Six-dimensional Supergravity Research Articles

Romans type IIA theory and the heterotic strings

In this Letter we study T2 compactification of six-dimensional massive type IIA supergravity in presence of Ramond–Ramond background fluxes. The resulting theory in four dimensions is shown to possess SL(2,R)×SL(2,R)×O(4,20) duality symmetry. It is shown that specific elements of this symmetry relate massive type IIA compactified on K3×T2 (fluxes along K3) to the ordinary type IIA compactified on K3×T2 (fluxes along T2). In turn, this relationship is exploited to relate Romans theory to heterotic strings. The D8-brane (domain-wall) wrapped on K3×T2 is found to correspond to pure gravity heterotic solution which is a direct product of six-dimensional flat space and a four-dimensional Taub–NUT instanton.

Supersymmetric grand unification model with the orbifold symmetry breaking in the six-dimensional supergravity

We construct supersymmetric (SUSY) grand unification (GUT) models in the six dimensional space-time where the GUT symmetry is broken down to the standard-model gauge group by a simple orbifolding T^2/Z_4 or T^2/Z_6 and a pair of massless Higgs doublets in the SUSY standard model are naturally obtained. Since the background geometry here is simple compared with models using the Scherk-Schwarz mechanism, one might hope for an approximate gauge coupling unification in the present models. Here, the presence of the massless Higgs multiplets in the bulk is quite natural, since the anomaly cancellation in the six dimensional space-time requires N=2 hyper multiplets in the bulk, some of which are origins of the Higgs doublets. However, the origin of the quarks and leptons is still not clear at all.

The string dual of an N = (4,0) two-dimensional gauge theory

We find solutions of the six-dimensional maximal supergravity by adding a perturbation of vector fields to the solution AdS${}_3$ $\times$ S${}^3$. For certain perturbations the solution represents a dual description of an ${\cal N}=(4,0)$ field theory in two dimensions by the AdS/CFT correspondence.

All couplings of minimal six-dimensional supergravity

We describe the complete coupling of (1,0) six-dimensional supergravity to tensor, vector and hypermultiplets. The generalized Green–Schwarz mechanism implies that the resulting theory embodies factorized gauge and supersymmetry anomalies, to be disposed of by fermion loops. Consequently, the low-energy theory is determined by the Wess–Zumino consistency conditions, rather than by the requirement of supersymmetry. As already shown for the case without hypermultiplets, this procedure does not fix a quartic coupling for the gauginos. With respect to these previous results, the inclusion of charged hypermultiplets gives additional terms in the supersymmetry anomaly. We also consider the case in which abelian vectors are present. As in the absence of hypermultiplets, abelian vectors allow additional couplings. Finally, we apply the Pasti–Sorokin–Tonin prescription to this model.

Vertex operators in background with NS–NS flux using Berkovits–Vafa–Witten variables

String equations for the vertex operators of type IIB on $AdS_3\times S^3$ background with NS-NS flux are calculated using Berkovits-Vafa-Witten formalism. With suitable field definitions, the linearized field equations for six-dimensional supergravity and a tensor multiplet on $AdS_3\times S^3$ are recovered from these. We also discuss the massless degrees of freedom that survive the $S^3$ Kaluza-Klein compactification and how our vertex operators are related to the vertex operators introduced by Giveon, Kutasov and Seiberg.

Abelian vectors and self-dual tensors in six-dimensional supergravity

In this note we describe the most general coupling of abelian vector and tensor multiplets to six-dimensional (1,0) supergravity. As was recently pointed out, it is of interest to consider more general Chern-Simons couplings to abelian vectors of the type H r=dB r−1/2 c rabA adA b , with c r matrices that may not be simultaneously diagonalized. We show that these couplings can be related to Green-Schwarz terms of the form B r c r ab F a F b , and how the complete local Lagrangian, that embodies factorized gauge and supersymmetry anomalies (to be disposed of by fermion loops) is uniquely determined by Wess-Zumino consistency conditions, aside from an arbitrary quartic coupling for the gauginos.

