Starting from a sigma-model for a doubled target-space geometry, we show that the number of target-space dimensions can be reduced by half through a gauging procedure. We apply this formalism to a class of backgrounds relevant for double field theory, and illustrate how choosing different gaugings leads to string-theory configurations T-dual to each other. We furthermore discuss that given a conformal doubled theory, the reduced theories are conformal as well.
As an example we consider the three-dimensional SU(2) WZW model and show that the only possible reduced backgrounds are the cigar and trumpet CFTs in two dimensions, which are indeed T-dual to each other.
In arXiv:0902.4032 [hep-th] an O(D,D)-covariant sigma model describing the embedding of a closed world-sheet into the 2D-dimensional twisted torus was proposed. Such sigma models provide a universal description of string theory with target spaces related by the action of T-duality. In this article a six-dimensional toy example is studied in detail. Different polarisations of the six-dimensional target space give different three-dimensional string backgrounds including a nilmanifold with H-flux, a T-fold with R-flux and a new class of T-folds. Global issues and connections with the doubled toru…
We present affine Lie algebras generated by the supercovariant derivatives and the supersymmetry generators for the left and right moving modes in the doubled space. Chirality is manifest in our doubled space as well as the T-duality symmetry. We present gauge invariant bosonic and superstring actions preserving the two-dimensional diffeomorphism invariance and the kappa-symmetry where doubled spacetime coordinates are chiral fields. The doubled space becomes the usual space by dimensional reduction constraints.
We revisit the fundamental string (F1) solution in the doubled formalism. We show that the wave-like solution of double field theory (DFT) corresponding to the F1/pp-wave duality pair is more properly a solution of the DFT action coupled to a doubled sigma model action. The doubled string configuration which sources the pp-wave can be thought of as static gauge with the string oriented in a dual direction. We also discuss the DFT solution corresponding to a vibrating string, carrying both winding and momentum. We further show that the solution dual to the F1 in both time and space can be viewe…
T-duality is one of the essential elements of string theory. Recently, Hull has developed a formalism where the dimension of the target space is doubled so as to make T-duality manifest. This is then supplemented with a constraint equation that allows the connection to the usual string sigma model. This paper analyses the partition function of the doubled formalism by interpreting the constraint equation as that of a chiral scalar and then using holomorphic factorisation techniques to determine the partition function. We find there is quantum equivalence to the ordinary string once the topolog…
The section condition in double field theory has been shown to imply that a physical point should be one-to-one identified with a gauge orbit in the doubled coordinate space. Here we show the converse is also true, and continue to explore the idea of `spacetime being doubled yet gauged'. Introducing an appropriate gauge connection, we construct a string action, with an arbitrary generalized metric, which is completely covariant with respect to the coordinate gauge symmetry, generalized diffeomorphisms, world-sheet diffeomorphisms, world-sheet Weyl symmetry and O(D,D) T-duality. A topological t…
We propose a universal geometric formulation of gauged supergravity in terms of a twisted doubled torus. We focus on string theory (M-theory) reductions with generalized Scherk-Schwarz twists residing in the O(n,n) (E_{7(7)}) duality group. The set of doubled geometric fluxes, associated with the duality twists and identified naturally with the embedding tensor of gauged supergravity, captures all known fluxes, i.e. physical form fluxes, ordinary geometric fluxes, as well as their non-geometric counterparts. Furthermore, we propose a prescription for obtaining the effective geometry embedded i…
We construct a world-sheet action for Green-Schwarz superstring in terms of doubled-yet-gauged spacetime coordinates. For an arbitrarily curved NS-NS background, the action possesses $\mathbf{O}(10,10)$ T-duality, $\mathbf{Spin}(1,9)\times\mathbf{Spin}(9,1)$ Lorentz symmetry, coordinate gauge symmetry, spacetime doubled-yet-gauged diffeomorphisms, world-sheet diffeomorphisms and Weyl symmetry. Further, restricted to flat backgrounds, it enjoys maximal spacetime supersymmetry and kappa-symmetry. After the auxiliary coordinate gauge symmetry potential being integrated out, our action can consist…
