Supergravities in Diverse Dimensions

The following sections are included:

• PREFACE

• References

• CONTENTS

Since superalgebras form the algebraic basis of all supersymmetric theories it seems appropriate to devote a chapter to this subject…

One of the main requirements Imposed on quantum field theory is invariance of the theory to the Poincare group [1] …

Supergauge transformations are defined in four space-time dimensions. Their commutators are shown to generate γ₅ transformations and conformal transformations. Various kinds of multiplets are described and examples of their combinations to new representations are given. The relevance of supergauge transformations for Lagrangian field theory is explained. Finally, the abstract group theoretic structure is discussed.

A systematic method for constructing Wess–Zumino supergauge transformations is exhibited.

A method is given for constructing some of the unitary irreducible representations of the Wess–Zumino super-gauge symmetry. Application of this symmetry to the analysis of S-matrix elements is considered. A new super-gauge symmetry which includes isospin is introduced and some of its representations are constructed.

A generator of a symmetry or supersymmetry of the S-matrix has to have three simple properties (see sect. 2). Starting from these properties one can give a complete analysis of the possible structure of the pseudo Lie algebra of these generators. In a theory with non-vanishing masses one finds that the only extension of previously known relations is the possible appearance of “central charges” as anticommutators of Fermi charges. In the massless case (disregarding infrared problems and symmetry breaking) the Fermi charges may generate the conformal group together with a unitary internal symmetry group.

We give an elementary presentation of the Lie superalgebras, their classification and some properties of their representations. A sketch of the classical Lie super-group is also given.

We determine all manifest supersymmetries in more than 1 + 1 dimensions, including those with conformal or de Sitter space-time symmetry. For the supersymmetries in flat space we determine the structure of all representations and give formulae for an effective computation. In particular we show that at least for masses m² = 0, 1,2 the states of the spinning string form supersymmetry multiplets.

Supersymmetric extensions of the Poincaré algebra in D-dimensional space-time are reviewed and a catalogue of their representations is developed. This catalogue includes all supermultiplets whose states carry helicity ≤ 2 in the massless cases and ≤ 1 in the massive cases.

Section 1

1. Eleven is the maximal dimension which supergravity theory can exist, with fields carrying spin J≤ 2, as first shown by Nahm [2]. The d = 11 supergravity was constructed by Cremmer, Julia and Sherk [3]. It was in fact the first supergravity theory to be constructed in dimensions higher than four. One of the motivations of these authors was to reduce dimensionally the d = 11 super-gravity down to d = 4 on a 7-torus to obtain the complete d = 4, N= 8 supergravity. (At that time, this theory had been constructed by Noether procedure, only to order κ² [1]). Another motivation was the reduction of the d = 11 theory to d = 10 to obtain the zero slope limit of the closed string model. The dimensional reduction of d = 11, N = 1 supergravity down to d = 4 was achieved by Cremmer and Julia [4], while the dimensional reduction to d = 10 was carried out by Chamseddine [8]…

We present the action and transformation laws of supergravity in 11 dimensions which is expected to be closely related to the O(8) theory in 4 dimesions after dimensional reduction.

d= 10

1. The fact that the massless states of the Neveu-Schwarz-Ramond open string [1] form a d = 10 super Yang-Mills multiplet was first noted by Gliozzi, Scherk and Olive (GSO) in 1977 [2] …

We construct the interaction of supergravity with Yang-Mills supersymmetry in 10 dimensions. The formulation is only possible with the use of a six-index antisymmetric gauge field in place of the familiar two-index field in the supergravity sector. We are interested in breaking supersymmetry and thus apply the generalized dimensional reduction. The nonpropagating field strength F_{μvρσαβγ} plays a vital role in making the scalar potential nontrivial. It would be possible to test the predictions of this model once the internal-symmetry group is specified and solutions of the potential minima are obtained.

The d = 10, N = 1 Yang-Mills system is coupled to d = 10, N = 1 supergravity in a locally scale-invariant way. An analysis of the currents agrees with the Noether coupling results and reveals the existence of two ordinary axial and more low-dimension auxiliary fields. The coupling of the photon A_μ to antisymmetric tensors A_μv is consistent because the Maxwell transformation δA_μ = ∂_μΛ is extended to δA_μv = κΛF_μv.

We show how to generalize the coupling of n = 1 super-Maxwell theory and n = 1 supergravity in ten dimensions to the case of a non-abelian gauge group. We find that the supergravity 2-form potential a_μv is coupled to the Yang-Mills gauge potential A_μ via the Chern-Simons 3-form.

Even though chiral N = 2 D = 10 supergravity does not have a manifestly Lorentz-invariant action principle, it does have covariant field equations. The latter are obtained by first deriving the local supersymmetry transformations of the fields and then deducing the set of field equations that transform into one another and are required for closure of the algebra. One of the equations is the self-duality of a supercovariant fifth-rank field strength tensor. An SU(1,1) symmetry plays a crucial role in the analysis.

We construct a massive version of N = 2 supergravity in ten dimensions by compactification of the eleven-dimensional, N = 1 theory. This theory describes the usual N = 2 massless supermultiplet, in addition to which there is an infinite tower of massive, charged N = 2 supermultiplets. The appearance of these massive states as a consequence of the spontaneous breaking of part of the eleven-dimensional supersymmetry and the presence of Goldstone modes is clarified.

