The spacetime of double field theory: Review, remarks, and outlook
TL;DR: In this paper, the authors review double field theory with emphasis on the doubled spacetime and its generalized coordinate transformations, which unify diffeomorphisms and b-field gauge transformations.
Abstract: We review double field theory (DFT) with emphasis on the doubled spacetime and its generalized coordinate transformations, which unify diffeomorphisms and b-field gauge transformations. We illustrate how the composition of generalized coordinate transformations fails to associate. Moreover, in dimensional reduction, the O(d,d) T-duality transformations of fields can be obtained as generalized diffeomorphisms. Restricted to a half-dimensional subspace, DFT includes ‘generalized geometry’, but is more general in that local patches of the doubled space may be glued together with generalized coordinate transformations. Indeed, we show that for certain T-fold backgrounds with nongeometric fluxes, there are generalized coordinate transformations that induce, as gauge symmetries of DFT, the requisite O(d,d;Z) monodromy transformations. Finally we review recent results on the α ′ extension of DFT which, reduced to the half-dimensional subspace, yields intriguing modifications of the basic structures of generalized geometry.
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TL;DR: In this article, the Kaluza-Klein vector field acts as a gauge field for the E-bracket, requiring the presence of 2-forms akin to the tensor hierarchy of gauged supergravity.
Abstract: We present the details of the recently constructed ${\mathrm{E}}_{6(6)}$-covariant extension of 11-dimensional supergravity. This theory requires a $5+27$-dimensional spacetime in which the ``internal'' coordinates transform in the $\overline{\mathbf{27}}$ of ${\mathrm{E}}_{6(6)}$. All fields are ${\mathrm{E}}_{6(6)}$ tensors and transform under (gauged) internal generalized diffeomorphisms. The ``Kaluza-Klein'' vector field acts as a gauge field for the ${\mathrm{E}}_{6(6)}$-covariant ``E-bracket'' rather than a Lie bracket, requiring the presence of 2-forms akin to the tensor hierarchy of gauged supergravity. We construct the complete and unique action that is gauge invariant under generalized diffeomorphisms in the internal and external coordinates. The theory is subject to covariant section constraints on the derivatives, implying that only a subset of the extra 27 coordinates is physical. We give two solutions of the section constraints: the first preserves GL(6) and embeds the action of the complete (i.e. untruncated) 11-dimensional supergravity; the second preserves $\text{GL}(5)\ifmmode\times\else\texttimes\fi{}\text{SL}(2)$ and embeds complete type IIB supergravity. As a byproduct, we thus obtain an off-shell action for type IIB supergravity.
348 citations
TL;DR: In this paper, the authors developed exceptional field theory for E8p8q, defined on a (3+248)-dimensional gener- alized spacetime with extended coordinates in the adjoint representation of E8 p8q. The theory consistently comprises components of the dual graviton encoded in the 248 bein.
Abstract: We develop exceptional field theory for E8p8q, defined on a (3+248)-dimensional gener- alized spacetime with extended coordinates in the adjoint representation of E8p8q. The fields transform under E 8p8q generalized diffeomorphisms and are subject to covariant section constraints. The bosonic fields include an 'internal' dreibein and an E8p8q-valued 'zweihundertachtundvierzigbein' (248-bein). Crucially, the theory also features gauge vectors for the E8p8q E-bracket governing the generalized diffeomorphism algebra and covariantly constrained gauge vectors for a separate but constrained E8p8q gauge sym- metry. The complete bosonic theory, with a novel Chern-Simons term for the gauge vectors, is uniquely determined by gauge invariance under internal and external gen- eralized diffeomorphisms. The theory consistently comprises components of the dual graviton encoded in the 248-bein. Upon picking particular solutions of the constraints the theory reduces to D " 11 or type IIB supergravity, for which the dual graviton becomes pure gauge. This resolves the dual graviton problem, as we discuss in detail.
309 citations
TL;DR: In this paper, the exceptional field theory for the group E 7 (7 ), based on a ( 4 + 56 ) -dimensional spacetime subject to a covariant section condition, is introduced.
