Massless spinors in more than four dimensions

We consider fermions in theories of higher dimensional gravity where the four-dimensional gauge group is embedded in the invariance group of d dimensional (d>4) Lorentz and general co-ordinate transformations. It is a necessary condition for obtaining massless chiral fermions from dimensional reduction that the d dimensional spinor does not admit a mass term consistent with Lorentz and general co-ordinate transformations. This is the case for a Weyl spinor for d = 6 8 mod 8, a Majorana spinor for d = 9 mod 8 or a Majorana-Weyl spinor for d = 2 mod 8.     

Dimensional reduction of Weyl,Majorana and Majorana-Weyl spinors

We discuss the dimensional reduction for Weyl, Majorana, or Majorana-Weyl spinors coupled to pure d-dimensional (d ? 4) gravity. The only case where a realistic four-dimensional low-energy spectrum for the fermions may be obtained, is for a Majorana-Weyl spinor in d = 2 mod 8 dimensions. Chiral massless fermions are not excluded in this case. The minimal number of dimensions for the construction of a realistic theory out of pure gravity is d = 18.     

Discrete symmetries in Kaluza-Klein theories

We investigate discrete symmetries in theories of higher-dimensional (d > 4) gravity and their consequences for the reduced four-dimensional theory, obtained for a ground state which is a direct product of four-dimensional Minkowski space and a compact d ? 4 dimensional internal space. If the action of pure d-dimensional gravity coupled to spinors is invariant under time reversal or reflection of an odd number of spacelike co-ordinates, the reduced four-dimensional theory has a non-trivial parity or CT symmetry not consistent with observation. A non-trivial d-dimensional charge conjugation results in an unwanted doubling of the four-dimensional fermion spectrum. As a consequence, realistic theories can only be obtained for Majorana-Weyl spinors in d = 2 mod 8 dimensions. The constraints are less stringent if supplementary fields are introduced in d dimensions. For d = 11 supergravity, for example, parity and CT invariance can be broken by a non-vanishing field strength of the totally antisymmetric three-index tensor.A ground state invariant under reflections of “internal” co-ordinates often gives rise to a non-trivial charge conjugation in four dimensions. We find that the ground state of a realistic Kaluza-Klein theory should not be invariant under any non-trivial internal co-ordinate reflection (which cannot be obtained by a gauge transformation). We finally comment on a possible solution of the strong-CP problem from Kaluza-Klein theories and discuss prospectives for finding internal spaces admitting chiral fermions.     

On dimensional reduction of gauge theories: Symmetric fields and harmonic expansion

We compare the four-dimensional symmetric fields obtained by the coset space dimensional reduction scheme to the infinite tower of fields given by the harmonic expansion in a 4+N dimensional gauge theory coupled to fermions on a space-timeM ⁴ ×S/R.     

Linear system and conservation laws for d = 10 super-yang-mills

A new twistor-like parametrization of the linear system for d = 10 super-Yang-Mills is proposed. It is more suitable for power series expansion of conservation laws. We also perform dimensional reduction to obtain a much stronger four-dimensional linear system which includes SU(4) self-duality constraint.     

Hyper-Kähler manifolds and nonlinear supermultiplets

A method for constructing σ-model manifolds with geometries different from the conformally flat is proposed. It is based on dualization of the coupling constant and makes it possible to construct N = 4, 8 super-symmetric four-dimensional hyper-Kähler σ models in d = 1 with one triholomorphic isometry.     

4d fermionic superstrings with arbitrary twists

We present the rules for systematically constructing all consistent four-dimensional string theories, using free world-sheet fermions which pick up arbitrary phases when parallel transported around the string. These rules are necessary and sufficient for multi-loop modular invariance. They lead to theories with general Z_N (GSO-type) projections, whose merits for model-building we discuss. We classify all boundary conditions yielding massless space-time spinors. We show that, in contrast to the case of only real 2d fermions, all possible realizations of world-sheet supersymmetry are now allowed. This opens the way for the construction of a new class of supersymmetric string models.     

