Classical yang-baxter equationfrom duality covariant formulation of string theory

Abstract

The aim of the thesis is to study the homogeneous Yang-Baxter (YB) deformation proposed in the physics literature for a generic Green-Schwarz sigma model from a geometric point of view. It has been shown that these kind of deformations are generated by a certain kind of non-constant O(d,d) transformation, called β transformation, which acts as solution generating transformations in string theory. We study the construction of such an O(d,d) transformation from a bi-vector field related to the Poisson structure on the manifold. It is a well-known fact that there is a Lie algebroid structure on the cotangent bundle of the manifold when there is a Poisson structure on the manifold. Moreover, this Lie algebroid structure is compatible with the standard Lie algebroid structure on the tangent bundle, so that there is a Courant algebroid structure on the direct sum of the tangent and cotangent bundle (called the generalized tangent bundle) of the manifold. We also study Courant algebroid structures in order to understand and to generalize the transformation and the YB deformation. Given a Lie algebra with a non-degenerate inner product, if there exists an endomorphism R, which satisfies the classical Yang-Baxter equation (CYBE), then the direct sum of the Lie algebra and its dual has a natural Drinfel'd structure. Such an endomorphism can be extended to the tangent bundle of the integral Lie group by the help of the adjoint action. In this way, an automorphism called the dressed R-matrix can be constructed, which satisfies the CYBE since the adjoint action is an automorphism of the Lie bracket. It is possible to build a Poisson bi-vector field on the manifold from the dressed R-matrix. It can be shown that the Schouten-Nijenhuis bracket of the bi-vector field with itself vanishes following directly from the fact that the dressed R-matrix satisfies CYBE. The Lie algebroid structure on the cotangent bundle induced from the Poisson structure is compatible with the standard Lie algebroid structure on the tangent bundle. Then the tangent and cotangent bundles with the stated Lie algebroid structures form a Lie bialgebroid, which is an example of a triangular Lie bialgebroid. The Drinfel'd double of the resulting triangular Lie bialgebroid is a Courant algebroid with transversal Dirac structures. This geometrical structure plays a prominent role in the solution generating mechanism stated above. The dynamical fields in the universal sector of the low energy effection action of string theory are the Riemannian metric, a 2-form field called the B-field and a scalar field called the dilaton field. The first two of these fields become the constituents of the generalized metric, which is a tensor on the generalized tangent bundle T M ⊕ T^{∗}M that transforms naturally under O(d,d). There is a O(d,d) covariant version of string theory, called Double Field Theory (DFT), which is written in terms of the generalized metric and the generalized dilaton field. DFT provides a suitable framework to demonstrate the fact that YB deformation preserves the solutions of string theory. From a geometric point of view, the existence of a generalized metric is equivalent to the existence of a subbundle of the generalized tangent bundle on which the inner product is positive definite. If one starts with a generalized metric of a specific form that solves the field equations of DFT in the limit in which it reduces to the field equations of supergravity and transforms it with the O(d,d) matrix generating the YB deformation, the resulting generalized metric also solves the field equations of DFT in the same limit. In the physics literature, the proof of this is based on comparing the "fluxes" before and after the transformation and showing that these fluxes do not change. From a geometrical point of view the fluxes are just the "structure functions" of the Courant algebroid structure on the generalized tangent bundle, when a specific basis is chosen for the sections of tangent and cotangent bundles. In order to understand this "flux preservation" principle from a geometrical point of view, we also study the axioms defining a Courant algebraid in local coordinates. We also work out in detail the case where the anchor of the Courant algebroid is determined by a bi-vector field associated by the YB deformation.