T-duality Transformations Research Articles

O(9, 25) symmetry of heterotic string theory at orders α′, α′2

In a recent study, we have observed that by imposing a truncated T-duality transformation on the circular reduction of the bosonic couplings in the heterotic theory at four- and six-derivative orders, we can calculate these couplings in a particular YM gauge where the YM potential vanishes but its field strength remains non-zero. Importantly, the coupling constants are independent of the gauge choice, so these results are valid across different YM gauge choices.In this work, we explore the cosmological reduction of these couplings when the YM gauge fields belong to the Cartan subalgebra of SO(32) or E8 × E8. We demonstrate that after applying appropriate one-dimensional field redefinitions and total derivative terms, the couplings can be expressed in a proposed O(9, 25)-invariant canonical form, which is the extension of the canonical O(9, 9)-invariant form for just the NS-NS fields proposed by Hohm and Zwiebach. This O(9, 25)-invariant expression is in terms of the trace of the first time derivative of the generalized metric, which encompasses both the YM field and the NS-NS fields.

A hierarchy of WZW models related to super Poisson–Lie T-duality

Motivated by super Poisson–Lie (PL) symmetry of the Wess–Zumino–Witten (WZW) model based on the (C3+A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(C^3+A)$$\\end{document} Lie supergroup of our previous work (Eghbali et al., in J High Energy Phys 07:134, 2013. arXiv:1303.4069 [hep-th]), we first obtain and classify all Drinfeld superdoubles (DSDs) generated by the Lie superbialgebra structures on the (C3+A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\mathscr {C}}^3+ {\\mathscr {A}})$$\\end{document} Lie superalgebra as a theorem. Then, introducing a general formulation we find the conditions under which a two-dimensional σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document}-model may be equivalent to a WZW model. With the help of this formulation and starting the super PL symmetric (C3+A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(C^3+A)$$\\end{document} WZW model, we get a hierarchy of WZW models related to super PL T-duality, in such a way that it is different from the super PL T-plurality, because the DSDs are, in this process, non-isomorphic. The most interesting indication of this work is that the (C3+A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(C^3+A)$$\\end{document} WZW model does remain invariant under the super PL T-duality transformation, that is, the model is super PL self-dual.

Equivariant Topological T-Duality

Topological T-duality is a relationship between pairs (E, P) over a fixed space X, where E→X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E \\rightarrow X$$\\end{document} is a principal torus bundle and P→E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P \\rightarrow E$$\\end{document} is a twist, such as a gerbe for a principal PU(H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$PU({\\mathcal {H}})$$\\end{document}-bundle. This is of interest to topologists because of the T-duality transformation: a T-duality relation between pairs (E, P) and (E^,P^)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\hat{E}}, {\\hat{P}})$$\\end{document} comes with an isomorphism (with degree shift) between the twisted K-theory of E and the twisted K-theory of E^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\hat{E}}$$\\end{document}. We formulate topological T-duality for circle bundles in the equivariant setting, following the definition of Bunke, Rumpf, and Schick. We define the T-duality transformation in equivariant K-theory and show that it is an isomorphism for all compact Lie groups, equal to its own inverse and uniquely characterized by naturality and a normalization for trivial situations.

Worldsheet formalism for decoupling limits in string theory

We study the bosonic sector of a decoupling limit of type IIA superstring theory, where a background Ramond-Ramond one-form is fined tuned to its critical value, such that it cancels the associated background D0-brane tension. The light excitations in this critical limit are D0-branes, whose dynamics is described by the Banks-Fischler-Shenker-Susskind (BFSS) Matrix theory that corresponds to M-theory in the Discrete Light-Cone Quantization (DLCQ). We develop the worldsheet formalism for the fundamental string in the same critical limit of type IIA superstring theory. We show that the fundamental string develops singularities on its worldsheet, whose topology is described by nodal Riemann spheres as in ambitwistor string theory. We study the T-duality transformations of this string sigma model and provide a worldsheet derivation for the recently revived and expanded duality web that unifies a zoo of decoupling limits in type II superstring theories. By matching the string worldsheet actions, we demonstrate how some of these decoupling limits are related to tensionless (and ambitwistor) string theory, Carrollian string theory, the Spin Matrix limits of the AdS/CFT correspondence, and more.

