Do Three Dimensions tell us Anything about a Theory of Everything?

It has been conjectured that four-dimensional N=8 supergravity may provide a suitable framework for a `Theory of Everything', if its composite SU(8) gauge fields become dynamical. We point out that supersymmetric three-dimensional coset field theories motivated by lattice models provide toy laboratories for aspects of this conjecture. They feature dynamical composite supermultiplets made of constituent holons and spinons. We show how these models may be extended to include N=1 and N=2 supersymmetry, enabling dynamical conjectures to be verified more rigorously. We point out some special features of these three-dimensional models, and mention open questions about their relevance to the dynamics of N=8 supergravity.


I. INTRODUCTION
Once upon a time, four-dimensional N = 8 supergravity was touted as a possible 'Theory of Everything' [1]. This suggestion was sparked by the observation that N = 8 supergravity has a hidden, composite SU(8) gauge structure [2]. The interacting scalar fields of N = 8 supergravity have a non-compact E 7 (7) structure with 133 components that are reduced by the 63 generators of the SU (8) symmetry to the 70 physical, on-shell scalar fields. Attention was drawn by the realization that SU(8) is a gauge symmetry large enough to include a flavour symmetry as well as an SU(5) grand-unification group. This motivated the suggestion that the composite SU(8) gauge fields might become physical particles [1], as a result of some unspecified dynamical mechanism. Already in the first papers on this subject, attention was drawn to two-dimensional coset models in which the composite gauge fields became dynamical as a result of infrared singularities [2].
However, the suggestion that N = 8 supergravity was a promising framework for a 'Theory of Everything' rapidly became unfashionable, for several reasons. One was that no convincing mechanism for making the gauge fields dynamical came to mind, and another was that the SU(8) gauge symmetry was thought to be anomalous [3]. However, the big blow was the realization that ten-dimensional E 8 × E 8 and SO(32) heterotic string theories had no anomalies, and could provide suitable frameworks for grand unification, after compactification to four dimensions [4]. Subsequently, many promising four-dimensional models were derived from string theory as understanding of string dynamics deepened. This deepening understanding led to the realization that all string theories are linked, and to the discovery that they are related to eleven-dimensional supergravity [5]. In turn, this theory may be related to N = 8 supergravity by compactification to four dimensions [6]. However, there is no simple limit in which string theory can be compactified to four-dimensional N = 8 supergravity without additional low-mass fields [7].
Interest in N = 8 supergravity has recently revived, motivated by the realization that it is very well-behaved in the ultraviolet, and may well be finite in perturbation theory [8].
This development has brought back to the collective consciousness a 'forgotten' paper by Marcus [9], in which he showed that the SU(8) gauge symmetry of N = 8 supergravity is, in fact, anomaly-free. This paper had been overlooked in the euphoria surrounding string theory but now, when coupled with the good ultraviolet behaviour of N = 8 supergravity, it motivates a re-examination of the possibility that this might be a promising road towards a 'Theory of Everything'. This possibility is not in necessarily conflict with the candidacy of string theory as the 'Theory of Everything', at least as long as the relation of string theory to N = 8 supergravity remains to be elucidated.
In this paper we review the relevant aspects of three-dimensional lattice models that may serve as inspirations for modelling the dynamics of N = 8 supergravity. In particular, we display the role of dynamical composite supermultiplets made of constituent holons and spinons. As we show explicitly, supersymmetry is an inessential complication, in the sense that it does not alter the nature of the infrared behaviour. On the other hand, the elevation of N = 1 supersymmetry to N = 2 does enable dynamical results to be placed on a more rigorous basis. We finish by recalling some of the limitations of the three-dimensional models, and highlighting some of the questions that arise before the existence of a similar mechanism in N = 8 supergravity could be addressed.

