## 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

### Citations

356 |
Differential Topology
- Hirsch
- 1997
(Show Context)
Citation Context ...spaces, can be computed as homotopy groups in a certain complex. However, this goes beyond what we wish to examine here: a good introductory discussion suitable 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 motivatio... |

339 |
The geometry and topology
- Thurston
- 1978
(Show Context)
Citation Context ...s A0(B). This definition is different from the usual concept of a “G-bundle” equipped with a flat connection in terms of fibre bundles, but the two concepts are equivalent, as established by Thurston =-=[83]-=-. Generally, a flat G-bundle on B takes loops in B into elements G. For any loop γ in B, it assigns an element A(γ) ∈ G. This is the holonomy around the loop 136Chapter 7. Extended TQFTs as 2-Functor... |

319 |
Homotopy associativity of H-spaces
- Stasheff
- 1963
(Show Context)
Citation Context ... will in fact involve. In fact, this is understood to a considerable degree, but this point of view is awkward because it involves the categorified versions of associativity - Stasheff’s associahedra =-=[79]-=-. These play the role of Pachner moves in higher dimensions. We could proceed with this categorified version of the construction, when G is a finite group. It turns out that a natural generalization o... |

259 |
A categorification of the Jones polynomial
- Khovanov
(Show Context)
Citation Context ...n 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’S AND QUANTUM GRAVITY 15 Two manifolds S1, S2 are cobordant if there is a compact man... |

187 | Virtual knot theory
- Kauffman
- 1999
(Show Context)
Citation Context ...ions 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’S AND QUANTUM GRAVITY 15 Two manifolds S1, S2 are cobordant if there is a compact manifold with boundary, M, such that... |

182 |
State sum invariants of 3-manifolds and quantum 6j-symbols, Topology 31
- Turaev, Viro
- 1992
(Show Context)
Citation Context ... 220Chapter 8. Prospects for Quantum Gravity In particular, we expected to recover the Ponzano-Regge model of 3D quantum gravity, based on SU(2), as an extended TQFT. Now, the Turaev-Viro model (see =-=[85]-=- and [36]) is based on the q-deformed quantum groups SU(2)q, and in some respects is more convenient than the Ponzano-Regge model. In particular, there are infinitely many representations of SU(2), bu... |

149 |
Sketches of an elephant: a topos theory compendium
- Johnstone
- 2002
(Show Context)
Citation Context ...ued functions is a categorified analog. On the other hand, Set-valued presheaves on certain kinds of categories are generic examples of toposes, about which much is known (see, for example, Johnstone =-=[49]-=-, [50]). Some results about these can be shown for Vect-valued presheaves also, although there are significant differences resulting from the fact that Vect is an additive category, whereas Set is Car... |

140 |
Introduction to bicategories
- Bénabou
- 1967
(Show Context)
Citation Context ...ap is a morphism f : C → C ′ such that the diagram: (3) C ′ �������� ��� f ��� � A C B commutes. A cospan is defined in the same way, but with the arrows reversed. It is a classical result of Bénabou =-=[16]-=- that for any category C which has all limits, there is a bicategory Span(C) whose objects are objects of C, whose morphisms are spans in C, and whose 2-morphisms are span maps. The composition of mor... |

138 | Higher-dimensional algebra and topological quantum field theory
- Baez, Dolan
- 1995
(Show Context)
Citation Context ...dal categories, or Hopf algebras; and the appearance of “Hopf categories” in 4D TQFT’s. These illustrate the move to higher-categorical structures in higher-dimensional field theories. Baez and Dolan =-=[8]-=- summarize the connection between TQFT’s and higher category theory, in the form of the Extended TQFT Hypothesis, suggesting that all extended TQFT’s can be viewed as representations of a certain kind... |

133 |
Review of the elements of 2-categories
- Kelly, Street
(Show Context)
Citation Context ...ra layer of structure: objects, morphisms between objects, and 2-morphisms between morphisms: (2) x f g α � y The “strict” form of a bicategory is a 2-category, which are reviewed by Kelly and Street =-=[53]-=-, but we are really interested in the weak forms—here, all the axioms which must be satisfied by a category hold only “up to” certain higherdimensional morphisms. That is, what had been equations are ... |

