We show that three-dimensional General Relativity, augmented with two vector fields, allows for a non-relativistic limit, different from the standard limit leading to Newtonian gravity, that results into a well-defined action which is of the Chern-Simons type. We show that this three-dimensional `Extended Bargmann Gravity', after coupling to matter, leads to equations of motion allowing a wider class of background geometries than the ones that one encounters in Newtonian gravity. We give the supersymmetric generalization of these results and point out an important application in the context of calculating partition functions of non-relativistic field theories using localization techniques.
We present a novel three-dimensional non-relativistic Chern-Simons supergravity theory invariant under a Maxwellian extended Bargmann superalgebra. We first study the non-relativistic limits of the minimal and the $\mathcal{N}=2$ Maxwell superalgebras. We show that a well-defined Maxwellian extended Bargmann supergravity requires to construct by hand a supersymmetric extension of the Maxwellian extended Bargmann algebra by introducing additional fermionic and bosonic generators. The new non-relativistic supergravity action presented here contains the extended Bargmann supergravity as a sub-cas…
In this paper we show how the method of Lie algebra expansions may be used to obtain, in a simple way, both the extended Bargmann Lie superalgebra and the Chern-Simons action associated to it in three dimensions, starting from $D=3$, $\mathcal{N}=2$ superPoincaré and its corresponding Chern-Simons supergravity.
In this work we present an alternative method to construct diverse non-relativistic Chern-Simons supergravity theories in three spacetime dimensions. To this end, we apply the Lie algebra expansion method based on semigroups to a supersymmetric extension of the Nappi-Witten algebra. Two different families of non-relativistic superalgebras are obtained, corresponding to generalizations of the extended Bargmann superalgebra and extended Newton-Hooke superalgebra, respectively. The expansion method considered here allows to obtain known and new non-relativistic supergravity models in a systematic…
In this work we study a non-relativistic three dimensional Chern-Simons gravity theory based on an enlargement of the Extended Bargmann algebra. A finite non-relativistic Chern-Simons gravity action is obtained through the non-relativistic contraction of a particular $U(1)$ enlargement of the so-called AdS-Lorentz algebra. We show that the non-relativistic gravity theory introduced here reproduces the Maxwellian Exotic Bargmann gravity theory when a flat limit $\ell \rightarrow \infty$ is applied. We also present an alternative procedure to obtain the non-relativistic versions of the AdS-Loren…
We construct a supersymmetric extension of three-dimensional Newton-Cartan gravity by gauging a super-Bargmann algebra. In order to obtain a non-trivial supersymmetric extension of the Bargmann algebra one needs at least two supersymmetries leading to a N=2 super-Bargmann algebra. Due to the fact that there is a universal Newtonian time, only one of the two supersymmetries can be gauged. The other supersymmetry is realized as a fermionic Stueckelberg symmetry and only survives as a global supersymmetry. We explicitly show how, in the frame of a Galilean observer, the system reduces to a supers…
In this paper, we present novel non-relativistic superalgebras which correspond to supersymmetric extensions of the enlarged extended Bargmann algebra. The three-dimensional non-relativistic Chern-Simons supergravity actions invariant under the aforementioned superalgebras are constructed. The new non-relativistic superalgebras allow to accommodate a cosmological constant in a non-relativistic supergravity theory. Interestingly, we show that one of the non-relativistic supergravity theories presented here leads to the recently introduced Maxwellian exotic Bargmann supergravity when the flat li…
We provide a systematic analysis of three-dimensional N = 2 extended Bargmann superalgebra and its Newton-Hooke, Lifshitz and Schrödinger extensions. These algebras admit invariant non-degenerate bi-linear forms which we utilized to construct corresponding Chern-Simons supergravity actions.
We present a generalization of the so-called Maxwellian extended Bargmann algebra by considering a non-relativistic limit to a generalized Maxwell algebra defined in three spacetime dimensions. The non-relativistic Chern-Simons gravity theory based on this new algebra is also constructed and discussed. We point out that the extended Bargmann and its Maxwellian generalization are particular sub-cases of the generalized Maxwellian extended Bargmann gravity introduced here. The extension of our results using the semigroup expansion method is also discussed.
