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

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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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Authors and Affiliations

  1. Dipartimento di Matematica, Sapienza - Università di Roma, P.le Aldo Moro 2, 00185, Roma, Italy

    Domenico Fiorenza

  2. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Hisham Sati

  3. Department of Mathematics, University of Nijmegen, Nijmegen, The Netherlands

    Urs Schreiber

Authors
  1. Domenico Fiorenza
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  2. Hisham Sati
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  3. Urs Schreiber
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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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