# Generalized metric formulation of double field theory on group manifolds

TL;DR: In this paper, a generalized metric formulation of double field theory on group manifolds was proposed and shown to be invariant under generalized dieomorphisms and 2D-dieomorphisms.

Abstract: We rewrite the recently derived cubic action of Double Field Theory on group manifolds (1) in terms of a generalized metric and extrapolate it to all orders in the fields. For the resulting action, we derive the field equations and state them in terms of a generalized curvature scalar and a generalized Ricci tensor. Compared to the generalized metric formulation of DFT derived from tori, all these quantities receive additional contributions related to the non-trivial background. It is shown that the action is invariant under its generalized dieomorphisms and 2D-dieomorphisms . Imposing additional constraints relating the background and fluctuations around it, the precise relation between the proposed generalized metric formulation of DFTWZW and of original DFT from tori is clarified. Furthermore, we show how to relate DFTWZW of the WZW background with the flux formulation of original DFT.

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TL;DR: In this paper, the authors define generalized coordinate transformations in double field theory as generalized Lie derivatives of the coordinate maps of a field and use Courant bracketing to construct generalized coordinate transformation.

Abstract: Finite gauge transformations in double field theory can be defined by the exponential of generalized Lie derivatives. We interpret these transformations as ‘generalized coordinate transformations’ in the doubled space by proposing and testing a formula that writes large transformations in terms of derivatives of the coordinate maps. Successive generalized coordinate transformations give a generalized coordinate transformation that differs from the direct composition of the original two. Instead, it is constructed using the Courant bracket. These transformations form a group when acting on fields but, intriguingly, do not associate when acting on coordinates.

176 citations

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TL;DR: In this article, a general review of extended supergravities and their gauging using the duality-covariant embedding tensor formalism is given, and an overview of the gauging procedure and related tensor hierarchy in higher-dimensional models is given.

Abstract: We give a general review of extended supergravities and their gauging using the duality-covariant embedding tensor formalism. Although the focus is on four-dimensional theories, an overview of the gauging procedure and the related tensor hierarchy in the higher-dimensional models is given. The relation of gauged supergravities to flux compactifications is discussed and examples are worked out in detail.

146 citations

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TL;DR: In this article, a duality-invariant pseudo-action formulation for the U-duality group SO(5, 5) is presented, which after reduction gives the maximal D = 6 supergravity.

Abstract: We construct Exceptional Field Theory for the group SO(5, 5) based on the extended (6+16)-dimensional spacetime, which after reduction gives the maximal D = 6 supergravity. We present both a true action and a duality-invariant pseudo-action formulations. All the fields of the theory depend on the complete extended spacetime. The U-duality group SO(5, 5) is made a geometric symmetry of the theory by virtue of introducing the generalised Lie derivative that incorporates a duality transformation. Tensor hierarchy appears as a natural consequence of the algebra of generalised Lie derivatives that are viewed as gauge transformations. Upon truncating different subsets of the extra coordinates, maximal supergravities in D = 11 and D = 10 (type IIB) can be recovered from this theory.

125 citations

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TL;DR: In this paper, the authors construct an exceptional field theory for the group $SO(5,5)$ based on the extended (6+16)-dimensional spacetime, which after reduction gives the maximal $D=6$ supergravity.

Abstract: We construct Exceptional Field Theory for the group $SO(5,5)$ based on the extended (6+16)-dimensional spacetime, which after reduction gives the maximal $D=6$ supergravity. We present both a true action and a duality-invariant pseudo-action formulations. All the fields of the theory depend on the complete extended spacetime. The U-duality group $SO(5,5)$ is made a geometric symmetry of the theory by virtue of introducing the generalised Lie derivative that incorporates a local duality transformation. Tensor hierarchy appears as a natural consequence of the algebra of generalised Lie derivatives that are viewed as gauge transformations. Upon truncating different subsets of the extra coordinates, maximal supergravities in $D=11$ and $D=10$ (type IIB) can be recovered from this theory.

122 citations

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TL;DR: In this paper, the duality invariant theory for the metric, b-field, and dilaton is defined and the Friedmann equations for all orders in α′ are analyzed.

Abstract: While the classification of α′ corrections of string inspired effective theories remains an unsolved problem, we show how to classify duality invariant α′ corrections for purely time-dependent (cosmological) backgrounds. We determine the most general duality invariant theory to all orders in α′ for the metric, b-field, and dilaton. The resulting Friedmann equations are studied when the spatial metric is a time-dependent scale factor times the Euclidean metric and the b-field vanishes. These equations can be integrated perturbatively to any order in α′. We construct nonperturbative solutions and display duality invariant theories featuring string-frame de Sitter vacua.

114 citations

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TL;DR: In this paper, the concept of a generalized Kahler manifold has been introduced, which is equivalent to a bi-Hermitian geometry with torsion first discovered by physicists.

Abstract: Generalized complex geometry, as developed by Hitchin, contains complex and symplectic geometry as its extremal special cases. In this thesis, we explore novel phenomena exhibited by this geometry, such as the natural action of a B-field. We provide new examples, including some on manifolds admitting no known complex or symplectic structure. We prove a generalized Darboux theorem which yields a local normal form for the geometry. We show that there is an elliptic deformation theory and establish the existence of a Kuranishi moduli space.
We then define the concept of a generalized Kahler manifold. We prove that generalized Kahler geometry is equivalent to a bi-Hermitian geometry with torsion first discovered by physicists. We then use this result to solve an outstanding problem in 4-dimensional bi-Hermitian geometry: we prove that there exists a Riemannian metric on the complex projective plane which admits exactly two distinct Hermitian complex structures with equal orientation.
Finally, we introduce the concept of generalized complex submanifold, and show that such sub-objects correspond to D-branes in the topological A- and B-models of string theory.

