52 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
Nilpotence and descent in equivariant stable homotopy theory
Let be a finite group and let be a family of subgroups of
. We introduce a class of -equivariant spectra that we call
-nilpotent. This definition fits into the general theory of
torsion, complete, and nilpotent objects in a symmetric monoidal stable
-category, with which we begin. We then develop some of the basic
properties of -nilpotent -spectra, which are explored further
in the sequel to this paper.
In the rest of the paper, we prove several general structure theorems for
-categories of module spectra over objects such as equivariant real and
complex -theory and Borel-equivariant . Using these structure theorems
and a technique with the flag variety dating back to Quillen, we then show that
large classes of equivariant cohomology theories for which a type of
complex-orientability holds are nilpotent for the family of abelian subgroups.
In particular, we prove that equivariant real and complex -theory, as well
as the Borel-equivariant versions of complex-oriented theories, have this
property.Comment: 63 pages. Revised version, to appear in Advances in Mathematic
Modular forms, Schwarzian conditions, and symmetries of differential equations in physics
We give examples of infinite order rational transformations that leave linear
differential equations covariant. These examples are non-trivial yet simple
enough illustrations of exact representations of the renormalization group. We
first illustrate covariance properties on order-two linear differential
operators associated with identities relating the same hypergeometric
function with different rational pullbacks. We provide two new and more general
results of the previous covariance by rational functions: a new Heun function
example and a higher genus hypergeometric function example. We then
focus on identities relating the same hypergeometric function with two
different algebraic pullback transformations: such remarkable identities
correspond to modular forms, the algebraic transformations being solution of
another differentially algebraic Schwarzian equation that emerged in a paper by
Casale. Further, we show that the first differentially algebraic equation can
be seen as a subcase of the last Schwarzian differential condition, the
restriction corresponding to a factorization condition of some associated
order-two linear differential operator. Finally, we also explore
generalizations of these results, for instance, to , hypergeometric
functions, and show that one just reduces to the previous cases through
a Clausen identity.
In a hypergeometric framework the Schwarzian condition encapsulates
all the modular forms and modular equations of the theory of elliptic curves,
but these two conditions are actually richer than elliptic curves or
hypergeometric functions, as can be seen on the Heun and higher genus example.
This work is a strong incentive to develop more differentially algebraic
symmetry analysis in physics.Comment: 43 page
Cohomology of Finite Groups: Interactions and Applications
The cohomology of finite groups is an important tool in many subjects including representation theory and algebraic topology. This meeting was the third in a series that has emphasized the interactions of group
cohomology with other areas
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
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