$ sigma-models. Two of these sigma-models are the complex and the K\"ahler structures on the torus. The effective Lagrangians are invariant under two different $O(2,2)$ groups and by the successive applications of these groups the affine $\widehat{O}(2,2)$ Kac-Moody is emerged. The latter has also a non-zero central term which generates constant Weyl rescalings of the reduced 2-dimensional background. In addition, there exists a number of discrete symmetries relating the field content of the reduced effective Lagrangians.Comment: 13 pages, Late

Field Theoretical Quantum Effects on the Kerr Geometry

We study quantum aspects of the Einstein gravity with one time-like and one space-like Killing vector commuting with each other. The theory is formulated as a \coset nonlinear $\sigma$-model coupled to gravity. The quantum analysis of the nonlinear $\sigma$-model part, which includes all the dynamical degrees of freedom, can be carried out in a parallel way to ordinary nonlinear $\sigma$-models in spite of the existence of an unusual coupling. This means that we can investigate consistently the quantum properties of the Einstein gravity, though we are limited to the fluctuations depending only on two coordinates. We find the forms of the beta functions to all orders up to numerical coefficients. Finally we consider the quantum effects of the renormalization on the Kerr black hole as an example. It turns out that the asymptotically flat region remains intact and stable, while, in a certain approximation, it is shown that the inner geometry changes considerably however small the quantum effects may be.Comment: 16 pages, LaTeX. The hep-th number added on the cover, and minor typos correcte

Political and Legal Considerations Regarding the Negotiation of the Future UK-EU Security Relationship

In this chapter, the authors analyse potential transitional arrangements, discussing the possibilities of substantial delays in—or the no conclusion of—a security agreement, alternative models to the UK-EU relationship, dispute resolution and enforcement mechanisms in light of the UK Government’s red line on CJEU jurisdiction, and the UK’s potential reduced influence on future EU internal security