Boosted rotating dyonic strings in Salam-Sezgin model

We show that the bosonic sector of the N = (1, 0), 6D Salam-Sezgin gauged supergravity model possesses a T-duality symmetry upon a circle reduction to D = 5. We then construct a simple magnetic rotating string solution with two equal angular momenta. Applying the T-duality transformation to this solution, we obtain the general boosted rotating dyonic black string solutions whose global structures and thermodynamic quantities are also analyzed. Owing to the fact that the solutions are not asymptotically flat, we find that there are two distinct globally-different non-extremal solutions with two different sets of thermal dynamic variables, with both satisfying the thermodynamic first law and the corresponding Small relations. However, their BPS limit becomes the same and we show that it preserves one quarter of supersymmetry by directly solving the corresponding Killing spinor equations.

T-duality relations between hyperkähler and bi-hypercomplex structures

We investigate the T-duality relations between hyperkähler and bi-hypercomplex structures using the doubled formalism. In generalized geometry, both the hyperkähler and bi-hypercomplex structures are embedded in generalized hyperkähler structures that satisfy the split-bi-quaternion algebra. We write down the analogue of the Buscher rule, which is the T-duality transformation of the hyperkähler and bi-hypercomplex structures. As a practical example, we construct the bi-hypercomplex structure of the -brane, known as a T-fold, from the hyperkähler structure of the Taub-NUT space using the T-duality transformation. The bi-hypercomplex structures of the T-fold have non-trivial monodromies. This results in the fact that the worldsheet instantons on the T-fold are multi-valued. We comment on the resolution of this issue using the Born sigma model.

The data on the boundary at order alpha '

The least action principle indicates that for the open spacetime manifolds, there are data on the boundary. Recently, it has been proposed that the data for the effective actions at order alpha ' are the values of the massless fields and their first derivatives. These data should be respected by the T-duality transformations at order alpha '. Moreover, the T-duality transformations should not change the unit vector to the boundary which in turns implies that the base space metric should be also invariant. Assuming such restricted T-duality transformations, we show that the transformation of the circular reduction of the parity-odd part of the effective action of the heterotic string theory at order alpha ' under the Buscher rules is cancelled by some total derivative terms and by some restricted T-duality transformations at order alpha '. Using the Stokes’ theorem, we then show that the boundary terms in the base space corresponding to the total derivative terms are exactly cancelled by transformation of the circular reduction of the Gibbons–Hawking boundary term under the above restricted T-duality transformations. These calculations confirm the above proposal for the data on the boundary for the effective actions at order alpha '.

Cylinder quantum field theories at small coupling

We show that any 2D scalar field theory compactified on a cylinder and with a Fourier expandable potential V is equivalent, in the small coupling limit, to a 1D theory involving a massless particle in a potential V and an infinite tower of free massive Kaluza-Klein (KK) modes. Moving slightly away from the deep IR region has the effect of switching on interactions between the zero mode and the KK modes, whose strength is controlled by powers of the coupling, hence making the interactions increasingly suppressed. We take the notable example of Liouville field theory and, starting from its worldline version, we compute the torus (one-loop) partition function perturbatively in the coupling constant. The partition function at leading order is invariant under a T-duality transformation that maps the radius of the cylinder to its inverse and rescales it by the square of the Schwinger parameter of the cylinder. We show that this behavior is a universal feature of cylinder QFTs.

