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Extended TQFT arising from enriched multi-fusion categories

Hao Zheng

TL;DR

The paper develops an enriched multi-fusion framework to extend the Reshetikhin–Turaev TQFT to dimension zero by encoding modular tensor categories as centers of enriched multi-fusion categories. It constructs a symmetric monoidal (4,3)-category of enriched IUMFCs, showing that every object yields a fully extended 3D TQFT $\mathcal{Z}_{\mathfrak{C}}$, with duals at all levels. The 1-2-3D content of $\mathcal{Z}_{\mathfrak{C}}$ combines the RT TQFT $\mathcal{Z}^{RT}_{\mathcal{C}}$ and the 4D Crane–Yetter framework $\mathcal{A}_{\mathcal{C}}$, while the double theory recovers the Turaev–Viro theory $\mathcal{Z}^{TV}_{\mathcal{C}}$, aligning zero- and higher-dimensional invariants. The work provides evidence for an $SO(3)$-fixed point structure for all objects in the enriched setting and links to gapped/gapless edge descriptions, suggesting broad physical relevance for topological phases and Chern–Simons-type theories.

Abstract

We define a symmetric monoidal (4,3)-category with duals whose objects are certain enriched multi-fusion categories. For every modular tensor category $\mathcal{C}$, there is a self enriched multi-fusion category $\mathfrak{C}$ giving rise to an object of this symmetric monoidal (4,3)-category. We conjecture that the extended 3D TQFT given by the fully dualizable object $\mathfrak{C}$ extends the 1-2-3-dimensional Reshetikhin-Turaev TQFT associated to the modular tensor category $\mathcal{C}$ down to dimension zero.

Extended TQFT arising from enriched multi-fusion categories

TL;DR

The paper develops an enriched multi-fusion framework to extend the Reshetikhin–Turaev TQFT to dimension zero by encoding modular tensor categories as centers of enriched multi-fusion categories. It constructs a symmetric monoidal (4,3)-category of enriched IUMFCs, showing that every object yields a fully extended 3D TQFT , with duals at all levels. The 1-2-3D content of combines the RT TQFT and the 4D Crane–Yetter framework , while the double theory recovers the Turaev–Viro theory , aligning zero- and higher-dimensional invariants. The work provides evidence for an -fixed point structure for all objects in the enriched setting and links to gapped/gapless edge descriptions, suggesting broad physical relevance for topological phases and Chern–Simons-type theories.

Abstract

We define a symmetric monoidal (4,3)-category with duals whose objects are certain enriched multi-fusion categories. For every modular tensor category , there is a self enriched multi-fusion category giving rise to an object of this symmetric monoidal (4,3)-category. We conjecture that the extended 3D TQFT given by the fully dualizable object extends the 1-2-3-dimensional Reshetikhin-Turaev TQFT associated to the modular tensor category down to dimension zero.

Paper Structure

This paper contains 4 sections, 12 theorems, 11 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Unitary categories
  3. Enriched IUMFC's and bimodules
  4. A symmetric monoidal (4,3)-category

Key Result

Theorem 2.1

Let $\mathcal{C},\mathcal{D}$ be UMFC's and let $\mathcal{M}$ be a $\mathcal{C}$-$\mathcal{D}$-bimodule. Then the $\mathcal{D}$-$\mathcal{C}$-bimodule $\mathcal{M}^\mathrm{op}$ is right dual to $\mathcal{M}$ with two duality maps $u$ and $v$ defined as follows: Since $(\mathcal{M}^\mathrm{op})^\mathrm{op} \simeq \mathcal{M}$ as $\mathcal{C}$-$\mathcal{D}$-bimodules, $\mathcal{M}^\mathrm{op}$ is a

Figures (7)

  • Figure 1: (a) depicts an enriched IUMFC $\mathfrak{C}=(\mathcal{C}_1,\mathcal{C}_2)$ where $\mathcal{C}_1$ admits a central action by $\mathcal{C}_2$. (b) depicts its reverse $\mathfrak{C}^\mathrm{rev}=(\mathcal{C}_1^\mathrm{rev},\bar{\mathcal{C}}_2)$. Keep in mind that the categorical constructions $\mathcal{C}\mapsto\bar{\mathcal{C}}$, $\mathcal{C}\mapsto\mathcal{C}^\mathrm{rev}$, $\mathcal{C}\mapsto\mathcal{C}^\mathrm{op}$ correspond to changing the orientations of 2,1,0-cells in figures.
  • Figure 2: (a) depicts a $\mathfrak{C}$-$\mathfrak{D}$-bimodule $\mathfrak{M}=(\mathcal{M}_0,\mathcal{M}_1)$ where $\mathcal{M}_1$ admits actions by $\mathcal{C}_2,\mathcal{D}_2$, and $\mathcal{M}_0$ admits actions by $\mathcal{C}_2,\mathcal{M}_1,\mathcal{D}_2,\mathcal{C}_1,\mathcal{D}_1$. (b) depicts the opposite $\mathfrak{D}$-$\mathfrak{C}$-bimodule $\mathfrak{M}^\mathrm{op}=(\mathcal{M}_0^\mathrm{op},\mathcal{M}_1^\mathrm{rev})$.
  • Figure 3:
  • Figure 4: This figure depicts a $\mathfrak{C}$-$\mathfrak{D}$-bimodule functor $\mathfrak{F}=(F,\mathcal{F})$.
  • Figure 5: A $\mathfrak{C}$-$\mathfrak{D}$-bimodule $\mathfrak{M}=(\mathcal{C}_1\boxtimes_{\mathcal{C}_2}\mathcal{M}_1,\mathcal{M}_1)$ is depicted on the left where $\mathfrak{C}=(\mathcal{C}_1,\mathcal{C}_2)$ and $\mathfrak{D}=(\mathcal{C}_1\boxtimes_{\mathcal{C}_2}\mathcal{M}_1,\mathcal{D}_2)$. The $\mathfrak{D}$-$\mathfrak{C}$-bimodule $\mathfrak{M}^\mathrm{op} \simeq (\mathcal{M}_1^\mathrm{op}\boxtimes_{\mathcal{C}_2}\mathcal{C}_1^\mathrm{op},\mathcal{M}_1^\mathrm{rev})$ is depicted on the right.
  • ...and 2 more figures

Theorems & Definitions (39)

  • Theorem 2.1
  • Theorem 2.2: KZ1 Theorem 3.3.6
  • Theorem 2.3: KZ1 Theorem 3.3.7
  • Remark 2.4
  • Corollary 2.5
  • proof
  • Corollary 2.6
  • proof
  • Corollary 2.7
  • proof
  • ...and 29 more