Non-Semisimple and Complex Gaugings of <Emphasis Type="Italic">N</Emphasis>=16 Supergravity

Maximal and non-maximal supergravities in three dimensions allow for a large variety of semisimple (Chern-Simons) gauge groups. In this paper, we analyze non-semisimple and complex gauge groups that satisfy the pertinent consistency relations for a maximal (N=16) gauged supergravity to exist. We give a general procedure how to generate non-semisimple gauge groups from known admissible semisimple gauge groups by a singular boost within E₈₍₈₎. Examples include the theories with gauge group SO(8)×T₂₈ that describe the reduction of IIA/IIB supergravity on the seven-sphere. In addition, we exhibit two strange embeddings of the complex gauge group into (real) E₈₍₈₎ and prove that both can be consistently gauged. We discuss the structure of the associated scalar potentials as well as their relation to those of D4 gauged supergravities.This work is partly supported by EU contract HPRN-CT-2000-00122 and HPRN-CT-2000-00131.     

O(5)×U(1) electroweak gauge theory and the neutrino oscillations in matter

A study is made of the groupO(5)×U(1). The group is economical in the number of gauge bosons, which we associate with each of its generators, and is anomaly-free. The left-handed leptonsL _L ^T (v _e ,e,,v ) _L are assigned to the four-dimensional spinorial representations ofO(5). The right-handed particles are taken to be the singlets of the group. The theory has three sets of gauge bosons: (1) analogues of the GWS model, (2) additional charged gauge bosons, and (3) a set of three additional neutral gauge bosons as compared to the GWS model. We introduce neutrino mixing by mixing the additional charged gauge bosons. We develope a theory of neutrino oscillations in matter in such a way that in the absence of matter the scattering length reduces to the usual scattering length in vacuum. Even if the neutrino masses are equal or the neutrinos are massless, we still have neutrino oscillations in matter, a result already noted by Wolfenstein.     

Nongauge local symmetry in gravity

It is shown that there exists a class of nongauge local symmetry in gravity. It is the combination of the known gauge symmetries and _{off shell} transformation. _{off shell} is the transformation which vanishes on shell. _{off shell} invariance imposes some restriction on model Lagrangians besides gauge symmetry, though it does not reduce the physical degrees of freedom.     

Free states of the canonical anticommutation relations

Each gauge invariant generalized free state _A of the anticommutation relation algebra over a complex Hilbert spaceK is characterized by an operatorA onK. It is shown that the cyclic representations induced by two gauge invariant generalized free states _A and _B are quasi-equivalent if and only if the operatorsA ^1/2–B ^1/2 and (I–A)^1/2–(I–B)^1/2 are of Hilbert-Schmidt class. The combination of this result with results from the theory of isomorphisms of von Neumann algebras yield necessary and sufficient conditions for the unitary equivalence of the cyclic representations induced by gauge invariant generalized free states.Work supported in part by US Atomic Energy Commission, under Contract AT (30-1)-2171 and by the National Science Foundation.     

High scale perturbative gauge coupling in R-parity conserving SUSY <Emphasis Type="Italic">SO</Emphasis>(10) with longer proton lifetime

It is well known that in single step breaking of R-parity conserving SUSY SO(10) that needs the Higgs representations , the GUT gauge coupling violates the perturbative constraint at mass scales a few times larger than the GUT scale. Therefore, if the SO(10) gauge coupling is to remain perturbative up to the Planck scale ( GeV), the scale M_U of the GUT symmetry breaking is to be bounded from below. The bound depends upon specific Higgs representations used for SO(10) symmetry breaking but, as we find, cannot be lower than $1.5 \times 10$¹⁷ GeV. In order to obtain such a high unification scale we propose a two-step SO(10) breaking through SU(2)_L $\times$ SU(2)_R $\times$ U(1)_B-L $\times$SU(3)_C ( ) intermediate gauge symmetry. We estimate the potential threshold and gravitational corrections to the gauge coupling running and show that they can make the picture of perturbative gauge coupling running consistent at least up to the Planck scale. We also show that when by the Higgs representations , gravitational corrections alone with negligible threshold effects may guarantee such perturbative gauge coupling. The lifetime of the proton is found to increase by nearly 6 orders over the present experimental limit for . For the proton decay mediated by a dim = 5 operator a wide range of lifetimes is possible, extending from the current experimental limit up to values 2-3 orders longer. Received: 1 July 2005, Revised: 21 August 2005, Published online: 11 October 2005     

