Double Field Theory Research Articles

The different faces of branes in double field theory

We show how the Wess-Zumino terms of the different branes in string theory can be embedded within double field theory. Crucial ingredients in our construction are the identification of the correct brane charge tensors and the use of the double field theory potentials that arise from dualizing the standard double field theory fields. This leads to a picture where under T-duality the brane does not change its worldvolume directions but where, instead, it shows different faces depending on whether some of the worldvolume and/or transverse directions invade the winding space. As a non-trivial by-product we show how the different Wess-Zumino terms are modified when the brane propagates in a background with a non-zero Romans mass parameter. Furthermore, we show that for non-zero mass parameter the brane creation process, when one brane passes through another brane, gets generalized to brane configurations that involve exotic branes as well.

Non-Abelian T-duality as a transformation in Double Field Theory

Non-Abelian T-duality (NATD) is a solution generating transformation for supergravity backgrounds with non-Abelian isometries. We show that NATD can be de-scribed as a coordinate dependent O(d,d) transformation, where the dependence on the coordinates is determined by the structure constants of the Lie algebra associated with the isometry group. Besides making calculations significantly easier, this approach gives a natural embedding of NATD in Double Field Theory (DFT), a framework which provides an O(d,d) covariant formulation for effective string actions. As a result of this embedding, it becomes easy to prove that the NATD transformed backgrounds solve supergravity equations, when the isometry algebra is unimodular. If the isometry algebra is non-unimodular, the generalized dilaton field is forced to have a linear dependence on the dual coordinates, which implies that the resulting background solves generalized supergravity equations.

Type II DFT solutions from Poisson–Lie $T$-duality/plurality

Abstract String theory has $T$-duality symmetry when the target space has Abelian isometries. A generalization of $T$-duality, where the isometry group is non-Abelian, is known as non-Abelian $T$-duality, which works well as a solution-generating technique in supergravity. In this paper we describe non-Abelian $T$-duality as a kind of $\text{O}(D,D)$ transformation when the isometry group acts without isotropy. We then provide a duality transformation rule for the Ramond–Ramond fields by using the technique of double field theory (DFT). We also study a more general class of solution-generating technique, the Poisson–Lie (PL) $T$-duality or $T$-plurality. We describe the PL $T$-plurality as an $\text{O}(n,n)$ transformation and clearly show the covariance of the DFT equations of motion by using the gauged DFT. We further discuss the PL $T$-plurality with spectator fields, and study an application to the $\text{AdS}_5\times\text{S}^5$ solution. The dilaton puzzle known in the context of the PL $T$-plurality is resolved with the help of DFT.

Heterotic Kerr-Schild double field theory and classical double copy

We discuss the generalization of the Kerr-Schild (KS) formalism for general relativity and double field theory (DFT) to the heterotic DFT and supergravity. We first introduce a heterotic KS ansatz by introducing a pair of null O (d, d+G) generalized tangent vectors. The pair of null vectors are represented by a pair of d-dimensional vector fields, and one of the vector fields is not a null vector. This implies that the null property of the usual KS formalism, which plays a crucial role in linearizing the field equations, can be partially relaxed in a consistent way. We show that the equations of motion under the heterotic KS ansatz in a flat background can be reduced to linear equations. Using the heterotic KS equations, we establish the single and zeroth copy for heterotic supergravity and derive the Maxwell and Maxwell-scalar equations. This agrees with the KLT relation for heterotic string theory.

Non-Riemannian geometry of M-theory

We construct a background for M-theory that is moduli free. This background is then shown to be related to a topological phase of the E8(8) exceptional field theory (ExFT). The key ingredient in the construction is the embedding of non-Riemannian geometry in ExFT. This allows one to describe non-relativistic geometries, such as Newton-Cartan or Gomis-Ooguri-type limits, using the ExFT framework originally developed to describe maximal supergravity. This generalises previous work by Morand and Park in the context of double field theory.

Higher Gauge Structures in Double and Exceptional Field Theory

AbstractWe review the higher gauge symmetries in double and exceptional field theory from the viewpoint of an embedding tensor construction. This is based on a (typically infinite‐dimensional) Lie algebra and a choice of representation R. The embedding tensor is a map from the representation space R into satisfying a compatibility condition (‘quadratic constraint'). The Lie algebra structure on is transported to a Leibniz–Loday algebra on R, which in turn gives rise to an ‐structure. We review how the gauge structures of double and exceptional field theory fit into this framework.

Pre‐NQ Manifolds and Correspondence Spaces: the Nilmanifold Example

AbstractCourant algebroids correspond to degree‐2 symplectic differential graded manifolds or NQ‐manifolds for short. We review how a similar construction shows that locally the gauge structure of Double Field Theory corresponds to degree‐2 symplectic pre‐NQ manifolds. To illustrate first steps towards a global understanding of the pre‐NQ case, we apply the local constructions to 3‐dimensional nilmanifolds carrying an abelian gerbe. These are among prime examples where T‐duality is well‐understood and allow us to investigate classic results in the graded language.

