State Sum Construction of Two-Dimensional Open-Closed Topological Quantum Field Theories

Abstract

We present a state sum construction of two-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, which generalizes the state sum of Fukuma-Hosono-Kawai from triangulations of conventional two-dimensional cobordisms to those of open-closed cobordisms, i.e. smooth compact oriented 2-manifolds with corners that have a particular global structure. This construction reveals the topological interpretation of the associative algebra on which the state sum is based, as the vector space that the TQFT assigns to the unit interval. Extending the notion of a two-dimensional TQFT from cobordisms to suitable manifolds with corners therefore makes the relationship between the global description of the TQFT in terms of a functor into the category of vector spaces and the local description in terms of a state sum fully transparent. We also illustrate the state sum construction of an open-closed TQFT with a finite set of D-branes using the example of the groupoid algebra of a finite groupoid.Mathematics Subject Classification (2000): 57R56, 57M99, 81T40, 57Q20.

Topological higher gauge theory: From BF to BFCG theory

“…Note that the transformations (24) and (25) are chosen such as to make the action gauge invariant in view of the transformations (22) and (23). Similar to the traditional BF action in (1), the BF CG action (26) also exhibits an extended local symmetry, in the sense that it is also invariant under the additional infinitesimal gauge transformations:…”

Finitely semisimple spherical categories and modular categories are self-dual

“…This construction would categorify the TQFTs of [16], based on the idea that the double of a monoidal category categorifies the notion of the center of an algebra. We are interested in the self-duality of our spherical categories because of the following observation.…”

Open-Closed TQFTS Extend Khovanov Homology From Links to Tangles

“…The answer to the second question, namely to find an algebraic operation that represents the composition of tangles, is inspired by our recent work [17] (1)), C = Z((0)), etc., for which A is strongly separable 5 and for which C is the centre of A. If the algebra A is strongly separable, then the element…”

“…We would like to glue the two open-cobordisms along their coloured boundaries like this, , ( 20) in order to obtain a cobordism from the 0-smoothing (É) of (È) to the 1-smoothing (Ê) of (È We can now employ the state sum construction presented in [17] This gives the chain complex 0 → A ⊗ C → A → 0 which relates the two smoothings (É) and (Ê) of the composite tangle diagram (È). We now present an equivalent way of computing the linear map A ⊗ C → A which makes transparent how this linear map can be computed from the two constituents of (1.19).…”

“…Given an Euler-graded [Euler-filtered] open-closed TQFT Z, one simply computes (2.37) for the generators of (1.12) in order to obtain (2.38). Triangulations of the generators are displayed, for example, in equation (4.15) of [17].…”