Double Field Theory Research Articles

Local \u03b2-deformations and Yang-Baxter sigma model

Homogeneous Yang-Baxter (YB) deformation of AdS5 × S5 superstring is revisited. We calculate the YB sigma model action up to quadratic order in fermions and show that homogeneous YB deformations are equivalent to β-deformations of the AdS5 ×S5 background when the classical r-matrices consist of bosonic generators. In order to make our discussion clearer, we discuss YB deformations in terms of the double-vielbein formalism of double field theory. We further provide an O(10, 10)-invariant string action that reproduces the Green-Schwarz type II superstring action up to quadratic order in fermions. When an AdS background contains a non-vanishing H-flux, it is not straightforward to perform homogeneous YB deformations. In order to get any hint for such YB deformations, we study β-deformations of H-fluxed AdS backgrounds and obtain various solutions of (generalized) type II supergravity.

Generalised fluxes, Yang-Baxter deformations and the O(d,d) structure of non-abelian T -duality

Based on the construction of Poisson-Lie T -dual σ-models from a common parent action we study a candidate for the non-abelian respectively Poisson-Lie T -duality group. This group generalises the well-known abelian T -duality group O(d, d) and we explore some of its subgroups, namely factorised dualities, B- and β-shifts. The corresponding duality transformed σ-models are constructed and interpreted as generalised (non-geometric) flux backgrounds.We also comment on generalisations of results and techniques known from abelian T -duality. This includes the Lie algebra cohomology interpretation of the corresponding non-geometric flux backgrounds, remarks on a double field theory based on non-abelian T -duality and an application to the investigation of Yang-Baxter deformations. This will show that homogeneously Yang-Baxter deformed σ-models are exactly the non-abelian T -duality β-shifts when applied to principal chiral models.

T-duality and non-geometric fluxes in double field theory

T-duality and non-geometric fluxes in double field theory

The topology of Double Field Theory

We describe the doubled space of Double Field Theory as a group manifold G with an arbitrary generalized metric. Local information from the latter is not relevant to our discussion and so G only captures the topology of the doubled space. Strong Constraint solutions are maximal isotropic submanifold M in G. We construct them and their Generalized Geometry in Double Field Theory on Group Manifolds. In general, G admits different physical subspace M which are Poisson-Lie T-dual to each other. By studying two examples, we reproduce the topology changes induced by T-duality with non-trivial H-flux which were discussed by Bouwknegt, Evslin and Mathai [1].

Extended Riemannian geometry II: local heterotic double field theory

We continue our exploration of local Double Field Theory (DFT) in terms of symplectic graded manifolds carrying compatible derivations and study the case of heterotic DFT. We start by developing in detail the differential graded manifold that captures heterotic Generalized Geometry which leads to new observations on the generalized metric and its twists. We then give a symplectic pre-NQ-manifold that captures the symmetries and the geometry of local heterotic DFT. We derive a weakened form of the section condition, which arises algebraically from consistency of the symmetry Lie 2-algebra and its action on extended tensors. We also give appropriate notions of twists — which are required for global formulations — and of the torsion and Riemann tensors. Finally, we show how the observed α′-corrections are interpreted naturally in our framework.

Point particle motion in double field theory and a singularity-free cosmological solution

We generalize the action for point particle motion to a double field theory background. After deriving the general equations of motion for these particle geodesics, we specialize to the case of a cosmological background with vanishing antisymmetric tensor field and constant dilaton. We then show that the geodesics can be extended to infinity in both time directions once we define the appropriate physical clock. Following this prescription, we also show the existence of a singularity-free cosmological solution.

Five-brane actions in double field theory

We construct an action for NSNS 5-branes which is manifestly covariant under O(d, d). This is done by doubling d of the spacetime coordinates which appear in the worldvolume action. By formulating the DBI part of the action in a manner similar to a “gauged sigma model”, only half the doubled coordinates genuinely appear. Our approach allows one to describe the full T-duality orbit of the IIB NS5 brane, the IIA KKM and their exotic relations in one formalism. Furthermore, by using ideas from double field theory, our action can be said to describe various aspects of non-geometric five-branes.

Massive deformations of Type IIA theory within double field theory

We obtain massive deformations of Type IIA supergravity theory through duality twisted reductions of Double Field Theory (DFT) of massless Type II strings. The mass deformation is induced through the reduction of the DFT of the RR sector. Such reductions are determined by a twist element belonging to Spin+(10, 10), which is the duality group of the DFT of the RR sector. We determine the form of the twists and give particular examples of twists matrices, for which a massive deformation of Type IIA theory can be obtained. In one of the cases, requirement of gauge invariance of the RR sector implies that the dilaton field must pick up a linear dependence on one of the dual coordinates. In another case, the choice of the twist matrix violates the weak and the strong constraints explicitly in the internal doubled space.

