A connection between ν-dimensional Yang-Mills theory and (ν−1)-dimensional,non-linear σ-models

We study non-linear σ-models and Yang-Mills theory. Yang-Mills theory on the ν-dimensional lattice ? ^v can be obtained as an integral of a product over all values of one coordinate of non-linear σ-models on ? ^v?1 in random external gauge fields. This exhibits two possible mechanisms for confinement of static quarks one of which is that clustering of certain two-point functions of those σ-models implies confinement of static quarks in the corresponding Yang-Mills theory. Clustering is proven for all one-dimensional σ-models, for theU(n) ×U(n) σ-models,n=1, 2, 3, ..., in two dimensions, and for the SU(2) × SU(2) σ-models for a large range of couplingsg ² ? O(ν). Arguments pertinent to the construction of the continuum limit are discussed. A representation of the expectation of Wilson loops in terms of expectations of random surfaces bounded by the loops is derived when the gauge group is SU(2),U(n) or O(n),n=1, 2, 3, ..., and connections to the theory of dual strings are sketched.     

On renormalizable two-dimensional σ-models with dynamical charges

(a) The general conditions are stated under which a non-linear σ-model in two dimensions has dynamical charges of the type found by Lüscher and Pohlmeyer for the O (N)-model. (b) he renormalization of a large class of σ-models including CPⁿ-models and chiral fields on Grassmann manifolds is discussed.     

Four-dimensional σ-model on quaternionic projective space

A four-dimensional σ-model based on the quaternionic projective space is constructed. The model possesses SU(2) gauge invariance and has one-instanton solutions analogous to those in the CPⁿ model in D=2.     

Nonlinear σ-models and their gauging in and out of superspace

We analyze and generalize bosonic nonlinear σ-models and their N = 1,2 supersymmetric extensions in (4 spacetime-dimensional) N = 1 superspace. We give a general construction of nonminimal kinetic terms for gauge fields and of N = 1,2 gauging of isometries on Kähler and hyper-Kähler manifolds. In particular, we study the gauging of noncompact groups. We derive the complete component action and supertrace formula. For N = 2 models, the supertrace always vanishes.     

S-matrix regularities of two-dimensional σ-models of stiefel manifolds

The S-matrices of the two-dimensional nonlinear O(n+m)/O(n) and O(n+m)/O(n) XO(m) σ-models corresponding to Stiefel and Grassmann manifolds, respectively, are compared in leading order in $\text{1}\text{n}$. It is shown, that after averaging over O(m) labels of the incoming and outgoing particles, the S-matrices of both models become identical. This result explains why commonly expected regularities of the Grassmann models, in particular absence of particle production, are found, modulo an O(m) average, also in Stiefel models.     

Spontaneous symmetry breaking with quaternionic scalar fields and electron-muon mass ratio

We investigate the problem of using quaternionic scalar fields as Higg's mesons in theories of spontaneously broken symmetries. We are led to the symplecticSp(1,Q) U(1) as a possible gauge group for a unified theory of electromagnetic and weak interactions. The features of this model are worked out and compared with those of Weinberg'sSU(2) U(1) model.     

Generalizations of σ-models and CpN models,and instantons

Generalized σ-models, called U(N,r) σ-models, are constructed. Fields are determined which allow simple discussion of their instanton solutions and symmetry properties, generalizing known results for Cp^N or SU(N,1) σ-models.     

The low lying spectrum of the 2d grassmannian non-linear models

We present analytic results for the mass gap values, of the two-dimensional grassmannian models $\text{SO}\text{(p + q)}\text{SO}\text{(p)}\text{×}\text{SO}\text{(q),}\text{SU}\text{(p + q)}\text{SU}\text{(p)}\text{×}\text{SU}\text{(q) ×}\text{U}\text{(1)}\text{and}\text{Sp}\text{(p + q)}\text{Sp}\text{(p) ×}\text{Sp}\text{(q)}$. The finite volume effective hamiltonian of these models is constructed to one-loop order.     

Unitary quantum groups,quantum projective spaces andq-oscillators

We discuss the parametrization of quantum groups in terms of independent operators. We find that this consideration leads to the parametrization ofSU _q(2) in terms of aq-oscillator plus a commuting phase. The commuting phase is naturally identified with the subgroupU(1) and the remaining cosetSU _q(2)/U(1)=CP _q(1) consists of aq-oscillator. For unitary quantum groupsSU _q (n), the analogous construction results in the quantum projective spaceSU _q(n+1)/U _q (n)=CP _q (n) being identified with then-dimensionalq-oscillator. This yields a nonlinear action of the quantum groupSU _q(n+1) on then-dimensionalq-oscillator.     

Beta-functions and superstring compactifications

It is shown that quartic curvature modifications to the gravitational effective action derived from superstrings agree on Ricci-flat Kähler backgrounds with a recently-proposed four-loop counterterm in N = 2 Ricci-flat σ-models. These results seem to suggest that Calabi-Yau manifolds may not provide consistent compactifications of the superstring.     

