Abstract
Four-graviton couplings in the low energy effective action of type II string vacua compactified on tori are strongly constrained by supersymmetry and U-duality. While the ℛ4 and D4ℛ4 couplings are known exactly in terms of Langlands-Eisenstein series of the U-duality group, the D6ℛ4 couplings are not nearly as well understood. Exploiting the coincidence of the U-duality group in D = 6 with the T-duality group in D = 5, we propose an exact formula for the D6ℛ4 couplings in type II string theory compactified on T4, in terms of a genus-two modular integral plus a suitable Eisenstein series. The same modular integral computes the two-loop correction to D6ℛ4 in 5 dimensions, but here provides the non-perturbative completion of the known perturbative terms in D = 6. This proposal hinges on a systematic re-analysis of the weak coupling and large radius of the D6ℛ4 in all dimensions D ≥ 3, which fills in some gaps and resolves some inconsistencies in earlier studies.
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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].
E. Cremmer and B. Julia, The N = 8 supergravity theory. 1. The Lagrangian, Phys. Lett. B 80 (1978) 48 [INSPIRE].
C.M. Hull and P.K. Townsend, Unity of superstring dualities, Nucl. Phys. B 438 (1995) 109 [hep-th/9410167] [INSPIRE].
E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B 443 (1995) 85 [hep-th/9503124] [INSPIRE].
N.A. Obers and B. Pioline, U duality and M-theory, Phys. Rept. 318 (1999) 113 [hep-th/9809039] [INSPIRE].
B. Pioline, A note on nonperturbative R 4 couplings, Phys. Lett. B 431 (1998) 73 [hep-th/9804023] [INSPIRE].
M.B. Green and S. Sethi, Supersymmetry constraints on type IIB supergravity, Phys. Rev. D 59 (1999) 046006 [hep-th/9808061] [INSPIRE].
M.B. Green, J.G. Russo and P. Vanhove, Automorphic properties of low energy string amplitudes in various dimensions, Phys. Rev. D 81 (2010) 086008 [arXiv:1001.2535] [INSPIRE].
A. Basu, Supersymmetry constraints on the R 4 multiplet in type IIB on T 2, Class. Quant. Grav. 28 (2011) 225018 [arXiv:1107.3353] [INSPIRE].
G. Bossard and V. Verschinin, Minimal unitary representations from supersymmetry, JHEP 10 (2014) 008 [arXiv:1406.5527] [INSPIRE].
G. Bossard and V. Verschinin, ℰ∇4 R 4 type invariants and their gradient expansion, JHEP 03 (2015) 089 [arXiv:1411.3373] [INSPIRE].
M.B. Green and P. Vanhove, D instantons, strings and M-theory, Phys. Lett. B 408 (1997) 122 [hep-th/9704145] [INSPIRE].
E. Kiritsis and B. Pioline, On R 4 threshold corrections in IIB string theory and (p, q) string instantons, Nucl. Phys. B 508 (1997) 509 [hep-th/9707018] [INSPIRE].
B. Pioline and E. Kiritsis, U duality and D-brane combinatorics, Phys. Lett. B 418 (1998) 61 [hep-th/9710078] [INSPIRE].
N.A. Obers and B. Pioline, Eisenstein series and string thresholds, Commun. Math. Phys. 209 (2000) 275 [hep-th/9903113] [INSPIRE].
A. Basu, The D 4 R 4 term in type IIB string theory on T 2 and U-duality, Phys. Rev. D 77 (2008) 106003 [arXiv:0708.2950] [INSPIRE].
B. Pioline, R 4 couplings and automorphic unipotent representations, JHEP 03 (2010) 116 [arXiv:1001.3647] [INSPIRE].
M.B. Green, S.D. Miller, J.G. Russo and P. Vanhove, Eisenstein series for higher-rank groups and string theory amplitudes, Commun. Num. Theor. Phys. 4 (2010) 551 [arXiv:1004.0163] [INSPIRE].
M.B. Green, S.D. Miller and P. Vanhove, Small representations, string instantons and Fourier modes of Eisenstein series (with an appendix by D. Ciubotaru and P. Trapa), arXiv:1111.2983 [INSPIRE].
