Eight years ago, I got interested in two seemingly quite different problems: quantum gauge theories in three space-time dimensions with a Chern-Simons term in the action functional, and the theory of the fractional quantum Hall efect. These two circles of problems have attracted a great deal of interest, during the past decade. The study of Chern-Simons gauge theory and of related quantum field theores in two space-time dimensions gave rise to the theory of graid statistics and of quantum symmetries in low-dimensional quantum field theory and its connections to knot theory; (see [1] and refs. given there). From the point of view of pure mathematics, the work on Chern-Simons theory culminated in Witten's paper [2] on the connection between pure, nonabelian Chern-Simons theory and the Jones polynomial. (For a somehwat different approach see also [3].)
The theory of braid statistics and of quantum symmetries led to new insights into the structure of braided tensor categories and the realization that a braided tensor category naturally gives rise to a projective representation of the modular group ${\rm SL}_2({\bf Z})$; see [4] and refs. given there.
The fractional quantum Hall effect is observed in the study of two-dimensional gases of electrons at very low temperatures subject to a strong, uniform magnetic field transversal to the plane of the system. When the density of electrons is varied, keeping the magnetic field constant, the longitudinal resistance of the system drops to zero at certain special values of the electron density. At these values of the density, the so-called Hall conductivity of the system is observed to be a rational multiple of a fundamental constant of nature, $\frac{e^2}h$, where $e$ is the elementary electric charge and $h$ is Planck's constant. The system is then called an incompressible quantum Hall fluid.
It has been noted in [5] that the electrodynamics of an incompressible quantum Hall fluid, in a regime of large distances and low frequencies (scaling limit), is described by a pure, abelian Chern-Simons theory. Physically, the gauge fields of this theory are the vector potentials of conserved currents on a three-dimensional space-time. These currents are quantum-mechanical, operator-valued distributions. Thus, the electrodynamics of an incompressible quantum Hall fluid in the scaling limit is described by a quantized, pure, abelian Chern-Simons theory. One must then study the Hilbert space of physical states of this theory. These states must obey certain natural constraints derived from the physics of the system. These states musy obey certain natural constraints derived from the physics of the system. Simplifying things somewhat, one then proves that these constraints imply that physical states can be labelled by quantum mnumbers which are points of a lattice, $\Gamma^*$ dual to an integral, odd Euclidean lattice $\Gamma$. In fact, the properties of an incompressible quantum Hall fluid in the scaling limit can be derived completely from the lattice $\Gamma$ and a primitive vector, $Q$, in the dual lattice $\Gamma^*$. For example, the Hall conductivity is given by the square length $\langle Q,Q\rangle$, of the vector $Q$, in units where $\frac{e^2}h=1$, and this explains why it is rational.
The task is then to classify pairs, $(\Gamma,Q)$, of an integral, odd (non-unimodular) Euclidean lattice $\Gamma$ and a primitive vector $Q$ in the dual lattice satisfying $\langle Qq\rangle\equiva \langle q,q\rangle(\mod 2)$, for all vectors $qe\Gamma$; see [6, 7]. This is an old and vast problem in pure mathematics which has appeared in many contexts, (e.g. in the topology of algebraic surfaces in algebraic four-manifolds, in arithmetics, etc.). Ideas from the physics of the quantum hall effect help in selecting an interesting subclass of data $(\Gamma,Q)$ relevant for the theoretical description of incompressible quantum Hall fluids. This subclass is understood rather well [7]. Its study reproduces available experimental data with astonishing precision and leads to interesting predictions.
Apart from making some comments on the mathematical and physical context of these problems, the main purpose of this lecture is to explain how the physics of the fractional quantum Hall effect leads one to the study of certain pure, abelian Chern-Simons gauge theories which, in turn, can be understood in terms of the data $(\Gamma,Q)$ described above. A partial classification of these data is then sketched. Finally, the role of ideas from non-commutative geometry in generalizing the notion of Chern-Simons theories and the construction of topological quantum field theories from generalized Chern-Simons theories will be indicated.
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