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Two-dimensional topological quantum field theories and Frobenius algebras. (English) Zbl 0897.57015

The relationship between two-dimensional (2-D) topological quantum field theories and Frobenius algebras \(A\) (characterized as algebras having a comultiplication which is a map of \(A\)-modules) is formulated as an equivalence of categories. A classification of the indecomposable 2-D topological quantum field theories as a result of classifying of the indecomposable Frobenius algebras (either ‘annihilator algebras’ whose socle is a principal ideal or field extensions). Eight examples are considered in the conclusion.

MSC:

57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
57R90 Other types of cobordism
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