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The E 8 Moduli 3-Stack of the C-Field in M-Theory

Abstract

The higher gauge field in 11-dimensional supergravity—the C-field—is constrained by quantum effects to be a cocycle in some twisted version of differential cohomology. We argue that it should indeed be a cocycle in a certain twisted nonabelian differential cohomology. We give a simple and natural characterization of the full smooth moduli 3-stack of configurations of the C-field, the gravitational field/background, and the (auxiliary) E 8-field. We show that the truncation of this moduli 3-stack to a bare 1-groupoid of field configurations reproduces the differential integral Wu structures that Hopkins–Singer had shown to formalize Witten’s argument on the nature of the C-field. We give a similarly simple and natural characterization of the moduli 2-stack of boundary C-field configurations and show that it is equivalent to the moduli 2-stack of anomaly free heterotic supergravity field configurations. Finally, we show how to naturally encode the Hořava–Witten boundary condition on the level of moduli 3-stacks, and refine it from a condition on 3-forms to a condition on the corresponding full differential cocycles.

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References

  1. Baez, J., Crans, A., Schreiber, U., Stevenson, D.: From loop groups to 2-groups. Homol. Homotopy Appl. 9(2), 101–135 (2007). arxiv:math/0504123

  2. Belov, D., Moore, G.W.: Holographic action for the self-dual field. arXiv:hep-th/0605038

  3. Brown R., Higgins P., Sivera R.: Nonabelian Algebraic Topology, Tracts in Mathematics 15. European Mathematical Society, Krakow (2011)

    Book  Google Scholar 

  4. Carey, A.L., Mickelsson, J., Wang, B.-L.: Differential twisted K-theory and applications, J. Geom. Phys. 59, 632–653 (2009). arXiv:0708.3114 [math.KT]

  5. Chern, S.S., Simons, J.: Characteristic forms and geometric invariants. Ann. Math. 99(2), 48–69 (1974)

  6. Diaconescu, E., Freed, D., Moore, G.: The M-theory 3-form and E 8 gauge theory. In: Elliptic Cohomology. 44–88, London Mathematical Society, Lecture Note Series, 342, Cambridge University Press, Cambridge (2007). arXiv:hep-th/0312069

  7. Disconescu, E., Moore, G., Witten, E.: E 8 gauge theory, and a derivation of K-theory from M-theory. Adv. Theor. Math. Phys. 6, 1031–1134 (2003). arXiv:hep-th/0005090 [hep-th]

  8. Falkowski, A.: Five dimensional locally supersymmetric theories with branes. Master thesis, Warsaw. http://www.fuw.edu.pl/~afalkows/Work/Files/msct.ps.gz

  9. Fiorenza, D., Sati, H., Schreiber, U.: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern–Simons theory. Advances in Theoretical and Mathematical Physics, vol. 18-2 (March–April 2014). arXiv:1201.5277 [hep-th]

  10. Fiorenza, D., Schreiber, U., Stasheff, J. Čech-cocycles for differential characteristic classes. Adv. Theor. Math. Phys. 16(1), 149–250 (2012). arXiv:1011.4735

  11. Freed, D.S.: Dirac Charge Quantization and Generalized Differential Cohomology, in Surv. Differ. Geom. vol. VII, pp. 129–194. Int. Press, Somerville (2000) arXiv:hep-th/0011220

  12. Freed, D.S., Moore, G.W.: Setting the quantum integrand of M-theory. Commun. Math. Phys. 263, 89–132 (2006). arXiv:hep-th/0409135

  13. Hopkins, M.J., Singer, I.M.: Quadratic functions in geometry, topology, and M-theory. J. Differ. Geom. 70(3), 329–452 (2005). arXiv:math/0211216 [math.AT]

  14. Horava, P., Witten, E.: Heterotic and Type I string dynamics from eleven dimensions. Nucl. Phys. B460, 506 (1996). arXiv:hep-th/9510209 and Eleven dimensional supergravity on a manifold with boundary. Nucl. Phys. B475, 94 (1996). arXiv:hep-th/9603142