The Bargmann algebra and centrally-extended Newton-Hooke algebras describe the non-relativistic symmetries of massive particles in flat and curved spacetimes, respectively. These three algebras all arise as deformations of the universal central-extension of the static kinematical Lie algebra. In this paper, we classify the N=1 super-extensions for each of these algebras in (3+1)-dimensions, up to isomorphism. We then identify the non-empty branches of the algebraic variety describing the N=2 super-extensions of these algebras. We find 9 isomorphism classes in the N=1 case and 22 branches in th…
We show how the Newton-Cartan formulation of Newtonian gravity can be obtained from gauging the Bargmann algebra, i.e., the centrally extended Galilean algebra. In this gauging procedure several curvature constraints are imposed. These convert the spatial (time) translational symmetries of the algebra into spatial (time) general coordinate transformations, and make the spin connection gauge fields dependent. In addition we require two independent Vielbein postulates for the temporal and spatial directions. In the final step we impose an additional curvature constraint to establish the connecti…
We treat the N-extended supergravity in 2+1 space-time dimensions as a Yang-Mills gauge field with Chern-Simons action associated to the N-extended Poincaré supergroup. We fix the gauge of this theory within the Batalin-Vilkovisky scheme.
We consider the interplay of duality symmetries and gauged isometries of supergravity models giving N-extended, spontaneously broken supergravity with a no-scale structure. Some examples, motivated by superstring and M-theory compactifications are described.
We discuss toroidal compactifications of maximal supergravity coupled to an extended configuration of BPS states which transform consistently under the U-duality group. Under certain conditions this leads to theories that live in more than eleven spacetime dimensions, with maximal supersymmetry but only partial Lorentz invariance. We demonstrate certain features of this construction for the case of nine-dimensional N=2 supergravity.
Recently a correspondence between non-propagating degrees of freedom in maximal supergravity and the very extended algebra E_11 has been found. We perform a similar analysis for a supergravity theory with eight supercharges and very extended G_2. In particular, in the context of d=5 minimal supergravity, we study whether supersymmetry can be realised on higher-rank tensors with no propagating degrees of freedom. We find that in this case the very extended algebra fails to capture these possibilities.
The bosonic sector of various supergravity theories reduces to a homogeneous space G/H in three dimensions. The corresponding algebras g are simple for (half-)maximal supergravity, but can be semi-simple for other theories. We extend the existing literature on the Kac-Moody extensions of simple Lie algebras to the semi-simple case. Furthermore, we argue that for N=2 supergravity the simple algebras have to be augmented with an su(2) factor.
A new approach for obtaining the three-dimensional Chern-Simons supergravity for the Poincaré algebra is presented. The $\mathcal{N}$-extended Poincaré supergravity is obtained by expanding the super Lorentz theory. We extend our procedure to their respective asymptotic symmetries and show that the $\mathcal{N}=(1,2,4)$ super-BMS$_{3}$ appear as expansions of one Virasoro superalgebra. Interestingly, the $\mathcal{N}$-extended super-BMS$_{3}$ obtained here are not only centrally extended but also endowed with internal symmetry. We also show that the $\mathcal{N}$-extended super Poincaré algebr…
We show that known de Sitter solutions in extended gauged supergravity theories are interrelated via a web of supersymmetry-breaking truncations. In particular, all N=8 models reduce to a subset of the N=4 possibilities. Furthermore, a different subset of the N=4 models can be truncated to stable de Sitter vacua in N=2 theories. In addition to relations between the known cases, we also find new (un)stable models.