Poincaré invariance for the master field on quantum chromodynamics

I construct allSU(N _c ) gauge fields with the property that Euclidean Poincaré transformations can be compensated by gauge transformations. Linear Abelian components are shown to be forbidden by Lorentz invariance. In a suitable gauge, the result is a set of constant potentials parametrized by Lorentz scalars. These scalars are constrained by the equation of motion atN _c =. A special solution is exhibited.Work supported in part by Schweizerischer Nationalfonds.Invited talk presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.I thank H. Leutwyler for drawing my attention to the configuration (35), and M. Lüscher, P. Schwab, P. Sorba and J. Stern for their comments.     

Bäcklund Transformation for Supersymmetric Self-Dual Theories for Semisimple Gauge Groups and a Hierarchy of Ai Solutions

We present a Bdcklund transformation (a discrete symmetry transformation) for the self-duality equations for supersymmetric gauge theories in N-extended super-Minkowski space ^4|4N for an arbitrary semisimple gauge group. For the case of anA ₁ gauge algebra we integrate the transformation starting with a given solution and iterating the process we construct a hierarchy of explicit solutions.     

W-Geometry

The geometric structure of theories with gauge fields of spins two and higher should involve a higher spin generalisation of Riemannian geometry. Such geometries are discussed and the case ofW -gravity is analysed in detail. While the gauge group for gravity ind dimensions is the diffeomorphism group of the space-time, the gauge group for a certainW-gravity theory (which isW -gravity in the cased=2) is the group of symplectic diffeomorphisms of the cotangent bundle of the space-time. Gauge transformations forW-gravity gauge fields are given by requiring the invariance of a generalised line element. Densities exist and can be constructed from the line element (generalising ) only ifd=1 ord=2, so that only ford=1,2 can actions be constructed. These two cases and the correspondingW-gravity actions are considered in detail. Ind=2, the gauge group is effectively only a subgroup of the symplectic diffeomorphism group. Some of the constraints that arise ford=2 are similar to equations arising in the study of self-dual four-dimensional geometries and can be analysed using twistor methods, allowing contact to be made with other formulations ofW-gravity. While the twistor transform for self-dual spaces with one Killing vector reduces to a Legendre transform, that for two Killing vectors gives a generalisation of the Legendre transform.     

On the mass spectrum of the 2+1 gauge-Higgs lattice quantum field theory

We investigate the mass spectrum of a 2+1 lattice gauge-Higgs quantum field theory with Wilson action A _P+A _H, whereA _P(A _H) is the gauge (gauge-Higgs) interaction. We determine the complete spectrum exactly for all , >0 by an explicit diagonalization of the gauge invariant transfer matrix in the approximation that the interaction terms in the spatial directions are omitted; all gauge invariant eigenfunctions are generated directly. For fixed momentum the energy spectrum is pure point and disjoint simple planar loops and strings are energy eigenfunctions. However, depending on the gauge group and Higgs representations, there are bound state energy eigenfunctions not of this form. The approximate model has a rich particle spectrum with level crossings and we expect that it provides an intuitive picture of the number and location of bound states and resonances in the full model for small , >0. We determine the mass spectrum, obtaining convergent expansions for the first two groups of masses above the vacuum, for small , and confirm our expectations.Research partially supported by CNPq, Brasil     

Topological electric charge

By treating magnetic charge as a gauge symmetry through the introduction of a magnetic pseudo four-vector potential, we show that it is possible to construct a topological electric charge from a theory which originally contains gauge magnetic charge. This is an explicit realization of the Montonen-Olive conjecture that there should exist a dual theory to the usual 't Hooft-Polyakov monopole theory in which the roles of the gauge and topological charges are reversed. The physical distinction between 't Hooft-Polyakov monopoles and the dual theory with electric charge is that the strong and weak coupling regimes are reversed. Physically this leads to the mass of the electrically charged soliton being on the order of (1/137)M _W as opposed to the much larger mass (on the order of 137M _W) of the magnetically charged soliton. Thus even forM _W in the TeV range such an electrically charged particle could be observed at some future accelerator.     