II. 2+1-DIMENSIONAL CONDENSED-MATTER MODELS WITH DYNAMICAL GAUGE BOSONS
The understanding of field theories in three space-time dimensions has largely been driven by the impetus to study condensed-matter systems formulated on a lattice. These field theories become relativistic near nodes of the Fermi surface, see Fig. 1, and may exhibit dynamical gauge bosons and supersymmetry, as we discuss below.
A classic model studied with antiferromagnetism and high-temperature superconductivity has been the t − J Hamiltonian [13,15], which is expressed in terms of electron creation operators c † a,i and their conjugates. As shown in [11,12], when inter-sublattice hopping is included, this Hamiltonian can be expressed in terms of field operators χ, whose structure we discuss below. For example, the hopping term reads where i denotes a lattice site, σ 3 is a 2 ×2 Pauli matrix, and summation over the spin indices is implied. To this may be added Heisenberg interaction terms, which can be written in the following convenient form [13]  The antiferromagnetic nature of such systems translates into the existence of more than one sublattice (see Fig. 1), associated with the spin states of the hole excitations, with intersublattice hopping non-zero but suppressed relative to the hopping inside a single sublattice.
As we review below, this results in internal gauge degrees of freedom for the hole excitations, which becomes a fully-fledged dynamical non-Abelian gauge symmetry in the effective field theory. There is an even number of sublattices or nodes in the Fermi surface of such systems, and hence an even number of spinor degrees of freedom, which results in parity remaining unbroken. (We remind the reader that in 2+1 dimensions a single fermion mass term breaks parity, but this can be restored if there is an even number of fermionic species, with mass terms in pairs of opposite signs.) To see this, as discussed in [11,12], one may represent the field χ via a spin-charge sepa-ration Ansatz in the case of a planar antiferromagnetic lattice with intersublattice hopping for a particle-hole symmetric formulation away from half-filling: The fields z a obey canonical bosonic commutation relations, and are associated with the spin degrees of freedom ('spinons'), whilst the fields ψ a are Grassmann variables on the lattice, which obey fermionic statistics, and are associated with the electric charge degrees of freedom ('holons').
The ansatz (3) admits a hidden non-Abelian local SU(2) spin symmetry at the constituent level, as discussed in [11,12]. This may readily be seen by considering the invariance of the χ variable under simultaneous local SU(2) rotations of the spinon and holon components: where h i ∈ SU(2). In three space-time dimensions, due to the fractional statistics of the planar excitations, one has an additional Abelian phase rotation, forming a U S (1) 'statistical' Abelian gauge group (which should not be confused with the U(1) of electromagnetism).
Hence, the full local gauge group of the spin-separation ansatz is SU(2) ⊗ U S (1). In terms of the spin and charge excitations appearing in (3), the hopping term in the Hamiltonian (1) may be written as which has a trivial local SU(2) symmetry.
The Heisenberg interaction term (2) can be linearized in terms of the fermion bilinears if one introduces in the path integral a Hubbard-Stratonovich field ∆ ij , in a standard fashion.
The result of the linearization for the combined hopping and interaction Hamiltonian is [11,12] had a Hartree-Fock form: This Hamiltonian determines the nature of any spontaneous symmetry breaking that occurs, and the associated phase structure of the low-energy theory near the nodes of the Fermi surface. Using the gauge symmetries of (3), we can write where V ∈ SU(2) and U ∈ U S (1) are group elements. The group U S (1) expresses the fractional statistics of the three-dimensional excitations of spinons Z and holons Ψ. The fact that apparently gauge non-invariant correlators are non-zero on the lattice is standard in gauge theories, and does not violate Elitzur's theorem [14], due to the fact that in order to evaluate the physical correlators one must follow a gauge-fixing procedure, which should be done prior to any computation. The amplitudes |A 1 | and |A 2 | ≡ K > 0 are considered frozen, which is a standard assumption in the gauge theory approach to strongly correlated electron systems [13,15]. The group elements V and U are phases of the above field bilinears and, to a first (mean-field) approximation, can be considered as composites of the constituent spinons z and holons ψ. Fluctuations about such ground states can then be considered by integrating over the constituent fields.
