Lorentz Center - Integrability and Isomonodromy in Mathematical Physics from 7 Jul 2014 through 11 Jul 2014
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    Integrability and Isomonodromy in Mathematical Physics
    from 7 Jul 2014 through 11 Jul 2014




Title: Painlevé equations and conformal field theory

Abstract: It will be explained how the Riemann-Hilbert problem associated to isomonodromic deformations of rank $2$ linear systems with $n$ regular singular points on $\mathbb{P}^1$ can be solved by taking suitable linear combinations of conformal blocks of the Virasoro algebra at $c=1$. This implies a similar representation for the isomonodromic tau function. In the case $n=4$, it provides the general solution of the Painlevé VI equation in the form of combinatorial sum over pairs of Young diagrams. Analogous solutions of Painlevé V and III may be formulated in terms of irregular limits of conformal blocks corresponding to decoupling of the matter hypermultiplets on the gauge side of the AGT correspondence.


Global Weyl groups and wild mapping class groups

In the classical theory of Painleve equations and isomonodromy there
are two types of discrete groups: braid/mapping class groups
controlling the global nonlinear monodromy, and Okamoto's affine Weyl
group symmetries giving equivalences between Painleve equations at
different values of the parameters. In this talk I will describe how
to generalise both of these groups. This involves for example all the
G-braid groups, and many non-affine Kac-Moody Weyl groups.



Title: The bosonic representation of tau-functions

The Riemann-Hilbert problem to find multivalued analytic functions
with SL(2,C)-valued monodromy on Riemann surfaces of genus zero
with n punctures can be solved by taking suitable linear combinations of the conformal blocks of Liouville theory at c=1. This implies a similar representation for the isomonodromic tau-functions. In the case n=4 we thereby get a proof of the relation between tau-functions and conformal blocks discovered by Gamayun, Iorgov and Lisovyy. These results can be understood as a bosonization of the fermionic constructions of isomonodromic tau-function due to Sato, Jimbo and Miwa.

Dynamics in one dimension: from fractional excitations to new out-of-equilibrium states of matter

This talk will review a number of integrability-based methods to investigate the dynamics of low-dimensional systems such as interacting atomic gases and quantum spin chains. In the equilibrium case, space- and time-dependent correlations will be considered. For out-of-equilibrium situations, a number of recent results will be reviewed, including a new method for explicitly calculating the relaxation of observables after a quantum quench. Exact solutions to the interaction turn-on quench in the Lieb-Liniger model and to the Néel-to-XXZ quench in spin chains  will be presented. Particular emphasis will be given to interesting open issues from a mathematical point of view.


Mobile impurity propagation in a one-dimensional quantum gas


We investigate the time evolution of an impurity atom injected into a gas

of impenetrable bosons (Tonks-Girardeau gas). The interaction between the

gas and impurity is assumed to be contact. If masses of the impurity and gas

particles coincide then the system is integrable by means of Bethe Ansatz

and we are able to obtain compact answer for the asymptotic momentum of

the impurity. At weak coupling we develop kinetic theory that describes impurity

behavior both in integrable and non-integrable cases and find striking

differences between them. The asymptotic momentum as a function of the

final momentum is found explicitly and the case of the external force applied

to the impurity is analyzed as well.



Title: Boundary qKZB equations.

Abstract: In this talk I introduce an explicit 9-parameter elliptic family of boundary qKZB equations. They are expected to describe the consistency conditions for correlation functions of the 8-vertex solid-on-solid model with reflecting boundaries. The family is constructed using a new 4-parameter elliptic family of solutions 
of a dynamical reflection equation, obtained by computing wall crossing formulas for the boundary qKZ equation associated to the XXZ spin chain.



Title: Painleve' equations and q-Askey scheme

Abstract: In this talk we will give a quantisation of the monodromy manifold associated to the Painleve' equations. We will show that the so obtained quantum algebras admit a representation in the space of (Laurent) polynomials by q-difference operators. We will show that special elements of the q-Askey scheme span such representations.



Title: Integrating the AdS_5 x S^5 superstring

Abstract: I review the integrability approach to the energy spectrum  of the AdS_5 x S^5 superstring and, via the gauge-string correspondence, 
to the spectrum of primary operators in planar N=4 super Yang-Mills theory. I will start from classical integrability of the string sigma-model 
and end up with the construction of the mirror Thermodynamic Bethe Ansatz which encodes the spectrum of the corresponding quantum theory.   


Bäcklund transformations for certain rational solutions of Painlevé VI

We introduce certain Bäcklund transformations for rational solutions of the Painlevé VI equation. These transformations act on a family of Painlevé VI tau functions. They are obtained from reducing the Hirota bilinear equations that describe the relation between certain points in the 3 component polynomial KP Grassmannian. In this way we obtain transformations that act on the root lattice of  A5. We also show that this A5 root lattice can be related to the F4(1) root lattice. We thus obtain Bäcklund transformations that relate Painlevé VI tau functions, parametrized by the elements of this F4(1) root lattice. This is based on joined work with Henrik Aratyn.


Title: Bihamiltonian cohomology of KdV Poisson brackets and spectral sequences.

The dispersive deformations of a Poisson pencil of hydrodynamic type are controlled by certain bihamiltonian cohomology groups. Liu and Zhang (2013) computed the first three bihamiltonian cohomology groups associated with the dispersionless KdV Poisson pencil and conjectured that all remaining bihamiltonian cohomology groups vanish.

After reviewing some of the theory, we outline a new method to derive the bihamiltonian cohomology in the KdV case. This is based on the introduction of a filtration of a related polynomial complex and on the computation of the associated spectral sequence. Using the convergence theorem for spectral sequences we obtain the cohomology of the polynomial complex and, by a long exact sequence argument, the bihamiltonian cohomology for the KdV Poisson pencil.

In particular we rederive the Liu-Zhang results and show that the remaining cohomology groups vanish, hence proving their conjecture.

Based on: G. Carlet, H. Posthuma, S. ShadrinBihamiltonian
cohomology of KdV brackets”, arXiv:1406.5595.