Interpretation of the Determinant Formulae for the Chiral Representations of the N=2 Superconformal Algebra

TL;DR

This paper interprets the determinant formulae for N=2 superconformal algebra representations, applying them to incomplete modules built on special states, and characterizes the spectrum of singular states including novel non-highest weight states.

Contribution

It provides a new interpretation of the N=2 determinant formulae for incomplete modules and characterizes the spectrum of singular states on chiral and Ramond ground states.

Findings

Zeroes of the determinant formulae correspond to singular states with modified level and charge interpretation.

Half of the zeroes relate to uncharged singular states, the other half to charged states.

Identifies non-highest weight singular states not derived from highest weight states.

Abstract

We show that the N=2 determinant formulae of the Aperiodic NS algebra and the Periodic R algebra can be applied directly to incomplete Verma modules built on chiral primary states and on Ramond ground states, respectively, provided one modifies the interpretation of the zeroes in an appropriate way. That is, the zeroes of the determinat formulae account for the highest weight singular states built on chiral primaries and on Ramond ground states, but the identification of the levels and relative U(1) charges of the singular states is different than for complete Verma modules. In particular, half of the zeroes of the quadratic vanishing surfaces $f^A_{r,s}=0$ and $f^P_{r,s}=0$ correspond to uncharged singular states, and the other half correspond to charged singular states. We derive the spectrum of singular states built on chiral primaries, including the singular states of the Twisted Topological algebra, and the spectrum of singular states built on the Ramond ground states. We also uncover the existence of non-highest weight singular states which are not secondary of any highest weight singular state.