L-Algebra Connections and Applications to String- and Chern-Simons n-Transport

  • Hisham Sati
  • Urs Schreiber
  • Jim Stasheff


We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L-algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals this way and describe aspects of their parallel transport and quantization.

It is known that over a D-brane the Kalb-Ramond background field of the string restricts to a 2-bundle with connection (a gerbe) which can be seen as the obstruction to lifting the PU(H)-bundle on the D-brane to a U(H)-bundle. We discuss how this phenomenon generalizes from the ordinary central extension U(1) → U(H) → PU(H) to higher categorical central extensions, like the String-extension BU(1) → String(G) → G. Here the obstruction to the lift is a 3-bundle with connection (a 2-gerbe): the Chern-Simons 3-bundle classified by the first Pontrjagin class. For G = Spin(n) this obstructs the existence of a String-structure. We discuss how to describe this obstruction problem in terms of Lie n-algebras and their corresponding categorified Cartan-Ehresmann connections. Generalizations even beyond String-extensions are then straightforward. For G = Spin(n) the next step is “Fivebrane structures” whose existence is obstructed by certain generalized Chern-Simons 7-bundles classified by the second Pontrjagin class.


Cartan-Ehresman connection L-algebra Chern-Simons theory BF-theory 2-bundles Eilenberg-MacLane spaces differential greded algebras branes strings 

Mathematics Subject Classification (2000)