Abstract: We introduce the exceptional field theory for the group E 7 ( 7 ) , based on a ( 4 + 56 ) -dimensional spacetime subject to a covariant section condition. The “internal” generalized diffeomorphisms of the coordinates in the fundamental representation of E 7 ( 7 ) are governed by a covariant “E-bracket,” which is gauged by 56 vector fields. We construct the complete and unique set of field equations that is gauge invariant under generalized diffeomorphisms in the internal and external coordinates. Among them are featured the non-Abelian twisted self-duality equations for the 56 gauge vectors. We discuss the explicit solutions of the section condition describing the embedding of the full, untruncated 11-dimensional and type IIB supergravity, respectively. As a new feature compared to the previously constructed E 6 ( 6 ) formulation, some components among the 56 gauge vectors descend from the 11-dimensional dual graviton but nevertheless allow for a consistent coupling by virtue of a covariantly constrained compensating 2-form gauge field.
273 citations
Cites background from "The spacetime of double field theor..."
... are subject to a covariant section constraint which implies that only a subset of the 56 internal coordinates is physical. The constraint can be written in terms of the E7p7q generators ptαq MN 1See [9] for a review and further references. 2 Such generalized spacetimes also appear in the proposal of [12]. 1 in the fundamental representation, and the invariant symplectic form ΩMN of E7p7q Ă Spp56q, a...
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TL;DR: In this article, the generalized Scherk-Schwarz reduction ansatz for the full supersymmetric exceptional eld theory in terms of group valued twist matrices subject to consistency equations is presented.
Abstract: We present the generalized Scherk-Schwarz reduction ansatz for the full supersymmetric exceptional eld theory in terms of group valued twist matrices subject to consistency equations. With this ansatz the eld equations precisely reduce to those of lowerdimensional gauged supergravity parametrized by an embedding tensor. We explicitly construct a family of twist matrices as solutions of the consistency equations. They induce gauged supergravities with gauge groups SOpp;qq and CSOpp;q;rq. Geometrically, they describe compactications on internal spaces given by spheres and (warped) hyperboloides H p;q , thus extending the applicability of generalized Scherk-Schwarz reductions beyond homogeneous spaces. Together with the dictionary that relates exceptional eld
193 citations
TL;DR: In this article, the degrees of freedom in a d-dimensional CFT can be reorganized in an insightful way by studying observables on the moduli space of causal diamonds (or equivalently, the space of pairs of timelike separated points).
Abstract: We argue that the degrees of freedom in a d-dimensional CFT can be reorganized in an insightful way by studying observables on the moduli space of causal diamonds (or equivalently, the space of pairs of timelike separated points). This 2d-dimensional space naturally captures some of the fundamental nonlocality and causal structure inherent in the entanglement of CFT states. For any primary CFT operator, we construct an observable on this space, which is defined by smearing the associated one-point function over causal diamonds. Known examples of such quantities are the entanglement entropy of vacuum excitations and its higher spin generalizations. We show that in holographic CFTs, these observables are given by suitably defined integrals of dual bulk fields over the corresponding Ryu-Takayanagi minimal surfaces. Furthermore, we explain connections to the operator product expansion and the first law of entanglemententropy from this unifying point of view. We demonstrate that for small perturbations of the vacuum, our observables obey linear two-derivative equations of motion on the space of causal diamonds. In two dimensions, the latter is given by a product of two copies of a two-dimensional de Sitter space. For a class of universal states, we show that the entanglement entropy and its spin-three generalization obey nonlinear equations of motion with local interactions on this moduli space, which can be identified with Liouville and Toda equations, respectively. This suggests the possibility of extending the definition of our new observables beyond the linear level more generally and in such a way that they give rise to new dynamically interacting theories on the moduli space of causal diamonds. Various challenges one has to face in order to implement this idea are discussed.
175 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
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
TL;DR: In this paper, target space duality and discrete symmetries in string theory are reviewed in different settings, and the authors present a generalization of the duality to string theory.
1,084 citations
"The spacetime of double field theor..." refers background in this paper
...In closed string theory, however, the field content is uniquely determined and closely related to the T-duality group O(d, d;Z) that is well known to emerge [1] when the theory is put on a torus background T ....
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TL;DR: In this article, the effect of duality transformations on the geometries of a subclass of two-dimensional sigma-models was investigated, and it was shown that duality preserves quantum conformal invariance at order [α′]0, where α′ is the string tension parameter, provided the change induced by duality is accompanied by a shift in the dilaton field.
1,060 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
"The spacetime of double field theor..." refers methods in this paper
...In fact, the gauge transformations to cubic order have been derived from closed string field theory [4], and there is a unique way to make them background independent as transformations of gij and bij [6]....
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