## 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.

### Citations

186 |
Introduction to piecewise-linear topology, volume 69 of Ergebnisse der Mathematik und ihrer Grenzgebiete
- Rourke, Sanderson
- 1972
(Show Context)
Citation Context ...sely the equivalence classes of the open-closed cobordisms depicted in (1.3). 3.2 Combinatorial open-closed cobordisms Open-closed cobordisms can be triangulated as follows. We use the terminology of =-=[32]-=-. Given an open-closed cobordism M, the underlying topological manifold is a compact oriented 2-manifold with boundary. We therefore have a finite simplicial complex K whose underlying polyhedron we d... |

182 |
State sum invariants of 3-manifolds and quantum 6j-symbols, Topology 31
- Turaev, Viro
- 1992
(Show Context)
Citation Context ...rder if not impossible. On the other hand, there are some generalizations of the state sum construction of Fukuma– Hosono–Kawai to higher dimensions, notably the 3-dimensional TQFT of Turaev and Viro =-=[15]-=-, extended by Barrett and Westbury [16], which produces a 3-dimensional TQFT for any given modular category or, more generally, for suitable spherical categories [17]. The step from dimension 2 to 3, ... |

144 |
The geometry of tensor calculus
- Joyal, Street
- 1991
(Show Context)
Citation Context ...Y ), the morphism, f ∗ is called the dual of f. := λX∗ ◦ (evY ⊗ idX∗) ◦ ((idY ∗ ⊗ f) ⊗ idX∗) ◦α −1 Y ∗ ,X,X ∗ ◦ (idY ∗ ⊗ coevX) ◦ ρ −1 Y ∗ : Y ∗ → X ∗ , (2.4) In the following, we use string diagrams =-=[25,26]-=- to visualize morphisms of a given symmetric monoidal category C and the identities between them. The diagrams are read from top to bottom. For each object X ∈ |C|, the identity morphism idX is denote... |

101 |
Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases
- Crane, Frenkel
- 1994
(Show Context)
Citation Context ...e of categorification which means replacing algebraic structures based on sets and maps by analogues that are rather based on categories and functors [18]. The dimensional ladder of Crane and Frenkel =-=[19]-=- sketches which sort of algebraic structures one would need in order to 3 It turns out that for a field of arbitrary characteristic, the appropriate class of algebras is that of the strongly separable... |

85 |
Bulk and boundary operators in conformal field theory, Phys
- Cardy, Lewellen
- 1991
(Show Context)
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... |

77 |
Frobenius Algebras and 2D Topological Quantum Field Theories
- Kock
- 2003
(Show Context)
Citation Context ...and corresponds to the field k. For n = 2, the category nCob is well understood, and so there are strong results about 2dimensional TQFTs. For these classic results, we refer to [2–4] and to the book =-=[5]-=-. It is known, for example, that 2-dimensional TQFTs are characterized by commutative Frobenius algebras. The objects of 2Cob are compact 1-manifolds without boundary, i.e. disjoint unions of circles ... |

63 |
PL homeomorphic manifolds are equivalent by elementary shellings
- Pachner
- 1991
(Show Context)
Citation Context ...om which one can construct n-dimensional state sum TQFTs for generic n. Whereas the algebraic structures of (1.4) that are relevant to the state sum construction, are closely related to Pachner moves =-=[20]-=- and to the coherence conditions in higher categories, they have no obvious relationship to the global description of the TQFT as a functor Z : nCob → Vectk. By showing that the associative algebra A ... |

59 | Two-dimensional topological quantum field theories and Frobenius algebras - Abrams - 1996 |

