Rationality and dual-pair property of the commutant

Determine whether the commutant algebra \(C_N=\operatorname{Com}(P(N),V)\) is rational and whether the parafermion algebra \(P(N)\) and \(C_N\) form a dual pair of commuting subalgebras of \(V\), equivalently whether \(\operatorname{Com}(C_N,V)=P(N)\).

Background

The paper constructs parafermion embeddings P(N)⊂VP(N)\subset V and studies their commutants CN=Com⁡(P(N),V)C_N=\operatorname{Com}(P(N),V). The commutant is expected to have a representation theory closely related to that of the parafermion algebra, but the general structural properties required for this expectation are not established.

Specifically, the authors identify two unresolved issues: whether CNC_N is a rational VOA and whether taking the commutant twice recovers the original parafermion algebra. The latter would establish that P(N)P(N) and CNC_N constitute a dual pair of mutually commuting subalgebras.

References

There are two subtle questions about $C_N$ that we are not able to answer in general in this work: (1) whether $C_N$ is rational, and (2) whether $P(N)$ and $C_N$ form a dual pair of commuting subalgebras of $V$, namely if it is also true that $\Com(C_N, V)=P(N)$. The second question can be answered in particular cases, using some of the tools described in this section.

— Monstrous parafermionic defects and other non-invertible symmetries in chiral CFTs  (2609.03043 - Volpato, 2 Sep 2026) in Section 2.3, “Characters and representations of the commutant algebra”