Lie 2-algebra models

  • Patricia Ritter
  • Christian Sämann
Open Access


In this paper, we begin the study of zero-dimensional field theories with fields taking values in a semistrict Lie 2-algebra. These theories contain the IKKT matrix model and various M-brane related models as special cases. They feature solutions that can be interpreted as quantized 2-plectic manifolds. In particular, we find solutions corresponding to quantizations of \( \mathbb{R} \) 3, S 3 and a five-dimensional Hpp-wave. Moreover, by expanding a certain class of Lie 2-algebra models around the solution corresponding to quantized \( \mathbb{R} \) 3, we obtain higher BF-theory on this quantized space.


Non-Commutative Geometry M(atrix) Theories M-Theory Matrix Models 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, A large-N reduced model as superstring, Nucl. Phys. B 498 (1997) 467 [hep-th/9612115] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  2. [2]
    J. Bagger and N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes, Phys. Rev. D 77 (2008) 065008 [arXiv:0711.0955] [INSPIRE].ADSMathSciNetGoogle Scholar
  3. [3]
    A. Gustavsson, Algebraic structures on parallel M2-branes, Nucl. Phys. B 811 (2009) 66 [arXiv:0709.1260] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  4. [4]
    O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, \( \mathcal{N} \) = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  5. [5]
    P. Di Francesco, P.H. Ginsparg and J. Zinn-Justin, 2D Gravity and random matrices, Phys. Rept. 254 (1995) 1 [hep-th/9306153] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    H. Aoki et al., IIB matrix model, Prog. Theor. Phys. Suppl. 134 (1999) 47 [hep-th/9908038] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    H. Aoki et al., Noncommutative Yang-Mills in IIB matrix model, Nucl. Phys. B 565 (2000) 176 [hep-th/9908141] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from N =4 super Yang-Mills, JHEP 04(2002) 013[hep-th/0202021][INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  9. [9]
    J. DeBellis, C. Sämann and R.J. Szabo, Quantized Nambu-Poisson manifolds in a 3-Lie algebra reduced model, JHEP 04 (2011) 075 [arXiv:1012.2236] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    H. Steinacker, Emergent geometry and gravity from matrix models: an introduction, Class. Quant. Grav. 27 (2010) 133001 [arXiv:1003.4134] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  11. [11]
    B. Zwiebach, Closed string field theory: quantum action and the B-V master equation, Nucl. Phys. B 390 (1993) 33 [hep-th/9206084] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  12. [12]
    H. Kajiura and J. Stasheff, Homotopy algebras inspired by classical open-closed string field theory, Commun. Math. Phys. 263 (2006) 553 [math/0410291] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    C. Hofman and J.-S. Park, Topological open membranes, hep-th/0209148 [INSPIRE].
  14. [14]
    S. Palmer and C. Sämann, M-brane models from non-abelian gerbes, JHEP 07 (2012) 010 [arXiv:1203.5757] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    H. Sati, U. Schreiber and J. Stasheff, L algebra connections and applications to String- and Chern-Simons n-transport, arXiv:0801.3480 [INSPIRE].
  16. [16]
    J.C. Baez and J. Huerta, An invitation to higher gauge theory, arXiv:1003.4485 [INSPIRE].
  17. [17]
    C. Sämann and M. Wolf, Non-abelian tensor multiplet equations from twistor space, arXiv:1205.3108 [INSPIRE].
  18. [18]
    C. Sämann and M. Wolf, Six-dimensional superconformal field theories from principal 3-bundles over twistor space, arXiv:1305.4870 [INSPIRE].
  19. [19]
    M. Sato, Covariant formulation of M-theory I, Int. J. Mod. Phys. A 24 (2009) 5019 [arXiv:0902.1333] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    K. Lee and J.-H. Park, Three-algebra for supermembrane and two-algebra for superstring, JHEP 04 (2009) 012 [arXiv:0902.2417] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    M. Sato, Covariant formulation of M-theory II, arXiv:0902.4102 [INSPIRE].
