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Twistor Actions for Self-Dual Supergravities

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Abstract

We give holomorphic Chern-Simons-like action functionals on supertwistor space for self-dual supergravity theories in four dimensions, dealing with \({\mathcal{N} = 0,\ldots,8}\) supersymmetries, the cases where different parts of the R-symmetry are gauged, and with or without a cosmological constant. The gauge group is formally the group of holomorphic Poisson transformations of supertwistor space where the form of the Poisson structure determines the amount of R-symmetry gauged and the value of the cosmological constant. We give a formulation in terms of a finite deformation of an integrable \({\bar\partial}\) -operator on a supertwistor space, i.e., on regions in \({\mathbb{CP}^{3|8}}\) . For \({\mathcal{N}=0}\) , we also give a formulation that does not require the choice of a background.

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References

  1. Abou-Zeid M., Hull C.M.: A chiral perturbation expansion for gravity. JHEP 0602, 057 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  2. Abou-Zeid M., Hull C.M., Mason L.J.: Einstein supergravity and new twistor string theories. Commun. Math. Phys. 282, 519 (2008)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Atiyah M.F., Hitchin N.J., Singer I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. Lond. A 362, 425 (1978)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. Bailey T.N., Eastwood M.G.: Complex paraconformal manifolds— their differential geometry and twistor theory. Forum. Math. 3, 61 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  5. Batchelor M.: The structure of supermanifolds. Trans. Amer. Math. Soc. 253, 329 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bergshoeff E., Sezgin E.: Self-dual supergravity theories in (2 + 2)-dimensions. Phys. Lett. B 292, 87 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  7. Berkovits N.: An alternative string theory in twistor space for \({\mathcal{N} = 4}\) super Yang-Mills. Phys. Rev. Lett. 93, 011601 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  8. Berkovits N., Witten E.: Conformal supergravity in twistor-string theory. JHEP 0408, 009 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  9. Bern Z., Dixon L.J., Roiban R.: Is \({\mathcal{N} = 8}\) supergravity ultraviolet finite?. Phys. Lett. B 644, 265 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  10. Bershadsky M., Cecotti S., Ooguri H., Vafa C.: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. Bjerrum-Bohr N.E.J., Dunbar D.C., Ita H., Perkins W.B., Risager K.: The no-triangle hypothesis for \({\mathcal{N} = 8}\) supergravity. JHEP 0612, 072 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  12. Boels R., Mason L.J., Skinner D.: Supersymmetric gauge theories in twistor space. JHEP 0702, 014 (2007a)

    Article  ADS  MathSciNet  Google Scholar 

  13. Boels R., Mason L.J., Skinner D.: From twistor actions to MHV diagrams. Phys. Lett. B 648, 90 (2007b)

    Article  ADS  MathSciNet  Google Scholar 

  14. Cap, A., Eastwood, M.G.: Some special geometry in dimension six. In: Proc. of the 22nd Winter School, Geometry and physics (Srni 2002), Rend. Circ. Mat. Palermo (2) Suppl. No. 71, 93 (2003)

  15. Christensen S.M., Deser S., Duff M.J., Grisaru M.T.: Chirality, self-duality, and supergravity counterterms. Phys. Lett. B 84, 411 (1979)

    Article  ADS  Google Scholar 

  16. Dijkgraaf R., Gukov S., Neitzke A., Vafa C.: Topological M-theory as unification of form theories of gravity. Adv. Theor. Math. Phys. 9, 603 (2005)

    MATH  MathSciNet  Google Scholar 

  17. Green M.B., Russo J.G., Vanhove P.: Ultraviolet properties of maximal supergravity. Phys. Rev. Lett. 98, 131602 (2007)

    Article  ADS  Google Scholar 

  18. Kallosh, R.E.: Super self-duality. JETP Lett. 29, 172 [Pisma Zh. Eksp. Teor. Fiz. 29, 192] (1979)

    Google Scholar 

  19. Kallosh R.E.: Self-duality in superspace. Nucl. Phys. B 165, 119 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  20. Karnas S., Ketov S.V.: An action of \({\mathcal{N} = 8}\) self-dual supergravity in ultra-hyperbolic harmonic superspace. Nucl. Phys. B 526, 597 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  21. Ketov, S.V., Nishino, H., Gates, S.J.J.: Self-dual supersymmetry and supergravity in Atiyah-Ward space-time. Nucl. Phys. B 393, 149 (1992). See also Phys. Lett. B 297, 323 (1992), Phys. Lett. B 307, 331 (1993), Phys. Lett. B 307, 323 (1993)

  22. Lechtenfeld O., Sämann C.: Matrix models and D-branes in twistor string theory. JHEP 0603, 002 (2006)

    Article  ADS  Google Scholar 

  23. Manin, Yu.I.: Gauge field theory and complex geometry. New York: Springer Verlag, 1988 [Russian: Moscow: Nauka, 1984]

