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Advancements in double & exceptional field theory on group manifolds
Advancements in double & exceptional field theory on group manifolds
This thesis deals with new backgrounds and concepts in Double Field Theory (DFT), a T-Duality invariant reformulation of supergravity (SUGRA). It is an effective theory capturing the dynamics of a closed string on a torus. For a consistent framework, the theory requires to add D winding coordinates to the D physical spacetime coordinates and gives rise to a doubled space. An important constraint for the consistency of the theory is the strong constraint. After imposing this constraint, all fields are only allowed to depend on half the coordinates. We begin by reviewing the basic concepts and notions of DFT. With regard to this context, we consider generalized diffeomorphisms, implementing the local diffeomorphisms and gauge transformations from SUGRA, and their associated gauge algebra which is governed by the C-bracket. In this setting, we examine the importance of the strong constraint for the closure of the gauge algebra. Subsequently, we investigate the action, in both the generalized metric formulation and the flux formulation, and its underlying symmetries. Afterwards, we turn to Double Field Theory on group manifolds (DFT_{WZW}),a generalization of DFT, whose worldsheet description is governed by a Wess-Zumino-Witten model on a group manifold. In order to obtain an action and the gauge transformations, Closed String Field Theory (CSFT) computations at tree level up to cubic order in fields and leading order in alpha' have to be performed. Again, we consider generalized diffeomorphisms and their gauge algebra, which closes under a modified strong constraint. From this setup, it is going to become clear that original DFT and DFT_{WZW} differ on a very fundamental level. However, they are related to each other. All these steps allow us to recast DFT_{WZW} in terms of doubled generalized objects by extrapolating it to all orders in fields. It yields a generalized metric formulation and a flux formulation of the theory. Although, in contrast to original DFT the fluxes split into a background and a fluctuation part, while the background generalized vielbein takes on the role of the twist in the usual generalized Scherk-Schwarz ansatz. In this thesis, we are going to study the underlying symmetries and field equations for both formulations. A striking difference between between DFT_{WZW} and original DFT lies in the appearance of an additional 2D-diffeomorphism invariance under the standard Lie derivative. On top of this, we observe the emergence of an additional extended strong constraint, which when imposed, reduces DFT_{WZW} to original DFT and both theories become equivalent while the 2D-diffeomorphism invariance breaks down. Following these steps, one can perform a generalized Scherk-Schwarz compactification ansatz to recover the bosonic subsector of half-maximal, electrically gauged supergravities. Moreover, we are going to solve the long standing problem of constructing a twist for each embedding tensor solution by using Maurer-Cartan forms to derive an appropriate background vielbein. Last but not least, we generalize the ideas and notions from DFT_{WZW} to geometric Exceptional Field Theory (gEFT). Subsequently, we show a procedure which allows for the construction of generalized parallelizable spaces in dim M = 4 SL(5) Exceptional Field Theory (EFT). These spaces permit a unified treatment of consistent maximally supersymmetric truncations of ten- and eleven-dimensional supergravity in Generalized Geometry (GG) and their construction has always been an open question. Furthermore, they admit a generalized frame field over the coset M = G/H reproducing the Lie algebra g of G under the generalized Lie derivative. Therefore, we identify the group manifold G with the extended space of the EFT. In the next step, the section condition (SC) needs to be solved to remove unwanted, unphysical directions from this extended space. Finally, we construct the generalized frame field using a left invariant Maurer Cartan form on G. All of these steps cast additional constraints on the groups G and H.
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Bosque, Pascal du
2017
Englisch
Universitätsbibliothek der Ludwig-Maximilians-Universität München
Bosque, Pascal du (2017): Advancements in double & exceptional field theory on group manifolds. Dissertation, LMU München: Fakultät für Physik
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Abstract

This thesis deals with new backgrounds and concepts in Double Field Theory (DFT), a T-Duality invariant reformulation of supergravity (SUGRA). It is an effective theory capturing the dynamics of a closed string on a torus. For a consistent framework, the theory requires to add D winding coordinates to the D physical spacetime coordinates and gives rise to a doubled space. An important constraint for the consistency of the theory is the strong constraint. After imposing this constraint, all fields are only allowed to depend on half the coordinates. We begin by reviewing the basic concepts and notions of DFT. With regard to this context, we consider generalized diffeomorphisms, implementing the local diffeomorphisms and gauge transformations from SUGRA, and their associated gauge algebra which is governed by the C-bracket. In this setting, we examine the importance of the strong constraint for the closure of the gauge algebra. Subsequently, we investigate the action, in both the generalized metric formulation and the flux formulation, and its underlying symmetries. Afterwards, we turn to Double Field Theory on group manifolds (DFT_{WZW}),a generalization of DFT, whose worldsheet description is governed by a Wess-Zumino-Witten model on a group manifold. In order to obtain an action and the gauge transformations, Closed String Field Theory (CSFT) computations at tree level up to cubic order in fields and leading order in alpha' have to be performed. Again, we consider generalized diffeomorphisms and their gauge algebra, which closes under a modified strong constraint. From this setup, it is going to become clear that original DFT and DFT_{WZW} differ on a very fundamental level. However, they are related to each other. All these steps allow us to recast DFT_{WZW} in terms of doubled generalized objects by extrapolating it to all orders in fields. It yields a generalized metric formulation and a flux formulation of the theory. Although, in contrast to original DFT the fluxes split into a background and a fluctuation part, while the background generalized vielbein takes on the role of the twist in the usual generalized Scherk-Schwarz ansatz. In this thesis, we are going to study the underlying symmetries and field equations for both formulations. A striking difference between between DFT_{WZW} and original DFT lies in the appearance of an additional 2D-diffeomorphism invariance under the standard Lie derivative. On top of this, we observe the emergence of an additional extended strong constraint, which when imposed, reduces DFT_{WZW} to original DFT and both theories become equivalent while the 2D-diffeomorphism invariance breaks down. Following these steps, one can perform a generalized Scherk-Schwarz compactification ansatz to recover the bosonic subsector of half-maximal, electrically gauged supergravities. Moreover, we are going to solve the long standing problem of constructing a twist for each embedding tensor solution by using Maurer-Cartan forms to derive an appropriate background vielbein. Last but not least, we generalize the ideas and notions from DFT_{WZW} to geometric Exceptional Field Theory (gEFT). Subsequently, we show a procedure which allows for the construction of generalized parallelizable spaces in dim M = 4 SL(5) Exceptional Field Theory (EFT). These spaces permit a unified treatment of consistent maximally supersymmetric truncations of ten- and eleven-dimensional supergravity in Generalized Geometry (GG) and their construction has always been an open question. Furthermore, they admit a generalized frame field over the coset M = G/H reproducing the Lie algebra g of G under the generalized Lie derivative. Therefore, we identify the group manifold G with the extended space of the EFT. In the next step, the section condition (SC) needs to be solved to remove unwanted, unphysical directions from this extended space. Finally, we construct the generalized frame field using a left invariant Maurer Cartan form on G. All of these steps cast additional constraints on the groups G and H.