Primary 83E30 Secondary 55P20 81T30 55R45 


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  1. [1]
    M. Alexandrov, M. Kontsevich, A. Schwarz and O. Zaboronsky, The Geometry of the Master Equation and Topological Quantum Field Theory, [http: //arxiv. org/refs/hep-th/0608150]Google Scholar
  2. [2]
    A. Asada, Characteristic classes of loop group bundles and generalized string classes, Differential geometry and its applications (Eger, 1989), 33–66, Colloq. Math. Soc. János Bolyai, 56, North-Holland, Amsterdam, (1992)Google Scholar
  3. [3]
    P. Aschieri and B. Jurco, Gerbes, M5-brane anomalies and E 8 gauge theory J. High Energy Phys. 0410 (2004) 068, [arXiv:hep-th/0409200v1].CrossRefADSMathSciNetGoogle Scholar
  4. [4]
    R. D’Auria and P. Fré, Geometric supergravity in D = 11 and its hidden supergroup, Nuclear Physics B, 201 (1982)Google Scholar
  5. [5]
    J. Baez and A. Crans, Higher-dimensional algebra VI: Lie 2-algebras, Theory and Applications of Categories 12 (2004) 492–528, [arXiv:math/0307263v5].MATHMathSciNetGoogle Scholar
  6. [6]
    T. Bartels, 2-Bundles, [arXiv:math/0410328v3]Google Scholar
  7. [7]
    J. Baez, A. Crans, U. Schreiber and D. Stevenson, From Loop Groups to 2-Groups, Homology, Homotopy and Applications, Vol. 9 (2007), No. 2, pp.101–135, [arXiv: math/0504123v2].MATHMathSciNetGoogle Scholar
  8. [8]
    J. Baez and D. Stevenson, The Classifying Space of a Topological 2-Group, [arXiv:0801.3843v1]Google Scholar
  9. [9]
    J. Baez and U. Schreiber, Higher gauge theory, in Contemporary Mathematics, 431, Categories in Algebra, Geometry and Mathematical Physics, [arXiv: math/0511710].Google Scholar
  10. [10]
    E. Bergshoeff, R. Percacci, E. Sezgin, K.S. Stelle, and P.K. Townsend, U(1)-extended gauge algebras in p-loop space, Nucl. Phys. B 398 (1993) 343, [arXiv: hep-th/9212037v1].CrossRefADSMathSciNetGoogle Scholar
  11. [11]
    E. Bergshoeff, E. Sezgin, and P.K. Townsend, Properties of the eleven-dimensional super membrane theory, Annals Phys.185 (1988) 330.CrossRefADSMathSciNetGoogle Scholar
  12. [12]
    L. Bonora, P. Cotta-Ramusino, M. Rinaldi, and J. Stasheff, The evaluation map in field theory, sigma-models and strings I, Commun. Math. Phys. 112 (1987) 237.MATHCrossRefADSMathSciNetGoogle Scholar
  13. [13]
    A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogńes de groupes de Lie compacts, Ann. Math. (2) 57 (1953), 115–207.CrossRefMathSciNetGoogle Scholar
  14. [14]
    A.K. Bousfield and V.K.A.M. Gugenheim, On PL de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 (1976), no. 179.Google Scholar
  15. [15]
    P. Bouwknegt and V. Mathai, D-branes, B-fields and twisted K-theory, J. High Energy Phys. 03 (2000) 007, [arXiv:hep-th/0002023].CrossRefADSMathSciNetGoogle Scholar
  16. [16]
    P. Bouwknegt, A. Carey, V. Mathai, M. Murray and D. Stevenson, Twisted K-theory and K-theory of bundle gerbes, Commun. Math. Phys. 228 (2002) 17, [arXiv: hep-th/0106194].MATHCrossRefADSMathSciNetGoogle Scholar
  17. [17]
    L. Breen and W. Messing, Differential geometry of gerbes, Adv. Math. 198 (2005), no. 2, 732–846, [arXiv:math/0106083v3]. MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    J.-L. Brylinski and D.A. McLaughlin, Čech cocycles for characteristic classes, Comm. Math. Phys. 178, 225–236 (1996)MATHCrossRefADSMathSciNetGoogle Scholar
  19. [19]
    J.-L. Brylinski and D.A. McLaughlin, A geometric construction of the first Pontryagin class, Quantum Topology, 209–220, World Scientific (1993)Google Scholar
  20. [20]
    A. Carey, S. Johnson, and M. Murray, Holonomy on D-branes, J. Geom. Phys. 52 (2004), no. 2, 186–216, [arXiv:hep-th/0204199v3].MATHCrossRefADSMathSciNetGoogle Scholar
  21. [21]
    A. Carey, S. Johnson, M. Murray, D. Stevenson, and B.-L. Wang, Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories, Commun. Math. Phys. 259 (2005) 577, [arXiv:math/0410013].MATHCrossRefADSMathSciNetGoogle Scholar
  22. [22]
    H. Cartan, Notions d’algébre différentielle; application aux groupes de Lie et aux variétés ou opére un groupe de Lie, Colloque de topologie (espaces fibrs), Bruxelles, 1950, pp. 15–27.Google Scholar
  23. [23]
    H. Cartan, Cohomologie reélle d’un espace fibré principal diffrentielle I, II, Séminaire Henri Cartan, 1949/50, pp. 19-01 – 19-10 and 20-01 – 20-11, CBRM, 1950.Google Scholar
  24. [24]
    L. Castellani, R. D’Auria and P. Fré, Supergravity and superstrings: a geometric perspective, World Scientific, Singapore (1991)Google Scholar
