Maximal Supergravity Theories Research Articles

Quaternionic quantum mechanics for N\u2009=\u20091, 2, 4 supersymmetry

BackgroundQuaternions have emerged as powerful tools in higher-dimensional quantum mechanics as they provide homogeneous four-dimensional structure in quantum field theories, offer compact representations, and incorporate spin naturally. Quantum field theories then lead to the unification of fundamental interactions so the use of quaternion becomes necessary when we are dealing with higher-dimensional theories. On the other hand, supersymmetry is the theory of bosons and fermions and is an essential constituent of grand unified theories. The use of quaternion in supersymmetric field theories provides an excellent framework for higher-dimensional unification theories.ResultA complete theory for supersymmetric quaternionic quantum mechanics has been constructed for N = 1, 2, 4 supersymmetry in terms of one, two, and four supercharges and Hamiltonians, respectively. It has been shown that N = 4 SUSY is the quaternionic extension of the N = 2 complex SUSY and N = 1 real SUSY; also spin is the natural outcome of using quaternion units. Pauli and Dirac Hamiltonian and their relationship have also been obtained in quaternion space. It has been shown that quaternionic quantum mechanics are superior to ordinary and complex quantum mechanics because in the quaternion framework we do not need three different theories for N = 1,2,4 SQM but a single theory only.ConclusionsIt has been concluded that N = 1 real SUSY is equal to N = 2 complex SUSY which in turn is equal to N = 4 quaternion SUSY so one can arrive at higher-dimensional quantum field theories starting from lower-dimensional quantum theories. Higher-dimensional quaternion field theories are suitable for nonphotonic light cone particles which are not allowed in complex QFT, also noncommutative nature of quaternion gives an extra degree of freedom and may provide the possibility of some new particle, dark matter, or new phenomenon. Though quaternions provide an excellent framework in higher-dimensional field theories, there are certain challenges due to their noncommutativity as calculations become tedious where large terms are involved. Keeping in view the noble features of quaternion, we expect some development to get a better understanding of N = 8 supergravity, maximal supergravity (D = 11 − n), and maximal supersymmetry theories (N = 10) in terms of quaternion operators.

New mathcal{N} = 1 AdS4 solutions of type IIB supergravity

We construct analytically a new family of supersymmetric AdS4 solutions of IIB supergravity, with the internal space provided by a deformed S5 × S1. The solutions preserve mathcal{N} = 1 supersymmetry and an SO(3) subgroup of isometries of S5, which is broken to U(1) along a flat direction. They are further parametrised by a winding number and a choice of SL(2) duality twist along the circle in an elliptic conjugacy class, thus including both globally geometric and S-fold configurations. We identify these solutions by first constructing a new family of vacua of D = 4, U(4) ⋉ ℝ12 gauged maximal supergravity and use exceptional field theory to perform the uplift to ten dimensions. We discuss the relevance of D = 5 Wilson loops associated to preserved and broken gauge symmetries in the construction of these classes of solutions.

Canonical analysis of E6(6)(R) invariant five dimensional (super-)gravity

We investigate the canonical structure of the bosonic sector of the unique maximal supergravity theory in five dimensions that is manifestly invariant under the global action of [Formula: see text]. Starting from the Lagrangian formulation of the theory, we construct the Hamiltonian formulation and the full set of canonical constraints. We determine all gauge transformations and compute the algebra formed by the canonical constraints under the Poisson bracket. We re-derive the number of physical degrees of freedom and construct the extended Hamiltonian, describing the most general time evolution of the theory, where the full gauge freedom is manifest.

Branes and polytopes

We investigate the hierarchies of half-supersymmetric branes in maximal supergravity theories. By studying the action of the Weyl group of the U-duality group of maximal supergravities we discover a set of universal algebraic rules describing the number of independent 1/2-BPS p-branes, rank by rank, in any dimension. We show that these relations describe the symmetries of certain families of uniform polytopes. This induces a correspondence between half-supersymmetric branes and vertices of opportune uniform polytopes. We show that half-supersymmetric 0-, 1- and 2-branes are in correspondence with the vertices of the k21, 2k1 and 1k2 families of uniform polytopes, respectively, while 3-branes correspond to the vertices of the rectified version of the 2k1 family. For 4-branes and higher rank solutions we find a general behavior. The interpretation of half-supersymmetric solutions as vertices of uniform polytopes reveals some intriguing aspects. One of the most relevant is a triality relation between 0-, 1- and 2-branes.

