×

Tree-level amplitudes from the pure spinor superstring. (English) Zbl 1530.81099

Summary: We give a comprehensive review of recent developments on using the pure spinor formalism to compute massless superstring scattering amplitudes at tree level. The main results of the pure spinor computations are placed into the context of related topics including the color-kinematics duality in field theory and the mathematical structure of \(\alpha^\prime\)-corrections.

MSC:

81R25 Spinor and twistor methods applied to problems in quantum theory
81U05 \(2\)-body potential quantum scattering theory
81Q60 Supersymmetry and quantum mechanics
81P68 Quantum computation
65B05 Extrapolation to the limit, deferred corrections

Software:

GammaMaP; LiE; FORM; PSS; GAMMA; LieART

References:

[1] Berkovits, N., Super Poincaré covariant quantization of the superstring, J. High Energy Phys., 04, 018 (2000), arXiv:hep-th/0001035 · Zbl 0959.81065
[2] Berkovits, N., Multiloop amplitudes and vanishing theorems using the pure spinor formalism for the superstring, J. High Energy Phys., 09, 047 (2004), arXiv:hep-th/0406055
[3] Berkovits, N., Pure spinor formalism as an N=2 topological string, J. High Energy Phys., 10, 089 (2005), arXiv:hep-th/0509120
[4] Friedan, D.; Martinec, E. J.; Shenker, S. H., Conformal invariance, supersymmetry and string theory, Nuclear Phys. B, 271, 93-165 (1986)
[5] D’Hoker, E.; Phong, D. H., The geometry of string perturbation theory, Rev. Modern Phys., 60, 917 (1988)
[6] D’Hoker, E.; Phong, D. H., Lectures on two loop superstrings, Conf. Proc. C, 0208124, 85-123 (2002), arXiv:hep-th/0211111
[7] Witten, E., Superstring perturbation theory revisited (2012), arXiv:1209.5461
[8] Green, M. B.; Schwarz, J. H., Covariant description of superstrings, Phys. Lett. B, 136, 367-370 (1984)
[9] Green, M. B.; Schwarz, J. H., Properties of the covariant formulation of superstring theories, Nuclear Phys. B, 243, 285-306 (1984)
[10] Green, M. B.; Schwarz, J. H.; Witten, E., (Superstring Theory. Vol. 1: Introduction. Superstring Theory. Vol. 1: Introduction, Cambridge Monographs on Mathematical Physics (1988), Cambridge University Press)
[11] Green, M. B.; Schwarz, J. H.; Witten, E., Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies and Phenomenology (1988), Cambridge University Press
[12] Polchinski, J., (String Theory. Vol. 1: An Introduction to the Bosonic String. String Theory. Vol. 1: An Introduction to the Bosonic String, Cambridge Monographs on Mathematical Physics (2007), Cambridge University Press)
[13] Polchinski, J., (String Theory. Vol. 2: Superstring Theory and beyond. String Theory. Vol. 2: Superstring Theory and beyond, Cambridge Monographs on Mathematical Physics (2007), Cambridge University Press)
[14] Zwiebach, B., A First Course in String Theory (2006), Cambridge University Press
[15] Becker, K.; Becker, M.; Schwarz, J. H., String Theory and M-Theory: A Modern Introduction (2006), Cambridge University Press
[16] Blumenhagen, R.; Lüst, D.; Theisen, S., (Basic Concepts of String Theory. Basic Concepts of String Theory, Theoretical and Mathematical Physics (2013), Springer: Springer Heidelberg, Germany) · Zbl 1262.81001
[17] Kiritsis, E., String Theory in a Nutshell (2019), Princeton University Press: Princeton University Press USA · Zbl 1411.81006
[18] Berkovits, N., Relating the RNS and pure spinor formalisms for the superstring, J. High Energy Phys., 08, 026 (2001), arXiv:hep-th/0104247
[19] Berkovits, N., Untwisting the pure spinor formalism to the RNS and twistor string in a flat and \(A d S_5 \times S^5\) background, J. High Energy Phys., 06, 127 (2016), arXiv:1604.04617
[20] Berkovits, N., Manifest spacetime supersymmetry and the superstring, J. High Energy Phys., 10, 162 (2021), arXiv:2106.04448 · Zbl 1476.83150
[21] Mafra, C. R.; Schlotterer, O.; Stieberger, S., Complete N-point superstring disk amplitude I. Pure spinor computation, Nuclear Phys., B873, 419-460 (2013), arXiv:1106.2645 · Zbl 1282.81151
[22] Siegel, W., Classical superstring mechanics, Nuclear Phys. B, 263, 93-104 (1986)
[23] Hoogeveen, J.; Skenderis, K., BRST quantization of the pure spinor superstring, J. High Energy Phys., 11, 081 (2007), arXiv:0710.2598 · Zbl 1245.81179
[24] Rosly, A. A.; Selivanov, K. G., On amplitudes in selfdual sector of Yang-Mills theory, Phys. Lett. B, 399, 135-140 (1997), arXiv:hep-th/9611101
[25] Rosly, A. A.; Selivanov, K. G., Gravitational SD perturbiner (1997), arXiv:hep-th/9710196
[26] Selivanov, K. G., On tree form-factors in (supersymmetric) Yang-Mills theory, Comm. Math. Phys., 208, 671-687 (2000), arXiv:hep-th/9809046 · Zbl 1052.81070
[27] Selivanov, K. G., Post-classicism in tree amplitudes, (34th Rencontres de Moriond: Electroweak Interactions and Unified Theories (1999)), arXiv:hep-th/9905128
[28] Berends, F. A.; Giele, W. T., Recursive calculations for processes with n gluons, Nuclear Phys. B, 306, 759-808 (1988)
[29] Kleiss, R.; Kuijf, H., Multi-gluon cross-sections and five jet production at hadron colliders, Nuclear Phys., B312, 616-644 (1989)
[30] Bern, Z.; Carrasco, J. J.M.; Johansson, H., New relations for gauge-theory amplitudes, Phys. Rev. D, 78, Article 085011 pp. (2008), arXiv:0805.3993
[31] Alencar, G., Type I supergravity effective action from pure spinor formalism, J. High Energy Phys., 02, 025 (2009), arXiv:0812.4201 · Zbl 1245.81108
[32] Alencar, G.; Tahim, M. O.; Landim, R. R.; Costa Filho, R. N., RNS and pure spinors equivalence for type I tree level amplitudes involving up to four Fermions (2011), arXiv:1104.1939
[33] Bischof, A.; Haack, M., Closed string disk amplitudes in the pure spinor formalism, J. High Energy Phys., 02, 206 (2021), arXiv:2011.10392 · Zbl 1460.83089
[34] Stieberger, S., Open & closed vs. Pure open string disk amplitudes (2009), arXiv:0907.2211
[35] Stieberger, S.; Taylor, T. R., Disk scattering of open and closed strings (I), Nuclear Phys. B, 903, 104-117 (2016), arXiv:1510.01774 · Zbl 1332.81193
[36] Bern, Z.; Carrasco, J. J.; Chiodaroli, M.; Johansson, H.; Roiban, R., The duality between color and kinematics and its applications (2019), arXiv:1909.01358
[37] Bern, Z.; Carrasco, J. J.; Chiodaroli, M.; Johansson, H.; Roiban, R., The SAGEX review on scattering amplitudes, chapter 2: An invitation to color-kinematics duality and the double copy (2022), arXiv:2203.13013 · Zbl 1520.81124
[38] Adamo, T.; Carrasco, J. J.M.; Carrillo-González, M.; Chiodaroli, M.; Elvang, H.; Johansson, H.; O’Connell, D.; Roiban, R.; Schlotterer, O., Snowmass white paper: the double copy and its applications, (2022 Snowmass Summer Study (2022)), arXiv:2204.06547
[39] Cachazo, F.; He, S.; Yuan, E. Y., Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D, 90, 6, Article 065001 pp. (2014), arXiv:1306.6575
[40] Cachazo, F.; He, S.; Yuan, E. Y., Scattering of massless particles in arbitrary dimensions, Phys. Rev. Lett., 113, 17, Article 171601 pp. (2014), arXiv:1307.2199
[41] Cachazo, F.; He, S.; Yuan, E. Y., Scattering of massless particles: Scalars, gluons and gravitons, J. High Energy Phys., 07, 033 (2014), arXiv:1309.0885 · Zbl 1391.81198
[42] Witten, E., Perturbative gauge theory as a string theory in twistor space, Comm. Math. Phys., 252, 189-258 (2004), arXiv:hep-th/0312171 · Zbl 1105.81061
[43] Roiban, R.; Spradlin, M.; Volovich, A., On the tree level S matrix of Yang-Mills theory, Phys. Rev. D, 70, Article 026009 pp. (2004), arXiv:hep-th/0403190
[44] Cachazo, F.; Skinner, D., Gravity from rational curves in twistor space, Phys. Rev. Lett., 110, 16, Article 161301 pp. (2013), arXiv:1207.0741
[45] Mason, L.; Skinner, D., Ambitwistor strings and the scattering equations, J. High Energy Phys., 07, 048 (2014), arXiv:1311.2564
[46] Adamo, T.; Casali, E.; Skinner, D., Ambitwistor strings and the scattering equations at one loop, J. High Energy Phys., 04, 104 (2014), arXiv:1312.3828
[47] Berkovits, N., Infinite tension limit of the pure spinor superstring, J. High Energy Phys., 03, 017 (2014), arXiv:1311.4156 · Zbl 1333.83174
[48] Adamo, T.; Casali, E., Scattering equations, supergravity integrands, and pure spinors, J. High Energy Phys., 05, 120 (2015), arXiv:1502.06826 · Zbl 1388.83707
[49] Geyer, Y.; Mason, L., The SAGEX review on scattering amplitudes, chapter 6: Ambitwistor strings and amplitudes from the worldsheet (2022), arXiv:2203.13017 · Zbl 1520.81101
[50] Gomez, H.; Yuan, E. Y., N-point tree-level scattering amplitude in the new Berkovits‘ string, J. High Energy Phys., 04, 046 (2014), arXiv:1312.5485
[51] Erler, T., Four lectures on closed string field theory, Phys. Rep., 851, 1-36 (2020), arXiv:1905.06785 · Zbl 1459.81088
[52] Erler, T., Four lectures on analytic solutions in open string field theory, Phys. Rep., 980, 1-95 (2022), arXiv:1912.00521 · Zbl 1507.81152
[53] Erbin, H., (String Field Theory: A Modern Introduction. String Field Theory: A Modern Introduction, Lecture Notes in Physics, Vol. 980 (2021)) · Zbl 1457.81002
[54] Sen, A., Tachyon condensation on the brane anti-brane system, J. High Energy Phys., 08, 012 (1998), arXiv:hep-th/9805170 · Zbl 0955.81038
[55] Sen, A.; Zwiebach, B., Tachyon condensation in string field theory, J. High Energy Phys., 03, 002 (2000), arXiv:hep-th/9912249 · Zbl 0959.81047
