## Homotopy quantum field theories and the homotopy cobordism category in dimension 1+1

Citations: | 16 - 0 self |

### BibTeX

@MISC{Rodrigues_homotopyquantum,

author = {Gonçalo Rodrigues},

title = {Homotopy quantum field theories and the homotopy cobordism category in dimension 1+1},

year = {}

}

### OpenURL

### Abstract

Abstract. We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n, X) such that an HQFT is a functor

### Citations

515 |
Topological quantum field theory
- Witten
- 1988
(Show Context)
Citation Context ...ough intersection theory in the Instanton moduli space, were obtained, at least formally, as correlation functions for a quantum field theory with a twisted supersymmetric topological lagrangian (see =-=[Wit88]-=-). This work was done with the support of the grant PRAXIS XXI/BD/17226/98 from Fundação para a Ciência e Tecnologia. This work was also supported by the programme Programa Operacional “Ciência, Tecno... |

250 |
Braided tensor categories
- Joyal, Street
- 1993
(Show Context)
Citation Context ...eans that there are natural isomorphisms c (M,g),(N,h) : τ((M, g) ∐ (N, h)) ∼ = τ(M, g) ⊗ τ(N, h) and an isomorphism u : τ(∅) ∼ = R that satisfy the usual axioms for a symmetric monoidal functor (see =-=[JS93]-=- for the actual diagrams).�� �� � �� �� HOMOTOPY QUANTUM FIELD THEORIES . .. 5 (3) For X-cobordisms (W, F) : (M, g) −→ (N, h) and (V, G) : (N ′ , h ′ ) −→ (P, j) glued along ψ : (N, h) −→ (N ′ , h ′ ... |

158 |
Categories for the Working Mathematician. Graduate Texts in Mathematics
- MacLane
- 1998
(Show Context)
Citation Context ...sociators ∆W,V,U, use Maclane’s coherence theorem to show that it is thin and then apply the above constructionlemma. Unfortunately the coherence theorem cannot be applied as is because, and we quote =-=[Mac71]-=-, “two apparently or formally different vertices of such a diagram might become equal in a particular monoidal category, in such a way as to spoil commutativity.” So we have to resort to some formal t... |

145 | Higher-dimensional algebra and topological quantum field theory
- Baez, Dolan
- 1995
(Show Context)
Citation Context ...isfying the additional condition τ(f † ) = (τf) † is called a Unitary homotopy quantum field theory. In this paper we will not discuss such unitary field theories and instead urge the reader to go to =-=[BD95]-=- to see their relevance in physics. Now let Ψ : X −→ Y be a continuous map. If (M, g) is an X-manifold then (M, gΨ) is a Y -manifold. In the same way, if (W, F) : (M, g) −→ (N, h) is an Xcobordism the... |

131 | Relativistic spin networks and quantum gravity. J.Math.Phys
- Barrett, Crane
- 1998
(Show Context)
Citation Context ...racional “Ciência, Tecnologia, Inovação” (POCTI) of the Fundação para a Ciência e Tecnologia (FCT), cofinanced by the European Community fund FEDER. 12 GONÇALO RODRIGUES Also on the physics side, in =-=[BC98]-=- J. Barrett and L. Crane gave a model for euclidean quantum gravity, by starting with a state sum model for a TQFT using the monoidal category of representations of SO(4) and then imposing a constrain... |

122 | Topological gauge theories and group cohomology
- Dijkgraaf, Witten
- 1990
(Show Context)
Citation Context ...ls for these HQFT’s. Introduction. The subject of Topological Quantum Field Theories is well established by now. Very early, homotopy theory methods were used to build examples of TQFT’s. We can cite =-=[DW90]-=- and the works of D. Yetter, [Yet92] and [Yet93] (see also their reformulation and generalization by T. Porter, [Por96] and [Por98]). Essentially, what all these theories do is fix a background space ... |

116 |
Differential Topology. Graduate Texts
- Hirsch
- 1976
(Show Context)
Citation Context ... equal to a cylinder cobordism. Proof. For the implication ⇐=, note that if W is a cobordism with a Morse function f that has no critical values then by the cylinder recognition theorem (see the book =-=[Hir76]-=-, page 153) there is a diffeomorphism Ψ : M ×I −→ W such that the triangle in figure 6 is commutative.HOMOTOPY QUANTUM FIELD THEORIES . .. 9 M × I� ���� � πI ���� � Ψ �� W �� I Figure 6. Since Ψ is t... |