The above lattice action does not contain kinetic terms for the SU(2) and U S (1) groups, which would take the forms: where the sum is over the plaquettes p of the lattice, and the coefficients are the inverse square couplings of the U S (1) and SU(2) groups, respectively. The specific relation between the coefficients of the SU(2) and U S (1) terms is a consequence of the appropriate normalizations of the generators of the groups. The absence of gauge kinetic terms in our case implies that we are in the limit of infinitely strong coupling for both gauge groups: However, as discussed in detail in [11], integration of the Z-spinon fields results in kinetic terms for gauge fields, which couple to the spinons Ψ only. The analysis of [11] assumes that the Z-spinons have a mass gap that is larger than the dynamical mass gap generated by the holon fields through their interactions with the Abelian gauge group, so that one can define the appropriate effective theory of the light degrees of freedom by integrating out the heavy ones. (We recall that the kinetic terms are the lowest terms in a derivative expansion, in which terms with more than two derivatives are irrelevant operators in the infrared, and hence can be ignored when one considers the low-energy continuum limit of interest here.) In such a case, one obtains [11]: where a is the (sub)lattice spacing, η denotes the doping concentration in the sample, and J is the Heisenberg interaction in the condensed-matter model (6), and group theory maintains the relationship (9) between the SU(2) and U S (1) couplings. The strong-U S (1) coupling corresponds formally to Jηa → ∞.
We now turn to the dynamical breaking of the SU(2) group [11], using a Schwinger-Dyson treatment of dynamical symmetry breaking based on a large-N approximation, in which the SU(2) spin group is replaced by SU(N) with N large. In this case, the non-Abelian coupling is related to the Abelian one through This implies that, even in the case of strong U S (1) coupling, β 1 → 0, the large-N (large-spin) limit may be implemented in such a way so that β SU (N ) is finite, and may be assumed weak.
As discussed in detail in [11] and references therein, the lattice hamiltonian (6) after integrating out the massive spinon Z fields, can be expressed in the infrared limit as a lattice spinor hamiltonian for the holon fields Ψ coupled to the U S (1) and SU(2) gauge fields: where is the SU(2) gauge field, and the plaquette terms appear as a result of the z (spinon) integration, as mentioned previously. Working in the large N-fermion flavour limit (N even sublattices, N → ∞) and keeping the SU(N) coupling β SU(N) finite, according to (12), the SU(2) gauge field becomes dynamical. The U S (1) field, on the other hand, is assumed strongly coupled: β 1 → ∞, and hence its kinetic term is absent. The fact that kinetic terms for the SU(2) gauge fields can be induced in the effective action by quantum corrections reflects the existence of a non-trivial infrared fixed point in the (2+1)-dimensional Abelian gauge theory, after the dynamical breaking of the SU(2). This was discussed in detail in [16], and has also been discussed in the N = 1 supersymmetric case [17].
After integrating the effective spinon-gauge-field Lagrangian over the strongly-coupled statistical U S (1) dynamical gauge group, one finds the following effective partition function: where Here, K is the amplitude |A 2 | of the fermionic bilinears in the Hartree-Fock lattice action (6) and I 0 is the zeroth-order Bessel function. The quantity y iµ may be written in terms of the bilinears with the result: In the language of particle physics, quantities analogous to the M (i) would represent physical meson states. Converting from fermionic to bosonic variables, the low-energy (longwavelength) effective action may written as a path integral in terms of gauge-boson and meson fields [11] where the meson-dependent term in (18) comes from the Jacobian that arises when converting the integral from fermion to meson variables.
