Skip to main content
SpringerLink
Log in

Selberg super-trace formula for super Riemann surfaces II: Elliptic and parabolic conjugacy classes, and Selberg super-zeta functions

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Further contributions developing a super analogue of the classical Selberg trace formula, the Selberg super-trace formula, are presented. This paper deals with the calculation of contributions arising from elliptic and parabolic conjugacy classes to the Selberg super-trace formula for super Riemann surfaces. Analytic properties and the functional equation for the corresponding Selberg super-zeta functionR 0,R 1 andZ S , respectively, are derived and discussed. In particular, the elliptic contributions to a super Fuchsian group only alter the multiplicities of the “trivial” zeros and poles of the Selberg super-zeta functionR 0,R 1 andZ S , respectively, already due to the hyperbolic conjugacy classes. The parabolic conjugacy classes introduce new features in the analytical structure.

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. Aoki, K.: Heat Kernels and Super Determinants of Laplace Operators on Super Riemann Surfaces. Commun. Math. Phys.117, 405–429 (1988)

    Google Scholar 

  2. Arthur, J.: The Selberg Trace Formula for Groups of F-Rank One. Ann. Math.100, 326–385 (1974)

    Google Scholar 

  3. Aurich, R., Bogomolny, E.B. Steiner F.: Periodic Orbits on the Regular Hyperbolic Octagon. PhysicaD48, 91–101 (1991); Aurich, R., Sieber, M., Steiner, F.: Quantum Chaos of the Hadamard-Gutzwiller Model. Phys. Rev. Lett.61, 483–487 (1988)

    Google Scholar 

  4. Aurich, A., Steiner, F.: On the Periodic Orbits for a Strongly Chaotic System. PhysicaD32, 451–460 (1988); Periodic-Orbit Sum Rules for the Hadamard-Gutzwiller Model. PhysicaD39, 169–193 (1989); Energy-Level Statistics of the Hadamard-Gutzwiller Ensemble. PhysicaD43, 155–180 (1990); Exact Theory for the Quantum Eigenstates of the Hadamard-Gutzwiller Model. PhysicaD48, 445–470 (1991); From Classical Periodic Orbits to the Quantization of Chaos. Proc. Roy. Soc. LondonA437, 693–719 (1992)

    Google Scholar 

  5. Balazs, N.L., Voros, A.: Chaos on the Pseudosphere. Phys. Rep.143, 109–240 (1986)

    Google Scholar 

  6. Baranov, A.M., Manin, Yu.I., Frolov, I.V., Schwarz, A.S.: The Multiloop Contribution in the Fermionic String. Sov. J. Nucl. Phys.43, 670–671 (1986)

    Google Scholar 

  7. Baranov, A.M., Manin, Yu.I., Frolov, I.V., Schwarz, A.S.: A Superanalog of the Selberg Trace Formula and Multiloop Contributions for Fermionic Strings. Commun. Math. Phys.111, 373–392 (1987)

    Google Scholar 

  8. Baranov, A.M., Frolov, I.V., Shvarts, A.S.: Geometry of Two-Dimensional Superconformal Field Theories. Theor. Math. Phys.70, 64–72 (1987)

    Google Scholar 

  9. Baranov, A.M., Schwarz, A.S.: Multiloop Contribution to String Theory. JETP Lett.42, 419–421 (1985); On the Multiloop Contributions to the String Theory. Int. J. Mod. Phys.A2, 1773–1796 (1987)

    Google Scholar 

  10. Batchelor, M.: The Structure of Supermanifolds. Trans. Am. Math. Soc.253, 329–338 (1979); Two Approaches to Supermanifolds. Trans. Am. Math. Soc.258, 257–270 (1980)

    Google Scholar 

  11. Batchelor, M., Bryant, P.: Graded Riemann Surfaces. Commun. Math. Phys.114, 243–255 (1988)

    Google Scholar 

  12. Bolte, J., Steiner, F.: Determinants of Laplace-like Operators on Riemann Surfaces. Commun. Math. Phys.130, 581–597 (1990)

    Google Scholar 

  13. Bolte, J., Steiner, F.: The Selberg Trace Formula for Bordered Riemann Surfaces. DESY preprint, DESY 90-082, July 1990, 1–14

