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String-net model of Turaev-Viro invariants

Alexander Kirillov

TL;DR

The paper proves that for a spherical fusion category $\mathcal{A}$, the Turaev–Viro TV state spaces $Z_{TV}(\Sigma)$ for closed surfaces are canonically isomorphic to the string-net spaces $H^{string}(\Sigma)$ of Levin–Wen, establishing a concrete equivalence between TV TQFT and the Levin–Wen string-net model. It then extends this equivalence to surfaces with boundary, showing that boundary conditions are governed by the Drinfeld center $Z(\mathcal{A})$, and introduces an extended theory of excited boundary states labeled by objects of $Z(\mathcal{A})$ via a boundary category $\mathcal{C}(\partial\Sigma)$. The work relies on a detailed analysis of the Drinfeld center, adjunctions, and projector constructions to relate TV cylinders to string-net boundary operators. Together, these results provide a complete mathematical bridge between TV invariants and string-net constructions, including boundary excitations and edge theories with applications to topological phases and anyon models.

Abstract

In this paper, we describe the relation between the Turaev--Viro TQFT and the string-net space introduced in the papers of Levin and Wen. In particular, the case of surfaces with boundary is considered in detail.

String-net model of Turaev-Viro invariants

TL;DR

The paper proves that for a spherical fusion category , the Turaev–Viro TV state spaces for closed surfaces are canonically isomorphic to the string-net spaces of Levin–Wen, establishing a concrete equivalence between TV TQFT and the Levin–Wen string-net model. It then extends this equivalence to surfaces with boundary, showing that boundary conditions are governed by the Drinfeld center , and introduces an extended theory of excited boundary states labeled by objects of via a boundary category . The work relies on a detailed analysis of the Drinfeld center, adjunctions, and projector constructions to relate TV cylinders to string-net boundary operators. Together, these results provide a complete mathematical bridge between TV invariants and string-net constructions, including boundary excitations and edge theories with applications to topological phases and anyon models.

Abstract

In this paper, we describe the relation between the Turaev--Viro TQFT and the string-net space introduced in the papers of Levin and Wen. In particular, the case of surfaces with boundary is considered in detail.

Paper Structure

This paper contains 8 sections, 22 theorems, 61 equations, 15 figures.

Table of Contents

  1. Spherical categories: an overview
  2. Colored graphs
  3. String nets
  4. Turaev-Viro model
  5. TV=string nets
  6. Category of boundary values
  7. Extended theory: excited states
  8. Drinfeld center: technical lemmas

Key Result

Theorem 2.3

There is a unique way to assign to every colored planar graph $\Gamma$ in a disk $D\subset \mathbb{R}^2$ a vector where $\mathbf{e}_1,\dots, \mathbf{e}_n$ are the edges of $\Gamma$ meeting the boundary of $D$ (legs), taken in counterclockwise order and with outgoing orientation, so that that following conditions are satisfied: Moreover, so defined $\langle\Gamma\rangle$ satisfies the following p

Figures (15)

  • Figure 1: Labeling of colored graphs
  • Figure 2: Colored graph as a tangle
  • Figure 5: Spherical property
  • Figure 6: Operator $B_p$
  • Figure 7: State space for a cell
  • ...and 10 more figures

Theorems & Definitions (42)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Definition 3.1
  • Remark 3.2
  • Example 3.3
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • proof
  • ...and 32 more