Skip to main content
SpringerLink
Log in

Finite strings from non-chiral Mumford forms

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We show that there is an infinite class of partition functions with world-sheet metric, space-time coordinates and first order systems, that correspond to volume forms on the moduli space of Riemann surfaces and are free of singularities at the Deligne-Mumford boundary. An example is the partition function with 4 = 2(c 2 + c 3 + c 4 − c 5) space-time coordinates, a b-c system of weight 3, one of weight 4 and a β-γ system of weight 5. Such partition functions are derived from the mapping of the Mumford forms to non-factorized scalar forms on M g introduced in arXiv:1209.6049.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Matone, Modular invariant regularization of string determinants and the Serre GAGA principle, arXiv:1209.6049 [INSPIRE].

  2. M. Matone, Extending the Belavin-Knizhnik ‘wonderful formula’ by the characterization of the Jacobian, JHEP 10 (2012) 175 [arXiv:1208.5994] [INSPIRE].

    Article  ADS  Google Scholar 

  3. Y. Manin, The partition function of the Polyakov string can be expressed in terms of theta functions, Phys. Lett. B 172 (1986) 184 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  4. A. Beilinson and Y. Manin, The Mumford form and the Polyakov measure in string theory, Commun. Math. Phys. 107 (1986) 359 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  5. A.M. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981) 207 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  6. D. Mumford, Stability of projective varieties, Enseign. Math. 23 (1977) 39.

    MathSciNet  MATH  Google Scholar 

  7. N. Berkovits, Super Poincaré covariant quantization of the superstring, JHEP 04 (2000) 018 [hep-th/0001035] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  8. M. Matone, L. Mazzucato, I. Oda, D. Sorokin and M. Tonin, The superembedding origin of the Berkovits pure spinor covariant quantization of superstrings, Nucl. Phys. B 639 (2002) 182 [hep-th/0206104] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  9. M. Matone and R. Volpato, Linear relations among holomorphic quadratic differentials and induced Siegel’s metric on M g , J. Math. Phys. 52 (2011) 102305 [math/0506550] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  10. M. Matone and R. Volpato, Determinantal characterization of canonical curves and combinatorial theta identities, Math. Ann. (2012) [math.AG/0605734] [INSPIRE].

  11. M. Matone and R. Volpato, The singular locus of the theta divisor and quadrics through a canonical curve, arXiv:0710.2124 [INSPIRE].

  12. M. Matone and R. Volpato, Vector-valued modular forms from the Mumford form, Schottky-Igusa form, product of Thetanullwerte and the amazing Klein formula, to appear in Proc. Amer. Math. Soc. [arXiv:1102.0006] [INSPIRE].

  13. J. Fay, Theta functions on Riemann surfaces, Springer Lect. Notes Math. 352 (1973) 1.

    Article  MathSciNet  Google Scholar 

  14. J. Fay, Kernel functions, analytic torsion and moduli spaces, Mem. Amer. Math. Soc. 96 (1992)1.

    MathSciNet  Google Scholar 

  15. N. Shepherd-Barron, Thomae’s formulae for non-hyperelliptic curves and spinorial square roots of theta-constants on the moduli space of curves, arXiv:0802.3014 [INSPIRE].

  16. A. Belavin and V. Knizhnik, Algebraic geometry and the geometry of quantum strings, Phys. Lett. B 168 (1986) 201 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  17. V. Knizhnik, Multiloop amplitudes in the theory of quantum strings and complex geometry, Sov. Phys. Usp. 32 (1989) 945 [Usp. Fiz. Nauk 159 (1989) 401] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  18. E.P. Verlinde and H.L. Verlinde, Chiral bosonization, determinants and the string partition function, Nucl. Phys. B 288 (1987) 357 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  19. E. D’Hoker and D. Phong, On determinants of Laplacians on Riemann surfaces, Commun. Math. Phys. 104 (1986) 537 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. E. D’Hoker and D. Phong, Multiloop amplitudes for the bosonic Polyakov string, Nucl. Phys. B 269 (1986) 205 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  21. E. D’Hoker and D. Phong, The geometry of string perturbation theory, Rev. Mod. Phys. 60 (1988) 917 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  22. L. Álvarez-Gaumé, G.W. Moore and C. Vafa, Theta functions, modular invariance and strings, Commun. Math. Phys. 106 (1986) 1 [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  23. L. Álvarez-Gaumé, G.W. Moore, P.C. Nelson, C. Vafa and J. Bost, Bosonization in arbitrary genus, Phys. Lett. B 178 (1986) 41 [INSPIRE].

