203 research outputs found
Periods for flat algebraic connections
In previous work, we established a duality between the algebraic de Rham
cohomology of a flat algebraic connection on a smooth quasi-projective surface
over the complex numbers and the rapid decay homology of the dual connection
relying on a conjecture by C. Sabbah, which has been proved recently by T.
Mochizuki for algebraic connections in any dimension. In the present article,
we verify that Mochizuki's results allow to generalize these duality results to
arbitrary dimensions also
Levi problem and semistable quotients
A complex space is in class if it is a semistable
quotient of the complement to an analytic subset of a Stein manifold by a
holomorphic action of a reductive complex Lie group . It is shown that every
pseudoconvex unramified domain over is also in .Comment: Version 2 - minor edits; 8 page
Moment determinants as isomonodromic tau functions
We consider a wide class of determinants whose entries are moments of the
so-called semiclassical functionals and we show that they are tau functions for
an appropriate isomonodromic family which depends on the parameters of the
symbols for the functionals. This shows that the vanishing of the tau-function
for those systems is the obstruction to the solvability of a Riemann-Hilbert
problem associated to certain classes of (multiple) orthogonal polynomials. The
determinants include Haenkel, Toeplitz and shifted-Toeplitz determinants as
well as determinants of bimoment functionals and the determinants arising in
the study of multiple orthogonality. Some of these determinants appear also as
partition functions of random matrix models, including an instance of a
two-matrix model.Comment: 24 page
On Gauge Invariance and Spontaneous Symmetry Breaking
We show how the widely used concept of spontaneous symmetry breaking can be
explained in causal perturbation theory by introducing a perturbative version
of quantum gauge invariance. Perturbative gauge invariance, formulated
exclusively by means of asymptotic fields, is discussed for the simple example
of Abelian U(1) gauge theory (Abelian Higgs model). Our findings are relevant
for the electroweak theory, as pointed out elsewhere.Comment: 13 pages, latex, no figure
A construction of Frobenius manifolds with logarithmic poles and applications
A construction theorem for Frobenius manifolds with logarithmic poles is
established. This is a generalization of a theorem of Hertling and Manin. As an
application we prove a generalization of the reconstruction theorem of
Kontsevich and Manin for projective smooth varieties with convergent
Gromov-Witten potential. A second application is a construction of Frobenius
manifolds out of a variation of polarized Hodge structures which degenerates
along a normal crossing divisor when certain generation conditions are
fulfilled.Comment: 46 page
Analytic geometry of semisimple coalescent Frobenius structures
We present some results of a joint paper with Dubrovin (see references), as exposed at the Workshop "Asymptotic and Computational Aspects of Complex Differential Equations" at the CRM in Pisa, in February 2017. The analytical description of semisimple Frobenius manifolds is extended at semisimple coalescence points, namely points with some coalescing canonical coordinates although the corresponding Frobenius algebra is semisimple. After summarizing and revisiting the theory of the monodromy local invariants of semisimple Frobenius manifolds, as introduced by Dubrovin, it is shown how the definition of monodromy data can be extended also at semisimple coalescence points. Furthermore, a local Isomonodromy theorem at semisimple coalescence points is presented. Some examples of computation are taken from the quantum cohomologies of complex Grassmannians
Blowing up generalized Kahler 4-manifolds
We show that the blow-up of a generalized Kahler 4-manifold in a
nondegenerate complex point admits a generalized Kahler metric. As with the
blow-up of complex surfaces, this metric may be chosen to coincide with the
original outside a tubular neighbourhood of the exceptional divisor. To
accomplish this, we develop a blow-up operation for bi-Hermitian manifolds.Comment: 16 page
Can billiard eigenstates be approximated by superpositions of plane waves?
The plane wave decomposition method (PWDM) is one of the most popular
strategies for numerical solution of the quantum billiard problem. The method
is based on the assumption that each eigenstate in a billiard can be
approximated by a superposition of plane waves at a given energy. By the
classical results on the theory of differential operators this can indeed be
justified for billiards in convex domains. On the contrary, in the present work
we demonstrate that eigenstates of non-convex billiards, in general, cannot be
approximated by any solution of the Helmholtz equation regular everywhere in
(in particular, by linear combinations of a finite number of plane waves
having the same energy). From this we infer that PWDM cannot be applied to
billiards in non-convex domains. Furthermore, it follows from our results that
unlike the properties of integrable billiards, where each eigenstate can be
extended into the billiard exterior as a regular solution of the Helmholtz
equation, the eigenstates of non-convex billiards, in general, do not admit
such an extension.Comment: 23 pages, 5 figure
The Hamiltonian Structure of the Second Painleve Hierarchy
In this paper we study the Hamiltonian structure of the second Painleve
hierarchy, an infinite sequence of nonlinear ordinary differential equations
containing PII as its simplest equation. The n-th element of the hierarchy is a
non linear ODE of order 2n in the independent variable depending on n
parameters denoted by and . We introduce new
canonical coordinates and obtain Hamiltonians for the and
evolutions. We give explicit formulae for these Hamiltonians showing that they
are polynomials in our canonical coordinates
The Ernst equation and ergosurfaces
We show that analytic solutions \mcE of the Ernst equation with non-empty
zero-level-set of \Re \mcE lead to smooth ergosurfaces in space-time. In
fact, the space-time metric is smooth near a "Ernst ergosurface" if and
only if \mcE is smooth near and does not have zeros of infinite order
there.Comment: 23 pages, 4 figures; misprints correcte
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