78 |
Braided compact closed categories with applications to low-dimensional topology
- Freyd, Yetter
- 1989
(Show Context)
Citation Context ...egory with duals. There are various definitions of what could be a category with duals, with varying degrees of laxness built into them, but the definition more suitable to our purposes is the one in =-=[FY89]-=-. In our discussion of duality we will make several statements whose proof is omitted. To see those proofs, as well as other details on the subject of duality in categories, the reader can look at, fo... |

59 | Two-dimensional topological quantum field theories and Frobenius algebras
- Abrams
(Show Context)
Citation Context ...asic morphisms that generate the whole category HCobord(1, K(G, 2)). It is well-known that the pants P−−+ : S 1 + ∐ S1 + −→ S1 + the disk D+ and the cylinder B make S 1 + into a Frobenius object (see =-=[Abr96]-=- for example). The new structure is related to the cylinders C−+(g) which are endomorphisms S 1 + −→ S 1 +. An application of theorem 2.2 shows that the assignment g ↦−→ C−+(g) gives a homomorphism G ... |

55 | Holonomy and parallel transport for abelian gerbes
- Mackaay, Picken
(Show Context)
Citation Context ...e higher-dimensional fundamental groupoids into the (generalized) center of these categories. Besides going up the dimensional ladder, we wish to mention two further avenues of research. In the paper =-=[MP]-=- the authors have shown that when X is simply connected, gerbes are essentially the same thing as morphisms from the thin fundamental group of the loop space into the structure group of the gerbe. Thi... |

36 |
Lattice topological field theory in two dimensions, Cornell preprint
- Fukuma, Hosono, et al.
(Show Context)
Citation Context ...ne Frobenius object. 3. State-sum models for 2d-HQFT’s We want to modify the state sum models for a TQFT by incorporating homotopical data. To do that we will in the first place review the results in =-=[FHK94]-=- since they are a necessary springboard for our computations. Let (M, T) be a closed triangulated surface. We will work with the dual graph T ∗ of the triangulation T, and instead of explaining what t... |

33 | Homotopy field theory in dimension 2 and group-algebras, preprint arXiv:math.QA/9910010
- Turaev
(Show Context)
Citation Context ...a complete characterization of HCobord(n, X) for n = 1 (or the 1 + 1 case) and X the Eilenberg-Maclane space K(G,2). In the final section we derive state sum models for these HQFT’s. Introduction. In =-=[Tur]-=-, V. Turaev introduced Homotopy quantum field theories (HQFT) in dimension n + 1 as Topological quantum field theories (TQFT) for n-manifolds and (n + 1)-cobordisms with a continuous map into some tar... |

32 | Spherical 2-categories and 4-manifold invariants, Adv
- Mackaay
- 1999
(Show Context)
Citation Context ...t in three dimensions topological state sums are built from certain monoidal categories (see for example [Tur94]) and that in four dimensions this role is played by certain monoidal 2-categories (see =-=[Mac99]-=-). To enrich these state sum models to homotopical ones we expect that what is needed is a functor from the higher-dimensional fundamental groupoids into the (generalized) center of these categories. ... |

28 |
Quantum invariants of knots and 3-manifolds
- Turaev
- 1994
(Show Context)
Citation Context ... make several statements whose proof is omitted. To see those proofs, as well as other details on the subject of duality in categories, the reader can look at, for example, [JS93], [BW99], [FY92] and =-=[Tur94]-=-, besides the reference cited above. Definition 1.5. A monoidal category (A, ⊗, K) has a duality structure (or is a category with duals) iff for each object a there is an object a ∗ , the dual object,... |

27 | Geometry of Deligne cohomology
- Gajer
(Show Context)
Citation Context ...n of wether it is possible to define Thin Homotopy Quantum Field theories, by using thin homotopy classes of maps. The second avenue is that as there is a classifying space BG of G-bundles, P. Gajer (=-=[GA97]-=-) has introduced a classifying space for (abelian) gerbes. Starting with these classifying spaces, the constructions made in [Qui] for TQFT’s should be generalized to HQFT’s. Appendix: The category HC... |

