The variety of consistent "gauging" deformations of supergravity theories in four dimensions depends on the choice of Lagrangian formulation. One important goal is to get the most general deformations without making hidden assumptions. Ignoring supersymmetry we consider in this paper $n_v$ abelian vector potentials in four spacetime dimensions with non-minimal kinetic coupling to $n_s$ uncharged (possibly nonlinear) scalar fields. As in the case of extended supergravities, one model may possess different formulations related by $Sp(2n_v,\mathbb{R})$. The symplectic group mixes its electric and magnetic potentials. The model admits a global duality symmetry subgroup $G$ which acts also on the scalars. We recall first how the general second order Lagrangian, its local deformations and those of its abelian gauge group will depend on the choice of $2n_v$ directions (choice of "Darboux frame"). We start from a general frame defined by the symplectic transformation relating it to a fixed"reference" one. Combinations of symplectic matrix coefficients appear then as constant parameters in the second order Lagrangians. Another gauging method uses an "embedding tensor" that characterizes the realization of the gauge group via the global duality group. It involves additional 2-form gauge fields. A suitable zero charge limit of this realization has abelian gauge group and the "gauging" can be viewed as a consistent deformation of that limit. We show that the two methods applied to the corresponding ungauged models have equivalent local deformations -- and more generally, have isomorphic local BRST cohomology at all ghost numbers. We finally consider manifestly duality invariant first order actions with abelian gauge group. We point out that obstructions to non-abelian deformations of the Yang-Mills type exhibited in a previous work remain present when couplings to scalar fields are included.
We consider no-scale extended supergravity models as they arise from string and M-theory compactifications in presence of fluxes. The special role of gauging axion symmetries for the Higgs and superHiggs mechanism is outlined.
We obtain new duality transformations relating some exact string backgrounds, by defining the nilpotent duality. We show that the ungauged $SL(2, R)$ WZW model transforms by its action into the three dimensional plane wave geometry. We also give the inverse transformation from the plane wave to the $SL(2, R)$ model and discuss the implications of the results.
We study duality transformations for two-dimensional sigma models with abelian chiral isometries and prove that generic such transformations are equivalent to integrated marginal perturbations by bilinears in the chiral currents, thus confirming a recent conjecture by Hassan and Sen formulated in the context of Wess-Zumino-Witten models. Specific duality transformations instead give rise to coset models plus free bosons.
T-Duality invariant worldsheet string actions, recently written down by Schwarz and Sen, are coupled to the worldsheet gauge fields. It is shown that the integration of the dual coordinates gives the conventional, vector, axial and chiral, gauged string actions for the appropriate choice of the gauged isometries. Alternatively, the gauge field integration is shown to give a T-duality invariant action which matches with the corresponding results known earlier.
We present a systematic and 'from the ground up' analysis of the 'minimal coupling' type of gauging of Yang-Mills symmetries in (2,2)-supersymmetric 1+1-dimensional spacetime. Unlike in the familiar 3+1-dimensional N=1 supersymmetric case, we find several distinct types of minimal coupling symmetry gauging, and so several distinct types of gauge (super)fields, some of which entirely novel. Also, we find that certain (quartoid) constrained superfields can couple to no gauge superfield at all, others (haploid ones) can couple only very selectively, while still others (non-minimal, i.e., linear o…
We demonstrate that the technique of abelian bosonization through duality transformations can be extended to gauging anomalous symmetries. The example of a Dirac fermion theory is first illustrated. This idea is then also applied to bosonize a chiral fermion by gauging its chiral phase symmetry.
Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京国际数学研究中心）；Dipartimento di Matematica “Giuseppe Peano”, Università di Torino（意大利国家核物理研究所都灵分部）
We construct the most general gaugings of the maximal D=6 supergravity. The theory is (2,2) supersymmetric, and possesses an on-shell SO(5,5) duality symmetry which plays a key role in determining its couplings. The field content includes 16 vector fields that carry a chiral spinor representation of the duality group. We utilize the embedding tensor method which determines the appropriate combinations of these vectors that participate in gauging of a suitable subgroup of SO(5,5). The construction also introduces the magnetic duals of the 5 two-form potentials and 16 vector fields.
We study the general formulation of gauged supergravity in seven dimensions with sixteen supercharges keeping duality covariance by means of the embedding tensor formalism. We first classify all inequivalent duality orbits of consistent deformations. Secondly, we analyse the complete set of critical points in a systematic way. Interestingly, we find the first examples of stable de Sitter solutions within a theory with such a large amount of supersymmetry.
