Semistrict higher gauge theory

Open Access
Regular Article - Theoretical Physics

Abstract

We develop semistrict higher gauge theory from first principles. In particular, we describe the differential Deligne cohomology underlying semistrict principal 2-bundles with connective structures. Principal 2-bundles are obtained in terms of weak 2-functors from the Čech groupoid to weak Lie 2-groups. As is demonstrated, some of these Lie 2-groups can be differentiated to semistrict Lie 2-algebras by a method due to Ševera. We further derive the full description of connective structures on semistrict principal 2-bundles including the non-linear gauge transformations. As an application, we use a twistor construction to derive superconformal constraint equations in six dimensions for a non-Abelian \( \mathcal{N}=\left(2,0\right) \) tensor multiplet taking values in a semistrict Lie 2-algebra.

Keywords

Extended Supersymmetry Differential and Algebraic Geometry M-Theory Integrable Field Theories 

References

  1. [1]
    L. Breen and W. Messing, Differential geometry of gerbes, Adv. Math. 198 (2005) 732 [math/0106083] [INSPIRE].CrossRefMATHMathSciNetGoogle Scholar
  2. [2]
    P. Aschieri, L. Cantini and B. Jurčo, Non-Abelian bundle gerbes, their differential geometry and gauge theory, Commun. Math. Phys. 254 (2005) 367 [hep-th/0312154] [INSPIRE].CrossRefADSMATHGoogle Scholar
  3. [3]
    T. Bartels, Higher gauge theory I: 2-bundles, math/0410328.
  4. [4]
    J.C. Baez and A.D. Lauda, Higher-dimensional algebra V: 2-groups, Theory Appl. Categ. 12 (2004) 423, http://www.kurims.kyoto-u.ac.jp/EMIS/journals/TAC/volumes/12/14/12-14.pdf [math/0307200].
  5. [5]
    J.C. Baez, D. Stevenson, A.S. Crans and U. Schreiber, From loop groups to 2-groups, Homology Homotopy Appl. 9 (2007) 101 [math/0504123] [INSPIRE].CrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    E. Witten, Some comments on string dynamics, in proceedings of Strings95, Los Angeles U.S.A. (1995) [hep-th/9507121] [INSPIRE].
  7. [7]
    J. Bagger, N. Lambert, S. Mukhi and C. Papageorgakis, Multiple membranes in M-theory, Phys. Rept. 527 (2013) 1 [arXiv:1203.3546] [INSPIRE].CrossRefADSMATHMathSciNetGoogle Scholar
  8. [8]
    D. Fiorenza, H. Sati and U. Schreiber, Multiple M5-branes, string 2-connections and 7d nonabelian Chern-Simons theory, Adv. Theor. Math. Phys. 18 (2014) 229 [arXiv:1201.5277] [INSPIRE].CrossRefMATHMathSciNetGoogle Scholar
  9. [9]
    C. Sämann and M. Wolf, Non-Abelian tensor multiplet equations from twistor space, Commun. Math. Phys. 328 (2014) 527 [arXiv:1205.3108] [INSPIRE].CrossRefADSMATHGoogle Scholar
  10. [10]
    C. Säemann and M. Wolf, Six-dimensional superconformal field theories from principal 3-bundles over twistor space, Lett. Math. Phys. 104 (2014) 1147 [arXiv:1305.4870] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  11. [11]
    C. Sämann and M. Wolf, On twistors and conformal field theories from six dimensions, J. Math. Phys. 54 (2013) 013507 [arXiv:1111.2539] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  12. [12]
    L.J. Mason, R.A. Reid-Edwards and A. Taghavi-Chabert, Conformal field theories in six-dimensional twistor space, J. Geom. Phys. 62 (2012) 2353 [arXiv:1111.2585] [INSPIRE].CrossRefADSMATHMathSciNetGoogle Scholar
  13. [13]
    H. Samtleben, E. Sezgin and R. Wimmer, (1, 0) superconformal models in six dimensions, JHEP 12 (2011) 062 [arXiv:1108.4060] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  14. [14]
    H. Samtleben, E. Sezgin, R. Wimmer and L. Wulff, New superconformal models in six dimensions: gauge group and representation structure, PoS(CORFU2011)071 [arXiv:1204.0542] [INSPIRE].
  15. [15]
    H. Samtleben, E. Sezgin and R. Wimmer, Six-dimensional superconformal couplings of non-abelian tensor and hypermultiplets, JHEP 03 (2013) 068 [arXiv:1212.5199] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  16. [16]