[56] Berkovits, N.; Sen, A.; Zwiebach, B., Tachyon condensation in superstring field theory, Nuclear Phys. B, 587, 147-178 (2000), arXiv:hep-th/0002211 · Zbl 1043.81710
[57] Schnabl, M., Analytic solution for tachyon condensation in open string field theory, Adv. Theor. Math. Phys., 10, 4, 433-501 (2006), arXiv:hep-th/0511286 · Zbl 1101.81344
[58] Erler, T.; Maccaferri, C., String field theory solution for any open string background. Part II, J. High Energy Phys., 01, 021 (2020), arXiv:1909.11675 · Zbl 1434.81082
[59] Pius, R.; Rudra, A.; Sen, A., Mass renormalization in string theory: Special states, J. High Energy Phys., 07, 058 (2014), arXiv:1311.1257
[60] Pius, R.; Rudra, A.; Sen, A., Mass renormalization in string theory: General states, J. High Energy Phys., 07, 062 (2014), arXiv:1401.7014 · Zbl 1333.83199
[61] Sen, A., One loop mass renormalization of unstable particles in superstring theory, J. High Energy Phys., 11, 050 (2016), arXiv:1607.06500 · Zbl 1390.81468
[62] Gopakumar, R.; Vafa, C., On the gauge theory / geometry correspondence, Adv. Theor. Math. Phys., 3, 1415-1443 (1999), arXiv:hep-th/9811131 · Zbl 0972.81135
[63] Gaiotto, D.; Rastelli, L., A paradigm of open / closed duality: Liouville D-branes and the kontsevich model, J. High Energy Phys., 07, 053 (2005), arXiv:hep-th/0312196
[64] Berkovits, N., Perturbative super-yang-mills from the topological \(A d S_5 \times S^5\) sigma model, J. High Energy Phys., 09, 088 (2008), arXiv:0806.1960 · Zbl 1245.81082
[65] Okawa, Y., Nonperturbative definition of closed string theory via open string field theory (2020), arXiv:2006.16449
[66] Berkovits, N., Covariant quantization of the Green-Schwarz superstring in a Calabi-Yau background, Nuclear Phys. B, 431, 258-272 (1994), arXiv:hep-th/9404162 · Zbl 1020.81760
[67] Berkovits, N.; Vafa, C., N=4 topological strings, Nuclear Phys. B, 433, 123-180 (1995), arXiv:hep-th/9407190 · Zbl 1020.81761
[68] Berkovits, N., A new description of the superstring, (8th Jorge Andre Swieca Summer School: Particles and Fields (1996)), 390-418, arXiv:hep-th/9604123
[69] Berkovits, N.; Vafa, C.; Witten, E., Conformal field theory of AdS background with Ramond-Ramond flux, J. High Energy Phys., 03, 018 (1999), arXiv:hep-th/9902098 · Zbl 0965.81090
[70] Berkovits, N., Quantization of the superstring with manifest U(5) superPoincaré invariance, Phys. Lett. B, 457, 94-100 (1999), arXiv:hep-th/9902099 · Zbl 0987.81078
[71] Berkovits, N.; Vallilo, B. C., One loop N point superstring amplitudes with manifest d=4 supersymmetry, Nuclear Phys. B, 624, 45-62 (2002), arXiv:hep-th/0110168 · Zbl 0985.81088
[72] Berkovits, N.; Bershadsky, M.; Hauer, T.; Zhukov, S.; Zwiebach, B., Superstring theory on \(A d S_2 \times S^2\) as a coset supermanifold, Nuclear Phys. B, 567, 61-86 (2000), arXiv:hep-th/9907200 · Zbl 0951.81040
[73] Dolan, L.; Witten, E., Vertex operators for \(A d S_3\) background with Ramond-Ramond flux, J. High Energy Phys., 11, 003 (1999), arXiv:hep-th/9910205 · Zbl 0957.81050
[74] Gaberdiel, M. R.; Gerigk, S., The massless string spectrum on \(A d S_3 \times S^3\) from the supergroup, J. High Energy Phys., 10, 045 (2011), arXiv:1107.2660 · Zbl 1303.81163
[75] Gerigk, S., String states on \(A d S_3 \times S^3\) from the supergroup, J. High Energy Phys., 10, 084 (2012), arXiv:1208.0345
[76] Dei, A.; Gaberdiel, M. R.; Gopakumar, R.; Knighton, B., Free field world-sheet correlators for \(A d S_3\), J. High Energy Phys., 02, 081 (2021), arXiv:2009.11306
[77] Gaberdiel, M. R.; Naderi, K., The physical states of the Hybrid Formalism, J. High Energy Phys., 10, 168 (2021), arXiv:2106.06476 · Zbl 1476.83157
[78] Bobkov, K.; Dolan, L., Three graviton amplitude in Berkovits-Vafa-Witten variables, Phys. Lett. B, 537, 155-164 (2002), arXiv:hep-th/0201027 · Zbl 0995.81111
[79] Eberhardt, L.; Gaberdiel, M. R.; Gopakumar, R., The worldsheet dual of the symmetric product CFT, J. High Energy Phys., 04, 103 (2019), arXiv:1812.01007 · Zbl 1415.83055
[80] Eberhardt, L.; Gaberdiel, M. R.; Gopakumar, R., Deriving the \(A d S_3\)/CFT \({}_2\) correspondence, J. High Energy Phys., 02, 136 (2020), arXiv:1911.00378
[81] Knighton, B., Higher genus correlators for tensionless \(A d S_3\) strings, J. High Energy Phys., 04, 211 (2021), arXiv:2012.01445 · Zbl 1462.83075
[82] Berkovits, N., Quantum consistency of the superstring in \(A d S_5 \times S^5\) background, J. High Energy Phys., 03, 041 (2005), arXiv:hep-th/0411170
[83] Metsaev, R. R.; Tseytlin, A. A., Type IIB superstring action in \(A d S_5 \times S^5\) background, Nuclear Phys. B, 533, 109-126 (1998), arXiv:hep-th/9805028 · Zbl 0956.81063
[84] Mazzucato, L., Superstrings in AdS, Phys. Rep., 521, 1-68 (2012), arXiv:1104.2604
[85] Berkovits, N.; D’Hoker, E.; Green, M. B.; Johansson, H.; Schlotterer, O., Snowmass white paper: String perturbation theory, (2022 Snowmass Summer Study (2022)), arXiv:2203.09099
[86] Gopakumar, R.; Perlmutter, E.; Pufu, S. S.; Yin, X., Snowmass white paper: Bootstrapping string theory (2022), arXiv:2202.07163
[87] Brink, L.; Schwarz, J. H.; Scherk, J., Supersymmetric Yang-Mills theories, Nuclear Phys. B, 121, 77-92 (1977)
[88] Green, M. B.; Schwarz, J. H.; Brink, L., N=4 Yang-Mills and N=8 supergravity as limits of string theories, Nuclear Phys. B, 198, 474-492 (1982)
[89] Siegel, W., Superfields in higher dimensional space-time, Phys. Lett. B, 80, 220-223 (1979)
[90] Witten, E., Twistor - like transform in ten-dimensions, Nuclear Phys. B, 266, 245-264 (1986) · Zbl 0608.53068
[91] Mafra, C. R.; Schlotterer, O., Multiparticle SYM equations of motion and pure spinor BRST blocks, J. High Energy Phys., 07, 153 (2014), arXiv:1404.4986
[92] Lee, S.; Mafra, C. R.; Schlotterer, O., Non-linear gauge transformations in \(D = 10\) SYM theory and the BCJ duality, J. High Energy Phys., 03, 090 (2016), arXiv:1510.08843 · Zbl 1388.81572
[93] Bridges, E.; Mafra, C. R., Algorithmic construction of SYM multiparticle superfields in the BCJ gauge, J. High Energy Phys., 10, 022 (2019), arXiv:1906.12252 · Zbl 1427.83097
[94] Gomez, H.; Mafra, C. R., The closed-string 3-loop amplitude and S-duality, J. High Energy Phys., 10, 217 (2013), arXiv:1308.6567 · Zbl 1342.83103
[95] Mafra, C. R.; Schlotterer, O., Solution to the nonlinear field equations of ten dimensional supersymmetric Yang-Mills theory, Phys. Rev. D, 92, 6, Article 066001 pp. (2015), arXiv:1501.05562
[96] Harnad, J. P.; Shnider, S., Constraints and field equations for ten-dimensional Super-Yang-Mills Theory, Comm. Math. Phys., 106, 183 (1986) · Zbl 0601.53071
[97] Policastro, G.; Tsimpis, D., \( R^4\), purified, Classical Quantum Gravity, 23, 4753-4780 (2006), arXiv:hep-th/0603165 · Zbl 1103.83018
[98] Ooguri, H.; Rahmfeld, J.; Robins, H.; Tannenhauser, J., Holography in superspace, J. High Energy Phys., 07, 045 (2000), arXiv:hep-th/0007104 · Zbl 0965.81070
[99] Berkovits, N., Super-Poincaré covariant two-loop superstring amplitudes, J. High Energy Phys., 01, 005 (2006), arXiv:hep-th/0503197
[100] Berkovits, N., ICTP lectures on covariant quantization of the superstring, ICTP Lect. Notes Ser., 13, 57-107 (2003), arXiv:hep-th/0209059 · Zbl 1069.81570
[101] Mafra, C. R., Superstring Scattering Amplitudes with the Pure Spinor Formalism (2008), São Paulo, IFT, arXiv:0902.1552
[102] Schlotterer, O., Scattering amplitudes in open superstring theory, Fortschr. Phys., 60, 373-691 (2012) · Zbl 1250.81089
[103] Green, M. B.; Schwarz, J. H., Supersymmetrical dual string theory, Nuclear Phys. B, 181, 502-530 (1981)
[104] Green, M. B.; Schwarz, J. H., Supersymmetrical string theories, Phys. Lett. B, 109, 444-448 (1982)
[105] Gross, D. J.; Harvey, J. A.; Martinec, E. J.; Rohm, R., The heterotic string, Phys. Rev. Lett., 54, 502-505 (1985)
[106] Bengtsson, I.; Cederwall, M., Covariant Superstrings do not admit covariant gauge fixing (1984)
[107] Dirac, P., (Lectures on Quantum Mechanics. Lectures on Quantum Mechanics, Belfer Graduate School of Science, monograph series (2001), Dover Publications), URL https://books.google.co.uk/books?id=GVwzb1rZW9kC
[108] Green, M. B.; Schwarz, J. H., Supersymmetrical dual string theory. 2. Vertices and trees, Nuclear Phys. B, 198, 252-268 (1982)
[109] Green, M. B.; Schwarz, J. H., Supersymmetrical dual string theory. 3. Loops and renormalization, Nuclear Phys. B, 198, 441-460 (1982)
[110] Francesco, P.; Mathieu, P.; Sénéchal, D., Conformal Field Theory (1997), Springer New York, NY · Zbl 0869.53052
[111] Essler, F.; Hatsuda, M.; Laenen, E.; Siegel, W.; Yamron, J. P.; Kimura, T.; Mikovic, A. R., Covariant quantization of the first ilk superparticle, Nuclear Phys. B, 364, 67-84 (1991)
[112] Essler, F.; Laenen, E.; Siegel, W.; Yamron, J. P., BRST operator for the first ilk superparticle, Phys. Lett. B, 254, 411-416 (1991) · Zbl 1176.81126
[113] Berkovits, N.; Nekrasov, N., The Character of pure spinors, Lett. Math. Phys., 74, 75-109 (2005), arXiv:hep-th/0503075 · Zbl 1101.81084
[114] Cartan, E., The Theory of Spinors (1966), Hermann · Zbl 0147.40101
[115] Furlan, P.; Raczka, R., Nonlinear spinor representations, J. Math. Phys., 26, 3021 (1985) · Zbl 0591.22013
[116] Berkovits, N.; Cherkis, S. A., Higher-dimensional twistor transforms using pure spinors, J. High Energy Phys., 12, 049 (2004), arXiv:hep-th/0409243