Irreducible representations of E theory

We construct the [Formula: see text] theory analogue of the particles that transform under the Poincaré group, that is, the irreducible representations of the semi-direct product of the Cartan involution subalgebra of [Formula: see text] with its vector representation. We show that one such irreducible representation has only the degrees of freedom of 11-dimensional supergravity. This representation is most easily discussed in the light cone formalism and we show that the duality relations found in [Formula: see text] theory take a particularly simple form in this formalism. We explain that the mysterious symmetries found recently in the light cone formulation of maximal supergravity theories are part of [Formula: see text]. We also argue that our familiar space–times have to be extended by additional coordinates when considering extended objects such as branes.

Space-filling branes & gaugings

We consider in any dimension the supersymmetric ℤ2 truncations of the maximal supergravity theories. In each dimension and for each truncation we determine all the sets of 1/2-BPS space-filling branes, i.e. branes whose world-volume invades the whole of space-time, that preserve the supersymmetry of the truncated theory and the representations of the symmetry of such theory to which they belong. We show that in any dimension below eight these sets always contain exotic branes, that are objects that do not have a ten-dimensional origin. We repeat the same analysis for half-maximal theories and for the quarter-maximal theories in four and three dimensions. We then discuss all the possible gaugings of these theories as described in terms of the embedding tensor. In general, the truncation acts on the quadratic constraints of the embedding tensor in such a way that some representations survive the truncation although they are not required by the supersymmetry of the truncated theory. We show that for any theory, among these representations, the highest-dimensional ones are precisely those of the 1/2-BPS space-filling branes that preserve the same supersymmetry of the truncated theory, and we interpret this result as the fact that these quadratic constraints after the truncation become tadpole conditions for such branes.

Subsectors, Dynkin diagrams and new generalised geometries

We examine how generalised geometries can be associated with a labelled Dynkin diagram built around a gravity line. We present a series of new generalised geometries based on the groups Spin(d, d) × ℝ+ for which the generalised tangent space transforms in a spinor representation of the group. In low dimensions these all appear in subsectors of maximal supergravity theories. The case d = 8 provides a geometry for eight-dimensional backgrounds of M theory with only seven-form flux, which have not been included in any previous geometric construction. This geometry is also one of a series of “half-exceptional” geometries, which “geometrise” a six-form gauge field. In the appendix, we consider exam-ples of other algebras appearing in gravitational theories and give a method to derive the Dynkin labels for the “section condition” in general. We argue that generalised geometry can describe restrictions and subsectors of many gravitational theories.

Nonsupersymmetric magic theories and Ehlers truncations

We consider the nonsupersymmetric “magic” theories based on the split quaternion and the split complex division algebras. We show that these theories arise as “Ehlers” [Formula: see text] and [Formula: see text] truncations of the maximal supergravity theory, exploiting techniques related to the very-extended Kac–Moody algebras. We also generalize the procedure to other [Formula: see text] truncations, resulting in additional classes of nonsupersymmetric theories, as well as to truncations of nonmaximal theories. Finally, we discuss duality orbits of extremal black hole solutions in some of these nonsupersymmetric theories.

A brief review of E theory

I begin with some memories of Abdus Salam who was my PhD supervisor. After reviewing the theory of nonlinear realisations and Kac–Moody algebras, I explain how to construct the nonlinear realisation based on the Kac–Moody algebra [Formula: see text] and its vector representation. I explain how this field theory leads to dynamical equations which contain an infinite number of fields defined on a space–time with an infinite number of coordinates. I then show that these unique dynamical equations, when truncated to low level fields and the usual coordinates of space–time, lead to precisely the equations of motion of 11-dimensional supergravity theory. By taking different group decompositions of [Formula: see text] we find all the maximal supergravity theories, including the gauged maximal supergravities, and as a result the nonlinear realisation should be thought of as a unified theory that is the low energy effective action for type II strings and branes. These results essentially confirm the [Formula: see text] conjecture given many years ago.

