---
title: "All once-extended 3D TQFTs are Reshetikhin--Turaev theories"
authors: ["Glen Lim"]
year: 2026
venue: "arXiv:math.AT"
arxiv: "2608.26068"
doi: null
source: "https://arxiv.org/abs/2608.26068"
topics: ["curiosity", "maths-foundations"]
seed_rank: 1546
seed_batch: "curiosity-2026-09-03"
reviewed: "2026-09-03"
pool: "maths-foundations"
relevance_score: 10
lineage: tqft
cites:
  - title: "All once-extended 3D TQFTs are Reshetikhin-Turaev theories"
    url: "https://arxiv.org/abs/2608.26068"
    year: 2026
    arxiv: "2608.26068"
    doi: null
see:
  - "1236-factorization-homology-of-topological-manifolds"
  - "1148-fault-tolerant-quantum-computation-by-anyons"
---

# All once-extended 3D TQFTs are Reshetikhin-Turaev theories

## One-sentence takeaway

A direct geometric circle-extension of Reshetikhin–Turaev matches Bartlett–Douglas–Schommer-Pries–Vicary, so every once-extended 3D TQFT is an RT theory.

## Why it's lovely

Why you might love this: the generators-and-relations classification of once-extended 3D TQFTs leaned on heavy Cerf theory. Lim gives a geometric RT-to-circles construction that agrees with BDSPV and closes the "are there exotic ones?" question — there are not. Classification by construction, not by exhaustion of handle slides.

## Key ideas

- Direct geometric extension of Reshetikhin–Turaev TQFT from surfaces to circles.
- Agreement with the Bartlett–Douglas–Schommer-Pries–Vicary generators-and-relations approach.
- Bypasses the Cerf-theoretic half of the BDSPV classification argument.
- Corollary: every once-extended 3-dimensional TQFT arises from this RT construction.

## Caveats

Once-extended (values on surfaces and circles), not fully extended to points. Oriented / anomaly-free setting of RT; does not invent new modular tensor categories. Read against BDSPV for the precise 2-categorical target.

## Implications for Broadside

Closes a folklore gap on the TQFT shelf next to factorization homology (1236) and anyon codes (1148). Diagrammatic / modular-tensor thinking remains the whole story in 3D once-extended land.

## Links

- arXiv abs: https://arxiv.org/abs/2608.26068
- PDF: https://arxiv.org/pdf/2608.26068.pdf