A method was presented in [11] for identifying the symmetry-breaking patterns of the gauge theory (18), by studying the dynamically-generated mass spectrum. The method consists of expanding i,µ lnI 0 ( √ y i,µ ) in powers of y iµ and concentrating on the lowest orders, which yield the gauge boson masses, whilst higher orders describe interactions. Keeping only the linear term in the expansion yields It is evident that the pattern of SU(2) breaking is determined by non-zero VEVs for the meson matrices M (i) . One has the following expansion for the meson states in terms of SU(2) bilinears [11]: with µ = 0, 1, 2, and Here, A 1 , A 2 , A 3 , B 1 , B 2 , B 3 transform as triplets under SU(2), and A 4 , B 4 transform as SU (2) singlets, the γ µ are (antihermitean) Dirac (space-time) 2 × 2 matrices, and the σ i , i = 1, 2, 3 are the (hermitean) 2 × 2 SU(2) Pauli matrices.
Naively, the composite meson fields A i and B i transform under a global SU (2) group [11,18]. However, because of the Hartree-Fock form of the lattice Hamiltonian (6), these composites realize a strongly-coupled local (gauged) SU(2) ⊗ U S (1) group, which (following the above discussion) enables the representation of the SU(2) gauge link variables in terms of the B i composites [11]: where a = 1, 2, 3 are SU(2) indices, and bold face letters indicate vectors in SU(2) space.
(The reader should bear in mind the trick (12), of working with large SU(N) instead of the SU(2) group, so as to guarantee a finite β SU (N ) coupling, while working with a strong U S (1) group.) The VEV of the matrix < M (i) >= uσ 3 is proportional to the chiral condensate u, which is responsible for generating dynamically a (parity-conserving) mass gap for the holons [11].
Substituting (20) into (19), and performing a naive perturbative expansion over the fields B one finds: From this it follows that two of the SU(2) gauge bosons, namely the B 1 ,B 2 , become massive, with masses proportional to Ku, whereas the gauge boson B 3 remains massless. These mass terms break SU(2) down to a U(1) subgroup.
We draw the reader's attention to the similarity of the above mechanism for symmetry breaking with the situation in the adjoint gauge-Higgs model [19]. There, the SU(2) symmetry is also broken down to a U(1) subgroup whenever the constant multiplying the Higgsgauge interaction is larger than a critical value. In our case the rôle of this constant is played by K 2 , as can be seen by the formal analogy between the adjoint-Higgs-gauge interaction terms and (19). In our approach symmetry breaking was achieved via the infinitely-strong U S (1) coupling. In view of the above analogy with the adjoint-Higgs model, however, one may speculate that an interesting phase diagram for the symmetry breaking of SU(2) could also emerge due to the K 2 coupling, whatever the U S (1) coupling.
At this point we would like to clarify certain points concerning the above-mentioned models: the lattice systems we examined above are simplifications of the actual situation encountered in realistic condensed matter systems, where in some regions of their parameter space the Fermi surface is characterized by pockets of finite size rather than points. In such models, the spin-charge separation ansatz is more subtle than the one presented here. The interested reader can find more details on such issue in the recent literature [20], where detailed phase diagrams are presented. However, for our purposes here, we are simply interested in the compositeness aspects of the above decomposition of spin and charge degrees of freedom, which are merely used here as a motivation for our particle-physics oriented analysis rather than a rigorous study of realistic condensed matter systems and their properties.
In this respect, in what follows we shall make use of the above formalism in order to draw some useful lessons and thus be able to discuss the notion of composite operators and their relevance to N=8 supergravity phenomenology.
We close this Section by mentioning the possibility of extending the above results to other gauge groups, provided the ansatz (3) for spin-charge separation in the simple SU (2) case that represents the spin degree of freedom in a antiferromagnetic system is modified accordingly.

