425 research outputs found
Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces
The purpose of this paper is to compute determinant index bundles of certain
families of Real Dirac type operators on Klein surfaces as elements in the
corresponding Grothendieck group of Real line bundles in the sense of Atiyah.
On a Klein surface these determinant index bundles have a natural holomorphic
description as theta line bundles. In particular we compute the first
Stiefel-Whitney classes of the corresponding fixed point bundles on the real
part of the Picard torus. The computation of these classes is important,
because they control to a large extent the orientability of certain moduli
spaces in Real gauge theory and Real algebraic geometry.Comment: LaTeX, 44 pages, to appear in Comm. Math. Phy
Cohomological Partition Functions for a Class of Bosonic Theories
We argue, that for a general class of nontrivial bosonic theories the path
integral can be related to an equivariant generalization of conventional
characteristic classes.Comment: 9 pages; standard LATEX fil
Gravitational Chern-Simons and the adiabatic limit
We compute the gravitational Chern-Simons term explicitly for an adiabatic
family of metrics using standard methods in general relativity. We use the fact
that our base three-manifold is a quasi-regular K-contact manifold heavily in
this computation. Our key observation is that this geometric assumption
corresponds exactly to a Kaluza-Klein Ansatz for the metric tensor on our three
manifold, which allows us to translate our problem into the language of general
relativity. Similar computations have been performed in a paper of Guralnik,
Iorio, Jackiw and Pi (2003), although not in the adiabatic context.Comment: 17 page
Index Theorems and Loop Space Geometry
We investigate the evaluation of the Dirac index using symplectic geometry in
the loop space of the corresponding supersymmetric quantum mechanical model. In
particular, we find that if we impose a simple first class constraint, we can
evaluate the Callias index of an odd dimensional Dirac operator directly from
the quantum mechanical model which yields the Atiyah-Singer index of an even
dimensional Dirac operator in one more dimension. The effective action obtained
by BRST quantization of this constrained system can be interpreted in terms of
loop space symplectic geometry, and the corresponding path integral for the
index can be evaluated exactly using the recently developed localization
techniques.Comment: 15 pages, report CERN-TH-6471 and HU-TFT-92-1
Conformal Motions and the Duistermaat-Heckman Integration Formula
We derive a geometric integration formula for the partition function of a
classical dynamical system and use it to show that corrections to the WKB
approximation vanish for any Hamiltonian which generates conformal motions of
some Riemannian geometry on the phase space. This generalizes previous cases
where the Hamiltonian was taken as an isometry generator. We show that this
conformal symmetry is similar to the usual formulations of the
Duistermaat-Heckman integration formula in terms of a supersymmetric Ward
identity for the dynamical system. We present an explicit example of a
localizable Hamiltonian system in this context and use it to demonstrate how
the dynamics of such systems differ from previous examples of the
Duistermaat-Heckman theorem.Comment: 13 pages LaTeX, run twice. Uses epsf.tex, 2 postscript files read
directly into LaTeX file from director
Sufficient stochastic maximum principle in a regime-switching diffusion model
We prove a sufficient stochastic maximum principle for the optimal control of
a regime-switching diffusion model. We show the connection to dynamic
programming and we apply the result to a quadratic loss minimization problem,
which can be used to solve a mean-variance portfolio selection problem
Szeg\"o kernel asymptotics and Morse inequalities on CR manifolds
We consider an abstract compact orientable Cauchy-Riemann manifold endowed
with a Cauchy-Riemann complex line bundle. We assume that the manifold
satisfies condition Y(q) everywhere. In this paper we obtain a scaling
upper-bound for the Szeg\"o kernel on (0, q)-forms with values in the high
tensor powers of the line bundle. This gives after integration weak Morse
inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a
refined spectral analysis we obtain also strong Morse inequalities which we
apply to the embedding of some convex-concave manifolds.Comment: 40 pages, the constants in Theorems 1.1-1.8 have been modified by a
multiplicative constant 1/2 ; v.2 is a final updat
Vector Bundle Moduli Superpotentials in Heterotic Superstrings and M-Theory
The non-perturbative superpotential generated by a heterotic superstring
wrapped once around a genus-zero holomorphic curve is proportional to the
Pfaffian involving the determinant of a Dirac operator on this curve. We show
that the space of zero modes of this Dirac operator is the kernel of a linear
mapping that is dependent on the associated vector bundle moduli. By explicitly
computing the determinant of this map, one can deduce whether or not the
dimension of the space of zero modes vanishes. It is shown that this
information is sufficient to completely determine the Pfaffian and, hence, the
non-perturbative superpotential as explicit holomorphic functions of the vector
bundle moduli. This method is illustrated by a number of non-trivial examples.Comment: 81 pages, LaTeX, corrected typo
The Bismut-Elworthy-Li type formulae for stochastic differential equations with jumps
Consider jump-type stochastic differential equations with the drift,
diffusion and jump terms. Logarithmic derivatives of densities for the solution
process are studied, and the Bismut-Elworthy-Li type formulae can be obtained
under the uniformly elliptic condition on the coefficients of the diffusion and
jump terms. Our approach is based upon the Kolmogorov backward equation by
making full use of the Markovian property of the process.Comment: 29 pages, to appear in Journal of Theoretical Probabilit
Exact Path Integrals by Equivariant Cohomology
It is a common belief among field theorists that path integrals can be
computed exactly only in a limited number of special cases, and that most of
these cases are already known. However recent developments, which generalize
the WKBJ method using equivariant cohomology, appear to contradict this folk
wisdom. At the formal level, equivariant localization would seem to allow exact
computation of phase space path integrals for an arbitrary partition function!
To see how, and if, these methods really work in practice, we have applied them
in explicit quantum mechanics examples. We show that the path integral for the
1-d hydrogen atom, which is not WKBJ exact, is localizable and computable using
the more general formalism. We find however considerable ambiguities in this
approach, which we can only partially resolve. In addition, we find a large
class of quantum mechanics examples where the localization procedure breaks
down completely.Comment: LATE
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