4 - Conformal field theory: the physics of Moonshine  pp. 226-310

Conformal field theory: the physics of Moonshine

By Terry Gannon

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This chapter presents the physical context for Moonshine. Rather than diving into a conventional discourse of conformal field theory (CFT), it might be more helpful to take several steps back and begin with Galileo. Physics even more than mathematics is interwoven with history. Our treatment of CFT is sketchy but should supply the reader with all that is necessary to appreciate the absolutely profound role physics has played in Moonshine and other aspects of ‘pure’ mathematics in recent years. It is hoped that this chapter will make it easier for the interested reader to pursue more standard treatments of CFT and string theory. It is written primarily with the mathematician in mind.

The third section explores the physics of CFT, and the fourth describes some mathematical formulations. CFT is to a generic quantum field theory what finite-dimensional semi-simple Lie algebras are to generic Lie algebras. Background for both sections is provided by the review of classical and quantum physics sketched in the first two sections.

For a mathematician studying physics, important to keep in mind is that physics has been driven historically more by its predictive power than by conceptual concerns (with a few remarkable exceptions, such as Einstein's general relativity). Given enough time, however, the theory becomes polished to a state of pristine mathematical elegance, as classical mechanics amply demonstrates. In particular, one has the sense that quantum theory is ad hoc and rather unsound – and it is both – but these features are due to the historical accident that we were born too close to its inception. Much more important is what it can teach mathematics, which is considerable.