On Determinants of Laplacians on Compact Riemann Surfaces Equipped with Pullbacks of Conical Metrics by Meromorphic Functions

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Abstract

Let \({\mathsf {m}}\) be any conical (or smooth) metric of finite volume on the Riemann sphere \({\mathbb {C}}P^1\). On a compact Riemann surface X of genus g consider a meromorphic function \(f: X\rightarrow {{\mathbb {C}}}P^1\) such that all poles and critical points of f are simple and no critical value of f coincides with a conical singularity of \({\mathsf {m}}\) or \(\{\infty \}\). The pullback \(f^*{\mathsf {m}}\) of \({\mathsf {m}}\) under f has conical singularities of angles \(4\pi \) at the critical points of f and other conical singularities that are the preimages of those of \({\mathsf {m}}\). We study the \(\zeta \)-regularized determinant \({\text {Det}}^\prime \Delta _F\) of the (Friedrichs extension of) Laplace–Beltrami operator on \((X,f^*{\mathsf {m}})\) as a functional on the moduli space of pairs (Xf) and obtain an explicit formula for \({\text {Det}}^\prime \Delta _F\).

Keywords

Conical metric Determinants of Laplacians Moduli space 

Mathematics Subject Classification

58J52 

Notes

Acknowledgements

It is a pleasure to thank Alexey Kokotov for helpful discussions and important remarks.

References

  1. 1.
    Aurell, E., Salomonson, P.: On functional determinants of Laplacians in polygons and simplicial complexes. Comm. Math. Phys. 165(2), 233–259 (1994)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brüning, J., Seeley, R.: Regular singular asymptotics. Adv. Math. 58, 133–148 (1985)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brüning, J., Seeley, R.: The resolvent expansion for second order regular singular operators. JFA 73, 369–429 (1987)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    D’Hoker, E., Phong, D.H.: On determinants of Laplacians on Riemann surfaces. Commun. Math. Phys. 104, 537–545 (1986)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Efrat, I.: Determinants of Laplacians on surfaces of finite volume. Comm. Math. Phys. 119(3), 443–451 (1988). [Erratum: Comm. Math. Phys. 138 (1991), no. 3, 607]MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Hillairet, L., Kokotov, A.: Krein formula and \(S\)-matrix for Euclidean surfaces with conical singularities. J. Geom. Anal. 23(N3), 1498–1529 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hillairet, L., Kokotov, A.: Isospectrality, comparison formulas for determinants of Laplacian and flat metrics with non-trivial holonomy. Proc. Am. Math. Soc. 145(9), 3915–3928 (2017)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hillairet, L., Kalvin, V., Kokotov, A.: Spectral determinants on Mandelstam diagrams. Commun. Math. Phys. 343(2), 563–600 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hillairet, L., Kalvin, V., Kokotov, A.: Moduli spaces of meromorphic functions and determinant of Laplacian, arXiv:1410.3106 [math.SP], Transactions of the American Mathematical Society.  https://doi.org/10.1090/tran/7430
  10. 10.
    Kalvin, V., Kokotov, A.: Metrics of curvature \(1\) with conical singularities, Hurwitz spaces, and determinants of Laplacians. In: International Mathematics Research Notices  https://doi.org/10.1093/imrn/rnx224 (2017)
  11. 11.
    Kalvin, V., Kokotov, A.: Determinant of Laplacian for tori of constant positive curvature with one conical point. arXiv:1712.04588 (2017)
  12. 12.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)CrossRefMATHGoogle Scholar
  13. 13.
    Khuri, H.H.: Determinants of Laplacians on the moduli space of Riemannian surfaces. ProQuest LLC, Ann Arbor, Thesis (Ph.D.) Stanford University (1990)Google Scholar
  14. 14.
    Klevtsov, S.: Lowest Landau level on a cone and zeta determinants. J. Phys. A 50, 234003 (2017)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kokotov, A., Korotkin, D.: Tau-functions on spaces of abelian differentials and higher genus generalizations of Ray–Singer formula. J. Differ. Geom. 82(1), 35–100 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kokotov, A., Korotkin, D.: Tau-functions on Hurwitz spaces. Math. Phys. Anal. Geom. 7(1), 47–96 (2004)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kokotov, A., Strachan Ian, A.B.: On the isomonodromic tau-function for the Hurwitz spaces of branched coverings of genus zero and one. Math. Res. Lett. 12(5–6), 857–875 (2005)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kokotov, A.: Polyhedral surfaces and determinant of Laplacian. Proc. Am. Math. Soc. 141(2), 725–735 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kokotov, A., Korotkin, D.: Isomonodromic tau-function of Hurwitz Frobenius manifolds and its appplications. Int. Math. Res. Not. N18746, 1–34 (2006)MATHGoogle Scholar
  20. 20.
    Kokotov, A., Korotkin, D., Zograf, P.: Isomonodtromic tau function on the space of admissible covers. Adv. Math. 227(1), 586–600 (2011)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kraus, D., Roth, O., Sugawa, T.: Metrics with conical singularities on the sphere and sharp extensions of the theorems of Landau and Schottky. Math. Z. 267(3–4), 851–868 (2011)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Osgood, B., Phillips, R., Sarnak, P.: Extremals of determinants of Laplacians. J. Funct. Anal. 80, 148–211 (1988)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Sarnak, P.: Determinants of Laplacians. Commun. Math. Phys. 110, 113–120 (1987)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Shubin, M.: Pseudodifferential Operators and Spectral Theory. Springer, Berlin (2001)CrossRefMATHGoogle Scholar
  25. 25.
    Seeley, R.: The resolvent of an elliptic boundary problem. Am. J. Math. 91, 889–920 (1969)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Spreafico, M., Zerbini, S.: Spectral analysis and zeta determinant on the deformed spheres. Commun. Math. Phys. 273, 677–704 (2007). arXiv:math-ph/0610046 MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Troyanov, M.: Coordonnées polaires sur les surfaces Riemanniennes singulières. Ann. Inst. Fourier Grenoble 40(4), 913–937 (1990)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsConcordia UniversityMontrealCanada

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