Skip to main content
SpringerLink
Log in

Teleparallel Gravity as a Higher Gauge Theory

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

Abstract

We show that general relativity can be viewed as a higher gauge theory involving a categorical group, or 2-group, called the teleparallel 2-group. On any semi-Riemannian manifold M, we first construct a principal 2-bundle with the Poincaré 2-group as its structure 2-group. Any flat metric-preserving connection on M gives a flat 2-connection on this 2-bundle, and the key ingredient of this 2-connection is the torsion. Conversely, every flat strict 2-connection on this 2-bundle arises in this way if M is simply connected and has vanishing 2nd deRham cohomology. Extending from the Poincaré 2-group to the teleparallel 2-group, a 2-connection includes an additional piece: a coframe field. Taking advantage of the teleparallel reformulation of general relativity, which uses a coframe field, a flat connection and its torsion, this lets us rewrite general relativity as a theory with a 2-connection for the teleparallel 2-group as its only field.

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. Aldrovandi R., Pereira J.G.: Teleparallel Gravity: An Introduction, Fundamental Theories of Physics, vol. 173. Springer, Berlin (2013)

    Book  Google Scholar 

  2. de Andrade, V.C., Guillen, L.C.T., Pereira, J.G.: Teleparallel gravity: an overview. arXiv:gr-qc/0011087

  3. Arcos, H.I., Pereira, J.G.: Torsion gravity: a reappraisal. Int. J. Mod. Phys. D13, 2193–2240 (2004). arXiv:gr-qc/0501017

  4. Baez, J.: An introduction to spin foam models of BF theory and quantum gravity. In: Gausterer, H., Grosse, H. (eds.) Geometry and Quantum Physics, pp. 25–93. Springer, Berlin (2000). arXiv:gr-qc/9905087

  5. Baez, J., Baratin, A., Freidel, L., Wise, D.: Infinite-dimensional representations of 2-groups. Mem. Am. Math. Soc. 219, 1032 (2012). arXiv:0812.4969

  6. Baez, J., Crans, A.: Higher dimensional algebra VI: Lie 2-algebras. Theory Appl. Categ. 12, 492–538 (2004). arXiv:math.0307263

  7. Baez, J., Crans, A., Schreiber, U., Stevenson, D.: From loop groups to 2-groups. HHA 9, 101–135 (2007). arXiv:math.QA/0504123

  8. Baez, J., Huerta, J.: An invitation to higher gauge theory. Gen. Relat. Gravit. 43, 2335–2392 (2011). arXiv:1003.4485

  9. Baez, J., Lauda, A.: Higher dimensional algebra V: 2-groups. Theory Appl. Categ. 12, 423–491 (2004). arXiv:math.0307200

  10. Baez, J., 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). arXiv:math.0511710

  11. Baratin, A., Freidel, L.: Hidden quantum gravity in 4d Feynman diagrams: emergence of spin foams. Class. Quant. Grav. 24, 2027–2060 (2007). arXiv:hep-th/0611042

  12. Baratin, A., Wise, D.: 2-Group representations for spin foams. AIP Conf. Proc. 1196, 28–35 (2009). arXiv:0910.1542

  13. Bartels, T.: Higher gauge theory: 2-bundles. arXiv:math.CT/0410328

  14. Behrend, K., Xu, P.: Differentiable stacks and gerbes. J. Symplectic Geom. 9, 285–341 (2011). arXiv:math.0605694

  15. Breen, L.: Differential geometry of gerbes and differential forms. arXiv:0802.1833

  16. Breen, L., Messing, W.: Differential geometry of gerbes. Adv. Math. 198, 732–846 (2005). arXiv:math.AG/0106083

  17. Crane, L., Sheppeard, M.D.: 2-Categorical Poincaré representations and state sum applications. arXiv:math.0306440

  18. Crane, L., Yetter, D.N.: Measurable categories and 2-groups. Appl. Categor. Struct. 13, 501–516 (2005). arXiv:math.0305176

  19. Debever R.: Élie Cartan and Albert Einstein: Letters on Absolute Parallelism, 1929–1932. Princeton University Press, Princeton (1979)

    Google Scholar 

  20. Engle, J., Livine, E., Pereira, R., Rovelli, C.: LQG vertex with finite Immirzi parameter. Nucl. Phys. B 799, 136–149 (2008). arXiv:0711.0146

  21. Freidel, L., Krasnov, K.: A new spin foam model for 4d gravity. Class. Quant. Grav. 25, 125018 (2008). arXiv:0708.1595

