CRM : Mathematical Physics Laboratory seminars

# Mathematical
Physics Laboratory seminars

### Fall 2010 -
Winter 2011

CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle
4336

### Ward identities and integrable
differential equations for quantum correlation functions

#### Benjamin
Doyon, Department of Mathematics, King's College London

Some time
ago, Fonseca and Zamolodchikov proposed a method for deriving the known
differential equations for correlation functions in the thermally
perturbed Ising model. The main idea of the method is to consider extra
symmetries emerging after ``doubling' the model, and the associated Ward
identities. Although the derivation used particularities of the Ising
conformal field theory, similar-looking methods in the theory of
classical integrability are known to generally produce bilinear
differential equations for tau functions. In this talk, I will explain,
in the context of QFT, how the main idea may be applied to correlation
functions of twist fields in arbitrary quadratic density matrices of
free massive (1+1-dimensional) fermionic models. To show how it works, I
will develop the case of the U(1)-twist fields in the massive Dirac
theory. This work was done in collaboration with my present PhD student
James Silk (who did most of the calculations).

### Structure of Levi extensions of
nilpotent algebras

#### Libor Snobl, Department of Physics, Czech
Technical University in Prague

We consider Lie algebras which can be
decomposed into a nontrivial semidirect sum $mathfrak{g}= mathfrak{p}
dotplus mathfrak{r}$ of a semisimple subalgebra $mathfrak{p}$ and the
radical $mathfrak{r}$. We assume that the nilradical of $mathfrak{r}$ is
given and investigate restrictions on possible Levi factors
$mathfrak{p}$ coming from the structure of characteristic ideals of the
nilradical. We present a new perspective on Turkowski's classification
of Levi decomposable algebras up to dimension 9. In particular we
explain, mostly by a simple dimensional analysis, why the majority of 5-
and 6-dimensional nilpotent algebras do not possess any Levi extension.

### Differential Geometry, Lie
Algebras and Lie Groups with Maple

#### Ian Anderson, Dept. of
mathematics and statistics, Utah State University

In this talk I
will describe a new suite of Maple packages for symbolic computations in
differential geometry, Lie groups and Lie algebras. The talk will begin
with an overview of the functionality of this software. I will then
give a brief demonstration of the basic features of the software and
conclude by presenting some recent applications to the integration of
ordinary differential equations and to the equivalence problem for
space-times and solutions to the Einstein equations.

### The Excitation Spectrum for Weakly
Interacting Bosons

#### Robert Seiringer, Dept. of mathematics, McGill

We
investigate the low energy excitation spectrum of a Bose gas with weak,
long range repulsive interactions. In particular, we prove that the
Bogoliubov
spectrum of elementary excitations with linear dispersion relation for
small
momentum becomes exact in the mean-field limit.

### Families
of Superintegrable Systems

#### Sarah POST, CRM

In this talk, I
will discuss several new results stemming from the recent
discovery of a family of exactly-solvable deformations of the simple
harmonic
oscillator by Turbiner, Tremblay and Winternitz (J. Phys. A, 2009, 42,
242001 )
(denoted TTW). The conjecture of superintegrability lead to several new
methods of
analysis of superintegrable systems which were successfully applied to
the TTW
system. I will also describe a transformation of integrable systems,
known as
coupling constant metamorphosis, and its application to the TTW system
to obtain a
new family of superintegrable deformations, this time of the
Kepler-Coulomb
potential.

### Two-variable
Wilson polynomials and the generic superintegrable system on
the 3-sphere

#### Willard Miller, University of Minnesota

We
show that the symmetry operators for the quantum superintegrable system
on the 3-sphere with
generic 4-parameter potential form a closed quadratic algebra with 6
linearly
independent generators that closes at order 6 (as differential
operators). Further
there is a algebraic relation at order 8 expressing the fact that there
are only 5
algebraically independent generators. We work out the details of
modeling
physically relevant irreducible representations of the quadratic algebra
in terms of
divided difference operators in two variables The representation spaces
turn out to
be those of two variable Wilson and Racah polynomials with arbitrary
parameters.
The quadratic algebra structure breaks the degeneracy of the spaces of
these
polynomials. In an earlier paper we found a similar characterization of
one variable
Wilson and Racah polynomials in terms of irreducible representations of
the
quadratic algebra for the quantum superintegrable system on the 2-sphere
with
generic 3-parameter potential. This indicates a general relationship
between 2nd
order superintegrable systems in n dimensions and multivariable
orthogonal
polynomials.
Joint work with Ernie Kalnins and Sarah Post.

### A Missing
System of Classical Orthogonal Polynomials

#### Alexei Zhedanov,
Donetsk Institute for Physics and Technology

We study a new class
of "classical" orthogonal polynomials which satisfy
(apart from 3-term recurrence relation) an eigenvalue problem with
Dunkl-type
differential operator. These polynomials thus satisfy the same
bispectrality
property as classical orthogonal polynomials. Explicit expression for
these
polynomials as well as their recurrence coefficients and the weight
function are
obtained. We also show that these polynomials provide a nontrivial
realization of
the Askey-Wilson algebra for q=-1.

### The
groundstate of the XXZ model and the quantum Knizhnik-Zamolodchikov
equation

#### Tiago Fonseca, LPTHE, Paris 6 et maintenant CRM

In
2000, Razumov and Stroganov conjectured an intriguing relation between
two statistical integrable models, the XXZ spin chain and the 6-Vertex
model.
Naturally, we started computing the XXZ groundstate (using the quantum
Knizhnik-Zamolodchikov equation). This led us to the proof of some old
results and
to new conjectures.

### The
Riemann-Hilbert approach to the transition from the Pearcey to the Airy
process

#### Mattia Cafasso, CRM

I will consider the gap
probability for the Airy and the Pearcey processes.
Using a Riemann-Hilbert approach I will show that the Pearcey process,
in a large
gap/large time limit, factorizes in two independent Airy processes. This
talk is
based on a joint work with Marco Bertola.

For further information: *SAINT@CRM.UMontreal.CA*