Global symmetries in four and higher dimensions

Knowing that a four-dimensional theory with gauge group G₀ is unified in theory with gauge group G puts restrictions on what global symmetries are possible in the low-energy world. Here we analyze those restrictions assuming that unification in G occurs inn four dimensions and assuming that unification occurs only in a higher-dimensional theory. There are possibilities for global symmetries which are not possible in the former case, so in principle indirect evidence for higher dimensions might be found by finding peculiar global symmetries in the low-energy world.     

Discrete symmetries in higher dimensions

We consider the constraints imposed by causality on the transformations of time reversal ?, charge conjugationC, and parityP in higher dimensional space-time.     

Berezinskii-Kosterlitz-Thouless phase transitions in two-dimensional systems with internal symmetries

The Berezinskii-Kosterlitz-Thouless (BKT) phase transitions in two-dimensional systems with internal continuous Abelian symmetries are investigated. In order for phase transitions to occur, the kinetic part of the action of the system must have conformal invariance, and the vacuum manifold must be degenerate and have a discrete Abelian homotopy group π ₁. In this case topological excitations have a logarithmically divergent energy and can be described by effective theories that generalize the two-dimensional Euclidean sine-Gordon theory, which is an effective theory of the original XY model. In particular, the effective actions are found for chiral models on the maximal Abelian tori T _G of the simple compact Lie groups G. The critical properties of the possible effective theories are found, and it is shown that they are characterized by the Coxeter numbers h _G of lattices of the and ℤ series and can be interpreted as properties of conformal theories with an integer central charge C=n, where n is the rank of the groups π ₁ and G. The possibility of reconstructing the complete symmetry of G in the massive phase is also discussed. Zh. éksp. Teor. Fiz. 116, 2045–2057 (December 1999)     

Frustrated symmetries in multi-Higgs-doublet models

Within multi-Higgs-doublet models, one can impose symmetries on the Higgs potential, either discrete or continuous, that mix several doublets. In two-Higgs-doublet model any such symmetry can be conserved or spontaneously violated after the electroweak symmetry breaking (EWSB), depending on the coefficients of the potential. With more than two doublets, there exist symmetries which are always spontaneously violated after EWSB. We discuss the origin of this phenomenon and show its similarity to frustration in condensed matter physics.     

Internal symmetries from field theories in higher dimensions

It is suggested that a dynamical space-time symmetry group of a given non-linear relativistic theory is able to generate an internal symmetry group. It is shown that onlyO(2N-1, 1) andO(2N, 1) groups are admitted as simple dynamical symmetry groups, and they implyU(2 ^N –2) as the largest internal symmetry groups. The boson and fermion content of the considered models is determined and a new dynamical symmetry breaking mechanism, different from the spontaneous one, is introduced.Presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.The author would like to thank Professor J. Niederle for the kind hospitality extended to him during his stay at Bechyn.     

Infinite-parameter symmetries and the Higgs effect in gravity-coupled sigma models in D+4 dimensions

We compute the mass spectra of small fluctuations of four-dimensional fields for Kaluza-Klein models in which the compactification from D+4 to 4 (flat) dimensions is induced by the scalar fields of a nonlinear sigma model defined on an S^N or CP^N manifold. The compactifications are stable for all values of N. The fact that the spectra contain no massless vector fields is traced to the absence of a local gauge invariance for the sigma-model action. We introduce a complete basis for the infinite-parameter symmetries that arise from the harmonic analysis of the higher-dimensional dynamical invariances. The spectrum of spin-one and spin-two fields is consistent with the Higgs effect associated with the breaking of the local symmetries corresponding to these generators. The commutation relations of the infinite parameter algebra for the case of CP¹ are also given. The algebra includes the spectrum-generating algebra SO(1,3) of Salam and Strathdee.     

Gauge theories with strong coupling in four dimensions

The long distance behavior of singlet current correlation functions of gauge theories on a lattice are found approximately using the renormalization group. For strong coupling the behavior is similar to that found in the lattice massive Schwinger model.     

Continuum symmetries of lattice models with staggered fermions

The validity of the flavour interpretation of staggered fermions is discussed in terms of the discrete symmetries of the interaction terms. Some aspects of the embedding of these symmetries in the symmetry group of the continuum limit are clarified. An explicit calculation, at first non-trivial order in 1/N, of the four-point function for a latticized Gross-Neveu model yields the same result in the continuum limit as the continuum theory for 2N fermions. A proof is then given that flavour and C, P, and T symmetries are restored in the continuum limit of 2-point correlation functions, for interactions, including the case of 4-dimensional QCD, which respect the discrete symmetries of the free action.     

A gauge model with confinement in four dimensions

It is shown that the correctly written “1/k⁴” propagator in four space-time dimensions leads to a consistent pertubative expansion of a superrenormalizable (asymptotically free) abelian gauge model. Confinement is found in any order of perturbation theory. A discussion of the physical space is sketched and the violation of asymptotic completeness is pointed out.     

Composite symmetries and sub-constituent models

Assuming the internal A-spin, B-spin and C-spin of particles from basic symmetry SO(4), the color SU(3), horizontal SU(3)', electroweak SU(2)'_w×U(1) and other higher composite symmetries are derived.     

Topological matter in four dimensions

Topological models involving matter couplings to Donaldson-Witten theory are presented. The construction is carried out using both the topological algebra and its central extension, which arise from the twisting of N = 2 supersymmetry in four dimensions. The framework on which the construction is based is constituted by the superspace associated to these algebras. The models show new features of topological quantum field theories which could provide either a mechanism for topological symmetry breaking, or the analog of two-dimensional mirror symmetry in four dimensions.     

Dimensional regularization in four dimensions

A new gauge invariant ultraviolet regularization proposed recently is discussed for Abelian gauge theories. This cut-off scheme resembles closely the canonical dimensional regularization (CDR), but it is formulated strictly in four dimensions. In a sense, it may be conceived as a continuous superposition of Pauli-Villars cut-offs. Although it differs from the CDR for a general graph, for some closed fermion loops the two schemes coincide. The new cut-off procedure is also well-suited for practical calculations of Feynman diagrams in the -parametric representation; in this respect it preserves the merits of CDR. Such a dimensional regularization in four dimensions obviously does not suffer from the conceptual problems connected with the definition of the ₅ matrix, which plague conventional DR schemes. Nevertheless, some spurious anomalies in the closed fermion loops do occur, similarly to the Pauli-Villars scheme.Dedicated to Academician Václav Votruba on the occasion of his eightieth birthday.     

Supersymmetric skyrmions in four dimensions

Possible topological solitons (skyrmions) of four-dimensional supersymmetric nonlinear σ-models are investigated. The requirements of supersymmetry limit our study to the CP¹ model. A stable soliton seems possible, but in the absence of a demonstrated lower-bound for the mass, the stability of the soliton is unproved. The semi-classical spectrum of the CP¹ skyrmion, as well as its supersymmetric extension, is studied.