We revisit partition functions of closed strings on toroidal backgrounds, including their $\mathbb{Z}_N$ shift orbifolds in the formalism where the dimension of the target space is doubled to make T-duality manifest. In such a T-duality covariant formalism, the constraint equation imposes a form of chiral factorization. Our computation furnishes a non-trivial consistency check for the quantum worldsheet theory of the doubled sigma model, when strings are placed on general toroidal backgrounds. The topological term that mixes the physical space and its T-dual is crucial in demonstrating that ch…
We provide some evidence that closed string coordinates will become non-commutative turning on H-field flux background in closed string compactifications. This is in analogy to open string non-commutativity on the world volume of D-branes with B- and F-field background. The class of 3-dimensional backgrounds we are studying are twisted tori (fibrations of a 2-torus over a circle) and the their T-dual H-field, 3-form flux backgrounds (T-folds). The spatial non-commutativity arises due to the non-trivial monodromies of the toroidal Kahler resp. complex structure moduli fields, when going around …
We derive and analyze the conditions for quantum conformal and Lorentz invariance of the duality symmetric interacting chiral boson sigma-models, which are conjectured to describe non-geometric string theory backgrounds. The one-loop Weyl and Lorentz anomalies are computed for the general case using the background field method. Subsequently, our results are applied to a class of (on-shell) Lorentz invariant chiral boson models which are based on twisted doubled tori. Our findings are in agreement with those expected from the effective supergravity approach, thereby firmly establishing that the…
We construct a (locally) supersymmetric worldsheet action for a string in a non-relativistic Newton-Cartan background. We do this using a doubled string action, which describes the target space geometry in an $O(D,D)$ covariant manner using a doubled metric and doubled vielbeins. By adopting different parametrisations of these doubled background fields, we can describe both relativistic and non-relativistic geometries. We focus on the torsional Newton-Cartan geometry which can be obtained by null duality/reduction (such null duality is particularly simple for us to implement). The doubled acti…
We use a T-duality invariant action to investigate the behaviour of a string in non-geometric backgrounds, where there is a non-trivial global $O(D,D)$ patching or monodromy. This action leads to a set of Dirac brackets describing the dynamics of the doubled string, with these brackets determined only by the monodromy. This allows for a simple derivation of non-commutativity and non-associativity in backgrounds which are (even locally) non-geometric. We focus here on the example of the three-torus with H-flux, finding non-commutativity but not non-associativity, and also comment on the relatio…
We develop doubled-coordinate field theory to determine the α' corrections to the massless sector of oriented bosonic closed string theory. Our key tool is a string current algebra of free left-handed bosons that makes O(D,D) T-duality manifest. While T-dualities are unchanged, diffeomorphisms and b-field gauge transformations receive corrections, with a gauge algebra given by an α'-deformation of the duality-covariantized Courant bracket. The action is cubic in a double metric field, an unconstrained extension of the generalized metric that encodes the gravitational fields. Our approach provi…
The Kaluza-Klein monopole has been recognized as a string background with significant non-geometric features: it must appear "localized" to winding strings to match the NS5-brane's localization on the T-dual circle. In this work, we explicitly construct this T-dual system in the doubled geometry formalism, which proves to successfully describe the duality despite a broken isometry on one side of the duality pair. We further suggest an extension of the doubled formalism to the gauged linear sigma models describing this system (both bosonic and supersymmetric) and show that previous calculations…
String compactifications with T-duality twists are revisited and the gauge algebra of the dimensionally reduced theories calculated. These reductions can be viewed as string theory on T-fold backgrounds, and can be formulated in a `doubled space' in which each circle is supplemented by a T-dual circle to construct a geometry which is a doubled torus bundle over a circle. We discuss a conjectured extension to include T-duality on the base circle, and propose the introduction of a dual base coordinate, to give a doubled space which is locally the group manifold of the gauge group. Special cases …
We clarify the relation between various approaches to the manifestly T-duality symmetric string. We explain in detail how the PST covariant doubled string arises from an unusual gauge fixing. We pay careful attention to the role of "spectator" fields in this process and also show how the T-duality invariant doubled dilaton emerges naturally. We extend these ideas to non-Abelian T-duality and show they give rise to the duality invariant formalism based on the semi-Abelian Drinfeld Double. We then develop the N=(0,1) supersymmetric duality invariant formalism.
