110 research outputs found
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
Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity
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, are actually diagonals of rational functions. As a consequence, the
power series expansions of these solutions of 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. Besides, in a more enumerative combinatorics
context, we show that 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 give a large set of results illustrating
the fact that the unique analytical solution of Calabi-Yau ODEs, and more
generally of MUM ODEs, is, almost always, diagonal of rational functions. We
revisit Christol's conjecture that globally bounded series of G-operators are
necessarily diagonals of rational functions. We provide a large set of examples
of globally bounded series, or series with integer coefficients, associated
with modular forms, or Hadamard product of modular forms, or associated with
Calabi-Yau ODEs, underlying the concept of modularity. 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 (in particular integrality of the Taylor coefficients of mirror
map), introducing new representations of Yukawa couplings.Comment: 100 page
The colored Jones function is q-holonomic
A function of several variables is called holonomic if, roughly speaking, it
is determined from finitely many of its values via finitely many linear
recursion relations with polynomial coefficients. Zeilberger was the first to
notice that the abstract notion of holonomicity can be applied to verify, in a
systematic and computerized way, combinatorial identities among special
functions. Using a general state sum definition of the colored Jones function
of a link in 3-space, we prove from first principles that the colored Jones
function is a multisum of a q-proper-hypergeometric function, and thus it is
q-holonomic. We demonstrate our results by computer calculations.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper29.abs.htm
The Quantum Dynamics of the Compactified Trigonometric Ruijsenaars-Schneider Model
We quantize a compactified version of the trigonometric
Ruijse\-naars-Schneider particle model with a phase space that is
symplectomorphic to the complex projective space CP^N. The quantum Hamiltonian
is realized as a discrete difference operator acting in a finite-dimensional
Hilbert space of complex functions with support in a finite uniform lattice
over a convex polytope (viz., a restricted Weyl alcove with walls having a
thickness proportional to the coupling parameter). We solve the corresponding
finite-dimensional (bispectral) eigenvalue problem in terms of discretized
Macdonald polynomials with q (and t) on the unit circle. The normalization of
the wave functions is determined using a terminating version of a recent
summation formula due to Aomoto, Ito and Macdonald. The resulting eigenfunction
transform determines a discrete Fourier-type involution in the Hilbert space of
lattice functions. This is in correspondence with Ruijsenaars' observation
that---at the classical level---the action-angle transformation defines an
(anti)symplectic involution of CP^N. From the perspective of algebraic
combinatorics, our results give rise to a novel system of bilinear summation
identities for the Macdonald symmetric functions
Rarefied elliptic hypergeometric functions
Two exact evaluation formulae for multiple rarefied elliptic beta integrals
related to the simplest lens space are proved. They generalize evaluations of
the type I and II elliptic beta integrals attached to the root system . In
a special case, the simplest limit is shown to lead to a new
class of -hypergeometric identities. Symmetries of a rarefied elliptic
analogue of the Euler-Gauss hypergeometric function are described and the
respective generalization of the hypergeometric equation is constructed. Some
extensions of the latter function to and root systems and
corresponding symmetry transformations are considered. An application of the
rarefied type II elliptic hypergeometric function to some eigenvalue
problems is briefly discussed.Comment: 41 pp., corrected numeration of formula
Calogero-Sutherland Approach to Defect Blocks
Extended objects such as line or surface operators, interfaces or boundaries
play an important role in conformal field theory. Here we propose a systematic
approach to the relevant conformal blocks which are argued to coincide with the
wave functions of an integrable multi-particle Calogero-Sutherland problem.
This generalizes a recent observation in 1602.01858 and makes extensive
mathematical results from the modern theory of multi-variable hypergeometric
functions available for studies of conformal defects. Applications range from
several new relations with scalar four-point blocks to a Euclidean inversion
formula for defect correlators.Comment: v2: changes for clarit
Collapsing D-Branes in Calabi-Yau Moduli Space: I
We study the quantum volume of D-branes wrapped around various cycles in
Calabi-Yau manifolds, as the manifold's moduli are varied. In particular, we
focus on the behaviour of these D-branes near phase transitions between
distinct low energy physical descriptions of the resulting string theory.
Whereas previous studies have solely considered quantum volumes in the context
of two-cycles in perturbative string theory or D-branes in the specific example
of the quintic hypersurface, we work more generally and find qualitatively new
features. On the mathematical side, as we briefly note, our work has some
interesting implications for certain issues in arithmetics.Comment: 77 pages, 15 figure
- …