Quaternionic multi-S4 instantons in general covariant SU(2) Yang-Mills and HP(n) σ-models

It is shown that the quaternionic maps xⁿ = S⁴ → S⁴ are realized by the n-instanton solutions of two related types of general covariant field theories in four dimensions. They are, respectively, the Einstein-SU(2) Yang-Mills theory with a cosmological term and the new HP(m) σ-models     

Cohomogeneity-one Einstein–Weyl structures: a local approach

We analyse in a systematic way the (non-) compact n-dimensional Einstein–Weyl spaces equipped with a cohomogeneity-one metric. In that context, with no compactness hypothesis for the manifold on which lives the Einstein–Weyl structure, we prove that, as soon as the (n−1)-dimensional space is a homogeneous reductive Riemannian space with a unimodular group of left-acting isometries G:

• •there exists a Gauduchon gauge such that the Weyl-form is co-closed and its dual is a Killing vector for the metric;
• •in that gauge, a simple constraint on the conformal scalar curvature holds;
• •a non-exact Einstein–Weyl structure may exist only if the (n−1)-dimensional homogeneous space G/H contains a non-trivial subgroup H′ that commutes with the isotropy subgroup H;
• •the extra isometry due to this Killing vector corresponds to the right-action of one of the generators of the algebra of the subgroup H′.

The first two results are well known when the Einstein–Weyl structure lives on a compact manifold, but our analysis gives the first hints on the enlargement of the symmetry due to the Einstein–Weyl constraint.We also prove that the subclass with G compact, a one-dimensional subgroup H′ and the (n−2)-dimensional space G/(H×H′) being an arbitrary compact symmetric Kähler coset space, corresponds to n-dimensional Riemannian locally conformally Kähler metrics. The explicit family of structures of cohomogeneity-one under SU(n/2) being, thanks to our results, invariant under U(1)×SU(n/2), it coincides with the one first studied by Madsen; our analysis allows us to prove most of his conjectures.     

The O(5) nonlinear σ-model as a SU(2) gauge theory

We show how the O(5) nonlinear σ-model, coupled with “gravity” in a generally covariant way, can be translated into the quaternionic model HP¹, invariant under a local SU (2) gauge, and we discuss the classical solutions.     

CNM models,holomorphic functions and projective superspace c-maps

Continuing the investigation of CNM (chiral-non-minimal) hypermultiplet non-linear σ-models, we propose extensions of the concept of the c-map which relate holomorphic functions to hyper-Kähler geometrics. In particular, we show that a whole series of hyper-Kähler potentials can be derived by replacing the role of the 4D, N = 1 tensor multiplet in the original c-map by 4D, N = 1 non-minimal multiplets and auxiliary superfields. The resulting N = 2 models appear to have interesting connections to Calabi-Yau manifolds and algebraic varieties. These models also emphasize the fact that special hyper-Kähler manifolds (the analogs of special Kähler manifolds) without isometries exist.     

Vortex Equations in Abelian Gauged σ-Models

We consider nonlinear gauged σ-models with Kähler domain and target. For a special choice of potential these models admit Bogomolny (or self-duality) equations — the so-called vortex equations. Here we describe the space of solutions and energy spectrum of the vortex equations when the gauge group is a torus T ⁿ , the domain is compact, and the target is We also obtain a large family of solutions when the target is a compact Kähler toric manifold.     

Critical behaviour of SU(n) quantum chains and topological non-linear σ-models

The critical behaviour of SU(n) quantum “spin” chains, Wess-Zumino-Witten σ-models and grassmanian σ-models at topological angle θ = π (of possible relevance to the quantum Hall effect) is reexamined. It is argued that an additional Z_n symmetry is generally necessary to stabilize the massless phase. This symmetry is not present for the σ-models for n > 2 and is only present for certain representations of “spin” chains.     

Oscillation and CP violation in the “horizontal” superweak gauge scheme

A model, in a framework with CP violation processes and K°- $\text{K}$°, D° -$\text{D}$°, … mixing, caused by “Horizontal” gauge interactions, is considered.     

Anomaly-free sequential superunification

The supergroup $\text{SU}\text{(}\text{7}\text{1}\text{)}$ defines a model with 8 generations of quarks and leptons. One half of these are chiral-inverted. The model represents an anomaly-free $\text{SU}\text{(}\text{7}\text{1}\text{) ×}\text{U}\text{(1)}$ gauge unification theory (GUT).     

On Weinberg's model of CP violation in gauge theories

The consequences of the Weinberg model of CP violation in gauge theories (ref. [1]) are discussed in detail. It is shown (using the valence quark hypothesis) that the contribution of the induced superweak interaction to the experimentally measured quantities η+? and η₀₀ are ≈m_K²/m_π² times larger than that of the direct milliweak decay K₂⁰→2π. The correction to the relation |η+?|/|η₀₀|=1 is also calculated in the model. We estimate the neutron dipole moment $\text{(D}{}_{\mathrm{<mtext>n</mtext>}}\text{??2.8·10}{}^{\mathrm{?25}}\text{e.cm.}\text{)}$ and note that the dipole moments of particles with strange quark constituents should be about three orders of magnitude larger $\text{(}\text{e.g.}\text{D}{}_{\mathrm{\Lambda }}\text{??1.15·10}{}^{\mathrm{?22}}\text{e·}\text{cm}$. The CP violating vertex d?s + gluon is also discussed.All physical quantities appear to depend crucially on the values of quark masses, so that CP violating effects in the models of the type considered may elucidate the nature of quark masses. We argue that the bare (or “mechanical”) values should be used in the calculations and this leads to a considerable difference in the estimates of various effects as compared to those of ref. [1].