The affine geometry of the Lanczos H-tensor formalism

We identify the fiber-bundle-with-connection structure that underlies the Lanczos H-tensor formulation of Riemannian geometrical structure. We consider linear connections to be type (1,2) affine tensor fields, and we sketch the structure of the appropriate fiber bundle that is needed to describe the differential geometry of such affine tensors, namely the affine frame bundleA ₁ ² M with structure groupA ₁ ² (4) =GL(4) T ₁ ^{2 4} over spacetimeM. Generalized affine connections on this bundle are in 1-1 correspondence with pairs(, K) onM, where thegl(4)-component denotes a linear connection and the T ₁ ^{2 4}-componentK is a type (1,3) tensor field onM. We show that the Lanczos H-tensor arises from a gauge fixing condition on this geometrical structure. The resulting translation gauge, theLanczos gauge, is invariant under the transformations found earlier by Lanczos. The other Lanczos variablesQ _mandq are constructed in terms of the translational component of the generalized affine connection in the Lanczos gauge. To complete the geometric reformulation we reconstruct the Lanczos Lagrangian completely in terms of affine invariant quantities. The essential field equations derived from ourA ₁ ² (4)-invariant Lagrangian are the Bianchi and Bach-Lanczos identities for four-dimensional Riemannian geometry.     

Lepton flavour symmetry and the neutrino magnetic moment

With the standard model gauge group and the three standard left-handed Weyl neutrinos, two minimal scenarios are investigated where an arbitrary non-Abelian lepton flavour symmetry groupG _H is responsible for a light neutrino with a large magnetic moment. In the first case, with scalar fields carrying lepton flavour, some finetuning is necessary to get a small enough neutrino mass for _v =O(10)^–11 _B. In the second scenario, the introduction of heavy charged gauge singlet fermions with lepton flavour allows for a strictly massless neutrino to one-loop order. In both cases, the interference mechanism for smallm and large _v is unique, independently ofG _H . In explicit realizations of the two scenarios, the horizontal groups are found to be non-Abelian extensions of a Zeldovich-Konopinski-Mahmoud lepton number symmetry. Only a discrete part ofG _H is spontaneously broken leading to a light Dirac neutrino with a large magnetic moment.     

On vector-tensor minimally coupled field theories

We consider vector-tensor minimally coupled Lagrangians, i.e., scalar densities of the form = g ^1/2 R +L(g _ij ; _i ; _i,j ). We prove that the gauge invariance of any of the sets of Euler-Lagrange expressions implies the gauge invariance of the Lagrangian itself forn even, and an almost gauge invariance forn odd. We also find those for whichE ⁱ () = 0 orE ^ij (L) = 0, generalizing well-known results by Lovelock and a result by the authors.     

Electrodynamics in Dirac's Gauge: A Geometrical Equivalence

It is shown that, when classical non-relativistic electrodynamics is formulated in Dirac's gauge A A =const. and the vector potential A interpreted as a velocity field of the vacuum, the motion of a charged particle results from purely inertial effects. A metric is given for particles of a fixed charge to mass ratio.     

q-Deformation of the Double Complex Ernst Equation

From a known three-dimensional representation of the double quantum algebra su _q ((J)), a nonlinear q-deformed Ernst equation system is obtained. By using a gauge covariant form, the deformation effects are found to generate a torsion in the field and to form a gauge field with source.     

A natural framework for the minimal supersymmetric gauge theories

We show that the sequence of Jordan algebras M _inf3 ^sup1 , M _inf3 ^sup2 , M _inf3 ^sup4 , and M _inf3 ^sup8 , whose elements are in the 3×3 Hermitean matrices over , , , and O, respectively, provide an elegant and natural framework in which to describe supersymmetric gauge theories. The four minimal supersymmetric gauge theories are in a one-to-one correspondence with these four Jordan algebras and, hence, with the four division algebras.     