A new version of the non-chiral N = 2a supergravity in ten dimensions is obtained, in which the two-index tensor field of the theory “eats” the single vector field and acquires a mass in a Higgs-type mechanism. The new theory, although it contains no fundamental vectors, bears many formal resemblances to gauged supergravities (in particular, the recently constructed F(4) theory in six dimensions). The scalar potential has no extrema, but nevertheless the classical equations of motion admit a wide variety of spontaneous compactifications, many to four dimensions.

d = 9

1. The maximal supergravity in d = 9 has N = 2 supersymmetry, and contains the fields …

N = 1 supergravity in d = 9 coupled to n vector multiplets, each containing a real scalar, is constructed by the Noether method. It is shown that the scalars parametrise the coset space Hⁿ = SO(n, l)/SO(n).

d = 8

1. The maximal d = 8, N = 2 supergravity is vectorlike, and has no matter couplings …

SU(2) gauged N = 2 supergravity in d = 8 is constructed by generalized dimensional reduction of d = 11 supergravity on SU(2) group manifold. The relation between the field equations of the d = 8 and those of d = 11 supergravities is established. As a byproduct of this, it is shown that certain compactifications of d = 11 supergravity give rise to anti-de Sitter space-time (AdS)⊗S⁴ or AdS⊗CP² (with or without SU(2) instanton) or AdS⊗S²⊗S² compactifications of d = 8 supergravity.

We couple d = 8, N = 1 supergravity to n vector multiplets. The 2n scalars of the theory parametrize the Kähler manifold SO(n, 2)/SO(n)×SO(2). The n+2 vector fields are used to gauge the [SO(l, 2)×H] subgroup of SO(n, 2) where H ⊂ SO(n − 1) and dim H = n − 1. It is shown that the theory compactifies to (Minkowski)₆ × S² by a monopole configuration which is embedded in SO(l, 2). The field equations fix the monopole charge to be ±1, which implies a stable, chiral N = 2 supergravity in d = 6.

d = 7

1. There exist N = 4 and N = 2 supergravities in seven dimensions …

We construct Yang–Mills theory and simple supergravity in seven dimensions. We gauge the rigid SU(2) symmetry of the latter. The potential for the scalar field φ is of the form exp(φ) and has no extremum. Possible improvement due to a “topological mass term” is discussed.

We obtain N = 2 supergravity theory coupled to N = 2 matter by truncation of the N = 4 theory in seven dimensions. The truncated theory possesses global GL(4, R) ⊗ local composite SO(4) invariance. We then gauge the global SO(4) subgroup of GL(4, R), and preserve the composite SO(4) symmetry, following the method of de Wit and Nicolai in d = 4. The resulting action has a potential which contains all ten scalars of the theory. The single second rank antisymmetric tensor field in the theory has a generalized field strength, which contains the Chern-Simons three-form.

The complete nonlinear gauged N = 4 ,d = 7 supergravity action and supersymmetry transformation laws (without four- and three-fermion terms) are obtained by the Noether method, starting from the linearized gauged N = 4, d = 7 model, as previously found by spontaneous compactification of d = 11 supergravity on S₄. The model has a local Yang-Mills SO(5) and a local composite SO(5) symmetry. Essential in the construction is the selfdual action for the five third-rank antisymmetric tensors.

We construct couplings of n vector multiplets to seven-dimensional N = 2 supergravity. The 3n scalars of the theory parametrize the coset SO(n,3)/SO(n)×SO(3). The (n +3) vector fields are used to gauge either an SO(3)×H (dimension H = n) or an SO(3,l)×H (dim H = n − 3) subgroup of SO(n,3). The theory has an indefinite potential which triggers compactification into (Minkowski)₄ ×S³, with surviving N = 1 supersymmetry.

The following sections are included :

• d = 6

• d = 6, N = 2

• Non chiral d = 6, N = 4

• Chiral d = 6, N = 4

• d = 6, N = 8

• References

A class of theories for which the Lorentz algebra closes only on the mass-shell, and which therefore cannot be written in a manifestly Lorentz-invariant form, is presented. Examples include certain supergravity theories in six and ten dimensions. These results make the corresponding phenomenon in the case of supersymmetry algebras less surprising. They also shed light on the problem of finding off-mass-shell dual string model amplitudes.

We show by an explicit example with a model of D = 6 supergravity, containing the sechsbein, a Weyl gravitino and a two-index photon, that a consistent theory on the group-manifold might have no counterpart in the usual Noether approach In our model the self-duality of the Maxwell field strength F_abc necessary to match the Bose–Fermi on-shell degrees of freedom, follows from group-manifold variational equations but not from their x space restriction. As a consequence, the theory is consistent, although the x space lagrangian is not supersymmetric invariant.

We derive all the quartic fermion terms in the action, and all the cubic fermion terms in the transformation rules of the N = 2, d = 6 supergravity plus matter-coupled Yang-Mills system constructed by the authors in an earlier paper. We also show how compactification to the 4-dimensional Minkowski spacetime is always automatically realized even with fermionic condensates, based on the argument of scale covariance by Witten.

We construct the lagrangian of N = 8 supergravity in six dimensions up to the quartic fermion terms. The non-polynomial interactions of the scalar fields are determined by the SO(5,5)/SO(5) × SO(5) coset structure and the SO(5,5) duality invariance for the antisymmetric tensor fields.