We consider the conformal gauging of non-abelian groups. In such cases there are inequivalent ways of gauging (generalizing the axial and vector cases for abelian groups) corresponding to external automorphisms of the group. Different $\s$-models obtained this way correspond to the same conformal field theory. We use the method of quotients to formulate this equivalence as a new duality symmetry.
We study the class of four-dimensional N=4 supergravities obtained by gauging the axionic shift and axionic rescaling symmetries. We formulate these theories using the machinery of embedding tensors, characterize the full gauge algebras and discuss several specific features of this family of gauged supergravities. We exhibit in particular a generalized duality between massive vectors and massive two-forms in four dimensions, inherited from the gauging of the shift symmetry. We show that these theories can be deduced from higher dimensions by a Scherk-Schwarz reduction, where a twist with respe…
We consider gauged WZW models based on a four dimensional non-semi-simple group. We obtain conformal $\s$-models in $D=3$ spacetime dimensions (with exact central charge $c=3$) by axially and vectorially gauging a one-dimensional subgroup. The model obtained in the axial gauging is related to the $3D$ black string after a correlated limit is taken in the latter model. By identifying the CFT corresponding to these $\s$-models we compute the exact expressions for the metric and dilaton fields. All of our models can be mapped to flat spacetimes with zero antisymmetric tensor and dilaton fields vi…
A consistent gauging of maximal supergravity requires that the T-tensor transforms according to a specific representation of the duality group. The analysis of viable gaugings is thus amenable to group-theoretical analysis, which we explain and exploit for a large variety of gaugings. We discuss the subtleties in four spacetime dimensions, where the ungauged Lagrangians are not unique and encoded in an E_7(7)\Sp(56,R)/GL(28) matrix. Here we define the T-tensor and derive all relevant identities in full generality. We present a large number of examples in d=4,5 spacetime dimensions which includ…
From gauging to duality in one-dimensional quantum lattice models
Ghent University（根特大学）；University of Vienna（维也纳大学）；Instituto de Física Teórica, UAM-CSIC（马德里自治大学-西班牙国家科研委员会理论物理研究所）；University of Cambridge（剑桥大学）；The University of Sydney（悉尼大学）
In this paper we reopen the discussion of gauging the two-dimensional off-shell (2,2) supersymmetric sigma models written in terms of semichiral superfields. The associated target space geometry of this particular sigma model is generalized Kahler (or bi-hermitean with two non-commuting complex structures). The gauging of the isometries of the sigma model is now done by coupling the semichiral superfields to the new (2,2) semichiral vector multiplet. We show that the two moment maps together with a third function form the complete set of three Killing potentials which are associated with this …
In the first part of the talk we discuss T-duality for a free boson on a world sheet with boundary in a setting suitable for the generalization to non-trivial backgrounds. The gauging method as well as the canonical transformation are considered. In both cases Dirichlet strings as T-duals of Neumann strings arise in a generic way. In the second part the gauging method is employed to construct the T-dual of a model with non-Abelian isometries.
We show that N=8 spontaneously broken supergravity in four dimensions obtained by Scherk-Schwarz generalized dimensional reduction can be obtained from a pure four dimensional perspective by gauging a suitable electric subgroup of E_{7,7}. Owing to the fact that there are non isomorphic choices of maximal electric subgroups of the U-duality group their gaugings give rise to inequivalent theories. This in particular shows that the Scherk-Schwarz gaugings do not fall in previous classifications of possible gauged N=8 supergravities. Gauging of flat groups appear in many examples of string compac…
We argue that in the tensionless phase of string theory where the stringy gauge symmetries are unbroken, (at least some) cosmological singularities can be understood as gauge artefacts. We present two conceptually related, but distinct, pieces of evidence: one relying on spacetime and the other on worldsheet.
