## Higher gauge theory

Citations: | 43 - 13 self |

### BibTeX

@MISC{Baez_highergauge,

author = {John Baez and Urs Schreiber},

title = {Higher gauge theory},

year = {}

}

### OpenURL

### Abstract

Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings) using 2-connections on 2-bundles. A 2-bundle is a categorified version of a bundle: that is, one where the fiber is not a manifold but a category with a suitable smooth structure. Where gauge theory uses Lie groups and Lie algebras, higher gauge theory uses their categorified analogues: Lie 2-groups and Lie 2-algebras. We describe a theory of 2-connections on principal 2-bundles and explain how this is related to Breen and Messing’s theory of connections on nonabelian gerbes. The distinctive feature of our theory is that a 2-connection allows parallel transport along paths and surfaces in a parametrization-independent way. In terms of Breen and Messing’s framework, this requires that the ‘fake curvature ’ must vanish. In this paper we summarize the main results of our theory without proofs. 1

### Citations

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128 |
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Citation Context ...paces’ that has pullbacks and includes path spaces. Various categories of this sort have been proposed. It is unclear which one is best, but we shall use the last of several variants proposed by Chen =-=[34, 35]-=-. In what follows, we use convex set to mean a convex subset of Rn , where n is arbitrary (not fixed). Any convex set inherits a topology from its inclusion in Rn . We say a map f between convex sets ... |

106 |
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Citation Context ...e other pullbacks needed to write the rules of a category as commutative diagrams. We can similarly define a functor in K and a natural transformation in K; details can be found in Borceux’s handbook =-=[32]-=-. There is a 2-category KCat whose objects, morphisms and 2-morphisms are categories, functors and natural transformations in K. To study symmetries in higher gauge theory, we need these examples: Def... |

95 | Bundle gerbes - Murray - 1996 |

60 |
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Citation Context ...e spaces’ are topological spaces equipped with extra structure. Unfortunately, spaces of continuous maps are ill-behaved in Top: this category is not cartesian closed. Ever since the work of Steenrod =-=[35]-=-, algebraic topologists have dealt with this problem by replacing Top with some ‘convenient category’ of topological spaces. For example, they may use compactly generated Hausdorff spaces and continuo... |

57 | Holonomy and parallel transport for abelian gerbes
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Citation Context ...h that Σ(s, t) is constant near s = 0 and s = 1, and independent of t near t = 0 and t = 1: x • γ1 �� γ2 Σ • y 6The ‘thin homotopy’ equivalence relation, borrowed from the work of Mackaay and Picken =-=[7, 22]-=-, guarantees that two maps differing only by a reparametrization define the same bigon. This is important because we seek a reparametrizationinvariant notion of surface holonomy. We define a ‘2-connec... |

53 |
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Citation Context ...group that can also be obtained by killing the 3rd homotopy group of G. When G = Spin(n), this topological group goes by the name of String(n), since it plays a role in defining spinors on loop space =-=[38]-=-. The group String(n) also shows up in Stolz and Teichner’s work on elliptic cohomology, which involves a notion of parallel transport over surfaces [39]. So, we expect that PkG will be an especially ... |

39 |
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Citation Context ... × F P |Ui called local trivializations such that these diagrams: P |Ui � �� � �� � � p � �� � �� ti Ui ��� ������������� �� Ui × F commute for all i ∈ I. Readers wise in the ways of categorification =-=[40]-=- may ask why we did not require that these diagrams commute up to natural isomorphism. The reason is that Ui, as an ordinary space, has only identity morphisms when we regard it as a 2-space. Thus, fo... |

33 |
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Citation Context ...plays a role in defining spinors on loop space [38]. The group String(n) also shows up in Stolz and Teichner’s work on elliptic cohomology, which involves a notion of parallel transport over surfaces =-=[39]-=-. So, we expect that PkG will be an especially interesting structure 2-group for applications of 2-bundles to string theory. Finally, here are some easy examples of smooth 2-spaces: Example 4. Any smo... |

31 |
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Citation Context ...n elaboration of the ideas in Example 6, and all the 2-category operations are then smooth maps. We thus say P2(M) is a smooth 2-category: that is, a 2-category in C ∞ . Indeed, the usual definitions =-=[41]-=- of 2-category, 2-functor, pseudonatural transformation, and modification can all be internalized in C ∞ , 21and we use these ‘smooth’ notions what follows. Furthermore, both morphisms and 2-morphism... |

28 |
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Citation Context ...certain equivalence classes of smooth paths in M as morphisms. There are various ways to work out the technical details and make P1(M) into a ‘smooth groupoid’; here we follow the approach of Barrett =-=[16]-=-, who uses ‘thin homotopy classes’ of paths. Technical details aside, the basic idea is that a connection on a trivial G-bundle gives a well-behaved map assigning to each path γ in the base space the ... |

22 |
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Citation Context ...ie group — that is, some gadget resembling a Lie group, but which is a category rather than a set. We call this ‘Lie 2-group’. In fact, Lie 2-groups and their Lie 2-algebras have already been studied =-=[12, 13]-=- and interesting examples have been constructed using the mathematics of string theory: central extensions of loop groups [14]. But even without knowing this, we could be led to the definition of a Li... |

