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

  
  

  based on Exposition of Higher Gauge Theory

  
  

  talk at:

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

    2-4 Dec 2024, Tromsø

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Abstract. The classical notion of principal connections is fundamental in mathematics (Lie theory, Chern-Weil theory) and (quantum) physics (gauge theory, Dyson series, Berry phases). 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 nominally different higher generalizations in use, modeled either on (1.) Maurer-Cartan theory of Lie-algebra valued forms (connections), or (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 review 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 other attack via non-abelian flux quantization).

Related expositions:

• Flux Quantization, Enc. Math Phys 2nd ed (2025)

• Higher Topos Theory in Physics, Enc. Math Phys 2nd ed (2025)