Description
This lecture series introduces structures in higher differential geometry with a focus on applications in mathematical and high-energy physics. The main goal is to define higher principal bundles with an interesting notion of connection and to discuss several applications of these structures.
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Higher principal bundles. We begin by motivating the need for a higher notion of parallel transport, drawing on both physical and mathematical perspectives. This leads naturally to the concept of categorification, which we briefly review. After giving a concise overview of various descriptions of higher (Lie) groups, illustrated with examples, we conclude the first lecture with the definition of higher principal bundles and a number of important examples.
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Connections on higher principal bundles — locally. By differentiating higher Lie groups, we obtain higher Lie algebras, which are closely related to homotopy Lie algebras. We recall the local description of principal bundle connections in terms of Lie-algebra–valued differential forms. Together with linearized bundle isomorphisms, these assemble into an action algebroid—the BRST-algebroid. A naive categorification of these structures encounters obstructions, and additional algebraic data ("adjustments") are required to arrive at a generally useful notion of higher connections. As an example, we study the particularly rich BRST-algebroid underlying the tensor hierarchies of gauged supergravity.
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Connections on higher principal bundles — globally. We discuss the integration of (higher) BRST-algebroids to the corresponding BRST-groupoids. Stackification of these groupoids then yields the definition of higher principal bundles with connections. A key example is given by string structures, which are higher analogues of spin structures.
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Applications. We explore several examples of interest in both mathematics and physics: higher monopoles and instantons, T-duality in higher geometric settings, the Penrose–Ward transform, and six-dimensional superconformal field theories. If time permits, we also review the case of Lie groupoid bundles.
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Christian Saemann (Heriot-Watt University Edinburgh)
We begin by motivating the need for a higher notion of parallel transport, drawing on both physical and mathematical perspectives. This leads naturally to the concept of categorification, which we briefly review. After giving a concise overview of various descriptions of higher (Lie) groups, illustrated with examples, we conclude the first lecture with the definition of higher principal...
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