  We give a leisurely, albeit woefully incomplete, overview of quantum field
theory, its relevance to condensed matter systems, and spin systems, which
proceeds via a series of illustrative examples. The goal is to provide readers
from the mathematics community a swift route into recent condensed matter
literature that makes use of topological quantum field theory and ideas from
stable homotopy theory to attack the problem of classification of topological
(or SPT) phases of matter. The toric code and Heisenberg spin chain are briefly
discussed; important conceptual ideas in physics, that may have somehow evaded
discussion for those with purely mathematical training, are also reviewed.
Emphasis is placed on the connection between (algebras of) nonlocal operators
and the appearance of nontrivial TQFTs in the infrared.
