Abstract
Recently, we introduced a new procedure for computing a class of one-loop BPS-saturated amplitudes in String Theory, which expresses them as a sum of one-loop contributions of all perturbative BPS states in a manifestly T-duality invariant fashion. In this paper, we extend this procedure to all BPS-saturated amplitudes of the form ∫ F Γ d+k,d Φ, with Φ being a weak (almost) holomorphic modular form of weight − k/2. We use the fact that any such Φ can be expressed as a linear combination of certain absolutely convergent Poincaré series, against which the fundamental domain F can be unfolded. The resulting BPS-state sum neatly exhibits the singularities of the amplitude at points of gauge symmetry enhancement, in a chamber-independent fashion. We illustrate our method with concrete examples of interest in heterotic string compactifications.
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References
E. Kiritsis, Duality and instantons in string theory, hep-th/9906018 [INSPIRE].
W. Lerche, B. Nilsson, A. Schellekens and N. Warner, Anomaly cancelling terms from the elliptic genus, Nucl. Phys. B 299 (1988) 91 [INSPIRE].
D. Zagier, The Rankin-Selberg method for automorphic functions which are not of rapid decay, J. Fac. Sci. Univ. Tokyo A 28 (1982) 415.
D. Kutasov and N. Seiberg, Number of degrees of freedom, density of states and tachyons in string theory and CFT, Nucl. Phys. B 358 (1991) 600 [INSPIRE].
C. Angelantonj, M. Cardella, S. Elitzur and E. Rabinovici, Vacuum stability, string density of states and the Riemann zeta function, JHEP 02 (2011) 024 [arXiv:1012.5091] [INSPIRE].
M.A. Cardella, Error Estimates in Horocycle Averages Asymptotics: Challenges from String Theory, arXiv:1012.2754 [INSPIRE].
M. Cardella, A Novel method for computing torus amplitudes for ZN orbifolds without the unfolding technique, JHEP 05 (2009) 010 [arXiv:0812.1549] [INSPIRE].
B. McClain and B.D.B. Roth, Modular invariance for interacting bosonic strings at finite temperature, Commun. Math. Phys. 111 (1987) 539 [INSPIRE].
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].
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].
P. Mayr and S. Stieberger, Threshold corrections to gauge couplings in orbifold compactifications, Nucl. Phys. B 407 (1993) 725 [hep-th/9303017] [INSPIRE].
C. Bachas, C. Fabre, E. Kiritsis, N. Obers and P. Vanhove, Heterotic/type-I duality and D-brane instantons, Nucl. Phys. B 509 (1998) 33 [hep-th/9707126] [INSPIRE].
W. Lerche and S. Stieberger, Prepotential, mirror map and F-theory on K3, Adv. Theor. Math. Phys. 2 (1998) 1105 [Erratum ibid. 3 (1999) 1199-2000] [hep-th/9804176] [INSPIRE].
K. Foerger and S. Stieberger, Higher derivative couplings and heterotic type-I duality in eight-dimensions, Nucl. Phys. B 559 (1999) 277 [hep-th/9901020] [INSPIRE].
E. Kiritsis and N. Obers, Heterotic type-I duality in D < 10-dimensions, threshold corrections and D instantons, JHEP 10 (1997) 004 [hep-th/9709058] [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. Mariño and G.W. Moore, Counting higher genus curves in a Calabi-Yau manifold, Nucl. Phys. B 543 (1999) 592 [hep-th/9808131] [INSPIRE].
J.A. Harvey and G.W. Moore, Algebras, BPS states and strings, Nucl. Phys. B 463 (1996) 315 [hep-th/9510182] [INSPIRE].
J.A. Harvey and G.W. Moore, On the algebras of BPS states, Commun. Math. Phys. 197 (1998)489 [hep-th/9609017] [INSPIRE].
C. Angelantonj, I. Florakis and B. Pioline, A new look at one-loop integrals in string theory, Commun. Num. Theor. Phys. 6 (2012) 1 [arXiv:1110.5318] [INSPIRE].
N. Obers and B. Pioline, Eisenstein series and string thresholds, Commun. Math. Phys. 209 (2000)275 [hep-th/9903113] [INSPIRE].
