Global conformal blocks via shadow formalism

Abstract

We study 𝔰𝔩₂ and 𝔰𝔩₃ global conformal blocks on a sphere and a torus, using the shadow formalism. These blocks arise in the context of Virasoro and 𝒲₃ conformal field theories in the large central charge limit. In the 𝔰𝔩₂ case, we demonstrate that the shadow formalism yields the known expressions in terms of conformal partial waves. Then, we extend this approach to the 𝔰𝔩₃ case and show that it allows to build simple integral representations for 𝔰𝔩₃ global blocks. We demonstrate this construction on two examples: the four-point block on the sphere and the one-point torus block.