    ADS  Google Scholar 

  24. L. Álvarez-Gaumé, J. Bost, G.W. Moore, P.C. Nelson and C. Vafa, Bosonization on higher genus Riemann surfaces, Commun. Math. Phys. 112 (1987) 503 [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  25. A. Morozov, Explicit formulae for one, two, three and four loop string amplitudes, Phys. Lett. B 184 (1987) 171 [Sov. J. Nucl. Phys. 45 (1987) 181] [Yad. Fiz. 45 (1987) 287] [INSPIRE].

    ADS  Google Scholar 

  26. P. Di Vecchia, M. Frau, A. Lerda and S. Sciuto, A simple expression for the multiloop amplitude in the bosonic string, Phys. Lett. B 199 (1987) 49 [INSPIRE].

    ADS  Google Scholar 

  27. P. Di Vecchia, M. Frau, A. Lerda and S. Sciuto, N string vertex and loop calculation in the bosonic string, Nucl. Phys. B 298 (1988) 527 [INSPIRE].

    ADS  Google Scholar 

  28. P. Di Vecchia, K. Hornfeck, M. Frau, A. Lerda and S. Sciuto, N string, G loop vertex for the fermionic string, Phys. Lett. B 211 (1988) 301 [INSPIRE].

    ADS  Google Scholar 

  29. K. Roland, Two and three loop amplitudes in covariant loop calculus, Nucl. Phys. B 313 (1989) 432 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  30. P. Di Vecchia, L. Magnea, A. Lerda, R. Russo and R. Marotta, String techniques for the calculation of renormalization constants in field theory, Nucl. Phys. B 469 (1996) 235 [hep-th/9601143] [INSPIRE].

    Article  ADS  Google Scholar 

  31. L. Álvarez-Gaumé, C. Gomez, G.W. Moore and C. Vafa, Strings in the operator formalism, Nucl. Phys. B 303 (1988) 455 [INSPIRE].

    Article  ADS  Google Scholar 

  32. L. Álvarez-Gaumé, C. Gomez, P.C. Nelson, G. Sierra and C. Vafa, Fermionic strings in the operator formalism, Nucl. Phys. B 311 (1988) 333 [INSPIRE].

    Article  ADS  Google Scholar 

  33. L. Bonora, A. Lugo, M. Matone and J. Russo, A global operator formalism on higher genus Riemann surfaces: b-c systems, Commun. Math. Phys. 123 (1989) 329 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. L. Bonora, M. Matone, F. Toppan and K. Wu, b-c system approach to minimal models. 1. The genus zero case, Phys. Lett. B 224 (1989) 115 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  35. L. Bonora, M. Matone, F. Toppan and K. Wu, Real weight b-c systems and conformal field theory in higher genus, Nucl. Phys. B 334 (1990) 717 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  36. E. D’Hoker and D. Phong, Two loop superstrings. 1. Main formulas, Phys. Lett. B 529 (2002) 241 [hep-th/0110247] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  37. E. D’Hoker and D. Phong, Two loop superstrings. 2. The chiral measure on moduli space, Nucl. Phys. B 636 (2002) 3 [hep-th/0110283] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  38. E. D’Hoker and D. Phong, Two loop superstrings. 3. Slice independence and absence of ambiguities, Nucl. Phys. B 636 (2002) 61 [hep-th/0111016] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  39. E. D’Hoker and D. Phong, Two loop superstrings. 4. The cosmological constant and modular forms, Nucl. Phys. B 639 (2002) 129 [hep-th/0111040] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  40. E. D’Hoker and D. Phong, Two-loop superstrings. 5. Gauge slice independence of the N -point function, Nucl. Phys. B 715 (2005) 91 [hep-th/0501196] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  41. E. D’Hoker and D. Phong, Asyzygies, modular forms and the superstring measure. I, Nucl. Phys. B 710 (2005) 58 [hep-th/0411159] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  42. E. D’Hoker and D. Phong, Asyzygies, modular forms and the superstring measure. II, Nucl. Phys. B 710 (2005) 83 [hep-th/0411182] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  43. E. D’Hoker and D. Phong, Two-loop superstrings. 6. Non-renormalization theorems and the 4-point function, Nucl. Phys. B 715 (2005) 3 [hep-th/0501197] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  44. E. D’Hoker and D.H. Phong, Complex geometry and supergeometry, Current Devel. Math. 2005 (2007) 1 [hep-th/0512197] [INSPIRE].

