Higher gauge theory

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In mathematical physics, higher gauge theory is the general study of counterparts of gauge theory that involve higher-degree differential forms instead of the traditional connection 1-forms of Yang-Mills gauge theories.

Contents

Frameworks for higher gauge theory

There are several distinct frameworks within which higher gauge theories have been developed. Alvarez et al. [1] extend the notion of integrability to higher dimensions in the context of geometric field theories. Several works [2] of John Baez, Urs Schreiber and coauthors have developed higher gauge theories heavily based on category theory. Arthur Parzygnat [3] has a detailed development of this framework. An alternative approach, [4] motivated by the goal of constructing geometry over spaces of paths and higher-dimensional objects, has been developed by Saikat Chatterjee, Amitabha Lahiri, and Ambar N. Sengupta.

The mathematical framework for traditional gauge theory places the gauge potential as a 1-form on a principal bundle over spacetime. Higher gauge theories provide geometric and category-theoretic, especially higher category theoretic, frameworks for field theories that involve multiple higher differential forms.

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References

  1. Alvarez, Orlando; Ferreira, Luiz A.; Guillén, J. Sánchez (1998). "A new approach to integrable theories in any dimension". Nuclear Physics B. 529: 689–736.
  2. Baez, John C.; Schreiber, Urs (2007). Categories in algebra, geometry and mathematical physics. Contemporary Mathematics. Vol. 431. Providence, RI: American Mathematical Society. pp. 7–30.
  3. Parzygnat, Arthur (2015). "Gauge invariant surface holonomy and monopoles". Theory and Applications of Categories. 30: 1319–1428.
  4. Chatterjee, Saikat; Lahiri, Ambitabha; Sengupta, Ambar N. (2017). "Connections on decorated path space bundles". Journal of Geometry and Physics. 112: 147–174.