On the Strong Unital Property for the Affine VOAs

TL;DR

This paper investigates the strong unital property of mode transition algebras in affine vertex operator algebras, providing explicit constructions for some cases and proving non-unitality at all levels except the critical one.

Contribution

It constructs explicit strong units for the affine VOA of sl_2 at level 1 and proves that such algebras are not strongly unital at any other level.

Findings

Explicit strong units constructed for L_{sl_2}(1,0)

Mode transition algebras are not strongly unital at levels other than -2

Implications for sheaves of conformal blocks as vector bundles

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 $-2$.