Abstract
We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on atwin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra(C,W,z,z∗)consists of a commutative Frobenius algebraC, a symmetric Frobenius algebraW, and an algebra homomorphismz:C→Wwith dualz∗:W→C, satisfying some extra conditions. We also introduce a generalized 2-dimensional Topological Quantum Field Theory defined on singular 2-dimensional cobordisms and show that it is equivalent to a twin Frobenius algebra in a symmetric monoidal category.
Highlights
A 2-dimensional Topological Quantum Field Theory (TQFT) is a symmetric monoidal functor from the category 2Cob of 2-dimensional cobordisms to the category Vectk of vector spaces over a field k
TQFTs defined on open-closed cobordisms. These cobordisms are certain smooth oriented 2-manifolds with corners that can be viewed as cobordisms between compact 1-manifolds with boundary, that is, between disjoint unions of circles S1 and unit intervals I = [0, 1]
Lauda and Pfeiffer showed that open-closed TQFTs are characterized by what they call knowledgeable Frobenius algebras (A, C, ι, ι∗), where the vector space C := Z(S1) associated with the circle has the structure of a commutative Frobenius algebra, the vector space A := Z(I) associated with the interval has the structure of a symmetric Frobenius algebra, and there are linear maps ι : C → A and ι∗ : A → C satisfying certain conditions
Summary
A 2-dimensional Topological Quantum Field Theory (TQFT) is a symmetric monoidal functor from the category 2Cob of 2-dimensional cobordisms to the category Vectk of vector spaces over a field k. Lauda and Pfeiffer showed that open-closed TQFTs are characterized by what they call knowledgeable Frobenius algebras (A, C, ι, ι∗), where the vector space C := Z(S1) associated with the circle has the structure of a commutative Frobenius algebra, the vector space A := Z(I) associated with the interval has the structure of a symmetric Frobenius algebra, and there are linear maps ι : C → A and ι∗ : A → C satisfying certain conditions This result was obtained by providing a description of the category of open-closed cobordisms in terms of generators and the Moore-Segal relations. They defined a normal form for such cobordisms, characterized by topological invariants, and proved the sufficiency of the relations by constructing a sequence of moves which transforms the given cobordism into the normal form They showed that the category 2Cobext of open-closed cobordisms is equivalent to the symmetric monoidal category freely generated by a knowledgeable Frobenius algebra.
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