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Towards a complexity-theoretic dichotomy for TQFT invariants

Nicolas Bridges, Eric Samperton

TL;DR

The paper establishes a sharp dichotomy for exact computations of (2+1)-D TQFT invariants: for fixed spherical or modular fusion categories, either the invariant on closed 3-manifolds is computable in polynomial time or the associated tensor contractions are \\#\\mathsf{P}-hard. This is achieved by reducing TVBW and RT state-sum computations to Cai-Chen's weighted #CSP dichotomy, thereby linking TQFT invariants to constraint satisfaction problems. While the results do not yet yield an outright 3-manifold level dichotomy, they provide a foundational step by showing that the relevant computational tasks are either easy or hard in a precise sense, with explicit constructions of the constraint families from category data. The work also outlines substantial future directions, including making the dichotomy effective for invariants themselves, extending the approach to higher dimensions, and exploring connections to holant problems and a broader anyon classification program.

Abstract

We show that for any fixed $(2+1)$-dimensional TQFT over $\mathbb{C}$ of either Turaev-Viro-Barrett-Westbury or Reshetikhin-Turaev type, the problem of (exactly) computing its invariants on closed 3-manifolds is either solvable in polynomial time, or else it is $\#\mathsf{P}$-hard to (exactly) contract certain tensors that are built from the TQFT's fusion category. Our proof is an application of a dichotomy result of Cai and Chen [J. ACM, 2017] concerning weighted constraint satisfaction problems over $\mathbb{C}$. We leave for future work the issue of reinterpreting the conditions of Cai and Chen that distinguish between the two cases (i.e. $\#\mathsf{P}$-hard tensor contractions vs. polynomial time invariants) in terms of fusion categories. We expect that with more effort, our reduction can be improved so that one gets a dichotomy directly for TQFTs' invariants of 3-manifolds rather than more general tensors built from the TQFT's fusion category.

Towards a complexity-theoretic dichotomy for TQFT invariants

TL;DR

The paper establishes a sharp dichotomy for exact computations of (2+1)-D TQFT invariants: for fixed spherical or modular fusion categories, either the invariant on closed 3-manifolds is computable in polynomial time or the associated tensor contractions are \\#\\mathsf{P}-hard. This is achieved by reducing TVBW and RT state-sum computations to Cai-Chen's weighted #CSP dichotomy, thereby linking TQFT invariants to constraint satisfaction problems. While the results do not yet yield an outright 3-manifold level dichotomy, they provide a foundational step by showing that the relevant computational tasks are either easy or hard in a precise sense, with explicit constructions of the constraint families from category data. The work also outlines substantial future directions, including making the dichotomy effective for invariants themselves, extending the approach to higher dimensions, and exploring connections to holant problems and a broader anyon classification program.

Abstract

We show that for any fixed -dimensional TQFT over of either Turaev-Viro-Barrett-Westbury or Reshetikhin-Turaev type, the problem of (exactly) computing its invariants on closed 3-manifolds is either solvable in polynomial time, or else it is -hard to (exactly) contract certain tensors that are built from the TQFT's fusion category. Our proof is an application of a dichotomy result of Cai and Chen [J. ACM, 2017] concerning weighted constraint satisfaction problems over . We leave for future work the issue of reinterpreting the conditions of Cai and Chen that distinguish between the two cases (i.e. -hard tensor contractions vs. polynomial time invariants) in terms of fusion categories. We expect that with more effort, our reduction can be improved so that one gets a dichotomy directly for TQFTs' invariants of 3-manifolds rather than more general tensors built from the TQFT's fusion category.

Paper Structure

This paper contains 14 sections, 3 theorems, 30 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Mapping the dichotomy
  4. Implications for "anyon classification"
  5. Outline
  6. Proof of Theorem \ref{['thm:main']}
  7. Cai and Chen's dichotomy for weighted CSPs
  8. Proof of Theorem \ref{['thm:main']}(a)
  9. Proof of Theorem \ref{['thm:main']}(b)
  10. Discussion
  11. Unitarity
  12. TQFTs in other dimensions
  13. Alternative proof strategies
  14. Towards Conjecture \ref{['con:main']}

Key Result

Theorem 1

Figures (11)

  • Figure 1: A 3j+1k-symbol
  • Figure 2: A trivalent graph in $\mathbb{R}^2$ which exhibits all three types of edges (it looks like a person waving!)
  • Figure 3: A portion of a crossed trivalent graph
  • Figure 4: Circle Removal
  • Figure 5: Tadpole Trim
  • ...and 6 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Conjecture 2
  • Theorem 3: Thm. 1, CaiChen-dichotomy
  • Lemma 4
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}(b)