Abstract
We investigate iterated integrals on an elliptic curve, which are a natural genus-one generalization of multiple polylogarithms. These iterated integrals coincide with the multiple elliptic polylogarithms introduced by Brown and Levin when constrained to the real line. At unit argument they reduce to an elliptic analogue of multiple zeta values, whose network of relations we start to explore. A simple and natural application of this framework are one-loop scattering amplitudes in open superstring theory. In particular, elliptic multiple zeta values are a suitable language to express their low energy limit. Similar to the techniques available at tree-level, our formalism allows to completely automatize the calculation.
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A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical polylogarithms for amplitudes and Wilson loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].
L.J. Dixon, C. Duhr and J. Pennington, Single-valued harmonic polylogarithms and the multi-Regge limit, JHEP 10 (2012) 074 [arXiv:1207.0186] [INSPIRE].
J. Broedel, O. Schlotterer and S. Stieberger, Polylogarithms, multiple zeta values and superstring amplitudes, Fortsch. Phys. 61 (2013) 812 [arXiv:1304.7267] [INSPIRE].
V. Del Duca, L.J. Dixon, C. Duhr and J. Pennington, The BFKL equation, Mueller-Navelet jets and single-valued harmonic polylogarithms, JHEP 02 (2014) 086 [arXiv:1309.6647] [INSPIRE].
A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math/0103059 [INSPIRE].
A.B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math. J. 128 (2005) 209 [math/0208144] [INSPIRE].
F. Brown, On the decomposition of motivic multiple zeta values, in Galois-Teichmüller theory and arithmetic geometry, Math. Soc. Japan, Tokyo Japan (2012), pg. 31 [arXiv:1102.1310] [INSPIRE].
C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP 08 (2012) 043 [arXiv:1203.0454] [INSPIRE].
L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise graph with arbitrary masses, J. Math. Phys. 54 (2013) 052303 [arXiv:1302.7004] [INSPIRE].
S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, J. Number Theor. 148 (2015) 328 [arXiv:1309.5865] [INSPIRE].
L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms, J. Math. Phys. 55 (2014) 102301 [arXiv:1405.5640] [INSPIRE].
F. Brown and O. Schnetz, A K3 in ϕ4, Duke Math. J. 161 (2012) 1817 [arXiv:1006.4064] [INSPIRE].
F. Brown and D. Doryn, Framings for graph hypersurfaces, arXiv:1301.3056 [INSPIRE].
S. Caron-Huot and K.J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [INSPIRE].
S. Bloch, M. Kerr and P. Vanhove, A Feynman integral via higher normal functions, arXiv:1406.2664 [INSPIRE].
F. Brown and A. Levin, Multiple elliptic polylogarithms, arXiv:1110.6917.
B. Enriquez, Analogues elliptiques des nombres multizétas (in French), arXiv:1301.3042.
A. Beilinson and A. Levin, The elliptic polylogarithm, in Proc. of Symp. in Pure Math. 55, Part II, J.-P.S.U. Jannsen and S.L. Kleiman eds., AMS, U.S.A. (1994), pg. 123.
S.J. Bloch, Higher regulators, algebraic K-theory, and zeta functions of elliptic curves, American Mathematical Society, Providence RI U.S.A. (2000), pg. 1.
A. Levin, Elliptic polylogarithms: an analytic theory, Compos. Math. 106 (1997) 267.
J. Wildeshaus, Realizations of polylogarithms, Springer, Germany (1997).
D. Zagier, The Bloch-Wigner-Ramakrishnan polylogarithm function, Math. Ann. 286 (1990) 613.
A. Weil, Elliptic functions according to Eisenstein and Kronecker, published in Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, Heidelberg Germany (1976).
B. Enriquez, Elliptic associators, Select. Math. (N.S.) 20 (2014) 491.
V. Drinfeld, Quasi Hopf algebras, Leningrad Math. J. 1 (1989) 1419.
V. Drinfeld, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with \( Gal\left(\overline{\mathbb{Q}}/\mathbb{Q}\right) \), Leningrad Math. J. 2 (1991) 829.
T. Le and J. Murakami, Kontsevich’s integral for the Kauffman polynomial, Nagoya Math. J. 142 (1996) 39.
S. Stieberger, Constraints on tree-level higher order gravitational couplings in superstring theory, Phys. Rev. Lett. 106 (2011) 111601 [arXiv:0910.0180] [INSPIRE].
O. Schlotterer and S. Stieberger, Motivic multiple zeta values and superstring amplitudes, J. Phys. A 46 (2013) 475401 [arXiv:1205.1516] [INSPIRE].
O. Schnetz, Graphical functions and single-valued multiple polylogarithms, Commun. Num. Theor. Phys. 08 (2014) 589 [arXiv:1302.6445] [INSPIRE].