Nonassociative differential geometry and gravity with non-geometric fluxes

We systematically develop the metric aspects of nonassociative differential geometry tailored to the parabolic phase space model of constant locally non-geometric closed string vacua, and use it to construct preliminary steps towards a nonassociative theory of gravity on spacetime. We obtain explicit expressions for the torsion, curvature, Ricci tensor and Levi-Civita connection in nonassociative Riemannian geometry on phase space, and write down Einstein field equations. We apply this formalism to construct R-flux corrections to the Ricci tensor on spacetime, and comment on the potential implications of these structures in non-geometric string theory and double field theory.

Extended geometries

We present a unified and completely general formulation of extended geometry, characterised by a Kac-Moody algebra and a highest weight coordinate module. Generalised diffeomorphisms are constructed, as well as solutions to the section constraint. Generically, additional (“ancillary”) gauge transformations are present, and we give a concrete criterion determining when they appear. A universal form of the (pseudo-)action determines the dynamics in all cases without ancillary transformations, and also for a restricted set of cases based on the adjoint representation of a finite-dimensional simple Lie group. Our construction reproduces (the internal sector of) all previously considered cases of double and exceptional field theories.

Generalised kinematics for double field theory

We formulate a kinematical extension of Double Field Theory on a 2d-dimensional para-Hermitian manifold left(mathcal{P},eta, omega right) where the O(d, d) metric η is supplemented by an almost symplectic two-form ω. Together η and ω define an almost bi-Lagrangian structure K which provides a splitting of the tangent bundle Tmathcal{P}=Loplus tilde{L} into two Lagrangian sub-spaces. In this paper a canonical connection and a corresponding generalised Lie derivative for the Leibniz algebroid on Tmathcal{P} are constructed. We find integrability conditions under which the symmetry algebra closes for general η and ω, even if they are not flat and constant. This formalism thus provides a generalisation of the kinematical structure of Double Field Theory. We also show that this formalism allows one to reconcile and unify Double Field Theory with Generalised Geometry which is thoroughly discussed.

Exceptional field theories, superparticles in an enlarged 11D superspace and higher spin theories

Recently proposed exceptional field theories (EFTs) making manifest the duality En(n) symmetry, first observed as nonlinearly realized symmetries of the maximal d=3,4,...,9 supergravity (n=11−d) and containing 11D and type IIB supergravity as sectors, were formulated in enlarged spacetimes. In the case of E7(7) EFT such an enlarged spacetime can be identified with the bosonic body of the d=4 central charge superspace Σ(60|32), the N=8d=4 superspace completed by 56 additional bosonic coordinates associated to central charges of the maximal d=4 supersymmetry algebra.In this paper we show how the hypothesis on the relation of all the known En(n) EFTs, including n=8, with supersymmetry leads to the conjecture on existence of 11D exceptional field theory living in 11D tensorial central charge superspace Σ(528|32) and underlying all the En(n) EFTs with n=2,...,8, and probably the double field theory (DFT). We conjecture the possible form of the section conditions of such an 11D EFT and show that quite generic solutions of these can be generated by superparticle models the ground states of which preserve from one half to all but one supersymmetry. The properties of these superparticle models are briefly discussed. We argue that, upon quantization, their quantum states should describe free massless non-conformal higher spin fields in D=11. We also discuss some relevant representations of the M-theory superalgebra which, in the present context, describes supersymmetry of the 11D EFT.

Double Field Theory description of heterotic gauge symmetry enhancing-breaking

A Double Field Theory (DFT) description of gauge symmetry enhancing-breaking in the heterotic string is presented. The construction, based on previous results for the bosonic string, relies on the extension of the tangent frame of DFT. The fluxes of a Scherk-Schwarz like generalized toroidal compactification are moduli dependent and become identified with the structure constants of the enhanced group at fixed “self-dual” points in moduli space. Slight displacements from such points provide the breaking of the symmetry, gauge bosons acquiring masses proportional to fluxes. The inclusion of fermions is also discussed.

Half‐Maximal Supersymmetry from Exceptional Field Theory

AbstractWe study‐dimensional half‐maximal flux backgrounds using exceptional field theory. We define the relevant generalised structures and also find the integrability conditions which give warped half‐maximal MinkowskiDand AdSDvacua. We then show how to obtain consistent truncations of type II / 11‐dimensional SUGRA which break half the supersymmetry. Such truncations can be defined on backgrounds admitting exceptional generalisedstructures, where, andNis the number of vector multiplets obtained in the lower‐dimensional theory. Our procedure yields the most general embedding tensors satisfying the linear constraint of half‐maximal gauged SUGRA. We use this to prove that allhalf‐maximal warped AdSDand MinkowskiDvacua of type II / 11‐dimensional SUGRA admit a consistent truncation keeping only the gravitational supermultiplet. We also show to obtain heterotic double field theory from exceptional field theory and comment on the M‐theory / heterotic duality. In five dimensions, we find a new SO(5,N) double field theory with a‐dimensional extended space. Its section condition has one solution corresponding to 10‐dimensionalsupergravity and another yielding six‐dimensionalSUGRA.

