Summary: Frobenius Algebra Structures
in Topological Quantum Field Theory
and Quantum Cohomology
A dissertation submitted to The Johns Hopkins University
in conformity with the requirements for the degree of
Doctor of Philosophy
We prove that a commutative finitedimensional algebra A is a Frobenius
algebra if and only if it has a cocommutative comultiplication with counit.
Based on this, we prove the onetoone correspondence between topological
quantum field theories and Frobenius algebras, formulated as an equivalence
of monoidal categories. For each Frobenius algebra A we define a canonical
``characteristic class,'' and show that this characteristic class is a unit if and
only if A is semisimple.
In quantum cohomology, the Frobenius algebra characteristic class, the