MODELS WITH DYNAMICAL GAUGE BOSONS
A. N = 1 supersymmetry A further step was taken in [21,22], where conditions were derived under which the continuum low-energy limit of the above lattice Hamiltonian becomes supersymmetric. The authors of [22] considered the lattice t − J models discussed in the previous section, but including non-nearest-neighbour hopping and interaction terms in the antiferromagnetic sublattices. Such models have been used in theoretical modeling of d-wave high-temperature superconductors [23]. The analysis led to the following N = 1 supersymmetric continuum Lagrangian in terms of component spinon ψ and holon z fields: where D µ denotes the gauge-covariant derivative with respect to the U S (1) gauge field, and f a is an auxiliary field. The field η is a Majorana fermion, which is viewed as the supersymmetric partner of the Abelian U S (1) gauge field, needed to reproduce (as an appropriate Lagrange multiplier field [22]), the CP 1 σ-model constraint, which in superfield notation reads 2 a=1 φ a φ a = 1, with the superfield φ a being given by This contains, for each colour, a complex scalar z a , a Dirac spinor ψ a , and a complex auxiliary field f a . We refer the reader to [22] for a detailed discussion how this formalism can actually z a ψ α + z a ψ a = 0, (27) and these results can be generalised to non-Abelian cases, such as the broken SU(2) case discussed previously.
In addition to supersymmetry between the constituent spinon and holons, it was suggested in [24], motivated by the Hartree-Fock form (6) and the associated spinon and holon bilinears moduli spaces of such theories have been studied extensively in [25], following the pioneering work of Seiberg and Witten [26] in four dimensions, with the result that there is a non-trivial non-perturbative infrared fixed point, induced by the matter multiplets. We recall that a (2+1)-dimensional gauge theory is superrenormalizable, without ultraviolet divergences, by elementary power counting. However, it has a non-trivial low-energy (infrared) structure.
There are increasing hints that four-dimensional N = 8 supergravity theory might also lack the expected ultraviolet divergences. However, the low-energy (infrared) structure of the four-dimensional N = 8 theory remains to be elucidated.
The non-trivial infrared structure of the (2+1)-dimensional theory was interpreted in [24] as indicating a deviation from the trivial Landau-liquid fixed point, implying several observable properties in the normal phase of the superconductor. In addition, these theories also exhibited a superconducting phase [10,24], associated with a phase in which the SU (2) non-Abelian symmetry is broken down to an Abelian subgroup. The superconducting phase corresponds to the Coulomb phase of the Abelian subgroup (not to be confused with electro-magnetic gauge symmetry), in which the corresponding gauge boson is massless. This leads to an anomalous current-current one loop diagram with a massless pole which, according to the anomaly mechanism of [10], results in superconductivity (according to the Landau criterion) upon coupling to a real external electromagnetic field. The Higgs phase of the Abelian subgroup of the original SU(2) gauge group, in which the gauge field is massive, corresponds to the pseudogap phase of the underdoped cuprates (in condensed-matter parlance), which occurs for low doping. In addition, non-perturbative (monopole) configurations have also been discussed [24], in various supersymmetric effective gauge theories, following [25], in an attempt to discuss the formation of domain-wall structures, such as the stripe phase and holon degrees of freedom. The latter are assumed to belong to N = 1 supermultiplets, but the supersymmetry may be extended to N = 2, if the gauge condition ∂ µ A µ = 0 is satisfied. The authors of [27] needed this gauge condition in the Abelian Higgs model in order to identify the gauge field with a topologically-conserved current, which was at the origin of their elevation of the supersymmetry. The same condition was found for the CP 1 model in [28], and was generalized to Yang-Mills theory in [29]. Its recurrence suggests that this gauge condition is necessary, model-independently, for supersymmetry elevation in 2+1 dimensions.
