Rankin-Selberg methods for closed strings on orbifolds

  • Carlo Angelantonj
  • Ioannis Florakis
  • Boris Pioline
Open Access


In recent work we have developed a new unfolding method for computing one-loop modular integrals in string theory involving the Narain partition function and, possibly, a weak almost holomorphic elliptic genus. Unlike the traditional approach, the Narain lattice does not play any role in the unfolding procedure, T-duality is kept manifest at all steps, a choice of Weyl chamber is not required and the analytic structure of the amplitude is transparent. In the present paper, we generalise this procedure to the case of Abelian \( {{\mathbb{Z}}_N} \) orbifolds, where the integrand decomposes into a sum of orbifold blocks that can be organised into orbits of the Hecke congruence subgroup Γ0(N). As a result, the original modular integral reduces to an integral over the fundamental domain of Γ0(N), which we then evaluate by extending our previous techniques. Our method is applicable, for instance, to the evaluation of one-loop corrections to BPS-saturated couplings in the low energy effective action of closed string models, of quantum corrections to the Kähler metric and, in principle, of the free-energy of superstring vacua.


Superstrings and Heterotic Strings Superstring Vacua 


  1. [1]
    K. O’Brien and C. Tan, Modular Invariance of Thermopartition Function and Global Phase Structure of Heterotic String, Phys. Rev. D 36 (1987) 1184 [INSPIRE].ADSGoogle Scholar
  2. [2]
    B. McClain and B.D.B. Roth, Modular invariance for interacting bosonic strings at finite temperature, Commun. Math. Phys. 111 (1987) 539 [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. [3]
    L.J. Dixon, V. Kaplunovsky and J. Louis, Moduli dependence of string loop corrections to gauge coupling constants, Nucl. Phys. B 355 (1991) 649 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    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].MathSciNetGoogle Scholar
  5. [5]
    C. Angelantonj, I. Florakis and B. Pioline, One-Loop BPS amplitudes as BPS-state sums, JHEP 06 (2012) 070 [arXiv:1203.0566] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    D. Niebur, A class of nonanalytic automorphic functions, Nagoya Math. J. 52 (1973) 133.MathSciNetzbMATHGoogle Scholar
  7. [7]
    D.A. Hejhal, The Selberg trace formula for PSL(2, \( \mathbb{R} \)), vol. 2, Springer (1983).Google Scholar
  8. [8]
    R. Rankin, Contributions to the theory of Ramanujans function τ(n) and similar arithmetical functions. I. The zeros of the function \( \sum {_{n=1}^{\infty}\frac{{\tau (n)}}{{{n^8}}}} \) on the line \( \Re (s)=\frac{13 }{2} \), Proc. Camb. Philos. Soc. 35 (1939) 351.MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    A. Selberg, Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid. B 43 (1940) 1.Google Scholar
  10. [10]
    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.MathSciNetzbMATHGoogle Scholar
  11. [11]
    I. Antoniadis, E. Gava and K. Narain, Moduli corrections to gauge and gravitational couplings in four-dimensional superstrings, Nucl. Phys. B 383 (1992) 93 [hep-th/9204030] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    J.A. Harvey and G.W. Moore, Algebras, BPS states and strings, Nucl. Phys. B 463 (1996) 315 [hep-th/9510182] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    J.A. Harvey and G.W. Moore, On the algebras of BPS states, Commun. Math. Phys. 197 (1998) 489 [hep-th/9609017] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. [14]
    P. Mayr and S. Stieberger, Threshold corrections to gauge couplings in orbifold compactifications, Nucl. Phys. B 407 (1993) 725 [hep-th/9303017] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    W. Lerche and S. Stieberger, Prepotential, mirror map and F-theory on K3, Adv. Theor. Math. Phys. 2 (1998) 1105 [Erratum ibid. 3 (1999) 1199] [hep-th/9804176] [INSPIRE].
  16. [16]
    W. Lerche, S. Stieberger and N. Warner, Quartic gauge couplings from K3 geometry, Adv. Theor. Math. Phys. 3 (1999) 1575 [hep-th/9811228] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  17. [17]
    C. Angelantonj, M. Cardella and N. Irges, An Alternative for Moduli Stabilisation, Phys. Lett. B 641 (2006) 474 [hep-th/0608022] [INSPIRE].MathSciNetADSGoogle Scholar
  18. [18]