121 | Topological quantum field theories - Atiyah - 1989 |

116 | Topological gauge theories and group cohomology
- Dijkgraaf, Witten
- 1990
(Show Context)
Citation Context ...egorified version of the construction, when G is a finite group. It turns out that a natural generalization of the FHK construction gives a theory equivalent to the (untwisted) Dijkgraaf-Witten model =-=[30]-=-. This is a topological22 JEFFREY MORTON gauge theory, which crucially involves a (flat) connection on a manifold. We will discuss this in more detail in Section 7.1, and explore how an extended TQFT... |

96 | The algebra of oriented simplexes - Street - 1987 |

90 |
2-categories and Zamolodchikov tetrahedra equations,” in: Algebraic groups and their generalizations: quantum and infinite-dimensional methods
- Kapranov, Voevodsky
- 1994
(Show Context)
Citation Context ... bilinear map. A�-linear functor between�-linear categories is one where morphism maps are�-linear. The standard example of this approach is the Kapranov-Voevodsky (KV) definition of a 2-vector space =-=[51]-=-, which is the form we shall use (at least when the situation is finite-dimensional). To motivate the KV definition, consider the idea that, in categorifying, one should replace the base field�with a ... |

77 |
Frobenius algebras and 2-d topological quantum field theories
- Kock
(Show Context)
Citation Context ...nal cobordisms. It has been known for some time that 2Cob can be seen as the free symmetric monoidal category on a commutative Frobenius object. (This is shown in the good development by Joachim Kock =-=[56]-=-.) This is a categorical formulation of the fact, shown by Abrams [1], that 2Cob is generated from four generators, called the unit, counit, multiplication, comultiplication, subject to some relations... |

64 | On Khovanov’s categorification of the Jones polynomial
- Bar-Natan
(Show Context)
Citation Context ... 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’S AND QUANTUM GRAVITY 15 Two manifolds S1, S2 are cobordant if there is a compact manifold with boundary, M, ... |

63 |
Higher Operads, Higher Categories
- LEINSTER
- 2004
(Show Context)
Citation Context ...er more carefully what kind of structure nCob2 must be. So we consider some background on higher category theory. This field of study is still developing, but there are good introductions by Leinster =-=[64]-=- and by Cheng and Lauda [23]. The essential idea of higher category theory is that as well as objects (represented in diagrams as zero-dimensional), and morphisms (or arrows) connecting them (which ar... |

61 |
Abelian categories. An introduction to the theory of functors
- Freyd
- 1964
(Show Context)
Citation Context ...s.�-linearity means that the set of morphisms are complex vector spaces. We should remark that these properties mean that 2vector spaces are closely related to abelian categories (introduced by Freyd =-=[42]-=-, and studied extensively as the general setting for homological algebra) have a structure on objects which is similar to addition for vectors. In particular, we are interested in the analog of “finit... |

60 |
Higher Algebraic Structures and Quantization
- Freed
(Show Context)
Citation Context ...olan [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]-=-, who referred to them as “higher algebraic” structures. The motivation for doing this is that this framework appears to allow us to find a new way of obtaining a known theory of quantum gravity in 3 ... |

59 | Two-dimensional topological quantum field theories and Frobenius algebras
- Abrams
- 1996
(Show Context)
Citation Context ...as the free symmetric monoidal category on a commutative Frobenius object. (This is shown in the good development by Joachim Kock [56].) This is a categorical formulation of the fact, shown by Abrams =-=[1]-=-, that 2Cob is generated from four generators, called the unit, counit, multiplication, comultiplication, subject to some relations. The generating cobordisms are the following: taking the empty set t... |

