Remarks on 2-dimensional HQFTs
An \(X\)-space \((M, f)\) consists of a closed oriented manifold \(M\) and a map \(f: M\to X\). The second author introduced the concept of \(X\)-Homotopy Quantum Field Theory (\(X\)-HQFT). A \(d\)-dimensional \(X\)-HQFT is basically a functor from the category of \(d\)-dimensional \(X\)-spaces and \(d+1\)-dimensional \(X\)-cobordisms to the category of vector spaces over a field. An \(X\)-HQFT reduces to a Topological Quantum Field Theory when the target space \(X\) is contractible. In the paper under review the authors define twisted Frobenius algebras (TF-algebras) and show that one can construct a TF-algebra from every 2-dimensional HQFT, unique up to coboundary equivalence. The authors expect that ``every TF-algebra can be realized by a 2-dimensional HQFT, which would yield a complete algebraic characterization of 2-dimensional HQFTs''. The last section computes the underlying TF-algebras of the cohomological 2-dimensional HQFTs.
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