Our starting point is to assume the existence of two complex N = 1 scalar supermultiplets (i is now a flavour indice) where the two-component Grassman coordinate θ is real, and the complex field components Based on these fundamental degrees of freedom, the following N = 1 composite supermultiplets were considered in [24], where D α ≡ ∂/∂θ α + i( / ∂θ) α is the supercovariant derivative: • scalar supermultiplets, forming an SU(2) triplet, • parity-conserving vector supermultiplets, forming an SU(2) triplet, • a parity-violating composite vector supermultiplet, forming an SU(2) singlet, The next step is is to gather the N = 2 supermultiplet degrees of freedom into the following complex superfields Each of these contains a real scalar field, a gauge field, a complex two-dimensional gaugino and a real auxiliary field. Let us denote by g the gauge coupling and by f abc the SU (2) structure constants. Using the Wess-Zumino gauge for V a α , it was shown in [29] that the following Lagrangian describes the resulting N = 2 supersymmetric SU(2) gauge theory: provided the gauge condition holds for any gauge index a. This gauge condition was found in [29] to be necessary to cancel unwanted contributions proportional to ∂ µ A a µ , when L is expanded in terms of the field components. Although we exhibit here an SU(2) model, it is clear that this construction may be extended to any SU(n) symmetry group, and is not restricted to the = 2 case presented here.
An important question is how the above-described composite ('meson-like') fields become dynamical. In Section II, we have seen that the SU(2) holon composites, which are obtained by integrating out a non-dynamical (strongly coupled) U(1), can become dynamical, as a result of their gauging, which is a direct consequence of the Hartree-Fock lattice hamiltonian (6) and the associated spinon and holon bilinears (7). A similar situation characterises the supersymmetric composites, where again there are lattice Hamiltonians of Hartree-Fock type, as we discussed extensively in Section III. However there are a few subtleties, which we now outline. Unlike the non-supersymmetric case, where the mass gaps of the spinon z and holons Ψ were different, in the supersymmetric case one is not allowed to integrate out the spinons, which now are degenerate in mass with the holons. Hence both the groups U S (1) and SU (2) appear strongly coupled at the constituent level. However, we can apply here again the trick of working with large-N SU(N), instead of the SU(2) group. This implies that, as in the non-supersymmetric case, one can consider finite couplings for the non-Abelian part, in which case one may demonstrate that the above-described N = 2 SU(N) supersymmetric composites become dynamical.
Finally, the addition of matter in the fundamental representation necessitates two new N = 1 scalar supermultiplets, made of composite field components, and containing different degrees of freedom from those considered so far: that we represent as a two-component superfield Q = (Q 1 , Q 2 ). We note that the matter superfield components must be complex, since they transform according to the fundamental representation of SU (2).
It is easy to see that the gauge-matter interaction then reads where the T a are the SU(2) generators in the fundamental representation.
In this respect, we note that, from the point of view of condensed-matter models, these matter fields describe interactions external to the two sublattice structures depicted in Fig. 1, e.g., by coupling the superconducting planes, etc. For our purposes though, such matter fields may be integrated out, which renders the gauge multiplets dynamical, by inducing kinetic terms for the gauge fields, in analogy with the z-spinon integration in the nonsupersymmetric models of Section II. In this way one still has formally a strongly-coupled SU(N) gauge theory, but the quantum corrections renormalise the effective coupling.
To summarize, starting from the two complex N = 1 scalar supermultiplets (28), we have the following construction: • Build three complex N = 1 scalar supermultiplets (29) and three N = 1 vector supermultiplets (30), all composed of more elementary degrees of freedom; • Gather these degrees of freedom into three N = 2 supermultiplets (32), so as to obtain an N = 2 supersymmetric SU(2) gauge theory, which is possible if the gauge condition (34) is imposed; • Add the composite matter N = 1 supermultiplets (35) so as to generate an N = 2 supersymmetric SU(2) theory interacting with matter in the fundamental representation.
The theory obtained in the Lagrangians (33) and (36) can of course be expressed in a more compact way using the N = 2 superspace formalism, but the point of the present derivation is to start from N = 1 supersymmetry, using the N = 1 formalism, to arrive at a theory invariant under N = 2 supersymmetry.
