Null vectors, 3-point and 4-point functions in conformal field theory

TL;DR

This work analyzes 3-point and 4-point functions in 2D CFTs with extended $W_3$ symmetry, contrasting them with the Virasoro case where descendant correlators are fixed by primaries. It shows that generic $W_3$ 3-point functions possess arbitrary fusion data unless null-vector degeneracies are present; doubly degenerate representations constrain these data to finite sets, and two such fields in a 4-point function yield finite intermediate channels and differential equations for chiral blocks, generalizing the BPZ framework. The findings clarify fusion rules and the role of degeneracy in minimal and non-minimal $W_3$ theories, and connect to known results by Bajnok et al. for differential equations in chiral blocks. Overall, the paper extends the structure of conformal blocks from Virasoro to $W_3$-based algebras and informs higher-point function analysis in theories with extended symmetry.

Abstract

We consider 3-point and 4-point correlation functions in a conformal field theory with a W-algebra symmetry. Whereas in a theory with only Virasoro symmetry the three point functions of descendants fields are uniquely determined by the three point function of the corresponding primary fields this is not the case for a theory with $W_3$ algebra symmetry. The generic 3-point functions of W-descendant fields have a countable degree of arbitrariness. We find, however, that if one of the fields belongs to a representation with null states that this has implications for the 3-point functions. In particular if one of the representations is doubly-degenerate then the 3-point function is determined up to an overall constant. We extend our analysis to 4-point functions and find that if two of the W-primary fields are doubly degenerate then the intermediate channels are limited to a finite set and that the corresponding chiral blocks are determined up to an overall constant. This corresponds to the existence of a linear differential equation for the chiral blocks with two completely degenerate fields as has been found in the work of Bajnok~et~al.