  22. [22]
    M. Sato, Model of M-theory with eleven matrices, JHEP 07 (2010) 026 [arXiv:1003.4694] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    K. Furuuchi and D. Tomino, Supersymmetric reduced models with a symmetry based on Filippov algebra, JHEP 05 (2009) 070 [arXiv:0902.2041] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  24. [24]
    J.-H. Park and C. Sochichiu, Taking off the square root of Nambu-Goto action and obtaining Filippov-Lie algebra gauge theory action, Eur. Phys. J. C 64 (2009) 161 [arXiv:0806.0335] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  25. [25]
    J. DeBellis, C. Sämann and R.J. Szabo, Quantized Nambu-Poisson manifolds and n-Lie algebras, J. Math. Phys. 51 (2010) 122303 [arXiv:1001.3275] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  26. [26]
    C.-S. Chu and G.S. Sehmbi, D1-strings in large RR 3-form flux, quantum Nambu geometry and M5-branes in C-field, J. Phys. A 45 (2012) 055401 [arXiv:1110.2687] [INSPIRE].ADSMathSciNetGoogle Scholar
  27. [27]
    S. Mukhi and C. Papageorgakis, M2 to D2, JHEP 05 (2008) 085 [arXiv:0803.3218] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  28. [28]
    F. Cantrijn, A. Ibort and M. de Leon, On the geometry of multisymplectic manifolds, J. Austral. Math. Soc. 66 (1999) 303.CrossRefzbMATHGoogle Scholar
  29. [29]
    J.C. Baez, A.E. Hoffnung and C.L. Rogers, Categorified symplectic geometry and the classical string, Commun. Math. Phys. 293 (2010) 701 [arXiv:0808.0246] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  30. [30]
    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].
  31. [31]
    J.C. Baez and A.S. Crans, Higher-Dimensional Algebra VI: Lie 2-Algebras, Theor. Appl. Categor. 12 (2004) 492 [math/0307263] [INSPIRE].
  32. [32]
    J.C. Baez, Higher Yang-Mills theory, hep-th/0206130 [INSPIRE].
  33. [33]
    R. Zucchini, AKSZ models of semistrict higher gauge theory, JHEP 03 (2013) 014 [arXiv:1112.2819] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  34. [34]
    M. Kontsevich, Feynman diagrams and low-dimensional topology, Birkhäuser, Boston U.S.A. (1992), pg. 173.Google Scholar
  35. [35]
    K. Igusa, Graph cohomology and Kontsevich cycles, math/0303157 [INSPIRE].
  36. [36]
    M. Markl, S. Shnider and J. Stasheff, Operads in Algebra, Topology and Physics (Mathematical Surveys and Monographs), American Mathematical Society, Providence U.S.A. (2007).CrossRefGoogle Scholar
  37. [37]
    J.F. Martins and A. Mikovic, Lie crossed modules and gauge-invariant actions for 2-BF theories, Adv. Theor. Math. Phys. 15 (2011) 1059 [arXiv:1006.0903] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  38. [38]
    S.A. Cherkis and C. Sämann, Multiple M2-branes and generalized 3-Lie algebras, Phys. Rev. D 78 (2008) 066019 [arXiv:0807.0808] [INSPIRE].ADSGoogle Scholar
  39. [39]
    J. Bagger and N. Lambert, Three-algebras and N = 6 Chern-Simons gauge theories, Phys. Rev. D 79 (2009) 025002 [arXiv:0807.0163] [INSPIRE].ADSMathSciNetGoogle Scholar
  40. [40]
    P. de Medeiros, J. Figueroa-O’Farrill, E. Mendez-Escobar and P. Ritter, On the Lie-algebraic origin of metric 3-algebras, Commun. Math. Phys. 290 (2009) 871 [arXiv:0809.1086] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  41. [41]
    H. Awata, M. Li, D. Minic and T. Yoneya, On the quantization of Nambu brackets, JHEP 02 (2001) 013 [hep-th/9906248] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  42. [42]
    P. Gautheron, Some remarks concerning Nambu mechanics, Lett. Math. Phys. 37 (1996) 103.ADSCrossRefzbMATHMathSciNetGoogle Scholar
  43. [43]
    C.L. Rogers, Courant algebroids from categorified symplectic geometry, arXiv:1001.0040 [INSPIRE].
  44. [44]
    D. Fiorenza, C.L. Rogers and U. Schreiber, L algebras of local observables from higher prequantum bundles, arXiv:1304.6292.