M.B. Green, J.G. Russo and P. Vanhove, String theory dualities and supergravity divergences, JHEP 06 (2010) 075 [arXiv:1002.3805] [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].
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].
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].
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, M. Gutperle and H.-h. Kwon, Sixteen fermion and related terms in M-theory on T 2, Phys. Lett. B 421 (1998) 149 [hep-th/9710151] [INSPIRE].
M.B. Green, H.-h. Kwon and P. Vanhove, Two loops in eleven-dimensions, Phys. Rev. D 61 (2000) 104010 [hep-th/9910055] [INSPIRE].
M.B. Green and P. Vanhove, Duality and higher derivative terms in M-theory, JHEP 01 (2006) 093 [hep-th/0510027] [INSPIRE].
M.B. Green, J.G. Russo and P. Vanhove, Non-renormalisation conditions in type-II string theory and maximal supergravity, JHEP 02 (2007) 099 [hep-th/0610299] [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].
A. Basu, The D 6 R 4 term from three loop maximal supergravity, Class. Quant. Grav. 31 (2014) 245002 [arXiv:1407.0535] [INSPIRE].
A. Basu, The D 6 R 4 term in type IIB string theory on T 2 and U-duality, Phys. Rev. D 77 (2008) 106004 [arXiv:0712.1252] [INSPIRE].
M.B. Green, S.D. Miller and P. Vanhove, SL(2, ℤ)-invariance and D-instanton contributions to the D 6 R 4 interaction, arXiv:1404.2192 [INSPIRE].
E. D’Hoker and M.B. Green, Zhang-Kawazumi invariants and superstring amplitudes, arXiv:1308.4597 [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].
H. Gomez and C.R. Mafra, The closed-string 3-loop amplitude and S-duality, JHEP 10 (2013) 217 [arXiv:1308.6567] [INSPIRE].
G. Bossard and A. Kleinschmidt, to appear.
N. Kawazumi, Johnson’s homomorphisms and the Arakelov-Green function, arXiv:0801.4218.
S.W. Zhang, Gross-Schoen cycles and dualising sheaves, Invent. Math. 179 (2010) 1.
B. Pioline and R. Russo, to appear.
M.B. Green and J.H. Schwarz, Supersymmetrical dual string theory. 3. Loops and renormalization, Nucl. Phys. B 198 (1982) 441 [INSPIRE].
D.J. Gross and E. Witten, Superstring modifications of Einstein’s equations, Nucl. Phys. B 277 (1986) 1 [INSPIRE].
C. Angelantonj, I. Florakis and B. Pioline, A new look at one-loop integrals in string theory, Commun. Num. Theor. Phys. 6 (2012) 159 [arXiv:1110.5318] [INSPIRE].
D. Zagier, The Rankin-Selberg method for automorphic functions which are not of rapid decay, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981) 415.
B. Pioline, Rankin-Selberg methods for closed string amplitudes, Proc. Symp. Pure Math. 88 (2014) 119 [arXiv:1401.4265] [INSPIRE].
Z. Bern, J.J.M. 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. D 78 (2008) 105019 [arXiv:0808.4112] [INSPIRE].
E. Kiritsis, N.A. Obers and B. Pioline, Heterotic/type-II triality and instantons on K(3), JHEP 01 (2000) 029 [hep-th/0001083] [INSPIRE].
P. Fleig and A. Kleinschmidt, Eisenstein series for infinite-dimensional U-duality groups, JHEP 06 (2012) 054 [arXiv:1204.3043] [INSPIRE].
M.A.A. van Leeuwen, A.M. Cohen and B. Lisser, LiE, a package for Lie group computations, http://wwwmathlabo.univ-poitiers.fr/∼maavl/LiE/.
J.G. Russo and A.A. Tseytlin, One loop four graviton amplitude in eleven-dimensional supergravity, Nucl. Phys. B 508 (1997) 245 [hep-th/9707134] [INSPIRE].
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Pioline, B. D6ℛ4 amplitudes in various dimensions. J. High Energ. Phys. 2015, 57 (2015). https://doi.org/10.1007/JHEP04(2015)057
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DOI: https://doi.org/10.1007/JHEP04(2015)057