Ivan G. Avramidi

A covariant technique for the calculation of the one-loop effective action

We develop a manifestly covariant technique for heat kernel calculation in the presence of arbitrary background fields in a curved space.The four lowest-order coefficients of Schwinger-De Witt asymptotic expansion are explicitly computed.We calculate also the heat kernel asymptotic expansion up to the terms of third order in the rapidly varying background fields (curvatures). This approximate series is summed up and the covariant nonlocal expressions for heat kernel,ζ-function and one-loop effective action are obtained.Other related problems are discussed. † The permanent address after 1 August, 1990: Nuclear Physics Department, Institute of Physics, Rostov State University, Stachki 194, Rostov-on-Don 344104, USSR I. G. Avramidi: Nuclear Physics B 355 (1991) 712-754 2

Heat Kernel Approach in Quantum Field Theory

Abstract We give a short overview of the effective action approach in quantum field theory and quantum gravity and describe various methods for calculation of the asymptotic expansion of the heat kernel for second-order elliptic partial differential operators acting on sections of vector bundles over a compact Riemannian manifold. We consider both Laplace type operators and non-Laplace type operators on manifolds without boundary as well as Laplace type operators on manifolds with boundary with oblique and non-smooth boundary conditions.

Gauge theories on manifolds with boundary

Abstract:The boundary-value problem for Laplace-type operators acting on smooth sections of a vector bundle over a compact Riemannian manifold with generalized local boundary conditions including both normal and tangential derivatives is studied. The condition of strong ellipticity of this boundary-value problem is formulated. The resolvent kernel and the heat kernel in the leading approximation are explicitly constructed. As a result, the previous work in the literature on heat-kernel asymptotics is shown to be a particular case of a more general structure. For a bosonic gauge theory on a compact Riemannian manifold with smooth boundary, the problem of obtaining a gauge-field operator of Laplace type is studied, jointly with local and gauge-invariant boundary conditions, which should lead to a strongly elliptic boundary-value problem. The scheme is extended to fermionic gauge theories by means of local and gauge-invariant projectors. After deriving a general condition for the validity of strong ellipticity for gauge theories, it is proved that for Euclidean Yang–Mills theory and Rarita–Schwinger fields all the above conditions can be satisfied. For Euclidean quantum gravity, however, this property no longer holds, i.e. the corresponding boundary-value problem is not strongly elliptic. Some non-standard local formulae for the leading asymptotics of the heat-kernel diagonal are also obtained. It is shown that, due to the lack of strong ellipticity, the heat-kernel diagonal is non-integrable near the boundary.

Covariant techniques for computation of the heat kernel

The heat kernel associated with an elliptic second-order partial differential operator of Laplace type acting on smooth sections of a vector bundle over a Riemannian manifold, is studied. A general manifestly covariant method for computation of the coefficients of the heat kernel asymptotic expansion is developed. The technique enables one to compute explicitly the diagonal values of the heat kernel coefficients, so called Hadamard–Minakshisundaram–De Witt–Seeley coefficients, as well as their derivatives. The elaborated technique is applicable for a manifold of arbitrary dimension and for a generic Riemannian metric of arbitrary signature. It is very algorithmic, and well suited to automated computation. The fourth heat kernel coefficient is computed explicitly for the first time. The general structure of the heat kernel coefficients is investigated in detail. On the one hand, the leading derivative terms in all heat kernel coefficients are computed. On the other hand, the generating functions in closed covariant form for the covariantly constant terms and some low-derivative terms in the heat kernel coefficients are constructed by means of purely algebraic methods. This gives, in particular, the whole sequence of heat kernel coefficients for an arbitrary locally symmetric space.

A new explicit expression for the Korteweg-De Vries hierarchy

Several constructions and an explicit expression for the right-hand side of the KdV hierarchy are presented.

A NEW ALGEBRAIC APPROACH FOR CALCULATING THE HEAT KERNEL IN GAUGE THEORIES

Abstract It is shown that the heat kernel for any Laplace-like operator on covariantly constant background in flat space may be presented in the form of an average over the corresponding Lie group with a gaussian measure. An explicit expression for the heat kernel is obtained using this representation. Related topics are discussed.

Heat Kernel Asymptotics of Operators with Non-Laplace Principal Part

We consider second-order elliptic partial differential operators acting on sections of vector bundles over a compact Riemannian manifold without boundary, working without the assumption of Laplace-like principal part -∇μ∇μ. Our objective is to obtain information on the asymptotic expansions of the corresponding resolvent and the heat kernel. The heat kernel and the Greens function are constructed explicitly in the leading order. The first two coefficients of the heat kernel asymptotic expansion are computed explicitly. A new semi-classical ansatz as well as the complete recursion system for the heat kernel of non-Laplace type operators is constructed. Some particular cases are studied in more detail.

A New algebraic approach for calculating the heat kernel in quantum gravity

It is shown that the heat kernel operator for the Laplace operator on any covariantly constant curved background, i.e., in symmetric spaces, may be presented in the form of an averaging over the Lie group of isometries with some nontrivial measure. Using this representation, the heat kernel diagonal, i.e., the heat kernel in coinciding points is obtained. Related topics concerning the structure of symmetric spaces and the calculation of the effective action are discussed.

Covariant algebraic method for calculation of the low‐energy heat kernel

Using our recently proposed covariant algebraic approach, the heat kernel for a Laplace‐like differential operator in a low‐energy approximation is studied. Neglecting all the covariant derivatives of the gauge field strength (Yang–Mills curvature) and the covariant derivatives of the potential term of third order and higher, a closed formula for the heat kernel as well as its diagonal is obtained. Explicit formulas for the coefficients of the asymptotic expansion of the heat kernel diagonal in terms of the Yang–Mills curvature, the potential term and its first two covariant derivatives are obtained.

The heat kernel on symmetric spaces via integrating over the group of isometries

Abstract A new algebraic approach for calculating the heat kernel for the Laplace operator on any Riemannian manifold with covariantly constant curvature is proposed. It is shown that the heat kernel operator can be obtained by an averaging over the Lie group of isometries. The heat kernel diagonal is obtained in form of an integral over the isotropy subgroup.