International Journal of Modern Physics A

The existence of a fundamental scale, a lower bound to any output of a position measurement, seems to be a model-independent feature of quantum gravity. In fact, different approaches to this theory lead to this result. The key ingredients for the appearance of this minimum length are quantum mechanics, special relativity and general relativity. As a consequence, classical notions such as causality or distance between events cannot be expected to be applicable at this scale. They must be replaced by some other, yet unknown, structure.

We consider a wide class of two-dimensional models as gauge theories, the Gross-Neveu model, O(N) and CP^N−1-like models using a formalism based on the introduction of bilocal fields that permits us to perform easily the large N expansion of this set of models in a unified and general way. We mainly discuss the SU(N) gauge field theory minimally coupled to fermionic plus bosonic matter in the fundamental representation, and we obtain within the path integral approach exact equations for the particle spectrum, also in the presence of renormalizable polynomial potentials. Finally, we discuss the correspondence between this new approach and the one previously used in the context of the O(N) vector models.

We derive, based on the real-time formalism (especially thermo-field-dynamics), the Schwinger-Dyson gap equation for the fermion propagator in QED and the four-fermion model at finite temperature and density. We discuss some advantages of the real-time formalism in solving the self-consistent gap equation, in comparison with the ordinary imaginary-time formalism. Once we specify the vertex function, we can write down the SD equation with only continuous variables without performing the discrete sum over Matsubara frequencies which cannot be performed in advance without further approximation in the imaginary-time formalism. By solving the SD equation obtained in this way, we find the chiral-symmetry-restoring transition at finite temperature and present the associated phase diagram of strong-coupling QED. In solving the SD equation, we consider two approximations: instantaneous-exchange and p₀-independent ones. The former has a direct correspondence in the imaginary-time formalism; the latter is a new approximation beyond the former, since it is able to incorporate new thermal effects which have been overlooked in the ordinary imaginary-time solution. However, the two approximations are shown to give qualitatively the same results on the finite-temperature phase transition.

We discuss the K3 surface and the complex torus from the viewpoint of “mirror symmetry.” We calculate the periods of some K3 surface and construct a mirror map for the two-point correlation function and the prepotential. We find that there are no instanton corrections of the Yukawa coupling for K3 (also torus), which is expected from the viewpoint of algebraic geometry.

We study classical N=2 super W₃ algebra and its interplay with N=2 supersymmetric extensions of the Boussinesq equation in the framework of the nonlinear realization method and the inverse Higgs-covariant reduction approach. These techniques have been previously used by us in the bosonic W₃ case to give a new geometric interpretation of the Boussinesq hierarchy. Here we deduce the most general N=2 super Boussinesq equation and two kinds of the modified N=2 super Boussinesq equations, as well as the super Miura maps relating these systems to each other, by applying the covariant reduction to certain coset manifolds of linear symmetry associated with N=2 super W₃. We discuss the integrability properties of the equations obtained and their correspondence with the formulation based on the notion of the second Hamiltonian structure.