82 research outputs found
Green functions of higher-order differential operators
The Green functions of the partial differential operators of even order
acting on smooth sections of a vector bundle over a Riemannian manifold are
investigated via the heat kernel methods. We study the resolvent of a special
class of higher-order operators formed by the products of second-order
operators of Laplace type defined with the help of a unique Riemannian metric
but with different bundle connections and potential terms. The asymptotic
expansion of the Green functions near the diagonal is studied in detail in any
dimension. As a by-product a simple criterion for the validity of the Huygens
principle is obtained. It is shown that all the singularities as well as the
non-analytic regular parts of the Green functions of such high-order operators
are expressed in terms of the usual heat kernel coefficients for a
special Laplace type second-order operator.Comment: 26 pages, LaTeX, 65 KB, no figures, some misprints and small mistakes
are fixed, final version to appear in J. Math. Phys. (May, 1998
Relation between chiral symmetry breaking and confinement in YM-theories
Spectral sums of the Dirac-Wilson operator and their relation to the Polyakov
loop are thoroughly investigated. The approach by Gattringer is generalized to
mode sums which reconstruct the Polyakov loop locally. This opens the
possibility to study the mode sum approximation to the Polyakov loop
correlator. The approach is re-derived for the ab initio continuum formulation
of Yang-Mills theories, and the convergence of the mode sum is studied in
detail. The mode sums are then explicitly calculated for the Schwinger model
and SU(2) gauge theory in a homogeneous background field. Using SU(2) lattice
gauge theory, the IR dominated mode sums are considered and the mode sum
approximation to the static quark anti-quark potential is obtained numerically.
We find a good agreement between the mode sum approximation and the static
potential at large distances for the confinement and the high temperature
plasma phase.Comment: 17 pages, 10 figures, typos corrected, references added, final
version to appear in PR
The C_2 heat-kernel coefficient in the presence of boundary discontinuities
We consider the heat-kernel on a manifold whose boundary is piecewise smooth.
The set of independent geometrical quantities required to construct an
expression for the contribution of the boundary discontinuities to the C_{2}
heat-kernel coefficient is derived in the case of a scalar field with Dirichlet
and Robin boundary conditions. The coefficient is then determined using
conformal symmetry and evaluation on some specific manifolds. For the Robin
case a perturbation technique is also developed and employed. The contributions
to the smeared heat-kernel coefficient and cocycle function are calculated.
Some incomplete results for spinor fields with mixed conditions are also
presented.Comment: 25 pages, LaTe
Estimates in Beurling--Helson type theorems. Multidimensional case
We consider the spaces of functions on the
-dimensional torus such that the sequence of the Fourier
coefficients belongs to
. The norm on is defined by
. We study the rate of
growth of the norms as
for -smooth real
functions on (the one-dimensional case was investigated
by the author earlier). The lower estimates that we obtain have direct
analogues for the spaces
Asymptotics of the Heat Kernel on Rank 1 Locally Symmetric Spaces
We consider the heat kernel (and the zeta function) associated with Laplace
type operators acting on a general irreducible rank 1 locally symmetric space
X. The set of Minakshisundaram- Pleijel coefficients {A_k(X)}_{k=0}^{\infty} in
the short-time asymptotic expansion of the heat kernel is calculated
explicitly.Comment: 11 pages, LaTeX fil
Coarse-Graining and Renormalization Group in the Einstein Universe
The Kadanoff-Wilson renormalization group approach for a scalar
self-interacting field theor generally coupled with gravity is presented. An
average potential that monitors the fluctuations of the blocked field in
different scaling regimes is constructed in a nonflat background and explicitly
computed within the loop-expansion approximation for an Einstein universe. The
curvature turns out to be dominant in setting the crossover scale from a
double-peak and a symmetric distribution of the block variables. The evolution
of all the coupling constants generated by the blocking procedure is examined:
the renormalized trajectories agree with the standard perturbative results for
the relevant vertices near the ultraviolet fixed point, but new effective
interactions between gravity and matter are present. The flow of the conformal
coupling constant is therefore analyzed in the improved scheme and the infrared
fixed point is reached for arbitrary values of the renormalized parameters.Comment: 18 pages, REVTex, two uuencoded figures. (to appear in Phys. Rev.
D15, July) Transmission errors have been correcte
Bose-Einstein Condensation on Product Manifolds
We investigate the phenomenon of Bose-Einstein condensation on manifolds
constructed as a product of a three-dimensional Euclidian space and a general
smooth, compact -dimensional manifold possibly with boundary. By using
spectral -function methods, we are able to explicitly provide
thermodynamical quantities like the critical temperature and the specific heat
when the gas of bosons is confined, in the three-dimensional manifold, by the
experimentally relevant anisotropic harmonic oscillator potential.Comment: 9 pages, LaTe
Relative entropy for compact Riemann surfaces
The relative entropy of the massive free bosonic field theory is studied on
various compact Riemann surfaces as a universal quantity with physical
significance, in particular, for gravitational phenomena. The exact expression
for the sphere is obtained, as well as its asymptotic series for large mass and
its Taylor series for small mass. One can also derive exact expressions for the
torus but not for higher genus. However, the asymptotic behaviour for large
mass can always be established-up to a constant-with heat-kernel methods. It
consists of an asymptotic series determined only by the curvature, hence common
for homogeneous surfaces of genus higher than one, and exponentially vanishing
corrections whose form is determined by the concrete topology. The coefficient
of the logarithmic term in this series gives the conformal anomaly.Comment: 20 pages, LaTeX 2e, 2 PS figures; to appear in Phys. Rev.
The Spectral Zeta Function for Laplace Operators on Warped Product Manifolds of the type
In this work we study the spectral zeta function associated with the Laplace
operator acting on scalar functions defined on a warped product of manifolds of
the type where is an interval of the real line and is a
compact, -dimensional Riemannian manifold either with or without boundary.
Starting from an integral representation of the spectral zeta function, we find
its analytic continuation by exploiting the WKB asymptotic expansion of the
eigenfunctions of the Laplace operator on for which a detailed analysis is
presented. We apply the obtained results to the explicit computation of the
zeta regularized functional determinant and the coefficients of the heat kernel
asymptotic expansion.Comment: 29 pages, LaTe
New Results in Sasaki-Einstein Geometry
This article is a summary of some of the author's work on Sasaki-Einstein
geometry. A rather general conjecture in string theory known as the AdS/CFT
correspondence relates Sasaki-Einstein geometry, in low dimensions, to
superconformal field theory; properties of the latter are therefore reflected
in the former, and vice versa. Despite this physical motivation, many recent
results are of independent geometrical interest, and are described here in
purely mathematical terms: explicit constructions of infinite families of both
quasi-regular and irregular Sasaki-Einstein metrics; toric Sasakian geometry;
an extremal problem that determines the Reeb vector field for, and hence also
the volume of, a Sasaki-Einstein manifold; and finally, obstructions to the
existence of Sasaki-Einstein metrics. Some of these results also provide new
insights into Kahler geometry, and in particular new obstructions to the
existence of Kahler-Einstein metrics on Fano orbifolds.Comment: 31 pages, no figures. Invited contribution to the proceedings of the
conference "Riemannian Topology: Geometric Structures on Manifolds"; minor
typos corrected, reference added; published version; Riemannian Topology and
Geometric Structures on Manifolds (Progress in Mathematics), Birkhauser (Nov
2008
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