July 2 -
July 7, 2001
Preliminary Program
Lectures:
Lecture 1. "Effective Action Approach in
Gauge Field Theories and Quantum Gravity"
Abstract: An overview of the effective
action approach in quantum field theory and quantum gravity.
Lecture 2. "Calculation
of the Heat Kernel Asymptotic Expansion"
Abstract: Review of basic techniques for calculation
of the heat kernel asymptotic expansion for Laplace type partial diffferential
operators acting o smooth sections of a vector bundle over a compact manifold
without boundary.
Lecture 3. "Covariant
Approximation Schemes for Calculation of the Heat Kernel"
Abstract: Development of various
approximation schemes for the heat kernel based on the behavior of the
"background fields". In the "high-energy" limit, which is
determined by rapidly varying background fields, the highest derivative terms
are computed and a partial summation is carried out leading to closed formulas
for the effective action. In the "low-energy" limit, which is
determined by slowly varying background fields, an algebraic approach is
developed that enables one to obtain closed non-perturbative formulas for the
heat kernel that generate the whole asymptotic expansion.
Lecture 4. "Heat-kernel
Asymptotics for Non-Laplace Type Operators"
Abstract: The most general class of second-order
operators (non-Laplace type operators), acting on sections of a vector bundle,
with positive definite leading symbol, is studied and the calculation of the
heat-kernel asymptotics is presented. The heat kernel and the resolvent are
constructed explicitly in the leading order.
Lecture 5. "Heat-Kernel
Asymptotics for the Grubb - Gilkey - Smith Boundary-Value Problem"
Abstract: Laplace type partial differential
operator acting on sections of a vector bundle over a compact Riemannian
manifold with smooth boundary with the Grubb - Gilkey - Smith boundary
conditions, which involve both normal and tangential derivatives on the
boundary. The criterion of strong ellipticity is discussed and the first
non-trivial coefficient of the asymptotic expansion of the trace of the heat
kernel is computed.
Lecture 6. "Heat-Kernel
Asymptotics for Non-Smooth Boundary Conditions"
Abstract: The boundary-value problem for
Laplace-type operators acting on smooth sections of a vector bundle over a
compact Riemannian manifold with nonstandard singular boundary conditions,
which include Dirichlet conditions on one part of the boundary and Neumann ones
on another part of the boundary, is studied. The resolvent kernel and the heat
kernel in the leading approximation are explicitly constructed and the leading
heat-kernel asymptotics are computed.
Abstract: The aim of these lectures is to
discuss mathematical methods in statistical mechanics in external stationary
gravitational and gauge fields, as well as their applications in black hole
physics. We will also discuss how summation over the modes in the partition
function on a stationary background can be related to the path integral in
Euclidean gravity.
Preliminary plan of the lectures:
Part I. Introduction
Part II. Canonical and Euclidean Methods in
Statistical Mechanics on Stationary Backgrounds
This part includes discussion of the following topics:
Part III. Some Applications
One of the applications which will be discussed here is thermodynamics
and statistical mechanics of black holes.
Notational conventions: Let M be a smooth compact Riemannian
manifold with smooth boundary ¶ M. Let D be an
operator of Laplace type, let f be the initial temperature
distribution, and let B be Dirichlet or Robin boundary conditions. Let uf be
the resulting temperature distribution for t > 0.
Lecture 1. "Heat
Content Asymptotics"
Abstract: Let r be the specific heat. Let
b (f, r, D, B)
(t) : = ò M uf r dx
be the total heat energy content. As t¯ 0,
there is a complete asymptotic expansion
b
~ S n³
0 bn
tn/2
where the coefficients bn
are locally computable in terms of geometric data. We
discuss formulas for these invariants. This problem can be generalized by
considering time-dependent families of operators of Laplace type. One can also
examine inhomogeneous boundary conditions which are time dependent and permit
time-dependent source terms. We discuss the resulting formulas for these
invariants. This is joint work with M. van den Berg, S. Desjardins, and J.H.
Park.
Lecture 2. "Heat Trace Asymptotics"
Abstract: Let the temperature distribution
be given by the fundamental solution of the heat equation: uf (x) =
ò M K (t,x,y,D,B) f (y) dy.
The kernel function K is smooth. Let f(x) be a localizing
function and define the heat trace asymptotics
a(f,D,B)(t):= ò M
Tr K(t,x,x,D,B) f(x) dx ;
there is an asymptotic series as t¯ 0
of the form:
a(f,D,B)(t) ~ S n³ 0 t(n-m)/2 an
(f,D,B)
where once again the coefficients an are locally
computable. We discuss formulas for these invariants in terms of geometric
data. This is joint work with T. Branson and K. Kirsten. These asymptotics can
be generalized by considering time-dependent families of operators of Laplace
type and time-dependent boundary conditions.
