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.     

On finite action solutions of the nonlinear σ-model

The instanton and anti-instanton solutions of the two-dimensional O(3) σ-model are special examples of harmonic maps, which have been studied extensively in the mathematical literature. We give an elementary and self-contained proof that these solutions are the only continuous maps for which the action is finite and stationary under variations, without assuming any additional boundary conditions at infinity. An element of the proof is the vanishing of the stress tensor for a finite action solution, which actually holds true for the general O(N) σ-model. For the two-dimensional O(2l + 1) σ-model we exhibit explicit finite action solutions that do not lie in any lower dimensional sphere; the existence of such solutions has been pointed out in the mathematical literature. We also present a rigorous proof, based on Derrick's scaling argument, that there are no nonconstant finite action solutions in more than two dimensions.     

Complex and quaternionic analyticity in chiral and gauge theories,I

A comparative and systematic study is made of 2-dimensional CP(n) σ-models and new 4-dimensional HP(n) σ-models and their respective embedded U(1) and Sp(1) holonomic gauge field structures. The central theme is complex versus quaternionic analyticity. A unified formulation is achieved by way of Cartan's method of moving frames adapted to the hypercomplex geometries of the harmonic symmetric spaces $\text{CP(n) ≈ SU(n + 1)}\text{SU(n) × U(1)}$ and $\text{HP(n) ≈ Sp(n + 1)}\text{Sp(n) × Sp(1)}$ respectively. Elements of complex Kähler manifolds are applied to a detailed analysis of the CP(n) σ-model and its instanton sector. Generalization to any Kählerian σ-model is manifest. On the basis of Cauchy-Riemann analyticity, Kählerian models are shown to have an infinite number of local continuity equations. In a parallel manner, new 4-dimensional conformally invariant HP(n) σ-models are constructed. Focus is on the latter's hidden local gauge invariance in their holonomy group Sp(n) × Sp(1) which allows a natural embedding of the Sp(1) ≈ SU(2) pure Yang-Mills theory. The associated quaternionic structure is discussed in light of both quaternionic quantum mechanics and Kählerian geometry. In this chiral setting, the SU(2) Yang-Mills duality equations are cast into quaternionic Cauchy-Riemann equations over S⁴ ≈ HP(1), the conformal spacetime. In analogy to the CP(n) case, their rational solutions are the most general (8n ? 3) parameter instantons where the associated algebraic nonlinear equations of the type of Atiyah, Drinfeld, Hitchin, and Manin are now expressed in a new conformally invariant form. Geometrically, the SU(2) instantons solve the Frenet-Serret equations for quaternionic holomorphic curves; they are conformal maps from HP(1) into HP(n) with n their second Chern index. Fueter's quaternionic analysis is presented, then applied: Fueter functions are particularly suited for the solutions of 't Hooft, of Jackiw, Nohl and Rebbi, and of Witten and Peng, as well as the self-dual finite action per unit time solution of Bogomol'nyi, Prasad and Sommerfield. Generalizing the latter, a new solution with unit Chern index and finite action per unit spacetime cell is found. It is expressed in terms of the quaternionic fourfold quasi-periodic Weierstrass Zeta function. Finally the essence of our method is revealed in terms of universal connections over Stiefel bundles; generalization to real, complex and quaternionic classifying Grassmanian σ-models with their embedded SO(m), SU(m) and Sp(m) gauge fields is outlined in terms of gauge invariant projector valued chiral fields. Other outstanding problems are briefly discussed.     

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.     

J/gy suppression as evidence for high densities in nuclear collisions

It is shown that the scattering amplitudes of some degenerate massive string states can be expressed in terms of that of higher spin states at the same mass level. This phenomenon is discussed from σ-model point of view and demonstrated explicitly by using massive Ward identities generated by zero-norm states atD=26. The subtlety of the scalar propagating state is briefly pointed out. We also compare our on-shell gauge symmetry derived from σ-model with off-shell gauge symmetry of covariant string field theory.     

The cosmological constant,broken gauge theories and 3 K black-body radiation

It is suggested that the cosmological constant today be identified with the one provided by gauge theories at T = 3 K and not T = 0 K or T = 10¹⁵ K as done so far. We then calculate Λ(3 K) in the σ-model, Weinberg-Salam model and Mohapatra and Sidhu model. While an acceptable value is obtained with the rather ideal σ-model, a very small Λ is obtained with the WS model. For the MS model, Λ can be fitted only if at least one of the Higgs mesons has a very light mass (≈3 K).     

Solutions of the supersymmetricU(N) σ models with the Wess-Zumino-Witten term

We present finite action classical solutions of the supersymmetric (Susy)U(N) sigma models with and without the Wess-Zumino-Witten (WZW) term. We derive solutions of the model with the WZW-term from the solutions of the Lax-pair problem for the corresponding model without this term. We study some properties of these solutions such as the value of their action and show that this value is related to that of the purely bosonicU(N) σ-model. Moreover, we prove that all derived solutions are unstable.     

Generalized non-linear σ-models with gauge invariance

Generalizations of the non-linear σ-model and of the CP^N model are considered. They are invariant under a global group G and under a local gauge subgroup H. Whenever G/H defines a symmetric space, we can show that these models are asymptotically free. Furthermore, in two dimensions the local gauge invariance leads to confinement.     

