×

Perturbative gauge theory as a string theory in twistor space. (English) Zbl 1105.81061

This paper opens a new perspective on explaining the unexpected simplicity of perturbative scattering amplitudes in Yang-Mills theory, in particular the holomorphy of the maximally helicity violating amplitudes. The momentum space scattering amplitudes are Fourier transformed to Penrose’s twistor space and it is argued that in twistor space they are supported on certain holomorphic curves. This property is interpreted as a consequence of an equivalence between the perturbative expansion of \(\mathcal N=4\) super Yang-Mills theory and the D-instanton expansion of a certain string theory, namely the topological B model whose target space is the Calabi-Yau supermanifold \(\mathbb{C} \mathbb{P}^{3| 4}\) (which has been known to be a supersymmetric version of twistor space for a long time).
This representation of weakly coupled \(\mathcal N=4\) super Yang-Mills theory is an interesting counterpart to the well-known equivalence of the strongly coupled regime of the same theory with type IIB superstring theory on AdS\(_5 \times S_5\). Among the many open questions discussed by the author is the possibility of describing the perturbative expansion of \(\mathcal N =8\) supergravity by some string theory. Whereas the description of perturbative Yang-Mills amplitudes in twistor space is self-contained and very pedagogical, more background knowledge (or consultation of the numerous references) on topological string theory will be required of the reader.

MSC:

81T45 Topological field theories in quantum mechanics
81T13 Yang-Mills and other gauge theories in quantum field theory
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Parke, S., Taylor, T.: An Amplitude for N Gluon Scattering. Phys. Rev. Lett. 56, 2459 (1986) · doi:10.1103/PhysRevLett.56.2459
[2] Berends, F.A., Giele, W.T.: Recursive Calculations for Processes with N. Gluons. Nucl. Phys. B306, 759 (1988)
[3] DeWitt, B.: Quantum Theory of Gravity, III: Applications of the Covariant Theory. Phys. Rev. 162, 1239 (1967) · Zbl 0161.46501 · doi:10.1103/PhysRev.162.1239
[4] Bern, Z., Dixon, L., Kosower, D.: Progress in One-Loop QCD Computations. Ann. Rev. Nucl. Part. Sci. 46, 109 (1996) · doi:10.1146/annurev.nucl.46.1.109
[5] Brink, L., Scherk, J., Schwarz, J.H.: Supersymmetric Yang-Mills Theories. Nucl. Phys. B121, 77 (1977)
[6] Anastasiou, C., Bern, Z., Dixon, L., Kosower, D.: Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory. Phys. Rev. Lett. 91, 251-602 (2003); Bern, Z., De Freitas, A., Dixon, L.: Two Loop Helicity Amplitudes for Quark Gluon Scattering in QCD and Gluino Gluon Scattering in Supersymmetric Yang-Mills Theory. JHEP 0306, 028 (2003)
[7] Penrose, R.: Twistor Algebra. J. Math. Phys. 8, 345 (1967) · Zbl 0163.22602 · doi:10.1063/1.1705200
[8] Penrose, R.: The Central Programme of Twistor Theory. Chaos, Solitons, and Fractals 10, 581 (1999) · Zbl 0994.81049 · doi:10.1016/S0960-0779(98)00333-6
[9] Ferber, A.: Supertwistors and Conformal Supersymmetry. Nucl. Phys. B132, 55 (1978)
[10] Witten, E.: An Interpretation of Classical Yang-Mills Theory. Phys. Lett. B77, 394 (1978)
[11] Isenberg, J., Yasskin, P.B., Green, P.S.: Nonselfdual Gauge Fields. Phys. Lett. B78, 464 (1978)
[12] Nair, V.: A Current Algebra for Some Gauge Theory Amplitudes. Phys. Lett. B214, 215 (1988)
[13] Maldacena, J.: The Large N Limit of Superconformal Field Theories and Supergravity. Adv. Theor. Math. Phys. 2, 231 (1998) · Zbl 0914.53047
[14] ?t Hooft, G.: A Planar Diagram Theory for Strong Interactions. Nucl. Phys. B72, 461 (1974)
[15] MacCallum, M.A.H., Penrose, R.: Twistor Theory: An Approach to the Quantization of Fields and Space-Time. Phys. Rept. 6C, 241 (1972)
