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.     

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.     

ALGELRAIC PROPERTIES OF NONLINEAR σ MODEL OF SU(2) WITH W-Z TERM IN 1+1 DIMENSIONS

Taking three Euler angles as the field variables of nonlinear σ model of SU(2),we calculated the Wess-Zumino term and showed by using canonical quantization that the chiral cuirents of this model satisfy a Kac-Moody alqebra at n equal an even intsger for arbitrary value of λ^o.     

Axially symmetric static solutions of four-dimensional SU(N) σ-models by the inverse scattering method

A systematic framework is derived for constructing a superpotential in static, axially symmetric four-dimensional SU(N) principal σ-models by applying an inverse scattering method.     

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.     

Inhomogeneous q-Differential Realization,Inhomogeneous q-Boson Realization of Quantum Group SU(n)q and Its Indecomposable Representations

An inhomogeneous q-differential realization (q-IHDR) and its corresponding inhomogeneous q-boson realization (q-IHBR) of quantum group SU(n)_q are studied. By making use of the q-IHBR a kind of infinite dimensional indecomposable and irreducible representations of SU(n)_q is studied on the q-Fcck space. The finite dimensional irreducible representations of SU(n), are obtained on the finite dimensional subspaces of q-Fock space. As an example, these representations of SU(2)_q are studied in detail and Jimbo standard irreducible representations are obtained.     

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.     

Lagrangian Form of the Self-Dual Equations for SU(N) Gauge Fields on Four-Dimensional Euclidean Space

By compactifying the four-dimensional Euclidean space into S₂×S₂ manifold and introducing two topological relevant Wess-Zumino terms to H_n≡SL(n, c)/SU(n) nonlinear sigma model, we construct a Lagrangian form for SU(n) self-dual Yang-Mills field, from which the self-dual equations follow as the Euler-Lagrange equations.     

Supersymmetric σ-models,twistors, and the Atiyah-Hitchin metric

The Legendre transform and its generalizations, originally found in supersymmetric σ-models, are techniques that can be used to givelocal constructions of hyperkähler metrics. We give a twistor space interpretation to the generalizations of the Legendre transform construction. The Atiyah-Hitchin metric on the moduli space of two monopoles is used as a detailed example.     

Kinematics of Yang-Mills solitons

In SU(2) Yang-Mills theory, theN-monopole configuration space is a bundle with fibers isomorphic to U(1)×...×U(1), and state vectors for which each monopole has chargen e/2 are homogeneous of degreen with respect to each U(1). Translations and rotations are defined for individual monopoles in theN-monopole space. The commutator of two translations is found to be a U(1) transformation that agrees for large monopole separation with the analogous phase change accompanying the translation of a charged point particle in an external magnetic field. The theory developed here is applied in a companion paper to prove a spin-statistics theorem for monopoles in SU(2) Yang-Mills theory.     

Classification of Gauge Orbit Types for SU(n)-Gauge Theories

A method for determining the orbit types of the action of the group of gauge transformations on the space of connections for gauge theories with gauge group SU(n) in spacetime dimension d4 is presented. The method is based on the one-to-one correspondence between orbit types and holonomy-induced reductions of the underlying principal SU(n)-bundle. It is shown that the orbit types are labelled by certain cohomology elements of spacetime satisfying two relations. Thus, for every principal SU(n)-bundle the corresponding stratification of the gauge orbit space can be explicitly determined. As an application, a criterion characterizing kinematical nodes for physical states in Yang–Mills theory with the Chern–Simons term proposed by Asorey et al. is discussed.     

SCATTERING OF u TO ū ON THE SU(5) COLORLESS MAGNETIC MONOPOLE

The scattering of quark u to antiquark u^C on the SU(5) colorless magnetic monopole is discussed in some detail.The baryon number non-conserved processes occur to the state j=1/2 and j=3/2 of the third type solutions. Solving the radial equation with j=1/2 and j=3/2 numerically,we show that the plane wave of quark u with helicity-1 can be scattered by the colorless monopole into antiquark u^C with the cross section σ=2π/p2.     

A geometric notion of complete integrability

A system of partial differential equations which can be described as a harmonic mapping of riemannian manifolds is called completely integrable when the corresponding n-dimensional manifold of fields admits 2n?1 independent Killing vector fields. It is conjectured that, for systems of two independent variables, complete integrability in the present sense implies the existence of a Lax pair for the system, for which the theory of the inverse scattering method is applicable. The stationary axisymmetric Einstein and Einstein-Maxwell equations, the SU(n) self-dual Yang-Mills fields in 1+1 dimensions, and the two-dimensional non-linear σ-models are shown to satisfy the conjecture; the conjecture is also proved for any system of n = 2 and n = 3 partial differential equations for n unknown scalar fields.     

The Geometry of Antiferromagnetic Spin Chains

It has been a conjecture of F.D.M.Haldane that long-wavelength excitations around antiferromagnetic vacua of certain spin chains are related to relativistic two-dimensional σ-models. We construct a class of such spin chains and associated σ-models, using symplectic geometry as a main tool. The target space of the σ-model can be an arbitrary complex flag manifold, and we find universal expressions for the metric and θ-term. The derivation relies on the fact that the flag manifold is a Lagrangian submanifold in a direct product of Grassmannians associated (via geometric quantization) to the representations of spins at consecutive sites of the spin chain. The true goal is not to reach the uttermost limits, but to discover a completeness that knows no boundaries. Rabindranath Tagore      

Continuity equations for the classical Euclidean two-dimensional non-linear σ-models

We derive an infinite set of independent covariant local non-polynomial continuity equations for the classical non-linear chiral O _n-models in two-dimensional Euclidean space.     

NEB NONLINEAR σ MODELS ON SYMMETRIC SPACES

Two new nonlinear σ models, defined on the symmetric coset spaces GL (n,c). ⊕ GL(n,c)/GL (n,c) and GL (n,c)/U(n) respectively, are formulated in this paper.The latter may be useful in discus-sing the four-dimensional Yang-Mills fields.     

Su (2) Poisson–Lie T Duality

Poisson–Lie target space duality is a framework where duality transformations are properly defined. In this Letter, we investigate the dual pair of -models defined by the double SO(3,1) in the Iwasawa decomposition.     

Renormalization group flow for general σ-models

The renormalization group flow for σ-models with base space of dimension 1 or 2 is investigated. In two dimensions it is shown that the flow is singular towards the UV for a generic target space. In one dimension it is shown that there are IR fixed points coming from negatively curved symmetric spaces.     

Theta-terms in nonlinear sigma-models

We trace the origin of θ-terms in nonlinear σ-models as a nonperturbative anomaly of current algebras. The nonlinear σ-models emerge as a low energy limit of fermionic σ-models. The latter describe Dirac fermions coupled to chiral bosonic fields. We discuss the geometric phases in three hierarchies of fermionic σ-models in space-time dimension (d+1) with chiral bosonic fields taking values on d-, d+1-, and d+2-dimensional spheres. The geometric phases in the first two hierarchies are θ-terms. We emphasize a relation between θ-terms and quantum numbers of solitons.     

Exact critical exponents for quantum spin chains,non-linear σ-models at θ=π and the quantum hall effect

We use non-abelian bosonization to predict critical exponents for quantum chains of arbitrary spin and arbitrary symmetry SU(n). Passing to the large representation limit gives non-linear σ-models on the manifolds U(2n)/U(n)×U(n) at topological angle θ=π. For n=1 this is the familiar “O(3)” model; taking the replica limit n→0 given critical exponents for the localization transition in the quantum Hall effect. Given certain assumptions, these exponents should be exact.