  14. De Witt, B.: Supermanifolds. Cambridge: Cambridge University Press 1984

    Google Scholar 

  15. D'Hoker, E., Phong, D.H.: Loop Amplitudes for the Bosonic Polyakov String. Nucl. Phys.B269, 205–234 (1986)

    Google Scholar 

  16. D'Hoker, E., Phong, D.H.: On Determinants of Laplacians on Riemann Surfaces. Commun. Math. Phys.104, 537–545 (1986)

    Google Scholar 

  17. D'Hoker, E., Phong, D.H.: Loop Amplitudes for the Fermionic String. Nucl. Phys.B278, 225–241 (1986)

    Google Scholar 

  18. D'Hoker, E., Phong, D.H.: The Geometry of String Perturbation Theory. Rev. Mod. Phys.60, 917–1065 (1988)

    Google Scholar 

  19. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G. (eds.): Tables of Integral Transforms, Vol. I&II. New York: McGraw Hill 1954

    Google Scholar 

  20. Gliozzi, F., Scherk, J., Olive, D.: Supergravity and the Spinor Dual Model. Phys. Lett.B65, 282–286 (1976); Supersymmetry, Supergravity Theories and the Spinor Dual Model. Nucl. Phys.B122, 253–290 (1977)

    Google Scholar 

  21. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. New York: Academic Press 1980

    Google Scholar 

  22. Green, M.B.: Supersymmetrical Dual String Theories and Their Field Theory Limits. Surveys in High Energy Physics3, 127–160 (1983)

    Google Scholar 

  23. Green, M.B., Schwarz, J.H.: Supersymmetrical Dual String Theory, I–III. Nucl. Phys.B181, 502–530 (1982),B198, 252–268, 441–460 (1982)

    Google Scholar 

  24. Green, M.B., Schwarz, J.H.: Anomaly Cancellations in SupersymmetricD=10 Gauge Theory and Superstring Theory. Phys. Lett.B149, 117–122 (1984); Infinity Cancellations in SO(32) Superstring Theory. Phys. Lett.B151, 21–25 (1985); The Hexagon Gauge Anomaly in Type I Superstring Theory. Nucl. Phys.B255, 93–114 (1985)

    Google Scholar 

  25. Green, M.B., Schwarz, J.H., Witten, E.: Superstring Theory I&II. Cambridge: Cambridge University Press 1988

    Google Scholar 

  26. Grosche, C.: The Path Integral on the Poincaré Upper Half-Plane With a Magnetic Field and for the Morse Potential. Ann. Phys. (N.Y.)187, 110–134 (1988); The Path Integral on the Poincaré Disc, the Poincaré Upper Half-Plane and on the Hyperbolic Strip. Fortschr. Phys.38, 531–569 (1990); Path Integration on the Hyperbolic Plane With a Magnetic Field. Ann. Phys. (N.Y.)201, 258–284 (1990); Path Integration on Hyperbolic Spaces. J. Phys. A.: Math. Gen.25, 4211–4244 (1992)

    Google Scholar 

  27. Grosche, C.: Selberg Supertrace Formula for Super Riemann Surfaces, Analytic Properties of the Selberg Super-zeta-Functions and Multiloop Contributions to the Fermionic String. DESY Preprint DESY 89-010, February 1989, 1–102, and Commun Math. Phys.133, 433–485 (1990)

    Google Scholar 

  28. Grosche, C., Steiner, F.: The Path Integral on the Poincaré Upper Half Plane and for Liouville Quantum Mechanics. Phys. Lett.A123, 319–328 (1987)

    Google Scholar 

  29. Gross, D.J., Periwal, V.: String Perturbation Theory Diverges. Phys. Rev. Lett.60, 2105–2108 (1988)

    Google Scholar 

  30. Gutzwiller, M.C.: Phase-Integral Approximation in Momentum Space and the Bound States of an Atom. J. Math. Phys.8, 1979–2000 (1967); Phase-Integral Approximation in Momentum Space and the Bound States of an Atom. II. J. Math. Phys.10, 1004–1020 (1969); Energy Spectrum According to Classical Mechanics. J. Math. Phys.11, 1791–1806 (1970); Periodic Orbits and Classical Quantization Conditions. J. Math. Phys.12, 343–358 (1971).