Action with Manifest Duality for Maximally Supersymmetric Six-dimensional Supergravity

We perform explicitly the toroidal compactification of eleven dimensional supergravity to six dimensions and present its action in a manifestly $SO(5,5)\over SO(5)\times SO(5)$ invariant form using the recently proposed covariant formulation of theories involving chiral fields.

Gauged Six-Dimensional Supergravity from Massive Type IIA String Theory

We obtain the complete nonlinear Kaluza-Klein ansatz for the reduction of the bosonic sector of massive type IIA supergravity to the Romans F(4) gauged supergravity in six dimensions. The latter arises as a consistent warped ${S}^{4}$ reduction.

Six-dimensional supergravity on S3 × AdS 3 and 2d conformal field theory

In this paper we study the relation between six-dimensional supergravity compactified on S 3 × AdS 3 and certain two-dimensional conformal field theories. We compute the Kaluza-Klein spectrum of supergravity using representation theory; these methods are quite general and can also be applied to other compactifications involving anti-de Sitter spaces. A detailed comparison between the spectrum of the two-dimensional conformal field theory and supergravity is made, and we find complete agreement. This applies even at the level of certain non-chiral primaries, and we propose a resolution to the puzzle of the missing states recently raised by Vafa. As a further illustration of the method the Kaluza-Klein spectra of F-theory on M 6 × S 3 × AdS 3 and of M-theory on M 6 × S 2 × AdS 3 are computed, with M 6 some Calabi-Yau manifold.

Non commutative gravity from the ADS/CFT correspondence

The exclusion principle of Maldacena and Strominger is seen to follow from deformed Heisenberg algebras associated with the chiral rings of S_N orbifold CFTs. These deformed algebras are related to quantum groups at roots of unity, and are interpreted as algebras of space-time field creation and annihilation operators. We also propose, as space-time origin of the stringy exclusion principle, that the $ADS_3 \times S^3$ space-time of the associated six-dimensional supergravity theory acquires, when quantum effects are taken into account, a non-commutative structure given by $SU_q(1,1) \times SU_q (2)$. Both remarks imply that finite N effects are captured by quantum groups $SL_q(2)$ with $q= e^{{i \pi \over {N + 1}}}$. This implies that a proper framework for the theories in question is given by gravity on a non-commutative spacetime with a q-deformation of field oscillators. An interesting consequence of this framework is a holographic interpretation for a product structure in the space of all unitary representations of the non-compact quantum group $SU_q(1,1)$ at roots of unity.

Consistent and covariant anomalies in six-dimensional supergravity

In this note we clarify some issues in six-dimensional (1,0) supergravity coupled to vector and tensor multiplets. In particular, we show that, while the low-energy equations embody tensor-vector couplings that contribute only to gauge anomalies, the divergence of the energy-momentum tensor is properly non-vanishing. In addition, we show how to revert to a supersymmetric formulation in terms of covariant non-integrable field equations that embody corresponding covariant anomalies.

Gauge dyonic strings and their global limit

We show that six-dimensional supergravity coupled to tensor and Yang-Mills multiplets admits not one but two different theories as global limits, one of which was previously thought not to arise as a global limit and the other of which is new. The new theory has the virtue that it admits a global anti-self-dual string solution obtained as the limit of the curved-space gauge dyonic string, and can, in particular, describe tensionless strings. We speculate that this global model can also represent the world-volume theory of coincident branes. We also discuss the Bogomol'nyi bounds of the gauge dyonic string and show that, contrary to expectations, zero eigenvalues of the Bogomol'nyi matrix do not lead to enhanced supersymmetry and that negative tension does not necessarily imply a naked singularity.

Some properties of tensor multiplets in six-dimensional supergravity

We review some results on the complete coupling between tensor and vector multiplets in six-dimensional (1,0) supergravity.

Tensor and vector multiplets in six-dimensional supergravity

We construct the complete coupling of (1,0) supergravity in six dimensions to n tensor multiplets, extending previous results to all orders in the fermi fields. We then add couplings to vector multiplets, as dictated by the generalized Green-Schwarz mechanism. The resulting theory embodies factorized gauge and supersymmetry anomalies, to be disposed of by fermion loops, and is determined by corresponding Wess-Zumino consistency conditions, aside from a quartic coupling for the gauginos. The supersymmetry algebra contains a corresponding extension that plays a crucial role for the consistency of the construction. We leave aside gravitational and mixed anomalies, that would only contribute to higher-derivative couplings.

