3:00 p.m., Friday (September 5, 2008)
Topological field theory in two dimensions and Teichmueller space
Abstract: A topological field theory in d dimensions associates to each (d-1)-dimensional
closed manifold M an inner-product space V(M), and to each d-dimensional manifold W with boundary
M a vector v(W) in V(M), satisfying certain natural axioms; for example, V(-) takes disjoint unions
to tensor products, and behaves well under diffeomorphisms. There are many flavours of topological
field theories - one may for example assume that all of the manifolds are oriented, or spin, or
carry a free action of a finite group G.
It turns out that the two-dimensional case is especially simple: two-dimensional topological
field theories are equivalent to commutative algebras with inner product (also known as
commutative Frobenius algebras). In this talk, we relate this to a result in topology. Harvey
has introduced a manifold with boundary containing the (6g-6)-dimensional Teichmueller space of
genus g closed Riemann surfaces as its interior, and we define a filtration F(i) of this space
such that the inclusion of F(i) into F(i+1) is i-connected. (The proof is an application of a
triangulation of Teichmueller space constructed by Harer.) This result and its generalizations
explain many pheonomena in topological field theory, including theorems of Moore and Seiberg,
Moore and Segal, and Turaev.
Refreshments will be served at 2:45 p.m. (Math Lounge, MATX 1115).