Gauged N = 4 supergravity theories with Yang-Mills symmetry SU(2) are constructed in six dimensions. There are four distinct theories, determined by the values of the SU(2) coupling constant g and a mass parameter m for the two-index tensor field contained in the theories. One of the theories has a scalar potential with two extrema; one extremum leads to a ground state exhibiting the full anti-de Sitter supersymmetry F(4), while the other breaks the supersymmetry completely. In this theory, and also in two of the remaining three theories, the two-index tensor “eats” an abelian vector and becomes massive, acquiring a cubic self-coupling in the process. The last theory, in which the tensor field remains massless, coincides with one previously obtained by dimensional reduction from seven dimensions. We obtain a variety of compactifications for all the theories, many supersymmetric and many to four dimensions. Finally, we comment on the geometrical structure of the theories, and compare them to ten-dimensional supergravities.

We determine the supersymmetric coupling of chiral N = 4b supergravity in six dimensions to an arbitrary number n of anti-self-dual tensor multiplets. In a novel phenomenon, the five self-duality and n anti-self-duality conditions for the two-index tensors are modified by the scalar interactions to be non-diagonal in terms of the elementary tensor gauge fields, which transform irreducibly under a global SO(5, n) invariance. The 5n scalar fields parametrize the coset manifold SO(5, n)/[SO(5) × so(n)]. The truncated N = 2 systems, in which the n scalars parametrize the space so(n,1)/so(n), are derived and found to have analogous properties. In general the theories do not admit action principles possessing manifest linearly realized Lorentz invariance, so the construction is effected at the level of the supersymmetry transformations and field equations. Finally, we speculate on generalizations including N = 2 vector matter, and comment on the possible implications of our results for chiral superstring models.

The following sections are included :

• d = 5

• d = 5, N = 2

• d = 5, N = 4

• d = 5, N = 8

The following sections are included:

• WHY 5 DIMENSIONS ?

• SYMPLECTIC SPINORS

• SUPERGRAVITIES IN 5 DIMENSIONS

• GLOBAL AND LOCAL SYMMETRIES IN SUPERGRAVITY

• N=8 SUPERGRAVITY IN 5 DIMENSIONS

• SUPERGRAVITIES N=6, 4, 2
  • N=6 Supergravity
  • N=4 Supergravity
  • N=2 Supergravity

• REFERENCES

We discuss in detail the possible gaugings, abelian and non-abelian, of a class of d = 5 Maxwell/Einstein supergravity theories for which the manifold of scalar fields is a symmetric coset space. We show that a U(l) “gauged” d = 5 supergravity theory is possible with a vanishing scalar field potential, and we give necessary and sufficient conditions for this to occur. We discuss the d = 5 Yang-Mills/Einstein supergravity theories for both compact and non-compact gauge groups. We show that the “irreducibility” of the “magical” subclass of d = 5 Maxwell/Einstein supergravity theories is preserved if and only if the Yang-Mills gauge group is SU(3,1). We expect that the irreducibility property of the “exceptional” d = 4 Maxwell/Einstein supergravity theory can similarly be preserved by gauging an SO*(8) subgroup of its symmetry group E₇₍₋₂₅₎. Throughout we make extensive use of the underlying Jordan algebraic structure of N = 2 supergravity which we have established in previous work.

We construct gauged N = 8 supergravity theories in five dimensions. Instead of the twenty-seven vector fields of the ungauged theory, the gauged theories contain fifteen vector fields and twelve second-rank antisymmetric tensor fields satisfying self-dual field equations. The fifteen vector fields can be used to gauge any of the fifteen-dimensional semisimple subgroups of SL(6, R), specifically SO(p, 6− p) for p = 0,1, 2, 3. The gauged theories also have a physical global SU(1,1) symmetry which survives from the E₆₍₆₎ symmetry of the ungauged theory. This SU(1,1) for the SO(6) gauging is presumably related to that of the chiral N = 2 theory in ten dimensions. In our formalism we maintain a composite local USp(8) symmetry analogous to SU(8) in four dimensions.

We consider the coupling of the N = 2 Maxwell and matter multiplets to N = 2 supergravity in d = 5. It is shown that the scalar manifold factorizes into a quaternionic manifold for the matter scalar and a cubic hypersurface for the Maxwell scalar fields. This factorization is carried out to d = 4 and 3 through ordinary dimensional reduction. We study also the N = 2 theories obtained from the truncation of the N = 8 and N = 4 theories in those dimensions. In d = 3 we find a remarkable N = 4 theory described by the coset space (E₈(− 24)/(E₇ × SU(2)))×(E₈(− 24)/(E₇ × SU(2))) which would yield a Kac-Moody algebra of the type when dimensionally reduced to d = 2.

The following sections are included :

• d = 4

• d = 4, N = 1

• d = 4, N = 2

• d = 4, N = 3

• d = 4, N = 4

• d = 4, N = 8

• Phenomenological Prospects of Matter Coupled N = 1, d = 4 Supergravity

As a new approach to supergravity, an action containing only vierbein and Rarita-Schwinger fields (Vaμ and ψ_μ) is presented together with supersymmetry transformations for these fields. The action is explicitly shown to be invariant except for a ψ⁵ term in its variation. This term may also vanish, depending on a complicated calculation. (Added note: This term has now been shown to vanish by a computer calculation, so that the action presented here does possess full local supersymmetry.)

