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Restricted (2+1)-TQFTs supported by thickened and solid tori

Dušan Đorđević, Danica Kosanović, Jovana Nikolić, Zoran Petrić

TL;DR

The paper investigates the existence of faithful $(2+1)$-TQFTs by restricting to subcategories of $3\mathsf{Cob}$ generated by tori, enabling linear representations via $SL(2,\mathbb{Z})$ data to bypass known obstructions from the full mapping class group. It develops two subcategories, $\beta\mathsf{MCG}$ and $\beta\varepsilon\mathsf{MCG}$, and provides a universal framework for torus-related objects through $T^{\beta}$- and $T^{\beta\varepsilon}$-objects, with explicit generators and relations. The authors construct concrete restricted TQFTs $F_1$, $F_2$, and $F_3$ from low-dimensional $SL(2,\mathbb{Z})$ representations, and demonstrate that these theories distinguish pairs of torus bundles and certain lens spaces that are indistinguishable by standard quantum invariants. While these restricted theories reveal nontrivial separation power (including across Funar-type examples), some are not faithful due to inherent representation-theoretic limitations, and the authors outline future work toward broader classes of 3-manifolds, such as graph manifolds. Overall, the work provides a concrete, representation-theoretic pathway to realizing informative, restricted $(2+1)$-TQFTs with clear topological distinguishing power.

Abstract

A faithful $(1+1)$ TQFT has recently been constructed, but the existence of a faithful $(2+1)$ TQFT remains an open question, that subsumes the hard problem of linearity of mapping class groups of surfaces. To circumvent the latter problem we construct a subcategory of the category of 3-cobordisms, containing disjoint unions of tori and simplest cobordisms between them. On this we define TQFTs that are able to distinguish pairs of torus bundles and lens spaces, previously shown not to be distinguishable by quantum invariants.

Restricted (2+1)-TQFTs supported by thickened and solid tori

TL;DR

The paper investigates the existence of faithful -TQFTs by restricting to subcategories of generated by tori, enabling linear representations via data to bypass known obstructions from the full mapping class group. It develops two subcategories, and , and provides a universal framework for torus-related objects through - and -objects, with explicit generators and relations. The authors construct concrete restricted TQFTs , , and from low-dimensional representations, and demonstrate that these theories distinguish pairs of torus bundles and certain lens spaces that are indistinguishable by standard quantum invariants. While these restricted theories reveal nontrivial separation power (including across Funar-type examples), some are not faithful due to inherent representation-theoretic limitations, and the authors outline future work toward broader classes of 3-manifolds, such as graph manifolds. Overall, the work provides a concrete, representation-theoretic pathway to realizing informative, restricted -TQFTs with clear topological distinguishing power.

Abstract

A faithful TQFT has recently been constructed, but the existence of a faithful TQFT remains an open question, that subsumes the hard problem of linearity of mapping class groups of surfaces. To circumvent the latter problem we construct a subcategory of the category of 3-cobordisms, containing disjoint unions of tori and simplest cobordisms between them. On this we define TQFTs that are able to distinguish pairs of torus bundles and lens spaces, previously shown not to be distinguishable by quantum invariants.

Paper Structure

This paper contains 20 sections, 51 theorems, 116 equations, 2 figures.

Table of Contents

  1. Preliminaries
  2. Conventions
  3. Useful facts about (self-)gluing
  4. The mapping class group
  5. Torus bundles
  6. Some cobordisms and relations between them
  7. Cylinders as cobordisms
  8. Interactions of cylinders and (co)pairings
  9. Torus bundles as cobordisms
  10. Solid tori as cobordisms
  11. Lens spaces as cobordisms
  12. Two subcategories of Cob
  13. Tb-objects
  14. Tbe-objects
  15. Universality
  16. ...and 5 more sections

Key Result

Theorem 1

There is a subcategory $\beta\varepsilon\mathsf{MCG}<3\mathsf{Cob}$ that contains all torus bundles and lens spaces, and such that a symmetric monoidal functor $F:\beta\varepsilon\mathsf{MCG}\to\mathsf{Vec}_{\mathbb{K}}$ is completely described by the data: that satisfy: for the identity matrix $E_{n_F}\in GL(n_F,\mathbb{K})$ and $\tau=\left( \right)\in GL(2n_F,\mathbb{K})$.

Figures (2)

  • Figure 1: The right-handed Dehn twists on $a$ and $b$.
  • Figure 2: A diagrammatic proof of \ref{['eq:Tb-object-gamma']}, with red $\rho^\mathbb{O}_{D_a}$ and blue $\rho^\mathbb{O}_{D_b}$.

Theorems & Definitions (107)

  • Theorem 1: Corollary \ref{['cor:TQFT']}
  • Theorem 2: Theorem \ref{['thm:F3']}
  • Lemma 1.1: g-Collar
  • proof
  • Corollary 1.2
  • Corollary 1.3
  • Lemma 1.4: Self-Gluing
  • proof
  • Proposition 1.5
  • Remark 1.6
  • ...and 97 more