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Communications in Mathematical Physics

, Volume 333, Issue 1, pp 117–151 | Cite as

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

  • Domenico Fiorenza
  • Hisham Sati
  • Urs SchreiberEmail author
Article

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.

Keywords

Gauge Transformation Simons Theory Canonical Morphism High Gauge Bundle Gerbe 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. BCSS.
    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. BM.
    Belov, D., Moore, G.W.: Holographic action for the self-dual field. arXiv:hep-th/0605038
  3. BHS.
    Brown R., Higgins P., Sivera R.: Nonabelian Algebraic Topology, Tracts in Mathematics 15. European Mathematical Society, Krakow (2011)CrossRefGoogle Scholar
  4. CMW.
    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. CS.
    Chern, S.S., Simons, J.: Characteristic forms and geometric invariants. Ann. Math. 99(2), 48–69 (1974)Google Scholar
  6. DFM.
    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. DMW.
    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. Fa.
    Falkowski, A.: Five dimensional locally supersymmetric theories with branes. Master thesis, Warsaw. http://www.fuw.edu.pl/~afalkows/Work/Files/msct.ps.gz
  9. FSaSc.
    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. FScSt.
    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. Fr.
    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. FM.
    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. HS.
    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. HW.
    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. KV.
    Kahle, A., Valentino, A.: T-duality and differential K-theory. Commun. Contemp. Math. 16, 1350014 (2014). arXiv:0912.2516 [math.KT]
  16. RoSt.
    Roberts, D., Stevenson, D.: Simplicial principal bundles in parametrized spaces. arXiv:1203.2460
  17. Sa10a.
    Sati, H.: E 8 Gauge theory and gerbes in string theory. Adv. Theor. Math. Phys. 14, 1–39 (2010). arXiv:hep-th/0608190
  18. Sa10b.
    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. Sa10c.
    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. Sa11a.
    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. Sa11b.
    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. Sa11c.
    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. Sa12a.
    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. Sa12b.
    Sati, H.: Duality and cohomology in M-theory with boundary. J. Geom. Phys. 62 1284–1297 (2012). arXiv:1012.4495 [hep-th]
  25. SSS09a.
    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. SSS09b.
    Sati, H., Schreiber, U., Stasheff, J.: Twisted differential String- and Fivebrane structures. Commun. Math. Phys. 315, 169–213 (2012). arXiv:0910.4001
  27. Sch.
    Schreiber, U.: Differential cohomology in a cohesive \({\infty}\) -topos. Habilitation, Hamburg University (2011). arXiv:1310.7930
  28. SSW.
    Schreiber, U., Schweigert, C., Waldorf, K.: Unoriented WZW models and holonomy of bundle gerbes. Commun. Math. Phys. 274, 31–64 (2007). arXiv:0512283
  29. SWI.
    Schreiber, U., Waldorf, K.: Smooth 2-functors vs. differential forms. Homol. Homotopy Appl. 13(1), 143–203 (2011). arXiv:0802.0663 [math.DG]
  30. SWII.
    Schreiber, U., Waldorf, K.: Connections on nonabelian gerbes. Theory Appl. Categories 28(17), 476–540 (2013). arXiv:0802.0663 [math.DG].
  31. Wa.
    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. Wi96.
    Witten, E.: Five-brane effective action in M-theory. J. Geom. Phys. 22, 103–133 (1997). arXiv:hep-th/9610234
  33. Wi97.
    Witten, E.: On flux quantization in M-theory and the effective action. J. Geom. Phys. 22, 1–13 (1997). arXiv:hep-th/9609122

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Dipartimento di MatematicaSapienza - Università di RomaRomaItaly
  2. 2.Department of MathematicsUniversity of PittsburghPittsburghUSA
  3. 3.Department of MathematicsUniversity of NijmegenNijmegenThe Netherlands

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