H. Ooguri and C. Vafa, Geometry of N = 2 strings, Nucl. Phys. B 361 (1991) 469 [INSPIRE].
S. Ferrara, C. Kounnas, D. Lüst and F. Zwirner, Duality invariant partition functions and automorphic superpotentials for (2, 2) string compactifications, Nucl. Phys. B 365 (1991) 431 [INSPIRE].
G. Lopes Cardoso, D. Lüst and T. Mohaupt, Threshold corrections and symmetry enhancement in string compactifications, Nucl. Phys. B 450 (1995) 115 [hep-th/9412209] [INSPIRE].
G. Lopes Cardoso, G. Curio, D. Lüst, T. Mohaupt and S.-J. Rey, BPS spectra and nonperturbative gravitational couplings in N = 2, N = 4 supersymmetric string theories, Nucl. Phys. B 464 (1996) 18 [hep-th/9512129] [INSPIRE].
W. Lerche and S. Stieberger, 1/4 BPS states and nonperturbative couplings in N = 4 string theories, Adv. Theor. Math. Phys. 3 (1999) 1539 [hep-th/9907133] [INSPIRE].
M.-F. Vignéras, Séries Théta des formes quadratiques indéfinies, in proceedings of International Summer School on Modular Functions, Bonn Germany (1976).
D. Niebur, Construction of automorphic forms and integrals, Trans. Am. Math. Soc. 191 (1974)373 .
M.I. Knopp, Rademacher on J (τ ), Poincaré series of nonpositive weights and the Eichler cohomology, Notices Am. Math. Soc. 37 (1990) 385.
J. Manschot and G.W. Moore, A Modern Farey Tail, Commun. Num. Theor. Phys. 4 (2010) 103 [arXiv:0712.0573] [INSPIRE].
A. Selberg, Proceedings of Symposia in Pure Mathematics. Vol. 8: On the estimation of Fourier coefficients of modular forms, American Mathematical Society, Providence U.S.A. (1965).
D. Niebur, A class of nonanalytic automorphic functions, Nagoya Math. J. 52 (1973) 133.
D.A. Hejhal, Lecture Notes in Math. Vol. 1001: The Selberg trace formula for PSL(2, R), Springer, Berlin Germany (1983).
J.H. Bruinier, Borcherds products on O(2, l) and Chern classes of Heegner divisors, Springer, Berlin Germany (2002).
J. Bruinier and K. Ono, “Heegner divisors, L-functions and harmonic weak Maass forms, Ann. Math. 172 (2010) 2135 .
K. Bringmann and K. Ono, Arithmetic properties of coefficients of half-integral weightMaass-Poincaré series, Math. Ann. 337 (2007) 591 .
K. Ono, A mock theta function for the delta-function, in Combinatorial Number Theory,W. de Gruyter, Berlin Germany (2009).
R.E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998)491 .
D. Goldfeld and P. Sarnak, Sums of Kloosterman sums, Invent. Math. 71 (1983) 243 .
J. Fay, Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math. 293/294 (1977) 143 .
J.H. Bruinier and J. Funke, On two geometric theta lifts., Duke Math. J. 125 (2004) 45 .
R. Dijkgraaf, J.M. Maldacena, G.W. Moore and E.P. Verlinde, A Black hole Farey tail, hep-th/0005003 [INSPIRE].
H. Iwaniec, Graduate Studies in Mathematics. Vol. 53: Spectral methods of automorphic forms, second edition, American Mathematical Society, Providence U.S.A. and Revista Matemática Iberoamericana, Madrid Spain (2002).
E. Yoshida, On Fourier coefficients of non-holomorphic Poincaré series, Mem. Fac. Sci. Kyushu Univ. A 45 (1991) 1 .
L. Carlevaro E. Dudas and D. Israel, Gauge threshold corrections for N = 2 heterotic local models with flux, and Mock modular forms, in progress.
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Angelantonj, C., Florakis, I. & Pioline, B. One-loop BPS amplitudes as BPS-state sums. J. High Energ. Phys. 2012, 70 (2012). https://doi.org/10.1007/JHEP06(2012)070
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DOI: https://doi.org/10.1007/JHEP06(2012)070