Reduction Of Supergravity Research Articles

Twistor Actions for Integrable Systems

Many integrable systems can be reformulated as holomorphic vector bundles on twistor space. This is a powerful organizing principle in the theory of integrable systems. One shortcoming is that it is formulated at the level of the equations of motion. From this perspective, it is mysterious that integrable systems have Lagrangians. In this paper, we study a Chern-Simons action on twistor space and use it to derive the Lagrangians of some integrable sigma models. Our focus is on examples that come from dimensionally reduced gravity and supergravity. The dimensional reduction of general relativity to two spacetime dimensions is an integrable coset sigma model coupled to a dilaton and 2d gravity. The dimensional reduction of supergravity to two spacetime dimensions is an integrable coset sigma model coupled to matter fermions, a dilaton, and 2d supergravity. We derive Lax operators and Lagrangians for these 2d integrable systems using the Chern-Simons theory on twistor space. In the supergravity example, we use an extended setup in which twistor Chern-Simons theory is coupled to a pair of matter fermions.

Hitchhiker’s guide to Courant algebroid relations

Courant algebroids provide a useful mathematical tool (not only) in string theory. It is thus important to define and examine their morphisms. To some extent, this was done before using an analogue of canonical relations known from symplectic geometry. However, it turns out that applications in physics require a more general notion. We aim to provide a self-contained and detailed treatment of Courant algebroid relations and morphisms. A particular emphasis is placed on providing enough motivating examples. In particular, we show how Poisson–Lie T-duality and Kaluza–Klein reduction of supergravity can be interpreted as Courant algebroid relations compatible with generalized metrics (generalized isometries).

Prepotentials for linearized supergravity

Linearized supergravity in arbitrary dimension is reformulated into a first order formalism which treats the graviton and its dual on the same footing at the level of the action. This generalizes previous work by other authors in two directions: (1) we work in arbitrary space-time dimension, and (2) the gravitino field and supersymmetry are also considered. This requires the construction of conformally invariant curvatures (the Cotton fields) for a family of mixed symmetry tensors and tensor-spinors, whose properties we prove (invariance; completeness; conformal Poincaré lemma). We use these geometric tools to solve the Hamiltonian constraints of the graviton and gravitino through the introduction of prepotentials (two tensor fields for the graviton, one tensor-spinor for the gravitino) enjoying (linearized) conformal invariance. The action and equations of motion take a geometrically simple form in terms of the Cotton tensor(-spinors) of the prepotentials. In particular, the equations of motion of the graviton are equivalent to twisted self-duality conditions. We express the supersymmetric transformations of the graviton and gravitino into each other in terms of the prepotentials. We also reproduce the dimensional reduction of supergravity within the prepotential formalism. Finally, our formulas in dimension five are recovered from the dimensional reduction of the already known prepotential formulation of the 6D maximally supersymmetric theory.

Pauli reductions of supergravities in six and five dimensions

The dimensional reduction of a generic theory on a curved internal space such as a sphere does not admit a consistent truncation to a finite set of fields that includes the Yang-Mills gauge bosons of the isometry group. In rare cases, for example the $S^7$ reduction of eleven-dimensional supergravity, such a consistent "Pauli reduction" does exist. In this paper we study this existence question in two examples of $S^2$ reductions of supergravities. We do this by making use of a relation between certain $S^2$ reductions and group manifold $S^3=SU(2)$ reductions of a theory in one dimension higher. By this means we establish the non-existence of a consistent $S^2$ Pauli reduction of five-dimensional minimal supergravity. We also show that a previously-discovered consistent Pauli reduction of six-dimensional Salam-Sezgin supergravity can be elegantly understood via a group-manifold reduction from seven dimensions.

Heterotic effective action and duality symmetries revisited

The dimensional reduction of heterotic supergravity with gauge fields truncated to the Cartan subalgebra exhibits a continuous O(d,d+16;R) global symmetry, related to the O(d,d+16;Z) T-duality of heterotic strings on a d-torus. The O(d,d+16;R) symmetry is not present, however, if the supergravity reduction is done including the full set of E_8 x E_8 or SO(32) gauge fields. We analyze which duality symmetries are realized to all orders in $\alpha'$ in the proper effective field theories for the massless string states. We find a universal O(d,d;R) symmetry, also predicted by double field theory. We confirm this by giving a novel formulation of the dimensionally reduced supergravity in terms of O(d,d) multiplets, and we discuss cases of symmetry enhancement.