Particle actions and brane tensions from double and exceptional geometry

Massless particles in n + 1 dimensions lead to massive particles in n dimensions on Kaluza-Klein reduction. In string theory, wrapped branes lead to multiplets of massive particles in n dimensions, in representations of a duality group G. By encoding the masses of these particles in auxiliary worldline scalars, also transforming under G, we write an action which resembles that for a massless particle on an extended spacetime. We associate this extended spacetime with that appearing in double field theory and exceptional field theory, and formulate a version of the action which is invariant under the generalised diffeomorphism symmetry of these theories. This provides a higher-dimensional perspective on the origin of mass and tension in string theory and M-theory. Finally, we consider the reduction of exceptional field theory on a twisted torus, which is known to give the massive IIA theory of Romans. In this case, our particle action leads naturally to the action for a D0 brane in massive IIA. Here an extra vector field is present on the worldline, whose origin in exceptional field theory is a vector field introduced to ensure invariance under generalised diffeomorphisms.

Classification of non-Riemannian doubled-yet-gauged spacetime

Assuming mathbf {O}(D,D) covariant fields as the ‘fundamental’ variables, double field theory can accommodate novel geometries where a Riemannian metric cannot be defined, even locally. Here we present a complete classification of such non-Riemannian spacetimes in terms of two non-negative integers, (n,bar{n}), 0le n+bar{n}le D. Upon these backgrounds, strings become chiral and anti-chiral over n and bar{n} directions, respectively, while particles and strings are frozen over the n+bar{n} directions. In particular, we identify (0, 0) as Riemannian manifolds, (1, 0) as non-relativistic spacetime, (1, 1) as Gomis–Ooguri non-relativistic string, (D{-1},0) as ultra-relativistic Carroll geometry, and (D, 0) as Siegel’s chiral string. Combined with a covariant Kaluza–Klein ansatz which we further spell, (0, 1) leads to Newton–Cartan gravity. Alternative to the conventional string compactifications on small manifolds, non-Riemannian spacetime such as D=10, (3, 3) may open a new scheme for the dimensional reduction from ten to four.

O(d+1, d+1) enhanced double field theory

Double field theory yields a formulation of the low-energy effective action of bosonic string theory and half-maximal supergravities that is covariant under the T-duality group O(d, d) emerging on a torus Td. Upon reduction to three spacetime dimensions and dualisation of vector fields into scalars, the symmetry group is enhanced to O(d+1, d+1). We construct an enhanced double field theory with internal coordinates in the adjoint representation of O(d + 1, d + 1). Its section constraints admit two inequivalent solutions, encoding in particular the embedding of D = 6 chiral and non-chiral theories, respectively. As an application we define consistent generalized Scherk-Schwarz reductions using a novel notion of generalized parallelization. This allows us to prove the consistency of the truncations of D = 6, mathcal{N}=left(1,1right) and D = 6, mathcal{N}=left(2,0right) supergravity on {mathrm{AdS}}_3times {mathbb{S}}^3 .

On Inflation and de Sitter in Non‐Geometric String Backgrounds

AbstractWe study the problem of obtaining de Sitter and inflationary vacua from dimensional reduction of double field theory (DFT) on nongeometric string backgrounds. In this context, we consider a new class of effective potentials that admit Minkowski and de Sitter minima. We then construct a simple model of chaotic inflation arising from T‐fold backgrounds and we discuss the possibility of trans‐Planckian field range from nongeometric monodromies as well as the conditions required to get slow roll.

U-duality in three and four dimensions

Using generalised geometry we study the action of U-duality acting in three and four dimensions on the bosonic fields of 11-dimensional supergravity. We compare the U-duality symmetry with the T-duality symmetry of double field theory and see how the [Formula: see text] and [Formula: see text] U-duality groups reduce to the [Formula: see text] and [Formula: see text] T-duality symmetry groups of the type IIA theory. As examples we dualise M2-branes, both black and extreme. We find that uncharged black M2-branes become charged under U-duality, generalising the Harrison transformation, while extreme M2-branes will become new extreme M2-branes. The resulting tension and charges are quantised appropriately if we use the discrete U-duality group [Formula: see text].

Duality twisted reductions of Double Field Theory of Type II strings

We study duality twisted reductions of the Double Field Theory (DFT) of the RR sector of massless Type II theory, with twists belonging to the duality group Spin+(10, 10). We determine the action and the gauge algebra of the resulting theory and determine the conditions for consistency. In doing this, we work with the DFT action constructed by Hohm, Kwak and Zwiebach, which we rewrite in terms of the Mukai pairing: a natural bilinear form on the space of spinors, which is manifestly Spin(n, n) invariant. If the duality twist is introduced via the Spin+(10, 10) element S in the RR sector, then the NS-NS sector should also be deformed via the duality twist U = ρ(S), where ρ is the double covering homomorphism between Pin(n, n) and O(n, n). We show that the set of conditions required for the consistency of the reduction of the NS-NS sector are also crucial for the consistency of the reduction of the RR sector, owing to the fact that the Lie algebras of Spin(n, n) and SO(n, n) are isomorphic. In addition, requirement of gauge invariance imposes an extra constraint on the fluxes that determine the deformations.