Abstract
In any type II superstring background, the supergravity vertex operators in the pure spinor formalism are described by a gauge superfield. In this paper, we obtain for the first time an explicit expression for this superfield in an AdS5 × S5 background. Previously, the vertex operators were only known close to the boundary of AdS5 or in the minus eight picture. Our strategy for the computation was to apply eight picture raising operators in the minus eight picture vertices. In the process, a huge number of terms are generated and we have developed numerical techniques to perform intermediary simplifications. Alternatively, the same numerical techniques can be used to compute the vertices directly in the zero picture by constructing a basis of invariants and fitting for the coefficients. One motivation for constructing the vertex operators is the computation of AdS5 × S5 string amplitudes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Berkovits, Super Poincaré covariant quantization of the superstring, JHEP 04 (2000) 018 [hep-th/0001035] [INSPIRE].
R.R. Metsaev and A.A. Tseytlin, Type IIB superstring action in AdS5 × S5 background, Nucl. Phys. B 533 (1998) 109 [hep-th/9805028] [INSPIRE].
L. Mazzucato, Superstrings in AdS, Phys. Rept. 521 (2012) 1 [arXiv:1104.2604] [INSPIRE].
G. Arutyunov and S. Frolov, Foundations of the AdS5 × S5 Superstring. Part I, J. Phys. A 42 (2009) 254003 [arXiv:0901.4937] [INSPIRE].
M. Cho, S. Collier and X. Yin, Strings in Ramond-Ramond Backgrounds from the Neveu-Schwarz-Ramond Formalism, JHEP 12 (2020) 123 [arXiv:1811.00032] [INSPIRE].
K. Roehrig and D. Skinner, Ambitwistor Strings and the Scattering Equations on AdS3 ×S3, arXiv:2007.07234 [INSPIRE].
L. Eberhardt, S. Komatsu and S. Mizera, Scattering equations in AdS: scalar correlators in arbitrary dimensions, JHEP 11 (2020) 158 [arXiv:2007.06574] [INSPIRE].
N. Berkovits and T. Fleury, Harmonic Superspace from the AdS5 × S5 Pure Spinor Formalism, JHEP 03 (2013) 022 [arXiv:1212.3296] [INSPIRE].
T. Azevedo, On the \( \mathcal{N} \) = 4, d = 4 pure spinor measure factor, JHEP 03 (2015) 136 [arXiv:1412.5927] [INSPIRE].
T. Azevedo and N. Berkovits, Open-closed superstring amplitudes using vertex operators in AdS5 × S5 , JHEP 02 (2015) 107 [arXiv:1412.5921] [INSPIRE].
N. Berkovits, Half-BPS vertex operators of the AdS5 × S5 superstring, JHEP 07 (2019) 084 [arXiv:1904.06564] [INSPIRE].
O. Chandía and B.C. Vallilo, Vertex operators for the plane wave pure spinor string, JHEP 10 (2018) 088 [arXiv:1807.05149] [INSPIRE].
O.A. Bedoya, L.I. Bevilaqua, A. Mikhailov and V.O. Rivelles, Notes on β-deformations of the pure spinor superstring in AdS5 × S5, Nucl. Phys. B 848 (2011) 155 [arXiv:1005.0049] [INSPIRE].
M.F. Sohnius, Bianchi Identities for Supersymmetric Gauge Theories, Nucl. Phys. B 136 (1978) 461.
A. Galperin, E. Ivanov, S. Kalitsyn, V. Ogievetsky and E. Sokatchev, Unconstrained N=2 Matter, Yang-Mills and Supergravity Theories in Harmonic Superspace, Class. Quant. Grav. 1 (1984) 469 [erratum: Class. Quant. Grav. 2 (1985), 127].
P.S. Howe and P.C. West, Nonperturbative Green’s functions in theories with extended superconformal symmetry, Int. J. Mod. Phys. A 14 (1999) 2659 [hep-th/9509140] [INSPIRE].
