The affine vertex algebra–small quantum group correspondence conjecture

Let

be an admissible affine vertex algebra at a principal admissible level $k$. Write

for its category of smooth weight representations with finite-dimensional weight spaces. Let

$$be the Langlands dual of$$

, and let

$$t=e^{\pi \sqrt{-1}(k+h^{\vee})},$$

where $h^{\vee}$ is the dual Coxeter number of

$$. **Affine vertex algebra–small quantum group correspondence conjecture.** The category$$

has enough projectives; more precisely, every simple object has a finite-length projective cover. Moreover, its principal block is equivalent to the principal block of the unrolled small quantum group

$$associated with$$

.

This conjecture proposes a precise relationship between weight representations of admissible affine vertex algebras and representations of unrolled small quantum groups. The surrounding discussion motivates it through the structure of weight-module categories for integrable and admissible affine vertex algebras; its resolution is not specified in the source.