## State-sum construction of two-dimensional open-closed TQFTs

Venue: | In preparation |

Citations: | 11 - 5 self |

### BibTeX

@INPROCEEDINGS{Lauda_state-sumconstruction,

author = {Aaron D. Lauda and Hendryk Pfeiffer},

title = {State-sum construction of two-dimensional open-closed TQFTs},

booktitle = {In preparation},

year = {}

}

### OpenURL

### 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.

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Citation Context ...he example of the groupoid algebra of a finite groupoid. Mathematics Subject Classification (2000): 57R56, 57M99, 81T40, 57Q20. 1 Introduction An n-dimensional Topological Quantum Field Theory (TQFT) =-=[1]-=- is a symmetric monoidal functor from the category nCob of n-dimensional cobordisms to the category Vectk of vector spaces over a given field k. The objects of the category nCob are smooth compact ori... |

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Citation Context ...iption of the category 2Cob ext in terms of generators and relations goes back to work on boundary conformal field theory by Cardy and Lewellen [7,8], Lazaroiu [9], and to the work of Moore and Segal =-=[10]-=- and Alexeevski and Natanzon [11]. We have shown the sufficiency of the relations in [12]. In order to get some intuition for the extended cobordism category 2Cob ext , we 21 INTRODUCTION 3 here disp... |

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Citation Context ...ector space associated with the unit interval. The description of the category 2Cob ext in terms of generators and relations goes back to work on boundary conformal field theory by Cardy and Lewellen =-=[7,8]-=-, Lazaroiu [9], and to the work of Moore and Segal [10] and Alexeevski and Natanzon [11]. We have shown the sufficiency of the relations in [12]. In order to get some intuition for the extended cobord... |

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Citation Context ...e sum, A therefore turns out to be the algebra associated to the open string. The state sum of Fukuma–Hosono–Kawai is also relevant to recent work on boundary conformal field theory, see, for example =-=[13,14]-=- where the algebra A already appears in connection with the boundary conditions, and so the present article is immediately relevant in this context. Another reason for better understanding the topolog... |

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Citation Context ...ing. 9�� �� �� �� �� �� 2 FROBENIUS ALGEBRAS 10 2. µ(e) = 1. A is called strongly separable if A is separable with a separability idempotent that satisfies τA,A(e) = e. Theorem 2.6 (see, for example =-=[29]-=-). Let A be an algebra over any field k. Then the following are equivalent: 1. A is finite-dimensional over k, and the canonical bilinear form is non-degenerate. 2. A is strongly separable. Every stro... |

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Citation Context ...ation is unique up to combinatorial equivalence, i.e. Pachner moves, by the validity of the Combinatorial Triangulation Conjecture and the Hauptvermutung for 2-dimensional manifolds, see, for example =-=[33]-=-. We have then dealt with the corner points ‘by hand’. The other direction, a solution to the smoothing problem, is not needed if one is just interested in a combinatorial construction of open-closed ... |