L. Andrianopoli and S. Ferrara, K-K excitations on AdS5 × S5 as N = 4 ‘primary’ superfields, Phys. Lett. B 430 (1998) 248 [hep-th/9803171] [INSPIRE].
P.S. Howe and P.C. West, The Complete N=2, D=10 Supergravity, Nucl. Phys. B 238 (1984) 181.
P. Heslop and P.S. Howe, Chiral superfields in IIB supergravity, Phys. Lett. B 502 (2001) 259 [hep-th/0008047] [INSPIRE].
K. Costello, E. Witten and M. Yamazaki, Gauge Theory and Integrability, I, ICCM Not. 06 (2018) 46 [arXiv:1709.09993] [INSPIRE].
K. Costello and B. Stefański, Chern-Simons Origin of Superstring Integrability, Phys. Rev. Lett. 125 (2020) 121602 [arXiv:2005.03064] [INSPIRE].
N. Berkovits, Simplifying and Extending the AdS5 × S5 Pure Spinor Formalism, JHEP 09 (2009) 051 [arXiv:0812.5074] [INSPIRE].
N. Berkovits and O. Chandía, Superstring vertex operators in an AdS5 × S5 background, Nucl. Phys. B 596 (2001) 185 [hep-th/0009168] [INSPIRE].
O. Chandía, General fluctuations of the type-II pure spinor string on curved backgrounds, JHEP 04 (2019) 073 [arXiv:1902.02289] [INSPIRE].
N. Berkovits and P.S. Howe, Ten-dimensional supergravity constraints from the pure spinor formalism for the superstring, Nucl. Phys. B 635 (2002) 75 [hep-th/0112160] [INSPIRE].
N. Berkovits, ICTP lectures on covariant quantization of the superstring, ICTP Lect. Notes Ser. 13 (2003) 57 [hep-th/0209059] [INSPIRE].
A. Mikhailov and S. Schäfer-Nameki, Algebra of transfer-matrices and Yang-Baxter equations on the string worldsheet in AdS5 × S5, Nucl. Phys. B 802 (2008) 1 [arXiv:0712.4278] [INSPIRE].
V.G.M. Puletti, Operator product expansion for pure spinor superstring on AdS5 * S5, JHEP 10 (2006) 057 [hep-th/0607076] [INSPIRE].
O.A. Bedoya, D.Z. Marchioro, D.L. Nedel and B. Carlini Vallilo, Quantum Current Algebra for the AdS5 × S5 Superstring, JHEP 08 (2010) 026 [arXiv:1003.0701] [INSPIRE].
L.N.S. Martins, Type IIB superstring vertex operator from the -8 picture, arXiv:1912.06498 [INSPIRE].
A. Mikhailov and D. Zavaleta, Geometrical framework for picture changing operators in the pure spinor formalism, JHEP 09 (2020) 108 [arXiv:2003.13995] [INSPIRE].
Work in progress.
C.R. Mafra, Superstring Scattering Amplitudes with the Pure Spinor Formalism, Ph.D. thesis, Sao Paulo, IFT, 2008. arXiv:0902.1552 [INSPIRE].
D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Correlation functions in the CFT(d)/AdS(d+1) correspondence, Nucl. Phys. B 546 (1999) 96 [hep-th/9804058] [INSPIRE].
E. D’Hoker, D.Z. Freedman and W. Skiba, Field theory tests for correlators in the AdS/CFT correspondence, Phys. Rev. D 59 (1999) 045008 [hep-th/9807098] [INSPIRE].
M. Baggio, J. de Boer and K. Papadodimas, A non-renormalization theorem for chiral primary 3-point functions, JHEP 07 (2012) 137 [arXiv:1203.1036] [INSPIRE].
O. Chandía, A. Mikhailov and B.C. Vallilo, A construction of integrated vertex operator in the pure spinor sigma-model in AdS5 × S5, JHEP 11 (2013) 124 [arXiv:1306.0145] [INSPIRE].