Inconsistency of breathing mode extensions of maximal five-dimensional supergravity embedding

Recent work on consistent Kaluza-Klein reductions on Einstein-Sasaki spaces prompted an intriguing conjecture that there might exist a consistent $S^5$ reduction of type IIB supergravity to give five-dimensional ${\cal N}=8$ gauged supergravity coupled to a massive supermultiplet that includes the breathing-mode scalar. Motivated by this, we investigate the possibility of augmenting the usual ${\cal N}=8$ supergravity reduction to include a breathing-mode scalar, and we show that this is in fact inconsistent. The standard reduction to the massless ${\cal N}=8$ supermultiplet depends for its consistency on a delicate interplay between properties of the ten-dimensional type IIB theory and properties of the Killing vectors on $S^5$. Our calculations show that turning on the breathing-mode is sufficient to destroy the balance, and hence render the reduction inconsistent.

Integrability of supergravity black holes and new tensor classifiers of regular and nilpotent orbits

In this paper we apply in a systematic way a previously developed integration algorithm of the relevant Lax equation to the construction of spherical symmetric, asymptotically flat black hole solutions of $ \mathcal{N} = 2 $ supergravities with symmetric Special Geometry. Our main goal is the classification of these black-holes according to the H⋆-orbits in which the space of possible Lax operators decomposes. By H⋆ one denotes the isotropy group of the coset UD=3/H⋆ which appears in the time-like dimensional reduction of supergravity from D = 4 to D = 3 dimensions. The main result of our investigation is the construction of three universal tensors, extracted from quadratic and quartic powers of the Lax operator, that are capable of classifying both regular and nilpotent H⋆-orbits of Lax operators. Our tensor based classification is compared, in the case of the simple one-field model S 3, to the algebraic classification of nilpotent orbits and it is shown to provide a simple discriminating method. In particular we present a detailed analysis of the S 3 model, constructing explicitly its solutions and discussing the Liouville integrability of the corresponding dynamical system. By means of the Kostant-representation of a generic Lie algebra element, we were able to develop an algorithm which produces the necessary number of hamiltonians in involution required by Liouville integrability of generic orbits. The degenerate orbits correspond to extremal black-holes and are nilpotent. We present an in depth discussion of their identification and of the construction of the corresponding supergravity solutions. We dwell on the relation between H⋆ orbits and critical points of the geodesic potential showing that there is correspondence yet not one-to-one.

Group reduction of heterotic supergravity

The reduction of ten-dimensional heterotic supergravity with Yang–Mills symmetry group K is performed on an arbitrary n-dimensional group manifold G. The reduction involves a non-vanishing 3-form flux, and the Lie algebra of G must have traceless structure constants to ensure the consistency of the reduction at the level of the action. A large class of gauged supergravities in d = 10 − n with (non)compact gaugings is obtained. The resulting models describe half-maximal gauged supergravities coupled to ( n + dim K ) vector multiplets. We uncover their hidden SO ( n , n + dim K ) duality symmetry, and the SO ( n , n + dim K ) / SO ( n ) × SO ( n + dim K ) coset structure that governs the couplings of the scalar fields. We find that the local gauge symmetry of the d-dimensional theory is K × G ⋉ R n . Differences from the existing gauged supergravities are highlighted. The consistent truncation to pure half-maximal gauged supergravity in any dimension is shown, and the obstacle to performing a chiral truncation of the theory in d = 6 dimensions is found. Among the results obtained are the complete diagonalisation of the fermionic kinetic terms, and other reduction formulae that are applicable to group reductions of supergravities in arbitrary dimensions.

Coset symmetries in dimensionally reduced heterotic supergravity

We investigate the coset structures which appear in the dimensional reduction of supergravity theories. Especially we investigate how to recognize the global symmetry groups if the coset is non-split. As an example we apply our analysis to the theories emanating from the dimensional reduction of Heterotic supergravity.