50 | Category theory for conformal boundary conditions
- Fuchs, Schweigert
(Show Context)
Citation Context ... C is called symmetric if ε ◦ µ = ε ◦ µ ◦ τ. (2.19) It is called commutative if µ = µ ◦ τ. (2.20) 5. Let C be locally small. A Frobenius algebra object (A, µ, η,∆, ε) in C is called special (also see =-=[28]-=-) if ε ◦ η = ξ · id and µ ◦ ∆ = ξA · idA. (2.21) for some ξ , ξA ∈ Hom( , ) that are invertible in the monoid Hom( , ). The string diagrams for the operations of a Frobenius algebra (A, µ, η,∆, ε) are... |

48 | Yetter D.: On algebraic structures implicit in topological quantum field theories, preprint - Crane - 1994 |

40 |
Invariants of piecewise-linear 3-manifolds, hep-th
- Barrett, Westbury
- 1996
(Show Context)
Citation Context ...nd, there are some generalizations of the state sum construction of Fukuma– Hosono–Kawai to higher dimensions, notably the 3-dimensional TQFT of Turaev and Viro [15], extended by Barrett and Westbury =-=[16]-=-, which produces a 3-dimensional TQFT for any given modular category or, more generally, for suitable spherical categories [17]. The step from dimension 2 to 3, i.e. from the state sum of Fukuma–Hoson... |

39 | Involutory Hopf algebras and 3-manifold invariants
- Kuperberg
- 1991
(Show Context)
Citation Context ...ence for such a relationship is provided by the Hopf algebra object in 3-dimensional extended TQFTs [21–23] in connection with Kuperberg’s 3-manifold invariant which is based on certain Hopf algebras =-=[24]-=-. In the present article, we consider Frobenius algebras not only in the category Vectk of vector spaces, but in any symmetric monoidal Abelian category C. This extends our results without any additio... |

37 |
Lattice Topological Field Theory in two Dimensions, Commun.Math.Phys
- Fukuma, Hosono, et al.
- 1994
(Show Context)
Citation Context ...pace C associated with the circle, the linear maps µ, η, ∆, and ε associated with the generators (1.1), and in terms of the relations among the morphisms of 2Cob. The state sum of Fukuma–Hosono–Kawai =-=[6]-=- forms a different and a priori independent way of defining a 2-dimensional TQFT. This construction starts with a finite-dimensional semisimple algebra (A, µ, η) over a field k of characteristic zero.... |

29 |
Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras,” math.at/0510664
- Lauda, Pfeiffer
(Show Context)
Citation Context ... 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 display the generators for its morphisms: µA ∆A ηA εA µC ∆C ηC εC ı ı ∗ An open-closed TQFT i... |

28 | A geometric approach to two dimensional conformal field theory - Dijkgraaf - 1989 |

27 |
of Categorical Algebra I. Basic Category Theory. Cambridge Univ
- BORCEUX
- 1994
(Show Context)
Citation Context ... closed to a circle, i.e. it is closely related with the generators ı and ı∗ of (1.3). The image of an idempotent can be defined in any Abelian category as follows. Proposition 2.21 (see, for example =-=[31]-=-). Let C be an Abelian category and p: A → A be an idempotent. The image factorization of p yields an object p(A), called the image of p, which is unique up to isomorphism, together with morphisms coi... |

24 | Spherical categories
- Barrett, Westbury
- 1999
(Show Context)
Citation Context ...mensional TQFT of Turaev and Viro [15], extended by Barrett and Westbury [16], which produces a 3-dimensional TQFT for any given modular category or, more generally, for suitable spherical categories =-=[17]-=-. The step from dimension 2 to 3, i.e. from the state sum of Fukuma–Hosono–Kawai to that of Turaev–Viro, can be understood as an example of categorification which means replacing algebraic structures ... |

22 |
Associative Algebras. Graduate texts in mathematics
- Pierce
- 1982
(Show Context)
Citation Context ...such result for the more general case of modules over a commutative ring. In order to illustrate how strong the condition of strong separability is, we include the following results and examples from =-=[29,30]-=-. Theorem 2.9. Let A be an algebra over some field k. 1. If A is strongly separable, then A is finite-dimensional, separable, and semisimple. 2. If A is separable and commutative, then A is strongly s... |