When gauging a $(d-1)$-form symmetry in $d$ spacetime dimensions, one formally expects the gauged theory to carry a dual $-1$-form symmetry. This work focuses on the study of such symmetries, in particular via the spacetime-filling topological operators that implement them, in two-dimensional field theory (with an eye towards general quantum field theory). As theories with $(d-1)$-form symmetries are known to be equivalent to direct sums of local theories, we review how gauging a $(d-1)$-form symmetry projects onto a single component in this sum, and explain how gauging the resulting $-1$-form…
We discuss duality orbits and symplectic deformations of D=4 gauged supergravity theories, with focus on N$\ge$2. We provide a general constructive framework for computing symplectic deformations starting from a reference gauging, and apply it to many interesting examples. We prove that no continuous deformations are allowed for Fayet-Iliopoulos gaugings of the N=2 STU model and in particular that any $ω$ deformation is classically trivial. We further show that although in the N=6 truncation of SO(8) maximal supergravity the $ω$ parameter can be dualized away, in the 'twin' N=2 truncation $ω$ …
In this review articel we study the gaugings of extended supergravity theories in various space-time dimensions. These theories describe the low-energy limit of non-trivial string compactifications. For each theory under consideration we review all possible gaugings that are compatible with supersymmetry. They are parameterized by the so-called embedding tensor which is a group theoretical object that has to satisfy certain representation constraints. This embedding tensor determines all couplings in the gauged theory that are necessary to preserve gauge invariance and supersymmetry. The conce…
We find the T-duality transformation rules for 2-dimensional (2,1) supersymmetric sigma-models in (2,1) superspace. Our results clarify certain aspects of the (2,1) sigma model geometry relevant to the discussion of T-duality. The complexified duality transformations we find are equivalent to the usual Buscher duality transformations (including an important refinement) together with diffeomorphisms. We use the gauging of sigma-models in (2,1) superspace, which we review and develop, finding a manifestly real and geometric expression for the gauged action. We discuss the obstructions to gauging…
We generalize S-duality to N=2 superconformal field theories (SCFTs) with Coulomb branch operators of non-integer scaling dimension. As simple examples, we find minimal generalizations of the S-dualities discovered in SU(2) gauge theory with four fundamental flavors by Seiberg and Witten and in SU(3) gauge theory with six fundamental flavors by Argyres and Seiberg. Our constructions start by weakly gauging diagonal SU(2) and SU(3) flavor symmetry subgroups of two copies of a particular rank-one Argyres-Douglas theory (along with sufficient numbers of hypermultiplets to guarantee conformality o…
A class of 4d $\mathcal{N}=3$ SCFTs can be obtained from gauging a discrete subgroup of the global symmetry group of $\mathcal{N}=4$ Super Yang-Mills theory. This discrete subgroup contains elements of both the $SU(4)$ R-symmetry group and the $SL(2,\mathbb{Z})$ S-duality group of $\mathcal{N}=4$ SYM. We give a prescription for how to perform the discrete gauging at the level of the superconformal index and Higgs branch Hilbert series. We interpret and match the information encoded in these indices to known results for rank one $\mathcal{N}=3$ theories. Our prescription is easily generalised f…
We study the Seiberg dualities with an adjoint matter for the $U(N)$ and $SU(N)$ gauge groups in three- and four-dimensions with four supercharges. The relation between three- and four-dimensional dualities is investigated. We especially derive the three-dimensional duality from four-dimensional one by the dimensional reduction including the non-perturbative effect of the $\mathbb{S}^1$-compactification. In the $U(N)$ case, we obtain the Kim-Park duality, which is known as a generalization of the Aharony duality including an adjoint matter. In the $SU(N)$ case, we obtain the duality which foll…
We investigate non-abelian gaugings of WZNW models. When the gauged group is semisimple we are able to present exact formulae for the dual conformal field theory, for all values of the level $k$. The results are then applied to non-abelian target space duality in string theory, showing that the standard formulae are quantum mechanically well defined in the low energy limit if the gauged group is semisimple.