  45. [45]
    C. Sämann and R.J. Szabo, Groupoids, loop spaces and quantization of 2-plectic manifolds, Rev. Math. Phys. 25 (2013) 1330005 [arXiv:1211.0395] [INSPIRE].CrossRefMathSciNetGoogle Scholar
  46. [46]
    S. Palmer and C. Sämann, Constructing generalized self-dual strings, JHEP 10 (2011) 008 [arXiv:1105.3904] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    C.-S. Chu and D.J. Smith, Towards the quantum geometry of the m5-brane in a constant C-field from multiple membranes, JHEP 04 (2009) 097 [arXiv:0901.1847] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032 [hep-th/9908142] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  49. [49]
    C. Sämann and R.J. Szabo, Groupoid quantization of loop spaces, PoS(CORFU2011)046 [arXiv:1203.5921] [INSPIRE].
  50. [50]
    M. Blau, J.M. Figueroa-O’Farrill and G. Papadopoulos, Penrose limits, supergravity and brane dynamics, Class. Quant. Grav. 19 (2002) 4753 [hep-th/0202111] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  51. [51]
    P. Ševera and A. WEinstein, Poisson geometry with a 3 form background, Prog. Theor. Phys. Suppl. 144 (2001) 145 [math/0107133] [INSPIRE].ADSGoogle Scholar
  52. [52]
    F. Petalidou, On the geometric quantization of twisted Poisson manifolds, J. Math. Phys. 48 (2007) 083502 [arXiv:0704.2989] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  53. [53]
    D. Mylonas, P. Schupp and R.J. Szabo, Membrane σ-models and Quantization of Non-Geometric Flux Backgrounds, JHEP 09 (2012) 012 [arXiv:1207.0926] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  54. [54]
    J. Nuiten, Cohomological quantization of local prequantum boundary field theory, MSc Thesis, Utrecht University, Utrecht The Netherlands (2013),
  55. [55]
    C.L. Rogers, Higher symplectic geometry, arXiv:1106.4068 [INSPIRE].
  56. [56]
    T. Voronov, Higher derived brackets and homotopy algebras, J. Pure Appl. Alg. 202 (2005) 133 [math/0304038].CrossRefzbMATHMathSciNetGoogle Scholar
  57. [57]
    T. Lada and J. Stasheff, Introduction to SH Lie algebras for physicists, Int. J. Theor. Phys. 32 (1993) 1087 [hep-th/9209099] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  58. [58]
    T. Banks, W. Fischler, S. Shenker and L. Susskind, M theory as a matrix model: a conjecture, Phys. Rev. D 55 (1997) 5112 [hep-th/9610043] [INSPIRE].ADSMathSciNetGoogle Scholar
  59. [59]
    Y. Kimura, Noncommutative gauge theories on fuzzy sphere and fuzzy torus from matrix model, Prog. Theor. Phys. 106 (2001) 445 [hep-th/0103192] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  60. [60]
    S. Iso, Y. Kimura, K. Tanaka and K. Wakatsuki, Noncommutative gauge theory on fuzzy sphere from matrix model, Nucl. Phys. B 604 (2001) 121 [hep-th/0101102] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  61. [61]
    A. Hammou, M. Lagraa and M. Sheikh-Jabbari, Coherent state induced star product on \( R_{\lambda}^3 \) and the fuzzy sphere, Phys. Rev. D 66 (2002) 025025 [hep-th/0110291] [INSPIRE].ADSMathSciNetGoogle Scholar
  62. [62]
    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].ADSCrossRefMathSciNetGoogle Scholar
  63. [63]
    M. Fukuma, H. Kawai, Y. Kitazawa and A. Tsuchiya, String field theory from IIB matrix model, Nucl. Phys. B 510 (1998) 158 [hep-th/9705128] [INSPIRE].CrossRefMathSciNetGoogle Scholar
  64. [64]
    T. Lada and M. Markl, Strongly homotopy Lie algebras, Commun. Alg. 23 (1995) 2147 [hep-th/9406095] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  65. [65]
    Y. Nambu, Generalized Hamiltonian dynamics, Phys. Rev. D 7 (1973) 2405 [INSPIRE].ADSMathSciNetGoogle Scholar
  66. [66]
    L. Takhtajan, On foundation of the generalized Nambu mechanics (second version), Commun. Math. Phys. 160 (1994) 295 [hep-th/9301111] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  67. [67]
    V.T. Filippov, n-Lie algebras, Sib. Mat. Zh. 26 (1985) 126.zbMATHGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Centro de Estudios Científicos (CECs)ValdiviaChile
  2. 2.Maxwell Institute for Mathematical Sciences, Department of MathematicsHeriot-Watt UniversityEdinburghU.K.

Personalised recommendations