$(1,k)$ CFT and RH problem with the $c=-2$ case

Following approach of Iorgov--Lisovyy--Teschner, we construct solutions of the (modified) Riemann--Hilbert problem using conformal blocks of $(1,k)$ Virasoro models.

For $k>1$ case, the solution of this Riemann--Hilbert problem is not unique due to more singular behavior at punctures.

On the CFT side the dimension of the space of conformal blocks also increases.

We specifically study the $k=2$ case, which corresponds to the central charge $c=-2$ and symplectic fermions.

We explicitly construct a corresponding solution of the modified Riemann--Hilbert problem in the case of 3 punctures and prove its uniqueness under suitable initial data conditions.

We also obtain new bilinear relations for $c=-2$ tau functions.