[16] Hodges, A.P., Huggett, S.: Twistor Diagrams. Surveys in High Energy Physics 1, 333 (1980); Hodges, A.: Twistor Diagrams. Physica 114A, 157 (1982); Twistor Diagrams. In: S.A. Huggett, et al. (eds.), The Geometric Universe: Science, Geometry, and the Work of Roger Penrose, Oxford: Oxford University Press, 1998
[17] Bjorken, J.D., Chen, M.C.: High Energy Trident Production with Definite Helicities. Phys. Rev. 154, 1335 (1966) · doi:10.1103/PhysRev.154.1335
[18] Reading Henry, G.: Trident Production with Nuclear Targets. Phys. Rev. 154, 1534 (1967) · doi:10.1103/PhysRev.154.1534
[19] Berends, F.A., Kleiss, R., De Causmaecker, P., Gastmans, R., Wu, T.T.: Single Bremsstrahlung Processes in Gauge Theory. Phys. Lett. 103B, 124 (1981); De Causmaecker, P., Gastmans, R., Troost, W., Wu, T.T.: Helicity Amplitudes for Massless QED. Phys. Lett. B105, 215 (1981); Nucl. Phys. B206, 53 (1982); Berends, F.A., Kleiss, R., De Causmaecker, P., Gastmans, R., Troost, W., Wu, T.T.: Multiple Bremsstrahlung in Gauge Theories at High-Energies. 2. Single Bremsstrahlung. Nucl. Phys. B206, 61 (1982)
[20] Kleis, R., Stirling, W.J.: Spinor Techniques for Calculating Proton Anti-Proton to W or Z Plus Jets. Nucl. Phys. B262, 235 (1985)
[21] Gunion, J.F., Kunzst, Z.: Improved Analytic Techniques for Tree Graph Calculations and the Process. Phys. Lett. 161B, 333 (1985)
[22] Berends, F.A., Giele, W-T.: The Six Gluon Process as an Example of Weyl-van der Waerden Spinor Calculus. Nucl. Phys. B294, 700 (1987)
[23] Xu, Z., Zhang, D.-H., Chang, L.: Helicity Amplitudes for Multiple Bremsstrahlung in Massless Nonabelian Gauge Theories. Nucl. Phys. B291, 392 (1987)
[24] Chalmers, G., Siegel, W.: Simplifying Algebra in Feynman Graphs, Part I: Spinors. Phys. Rev. D59, 045012 (1999), Part II: Spinor Helicity from the Light Cone. Phys. Rev. D59, 045013 (1999), Part III: Massive Vectors. Phys. Rev. D63, 125027 (2001)
[25] Mangano, M.L., Parke, S.J.: Multiparton Amplitudes in Gauge Theories. Phys. Rept. 200, 301 (1991) · doi:10.1016/0370-1573(91)90091-Y
[26] Dixon, L.: Calculating Scattering Amplitudes Efficiently. TASI Lectures, 1995, http:// arxiv.org/abs/hep-ph/9601359, 1996
[27] Witten, L.: Invariants of General Relativity and the Classification of Spaces. Phys. Rev. (2) 113, 357 (1959) · Zbl 0083.42701
[28] Penrose, R., Rindler, W.: Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields, Volume 2, Spinor and Twistor Methods in Spacetime Geometry. Cambridge: Cambridge University Press, 1986 · Zbl 0602.53001
[29] Ward, R.S., Wells, R.O.Jr.: Twistor Geometry and Field Theory. Cambridge: Cambridge University Press, 1991 · Zbl 0729.53068
[30] Hughston, L.P.: Twistors and Particles. Lecture Notes in Physics 97, Berlin: Springer-Verlag, 1989
[31] Bailey, T.N., Baston, R.J. (eds.): Twistors in Mathematics and Physics. London: London Mathematical Society Lecture Notes Series, 156, 1990 · Zbl 0702.53003
[32] Huggett, S.A., Tod, K.P.: An Introduction to Twistor Theory. London Mathematical Society Student Texts 4, New York: Cambridge Univ. Press, 1985 · Zbl 0573.53001
[33] Penrose, R.: The Nonlinear Graviton. Gen. Rel. Grav. 7, 171 (1976) · Zbl 0354.53025 · doi:10.1007/BF00763433
[34] Ward, R.: On Self-Dual Gauge Fields. Phys. Lett. 61A, 81 (1977) · Zbl 0964.81519 · doi:10.1016/0375-9601(77)90842-8
[35] Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-Duality in Four-Dimensional Riemannian Geometry. Proc. Roy. Soc. London Ser. A 362, 425 (1978) · Zbl 0389.53011 · doi:10.1098/rspa.1978.0143
[36] Atiyah, M.F.: Geometry of Yang-Mills Fields. Lezioni Fermiane, Pisa: Academia Nazionale dei Lincei and Scuola Normale Superiore, 1979
