ABSTRACT OF THE DISSERTATION LES ÉTUDES IN ISING FIELD THEORY AND RELATED PROBLEMS by

The main subject of this thesis is the Ising field theory, the field theory describing the scaling limit of the two-dimensional Ising model near its critical point. We study the Ising field theory in low- and high-temperature regimes. In the low-temperature regime we address questions related to the mass spectrum of the bound states that exist in the theory. The overall goal is to study analytic properties of the masses taken as functions of the scaling variable in the theory. At this stage, we are primarily interested in the asymptotic series in the coupling parameter. We describe methods used for developing expansions in the coupling parameter when this parameter is sufficiently small. This gives rise to the low-energy and semiclassical series. Methods developed are then applied to a different model, a model that has many things in common with the Ising field theory, the two-dimensional quantum chromodynamics with infinite number of colors, also known as the ’t Hooft model. In the second part of the thesis we turn to the Ising field theory in the high-temperature regime. Here we study the problem of scattering in a weak magnetic field. We compute the leading perturbative correction to the scattering amplitude for the 2 → 2 process. This is done first by means of standard methods based on the direct ii intermediate-state decomposition and dispersion relations. Then we compute the large energy asymptotic of the amplitude for this process directly using a known relation be-tween the Ising field theory at zero magnetic field and classical sinh-Gordon equation. We obtain consistent results that show the logarithmic growth of the amplitude with energy at this order in magnetic field. Going beyond the leading order we argue that at large energies for a sufficiently weak magnetic field the amplitude exhibits a power-like decay. ii