F. Brown, Single-valued motivic periods and multiple zeta values, SIGMA 2 (2014) e25 [arXiv:1309.5309] [INSPIRE].
S. Stieberger, Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator, J. Phys. A 47 (2014) 155401 [arXiv:1310.3259] [INSPIRE].
S. Stieberger and T.R. Taylor, Closed string amplitudes as single-valued open string amplitudes, Nucl. Phys. B 881 (2014) 269 [arXiv:1401.1218] [INSPIRE].
D. Oprisa and S. Stieberger, Six gluon open superstring disk amplitude, multiple hypergeometric series and Euler-Zagier sums, hep-th/0509042 [INSPIRE].
S. Stieberger and T.R. Taylor, Multi-gluon scattering in open superstring theory, Phys. Rev. D 74 (2006) 126007 [hep-th/0609175] [INSPIRE].
T. Terasoma, Selberg integrals and multiple zeta values, Compos. Math. 133 (2002) 1 [math/9908045].
J.M. Drummond and É. Ragoucy, Superstring amplitudes and the associator, JHEP 08 (2013) 135 [arXiv:1301.0794] [INSPIRE].
J. Broedel, O. Schlotterer, S. Stieberger and T. Terasoma, All order α′-expansion of superstring trees from the Drinfeld associator, Phys. Rev. D 89 (2014) 066014 [arXiv:1304.7304] [INSPIRE].
α′-expansion of open superstring amplitudes website, http://mzv.mpp.mpg.de.
M.B. Green, J. Schwarz and E. Witten, Superstring theory. Vol. 2: loop amplitudes, anomalies and phenomenology, Cambridge Monographs on Mathematical Physics, Cambridge Univ. Pr., Cambridge U.K. (1987).
C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].
J. Ablinger, J. Blümlein and C. Schneider, Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms, J. Math. Phys. 54 (2013) 082301 [arXiv:1302.0378] [INSPIRE].
J. Ablinger and J. Blümlein, Harmonic sums, polylogarithms, special numbers and their generalizations, arXiv:1304.7071 [INSPIRE].
J.M. Borwein, D.M. Bradley, D.J. Broadhurst and P. Lisonek, Special values of multiple polylogarithms, Trans. Amer. Math. Soc. 353 (2001) 907 [math/9910045] [INSPIRE].
F. Brown, Mixed Tate motives over \( \mathbb{Z} \), Ann. Math. 175 (2012) 949.
F.C.S. Brown, Multiple zeta values and periods of moduli spaces \( {\mathfrak{M}}_{0,n} \), Annales Sci. Ecole Norm. Sup. 42 (2009) 371 [math/0606419] [INSPIRE].
C. Bogner and F. Brown, Feynman integrals and iterated integrals on moduli spaces of curves of genus zero, Commun. Num. Theor. Phys. 09 (2015) 189 [arXiv:1408.1862] [INSPIRE].
J. Blumlein, D.J. Broadhurst and J.A.M. Vermaseren, The multiple zeta value data mine, Comput. Phys. Commun. 181 (2010) 582 [arXiv:0907.2557] [INSPIRE].
J. Broedel, N. Matthes and O. Schlotterer, Relations between elliptic multiple zeta values and a special derivation algebra, arXiv:1507.02254 [INSPIRE].
A. Levin and G. Racinet, Towards multiple elliptic polylogarithms, math/0703237.
L. Kronecker, Zur Theorie der elliptischen Funktionen (in German), Mathematische Werke IV (1881) 313.
D. Zagier, Periods of modular forms and Jacobi theta functions, Invent. Math. 104 (1991) 449.
D. Mumford, M. Nori and P. Norman, Tata lectures on theta I, Birkhäuser, U.S.A. (1983).
D. Mumford, M. Nori and P. Norman, Tata lectures on theta II, Birkhäuser, U.S.A. (1984).
R. Hain, Notes on the universal elliptic KZB equation, arXiv:1309.0580.
M.B. Green and J.H. Schwarz, Infinity cancellations in SO(32) superstring theory, Phys. Lett. B 151 (1985) 21 [INSPIRE].
M.B. Green and J.H. Schwarz, Anomaly cancellation in supersymmetric D = 10 gauge theory and superstring theory, Phys. Lett. B 149 (1984) 117 [INSPIRE].
M.B. Green and J.H. Schwarz, The hexagon gauge anomaly in type I superstring theory, Nucl. Phys. B 255 (1985) 93 [INSPIRE].
M.B. Green and J.H. Schwarz, Supersymmetrical dual string theory. 3. Loops and renormalization, Nucl. Phys. B 198 (1982) 441 [INSPIRE].
J.H. Schwarz, Superstring theory, Phys. Rept. 89 (1982) 223 [INSPIRE].
