The mirror extension conjecture for vertex operator algebras

Let $V$ be a rational and $C_2$-cofinite vertex operator algebra, and let $U$ be a rational and $C_2$-cofinite vertex operator subalgebra of $V$. Denote by $U^c$ the commutant vertex operator algebra of $U$ in $V$. Assume that

$$(U^c)^c=U.$$

Suppose

$$V=U\otimes U^c\bigoplus \left(\oplus_{i=1}^n U_i\otimes U^c_i\right),$$

where the $U_i$ and $U^c_i$ are irreducible modules for $U$ and $U^c$, respectively. If

$$U^e=U\bigoplus \left(\oplus_{i=1}^n m_iU_i\right)$$

is a rational vertex operator algebra, where $m_i\geq 0$, then

$$(U^c)^e=U^c\bigoplus \left(\oplus_{i=1}^n m_iU^c_i\right)$$

is also a rational vertex operator algebra.

Mirror extension conjecture. Under these hypotheses, rationality of the extension $U^e$ implies rationality of the mirror extension $(U^c)^e$.

This is the vertex operator algebra analogue of a mirror-extension theorem for completely rational conformal nets. The conjecture concerns transferring rationality across the commutant pair $U$ and $U^c$ using the same multiplicities $m_i$ in the two extensions; its resolution is not supplied here.