## An Invitation to Higher Gauge Theory (2010)

Citations: | 3 - 2 self |

### BibTeX

@MISC{Baez10aninvitation,

author = {John C. Baez and John Huerta},

title = {An Invitation to Higher Gauge Theory},

year = {2010}

}

### OpenURL

### Abstract

In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2-group’. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a ‘tangent 2-group’, which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an ‘inner automorphism 2-group’, which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an ‘automorphism 2-group’, which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a ‘string 2-group’. We also touch upon higher structures such as the ‘gravity 3-group’, and the Lie 3-superalgebra that governs 11-dimensional supergravity. 1

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Citation Context .... By Theorem 3, a connection on a trivial U(1) gerbe is just an ordinary real-valued 2-form B. Its holonomy is given by: γ hol: • • ↦→ 1 hol: • Σ • ↦→ ( ∫ ) exp i B ∈ U(1). Σ 28The book by Brylinski =-=[31]-=- gives a rather extensive introduction to U(1) gerbes and their applications. Murray’s theory of ‘bundle gerbes’ gives a different viewpoint [74, 89]. Here let us discuss two places where U(1) gerbes ... |

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Citation Context ...22�� � �� This result was announced by Baez and Schreiber [16], and a proof can be be found in the work of Schreiber and Waldorf [88]. This work was deeply inspired by the ideas of Breen and Messing =-=[29, 30]-=-, who considered a special class of 2-groups, and omitted the equation t(B) = dA + A ∧ A, since their sort of connection did not assign holonomies to surfaces. One should also compare the closely rela... |

30 |
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Citation Context ...s does the work of Waldorf [91]. In fact, the string Lie 2-group had lived through many previous incarnations before being constructed as an infinite-dimensional Lie 2-group. Brylinski and McLaughlin =-=[33]-=- thought of it as a U(1) gerbe over the group G. The fact that this gerbe is ‘multiplicative’ makes it something like a group in its own right [32]. This viewpoint was also been explored by Murray and... |

30 |
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Citation Context ...ts structure group from Spin(n) to this group [94]. The string group also plays a role in Stolz and Teichner’s work on elliptic cohomology, which involves a notion of parallel transport over surfaces =-=[90]-=-. There is a lot of sophisticated mathematics involved here, but ultimately much of it should arise from the way string 2-groups are involved in the parallel transport of strings! The work of Sati, Sc... |

23 |
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Citation Context ...on 4.5. 4.4 Inner Automorphism 2-Groups There is also a Lie 2-group where: • G is any Lie group, • H = G, • t is the identity map, • α is conjugation: α(g)h = ghg −1 . Following Roberts and Schreiber =-=[79]-=- we call this the inner automorphism 2-group of G, and denote it by IN N (G). We explain this terminology in the next section. A 2-connection on the trivial IN N (G)-2-bundle over a manifold consists ... |

22 |
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Citation Context ...its target, • the action α of G on H is given by α(g)h = 1g ◦ h ◦ 1 g −1 . Indeed, these two processes set up an equivalence between 2-groups and crossed modules, as described more formally elsewhere =-=[13, 51]-=-. It thus makes sense to define a Lie 2-group to be a 2-group for which the groups G and H in its crossed module are Lie groups, with the maps t: H → G and α: G → Aut(H) being smooth. It is worth emph... |

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20 |
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Citation Context ...o by Gawedski and Reis [55]. In short, electromagnetism has a ‘higher version’. What about symplectic geometry? This also has a higher version, which dates back to 1935 work by DeDonder [39] and Weyl =-=[92]-=-. The idea here is that an n-dimensional classical field theory has a kind of finite-dimensional phase space equipped with a closed (n + 2)-form ω which is nondegenerate in the following sense: ∀v1, .... |

19 |
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Citation Context ..., since their sort of connection did not assign holonomies to surfaces. One should also compare the closely related work of Mackaay, Martins, and Picken [67, 69], and the work of Pfeiffer and Girelli =-=[76, 58]-=-. In the above theorem, the first item mentions ‘2-connections’ and ‘2-bundles’— concepts that we have not defined. But since we are only talking about 2connections on trivial 2-bundles, we do not nee... |

18 | Nonabelian bundle gerbes, their differential geometry and gauge theory, textslComm
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Citation Context ...ot be smooth: there could be a ‘kink’ at the point y. There are different ways to get around this problem. One is to work with piecewise smooth paths. But here is another approach: say that a path γ: =-=[0, 1]-=- → M is lazy if it is smooth and also constant in a neighborhood of t = 0 and t = 1. The idea is that a lazy hiker takes a rest before starting a hike, and also after completing it. Suppose γ and δ ar... |