    I. Bandos, H. Samtleben and D. Sorokin, Duality-symmetric actions for non-Abelian tensor fields, Phys. Rev. D 88 (2013) 025024 [arXiv:1305.1304] [INSPIRE].ADSGoogle Scholar
  17. [17]
    S.-L. Ko, D. Sorokin and P. Vanichchapongjaroen, The M5-brane action revisited, JHEP 11 (2013) 072 [arXiv:1308.2231] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  18. [18]
    C.-S. Chu, A theory of non-Abelian tensor gauge field with non-Abelian gauge symmetry G × G, Nucl. Phys. B 866 (2013) 43 [arXiv:1108.5131] [INSPIRE].CrossRefADSGoogle Scholar
  19. [19]
    C.-S. Chu and S.-L. Ko, Non-Abelian action for multiple five-branes with self-dual tensors, JHEP 05 (2012) 028 [arXiv:1203.4224] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  20. [20]
    S. Palmer and C. Sämann, Six-dimensional (1, 0) superconformal models and higher gauge theory, J. Math. Phys. 54 (2013) 113509 [arXiv:1308.2622] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  21. [21]
    E. Getzler, Lie theory for nilpotent L -algebras, Ann. Math. 170 (2009) 271 [math/0404003].CrossRefMATHMathSciNetGoogle Scholar
  22. [22]
    A. Henriques, Integrating L -algebras, Comp. Math. 144 (2008) 1017 [math/0603563].CrossRefMATHMathSciNetGoogle Scholar
  23. [23]
    P. Severa, L -algebras as 1-jets of simplicial manifolds (and a bit beyond), math/0612349.
  24. [24]
    R. Zucchini, AKSZ models of semistrict higher gauge theory, JHEP 03 (2013) 014 [arXiv:1112.2819] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  25. [25]
    Y. Sheng and C. Zhu, Integration of Lie 2-algebras and their morphisms, Lett. Math. Phys. 102 (2012) 223 [arXiv:1109.4002].CrossRefADSMATHMathSciNetGoogle Scholar
  26. [26]
    S. Palmer and C. Sämann, Self-dual string and higher instanton solutions, Phys. Rev. D 89 (2014) 065036 [arXiv:1312.5644] [INSPIRE].ADSGoogle Scholar
  27. [27]
    D. Fiorenza, U. Schreiber and J. Stasheff, Čech cocycles for differential characteristic classes: an-Lie theoretic construction, Adv. Theor. Math. Phys. 16 (2012) 149 [arXiv:1011.4735] [INSPIRE].CrossRefMATHMathSciNetGoogle Scholar
  28. [28]
    A. Kotov and T. Strobl, Characteristic classes associated to Q-bundles, Int. J. Geom. Meth. Mod. Phys. 12 (2014) 1550006 [arXiv:0711.4106] [INSPIRE].CrossRefMathSciNetGoogle Scholar
  29. [29]
    J. Bénabou, Lecture Notes in Mathematics. Vol. 47: Introduction to bicategories, Springer, Heidelberg Germany (1967).Google Scholar
  30. [30]
    T. Leinster, Basic bicategories, math/9810017.
  31. [31]
    T. Leinster, Higher operads, higher categories, Cambridge University Press, Cambridge U.K. (2004).CrossRefMATHGoogle Scholar
  32. [32]
    G. Kelly, On MacLanes conditions for coherence of natural associativities, commutativities, etc., J. Algebra 1 (1964) 397.CrossRefMATHMathSciNetGoogle Scholar
  33. [33]
    D. Verity, Enriched categories, internal categories and change of base, Reprints Theory Appl. Categ. 20 (2011) 1.MathSciNetGoogle Scholar
  34. [34]
    K. Hardie, K. Kamps and R. Kieboom, A homotopy bigroupoid of a topological space, Appl. Categ. Struct. 9 (2001) 311.CrossRefMATHMathSciNetGoogle Scholar
  35. [35]
    S. Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics, second edition, Springer, Heidelberg Germany (1998).Google Scholar
  36. [36]
    D. Roytenberg, On weak Lie 2-algebras, in: XXVI Workshop on Geometrical Methods in Physics 2007, P. Kielanowski et al. eds., AIP Conf. Proc. 956 (2007) 180 [arXiv:0712.3461].
  37. [37]
    J.C. Baez and A.S. Crans, Higher-dimensional algebra VI: Lie 2-algebras, Theory Appl. Categor. 12 (2004) 492 [math/0307263] [INSPIRE].MATHMathSciNetGoogle Scholar
  38. [38]
    G. Segal, Classifying spaces and spectral sequences, Publ. Math. IHÉS 34 (1968) 105.CrossRefMATHGoogle Scholar
  39. [39]
    B. Jurčo, From simplicial Lie algebras and hypercrossed complexes to differential graded Lie algebras via 1-jets, J. Geom. Phys. 62 (2012) 2389 [arXiv:1110.0815].CrossRefADSMATHMathSciNetGoogle Scholar
  40. [40]
