Ivan G. Avramidi (Department of Mathematics, The University of Iowa) and
Giampiero Esposito (Istituto Nazionale di Fisica Nucleare, Sezione di Napoli)

ABSTRACT:

The formulation of the gauge theories on compact manifolds with boundary leads to partial differential operators with the boundary conditions that involve both the normal and tangential derivatives on the boundary, so called Gilkey-Smith boundary conditions. In contrary to the standard Dirichlet or Neumann boundary conditions this boundary-value problem is not automatically elliptic but becomes elliptic under certain condition on the boundary operator. We study the Gilkey-Smith boundary-value problem for Laplace-type operators and find a simple criterion of ellipticity. The first non-trivial coefficient of the asymptotic expansion of the trace of the heat kernel is computed and the local leading asymptotics of the heat kernel diagonal is also obtained. It is shown that in the non-elliptic case the heat-kernel diagonal is non-integrable near the boundary, which reflects the fact that the heat kernel is not of trace class. We apply this analysis in general linear bosonic gauge theories and find an explicit condition of ellipticity.
 

Slides of the lecture: (PDF)

Proceedings of the Conference: Heat kernel asymptotics of Gilkey-Smith boundary value problem, in: Trends in Mathematical Physics, Eds. V. Alexiades and G. Siopsis, AMS/IP Studies in Advanced Mathematics, vol. 13, (American Mathematical Society and International Press, 1999), pp. 15-34;  

Preprint: math-ph/9812010