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Published January 2019 | Submitted + Published
Journal Article Open

Modular forms in the spectral action of Bianchi IX gravitational instantons


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.

Additional Information

¬© 2019 The Author(s). 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. Article funded by SCOAP3. Received: December 13, 2018; Revised: January 6, 2019; Accepted: January 22, 2019; Published: January 31, 2019. The first author was supported by a Summer Undergraduate Research Fellowship at Caltech. The second author acknowledges the support from the Marie Curie/SER Cymru II Cofund Research Fellowship 663830-SU-008, and thanks the Institut des Hautes √Čtudes Scientifiques (I.H.E.S.) for an excellent environment and their hospitality in the Summer of 2015, where this work was partially carried out. The third author was partially supported by NSF grants DMS-1201512, PHY-1205440, DMS-1707882, by NSERC Discovery Grant RGPIN-2018-04937 and Accelerator Supplement Grant RGPAS-2018-522593, and by the Perimeter Institute for Theoretical Physics.

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Published - Fan2019_Article_ModularFormsInTheSpectralActio.pdf

Submitted - 1511.05321.pdf


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