Nesting behind $\hat{Z}$-invariants

TL;DR

This paper introduces an abelian categorification of $\

Contribution

It offers a new categorical framework linking $\

Findings

Provides a recursive method to compute $\

Connects $\

Suggests a deep relationship between 3-manifold invariants and log VOAs

Abstract

In the spirit of arXiv:2501.12985, we propose an abelian categorification of $\hat{Z}$-invariants for negative definite plumbed 3-manifolds. It provides a blueprint for the expected dictionary between these $3$-manifolds and log VOAs; that is, the contribution from 3d $\mathcal{N}=2$ theory via 3d-3d correspondence is encoded as recursive and binary deviations from semisimplicity in the abelian category of modules over the hypothetical log VOA, and is decoded by the recursive application of the theory of Feigin--Tipunin construction. In particular, the nested Weyl-type character formulas provide virtual generalized characters reconstructing the $\hat{Z}$-invariants.