Open–closed strings: Two-dimensional extended TQFTs and Frobenius algebras

Abstract

We study a special sort of 2-dimensional extended Topological Quantum Field Theories (TQFTs). These are defined on open-closed cobordisms by which we mean smooth compact oriented 2-manifolds with corners that have a particular global structure in order to model the smooth topology of open and closed string worldsheets. We show that the category of open-closed TQFTs is equivalent to the category of knowledgeable Frobenius algebras. A knowledgeable Frobenius algebra (A, C, ı, ı * ) consists of a symmetric Frobenius algebra A, a commutative Frobenius algebra C, and an algebra homomorphism ı : C → A with dual ı * : A → C, subject to some conditions. This result is achieved by providing a description of the category of open-closed cobordisms in terms of generators and the well-known Moore-Segal relations. In order to prove the sufficiency of our relations, we provide a normal form for such cobordisms which is characterized by topological invariants. Starting from an arbitrary such cobordism, we construct a sequence of moves (generalized handle slides and handle cancellations) which transforms the given cobordism into the normal form. Using the generators and relations description of the category of open-closed cobordisms, we show that it is equivalent to the symmetric monoidal category freely generated by a knowledgeable Frobenius algebra. Our formalism is then generalized to the context of open-closed cobordisms with labeled free boundary components, i.e. to open-closed string worldsheets with D-brane labels at their free boundaries. (2000): 57R56, 57M99, 81T40, 58E05, 19D23, 18D35. Mathematics Subject Classification

Cardy Condition for Open-Closed Field Algebras

“…In particular, the left hand side of (3.73) viewed as a morphism in Hom V (V op (V op Remark 5.14. Notice that, in the case V = C, the Cardy C V |C V ⊗V -algebra (using (5.31)) exactly coincides with the usual algebraic formulation of 2-dimensional open-closed topological field theory [La,Mo1,Mo2,Se2,MSeg,AN,LP]. As we discussed in the introduction, we believe that open-closed partial conformal field theories of all genus satisfying the V -invariant boundary condition [Ko2] are classified by Cardy C V |C V ⊗V -algebras.…”

“…An Ishibashi state is a coherent state ψ L. Kong Other surfaces which are not included in S only provide additional compatibility conditions. In 2-d topological field theories, only three additional compatibility conditions are needed to ensure the consistency of a theory of all genus [La,Mo1,Se2,MSeg,AN,LP]. The first compatibility condition says that both V op and V cl are finite dimensional.…”

Uniqueness of open/closed rational CFT with given algebra of open states

“…(ii) What we refer to as physical boundaries ofẊ are called 'boundary sector boundaries' in [21], 'free boundaries' in [22,23], 'coloured boundaries' in [24], and 'constrained boundaries' in [25].…”

“…(iii) WSh is different from the category of open/closed two-dimensional cobordisms considered in the context of two-dimensional open/closed topological field theory in [21,22,23,24,25]. There, objects are disjoint unions of circles and intervals, and morphisms are equivalence classes of cobordisms between these unions.…”

“…Every world sheet X can be obtained by applying sewing to a small collection of fundamental world sheets [37,38,24,39]. In terms of the category WSh this means that for every world sheet X there is a (non-unique) morphism…”