Higher-derivative field redefinitions in the presence of boundary

Recently it has been proposed that the consistency with T-duality requires the effective action of string theory at order alpha '^n to satisfy the least action principle provided that the values of the massless fields and their derivatives up to order n are known on the boundary. In this paper we speculate that this boundary condition constrains the field redefinitions and the corrections to the T-duality transformations in the presence of boundary, e.g., at order alpha ', the metric does not change, and all other massless fields should change to include only the first derivative of the massless fields. Using the above restricted field redefinitions, we write all gauge invariant bulk and boundary couplings in the bosonic string theory at order alpha ' in a minimal scheme. Then using the assumption that the effective action of string theory at the critical dimension is background independent, we fix the coefficients of the tree-level gauge invariant couplings by imposing O(1, 1) symmetry when the background has a circle and by imposing O(d, d) symmetry when the background has T^d. These constraints fix the bulk action up to an overall factor, and the boundary action up to two parameters. By requiring the gravity couplings in the boundary action to be consistent with those in the Chern-Simons gravity, the two boundary parameters are also fixed. Up to a restricted field redefinition, the bulk and boundary couplings are exactly those in the K.A. Meissner action and its corresponding boundary action.

Non-Abelian T-duality of AdS_{dle 3} families by Poisson-Lie T-duality

We proceed to investigate the non-Abelian T-duality of AdS_{2}, AdS_{2}times S^1 and AdS_{3} physical backgrounds, as well as the metric of the analytic continuation of AdS_{2} from the point of view of Poisson-Lie (PL) T-duality. To this end, we reconstruct these metrics of the AdS families as backgrounds of non-linear sigma -models on two- and three-dimensional Lie groups. By considering the Killing vectors of these metrics and by taking into account the fact that the subgroups of isometry Lie group of the metrics can be taken as one of the subgroups of the Drinfeld double (with Abelian duals) we look up the PL T-duality. To construct the dualizable metrics by the PL T-duality we find all subalgebras of Killing vectors that generate subgroup of isometries which acts freely and transitively on the manifolds defined by aforementioned AdS families. We then obtain the dual backgrounds for these families of AdS in such a way that we apply the usual rules of PL T-duality without further corrections. We have also investigated the conformal invariance conditions of the original backgrounds (AdS families) and their dual counterparts. Finally, by using the T-duality rules proposed by Kaloper and Meissner (KM) we calculate the Abelian T-duals of BTZ black hole up to two-loop by dualizing on the coordinates varphi and t. When the dualizing is implemented by the shift of direction varphi , we show that the horizons and singularity of the dual spacetime are the same as in charged black string derived by Horne and Horowitz without alpha '-corrections, whereas in dualizing on the coordinate t we find a new three-dimensional black string whose structure and asymptotic nature are clearly determined. For this case, we show that the T-duality transformation changes the asymptotic behavior from AdS_3 to flat.

Hyperkähler, bi-hypercomplex, generalized hyperkähler structures and T-duality

We exploit the doubled formalism to study comprehensive relations among T-duality, complex and bi-hermitian structures (J+,J−) in two-dimensional N=(2,2) sigma models with/without twisted chiral multiplets. The bi-hermitian structures (J+,J−) embedded in generalized Kähler structures (J+,J−) are organized into the algebra of the tri-complex numbers. We write down an analogue of the Buscher rule by which the T-duality transformation of the bi-hermitian and Kähler structures are apparent. We also study the bi-hypercomplex and hyperkähler cases in N=(4,4) theories. They are expressed, as a T-duality covariant fashion, in the generalized hyperkähler structures and form the split-bi-quaternion algebras. As a concrete example, we show the explicit T-duality relation between the hyperkähler structures of the KK-monopole (Taub-NUT space) and the bi-hypercomplex structures of the H-monopole (smeared NS5-brane). Utilizing this result, we comment on a T-duality relation for the worldsheet instantons in these geometries.

