Comments on Determinant Formulas for General CFTs

TL;DR

The paper argues that a determinant formula for parabolic Verma modules provides a general framework for higher-dimensional (super)conformal field theories, mirroring the Kac determinant in 2d CFT. It presents the explicit bosonic determinant for $M_{\mathfrak{p}}(\lambda)$, details the requisite inner product structure, and derives a simplicity criterion based on Jantzen’s analysis. It then extends the discussion to superconformal algebras, proposing a conjectured determinant formula that incorporates bosonic and fermionic roots and a modified Weyl vector, consistent with known reductions to the bosonic case. The work shows how these representation-theoretic inputs constrain conformal-block poles and unitarity bounds, inform recursion relations, and hint at higher-dimensional analogs of minimal models, highlighting a path toward a more systematic treatment of general correlators in (super)conformal theories.

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.