O. Chandía and B.C. Vallilo, A superfield realization of the integrated vertex operator in an AdS5 × S5 background, JHEP 10 (2017) 178 [arXiv:1709.05517] [INSPIRE].
N. Berkovits and L. Mazzucato, Taming the b antighost with Ramond-Ramond flux, JHEP 11 (2010) 019 [arXiv:1004.5140] [INSPIRE].
M.B. Green, J.H. Schwarz and L. Brink, Superfield Theory of Type II Superstrings, Nucl. Phys. B 219 (1983) 437 [INSPIRE].
D. Friedan, E.J. Martinec and S.H. Shenker, Conformal Invariance, Supersymmetry and String Theory, Nucl. Phys. B 271 (1986) 93.
N. Berkovits, Sketching a Proof of the Maldacena Conjecture at Small Radius, JHEP 06 (2019) 111 [arXiv:1903.08264] [INSPIRE].
V.A. Kostelecky, O. Lechtenfeld, W. Lerche, S. Samuel and S. Watamura, Conformal Techniques, Bosonization and Tree Level String Amplitudes, Nucl. Phys. B 288 (1987) 173 [INSPIRE].
N. Berkovits, A New Limit of the AdS5 × S5 Sigma Model, JHEP 08 (2007) 011 [hep-th/0703282] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
C.A. Bayona and N.R.F. Braga, Anti-de Sitter boundary in Poincaré coordinates, Gen. Rel. Grav. 39 (2007) 1367 [hep-th/0512182] [INSPIRE].
D. Chicherin et al., Correlation functions of the chiral stress-tensor multiplet in \( \mathcal{N} \) = 4 SYM, JHEP 06 (2015) 198 [arXiv:1412.8718] [INSPIRE].
A.V. Belitsky, S. Hohenegger, G.P. Korchemsky and E. Sokatchev, N = 4 superconformal Ward identities for correlation functions, Nucl. Phys. B 904 (2016) 176 [arXiv:1409.2502] [INSPIRE].
C.R. Mafra, PSS: A FORM Program to Evaluate Pure Spinor Superspace Expressions, arXiv:1007.4999 [INSPIRE].
L. Rastelli and X. Zhou, Mellin amplitudes for AdS5 × S5 , Phys. Rev. Lett. 118 (2017) 091602 [arXiv:1608.06624] [INSPIRE].
V. Gonçalves, Four point function of \( \mathcal{N} \) = 4 stress-tensor multiplet at strong coupling, JHEP 04 (2015) 150 [arXiv:1411.1675] [INSPIRE].
L.F. Alday, A. Bissi and E. Perlmutter, Genus-One String Amplitudes from Conformal Field Theory, JHEP 06 (2019) 010 [arXiv:1809.10670] [INSPIRE].
D.J. Binder, S.M. Chester, S.S. Pufu and Y. Wang, \( \mathcal{N} \) = 4 Super-Yang-Mills correlators at strong coupling from string theory and localization, JHEP 12 (2019) 119 [arXiv:1902.06263] [INSPIRE].
J.M. Drummond, D. Nandan, H. Paul and K.S. Rigatos, String corrections to AdS amplitudes and the double-trace spectrum of \( \mathcal{N} \) = 4 SYM, JHEP 12 (2019) 173 [arXiv:1907.00992] [INSPIRE].
J.M. Drummond, H. Paul and M. Santagata, Bootstrapping string theory on AdS5 × S5, arXiv:2004.07282 [INSPIRE].
F. Aprile and P. Vieira, Large p explorations. From SUGRA to big STRINGS in Mellin space, JHEP 12 (2020) 206 [arXiv:2007.09176] [INSPIRE].
L.F. Alday and A. Bissi, Loop Corrections to Supergravity on AdS5 × S5, Phys. Rev. Lett. 119 (2017) 171601 [arXiv:1706.02388] [INSPIRE].
F. Aprile, J.M. Drummond, P. Heslop and H. Paul, Quantum Gravity from Conformal Field Theory, JHEP 01 (2018) 035 [arXiv:1706.02822] [INSPIRE].