Tensor multiplets in six-dimensional (2,0) supergravity

We construct the complete coupling of (2,0) supergravity in six dimensions to n tensor multiplets, extending previous results to all orders in the fermi fields. The truncation to (1,0) supergravity coupled to tensor multiplets exactly reproduces the complete couplings recently obtained.

N = 1, D = 6 supergravity: Duality and non-minimal couplings

Six-dimensional supergravity theories and their duality properties play a central role in the context of string duality and string compactifications. Lowering dimensions usually leads to an increasing complexity of theories; in this respect six dimensions seem to constitute an appropriate compromise between the physical four and the presumably more fundamental ten or eleven dimensions. In this paper we present a superspace formulation of N = l, D = 6 supergravity with one tensor multiplet and an arbitrary number of vector- and hypermultiplets, in which the bosonic abelian superforms of the theory, the dilaton, the abelian gauge fields and the two-form are replaced by their S-duals i.e. four-, three- and two-superforms respectively, in compatibility with supersymmetry. As usual this replacement interchanges Bianchi identities with equations of motion. This formulation holds in the presence of one tensor multiplet and arbitrary numbers of hypermultiplets and abelian super-Maxwell multiplets if all couplings are minimal. We determine the consistency conditions for non-minimal couplings in N = 1, D = 6 supergravity, for which we present a particularly significant solution, namely the one associated with the Chem-SimonsLorentz three-form which entails the Green-Schwarz anomaly cancellation mechanism. In the case of non-minimal couplings it is found that the gauge fields and the two-form can still be dualized while the dilaton has to remain a zero-form.

New couplings of six-dimensional supergravity

We describe the couplings of six-dimensional supergravity, which contain a self-dual tensor multiplet, to n T anti-self-dual tensor matter multiplets, nv vector multiplets and n H hypermultiplets. The scalar fields of the tensor multiplets form a coset SO( n T, 1)/ SO( n T), while the scalars in the hypermultiplets form quarernionic Kähler-symmetric spaces, the generic example being Sp( n H, 1)/ Sp( n H) ⊗ Sp (1). The gauging of the compact subgroup Sp( n H) ⊗ Sp(1) is also described. These results generalize previous ones in the literature on matter couplings of N = 1 supergravity in six dimensions.

Strong/weak coupling duality from the dual string

On compactification to six spacetime dimensions, the fundamental heterotic string admits as a soliton a dual string whose effective worldsheet action couples to the background fields of the dual formulation of six-dimensional supergravity. On further toroidal compactification to four spacetime dimensions, the dual string acquires an $O(2,2;Z)$ target-space duality. This contains as a subgroup the axion-dilaton $SL(2,Z)$ which corresponds to a strong/weak coupling duality for the fundamental string. The dual string also provides a new non-perturbative mechanism for enhancement of the gauge symmetry.

Anomalies of Kaluza-Klein theories in six dimensions

The author calculates the gauge, gravitational and mixed (perturbative) anomalies in general six-dimensional Kaluza-Klein theories, including supergravity models, from the Atiyah-Singer index theorems and discuss the conditions for anomaly cancellation. The topological meaning of the index polynomials, when used in computing anomalies, is emphasised. The absence of anomalies in the compactified heterotic superstring theory in six dimensions is reviewed. The existence of matter and gauge coupled anomaly-free U(1)-gauged six-dimensional supergravities with E8(X)E7, E8(X)E6(X)SU(2), E7(X)E6(X)SU(2)(X)U(1), E7(X)E7 and E6(X)E6(X) SU(2)(X)SU(2)(X)U(1) gauge groups is proved without any reference to the compactification procedure. It realises a highly non-trivial check of the anomaly cancellation in some K3 compactified heterotic superstrings. Some comments on global anomalies in six dimensions are made.

Invariant action principle for six-dimensional chiral supergravities

Complete invariant lagrangians for N = 4 b and N = 2 six-dimensional chiral supergravities which contain (anti)self-duality equations among their equations of motion are constructed, and the algebra of symmetries of these lagrangians is considered.