Advertisement

Supersymmetric Yang-Mills theory as higher Chern-Simons theory

  • Christian Sämann
  • Martin Wolf
Open Access
Regular Article - Theoretical Physics

Abstract

We observe that the string field theory actions for the topological sigma models describe higher or categorified Chern-Simons theories. These theories yield dynamical equations for connective structures on higher principal bundles. As a special case, we consider holomorphic higher Chern-Simons theory on the ambitwistor space of four-dimensional space-time. In particular, we propose a higher ambitwistor space action functional for maximally supersymmetric Yang-Mills theory.

Keywords

Chern-Simons Theories Differential and Algebraic Geometry String Field Theory Superspaces 

Notes

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.

References

  1. [1]
    E. Witten, Chern-Simons gauge theory as a string theory, Prog. Math. 133 (1995) 637 [hep-th/9207094] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  2. [2]
    E. Witten, Interacting field theory of open superstrings, Nucl. Phys. B 276 (1986) 291 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    C. Wendt, Scattering amplitudes and contact interactions in Witten’s superstring field theory, Nucl. Phys. B 314 (1989) 209 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    T. Erler, S. Konopka and I. Sachs, Resolving Witten’s superstring field theory, JHEP 04 (2014) 150 [arXiv:1312.2948] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Covariant string field theory, Phys. Rev. D 34 (1986) 2360 [INSPIRE].ADSMathSciNetGoogle Scholar
  6. [6]
    H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Covariant string field theory. II, Phys. Rev. D 35 (1987) 1318 [INSPIRE].
  7. [7]
    M.R. Gaberdiel and B. Zwiebach, Tensor constructions of open string theories. I: Foundations, Nucl. Phys. B 505 (1997) 569 [hep-th/9705038] [INSPIRE].
  8. [8]
    B. Zwiebach, Oriented open-closed string theory revisited, Annals Phys. 267 (1998) 193 [hep-th/9705241] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    T. Nakatsu, Classical open-string field theory: A -algebra, renormalization group and boundary states, Nucl. Phys. B 642 (2002) 13 [hep-th/0105272] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Y. Iimori, T. Noumi, Y. Okawa and S. Torii, From the Berkovits formulation to the Witten formulation in open superstring field theory, JHEP 03 (2014) 044 [arXiv:1312.1677] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    B. Zwiebach, Closed string field theory: quantum action and the Batalin-Vilkovisky master equation, Nucl. Phys. B 390 (1993) 33 [hep-th/9206084] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    E. Witten, Mirror manifolds and topological field theory, in Essays on mirror manifolds, S.-T. Yau ed., International Press (1992) [hep-th/9112056] [INSPIRE].
  13. [13]
    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 (2009), pg. 303 [arXiv:0801.3480] [INSPIRE].
  14. [14]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    D. Fiorenza, C.L. Rogers and U. Schreiber, A higher Chern-Weil derivation of AKSZ σ-models, Int. J. Geom. Meth. Mod. Phys. 10 (2013) 1250078 [arXiv:1108.4378] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    E. Soncini and R. Zucchini, 4D semistrict higher Chern-Simons theory I, JHEP 10 (2014) 079 [arXiv:1406.2197] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  17. [17]
    R. Zucchini, A Lie based 4-dimensional higher Chern-Simons theory, J. Math. Phys. 57 (2016) 052301 [arXiv:1512.05977] [INSPIRE].
  18. [18]
    P. Ritter and C. Sämann, L -algebra models and higher Chern-Simons theories, Rev. Math. Phys. 28 (2016) 1650021 [arXiv:1511.08201] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    A.S. Schwarz, A-model and generalized Chern-Simons theory, Phys. Lett. B 620 (2005) 180 [hep-th/0501119] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    M. Movshev and A.S. Schwarz, On maximally supersymmetric Yang-Mills theories, Nucl. Phys. B 681 (2004) 324 [hep-th/0311132] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    M. Movshev and A.S. Schwarz, Algebraic structure of Yang-Mills theory, Prog. Math. 244 (2006) 473 [hep-th/0404183] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    L.J. Mason, Twistor actions for non-self-dual fields: a derivation of twistor-string theory, JHEP 10 (2005) 009 [hep-th/0507269] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    R. Boels, L.J. Mason and D. Skinner, Supersymmetric gauge theories in twistor space, JHEP 02 (2007) 014 [hep-th/0604040] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    L.J. Mason and D. Skinner, An ambitwistor Yang-Mills Lagrangian, Phys. Lett. B 636 (2006) 60 [hep-th/0510262] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    B. Jia, Topological σ-models on supermanifolds, Nucl. Phys. B 915 (2017) 84 [arXiv:1608.00597] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    B. Jia, Topological string theory revisited I: the stage, Int. J. Mod. Phys. A 31 (2016) 1650135 [arXiv:1605.03207] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    J. Stasheff, Differential graded Lie algebras, quasi-Hopf algebras and higher homotopy algebras, in Quantum groups, Lecture Notes in Mathematics, vol. 1510, Springer, Berlin Germany (1992), pp. 120-137.Google Scholar
  29. [29]
    J.D. Stasheff, Homotopy associativity of H-spaces. I, Trans. Am. Math. Soc. 108 (1963) 275.Google Scholar
  30. [30]
