Research Plans


Ivan Avramidi


Objectives

The main subjects of my research are:

  • development of general effective methods for the calculation of the heat kernel and its asymptotics;

  • explicit computation of heat kernel coefficients in special cases;

  • analysis of deformations of spectral invariants and their relation to integrable systems;

  • applications of these methods to various problems of mathematical physics and differential geometry as well as in quantum field theory and quantum gravity.


Significance

The heat kernel is one of the most powerful tools in modern theoretical and mathematical physics, analysis on manifolds and differential geometry. In particular, it gives a general framework for the calculation of the effective action and Green functions in quantum field theory, especially in quantum gravity and gauge theories. Of special interest and great importance is the asymptotic expansion of the heat kernel. It is very closely related to the semi-classical approximation in the quantum theory, the high-temperature expansion in statistical physics and the renormalization procedure in quantum field theory. The coefficients of the asymptotic expansion of the heat kernel describe the asymptotic properties of the spectrum of the corresponding differential operator and are of central interest also in spectral geometry and closely allied to the non-linear completely integrable systems, such as Korteweg-de Vries hierarchy.

The planned activities are furhter development of the prior work and consitute a natural part of my long-term research plans. The progress in this area will be a significant contribution to the spectral theory of natural geometric differential operators, conformal geometry, spectral geometry, quantum field theory in curved spacetime and quantum gravity, mathematical physics as well as applied mathematics and engineering.


Plan of work

My research plans can be separated into the following categories:

  • Non-Laplace type operators

    • Further study of the heat kernel asymptotics for second order non-Laplace type operators. Despite the importance of second-order operators with non-Laplace principal part in gauge field theory and quantum gravity, their study is still quite new, and the available methodology is still underdeveloped in comparison with the Laplace type theory. We hope to lay the groundwork for a systematic attack on the spectral asymptotics of this larger class of operators.

  • Boundary value problems.

    • study the heat kernel asymptotics for Laplace type operators with non-smooth boundary conditions;

    • study the heat kernel asymptotics with a piece-wise smooth boundary; calculate some first coefficients;

    • study the relationship between the heat kernel asymptotics for a Laplace type operator with different boundary conditions; find some reduction formulas, which reduce the problem to a standard (Dirichlet or Neumann) one;

    • develop a method for calculation of the heat kernel asymptotics for manifolds with boundary with oblique boundary conditions which involve tangential differential operators on the boundary; calculate some first coefficients of this expansion and as a result the zeta-function at the origin $\zeta(0)$ for operators appearing in quantum gravity;

    • develop a method for calculation of the coefficients of the heat kernel asymptotic expansion for non-Laplace type operators on manifolds with boundary;

    These problems present a significult difficulty in comparison with standard smooth category. It is a new intriguing area and we hope to understand the general structure of the asymptotic expansion as well as to compute explicitly some low-order coefficients.

  • Functorial properties of the heat kernel.

    • Study the asymptotic expansion of the combined trace ${\rm Tr}\, P\exp(-tF)$ for a second-order differential operator P and a Laplace type operator F on a manifold without boundary. The coefficients of this asymptotic expansion Ek(P,F) were not studied as thoroughly as the heat invariants Ak(F). They present important and interesting objects of study. As a result calculate the vacuum expectation value of the energy-momentum tensor for various cases (e.g. for symmetric spaces);

    • investigate various deformations of the heat kernel ${\rm Tr}\,\exp(-tF(\tau))$ with a real deformation parameter $\tau$ and its asymptotics. Try to find some new isospectral deformations and the corresponding Lax representation $\dot F=[L,F]$ and, as a by-product, a new non-linear higher-dimensional integrable dynamical model (analogous to KdV hierarchy), such that the coefficients of the heat kernel asymptotic expansion would give the infinite series of integrals of motions of the system. If this is impossible we will try to understand why.

  • Heat kernel for specific backgrounds.

    • study the heat kernel asymptotics for a Laplace type operator on symmetric spaces and bundles with parallel curvature, and on Kaehler manifolds and line bundles with parallel curvature. An interesting question is also the relationship of the heat kernel asymptotics for dual (compact versus noncompact) symmetric spaces.

    • Develop a perturbation theory for computation of the heat kernel for the slightly perturbed (locally) situation described above, i.e. almost symmetric spaces and almost parallel bundle curvature. In the language of quantum field theory this is equivalent to the covaraintly constant background fields, which determine the so-called effective potential in quantum field theory and quantum gravity. This study is also important in string theory.

  • Related problems in differential geometry.

    • study the problem of existence of a metric with positive scalar curvature on a given manifold and its relation to the Dirac operator

    • develop a consistent approximate method to solve the Killing equations on quasihomogeneous manifolds, when the Riemannian curvature is strong but has small (if not zero) covariant derivatives.

  • Applications in quantum field theory.

    • investigate the dynamics of the vacuum of the chromomagnetic type in non-Abelian gauge theories in higher dimensions; find a nontrivial stable vacuum configuration;

    • calculate the low-energy effective action in quantum gravity using the results for the heat kernel; analyze the stability of the vacuum in quantum gravity via the low-energy effective action; study the Hawking radiation by means of the effective action technique;

    • analyze the vacuum structure of the higher-derivative quantum gravity; find a curved stable vacuum field configuration for the case of positive Euclidean action;

  • Book project

    • I was invited by the World Scientific Publishing to write a new book ``Heat Kernel in Quantum Field Theory and Mathematical Physics''.

 

Ivan Avramidi
8/7/2001