Inequivalent quantizations of the rational Calogero model with a Coulomb type interaction

We consider the inequivalent quantizations of a N-body rational Calogero model with a Coulomb type interaction. It is shown that for a certain range of the coupling constants, this system admits a one-parameter family of self-adjoint extensions. We analyze both the bound and scattering state sectors and find novel solutions of this model. We also find the ladder operators for this system, with which the previously found solutions can be constructed.     

Integrable non-linear σ models with fermions

The two-dimensional non-linear model on a Riemannian symmetric spaceM=G/H is coupled to fermions with quartic self-interactions. The resulting hybrid model is presented in a gauge-dependent formulation, with a bosonic field taking values inG and a fermionic field transforming under a given representation of the gauge groupH. General criteria for classical integrability are presented: they essentially fix the Lagrangian of the model but leave the fermion representation completely arbitrary. It is shown that by a special choice for the fermion representation (derived from the adjoint representation ofG by an appropriate reduction) one arrives naturally at the supersymmetric non-linear model onM=G/H. The issue of quantum integrability is also discussed, though with less stringent results.Work partially supported by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), Brazil, and KFA Jülich, Federal Republic of GermanyOn leave of absence from Fakultät für Physik der Universität Freiburg, Federal Republic of Germany     

Quantal symmetries in the non-linear σ model with Maxwell and non-Abelian Chern-Simons terms

The quantal symmetry property of the CP1 nonlinear （y model with Maxwell non-Abelian Chern- Simons terms in （2＋1） dimension is studied. In the Coulomb gauge, the system is quantized by using the Faddeev-Senjanovic （FS） path-integral formalism. Based on the quantaum Noether theorem, the quantal conserved angular momentum is derived and the fractional spin at the quantum level in this system is presented.     

Quantal symmetries in the non-linear σ model with Maxwell and non-Abelian Chern-Simons terms

The quantal symmetry property of the CP1 nonlinear σ model with Maxwell non-Abelian ChernSimons terms in(2+1) dimension is studied.In the Coulomb gauge,the system is quantized by using the Faddeev-Senjanovic(FS) path-integral formalism.Based on the quantaum Noether theorem,the quantal conserved angular momentum is derived and the fractional spin at the quantum level in this system is presented.     

Three-loop β-function for the bosonic non-linear sigma model

We present a calculation of the three-loop metric β-function for the bosonic non-linear sigma model. We also construct the action whose equations of motion coincide with the vanishing of the β-function. It is found to be the effective action derived from string scattering amplitudes.     

Monopoles,non-linear σ models,and two-fold loop spaces

In this paper we study the topology of , the moduli spaces ofSU(2) monopoles associated with the Yang-Mills-Higgs and Bogomol'nyi equations, and (m) _k , non-linear models from quantum field theory. Beautiful work of Donaldson [18, 19], Hitchin [24, 25] and Taubes [37, 39, 40] shows that gauge equivalence classes of monopoles correspond to based rational self-maps of the Riemann sphere. Similarly, the non-linear models we consider here are based harmonic maps from the Riemann sphere to complex projectivem space. In seminal work, Segal [35] studied (m) _k , the space of based rational maps from the Riemann sphere to complex projectivem space of a fixed degreek. Any element of (m) _k is clearly an element of _k ² CP(m), the space of all based continuous maps from the Riemann sphere to complex projectivem space of a fixed degreek, and this assignment gives rise to the natural inclusion of (m) _k in _k ² CP(m). Segal showed that these natural inclusions are homotopy equivalences through dimensionk(2m – 1). As the topology of the two-fold loop space ² CP(m) is well understood, Segal's result gives a very efficient way to explicitly determine the low dimensional topology of (m) _k . Thus iterated loop spaces have much to say about the topology of monopoles and non-linear models.Partially supported by NSF grant DMS-8508950Partially supported by NSF grant DMS-8701539     

Cancellations of infrared divergences in the two-dimensional non-linear σ models

In the two-dimensionalO(N) nonlinear models, the expectation value of anyO(N) invariant observable is shown to have an infrared finite weak coupling perturbative expansion, although it is computed in the wrong spontaneously broken symmetry phase. This result is proved by extracting all infrared divergences of any bare Feynman amplitude atD=2– dimension. The divergences cancel at any order only for invariant observables. The renormalization atD=2 preserves the infrared finiteness of the theory.     

