Schreiber Two Notions of Nonabelian Differential Cohomology

A talk that I will have given:

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• Urs Schreiber:

  
  

  Two Notions of Nonabelian Differential Cohomology

  
  

  talk at 2024 Lie-Størmer Colloquium – Foundational and computational aspects of symmetry

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Abstract. The classical notion of principal connections is fundamental in mathematics (Lie theory, Chern-Weil theory) and physics (quantum dynamics, gauge theory). Now that higher-structure variants are increasingly finding attention (higher dimensional holonomy, categorified symmetries, higher gauge fields), this talk is to highlight that there are two different higher generalizations in use: One generalizes (1.) Maurer-Cartan theory of Lie-algebra valued forms (connections), the other generalizes (2.) the Chern-Dold character (curvature invariants) on generalized cohomology. Beyond the case of abelian ordinary differential cohomology (Deligne cohomology, Cheeger-Simons characters), the relation between the two is only quite partially understood.

I will speak about what I do and do not understand in this regard, with reference to our models of (1.) Čech Cocycles for Differential Characteristic Classes (which underlies our original attack on stringy gauge fields and branes) and (2.) of The Character Map in Non-Abelian Cohomology (which underlies our new attack via non-abelian flux quantization).

• slides/notes to appear here