A combined spin 2 – spin 3/2 extension of general relativity is given which is both free of the usual higher spin inconsistencies and invariant under local supersymmetry transformations.

The multiplication rule for multiplets of Poincaré and conformal supergravity is obtained. It is shown how to construct locally supersymmetric densities and hence actions out of these multiplets. Various examples illustrate these general results.

Rules are given for constructing a local vector multiplet from a scalar and a local scalar multiplet from a vector. Applications are given. The question of gauge invariance in couplings to supergravity is discussed and a local version of the Fayet-Iliopoulos term is constructed. This term achieves the gauge invariances of the Maxwell supermultiplet through the occurrence of compensating superconformal transformations.

We derive the lagrangian and transformation laws of the coupled Yang-Mills-matter-supergravity system for unextended n = 1 local supersymmetry. We study the super-Higgs effect and the normal Higgs effect of the Yang-Mills gauge group G. In the case of N chiral multiplets “minimally” coupled to supergravity, transforming according to some N-dimensional, generally reducible representation of G, we find a model-independent mass formula:

We begin to construct the most general supersymmetric Lagrangians in one, two and four dimensions. We find that the matter couplings have a natural interpretation in the language of the nonlinear sigma model.

We construct general matter couplings of N = 1 four-dimensional supergravity for theories with curvature-squared terms, suggested by D = 10 anomaly-free supergravity and superstrings. The scalar potential induced by the supersymmetric completion of the Euler characteristic is derived and its positivity properties discussed. Some implications for the nonpolynOmial structure of ten-dimensional supergravity are drawn.

We present explicit expressions for general actions of vector and scalar multiplets coupled to N = 2 supergravity. We outline their construction which is based on the superconformal tensor calculus. The vector multiplets may be associated with a gauge group G which may also act on the scalar multiplets. The latter are naturally described in terms of quaternions; in the simplest case their kinetic terms define a nonlinear sigma model of a quaternionic projective space. We give an extension of the vector multiplet action which is not obtained from a chiral superspace density, and contains a Chera-Simons-type term. Transformation rules are given and the conditions for supersymmetry breaking are defined.

We develop a technique to compute scalar potentials in extended supergravities in any dimension on a pure group-theoretical basis. The fermionic supersymmetry variations, which determine the potential, only depend on boosted structure constants of a gauge subgroup K ⊂ G of the isometry group G of the scalar manifold G/H. These variations follow from the Bianchi identities of the underlying gauge superalgebra. We apply this method to construct the coupling of N = 3 matter to supergravity.

We couple N = 4, d = 4 supersymmetric Yang-Mills theory to supergravity. The scalars of the theory parametrize the coset (SO(n,6)/[SO(n)×SO(6)])x(SU(l,l)/U(l)). Keeping the composite local SO(n)×SO(6)×U(l) invariance intact, we gauge an (n +6) parameter subgroup of SO(n,6) which is either (i) SU(2)×SU(2)×H (dim H = n), (ii) SO(4,l)×H (dim H = n − 4) or (iii) SO(6,l)×H (dim H = n − 15). In all these cases the theory has an indefinite potential.

Gauged N = 4 supergravity with an arbitrary number of matter multiplets is constructed from a superconformal starting point. It includes both the SO(4) and SU(4) symmetric N = 4 supergravity theories, and all their gaugings. Noncompact Yang-Mills symmetries may mix the matter and supergravity vector fields. We establish that in matter coupled N = 4 supergravity theories the super-Higgs effect can occur with a vanishing cosmological constant. An example is given with gauged SO(3)×SO(2,1) symmetry in which the scalar potential vanishes completely, and all four supersymmetries are broken.

The SO(8) supergravity action is constructed in closed form. A local SU(8) group as well as the exceptional group E₇ are invariances of the equations of motion and of a new first order lagrangian.

We present an extension of N = 8 supergravity in which the natural symmetry group SO(8) is gauged. Local SO(8) invariance is shown to be consistent with the dynamically realized SU(8) symmetry. We mention possible implications of this result for superunification.

A one-parameter family of gaugings of N = 8 supergravity is given. For positive values of the real parameter ξ, the SO(8) gauging is recovered, ξ = 0 gives the ISO(7) gauging, while for ξ < 0 a new SO(7,1) gauging is obtained. Despite the non-compact gauge symmetry for , the theory is ghost-free.

The following sections are included :

• d = 3

• d = 2

• d = 1

The locally supersymmetric extension of three-dimensional topologically massive gravity is constructed. Its fermionic part is the sum of the (dynamically trivial) Rarita–Schwinger action and a gauge-invariant topological term, of second derivative order, analogous to the gravitational one. It is ghost free and represents a single massive spin 3/2 excitation. The fermion–gravity coupling is minimal and the invariance is under the usual supergravity transformations. The system’s energy, as well as that of the original topological gravity, is therefore positive.

Two three-dimensional supergravity theories are constructed. The first one has eight local supersymmetries as well as an SO(8, n) global symmetry, where n is an arbitrary positive integer. The physical spectrum consists of 8n scalars and 8n spinors. The second theory has sixteen local supersymmetries and an E_8.8 global symmetry. The 128 bosons and 128 fermions transform as inequivalent spinors of the SO(16) subalgebra. The theories are formulated without auxiliary scalar fields, which requires that the non-compact symmetries act non-linearly.