A class of special holonomy spaces arise as nilmanifolds fibred over a line interval and are dual to intersecting brane solutions of string theory. Further dualities relate these to T-folds, exotic branes, essentially doubled spaces and spaces with R-flux. We develop the doubled geometry of these spaces, with the various duals arising as different slices of the doubled space.
String backgrounds with a local torus fibration such as T-folds are naturally formulated in a doubled formalism in which the torus fibres are doubled to include dual coordinates conjugate to winding number. Here we formulate and explore a generalisation of this construction in which all coordinates are doubled, so that the doubled space is a twisted torus, i.e. a compact space constructed from identifying a group manifold under a discrete subgroup. This incorporates reductions with duality twists, T-folds and a class of flux compactifications, together with the non-geometric backgrounds expect…
The doubled formulation of string theory, which is T-duality covariant and enlarges spacetime with extra coordinates conjugate to winding number, is reformulated and its geometric and topological features examined. It is used to formulate string theory in T-fold backgrounds with T-duality transition functions and a quantum implementation of the constraints of the doubled formalism is presented. This establishes the quantum equivalence to the usual sigma-model formalism for world-sheets of arbitrary genus, provided a topological term is added to the action. The quantisation involves a local cho…
This paper describes the background field equations for strings in T-duality symmetric formalisms in which the dimension of target space is doubled and the sigma model supplemented with constraints. These are calculated by demanding the vanishing of the beta-functional of the sigma model couplings in the doubled target space. We demonstrate the equivalence with the background field equations of the standard string sigma model.
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 the 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 L_0 - \bar L_0=0 condition in closed string theory. The symmetry of this double field theory includes usual and …
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, $(n,\bar{n})$, $0\leq n+\bar{n}\leq D$. 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-…
The SU(2) WZW model at large level N can be interpreted semiclassically as string theory on S^3 with N units of Neveu-Schwarz H-flux. While globally geometric, the model nevertheless exhibits an interesting doubled geometry possessing features in common with nongeometric string theory compactifications, for example, nonzero Q-flux. Therefore, it can serve as a fertile testing ground through which to improve our understanding of more exotic compactifications, in a context in which we have a firm understanding of the background from standard techniques. Three frameworks have been used to systema…
In this letter we describe how to string together the doubled field approach by Cremmer, Julia, Lu and Pope with Pasti-Sorokin-Tonin technique to construct the sigma-model-like action for type IIA supergravity. The relation of the results with that of obtained in the context of searching for Superstring/M-theory hidden symmetry group is discussed.
It has recently been shown that a set of the generalized type IIB supergravity equations follows from the requirement of kappa symmetry of the type IIB Green-Schwarz superstring theory defined on an arbitrary background. In this paper, we show that the whole bosonic part of the generalized type II supergravity equations can be reproduced from the T-duality covariant equations of motion of the double field theory by choosing a non-standard solution of the strong constraint. Then, by using the doubled formalism, we show the Weyl invariance of the bosonic string sigma model on a generalized gravi…
We construct an action for double field theory using a metric connection that is compatible with both the generalised metric and the O_{D,D} structure. The connection is simultaneously torsionful and flat. Using this connection one may construct a proper covariant derivative for double field theory. We then write the doubled action in terms of the generalised torsion of this connection. This action then exactly reproduces that required for doubled field theory and gauged supergravity.
