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Journal of High Energy Physics

, 2010:82 | Cite as

Classical theta constants vs. lattice theta series, and super string partition functions

  • Francesco Dalla Piazza
  • Davide Girola
  • Sergio L. Cacciatori
Article

Abstract

Recently, various possible expressions for the vacuum-to-vacuum superstring amplitudes has been proposed at genus g = 3, 4, 5. To compare the different proposals, here we will present a careful analysis of the comparison between the two main technical tools adopted to realize the proposals: the classical theta constants and the lattice theta series. We compute the relevant Fourier coefficients in order to relate the two spaces. We will prove the equivalence up to genus 4. In genus five we will show that the solutions are equivalent modulo the Schottky form and coincide if we impose the vanishing of the cosmological constant.

Keywords

Superstrings and Heterotic Strings Differential and Algebraic Geometry 

References

  1. [1]
    A.N. Andrianov and V.G. Zhuravlev, Modular Forms and Hecke Operators, American Mathematical Society, (1995).Google Scholar
  2. [2]
    J.J. Atick, G.W. Moore and A. Sen, Some global issues in string perturbation theory, Nucl. Phys. B 308 (1988) 1 [SPIRES].MathSciNetADSGoogle Scholar
  3. [3]
    J.J. Atick, J.M. Rabin and A. Sen, An ambiguity in fermionic string perturbation theory, Nucl. Phys. B 299 (1988) 279 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  4. [4]
    S.L. Cacciatori and F. Dalla Piazza, Two loop superstring amplitudes and S 6 representations, Lett. Math. Phys. 83 (2008) 127 [arXiv:0707.0646] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  5. [5]
    S.L. Cacciatori and F. Dalla Piazza, Modular forms and superstrings amplitudes, to appear in Superstring Theory in the 21st Century. Horizons in W orld Physics Volume 270, G.B. Charney ed., Nova Publishers.Google Scholar
  6. [6]
    S.L. Cacciatori, F. Dalla Piazza and B. van Geemen, Modular Forms and Three Loop Superstring Amplitudes, Nucl. Phys. B 800 (2008) 565 [arXiv:0801.2543] [SPIRES].CrossRefADSGoogle Scholar
  7. [7]
    S.L. Cacciatori, F.D. Piazza and B. van Geemen, Genus four superstring measures, Lett. Math. Phys. 85 (2008) 185 [arXiv:0804.0457] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  8. [8]
    J.H. Conway and N.J.A. Sloane, Sphere packings, lattice and groups, Springer-Verlag, (1998).Google Scholar
  9. [9]
    F. Dalla Piazza, More on superstring chiral measures, accepted in Nucl. Phys. B arXiv:0809.0854 [SPIRES].
  10. [10]
    F. Dalla Piazza and B. van Geemen, Siegel modular forms and finite symplectic groups, Adv. Theor. Math. Phys. 13 (2009) no. 6 (in press).Google Scholar
  11. [11]
    S.L. Cacciatori, F.D. Piazza and B. van Geemen, Genus four superstring measures, Lett. Math. Phys. 85 (2008) 185 [arXiv:0804.0457] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  12. [12]
    E. D’Hoker and D.H. Phong, Two-Loop Superstrings I, Main Formulas, Phys. Lett. B 529 (2002) 241 [hep-th/0110247] [SPIRES].MathSciNetADSGoogle Scholar
  13. [13]
    E. D’Hoker and D.H. Phong, Two-Loop Superstrings II, The Chiral Measure on Moduli Space, Nucl. Phys. B 636 (2002) 3 [hep-th/0110283] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  14. [14]
    E. D’Hoker and D.H. Phong, Two-Loop Superstrings III, Slice Independence and Absence of Ambiguities, Nucl. Phys. B 636 (2002) 61 [hep-th/0111016] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  15. [15]
    E. D’Hoker and D.H. Phong, Two-Loop Superstrings IV, The Cosmological Constant and Modular Forms, Nucl. Phys. B 639 (2002) 129 [hep-th/0111040] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  16. [16]
    E. D’Hoker and D.H. Phong, Asyzygies, modular forms and the superstring measure. I, Nucl. Phys. B 710 (2005) 58 [hep-th/0411159] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  17. [17]
    E. D’Hoker and D.H. Phong, Asyzygies, modular forms and the superstring measure. II, Nucl. Phys. B 710 (2005) 83 [hep-th/0411182] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  18. [18]
    P. Dunin-Barkowski, A. Morozov and A. Sleptsov, Lattice Theta Constants vs Riemann Theta Constants and NSR Superstring Measures, JHEP 10 (2009) 072 [arXiv:0908.2113] [SPIRES]. CrossRefMathSciNetADSGoogle Scholar
  19. [19]
    E. Freitag, Singular modular forms and theta relations, Lecture Notes in Mathematics, 1487, Springer-Verlag, Berlin Germany (1991).zbMATHGoogle Scholar
  20. [20]
    S. Grushevsky, Superstring scattering amplitudes in higher genus, Commun. Math. Phys. 287 (2009) 749 [arXiv:0803.3469] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  21. [21]
    S. Grushevsky and R.S. Manni, The superstring cosmological constant and the Schottky form in genus 5, arXiv:0809.1391 [SPIRES].
  22. [22]
    M. Matone and R. Volpato, Getting superstring amplitudes by degenerating Riemann surfaces, Nucl. Phys. B 839 (2010) 21 [arXiv:1003.3452] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  23. [23]
    G.W. Moore and A. Morozov, Some remarks on two loop superstring calculations, Nucl. Phys. B 306 (1988) 387 [SPIRES]. CrossRefMathSciNetADSGoogle Scholar
  24. [24]
    A. Morozov and A. Perelomov, A Note on Multiloop Calculations for Superstrings in the NSR Formalism, Int. J. Mod. Phys. A 4 (1989) 1773 [SPIRES].MathSciNetADSGoogle Scholar
  25. [25]
    M. Oura, C. Poor, R. Salvati Manni and D.S. Yuen, Modular form of weight 8 for Γg (1, 2), Math. Ann. 346 (2010) 477.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    R. Salvati-Manni, Remarks on Superstring amplitudes in higher genus, Nucl. Phys. B 801 (2008) 163 [arXiv:0804.0512] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  27. [27]
    R. Salvati Manni, Thetanullwerte and stable modular forms, Amer. J. Math. 111 (1989) 435.zbMATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    R. Salvati Manni, Thetanullwerte and stable modular forms II, Amer. J. Math. 113 (1991) 733.zbMATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    R. Salvati Manni, Thetanullwerte and stable modular forms for Hecke groups, Math. Z. 216 (1994) 529.zbMATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    E.P. Verlinde and H.L. Verlinde, Multiloop Calculations in Covariant Superstring Theory, Phys. Lett. B 192 (1987) 95 [SPIRES].MathSciNetADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  • Francesco Dalla Piazza
    • 1
    • 2
  • Davide Girola
    • 1
  • Sergio L. Cacciatori
    • 1
    • 2
  1. 1.Dipartimento di Fisica e MatematicaUniversità degli Studi dell’InsubriaComoItaly
  2. 2.INFNMilanoItaly

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