1. Hypercubic structures behind $\hat{Z}$-invariants
- Author
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Sugimoto, Shoma
- Subjects
Mathematics - Representation Theory - Abstract
We propose an abelian categorification of $\hat{Z}$-invariants for Seifert $3$-manifolds. First, we give a recursive combinatorial derivation of these $\hat{Z}$-invariants using graphs with certain hypercubic structures. Next, we consider such graphs as annotated Loewy diagrams in an abelian category, allowing non-split extensions by the ambiguity of embedding of subobjects. If such an extension has good algebraic group actions, then the above derivation of $\hat{Z}$-invariants in the Grothendieck group of the abelian category can be understood in terms of the theory of shift systems, i.e., Weyl-type character formula of the nested Feigin-Tipunin constructions. For the project of developing the dictionary between logarithmic CFTs and 3-manifolds, these discussions give a glimpse of a hypothetical and prototypical, but unified construction/research method for the former from the new perspective, reductions of representation theories by recursive structures., Comment: 18 pages
- Published
- 2025