Non-stationary Lamé equation and WZW/Virasoro conformal blocks

Question

The non-stationary Lamé equation

$\left(2i \pi \kappa \partial_\tau -\partial_x^2+g(g-1)\wp(x)\right)\psi(x,\tau)=E \psi(x,\tau)$

appears as a BPZ type equation (for the 2 points Virasoro conformal blocks with one degenerate field on the torus), and as a KZ type equation (for the one point conformal blocks on the torus of a WZW model).

1) I am not familiar with the WZW model. Why does this equation appear twice? Is it just a coincidence?

2) Do these conformal blocks give different kind of solution to this equation?

Answer 1

There are known relations between KZ and BPZ equations. For the torus, see https://arxiv.org/abs/0706.1030 . That article describes relations between correlators of the H3+ model and Liouville theory from a path integral point of view.

This implies relations between differential equations. In Appendix A genus one differential equations are worked out.

Of course, this also implies relations between conformal blocks. Virasoro conformal blocks provide a particular basis of affine conformal blocks, for a particular choice of isospin variables.