E11 must be a symmetry of strings and branes

We construct the non-linear realisation of the semi-direct product of E11 and its vector representation in five and eleven dimensions and find the dynamical equations it predicts at low levels. Restricting these results to contain only the usual fields of supergravity and the generalised space-time to be the usual space-time we find the equations of motion of the five and eleven dimensional maximal supergravity theories. Since this non-linear realisation contains effects that are beyond the supergravity approximation and are thought to be present in an underlying theory we conclude that the low energy effective action of string and branes must possess an E11 symmetry.

Exceptional versus superPoincaré algebra as the defining symmetry of maximal supergravity

We describe how one may use either the superPoincar\'e algebra or the exceptional algebra to construct maximal supergravity theories in the light-cone formalism. The d=4 construction shows both symmetries albeit in a non-linearly realized manner. In d=11, we find that we have to choose which of these two symmetries to use, in constructing the theory. In order to understand the other "unused" symmetry, one has to perform a highly non-trivial field redefinition. We argue that this shows that one cannot trust counterterm arguments that do not take the full symmetry of the theory into account. Finally we discuss possible consequences for Superstring theory and M-theory.

Forms and algebras in (half-)maximal supergravity theories

The forms in D-dimensional (half-)maximal supergravity theories are discussed for 3 $\leq$ D $\leq$ 11. Superspace methods are used to derive consistent sets of Bianchi identities for all the forms for all degrees, and to show that they are soluble and fully compatible with supersymmetry. The Bianchi identities determine Lie superalgebras that can be extended to Borcherds superalgebras of a special type. It is shown that any Borcherds superalgebra of this type gives the same form spectrum, up to an arbitrary degree, as an associated Kac-Moody algebra. For maximal supergravity up to D-form potentials, this is the very extended Kac-Moody algebra E11. It is also shown how gauging can be carried out in a simple fashion by deforming the Bianchi identities by means of a new algebraic element related to the embedding tensor. In this case the appropriate extension of the form algebra is a truncated version of the so-called tensor hierarchy algebra.

The tensor hierarchy simplified

A compact formulation of the field-strengths, Bianchi identities and gauge transformations for tensor hierarchies in gauged maximal supergravity theories is given. A key role in the construction is played by the recently-introduced tensor hierarchy algebra.

E 11, Borcherds algebras and maximal supergravity

Abstract The dynamical p-forms of torus reductions of maximal supergravity theory have been shown some time ago to possess remarkable algebraic structures. The set (“dynamical spectrum”) of propagating p-forms has been described as a (truncation of a) real Borcherds superalgebra "Image missing" D that is characterized concisely by a Cartan matrix which has been constructed explicitly for each spacetime dimension 11 ≥ D ≥ 3. In the equations of motion, each differential form of degree p is the coefficient of a (super-) group generator, which is itself of degree p for a specific gradation (the "Image missing"-gradation). A slightly milder truncation of the Borcherds superalgebra enables one to predict also the “spectrum” of the non-dynamical (D − 1) and D-forms. The maximal supergravity p-form spectra were reanalyzed more recently by truncation of the field spectrum of E 11 to the p-forms that are relevant after reduction from 11 to D dimensions. We show in this paper how the Borcherds description can be systematically derived from the split (“maximally non compact”) real form of E 11 for D ≥ 1. This explains not only why both structures lead to the same propagating p-forms and their duals for p ≤ (D − 2), but also why one obtains the same (D−1)-forms and “top” D-forms. The Borcherds symmetries "Image missing"2 and "Image missing"1 are new too. We also introduce and use the concept of a presentation of a Lie algebra that is covariant under a given subalgebra.

Superconformal symmetry and maximal supergravity in various dimensions

In this paper we explore the relation between conformal superalgebras with 64 supercharges and maximal supergravity theories in three, four and six dimensions using twistorial oscillator techniques. The massless fields of N=8 supergravity in four dimensions were shown to fit into a CPT-self-conjugate doubleton supermultiplet of the conformal superalgebra SU(2,2|8) a long time ago. We show that the fields of maximal supergravity in three dimensions can similarly be fitted into the super singleton multiplet of the conformal superalgebra OSp(16|4,R), which is related to the doubleton supermultiplet of SU(2,2|8) by dimensional reduction. Moreover, we construct the ultra-short supermultiplet of the six-dimensional conformal superalgebra OSp(8*|8) and show that its component fields can be organized in an on-shell superfield. The ultra-short OSp(8*|8) multiplet reduces to the doubleton supermultiplet of SU(2,2|8) upon dimensional reduction. We discuss the possibility of a chiral maximal (4,0) six-dimensional supergravity theory with USp(8) R-symmetry that reduces to maximal supergravity in four dimensions and is different from six-dimensional (2,2) maximal supergravity, whose fields cannot be fitted into a unitary supermultiplet of a simple conformal superalgebra. Such an interacting theory would be the gravitational analog of the (2,0) theory.

