State sum construction of two-dimensional open-closed Topological Quantum Field Theories
Aaron D. Lauda, Hendryk Pfeiffer
TL;DR
This work extends the classic Fukuma–Hosono–Kawai state-sum construction of 2D TQFTs to open-closed cobordisms by working in the extended category 2Cob ext. The authors show that a strongly separable symmetric Frobenius algebra A underpins a state-sum that yields an open-closed TQFT Z, with the unit interval I mapping to A and the circle S^1 mapping to Z(A) via a knowledgeable Frobenius algebra (A,Z(A),i,i*). They develop the combinatorial framework using triangulations, Pachner moves, and smoothing arguments, proving invariance of Z(M) under moves and boundary triangulations, and they connect the local state-sum data to the global functorial description. A key insight is that the state-sum A encodes the open-string sector while Z(S^1) encodes the closed-string sector, and the structure (A,C,i,i*) clarifies how D-brane data and center-like structures arise in open-closed theories. The paper also provides explicit examples, notably the groupoid algebra, to illustrate D-brane labeling and the interpretation of Z(A) in topological terms. Overall, the work links the local, algebraic state-sum data to the global open-closed TQFT, advancing the understanding of how higher-level topological quantum field theories can be constructed from algebraic inputs. It also clarifies the role of A as Z(I) and offers pathways toward higher-dimensional/state-sum generalizations via categorification ideas linked to the 3D Turaev–Viro framework and beyond.
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.