The large central charge limit of conformal blocks

TL;DR

The paper analyzes the large central charge limit of $W_3$ conformal blocks on the sphere, showing they reduce to $s\ell_3$-invariant functions. It develops integral representations for these large-$c$ blocks, derives a sixth-order differential equation for the two semi-degenerate case, and reveals a novel singularity at $z=-1$ in general. The authors compare the large-$c$ blocks with a proposed combinatorial (Young diagram) expansion and confirm agreement up to $z^5$, supporting the proposed expansion under appropriate degeneracy conditions. They also discuss the quantum-mechanical interpretation via $SL_N(\C)$ and outline generalizations to $W_N$ and finite-$c$ regimes, highlighting the role of infinite fusion multiplicities. These results provide a framework for computing and understanding $W_N$ blocks in the large-$c$ limit and motivate further connections to $s\ell_N$-invariant structures.

Abstract

We study conformal blocks of conformal field theories with a W3 symmetry algebra in the limit where the central charge is large. In this limit, we compute the four-point block as a special case of an sl3-invariant function. In the case when two of the four fields are semi-degenerate, we check that our results agree with the block's combinatorial expansion as a sum over Young diagrams. We also show that such a block obeys a sixth-order differential equation, and that it has an unexpected singularity at z=-1, in addition to the expected singularities at z=0,1,infinity.