The Lie-Poisson structure of integrable classical non-linear sigma models

The canonical structure of classical non-linear sigma models on Riemannian symmetric spaces, which constitute the most general class of classical non-linear sigma models known to be integrable, is shown to be governed by a fundamental Poisson bracket relation that fits into ther-s-matrix formalism for non-ultralocal integrable models first discussed by Maillet. The matricesr ands are computed explicitly and, being field dependent, satisfy fundamental Poisson bracket relations of their own, which can be expressed in terms of a new numerical matrixc. It is proposed that all these Poisson brackets taken together are, representation conditions for a new kind of algebra which, for this class of models, replaces the classical Yang-Baxter algebra governing the canonical structure of ultralocal models. The Poisson brackets for the transition matrices are also computed, and the notorious regularization problem associated with the definition of the Poisson brackets for the monodromy matrices is discussed.Suported by the Deutsche Forschungsgemeinschaft, Contract No. Ro 864/1-1Supported by the Studienstiftung des Deutschen Volkes     

Classical integrability of two-dimensional non-linear sigma models

The conditions under which a general two-dimensional non-linear sigma model is classically integrable are given. These requirements are found by demanding that the equations of motion of the theory are expressible as a zero curvature relation. Some new integrable two-dimensional sigma models are then presented.     

On the dual symmetry of the non-linear sigma models

We analyze the dual symmetry which is reponsible for the existence of infinitely many conserved non-local charges in the classical two-dimensional non-linear σ models. For compact global symmetry groups, we prove that the σ model has the dual symmetry if and only if the field takes values in a symmetric space.     

Current algebra of classical non-linear sigma models

The current algebra of classical non-linear sigma models on arbitrary Riemannian manifolds is analyzed. It is found that introducing, in addition to the Noether currentj associated with the global symmetry of the theory, a composite scalar fieldj, the algebra closes under Poisson brackets.Address after September 1, 1991: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge MA 02138, USA. Supported by the Studienstiftung des Deutschen Volkes     

Ultra-violet finiteness of supersymmetric non-linear sigma models

It is shown that N = 4 supersymmetric non-linear sigma models in two spacetime dimensions are ultra-violet finite to all orders in perturbation theory.     

New integrable canonical structures in two-dimensional models

We develop a new canonical r-s matrix type approach for integrable two-dimensional models of non-ultralocal type. The L-matrices algebra and the monodromy matrices' algebras are given in terms of the usual r-matrix and of the new s-matrix, which, for consistency (Jacobi identity) have to obey an extended, dynamical Yang-Baxter type equation. The possible violation of the Jacobi identity arising in the (naive) equal-point limit of the monodromy matrices' algebras is discussed and a general, consistent procedure, i.e. satisfying the Jacobi identity, is defined. The method is applied to the complex sine-Gordon model.     

Semiclassical spectrum of the two-dimensional non-linear σ models

The spectrum of O(N) invariant two-dimensional non-linear σ models is analyzed for large N by the methods of Dashen, Hasslacher and Neveu. Calculations to $\text{O}\text{(}\text{1}\text{N}\text{)}$ are carried out and the spectrum is shown to consist solely of N degenerate massive mesons. This is consistent with strong coupling lattice calculations and indicates that for sufficiently large N there is no phase transition between the weak and strong coupling regions.     

Noncommutative geometry and a class of completely integrable models

We introduce a Hodge operator in a framework of noncommutative geometry. The complete integrability of 2-dimensional classical harmonic maps into groups (-models or principal chiral models) is then extended to a class of noncommutative harmonic maps into matrix algebras.     

The algebra of non-local charges in non-linear sigma models

We obtain the exact classical algebra obeyed by the conserved non-local charges in bosonic non-linear sigma models. Part of the computation is specialized for a symmetry groupO(N). As it turns out the algebra corresponds to a cubic deformation of the Kac-Moody algebra. We generalize the results for the presence of a Wess-Zumino term. The algebra is very similar to the previous one, now containing a calculable correction of order one unit lower. The relation with Yangians and the role of the results in the context of Lie-Poisson algebras are also discussed.Supported by FAPESP.Supported by CNPq.Supported by CNPq     

On the origin of anomalies in the quantum non-local charge for the generalized non-linear sigma models

A general criterion for the absence or presence of anomalies in the quantum non-local charge of the non-linear σ-model on a riemannian symmetric space is presented.     

An explicit three-loop calculation for the purely metric two-dimensional non-linear sigma model

We present the details of an explicit calculation of the UV divergences to three loops of the two-dimensional non-linear sigma model. The beta function is shown to derive from the low-energy string effective action.     

The effective potential in extended supersymmetric non-linear sigma models

It is proved that for a certain class of off-shell formulations the effective potential is either renormalizable, or extended supersymmetry is broken by the effects of renormalization. Examples of the latter are given and possibilities for the former are discussed. Explicit two loop calculations support the general results.     

Numerical investigation of multidimensional cosmological models based on the Weyl integrable geometry

The dynamics of multidimensional cosmological models based on the Weyl integrable geometry are investigated by means of numerical methods. Models are considered in space and in the presence of matter, the latter modeled by an ideal liquid and a nonminimal scalar field. Sufficient conditions are obtained under which cosmological singularity is absent and the scenario of dynamic dimensional reduction is realized.Scientific Research Center of PV. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 107–113, May, 1995.     

Monte Carlo simulations for the two-dimensional O(3) non-linear sigma model

We report on the results of Monte Carlo simulations for the two-dimensional O(3) non-linear sigma model. The estimates based on the combined use of the renormalization group and of the high temperature expansion are found to be in agreement with our “data”. We present good experimental evidence for the absence of any phase transition, as expected on theoretical grounds.     

On the stability of nonlinear waves in integrable models

A new method of stability investigation is presented for solutions of nonlinear equations integrable with the help of the inverse scattering transform (IST). The stability problem for periodic nonlinear waves in weakly dispersive media is solved with respect to transverse perturbations. It is shown that for positive dispersion media one-dimensional waves are unstable, and for negative dispersion such waves are stable.     

On the Mathai-Quillen formalism of topological sigma models

We present a Mathai-Quillen interpretation of topological sigma models. The key to the construction is a natural connection in a suitable infinite-dimensional vector bundle over the space of maps from a Riemann surface (the world sheet) to an almost complex manifold (the target). We show that the covariant derivative of the section defined by the differential that appears in the equation for pseudo-holomorphic curves is precisely the linearization of the operator itself. We also discuss the Mathai-Quillen formalism of gauged topological sigma models.     

On classicalization in nonlinear sigma models

We consider the phenomenon of classicalization in nonlinear sigma models with both positive and negative target space curvature and with any number of derivatives. We find that the theories with only two derivatives exhibit a weak form of classicalization, and that the quantitative results depend on the sign of the curvature. Nonlinear sigma models with higher derivatives show a strong form of the phenomenon which is independent of the sign of curvature. We argue that weak classicalization may actually be equivalent to asymptotic safety, whereas strong classicalization seems to be a genuinely different phenomenon. We also discuss possible ambiguities in the definition of the classical limit.