Heat Kernel Expansion for Operators of the Type of the Square Root of the Laplace Operator

A method is suggested for the calculation of the DeWitt-Seeley-Gilkey (DWSG) coefficients for the operator $\sqrt{-\nabla^2 + V(x)}$ basing on a generalization of the pseudodifferential operator technique. The lowest DWSG coefficients for the operator $\sqrt{-\nabla^2} + V(x)$ are calculated by using the method proposed. It is shown that the method admits a generalization to the case of operators of the type $(-\nabla^2 + V(X))^{1/{\rm m}}$, where m is an arbitrary rational number. A more simple method is proposed for the calculation of the DWSG coefficients for the case of strictly positive operators under the sign of root. By using this method, it is shown that the problem of the calculation of the DWSG coefficients for such operators is exactly solvable. Namely, an explicit formula expressing the DWSG coefficients for operators with root through the DWSG coefficients for operators without root is deduced.Comment: 17 pages, LaTeX, no figure

New Heat Kernel Method in Lifshitz Theories

We develop a new heat kernel method that is suited for a systematic study of the renormalization group flow in Horava gravity (and in Lifshitz field theories in general). This method maintains covariance at all stages of the calculation, which is achieved by introducing a generalized Fourier transform covariant with respect to the nonrelativistic background spacetime. As a first test, we apply this method to compute the anisotropic Weyl anomaly for a (2+1)-dimensional scalar field theory around a z=2 Lifshitz point and corroborate the previously found result. We then proceed to general scalar operators and evaluate their one-loop effective action. The covariant heat kernel method that we develop also directly applies to operators with spin structures in arbitrary dimensions.Comment: 47 pages, 1 figure; v2: appendix C updated, minor typos corrected, references adde

Curvature in Noncommutative Geometry

Our understanding of the notion of curvature in a noncommutative setting has progressed substantially in the past ten years. This new episode in noncommutative geometry started when a Gauss-Bonnet theorem was proved by Connes and Tretkoff for a curved noncommutative two torus. Ideas from spectral geometry and heat kernel asymptotic expansions suggest a general way of defining local curvature invariants for noncommutative Riemannian type spaces where the metric structure is encoded by a Dirac type operator. To carry explicit computations however one needs quite intriguing new ideas. We give an account of the most recent developments on the notion of curvature in noncommutative geometry in this paper.Comment: 76 pages, 8 figures, final version, one section on open problems added, and references expanded. Appears in "Advances in Noncommutative Geometry - on the occasion of Alain Connes' 70th birthday

Modular forms in the spectral action of Bianchi IX gravitational instantons

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