An abelian categorification of $\hat{Z}$-invariants

The $\hat{Z}$-invariant is a $q$-series valued quantum invariant for (negative definite plumbed) 3-manifolds introduced by Gukov–Pei–Putrov–Vafa in 2017. It provides not only a $q$-expansion of the Witten–Reshetikhin–Turaev invariant, but also rich examples of "spoiled" modular forms such as mock/false theta functions. The latter fact suggests the existence of non-rational vertex operator algebras (log VOAs) with $\hat{Z}$-invariants as their $q$-characters. However, the study of log VOAs is still underdeveloped, and no examples of such log VOAs have been found so far except for the two easiest cases (3- or 4-leg star graphs). This talk will outline the "nested Feigin–Tipunin construction" introduced and developed by the speaker to provide a unified construction/research methodology of the above correspondence between log VOAs and (negative definite plumbed) 3-manifolds. It enables us to construct and study the abelian category of modules over the hypothetical log VOAs via the recursive application of the purely Lie algebraic geometric representation theory of FT construction. In particular, the corresponding $\hat{Z}$-invariants are reconstructed in the Grothendieck group via the recursive application of the Weyl-type character formula. From a theoretical physics perspective, the nested FT construction can be viewed as the algebraic counterpart to the contribution from 3d $\mathcal{N}=2$ theory in the $\hat{Z}$-invariants.