51 | Clock and category: is quantum gravity algebraic
- Crane
- 1995
(Show Context)
Citation Context ...FT’s. This is described in terms of higher category theory. The idea that category theory could play a role in clarifying problems in quantum gravity seems to have been first expressed by Louis Crane =-=[24]-=-, who coined the term “categorification” . Categorification is a process of replacing set-based concepts by category-based concepts. Categories are structures which have not only elements (that is, ob... |

50 | From finite sets to Feynman diagrams
- Baez, Dolan
- 2001
(Show Context)
Citation Context ...asses, but a colimit. In fact, the correct prescription involves the groupoid cardinality of the groupoid of those connections which contribute to the sum. This concept is described by Baez and Dolan =-=[10]-=-, and related to Leinster’s [63] concept of the Euler characteristic of a category. For a more in-depth discussion of groupoid cardinality, and also of its role (closely related to the role it plays h... |

48 | Yetter D.: On algebraic structures implicit in topological quantum field theories, preprint
- Crane
- 1994
(Show Context)
Citation Context ...ith corners. Here we shall present a formalism for describing the ways such cobordisms can be glued together. Louis Crane has written a number of papers on this issue, including one with David Yetter =-=[27]-=- which gives a bicategory of such cobordisms. We want to define a structure nCob2, whose objects are (n − 2)-manifolds, whose morphisms are (n − 1)-cobordisms, and whose 2-morphisms are n-cobordisms w... |

43 | Higher-dimensional algebra VI. Lie 2-algebras
- Baez, Crans
(Show Context)
Citation Context ... target, composition, etc. are linear maps. This is a useful concept for some purposes—it was developed to give a categorification of Lie algebras. The reader may refer to the paper of Baez and Crans =-=[6]-=- for more details. However, a BC 2-vector space turns out to be equivalent to a 2-term chain complex and, this is not the concept of 2-vector space which concerns us here. The other, and prior, approa... |

42 |
Catégories et structures
- Ehresmann
- 1965
(Show Context)
Citation Context ...of category theory since its inception by MacLane and Eilenberg (see, for instance, [66]), and are important features of higher categorical structures. Double categories, introduced by Ehresmann [32] =-=[33]-=-, may be seen as “internal” categories in Cat. That is, a double category is a structure with a category of objects and a category of morphisms. Less abstractly, it has objects, horizontal and vertica... |

41 | Higher Gauge Theory
- Baez, Schreiber
- 2005
(Show Context)
Citation Context ...auge group G with a categorical group gives a theory based 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... |

40 |
BF description of higher-dimensional gravity theories
- Freidel, Krasnov, et al.
- 1999
(Show Context)
Citation Context ...sively by Wise [87]. What is true in 4 dimensions is that the purely topological theory corresponds to a theory of flat connections on a manifold known as BF theory. and by Freidel, Krasnov and Puzio =-=[38]-=-). To describe a theory of gravity would need something more than what is discussed here. In Section 8.3 we briefly consider some possible approaches to this problem. 8.2. Ponzano-Regge with Matter. I... |

38 |
An introduction to spin foam models of BF theory and quantum gravity
- Baez
- 2000
(Show Context)
Citation Context ...which have already been studied. One of these which is particularly relevant involves so-called spin foam models. A self-contained description of such models for BF theory and quantum gravity by Baez =-=[4]-=-. Spin foam models are a generalization of the spin networks of Penrose [76]. A spin network is a network in the sense of a graph—a collection of nodes, connected by edges. In a spin network, the edge... |

38 |
Non-Semisimple Topological Quantum Field Theories for
- Kerler, Lyubashenko
- 2001
(Show Context)
Citation Context ...uss the application of double categories to mathematical physics, particularly TQFT’s, and dynamical systems with changing boundary conditions—that is, with inputs and outputs. Kerler and Lyubashenko =-=[54]-=- describe extended TQFT’s as “double pseudofunctors” between double categories. This formulation involves, among other things, a double category of cobordisms with corners—we return to a weakening of ... |

