Skip to main content
SpringerLink
Log in

Canonical Gauges in Higher Gauge Theory

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study the problem of finding good gauges for connections in higher gauge theories. We find that, for 2-connections in strict 2-gauge theory and 3-connections in 3-gauge theory, there are local “Coulomb gauges” that are more canonical than in classical gauge theory. In particular, they are essentially unique, and no smallness of curvature is needed in the critical dimensions. We give natural definitions of 2-Yang–Mills and 3-Yang–Mills theory and find that the choice of good gauges makes them essentially linear. As an application, (anti-)selfdual 2-connections over \(B^6\) are always 2-Yang–Mills, and (anti-)selfdual 3-connections over \(B^8\) are always 3-Yang–Mills.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arvasi, Z., Kuzpinari, T.S., Uslu, E.Ö.: Three-crossed modules. Homol. Homotopy Appl. 11, 161–187 (2009)

    Article  MathSciNet  Google Scholar 

  2. Baez, J.C.: Higher Yang–Mills theory (2002). arXiv:hep-th/0206130

  3. Baez, J.C., Huerta, J.: An invitation to higher gauge theory. Gen. Relativ. Gravit. 43, 2335–2392 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  4. Baez, J.C., Schreiber, U.: Higher gauge theory. In: Davydov, A., et al. (eds.) Categories in Algebra, Geometry and Mathematical Physics. Contemporary Mathematics, vol. 431, pp. 7–30. AMS, Providence (2007)

    Chapter  Google Scholar 

  5. Bartels, T.: Higher gauge theory: 2-bundles (2004). arXiv:math.CT/0410328

  6. Breen, L., Messing, W.: Differential geometry of gerbes. Adv. Math. 198, 732–846 (2005)

    Article  MathSciNet  Google Scholar 

  7. Demessie, G.A., Sämann, C.: Higher Poincaré lemma and integrability. J. Math. Phys. 56, 082902 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  8. Iwaniec, T., Scott, C., Stroffolini, B.: Nonlinear Hodge theory on manifolds with boundary. Ann. Mat. Pura Appl. (4) 177, 37–115 (1999)

    Article  MathSciNet  Google Scholar 

  9. Jurco, B., Raspollini, L., Sämann, C., Wolf, M.: \(L_\infty \)-algebras of classical field theories and the Batalin–Vilkovisky formalism (2018). arXiv:1809.09899

  10. Martins, J.F., Mikovic, A.: Lie crossed modules and gauge-invariant actions for 2-\(BF\) theories. Adv. Theor. Math. Phys. 15, 1059–1084 (2011)

    Article  MathSciNet  Google Scholar 

  11. Sämann, C., Wolf, M.: Six-dimensional superconformal field theories from principal \(3\)-bundles over twistor space. Lett. Math. Phys. 104, 1147–1188 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  12. Schwarz, G.: Hodge Decomposition—A Method for Solving Boundary Value Problems. Springer Lecture Notes in Mathematics, vol. 1607. Springer, Berlin (1995)

    Book  Google Scholar 

  13. Uhlenbeck, K.: Connections with \(L^p\) bounds on curvature. Commun. Math. Phys. 83, 31–42 (1982)

    Article  ADS  Google Scholar 

  14. Wang, W.: On 3-gauge transformations, 3-curvatures, and Gray-categories. J. Math. Phys. 55, 043506, 32 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Wehrheim, K.: Uhlenbeck Compactness. EMS Series of Lectures in Mathematics. EMS, Zürich (2004)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Universität Duisburg-Essen, 45117, Essen, Germany

    Andreas Gastel

Authors
  1. Andreas Gastel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andreas Gastel.

Additional information

Communicated by C. Schweigert

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gastel, A. Canonical Gauges in Higher Gauge Theory. Commun. Math. Phys. 376, 1053–1071 (2020). https://doi.org/10.1007/s00220-019-03530-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-019-03530-4

Search

Navigation