Extended Riemannian geometry III: global Double Field Theory with nilmanifolds

We describe the global geometry, symmetries and tensors for Double Field Theory over pairs of nilmanifolds with fluxes or gerbes. This is achieved by a rather straightforward application of a formalism we developed previously. This formalism constructs the analogue of a Courant algebroid over the correspondence space of a T-duality, using the language of graded manifolds, derived brackets and we use the description of nilmanifolds in terms of periodicity conditions rather than local patches. The strong section condition arises purely algebraically, and we show that for a particularly symmetric solution of this condition, we recover the Courant algebroids of both nilmanifolds with fluxes. We also discuss the finite, global symmetries of general local Double Field Theory and explain how this specializes to the case of T-duality between nilmanifolds.

General relativity as a biconformal gauge theory

We consider the conformal group of a space of dimn=p+q, with SO(p,q) metric. The quotient of this group by its homogeneous Weyl subgroup gives a principal fiber bundle with 2n-dim base manifold and Weyl fibers. The Cartan generalization to a curved 2n-dim geometry admits an action functional linear in the curvatures. Because symmetry is maintained between the translations and the special conformal transformations in the construction, these spaces are called biconformal; this same symmetry gives biconformal spaces overlapping structures with double field theories, including manifest T-duality. We establish that biconformal geometry is a form of double field theory, showing how general relativity with integrable local scale invariance arises from its field equations. While we discuss the relationship between biconformal geometries and the double field theories of T-dual string theories, our principal interest is the study of the gravity theory. We show that vanishing torsion and vanishing co-torsion solutions to the field equations overconstrain the system, implying a trivial biconformal space. With co-torsion unconstrained, we show that (1) the torsion-free solutions are foliated by copies of an n-dim Lie group, (2) torsion-free solutions generically describe locally scale-covariant general relativity with symmetric, divergence-free sources on either the co-tangent bundle of n-dim (p,q)-spacetime or the torus of double field theory, and (3) torsion-free solutions admit a subclass of spacetimes with n-dim non-abelian Lie symmetry. These latter cases include the possibility of a gravity-electroweak unification. It is notable that the field equations reduce all curvature components to dependence only on the solder form of an n-dim Lagrangian submanifold, despite the increased number of curvature components and doubled number of initial independent variables.

A Kaluza–Klein Approach to Double and Exceptional Field Theory

AbstractWe examine the challenge of viewing all the fields in supergravity as arising from a Kaluza–Klein like dimensional reduction of some higher‐dimensional theory. This gives rise to what is known as exceptional field theory or double field theory. A particular emphasis is placed on following the Kaluza–Klein intuition leading to the identification of charged states and a reinterpretation of the central charges. We further give a description of the novel extended geometry as a generalised phase space and the relationship to string and M‐theory theory and the notion of quantization.

Kaluza–Klein reduction on a maximally non-Riemannian space is moduli-free

We propose a novel Kaluza–Klein scheme which assumes the internal space to be maximally non-Riemannian, meaning that no Riemannian metric can be defined for any subspace. Its description is only possible through Double Field Theory but not within supergravity. We spell out the corresponding Scherk–Schwarz twistable Kaluza–Klein ansatz, and point out that the internal space prevents rigidly any graviscalar moduli. Plugging the same ansatz into higher-dimensional pure Double Field Theory and also to a known doubled-yet-gauged string action, we recover heterotic supergravity as well as heterotic worldsheet action. In this way, we show that 1) supergravity and Yang–Mills theory can be unified into higher-dimensional pure Double Field Theory, free of moduli, and 2) heterotic string theory may have a higher-dimensional non-Riemannian origin.

T -dual cosmological solutions in double field theory. II.

In this paper we present cosmological solutions of Double Field Theory in the supergravity frame and in the winding frame which are related via T-duality. In particular, we show that the solutions can be viewed without the need of complexifying the cosmological scale factor.

A Unique Connection for Born Geometry

It has been known for a while that the effective geometrical description of compactified strings on $d$-dimensional target spaces implies a generalization of geometry with a doubling of the sets of tangent space directions. This generalized geometry involves an $O(d,d)$ pairing $\eta$ and an $O(2d)$ generalized metric $\mathcal{H}$. More recently it has been shown that in order to include T-duality as an effective symmetry, the generalized geometry also needs to carry a phase space structure or more generally a para-Hermitian structure encoded into a skew-symmetric pairing $\omega$. The consistency of string dynamics requires this geometry to satisfy a set of compatibility relations that form what we call a Born geometry. In this work we prove an analogue of the fundamental theorem of Riemannian geometry for Born geometry. We show that there exists a unique connection which preserves the Born structure $(\eta,\omega,\mathcal{H})$ and which is torsionless in a generalized sense. This resolves a fundamental ambiguity that is present in the double field theory formulation of effective string dynamics.