We consider partial supersymmetry breaking in ${\cal N}=2$ supergravity coupled to a single vector and a single hypermultiplet. This breaking pattern is in principle possible if the quaternion-Kähler space of the hypermultiplet admits (at least) one pair of commuting isometries. For this class of manifolds, explicit metrics exist and we analyse a generic electro-magnetic (dyonic) gauging of the isometries. An example of partial breaking in Minkowski spacetime has been found long ago by Ferrara, Girardello and Porrati, using the gauging of two translation isometries on $SO(4,1)/SO(4)$. We demon…
We construct a duality between several simple physical systems by showing that they are different aspects of the same quantum theory. Examples include the free relativistic massless particle and the hydrogen atom in any number of dimensions. The key is the gauging of the Sp(2) duality symmetry that treats position and momentum (x,p) as a doublet in phase space. As a consequence of the gauging, the Minkowski space-time vectors (x^μ, p^μ) get enlarged by one additional space-like and one additional time-like dimensions to (x^M,p^M). A manifest global symmetry SO(d,2) rotates (x^M,p^M) like d+2 d…
Deformations of maximal supergravity theories induced by gauging non-abelian subgroups of the duality group reveal the presence of charged M-theory degrees of freedom that are not necessarily contained in supergravity. The relation with M-theory degrees of freedom is confirmed by the representation assignments under the duality group of the gauge charges and the ensuing vector and tensor gauge fields. The underlying hierarchy of these gauge fields is required for consistency of general gaugings. As an example gauged maximal supergravity in three space-time dimensions is presented in a version …
Scattering amplitudes have the potential to provide new insights to the study of supergravity theories with gauged R-symmetry and Minkowski vacua. Such gaugings break supersymmetry spontaneously, either partly or completely. In this paper, we develop a framework for double-copy constructions of Abelian and non-Abelian gaugings of N=8 supergravity with these properties. They are generally obtained as the double copy of a spontaneously-broken (possibly supersymmeric) gauge theory and a theory with explicitly-broken supersymmetry. We first identify purely-adjoint deformations of N=4 super-Yang-Mi…
In this paper we obtain the complete classification of the inequivalent classes of M2-brane symplectic torus bundles with monodromy in $SL(2,Z)$ and their precise U-duality relations among them. There are eight inequivalent classes of bundles whose monodromy groups, at low energies, are in correspondence with the gauging groups of the eight type II gauged supergravities in nine dimensions. Four of those have been previously found and they correspond to the 'type IIB side'. In this paper we provide the explicit realization of the remaining four classes associated to the 'type IIA side'. The pre…
We consider the reduction of the duality invariant approach to M-theory by a U-duality group valued Scherk-Schwarz twist. The result is to produce potentials for gauged supergravities that are normally associated with non-geometric compactifications. The local symmetry reduces to gauge transformations with the gaugings exactly matching those of the embedding tensor approach to gauged supergravity. Importantly, this approach now includes a nontrivial dependence of the fields on the extra coordinates of the extended space.
In this letter we implement a recently proposed {\it spacetime duality} approach to dualize a two dimensional, Abelian, gauge field theory, which has no dual version under $p$--duality. Our result suggests that spacetime duality spans a new, wider, class of dual theories, which cannot be related one to another by $p$--duality transformations.
We define a nonabelian version of particle-vortex duality, by dimensionally extending usual (1+1)-dimensional nonabelian T-duality to (2+1) dimensions. While we will explicitly describe a global $SU(2)$ symmetry, our methods can also be applied to a larger group $G$, by gauging an appropriate subgroup. We will exemplify our duality with matter in both adjoint and fundamental representations by considering a modification of ${\cal N}=2$ supersymmetric Yang-Mills theory (Seiberg-Witten theory reduced to (2+1) dimensions), and an $SU(2)\times U(1)$ color-flavor locked theory that exhibits nonabel…
We study the gauging of a discrete $\mathbb{Z}_3$ symmetry in the five-dimensional superconformal $T_N$ theories. We argue that this leads to an infinite sequence of five-dimensional superconformal theories with either $E_6 \times SU(N)$ or $SU(3)\times SU(N)$ global symmetry group. In the $M$-theory realisation of $T_N$ theories as residing at the origin in the Calabi-Yau orbifolds ${\mathbb{C}^3 \over {\mathbb{Z}_N \times \mathbb{Z}_N}}$ we identify the $\mathbb{Z}_3$ symmetry geometrically and the new theories arise from $M$-theory on the non-Abelian orbifolds $({\mathbb{C}^3 \over {\mathbb…
We examine non-abelian duality transformations in the open string case. After gauging the isometries of the target space and developing the general formalism, we study in details the duals oftarget spaces with SO(N) isometries which, for the SO(2) case, reduces to the known abelian T-duals. We apply the formalism to electrically and magnetically charged 4D black hole solutions and, as in the abelian case, dual coordinates satisfy Dirichlet conditions.