Radial and Regge excitations in unified,grand unified and subconstituent models

Necessary group theoretic conditions for all elementary gauge bosons and fermions of an arbitrary renormalizable gauge theory to lie on Regge trajectories are reviewed. It is then argued that in properly unified gauge theories all particles of a given spin lie on Regge trajectories. This then implied that a properly unified gauge theory has no local U(1) factor groups, and no massive fermion singlets. A consideration of the general pattern of Regge and radial recurrences to be expected in quantum field theories suggests that the presence or absence of spin $\text{3}\text{2}$ quarks and/or leptons in the TeV region will provide crucial clues to enable one to distinguish between various classes of unified, grand unified, and subconstituent models. The correct interpretation of such excited fermions will require correlation with the Higgs boson mass and possible radial and Regge excitations of the weak vector bosons.     

Non-perturbative renormalization group for field theories with scalars and fermions

An approximate, but non-perturbative, RG equation is derived for theories involving scalars and fermions, ind dimensions withn _f flavours. The approximation consists in restricting the parameter space to interactions without derivatives. In a numerical study of the equation ind=3 andd=4, in the range of parameter space explored, no evidence is found of new fixed points generated by the inclusion of fermions.     

Lower dimension vacuum defects in lattice Yang-Mills theory

We overview lattice data on d = 0, 1, 2, 3 dimensional vacuum defects in lattice four-dimensional SU(2) (SU(3)) gluodynamics. In all the cases, defects have a total volume which scales in physical units (with zero fractal dimension). In the case of d = 1, 2, the defects are distinguished by ultraviolet divergent non-Abelian action as well. This sensitivity to the ultraviolet scale allows us to derive strong constraints from the continuum theory on the properties of the defects, which turn out to be satisfied by the lattice data. We discuss a classification scheme of the defects which allows us to (at least) visualize the defect properties in a simple and unified way. A not-yet-checked relation of the defects to the spontaneous chiral symmetry breaking is suggested by the scheme. Finally, we present some arguments that the defects considered could become fundamental variables of a dual formulation of the theory.     

Two dimensional gases of weight charges and SU(N) fermino theories

We analyze two dimensional gases composed of particles interacting via a Coulomb or Yakawa potential through their “non-Abelian” charges. These charges are taken to be elementary weight or root vectors of SU(N). The grand partition function of these gases is shown to be equivalent to the generating functional of sine-Gordon models with weight vectors and hence to that of SU(N) fermion models. The fermion field creates or annihilates topological solitons which have elementary weight vectors as topological quantum numbers. Then, we discuss the confinement of fermions in the SU(N) Higgs models, where instantons (Z_N vortices) constitute a Yukawa gas of weight charges. We prove that fermions are confirmed by the effects of instantons in the SU(N) Higgs models in contrast with the Abelian Higgs model.     

Cosmological constraints on heavy weakly interacting fermions

The masses and lifetimes of very heavy weakly interacting fermions which appear in many grand unified gauge models are constrained by the requirement that their decays in the hot big bang early universe should not generate excessive entropy which would dilute n_B/n_γ below its observed value.     

From the Dirac operator to Wess-Zumino models on spatial lattices

We investigate two-dimensional Wess-Zumino models in the continuum and on spatial lattices in detail. We show that a non-antisymmetric lattice derivative not only excludes chiral fermions but in addition introduces supersymmetry breaking lattice artifacts. We study the non-local and antisymmetric SLAC derivative which allows for chiral fermions without doublers and minimizes those artifacts. The supercharges of the lattice Wess-Zumino models are obtained by dimensional reduction of Dirac operators in high-dimensional spaces. The normalizable zero modes of the models with N=1 and N=2 supersymmetry are counted and constructed in the weak- and strong-coupling limits. Together with known methods from operator theory this gives us complete control of the zero mode sector of these theories for arbitrary coupling.     

Spontaneous compactification and CPN: SU(3) × SU(2) × U(1), sin2θW, g3/g2 and SU(3)-triplet chiral fermions in four dimensions

We examine the compactification of D=4+2N, Einstein-Maxwell-Dirac theory. It is shown that the manifold CP_N × M₄ is a solution of the equations of motion. The structure of the fermions, gauge bosons and their couplings in the four-dimensional effective theory is investigated. The scale of CP_N is quantized by a generalized Dirac condition. When the results are applied to the solution with internal space CP₁×CP₂, the weak mixing angle and the ratio of the couplings of SU(3) (g₃) and SU(2) (g₂) are defined by two integers and a hypercharge. An SU(3)-triplet chiral fermion can appear in four-dimensional effective theory.     

