Algebraic methods for the `low-energy' heat kernel

In the papers [6,7,8,9] I investigated the heat kernel in the approximation of slowly varying but strong background fields (so called `low-energy' approximation). This approximation is related to the background fields with small covariant derivatives, i.e. the curvatures R and ${\cal R}$ and the endomorphism Q are covariantly constant in the zeroth order. I developed a new original algebraic approach for calculating the heat kernel for a Laplace type operator [6]. It is shown that in the case of flat manifold and covariantly constant curvature of the bundle connection and endomorphism Q, i.e., $R=\nabla{\cal R}=\nabla Q=0$, the heat kernel may be presented in form of an average over a corresponding Lie group with a Gaussian measure and explicit expression for the heat kernel diagonal is obtained in closed form. In the paper [7] I calculated the heat kernel diagonal in a closed form in the framework of this approach in the next order of the `low-energy' approximation when additionally the second derivatives of the potential term Q are taken into account, i.e. only under the assumptions $R=\nabla{\cal R}=\nabla\nabla\nabla Q=0$.

In [8,9] this algebraic approach is generalized to the case of arbitrary Riemannian manifold with covariantly constant curvature, i.e. $\nabla R={\cal R}=\nabla Q=0$. I showed that the heat kernel operator for the Laplace operator on any Riemannian manifold with covariantly constant curvature, i.e. on locally symmetric spaces, can be presented in the form of an average over the group of isometries with some nontrivial measure. Using this representation I obtained the explicit covariant formula for the heat kernel diagonal in symmetric spaces. This formula serves as a generating function for the terms without covariant derivatives in all HMDS-coefficients, $a^{\rm diag}_k$, for arbitrary manifold.