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Spin Modular Categories

Anna Beliakova, Christian Blanchet, Eva Contreras

TL;DR

This work develops an algebraic framework to refine WRT quantum 3-manifold invariants using spin and cohomological data encoded by invertible objects in modular categories. It introduces Spin^c_d and generalized Spin_d structures, with combinatorial models and refined Kirby moves, and shows how these refinements yield topological invariants for manifolds equipped with extra structures. A central result is a decomposition formula: refined invariants factor into a reduced-category invariant and a Murakami–Ohtsuki–Okada-type factor, generalizing known decompositions for quantum invariants. The paper further connects ribbon category theory to modular categories through invertible-object gradings, defines spin and complex-spin refinements in various settings, and discusses potential extensions to spin-type TQFTs.

Abstract

Modular categories are a well-known source of quantum 3-manifold invariants. In this paper we study structures on modular categories which allow to define refinements of quantum 3-manifold invariants involving cohomology classes or generalized spin and complex spin structures. A crucial role in our construction is played by objects which are invertible under tensor product. All known examples of cohomological or spin type refinements of the Witten-Reshetikhin-Turaev 3-manifold invariants are special cases of our construction. In addition, we establish a splitting formula for the refined invariants, generalizing the well-known product decomposition of quantum invariants into projective ones and those determined by the linking matrix.

Spin Modular Categories

TL;DR

This work develops an algebraic framework to refine WRT quantum 3-manifold invariants using spin and cohomological data encoded by invertible objects in modular categories. It introduces Spin^c_d and generalized Spin_d structures, with combinatorial models and refined Kirby moves, and shows how these refinements yield topological invariants for manifolds equipped with extra structures. A central result is a decomposition formula: refined invariants factor into a reduced-category invariant and a Murakami–Ohtsuki–Okada-type factor, generalizing known decompositions for quantum invariants. The paper further connects ribbon category theory to modular categories through invertible-object gradings, defines spin and complex-spin refinements in various settings, and discusses potential extensions to spin-type TQFTs.

Abstract

Modular categories are a well-known source of quantum 3-manifold invariants. In this paper we study structures on modular categories which allow to define refinements of quantum 3-manifold invariants involving cohomology classes or generalized spin and complex spin structures. A crucial role in our construction is played by objects which are invertible under tensor product. All known examples of cohomological or spin type refinements of the Witten-Reshetikhin-Turaev 3-manifold invariants are special cases of our construction. In addition, we establish a splitting formula for the refined invariants, generalizing the well-known product decomposition of quantum invariants into projective ones and those determined by the linking matrix.

Paper Structure

This paper contains 31 sections, 31 theorems, 94 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Complex spin structures mod d
  3. ${\operatorname{Spin}}^c$--structures
  4. The Chern map
  5. $d$--complex spin structures
  6. Restriction to the boundary
  7. From ${\operatorname{Spin}}^c$ to ${\operatorname{Spin}}^c_d$
  8. Combinatorial description of ${\operatorname{Spin}}^c_d$--structures
  9. ${\operatorname{Spin}}^c_d$ Kirby moves
  10. Generalized spin structures
  11. Combinatorial description of ${\operatorname{Spin}}_d$--structures
  12. $(K,v)$-spin structures
  13. ${\operatorname{Spin}}_d$ Kirby moves
  14. From ribbon to modular categories
  15. Ribbon categories
  16. ...and 16 more sections

Key Result

Theorem 1

Any $H$--spin modular category ${\mathcal{C}}$ with associated spin character $v\in \mathrm{Tor}_2(\widehat{H})$ provides a topological invariant $\tau_{\mathcal{C}}(M,\sigma)$ of a pair $(M,\sigma)$, where $\sigma$ is a $(\widehat{H},v)$ generalized spin structure on $M$. Moreover,

Figures (6)

  • Figure 2.1: Refined ${\operatorname{Spin}}^c_d$--Kirby moves (a) Stabilization (b) Handle slide (c) Orientation reversal. Note that we use the blackboard framing and the labels refer to modulo $d$ Chern vectors.
  • Figure 3.1: Refined ${\operatorname{Spin}}_d$--Kirby moves (a) Stabilization (b) Handle slide (c) Orientation reversal. Note that we use the blackboard framing and the labels refer to modulo $d$ characteristic solutions of $L$.
  • Figure 4.1: Morphisms in $\operatorname{Rib}_{\mathcal{C}}$
  • Figure 4.2: $(d_V\otimes {\operatorname{id}}_{V^*})({\operatorname{id}}_{V^*}\otimes c_V)={\operatorname{id}}_{V^*}$
  • Figure 4.3: The Hopf link with linking number $+1$ and colors $\lambda$ and $\mu$
  • ...and 1 more figures

Theorems & Definitions (65)

  • Definition
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Example
  • Example
  • Definition : Ozbagci:2004
  • Lemma
  • Definition
  • Proposition
  • ...and 55 more