Revisiting the Ramond sector of the $\mathcal{N}=1$ superconformal minimal models

Key to the exact solubility of the unitary minimal models in two-dimensional conformal field theory is the organization of their Hilbert space into Verma modules, whereby all eigenstates of the Hamiltonian are obtained by the repeated action of Virasoro lowering operators onto a finite set of highest-weight states. The usual representation-theoretic approach to removing from all modules zero-norm descendant states generated in such a way is based on the assumption that those states form a nested sequence of Verma submodules built upon singular vectors, i.e., descendant highest-weight states. We show that this fundamental assumption breaks down for the Ramond-sector Verma module with highest weight $c/24$ in the even series of $\mathcal{N}=1$ superconformal minimal models with central charge $c$. To resolve this impasse, we conjecture, and prove at low orders, the existence of a nested sequence of linear-dependence relations that enables us to compute the character of the irreducible $c/24$ module. Based on this character formula, we argue that imposing modular invariance of the torus partition function requires the introduction of a non-null odd-parity Ramond-sector ground state. This symmetrization of the ground-state manifold allows us to uncover a set of conformally invariant boundary conditions not previously discussed and absent in the odd series of superconformal minimal models, and to derive for the first time a complete set of fusion rules for the even series of those models.

Figure 1

Examples illustrating four types of Kac tables that characterize unitary minimal models and their $\mathcal{N}=1$ supersymmetric generalization. The first three types have an even number of entries, half of which are redundant (white boxes), while the Kac table for even-$m$ SMMs has an odd number of entries and no redundancy.

Figure 2

Schematic Hilbert-space structure for even-$m$ SMMs. The commonly held viewpoint (left) leads to inconsistency with modular invariance, which can be resolved by introducing an additional, odd-parity Ramond ground state (right).