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Fully local Reshetikhin-Turaev theories

Daniel S. Freed, Claudia I. Scheimbauer, Constantin Teleman

TL;DR

This work builds a rigorous framework to fully localize Reshetikhin–Turaev theories by enlarging the ambient 3-category of fusion categories. The core construction introduces a universal symmetric tensor 3-category $ ext{E} ext{F}$ (bosonic) and its fermionic analogue $ ext{E}S ext{F}$, obtained as direct sums over a $oldsymbol{ u}_6$-extension of the Witt group (and its super-version), so that every RT theory admits a fully local realization. Key features include full dualizability, a finite set of unit invertibles tied to tangential structures (notably $p_1$-structures), and a precise account of central charges modulo $24$, with Spin and SO invariance linked to modular and spherical structures. The paper also develops a spin-invariant formulation for fusion supercategories, analyzes central-charge liftings via complex $p_1$-structures, and discusses higher-categorical implications for 4D TQFTs, as well as the role of anomalous theories and boundary theories in this universal localization. Together, these results provide a robust, unified route to fully local RT theories and illuminate how tangential structures, spin data, and modularity cohere in fully local topological quantum field theories with potential 4D extensions.

Abstract

We define a symmetric tensor enhancement $\mathrm{E}\mathbb{F}$ with full duals of the 3-category $\mathbb{F}$ of fusion categories in which every Reshetikhin--Turaev theory has a fully local realization. Our $\mathrm{E}\mathbb{F}$ is a direct sum of invertible $\mathbb{F}$-modules, indexed by a $μ_6$-extension of the Witt group $W$ of non-degenerate braided fusion categories. Similarly, we enhance the 3-category $S\mathbb{F}$ of fusion super-categories to a symmetric tensor 3-category $\mathrm{E} S\mathbb{F}$ with full duals, which is a sum of invertible $S\mathbb{F}$-modules, indexed by an extension of the super-Witt group $SW$ with kernel the Pontrjagin dual of the stable stem $π_3^s$. The unit spectrum of $\mathrm{E}S\mathbb{F}$ is the connective cover of the Pontrjagin dual of $\mathbb{S}^{-3}$. We discuss tangential structures and central charges of the resulting TQFTs. We establish Spin-invariance of fusion supercategories and relate SO-invariance structures to modular and spherical structures. This confirms some conjectures from arxiv:1312.7188.

Fully local Reshetikhin-Turaev theories

TL;DR

This work builds a rigorous framework to fully localize Reshetikhin–Turaev theories by enlarging the ambient 3-category of fusion categories. The core construction introduces a universal symmetric tensor 3-category (bosonic) and its fermionic analogue , obtained as direct sums over a -extension of the Witt group (and its super-version), so that every RT theory admits a fully local realization. Key features include full dualizability, a finite set of unit invertibles tied to tangential structures (notably -structures), and a precise account of central charges modulo , with Spin and SO invariance linked to modular and spherical structures. The paper also develops a spin-invariant formulation for fusion supercategories, analyzes central-charge liftings via complex -structures, and discusses higher-categorical implications for 4D TQFTs, as well as the role of anomalous theories and boundary theories in this universal localization. Together, these results provide a robust, unified route to fully local RT theories and illuminate how tangential structures, spin data, and modularity cohere in fully local topological quantum field theories with potential 4D extensions.

Abstract

We define a symmetric tensor enhancement with full duals of the 3-category of fusion categories in which every Reshetikhin--Turaev theory has a fully local realization. Our is a direct sum of invertible -modules, indexed by a -extension of the Witt group of non-degenerate braided fusion categories. Similarly, we enhance the 3-category of fusion super-categories to a symmetric tensor 3-category with full duals, which is a sum of invertible -modules, indexed by an extension of the super-Witt group with kernel the Pontrjagin dual of the stable stem . The unit spectrum of is the connective cover of the Pontrjagin dual of . We discuss tangential structures and central charges of the resulting TQFTs. We establish Spin-invariance of fusion supercategories and relate SO-invariance structures to modular and spherical structures. This confirms some conjectures from arxiv:1312.7188.
Paper Structure (48 sections, 18 theorems, 61 equations, 4 figures, 2 tables)

This paper contains 48 sections, 18 theorems, 61 equations, 4 figures, 2 tables.

Table of Contents

  1. Background.
  2. RT theories from categories.
  3. Formulation of the problem.
  4. Results.
  5. Commentary.
  6. Super-Results.
  7. Caution on central charges.
  8. Four-categorical aspects.
  9. Orientations and spherical structures.
  10. Related prior work.
  11. Extended summary.
  12. Main result: bosonic case
  13. Main result: fermionic case
  14. Invertible TQFTs.
  15. Anomalous TQFTs and reduced central charge
  16. ...and 33 more sections

Key Result

Theorem 1

There exists a symmetric monoidal $3$-category $\mathrm{E}\mathbb{F} \equiv \bigoplus_{\widetilde{W}} \mathbb{F}_w$ satisfying properties (i)-(vii) in §results, unique up to isomorphism.

Figures (4)

  • Figure 1: The end $\partial S^2_F$ of the bulk squared-Serre reaching the boundary and its value in $Z(F)$
  • Figure 2: We have $\theta'\equiv 1$ on $\kappa\cdot W$ because of strict rotation-invariance
  • Figure 3: Trivializing the boundary Serre functor $S_M$
  • Figure 4: The unlabeled arm of the loop can be read as either ${}^* x$ or $x^*$.

Theorems & Definitions (62)

  • Remark 1
  • Theorem 1
  • Corollary 1.1
  • proof
  • Remark 1.7
  • Remark 1.8: Symmetry
  • Remark 1.9: Universality
  • Remark 1.10: Decomposition into simples
  • Remark 1.11: 'Fake fusion' calculus
  • Remark 1.12: $k$-invariant of $\mathrm{GL}_1(\mathrm{E}\mathbb{F})$
  • ...and 52 more