18 | Functional integration on spaces of connections - Baez, Sawin - 1997 |

18 |
Higher gauge theory and a non-abelian generalization of p-form electromagnetism
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Citation Context ..., since their sort of connection did not assign holonomies to surfaces. One should also compare the closely related work of Mackaay, Martins, and Picken [67, 69], and the work of Pfeiffer and Girelli =-=[76, 58]-=-. In the above theorem, the first item mentions ‘2-connections’ and ‘2-bundles’— concepts that we have not defined. But since we are only talking about 2connections on trivial 2-bundles, we do not nee... |

17 |
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Citation Context ...vertical composition must be equal when G is trivial. This proof is called the ‘Eckmann–Hilton argument’, since Eckmann and Hilton used it to show that the second homotopy group of a space is abelian =-=[43]-=-. So, we can build a 2-group where: • G is the trivial group, • H is any abelian Lie group, • α is trivial, and • t is trivial. This is called the shifted version of H, and denoted bH. In applications... |

17 | Catégories topologiques et catégories différentiables - Ehresmann |

17 | Higgs fields, bundle gerbes and string structures
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Citation Context ... it as a U(1) gerbe over the group G. The fact that this gerbe is ‘multiplicative’ makes it something like a group in its own right [32]. This viewpoint was also been explored by Murray and Stevenson =-=[75]-=-. Later, Baez and Crans [7] constructed a Lie 2-algebra stringk(g) corresponding to the string Lie 2-group. For pedagogical purposes, our discussion of Lie 2-groups has focused solely on ‘strict’ 2-gr... |

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

15 | Categorical representation of categorical groups
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Citation Context ...he Poincaré 2-group good for? It is not clear, but there are some clues. Just as we can study representations of groups on vector spaces, we can study representations of 2-groups on ‘2-vector spaces’ =-=[6, 24, 38, 48]-=-. The representations of a group are the objects of a category, and this sort of category can be used to build ‘spin foam models’ of background-free quantum field theories [5]. This endeavor has been ... |

15 |
L∞-connections and applications to String- and Chern-Simons n-transport
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Citation Context ... to build a theory sufficiently rich to describe the quantum geometry of spacetime. Indeed, many existing ideas from string theory and supergravity have recently been clarified by higher gauge theory =-=[82, 83]-=-. But we may also hope for applications of higher gauge theory to other less speculative branches of physics, such as condensed matter physics. Of course, for this to happen, more physicists need to l... |

15 | Parallel transport and functors
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Citation Context ...ds, and maps hol: P1(M) → G of this more general sort correspond to connections on not necessarily trivial principal G-bundles over M. For details, see the work of Bartels [25], Schreiber and Waldorf =-=[87]-=-. But if this sounds like too much work, we can take the following shortcut. Suppose we have a smooth function F : [0, 1] n × [0, 1] → M, which we think of as a parametrized family of paths. And suppo... |

12 | 2-Categorical Poincaré representations and state sum applications, available as arXiv:math/0306440
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Citation Context ...oup are the objects of a 2-category, and Crane and Sheppeard outlined a program for building a 4-dimensional spin foam 32model starting from the 2-category of representations of the Poincaré 2-group =-=[36]-=-. Crane and Sheppeard hoped their model would be related to quantum gravityin 4 spacetime dimensions. This has not come to pass, at least not yet—but this spin foam model does have interesting connect... |

11 | Quantization of strings and branes coupled to BF theory - Baez, Perez |

11 | Measurable categories and 2-groups
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(Show Context)
Citation Context ...he Poincaré 2-group good for? It is not clear, but there are some clues. Just as we can study representations of groups on vector spaces, we can study representations of 2-groups on ‘2-vector spaces’ =-=[6, 24, 38, 48]-=-. The representations of a group are the objects of a category, and this sort of category can be used to build ‘spin foam models’ of background-free quantum field theories [5]. This endeavor has been ... |

11 | Note on a previous paper entitled On adding relations to homotopy groups - Whitehead - 1946 |