We discuss a component formalism of N = 1 supergravity theories in 2 and 3 spacetime dimensions. Starting from gauge theories of the superconformal group, we derive the tensor calculus for conformal and Poincaré supergravity theories. A supersymmetric extension of the non-trivial analog of Einstein’s equation for 2 dimensions is given in terms of the scalar curvature multiplet.

We construct the (p, q)-type anti-de Sitter (adS) supergravity theories associated with the three-dimensional adS supergroups OSp(p|2; R)⊗OSp(q|2; R). They are distinguished, for fixed N=p+q, by the gravitini mass-matrix, which has signature p−q. We show that the action can be written as the integral of the Chern-Simons three-form associated with the (adS) supergroup.

For the maximally extended N = 16 supergravity theory in two dimensions, we explicitly construct a linear system whose integrability conditions are equivalent to the full nonlinear field equations of this theory. All the (on-shell) information contained in it can thus be encoded into a single E₈ matrix and its dependence on a spectral parameter; the invariance of the equations of motion under E₉ is manifest. Possible consequences and further developments are briefly discussed.

First and second order forms of the covariant action for a spinning particle are exhibited. The action consistently incorporates the necessary constraints and is invariant under both local supersymmetry and general time parameter transformations, and provides a simple one-dimensional model for the interaction between matter and supergravity. A formulation invariant under general co-ordinate transformations in superspace is also given and shown to be equivalent to the locally supersymmetric one.

Here we briefly collect together the special features concerning the interactions of scalar, vector and antisymmetric tensor fields (occuring in supergravities), which are scattered in the previous sections (the references are provided at the end of the introductions in Secs. 2-8, and will not be repeated here) …

The following sections are included:

• CONTENTS

A. In the first volume we gave an overview of Poincare and anti de Sitter supergravity theories. In this first chapter of the second volume we review conformal supergravities in d = 10, 6, 4, 3 and 2 …

We present the complete off-shell structure of conformal supergravity in ten dimensions. It is based on 128 + 128 degrees of freedom and its formulation requires differential constraints. We study how these constraints are resolved in four and five dimensions. Covariant conditions are given that restrict conformal supergravity to its on-shell Poincaré counterpart. In ten dimensions the relationship between the two theories has new and unusual aspects, which we explore in a variety of ways. We rewrite on-shell Poincaré supergravity in a superconformally invariant form, from which we deduce that its off-shell version must contain at least a scalar (chiral) multiplet. We analyze some aspects of the non-linear structure of the field representation based on the conformal fields combined with one scalar multiplet.

Using superconformal tensor calculus we construct general interactions of N = 2, d = 6 supergravity with a tensor multiplet and a number of scalar, vector and linear multiplets. We start from the superconformal algebra which we realize on a 40+40 Weyl multiplet and on several matter multiplets. A special role is played by the tensor multiplet, which cannot be treated as an ordinary matter multiplet, but leads to a second 40+40 version of the Weyl multiplet. We also obtain a 48 + 48 off-shell formulation of Poincaré supergravity coupled to a tensor multiplet.

The superconformal extension of (R_µνab)² is given in six dimensions. It is shown that in a superconformal gauge the three-form field H_μνρ has a natural torsion interpretation. Also partial results are given on the superconformal extension of the Gauss–Bonnet combination: .

We complete our program of constructing the gauge theory of the superconformal group, and show that the previously proposed action is completely invariant under both local supersymmetries. The gauge algebra closes off-shell as well as on-shell. A flat-space model with a local supersymmetry is also presented.

The supersymmetric extension of the gravitational Chern-Simons term in three-dimensional spacetime coincides with three-dimensional conformal supergravity. The action reads , with γ_AB the Killing supermetric and f_ABC the structure constants of Osp(1/4). The constraints read , and R_μν^mn(M)=0. Even when auxiliary fields close the super-Poincaré gauge algebra, I is invariant under local Poincaré sypersymmetry and independent of auxiliary fields.

We present an action for the Neveu–Schwarz–Ramond model from which follow both the field equations and the gauge and supergauge constraints. This is done by coupling the free-field action to two-dimensional supergravity in a geometrically clear way. The constraints arise as the supergravity field equations, the supergravity fields playing the role of Lagrange multipliers. The action is invariant under local supersymmetry transformations and, as a consequence, the field equations and the constraints are consistent. The commutator structure of the local supersymmetry algebra is exhibited. It is also shown that there exists a special gauge in which the action, the field equations and die constraints take the free-field form of the usual formulation of the Neveu–Schwarz–Ramond model.

We construct an action for the spinning string which is locally supersymmetric and reparametrization invariant using the techniques of supergravity. In a special gauge it is shown that the equations of motion and the constraints are those of the Neveu-Schwarz-Ramond model.

The (1, 0) and (2, 0) type heterotic σ-models with Wess–Zumino term are coupled to conformal supergravity in two dimensions. There are no new restrictions on the σ-model manifolds in addition to those which arise in the globally supersymmetric cases. In the (1, 0) case possible isometrics of the scalar manifold are gauged. A derivation of d = 2 conformal supergravity based on the super Lie algebra OSp(2, N)⊕OSp(2, N) (N = 1, 2) is given.