L.F. Alday and S. Caron-Huot, Gravitational S-matrix from CFT dispersion relations, JHEP 12 (2018) 017 [arXiv:1711.02031] [INSPIRE].
F. Aprile, J.M. Drummond, P. Heslop and H. Paul, Loop corrections for Kaluza-Klein AdS amplitudes, JHEP 05 (2018) 056 [arXiv:1711.03903] [INSPIRE].
F. Aprile, J. Drummond, P. Heslop and H. Paul, One-loop amplitudes in AdS5 × S5 supergravity from \( \mathcal{N} \) = 4 SYM at strong coupling, JHEP 03 (2020) 190 [arXiv:1912.01047] [INSPIRE].
L.F. Alday and X. Zhou, Simplicity of AdS Supergravity at One Loop, JHEP 09 (2020) 008 [arXiv:1912.02663] [INSPIRE].
L.F. Alday, On genus-one string amplitudes on AdS5 × S5, JHEP 04 (2021) 005 [arXiv:1812.11783] [INSPIRE].
J.M. Drummond and H. Paul, One-loop string corrections to AdS amplitudes from CFT, JHEP 03 (2021) 038 [arXiv:1912.07632] [INSPIRE].
J.M. Drummond, R. Glew and H. Paul, One-loop string corrections for AdS Kaluza-Klein amplitudes, arXiv:2008.01109 [INSPIRE].
S. Caron-Huot and A.-K. Trinh, All tree-level correlators in AdS5 × S5 supergravity: hidden ten-dimensional conformal symmetry, JHEP 01 (2019) 196 [arXiv:1809.09173] [INSPIRE].
T. Abl, P. Heslop and A.E. Lipstein, Towards the Virasoro-Shapiro amplitude in AdS5 × S5, JHEP 04 (2021) 237 [arXiv:2012.12091] [INSPIRE].
F. Aprile, J.M. Drummond, H. Paul and M. Santagata, The Virasoro-Shapiro amplitude in AdS5 ×S5 and level splitting of 10d conformal symmetry, arXiv:2012.12092 [INSPIRE].
A. Bissi, G. Fardelli and A. Georgoudis, Towards All Loop Supergravity Amplitudes on AdS5 × S5, arXiv:2002.04604 [INSPIRE].
A. Bissi, G. Fardelli and A. Georgoudis, All loop structures in Supergravity Amplitudes on AdS5 × S5 from CFT, arXiv:2010.12557 [INSPIRE].
V. Gonçalves, R. Pereira and X. Zhou, 20′ Five-Point Function from AdS5 × S5 Supergravity, JHEP 10 (2019) 247 [arXiv:1906.05305] [INSPIRE].
M.B. Green and C. Wen, Maximal U(1)Y-violating n-point correlators in \( \mathcal{N} \) = 4 super-Yang-Mills theory, JHEP 02 (2021) 042 [arXiv:2009.01211] [INSPIRE].
D. Dorigoni, M.B. Green and C. Wen, Novel Representation of an Integrated Correlator in \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 126 (2021) 161601 [arXiv:2102.08305] [INSPIRE].
B. Basso, S. Komatsu and P. Vieira, Structure Constants and Integrable Bootstrap in Planar N = 4 SYM Theory, arXiv:1505.06745 [INSPIRE].
T. Fleury and S. Komatsu, Hexagonalization of Correlation Functions, JHEP 01 (2017) 130 [arXiv:1611.05577] [INSPIRE].
B. Eden and A. Sfondrini, Tessellating cushions: four-point functions in \( \mathcal{N} \) = 4 SYM, JHEP 10 (2017) 098 [arXiv:1611.05436] [INSPIRE].
B. Basso, F. Coronado, S. Komatsu, H.T. Lam, P. Vieira and D.-l. Zhong, Asymptotic Four Point Functions, JHEP 07 (2019) 082 [arXiv:1701.04462] [INSPIRE].