  22. Girelli, F., Pfeiffer, H., Popescu, E. M.: Topological higher gauge theory—from BF to BFCG theory. J. Math. Phys. 49, 032503, (2008). arXiv:0708.3051

  23. Heinloth, J.: Some notes on differentiable stacks. http://www.uni-due.de/~hm0002/stacks.pdf

  24. Henriques, A.: Integrating L ∞-algebras. Compositio Math. 144, 1017–1045 (2008). arXiv:math.0603563

  25. Itin, Y.: Energy-momentum current for coframe gravity. Class. Quant. Grav. 19, 173–190 (2002). arXiv:gr-qc/0111036

  26. Klein, F.: A comparative review of recent researches in geometry. trans. In: M.W. Haskell (ed.)Bulletin of the American Mathematical Society, vol. 2, pp. 215–249 (1892–1893). arXiv:0807.3161

  27. Mackenzie K.C.H.: General Theory of Lie Groupoids and Lie Algebroids. Cambridge University Press, Cambridge (2005)

    Book  MATH  Google Scholar 

  28. Maluf J.: Hamiltonian formulation of the teleparallel description of general relativity. J. Math. Phys. 35, 335–343 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Martins, J.F., Miković, A.: Lie crossed modules and gauge-invariant actions for 2-BF theories. Adv. Theor. Math. Phys. 15(4), 1059–1084 (2011). arXiv:1006.0903

  30. Martins, J.F., Picken, R.: On two-dimensional holonomy. Trans. Am. Math. Soc. 362, 5657–5695 (2010). arXiv:0710.4310

  31. Martins, J.F., Picken, R.: Surface holonomy for non-abelian 2-bundles via double groupoids. Adv. Math. 226, 3309–3366 (2011). arXiv:0808.3964 (under a different title)

  32. Miković, A., Vojinović, M.: Poincaré 2-group and quantum gravity. Class. Quant. Grav. 29, 165003 (2012). arXiv:1110.4694

  33. Oriti, D.: Spin foam models of quantum spacetime, Ph.D. Thesis, Cambridge (2003). arXiv:gr-qc/0311066

  34. Roytenberg, D.: On weak Lie 2-algebras. AIP Conference Proceedings 956, 180 (2007). arXiv:0712.3461

  35. Sauer, T.: Field equations in teleparallel spacetime: Einstein’s Fernparallelismus approach towards unified field theory. Historia Math. 33, 399–439 (2006). arXiv:physics/0405142

  36. Schommer-Pries, C.: Central extensions of smooth 2-groups and a finite-dimensional string 2-group. Geom. Topol. 15, 609–676 (2011). arXiv:0911.2483

  37. Sati, H., Schreiber, U., Stasheff, J.: L ∞-algebras and applications to string- and Chern–Simons n-transport. In: Fauser, B., Tolksdorf, J., Zeidler, E. (eds.) Quantum Field Theory: Competitive Models, pp. 303–424. Springer (2009). arXiv:0801.3480

  38. Schreiber, U., Waldorf, K.: Connections on non-abelian gerbes and their holonomy. Theory Appl. Categ. 28, 476–540 (2013). arXiv:0808.1923

  39. Segal G.B.: Classifying spaces and spectral sequences. Publ. Math. IHES 34, 105–112 (1968)

    Article  MATH  Google Scholar 

  40. Sharpe R.W.: Differential Geometry: Cartan’s Generalization of Klein’s Erlangen Program. Springer, Berlin (1997)

    MATH  Google Scholar 

  41. Wise, D.K.: Symmetric space Cartan connections and gravity in three and four dimensions. SIGMA 5, 080 (2009). arXiv:0904.1738

  42. Wise, D.K.: MacDowell–Mansouri gravity and Cartan geometry. Class. Quant. Grav. 27, 155010 (2010). arXiv:gr-qc/0611154

  43. Yetter, D.: Measurable categories. Appl. Cat. Str. 13, 469–500 (2005). arXiv:math/0309185

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Riverside, CA, 92521, USA

    John C. Baez

  2. Centre for Quantum Technologies, National University of Singapore, Singapore, 117543, Singapore

    John C. Baez

  3. Department of Mathematics, University of Erlangen-Nürnberg, Cauerstr. 11, 91058, Erlangen, Germany

    Derek K. Wise

Authors
  1. John C. Baez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Derek K. Wise
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to John C. Baez.

Additional information

Communicated by A. Connes

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baez, J.C., Wise, D.K. Teleparallel Gravity as a Higher Gauge Theory. Commun. Math. Phys. 333, 153–186 (2015). https://doi.org/10.1007/s00220-014-2178-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-2178-7

Keywords

Search

Navigation