Topics in planar physics

Abstract

While the evident goal of physics is to explain phenomena in four-dimensional space-time, where physical Nature resides, and perhaps even to explain why Nature resides in four dimensions, the means that we have come to employ in reaching this goal are sufficiently intricate that it has proven useful to make a detour from the direct path, and to wander into lower-dimensional worlds, with the hope that in the simpler setting we can learn useful things about the agreed upon four-dimensional problem. This indeed has happened, initially in two dimensions, where we first encountered spontaneous gauge symmetry breaking, anomalies, the soliton phenomenon, to name three important examples. Moreover, when it was appreciated that there exist physical environments — not in particle physics but in condensed matter and statistical systems — which are properly described by two-dimensional field theories — e.g. linear chains whose time evolution gives rise to two-dimensional dynamics, or planar arrays in equilibrium whose static properties are governed by two-dimensional Euclidean field theory — a physical application of the pedagogical investigations could be made, for example to solitons and fractional charge in polyacetylene or to conformally invariant critical phenomena. Additionally, mathematical and speculative uses for two-dimensional field theories were found in the string program.