Tau-functions and Dressing Transformations for Zero-Curvature Affine Integrable Equations

The solutions of a large class of hierarchies of zero-curvature equations that includes Toda and KdV type hierarchies are investigated. All these hierarchies are constructed from affine (twisted or untwisted) Kac-Moody algebras~$\ggg$. Their common feature is that they have some special ``vacuum solutions'' corresponding to Lax operators lying in some abelian (up to the central term) subalgebra of~$\ggg$; in some interesting cases such subalgebras are of the Heisenberg type. Using the dressing transformation method, the solutions in the orbit of those vacuum solutions are constructed in a uniform way. Then, the generalized tau-functions for those hierarchies are defined as an alternative set of variables corresponding to certain matrix elements evaluated in the integrable highest-weight representations of~$\ggg$. Such definition of tau-functions applies for any level of the representation, and it is independent of its realization (vertex operator or not). The particular important cases of generalized mKdV and KdV hierarchies as well as the abelian and non abelian affine Toda theories are discussed in detail.Comment: 27 pages, plain Te

The Hamiltonian Structure of Soliton Equations and Deformed W-Algebras

The Poisson bracket algebra corresponding to the second Hamiltonian structure of a large class of generalized KdV and mKdV integrable hierarchies is carefully analysed. These algebras are known to have conformal properties, and their relation to $\cal W$-algebras has been previously investigated in some particular cases. The class of equations that is considered includes practically all the generalizations of the Drinfel'd-Sokolov hierarchies constructed in the literature. In particular, it has been recently shown that it includes matrix generalizations of the Gelfand-Dickey and the constrained KP hierarchies. Therefore, our results provide a unified description of the relation between the Hamiltonian structure of soliton equations and $\cal W$-algebras, and it comprises almost all the results formerly obtained by other authors. The main result of this paper is an explicit general equation showing that the second Poisson bracket algebra is a deformation of the Dirac bracket algebra corresponding to the $\cal W$-algebras obtained through Hamiltonian reduction.Comment: 41 pages, plain TeX, no figures. New introduction and references added. Version to be published in Annals of Physics (N.Y.

Semi-classical spectrum of the Homogeneous sine-Gordon theories

The semi-classical spectrum of the Homogeneous sine-Gordon theories associated with an arbitrary compact simple Lie group G is obtained and shown to be entirely given by solitons. These theories describe quantum integrable massive perturbations of Gepner's G-parafermions whose classical equations-of-motion are non-abelian affine Toda equations. One-soliton solutions are constructed by embeddings of the SU(2) complex sine-Gordon soliton in the regular SU(2) subgroups of G. The resulting spectrum exhibits both stable and unstable particles, which is a peculiar feature shared with the spectrum of monopoles and dyons in N=2 and N=4 supersymmetric gauge theories.Comment: 28 pages, plain TeX, no figure

S-matrices and quantum group symmetry of k-deformed sigma models

Recently, several kinds of integrable deformations of the string world sheet theory in the gauge/gravity correspondence have been constructed. One class of these, the k deformations associated to the more general q deformations but with q=exp(i pi/k) a root of unity, has been shown to be related to a particular discrete deformation of the principal chiral models and (semi-)symmetric space sigma models involving a gauged WZW model. We conjecture a form for the exact S-matrices of the bosonic integrable field theories of this type. The S-matrices imply that the theories have a hidden infinite dimensional affine quantum group symmetry. We provide some evidence, via quantum inverse scattering techniques, that the theories do indeed possess the finite-dimensional part of this quantum grou

The origin of power-law distributions in deterministic walks: the influence of landscape geometry

We investigate the properties of a deterministic walk, whose locomotion rule is always to travel to the nearest site. Initially the sites are randomly distributed in a closed rectangular ($A/L \times L)$ landscape and, once reached, they become unavailable for future visits. As expected, the walker step lengths present characteristic scales in one ($L \to 0$) and two ($A/L \sim L$) dimensions. However, we find scale invariance for an intermediate geometry, when the landscape is a thin strip-like region. This result is induced geometrically by a dynamical trapping mechanism, leading to a power law distribution for the step lengths. The relevance of our findings in broader contexts -- of both deterministic and random walks -- is also briefly discussed.Comment: 7 pages, 11 figures. To appear in Phys. Rev.

Oral administration of melatonin counteracts several of the effects of chronic stress in rainbow trout

To assess a possible antistress role of melatonin in fish, we orally administered melatonin to rainbow trout for 10 d and then kept the fish under normal or high stocking density conditions during the last 4 d. Food intake; biochemical parameters in plasma (cortisol, glucose, and lactate concentrations); liver (glucose and glycogen concentrations, and glycogen synthase activity); enzyme activities of amylase, lipase, and protease in foregut and midgut; and content of the hypothalamic neurotransmitters dopamine and serotonin, as well as their oxidized metabolites, 3,4-dihydroxyphenylacetic acid and 5-hydroxy-3-indoleacetic acid, were evaluated under those conditions. High stocking density conditions alone induced changes indicative of stress conditions in plasma cortisol concentrations, liver glycogenolytic potential, the activities of some digestive enzymes, and the 3,4-dihydroxyphenylacetic acid-to-dopamine and 5-hydroxy-3-indoleacetic acid-to-serotonin ratios in the hypothalamus. Melatonin treatment in nonstressed fish induced an increase in liver glycogenolytic potential, increased the activity of some digestive enzymes, and enhanced serotoninergic and dopaminergic metabolism in hypothalamus. The presence of melatonin in stressed fish resulted in a significant interaction with cortisol concentrations in plasma, glycogen content, and glycogen synthase activity in liver and dopaminergic and serotoninergic metabolism in the hypothalamus. In general, the presence of melatonin mitigated several of the effects induced by stress, supporting an antistress role for melatonin in rainbow trout.Ministerio de Ciencia e Innovación | Ref. AGL2010-22247-C03-03Xunta de Galicia | Ref. CN2012/ 00

Double Scaling Limits in Gauge Theories and Matrix Models

We show that $\N=1$ gauge theories with an adjoint chiral multiplet admit a wide class of large-N double-scaling limits where $N$ is taken to infinity in a way coordinated with a tuning of the bare superpotential. The tuning is such that the theory is near an Argyres-Douglas-type singularity where a set of non-local dibaryons becomes massless in conjunction with a set of confining strings becoming tensionless. The doubly-scaled theory consists of two decoupled sectors, one whose spectrum and interactions follow the usual large-N scaling whilst the other has light states of fixed mass in the large-N limit which subvert the usual large-N scaling and lead to an interacting theory in the limit. $F$-term properties of this interacting sector can be calculated using a Dijkgraaf-Vafa matrix model and in this context the double-scaling limit is precisely the kind investigated in the "old matrix model'' to describe two-dimensional gravity coupled to $c<1$ conformal field theories. In particular, the old matrix model double-scaling limit describes a sector of a gauge theory with a mass gap and light meson-like composite states, the approximate Goldstone boson of superconformal invariance, with a mass which is fixed in the double-scaling limit. Consequently, the gravitational $F$-terms in these cases satisfy the string equation of the KdV hierarchy.Comment: 38 pages, 1 figure, reference adde