On the Strong Unital Property for the Affine VOAs
Angela Cai
TL;DR
The paper investigates when mode-transition algebras $\mathfrak{A}_d$ of a VOA admit strong units, linking this property to smoothing of conformal-block sheaves and to an equivalence with Zhu’s algebra. It shows that for the vacuum module VOA $V_{\hat{\mathfrak{sl}}_2}(k,0)$ there is no strong unit in $\mathfrak{A}_1$ for any non-critical level $k \neq -2$, while the rational simple quotient $L_{\hat{\mathfrak{sl}}_2}(k,0)$ remains compatible with strong unitality. The authors establish a continuous isomorphism $\mathscr{U}(L_{\hat{\mathfrak{sl}}_2}(1,0)) \cong \widetilde{U}(\widehat{\mathfrak{sl}}_2,1)/\overline{\langle e(-1)e(-1)\rangle}$, enabling explicit presentations of the enveloping algebra and its relations. Finally, they construct explicit strong units $\mathscr{I}_d$ for the $d$-th mode-transition algebras of $L_{\hat{\mathfrak{sl}}_2}(1,0)$ and prove their two-sided action, establishing strong unitality in this affine module VOA case. This advances understanding of when mode-transition algebras behave well enough to yield vector-bundle conformal-block structures and clarifies the role of quotients in restoring strong unitality.
Abstract
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 $-2$.