2,732 research outputs found
Globally nilpotent differential operators and the square Ising model
We recall various multiple integrals related to the isotropic square Ising
model, and corresponding, respectively, to the n-particle contributions of the
magnetic susceptibility, to the (lattice) form factors, to the two-point
correlation functions and to their lambda-extensions. These integrals are
holonomic and even G-functions: they satisfy Fuchsian linear differential
equations with polynomial coefficients and have some arithmetic properties. We
recall the explicit forms, found in previous work, of these Fuchsian equations.
These differential operators are very selected Fuchsian linear differential
operators, and their remarkable properties have a deep geometrical origin: they
are all globally nilpotent, or, sometimes, even have zero p-curvature. Focusing
on the factorised parts of all these operators, we find out that the global
nilpotence of the factors corresponds to a set of selected structures of
algebraic geometry: elliptic curves, modular curves, and even a remarkable
weight-1 modular form emerging in the three-particle contribution
of the magnetic susceptibility of the square Ising model. In the case where we
do not have G-functions, but Hamburger functions (one irregular singularity at
0 or ) that correspond to the confluence of singularities in the
scaling limit, the p-curvature is also found to verify new structures
associated with simple deformations of the nilpotent property.Comment: 55 page
Non-canonical extension of theta-functions and modular integrability of theta-constants
This is an extended (factor 2.5) version of arXiv:math/0601371 and
arXiv:0808.3486. We present new results in the theory of the classical
-functions of Jacobi: series expansions and defining ordinary
differential equations (\odes). The proposed dynamical systems turn out to be
Hamiltonian and define fundamental differential properties of theta-functions;
they also yield an exponential quadratic extension of the canonical
-series. An integrability condition of these \odes\ explains appearance
of the modular -constants and differential properties thereof.
General solutions to all the \odes\ are given. For completeness, we also solve
the Weierstrassian elliptic modular inversion problem and consider its
consequences. As a nontrivial application, we apply proposed techni\-que to the
Hitchin case of the sixth Painlev\'e equation.Comment: Final version; 47 pages, 1 figure, LaTe
Analytic connections on Riemann surfaces and orbifolds
We give a differentially closed description of the uniformizing
representation to the analytical apparatus on Riemann surfaces and orbifolds of
finite analytic type. Apart from well-known automorphic functions and Abelian
differentials it involves construction of the connection objects. Like
functions and differentials, the connection, being also the fundamental object,
is described by algorithmically derivable ODEs. Automorphic properties of all
of the objects are associated to different discrete groups, among which are
excessive ones. We show, in an example of the hyperelliptic curves, how can the
connection be explicitly constructed. We study also a relation between
classical/traditional `linearly differential' viewpoint (principal Fuchsian
equation) and uniformizing -representation of the theory. The latter is
shown to be supplemented with the second (to the principal) Fuchsian equation.Comment: Final version. LaTeX, 16 pages. No figure
The Painlev\'e methods
This short review is an introduction to a great variety of methods, the
collection of which is called the Painlev\'e analysis, intended at producing
all kinds of exact (as opposed to perturbative) results on nonlinear equations,
whether ordinary, partial, or discrete.Comment: LaTex 2e, subject index, Nonlinear integrable systems: classical and
quantum, ed. A. Kundu, Special issue, Proceedings of Indian Science Academy,
The sixth Painleve transcendent and uniformization of algebraic curves
We exhibit a remarkable connection between sixth equation of Painleve list
and infinite families of explicitly uniformizable algebraic curves. Fuchsian
equations, congruences for group transformations, differential calculus of
functions and differentials on corresponding Riemann surfaces, Abelian
integrals, analytic connections (generalizations of Chazy's equations), and
other attributes of uniformization can be obtained for these curves. As
byproducts of the theory, we establish relations between Picard-Hitchin's
curves, hyperelliptic curves, punctured tori, Heun's equations, and the famous
differential equation which Apery used to prove the irrationality of Riemann's
zeta(3).Comment: Final version. Numerous improvements; English, 49 pages, 1 table, no
figures, LaTe
Ising n-fold integrals as diagonals of rational functions and integrality of series expansions
We show that the n-fold integrals of the magnetic susceptibility
of the Ising model, as well as various other n-fold integrals of the "Ising
class", or n-fold integrals from enumerative combinatorics, like lattice Green
functions, correspond to a distinguished class of function generalising
algebraic functions: they are actually diagonals of rational functions. As a
consequence, the power series expansions of the, analytic at x=0, solutions of
these linear differential equations "Derived From Geometry" are globally
bounded, which means that, after just one rescaling of the expansion variable,
they can be cast into series expansions with integer coefficients. We also give
several results showing that the unique analytical solution of Calabi-Yau ODEs,
and, more generally, Picard-Fuchs linear ODEs, with solutions of maximal
weights, are always diagonal of rational functions. Besides, in a more
enumerative combinatorics context, generating functions whose coefficients are
expressed in terms of nested sums of products of binomial terms can also be
shown to be diagonals of rational functions. We finally address the question of
the relations between the notion of integrality (series with integer
coefficients, or, more generally, globally bounded series) and the modularity
of ODEs.Comment: This paper is the short version of the larger (100 pages) version,
available as arXiv:1211.6031 , where all the detailed proofs are given and
where a much larger set of examples is displaye
Lattice Green Functions: the seven-dimensional face-centred cubic lattice
We present a recursive method to generate the expansion of the lattice Green
function of the d-dimensional face-centred cubic (fcc) lattice. We produce a
long series for d =7. Then we show (and recall) that, in order to obtain the
linear differential equation annihilating such a long power series, the most
economic way amounts to producing the non-minimal order differential equations.
We use the method to obtain the minimal order linear differential equation of
the lattice Green function of the seven-dimensional face-centred cubic (fcc)
lattice. We give some properties of this irreducible order-eleven differential
equation. We show that the differential Galois group of the corresponding
operator is included in . This order-eleven operator is
non-trivially homomorphic to its adjoint, and we give a "decomposition" of this
order-eleven operator in terms of four order-one self-adjoint operators and one
order-seven self-adjoint operator. Furthermore, using the Landau conditions on
the integral, we forward the regular singularities of the differential equation
of the d-dimensional lattice and show that they are all rational numbers. We
evaluate the return probability in random walks in the seven-dimensional fcc
lattice. We show that the return probability in the d-dimensional fcc lattice
decreases as as the dimension d goes to infinity.Comment: 19 page
A note on Chudnovsky's Fuchsian equations
We show that four exceptional Fuchsian equations, each determined by the four
parabolic singularities, known as the Chudnovsky equations, are transformed
into each other by algebraic transformations. We describe equivalence of these
equations and their counterparts on tori. The latter are the Fuchsian equations
on elliptic curves and their equivalence is characterized by transcendental
transformations which are represented explicitly in terms of elliptic and theta
functions.Comment: Final version; LaTeX, 27 pages, 1 table, no figure
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