[117] Mafra, C. R., Introdução aos Formalismos de Green-Schwarz e Espinores Puros da Supercorda (2005), São Paulo, IFT, (Master’s thesis)
[118] Hoogeveen, J., Fundamentals of the pure spinor formalism (2010), Amsterdam U., Inst. Math., (Ph.D. thesis)
[119] Mafra, C. R., Simplifying the tree-level superstring massless five-point amplitude, J. High Energy Phys., 01, 007 (2010), arXiv:0909.5206 · Zbl 1269.81165
[120] Gomez, H., One-loop superstring amplitude from integrals on pure spinors space, J. High Energy Phys., 12, 034 (2009), arXiv:0910.3405
[121] Gomez, H.; Mafra, C. R., The overall coefficient of the two-loop superstring amplitude using pure spinors, J. High Energy Phys., 05, 017 (2010), arXiv:1003.0678 · Zbl 1288.81105
[122] Berkovits, N., Twistor origin of the superstring, J. High Energy Phys., 03, 122 (2015), arXiv:1409.2510 · Zbl 1388.83515
[123] Matone, M.; Mazzucato, L.; Oda, I.; Sorokin, D.; Tonin, M., The Superembedding origin of the Berkovits pure spinor covariant quantization of superstrings, Nuclear Phys. B, 639, 182-202 (2002), arXiv:hep-th/0206104 · Zbl 0997.81082
[124] Aisaka, Y.; Kazama, Y., Origin of pure spinor superstring, J. High Energy Phys., 05, 046 (2005), arXiv:hep-th/0502208
[125] Berkovits, N., Explaining the pure spinor formalism for the superstring, J. High Energy Phys., 01, 065 (2008), arXiv:0712.0324
[126] Berkovits, N., Pure spinors, twistors, and emergent supersymmetry, J. High Energy Phys., 12, 006 (2012), arXiv:1105.1147 · Zbl 1397.81225
[127] Howe, P. S., Pure spinors lines in superspace and ten-dimensional supersymmetric theories, Phys. Lett. B. Phys. Lett. B, Phys. Lett. B, 259, 511-144 (1991), Addendum
[128] Nilsson, B. E., Pure spinors as auxiliary fields in the ten-dimensional supersymmetric Yang-Mills theory, Classical Quantum Gravity, 3, 2, L41 (1986) · Zbl 0587.53019
[129] Cederwall, M., Pure spinor superfields – an overview, Springer Proc. Phys., 153, 61-93 (2014), arXiv:1307.1762
[130] Cederwall, M., Pure spinors in classical and quantum supergravity (2022), arXiv:2210.06141
[131] Hughston, L. P.; Shaw, W. T., Classical strings in ten dimensions, Proc. Roy. Soc. Lond. A, 414, 423 (1987) · Zbl 0635.53079
[132] Berkovits, N.; Hatsuda, M. T.; Siegel, W., The Big picture, Nuclear Phys. B, 371, 434-466 (1992), arXiv:hep-th/9108021
[133] Oda, I.; Tonin, M., Y-formalism and b ghost in the non-minimal pure spinor formalism of superstrings, Nuclear Phys. B, 779, 63-100 (2007), arXiv:0704.1219 · Zbl 1200.81129
[134] Lipinski Jusinskas, R., Nilpotency of the b ghost in the non-minimal pure spinor formalism, J. High Energy Phys., 05, 048 (2013), arXiv:1303.3966 · Zbl 1342.81439
[135] Chandia, O., The b ghost of the pure spinor formalism is nilpotent, Phys. Lett. B, 695, 312-316 (2011), arXiv:1008.1778
[136] Jusinskas, R. L., Notes on the pure spinor b ghost, J. High Energy Phys., 07, 142 (2013), arXiv:1306.1963 · Zbl 1342.81440
[137] Chandia, O.; Vallilo, B. C., Relating the \(b\) ghost and the vertex operators of the pure spinor superstring, J. High Energy Phys., 03, 165 (2021), arXiv:2101.01129 · Zbl 1461.81056
[138] Berkovits, N., Cohomology in the pure spinor formalism for the superstring, J. High Energy Phys., 09, 046 (2000), arXiv:hep-th/0006003 · Zbl 0989.81587
[139] Berkovits, N.; Chandia, O., Lorentz invariance of the pure spinor BRST cohomology for the superstring, Phys. Lett. B, 514, 394-400 (2001), arXiv:hep-th/0105149 · Zbl 0971.81127
[140] Aisaka, Y.; Arroyo, E. A.; Berkovits, N.; Nekrasov, N., Pure spinor partition function and the massive superstring spectrum, J. High Energy Phys., 08, 050 (2008), arXiv:0806.0584
[141] Berkovits, N.; Chandia, O., Massive superstring vertex operator in D=10 superspace, J. High Energy Phys., 08, 040 (2002), arXiv:hep-th/0204121 · Zbl 1226.81158
[142] Chakrabarti, S.; Kashyap, S. P.; Verma, M., Integrated massive vertex operator in pure spinor formalism, J. High Energy Phys., 10, 147 (2018), arXiv:1802.04486 · Zbl 1402.83093
[143] Chakrabarti, S.; Kashyap, S. P.; Verma, M., Amplitudes involving massive states using pure spinor formalism, J. High Energy Phys., 12, 071 (2018), arXiv:1808.08735 · Zbl 1405.83059
[144] Bjerrum-Bohr, N. E.J.; Damgaard, P. H.; Vanhove, P., Minimal basis for gauge theory amplitudes, Phys. Rev. Lett., 103, Article 161602 pp. (2009), arXiv:0907.1425
[145] Broedel, J.; Dixon, L. J., Color-kinematics duality and double-copy construction for amplitudes from higher-dimension operators, J. High Energy Phys., 10, 091 (2012), arXiv:1208.0876
[146] Stieberger, S.; Taylor, T. R., Closed string amplitudes as single-valued open string amplitudes, Nuclear Phys., B881, 269-287 (2014), arXiv:1401.1218 · Zbl 1284.81245
[147] Mafra, C. R., KK-like relations of \(\alpha \)’ corrections to disk amplitudes, J. High Energy Phys., 03, 012 (2022), arXiv:2108.01081 · Zbl 1522.81387
[148] Solomon, L., A Mackey formula in the group ring of a Coxeter group, J. Algebra, 41, 2, 255-264 (1976) · Zbl 0355.20007
[149] Garsia, A. M.; Reutenauer, C., A decomposition of Solomon’s descent algebra, Adv. Math., 77, 2, 189-262 (1989) · Zbl 0716.20006
[150] Schocker, M., (The Descent Algebra of the Symmetric Group (2004)), 145-161 · Zbl 1072.20004
[151] Thibon, J.-Y., Lie idempotents in descent algebras (2016), URL https://igm.univ-mlv.fr/jyt/TALKS/lieids.ps
[152] Reutenauer, C., Free Lie algebras, (Handbook of Algebra. Handbook of Algebra, Handb. Algebr., Vol. 3 (2003), Elsevier/North-Holland, Amsterdam), 887-903 · Zbl 1071.17003
[153] Azevedo, T., On the \(\mathcal{N} =4\), d=4 pure spinor measure factor, J. High Energy Phys., 03, 136 (2015), arXiv:1412.5927 · Zbl 1388.81762
[154] Berkovits, N.; Vallilo, B. C., Consistency of superPoincaré covariant superstring tree amplitudes, J. High Energy Phys., 07, 015 (2000), arXiv:hep-th/0004171 · Zbl 0989.81098
[155] Leeuwen, A.; Cohen, M.; Lisser, B., LiE, A Package for Lie group computations, (Computer Algebra Nederland, Amsterdam (1992)), ISBN 90-74116-02-7, URL http://www-math.univ-poitiers.fr/maavl/LiE/
[156] Scherk, J., Zero-slope limit of the dual resonance model, Nuclear Phys. B, 31, 222-234 (1971)
[157] Neveu, A.; Scherk, J., Connection between Yang-Mills fields and dual models, Nuclear Phys. B, 36, 155-161 (1972)
[158] Gliozzi, F.; Scherk, J.; Olive, D. I., Supergravity and the spinor dual model, Phys. Lett. B, 65, 282-286 (1976)
[159] Gliozzi, F.; Scherk, J.; Olive, D. I., Supersymmetry, supergravity theories and the dual spinor model, Nuclear Phys. B, 122, 253-290 (1977)
[160] Berkovits, N., Explaining pure spinor superspace (2006), arXiv:hep-th/0612021
[161] Vermaseren, J. A.M., New features of FORM (2000), arXiv:math-ph/0010025
[162] Mafra, C. R., PSS: A FORM program to evaluate pure spinor superspace expressions (2010), arXiv:1007.4999
[163] Sun, K.-S.; Ding, X.-M.; Sun, F.; Zhang, H.-B., Computations of superstring amplitudes in pure spinor formalism via Cadabra (2016), arXiv:1607.00463
[164] Giele, W.; Stavenga, G.; Winter, J.-C., Thread-scalable evaluation of multi-jet observables, Eur. Phys. J. C, 71, 1703 (2011), arXiv:1002.3446
[165] Mafra, C. R., Planar binary trees in scattering amplitudes, ((2020)), arXiv:2011.14413
[166] Frost, H.; Mafra, C. R.; Mason, L., A Lie bracket for the momentum kernel (2020), arXiv:2012.00519
[167] Mafra, C. R., Pure spinor superspace identities for massless four-point kinematic factors, J. High Energy Phys., 04, 093 (2008), arXiv:0801.0580 · Zbl 1246.81284
[168] Tong, D., String theory (2009), arXiv:0908.0333
[169] Mafra, C. R.; Schlotterer, O.; Stieberger, S.; Tsimpis, D., Six open string disk amplitude in pure spinor superspace, Nuclear Phys., B846, 359-393 (2011), arXiv:1011.0994 · Zbl 1208.81167
[170] Mafra, C. R., Towards field theory amplitudes from the cohomology of pure spinor superspace, J. High Energy Phys., 11, 096 (2010), arXiv:1007.3639 · Zbl 1294.81218
[171] Mafra, C. R.; Schlotterer, O.; Stieberger, S.; Tsimpis, D., A recursive method for SYM n-point tree amplitudes, Phys. Rev., D83, Article 126012 pp. (2011), arXiv:1012.3981
[172] Blessenohl, D.; Laue, H., Generalized Jacobi identities, Note Mat., 8, 1, 111-121 (1988) · Zbl 0715.17005
[173] Mafra, C. R.; Schlotterer, O., The structure of n-point one-loop open superstring amplitudes, J. High Energy Phys., 08, 099 (2014), arXiv:1203.6215
[174] Ahmadiniaz, N.; Balli, F. M.; Lopez-Arcos, C.; Velez, A. Q.; Schubert, C., Color-kinematics duality from the Bern-Kosower formalism, Phys. Rev. D, 104, 4, L041702 (2021), arXiv:2105.06745
[175] Mafra, C. R.; Schlotterer, O., Berends-Giele recursions and the BCJ duality in superspace and components, J. High Energy Phys., 03, 097 (2016), arXiv:1510.08846 · Zbl 1388.81581
[176] Broedel, J.; Schlotterer, O.; Stieberger, S., Polylogarithms, multiple zeta values and superstring amplitudes, Fortschr. Phys., 61, 812-870 (2013), arXiv:1304.7267 · Zbl 1338.81316
[177] Ree, R., Lie elements and an algebra associated with shuffles, Ann. of Math., 210-220 (1958) · Zbl 0083.25401
[178] Green, M. B.; Mafra, C. R.; Schlotterer, O., Multiparticle one-loop amplitudes and S-duality in closed superstring theory, J. High Energy Phys., 10, 188 (2013), arXiv:1307.3534 · Zbl 1342.83372
[179] Mafra, C. R.; Schlotterer, O., Towards one-loop SYM amplitudes from the pure spinor BRST cohomology, Fortschr. Phys., 63, 2, 105-131 (2015), arXiv:1410.0668 · Zbl 1338.81291
