## Extended TQFT’s and Quantum Gravity (2007)

Citations: | 2 - 1 self |

### BibTeX

@TECHREPORT{Morton07extendedtqft’s,

author = {Jeffrey Morton},

title = {Extended TQFT’s and Quantum Gravity},

institution = {},

year = {2007}

}

### OpenURL

### Abstract

Abstract. This paper gives a definition of an extended topological quantum field theory (TQFT) as a weak 2-functor Z: nCob2→2Vect, by analogy with the description of a TQFT as a functor Z: nCob→Vect. We also show how to obtain such a theory from any finite group G. This theory is related to a topological gauge theory, the Dijkgraaf-Witten model. To give this definition rigorously, we first define a bicategory of cobordisms between cobordisms. We also give some explicit description of a higher-categorical version of Vect, denoted 2Vect, a bicategory of 2-vector spaces. Along the way, we prove several results showing how to construct 2-vector spaces of Vect-valued presheaves on certain kinds of groupoids. In particular, we use the case when these are groupoids whose objects are connections, and whose morphisms are gauge transformations, on the manifolds on which the extended TQFT is to be defined. On cobordisms between these manifolds, we show how a construction of “pullback and pushforward ” of presheaves gives both the morphisms and 2-morphisms in 2Vect for the extended TQFT, and that these

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Citation Context ...ut with weakened axioms, just as bicategories were defined by weakening those for a category. The concept of a “weak double category” has been defined (for instance, see Marco Grandis and Robert Paré =-=[45]-=-, and Martins-Ferreira’s [71] discussion of them as “pseudocategories”). Thomas Fiore [35] describes these as “Pseudo Double Categories”, arising by “categorification” of the theory of categories, and... |

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Citation Context ...ased not on connections, but on 2-connections. There is extensive work on this topic, but a good overview is the discussion by Baez and Schreiber [12] (see also the definition of 2-bundles by Bartels =-=[14]-=-). The extension of the Dijkgraaf-Witten model to categorical groups is discussed in a somewhat different framework by Martins and Porter [70]. An extension of these ideas to quantum groups is less we... |

11 |
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Citation Context ...” (i.e. the punctures in space are 1-dimensional manifolds, namely circles, and in “spacetime” are 2-dimensional, namely “worldsheets”) the dynamics for such matter has been studied by Baez and Perez =-=[11]-=-. In94 JEFFREY MORTON terms of our extended TQFT setting, the dynamics are described by the action of ZG on cobordisms of cobordisms. In particular, suppose we have a cobordism with corners M : S → S... |

11 |
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8 |
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8 |
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8 |
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6 |
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Citation Context ...mple, Freidel, Livine, and Louapré [40] [41][39], discussing the Ponzano-Regge model coupled to matter, by Noui [74], and Noui and Perez [75] on 3D quantum gravity with matter). Baez, Crans, and Wise =-=[7]-=- describe how conjugacy classes of gauge groups can be construed as “particle types”: an “elementary” particle corresponds with an irreducible 2-vector in ZG(B). This associates to a hole—whose bounda... |

5 | The Impact of Thom’s Cobordism Theory
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Citation Context ...uitable for our needs is found, e.g. in Hirsch [47]. There is substantial research on many questions in, and applications of, cobordism theory: a brief survey of some has been given by Michael Atiyah =-=[3]-=-. Some further examples related to our motivation here include Khovanov homology [55] (also discussed in [13] and [52]), and Turaev’s recent work on cobordism of knots on surfaces [84].EXTENDED TQFT’... |

4 |
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Citation Context ...of coproduct (disjoint union) and product (cartesian product), which have purely arrow-based descriptions. For some further background on the concept of categorification, see work by Crane and Yetter =-=[26]-=-, or Baez and Dolan [9]. So what we study here are categorified topological quantum field theories (TQFT’s). The program of applying categorical notions to field theories was apparently first describe... |

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Citation Context ...ives rise to a bicategory of spans (or cospans). This relies in part on the fact that the pullback is a universal construction (universal properties of Span(C) are discussed by Dawson, Paré and Pronk =-=[29]-=-). Remark 1. [16], ex. 2.6 Given any category C with all limits, there is a bicategory Span(C), whose objects are the objects of C, whose hom-sets of morphisms Span(C)(X1, X2) consist of all spans bet... |

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Citation Context ...f(x) This is just the canonical map into the colimit. Now, in Section 6.3 we discuss a generalization of 2-vector spaces based on the fields of measurable Hilbert spaces discussed by Crane and Yetter =-=[28]-=-. This generalization has much in common with KV 2-vector spaces, but corresponds to infinite dimensional Hilbert spaces in the way that they correspond to the finite dimensional case. 6.3. 2-Hilbert ... |

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Citation Context ...s there is a categorified analog of these operations, as wel shall see. Indeed, there are different sensible generalizations of vector space even within this second philosophy, however. Josep Elgueta =-=[34]-=- shows several different types of “generalized” 2-vector spaces, and relationships among them. In particular, while KV 2-vector spaces can be thought of as having a set of basis elements, a generalize... |

2 |
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Citation Context ...nion) and product (cartesian product), which have purely arrow-based descriptions. For some further background on the concept of categorification, see work by Crane and Yetter [26], or Baez and Dolan =-=[9]-=-. So what we study here are categorified topological quantum field theories (TQFT’s). The program of applying categorical notions to field theories was apparently first described by Dan Freed [37], wh... |

2 |
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Citation Context ... across the second. Note, however, that we do not expect this to be invertible. When it is, the square is said to satisfy the Beck-Chevalley (BC) condition. This is discussed by Bénabou and Streicher =-=[17]-=-, MacLane and Moerdijk [67], and by Dawson, Paré and Pronk [29]. Remark 15. It is useful to consider a description of the two functors between which we have found this natural isomorphism βS,S ′—namel... |

2 |
Les Algeèbres d’Opérateurs dans l’Espace Hilbertien (Algèbres de von
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Citation Context ...omewhat simplified definition should suffice for our later discussion, since we return to these ideas only briefly in Chapter 8. The construction of fields of Hilbert spaces is due to Jacques Dixmier =-=[31]-=-, although he described them, not as categories, but merely as Hilbert spaces with a particular decomposition in terms of the measurable space X. As with L2 spaces, to get what we will call a 2-Hilber... |

1 |
ways to quantize (2+1)-dimensional gravity. U.C. Davis preprint UCD-93-15 and Santa Barbara ITP preprint NSF-ITP-93-63
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(Show Context)
Citation Context ... quantum gravity, since in that case, gravity is a purely topological theory. (For more background on 3D quantum gravity, particularly in the case of signature (2, 1), see work by Steven Carlip [22], =-=[21]-=-). However, in 4 dimensions, a theory of flat connections does not describe gravity, but rather a limiting case of Einsteinian gravity as Newton’s constant G → 0. The subject of this limit, and in gen... |

1 |
Adjoints for double categories. Cah. Topologie Géom. Différentielle Catég
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(Show Context)
Citation Context ...from a larger structure, 2Cosp(C), which is a double bicategory, as we shall show shortly. It is analogous to the “profunctor-based examples” of pseudo-double categories described by Grandis and Paré =-=[46]-=-. The Verity double bicategory described above is derived from it. To see these facts, we first define 2Cosp(C) explicitly:� � � �� �� � � � �� �� �� �� � � � �� � � � � � � � �� �� � � � � � � �� ��... |