37 |
Lattice Topological Field Theory in two Dimensions, Commun.Math.Phys
- Fukuma, Hosono, et al.
- 1994
(Show Context)
Citation Context ...space of states to each manifold, and a linear transformation between states to cobordisms.EXTENDED TQFT’S AND QUANTUM GRAVITY 5 Section 2.3 discusses a construction due to Fukuma, Hosono, and Kawai =-=[43]-=- for constructing a TQFT explicitly in dimension n = 2 starting from any finite group G. The FHK construction is an example of how this quantum theory intimately involves a relation between smooth and... |

33 |
Ponzano-Regge model revisited I: Gauge fixing, observables and interacting spinning particles
- Freidel, Louapre
(Show Context)
Citation Context ...s derived from a finite group, but if G = SU(2) the label is a mass and spin) moving on a background described by Ponzano-Regge quantum gravity (see work by, for example, 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 g... |

29 |
Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras,” math.at/0510664
- Lauda, Pfeiffer
(Show Context)
Citation Context ...ightforward description of an ndimensional TQFT is known. To provide one would require a presentation of nCob in terms of generators and relations (for both objects and morphisms). Lauda and Pfeiffer =-=[59]-=- do provide such a presentation a similar, though more complicated, characterization of 2-dimensional →� open-closed TQFT’s. In these, we do not assume that the manifolds representing spaces have no b... |

25 | Ponzano–Regge model revisited. III: Feynman diagrams and effective field theory”, Class
- Freidel, Livine
- 2006
(Show Context)
Citation Context ... from a finite group, but if G = SU(2) the label is a mass and spin) moving on a background described by Ponzano-Regge quantum gravity (see work by, for example, 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 grou... |

22 |
Higher-dimensional categories: an illustrated guide book
- Cheng, Lauda
- 2004
(Show Context)
Citation Context ...of structure nCob2 must be. So we consider some background on higher category theory. This field of study is still developing, but there are good introductions by Leinster [64] and by Cheng and Lauda =-=[23]-=-. The essential idea of higher category theory is that as well as objects (represented in diagrams as zero-dimensional), and morphisms (or arrows) connecting them (which are one-dimensional), there al... |

20 | Higher Yang-Mills Theory
- Baez
(Show Context)
Citation Context ...ion of G on H, such that t and α satisfy some compatibility conditions, which turn out to be equivalent to the category axioms in the 2-group described above. The Poincare 2-group, introduced by Baez =-=[5]-=-, is an example of an automorphic 2-group. It has been a subject of interest as a source of a new class of spin foam models, first suggested by Crane and Sheppeard [25]. Such models are based on the r... |

20 |
On cobordism of manifolds with corners
- Laures
(Show Context)
Citation Context ...can be reduced by taking isomorphism classes in order to obtain a Verity double bicategory. We describe more specifically the geometric framework for cobordisms with corners in Chapter 5. Gerd Laures =-=[60]-=- discusses the general theory of cobordisms of manifolds with corners. In the terminology used there, introduced by Jänich [48], what we primarily discuss in this work are 〈2〉-manifolds. This describe... |

20 |
Functorial Semantics of Algebraic Theories and Some Algebraic Problems in the context of Functorial Semantics of Algebraic Theories
- Lawvere
- 1963
(Show Context)
Citation Context ...sm α : p2 ◦ PS −→ p ′ 1 ◦ PS ′. In the case of groupoids, a weak pullback can be seen as an example of a comma category (the concept, though not the name, introduced by Lawvere in his doctoral thesis =-=[62]-=-). We briefly discuss this next before stating the theorem regarding composition. Remark 13. In general, suppose we have a diagram of categories A F →C G ←B. Then an object in the comma category (F ↓ ... |

19 |
Quantum Gravity in 2+1 Dimensions (Cambridge
- Carlip
- 1998
(Show Context)
Citation Context ...for 3D 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 ... |

18 |
Representation theory of finite groups
- Burrow
- 1965
(Show Context)
Citation Context ... of groups, this is usually called the “restricted representation”. The adjoint process to the restriction of representations is generally called finding the induced representation (see, e.g. Burrows =-=[20]-=- for a classical discussion of this when f is an inclusion). We will use the same term for the general case when f is just a homomorphism, and slightly generalize the usual description. The pushforwar... |