The possibility of new fermions with ΔI = 0 mass

In the Glashow-Weinberg-Salam model the fermions have $\text{ΔI =}\text{1}\text{2}$ masses from the breaking of the weak SU(2) gauge symmetry. In many enlarged models, such as those from grand unified and/or supersymmetric theories, there are additional fermions with undetermined ΔI = 0 (SU(2) invariant) masses. We study these new fermions. They induce flavour changing neutral currents. We show that the mixing angles of ΔI = 0 fermions of mass order M with normal $\text{ΔI =}\text{1}\text{2}$ fermions of mass order m ? M are order η or η², where η = m/M. If M > 150 δ, δ being a model-dependent mixing parameter of order a normal fermion mass, the amplitudes of all FCNC processes are below the experimental sizes and limits. Thus for δ?0.1 GeV, M can be as low as 20 GeV, close to the present experimental lower bound. δ is fixed, and less than 0.1 GeV for all relevant cases, if we assume the mass hierarchy of the known fermions is not the result of a particular choice of ΔI = 0 mass parameters. If produced ΔI = 0 mass fermions will be noticeable by the mass degeneracy within their isospin multiplets. There will be an enhanced ratio of neutral to charged decays over the normal fermions. Standard GUT predictions are changed little.     

Unification of 3-3-1 models

A 3-3-1 model is constructed for three families that can be embedded into a single SU(8) unified model. Assuming appropriate branching rules and symmetry-breaking pattern, a complete fermion content is found within irreducible representations of SU(8), where light Standard Model fermions, heavy 3-3-1 fermions and superheavy fermions may be distinguished.     

Improved renormalization group equations with dimensional regularization and the ε expansions

Due to the absence of dimensional cut-off parameters in the dimensional regularization scheme, vanishing of the renormalized mass of the scalar boson implies vanishing of its renormalized mass; thus the masses of both bosons and fermions in renormalizable field theories can be made finite by multiplicative mass renormalizations. The improved renormalization group equations in D dimensions are derived in such a way that both the large (or the small) momentum limits and the Wilson ? expansions can be uniformly treated for the fermion as well as the boson cases. We discuss the improved equations for φ₆³ theory, φ₄⁴ theory, quantumelectrodynamics, massive vector-gluon model, and non-Abelian guage theories incorporating fermions. For the latter three classes of theories, the gauge dependent problem of the coefficient functions in the improved renormalization group equations is discussed.     

Finite-size crossover in systems with slab geometry

A renormalization group study of the finite-size (dimensional) crossover is carried out with the help pf ε = 4 − d and ε₀ = 3 − d expansion techniques. The finite-size crossover and the invariance relation for the length scale transformation are proven up to the two-loop approximation. The formal equivalence between the finite-size crossover in classical systems and the quantum-to-classical dimensional crossover in certain quantum statistical models is emphasized and exploited. The finite-size corrections to the fluctuation shift of the critical temperature and the width of the critical region are investigated. It is shown that the shift exponent λ describing the fractional rounding of the critical temperature obeys the relation λ = D − 2, where D is the dimensionality of the system.     

A heterotic N=2 string with space–time supersymmetry

We reconsider the issue of embedding space–time fermions into the four-dimensional N=2 worldsheet supersymmetric string. A new heterotic theory is constructed, taking the right-movers from the N=4 topological extension of the conventional N=2 string but a c=0 conformal field theory supporting target-space supersymmetry for the left-moving sector. The global bosonic symmetry of the full formalism proves to be U(1,1), just as in the usual N=2 string. Quantization reveals a spectrum of only two physical states, one boson and one fermion, which fall in a multiplet of (1,0) supersymmetry.     

O(d,d) symmetry in quantum cosmology

We analyze the quantum cosmology of one-loop string effective models which exhibit an O(d, d) symmetry. It is shown that due to the large symmetry of these models the Wheeler-de Witt equation can completely be solved. As a result, we find a basis of solutions with well-defined transformation properties under O(d, d) and under scale factor duality in particular. The general results are explicitly applied to 2-dimensional target spaces while some aspects of higher dimensional cases are also discussed. Moreover, a semiclassical wave function for the 2-dimensional black hole is constructed as a superposition of our basis.