Hypercubic structures behind -invariants
arXiv:2501.12985
Abstract
We propose an abelian categorification of -invariants for Seifert -manifolds. First, we give a recursive combinatorial derivation of these -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 -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.
18 pages