We show that there exists an alternative procedure in order to extract differential hierarchies, such as the KdV hierarchy, from one--matrix models, without taking a continuum limit. To prove this we introduce the Toda lattice and reformulate it in operator form. We then consider the reduction to the systems appropriate for one--matrix model.Comment: 18

In the context of hermitean one-matrix models we show that the emergence of the NLS hierarchy and of its reduction, the KdV hierarchy, is an exact result of the lattice characterizing the matrix model. Said otherwise, we are not obliged to take a continuum limit to find these hierarchies. We interpret this result as an indication of the topological nature of them. We discuss the topological field theories associated with both and discuss the connection with topological field theories coupled to topological gravity already studied in the literature.

A heavy fourth generation with a mass of the order of 400 GeV or more could trigger dynamical electroweak symmetry breaking by forming condensates through the exchange of a fundamental Higgs scalar doublet. The dynamics leading to these condensates is studied within the framework of the Schwinger-Dyson equation. This scenario leads to the presence of three (two composite and one fundamental) Higgs doublets, with interesting phenomenological implications. In addition, this dynamical phenomenon occurs in the vicinity of the energy scale where the restoration of scale symmetry might happen.

We investigate superfluidity, and the mechanism for creation of quantized vortices, in the relativistic regime. The general framework is a nonlinear Klein-Gordon equation in curved spacetime for a complex scalar field, whose phase dynamics gives rise to superfluidity. The mechanisms discussed are local inertial forces (Coriolis and centrifugal), and current-current interaction with an external source. The primary application is to cosmology, but we also discuss the reduction to the nonrelativistic nonlinear Schrödinger equation, which is widely used in describing superfluidity and vorticity in liquid helium and cold-trapped atomic gases.