  25. [25]
    J. Cheeger and J. Simons, Differential characters and geometric invariants, in Geometry and Topology, 50–80, Lecture Notes in Math., 1167, Springer-Verlag, 1985.CrossRefMathSciNetGoogle Scholar
  26. [26]
    A. Clingher, Heterotic string data and theta functions, Adv. Theor. Math. Phys. 9 (2005) 173, [arXiv:math/0110320v2].MATHMathSciNetGoogle Scholar
  27. [27]
    R. Coquereaux and K. Pilch, String structures on loop bundles, Commun. Math. Phys. 120 (1989) 353.MATHCrossRefADSMathSciNetGoogle Scholar
  28. [28]
    E. Diaconescu, D. Freed, and G. Moore, The M theory three form and E 8 gauge theory, [arXiv: hep-th/0312069].Google Scholar
  29. [29]
    E. Diaconescu, G. Moore and E. Witten, E 8 gauge theory, and a derivation of K-theory from M-theory Adv. Theor. Math. Phys. 6 (2003) 1031, [arXiv: hep-th/0005090].MathSciNetGoogle Scholar
  30. [30]
    J.A. Dixon, M.J. Duff, and E. Sezgin, The coupling of Yang-Mills to extended objects, Phys. Lett. B 279 (1992) 265, [arXiv:hep-th/9201019v1].CrossRefADSMathSciNetGoogle Scholar
  31. [31]
    D.S. Freed, Classical Chern-Simons Theory, Part I, [hep-th/9206021v1]Google Scholar
  32. [32]
    D.S. Freed, Higher algebraic structures and quantization, [arXiv: hep-th/9212115v2]Google Scholar
  33. [33]
    D.S. Freed, Quantum groups from path integrals, [arXiv:q-alg/9501025v1]Google Scholar
  34. [34]
    D.S. Freed and E. Witten, Anomalies in string theory with D-Branes, Asian J. Math. 3 (1999) 819, [arXiv:hep-th/9907189].MATHMathSciNetGoogle Scholar
  35. [35]
    E. Getzler, Lie theory for nilpotent L -algebras, Ann. Math. [arXi v: math/0404003v4]Google Scholar
  36. [36]
    G. Ginot and M. Stiénon, Groupoid extensions, principal 2-group bundles and characteristic classes, [arXiv:0801.1238v1].Google Scholar
  37. [37]
    G. Ginot and P. Xu Cohomology of Lie 2-groups, [arXiv:0712.2069v1].Google Scholar
  38. [38]
    F. Girelli and H. Pfeiffer, Higher gauge theory-differential versus integral formulation, J. Math. Phys. 45 (2004) 3949–3971, [arXiv:hep-th/0309173v2].MATHCrossRefADSMathSciNetGoogle Scholar
  39. [39]
    F. Girelli, H. Pfeiffer, and E. Popescu, Topological higher gauge theory-from BF to BFCG theory, [arXiv:0708.3051v1].Google Scholar
  40. [40]
    M.B. Green and J.H. Schwarz, Anomaly cancellation in supersymmetric D = 10 gauge theory and superstring theory, Phys. Lett. B149 (1984) 117.ADSMathSciNetGoogle Scholar
  41. [41]
    W. Greub, S. Halperin, and R. Vanstone, Connections, curvature, and cohomology. Vol. II: Lie groups, principal bundles, and characteristic classes, Academic Press, New York-London, 1973.MATHGoogle Scholar
  42. [42]
    S. Halperin and J.-C. Thomas, Rational equivalence of fibrations with fibre G/K, Canad. J. Math. 34 (1982), no. 1, 31–43.MATHMathSciNetCrossRefGoogle Scholar
  43. [43]
    A. Henriques, Integrating L algebras, [arXiv:math/0603563v2].Google Scholar
  44. [44]
    K. Hess, Rational Homotopy Theory: A Brief Introduction, [http: //www.math.uic.edu/~bshipley/hess_ratlhtpy. pdf]Google Scholar
  45. [45]
    M.J. Hopkins, Topological aspects of topological field theories, Andrejewski lecture, Göttingen (2006)Google Scholar
  46. [46]
    M.J. Hopkins, I.M. Singer, Quadratic functions in geometry, topology, and M-theory, J. Diff. Geom. 70 (2005) 329–452, [arXiv:math.AT/0211216].MATHMathSciNetGoogle Scholar
  47. [47]
    P. Horava and E. Witten, Eleven-dimensional supergravity on a manifold with boundary, Nucl.Phys. B475 (1996) 94, [arXiv:hep-th/9603142].CrossRefADSMathSciNetGoogle Scholar
  48. [48]
    P.T. Johnstone, Stone Spaces, Cambridge studies in advanced mathematica (1986)Google Scholar
  49. [49]
    J. Kalkman, BRST model for equivariant cohomology and representatives for the equivariant Thom class, Comm. Math. Phys. 153 (1993), no. 3, 447–463.MATHCrossRefADSMathSciNetGoogle Scholar
  50. [50]
    A. Kapustin, D-branes in a topologically nontrivial B-field, Adv. Theor. Math. Phys. 4 (2000) 127, [arXiv:hep-th/9909089].MATHMathSciNetGoogle Scholar
  51. [51]
    L.H. Kauffman, Vassiliev invariants and functional integration without integration, in Stochastic analysis and mathematical physics, p 91–114Google Scholar
  52. [52]
    T.P. Killingback, World-sheet anomalies and loop geometry, Nucl. Phys. B 288 (1987) 578.CrossRefADSMathSciNetGoogle Scholar
  53. [53]
