Abstract
In type II string theory compactified on a d-dimensional torus T d down to D = 10−ddimensions,the ℛ4 and ∇4ℛ4 four-gravitoncouplingsareknownexactly,forall values of the moduli, in terms of certain Eisenstein series of the U-duality group E d (ℤ). In the limit where one circle in the torus becomes large, these couplings are expected to reduce to their counterpart in dimension D +1, plus threshold effects and exponentially suppressed corrections corresponding to BPS black holes in dimension D + 1 whose worldline winds around the circle. By combining the weak coupling and large radius limits, we determine these exponentially suppressed corrections exactly, and demonstrate that the contributions of 1/4-BPS black holes to the ∇4ℛ4 coupling are proportional to the appropriate helicity supertrace. Mathematically, our results provide the complete Fourier expansion of the next-to-minimal theta series of E d + 1(ℤ) with respect to the maximal parabolic subgroup with Levi component E d for d ≤ 6, and the complete Abelian part of the Fourier expansion of the same for d = 7.
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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 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].
M.B. Green, M. Gutperle and P. Vanhove, One loop in eleven-dimensions, Phys. Lett. B 409 (1997) 177 [hep-th/9706175] [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].
N.A. Obers and B. Pioline, Eisenstein series and string thresholds, Commun. Math. Phys. 209 (2000) 275 [hep-th/9903113] [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].
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].
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].
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, J. Number Theor. 146 (2015) 187 [arXiv:1111.2983] [INSPIRE].
G. Bossard and V. Verschinin, Minimal unitary representations from supersymmetry, JHEP 10 (2014) 008 [arXiv:1406.5527] [INSPIRE].
G. Bossard and V. Verschinin, ε∇4 R 4 type invariants and their gradient expansion, JHEP 03 (2015) 089 [arXiv:1411.3373] [INSPIRE].
M.B. Green and P. Vanhove, Duality and higher derivative terms in M-theory, JHEP 01 (2006) 093 [hep-th/0510027] [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, Commun. Num. Theor. Phys. 09 (2015) 307 [arXiv:1404.2192] [INSPIRE].
B. Pioline, D 6 R 4 amplitudes in various dimensions, JHEP 04 (2015) 057 [arXiv:1502.03377] [INSPIRE].
G. Bossard and A. Kleinschmidt, Loops in exceptional field theory, JHEP 01 (2016) 164 [arXiv:1510.07859] [INSPIRE].
M. Günaydin, A. Neitzke, B. Pioline and A. Waldron, BPS black holes, quantum attractor flows and automorphic forms, Phys. Rev. D 73 (2006) 084019 [hep-th/0512296] [INSPIRE].
B. Pioline, Lectures on black holes, topological strings and quantum attractors, Class. Quant. Grav. 23 (2006) S981 [hep-th/0607227] [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].
H. Matumoto, Whittaker vectors and associated varieties, Invent. Math. 89 (1987) 219.
S.D. Miller and S. Sahi, Fourier coefficients of automorphic forms, character variety orbits, and small representations, J. Number Theory 132 (2012) 3070 [arXiv:1202.0210].
G. Bossard and V. Verschinin, The two ∇6 R 4 type invariants and their higher order generalisation, JHEP 07 (2015) 154 [arXiv:1503.04230] [INSPIRE].
G. Bossard, H. Nicolai and K.S. Stelle, Universal BPS structure of stationary supergravity solutions, JHEP 07 (2009) 003 [arXiv:0902.4438] [INSPIRE].
N. Dorey and A. Parnachev, Instantons, compactification and S duality in N = 4 SUSY Yang-Mills theory. 2, JHEP 08 (2001) 059 [hep-th/0011202] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys. 299 (2010) 163 [arXiv:0807.4723] [INSPIRE].
S. Alexandrov, G.W. Moore, A. Neitzke and B. Pioline, ℝ 3 Index for Four-Dimensional N = 2 Field Theories, Phys. Rev. Lett. 114 (2015) 121601 [arXiv:1406.2360] [INSPIRE].
