Lattice regularizations of supersymmetric and chiral gauge theory

Abstract

Weakly coupled quantum field theories are well understood in terms of perturbative expansions. Various regularization schemes such as Pauli-Villars and dimensional regularization get rid of the ultraviolet divergences without ruining the basic symmetry properties that are an integral part of the theory. However, as we know many important phenomena such as quark confinement and dynamical symmetry breaking are essentially nonperturbative, where in general the low energy (a.k.a. infrared or IR) degrees of freedom appear drastically different from the high energy (a.k.a. ultraviolet or UV) degrees of freedom as a result of strong interactions. Clearly, we can not claim that we fully understand all these phenomena without finding tools that can treat all of them systematically. If quantum field theory is believed to be the right theory describing all of these interactions, then a systematic tool means a way to predict everything even when the coupling constant is too large for a perturbative expansion. Yet, up to now, the only type of quantum field theory that is well defined nonperturbatively are nonsupersymmetric vector-like theories such as quantum chromodynamics, in which gauge fields only couple to a vector current. In many other interesting models, such as chiral gauge theories or theories with supersymmetries (could be either chiral or vectorlike), quantitative studies of their nonperturbative parameter region still remains a barely touched territory.

The study in this thesis consists of two parts. The first part is a very primitive investigation of the lattice regularization of four dimensional supersymmetric quantum field theories. The second part is about the formulation of chiral gauge theories on the lattice. Both parts are about trying to study the nonperturbative regions of two interesting models by formulating them on a spacetime lattice, similar to what has been done with nonsupersymmetric vector-like theories, where much successes has been achieved.