1,380 citations

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TL;DR: A geometrical structure on even-dimensional manifolds is defined in this paper, which generalizes the notion of a Calabi-Yau manifold and also a symplectic manifold.

Abstract: A geometrical structure on even-dimensional manifolds is defined which generalizes the notion of a Calabi–Yau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action both of diffeomorphisms and closed 2-forms. In the special case of six dimensions we characterize them as critical points of a natural variational problem on closed forms, and prove that a local moduli space is provided by an open set in either the odd or even cohomology. We introduce in this paper a geometrical structure on a manifold which generalizes both the concept of a Calabi–Yau manifold—a complex manifold with trivial canonical bundle—and that of a symplectic manifold. This is possibly a useful setting for the background geometry of recent developments in string theory; but this was not the original motivation for the author’s first encounter with this structure. It arose instead as part of a programme (following the papers [ 11, 12]) for characterizing special geometry in low dimensions by means of invariant functionals of differential forms. In this respect, the dimension six is particularly important. This paper has two aims, then: first to introduce the general concept, and then to look at the variational and moduli space problem in the special case of six dimensions. We begin with the definition in all dimensions of what we call generalized complex manifolds and generalized Calabi–Yau manifolds .

1,275 citations

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TL;DR: In this article, a theory of massless fields on the doubled torus was constructed, which includes usual and dual diffeomorphisms, together with a T-duality acting on fields that have explicit dependence on the torus coordinates and the dual coordinates.

Abstract: The zero modes of closed strings on a torus — the torus coordinates plus dual coordinates conjugate to winding number — parameterize a doubled torus. In closed string field theory, the string field depends on all zero-modes and so can be expanded to give an infinite set of fields on the doubled torus. We use string field theory to construct a theory of massless fields on the doubled torus. Key to the consistency is a constraint on fields and gauge parameters that arises from the L0−0 = 0 condition in closed string theory. The symmetry of this double field theory includes usual and `dual diffeomorphisms', together with a T-duality acting on fields that have explicit dependence on the torus coordinates and the dual coordinates. We find that, along with gravity, a Kalb-Ramond field and a dilaton must be added to support both usual and dual diffeomorphisms. We construct a fully consistent and gauge invariant action on the doubled torus to cubic order in the fields. We discuss the challenges involved in the construction of the full nonlinear theory. We emphasize that the doubled geometry is physical and the dual dimensions should not be viewed as an auxiliary structure or a gauge artifact.

952 citations

### "Generalized metric formulation of d..." refers background in this paper

...derivation [3] and it states that winding and momentum excitations in the same direction are not allowed....

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...The calculations shown in this subsection generalize in some sense the endeavor of [5] to find a background independent version of the cubic DFT action derived in [3]....

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TL;DR: This work extends spacetime duality to superspace, including fermions in the low-energy limits of superstrings, and the tangent space is a curved, extended superspace based on an enlarged coordinate space where the vanishing of the d'Alembertian is as fundamental as the disappearing of the curl of a gradient.

Abstract: We extend spacetime duality to superspace, including fermions in the low-energy limits of superstrings. The tangent space is a curved, extended superspace. The geometry is based on an enlarged coordinate space where the vanishing of the d'Alembertian is as fundamental as the vanishing of the curl of a gradient.

869 citations

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TL;DR: In this article, a T-fold background with a local $n$-torus fibration and T-duality transition functions in O(n,n:\Z) space is formulated in an enlarged space with a $T^{2n}$ fibration which is geometric, with spacetime emerging locally from a choice of a Ωn$ submanifold of each fiber, so that it is embedded in the enlarged space.

Abstract: A geometric string solution has background fields in overlapping coordinate patches related by diffeomorphisms and gauge transformations, while for a non-geometric background this is generalised to allow transition functions involving duality transformations. Non-geometric string backgrounds arise from T-duals and mirrors of flux compactifications, from reductions with duality twists and from asymmetric orbifolds. Strings in ` T-fold' backgrounds with a local $n$-torus fibration and T-duality transition functions in $O(n,n:\Z)$ are formulated in an enlarged space with a $T^{2n}$ fibration which is geometric, with spacetime emerging locally from a choice of a $T^n$ submanifold of each $T^{2n}$ fibre, so that it is a subspace or brane embedded in the enlarged space. T-duality acts by changing to a different $T^n$ subspace of $T^{2n}$. For a geometric background, the local choices of $T^n$ fit together to give a spacetime which is a $T^n$ bundle, while for non-geometric string backgrounds they do not fit together to form a manifold. In such cases spacetime geometry only makes sense locally, and the global structure involves the doubled geometry. For open strings, generalised D-branes wrap a $T^n$ subspace of each $T^{2n}$ fibre and the physical D-brane is the part of the part of the physical space lying in the generalised D-brane subspace.

816 citations

### "Generalized metric formulation of d..." refers background in this paper

...In this case so called non-geometric backgrounds [12, 13, 16, 17] arise....

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