We perform canonical analysis of non-relativistic string theory with non-relativistic world-sheet gravity. We determine structure of constraints and symplectic structure of canonical variables.
We show that twisted doubled tori can be used to construct a general class of worldsheet models describing non-geometric string backgrounds. By employing a first order formulation of interacting chiral bosons, we first refine the analysis on the general conditions of worldsheet Lorentz invariance and then prove that twisted doubled tori provide good duality symmetric backgrounds. Subsequently we apply our general analysis to several examples which enable us to gain new insight on the difference between geometric, locally geometric and genuine non-geometric backgrounds.
In this paper, we first construct a globally well-defined non-geometric background which contains several branes in type II string theory compactified on a 7-torus. One of these branes is called 5^2_2, which is a codimension-2 object and has a non-trivial monodromy given by a T-duality transformation. The geometry near the 5^2_2-brane is shown to approach the non-geometric background constructed in arXiv:1004.2521. We then construct the solution of a fundamental string rotating along a non-trivial cycle in the 5^2_2 background. Although the background is not axisymmetric in the usual sense, we…
Double Field Theory (DFT) is a low-energy effective theory of a manifestly $O(D,D)$ invariant formulation of the closed string theory when toroidally compactified dimensions are present. The theory is based on a doubled spacetime structure and, in order to preserve the gauge symmetry provided by the invariance under generalized diffeomorphisms, a constraint has to be imposed on fields and gauge parameters. In this paper, we propose a DFT-inspired Supergravity by using a suitable {\em star product} that implements such constraint and the corresponding algebraic structure is explored. We get a c…
This note is devoted to the study of the classical solutions on the bosonic string with euclidean world-sheet in AdS(5)xS(5). We generalize solutions presented in arXiv:0707.4254 [hep-th] to the case where we include the dynamics of the string on S(5) as well.
We define the open string version of the nonlinear sigma model on doubled geometry introduced by Hull and Reid-Edwards, and derive its boundary conditions. These conditions include the restriction of D-branes to maximally isotropic submanifolds as well as a compatibility condition with the Lie algebra structure on the doubled space. We demonstrate a systematic method to derive and classify D-branes from the boundary conditions, in terms of embeddings both in the doubled geometry and in the physical target space. We apply it to the doubled three-torus with constant H-flux and find D0-, D1-, and…
We study some properties of target space strings constructed from (2,1) heterotic strings. We argue that world-sheet complexification is a general property of the bosonic sector of such target world-sheets. We give a target space interpretation of this fact and relate it to the non-gaussian nature of free String Field Theory. We provide several one loop calculations supporting the stringy construction of critical world-sheets in terms of (2,1) models. Using finite temperature boundary conditions in the underlying (2,1) string we obtain non-chiral target space spin structures, and point out som…
The model of the bosonic string with the noncommutative world-sheet geometry is proposed in the framework of Fedosov's deformation quantization. The re-interpretation of the model in terms of bosonic string coupled to infinite multiplet of background fields is given and the link to the W-symmetry is discussed. For the case of d=4 Euclidean target space, the stringy counterparts of the Yang-Mills instantons are constructed in both commutative and noncommutative regimes.
We suggest a new model of string theory with world-sheet supersymmetry. It possesses an additional global fermionic symmetry which is similar in many ways to BRST symmetry. The spectrum consists of massless states of Rarita-Schwinger fields describing infinite tower of half-integer spins.
