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.
Highlights
The representation theory of the conformal algebra has a long history
We point out that the determinant formula for a parabolic Verma module plays a key role in the study ofconformal field theories and in particular theirconformal blocks
The determinant formula is known from the old work of Jantzen for bosonic conformal algebras, and we present a conjecture for superconformal algebras
Summary
The representation theory of the conformal (or superconformal) algebra has a long history (see e.g. [12,13,14,15,16,17] for an incomplete list). Starting with the conformal primary |Ωλ we can consider its descendants These are generated by the action of the subalgebra n−, which spans the parabolic Verma module: Mp(λ) := span g1g2 . The primary constraints ((2.4) and (2.5)) imply that |Ωλ is a representation of p Given such a representation Vλ (whose highest weight we denote by λ), we can define the associated parabolic Verma module by. Module reduces to the ordinary Verma module, and the parabolic subalgebra p coincides with the Borel subalgebra This case is familiar from the representation theory of the Virasoro algebra, in which case the decomposition (2.1) reads n+ = Ln>0 , l = Ln=0 , n− = Ln
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