Symmetry enhancement interpolation, non-commutativity and Double Field Theory

We present a moduli dependent target space effective field theory action for (truncated) heterotic string toroidal compactifications. When moving continuously along moduli space, the stringy gauge symmetry enhancement-breaking effects, which occur at particular points of moduli space, are reproduced.Besides the expected fields, originating in the ten dimensional low energy effective theory, new vector and scalar fields are included. These fields depend on “double periodic coordinates” as usually introduced in Double Field Theory. Their mode expansion encodes information about string states, carrying winding and KK momenta, associated to gauge symmetry enhancements. It is found that a non-commutative product, which introduces an intrinsic non-commutativity on the compact target space, is required in order to make contact with string theory amplitude results.

Doubled aspects of generalised dualities and integrable deformations

The worldsheet theories that describe Poisson-Lie T-dualisable σ-models on group manifolds as well as integrable η, λ and β-deformations provide examples of ℰ-models. Here we show how such ℰ-models can be given an elegant target space description within Double Field Theory by specifying explicitly generalised frame fields forming an algebra under the generalised Lie derivative. With this framework we can extract simple criteria for the R/R fields and the dilaton that extend the ℰ-model conditions to type II backgrounds. In particular this gives conditions for a type II background to be Poisson-Lie T-dualisable. Our approach gives rise to algebraic field equations for Poisson-Lie symmetric spacetimes and provides an effective tool for their study.

T -dual cosmological solutions in double field theory

We investigate the cosmological solutions coming from the double field theory equations of motion after coupling a matter source to them. Assuming constant dilaton and imposing the section condition with respect to the regular coordinates leads to a universe dominated by momentum modes while imposing the section condition with respect to the dual coordinates naturally leads to a universe dominated by momentum modes. We show that both regimes have asymptotic behaviours related by T-duality. This hints towards defining a duality between the clocks considered in each regime and interpreting winding modes as being radiation from the point of view of a Euclidean time.

L\u221e algebras and tensor hierarchies in Exceptional Field Theory and Gauged Supergravity

We show how the gauge and field structure of the tensor hierarchies in Double and E7(7) Exceptional Field Theory fits into L∞ algebras. Special attention is paid to redefinitions, the role of covariantly constrained fields and intertwiners. The results are connected to Gauged Supergravities through generalized Scherk-Schwarz reductions. We find that certain gauging-dependent parameters generate trivial gauge transformations, giving rise to novel symmetries for symmetries that are absent in their ungauged counterparts.

Algebroid structures on para-Hermitian manifolds

We present a global construction of a so-called D-bracket appearing in the physics literature of Double Field Theory (DFT) and show that if certain integrability criteria are satisfied, it can be seen as a sum of two Courant algebroid brackets. In particular, we show that the local picture of the extended space-time used in DFT fits naturally in the geometrical framework of para-Hermitian manifolds and that the data of an (almost) para-Hermitian manifold is sufficient to construct the D-bracket. Moreover, the twists of the bracket appearing in DFT can be interpreted in this framework geometrically as a consequence of certain deformations of the underlying para-Hermitian structure.

Exotic branes in Exceptional Field Theory: E7(7) and beyond

In recent years, it has been widely argued that the duality transformations of string and M-theory naturally imply the existence of so-called ‘exotic branes’ — low codimension objects with highly non-perturbative tensions, scaling as gsα for α ≤ −3. We argue that their intimate link with these duality transformations make them an ideal object of study using the general framework of Double Field Theory (DFT) and Exceptional Field Theory (EFT) — collectively referred to as ExFT. Parallel to the theme of dualities, we also stress that these theories unify known solutions in string- and M-theory into a single solution under ExFT. We argue that not only is there a natural unifying description of the lowest codimension objects, many of these exotic states require this formalism as a consistent supergravity description does not exist.

Semi-doubled gauged linear sigma model for five-branes of codimension two

We establish a double dualization in two-dimensional supersymmetric gauge theory. We construct a gauged linear sigma model (GLSM) which contains a complex twisted linear superfield coupled to two sets of Abelian vector superfields. In the IR regime, the GLSM provides a string sigma model whose target spaces are a defect NS5-brane, a Kaluza-Klein vortex and an exotic 522-brane. All of them are five-branes of codimension two and are related by T-duality. This model is a natural extension of the GLSM proposed by Tong which gives a sigma model for an H-monopole, i.e., a smeared NS5-brane of codimension three. This is also regarded as an alternative system of the GLSM for exotic five-branes proposed by the present authors. In this analysis, we confirm that the T-duality transformation procedure in terms of the complex twisted linear superfield is applicable to dualize both the real and imaginary parts of the twisted chiral superfield even at the UV level, beyond the IR limit. This indicates that the T-duality transformations at finite gauge couplings can be performed in terms of reducible superfields in the same way as irreducible (twisted) chiral superfields. Furthermore, we study quantum vortex corrections to the GLSM at the UV level. In the IR limit, these corrections are mapped to string worldsheet instanton corrections to the five-branes of codimension two. The result completely agrees with those in double field theory analysis.