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.     

Dynamical gauge fields in four dimensions from σ-models

We calculate the modification of the D = 4 CP(n) σ-model required by the existence of the effective action for composite fields which is finite in all orders in $\text{1}\text{N}$. We obtain the Higgs-like action with “free” bilinear terms for composite scalar and gauge fields. Contrary to the D = 2 case, due to the freedom of finite renormalization, the composite field excitations are not uniquely determined by the dynamics.     

Analytical results for the four-loop RG functions in the 2D non-linear O(n) σ-model on the lattice

We recalculate four-loop renormalization group functions in 2-dimensional nonlinear O(n) σ-model using coordinate-space method. The high accuracy of calculation allow us to find the analytical form of β- and γ-function (anomaluos dimension).     

Massive lepton pair production revisited with the optimized perturbative QCD

We discuss the interplay between the extrinsic geometry of strings and the symmetric KählerianG _2,n σ-model coupled with two-dimensional gravity. It is shown that the recently proposed Polyakov's action classically corresponds to a particular topological sector of theG _2,n model. The geometrical and physical meaning of the other topological sectors are also examined and shown to be related to the geometry of line bundles over Riemann surfaces (e.g. spinor bundles). The one-loop quantization of the model, around istanton-like solutions, provides scaling behaviour and asymptotic freedom for the adimensional coupling.     

Background field method for nonlinear σ-model in stochastic quantization

We formulate the background field method for the nonlinear σ-model in stochastic quantization. We demonstrate a one-loop calculation for a two-dimensional non-linear σ-model on a general riemannian manifold based on our formulation. The formulation is consistent with the known results in ordinary quantization. As a simple application, we also analyse the multiplicative renormalization of the O(N) nonlinear σ-model.     

A four-dimensional non-linear σ-model

We study a four-dimensional version of the non-linear O(n) σ-model in Euclidean space, introducing a suitable kinetic term in order to have a theory with many of the properties of the two-dimensional model. The outcoming theory is shown to be ultraviolet free for n > 2. Its behaviour is then discussed in the large-n expansion. It is found that in general two phases exist, in one symmetry is broken, in the other preserved, and both are possible in four dimensions. For a range of values of the coupling constant both asymptotic freedom and the perturbative expansion are valid, contrary to the situation of the two-dimensional σ-model. The infrared behaviour is found to be regular and a singularity of the running coupling constant to be of no relevance for the Green functions.     

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.     

Precession of classical spin and spin-orbit splitting

We present solutions of a fermion-boson model based on the supersymmetric (Susy) extension of theU(N) σ-models with the Wess-Zumino-Witten (WZW) term. We study some properties of these solutions. We point out that the obtained solutions are related to the components of the energy-momentum tensor of the purely bosonicU(N)σ-model and that some classes of these solutions are traceless.     

Canonical quantization of the σ-model with torsion in superspace

We use the Dirac quantization method extended to superspace to quantize the (1+1)-dimensional σ-model with torsion. We consider a superspaceN=1 and ann-dimensional Riemannian manifoldM.     

Classical solutions of CPn non linear σ -models; an algebraic geometrical description

The classical SO (3)-invariant σ-model and its suitably generalized versions are studied from the geometrical point of view. Known mathematical results concerning harmonic and holomorphic maps of a Riemann surface into the n-dimensional complex projective space are briefly reviewed. These are used to give a classification of classical solutions (both instantons and a certain subclass of unstable solutions) of $\text{C}$Pⁿ models and to study the properties of their energy spectrum.     

Renormalizations and Operator Expansion in σ-Model

The operator expansion (OPE) is studied for the Green function n(0) n(x)) at x² → 0 (n(x) is the dynamical field of σ-model) in the framework of the two-dimensional σ-model with the O(N) symmetry group at large N. As a preliminary step we formulate the ronormalization scheme which permits introduction of an arbitrary intermediate scale μ² in the framework of 1/N expansion and discuss factorization (separation) of small (p < μ ) and large (p > μ ) momentum region. It is shown that definition of composite local operators and coefficient functions figuring in OPE is unambiguous only in the leading order in 1/N expansion when dominant are the solutions with exrtemum of action. Corrections of order f(μ²)/N (here /(μ²) is the effective interaction constant at the point μ²) in composite operators and coefficient functions essentially depend on factorization method of high and low momentum regions. It is shown also that contributions to the power corrections of order m²x²/(μ²)/N in the Green function (here m is the dynamical mass-scale factor in σ-model) arise simultaneously from two sources: from the mean vacuum value of the composite operator n ∂²n and from the hard particle contributions in the coefficient function of unite operator. Due to the analogy between σ-model and QCD the obtained result indicates theoretical limitations to the sum rule method in QCD.     

Massive inter-particle symmetry and Ward identity in fermionic string theory

We show that there exist enlarged stringy (α′→∞) symmetries for (evenG-parity) massive modes in the 10D fermionic string theory. These symmetries are derived from on-shell Ward identities corresponding to the decoupling of massive gauge states in the spectrum. In the generalized massive supersymmetric σ-model formalism, some symmetry transformations relate particles with different spins in the first order weak field approximation.