Journal of High Energy Physics

, 2019:234 | Cite as

Modular forms in the spectral action of Bianchi IX gravitational instantons

  • Wentao Fan
  • Farzad FathizadehEmail author
  • Matilde Marcolli
Open Access
Regular Article - Theoretical Physics


We prove a modularity property for the heat kernel and the Seeley-deWitt coefficients of the heat kernel expansion for the Dirac-Laplacian on the Bianchi IX gravitational instantons. We prove, via an isospectrality result for the Dirac operators, that each term in the expansion is a vector-valued modular form, with an associated ordinary (meromorphic) modular form of weight 2. We discuss explicit examples related to well known modular forms. Our results show the existence of arithmetic structures in Euclidean gravity models based on the spectral action functional.


Models of Quantum Gravity Non-Commutative Geometry 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Wentao Fan
    • 1
    • 2
  • Farzad Fathizadeh
    • 2
    • 3
    • 4
    Email author
  • Matilde Marcolli
    • 2
    • 5
    • 6
  1. 1.Department of PhysicsPrinceton UniversityPrincetonU.S.A.
  2. 2.Division of Physics, Mathematics and AstronomyCalifornia Institute of TechnologyPasadenaU.S.A.
  3. 3.Department of Mathematics, Computational FoundrySwansea UniversitySwanseaU.K.
  4. 4.Max Planck Institute for Biological CyberneticsTübingenGermany
  5. 5.Department of MathematicsUniversity of TorontoTorontoCanada
  6. 6.Perimeter Institute for Theoretical PhysicsWaterlooCanada

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