22 | Combinatorics of non-abelian gerbes with connection and curvature, available as math-ph/0203056 - Attal |

21 | Gerbes, M5-brane anomalies and E8 gauge theory. Also available as arXiv:hep-th/0409200
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Citation Context ...sis develops 7the general theory of 2-bundles. The second author’s thesis investigates the relationship between nonabelian higher gauge theory and the physics of strings. Aschieri, Cantini and Jurčo =-=[28, 29]-=- have also studied this subject, using connections on nonabelian gerbes. Other physicists, including Chepelev [30] and Hofman [31], have also studied nonabelian higher gauge fields. 2 Internalization ... |

20 |
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Citation Context ...g fake curvature in the framework of lattice gauge theory was already emphasized by Girelli and Pfeiffer [21]. The special case where also FA = 0 was studied by Alvarez, Ferreira, Sanchez and Guillen =-=[24]-=-. The case where G = H has been studied already by the second author of this paper [25]. Our result subsumes these cases in a common framework. This paper is an introduction to work in progress [23], ... |

20 | Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory
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Citation Context ...sis develops 7the general theory of 2-bundles. The second author’s thesis investigates the relationship between nonabelian higher gauge theory and the physics of strings. Aschieri, Cantini and Jurčo =-=[28, 29]-=- have also studied this subject, using connections on nonabelian gerbes. Other physicists, including Chepelev [30] and Hofman [31], have also studied nonabelian higher gauge fields. 2 Internalization ... |

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19 |
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Citation Context ...re vanishes, one obtains a well-defined 2-connection hol: P2(M) → G. The importance of vanishing fake curvature in the framework of lattice gauge theory was already emphasized by Girelli and Pfeiffer =-=[21]-=-. The special case where also FA = 0 was studied by Alvarez, Ferreira, Sanchez and Guillen [24]. The case where G = H has been studied already by the second author of this paper [25]. Our result subsu... |

17 |
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(Show Context)
Citation Context ...tural to assign a group element 1s�� �� �� we must have (g1g ′ 1)(g2g ′ 2) = (g1g2)(g ′ 1g ′ 2). This forces G to be abelian! In fact, this argument goes back to a classic paper by Eckmann and Hilton =-=[1]-=-. They showed that even if we allow G to be equipped with two products, say gg ′ for vertical composition and g ◦ g ′ for horizontal, so long as both products share the same unit and satisfy this ‘int... |

17 |
Catégories topologiques et catégories différentiables. (French) 1959 Colloque Géom. Diff. Globale (Bruxelles
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Citation Context ...laws hold only up to isomorphism [12, 13]. We could also consider categories in Diff, the category whose objects are finite-dimensional smooth manifolds and whose morphisms are smooth maps. Ehresmann =-=[33]-=- introduced these in the late 1950’s under the name of ‘differentiable categories’. However, these are not quite what we want here, for two reasons. First, unlike LieGrp and LieAlg, Diff does not have... |

16 | The geometry of bundle gerbes - Stevenson - 2000 |

16 | Higher-dimensional algebra VI - Baez, Crans |

12 |
From Loop Space Mechanics to Nonabelian Strings
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Citation Context ...aper is an introduction to work in progress [23], which began in rudimentary form as an article by the first author [26], and overlaps to some extent with theses by Bartels [17] and the second author =-=[27]-=-. Bartels’ thesis develops 7the general theory of 2-bundles. The second author’s thesis investigates the relationship between nonabelian higher gauge theory and the physics of strings. Aschieri, Cant... |

12 |
Breathing Spaces
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Citation Context ...paces’ that has pullbacks and includes path spaces. Various categories of this sort have been proposed. It is unclear which one is best, but we shall use the last of several variants proposed by Chen =-=[34, 35]-=-. In what follows, we use convex set to mean a convex subset of Rn , where n is arbitrary (not fixed). Any convex set inherits a topology from its inclusion in Rn . We say a map f between convex sets ... |

12 |
Higher Gauge Theory: 2-Connections on 2-Bundles, eprint hep-th/0412325v1 at http://arxiv.org
- Baez, Schreiber
(Show Context)
Citation Context ...njugation in Mor(G) as follows: α(g)h = 1gh1g −1 . Differentiating all this data one obtains a ‘differential crossed module’ (g, h, dt, dα), which is just another way of talking about a Lie 2-algebra =-=[23]-=-. In these terms, a 2-connection on a trivial principal 2-bundle over M with structure 2-group G consists of a g-valued 1-form A together with an h-valued 2-form B on M. Translated into this framework... |

12 |
Iterated integrals, fundamental groups and covering spaces
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- 1975
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Citation Context ...ently large class of infinite-dimensional manifolds. Various categories of this sort have been proposed. It is unclear which one is best, but we shall use a slight variant of an idea proposed by Chen =-=[34]-=-. Chen’s ‘differentiable spaces’ are topological spaces equipped with extra structure. Unfortunately, spaces of continuous maps are ill-behaved in Top: this category is not cartesian closed. Ever sinc... |

