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Rennecke, Felix (2014): O(d,d) Target-space duality in string theory. Dissertation, LMU München: Fakultät für Physik



In this thesis various aspects of target-space duality in closed bosonic string theory are studied. It begins by introducing generalized geometry as the main mathematical framework. In analogy to general relativity with the Riemannian metric as dynamical quantity, a unified description for string backgrounds – Riemannian metrics together with Kalb-Ramond two-form fields – is approached via Courant algebroids on the generalized tangent bundle equipped with a generalized metric. The dual background configuration, i.e. a metric and a bivector field, is described by the generalized cotangent bundle. The absence of a conventional curvature tensor and consequently the problem of defining generalized gravity theories on Courant algebroids is investigated in detail. This leads to the introduction of Lie algebroids whose differential geometry is suitable for the formulation of gravity theories. Different such theories are shown to be interrelated by appropriate homomorphisms. This proves to be useful for describing non-geometric backgrounds. Target-space duality is introduced in terms of O(d,d)-duality which identifies two-dimensional non-linear sigma models for different string backgrounds as physically equivalent under certain conditions: The backgrounds and coordinates of the dual theories have to be related by certain O(d, d) transformations. In particular, integrability conditions of the dual coordinates are formulated in terms of Courant algebroids. Apart from (non-abelian) T-duality, O(d,d)-duality contains the novel Poisson-duality induced by Poisson structures. T- and Poisson-duality are applied to the three-torus with constant H-flux which shows the existence of non-geometric backgrounds. The latter exceed conventional conceptions of geometry as they cannot be described globally. The problem of describing non-geometric backgrounds is approached with generalizes geometry. A unified description of T-dual backgrounds is given in terms of proto-Lie bialgebroids – one for the geometric sector and another for the non-geometric one. They combine into a Courant algebroid whose anomalous Jacobi identity provides conditions for the concurrent appearance of dual fluxes. The absence of a gravity theory leads to the restriction to Lie algebroids. Their gravity theories allow for a global description of non-geometric backgrounds by an exact prescription for the patching of these backgrounds. The description extends to all possible supergravity theories. The question whether a unified description of dual backgrounds is possible is reconsidered in a manifestly T-duality invariant conformal field theory approach. Dual coordinates are treated on equal footing. Modular invariance of the one-loop partition function together with the premise of physical intermediate states in four-tachyon scattering inevitably leads to the appearance of the strong constraint of double field theory on non-compact spaces. Toroidally compactified directions do not require a constraint. This explains the appearance of the strong constraint and justifies possible attenuations.