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Hypercubic structures behind $\hat{Z}$-invariants

Shoma Sugimoto

TL;DR

The paper develops an abelian categorification of $\\hat{Z}$-invariants for Seifert 3-manifolds by embedding these invariants into hypercubic, recursively structured DAGs and annotated Loewy diagrams. It introduces a shift-system formalism and Felder complexes to build nested Feigin–Tipunin constructions, transforming combinatorial recursion into an abelian-categorical framework that mirrors LCFT structures. A bosonic, recursive derivation yields a concrete formula for the $\\hat{Z}$-invariants in terms of $q$-series and holomorphic characters, connecting to known singlet/triplet LCFTs in low-rank cases and pointing toward a generalizable dictionary for higher-rank/complex Seifert graphs. Taken together, the work provides a principled, abelian-categorical route toward unifying aspects of logarithmic CFTs, quantum topology, and 3-manifold invariants, and outlines a concrete program for constructing LCFTs from hypercube-based recursions.

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.

Hypercubic structures behind $\hat{Z}$-invariants

TL;DR

The paper develops an abelian categorification of -invariants for Seifert 3-manifolds by embedding these invariants into hypercubic, recursively structured DAGs and annotated Loewy diagrams. It introduces a shift-system formalism and Felder complexes to build nested Feigin–Tipunin constructions, transforming combinatorial recursion into an abelian-categorical framework that mirrors LCFT structures. A bosonic, recursive derivation yields a concrete formula for the -invariants in terms of -series and holomorphic characters, connecting to known singlet/triplet LCFTs in low-rank cases and pointing toward a generalizable dictionary for higher-rank/complex Seifert graphs. Taken together, the work provides a principled, abelian-categorical route toward unifying aspects of logarithmic CFTs, quantum topology, and 3-manifold invariants, and outlines a concrete program for constructing LCFTs from hypercube-based recursions.

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.
Paper Structure (13 sections, 5 theorems, 69 equations, 1 table)

This paper contains 13 sections, 5 theorems, 69 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Technical summary
  4. A recursive combinatorial derivation of $\hat{Z}$-invariants
  5. Directed acyclic graphs
  6. Fragmentation
  7. Derivation of $\hat{Z}$-invariants
  8. Horizontal extension
  9. An abelian categorification of $\hat{Z}$-invariants
  10. Annotated Loewy diagrams
  11. Defragmentation
  12. Recursive shift systems
  13. Glimpse of LCFTs for Seifert $3$-manifolds

Key Result

Lemma 2.2

For $\lambda,\mu\in\mathbb{Z}_2^{m}$, let $\hat{\lambda},\hat{\lambda}',\hat{\mu},\hat{\mu}'\in\{+,-,\pm\}^{m}$ be elements such that for each $1\leq i\leq m$, Then we have the isomorphism where $\hat{\lambda}\cup\hat{\lambda}'$ and $\hat{\mu}\cup\hat{\mu}'$ defines the corresponding elements in $\{+,-,\pm\}^{m+m'}$. In other words, for $m_1=|\{i|\hat{\lambda}_i=\pm\}|$ and $m_2=|\{i|\hat{\mu}_i

Theorems & Definitions (27)

  • Remark 1.1
  • Remark 1.2
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 17 more