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Gapped boundary theories in three dimensions

Daniel S. Freed, Constantin Teleman

TL;DR

<3-5 sentence high-level summary>The paper develops a fully extended 3-dimensional topological field theory framework to answer when a gapped (2+1)-dimensional quantum system can admit a gapped boundary. It proves a sharp obstruction: a Reshetikhin–Turaev theory has a nonzero boundary theory only if it is a TV theory, with the boundary data encoding a fusion category Φ via the Drinfeld center Z(Φ). This yields a concrete dualizability criterion for fusion categories and a Morita-invariant pathway to identify when RT theories arise from TV constructions; conversely, TV theories always admit a boundary theory built from their regular module. The results tie boundary existence to the algebraic structure of fusion categories and have physical implications for edge modes and central charge in gapped phases, providing a rigorous bridge between topological field theory and condensed-matter phenomenology.

Abstract

We prove a theorem in 3-dimensional topological field theory: a Reshetikhin-Turaev theory admits a nonzero boundary theory iff it is a Turaev-Viro theory. The proof immediately implies a characterization of fusion categories in terms of dualizability. The main theorem applies to physics, where it implies an obstruction to a gapped 3-dimensional quantum system admitting a gapped boundary theory. Appendices on bordism multicategories and on internal duals may be of independent interest.; v2 extensive revision: added theorem on dualizable 2-categories, material on natural transformations, reworked theorems and several proofs, and more.

Gapped boundary theories in three dimensions

TL;DR

<3-5 sentence high-level summary>The paper develops a fully extended 3-dimensional topological field theory framework to answer when a gapped (2+1)-dimensional quantum system can admit a gapped boundary. It proves a sharp obstruction: a Reshetikhin–Turaev theory has a nonzero boundary theory only if it is a TV theory, with the boundary data encoding a fusion category Φ via the Drinfeld center Z(Φ). This yields a concrete dualizability criterion for fusion categories and a Morita-invariant pathway to identify when RT theories arise from TV constructions; conversely, TV theories always admit a boundary theory built from their regular module. The results tie boundary existence to the algebraic structure of fusion categories and have physical implications for edge modes and central charge in gapped phases, providing a rigorous bridge between topological field theory and condensed-matter phenomenology.

Abstract

We prove a theorem in 3-dimensional topological field theory: a Reshetikhin-Turaev theory admits a nonzero boundary theory iff it is a Turaev-Viro theory. The proof immediately implies a characterization of fusion categories in terms of dualizability. The main theorem applies to physics, where it implies an obstruction to a gapped 3-dimensional quantum system admitting a gapped boundary theory. Appendices on bordism multicategories and on internal duals may be of independent interest.; v2 extensive revision: added theorem on dualizable 2-categories, material on natural transformations, reworked theorems and several proofs, and more.

Paper Structure

This paper contains 39 sections, 34 theorems, 113 equations, 30 figures.

Table of Contents

  1. Mathematical background and statement of main theorem
  2. RT theories and TV theories
  3. Definitions and terminology
  4. Relations between RT and TV theories
  5. Existence of boundary theories
  6. A characterization of fusion categories
  7. Preliminaries
  8. Bordism $n$-categories
  9. Arrows of time
  10. Constancy data
  11. Tangential structures
  12. Two conventions
  13. Duals and adjoints
  14. Coloring with a boundary theory
  15. Natural transformations
  16. ...and 24 more sections

Key Result

Theorem 1.4

$\mathop{\mathrm{Fus}}\nolimits$ has duals, i.e., $\mathop{\mathrm{Fus}}\nolimits=\mathop{\mathrm{Fus}}\nolimits^{\textnormal{fd}}$.

Figures (30)

  • Figure 1: A 2-morphism $C_0\amalg C_1\xrightarrow{\;Y\;}\emptyset ^1$ in $\mathop{\mathrm{Bord}}\nolimits_2$
  • Figure 2: A 1-morphism $P_0\amalg P_1\xrightarrow{\;C\;}\emptyset ^0$ in $\mathop{\mathrm{Bord}}\nolimits_2$
  • Figure 3: (a) a legal 2-morphism and (b) not a 2-morphism
  • Figure 4: A 2-framed 1-morphism
  • Figure 5: The standard points
  • ...and 25 more figures

Theorems & Definitions (107)

  • Remark 1.2
  • Theorem 1.4: Douglas-Schommer-Pries-Snyder DSS
  • Remark 1.6
  • Definition 1.7
  • Remark 1.8
  • Definition 1.10
  • Remark 1.11
  • Theorem 1.13
  • Remark 1.16
  • Theorem 1
  • ...and 97 more