[180] Mafra, C. R.; Schlotterer, O., Two-loop five-point amplitudes of super Yang-Mills and supergravity in pure spinor superspace, J. High Energy Phys., 10, 124 (2015), arXiv:1505.02746 · Zbl 1388.83860
[181] Mafra, C. R.; Schlotterer, O., Towards the n-point one-loop superstring amplitude. Part III. One-loop correlators and their double-copy structure, J. High Energy Phys., 08, 092 (2019), arXiv:1812.10971 · Zbl 1421.83120
[182] D’Hoker, E.; Mafra, C. R.; Pioline, B.; Schlotterer, O., Two-loop superstring five-point amplitudes. Part I. Construction via chiral splitting and pure spinors, J. High Energy Phys., 08, 135 (2020), arXiv:2006.05270 · Zbl 1454.83159
[183] Bjornsson, J., Multi-loop amplitudes in maximally supersymmetric pure spinor field theory, J. High Energy Phys., 01, 002 (2011), arXiv:1009.5906 · Zbl 1214.81134
[184] Bjornsson, J.; Green, M. B., 5 loops in 24/5 dimensions, J. High Energy Phys., 08, 132 (2010), arXiv:1004.2692 · Zbl 1290.81144
[185] Berends, F. A.; Giele, W. T., Multiple soft gluon radiation in parton processes, Nuclear Phys. B, 313, 595-633 (1989)
[186] Schocker, M., Lie elements and knuth relations, Canad. J. Math., 56, 4, 871-882 (2004) · Zbl 1089.17008
[187] Schubert, C., The Finite gauge transformations in closed string field theory, Lett. Math. Phys., 26, 259-264 (1992) · Zbl 0766.53065
[188] Elliot, N. B., BCJ Identities in Pure Spinor Superspace (2021), Southampton U., (Ph.D. thesis)
[189] Garsia, A. M., Combinatorics of the free Lie algebra and the symmetric group, (Analysis, Et Cetera (1990), Elsevier), 309-382 · Zbl 0709.17003
[190] Frost, H., Universal aspects of perturbative gauge theory (2020), University of Oxford, (Ph.D. thesis)
[191] Michaelis, W., Lie coalgebras, Adv. Math., 38, 1, 1-54 (1980) · Zbl 0451.16006
[192] Kawai, H.; Lewellen, D. C.; Tye, S. H.H., A relation between tree amplitudes of closed and open strings, Nuclear Phys. B, 269, 1-23 (1986)
[193] Bern, Z.; Dixon, L. J.; Perelstein, M.; Rozowsky, J. S., Multileg one loop gravity amplitudes from gauge theory, Nuclear Phys. B, 546, 423-479 (1999), arXiv:hep-th/9811140 · Zbl 0953.83006
[194] Bjerrum-Bohr, N. E.J.; Damgaard, P. H.; Feng, B.; Sondergaard, T., Gravity and Yang-Mills amplitude relations, Phys. Rev. D, 82, Article 107702 pp. (2010), arXiv:1005.4367
[195] Bjerrum-Bohr, N. E.J.; Damgaard, P. H.; Sondergaard, T.; Vanhove, P., The momentum kernel of gauge and gravity theories, J. High Energy Phys., 01, 001 (2011), arXiv:1010.3933 · Zbl 1214.81145
[196] Du, Y.-J.; Fu, C.-H., Explicit BCJ numerators of nonlinear simga model, J. High Energy Phys., 09, 174 (2016), arXiv:1606.05846 · Zbl 1390.81321
[197] Carrasco, J. J.M.; Mafra, C. R.; Schlotterer, O., Abelian Z-theory: NLSM amplitudes and \(\alpha \)’-corrections from the open string, J. High Energy Phys., 06, 093 (2017), arXiv:1608.02569 · Zbl 1380.83251
[198] Mizera, S., Combinatorics and topology of Kawai-Lewellen-Tye relations, J. High Energy Phys., 08, 097 (2017), arXiv:1706.08527 · Zbl 1381.83126
[199] Mizera, S., Scattering amplitudes from intersection theory, Phys. Rev. Lett., 120, 14, Article 141602 pp. (2018), arXiv:1711.00469
[200] Mizera, S., Aspects of Scattering Amplitudes and Moduli Space Localization (2020), Princeton, Inst. Advanced Study, arXiv:1906.02099 · Zbl 1457.81067
[201] Frost, H., The algebraic structure of the KLT relations for gauge and gravity tree amplitudes, SIGMA, 17, 101 (2021), arXiv:2111.07257 · Zbl 1480.81091
[202] Mafra, C. R.; Schlotterer, O.; Stieberger, S., Complete N-point superstring disk amplitude II. Amplitude and hypergeometric function structure, Nuclear Phys., B873, 461-513 (2013), arXiv:1106.2646 · Zbl 1282.81152
[203] Berends, F. A.; Giele, W., The six gluon process as an example of Weyl-Van Der Waerden spinor calculus, Nuclear Phys. B, 294, 700-732 (1987)
[204] Mangano, M. L.; Parke, S. J.; Xu, Z., Duality and multi - gluon scattering, Nuclear Phys. B, 298, 653-672 (1988)
[205] Del Duca, V.; Dixon, L. J.; Maltoni, F., New color decompositions for gauge amplitudes at tree and loop level, Nuclear Phys. B, 571, 51-70 (2000), arXiv:hep-ph/9910563
[206] Badger, S.; Biedermann, B.; Hackl, L.; Plefka, J.; Schuster, T.; Uwer, P., Comparing efficient computation methods for massless QCD tree amplitudes: Closed analytic formulas versus Berends-Giele recursion, Phys. Rev. D, 87, 3, Article 034011 pp. (2013), arXiv:1206.2381
[207] Duhr, C.; Hoeche, S.; Maltoni, F., Color-dressed recursive relations for multi-parton amplitudes, J. High Energy Phys., 08, 062 (2006), arXiv:hep-ph/0607057
[208] Medina, R.; Brandt, F. T.; Machado, F. R., The Open superstring five point amplitude revisited, J. High Energy Phys., 07, 071 (2002), arXiv:hep-th/0208121
[209] Oprisa, D.; Stieberger, S., Six gluon open superstring disk amplitude, multiple hypergeometric series and Euler-Zagier sums (2005), arXiv:hep-th/0509042
[210] C. Mafra, O. Schlotterer, PSS: From pure spinor superspace to components, URL http://www.southampton.ac.uk/crm1n16/pss.html. · Zbl 1388.83860
[211] Garozzo, L. M.; Queimada, L.; Schlotterer, O., Berends-Giele currents in Bern-Carrasco-Johansson gauge for \(F^3\)- and \(F^4\)-deformed Yang-Mills amplitudes, J. High Energy Phys., 02, 078 (2019), arXiv:1809.08103
[212] Berends, F. A.; Giele, W. T.; Kuijf, H., Exact and approximate expressions for multi - gluon scattering, Nuclear Phys. B, 333, 120-159 (1990)
[213] Bandiera, R., private communication (2020)
[214] Novelli, J.-C.; Thibon, J.-Y.; Toumazet, F., A noncommutative cycle index and new bases of quasi-symmetric functions and noncommutative symmetric functions, Ann. Comb., 24, 3, 557-576 (2020), arXiv:1804.01762 · Zbl 1460.16037
[215] Feng, B.; Huang, R.; Jia, Y., Gauge amplitude identities by on-shell recursion relation in S-matrix program, Phys. Lett. B, 695, 350-353 (2011), arXiv:1004.3417
[216] Chen, Y.-X.; Du, Y.-J.; Feng, B., A proof of the explicit minimal-basis expansion of tree amplitudes in gauge field theory, J. High Energy Phys., 02, 112 (2011), arXiv:1101.0009 · Zbl 1294.81273
[217] Berkovits, N., Covariant quantization of the superparticle using pure spinors, J. High Energy Phys., 09, 016 (2001), arXiv:hep-th/0105050
[218] Brown, L. S., Summing tree graphs at threshold, Phys. Rev. D, 46, R4125-R4127 (1992), arXiv:hep-ph/9209203
[219] Aomoto, K., Gauss-Manin connection of integral of difference products, J. Math. Soc. Japan, 39, 2, 191-208 (1987) · Zbl 0619.32010
[220] Huang, Y.-t.; Schlotterer, O.; Wen, C., Universality in string interactions, J. High Energy Phys., 09, 155 (2016), arXiv:1602.01674 · Zbl 1390.81441
[221] Azevedo, T.; Chiodaroli, M.; Johansson, H.; Schlotterer, O., Heterotic and bosonic string amplitudes via field theory, J. High Energy Phys., 10, 012 (2018), arXiv:1803.05452 · Zbl 1402.83092
[222] Veneziano, G., Construction of a crossing - symmetric, Regge behaved amplitude for linearly rising trajectories, Nuovo Cimento A, 57, 190-197 (1968)
[223] Kitazawa, Y., Effective Lagrangian for open superstring from five point function, Nuclear Phys. B, 289, 599-608 (1987)
[224] Barreiro, L. A.; Medina, R., 5-field terms in the open superstring effective action, J. High Energy Phys., 03, 055 (2005), arXiv:hep-th/0503182
[225] Stieberger, S.; Taylor, T. R., Amplitude for N-gluon superstring scattering, Phys. Rev. Lett., 97, Article 211601 pp. (2006), arXiv:hep-th/0607184 · Zbl 1228.81272
[226] Stieberger, S.; Taylor, T. R., Multi-gluon scattering in open superstring theory, Phys. Rev. D, 74, Article 126007 pp. (2006), arXiv:hep-th/0609175
[227] Stieberger, S.; Taylor, T. R., Supersymmetry relations and MHV amplitudes in superstring theory, Nuclear Phys. B, 793, 83-113 (2008), arXiv:0708.0574 · Zbl 1225.81103
[228] Stieberger, S.; Taylor, T. R., Complete six-gluon disk amplitude in superstring theory, Nuclear Phys. B, 801, 128-152 (2008), arXiv:0711.4354 · Zbl 1189.81190
[229] Aomoto, K., On the complex Selberg integral, Q. J. Math., 38, 4, 385-399 (1987) · Zbl 0639.33002
[230] Stieberger, S.; Taylor, T. R., Maximally helicity violating disk amplitudes, twistors and transcendental integrals, Phys. Lett. B, 716, 236-239 (2012), arXiv:1204.3848
[231] Parke, S. J.; Taylor, T. R., An amplitude for \(n\) gluon scattering, Phys. Rev. Lett., 56, 2459 (1986)
[232] Frampton, P., (Dual Resonance Models. Dual Resonance Models, Advance Book Program (1974), W.A. Benjamin), URL https://books.google.co.uk/books?id=lp6yAAAAIAAJ
[233] Brown, F.; Dupont, C., Single-valued integration and superstring amplitudes in genus zero, Comm. Math. Phys., 382, 2, 815-874 (2021), arXiv:1910.01107 · Zbl 1483.81117
[234] Mafra, C. R., Berends-Giele recursion for double-color-ordered amplitudes, J. High Energy Phys., 07, 080 (2016), arXiv:1603.09731 · Zbl 1390.81336
[235] Mafra, C. R.; Schlotterer, O., Non-abelian \(Z\)-theory: Berends-Giele recursion for the \(\alpha^\prime \)-expansion of disk integrals, J. High Energy Phys., 01, 031 (2017), arXiv:1609.07078 · Zbl 1373.83110
[236] Mafra, C. R.; Schlotterer, O.; Stieberger, S., Explicit BCJ numerators from pure spinors, J. High Energy Phys., 07, 092 (2011), arXiv:1104.5224 · Zbl 1298.81319
[237] Bern, Z.; Dennen, T.; Huang, Y.-t.; Kiermaier, M., Gravity as the square of gauge theory, Phys. Rev., D82, Article 065003 pp. (2010), arXiv:1004.0693