    I. Kolávr, J. Slovák and P.W. Michor, Natural operations in differential geometry, Springer-Verlag (1993)Google Scholar
  54. [54]
    A. Kotov and Th. Strobl, Characteristic classes associated to Q-bundles, [arXiv:0711.4106v1]Google Scholar
  55. [55]
    I. Kriz and H. Sati, M Theory, type IIA superstrings, and elliptic cohomology, Adv. Theor. Math. Phys. 8 (2004) 345, [arXiv:hep-th/0404013].MATHMathSciNetGoogle Scholar
  56. [56]
    K. Kuribayashi, On the vanishing problem of String classes, J. Austr. Math. Soc. (Series A) 61, 258–266 (1996)MATHCrossRefMathSciNetGoogle Scholar
  57. [57]
    T. Lada and J. Stasheff, Introduction to sh Lie algebras for physicists, Int. J. Theor. Phys. 32 (1993) 1087, [arXiv:hep-th/9209099].MATHCrossRefMathSciNetGoogle Scholar
  58. [58]
    T. Lada, M. Markl, Strongly homotopy Lie algebras, Comm. Algebra 23 (1995), no. 6, 2147–2161, [arXiv:hep-th/9406095v1].MATHCrossRefMathSciNetGoogle Scholar
  59. [59]
    V. Mathai and H. Sati, Some relations between twisted K-theory and E 8 gauge theory, J. High Energy Phys. 0403 (2004) 016, [arXiv:hep-th/0312033v4].CrossRefADSMathSciNetGoogle Scholar
  60. [60]
    J. Mickelsson and R. Percacci, Global aspects of p-branes, J. Geom. and Phys. 15 (1995) 369–380, [arXiv:hep-th/9304054v1].MATHCrossRefADSMathSciNetGoogle Scholar
  61. [61]
    I. Moerdijk and G. Reyes, Models for smooth infinitesimal analysis, Springer-Verlag (1991)Google Scholar
  62. [62]
    M. Murray, Bundle gerbes, J. London Math. Soc. (2) 54 (1996), no. 2, 403–416.MATHMathSciNetGoogle Scholar
  63. [63]
    M. Murray and D. Stevenson, Higgs fields, bundle gerbes and string structures, Commun. Math. Phys. 243 (2003) 541–555, [arXiv:math/0106179v1].MATHCrossRefADSMathSciNetGoogle Scholar
  64. [64]
    R. Percacci and E. Sezgin, Symmetries of p-branes, Int. J. Theor. Phys. A 8 (1993) 5367, [arXiv:hep-th/9210061v1].ADSMathSciNetGoogle Scholar
  65. [65]
    D. Roberts and U. Schreiber, The inner automorphism 3-group of a strict 2-group, to appear in Journal of Homotopy and Related Structures, [arXiv:0708.1741].Google Scholar
  66. [66]
    D. Roytenberg, AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories, [http://arxiv.org/abs/hep-th/0608150]Google Scholar
  67. [67]
    H. Sati, On Higher twists in string theory, [arXiv:hep-th/0701232v2].Google Scholar
  68. [68]
    H. Sati, U. Schreiber, J. Stasheff Fivebrane structures: topology, [arXiv: 0805.0564]Google Scholar
  69. [69]
    H. Sati, U. Schreiber, J. Stasheff Fivebrane structures: geometry, (in preparation)Google Scholar
  70. [70]
    U. Schreiber, On nonabelian differential cohomology, lecture notes, [http://www.math.uni-hamburg.de/home/schreiber/ndclecture.pdf]Google Scholar
  71. [71]
    U. Schreiber, On ∞-Lie theory, [http://www.math.uni-hamburg.de/home/schreiber/act ion.pdf]Google Scholar
  72. [72]
    U. Schreiber and K. Waldorf, Parallel transport and functors, [arXiv: 0705.0452v1].Google Scholar
  73. [73]
    U. Schreiber and K. Waldorf, 2-Functors vs. differential forms, [arXiv:0802.0663v1]Google Scholar
  74. [74]
    U. Schreiber and K. Waldorf, Connections on non-abelian Gerbes and their Holonomy, to appear.Google Scholar
  75. [75]
    J. Stasheff and M. Schlessinger, Deformation Theory and Rational Homotopy Type, unpublished.Google Scholar
  76. [76]
    St. Stolz and P. Teichner, What is an elliptic object? in Topology, geometry and quantum field theory, London Math. Soc. LNS 308, Cambridge University Press 2004, 247–343.MathSciNetGoogle Scholar
  77. [77]
    Sullivan, Infinitesimal computations in topology, Publications mathématique de l I.H.E.S., tome 47 (1977), p. 269–331MATHCrossRefMathSciNetGoogle Scholar
  78. [78]
    C. Teitelboim and M. Henneaux, Quantization of gauge systems Princeton University Press, 1992.Google Scholar
  79. [79]
    E. Witten, On Flux quantization in M-Theory and the effective action, J. Geom. Phys. 22 (1997) 1–13, [arXiv:hep-th/9609122].MATHCrossRefADSMathSciNetGoogle Scholar
  80. [80]
    A. Yekutieli, Central Extensions of Gerbes, [arXiv:0801.0083v1].Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Hisham Sati
    • 1
  • Urs Schreiber
    • 2
  • Jim Stasheff
    • 3
  1. 1.Department of MathematicsYale UniversityNew Haven
  2. 2.Fachbereich MathematikUniversität HamburgHamburg
  3. 3.Department of MathematicsUniversity of PennsylvaniaPhiladelphia

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