We probe doubled geometry with dual fundamental branes, i.e. solitons. Restricting ourselves first to solitonic branes with more than two transverse directions we find that the doubled geometry requires an effective wrapping rule for the solitonic branes which is dual to the wrapping rule for fundamental branes. This dual wrapping rule can be understood by the presence of Kaluza-Klein monopoles. Extending our analysis to supersymmetric solitonic branes with less than or equal to two transverse directions we show that such solitons are precisely obtained by applying the same dual wrapping rule …
We examine various properties of double field theory and the doubled string sigma model in the context of geometric quantisation. In particular we look at T-duality as the symplectic transformation related to an alternative choice of polarisation in the construction of the quantum bundle for the string. Following this perspective we adopt a variety of techniques from geometric quantisation to study the doubled space. One application is the construction of the double coherent state that provides the shortest distance in any duality frame and a stringy deformed Fourier transform.
The scaling limit used recently to derive matrix models, and a certain analyticity assumption, are invoked to argue that the agreement between some matrix model calculations and supergravity is a consequence of string world sheet duality.
It is reviewed how space-time supersymmetry is realized nonlinearly in open superstring theory without making the GSO projection. We show that the world-sheet string dualities, viz. dualities of open-closed strings and of open-open strings, play crucial roles for the existence of 10 dimensional N=2 supersymmetry in a spontaneously broken phase. We also speculate on a possible mechanism of the restoration of supersymmetry from the viewpoint of world-sheet dynamics.
The ten-dimensional supergravity theory is a geometric low-energy effective theory and the equations of motion for its fields can be obtained from string theory by computing $β$ functions. With $d$ compact dimensions, we can add to it an $O(d, d;\mathbb{Z})$ geometric structure and construct the supergravity theory inspired by double field theory through the use of a suitable commutative star product. The latter implements the weak constraint of the double field theory on its fields and gauge parameters in order to have a closed gauge symmetry algebra. The consistency of the action here propos…
We consider the (2, 0) supersymmetric theory of tensor multiplets and self-dual strings in six space-time dimensions. Space-time diffeomorphisms that leave the string world-sheet invariant appear as gauge transformations on the normal bundle of the world-sheet. The naive invariance of the model under such transformations is however explicitly broken by anomalies: The electromagnetic coupling of the string to the two-form gauge field of the tensor multiplet suffers from a classical anomaly, and there is also a one-loop quantum anomaly from the chiral fermions on the string world-sheet. Both of …
We study the stochastic motion of a relativistic trailing string in black hole AdS_5. The classical string solution develops a world-sheet horizon and we determine the associated Hawking radiation spectrum. The emitted radiation causes fluctuations on the string both above and below the world-sheet horizon. In contrast to standard black hole physics, the fluctuations below the horizon are causally connected with the boundary of AdS. We derive a bulk stochastic equation of motion for the dual string and use the AdS/CFT correspondence to determine the evolution a fast heavy quark in the strongly…
Most of the non-Abelian string-vortices studied so far are characterized by two-dimensional \cpn models with various degrees of supersymmetry on their world sheet. We generalize this construction to "composite" non-Abelian strings supporting the Grassmann $\mathcal{G}(L,M)$ models (here $L+M=N$). The generalization is straightforward and provides, among other results, a simple and transparent way for counting the number of vacua in ${\mathcal N}=(2,2)$ Grassmannian model.
This thesis investigates the quantum properties of T-duality invariant formalisms of String Theory. We introduce and review duality invariant formalisms of String Theory including the Doubled Formalism. We calculate the background field equations for the Doubled Formalism of Abelian T-duality and show how they are consistent with those of a conventional String Theory description of a toroidal compactification. We generalise these considerations to the case of Poisson--Lie T-duality and show that the system of renormalisation group equations obtained from the duality invariant parent theory are…
We discuss several closely related concepts in the NSR formulation of superstring theory. We demonstrated that recently proposed NSR model for superstrings on $AdS_5 \times S^5$ is described by the world-sheet logarithmic conformal field theory (LCFT). The origin of LCFT on a world-sheet is closely connected to the matter-ghost mixing in the structure of a brane-like vortex operators. We suggest a dynamical origin of M theory as a string theory with an extra dimension given by bosonised superconformal ghosts.