    MathSciNet  Google Scholar 

  45. E. D’Hoker and D. Phong, Two-loop superstrings. 7. Cohomology of chiral amplitudes, Nucl. Phys. B 804 (2008) 421 [arXiv:0711.4314] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  46. E. D’Hoker, M. Gutperle and D. Phong, Two-loop superstrings and S-duality, Nucl. Phys. B 722 (2005) 81 [hep-th/0503180] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  47. M. Matone and R. Volpato, Higher genus superstring amplitudes from the geometry of moduli space, Nucl. Phys. B 732 (2006) 321 [hep-th/0506231] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  48. S.L. Cacciatori and F. Dalla Piazza, Two loop superstring amplitudes and S 6 representations, Lett. Math. Phys. 83 (2008) 127 [arXiv:0707.0646] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  49. 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] [INSPIRE].

    Article  ADS  Google Scholar 

  50. S.L. Cacciatori, F.D. Piazza and B. van Geemen, Genus four superstring measures, Lett. Math. Phys. 85 (2008) 185 [arXiv:0804.0457] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  51. S. Grushevsky, Superstring scattering amplitudes in higher genus, Commun. Math. Phys. 287 (2009) 749 [arXiv:0803.3469] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  52. R. Salvati-Manni, Remarks on superstring amplitudes in higher genus, Nucl. Phys. B 801 (2008) 163 [arXiv:0804.0512] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  53. A. Morozov, NSR superstring measures revisited, JHEP 05 (2008) 086 [arXiv:0804.3167] [INSPIRE].

    Article  ADS  Google Scholar 

  54. F. Dalla Piazza and B. van Geemen, Siegel modular forms and finite symplectic groups, Adv. Theor. Math. Phys. 13 (2009) 1771 [arXiv:0804.3769] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  55. A. Morozov, NSR measures on hyperelliptic locus and non-renormalization of 1, 2, 3-point functions, Phys. Lett. B 664 (2008) 116 [arXiv:0805.0011] [INSPIRE].

    ADS  Google Scholar 

  56. F.D. Piazza, More on superstring chiral measures, Nucl. Phys. B 844 (2011) 471 [arXiv:0809.0854] [INSPIRE].

    Article  ADS  Google Scholar 

  57. F.D. Piazza, D. Girola and S.L. Cacciatori, Classical theta constants vs. lattice theta series and super string partition functions, JHEP 11 (2010) 082 [arXiv:1009.4133] [INSPIRE].

    Article  Google Scholar 

  58. M. Oura, C. Poor, R. Salvati Manni and D. Yuen, Modular forms of weight 8 for Γ g (1, 2), Math. Ann. 346 (2010) 477 [arXiv:0811.2259].

    Article  MathSciNet  MATH  Google Scholar 

  59. P. Dunin-Barkowski, A. Morozov and A. Sleptsov, Lattice theta constants versus Riemann theta constants and NSR superstring measures, JHEP 10 (2009) 072 [arXiv:0908.2113] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  60. S. Grushevsky and R. Salvati Manni, The vanishing of two-point functions for three-loop superstring scattering amplitudes, Commun. Math. Phys. 294 (2010) 343 [arXiv:0806.0354] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  61. M. Matone and R. Volpato, Superstring measure and non-renormalization of the three-point amplitude, Nucl. Phys. B 806 (2009) 735 [arXiv:0806.4370] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  62. S. Grushevsky and R.S. Manni, The superstring cosmological constant and the Schottky form in genus 5, Amer. J. Math. 133 (2011) 1007 [arXiv:0809.1391] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  63. C. Poor and D.S. Yuen, Binary forms and the hyperelliptic superstring ansatz, Math. Ann. 352 (2011) 941 [arXiv:0911.4545] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  64. M. Matone and R. Volpato, Getting superstring amplitudes by degenerating Riemann surfaces, Nucl. Phys. B 839 (2010) 21 [arXiv:1003.3452] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  65. G. Codogni and N.I. Shepherd-Barron, The absence of stable Schottky forms, arXiv:1112.6137.

  66. P. Dunin-Barkowski, A. Sleptsov and A. Stern, NSR superstring measures in genus 5, arXiv:1208.2324 [INSPIRE].

  67. E. Witten, Notes on supermanifolds and integration, arXiv:1209.2199 [INSPIRE].

  68. E. Witten, Notes on super Riemann surfaces and their moduli, arXiv:1209.2459 [INSPIRE].

  69. E. Witten, Superstring perturbation theory revisited, arXiv:1209.5461 [INSPIRE].

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Fisica “G. Galilei” and Istituto Nazionale di Fisica Nucleare, Università di Padova, Via Marzolo, 8, 35131, Padova, Italy

    Marco Matone

Authors
  1. Marco Matone
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco Matone.

Additional information

ArXiv ePrint: 1209.6351

Rights and permissions

Reprints and permissions

About this article

Cite this article

Matone, M. Finite strings from non-chiral Mumford forms. J. High Energ. Phys. 2012, 50 (2012). https://doi.org/10.1007/JHEP11(2012)050

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP11(2012)050

Keywords

Search

Navigation