## Holonomy and parallel transport for abelian gerbes, Preprint math.DG/0007053

Citations: | 57 - 10 self |

### BibTeX

@MISC{Mackaay_holonomyand,

author = {Marco Mackaay and Área Departamental De Matemática and Roger Picken},

title = {Holonomy and parallel transport for abelian gerbes, Preprint math.DG/0007053},

year = {}

}

### OpenURL

### Abstract

In this paper we establish a one-to-one correspondence between S 1-gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higher-order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The holonomy of a gerbe with presently working as a postdoc at the University of Nottingham, UK 1 group S 1 on a simply connected manifold M is a group morphism from the thin second homotopy group to S 1, satisfying a smoothness condition, where a homotopy between maps from [0,1] 2 to M is thin when its derivative is of rank ≤ 2. For the non-simply connected case, holonomy is replaced by a parallel transport functor between two monoidal Lie groupoids. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply connected case. Our approach to abelian gerbes with connections holds out prospects for generalizing to the non-abelian case via the theory of double Lie groupoids. 1

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Citation Context ...t see [11]. 3 Ordinary holonomy It is well-known that a principal G-bundle with connection over M allows one to define the notion of holonomy around any smooth closed curve on M (Kobayashi and Nomizu =-=[19]-=-). In particular, given a fixed basepoint ∗ in M, and a point in the fibre over ∗, this data induces an assignment of an element of G, to each loop in M, based at ∗. Such an assignment is called a hol... |

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Citation Context ...y of Nottingham, UK 1group S 1 on a simply connected manifold M is a group morphism from the thin second homotopy group to S 1 , satisfying a smoothness condition, where a homotopy between maps from =-=[0,1]-=- 2 to M is thin when its derivative is of rank ≤ 2. For the non-simply connected case, holonomy is replaced by a parallel transport functor between two monoidal Lie groupoids. The reconstruction of th... |

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Citation Context ...ions. Gerbes were first introduced by Giraud [15] in an attempt to understand non-Abelian cohomology. Ironically only Abelian gerbes have developed into a nice geometric theory so far. Several people =-=[8, 11, 13, 18]-=- have studied Abelian gerbes. In this paper when we say gerbe we always mean a U(1)gerbe. Just as a line-bundle on M can be defined by a set of transition functions on double intersections of open set... |

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Citation Context ...n gerbe-connections in general along arbitrary homotopies between arbitrary loops. For this discussion we assume some basic knowledge about groupoids and 2-groupoids, which can be obtained by reading =-=[2, 4, 22]-=- for example. We should warn the reader that this section is different in flavour from the rest of the paper. Whereas we have followed a down-to-earth approach in the rest of the paper, here some read... |

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Citation Context ... above. Another motivation for our approach is that we are trying to understand the differential geometry behind the four-dimensional state-sum models defined by the first author of the present paper =-=[21, 20]-=-. For the understanding of these state-sums it would be helpful to find a categorical way of perceiving the relation between homotopy theory and differential geometry. Concretely the motivation was to... |

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Citation Context .... For this kind of general remark about ncategories see [2, 4, 5] for example.) Without the smoothness condition this kind of groupoid goes under a variety of names in the homotopy literature. Yetter =-=[27]-=- calls them categorical groups, for example. This particular Lie 2-group we denote by G(G, A, M). Let us now define the Lie 2-group of thin cylinders, denoted by C2 2(M, ∗). We define C2 2 (M, ∗) as t... |

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Citation Context ... answer is that, for a 1-connected manifold M, group homomorphisms π2 2(M) → U(1) correspond bijectively to equivalence classes of U(1)-gerbes with connections. Gerbes were first introduced by Giraud =-=[15]-=- in an attempt to understand non-Abelian cohomology. Ironically only Abelian gerbes have developed into a nice geometric theory so far. Several people [8, 11, 13, 18] have studied Abelian gerbes. In t... |

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Citation Context ...might be that it can be found in earlier papers on monoidal categories and 2-categories. We suspect that this definition goes back to the time when monoidal categories were defined for the first time =-=[7]-=-, but we have been unable to find a precise written reference in the older literature. Lemma 6.8 The monoidal subgroupoid N(M, ∗) is normal in C(M, ∗). Proof The first condition is obviously satisfied... |

10 |
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Citation Context ...o a notion of n-gerbe connection, and the parallel transport of an n-gerbe connection is defined along n-dimensional paths. Some details of these notions remain to be worked out completely, but Gajer =-=[13, 14]-=- has worked out a considerable part already. The truly categorical nature of gerbe-connections comes to light when we study the general setting of their parallel transport. To understand the need for ... |

10 |
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Citation Context ...rove that the second condition holds. Denote the group of homotopies γ → γ in C(M, ∗) by C(M, ∗)(γ). This group is isomorphic to C(M, ∗)(γ⋆µ), for any µ ∈ Ω(M, ∗) (this is well known, see for example =-=[17]-=-). The isomorphism, which clearly preserves thinness, is given by φ: G ↦→ G ⋆ 1µ. 36Under φ the homotopy GHG −1 : γ → γ is mapped to (G ⋆ 1µ)(H ⋆ 1µ)(G −1 ⋆ 1µ): γ⋆µ → γ⋆µ. Clearly this is thin homot... |

8 | On a family of topological invariants similar to homotopy groups - Caetano, Picken - 1998 |

8 |
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Citation Context ...ions. Gerbes were first introduced by Giraud [15] in an attempt to understand non-Abelian cohomology. Ironically only Abelian gerbes have developed into a nice geometric theory so far. Several people =-=[8, 11, 13, 18]-=- have studied Abelian gerbes. In this paper when we say gerbe we always mean a U(1)gerbe. Just as a line-bundle on M can be defined by a set of transition functions on double intersections of open set... |

1 |
Introducing quaternionic gerbes. preprint available as math.DG/0009201
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Citation Context ... help to understand how the differential geometry of Abelian gerbes fits into a framework of 2-dimensional parallel transport which is not confined to the Abelian case. We should remark that Thompson =-=[26]-=- has worked out in detail the definition of quaternionic gerbes and some of their properties. Unfortunately his only examples, related to conformal 4-manifolds, are actually just Čech 1-cochains with ... |