M.B. Green, J. Schwarz and E. Witten, Superstring theory. Vol. 1: introduction, Cambridge Monographs on Mathematical Physics, Cambridge Univ. Pr., Cambridge U.K. (1987).
M.B. Green, J.H. Schwarz and L. Brink, N = 4 Yang-Mills and N = 8 supergravity as limits of string theories, Nucl. Phys. B 198 (1982) 474 [INSPIRE].
M.B. Green and P. Vanhove, The low-energy expansion of the one loop type-II superstring amplitude, Phys. Rev. D 61 (2000) 104011 [hep-th/9910056] [INSPIRE].
M.B. Green, J.G. Russo and P. Vanhove, Low energy expansion of the four-particle genus-one amplitude in type-II superstring theory, JHEP 02 (2008) 020 [arXiv:0801.0322] [INSPIRE].
D.M. Richards, The one-loop five-graviton amplitude and the effective action, JHEP 10 (2008) 042 [arXiv:0807.2421] [INSPIRE].
M.B. Green, C.R. Mafra and O. Schlotterer, Multiparticle one-loop amplitudes and S-duality in closed superstring theory, JHEP 10 (2013) 188 [arXiv:1307.3534] [INSPIRE].
N. Matthes, Elliptic double zeta values, in preparation.
N. Matthes, work in progress.
C.R. Mafra and O. Schlotterer, The structure of n-point one-loop open superstring amplitudes, JHEP 08 (2014) 099 [arXiv:1203.6215] [INSPIRE].
Z. Bern, J.J.M. Carrasco and H. Johansson, New relations for gauge-theory amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].
N. Berkovits, Super Poincaré covariant quantization of the superstring, JHEP 04 (2000) 018 [hep-th/0001035] [INSPIRE].
P. Ramond, Dual theory for free fermions, Phys. Rev. D 3 (1971) 2415 [INSPIRE].
A. Neveu and J.H. Schwarz, Factorizable dual model of pions, Nucl. Phys. B 31 (1971) 86 [INSPIRE].
A. Neveu and J.H. Schwarz, Quark model of dual pions, Phys. Rev. D 4 (1971) 1109 [INSPIRE].
A. Tsuchiya, More on one loop massless amplitudes of superstring theories, Phys. Rev. D 39 (1989) 1626 [INSPIRE].
A.G. Tsuchiya, On the pole structures of the disconnected part of hyper elliptic g-loop M -point super string amplitudes, arXiv:1209.6117 [INSPIRE].
L. Dolan and P. Goddard, Current algebra on the torus, Commun. Math. Phys. 285 (2009) 219 [arXiv:0710.3743] [INSPIRE].
M.A. Namazie, K.S. Narain and M.H. Sarmadi, On loop amplitudes in the fermionic string, RAL-86-051, (1986) [INSPIRE].
J. Igusa, Theta functions, Springer, Germany (1972).
J. Fay, Theta functions on Riemann surfaces, Springer, Germany (1973).
S. Stieberger and T.R. Taylor, Non-Abelian Born-Infeld action and type-I heterotic duality (I): heterotic F 6 terms at two loops, Nucl. Phys. B 647 (2002) 49 [hep-th/0207026] [INSPIRE].
S. Stieberger and T.R. Taylor, Non-Abelian Born-Infeld action and type-I heterotic duality (II): nonrenormalization theorems, Nucl. Phys. B 648 (2003) 3 [hep-th/0209064] [INSPIRE].
L. Clavelli, P.H. Cox and B. Harms, Parity violating one loop six point function in type-I superstring theory, Phys. Rev. D 35 (1987) 1908 [INSPIRE].
F. Brown, Motivic periods and the projective line minus three points, in Proceedings of the ICM 2014, Seoul Korea (2014) [arXiv:1407.5165].
F. Brown, Multiple modular values for SL2(Z), arXiv:1407.5167.
C.R. Mafra and O. Schlotterer, Cohomology foundations of one-loop amplitudes in pure spinor superspace, arXiv:1408.3605 [INSPIRE].
P.J. Cameron, Combinatorics. Topics, techniques, algorithms, Cambridge Univ. Pr., Cambridge U.K. (1994).
J. Riordan, Introduction to combinatorial analysis, Dover Publications, U.S.A. (2002).
R.P. Stanley, Enumerative combinatorics, second edition, Cambridge Univ. Pr., Cambridge U.K. (2012).
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Broedel, J., Mafra, C.R., Matthes, N. et al. Elliptic multiple zeta values and one-loop superstring amplitudes. J. High Energ. Phys. 2015, 112 (2015). https://doi.org/10.1007/JHEP07(2015)112
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DOI: https://doi.org/10.1007/JHEP07(2015)112