  15. Kahle, A., Valentino, A.: T-duality and differential K-theory. Commun. Contemp. Math. 16, 1350014 (2014). arXiv:0912.2516 [math.KT]

  16. Roberts, D., Stevenson, D.: Simplicial principal bundles in parametrized spaces. arXiv:1203.2460

  17. Sati, H.: E 8 Gauge theory and gerbes in string theory. Adv. Theor. Math. Phys. 14, 1–39 (2010). arXiv:hep-th/0608190

  18. Sati, H.: Geometric and topological structures related to M-branes. In: Proceedings of Symposium of Pure Mathematics, vol. 81, pp. 181–236 (2010). arXiv:1001.5020

  19. Sati, H.: Geometric and topological structures related to M-branes II: twisted String- and \({{\rm String}^c}\) -structures. J. Aust. Math. Soc. 90, 93–108 (2011). arXiv:1007.5419

  20. Sati, H.: Twisted topological structures related to M-branes. Int. J. Geom. Meth. Mod. Phys. 8, 1097–1116 (2011). arXiv:1008.1755 [hep-th]

  21. Sati, H.: Anomalies of E 8 gauge theory on String manifolds. Int. J. Mod. Phys. A26, 2177–2197 (2011). arXiv:0807.4940 [hep-th]

  22. Sati, H.: Twisted topological structures related to M-branes II: twisted Wu and Wuc structures. Int. J. Geom. Meth. Mod. Phys. 9, 1250056 (2012). arXiv:1109.4461 [hep-th]

  23. Sati, H.: Geometry of Spin and Spinc structures in the M-theory partition function. Rev. Math. Phys. 24 1250005 (2012). arXiv:1005.1700 [hep-th]

  24. Sati, H.: Duality and cohomology in M-theory with boundary. J. Geom. Phys. 62 1284–1297 (2012). arXiv:1012.4495 [hep-th]

  25. Sati, H., Schreiber, U., Stasheff, J.: \({{\rm L}_\infty}\) -Algebra connections and applications to String- and Chern–Simons n-transport. In: Recent Developments in Quantum Field Theory. Birkhäuser (2009) arXiv:0801.3480 [math.DG]

  26. Sati, H., Schreiber, U., Stasheff, J.: Twisted differential String- and Fivebrane structures. Commun. Math. Phys. 315, 169–213 (2012). arXiv:0910.4001

  27. Schreiber, U.: Differential cohomology in a cohesive \({\infty}\) -topos. Habilitation, Hamburg University (2011). arXiv:1310.7930

  28. Schreiber, U., Schweigert, C., Waldorf, K.: Unoriented WZW models and holonomy of bundle gerbes. Commun. Math. Phys. 274, 31–64 (2007). arXiv:0512283

  29. Schreiber, U., Waldorf, K.: Smooth 2-functors vs. differential forms. Homol. Homotopy Appl. 13(1), 143–203 (2011). arXiv:0802.0663 [math.DG]

  30. Schreiber, U., Waldorf, K.: Connections on nonabelian gerbes. Theory Appl. Categories 28(17), 476–540 (2013). arXiv:0802.0663 [math.DG].

  31. Wang, B.-L.: Geometric cycles, index theory and twisted K-homology. J. Noncommut. Geom. 2,(4), 497–552 (2008). arXiv:0710.1625 [math.KT]

  32. Witten, E.: Five-brane effective action in M-theory. J. Geom. Phys. 22, 103–133 (1997). arXiv:hep-th/9610234

  33. Witten, E.: On flux quantization in M-theory and the effective action. J. Geom. Phys. 22, 1–13 (1997). arXiv:hep-th/9609122

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Correspondence to Urs Schreiber.

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Communicated by H.-T. Yau

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Fiorenza, D., Sati, H. & Schreiber, U. The E 8 Moduli 3-Stack of the C-Field in M-Theory. Commun. Math. Phys. 333, 117–151 (2015). https://doi.org/10.1007/s00220-014-2228-1

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  • DOI: https://doi.org/10.1007/s00220-014-2228-1

Keywords

  • Gauge Transformation
  • Simons Theory
  • Canonical Morphism
  • High Gauge
  • Bundle Gerbe