We discuss theories containing higher-order forms in various dimensions. We explain how Chern--Simons-type theories of forms can be defined from TQFTs in one less dimension. We also exhibit new TQFTs with interacting Yang--Mills fields and higher--order forms. They are obtained by the dimensional reduction of TQFTs whose gauge functions are free self-duality equations. Interactions are due to the gauging of global internal symmetries after dimensional reduction. We list possible symmetries and give a brief discussion on the possible relation of such systems to interacting field theories.
We give a general review of extended supergravities and their gauging using the duality-covariant embedding tensor formalism. Although the focus is on four-dimensional theories, an overview of the gauging procedure and the related tensor hierarchy in the higher-dimensional models is given. The relation of gauged supergravities to flux compactifications is discussed and examples are worked out in detail.
An extended field theory is presented that captures the full SL(2) x O(6,6+n) duality group of four-dimensional half-maximal supergravities. The theory has section constraints whose two inequivalent solutions correspond to minimal D=10 supergravity and chiral half-maximal D=6 supergravity, respectively coupled to vector and tensor multiplets. The relation with O(6,6+n) (heterotic) double field theory is thoroughly discussed. Non-Abelian interactions as well as background fluxes are captured by a deformation of the generalised diffeomorphisms. Finally, making use of the SL(2) duality structure,…
In this series of lectures I present a review of the geometric structures of supergravity in diverse dimensions mostly relevant to p-brane physics and to pinpoint the correspondence between the macroscopic and microscopic description of branes. In particular I review duality transformations, coset manifold structures and the general steps involved by the process of gauging supergravity lagrangians both with respect to compact, non compact and non semisimple groups. I focus specifically on the issue of the Domain Wall field theory correspondence and its relation with the gaugings of supergravit…
We present the gauged N=4 (half-maximal) supergravities in four and five spacetime dimensions coupled to an arbitrary number of vector multiplets. The gaugings are parameterized by a set of appropriately constrained constant tensors, which transform covariantly under the global symmetry groups SL(2) x SO(6,n) and SO(1,1) x SO(5,n), respectively. In terms of these tensors the universal Lagrangian and the Killing Spinor equations are given. The known gaugings, in particular those originating from flux compactifications, are incorporated in the formulation, but also new classes of gaugings are fo…
The duality angles deformation developed by de Roo and Wagemans within the context of N=4 gauged supergravity is used in order to study certain classes of gaugings of N=8 supergravity, namely, those that are consistent when halving the maximal D=4 theory. After reviewing the truncation process from N=8 to N=4 supergravity in terms of the embedding tensor formalism, the de Roo-Wagemans phases method is implemented for solving the resulting constraints on the gauging parameters by means of the Schon-Weidner ansatz. In contrast with the twenty semisimple N=4 gaugings admitting more than a single …
I discuss the role of spacetime supersymmetry in the interplay between strong/weak coupling duality and target space duality in string theory which arises in string/string duality. This can be seen via the construction of string soliton solutions which in $N=4$ compactifications of heterotic string theory break more than $1/2$ of the spacetime supersymmetries but whose analogs in $N=2$ and $N=1$ compactifications break precisely $1/2$ of the spacetime supersymmetries. As a result, these solutions may be interpreted as stable solitons in the latter two cases, and correspond to Bogomol'nyi-satur…
We discuss how nonrelativistic spacetime symmetries can be gauged in the context of the coset construction. We consider theories invariant under the centrally extended Galilei algebra as well as the Lifshitz one, and we investigate under what conditions they can be supplemented by scale transformations. We also clarify the role of torsion in these theories.
We analyze in detail the possible breaking of spacetime supersymmetry under T-duality transformations. We find that when appropiate world-sheet effects are taken into account apparent puzzles concerning supersymmetry in spacetime are solved. We study T-duality in general heterotic $σ$-models analyzing possible anomalies, and we find some modifications of Buscher's rules. We then work out a simple but representative example which contains most of the difficulties. We also consider the issue of supersymmetry versus duality for marginal deformations of WZW models and present a mechanism that rest…
Freudenthal duality in N=2, D=4 ungauged supergravity is generated by an anti-involutive operator that acts on the electromagnetic fluxes, and results to be a symmetry of the Bekenstein-Hawking entropy. We show that, with a suitable extension, this duality can be generalized to the abelian gauged case as well, even in presence of hypermultiplets. By defining Freudenthal duality along the scalar flow, one can prove that two configurations of charges and gaugings linked by the Freudenthal operator share the same set of values of the scalar fields at the black hole horizon. Consequently, Freudent…