The general conformal coupling of an arbitrary number of scalar multiplets to d = 2, N = 4 supergravity is given. The supersymmetric nonlinear sigma models can be either hyperkahler or quaternionic, unlike in the analogous case in d = 4. This model is the long-ago predicted SU(2) spinning string.

The covariant SU(2) spinning string model of Pernici and van Nieuwenhuizen which has (4,4) supersymmetry is chirally truncated to a (4,0) model. The model is extended by the addition of a locally supersymmetric Wess-Zumino term, and heterotic fermions. This system is coupled to composite as well as fundamental Yang-Mills gauge fields.

The (8,0) conformal supergravity, and an action which describes its coupling to an arbitrary number of (8,0) scalar multiplets are constructed. The 64+64 components of the conformal supermultiplet occur as Lagrange multipliers which lead to differential and algebraic constraints on the fields of the scalar multiplets. Solving the algebraic constraints yields an (8,0) locally supersymmetric sigma model based on the manifold SO(8+n,m)/SO(8) ×SO(n,m), where n, .

1. By an anomaly one means a breakdown of a classical symmetry due to quantum corrections. This phenomenon was first noticed by Schwinger in 1951 when he tried to build a model with fermions coupled to axial currents [2] …

It was pointed out some time ago by Ferrara and Zumino¹⁾ that in a supersymmetric theory the axial current , the improved energy-momentum tensor θ_μν and the improved supersymmetry current S_μ can be identified with the components of a superfield V_μ, so that an intimate relation exists between them. At the classical level this superfield satisfies certain relations reflecting the conservation and trace relations (modulo mass terms)

We present a general algorithm for the construction of gravitational axial and conformal anomalies for fields of arbitrary spin. A variety of models is then displayed in which one or both of the anomalies vanish by cancellation. Our results are compared (and in the spin 3/2 case, contrasted) with previous calculations.

A new restriction on fermion quantum numbers in gauge theories is derived. For instance, it is shown that an SU(2) gauge theory with an odd number of left-handed fermion doublets (and no other representations) is mathematically inconsistent.

A general formula for global gauge and gravitational anomalies is derived. It is used to show that the anomaly free supergravity and superstring theories in ten dimensions are all free of global anomalies that might have ruined their consistency. However, it is shown that global anomalies lead to some restrictions on allowed compactifications of these theories. For example, in the case of O(32) superstring theory, it is shown that a global anomaly related to π₇(O(32)) leads to a Dirac-like quantization condition for the field strength of the antisymmetric tensor field.

Related to global anomalies is the question of the number of fermion zero modes in an instanton field. It is argued that the relevant gravitational instantons are exotic spheres. It is shown that the number of fermion zero modes in an instanton field is always even in ten dimensional supergravity.

Characteristic classes for the index of the Dirac family are computed in terms of differential forms on the orbit space of vector potentials under gauge transformations. They represent obstructions to the existence of a covariant Dirac propagator. The first obstruction is related to a chiral anomaly.

It is shown that in certain parity-violating theories in 4k + 2 dimensions, general covariance is spoiled by anomalies at the one-loop level. This occurs when Weyl fermions of spin or or self-dual antisymmetric tensor fields are coupled to gravity. (For Dirac fermions there is no trouble.) The conditions for anomaly cancellation between fields of different spin is investigated. In six dimensions this occurs in certain theories with a fairly elaborate field content. In ten dimensions there is a unique theory with anomaly cancellation between fields of different spin. It is the chiral n = 2 supergravity theory, which is the low-energy limit of one of the superstring theories. Beyond ten dimensions there is no way to cancel anomalies between fields of different spin.

It is shown how the form of the gauge and gravitational anomalies in quantum field theories may be derived from classical index theorems. The gravitational anomaly in both Einstein and Lorentz form is considered and their equivalence is exhibited. The formalism of gauge and gravitational theories is reviewed using the language of differential geometry, and notions from the theory of characteristic classes necessary for understanding the classical index theorems are introduced. The treatment of known topological results includes a pedagogical derivation of the Wess–Zumino effective Lagrangian in arbitrary even dimension. The relation between various forms of the anomaly present in the literature is also clarified.

Supersymmetric ten-dimensional Yang–Mills theory coupled to N = 1, D = 10 supergravity has gauge and gravitational anomalies that can be partially cancelled by the addition of suitable local interactions. The remaining pieces of all the anomalies cancel if the gauge group is SO (3 2) or E₈ × E₈. These cancellations are automatically incorporated in the type I superstring theory based on SO (32). A superstring theory for E₈ × E₈ has not yet been constructed.

We give criteria for anomaly cancellations in chiral Yang–Mills supergravities (including dual formulations of the theories) in 6, 8 and 10 dimensions.

We show that a gauged supergravity theory based on E₆ × E₇ × U(1) is free of gauge and gravitational anomalies in six dimensions. It compactifies to (Minkowski)⁴ × S² by the standard monopole mechanism. With a monopole of strength n in E₆, the resulting four-dimensional theory exhibits chiral SO(10) × U(1) with 2|n| families (and no antifamilies). Supersymmetry is broken.

