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.