[238] Bern, Z.; Carrasco, J. J.M.; Johansson, H., Perturbative quantum gravity as a double copy of gauge theory, Phys. Rev. Lett., 105, Article 061602 pp. (2010), arXiv:1004.0476
[239] Bjerrum-Bohr, N. E.J.; Damgaard, P. H.; Sondergaard, T.; Vanhove, P., Monodromy and Jacobi-like relations for color-ordered amplitudes, J. High Energy Phys., 06, 003 (2010), arXiv:1003.2403 · Zbl 1290.83015
[240] Dixon, L. J., Calculating scattering amplitudes efficiently, (QCD and beyond. Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics, TASI-95, Boulder, USA, June 4-30, 1995 (1996)), 539-584, arXiv:hep-ph/9601359, URL http://www-public.slac.stanford.edu/sciDoc/docMeta.aspx?slacPubNumber=SLAC-PUB-7106
[241] Brandhuber, A.; Brown, G. R.; Chen, G.; Gowdy, J.; Travaglini, G.; Wen, C., Amplitudes, Hopf algebras and the colour-kinematics duality (2022), arXiv:2208.05886 · Zbl 1536.81124
[242] Chen, G.; Lin, G.; Wen, C., Kinematic Hopf algebra for amplitudes and form factors (2022), arXiv:2208.05519
[243] Brandhuber, A.; Chen, G.; Johansson, H.; Travaglini, G.; Wen, C., Kinematic hopf algebra for Bern-Carrasco-Johansson numerators in heavy-mass effective field theory and Yang-Mills theory, Phys. Rev. Lett., 128, 12, Article 121601 pp. (2022), arXiv:2111.15649
[244] Ma, Q.; Du, Y.-J.; Chen, Y.-X., On primary relations at tree-level in string theory and field theory, J. High Energy Phys., 02, 061 (2012), arXiv:1109.0685 · Zbl 1309.81281
[245] Mizera, S., Inverse of the string theory KLT kernel, J. High Energy Phys., 06, 084 (2017), arXiv:1610.04230 · Zbl 1380.81424
[246] Cho, K.; Matsumoto, K., Intersection theory for twisted cohomologies and twisted Riemann’s period relations. I, Nagoya Math. J., 139, 67-86 (1995), https://projecteuclid.org:443/euclid.nmj/1118775097 · Zbl 0856.32015
[247] Green, M. B.; Gutperle, M., Symmetry breaking at enhanced symmetry points, Nuclear Phys., B460, 77-108 (1996), arXiv:hep-th/9509171 · Zbl 1003.81526
[248] Metsaev, R. R.; Rakhmanov, M.; Tseytlin, A. A., The Born-Infeld action as the effective action in the open superstring theory, Phys. Lett., B193, 207-212 (1987)
[249] Tseytlin, A. A., Born-Infeld action, supersymmetry and string theory, 417-452 (1999), arXiv:hep-th/9908105 · Zbl 1065.81594
[250] Kallosh, R., Volkov-Akulov theory and D-branes, Lecture Notes in Phys., 509, 49 (1998), arXiv:hep-th/9705118 · Zbl 0933.81024
[251] Bergshoeff, E.; Coomans, F.; Kallosh, R.; Shahbazi, C. S.; Van Proeyen, A., Dirac-Born-Infeld-Volkov-Akulov and deformation of supersymmetry, J. High Energy Phys., 08, 100 (2013), arXiv:1303.5662 · Zbl 1342.81549
[252] Cachazo, F.; He, S.; Yuan, E. Y., Scattering equations and matrices: From Einstein to Yang-Mills, DBI and NLSM, J. High Energy Phys., 07, 149 (2015), arXiv:1412.3479 · Zbl 1388.83196
[253] Cronin, J. A., Phenomenological model of strong and weak interactions in chiral \(U ( 3 ) \times U ( 3 )\), Phys. Rev., 161, 1483-1494 (1967)
[254] Weinberg, S., Dynamical approach to current algebra, Phys. Rev. Lett., 18, 188-191 (1967)
[255] Weinberg, S., Nonlinear realizations of chiral symmetry, Phys. Rev., 166, 1568-1577 (1968)
[256] Brown, L. S., Field theory of chiral symmetry, Phys. Rev., 163, 1802-1807 (1967)
[257] Chang, P.; Gursey, F., Unified formulation of effective nonlinear pion-nucleon Lagrangians, Phys. Rev., 164, 1752-1761 (1967)
[258] Osborn, H., Implications of adler zeros for multipion processes, Lett. Nuovo Cimento. Lett. Nuovo Cimento, Lett. Nuovo Cim., 2, 717-723 (1969)
[259] Susskind, L.; Frye, G., Algebraic aspects of pionic duality diagrams, Phys. Rev., D1, 1682-1686 (1970)
[260] Ellis, J. R.; Renner, B., On the relationship between chiral and dual models, Nuclear Phys., B21, 205-216 (1970)
[261] Kampf, K.; Novotny, J.; Trnka, J., Tree-level amplitudes in the nonlinear sigma model, J. High Energy Phys., 05, 032 (2013), arXiv:1304.3048 · Zbl 1392.81139
[262] Chen, G.; Du, Y.-J., Amplitude relations in non-linear sigma model, J. High Energy Phys., 01, 061 (2014), arXiv:1311.1133 · Zbl 1390.81194
[263] Carrasco, J. J.M.; Mafra, C. R.; Schlotterer, O., Semi-abelian Z-theory: NLSM+\( \phi^3\) from the open string, J. High Energy Phys., 08, 135 (2017), arXiv:1612.06446 · Zbl 1381.83121
[264] Cheung, C.; Shen, C.-H., Symmetry for flavor-kinematics duality from an action, Phys. Rev. Lett., 118, 12, Article 121601 pp. (2017), arXiv:1612.00868
[265] Cachazo, F.; Cha, P.; Mizera, S., Extensions of theories from soft limits, J. High Energy Phys., 06, 170 (2016), arXiv:1604.03893 · Zbl 1388.81203
[266] Gross, D. J.; Harvey, J. A.; Martinec, E. J.; Rohm, R., Heterotic string theory. 2. The interacting heterotic string, Nuclear Phys. B, 267, 75-124 (1986)
[267] Frenkel, I.; Zhu, M., Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J., 66, 123 (1992) · Zbl 0848.17032
[268] Chiodaroli, M.; Günaydin, M.; Johansson, H.; Roiban, R., Scattering amplitudes in \(\mathcal{N} = 2\) Maxwell-Einstein and Yang-Mills/Einstein supergravity, J. High Energy Phys., 01, 081 (2015), arXiv:1408.0764 · Zbl 1388.83772
[269] Schlotterer, O., Amplitude relations in heterotic string theory and Einstein-Yang-Mills, J. High Energy Phys., 11, 074 (2016), arXiv:1608.00130 · Zbl 1390.83354
[270] He, S.; Teng, F.; Zhang, Y., String amplitudes from field-theory amplitudes and vice versa, Phys. Rev. Lett., 122, 21, Article 211603 pp. (2019), arXiv:1812.03369
[271] He, S.; Teng, F.; Zhang, Y., String correlators: Recursive expansion, integration-by-parts and scattering equations, J. High Energy Phys., 09, 085 (2019), arXiv:1907.06041 · Zbl 1423.83083
[272] Johansson, H.; Nohle, J., Conformal gravity from gauge theory (2017), arXiv:1707.02965
[273] Chiodaroli, M.; Gunaydin, M.; Johansson, H.; Roiban, R., Explicit formulae for Yang-Mills-Einstein amplitudes from the double copy, J. High Energy Phys., 07, 002 (2017), arXiv:1703.00421 · Zbl 1380.83280
[274] Stieberger, S.; Taylor, T. R., New relations for Einstein-Yang-Mills amplitudes, Nuclear Phys. B, 913, 151-162 (2016), arXiv:1606.09616 · Zbl 1349.81162
[275] Cachazo, F.; He, S.; Yuan, E. Y., Einstein-Yang-Mills scattering amplitudes from scattering equations, J. High Energy Phys., 01, 121 (2015), arXiv:1409.8256 · Zbl 1388.81917
[276] Nandan, D.; Plefka, J.; Schlotterer, O.; Wen, C., Einstein-Yang-Mills from pure Yang-Mills amplitudes, J. High Energy Phys., 10, 070 (2016), arXiv:1607.05701 · Zbl 1390.83022
[277] Mazloumi, P.; Stieberger, S., Einstein Yang-Mills amplitudes from intersections of twisted forms, J. High Energy Phys., 06, 125 (2022), arXiv:2201.00837 · Zbl 1522.81235
[278] Du, Y.-J.; Feng, B.; Teng, F., Expansion of all multitrace tree level EYM amplitudes, J. High Energy Phys., 12, 038 (2017), arXiv:1708.04514 · Zbl 1383.83009
[279] Broedel, J.; Schlotterer, O.; Stieberger, S.; Terasoma, T., All order \(\alpha^\prime \)-expansion of superstring trees from the Drinfeld associator, Phys. Rev., D89, 6, Article 066014 pp. (2014), arXiv:1304.7304
[280] Stieberger, S., Constraints on tree-level higher order gravitational couplings in superstring theory, Phys. Rev. Lett., 106, Article 111601 pp. (2011), arXiv:0910.0180
[281] Schlotterer, O.; Stieberger, S., Motivic multiple zeta values and superstring amplitudes, J. Phys., A46, Article 475401 pp. (2013), arXiv:1205.1516 · Zbl 1280.81112
[282] Boels, R. H., On the field theory expansion of superstring five point amplitudes, Nuclear Phys. B, 876, 215-233 (2013), arXiv:1304.7918 · Zbl 1284.81226
[283] Puhlfürst, G.; Stieberger, S., Differential equations, associators, and recurrences for amplitudes, Nuclear Phys. B, 902, 186-245 (2016), arXiv:1507.01582 · Zbl 1332.81188
[284] Terasoma, T., Selberg integrals and multiple zeta values, Compos. Math., 133, 1, 1-24 (2002) · Zbl 1003.11042
[285] Goncharov, A. B., Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math. J., 128, 209 (2005), arXiv:math/0208144 · Zbl 1095.11036
[286] Brown, F., Multiple zeta values and periods of moduli spaces \(\mathcal{M}_{0 , n} ( \mathbb{R} )\), Ann. Sci. Ecole Norm. Sup., 42, 371 (2009), arXiv:math/0606419 · Zbl 1216.11079
[287] Brown, F., On the decomposition of motivic multiple zeta values, (Galois-Teichmüller Theory and Arithmetic Geometry. Galois-Teichmüller Theory and Arithmetic Geometry, Adv. Stud. Pure Math., Vol. 63 (2012), Math. Soc. Japan, Tokyo), 31-58, arXiv:1102.1310 · Zbl 1321.11087
[288] Schnetz, O., Graphical functions and single-valued multiple polylogarithms, Commun. Number Theory Phys., 08, 589-675 (2014), arXiv:1302.6445 · Zbl 1320.81075
[289] Brown, F., Single-valued motivic periods and multiple zeta values, SIGMA, 2, Article e25 pp. (2014), arXiv:1309.5309 · Zbl 1377.11099
[290] Brown, F.; Dupont, C., Single-valued integration and double copy, J. Reine Angew. Math., 2021, 775, 145-196 (2021), arXiv:1810.07682 · Zbl 1484.14043
[291] O. Schlotterer, The number theory of superstring amplitudes, in: J.I. Burgos Gil, K. Ebrahimi-Fard, H. Gangl (Eds.) Periods in Quantum Field Theory and Arithmetic, http://dx.doi.org/10.1007/978-3-030-37031-2. · Zbl 1444.81033
[292] S. Stieberger, Periods and superstring amplitudes, in: J.I. Burgos Gil, K. Ebrahimi-Fard, H. Gangl (Eds.) Periods in Quantum Field Theory and Arithmetic, arXiv:1605.03630.