    E. Kiritsis and C. Kounnas, Perturbative and nonperturbative partial supersymmetry breaking: N = 4 → N = 2 → N = 1, Nucl. Phys. B 503 (1997) 117 [hep-th/9703059] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    M. Trapletti, On the unfolding of the fundamental region in integrals of modular invariant amplitudes, JHEP 02 (2003) 012 [hep-th/0211281] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    S. Hohenegger and D. Persson, Enhanced Gauge Groups in N = 4 Topological Amplitudes and Lorentzian Borcherds Algebras, Phys. Rev. D 84 (2011) 106007 [arXiv:1107.2301] [INSPIRE].ADSGoogle Scholar
  21. [21]
    M. Cardella, A Novel method for computing torus amplitudes for Z(N) orbifolds without the unfolding technique, JHEP 05 (2009) 010 [arXiv:0812.1549] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    N. Obers and B. Pioline, Eisenstein series and string thresholds, Commun. Math. Phys. 209 (2000) 275 [hep-th/9903113] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  23. [23]
    G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, Princeton, U.S.A. (1971)zbMATHGoogle Scholar
  24. [24]
    H. Iwaniec, Topics in Classical Automorphic Forms, American Mathematical Society (2002).Google Scholar
  25. [25]
    H. Iwaniec, Spectral Methods of Automorphic Forms, American Mathematical Society (1997).Google Scholar
  26. [26]
    S.D. Gupta, On the Rankin-Selberg method for functions not of rapid decay on congruence subgroups, J. Number Theory 62 (1997) 115.MathSciNetADSzbMATHCrossRefGoogle Scholar
  27. [27]
    A. Gregori, E. Kiritsis, C. Kounnas, N. Obers, P. Petropoulos and B. Pioline, R 2 corrections and nonperturbative dualities of N = 4 string ground states, Nucl. Phys. B 510 (1998) 423 [hep-th/9708062] [INSPIRE].MathSciNetADSGoogle Scholar
  28. [28]
    J.H. Bruinier, Borcherds products on O(2, l) and Chern classes of Heegner divisors, Springer (2002).Google Scholar
  29. [29]
    K. Bringmann and K. Ono, Arithmetic properties of coefficients of half-integral weight Maass-Poincaré series, Math. Ann. 337 (2007) 591.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    K. Bringmann and K. Ono, Coefficients of harmonic Maass forms, in Partitions, q-Series, and Modular Forms 23 (2012) 23.Google Scholar
  31. [31]
    M. Henningson and G.W. Moore, Threshold corrections in K3 × T 2 heterotic string compactifications, Nucl. Phys. B 482 (1996) 187 [hep-th/9608145] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  32. [32]
    S. Stieberger, (0,2) heterotic gauge couplings and their M-theory origin, Nucl. Phys. B 541 (1999) 109 [hep-th/9807124] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  33. [33]
    I. Florakis, C. Kounnas and N. Toumbas, Marginal Deformations of Vacua with Massive boson-fermion Degeneracy Symmetry, Nucl. Phys. B 834 (2010) 273 [arXiv:1002.2427] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    I. Florakis, C. Kounnas, H. Partouche and N. Toumbas, Non-singular string cosmology in a 2d Hybrid model, Nucl. Phys. B 844 (2011) 89 [arXiv:1008.5129] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  35. [35]
    C. Kounnas, Massive Boson-Fermion Degeneracy and the Early Structure of the Universe, Fortsch. Phys. 56 (2008) 1143 [arXiv:0808.1340] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  36. [36]
    I. Florakis and C. Kounnas, Orbifold Symmetry Reductions of Massive Boson-Fermion Degeneracy, Nucl. Phys. B 820 (2009) 237 [arXiv:0901.3055] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  37. [37]
    D. Zagier, Introduction to modular forms, Springer-Verlag (1992).Google Scholar
  38. [38]
    N. Koblitz, Introduction to elliptic curves and modular forms, 2nd ed., Springer-Verlag (1993).Google Scholar
  39. [39]
    T.M. Apostol, Modular functions and Dirichlet series in number theory, 2nd ed., Springer-Verlag (1990).Google Scholar
  40. [40]
    D. Zagier, Elliptic modular forms and their applications, in The 1-2-3 of Modular Forms, Springer (2008), pg. 1–103.Google Scholar
  41. [41]
    W.A. Stein, An introduction to computing modular forms using modular symbols, Cambridge University Press, Cambridge, U.K. (2008).Google Scholar
  42. [42]
  43. [43]
  44. [44]
  45. [45]
  46. [46]
    J. Conway and S. Norton, Monstrous moonshine, Bull. Lond. Math. Soc. 11 (1979) 308.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© SISSA 2013

Authors and Affiliations

  • Carlo Angelantonj
    • 1
  • Ioannis Florakis
    • 2
  • Boris Pioline
    • 3
    • 4
  1. 1.Dipartimento di Fisica, Università di Torino, and INFN Sezione di TorinoTorinoItaly
  2. 2.Max-Planck-Institut für Physik, Werner-Heisenberg-InstitutMünchenGermany
  3. 3.CERN Dep PH-THGeneva 23Switzerland
  4. 4.Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, Université Pierre et Marie Curie, Paris 6Paris cedex 05France

Personalised recommendations