Built on

E. Gottschling, “Explizite Bestimmung der Randflächen des Fundamentalbereiches der Modulgruppe zweiten Grades.,” Math. Ann
1959
Earlier work this paper cites.
J. Fay, “Theta functions on Riemann surfaces”, Lectures Notes in Mathematics (Springer, Berlin), vol. 352. 1973
1973
Earlier work this paper cites.
M. B. Green, J. H. Schwarz, and L. Brink, “N=4 Yang-Mills and N=8 Supergravity as Limits of String Theories,” Nucl.Phys
1982
Earlier work this paper cites.
M. B. Green and J. H. Schwarz, “Supersymmetrical String Theories,” Phys. Lett
1982
Earlier work this paper cites.
D. Zagier, “The Rankin-Selberg method for automorphic functions which are not of rapid decay,” J. Fac. Sci. Univ. Tokyo Sect. IA Math
1982
Earlier work this paper cites.
D. J. Gross and E. Witten, “Superstring modifications of Einstein’s equations,” Nucl. Phys
1986
Earlier work this paper cites.
S. Wolpert, “Asymptotics of the Spectrum and the Selberg Zeta Function on the Space of Riemann Surfaces”, Commun. Math. Phys
1987
Earlier work this paper cites.
E. D’Hoker and D. Phong, “The Geometry of String Perturbation Theory,” Rev.Mod.Phys
1988
Earlier work this paper cites.
B. Zwiebach, “How Covariant Closed String Theory Solves A Minimal Area Problem,” Commun. Math. Phys
1991
Earlier work this paper cites.
C. M. Hull and P. K. Townsend, “Unity of superstring dualities,” Nucl. Phys
1995
Earlier work this paper cites.
Z. Bern, L. J. Dixon, D. C. Dunbar, M. Perelstein, and J. S. Rozowsky, “On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences,” Nucl. Phys
1998
Earlier work this paper cites.
M. B. Green and P. Vanhove, “The low energy expansion of the one-loop type II superstring amplitude,” Phys. Rev
2000
Earlier work this paper cites.
E. D’Hoker and D. H. Phong, “Two loop superstrings. 1. Main formulas,” Phys. Lett
2002
Earlier work this paper cites.
E. D’Hoker and D. Phong, “Lectures on two loop superstrings,” Conf.Proc
2002
Earlier work this paper cites.
E. D’Hoker and D. H. Phong, “Two-loop superstrings VI: Non-renormalization theorems and the 4-point function,” Nucl. Phys
2005
Earlier work this paper cites.

Similar

N. Berkovits, “Super-Poincare covariant two-loop superstring amplitudes,” JHEP
2006
Cited alongside, same era.
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
Original
2008
Cited alongside, same era.
M. B. Green, J. G. Russo, and P. Vanhove, “Modular properties of two-loop maximal supergravity and connections with string theory,” JHEP
Original
2008
Cited alongside, same era.
Z. Bern, J. Carrasco, L. J. Dixon, H. Johansson, and R. Roiban, “Manifest Ultraviolet Behavior for the Three-Loop Four-Point Amplitude of N=8 Supergravity,” Phys.Rev
Original
2008
Cited alongside, same era.
unpublished notes (2008)
D. Zagier, · 2008
Cited alongside, same era.

Then

E. D’Hoker, M. B. Green, and P. Vanhove, “On the modular structure of the genus-one Type II superstring low energy expansion,” JHEP
Original
2015
Later among the works it cites.
E. D’Hoker, M. B. Green, B. Pioline, and R. Russo, “Matching the D 6 R 4 D^{6}R^{4} interaction at two-loops,” JHEP
Original
2015
Later among the works it cites.
B. Pioline and R. Russo, “Infrared divergences and harmonic anomalies in the two-loop superstring effective action,” JHEP
Original
2015
Later among the works it cites.
E. D’Hoker, M. B. Green, B. Pioline, and R. Russo, “Matching the D 6 R 4 D^{6}R^{4} interaction at two-loops,” JHEP
Original
2015
Later among the works it cites.
G. Bossard and A. Kleinschmidt, “Supergravity divergences, supersymmetry and automorphic forms,” JHEP

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2018

String theory integrands and supergravity divergences

Pioline, Boris

Understand

At low energies, interactions of massless particles in type II strings compactified on a torus $T^d$ are described by an effective Wilsonian action $\mathcal{S}(\Lambda)$, consisting of the usual supergravity Lagrangian supplemented by an infinite series of higher-derivative vertices, including the much studied $\nabla^{4p+6q} \mathcal{R}^4$ gravitational interactions.

Using recent results on the asymptotics of the integrands governing four-graviton scattering at genus one and two, I determine the $\Lambda$-dependence of the coefficient of the above interaction, and show that the logarithmic terms appearing in the limit $\Lambda\to 0$ are related to UV divergences in supergravity amplitudes, augmented by stringy interactions.
This provides a strong consistency check on the expansion of the integrand near the boundaries of moduli space, in particular it elucidates the appearance of odd zeta values in these expansions.
I briefly discuss how these logarithms are reflected in non-analytic terms in the low energy expansion of the string scattering amplitude.

Built on

E. Gottschling, “Explizite Bestimmung der Randflächen des Fundamentalbereiches der Modulgruppe zweiten Grades.,” Math. Ann
1959
Earlier work this paper cites.
J. Fay, “Theta functions on Riemann surfaces”, Lectures Notes in Mathematics (Springer, Berlin), vol. 352. 1973
1973
Earlier work this paper cites.
M. B. Green, J. H. Schwarz, and L. Brink, “N=4 Yang-Mills and N=8 Supergravity as Limits of String Theories,” Nucl.Phys
1982
Earlier work this paper cites.
M. B. Green and J. H. Schwarz, “Supersymmetrical String Theories,” Phys. Lett
1982
Earlier work this paper cites.
D. Zagier, “The Rankin-Selberg method for automorphic functions which are not of rapid decay,” J. Fac. Sci. Univ. Tokyo Sect. IA Math
1982
Earlier work this paper cites.
D. J. Gross and E. Witten, “Superstring modifications of Einstein’s equations,” Nucl. Phys
1986
Earlier work this paper cites.
S. Wolpert, “Asymptotics of the Spectrum and the Selberg Zeta Function on the Space of Riemann Surfaces”, Commun. Math. Phys
1987
Earlier work this paper cites.
E. D’Hoker and D. Phong, “The Geometry of String Perturbation Theory,” Rev.Mod.Phys
1988
Earlier work this paper cites.
B. Zwiebach, “How Covariant Closed String Theory Solves A Minimal Area Problem,” Commun. Math. Phys
1991
Earlier work this paper cites.
C. M. Hull and P. K. Townsend, “Unity of superstring dualities,” Nucl. Phys
1995
Earlier work this paper cites.
Z. Bern, L. J. Dixon, D. C. Dunbar, M. Perelstein, and J. S. Rozowsky, “On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences,” Nucl. Phys
1998
Earlier work this paper cites.
M. B. Green and P. Vanhove, “The low energy expansion of the one-loop type II superstring amplitude,” Phys. Rev
2000
Earlier work this paper cites.
E. D’Hoker and D. H. Phong, “Two loop superstrings. 1. Main formulas,” Phys. Lett
2002
Earlier work this paper cites.
E. D’Hoker and D. Phong, “Lectures on two loop superstrings,” Conf.Proc
2002
Earlier work this paper cites.
E. D’Hoker and D. H. Phong, “Two-loop superstrings VI: Non-renormalization theorems and the 4-point function,” Nucl. Phys
2005
Earlier work this paper cites.