Coupled inflation and brane gases

We study an effective four-dimensional theory with an action with two scalar fields minimally coupled to gravity, and with a matter action which couples to the two scalar fields via an overall field-dependent coefficient in the action. Such a theory could arise from a dimensional reduction of supergravity coupled to a gas of branes winding the compactified dimensions. We show the existence of solutions corresponding to power-law inflation. The graceful exit from inflation can be obtained by postulating the decay of the branes, as would occur if the branes are unstable in the vacuum and stabilized at high densities by plasma effects. This construction provides an avenue for connecting string gas cosmology and the late-time universe.

N = 2 → N = 1 supergravity reduction in four dimensions

We discuss the reduction of N=2 supergravity to N=1, by a consistent truncation of the second gravitino multiplet.

Consistent reduction of N=2→ N=1 four-dimensional supergravity coupled to matter

We analyze the constraints which follow both on the geometry and on the gauge sector for a consistent supergravity reduction of a general matter-coupled N=2 supergravity theory in four dimensions. These constraints can be derived in an elegant way by looking at the fermionic sector of the theory.

Domain-wall supergravities from sphere reduction

Kaluza-Klein sphere reductions of supergravities that admit Ads × Sphere vacuum solutions are believed to be consistent. The examples include the S 4 and S 7 reductions of eleven-dimensional supergravity, and the S 5 reduction of ten-dimensional type IIB supergravity . In this paper we provide evidence that sphere reductions of supergravities that admit instead Domain-wallxSphere vacuum solutions are also consistent, where the background can be viewed as the near-horizon structure of a dilatonic p -brane of the theory. The resulting lower-dimensional theory is a gauged supergravity that admits a domain wall, rather than AdS, as a vacuum solution. We illustrate this consistency by taking the singular limits of certain modulus parameters, for which the original S n compactifying spheres ( n = 4, 5 or 7) becomes S p × R q , with p = n − q < n . The consistency of the S 4 , S 7 reductions then implies the consistency of the S p reductions of the lower-dimensional supergravities. In particular, we obtain explicit non-linear ansätze for the S 3 reduction of type IIA and heterotic supergravities, restricting to the U (1) 2 subgroup of the SO (4) gauge group of S 3 . We also study the black-hole solutions in the lower-dimensional gauged supergravities with domain-wall backgrounds. We find new domain-wall black holes which are not the singular-modulus limits of the AdS black holes of the original theories, and we obtain their Killing spinors.

General dimensional reduction of ten-dimensional supergravity and superstring

Dimensional reductions of supergravity theories are shown to yield to specific classes of four-dimensional no-scale models with N=4, 2 or 1 residual supersymmetry. N=1 “maximal” supergravity lagrangian, corresponding to the “untwisted” sector of orbifold compactification of superstrings, contains nine families and has a no-scale structure based on the Kähler manifold [SU(3, 3+3 n)/SU(3) ×SU(3+3 n)]×[SU(1,1)/U(1)]. The quantum consistency of the resulting theories give information on the non Kaluza-Klein (string) “twisted” sector.

Dimensional reduction of supergravity

Supergravity defined on a manifold of the form M 4 × S R , where S R is a homogeneous space and M 4 is ordinary space-time, is dimensionally reduced to M 4 by requiring all fields to be invariant under the action of the group S. Examples of the dimensional reduction of 11-dimensional supergravity and of 10-dimensional supergravity coupled to super-Yang-Mills fields are presented.

Electrovac ground state in gauged SU(2) × SU(2) supergravity

In one version of gauged N = 4 extended supergravity, the scalar potential is unbounded below and has no critical point. Thus four-dimensional anti-de Sitter space ((AdS) 4) which is the natural background geometry in most theories of this type is not a solution of the classical field equations. We show that there are background solutions with abelian electric and magnetic fields and space-time geometry which is a product of two-dimensional anti-de Sitter space and the 2-sphere-(AdS) 2 × S 2. For purely electric backgrounds the S 2 factor becomes the plane R 2 and the solution admits Killing spinors so that the background is supersymmetric. The Witten positive energy argument is applied to show that the supersymmetric background is stable. New techniques are required to specify the conserved Killing charges of an electrovac background and their contribution to the Witten-Nester integral. The product space background is an example of non-trivial dimensional reduction in supergravity.