Frobenius Algebras and 2D Topological Quantum Field Theories
An n-dimensional topological quantum field theory (TQFT) is a functorial rule that assigns to each (n-1)-dimensional closed manifold a vector space (the “state space”) and to each n-dimensional cobordism between (n-1)-dimensional manifolds a homomorphism of these state spaces (the “propagator”). Recall also that a Frobenius algebra is an associative algebra carrying a non-degenerate invariant bilinear form. The present book is a self-contained exposition of one of the most fundamental results in this field saying that there is a one-to-one correspondence between 2-dimensional TQFT's and Frobenius algebras. The author also demonstrates that this equivalence follows from the fact that the category of two-dimensional cobordisms is the free monoidal category containing a commutative Frobenius object. The book is very well written and organized. I warmly recommend it as an introduction to basic techniques of algebraic geometry. It requires only a preliminary knowledge of algebra, category theory and geometry.