Title: "Spin chain - Coulomb gas" correspondence


Abstract:

I present a one-to-one correspondence between certain smooth functions 
of several variables and vectors in a tensor product representation of 
the quantum sl_2. The correspondence has various nice properties. First, 
functions corresponding to highest weight vectors provide solutions to 
systems of linear partial differential equations. Second, asymptotic 
conditions posed on the functions transform to so called projection 
conditions for the corresponding vectors, that is, certain properties 
required from the projections of the vectors onto irreducible 
sub-representations. The asymptotic conditions allow one to sort out 
the solutions of the pde's satisfying desired boundary conditions, that is, 
solutions which are associated to physical problems.
I discuss two applications of the correspondence to questions related 
to Schramm Loewner evolution - the (believed, and proved for some models) 
scaling limit of interfaces in statistical mechanics models. 
The first application is the existence and uniqueness of multiple SLE 
pure partition functions, which should define the measures that span 
the convex hull of the probability measures of multiple SLE-processes. 
The second application concerns calculating the probability amplitudes 
of boundary visits of chordal SLE.
Both of the applications involve a system of linear partial differential 
equations, obtained from an SLE-martingale, for which solutions are 
provided by functions corresponding to the highest weight vectors of a 
suitable tensor product representation of the quantum sl_2.