    Google Scholar 

  31. Gutzwiller, M.C.: The Quantization of a Classically Ergodic System. PhysicaD5, 183–207 (1982); The Geometry of Quantum Chaos. Physica ScriptaT9, 184–192 (1985); Physics and Selberg's Trace Formula. Contemp. Math.53, 215–251 (1986); Mechanics on a Surface of Constant Curvature. In: Number Theory, Lect. Notes in Math., Vol.1240, pp. 230–258. Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  32. Hardy, G.H., Wright E.M.: The Theory of Numbers. Oxford: Oxford University Press 1960

    Google Scholar 

  33. Hejhal, D.A.: The Selberg Trace Formula and the Riemann Zeta Function. Duke. Math. J.43, 441–482 (1976)

    Google Scholar 

  34. Hejhal, D.A.: The Selberg Trace Formula for PSL(2, R), I&II. Lect. Notes in Math. Vols.548 & 1001. Berlin, Heidelberg, New York: Springer 1976, 1981

    Google Scholar 

  35. Howe, P.S.: Superspace and the Spinning String. Phys. Lett.B70, 453–456 (1977); Super Weyl Transformations in Two Dimensions. J. Phys. A: Math. Gen.12, 393–402 (1979)

    Google Scholar 

  36. Kubota, T.: Elementary Theory of Eisenstein Series, Tokio. Kodansha 1973

    Google Scholar 

  37. Littlejohn, R.G.: Semiclassical Structure of Trace Formulas. J. Math. Phys.31, 2952–2977 (1990)

    Google Scholar 

  38. McKean, H.P.: Selberg's Trace Formula as Applied to a Compact Riemann Surface. Commun. Pure Appl. Math.25, 225–246 (1972)

    Google Scholar 

  39. Magnus, W., Oberhettinger, F., Soni, R.: Formulas and Theorems for the Special Functions of Mathematical Physics. Berlin, Heidelberg, New York: Springer 1966

    Google Scholar 

  40. Manin, Yu.I.: The Partition Function of the Polyakov String Can Be Expressed in Terms of Theta-Functions. Phys. Lett.B172, 184–185 (1986)

    Google Scholar 

  41. Matsumoto, S., Yasui, Y.: Chaos on the Super Riemann Surface. Prog. Theor. Phys.79, 1002–1027 (1988)

    Google Scholar 

  42. Matsumoto, S., Uehara, S., Yasui, Y.: Hadamard Model on the Super Riemann Surface. Phys. Lett.A134, 81–86 (1988)

    Google Scholar 

  43. Matsumoto, S., Uehara, S., Yasui, Y.: A Superparticle on the Super Riemann Surface. J. Math. Phys.31, 476–501 (1990)

    Google Scholar 

  44. Matthies, C., Steiner, F.: Selberg's Zeta Function and the Quantization of Chaos. Phys. Rev.A44, R7877-R7880 (1991)

    Google Scholar 

  45. Moore, G., Nelson, P., Polchinski, J.: Strings and Supermoduli. Phys. LettB169, 47–53 (1986)

    Google Scholar 

  46. Neveu, A., Schwarz, J.H.: Quark Model of Dual Pions. Phys. Rev.D4, 1109–1111 (1971)

    Google Scholar 

  47. Ninnemann, H.: Holomorphe and Harmonische Formen auf Super-Riemannschen Flächen und ihre Anwendung auf den Fermionischen String. Diploma Thesis, Hamburg University 1989; Deformations of Super Riemann Surfaces. DESY preprint DESY 90-120, 1–23

  48. Oshima, K.: Completeness Relations for Maass Laplacians and Heat Kernels on the Super Poincaré Upper Half-Plane. J. Math. Phys.31, 3060–3063 (1990)

    Google Scholar 

  49. Polyakov, A.M.: Quantum Geometry of Bosonic Strings. Phys. Lett.B103, 207–210 (1981)

    Google Scholar 

  50. Polyakov, A.M.: Quantum Geometry of Fermionic Strings. Phys. Lett.B103, 211–213 (1981)

    Google Scholar 

  51. Rabin, J.M., Crane, L.: Global Properties of Supermanifolds. Commun. Math. Phys.100, 141–160 (1985); How Different are the Supermanifolds of Rogers and DeWitt? Commun. Math. Phys.102, 123–137 (1985); SuperRiemann Surfaces: Uniformization and Teichmüller Theory. Commun. Math. Phys.113, 601–623 (1988)

    Google Scholar 

  52. Ramond, P.: Dual Theory for Free Fermions. Phys. Rev.D3, 2415–2418 (1971)

    Google Scholar 

  53. Rogers, A.: A Global Theory of Supermanifolds. J. Math. Phys.21, 1352–1364 (1980); On the Existence of Gloal Integral Forms on Supermanifolds. J. Math. Phys.26, 2749–2753 (1985); Graded Manifolds, Supermanifolds and Infinite-Dimensional Grassmann Algebras. Commun. Math. Phys.105, 375–384 (1986)

