| 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.
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