Equivalence of dimensional reduction and dimensional regularisation

For some years there has been uncertainty over whether regularisation by dimensional reduction (DRED) is viable for non-supersymmetric theories. We resolve this issue by showing that DRED is entirely equivalent to standard dimensional regularisation (DREG), to all orders in perturbation theory and for a general renormalisable theory. The two regularisation schemes are related by an analytic redefinition of the couplings, under which the -functions calculated using DRED transform into those computed in DREG. TheS-matrix calculated using DRED is numerically equal to the DREG version, ensuring that both schemes give the same physics.     

Stability and dimensional reduction

With the help of dynamical system methods the stability ofR×S ³×S ^D solutions (whereS ³ andS ^D are maximally symmetric spaces) with the static microspace and the energy-momentum tensor determined by unifying theories is investigated. The general stability criterion is given and the significance of certain particular cosmological solutions is discussed.     

Dynamical dimensional reduction

We propose to call a dynamical dimensional reduction effective if the corresponding dynamical system possesses a single attracting critical point representing expanding physical space-time and static internal space. We show that theBV × T ^D multidimensional cosmological model with a hydrodynamic energy-momentum tensor provides an example of effective dimensional reduction. We also study the dynamics of the multidimensional cosmological model of typeBI × T ^D with an energy-momentum tensor representing low temperature quantum effects, monopole contribution and the cosmological constant. It turns out that anisotropy and the cosmological constant are crucial for the process of dimensional reduction to be effective. We argue that this is the general property of homogeneous multidimensional cosmological models.     

Spontaneous compactification and dimensional reduction

A connection between dimensional reduction and spontaneous compactification is studied. It is shown that the methods of the coset-space dimensional-reduction scheme can be effectively used for constructing solutions of multidimensional Einstein-Yang-Mills equations and for interpreting from the viewpoint of the four-dimensional theory.     

Phase transitions and dimensional reduction

Using field theoretic methods a formalism is presented within which the critical behaviour of a system undergoing a dimensional reduction may be investigated. As a paradigm we study an Ising-like system on S¹ × R^3−ε. If the size of the system is L, and the correlation length ξ, then as L/ξ varies it is possible to get critical behaviour associated with two different fixed points. By exploiting a set of renormalization schemes which lead to manifest dimensional reduction in the loop expansion, and utilizing the renormalization group and an expansion about the fixed point of the finite system, we quantitatively investigate such crossover behaviour in its entirety. In particular, effective susceptibility and correlation length exponents are defined and computed. These exponents interpolate between those associated with a (4 − ε)-dimensional and a (3 − ε)-dimensional Ising model.     

Non-abelian exclusion statistics

We introduce the notion of ‘order-k non-abelian exclusion statistics’. We derive the associated thermodynamic equations by employing the thermodynamic Bethe ansatz for specific non-diagonal scattering matrices. We make contact with results obtained by different methods and we point out connections with ‘fermionic sum formulas’ for characters in a conformal field theory. As an application, we derive thermodynamic distribution functions for quasi-holes over a class of non-abelian quantum Hall states recently proposed by Read and Rezayi.     

Non-abelian eikonal exponentiation

Recently, Gatheral [1] has generalized the well-known exponentiation properties of soft photons in QED to the case of soft gluons in QCD in the eikonal approximation. We add three things to his work. (i) We clarify the definition of the colour weight factors which appear in the exponent. (ii) We show how the eikonal approximation can be renormalized, consistently with the exponentiation. (iii) We derive a differential equation obeyed by the cross section for emission of soft gluons with given maximum total energy. We also discuss the relationship between the exponentiation theorem and expectation values of Wilson loops.     

Chirality index and dimensional reduction of fermions

The number of four-dimensional chiral fermions obtained from dimensional reduction of models with spinor matter fields coupled to pure gravity in d > 4 dimensions is linked to topological properties of the internal d ? 4 dimensional space. This gives important restrictions on possible ground states of such models consistent with a realistic four-dimensional unified theory. Connections with spontaneous symmetry breaking and Yukawa couplings of fermions in unified theories are discussed.     