In this work, we build the most general boundary conditions for $\mathcal N=\left(1,1\right)$ and $\mathcal N=\left(2,2\right)$ extended supergravity. We show that for the loosest set of boundary conditions, their asymptotic symmetry algebras are two copies of the $\mathfrak{osp}\left(1\mid 2\right)_k$ and $\mathfrak{osp}\left(2\mid 2\right)_k$ algebra, respectively. Then, we restrict the gauge fields on the boundary conditions in order to obtain supersymmetric extensions of the Brown-Henneaux and Troessaert boundary conditions. Furthermore, for $\mathcal N=1$ and $\mathcal N=2$ (extended) sup…
We give generalizations of extended Poincaré supergravity with {\it arbitrarily many} supersymmetries in the absence of central charges in three-dimensions by gauging its intrinsic global $~SO(N)$~ symmetry. We call these \alephnull (Aleph-Null) supergravity theories. We further couple a non-Abelian supersymmetric Chern-Simons theory and an Abelian topological BF theory to \alephnull supergravity. Our result overcomes the previous difficulty for supersymmetrization of Chern-Simons theories beyond $~N=4$. This feature is peculiar to the Chern-Simons and BF theories including supergravity in thr…
We consider generalized diffeomorphisms on an extended mega-space associated to the U-duality group of gauged maximal supergravity in four dimensions, E_7. Through the bein for the extended metric we derive dynamical (field-dependent) fluxes taking values in the representations allowed by supersymmetry, and obtain their quadratic constraints from gauge consistency conditions. A covariant generalized Ricci tensor is introduced, defined in terms of a connection for the generalized diffeomorphisms. We show that for any torsionless and metric-compatible generalized connection, the Ricci scalar rep…
Maximally $\mathcal{N}$ extended $2+1$ dimensional flat Supergravity theories exist for a class of super-Poincaré algebras for arbitrary $\mathcal{N}$. They have rich asymptotic structures and contain all interesting topological supergravity solutions in presence of non-trivial holonomy. For the asymptotic symmetry algebra being a suitable flat limit of the superconformal algebra $Osp(\mathcal{N}|2;R)$, we have found the $1+1$ dimensional chiral WZW model as the dual quantum field theory that describes the dynamics of these solutions. In the Hamiltonian analysis, the reduced phase space resemb…
Introducing an extended Lie derivative along the dual of A, the three-form field of d=11 supergravity, the full diffeomorphism algebra of d=11 supergravity is presented. This algebra suggests a new formulation of the theory, where the three-form field A is replaced by bivector B^{ab}, bispinor B^{αβ}, and spinor-vector η^{aβ} one-forms. Only the bivector one-form B^{ab} is propagating, and carries the same degrees of freedom of the three-form in the usual formulation, its curl \Dcal_{[μ} B^{ab}_{ν]} being related to the F_{μνa b} curl of the three-form. The other one-forms are auxiliary, and t…
We consider the reduction of supersymmetry in N-extended four dimensional supergravity via the super Higgs mechanism in theories without cosmological constant. We provide an analysis largely based on the properties of long and short multiplets of Poincare' supersymmetry. Examples of the super Higgs phenomenon are realized in spontaneously broken N=8 supergravity through the Scherk-Schwarz mechanism and in superstring compactification in presence of brane fluxes. In many models the massive vectors count the difference in number of the translation isometries of the scalar sigma-model geometries …
We consider the stability of non-supersymmetric critical points of general N=4 supergravities. A powerful method to analyse this issue based on the sGoldstino direction has been developed for minimal supergravity. We adapt this to the present case, and address the conceptually new features arising for extended supersymmetry. As an application, we investigate the stability when supersymmetry breaking proceeds via either the gravity or the matter sector. Finally, we outline the N=8 case.
We show that the Extended Bargmann and Newton-Hooke algebras in 2+1 dimensions can be obtained as expansions of the Nappi-Witten algebra. The result can be generalized to obtain two infinite families of non-relativistic symmetries, which include the Maxwellian Exotic Bargmann symmetry, its generalized Newton-Hooke counterpart, and its Hietarinta dual. In each case, the invariant bilinear form on the Nappi-Witten algebra leads to the invariant tensor on the expanded algebra, allowing one to construct the corresponding Chern-Simons gravity theory.
We define a level for a large class of Lorentzian Kac-Moody algebras. Using this we find the representation content of very extended $A_{D-3}$ and $E_8$ (i.e. $E_{11}$) at low levels in terms of $A_{D-1}$ and $A_{10}$ representations respectively. The results are consistent with the conjectured very extended $A_8$ and $E_{11}$ symmetries of gravity and maximal supergravity theories given respectively in hep-th/0104081 and hep-th/0107209. We explain how these results provided further evidence for these conjectures.
