Machine computation of topological field theories on 2-complexes

                           F.Quinn

Topological field theories have been proposed as the proper tools to probe
the bizarre mysteries of low-dimensional topology. However this idea has
yet to prove itself. The really successful developments -- particularly the
Donaldson and Seiberg-Witten gauge theories on 4-manifolds -- are obviously
related, but do not satisfy the current axioms for a field theory. And we
do not understand their structure well enough to propose more general
axioms that would include them. There are sophisticated constructions
(initiated by Reshetikhin and Turaev) that use categories or representation
theory to give genuine field theories. These have led to some very
interesting mathematics (especially "quantum groups"), but have not had
consequences comparable to those of gauge theory. It is not clear why. The
current axioms and constructions may somehow miss the mark. Or the
constructions may be good, but we can't yet use them effectively. Or it may
be that there is simply nothing wonderful to find in the areas they have
been most developed (knots, links, and 3-manifolds).

This lecture describes an exploration of "baby models"; field theories
defined on 2-dimensional CW complexes using mod p representations of simple
Lie algebras. It turns out that the only thing "baby" about these is our
grand plan about what to do next. The abstract definition is difficult, and
gives almost no information about actual numerical values. Direct
computation is possible in principle, but in practice requires elaborate
and careful use of the structure of representations of Lie algebras, the
structure of monoidal categories, and linear algebra with "sparse"
matrices. And even after examples can be computed, strange cyborg mixtures
of abstract structure and numerical computation are needed to make sense of
the output.