[293] Duhr, C., Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, J. High Energy Phys., 08, 043 (2012), arXiv:1203.0454 · Zbl 1397.16028
[294] Richards, D. M., The one-loop five-graviton amplitude and the effective action, J. High Energy Phys., 10, 042 (2008), arXiv:0807.2421
[295] Tseytlin, A. A., Vector field effective action in the open superstring theory, Nuclear Phys. B. Nuclear Phys. B, Nucl. Phys. B, 291, 876 (1987), Erratum
[296] Gross, D. J.; Witten, E., Superstring modifications of Einstein’s equations, Nuclear Phys., B277, 1 (1986)
[297] Barreiro, L. A.; Medina, R., Revisiting the S-matrix approach to the open superstring low energy effective lagrangian, J. High Energy Phys., 10, 108 (2012), arXiv:1208.6066 · Zbl 1397.81185
[298] Barreiro, L. A.; Medina, R., RNS derivation of N-point disk amplitudes from the revisited S-matrix approach, Nuclear Phys. B, 886, 870-951 (2014), arXiv:1310.5942 · Zbl 1325.81134
[299] Andreev, O. D.; Tseytlin, A. A., Partition function representation for the open superstring effective action: Cancellation of mobius infinities and derivative corrections to Born-infeld Lagrangian, Nuclear Phys. B, 311, 205-252 (1988) · Zbl 1232.81040
[300] Koerber, P.; Sevrin, A., The NonAbelian D-brane effective action through order \(\alpha{}^\prime^4\), J. High Energy Phys., 10, 046 (2002), arXiv:hep-th/0208044
[301] Howe, P. S.; Lindstrom, U.; Wulff, L., D=10 supersymmetric Yang-Mills theory at \(\alpha{}^\prime^4\), J. High Energy Phys., 07, 028 (2010), arXiv:1004.3466 · Zbl 1290.81070
[302] Tseytlin, A. A., On nonAbelian generalization of Born-Infeld action in string theory, Nuclear Phys. B, 501, 41-52 (1997), arXiv:hep-th/9701125 · Zbl 0939.81030
[303] Cederwall, M.; Nilsson, B. E.W.; Tsimpis, D., Spinorial cohomology of Abelian D=10 superYang-Mills at \(O( \alpha{}^\prime^3)\), J. High Energy Phys., 11, 023 (2002), arXiv:hep-th/0205165
[304] Peeters, K.; Plefka, J.; Stern, S., Higher-derivative gauge field terms in the M-theory action, J. High Energy Phys., 08, 095 (2005), arXiv:hep-th/0507178
[305] Liu, J. T.; Minasian, R.; Savelli, R.; Schachner, A., Type IIB at eight derivatives: insights from Superstrings, Superfields and Superparticles, J. High Energy Phys., 08, 267 (2022), arXiv:2205.11530 · Zbl 1522.81385
[306] Grisaru, M. T.; van de Ven, A. E.M.; Zanon, D., Four loop beta function for the N=1 and N=2 supersymmetric nonlinear sigma model in two-dimensions, Phys. Lett. B, 173, 423-428 (1986)
[307] Grisaru, M. T.; Zanon, D., \( \sigma\) Model superstring corrections to the Einstein-hilbert action, Phys. Lett. B, 177, 347-351 (1986)
[308] Freeman, M. D.; Pope, C. N.; Sohnius, M. F.; Stelle, K. S., Higher order \(\sigma\) model counterterms and the effective action for superstrings, Phys. Lett. B, 178, 199-204 (1986)
[309] Green, M. B.; Gutperle, M.; Kwon, H.-h., Sixteen fermion and related terms in M theory on \(T^2\), Phys. Lett. B, 421, 149-161 (1998), arXiv:hep-th/9710151
[310] Drummond, J. M.; Ragoucy, E., Superstring amplitudes and the associator, J. High Energy Phys., 08, 135 (2013), arXiv:1301.0794 · Zbl 1342.81420
[311] Bergshoeff, E.; Rakowski, M.; Sezgin, E., Higher derivative super Yang-Mills theories, Phys. Lett. B, 185, 371-376 (1987)
[312] Gates, S. J.; Vashakidze, S., On \(D = 10, N = 1\) supersymmetry, superspace geometry and superstring effects, Nuclear Phys. B, 291, 172 (1987)
[313] Cederwall, M.; Nilsson, B. E.W.; Tsimpis, D., The Structure of maximally supersymmetric Yang-Mills theory: Constraining higher order corrections, J. High Energy Phys., 06, 034 (2001), arXiv:hep-th/0102009
[314] Cederwall, M.; Nilsson, B. E.W.; Tsimpis, D., D=10 superYang-Mills at \(O ( \alpha{}^\prime^2 )\), J. High Energy Phys., 07, 042 (2001), arXiv:hep-th/0104236
[315] Green, M. B.; Sethi, S., Supersymmetry constraints on type IIB supergravity, Phys. Rev. D, 59, Article 046006 pp. (1999), arXiv:hep-th/9808061
[316] Koerber, P.; Sevrin, A., The NonAbelian Born-Infeld action through order \(\alpha{}^\prime^3\), J. High Energy Phys., 10, 003 (2001), arXiv:hep-th/0108169
[317] Collinucci, A.; De Roo, M.; Eenink, M. G.C., Supersymmetric Yang-Mills theory at order \(\alpha{}^\prime^3\), J. High Energy Phys., 06, 024 (2002), arXiv:hep-th/0205150
[318] Berkovits, N.; Pershin, V., Supersymmetric Born-Infeld from the pure spinor formalism of the open superstring, J. High Energy Phys., 01, 023 (2003), arXiv:hep-th/0205154 · Zbl 1226.81160
[319] Drummond, J. M.; Heslop, P. J.; Howe, P. S.; Kerstan, S. F., Integral invariants in N=4 SYM and the effective action for coincident D-branes, J. High Energy Phys., 08, 016 (2003), arXiv:hep-th/0305202
[320] Cederwall, M.; Karlsson, A., Pure spinor superfields and Born-Infeld theory, J. High Energy Phys., 11, 134 (2011), arXiv:1109.0809 · Zbl 1306.81297
[321] Wang, Y.; Yin, X., Constraining higher derivative supergravity with scattering amplitudes, Phys. Rev. D, 92, 4, Article 041701 pp. (2015), arXiv:1502.03810
[322] Zhao, J., Multiple Zeta Functions, Multiple Polylogarithms, and their Special Values (2016), World Scientific: World Scientific New Jersey · Zbl 1367.11002
[323] Gil, J. I.B.; Fresan, J., Multiple Zeta Values: From Numbers to Motives (2023), in press
[324] Bluemlein, J.; Broadhurst, D. J.; Vermaseren, J. A.M., The multiple zeta value data mine, Comput. Phys. Comm., 181, 582-625 (2010), arXiv:0907.2557 · Zbl 1221.11183
[325] Zagier, D., Values of zeta functions and their applications, (First European Congress of Mathematics, II (1992))
[326] Brown, F., Mixed tate motives over \(\mathbb{Z} \), Ann. of Math., 175, 949-976 (2012), arXiv:1102.1312 · Zbl 1278.19008
[327] Brown, F., Notes on motivic periods, Commun. Number Theory Phys., 11, 3, 557-655 (2015), arXiv:1512.06410 · Zbl 1390.14024
[328] Panzer, E., Feynman Integrals and Hyperlogarithms (2015), Humboldt U., arXiv:1506.07243 · Zbl 1344.81024
[329] Drinfeld, V., Quasi Hopf algebras, Leningrad Math., J. 1, 1419-1457 (1989) · Zbl 0718.16033
[330] Drinfeld, V., On quasitriangular quasi-Hopf algebras and on a group that is closely connected with \(\text{Gal} ( \overline{\mathbb{Q}} / \mathbb{Q} )\), Leningrad Math., J. 2 (4), 829-860 (1991) · Zbl 0728.16021
[331] Le, T.; Murakami, J., Kontsevich’s integral for the Kauffman polynomial, Nagoya Math J., 142, 39-65 (1996) · Zbl 0866.57008
[332] Brown, F., Polylogarithmes multiples uniformes en une variable, C. R. Acad. Sci. Paris, Ser. I 338, 527-532 (2004) · Zbl 1048.11053
[333] Stieberger, S., Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator, J. Phys. A, 47, Article 155401 pp. (2014), arXiv:1310.3259 · Zbl 1288.81121
[334] Abreu, S.; Britto, R.; Duhr, C.; Gardi, E., Algebraic structure of cut feynman integrals and the diagrammatic coaction, Phys. Rev. Lett., 119, 5, Article 051601 pp. (2017), arXiv:1703.05064
[335] Abreu, S.; Britto, R.; Duhr, C.; Gardi, E., Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case, J. High Energy Phys., 12, 090 (2017), arXiv:1704.07931 · Zbl 1383.81321
[336] Abreu, S.; Britto, R.; Duhr, C.; Gardi, E.; Matthew, J., From positive geometries to a coaction on hypergeometric functions, J. High Energy Phys., 02, 122 (2020), arXiv:1910.08358 · Zbl 1435.81075
[337] Abreu, S.; Britto, R.; Duhr, C.; Gardi, E.; Matthew, J., The diagrammatic coaction beyond one loop, J. High Energy Phys., 10, 131 (2021), arXiv:2106.01280
[338] Brown, F.; Dupont, C., Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions, Nagoya Math. J., 1-73 (2022), arXiv:1907.06603
[339] J. Broedel, O. Schlotterer, S. Stieberger, \( \alpha^\prime \)-expansion of open superstring amplitudes, URL https://wwwth.mpp.mpg.de/members/stieberg/mzv/index.html. · Zbl 1338.81316
[340] Bjerrum-Bohr, N. E.J.; Damgaard, P. H.; Johansson, H.; Sondergaard, T., Monodromy-like relations for finite loop amplitudes, J. High Energy Phys., 05, 039 (2011), arXiv:1103.6190 · Zbl 1296.81124
[341] Solomon, L., On the Poincaré-Birkhoff-Witt theorem, J. Combin. Theory, 4, 4, 363-375 (1968) · Zbl 0174.06601
[342] Reutenauer, C., Theorem of Poincaré-Birkhoff-Witt, logarithm and symmetric group representations of degrees equal to Stirling numbers, (Combinatoire Énumérative (1986), Springer), 267-284 · Zbl 0621.20004
[343] Loday, J.-L., Série de Hausdorff, Idempotents Eulériens et Algebres de Hopf (1992), Université Louis Pasteur, Département de mathématique
[344] Graham, R. L.; Knuth, D. E.; Patashnik, O.; Liu, S., Concrete mathematics: a foundation for computer science, Comput. Phys., 3, 5, 106-107 (1989) · Zbl 0668.00003
[345] Knuth, D. E., Two notes on notation, Amer. Math. Monthly, 99, 5, 403-422 (1992) · Zbl 0785.05014
[346] Mizera, S.; Skrzypek, B., Perturbiner methods for effective field theories and the double copy, J. High Energy Phys., 10, 018 (2018), arXiv:1809.02096 · Zbl 1402.81193
[347] C. Mafra, O. Schlotterer, BGap: \( \alpha^\prime \)-expansion of disk integrals, URL https://repo.or.cz/BGap.git. · Zbl 1373.83110
[348] Kaderli, A., A note on the Drinfeld associator for genus-zero superstring amplitudes in twisted de Rham theory, J. Phys. A, 53, 41, Article 415401 pp. (2020), arXiv:1912.09406 · Zbl 1519.81422
[349] Fleischer, J.; Kotikov, A. V.; Veretin, O. L., Analytic two loop results for selfenergy type and vertex type diagrams with one nonzero mass, Nuclear Phys. B, 547, 343-374 (1999), arXiv:hep-ph/9808242
[350] Kotikov, A. V.; Lipatov, L. N.; Onishchenko, A. I.; Velizhanin, V. N., Three loop universal anomalous dimension of the Wilson operators in \(N = 4\) SUSY Yang-Mills model, Phys. Lett. B. Phys. Lett. B, Phys. Lett. B, 632, 754-756 (2006), Erratum · Zbl 1247.81485
[351] Henn, J. M., Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett., 110, Article 251601 pp. (2013), arXiv:1304.1806
[352] Adams, L.; Weinzierl, S., The \(ɛ\)-form of the differential equations for Feynman integrals in the elliptic case, Phys. Lett. B, 781, 270-278 (2018), arXiv:1802.05020 · Zbl 1398.81097
[353] Kotikov, A. V., Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B, 254, 158-164 (1991)
[354] Arkani-Hamed, N.; Bourjaily, J. L.; Cachazo, F.; Trnka, J., Local integrals for planar scattering amplitudes, J. High Energy Phys., 06, 125 (2012), arXiv:1012.6032 · Zbl 1397.81428
[355] Broedel, J.; Duhr, C.; Dulat, F.; Penante, B.; Tancredi, L., Elliptic Feynman integrals and pure functions, J. High Energy Phys., 01, 023 (2019), arXiv:1809.10698 · Zbl 1409.81162
