Abstract
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.
Highlights
W1(μ), w2(μ), w3(μ) and with a general conformal factor F (μ)
We have shown that this SU(2) symmetry of the Bianchi IX gravity models provides the ground for the existence of arithmetic structures in the resulting spectral model of gravity, which can be identified explicitly with mathematical methods
We have taken the rationality phenomenon presented in Proposition 2.1, which is due to the symmetries, as a first strong indication of the presence of deeper arithmetic structures hidden in the Seeley-deWitt coefficients of the small time heat kernel expansions for these geometries
Summary
The focus of this paper is primarily on deriving an explicit form of the Seeley-deWitt coefficients for the heat kernel expansion of the Dirac Laplacian on Bianchi IX gravitational instantons and identifying the modular properties of these expressions in terms of vectorvalued modular forms. We will discuss briefly in the Conclusions section the relevance of these Seeley-deWitt coefficients in the asymptotic expansion of the spectral action, which is an action functional for a model of modified (Euclidean) gravity, where the terms of the expansion represent higher derivative corrections to the usual Einstein-Hilbert gravity with cosmological term. As we discuss in detail the modularity that we analyze in the coefficients of the heat kernel expansion on Bianchi IX gravitational instantons corresponds to modular transformations of the time coordinate. The fact that this can be viewed as a complexified coordinate reflects the fact that the Euclidean Bianchi IX metrics can be seen as Wick rotations of Lorentzian signature metrics. We will return to a more thorough discussion of these physical aspects in forthcoming work
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