17 |
Low-dimensional topology and higher-order categories
- Street
(Show Context)
Citation Context ...Y ), the morphism, f ∗ is called the dual of f. := λX∗ ◦ (evY ⊗ idX∗) ◦ ((idY ∗ ⊗ f) ⊗ idX∗) ◦α −1 Y ∗ ,X,X ∗ ◦ (idY ∗ ⊗ coevX) ◦ ρ −1 Y ∗ : Y ∗ → X ∗ , (2.4) In the following, we use string diagrams =-=[25,26]-=- to visualize morphisms of a given symmetric monoidal category C and the identities between them. The diagrams are read from top to bottom. For each object X ∈ |C|, the identity morphism idX is denote... |

12 |
The topology of the category of open and closed strings, Recent developments in algebraic topology
- Baas, Cohen, et al.
- 2006
(Show Context)
Citation Context ...ected open-closed cobordisms are determined up to orientationpreserving diffeomorphism preserving the black boundary by a set of topological invariants defined in the work of Baas, Cohen, and Ramírez =-=[34]-=-. These topological invariants are the genus (defined as the genus of the underlying topological 2-manifold), the window number, defined as the number of components of ∂1M diffeomorphic to S1 , and th... |

11 |
Natural associativity and commutativity, Rice Univ
- MacLane
- 1963
(Show Context)
Citation Context ...f ⊗ g = f ⊗ g = f g . (2.7) �� Y1 ⊗ Y2 �� Y1 �� Y2 5�� �� �� � 2 FROBENIUS ALGEBRAS 6 The symmetric braiding is denoted by, X �� Y τX,Y = . (2.8) Mac Lane’s coherence theorem for monoidal categories =-=[27]-=- then ensures that one can unambiguously translate any such string diagram into a morphism of C. One therefore chooses parentheses for all tensor products that occur in the source and target objects o... |

8 |
Atiyah: Topological quantum field theories
- F
- 1988
(Show Context)
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... |

8 | Noncommutative two-dimensional topological field theories and Hurwitz numbers for real algebraic curves
- Alexeevski, Natanzon
(Show Context)
Citation Context ...n 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 display the generators for its morphi... |

7 | Towards an algebraic characterization of 3-dimensional cobordisms - Kerler - 2003 |

5 |
Lectures on branes, K-theory and RR charges, in Lecture notes from the Clay Institute School on Geometry and String Theory held at the Isaac Newton Institute
- Moore, Segal
- 2001
(Show Context)
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... |

3 | Sawin: Direct sum decompositions and indecomposable TQFTs - F - 1995 |

3 | Conformal correlation functions, Frobenius algebras and triangulations - Schweigert |

2 |
Lewellen: Sewing constraints for conformal field theories on surfaces with boundaries
- C
- 1992
(Show Context)
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... |

2 |
Lazaroiu: On the structure of open-closed topological field theory in two dimensions
- I
(Show Context)
Citation Context ...ciated 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 cobordism category 2... |

1 |
Topological and conformal field theory as Frobenius algebras
- Schweigert
- 2005
(Show Context)
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... |

1 |
2d and 3d TQFTs and higher-dimensional algebra. Poster at the conference Loops ’05
- Morton
- 2005
(Show Context)
Citation Context ...hat of Turaev–Viro, can be understood as an example of categorification which means replacing algebraic structures based on sets and maps by analogues that are rather based on categories and functors =-=[18]-=-. The dimensional ladder of Crane and Frenkel [19] sketches which sort of algebraic structures one would need in order to 3 It turns out that for a field of arbitrary characteristic, the appropriate c... |

1 | Portrait of the handle as a Hopf algebra - Yetter - 1997 |

1 |
A note on strongly separable algebras
- Aguiar
(Show Context)
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... |

1 |
Ranicki (ed.): The Hauptvermutung Book. K-Monographs in Mathematics 1
- A
- 1996
(Show Context)
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 ... |