24 |
Coherence theorems via knot theory
- Freyd, Yetter
- 1992
(Show Context)
Citation Context ...ity we will make several statements whose proof is omitted. To see those proofs, as well as other details on the subject of duality in categories, the reader can look at, for example, [JS93], [BW99], =-=[FY92]-=- and [Tur94], besides the reference cited above. Definition 1.5. A monoidal category (A, ⊗, K) has a duality structure (or is a category with duals) iff for each object a there is an object a ∗ , the ... |

23 |
Topological quantum field theories associated to finite groups and crossed G-sets
- Yetter
- 1992
(Show Context)
Citation Context ...The subject of Topological Quantum Field Theories is well established by now. Very early, homotopy theory methods were used to build examples of TQFT’s. We can cite [DW90] and the works of D. Yetter, =-=[Yet92]-=- and [Yet93] (see also their reformulation and generalization by T. Porter, [Por96] and [Por98]). Essentially, what all these theories do is fix a background space X and compute a weighted sum over ho... |

19 | Representations of the homotopy surface category of a simply connected space
- Brightwell, Turner
- 2000
(Show Context)
Citation Context ...obordisms with a continuous map into some target space X. One of the important theorems in [Tur] was that the HQFT’s only depended on the n-homotopy type of X. Around the same time appeared the paper =-=[BT]-=- which discussed HQFT’s in dimension 1+1, but for a simply connected target space, and therefore are not covered by V. Turaev’s definition. Here we present a new and broader definition of a Homotopy q... |

16 |
TQFTs from homotopy 2-types
- Yetter
- 1993
(Show Context)
Citation Context ...of Topological Quantum Field Theories is well established by now. Very early, homotopy theory methods were used to build examples of TQFT’s. We can cite [DW90] and the works of D. Yetter, [Yet92] and =-=[Yet93]-=- (see also their reformulation and generalization by T. Porter, [Por96] and [Por98]). Essentially, what all these theories do is fix a background space X and compute a weighted sum over homotopy class... |

11 | Interpretations of Yetter’s notion of G-coloring: simplicial fibre bundles and non-abelian cohomology
- Porter
- 1996
(Show Context)
Citation Context ...early, homotopy theory methods were used to build examples of TQFT’s. We can cite [DW90] and the works of D. Yetter, [Yet92] and [Yet93] (see also their reformulation and generalization by T. Porter, =-=[Por96]-=- and [Por98]). Essentially, what all these theories do is fix a background space X and compute a weighted sum over homotopy classes of maps f : M −→ X for a closed manifold M. In all these cases the h... |

10 | Non–Abelian Gerbes from Strings on a Branched Space–Time, e–Print arXiv: hep–th/9910048
- Kalkkinen
(Show Context)
Citation Context ...In other words topological matter in 1 + 1 dimensions is the same thing as a gerbe in X. It is very interesting that gerbes are also appearing in string theory and related subjects, see, for example, =-=[Kal]-=- and [Zun]. In higher dimensions we expect to observe the same categorification pattern already recognized in TQFT’s. It is well known that in three dimensions topological state sums are built from ce... |

4 | TQFT’s from Homotopy n-Types
- Porter
(Show Context)
Citation Context ...opy theory methods were used to build examples of TQFT’s. We can cite [DW90] and the works of D. Yetter, [Yet92] and [Yet93] (see also their reformulation and generalization by T. Porter, [Por96] and =-=[Por98]-=-). Essentially, what all these theories do is fix a background space X and compute a weighted sum over homotopy classes of maps f : M −→ X for a closed manifold M. In all these cases the homotopy info... |

2 |
Group categories and their field theories, Preprint math.GT/9811047
- Quinn
(Show Context)
Citation Context ...e is that as there is a classifying space BG of G-bundles, P. Gajer ([GA97]) has introduced a classifying space for (abelian) gerbes. Starting with these classifying spaces, the constructions made in =-=[Qui]-=- for TQFT’s should be generalized to HQFT’s. Appendix: The category HCobord(n, X) In this appendix we give the details of the construction of the category of homotopy cobordisms HCobord(n, X). This co... |

1 |
p-gerbes and extended objects in string theory, Preprint hep-th/0002074. Centro de Matemática Aplicada, Departamento de
- Zunger
(Show Context)
Citation Context ...ords topological matter in 1 + 1 dimensions is the same thing as a gerbe in X. It is very interesting that gerbes are also appearing in string theory and related subjects, see, for example, [Kal] and =-=[Zun]-=-. In higher dimensions we expect to observe the same categorification pattern already recognized in TQFT’s. It is well known that in three dimensions topological state sums are built from certain mono... |