Generalised geometry, eleven dimensions and E11

We construct the non-linear realisation of E11 and its first fundamental representation in eleven dimensions at low levels. The fields depend on the usual coordinates of space-time as well as two form and five form coordinates. We derive the terms in the dynamics that contain the three form and six form fields and show that when we restricted their field dependence to be only on the usual space-time we recover the correct self-duality relation. Should this result generalise to the gravity fields then the non-linear realisation is an extension of the maximal supergravity theory, as previously conjectured. We also comment on the connections between the different approaches to generalised geometry.

Maximal supergravity in three dimensions: supergeometry and differential forms

The maximal supergravity theory in three dimensions, which has local SO(16) and rigid $E_8$ symmetries, is discussed in a superspace setting starting from an off-shell superconformal structure. The on-shell theory is obtained by imposing further constraints. It is essentially a non-linear sigma model that induces a Poincar\'e supergeometry that is described in detail. The possible $p$-form field strengths, for $p=2,3,4$, are explicitly constructed using supersymmetry and $E_8$. The gauged theory is also discussed.

On duality symmetries of supergravity invariants

The role of duality symmetries in the construction of counterterms for maximal supergravity theories is discussed in a field-theoretic context from different points of view. These are: dimensional reduction, the question of whether appropriate superspace measures exist and information about non-linear invariants that can be gleaned from linearised ones. The former allows us to prove that F-term counterterms cannot be E 7(7)-invariant in D = 4, N = 8 supergravity or E 6(6)-invariant in D = 5 maximal supergravity. This is confirmed by the two other methods which can also be applied to D = 4 theories with fewer supersymmetries and allow us to prove that N = 6 supergravity is finite at three and four loops and that N = 5 supergravity is three-loop finite.

The very-extended trombone

Starting from the very-extended Kac-Moody algebra $E_{11}$, we consider the algebra $E_{11,D}^{local}$, obtained by adding to the non-negative level $E_{11}$ generators the $D$-dimensional momentum operator and an infinite set of additional generators that promote the global $E_{11}$ symmetries to gauge ones. We determine all the possible trombone deformations of this algebra, that is the deformations that involve the $D$-dimensional scaling operator. The Jacobi identities imply that such deformations are uniquely determined by a single tensor belonging to the same representation of the internal symmetry group as the vector generators and satisfying additional quadratic constraints. The non-linear realisation of the deformed algebra gives the field strengths of the theory which are those of any possible maximal supergravity theory in which the global scaling symmetry is gauged in any dimension. All the possible deformed algebras are in one to one correspondence with all such maximal supergravity theories. The tensor that parametrises the deformation is identified with the embedding tensor that is used to parametrise all maximal supergravity theories with gauged scaling symmetry, and the quadratic constraints that we determine exactly coincide with the field theory results.

TheE11origin of all maximal supergravities. The hierarchy of field-strengths

Starting from $E_{11}$ and the space-time translations we construct an algebra that promotes the global $E_{11}$ symmetries to local ones, and consider all its possible massive deformations. The Jacobi identities imply that such deformations are uniquely determined by a single tensor that belongs to the same representation of the internal symmetry group as the $D-1$ forms specified by $E_{11}$. The non-linear realisation of the deformed algebra gives the field strengths of the theory which are those of any possible gauged maximal supergravity theory in any dimension. All the possible deformed algebras are in one to one correspondence with all the possible massive maximal supergravity theories. The hierarchy of fields inherent in the $E_{11}$ formulation plays an important role in the derivation. The tensor that determines the deformation can be identified with the embedding tensor used previously to parameterise gauged supergravities. Thus we provide a very efficient, simple and unified derivation of the bosonic sector, and in the absence of gravity, of all maximal gauged supergravities from $E_{11}$.