Current algebra Ward identities in the renormalized σ model

The many-current Ward identities corresponding to the Gell-Mann current algebra are discussed in the renormalized model. The Ward identities are verified in the case of the SU(2)×SU(2) chiral symmetry. In the SU(3)×SU(3) case the uniqueness of the Adler-Bardeen anomaly is proved using the Wess-Zumino consistency conditions.On leave of absence from the University of Genova (Italy).Chercheur Associé au C.N.R.S. (C.P.T./Marseille).     

Four-loop β-function for the N = 1 and N = 2 supersymmetric non-linear sigma model in two dimensions

A superspace calculation of the four-loop β-function is presented for two-dimensional N= 1,2 supersymmetric non-linear σ-models. It is concluded that besides the one-loop contribution which vanishes on Ricci-flat manifolds, the β-function receives four-loop contributions which do not vanish in the Ricci-flat case. Implications for the superstring are discussed.     

Nonlocal field correlation functions on a lattice in the HP1 σ model

Connected two-point field-strength correlation functions are measured on a lattice in the quaternionic projective σ model within pure SU(2) theory. The correlation lengths extracted from exponential fits for these correlation functions, λ₁⁻¹ = 1.40(3) GeV and λ⁻¹ = 1.51(3) GeV, are found to be in good agreement with the results of other known calculations. The dependence of bilocal functions on the connector shape is also studied.     

Wave operators and analytic solutions for systems of non-linear Klein-Gordon equations and of non-linear Schrödinger equations

We consider, in a 1+3 space time, arbitrary (finite) systems of nonlinear Klein-Gordon equations (respectively Schrödinger equations) with an arbitrary local and analytic non-linearity in the unknown and its first and second order space-time (respectively first order space) derivatives, having no constant or linear terms. No restriction is given on the frequency sign of the initial data. In the case of non-linear Klein-Gordon equations all masses are supposed to be different from zero.We prove, for such systems, that the wave operator (fromt= tot=0) exists on a domain of small entire test functions of exponential type and that the analytic Cauchy problem, in ⁺׳, has a unique solution for each initial condition (att=0) being in the image of the wave operator. The decay properties of such solutions are discussed in detail.Partially supported by the Swiss National Science FoundationOn leave from Institut de Physique Théorique, 32 Bd d'Ivoy, CH-1211 Geneve 4 Switzerland.     

Smarr’s formula for black holes with non-linear electrodynamics

It is known that for nonlinear electrodynamics the First Law of Black Hole Mechanics holds, however the Smarrs formula for the total mass does not. In this contribution we discuss the point and determine the corresponding expressions for the Bardeen black hole solution that represents a nonlinear magnetic monopole. The same is done for the regular black hole solution derived by Ayón–Beato and García [1], showing that in the case that variations of the electric charge are involved, the Smarrs formula is no longer valid.     

Decay of a scalar σ meson near the critical end point in the PNJL model

Properties of a scalar a meson are investigated in the two-flavor Nambu-Jona-Lasinio model with the Polyakov loop (PNJL). Model analysis of the phase diagram of strong interacting matter is performed. The temperature dependence of the σ → ππ decay width is studied at the zero chemical potential and near the critical end point. The calculated strong coupling constant g _σππ and the decay width are compared with available experimental data and other model results. Nonthermal enhancement of the total decay width is noted for the σ meson near the critical end point when the condition m _σ ≥ 2m _π is broken.     

On the coincidence of an Alday-Maldacena regularized σ model and Nambu-Goto areas of minimal surfaces

For the σ model and Nambu-Goto actions, values of the Alday-Maldacena regularized actions are calculated on the solutions of the equations of motion with constant unregularized Lagrangian. It turns out that these values coincide up to a factor independent of boundary conditions.     

Universality of the Bethe-ansatz solution for the chiral non-linear σ-model

A euclidean lattice action is constructed to test the idea of universality of the Bethe-ansatz solution for the two-dimensional principal chiral model. The response of the system to uniform external fields is studied using the Monte Carlo simulation. A transition to the gapless phase is observed at the critical external field predicted by the Bethe-ansatz method.     

Local and non-local conserved quantities for generalized non-linear Schrödinger equations

It is shown how to construct infinitely many conserved quantities for the classical non-linear Schrödinger equation associated with an arbitrary Hermitian symmetric spaceG/K. These quantities are non-local in general, but include a series of local quantities as a special case. Their Poisson bracket algebra is studied, and is found to be a realization of the half Kac-Moody algebrak _R [], consisting of polynomials in positive powers of a complex parameter which have coefficients in the compact real form ofk (the Lie algebra ofK).