F. Coronado, Perturbative four-point functions in planar \( \mathcal{N} \) = 4 SYM from hexagonalization, JHEP 01 (2019) 056 [arXiv:1811.00467] [INSPIRE].
F. Coronado, Bootstrapping the Simplest Correlator in Planar \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills Theory to All Loops, Phys. Rev. Lett. 124 (2020) 171601 [arXiv:1811.03282] [INSPIRE].
I. Kostov, V.B. Petkova and D. Serban, Determinant Formula for the Octagon Form Factor in N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 122 (2019) 231601 [arXiv:1903.05038] [INSPIRE].
T. Bargheer, F. Coronado and P. Vieira, Octagons II: Strong Coupling, arXiv:1909.04077 [INSPIRE].
A.V. Belitsky and G.P. Korchemsky, Exact null octagon, JHEP 05 (2020) 070 [arXiv:1907.13131] [INSPIRE].
A.V. Belitsky and G.P. Korchemsky, Octagon at finite coupling, JHEP 07 (2020) 219 [arXiv:2003.01121] [INSPIRE].
A.V. Belitsky and G.P. Korchemsky, Crossing bridges with strong Szegő limit theorem, JHEP 04 (2021) 257 [arXiv:2006.01831] [INSPIRE].
I. Kostov and V.B. Petkova, Octagon with finite BRIDGE: free fermions and determinant identities, JHEP 06 (2021) 098 [arXiv:2102.05000] [INSPIRE].
N. Berkovits and O. Chandía, Massive superstring vertex operator in D = 10 superspace, JHEP 08 (2002) 040 [hep-th/0204121] [INSPIRE].
S. Chakrabarti, S.P. Kashyap and M. Verma, Theta Expansion of First Massive Vertex Operator in Pure Spinor, JHEP 01 (2018) 019 [arXiv:1706.01196] [INSPIRE].
S. Chakrabarti, S.P. Kashyap and M. Verma, Integrated Massive Vertex Operator in Pure Spinor Formalism, JHEP 10 (2018) 147 [arXiv:1802.04486] [INSPIRE].
S. Chakrabarti, S.P. Kashyap and M. Verma, Amplitudes Involving Massive States Using Pure Spinor Formalism, JHEP 12 (2018) 071 [arXiv:1808.08735] [INSPIRE].
B.C. Vallilo and L. Mazzucato, The Konishi multiplet at strong coupling, JHEP 12 (2011) 029 [arXiv:1102.1219] [INSPIRE].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
N. Gromov, V. Kazakov and P. Vieira, Exact Spectrum of Anomalous Dimensions of Planar N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 103 (2009) 131601 [arXiv:0901.3753] [INSPIRE].
N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum Spectral Curve for Planar \( \mathcal{N} \) = 4 Super-Yang-Mills Theory, Phys. Rev. Lett. 112 (2014) 011602 [arXiv:1305.1939] [INSPIRE].
S. Weinberg, The quantum theory of fields. Vol. 2: Modern applications, Cambridge University Press (1996) [DOI].
R.L. Jusinskas, Spectrum generating algebra for the pure spinor superstring, JHEP 10 (2014) 022 [arXiv:1406.1902] [INSPIRE].
D.F.Z. Marchioro and D.L. Nedel, Quantum corrections to AdS5 × S5 left-invariant superstring current algebra, Phys. Rev. D 87 (2013) 126001 [arXiv:1305.4991] [INSPIRE].
R. Benichou, First-principles derivation of the AdS/CFT Y-systems, JHEP 10 (2011) 112 [arXiv:1108.4927] [INSPIRE].
A.S. Galperin, E.A. Ivanov, V.I. Ogievetsky and E.S. Sokatchev, Harmonic superspace, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2007) [DOI] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2104.03333
Rights and permissions
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.
About this article
Cite this article
Fleury, T., Martins, L.N.S. AdS 5 × S5 supergravity vertex operators. J. High Energ. Phys. 2021, 210 (2021). https://doi.org/10.1007/JHEP07(2021)210
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2021)210