The global U-dualities of extended supergravity have played a central role in differentiating the distinct classes of extremal black hole solutions. When the U-duality group satisfies certain algebraic conditions, as is the case for a broad class of supergravities, the extremal black holes enjoy a further symmetry known as Freudenthal duality (F-duality), which although distinct from U-duality preserves the Bekenstein-Hawking entropy. Here it is shown that, by adopting the doubled Lagrangian formalism, F-duality, defined on the doubled field strengths, is not only a symmetry of the black hole …
We show that relativistic strings of open and closed types in Minkowski space-time of dimension 3 and 4 have topologically stable singular points. This paper describes the structure of singularities, derives their normal forms, and introduces a local characteristic of singularity (topological charge), possessing a global law of conservation. Two other types of solutions (breaking and exotic strings) are also considered, which have singularities at arbitrary value of dimension.
A study of sigma models whose target space is a group G that admits a compatible Poisson structure is presented. The natural action of O(D,D;Z) on the generalised tangent bundle TG+T*G and a generalisation of the Courant bracket that appears are reviewed. This background provides a concrete example where the generalised geometry and doubled geometry descriptions are both well understood. Connections between the two formalisms are discussed and the world-sheet theory from Hamiltonian and Lagrangian perspectives is investigated. The comparisons between the approaches given by generalised geometr…
We study the CPT theorem for a two-dimensional conformal field theory on an arbitrary Riemann surface. On the sphere the theorem follows from the assumption that the correlation functions have standard hermiticity properties and are invariant under the transformation $z \rightarrow 1/z$. The theorem can then be extended to higher genus surfaces by sewing. We show that, as a consequence of the CPT theorem on the world-sheet, the scattering $T$-matrix in string theory is {\sl formally\/} hermitean at any loop order.
In Double Field Theory, the mass-squared of doubled fields associated with bosonic closed string states is proportional to $N_L+N_R-2$. Massless states are therefore not only the graviton, anti-symmetric, and dilaton fields with $(N_L=1, N_R=1)$ such theory is focused on, but also the symmetric traceless tensor and the vector field relative to the states $(N_L=2, N_R=0)$ and $(N_L=0, N_R=2)$ which are massive in the lower-dimensional non-compactified space. While they are not even physical in the absence of compact dimensions, they provide a sample of states for which both momenta and winding …
We investigate classical solutions of string field theory proposed by Takahashi and Tanimoto in the case of even order polynomial functions. The BRS charge and the Feynman propagator of open string field theory expanded around the solution are specified by Jenkins-Strebel quadratic differential, which describes geometry of the string worldsheet. We show that the solution becomes nontrivial when two second order poles of the quadratic differential coincide each other on the unit disk. In this case, an open string boundary shrinks to a point.
We are studying quantum corrections in the earlier proposed string theory based on world-sheet action which measures the linear sizes of the surfaces. At classical level the string tension is equal to zero and as it was demonstrated in the previous studies one loop correction to the classical world-sheet action generates Nambu-Goto area term, that is nonzero string tension. We extend this analysis computing the world-sheet effective action in the second order of the loop expansion.
We have argued previously that black holes may be represented in a D-brane approach by monopole and vortex defects in a sine-Gordon field theory model of Liouville dynamics on the world sheet. Supersymmetrizing this sine-Gordon system, we find critical behaviour in 11 dimensions, due to defect condensation that is the world-sheet analogue of D-brane condensation around an extra space-time dimension in M theory. This supersymmetric description of Liouville dynamics has a natural embedding within a 12-dimensional framework suggestive of F theory.
We show that the vanishing of the one-loop beta-functional of the doubled formalism (which describes string theory on a torus fibration in which the fibres are doubled) is the same as the equation of motion of the recently proposed generalised metric formulation of double field theory restricted to this background: both are the vanishing of a generalised Ricci tensor. That this tensor arises in both backgrounds indicates the importance of a new doubled differential geometry for understanding both constructions.
