Topological Aspects of QFT: Monopoles, Skirmions, Strings and All That

Abstract

The following discussions were in response to three lectures on the general subject of anomalies in quantum field theories. The first lecture reviewed the well-known results on the chiral and gauge anomalies in general dimensions, the anomaly descent equations which allow one to derive the gauge anomaly in 2n dimensions from the chiral anomaly in 2n + 2 dimensions. A simple “physical” interpretation of the gauge anomaly in terms of motion of the Dirac sea under the adiabatic changes in external gauge fields was also presented. The second lecture was devoted to a “physical” interpretation of the anomaly descent equations in terms of phenomena occurring in the presence of an axion string and a charged fermion which gets its mass from the axion. The key phenomenon is that in the presence of an electric field, the 3+1 dimensional chiral anomaly legislates a “transverse Hall current” inflow of charge which accumulates on the string — i.e., a gauge anomaly in the 1+1 dimensional world of the string and excitations living on it! The third lecture was devoted to anomalies in scale invariance of two-dimensional non-linear sigma models and their connection with string theory. The basic point is that strings propagating in curved spacetimes are described by sigma models with coupling constants given by the spacetime metric functions G μν (and other massless fields). The condition for consistent string propagation is that the coupling constants be chosen so that the quantum scale invariance anomaly (roughly speaking, the renormalization group beta function) vanish. When evaluated perturbatively, these conditions yield stringy generalizations of the Einstein equations and provide the framework for studying compactification of extra dimensions.