Nonrelativistic strings and the limits of mathcal{N} = 2 dualities

We carry out the formulation of nonrelativistic (NR) F-string theory for mathcal{N} = 2 Gaiotto-Maldacena class of geometries in type IIA supergravity. Considering the NS-NS sector, we reduce the type IIA theory into the NR sigma models those are formulated over the nonrelativistic target space geometries known as the torsional String Newton-Cartan (TSNC) backgrounds. Using these TSNC data, we write down the sigma model action and explore several properties in the semiclassical limit. In particular, we carry out the canonical analysis of the T-duality transformation rules in TSNC background. Following the standard canonical prescription of obtaining T-duality transformations, we map these TSNC sigma models to relativistic sigma models propagating over general relativity backgrounds. Finally, we conclude with a detailed discussion on the effects of adding flavour branes and thereby taking the corresponding TSNC limit in the bulk description of mathcal{N} = 2 quivers. We compute various field theory observables in the nonrelativistic limit and outline the structure of the NR Lagrangian following the light cone reduction of mathcal{N} = 2 SYM theory.

Non-supersymmetric black holes with α′ corrections

We study extremal 5- and 4-dimensional black hole solutions of the Heterotic Superstring effective action at first order in α′ with, respectively, 3 and 4 charges of arbitrary signs. For a particular choice of the relative signs of these charges the solutions are supersymmetric, but other signs give rise to non-supersymmetric extremal black holes. Although at zeroth order all these solutions are formally identical, we show that the α′ corrections are drastically different depending on how we break supersymmetry. We provide fully analytic mathcal{O} (α′) solutions and we compute their charges, masses and entropies and check that they are invariant under α′-corrected T-duality transformations. The masses of some of these black holes are corrected in a complicated way, but we show that the shift is always negative, in agreement with the Weak Gravity Conjecture. Our formula for the corrected entropy of the four-dimensional black holes generalizes previous results obtained from the entropy function method, as we consider a more general way in which supersymmetry can be broken.

D-brane actions in nonrelativistic string theory and T duality

We study D-brane actions in non-relativistic string theory. We consider single D-brane and analyse its properties under T-duality. We also derive an action for N D8-branes and also for lower dimensional D-branes through T-duality transformations.

The classical double copy for half-maximal supergravities and T-duality

We study the classical double copy for ungauged half-maximal supergravities using the Kaluza-Klein reduction of double field theory (DFT). We construct a general formula for the Kaluza-Klein (KK) reduction of the DFT Kerr-Schild ansatz. The KK reduction of the ansatz is highly nonlinear, but the associated equations of motion are linear. This linear structure implies that half-maximal supergravities admit a classical double copy. We show that their single copy is given by a pair of Maxwell-scalar theories, which are the KK reduction of a higher-dimensional single copy of DFT. We also investigate their T-duality transformations — both the Buscher rule and continuous O(D, D) rotations. Applying the Buscher rule to the Kerr BH, we obtain a solution with a nontrivial Kalb-Ramond field and dilaton. We also identify the single copy of Sen’s heterotic BH and the chiral null model and show that the chiral null model is self-dual under T-duality rotations.

Supergravity solution-generating techniques and canonical transformations of \u03c3-models from O(D, D)

Within the framework of the flux formulation of Double Field Theory (DFT) we employ a generalised Scherk-Schwarz ansatz and discuss the classification of the twists that in the presence of the strong constraint give rise to constant generalised fluxes interpreted as gaugings. We analyse the various possibilities of turning on the fluxes Hijk, Fijk, Qijk and Rijk, and the solutions for the twists allowed in each case. While we do not impose the DFT (or equivalently supergravity) equations of motion, our results provide solution-generating techniques in supergravity when applied to a background that does solve the DFT equations. At the same time, our results give rise also to canonical transformations of 2-dimensional σ-models, a fact which is interesting especially because these are integrability-preserving transformations on the worldsheet. Both the solution-generating techniques of supergravity and the canonical transformations of 2-dimensional σ-models arise as maps that leave the generalised fluxes of DFT and their flat derivatives invariant. These maps include the known abelian/non-abelian/Poisson-Lie T-duality transformations, Yang-Baxter deformations, as well as novel generalisations of them.

