Comments on Determinant Formulas for General CFTs

TL;DR

This paper discusses the importance of determinant formulas for parabolic Verma modules in conformal field theories, proposing a conjecture for superconformal cases and exploring their applications in deriving bounds and recursion relations.

Contribution

It highlights the role of determinant formulas in (super)conformal field theories and introduces a new conjecture for superconformal algebras.

Findings

Determinant formulas are crucial for understanding conformal blocks.

A conjecture for superconformal algebra determinant formulas is proposed.

Applications include deriving unitarity bounds and recursion relations.

Abstract

We point out that the determinant formula for a parabolic Verma module plays a key role in the study of (super)conformal field theories and in particular their (super)conformal blocks. The determinant formula is known from the old work of Jantzen for bosonic conformal algebras, and we present a conjecture for superconformal algebras. The application of the formula includes derivation of the unitary bound and recursion relations for conformal blocks.