    M. Kontsevich, Deformation quantization of Poisson manifolds. I, Lett. Math. Phys. 66 (2003) 157 [q-alg/9709040].CrossRefADSMATHMathSciNetGoogle Scholar
  41. [41]
    T. Lada and J. Stasheff, Introduction to sh Lie algebras for physicists, Int. J. Theor. Phys. 32 (1993) 1087 [hep-th/9209099] [INSPIRE].CrossRefMATHMathSciNetGoogle Scholar
  42. [42]
    H. Sati, U. Schreiber and J. Stasheff, L algebra connections and applications to String- and Chern-Simons n-transport, in: Quantum Field Theory, B. Fauser, J. Tolksdorf and E. Zeidler eds., Birkhäuser, Boston U.S.A. (2009), pg. 303 [arXiv:0801.3480] [INSPIRE].
  43. [43]
    J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization, Birkhäuser, Boston U.S.A. (2007).Google Scholar
  44. [44]
    U. Schreiber and K. Waldorf, Connections on non-Abelian gerbes and their holonomy, Theory Appl. Categ. 28 (2013) 476 [arXiv:0808.1923].MATHMathSciNetGoogle Scholar
  45. [45]
    W. Wang, On 3-gauge transformations, 3-curvatures and Gray-categories, J. Math. Phys. 55 (2014) 043506 [arXiv:1311.3796] [INSPIRE].CrossRefADSGoogle Scholar
  46. [46]
    B. Jurčo, Crossed module bundle gerbes: classification, string group and differential geometry, Int. J. Geom. Meth. Mod. Phys. 8 (2011) 1079 [math/0510078] [INSPIRE].CrossRefMATHGoogle Scholar
  47. [47]
    B. Jurčo, Nonabelian bundle 2-gerbes, arXiv:0911.1552 [INSPIRE].
  48. [48]
    M.K. Murray, A Penrose transform for the twistor space of an even dimensional conformally flat Riemannian manifold, Ann. Global Anal. Geom. 4 (1986) 71.CrossRefMATHMathSciNetGoogle Scholar
  49. [49]
    L.P. Hughston and W.T. Shaw, Minimal curves in six-dimensions, Class. Quant. Grav. 4 (1987) 869 [INSPIRE].CrossRefADSMATHMathSciNetGoogle Scholar
  50. [50]
    L.P. Hughston, The wave equation in even dimensions, Twistor Newsletter 9 (1979) 60.ADSGoogle Scholar
  51. [51]
    L.P. Hughston, Applications of SO(8) spinors, in: Gravitation and Geometry: a volume in honour of Ivor Robinson, W. Rindler and A. Trautman eds., Bibliopolis, Naples Italy (1987).Google Scholar
  52. [52]
    L.P. Hughston and L.J. Mason, A generalized Kerr-Robinson theorem, Class. Quant. Grav. 5 (1988) 275 [INSPIRE].CrossRefADSMATHMathSciNetGoogle Scholar
  53. [53]
    R.J. Baston and M.G. Eastwood, The Penrose transform, Oxford University Press, Oxford U.K. (1990).Google Scholar
  54. [54]
    L.J. Mason and R.A. Reid-Edwards, The supersymmetric Penrose transform in six dimensions, arXiv:1212.6173 [INSPIRE].
  55. [55]
    Y.I. Manin, Grundlehren der mathematischen Wissenschaften. Vol. 289: Gauge field theory and complex geometry, Springer, Berlin Germany (1988).Google Scholar
  56. [56]
    G.A. Demessie and C. Saemann, The Poincaré lemma in categorified differential geometry, work in progress.Google Scholar
  57. [57]
    T. Lada and M. Markl, Strongly homotopy Lie algebras, Commun. Alg. 23 (1995) 2147 [hep-th/9406095] [INSPIRE].CrossRefMATHMathSciNetGoogle Scholar
  58. [58]
    M. Markl, S. Shnider and J. Stasheff, Operads in algebra, topology and physics, Mathematical Surveys and Monographs, American Mathematical Society Press, Providence U.S.A. (2002).Google Scholar
  59. [59]
    J. Stasheff, Deformation theory and the Batalin-Vilkovisky master equation, Proceedings of the Conference on Deformation Theory, Ascona Switzerland (1996) [q-alg/9702012] [INSPIRE].
  60. [60]
    C. Lazaroiu, String field theory and brane superpotentials, JHEP 10 (2001) 018 [hep-th/0107162] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Branislav Jurčo
    • 1
  • Christian Sämann
    • 2
  • Martin Wolf
    • 3
  1. 1.Mathematical Institute, Faculty of Mathematics and PhysicsCharles University in PraguePragueCzech Republic
  2. 2.Maxwell Institute for Mathematical Sciences, Department of MathematicsHeriot-Watt UniversityEdinburghUnited Kingdom
  3. 3.Department of MathematicsUniversity of SurreyGuildfordUnited Kingdom

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