18 |
On the classification of O(n)-manifolds
- Jänich
- 1968
(Show Context)
Citation Context ...eometric framework for cobordisms with corners in Chapter 5. Gerd Laures [60] discusses the general theory of cobordisms of manifolds with corners. In the terminology used there, introduced by Jänich =-=[48]-=-, what we primarily discuss in this work are 〈2〉-manifolds. This describes the relation of “faces” of the manifold, but in particular in this case it is related to the8 JEFFREY MORTON fact that the c... |

16 |
Double groupoids and crossed modules
- Brown, Spencer
(Show Context)
Citation Context ...case, is when the horizontal and vertical categories on the objects are the same: this is the case of path-symmetric double categories, and the recovery of a bicategory was shown by Brown and Spencer =-=[19]-=-. Fiore [35] shows how their demonstration of this fact is equivalent to one involving folding structures. In this case we again can interpret squares as bigons by composing the top and right edges, a... |

16 | Pseudo algebras and pseudo double categories
- Fiore
- 2006
(Show Context)
Citation Context ...ry. 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 describes examples�� �� �� 30 JEFFREY MORTON motivated by conformal field theory. A deta... |

16 | Enriched categories, internal categories, and change of base
- Verity
- 1992
(Show Context)
Citation Context ...ms with corners we want. 61Part II Higher Categories for Extended TQFT’s 62Chapter 4 Verity Double Bicategories The term double bicategory seems to have been originally introduced by Dominic Verity =-=[86]-=-, and the structure it refers to is the one we want to use. There is some ambiguity here since the term double bicategory appears to describe is an internal bicategory in Bicat (the category of all bi... |

13 |
Limits in double categories, Cahiers Topologie Géom. Différentielle Catég
- Grandis, Paré
- 1999
(Show Context)
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... |

13 |
Quantum Groups: A Path to Current Algebra
- Street
- 2007
(Show Context)
Citation Context ...e groups to “finite quantum groups”, by which we mean finite-dimensional quasitriangular Hopf algebras. The idea behind quantum groups is described by Shahn Majid [68] and also notably by Ross Street =-=[81]-=-. The idea provides a way to speak of deforming topological groups, although there is no way of smoothly deforming the group action of a topological group to a family of other such groups. Instead, on... |

12 | 2-Categorical Poincaré representations and state sum applications, available as arXiv:math/0306440
- Crane, Sheppeard
(Show Context)
Citation Context ...care 2-group, introduced by Baez [5], is an example of an automorphic 2-group. It has been a subject of interest as a source of a new class of spin foam models, first suggested by Crane and Sheppeard =-=[25]-=-. Such models are based on the representation theory of 2-groups, which is 2-categorical in nature, since one must consider representations, intertwiners between representations, and 2-intertwiners be... |

12 | Frobenius algebras and ambidextrous adjunctions
- Lauda
(Show Context)
Citation Context ...d right adjoint are the same. This means that the “pushforward” map is an ambidextrous adjunction for the pullback (for much more on the relation between ambidextrous adjunctions and TQFTs, see Lauda =-=[57]-=-). It seems useful, then, to have another approach to the “pushforward” map than the matrix-dependent view of Theorem 5. Fortunately, there is a more instrinsic way to describe the 2-linear map f∗, th... |

11 |
Quantization of strings and branes coupled to BF theory
- Baez, Perez
(Show Context)
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 | Higher gauge theory I: 2-Bundles
- Bartels
- 2006
(Show Context)
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 |
Categories structurees
- Ehresmann
- 1963
(Show Context)
Citation Context ...heme of category theory since its inception by MacLane and Eilenberg (see, for instance, [66]), and are important features of higher categorical structures. Double categories, introduced by Ehresmann =-=[32]-=- [33], may be seen as “internal” categories in Cat. That is, a double category is a structure with a category of objects and a category of morphisms. Less abstractly, it has objects, horizontal and ve... |