10 | Higher holonomies, geometric loop groups and smooth Deligne cohomology - Gajer - 1998 |

7 | Non-abelian Wilson Surfaces
- Chepelev
(Show Context)
Citation Context ...elian higher gauge theory and the physics of strings. Aschieri, Cantini and Jurčo [28, 29] have also studied this subject, using connections on nonabelian gerbes. Other physicists, including Chepelev =-=[30]-=- and Hofman [31], have also studied nonabelian higher gauge fields. 2 Internalization The idea of internalization is simple: given a mathematical concept X defined solely in terms of sets, functions a... |

6 | Heterotic String Data and Theta Functions. math.DG/0110320 - Clingher |

6 |
Differential Geometry of Gerbes, available as math.AG/0106083
- Breen, Messing
(Show Context)
Citation Context ...r M. For another choice, they give nonabelian gerbes. This sets the stage for a result relating the 2-bundle approach to higher gauge theory to Breen and Messing’s approach based on nonabelian gerbes =-=[18]-=-. Just as a connection on a trivial principal bundle over M gives a functor from the path groupoid of M to the structure group, one might hope that a ‘2-connection’ on a trivial principal 2-bundle wou... |

3 |
Higher gauge theory I: 2-bundles, available as math.CT/0410328
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(Show Context)
Citation Context ...ie 2-group, and also to that of a ‘smooth 2space’: a category in Diff, or more generally in some category of smooth spaces that allows for infinite-dimensional examples. Using these concepts, Bartels =-=[17]-=- has defined a ‘principal 2-bundle E over M with structure 2-group G’. To arrive at this definition, the key steps are to replace the total space E and base space M of a principal bundle by smooth 2-s... |

3 |
Higher Yang{Mills theory, available at hep-th/0206130
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(Show Context)
Citation Context ...or of this paper [25]. Our result subsumes these cases in a common framework. This paper is an introduction to work in progress [23], which began in rudimentary form as an article by the first author =-=[26]-=-, and overlaps to some extent with theses by Bartels [17] and the second author [27]. Bartels’ thesis develops 7the general theory of 2-bundles. The second author’s thesis investigates the relationsh... |

2 | Holonomy on D-branes, available as hep-th-/0204199 - Carey, Johnson, et al. |

2 |
Higher gauge theory II: 2-connections. Available as arXiv:hep-th/0412325
- Baez, Schreiber
(Show Context)
Citation Context ...njugation in Mor(G) as follows: α(g)h = 1gh1g −1 . Differentiating all this data one obtains a ‘differential crossed module’ (g, h, dt, dα), which is just another way of talking about a Lie 2-algebra =-=[23]-=-. In these terms, a 2-connection on a trivial principal 2-bundle over M with structure 2-group G consists of a g-valued 1-form A together with an h-valued 2-form B on M. Translated into this framework... |

1 |
Catégories structurées, Ann. Ecole Norm
- Ehresmann
- 1963
(Show Context)
Citation Context ...mple, a Lie group is a group in Diff, the category of smooth manifolds and smooth maps. The general idea of an ‘internal category’ living within some other category was described by Charles Ehresmann =-=[15]-=- in the early 1960’s. In the next section we begin by reviewing this idea. In the rest of the paper, we use it to categorify the theory of Lie groups, Lie algebras, bundles and connections. Before we ... |

1 |
A note on the holonomy of connections in twisted bundles. Available as math.DG/0106019
- Mackaay
(Show Context)
Citation Context ...h that Σ(s, t) is constant near s = 0 and s = 1, and independent of t near t = 0 and t = 1: x • γ1 �� γ2 Σ • y 6The ‘thin homotopy’ equivalence relation, borrowed from the work of Mackaay and Picken =-=[7, 22]-=-, guarantees that two maps differing only by a reparametrization define the same bigon. This is important because we seek a reparametrizationinvariant notion of surface holonomy. We define a ‘2-connec... |

1 |
Nonabelian 2-forms. Also available as hep-th/0207017
- Hofman
(Show Context)
Citation Context ...ge theory and the physics of strings. Aschieri, Cantini and Jurčo [28, 29] have also studied this subject, using connections on nonabelian gerbes. Other physicists, including Chepelev [30] and Hofman =-=[31]-=-, have also studied nonabelian higher gauge fields. 2 Internalization The idea of internalization is simple: given a mathematical concept X defined solely in terms of sets, functions and commutative d... |

1 |
Compactly generated spaces. Available online at http:// www.shef.ac.uk/nps/courses/homotopy/cgwh.pdf
- Strickland
(Show Context)
Citation Context ... these, as in Steenrod’s original work. Or, following the more recent preference in algebraic topology, they may use compactly generated weakly Hausdorff spaces and continuous functions between these =-=[36]-=-. Both these categories are cartesian closed. They also have arbitrary limits and colimits. For the record, we recall that a topological space X is compactly generated if a subset of X is closed whene... |

1 | Streetfest lectures on higher gauge theory, 2005. Available online at http://math.ucr.edu/home/baez/street - Baez, Crans, et al. |