Quantum stabilizer codes, lattices, and CFTs

There is a rich connection between classical error-correcting codes, Euclidean lattices, and chiral conformal field theories. Here we show that quantum error-correcting codes, those of the stabilizer type, are related to Lorentzian lattices and non-chiral CFTs. More specifically, real self-dual stabilizer codes can be associated with even self-dual Lorentzian lattices, and thus define Narain CFTs. We dub the resulting theories code CFTs and study their properties. T-duality transformations of a code CFT, at the level of the underlying code, reduce to code equivalences. By means of such equivalences, any stabilizer code can be reduced to a graph code. We can therefore represent code CFTs by graphs. We study code CFTs with small central charge c = n ≤ 12, and find many interesting examples. Among them is a non-chiral E8 theory, which is based on the root lattice of E8 understood as an even self-dual Lorentzian lattice. By analyzing all graphs with n ≤ 8 nodes we find many pairs and triples of physically distinct isospectral theories. We also construct numerous modular invariant functions satisfying all the basic properties expected of the CFT partition function, yet which are not partition functions of any known CFTs. We consider the ensemble average over all code theories, calculate the corresponding partition function, and discuss its possible holographic interpretation. The paper is written in a self-contained manner, and includes an extensive pedagogical introduction and many explicit examples.

T-duality in nonrelativistic open string theory

Nonrelativistic open string theory is defined by a worldsheet theory that produces a Galilean invariant string spectrum and is described at low energies by a nonrelativistic Yang-Mills theory [1]. We study T-duality transformations in the path integral for the sigma model that describes nonrelativistic open string theory coupled to an arbitrary closed string background, described by a string Newton-Cartan geometry, Kalb-Ramond, and dilaton field. We prove that T-duality transformations map nonrelativistic open string theory to relativistic and noncommutative open string theory in the discrete light cone quantization (DLCQ), a quantization scheme relevant for Matrix string theory. We also show how the worldvolume dynamics of nonrelativistic open string theory described by the Dirac-Born-Infeld type action found in [1] maps to the Dirac-Born-Infeld actions describing the worldvolume theories of the DLCQ of open string theory and noncommutative open string theory.

Effective action of type II superstring theories at order \u03b1\u20323: NS-NS couplings

Recently, it has been shown that the minimum number of gauge invariant couplings for B-field, metric and dilaton at order α′3 is 872. These couplings, in a particular scheme, appear in 55 different structures. In this paper, up to an overall factor, we fix all parameters in type II supertirng theories by requiring the reduction of the couplings on a circle to be invariant under T-duality transformations. We find that there are 445 non-zero couplings which appear in 15 different structures. The couplings are fully consistent with the partial couplings that have been found in the literature by the four-point S-matrix element and by the non-linear Sigma model methods.

O-plane couplings at order \u03b1\u20322: one R-R field strength

It is known that the anomalous Chern-Simons (CS) coupling of Op-plane is not consistent with the T-duality transformations. Compatibility of this coupling with the T-duality requires the inclusion of couplings involving one R-R field strength. In this paper we find such couplings at order α′2. By requiring the R-R and NS-NS gauge invariances, we first find all independent couplings at order α′2. There are 1, 6, 28, 20, 19, 2 couplings corresponding to the R-R field strengths F(p−4), F(p−2), F(p), F(p+2), F(p+4) and F(p+6), respectively. We then impose the T-duality constraint on these couplings and on the CS coupling C(p−3) ∧ R ∧ R at order α′2 to fix their corresponding coefficients. The T-duality constraint fixes all coefficients in terms of the CS coefficient. They are fully consistent with the partial couplings that have been already found in the literature by the S-matrix method.