The coupled N = 1 Yang-Mills plus supergravity theory in ten dimensions can be made anomaly-free for SO(32) or E₈ × E₈. Only the case of SO(32) is known to correspond to a superstring theory, which is probably necessary for a fully consistent quantum theory. Anomaly-free chiral theories in lower dimensions can be obtained by considering nontrivial compactifications (involving nonzero background gauge fields) of the ten-dimensional theory that satisfy a topological consistency condition. This paper considers the compactification of four dimensions on the manifold K₃ without requiring that the equations of motion be satisfied. This leads to a large number of anomaly-free chiral supersymmetric six-dimensional theories, corresponding to various ways of embedding U(1) factors in SO(32) or E₈ × E₈.

Applying Witten’s formula for global gauge and gravitational anomalies to 6-dimensional supergravities, we find: (a) The perturbatively anomaly free N = 4 chiral supergravity coupled to 21 tensor multiplets is global anomaly free for any choice of space-time manifold with vanishing third Betti number (b₃); (b) The perturbatively anomaly free matter coupled N = 2 chiral supergravities with arbitrary number of tensor multiplets, whose Yang-Mills gauge groups do not include G₂, SU(2), or SU(3) are free of global anomalies if the theory is formulated on S⁶. In the case of 9 tensor multiplets coupled to supergravity, this result holds for any space-time with vanishing b₃. (c) The N = 6 chiral supergravity has perturbative gravitational anomalies and therefore the global anomalies need not be considered in this case.

The effective gauge field action due to an odd number of fermion species in three-dimensional SV(N) gauge theories is shown to change by ±π|n| under a homotopically non-trivial gauge transformation with winding number n. Gauge invariance can be restored by use of Pauli-Villars regularization, which, however, introduces parity nonconservation in the form of a parity-nonconserving, topological term in the effective action.

We develop a formalism for computing sums over random surfaces which arise in all problems containing gauge invariance (like QCD, three-dimensional Ising model etc.). These sums are reduced to the exactly solvable quantum theory of the two-dimensional Liouville lagrangian. At D = 26 the string dynamics is that of harmonic oscillators as was predicted earlier by dual theorists, otherwise it is described by the nonlinear integrable theory.

The formalism of the previous paper is extended to the case of supersymmetric strings. The effective theory which sums up fermionic surfaces is described by the supersymmetric Liouville equation. At D = 10 effective decoupling of the Liouville dilaton takes place and our theory coincides with the old ones. At D = 3 our theory is equivalent to the three-dimensional Ising model, which is thus reduced to the two-dimensional supersymmetric Liouville theory.

Two-dimensional supersymmetric models that contain only complex right moving scalars and fermions are shown to possess a supersymmetry anomaly. This supersymmetry anomaly lies in a supermultiplet with the gravitational and chiral anomalies.

Certain nonlinear sigma models with fermions suffer from an anomaly similar to the one in non-Abelian gauge theory. We exhibit this anomaly using both perturbative and global methods. The affected theories are ill defined and hence unsuitable for describing low-energy dynamics. They include certain supersymmetric models in four-space dimensions.

Anomalies in nonlinear sigma models can sometimes be cancelled by local counterterms. We show that these counterterms have a simple topological interpretation, and that the requirements for anomaly cancellation can be easily understood in terms of ’t Hooft’s anomaly matching conditions. We exhibit the anomaly cancellation on homogeneous spaces G/H and on general riemannian manifolds . We include external gauge fields on the manifolds and derive the generalized anomaly-cancellation conditions. Finally, we discuss the implications of this work for superstring theories.

We define the (1 + 1)-dimensional supersymmetry algebra of type (p,q) to be that generated by p right-handed Majorana–Weyl supercharges and q left-handed ones. We construct the non-linear sigma models with supersymmetry of type (1,0) and (2,0) and discuss their geometry and their relevance to compactifications of the heterotic superstring. The sigma-model anomalies can be cancelled by a mechanism closely related to that used by Green and Schwarz to cancel gravitational and Yang–Mills anomalies for the superstring.

A generalized method of dimensional reduction, applicable to theories in curved space, is described. As in previous works by other authors, the extra dimensions are related to the manifold of a Lie group. The new feature of this work is to define and study a class of Lie groups, called “flat groups”, for which the resulting theory has no cosmological constant, a well-behaved potential, and a number of arbitrary mass parameters. In particular, when the analysis is applied to the reduction of 11-dimensional supergravity to four dimensions it becomes possible to incorporate three arbitrary mass parameters in the resulting N = 8 theory. This shows that extended supersymmetry theories allow more possibilities for spontaneous symmetry breaking than was previously believed to be the case.

In d-dimensional unified theories that, along with gravity, contain an antisymmetric tensor field of rank s–1, preferential compactification of d–s or of s space-like dimensions is found to occur. This is the case in 11-dimensional supergravity where s = 4.

An attempt is made to construct a realistic model of particle physics based on eleven-dimensional supergravity with seven dimensions compactified. It is possible to obtain an SU(3) × SU(2) × U(1) gauge group, but the proper fermion quantum numbers are difficult to achieve.

Assuming the compactification of 4 + K-dimensional space-time implied in Kaluza–Klein-type theories, we consider the case in which the internal manifold is a quotient space, G/H. We develop normal mode expansions on the internal manifold and show that the conventional gravitational plus Yang–Mills theory (realizing local G symmetry) is obtained in the leading approximation. The higher terms in the expansions give rise to field theories of massive particles. In particular, for the original Kaluza–Klein 4 + 1-dimensional theory, the higher excitations describe massive, charged, purely spin-2 particles. These belong to infinite dimensional representations of an O(1, 2).