We derive a $2+1$ dimensional model with unconventional supersymmetry at the boundary of an ${\rm AdS}_4$ $\mathcal{N}$-extended supergravity, generalizing previous results. The (unconventional) extended supersymmetry of the boundary model is instrumental in describing, within a top-down approach, the electronic properties of graphene-like 2D materials at the two Dirac points, ${\bf K}$ and ${\bf K}'$. The two valleys correspond to the two independent sectors of the ${\rm OSp}(p|2)\times {\rm OSp}(q|2)$ boundary model in the $p=q$ case, which are related by a parity transformation. The Semenof…
We investigate systematically the asymptotic dynamics and symmetries of all three-dimensional extended AdS supergravity models. First, starting from the Chern-Simons formulation, we show explicitly that the (super)anti-de Sitter boundary conditions imply that the asymptotic symmetry algebra is the extended superconformal algebra with quadratic nonlinearies in the currents. We then derive the super-Liouville action by solving the Chern-Simons theory and obtain a realization of the superconformal algebras in terms of super-Liouville fields. Finally, we discuss the possible periodic conditions th…
The hyperbolic Kac-Moody algebra E10 has repeatedly been suggested to play a crucial role in the symmetry structure of M-theory. Recently, following the analysis of the asymptotic behaviour of the supergravity fields near a cosmological singularity, this question has received a new impulse. It has been argued that one way to exhibit the symmetry was to rewrite the supergravity equations as the equations of motion of the non-linear sigma model E10/K(E10). This attempt, in line with the established result that the scalar fields which appear in the toroidal compactification down to three spacetim…
We review the general gauged N=2 supergravity coupled to an arbitrary number of vector multiplets and hypermultiplets. We consider two different models where N=2 supersymmetry is broken to N=1 spontaneously, one has a U(1) vector multiplet and the other has a U(N) vector multiplet. In both cases, partial breaking of N=2 supersymmetry is accomplished by the Higgs and the super-Higgs mechanisms. The mass spectrum can be evaluated and we conclude that the resulting models have N=1 supersymmetry. This is based on master thesis submitted to Graduate School of Science, Osaka City University, in Marc…
Relying on the geometrical set up of Special Kähler Geometry and Quaternionic Geometry, which I discussed at length in my Lectures at the 1995 edition of this Spring School, I present here the recently obtained fully general form of N=2 supergravity with completely arbitrary couplings. This lagrangian has already been used in the literature to obtain various results: notably the partial breaking of supersymmetry and various extremal black--hole solutions. My emphasis, however, is only on providing the reader with a completely explicit and ready to use component expression of the supergravity a…
We determine the most general form of off-shell N=(1,1) supergravity field configurations in three dimensions by requiring that at least one off-shell Killing spinor exists. We then impose the field equations of the topologically massive off-shell supergravity and find a class of solutions whose properties crucially depend on the norm of the auxiliary vector field. These are spacelike-squashed and timelike-stretched AdS_3 for the spacelike and timelike norms, respectively. At the transition point where the norm vanishes, the solution is null warped AdS_3. This occurs when the coefficient of th…
We review some aspects of the attractor mechanism for extremal black holes of (not necessarily supersymmetric) theories coupling Einstein gravity to scalars and Maxwell vector fields. Thence, we consider N=2 and N=8, d=4 supergravities, reporting some recent advances on the moduli spaces associated to BPS and non-BPS attractor solutions supported by charge orbits with non-compact stabilizers.
Repulsive singularities (repulsons) in extended supergravity theories are investigated. These repulsive singularities are related to attractive singularities (black holes) in moduli space of extended supergravity vacua. In order to study these repulsive singularities a scalar test-particle in the background of a repulson is investigated. It is shown, using a partial wave expansion, that the wave function of the scalar particle vanishes at the curvature singularity at the origin. In addition the connection to higher dimensional p-brane solutions including anti-branes is discussed.
