Differential cohomology in a cohesive infinity-topos
Differential cohomology in a cohesive infinity-topos
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内聚无穷拓扑中的微分上同调
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发表时间：
2013
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通讯作者：
U. Schreiber
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作者：
U. Schreiber
我们制定微分上同调和陈-韦尔理论-纤维束和规范领域的理论连接-抽象的背景下，我们称之为“凝聚力”的某一类较高的toposes。在这个微分上同调中的上圈分类了具有内聚结构（拓扑的、光滑的、合成微分的、超几何的等）的高级主丛。并配备了连接，因此更高的规范场。我们将讨论公理的各种模型以及在量子场论和弦论中的基本概念和构造的应用。特别是，我们表明，凝聚力和微分细化的普遍特征上循环构成了一个更高的陈-韦伊同态细化从二级charteristic类的高模堆叠的高规范场的态射，并在同一时间构成了扩展的几何预量子化-在这个意义上的扩展/多层量子场论-层次的高维陈-西蒙斯型场论，它们的高阶Wess-Zumino-Witten型边界场论和所有更高阶余维亏损场论。最后，我们展望了这种高边界前量子场论的上同调量子化的一种凝聚动机。
We formulate differential cohomology and Chern-Weil theory -- the theory of connections on fiber bundles and of gauge fields -- abstractly in the context of a certain class of higher toposes that we call "cohesive". Cocycles in this differential cohomology classify higher principal bundles equipped with cohesive structure (topological, smooth, synthetic differential, supergeometric, etc.) and equipped with connections, hence higher gauge fields. We discuss various models of the axioms and applications to fundamental notions and constructions in quantum field theory and string theory. In particular we show that the cohesive and differential refinement of universal characteristic cocycles constitutes a higher Chern-Weil homomorphism refined from secondary caracteristic classes to morphisms of higher moduli stacks of higher gauge fields, and at the same time constitutes extended geometric prequantization -- in the sense of extended/multi-tiered quantum field theory -- of hierarchies of higher dimensional Chern-Simons-type field theories, their higher Wess-Zumino-Witten-type boundary field theories and all further higher codimension defect field theories. We close with an outlook on the cohomological quantization of such higher boundary prequantum field theories by a kind of cohesive motives.