    T. Lada and J. Stasheff, Introduction to SH Lie algebras for physicists, Int. J. Theor. Phys. 32 (1993) 1087 [hep-th/9209099] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    M. Kontsevich, Feynman diagrams and low-dimensional topology, in First European Congress of Mathematics, Paris France (1992), Progress in Mathematics, vol. 120, Birkhäuser, Basel Switzerland (1994), pp. 97-121.Google Scholar
  32. [32]
    H. Kajiura, Noncommutative homotopy algebras associated with open strings, Rev. Math. Phys. 19 (2007) 1 [math.QA/0306332] [INSPIRE].
  33. [33]
    M. Markl, S. Shnider and J. Stasheff, Operads in algebra, topology and physics, Mathematical Surveys and Monographs, American Mathematical Society, Providence U.S.A. (2002).Google Scholar
  34. [34]
    J.C. Baez and A.S. Crans, Higher-dimensional algebra VI: Lie 2-algebras, Theor. Appl. Categ. 12 (2004) 492, http://tac.mta.ca/tac/volumes/12/15/12-15.pdf [math.QA/0307263] [INSPIRE].
  35. [35]
    T. Lada and M. Markl, Strongly homotopy Lie algebras, Commun. Alg. 23 (1995) 2147 [hep-th/9406095] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    R. Zucchini, Algebraic formulation of higher gauge theory, arXiv:1702.01545 [INSPIRE].
  37. [37]
    B. Jurčo, C. Sämann and M. Wolf, Semistrict higher gauge theory, JHEP 04 (2015) 087 [arXiv:1403.7185] [INSPIRE].ADSMathSciNetGoogle Scholar
  38. [38]
    A. Henriques, Integrating L -algebras, Comp. Math. 144 (2008) 1017 [math.AT/0603563].
  39. [39]
    T. Nikolaus, U. Schreiber, and D. Stevenson, Principal-bundles: general theory, J. Homot. Relat. Struct. 10 (2015) 749 [arXiv:1207.0248].MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    B. Jurčo, C. Sämann and M. Wolf, Higher groupoid bundles, higher spaces and self-dual tensor field equations, Fortschr. Phys. 64 (2016) 674 [arXiv:1604.01639] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    E. Witten, Topological σ-models, Commun. Math. Phys. 118 (1988) 411 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    M. Roček and N. Wadhwa, On Calabi-Yau supermanifolds, Adv. Theor. Math. Phys. 9 (2005) 315 [hep-th/0408188] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. B 77 (1978) 394 [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    J. Isenberg, P.B. Yasskin and P.S. Green, Non-self-dual gauge fields, Phys. Lett. B 78 (1978) 462 [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    M. Eastwood, Supersymmetry, twistors, and the Yang-Mills equations, Trans. Am. Math. Soc. 301 (1987) 615.MathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    J.P. Harnad, J. Hurtubise, M. Legare and S. Shnider, Constraint equations and field equations in supersymmetric N = 3 Yang-Mills theory, Nucl. Phys. B 256 (1985) 609 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    J.P. Harnad and S. Shnider, Constraints and field equations for ten-dimensional super Yang-Mills theory, Commun. Math. Phys. 106 (1986) 183 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    A.D. Popov and C. Sämann, On supertwistors, the Penrose-Ward transform and \( \mathcal{N}=4 \) super-Yang-Mills theory, Adv. Theor. Math. Phys. 9 (2005) 931 [hep-th/0405123] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    Y.I. Manin, Gauge field theory and complex geometry, Grundlehren der Mathematischen Wissenschaften, vol. 289, Springer, Berlin Germany (1988).Google Scholar
  50. [50]
    A.D. Popov, Selfdual Yang-Mills: symmetries and moduli space, Rev. Math. Phys. 11 (1999) 1091 [hep-th/9803183] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    A.D. Popov, Holomorphic Chern-Simons-Witten theory: from 2D to 4D conformal field theories, Nucl. Phys. B 550 (1999) 585 [hep-th/9806239] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    M. Wolf, On supertwistor geometry and integrability in super gauge theory, Ph.D. Thesis, Universität Hannover (2006) [hep-th/0611013] [INSPIRE].
  53. [53]
    N.A. Rink, Complex geometry of vortices and their moduli spaces, Ph.D. Thesis, University of Cambridge (2012).Google Scholar
  54. [54]
    C. Sämann 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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    P. Ševera, L -algebras as 1-jets of simplicial manifolds (and a bit beyond), math.DG/0612349.
  56. [56]
    C. Sämann and M. Wolf, On twistors and conformal field theories from six dimensions, J. Math. Phys. 54 (2013) 013507 [arXiv:1111.2539] [INSPIRE].
  57. [57]
    C. Sämann and M. Wolf, Non-Abelian tensor multiplet equations from twistor space, Commun. Math. Phys. 328 (2014) 527 [arXiv:1205.3108] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    G.A. Demessie and C. Sämann, Higher Poincaré lemma and integrability, J. Math. Phys. 56 (2015) 082902 [arXiv:1406.5342] [INSPIRE].
  59. [59]
    T. Kadeishvili, On the homology theory of fibre spaces, Uspekhi Mat. Nauk 35 (1980) 183 [math.AT/0504437].
  60. [60]
    C. Sämann, R. Wimmer and M. Wolf, A twistor description of six-dimensional \( \mathcal{N}=\left(1,1\right) \) super Yang-Mills theory, JHEP 05 (2012) 020 [arXiv:1201.6285] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    L.J. Mason and R.A. Reid-Edwards, The supersymmetric Penrose transform in six dimensions, arXiv:1212.6173 [INSPIRE].

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Maxwell Institute for Mathematical Sciences, Department of MathematicsHeriot-Watt UniversityEdinburghUnited Kingdom
  2. 2.Department of MathematicsUniversity of SurreyGuildfordUnited Kingdom

Personalised recommendations