Lecture 3. "Nonlocal Boundary Conditions
and Discontinuous Boundary Conditions"
Abstract: If A is an operator of
Dirac type, one can impose spectral boundary conditions on the resulting
operator D=A*A of Laplace type. The coefficients of the heat
trace asymptotics, in contrast to those discussed in Lectures One and Two,
exhibit non-trivial dependence upon the dimension m. One can also
consider the D/N problem. Assume given a decomposition of ¶ M=CN
È
CD as the union of two closed subsets where CD
Ç CN
is a non-empty smooth codimension 1 submanifold of the
boundary. Take Dirichlet/Neumann boundary conditions on CD/CN.
Preliminary work with Dowker and Kirsten indicates that there does not exist a
classical asymptotic expansion in this setting and some results will be
discussed in this context.
Abstract: Trace expansions for functions of
partial differential and pseudodifferential operators, with or without boundary
conditions, are of interest for example in the study of the index and other
geometric invariants, as well as the noncommutative residue. The lectures will
give an exposition of the main techniques and some applications.
Lecture 1. In this lecture I follow Segal's
notes and show how the Grasmannian of the boundary conditions and the
determinant defined on this Grassmannian appear in the Topological Quantum
Field Theory.
Lecture 2. I discuss zeta determinant of the
Dirac operators and elliptic boundary problems.
Lecture 3. In this lecture I prove the
pasting formula for the phase of the zeta-determinant.
Lecture 4. In the final lecture I discuss
pasting of the modulus of the zeta-determinant. This is the most difficult case
from the analytical point of view and at present we only have an adiabatic
formula.
Colloquium Lectures:
Abstract: We consider the heat kernel and
the zeta function associated with Laplace type operators acting on a general
irreducible rank-one locally symmetric space. The set of Minakshisundaram -
Pleijel coefficients in the small-t asymptotic expansion of the kernel
is calculated explicitly. The Chern - Simons invariants of irreducible U(n)-flat
connections on compact hyperbolic 3-manifolds of the form G H3
are derived. The explicit formula for the Chern - Simons functional is given in
terms of Selberg type zeta functions related to the twisted eta invariants of
Atiyah - Patodi - Singer. The semiclassical limit for the partition function is
presented.
Talks:
M. de Gosson, J.S. Dowker, E. Elizalde, S. Fulling, R. Ivanova, A.Yu.
Kamenshchik, V.N. Marachevsky, D. Mauro, V. Moretti, I.G. Moss, O.V. Pavlovsky, D. Ratiba, D. Seminara, S. Sushkov, D.V.
Vassilevich, F. Williams, S. Zerbini.
Details available so far:
Abstract: The recent discovery in symplectic
topology of a phenomenon known as Gromov's non-squeezing principle makes
plausible the existence of a phase-space quantum mechanics.
Abstract: The zeta function method, for the
cases when the spectrum of the relevant operator is known either implicitly or
explicitly (making use in this case of the asymptotics of the function
determining the spectrum), will be described.
Abstract: We consider zeta regularization of
the basic commutators in string theories suggested by Hwang, Marnellius and
Saltsidis and apply it to string theories with non-trivial vacuums. It is shown
that the application of this regularization allows one to restore the Jacobi
identities in these theories.
Abstract: Path-integral and other approaches
to the calculation of Casimir energy of dielectrics will be analysed with
emphasis on the nature of divergences and comparison of different
regularization schemes. Dielectric ball and perfectly conducting wedge will be
considered as illustrative examples of general methods.
Abstract: Sometime ago it was shown that the
operatorial approach to classical mechanics, pioneered in the 30's by Koopman
and von Neumann, can have a functional version. In this talk we will extend
this functional approach to the case of classical field theories and in
particular to the Yang--Mills ones. We shall show that the issues of
gauge-fixing and Faddeev - Popov determinant arise also in this classical
formalism. We shall also indicate how to quantize the system by freezing to
zero the Grassmannian partners of time.
·
I.G. Moss: "The asymptotics of quasi-normal modes"
Abstract: In a first part of the work, Newtonian
gravity is considered within the context of a gauge theory of the de Sitter
group. For this, de Sitter fibre bundles over curved non-relativistic
space-time are used. In the case of Riemannian base manifolds, soldering for de
Sitter bundles in Drechsler framework proved to be a main concept permitting a
link between the Lorentz subsymmetry contained in the de Sitter group with the
Lorentz symmetry of a vierbein formulation of gravity. In the present case,
after defining a mathematical concept of partial soldering, the connection
representing Newtonian gravity is derived from that contained in the de Sitter
theory. In a second part, the quantization scheme, based on the method of
induced representation, is realised. The method is applied for the de Sitter
group in the fibre and the Bargmann semigroup of trajectories in the base space
when the latter is identified with the flat space-time of Newton. The
interaction appears in an ordered exponential generalising the pointlike
particle case in which the propagator of the particle in a gauge field was
derived by Mensky in the form of a path integral.
Abstract: Using the trace formula we analyze
the spectral zeta function for twisted p-forms on a compact real hyperbolic
space. As an application we obtain explicit small-time asymptotics for the
corresponding heat kernel.
Abstract: The multiplicative anomaly of
functional determinants will be introduced and evaluated in some simple cases. Some
physical applications will be discussed.