[37] Penrose, R.: Twistor Quantization and Curved Spacetime. Int. J. Theor. Phys. 1, 61 (1968) · doi:10.1007/BF00668831
[38] Mangano, M., Parke, S., Xu, Z.: Duality and Multi-Gluon Scattering. Nucl. Phys. B298, 653 (1988)
[39] Berends, F.A., Giele, W.T., Kuijf, H.: Exact and Approximate Expressions for Multi-Gluon Scattering. Nucl. Phys. B333, 120 (1990)
[40] Bern, Z., Dixon, L., Dunbar, D., Kosower, D.A.: One-Loop N-Point Gauge Theory Amplitudes. Unitarity and Collinear Limits, hep-ph/9403226 · Zbl 1049.81644
[41] Bern, Z., Dixon, L., Kosower, D.: One Gluon Corrections to Five Gluon Amplitudes. Phys. Rev. Lett. 70, 2677 (1993) · doi:10.1103/PhysRevLett.70.2677
[42] Bern, Z., Dixon, L., Kosower, D.A.: In: M.B. Halpern, et al. (eds.), Strings 1993, Singapore: World Scientific, 1995; Bern, Z., Chalmers, G., Koxon, L., Kosower, D.A.: Phys. Rev. Lett. 72, 2134 (1994) · doi:10.1103/PhysRevLett.72.2134
[43] Mahlon, G.: Multigluon Helicity Amplitudes Involving a Quark Loop. Phys. Rev. D49, 4438 (1994)
[44] Mahlon, G., Yan, T.-M.: Multi-Photon Production at High Energies in the Standard Model: I. Phys. Rev. D47, 1776 (1993)
[45] Bern, Z., Kosower, D.: The Computation of Loop Amplitudes in Gauge Theories. Nucl. Phys. B379, 451 (1992)
[46] Bern, Z., Dixon, L., Dunbar, D.C., Kosower, D.A.: One-loop Gauge Theory Amplitudes with an Arbitrary Number of External Legs, hep-ph/9405248 · Zbl 1049.81644
[47] Berends, F.A., Giele, W.T., Kuijf, H.: On Relations Between Multi-Gluon and Multi-Graviton Scattering. Phys. Lett. 211B, 91 (1988)
[48] Witten, E.: Mirror Manifolds and Topological Field Theory. hep-th/9112056, In: S.-T. Yau, (ed.), Mirror Symmetry, Cambridge, MA: International Press, 1998
[49] Hori, K., et al. (eds.): Mirror symmetry. Providence, RI: American Mathematical Society, 2003 · Zbl 1069.81562
[50] Witten, E.: Chern-Simons Gauge Theory as a String Theory. Prog. Math. 133, 637 (1995) · Zbl 0844.58018
[51] Dijkgraaf, R., Vafa, C.: Matrix Models, Topological Strings, and Supersymmetric Gauge Theories. Nucl. Phys. B644, 3 (2002) · Zbl 0999.81068
[52] Aganagic, M., Dijkgraaf, R., Klemm, A., Marino, M., Vafa, C.: Topological Strings and Integrable Hierarchies. hep-th/0312085 · Zbl 1095.81049
[53] D?Adda, A., Di Vecchia, P., Luscher, M.: Confinement and Chiral Symmetry Breaking in Models with Quarks. Nucl. PHys. B152, 125 (1979)
[54] Witten, E.: Instantons, the Quark Model, and the 1/N Expansion. Nucl. Phys. B149, 285 (1979)
[55] Zumino, B.: Supersymmetry and Kahler Manifolds. Phys. Lett. B87, 203 (1979)
[56] Movshev, M., Schwarz, A.: On Maximally Supersymmetric Yang-Mills Theory. hep-th/0311132 · Zbl 1044.81097
[57] Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Holomorphic Anomalies in Topological Field Theories. Nucl. Phys. B405, 279 (1993); Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes. Commun. Math. Phys. B165, 311 (1994) · Zbl 0908.58074
[58] Grisaru, M., Rocek, M., Siegel, W.: Superloops 3, Beta 0: A Calculation in N=4 Yang-Mills Theory. Phys. Rev. Lett. 45, 1063 (1980) · doi:10.1103/PhysRevLett.45.1063
[59] Avdeev, L.V., Tarasov, O.V.: The Three Loop Beta Function in the N=1, N=2, N=4 Supersymmetric Yang-Mills Theories. Phys. Lett. B112, 356 (1982)
[60] Mandelstam, S.: Light Cone Superspace and the Ultraviolet Finiteness of the N=4 Model. Nucl. Phys. B213, 149 (1983)
[61] Krasnov, K.: Twistors, CFT, and Holography. hep-th/0311162
[62] Atiyah, M.F., Hitchin, N.: The Geometry and Dynamics of Magnetic Monopoles. Princeton, NJ: Princeton University Press, 1988 · Zbl 0671.53001
[63] Polchinski, J.: Dirichlet Branes and Ramond-Ramond Charges. Phys. Rev. Lett. 75, 4724 (1995) · Zbl 1020.81797 · doi:10.1103/PhysRevLett.75.4724
[64] Lazaroiu, C.: Holomorphic Matrix Models. hep-th/0303008.