  24. Mason L.J.: Twistor actions for non-self-dual fields: A derivation of twistor string theory. JHEP 0510, 009 (2005)

    Article  ADS  Google Scholar 

  25. Mason L.J., Newman E.T.: A connection between the Einstein and Yang-Mills equations. Commun. Math. Phys. 121, 659 (1989)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. Mason L.J., Skinner D.: An ambitwistor Yang-Mills Lagrangian. Phys. Lett. B 636, 60 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  27. Mason L.J., Skinner D.: Heterotic twistor-string theory. Nucl. Phys. B 795, 105 (2008)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  28. Mason L.J., Woodhouse N.M.J.: Integrability, self-duality, and twistor theory. Clarendon Press, Oxford (1996)

    MATH  Google Scholar 

  29. Merkulov S.A.: Paraconformal supermanifolds and non-standard \({\mathcal{N}}\) -extended supergravity models. Class. Quant. Grav. 8, 557 (1991)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  30. Merkulov S.A.: Supersymmetric non-linear graviton. Funct. Anal. Appl. 26, 69 (1992a)

    Article  MathSciNet  Google Scholar 

  31. Merkulov S.A.: Simple supergravity, supersymmetric non-linear gravitons and supertwistor theory. Class. Quant. Grav. 9, 2369 (1992b)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  32. Merkulov S.A.: Quaternionic, quaternionic Kähler, and hyper-Kähler supermanifolds. Lett. Math. Phys. 25, 7 (1992c)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  33. Nair V.P.: A note on graviton amplitudes for new twistor string theories. Phys. Rev. D 78, 041501 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  34. Penrose R.: Twistor quantization and curved space-time. Int. J. Theor. Phys. 1, 61 (1968)

    Article  Google Scholar 

  35. Penrose R.: Non-linear gravitons and curved twistor theory. Gen. Rel. Grav. 7, 31 (1976)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  36. Popov A.D., Wolf M.: Topological B model on weighted projective spaces and self-dual models in four dimensions. JHEP 0409, 007 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  37. Penrose R., Sämann C.: On supertwistors, the Penrose-Ward transform and \({\mathcal{N} = 4}\) super Yang-Mills theory. Adv. Theor. Math. Phys. 9, 931 (2005)

    MathSciNet  Google Scholar 

  38. Penrose R., Sämann C., Wolf M.: The topological B model on a mini-supertwistor space and supersymmetric Bogomolny monopole equations. JHEP 0510, 058 (2005)

    ADS  Google Scholar 

  39. Sämann C.: The topological B model on fattened complex manifolds and subsectors of \({\mathcal{N} = 4}\) self-dual Yang-Mills theory. JHEP 0501, 042 (2005)

    Article  Google Scholar 

  40. Sämann, C.: Aspects of twistor geometry and supersymmetric field theories within superstring theory, Ph.D. thesis, Leibniz University of Hannover, available at http://arXiv.org/list/hep-th/0603098, 2006

  41. Siegel W.: Self-dual \({\mathcal{N} = 8}\) supergravity as closed N = 2 (N =  4) strings. Phys. Rev. D 47, 2504 (1992)

    Article  ADS  Google Scholar 

  42. Sokatchev E.S.: Action for \({\mathcal{N} = 4}\) supersymmetric self-dual Yang-Mills theory. Phys. Rev. D 53, 2062 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  43. Stelle, K.S.: Counterterms, holonomy and supersymmetry. In: Deserfest: A celebration of the Life and works of Stanley Deser, Ann Arbor Michigan, 2004, Liu, J.T., Duff, M.J., Stelle, K.S., Woodward, R.P., (eds.), River Edge, NJ: World Scientific, 2006, p. 303

  44. Waintrob, A.Yu.: Deformations and moduli of supermanifolds. In: Group theoretical methods in physics, Vol. 1, Moscow: Nauka, 1986

  45. Ward R.S.: Self-dual space-times with cosmological constants. Commun. Math. Phys. 78, 1 (1980)

    Article  MATH  ADS  Google Scholar 

  46. Ward R.S., Wells R.O.: Twistor geometry and field theory. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  47. Witten E.: Topology changing amplitudes in (2 + 1)-dimensional gravity. Nucl. Phys. B 323, 113 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  48. Witten E.: Perturbative gauge theory as a string theory in twistor space. Commun. Math. Phys. 252, 189 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  49. Wolf, M.: On supertwistor geometry and integrability in super gauge theory. Ph.D. thesis, Leibniz University of Hannover, available at http://arXiv.org/list/hep-th/0611013, 2006

  50. Wolf M.: Self-dual supergravity and twistor theory. Class. Quant. Grav. 24, 6287 (2007)

    Article  MATH  ADS  Google Scholar 

  51. Woodhouse N.M.J.: Real methods in twistor theory. Class. Quant. Grav. 2, 257 (1985)

    Article  MATH  ADS  MathSciNet  Google Scholar 

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Correspondence to Martin Wolf.

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Communicated by G. W. Gibbons

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Mason, L.J., Wolf, M. Twistor Actions for Self-Dual Supergravities. Commun. Math. Phys. 288, 97–123 (2009). https://doi.org/10.1007/s00220-009-0732-5

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