[356] Caola, F.; Chen, W.; Duhr, C.; Liu, X.; Mistlberger, B.; Petriello, F.; Vita, G.; Weinzierl, S., The path forward to \(N{}^3\) LO, (2022 Snowmass Summer Study (2022)), arXiv:2203.06730
[357] Bourjaily, J. L., Functions beyond multiple polylogarithms for precision collider physics, (2022 Snowmass Summer Study (2022)), arXiv:2203.07088
[358] Weinzierl, S., Feynman integrals (2022), arXiv:2201.03593 · Zbl 1493.81001
[359] Abreu, S.; Britto, R.; Duhr, C., The SAGEX review on scattering amplitudes, chapter 3: Mathematical structures in feynman integrals (2022), arXiv:2203.13014 · Zbl 1520.81123
[360] Blümlein, J.; Schneider, C., The SAGEX review on scattering amplitudes, chapter 4: Multi-loop feynman integrals (2022), arXiv:2203.13015 · Zbl 1520.81133
[361] Selberg, A., Remarks on a multiple integral, Norsk Matematisk Tidsskrift, 26, 71 (1944) · Zbl 0063.06870
[362] Britto, R.; Mizera, S.; Rodriguez, C.; Schlotterer, O., Coaction and double-copy properties of configuration-space integrals at genus zero, J. High Energy Phys., 05, 053 (2021), arXiv:2102.06206 · Zbl 1466.83115
[363] Felder, G.; Varchenko, A., Integral representation of solutions of the elliptic Knizhnik-Zamolodchikov-Bernard equations, Int. Math. Res. Not., 221 (1995), arXiv:9502165 · Zbl 0840.17020
[364] Mano, T.; Watanabe, H., Twisted cohomology and homology groups associated to the Riemann-Wirtinger integral, Proc. Amer. Math. Soc., 140, 11, 3867 (2012) · Zbl 1290.14030
[365] Mafra, C. R.; Schlotterer, O., All order \(\alpha \)’ expansion of one-loop open-string integrals, Phys. Rev. Lett., 124, 10, Article 101603 pp. (2020), arXiv:1908.09848
[366] Mafra, C. R.; Schlotterer, O., One-loop open-string integrals from differential equations: all-order \(\alpha \)’-expansions at \(n\) points, J. High Energy Phys., 03, 007 (2020), arXiv:1908.10830 · Zbl 1435.83183
[367] Broedel, J.; Kaderli, A., Amplitude recursions with an extra marked point, Commun. Number Theory Phys., 16, 1, 75-158 (2022), arXiv:1912.09927 · Zbl 1519.81417
[368] Broedel, J.; Kaderli, A.; Schlotterer, O., Two dialects for KZB equations: generating one-loop open-string integrals, J. High Energy Phys., 12, 036 (2020), arXiv:2007.03712 · Zbl 1457.83069
[369] Kaderli, A.; Rodriguez, C., Open-string integrals with multiple unintegrated punctures at genus one, J. High Energy Phys., 10, 159 (2022), arXiv:2203.09649 · Zbl 1534.81053
[370] Beilinson, A.; Varchenko, A.; Goncharov, A.; Shekhtman, V., Projective Geometry and K-theory, Leningrad Math J. 2, 3, 523-575 (1991) · Zbl 0728.14008
[371] Schnetz, O., Graphical hyperlogarithms, (Talk Within the Trimester “Periods in Number Theory, Algebraic Geometry and Physics”, HIM, Bonn, Feb. 27 (2018))
[372] Schlotterer, O.; Schnetz, O., Closed strings as single-valued open strings: A genus-zero derivation, J. Phys. A, 52, 4, Article 045401 pp. (2019), arXiv:1808.00713 · Zbl 1422.81155
[373] Vanhove, P.; Zerbini, F., Single-valued hyperlogarithms, correlation functions and closed string amplitudes, Adv. Theor. Math. Phys., 26, 2 (2022), arXiv:1812.03018 · Zbl 1517.81085
[374] Hohm, O.; Siegel, W.; Zwiebach, B., Doubled \(\alpha^\prime \)-geometry, J. High Energy Phys., 02, 065 (2014), arXiv:1306.2970 · Zbl 1333.83190
[375] Siegel, W., Amplitudes for left-handed strings (2015), arXiv:1512.02569
[376] Huang, Y.-t.; Siegel, W.; Yuan, E. Y., Factorization of chiral string amplitudes, J. High Energy Phys., 09, 101 (2016), arXiv:1603.02588 · Zbl 1390.81442
[377] Lipinski Jusinskas, R., Chiral strings, the sectorized description and their integrated vertex operators, J. High Energy Phys., 12, 143 (2019), arXiv:1909.04069 · Zbl 1431.83171
[378] Azevedo, T.; Jusinskas, R. L.; Lize, M., Bosonic sectorized strings and the \(( D F )^2\) theory, J. High Energy Phys., 01, 082 (2020), arXiv:1908.11371 · Zbl 1434.83141
[379] Guillen, M.; Johansson, H.; Jusinskas, R. L.; Schlotterer, O., Scattering massive string resonances through field-theory methods, Phys. Rev. Lett., 127, 5, Article 051601 pp. (2021), arXiv:2104.03314
[380] Berkovits, N.; Mafra, C. R., Pure spinor formulation of the superstring and its applications (2022), arXiv:2210.10510
[381] D’Hoker, E.; Phong, D. H., Conformal scalar fields and chiral splitting on superriemann surfaces, Comm. Math. Phys., 125, 469 (1989) · Zbl 0698.58008
[382] Berkovits, N., New higher-derivative \(R^4\) theorems, Phys. Rev. Lett., 98, Article 211601 pp. (2007), arXiv:hep-th/0609006 · Zbl 1228.83042
[383] Berkovits, N.; Green, M. B.; Russo, J. G.; Vanhove, P., Non-renormalization conditions for four-gluon scattering in supersymmetric string and field theory, J. High Energy Phys., 11, 063 (2009), arXiv:0908.1923
[384] Berkovits, N.; Mafra, C. R., Equivalence of two-loop superstring amplitudes in the pure spinor and RNS formalisms, Phys. Rev. Lett., 96, Article 011602 pp. (2006), arXiv:hep-th/0509234
[385] D’Hoker, E.; Phong, D. H., Two-loop superstrings VI: Non-renormalization theorems and the 4-point function, Nuclear Phys. B, 715, 3-90 (2005), arXiv:hep-th/0501197 · Zbl 1207.81111
[386] Mafra, C. R.; Stahn, C., The one-loop open superstring massless five-point amplitude with the non-minimal pure spinor formalism, J. High Energy Phys., 03, 126 (2009), arXiv:0902.1539
[387] Gomez, H.; Mafra, C. R.; Schlotterer, O., Two-loop superstring five-point amplitude and \(S\)-duality, Phys. Rev. D, 93, 4, Article 045030 pp. (2016), arXiv:1504.02759
[388] Mafra, C. R.; Schlotterer, O., One-loop superstring six-point amplitudes and anomalies in pure spinor superspace, J. High Energy Phys., 04, 148 (2016), arXiv:1603.04790 · Zbl 1388.83681
[389] Green, M. B.; Schwarz, J. H., The hexagon gauge anomaly in type I superstring theory, Nuclear Phys. B, 255, 93-114 (1985)
[390] Green, M. B.; Schwarz, J. H., Anomaly cancellation in supersymmetric D=10 gauge theory and superstring theory, Phys. Lett. B, 149, 117-122 (1984)
[391] Berkovits, N.; Mafra, C. R., Some superstring amplitude computations with the non-minimal pure spinor formalism, J. High Energy Phys., 11, 079 (2006), arXiv:hep-th/0607187
[392] Mafra, C. R.; Schlotterer, O., Towards the n-point one-loop superstring amplitude. Part I. Pure spinors and superfield kinematics, J. High Energy Phys., 08, 090 (2019), arXiv:1812.10969 · Zbl 1421.83118
[393] Mafra, C. R.; Schlotterer, O., Towards the n-point one-loop superstring amplitude. Part II. Worldsheet functions and their duality to kinematics, J. High Energy Phys., 08, 091 (2019), arXiv:1812.10970 · Zbl 1421.83119
[394] Mafra, C. R.; Schlotterer, O., Double-copy structure of one-loop open-string amplitudes, Phys. Rev. Lett., 121, 1, Article 011601 pp. (2018), arXiv:1711.09104
[395] D’Hoker, E.; Schlotterer, O., Two-loop superstring five-point amplitudes. Part III. Construction via the RNS formulation: even spin structures, J. High Energy Phys., 12, 063 (2021), arXiv:2108.01104 · Zbl 1521.81218
[396] Geyer, Y.; Monteiro, R.; Stark-Muchão, R., Superstring loop amplitudes from the field theory limit, Phys. Rev. Lett., 127, 21, Article 211603 pp. (2021), arXiv:2106.03968
[397] Carrasco, J. J.M.; Johansson, H., Five-point amplitudes in N=4 super-yang-mills theory and N=8 supergravity, Phys. Rev. D, 85, Article 025006 pp. (2012), arXiv:1106.4711
[398] Tourkine, P., Tropical amplitudes, Ann. Henri Poincaré, 18, 6, 2199-2249 (2017), arXiv:1309.3551 · Zbl 1387.81292
[399] He, S.; Monteiro, R.; Schlotterer, O., String-inspired BCJ numerators for one-loop MHV amplitudes, J. High Energy Phys., 01, 171 (2016), arXiv:1507.06288 · Zbl 1388.81544
[400] He, S.; Schlotterer, O.; Zhang, Y., New BCJ representations for one-loop amplitudes in gauge theories and gravity, Nuclear Phys. B, 930, 328-383 (2018), arXiv:1706.00640 · Zbl 1404.81176
[401] Geyer, Y.; Mason, L.; Monteiro, R.; Tourkine, P., Loop integrands for scattering amplitudes from the Riemann sphere, Phys. Rev. Lett., 115, 12, Article 121603 pp. (2015), arXiv:1507.00321
[402] He, S.; Schlotterer, O., New relations for gauge-theory and gravity amplitudes at loop level, Phys. Rev. Lett., 118, 16, Article 161601 pp. (2017), arXiv:1612.00417
[403] Bridges, E.; Mafra, C. R., Local BCJ numerators for ten-dimensional SYM at one loop, J. High Energy Phys., 07, 031 (2021), arXiv:2102.12943 · Zbl 1468.81104
[404] Ben-Shahar, M.; Guillen, M., 10D super-Yang-Mills scattering amplitudes from its pure spinor action, J. High Energy Phys., 12, 014 (2021), arXiv:2108.11708 · Zbl 1521.81373
[405] Aisaka, Y.; Berkovits, N., Pure spinor vertex operators in Siegel gauge and loop amplitude regularization, J. High Energy Phys., 07, 062 (2009), arXiv:0903.3443
[406] Grassi, P. A.; Vanhove, P., Higher-loop amplitudes in the non-minimal pure spinor formalism, J. High Energy Phys., 05, 089 (2009), arXiv:0903.3903
[407] Broedel, J.; Mafra, C. R.; Matthes, N.; Schlotterer, O., Elliptic multiple zeta values and one-loop superstring amplitudes, J. High Energy Phys., 07, 112 (2015), arXiv:1412.5535 · Zbl 1388.83190
[408] Tourkine, P.; Vanhove, P., Higher-loop amplitude monodromy relations in string and gauge theory, Phys. Rev. Lett., 117, 21, Article 211601 pp. (2016), arXiv:1608.01665
[409] Hohenegger, S.; Stieberger, S., Monodromy relations in higher-loop string amplitudes, Nuclear Phys. B, 925, 63-134 (2017), arXiv:1702.04963 · Zbl 1375.81202
[410] Tourkine, P., Integrands and loop momentum in string and field theory, Phys. Rev. D, 102, 2, Article 026006 pp. (2020), arXiv:1901.02432
[411] Casali, E.; Mizera, S.; Tourkine, P., Monodromy relations from twisted homology, J. High Energy Phys., 12, 087 (2019), arXiv:1910.08514 · Zbl 1431.83163
[412] Casali, E.; Mizera, S.; Tourkine, P., Loop amplitudes monodromy relations and color-kinematics duality, J. High Energy Phys., 03, 048 (2021), arXiv:2005.05329 · Zbl 1461.81081
[413] Stieberger, S., Open & closed vs. Pure open string one-loop amplitudes (2021), arXiv:2105.06888
[414] D’Hoker, E.; Phong, D. H., Momentum analyticity and finiteness of the one loop superstring amplitude, Phys. Rev. Lett., 70, 3692-3695 (1993), arXiv:hep-th/9302003 · Zbl 1050.81637
[415] D’Hoker, E.; Phong, D. H., Dispersion relations in string theory, Theoret. Math. Phys., 98, 306-316 (1994), arXiv:hep-th/9404128 · Zbl 0831.53054