As mentioned in the previous Section, one may use exact results on N = 2 gauge theories to understand the dynamical behaviour of this model.

IV. SUMMARY AND PROSPECTS
We have shown that supersymmetric models in 2+1 dimensions, formulated either at the lattice level or as coset field theories, exhibit the emergence of dynamical gauge bosons realizing an SU(N) gauge group. There are also composite scalar fields whose expectation values may break this composite gauge symmetry spontaneously to a subgroup that is realized in the Coulomb phase. Specifically, in the lattice models discussed in Sections II and III, the underlying constituents are spinons and holons, and the composite scalars are in the adjoint representation of the gauge group. In the coset field theories discussed in Section III, there may also appear scalar fields in the fundamental representation of the SU(N) gauge group.
By formulating N = 2 supersymmetric versions of these lattice and coset field theories, the derivation of these results may be placed on a rigorous basis.
An important feature of our models was that the gauge fields were strongly-coupled, in the sense of not having kinetic terms (i.e., plaquette terms in lattice models). However, we have argued that, by working in the large N-limit, where N is the number of species (i.e., sublattices, or nodes) which must be even for reasons of parity conservation, one could formally re-instate a finite value of the SU(N) gauge coupling, thus making the gauge fields dynamical.
Moreover, by using arguments based on a Hartree-Fock approximation in the microscopic models, we have supported the idea that composite gauge fields made out of spinons and holons can, also in the supersymmetric regions of the parameter space, become dynamical, leading to full-fledged N = 2 supersymmetric SU(N) gauge theories. In (2+1)-dimensions these are characterised by a non-trivial infrared fixed-point structure, which constitutes an additional, more rigorous for inferring the dynamical nature of the gauge fields from the effects of the quantum corrections. Even if the bare action is characterised by infinite gauge couplings, quantum loop corrections can generate such terms, thereby generating the dynamics of the gauge fields. This feature has been seen explicitly in non-supersymmetric models, by the integration of the spinon (magnon) fields z. In supersymmetric theories, formally one works with additional matter multiplets, which can be integrated out, thereby inducing finite (renormalised) gauge kinetic terms and hence rendering the strongly-coupled gauge theory dynamical.
Although in this work we have dealt explicitly with unitary SU(N) gauge theories, in which N must be even because of arguments based on the parity invariance that characterises the antiferromagnetic case, nevertheless other gauge groups are expected to lead to qualitatively similar conclusions.
These features are exactly what one might like to see [1] in N = 8 supergravity in 3+1 dimensions [2], which has a non-anomalous SU(8) gauge symmetry [9]. If this gauge symmetry would become dynamical, an adjoint scalar multiplet could break it spontaneously into a rank-7 subgroup such as SU(5) × U(1) 3 or SU(3) × SU(2) × U(1) 3 , and scalars in fundamental representations could in principle reduce the rank to the SU(2) × U(1) gauge group of the Standard Model. However, infrared behaviour in 3+1 dimensions is different from that in 2+1 dimensions, and we have no real understanding how the composite gauge fields might become dynamical in this case. In this connection, we recall that N = 8 supergravity cannot be obtained simply as some limit of string theory without the appearance of additional low-mass fields, and it is possible that these might play an essential role.
Our work is complementary to the mechanism for making dynamical the gauge fields of three-dimensional N = 8 supergravity based on the maximal supersymmetric effective low-energy theory of multiple M2 branes that was presented in [30]. There, the relevant non-Abelian gauge fields have no Maxwell term, but only mixed (parity-conserving) Chern-Simons structures. Our approach is to suggest that a kinetic term for the gauge field may be generated through renormalization-group effects in the infrared region of the effective theory, in direct analogy to what happens in planar condensed-matter models on the lattice and their continuum effective theories. It remains to be seen whether such an approach, which works so well in three space-time dimensions, can describe correctly the properties of the four-dimensional N = 8 supergravity theory.