Abstract
The low energy expansion of Type II superstring amplitudes at genus one is organized in terms of modular graph functions associated with Feynman graphs of a conformal scalar field on the torus. In earlier work, surprising identities between two-loop graphs at all weights, and between higher-loop graphs of weights four and five were constructed. In the present paper, these results are generalized in two complementary directions. First, all identities at weight six and all dihedral identities at weight seven are obtained and proven. Whenever the Laurent polynomial at the cusp is available, the form of these identities confirms the pattern by which the vanishing of the Laurent polynomial governs the full modular identity. Second, the family of modular graph functions is extended to include all graphs with derivative couplings and worldsheet fermions. These extended families of modular graph functions are shown to obey a hierarchy of inhomogeneous Laplace eigenvalue equations. The eigenvalues are calculated analytically for the simplest infinite sub-families and obtained by Maple for successively more complicated sub-families. The spectrum is shown to consist solely of eigenvalues s(s − 1) for positive integers s bounded by the weight, with multiplicities which exhibit rich representation-theoretic patterns.
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References
M.B. Green and M. Gutperle, Effects of D instantons, Nucl. Phys. B 498 (1997) 195 [hep-th/9701093] [INSPIRE].
M.B. Green, M. Gutperle and P. Vanhove, One loop in eleven-dimensions, Phys. Lett. B 409 (1997) 177 [hep-th/9706175] [INSPIRE].
M.B. Green and S. Sethi, Supersymmetry constraints on type IIB supergravity, Phys. Rev. D 59 (1999) 046006 [hep-th/9808061] [INSPIRE].
N.A. Obers and B. Pioline, Eisenstein series and string thresholds, Commun. Math. Phys. 209 (2000) 275 [hep-th/9903113] [INSPIRE].
E. D’Hoker and D.H. Phong, Lectures on two loop superstrings, Conf. Proc. C 0208124 (2002) 85 [hep-th/0211111] [INSPIRE].
E. D’Hoker and D.H. Phong, Two-loop superstrings VI: Non-renormalization theorems and the 4-point function, Nucl. Phys. B 715 (2005) 3 [hep-th/0501197] [INSPIRE].
E. D’Hoker, M. Gutperle and D.H. Phong, Two-loop superstrings and S-duality, Nucl. Phys. B 722 (2005) 81 [hep-th/0503180] [INSPIRE].
H. Gomez and C.R. Mafra, The closed-string 3-loop amplitude and S-duality, JHEP 10 (2013) 217 [arXiv:1308.6567] [INSPIRE].
E. D’Hoker, M.B. Green, B. Pioline and R. Russo, Matching the D 6 R 4 interaction at two-loops, JHEP 01 (2015) 031 [arXiv:1405.6226] [INSPIRE].
B. Pioline, \( {D}^6{\mathrm{\mathcal{R}}}^4 \) amplitudes in various dimensions, JHEP 04 (2015) 057 [arXiv:1502.03377] [INSPIRE].
M.B. Green, J.G. Russo and P. Vanhove, Modular properties of two-loop maximal supergravity and connections with string theory, JHEP 07 (2008) 126 [arXiv:0807.0389] [INSPIRE].
E. D’Hoker and M.B. Green, Zhang-Kawazumi Invariants and Superstring Amplitudes, arXiv:1308.4597 [INSPIRE].
S.W. Zhang, Gross-Schoen Cycles and Dualising Sheaves, Invent. Math. 179 1 [arXiv:0812.0371].
N. Kawazumi, Johnson’s homomorphisms and the Arakelov Green function, arXiv:0801.4218.
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].
E. D’Hoker, M.B. Green, O. Gurdogan and P. Vanhove, Modular Graph Functions, arXiv:1512.06779 [INSPIRE].
E. D’Hoker and M.B. Green, Identities between Modular Graph Forms, arXiv:1603.00839 [INSPIRE].
E. D’Hoker, M.B. Green and P. Vanhove, On the modular structure of the genus-one Type II superstring low energy expansion, JHEP 08 (2015) 041 [arXiv:1502.06698] [INSPIRE].
E. D’Hoker, M.B. Green and P. Vanhove, Proof of a modular relation between 1-, 2- and 3-loop Feynman diagrams on a torus, arXiv:1509.00363 [INSPIRE].
F. Zerbini, Single-valued multiple zeta values in genus 1 superstring amplitudes, arXiv:1512.05689 [INSPIRE].
E. D’Hoker, M.B. Green and P. Vanhove, unpublished notes (2016).
D. Zagier, Values of zeta functions and their application, First European Congress of Mathematics, Paris France (1992) [Progr. Math. 120 (1994) 497].
M.E. Hoffmann, Multiple harmonic series, Pacific J. Math. 152 (1992) 275.
M. Waldschmidt, Valeurs zêta multiples: une introduction, J. Théor. Nombres Bordeaux 12 (2000) 581.
J.M. Borwein, D.M. Bradley, D.J. Broadhurst and P. Lisonek, Special values of multiple polylogarithms, Trans. Am. Math. Soc. 353 (2001) 907 [math/9910045] [INSPIRE].
V.V. Zudilin, Algebraic relations for multiple zeta values, Uspekhi Mat. Nauk 58 (2003) 3 [Russ. Math. Surv. 58 (2003) 1].
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].
F. Brown, Single-valued Motivic Periods and Multiple Zeta Values, SIGMA 2 (2014) e25 [arXiv:1309.5309] [INSPIRE].
J. Broedel, O. Schlotterer and S. Stieberger, Polylogarithms, Multiple Zeta Values and Superstring Amplitudes, Fortsch. Phys. 61 (2013) 812 [arXiv:1304.7267] [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].
J. Broedel, N. Matthes and O. Schlotterer, Relations between elliptic multiple zeta values and a special derivation algebra, J. Phys. A 49 (2016) 155203 [arXiv:1507.02254] [INSPIRE].
O. Schlotterer and S. Stieberger, Motivic Multiple Zeta Values and Superstring Amplitudes, J. Phys. A 46 (2013) 475401 [arXiv:1205.1516] [INSPIRE].
E. D’Hoker and D.H. Phong, Momentum analyticity and finiteness of the one loop superstring amplitude, Phys. Rev. Lett. 70 (1993) 3692 [hep-th/9302003] [INSPIRE].
E. D’Hoker and D.H. Phong, The Box graph in superstring theory, Nucl. Phys. B 440 (1995) 24 [hep-th/9410152] [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].
D. Zagier, Notes on Lattice Sums, unpublished.
A. Basu, Poisson equation for the three loop ladder diagram in string theory at genus one, arXiv:1606.02203 [INSPIRE].
A. Basu, Proving relations between modular graph functions, arXiv:1606.07084 [INSPIRE].
A. Basu, Poisson equation for the Mercedes diagram in string theory at genus one, Class. Quant. Grav. 33 (2016) 055005 [arXiv:1511.07455] [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].
A. Basu, Simplifying the one loop five graviton amplitude in type IIB string theory, arXiv:1608.02056 [INSPIRE].
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ArXiv ePrint: 1608.04393
The research reported in this paper was supported in part by the National Science Foundation under the grants PHY-13-13986 and PHY-16-19926.
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D’Hoker, E., Kaidi, J. Hierarchy of modular graph identities. J. High Energ. Phys. 2016, 51 (2016). https://doi.org/10.1007/JHEP11(2016)051
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DOI: https://doi.org/10.1007/JHEP11(2016)051