[416] D’Hoker, E.; Phong, D. H., The Box graph in superstring theory, Nuclear Phys. B, 440, 24-94 (1995), arXiv:hep-th/9410152 · Zbl 0990.81655
[417] Green, M. B.; Russo, J. G.; Vanhove, P., Low energy expansion of the four-particle genus-one amplitude in type II superstring theory, J. High Energy Phys., 02, 020 (2008), arXiv:0801.0322
[418] D’Hoker, E.; Green, M. B.; Vanhove, P., On the modular structure of the genus-one Type II superstring low energy expansion, J. High Energy Phys., 08, 041 (2015), arXiv:1502.06698 · Zbl 1388.81515
[419] D’Hoker, E.; Green, M. B., Exploring transcendentality in superstring amplitudes, J. High Energy Phys., 07, 149 (2019), arXiv:1906.01652 · Zbl 1418.81090
[420] Edison, A.; Guillen, M.; Johansson, H.; Schlotterer, O.; Teng, F., One-loop matrix elements of effective superstring interactions: \( \alpha \)’-expanding loop integrands, J. High Energy Phys., 12, 007 (2021), arXiv:2107.08009 · Zbl 1521.81219
[421] Eberhardt, L.; Mizera, S., Unitarity cuts of the worldsheet (2022), arXiv:2208.12233
[422] Hull, C. M.; Townsend, P. K., Unity of superstring dualities, Nuclear Phys. B, 438, 109-137 (1995), arXiv:hep-th/9410167 · Zbl 1052.83532
[423] Green, M. B.; Gutperle, M., Effects of D instantons, Nuclear Phys. B, 498, 195-227 (1997), arXiv:hep-th/9701093 · Zbl 0979.81566
[424] Green, M. B.; Kwon, H.-h.; Vanhove, P., Two loops in eleven-dimensions, Phys. Rev. D, 61, Article 104010 pp. (2000), arXiv:hep-th/9910055
[425] D’Hoker, E.; Gutperle, M.; Phong, D. H., Two-loop superstrings and S-duality, Nuclear Phys. B, 722, 81-118 (2005), arXiv:hep-th/0503180 · Zbl 1128.81317
[426] Green, M. B.; Vanhove, P., Duality and higher derivative terms in M theory, J. High Energy Phys., 01, 093 (2006), arXiv:hep-th/0510027
[427] D’Hoker, E.; Green, M. B.; Pioline, B.; Russo, R., Matching the \(D^6 R^4\) interaction at two-loops, J. High Energy Phys., 01, 031 (2015), arXiv:1405.6226
[428] D’Hoker, E.; Mafra, C. R.; Pioline, B.; Schlotterer, O., Two-loop superstring five-point amplitudes. Part II. Low energy expansion and S-duality, J. High Energy Phys., 02, 139 (2021), arXiv:2008.08687 · Zbl 1460.83112
[429] Boels, R. H., Maximal R-symmetry violating amplitudes in type IIB superstring theory, Phys. Rev. Lett., 109, Article 081602 pp. (2012), arXiv:1204.4208
[430] Green, M. B.; Wen, C., Modular Forms and \(S L ( 2 , \mathbb{Z} )\)-covariance of type IIB superstring theory, J. High Energy Phys., 06, 087 (2019), arXiv:1904.13394 · Zbl 1416.83124
[431] Green, M. B., Interconnections between type II superstrings, M theory and N=4 supersymmetric Yang-Mills, Lecture Notes in Phys., 525, 22 (1999), arXiv:hep-th/9903124 · Zbl 0991.81103
[432] D’Hoker, E.; Green, M. B.; Gürdogan, Ö.; Vanhove, P., Modular graph functions, Commun. Number Theory Phys., 11, 165-218 (2017), arXiv:1512.06779 · Zbl 1426.11082
[433] D’Hoker, E.; Green, M. B., Identities between modular graph forms, J. Number Theory, 189, 25-80 (2018), arXiv:1603.00839 · Zbl 1450.11062
[434] Green, M. B.; Vanhove, P., The Low-energy expansion of the one loop type II superstring amplitude, Phys. Rev., D61, Article 104011 pp. (2000), arXiv:hep-th/9910056
[435] Gerken, J. E., Modular graph forms and scattering amplitudes in string theory (2020), arXiv:2011.08647
[436] D’Hoker, E.; Kaidi, J., Lectures on modular forms and strings (2022), arXiv:2208.07242
[437] Dorigoni, D.; Doroudiani, M.; Drewitt, J.; Hidding, M.; Kleinschmidt, A.; Matthes, N.; Schlotterer, O.; Verbeek, B., Modular graph forms from equivariant iterated Eisenstein integrals (2022), arXiv:2209.06772 · Zbl 1536.81067
[438] Brown, F., A class of non-holomorphic modular forms I, Res. Math. Sci., 5, 5:7 (2018), arXiv:1707.01230 · Zbl 1441.11103
[439] Brown, F., A class of non-holomorphic modular forms II: equivariant iterated Eisenstein integrals, Forum Math., Sigma, 8, 1 (2020), arXiv:1708.03354
[440] D’Hoker, E.; Green, M. B., Zhang-Kawazumi invariants and superstring amplitudes, J. Number Theory, 144, 111-150 (2014), arXiv:1308.4597 · Zbl 1298.81260
[441] Pioline, B., A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces, J. Number Theory, 163, 520-541 (2016), arXiv:1504.04182 · Zbl 1408.14096
[442] D’Hoker, E.; Green, M. B.; Pioline, B., Higher genus modular graph functions, string invariants, and their exact asymptotics, Comm. Math. Phys., 366, 3, 927-979 (2019), arXiv:1712.06135 · Zbl 1430.14072
[443] D’Hoker, E.; Green, M. B.; Pioline, B., Asymptotics of the \(D^8 R^4\) genus-two string invariant, Commun. Number Theory Phys., 13, 351-462 (2019), arXiv:1806.02691 · Zbl 1416.83116
[444] Basu, A., Eigenvalue equation for genus two modular graphs, J. High Energy Phys., 02, 046 (2019), arXiv:1812.00389 · Zbl 1411.83110
[445] D’Hoker, E.; Schlotterer, O., Identities among higher genus modular graph tensors, Commun. Number Theory Phys., 16, 1, 35-74 (2022), arXiv:2010.00924 · Zbl 1486.81152
[446] Broedel, J.; Matthes, N.; Richter, G.; Schlotterer, O., Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes, J. Phys. A, 51, 28, Article 285401 pp. (2018), arXiv:1704.03449 · Zbl 1401.81071
[447] Enriquez, B., Analogues elliptiques des nombres multizétas, Bull. Soc. Math. France, 144, 3, 395-427 (2016), arXiv:1301.3042 · Zbl 1407.11101
[448] Brown, F.; Levin, A., Multiple elliptic polylogarithms (2011), arXiv:1110.6917
[449] Zerbini, F., Single-valued multiple zeta values in genus 1 superstring amplitudes, Commun. Number Theory Phys., 10, 703-737 (2016), arXiv:1512.05689 · Zbl 1365.81095
[450] Broedel, J.; Schlotterer, O.; Zerbini, F., From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop, J. High Energy Phys., 01, 155 (2019), arXiv:1803.00527 · Zbl 1409.83178
[451] Zagier, D.; Zerbini, F., Genus-zero and genus-one string amplitudes and special multiple zeta values, Commun. Number Theory Phys., 14, 2, 413-452 (2020), arXiv:1906.12339 · Zbl 1447.11097
[452] Gerken, J. E.; Kleinschmidt, A.; Schlotterer, O., All-order differential equations for one-loop closed-string integrals and modular graph forms, J. High Energy Phys., 01, 064 (2020), arXiv:1911.03476 · Zbl 1434.83147
[453] Gerken, J. E.; Kleinschmidt, A.; Schlotterer, O., Generating series of all modular graph forms from iterated Eisenstein integrals, J. High Energy Phys., 07, 07, 190 (2020), arXiv:2004.05156 · Zbl 1451.83095
[454] Gerken, J. E.; Kleinschmidt, A.; Mafra, C. R.; Schlotterer, O.; Verbeek, B., Towards closed strings as single-valued open strings at genus one, J. Phys. A, 55, 2, Article 025401 pp. (2022), arXiv:2010.10558 · Zbl 1499.81080
[455] Gran, U., GAMMA: A Mathematica package for performing gamma matrix algebra and Fierz transformations in arbitrary dimensions (2001), arXiv:hep-th/0105086
[456] Guttenberg, S., Superstrings in General Backgrounds (2007), Vienna, Tech. U., arXiv:0807.4968
[457] Freedman, D. Z.; Van Proeyen, A., Supergravity (2012), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, UK · Zbl 1245.83001
[458] Mukhopadhyay, P., On D-brane boundary state analysis in pure-spinor formalism, J. High Energy Phys., 03, 066 (2006), arXiv:hep-th/0505157 · Zbl 1226.81217
[459] van Holten, J. W.; van Proeyen, A., N=1 supersymmetry algebras in d=2,3,4 mod 8, J. Phys. A: Math. Gen., 15, 12, 3763-3783 (1982)
[460] Kuusela, P., “GammaMaP” - a mathematica package for clifford algebras, Gamma matrices and spinors (2019), arXiv:1905.00429
[461] Mueck, W., General (anti-)commutators of gamma matrices (2007), arXiv:0711.1436
[462] Alexandrov, V.; Krotov, D.; Losev, A.; Lysov, V., On pure spinor superfield formalism, J. High Energy Phys., 10, 074 (2007), arXiv:0705.2191
[463] Georgi, H., (Lie Algebras in Particle Physics: From Isospin To Unified Theories. Lie Algebras in Particle Physics: From Isospin To Unified Theories, Frontiers in Physics (1999), Avalon Publishing), URL https://books.google.co.uk/books?id=g4yEuH5rBMUC · Zbl 0505.00036
[464] Grassi, P. A.; Policastro, G.; van Nieuwenhuizen, P., An Introduction to the covariant quantization of superstrings, Classical Quantum Gravity, 20, S395-S410 (2003), arXiv:hep-th/0302147 · Zbl 1056.83026
[465] Schiappa, R.; Wyllard, N., D-brane boundary states in the pure spinor superstring, J. High Energy Phys., 07, 070 (2005), arXiv:hep-th/0503123
[466] Aisaka, Y.; Kazama, Y., A New first class algebra, homological perturbation and extension of pure spinor formalism for superstring, J. High Energy Phys., 02, 017 (2003), arXiv:hep-th/0212316
[467] Brauer, R.; Weyl, H., Spinors in n dimensions, Amer. J. Math., 57, 2, 425-449 (1935) · JFM 61.1025.06
[468] Kaplan, D. B.; Unsal, M., A Euclidean lattice construction of supersymmetric Yang-Mills theories with sixteen supercharges, J. High Energy Phys., 09, 042 (2005), arXiv:hep-lat/0503039
[469] Pais, A., On spinors in n dimensions, J. Math. Phys., 3, 6, 1135-1139 (1962) · Zbl 0109.21202
[470] Lothaire, M., Combinatorics on Words (1997), Cambridge University Press · Zbl 0874.20040
[471] Melançon, G.; Reutenauer, C., Lyndon words, free algebras and shuffles, Canad. J. Math., 41, 4, 577-591 (1989) · Zbl 0694.17003
[472] Griffing, G., Dual Lie elements and a derivation for the cofree coassociative coalgebra, Proc. Amer. Math. Soc., 123, 11, 3269-3277 (1995) · Zbl 0847.17027
[473] Humphreys, J. E., Introduction to Lie Algebras and Representation Theory (2012), Springer Science & Business Media · Zbl 0254.17004
[474] Fulton, W.; Harris, J., Representation Theory: A First Course (2013), Springer Science & Business Media
[475] Feger, R.; Kephart, T. W.; Saskowski, R. J., LieART 2.0 - a mathematica application for Lie algebras and representation theory, Comput. Phys. Comm., 257, Article 107490 pp. (2020), arXiv:1912.10969 · Zbl 1515.17005
[476] Stahn, C., Fermionic superstring loop amplitudes in the pure spinor formalism, J. High Energy Phys., 05, 034 (2007), arXiv:0704.0015
[477] Trivedi, G., Correlation functions in Berkovits’ pure spinor formulation, Modern Phys. Lett. A, 17, 2239-2248 (2002), arXiv:hep-th/0205217 · Zbl 1083.81580
[478] de Azcarraga, J. A.; Izquierdo, J. M.; Perez Bueno, J. C., On the generalizations of Poisson structures, J. Phys. A, 30, L607-L616 (1997), arXiv:hep-th/9703019 · Zbl 0932.37056
[479] de Azcarraga, J. A.; Izquierdo, J. M., n-ary algebras: A Review with applications, J. Phys. A, 43, Article 293001 pp. (2010), arXiv:1005.1028 · Zbl 1202.81187
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.