We connect the attractor equations of a certain class of N=2, d=5 supergravities with their (1,0), d=6 counterparts, by relating the moduli space of non-BPS d=5 black hole/black string attractors to the moduli space of extremal dyonic black string d=6 non-BPS attractors. For d = 5 real special symmetric spaces and for N = 4,6,8 theories, we explicitly compute the flat directions of the black object potential corresponding to vanishing eigenvalues of its Hessian matrix. In the case N = 4, we study the relation to the (2,0), d=6 theory. We finally describe the embedding of the N=2, d=5 magic mod…
We present a new class of three-dimensional $\mathcal{N}$-extended supergravity theories based on the $\mathcal{N}$-extended Maxwell superalgebra with central charges and $\mathfrak{so}(\mathcal{N})$ internal symmetry generators. The presence of $\mathfrak{so}(\mathcal{N})$ generators is required in order to define a non-degenerate invariant inner product. Such symmetry allows us to construct an alternative supergravity action without cosmological constant term. Interestingly, the new theories can be obtained as a flat limit of a $\mathcal{N}$-extended AdS-Lorentz supergravity theories enlarge…
We construct a gauge theory based in the supergroup $G=SU(2,2|2)$ that generalizes MacDowell-Mansouri supergravity. This is done introducing an extended notion of Hodge operator in the form of an outer automorphism of $su(2,2|2)$-valued 2-form tensors. The model closely resembles a Yang-Mills theory -- including the action principle, equations of motion and gauge transformations -- which avoids the use of the otherwise complicated component formalism. The theory enjoys $H=SO(3,1)\times \mathbb{R} \times U(1)\times SU(2)$ off-shell symmetry whilst the broken symmetries $G/H$, translation-type s…
Recent results on the relation between hyper-Kahler geometry with torsion and solutions admitting Killing spinors in minimal de sitter supergravity are extended to more general supergravity models with vector multiplets.
A description of the bosonic sector of ten-dimensional N=1 supergravity as a non-linear realisation is given. We show that if a suitable extension of this theory were invariant under a Kac-Moody algebra, then this algebra would have to contain a rank eleven Kac-Moody algebra, that can be identified to be a particular real form of very-extended D_8. We also describe the extension of N=1 supergravity coupled to an abelian vector gauge field as a non-linear realisation, and find the Kac-Moody algebra governing the symmetries of this theory to be very-extended B_8. Finally, we discuss the related …
Eleven-dimensional supergravity reveals large exceptional symmetries upon reduction, in accordance with the U-duality groups of M-theory, but their higher-dimensional geometric origin has remained a mystery. In this letter, we show that D=11 supergravity can be extended to be fully covariant under the exceptional groups E_{n(n)}, n=6,7,8. Motivated by a similar formulation of double field theory we introduce an extended `exceptional spacetime'. We illustrate the construction by giving the explicit E_{6(6)} covariant form: the full D=11 supergravity, in a 5+6 splitting of coordinates but withou…
We propose a class of models in which extended objects are introduced in Chern-Simons supergravity in such a way that those objects appear on the same footing as the target space. This is motivated by the idea that branes are already first quantized object, so that it is desirable to have a formalism that treats branes and their target space in a similar fashion. Accordingly, our models describe interacting branes, as gauge systems for supergroups. We also consider the case in which those objects have boundaries, and discuss possible links to superstring theory and/or M-theory, by studying the…
We develop the superspace geometry of N-extended conformal supergravity in three space-time dimensions. General off-shell supergravity-matter couplings are constructed in the cases N=1,2,3,4.
Long ago, Achucarro and Townsend discovered that in three dimensions (3D) N-extended anti-de Sitter (AdS) supergravity exists in several incarnations, which were called the (p,q) AdS supergravity theories with non-negative integers p>=q such that N=p+q. Using the superspace approach to 3D N-extended supergravity developed in arXiv:1101.4013, we present three superfield formulations for N=2 supergravity that allow for well defined cosmological terms and supersymmetric AdS solutions. The conformal compensators corresponding to these theories are respectively: (i) a chiral scalar multiplet: (ii) …
The N-extended self-dual supergravity in the ultra-hyperbolic four-dimensional spacetime of kleinian signature (2+2) is given in the N-extended harmonic superspace. We reformulate the on-shell N-extended self-dual supergravity constraints of Siegel to a `zero-curvature' representation, and solve all of them but one in terms of a single superfield prepotential, by using a covariant Frobenius gauge in the Devchand-Ogievetsky approach. An off-shell superspace action, whose equation of motion yields the remaining constraint, is found. Our manifestly Lorentz-covariant action in harmonic superspace …
We provide a Lie algebra expansion procedure to construct three-dimensional higher-order Schrödinger algebras which relies on a particular subalgebra of the four-dimensional relativistic conformal algebra. In particular, we reproduce the extended Schrödinger algebra and provide a new higher-order Schrödinger algebra. The structure of this new algebra leads to a discussion on the uniqueness of the higher-order non-relativistic algebras. Especially, we show that the recent d-dimensional symmetry algebra of an action principle for Newtonian gravity is not uniquely defined but can accommodate thre…
The technique of generating new solutions to 4D gravity/matter systems by dimensional reduction to a sigma-model is extended to supersymmetric configurations of supergravity. The conditions required for the preservation of supersymmetry under isometry transformations in the sigma-model target space are found. Some examples illustrating the technique are given.
