On the Strong Unital Property for the Affine VOAs

Representations of vertex operator algebras $V$ (VOAs) have numerous applications, including the construction of sheaves of conformal blocks on moduli spaces of curves. For a $V$-module $W = \oplus W_d$, a sequence of associative algebras $\mathfrak{A}_d$ acts on each graded component $W_d$. When these $d$th-mode transition algebras $\mathfrak{A}_d$ are strongly unital - meaning they are unital with units acting as the identity on $W_d$ - the associated sheaves of conformal blocks are vector bundles rather than merely coherent sheaves. This strong unital property, while difficult to verify in practice, has other important implications as well. Here we construct explicit strong units for $L_{\widehat{\mathfrak{sl}_2}}(1,0)$, the simple affine VOA for $\mathfrak{sl}_2$ at level $1$, and establish that mode transition algebras for universal affine VOAs for $\mathfrak{sl}_2$ are never strongly unital at any level $k$ not equal to the critical level $-2$.