Schreiber Exposition of Higher Gauge Theory

An article that we are developing at CQTS:

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• Hisham Sati$\;$ and $\;$ Urs Schreiber:

  
  

  Exposition of Higher Gauge Theory

  and Nonabelian Differential Cohomology

  
  

  download: pdf

  
  

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Abstract. The classical notion of principal connections is fundamental in mathematics (Lie theory, Chern-Weil theory, Cartan geometry) and in quantum physics (gauge theory, Dyson series, Berry phases).

This survey reviews motivations and constructions for higher-structure enhancements of this notion that are finally finding attention (categorified symmetries, higher dimensional holonomy, higher gauge fields), amplifying that there are two nominally different higher generalizations in use, modeled alternatively on:

(i) Chern-Weil theory of connection forms,

(ii) Chern-Dold theory of character forms.

In the “ordinary” abelian case, these perspectives coincide and are well-studied (Deligne cohomology, Cheeger-Simons characters, ordinary differential cohomology), but crucial applications require their non-abelian generalizations which have received less attention.

We motivate and survey both directions of nonabelian higher connection theory, with reference to our models of

(i) Čech cocycles for differential characteristic classes ([FSS12], which underlies our original take on stringy gauge fields and branes) and

(ii) the character map on non-abelian cohomology ([FSS23], which underlies our current take via non-abelian flux quantization).

Related expositions.

• Higher Topos Theory in Physics

• Flux Quantization