We show that a generalised reduction of D=10 IIB supergravity leads, in a certain limit, to a maximally extended SO(2) gauged supergravity in D=9. We show the scalar potential of this model allows both Minkowski and a new type of domain wall solution to the Bogomol'nyi equations. We relate these vacua to type IIB D-branes.
The effective action for 2d-gravity in Weyl-invariant regularization is extended to supersymmetric case. The super area-preserving invariance and cocyclic properties under general supergravitational transformations of the last action is shown.
We propose a new off-shell formulation for N-extended conformal supergravity in three spacetime dimensions. Our construction is based on the gauging of the N-extended superconformal algebra in superspace. Covariant constraints are imposed such that the algebra of covariant derivatives is given in terms of a single curvature superfield which turns out to be the super Cotton tensor. An immediate corollary of this construction is that the curved superspace is conformally flat if and only if the super Cotton tensor vanishes. Upon degauging of certain local symmetries, our formulation is shown to r…
We report on recent results in the study of extremal black hole attractors in N=2, d=4 ungauged Maxwell-Einstein supergravities. For homogeneous symmetric scalar manifolds, the three general classes of attractor solutions with non-vanishing Bekenstein-Hawking entropy are discussed. They correspond to three (inequivalent) classes of orbits of the charge vector, which sits in the relevant symplectic representation R_{V} of the U-duality group. Other than the 1/2-BPS one, there are two other distinct non-BPS classes of charge orbits, one of which has vanishing central charge. The complete classif…
We review a construction, using the harmonic space method, of solutions to the superfield equations of motion for N-extended self-dual supergravity theories. A superspace gauge condition suitable for the performance of a component analysis is discussed.
We discuss membranes in four-dimensional N=1 superspace. The kappa-invariance of the Green-Schwarz action implies that there is a dual version of N=1 supergravity with a three-form potential. We formulate this new supergravity in terms of a three-form superfield in curved superspace, giving the relevant constraints on the field strength. We find the corresponding membrane soliton in the new supergravity and discuss how the extended supersymmetry algebra emerges from the symmetries of the flat superspace background.
We present the algebraic framework for the quantization of the classical bosonic charge algebra of maximally extended (N=16) supergravity in two dimensions, thereby taking the first steps towards an exact quantization of this model. At the core of our construction is the Yangian algebra $Y(e_8)$ whose RTT presentation we discuss in detail. The full symmetry algebra is a centrally extended twisted version of the Yangian double $DY(e_8)_c$. We show that there exists only one special value of the central charge for which the quantum algebra admits an ideal by which the algebra can be divided so a…
We consider (2+1)-dimensional (N, M)-extended higher-spin anti-de Sitter supergravity and study its asymptotic symmetries. The theory is described by the Chern-Simons action based on a real, infinite-dimensional higher-spin superalgebra. We specify consistent boundary conditions on the higher-spin super-gauge connection corresponding to asymptotically anti-de Sitter spacetimes. We then determine the residual gauge transformations that preserve these asymptotic conditions and compute their Poisson bracket algebra. We find that the asymptotic symmetry is enhanced from the higher-spin superalgebr…
Recent calculations have confirmed the belief that the ultraviolet behaviour of N-extended supergravities improves with increasing N. We give a comprehensive review.