    Google Scholar 

  54. Schwarz, J.H.: Superstring Theory. Phys. Rep.89, 223–322 (1982)

    Google Scholar 

  55. Selberg, A.: Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces With Application to Dirichlet Series. J. Indian Math. Soc.20, 47–87 (1956)

    Google Scholar 

  56. Sieber, M.: The Hyperbola Billiard: A Model for the Semiclassical Quantization of Chaotic Systems. DESY preprint, DESY 91-030, April 1991, 1–101; Sieber, M., Steiner, F.: Generalized Periodic-Orbit Sum Rules for Strongly Chaotic Systems; Phys. LettA144, 159–163 (1990); Quantization of Chaos; Phys. Rev. Lett.67, 1941–1944 (1991)

    Google Scholar 

  57. Steiner, F.: On Selberg's Zeta Function for Compact Riemann Surfaces Phys. Lett.B188, 447–454 (1987); Quantum Chaos and Geometry. In: Mitter, H., Pittner, L. (eds.) Recent Developments in Mathematical Physics.26. Internationale Universitäswochen, Schladming 1987, pp. 305–312. Berlin, Heidelberg, New York: Springer 1987

    Google Scholar 

  58. Subia, N.: Formule de Selberg et Formes D'Espace Hyperboliques Compactes. Analyse Harmonique sur les Groupes de Lie. Lect. Notes in Math., Vol.497, pp. 674–700. Berlin, Heidelberg, New York: Springer 1975

    Google Scholar 

  59. Takhtajan, L.A., Zograf, P.G.: A Local Index Theorem for Families of\(\bar \partial \)-Operators on Punctured Riemann Surfaces and a New Kähler Metric on Their Moduli Spaces. Commun. Math. Phys.137, 399–426 (1991)

    Google Scholar 

  60. Uehara, S., Yasui, Y.: A Superparticle on the “Super” Poincaré Upper Half Plane. Phys. Lett.B202, 530–534 (1988; Super-Selberg's Trace Formula from the Chaotic Model. J. Math. Phys.29, 2486–2490 (1988)

    Google Scholar 

  61. Venkov, A.B.: Expansion in Automorphic Eigenfunctions of the Laplace Operator and the Selberg Trace Formula in the SpaceSO 0 (n, 1)/SO(n). Soviet Math. Dokl.12, 1363–1366 (1971)

    Google Scholar 

  62. Venkov, A.B.: Expansion in Automorphic Eigenfunctions of the Laplace-Beltrami Operator in Classical Symmetric Spaces of Rank One, and the Selberg Trace Formula. Proc. Math. Inst. Steklov125, 1–48 (1973)

    Google Scholar 

  63. Venkov, A.B.: Selberg's Trace Formula and Non-Euclidean Vibrations of an Infinite Membrane. Sov. Math. Dokl.19, 708–712 (1979)

    Google Scholar 

  64. Venkov, A.B.: Spectral Theory of Automorphic Functions. Proc. Math. Inst. Steklov153, 1–163 (1981)

    Google Scholar 

  65. Venkov, A.B.: On the Selberg Trace Formula for the Automorphic Schroedinger Operator. Institut des Hautes Etudes Scientific preprint, IHES/M/90/98, December 1990, 1–18

  66. Weil, A.: Sur les ‘Formules Explicites’ de la Théorie des Nombres Premiere. Comm. Sém. Math. Lund (Medd. Lunds Univ. Mat. Sem., Suppl. 252–265) 1952

  67. Bolte, J., Grosche, C.: Selberg Trace Formula for Bordered Riemann Surfaces: Hyperbolic, Elliptic and parabolic Conjugacy Classes, and Determinants of Maass-Laplacians. DESY-preprint, DESY 92-118, 1–27, August 1992

Download references

Author information

Author notes

  1. C. Grosche

    Present address: International School for Advanced Studies, SISSA, Via Beirut 4, 34014, Trieste, Miramare, Italy

Authors and Affiliations

  1. The Blackett Laboratory, Imperial College of Science, Technology and Medicine, Prince Consort Road, SW7 2BZ, London, UK

    C. Grosche

Authors
  1. C. Grosche
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by A. Jaffe

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grosche, C. Selberg super-trace formula for super Riemann surfaces II: Elliptic and parabolic conjugacy classes, and Selberg super-zeta functions. Commun.Math. Phys. 151, 1–37 (1993). https://doi.org/10.1007/BF02096746

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02096746

Keywords

Search

Navigation