Symmetries of Einstein-Yang-Mills fields and dimensional reduction

LetE be a manifold on which a compact Lie groupS acts simply (all orbits of the same type);E can be written locally asM×S/I,M being the manifold of orbits (space-time) andI a typical isotropy group for theS action. We study the geometrical structure given by anS-invariant metric and anS-invariant Yang Mills field onE with gauge groupR. We show that there is a one to one correspondence between such structures and quadruplets of fields defined solely onM; _v is a metric onM,h are scalar fields characterizing the geometry of the orbits (internal spaces), ⁱ are other scalar fields (Higgs fields) characterizing theS invariance of the Lie(R)-valued Yang Mills field and is a Yang Mills field for the gauge groupN(I)|I×Z((I)),N(I) being the normalizer ofI inS, is a homomorphism ofI intoR associated to theS action, andZ((I)) is the centralizer of(I) inR. We express the Einstein-Yang-Mills Lagrangian ofE in terms of the component fields onM. Examples and model building recipes are given.     

Equivariant dimensional reduction and quiver gauge theories

We review recent applications of equivariant dimensional reduction techniques to the construction of Yang-Mills–Higgs–Dirac theories with dynamical mass generation and exactly massless chiral fermions.     

Two-loop parameter relations between dimensional regularization and dimensional reduction applied to SUSY-QCD

The two-loop relations between the running gluino–quark–squark coupling, the gluino and the quark mass defined in dimensional regularization (DREG) and dimensional reduction (DRED) in the framework of SUSY-QCD are presented. Furthermore, we verify with the help of these relations that the three-loop β-functions derived in the minimal subtraction scheme combined with DREG or DRED transform into each other. This result confirms the equivalence of the two schemes at the three-loop order, if applied to SUSY-QCD.     

On the dimensional reduction of gravity

Using the dimensional reduction characterizedM ₆ =M ₄×S² as an example, it is demonstrated that higher-dimensional Einstein-Cartan (EC) gravity reduces to EC gravity with a cosmological term in four dimensions. A new higher-dimensional theory of gravity is proposed that results in EC gravitywithout cosmological term in four dimensions.     

Non-abelian D = 11 supermembrane

We obtain a U(M) action for super membranes with central charges in the Light Cone Gauge (LCG). The theory realizes all of the symmetries and constraints of the supermembrane together with the invariance under a U(M) gauge group with M arbitrary. The worldvolume action has (LCG) N = 8 supersymmetry arid it corresponds to M parallel supermembranes minimally immersed on the target M g × T ² (MIM2). In order to ensure the invariance under the symmetries and to close the corresponding algebra, a star-product determined by the central charge condition is introduced. It is constructed with a nonconstant symplectic two-form where curvature terms are also present. The theory is in the strongly coupled gauge-gravity regime. At low energies, the theory enters in a decoupling limit and it is described by an ordinary N = 8 SYM in the IR phase for any number of M2-branes.     

Non-abelian bosonization in two dimensions

A non-abelian generalization of the usual formulas for bosonization of fermions in 1+1 dimensions is presented. Any fermi theory in 1+1 dimensions is equivalent to a local bose theory which manifestly possesses all the symmetries of the fermi theory.Supported in part by NSF Grant PHY-80-19754     

Group actions on principal bundles and dimensional reduction

Using the invariant geometrical interpretation of gauge and Higgs fields, a simple derivation is given of the dimensional reduction procedure. The underlying assumption with regard to the Riemannian structure, group orbits and invariant connection are clarified and the critical points of the Higgs potential are shown to have a natural geometrical interpretation.     

Fermions and parity violation in dimensional reduction schemes

Dimensional reduction, previously applied to Yang-Mills theories, is extended to gauge theories with spinor fields. It is shown that a fairly realistic model in Minkowski space can be obtained from the simplest initial lagrangian, defined in a space-time with extra, compact dimensions. Left-right asymmetry in the fermion sector in four dimensions is possible, and its occurrence is related to a non-vanishing Atiyah-Singer index.