Communications in Mathematical Physics

, Volume 174, Issue 1, pp 29–42 | Cite as

Modular invariance and characteristic numbers

  • Kefeng Liu


We prove that a general miraculous cancellation formula, the divisibility of certain characteristic numbers, and some other topological results related to the generalized Rochlin invariant, the η-invariant and the holonomies of certain determinant line bundles, are consequences of the modular invariance of elliptic operators on loop space.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Line Bundle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A] Atiyah, M.F.: Collected Works. Oxford: Oxford Science Publications, 1989Google Scholar
  2. [Al] Atiyah, M.F.: Circular Symmetry and Stationary-Phase Approximation. In: [A], Vol.4, pp. 664–685Google Scholar
  3. [AB] Atiyah, M.F., Bott, R.: The Lefschetz Fixed Point Theorems for Elliptic Complexes1, II. In: [A] Vol.3, pp. 91–170Google Scholar
  4. [APS] Atiyah, M.F., Patodi, V.K., Singer, I.: Spectral Asymmetry and Riemannian Geometry 1, Math. Proc. Camb. Phil. Soc.71, 43–69 (1975)Google Scholar
  5. [AS] Atiyah, M.F., Singer, I.: The Index of Elliptic Operators III. In: [A] Vol.3, pp. 239–300, (or in [A], Vol.3)Google Scholar
  6. [AW] Alvarez-Gaumé, L., Witten, E.: Gravitational Anomalies. Nucl. PhysicsB234, 269–330 (1983)Google Scholar
  7. [BF] Bismut, J.-M., Freed, D.: The Analysis of Elliptic Families I. Metrics and Connections on Determinant Line Bundles. Commun. Math. Phys.106, 159–176, (1986), II. Dirac Families, Eta Invariants and the Holonomy Theorem. Commun. Math. Phys.107, 103–163 (1986)Google Scholar
  8. [BH] Borel, A., Hirzebruch, F.: Characteristic Classes and Homogeneous Spaces II, Am. J. Math.81, 315–382 (1959)Google Scholar
  9. [BT] Bott, R., Taubes, C.: On the Rigidity Theorems of Witten. J. AMS.2, 137–186 (1989)Google Scholar
  10. [Br] Brylinski, J-L.: Representations of Loop Groups, Dirac Operators on Loop Spaces and Modular Forms. Topology29, 461–480 (1990)Google Scholar
  11. [Ch] Chandrasekharan, K.: Elliptic Functions. Berlin, Heidelberg, New York: Springer, 1985Google Scholar
  12. [De] Dessai, A.:SU(2)-actions and the Witten Genus. Preprint 1993Google Scholar
  13. [F] Finashin, S.M.: A Pin-cobordism Invariant and a Generalization of the Rochlin Signature Congruence. Leningrad Math. J.2, 917–924 (1991)Google Scholar
  14. [GS] Green, M.G., Schwarz, J.H.: Anomaly Cancellations in SupersymmetryD=10 Gauge Theory and Superstring Theory, Phys. Lett.148B, 117–122 (1982)Google Scholar
  15. [GSW] Green, M.G., Schwarz, J.H., Witten, E.: String Theory II. Cambridge: Cambridge University Press, 1987Google Scholar
  16. [H1] Hirzebruch, F.: Mannigfaltigkeiten und Modulformen. Jber. d. Dt. Math.-Verein. Jubiläumstagung (1990) pp. 20–38Google Scholar
  17. [H2] Hirzebruch, F.: Topological Methods in Algebraic Geometry. Berlin, Heidelberg, New York: Springer, 1966Google Scholar
  18. [HBJ] Hirzebruch, F., Berger, T., Jung, R.: Manifolds and Modular Forms, Vieweg, 1992Google Scholar
  19. [L] Landweber, P.S.: Elliptic Curves and Modular Forms in Algebraic Topology. Lecture Notes in Math.1326, Berlin, Heidelberg, New York: Springer-Verlag, 1988Google Scholar
  20. [Liu] Liu, K.: On Mod 2 and Higher Elliptic Genera. Commun. Math. Phys.149, 71–97 (1992)Google Scholar
  21. [Liu1] Liu, K.: On Modular Invariance and Rigidity Theorems. To appear in J. Diff. Geom. (1994)Google Scholar
  22. [Liu2] Liu, K.: OnSL 2(Z) and Topology. Math. Res. Lettera1, 53–64 (1994)Google Scholar
  23. [LZ] Liu, K., Zhang, W.: Elliptic Genus and η-invariant. Int. Math. Research Notices8, 319–328 (1994)Google Scholar
  24. [LM] Lee, R., Miller, E.: Some Invariants of Spin Manifolds. Topology and Its Applications25, 301–311 (1987)Google Scholar
  25. [LMW] Lee, R., Miller, E., Weintraub, S.: Rochlin Invariants, Theta-functions and Holonomies of Some Determinant Line Bundles. J. Reine Angew, Math.392, 187–218 (1988)Google Scholar
  26. [O] Ochanine, S.: Signature Modulo 16, Invariants de Kervaire Generalisés et Nombres Caracteristiques dans laK-theorie Reelle. Memoire de la Soc. Math. de France109, 1–141 (1981)Google Scholar
  27. [T] Taubes, C.:S 1-Actions and Elliptic Genera. Commun. Math. Phys.122, 455–520 (1989)Google Scholar
  28. [We] Weil, A.: Elliptic Functions According to Eisenstein and Kronecker. Berlin, Heidelberg, New York: Springer, 1976Google Scholar
  29. [W] Witten, E.: The Index of the Dirac Operator in Loop Space. In: [La] pp. 161–186Google Scholar
  30. [W1] Witten, E.: Elliptic Genera and Quantum Field Theory. Commun. Math. Phys.109, 525–536 (1987)Google Scholar
  31. [Z] Zhang, W.: Circle Bundles, Adiabatic Limits of η-invariants and Rokhlin Congruences. Ann. Inst. Fourier (Grenoble)44, 249–270 (1994)Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Kefeng Liu
    • 1
  1. 1.Mathematics DepartmentM.I.T.CambridgeUSA

Personalised recommendations