# Mini symposium

## Discrete conformal invariance, SLE and integrable lattice models

### Melbourne, 3 July 2009

Organiser: Jan De Gier
Contact: jdgier@unimelb.edu.au
Address: Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia

Partial sponsorship of the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems (MASCOS) is gratefully acknowledged.

Registration
If you plan to participate, please register (for catering purposes) by sending an email to jdgier@unimelb.edu.au.

Venue
Old Geology 1 (Buillding 155, G19 on the campus map)

Program
   9.15   Coffee [tearoom] 9.45   Vladimir Mangazeev The eight-vertex model, Painleve and 2D polymers 10.3   Break [tearoom] 11   Sergey Sergeev Manin matrices and Bethe algebra for U_q(sl_n) 11.45   Omar Foda A geometric Langlands correspondence in integrable lattice models 12.3   Lunch [Lunch provided, tearoom] 13.3   Murray Batchelor   SAW, integrable models and SLE 14.15   Jorgen Rasmussen Logarithmic minimal models, geometric exponents and SLE 15   Break [tearoom] 15.3   Bernard Nienhuis Integrability and discrete analyticity 16.15   Stas Smirnov Discrete analyticity and SLE 17   End

Participants
Richard Brak (University of Melbourne)
Chunhoa Chhenh (Australian National University)
Suhyoung Choi (KAIST, Korea)
Michael Couch (University of Melbourne)
Jan De Gier (University of Melbourne)
Caley Finn (University of Melbourne)
Peter Forrester (University of Melbourne)
Tim Garoni (University of Melbourne)
Yi Huay (University of Melbourne)
Barry Hughes (University of Melbourne)
Gye-Seon Lee (KAIST, Korea)
Chris Ormerod (University of Melbourne)
Anita Ponsaing (University of Melbourne)
Arun Ram (University of Melbourne)
Gus Schrader (University of Melbourne)
Mark Sorrell (University of Melbourne)
Michael Sun (University of Sydney)
Maria Tsarenko (University of Melbourne)
George Yiannakopoulos (DSTO Melbourne)
Craig Westerland (University of Melbourne)

Abstracts

I will discuss employing Schramm-Loewner Evolution to obtain intersection exponents for several chordal SLE_{8/3} curves in a wedge. As SLE_{8/3} is believed to describe the continuum limit of self-avoiding walks, these exponents correspond to those obtained by Cardy, Duplantier and Saleur for self-avoiding walks in an arbitrary wedge-shaped geometry using conformal invariance based arguments, as also obtained via the integrable loop model. The approach used builds on work by Werner, where the restriction property for SLE(\kappa,\rho) processes and an absolute continuity relation allow the calculation of such exponents in the half-plane. Furthermore, the method by which these results are extended is general enough to apply to the new class of hiding exponents introduced by Werner.

In a series of papers that started in 2002, Mukhin, Tarasov and Varchenko obtained a concrete example of a geometric Langlands correspondence in the context of Gaudin spin chains based on any semi-simple Lie algebra. At the core of the correspondence is a bijection between the Bethe eigenvectors and Fuchsian differential operators with no monodromy. I wish to derive (one side of) the analogous bijection in the context of sl(2) XXZ spin chains.

I will discuss a special case of the eight-vertex model which corresponds to the off-critical deformation of the \delta=-1/2 six-vertex model. In this case the ground state can be found exactly for all odd system sizes. The eigenvalues of the Q-operator satisfy a special deformation of the Lame equation. In the scaling limit this theory describes 2D dilute polymers on a cylinder where the free energy of a single incontractible polymer loop satisfies the Painlev´e III equation. Using the scaling limit technique I shall demonstrate how to obtain the Painlev'e VI equation on the lattice and present some results for the components of the ground state eigenvector.

Many integrable lattice models are critical and believed to have a conformal invariant scaling limit. Also there are many lattice models which are only integrable at their conformally invariant critical point. At the same time most integrable and conformally invariant lattice models have a massive and perturbation which is also integrable. It is not clear therefore if there is a relation between integrability and conformal invariance.

Recently Ikhlef and Cardy showed that in certain loop models one can identify correlators which show a lattice version of analyticity, but only at their critical point. This could be used to identify their critical point. In this presentation these findings will be presented.

Working in the context of a loop gas with loop fugacity beta=-2cos(4pi/kappa), we consider the fractal dimensions of various geometric objects such as paths and the generalizations of cluster mass, cluster hull, external perimeter and red bonds. Specializing to the case where the associated SLE parameter kappa=4p'/p is rational with p

A Manin matrix is a matrix with non-commutative entries but with well-defined determinant. The recent progress in Yangian- and Gaudin-type models is based on Manin properties of monodromy matrices. In my talk I give an extension of Manin theory to q ≠1 case of quantum groups.

Recently discrete analytic observables were found in some 2D lattice models at criticality: percolation, uniform spanning trees and the Ising model. This allowed to show that their interfaces converge to conformally invariant sclaing limits, described by Schramm's SLE curves, which in turn allows to rigorously compute scaling exponets and other parameters.

We will discuss how much information one needs to find a conformally invariant scaling limit (in fact, not much besides a discrete analytic observable), and what seems to be missing for other models, like SARW or Potts.