An invitation to higher gauge theory
An invitation to higher gauge theory
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DOI：
10.1007/s10714-010-1070-9
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发表时间：
2010-03
影响因子：
2.8
通讯作者：
J. Baez;John Huerta
中科院分区：
文献类型：
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作者：
J. Baez;John Huerta
在这篇关于更高规范理论的简单介绍中，我们用2-丛上的2-连接来描述粒子和弦的平行输运。正如普通规范理论涉及规范群一样，这种推广涉及规范‘2-群’。我们集中在6个例子上。首先，每个阿贝尔李群都有一个李2-群；U(1)的情形产生了U(1)细菌理论，它在弦理论和多辛几何中起着重要的作用。其次，每个群表示都给出一个Lie-2-群；Lorentz群在4D Minkowski时空上的表示给出了Poincaré2-群，从而得到了Minkowski时空的自旋泡沫模型。第三，利用李群在其李代数上的伴随表示，给出了一个在4dBF理论中作为规范2-群的“切2-群”，它具有拓扑引力作为特例。第四，每个李群都有一个“内自同构2-群”，它是4dBF理论中具有宇宙学常数项的规范群。第五，每个李群都有一个‘自同构2-群’，它在非交换细菌理论中起着重要的作用。第六，每个紧单李群都有一个‘弦2-群’。我们还涉及更高的结构，如‘重力3-群’，以及支配11维超引力的李3-超代数。
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 4dBFtheory, 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 4dBFtheory 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.