Higgs Field—Fermion Coupling in the Tensor Dirac Theory

In previous work, the Dirac and Einstein equations were unified in a tetradformulation of a Kaluza—Klein model which gives precisely the usualDirac—Einstein Lagrangian. In this model, the self-adjoint modes of the tetraddescribe gravity, whereas the isometric modes of the tetrad together with a scalarfield describe fermions. The tetrad Kaluza—Klein model is based on a constrainedYang—Mills formulation of the Dirac Lagrangian in which the bispinor field is mapped to a set of SL(2, R) × U(1) gauge potentialsA ^K _a and a complex scalarfield . In this paper we generalize the map (A ^K _a , ) to multiplets of nbispinor fields representing a fermion multiplet as in standard electroweak theory.We show that the Lagrangian for bispinor multiplets used in the Standard Modelbecomes a constrained Yang—Mills Lagrangian, for which the Higgs fielddetermines a noninvariant gauge metric, thereby breaking the full gauge symmetry.     

Poincaré gauge invariant spinor theory and the gravitational field. II

The nonlinear equation for an abstract noncanonical 2-component Weyl spinor field — as used with the inclusion of internal symmetries in Heisenberg's nonlinear spinor theory of elementary particles — which is invariant under scale, phase, and Poincaré transformations is modified in such a way as to become invariant under spacetime dependent phase gauge and Poincaré gauge transformations. In such an equation a phase gauge field B _m , six Lorentz gauge fields A[]m and four translation gauge fields g_m have to be introduced. It is demonstrated that all these fields can be identified as certain combinations of the Weyl spinor field, and hence should be considered in a rough sense as bound states of this spinor field. In particular the electromagnetic field B_m and the gravitational field g m appear as S-states and P-states of a spinor-antispinor system. The noncanonical property and the operator character of the spinor field is essential for this result. The relation between the translation gauge field and the spinor field involves a fundamental length. In a classical geometrical interpretation this relation leads to Einstein's equation of gravitation without cosmological term in a Riemannian space without torsion if the fundamental length is identified with Planck's length. It is shown that this equation is covariant under the larger symmetry group of phase gauge and Poincaré gauge transformations. The modified nonlinear equation constructed solely from a single 2-component Weyl field hence seems to incorporate in an extremely compact way electromagnetic and gravitational interaction in addition to non-mass-zero interactions. In this equation no arbitrary dimensionless constants enter. The considerations can be generalized to Dirac spinor fields and to spinor fields involving additional interior degress of freedom.An abridged version of this paper was presented at the International Conference on Gravitation and Relativity, Copenhagen, July 1971.     

The induced action ofW 3 gravity

We obtain the induced action [h, b] for chiralW ₃ gravity in thec± limit from the induced action of a gaugedSl(3,R) Wess-Zumino-Witten model by imposing constraints on the currents of the latter. In the process we find a closed gauge algebra for the gauge sector ofW ₃ gravity in which the currentsT andW become auxiliary fields. An explicit realization ofT andW in terms of the gauge fields is given. In terms of new fieldsr ands, which are a generalization of Polyakov'sf variable for ordinary gravity, the complete induced action [h, b; c±] becomes local.Work supported in part by NSF grant No. PHYS 89-08495Address after September 1, 1991: Physics Department, U.C. Berkeley, Berkeley, CA 94720, USA     

Theq-state Potts model in the standard Pirogov-Sinai theory: Surface tensions and Wilson loops

Theq-state Potts model (both scalar and gauge versions) is rewritten, with the help of the duality transformation, into a form of the Pirogov-Sinai theory with noninteracting contours that can be controlled by cluster expansions onceq is large enough. This is then used in a new proof of the existence of a unique transition (inverse) temperature _t , where the mean internal energy is discontinuous. Moreover, we prove for the scalar model (again forq large enough) that there are discontinuities at _t of the magnetization and of the mass gap, with the magnetization vanishing below _t and the mass gap vanishing above _t . We also show that the surface tensions between ordered stable phases are strictly positive up to _t , and the surface tension between an ordered phase and the disordered one is strictly positive at _t . For the three-dimensional gauge model, the Wilson parameter exhibits a direct transition from an area law decay (quark confinement) to a perimeter law decay (deconfinement).On leave from ENS Rabat, Morocco.