10 | Four-dimensional BF theory as a topological quantum field theory
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(Show Context)
Citation Context ... tr(e ∧ e ∧ F − λ e ∧ e ∧ e ∧ e). 2 M When we choose the bilinear form ‘tr’ correctly, this is the action for general relativity with a cosmological constant proportional to λ. There is some evidence =-=[4]-=- that BF theory with nonzero cosmological constant can be quantized to obtain the so-called Crane–Yetter model [35, 37], which is a spin foam model based on the category of representations of the quan... |

10 | Categorified symplectic geometry and the classical string
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(Show Context)
Citation Context ...lectic geometry, see the paper by Gotay, Isenberg, Marsden, and Montgomery [57]. The link between multisymplectic geometry and higher electromagnetism was made in a paper by Baez, Hoffnung and Rogers =-=[10]-=-. Everything is closely analogous to the story for point particles. For a classical bosonic string propagating on Minkowski spacetime of any dimension, say M, there is a finitedimensional manifold X w... |

10 | Representation theory of 2-groups on Kapranov and Voevodsky 2-vector spaces
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(Show Context)
Citation Context ...he Poincaré 2-group good for? It is not clear, but there are some clues. Just as we can study representations of groups on vector spaces, we can study representations of 2-groups on ‘2-vector spaces’ =-=[6, 24, 38, 48]-=-. The representations of a group are the objects of a category, and this sort of category can be used to build ‘spin foam models’ of background-free quantum field theories [5]. This endeavor has been ... |

10 |
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(Show Context)
Citation Context ...its target, • the action α of G on H is given by α(g)h = 1g ◦ h ◦ 1 g −1 . Indeed, these two processes set up an equivalence between 2-groups and crossed modules, as described more formally elsewhere =-=[13, 51]-=-. It thus makes sense to define a Lie 2-group to be a 2-group for which the groups G and H in its crossed module are Lie groups, with the maps t: H → G and α: G → Aut(H) being smooth. It is worth emph... |

10 |
Diffeomorphism invariant quantum field theories of connections in terms of webs
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- 1999
(Show Context)
Citation Context ...to be a functor hol: P1(M) → G. Generalized connections have long played an important role in loop quantum gravity, first in the context of real-analytic manifolds [3], and later for smooth manifolds =-=[17, 65]-=-. The reason is that if M is any manifold and G is a connected compact Lie group, there is a natural measure on the space of generalized connections. This means that you can define a Hilbert space of ... |

10 |
Smooth functors vs. differential forms, [arXiv:0802.0663] [math.DG
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(Show Context)
Citation Context ...tal composition for 2-morphisms, and identity 2-morphisms: F (α · β) = F (α) · F (β) F (α ◦ β) = F (α) ◦ F (β) F (1f ) = 1 F (f). There is a general theory of smooth 2-groupoids and smooth 2-functors =-=[16, 88]-=-. But here we prefer to take a more elementary approach. We already know that for any Lie 2-group G, the morphisms form a Lie group. In the next section we say that the 2-morphisms also form a Lie gro... |

7 |
Division algebras and supersymmetry II. Available as arXiv:1003.3436
- Baez, Huerta
(Show Context)
Citation Context ...n dimensions 4, 5, 7 and 11. These can be built via a systematic construction starting from the four normed division algebras: the real numbers, the complex numbers, the quaternions and the octonions =-=[11]-=-. These four algebras also give rise to Lie 2-superalgebras extending the Poincaré Lie superalgebra in dimensions 3, 4, 6, and 10. The Lie 2-superalgebras are related to superstring theories in dimens... |

7 |
Higher gauge theory I: 2-bundles. Available as arXiv:math.CT/0410328
- Bartels
(Show Context)
Citation Context ... maps between smooth groupoids, and maps hol: P1(M) → G of this more general sort correspond to connections on not necessarily trivial principal G-bundles over M. For details, see the work of Bartels =-=[25]-=-, Schreiber and Waldorf [87]. But if this sounds like too much work, we can take the following shortcut. Suppose we have a smooth function F : [0, 1] n × [0, 1] → M, which we think of as a parametrize... |

7 |
Differentiable cohomology of gauge groups
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(Show Context)
Citation Context ...te-dimensional Lie 2-group. Brylinski and McLaughlin [33] thought of it as a U(1) gerbe over the group G. The fact that this gerbe is ‘multiplicative’ makes it something like a group in its own right =-=[32]-=-. This viewpoint was also been explored by Murray and Stevenson [75]. Later, Baez and Crans [7] constructed a Lie 2-algebra stringk(g) corresponding to the string Lie 2-group. For pedagogical purposes... |