G. Bossard, C. Cosnier-Horeau and B. Pioline, Protected couplings and BPS dyons in half-maximal supersymmetric string vacua, Phys. Lett. B 765 (2017) 377 [arXiv:1608.01660] [INSPIRE].
S. Ferrara and J.M. Maldacena, Branes, central charges and U duality invariant BPS conditions, Class. Quant. Grav. 15 (1998) 749 [hep-th/9706097] [INSPIRE].
S. Ferrara and M. Günaydin, Orbits of exceptional groups, duality and BPS states in string theory, Int. J. Mod. Phys. A 13 (1998) 2075 [hep-th/9708025] [INSPIRE].
S. Ferrara, C.A. Savoy and L. Girardello, Spin Sum Rules in Extended Supersymmetry, Phys. Lett. B 105 (1981) 363 [INSPIRE].
E. Kiritsis, Introduction to superstring theory, hep-th/9709062.
A. Dabholkar and J.A. Harvey, Nonrenormalization of the Superstring Tension, Phys. Rev. Lett. 63 (1989) 478 [INSPIRE].
D. Shih, A. Strominger and X. Yin, Counting dyons in N = 8 string theory, JHEP 06 (2006) 037 [hep-th/0506151] [INSPIRE].
B. Pioline, BPS black hole degeneracies and minimal automorphic representations, JHEP 08 (2005) 071 [hep-th/0506228] [INSPIRE].
A. Sen, U-duality Invariant Dyon Spectrum in type-II on T 6, JHEP 08 (2008) 037 [arXiv:0804.0651] [INSPIRE].
A. Sen, Arithmetic of N = 8 Black Holes, JHEP 02 (2010) 090 [arXiv:0908.0039] [INSPIRE].
R. Kallosh and B. Kol, E 7 symmetric area of the black hole horizon, Phys. Rev. D 53 (1996) R5344 [hep-th/9602014] [INSPIRE].
A. Dabholkar, Exact counting of black hole microstates, Phys. Rev. Lett. 94 (2005) 241301 [hep-th/0409148] [INSPIRE].
A. Sinha and N.V. Suryanarayana, Extremal single-charge small black holes: Entropy function analysis, Class. Quant. Grav. 23 (2006) 3305 [hep-th/0601183] [INSPIRE].
P. Fleig, H.P.A. Gustafsson, A. Kleinschmidt and D. Persson, Eisenstein series and automorphic representations, arXiv:1511.04265 [INSPIRE].
R. Gomez, D. Gourevitch and S. Sahi, Whittaker supports for representations of reductive groups, arXiv:1610.00284.
I. Florakis and B. Pioline, On the Rankin-Selberg method for higher genus string amplitudes, arXiv:1602.00308 [INSPIRE].
S. Krutelevich, Jordan algebras, exceptional groups, and bhargava composition, J. Algebra 314 (2007) 924 [math/0411104].
J.M. Maldacena, G.W. Moore and A. Strominger, Counting BPS black holes in toroidal Type II string theory, hep-th/9903163 [INSPIRE].
B. Pioline, A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces, J. Number Theor. 163 (2016) 520 [arXiv:1504.04182] [INSPIRE].
E. D’Hoker and M.B. Green, Zhang-Kawazumi invariants and superstring amplitudes, J. Number Theory 144 (2014) 111.
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.
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].
B. Pioline, Rankin-Selberg methods for closed string amplitudes, Proc. Symp. Pure Math. 88 (2014) 119 [arXiv:1401.4265] [INSPIRE].
H. Freudenthal, Beziehungen der e7 und e8 zur oktavenebene. I-IV, in Indagationes Mathematicae (Proceedings), Elsevier (1955), vol. 58, pg. 277-285.
S. Krutelevich, Jordan algebras, exceptional groups, and higher composition laws, math/0411104.
G. Bossard and S. Katmadas, Duality covariant multi-centre black hole systems, JHEP 08 (2013) 007 [arXiv:1304.6582] [INSPIRE].
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Bossard, G., Pioline, B. Exact ∇4ℛ4 couplings and helicity supertraces. J. High Energ. Phys. 2017, 50 (2017). https://doi.org/10.1007/JHEP01(2017)050
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DOI: https://doi.org/10.1007/JHEP01(2017)050