Global conformal blocks via Shadow formalism

Abstract

We study ${\mathfrak{sl}}_2$ and ${\mathfrak{sl}}_3$ global conformal blocks on a sphere and a torus, using the shadow formalism. These blocks arise in the context of Virasoro and ${\mathcal{W}}_3$ conformal field theories in the large central charge limit. In the ${\mathfrak{sl}}_2$ case, we demonstrate that the shadow formalism yields the known expressions in terms of conformal partial waves. Then, we extend this approach to the ${\mathfrak{sl}}_3$ case and show that it allows to build simple integral representations for ${\mathfrak{sl}}_3$ global blocks. We demonstrate this construction on two examples: the four-point block on the sphere and the one-point torus block.