[65] Siegel, W.: The = 2(4) String is Self-Dual = 4 Yang-Mills. Phys. Rev. D46, R3235 (1992); Self-Dual = 8 Supergravity as Closed = 2(4) Strings, hep-th/9207043 · Zbl 0968.81544
[66] Chalmers, G., Siegel, W.: The Self-Dual Sector of QCD Amplitudes. hep-th/9606061
[67] Witten, E.: Small Instantons in String Theory. Nucl. Phys. B460, 541 (1996) · Zbl 0935.81052
[68] Douglas, M.R.: Gauge Fields and D-Branes. J. Geom. Phys. 28, 255 (1998) · Zbl 1057.81546 · doi:10.1016/S0393-0440(97)00024-7
[69] Fradkin, E.S., Tseytlin, A.A.: Conformal Supergravity. Phys. Rept. 119, 233 (1985) · Zbl 0967.81523 · doi:10.1016/0370-1573(85)90138-3
[70] Salam, A., Sezgin, E.: Supergravities in Diverse Dimensions. Vol. 2, Singapore: World Scientific, 1989 · Zbl 0686.53061
[71] Bena, I., Polchinski, J., Roiban, R.: Hidden Symmetries of the AdS5{\(\times\)} S5 Superstring. hep-th/0305116
[72] Dolan, L., Nappi, C.R., Witten, E.: A Relation Among Approaches to Integrability in Superconformal Yang-Mills Theory. JHEP 0310, 017 (2003) · doi:10.1088/1126-6708/2003/10/017
[73] Luscher, M., Pohlmeyer, K.: Scattering of Massless Lumps and Nonlocal Charges in the Two-Dimensional Classical Non-Linear Sigma Model. Nucl. Phys. B137, 46 (1978)
[74] de Vega, H.J., Eichenherr, H., Maillet, J.M.: Yang-Baxter Algebras of Monodromy Matrices in Integrable Quantum Field Theories. Nucl. Phys. B240, 377 (1984) · Zbl 0536.58013
[75] Berg, B., Weisz, P.: Exact S Matrix of the Adjoint SU(N) Representation. Commun. Math. Phys. 67, 241 (1979) · doi:10.1007/BF01238847
[76] Goldschmidt, Y., Witten, E.: Conservation Laws in Some Two-Dimensional Models. Phys. Lett. 91B, 392 (1980)
[77] Abdalla, E.: In: N. Sanchez, (ed.), Non-linear Equations in Classical and Quantum Field Theory. Springer Lectures in Physics, Vol. 226, Berlin-Heidelberg-New York: Springer, 1985, p. 140; Abdalla, E., Abdalla, M.C.B., Gomes, M.: Phys. Rev. D25, 452 (1982), D27, 825 (1983) · Zbl 0484.73036
[78] Berkovits, N.: Super Poincaré Covariant Quantization of the Superstring. JHEP 0004, 018 (2000), hep-th/0001035, Covariant Quantization of the Superparticle Using Pure Spinors. hep-th/0105050, JHEP 0109, 016 (2001) · Zbl 0959.81065 · doi:10.1088/1126-6708/2000/04/018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.