66,098 research outputs found
Moduli Spaces of Abelian Vortices on Kahler Manifolds
We consider the self-dual vortex equations on a positive line bundle L --> M
over a compact Kaehler manifold of arbitrary dimension. When M is simply
connected, the moduli space of vortex solutions is a projective space. When M
is an abelian variety, the moduli space is the projectivization of the
Fourier-Mukai transform of L. We extend this description of the moduli space to
the abelian GLSM, i.e. to vortex equations with a torus gauge group acting
linearly on a complex vector space. After establishing the Hitchin-Kobayashi
correspondence appropriate for the general abelian GLSM, we give explicit
descriptions of the vortex moduli space in the case where the manifold M is
simply connected or is an abelian variety. In these examples we compute the
Kaehler class of the natural L^2-metric on the moduli space. In the simplest
cases we compute the volume and total scalar curvature of the muduli space.
Finally, we note that for abelian GLSM the vortex moduli space is a
compactification of the space of holomorphic maps from M to toric targets, just
as in the usual case of M being a Riemann surface. This leads to various
natural conjectures, for instance explicit formulae for the volume of the space
of maps CP^m --> CP^n.Comment: v2: 48 pages; significant changes; description of the vortex moduli
spaces of the GLSM extended to allow general values of the parameters, beyond
the generic values of v
A constructive mean field analysis of multi population neural networks with random synaptic weights and stochastic inputs
We deal with the problem of bridging the gap between two scales in neuronal
modeling. At the first (microscopic) scale, neurons are considered individually
and their behavior described by stochastic differential equations that govern
the time variations of their membrane potentials. They are coupled by synaptic
connections acting on their resulting activity, a nonlinear function of their
membrane potential. At the second (mesoscopic) scale, interacting populations
of neurons are described individually by similar equations. The equations
describing the dynamical and the stationary mean field behaviors are considered
as functional equations on a set of stochastic processes. Using this new point
of view allows us to prove that these equations are well-posed on any finite
time interval and to provide a constructive method for effectively computing
their unique solution. This method is proved to converge to the unique solution
and we characterize its complexity and convergence rate. We also provide
partial results for the stationary problem on infinite time intervals. These
results shed some new light on such neural mass models as the one of Jansen and
Rit \cite{jansen-rit:95}: their dynamics appears as a coarse approximation of
the much richer dynamics that emerges from our analysis. Our numerical
experiments confirm that the framework we propose and the numerical methods we
derive from it provide a new and powerful tool for the exploration of neural
behaviors at different scales.Comment: 55 pages, 4 figures, to appear in "Frontiers in Neuroscience
Conformal Toda theory with a boundary
We investigate sl(n) conformal Toda theory with maximally symmetric
boundaries. There are two types of maximally symmetric boundary conditions, due
to the existence of an order two automorphism of the W(n>2) algebra. In one of
the two cases, we find that there exist D-branes of all possible dimensions 0
=< d =< n-1, which correspond to partly degenerate representations of the W(n)
algebra. We perform classical and conformal bootstrap analyses of such
D-branes, and relate these two approaches by using the semi-classical light
asymptotic limit. In particular we determine the bulk one-point functions. We
observe remarkably severe divergences in the annulus partition functions, and
attribute their origin to the existence of infinite multiplicities in the
fusion of representations of the W(n>2) algebra. We also comment on the issue
of the existence of a boundary action, using the calculus of constrained
functional forms, and derive the generating function of the B"acklund
transformation for sl(3) Toda classical mechanics, using the minisuperspace
limit of the bulk one-point function.Comment: 42 pages; version 4: added clarifications in section 2.2 and
footnotes 1 and
Qualitative properties of solutions for an integral system related to the Hardy-Sobolev inequality
This article carries out a qualitative analysis on a system of integral
equations of the Hardy--Sobolev type. Namely, results concerning Liouville type
properties and the fast and slow decay rates of positive solutions for the
system are established. For a bounded and decaying positive solution, it is
shown that it either decays with the slow rates or the fast rates depending on
its integrability. Particularly, a criterion for distinguishing integrable
solutions from other bounded and decaying solutions in terms of their
asymptotic behavior is provided. Moreover, related results on the optimal
integrability, boundedness, radial symmetry and monotonicity of positive
integrable solutions are also established. As a result of the equivalence
between the integral system and a system of poly-harmonic equations under
appropriate conditions, the results translate over to the corresponding
poly-harmonic system. Hence, several classical results on semilinear elliptic
systems are recovered and further generalized.Comment: 33 pages, final author versio
New tools for classifying Hamiltonian circle actions with isolated fixed points
For every compact almost complex manifold (M,J) equipped with a J-preserving
circle action with isolated fixed points, a simple algebraic identity involving
the first Chern class is derived. This enables us to construct an algorithm to
obtain linear relations among the isotropy weights at the fixed points. Suppose
that M is symplectic and the action is Hamiltonian. If the manifold satisfies
an extra "positivity condition" this algorithm determines a family of vector
spaces which contain the admissible lattices of weights.
When the number of fixed points is minimal, this positivity condition is
necessarily satisfied whenever dim(M)< 8, and, when dim(M)=8, whenever the
S^1-action extends to an effective Hamiltonian T^2-action, or none of the
isotropy weights is 1. Moreover there are no known examples with a minimal
number of fixed points contradicting this condition, and their existence is
related to interesting questions regarding fake projective spaces [Y]. We run
the algorithm for dim(M)< 10, quickly obtaining all the possible families of
isotropy weights. In particular, we simplify the proofs of Ahara and Tolman for
dim(M)=6 [Ah,T1] and, when dim(M)=8, we prove that the equivariant cohomology
ring, Chern classes and isotropy weights agree with the ones of C P^4 with the
standard S^1-action (thus proving the symplectic Petrie conjecture [T1] in this
setting).Comment: 59 Pages; 16 Figures; Please find accompanying software at page
http://www.math.ist.utl.pt/~lgodin/MinimalActions.htm
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