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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abou-Zeid M., Hull C.M.: A chiral perturbation expansion for gravity. JHEP 0602, 057 (2006)
Abou-Zeid M., Hull C.M., Mason L.J.: Einstein supergravity and new twistor string theories. Commun. Math. Phys. 282, 519 (2008)
Atiyah M.F., Hitchin N.J., Singer I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. Lond. A 362, 425 (1978)
Bailey T.N., Eastwood M.G.: Complex paraconformal manifolds— their differential geometry and twistor theory. Forum. Math. 3, 61 (1991)
Batchelor M.: The structure of supermanifolds. Trans. Amer. Math. Soc. 253, 329 (1979)
Bergshoeff E., Sezgin E.: Self-dual supergravity theories in (2 + 2)-dimensions. Phys. Lett. B 292, 87 (1992)
Berkovits N.: An alternative string theory in twistor space for \({\mathcal{N} = 4}\) super Yang-Mills. Phys. Rev. Lett. 93, 011601 (2004)
Berkovits N., Witten E.: Conformal supergravity in twistor-string theory. JHEP 0408, 009 (2004)
Bern Z., Dixon L.J., Roiban R.: Is \({\mathcal{N} = 8}\) supergravity ultraviolet finite?. Phys. Lett. B 644, 265 (2007)
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)
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)
Boels R., Mason L.J., Skinner D.: Supersymmetric gauge theories in twistor space. JHEP 0702, 014 (2007a)
Boels R., Mason L.J., Skinner D.: From twistor actions to MHV diagrams. Phys. Lett. B 648, 90 (2007b)
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)
Christensen S.M., Deser S., Duff M.J., Grisaru M.T.: Chirality, self-duality, and supergravity counterterms. Phys. Lett. B 84, 411 (1979)
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)
Green M.B., Russo J.G., Vanhove P.: Ultraviolet properties of maximal supergravity. Phys. Rev. Lett. 98, 131602 (2007)
Kallosh, R.E.: Super self-duality. JETP Lett. 29, 172 [Pisma Zh. Eksp. Teor. Fiz. 29, 192] (1979)
Kallosh R.E.: Self-duality in superspace. Nucl. Phys. B 165, 119 (1980)
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)
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)
Lechtenfeld O., Sämann C.: Matrix models and D-branes in twistor string theory. JHEP 0603, 002 (2006)
Manin, Yu.I.: Gauge field theory and complex geometry. New York: Springer Verlag, 1988 [Russian: Moscow: Nauka, 1984]
Mason L.J.: Twistor actions for non-self-dual fields: A derivation of twistor string theory. JHEP 0510, 009 (2005)
Mason L.J., Newman E.T.: A connection between the Einstein and Yang-Mills equations. Commun. Math. Phys. 121, 659 (1989)
Mason L.J., Skinner D.: An ambitwistor Yang-Mills Lagrangian. Phys. Lett. B 636, 60 (2006)
Mason L.J., Skinner D.: Heterotic twistor-string theory. Nucl. Phys. B 795, 105 (2008)
Mason L.J., Woodhouse N.M.J.: Integrability, self-duality, and twistor theory. Clarendon Press, Oxford (1996)
Merkulov S.A.: Paraconformal supermanifolds and non-standard \({\mathcal{N}}\) -extended supergravity models. Class. Quant. Grav. 8, 557 (1991)
Merkulov S.A.: Supersymmetric non-linear graviton. Funct. Anal. Appl. 26, 69 (1992a)
Merkulov S.A.: Simple supergravity, supersymmetric non-linear gravitons and supertwistor theory. Class. Quant. Grav. 9, 2369 (1992b)
Merkulov S.A.: Quaternionic, quaternionic Kähler, and hyper-Kähler supermanifolds. Lett. Math. Phys. 25, 7 (1992c)
Nair V.P.: A note on graviton amplitudes for new twistor string theories. Phys. Rev. D 78, 041501 (2008)
Penrose R.: Twistor quantization and curved space-time. Int. J. Theor. Phys. 1, 61 (1968)
Penrose R.: Non-linear gravitons and curved twistor theory. Gen. Rel. Grav. 7, 31 (1976)
Popov A.D., Wolf M.: Topological B model on weighted projective spaces and self-dual models in four dimensions. JHEP 0409, 007 (2004)
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)
Penrose R., Sämann C., Wolf M.: The topological B model on a mini-supertwistor space and supersymmetric Bogomolny monopole equations. JHEP 0510, 058 (2005)
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)
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
Siegel W.: Self-dual \({\mathcal{N} = 8}\) supergravity as closed N = 2 (N = 4) strings. Phys. Rev. D 47, 2504 (1992)
Sokatchev E.S.: Action for \({\mathcal{N} = 4}\) supersymmetric self-dual Yang-Mills theory. Phys. Rev. D 53, 2062 (1995)
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
Waintrob, A.Yu.: Deformations and moduli of supermanifolds. In: Group theoretical methods in physics, Vol. 1, Moscow: Nauka, 1986
Ward R.S.: Self-dual space-times with cosmological constants. Commun. Math. Phys. 78, 1 (1980)
Ward R.S., Wells R.O.: Twistor geometry and field theory. Cambridge University Press, Cambridge (1990)
Witten E.: Topology changing amplitudes in (2 + 1)-dimensional gravity. Nucl. Phys. B 323, 113 (1989)
Witten E.: Perturbative gauge theory as a string theory in twistor space. Commun. Math. Phys. 252, 189 (2004)
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
Wolf M.: Self-dual supergravity and twistor theory. Class. Quant. Grav. 24, 6287 (2007)
Woodhouse N.M.J.: Real methods in twistor theory. Class. Quant. Grav. 2, 257 (1985)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. W. Gibbons
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-009-0732-5