We report results on the particle spectrum of the d=11 supergravity compactified on AdS⊗S⁷. Our method is to use Salam-Strathdee type harmonic expansions on S⁴ ⊗S⁷ and to calculate the propagator. We then examine the poles in the propagator and exploit a close relationship between the SO(5) Casimir labels and those of SO(3,2), to interpret the physical states in terms of Fronsdal’s positive energy representations. The resulting spectrum is arranged into supermultiplets labelled by an integer . The massless supermultiplet corresponds to . In addition, a singleton representation seems to emerge for . It consists of an 8-plet of spin-0 and an 8-plet of spin-1/2 particles.

When d = 11 supergravity spontaneously compactifies to d = 4, the number of unbroken sypersymmetries, , is determined by the holonomy group, , of the d = 7 ground-state connection. Here we present a new solution: Minkowski spacetime × K3 × T³, for which and N = 4. The massless sector in d = 4 is given by N = 4 supergravity coupled to 22 N = 4 vector multiplets. Aside from its intrinsic interest, this example throws new light on Kaluza–Klein supergravity. In particular, we note that the 192 + 192 massless degrees of freedom obtained from K3 × T³ exceed the 128 + 128 of the N = 8 theory obtained from T⁷ or S⁷.

A discrete set of solutions to the classical Einstein–Maxwell equations in six-dimensional space-time is considered. These solutions have the form of a product of four-dimensional constant curvature space-time with a 2-sphere. The Maxwell field has support on the 2-sphere where it represents a monopole of magnetic charge, n = ±1, ±2,... The spectrum of massless and massive states is obtained for the special case of flat 4-space, and the solution is shown to be classically stable. The limiting case where the radius of the 2-sphere becomes small is considered and a dimensionally reduced effective lagrangian for the long range modes is derived. This turns out to be an SU(2) × U(1) gauge theory with chiral couplings.

The question of fermion chirality in Kaluza–Klein theories with coupling to Yang–Mills fields is discussed. The argument is illustrated in eight dimensions where an SU(2) Yang–Mills field assumes the one-instanton form on the internal space. This serves not only to trigger spontaneous compactification of the internal space but will ensure the emergence of zero modes in an irreducible eight-spinor belonging to the (2t + 1)-dimensional representation of SU(2).

The following sections are included:

• Introduction

• Preliminaries

• Operators on Homogeneous Spaces

• The Atiyah-Hirzebruch Theorem

• Rarita-Schwinger Fields

• Elementary Gauge Fields

• Other Contexts

• Massless Scalars

• The Cosmological Constant

• Acknowledgments

• References

We investigate the criteria which determine super-gravity-induced compactification of a supersymmetric Yang–Mills theory in ten dimensions down to spaces of the type (Minkowski) × (G/H).

It is shown that nontrivial spontaneous compactification of ten-dimensional N = 1 supergravity with or without Yang–Mills matter is not possible unless maximal symmetry (i.e. Lorentz invariance) is violated in the four-dimensional spacetime.

We establish the existence at the linearized level of a new chiral N = 4 extended supergravity theory in six-dimensional spacetime that could couple to the chiral N = 4 antisymmetric tensor multiplet. The unique anomaly free combination occurs for 21 of the latter and the spectrum then coincides with the zero-mass sector of the N = 2 ten-dimensional chiral supergravity theory compactified on K₃.

It is seen that a nonlinear sigma model based on the noncompact coset space SU(1,1)/U(1) can curl up two spatial dimensions into a topologically noncompact surface of finite area with a compact U(1) isometry group. This mechanism can be used for several higher-dimensional supergravity theories. In particular, chiral N = 2, D = 10 supergravity would reduce to an N = 1, D = 8 theory in which the masslessness of fermions does not depend only on supersymmetry. Further reduction to four dimensions is possible.

We show that all known solutions of the N = 2 non-chiral d = 10 supergravity theory can, by a simple procedure, be obtained from the known Freund–Rubin type solutions of d = 11 supergravity. This is a consequence of the fact that each solution of the d = 10 theory depends crucially on the presence of a topologically non-trivial U(1) field strength F_mn. The mass spectrum of a particular d = 10 solution is obtained by truncating the spectrum of the corresponding d = 11 solution to the subset of fields neutral under the non-trivial U(1) symmetry. We investigate the compactification on CP³, and show that this yields a theory with N = 6 or N = 0 supersymmetry according to the orientation, and we relate this to the S⁷ compactification of d = 11 supergravity. Starting from the squashed S⁷ in d = 11 we derive a new solution of the d = 10 theory, namely (Ads)₄ times CP³ with a squashed non-Einstein metric. The supersymmetry is either N = 1 or N = 0 depending on the orientation. We conjecture that none of the supergravity theories in 4<d<10 dimensions obtained from d =11 by dimensional reduction admit non-trivial compactifications to four dimensions.

We show that the U(1) gauged Einstein–Maxwell supergravity in six dimensions, spontaneously compactifies on Minkowski × S², with a monopole-valued Maxwell field on S². The bosonic symmetry of the background is SU(2) × U(1). The field equations fix the monopoie charge to be ± 1. The consequence of this is that the N = 2 supersymmetry breaks down to N = 1, and chiral fermions emerge.