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Topological Orders in (4+1)-Dimensions

Theo Johnson-Freyd, Matthew Yu

TL;DR

The paper develops a Morita/Witt-based framework to classify $(4+1)$-dimensional topological orders by condensing line operators to reduce to surface data and analyzing the resulting sylleptic 2-categories. It demonstrates a sharp dichotomy: all $(4+1)$-d super topological orders admit gapped boundaries (Morita trivial), while bosonic $(4+1)$-d orders form an infinite family of Morita-inequivalent phases, with many chiral ones that enforce gapless boundary modes unless fermionization or spin structure is applied. The analysis leverages symplectic structures on surface-operator groups, Lagrangian subgroups, and boundary-bulk correspondences via sylleptic centers and SH$^6$ cohomology, supported by spectral sequence calculations. Overall, the work clarifies how fermionic and bosonic higher-dimensional topological orders differ in their boundary behavior and Morita classifications, aligning fermionic results with spin-cobordism expectations while revealing a richly structured bosonic landscape.

Abstract

We investigate the Morita equivalences of (4+1)-dimensional topological orders. We show that any (4+1)-dimensional super (fermionic) topological order admits a gapped boundary condition -- in other words, all (4+1)-dimensional super topological orders are Morita trivial. As a result, there are no inherently gapless super (3+1)-dimensional theories. On the other hand, we show that there are infinitely many algebraically Morita-inequivalent bosonic (4+1)-dimensional topological orders.

Topological Orders in (4+1)-Dimensions

TL;DR

The paper develops a Morita/Witt-based framework to classify -dimensional topological orders by condensing line operators to reduce to surface data and analyzing the resulting sylleptic 2-categories. It demonstrates a sharp dichotomy: all -d super topological orders admit gapped boundaries (Morita trivial), while bosonic -d orders form an infinite family of Morita-inequivalent phases, with many chiral ones that enforce gapless boundary modes unless fermionization or spin structure is applied. The analysis leverages symplectic structures on surface-operator groups, Lagrangian subgroups, and boundary-bulk correspondences via sylleptic centers and SH cohomology, supported by spectral sequence calculations. Overall, the work clarifies how fermionic and bosonic higher-dimensional topological orders differ in their boundary behavior and Morita classifications, aligning fermionic results with spin-cobordism expectations while revealing a richly structured bosonic landscape.

Abstract

We investigate the Morita equivalences of (4+1)-dimensional topological orders. We show that any (4+1)-dimensional super (fermionic) topological order admits a gapped boundary condition -- in other words, all (4+1)-dimensional super topological orders are Morita trivial. As a result, there are no inherently gapless super (3+1)-dimensional theories. On the other hand, we show that there are infinitely many algebraically Morita-inequivalent bosonic (4+1)-dimensional topological orders.

Paper Structure

This paper contains 10 sections, 10 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. 5-dimensional Super Topological Orders
  3. Condensing out the lines
  4. Sylleptic and symplectic groups: bosonic case
  5. Sylleptic and symplectic groups: fermionic case
  6. 5d Topological order from the boundary
  7. Symplectic Reduction to Isotropic Subspaces
  8. Lagrangian subgroups as boundary theories
  9. Morita trivial 5d phases
  10. Bosonic 5-dimensional Topological Orders

Key Result

Theorem 2.2

If $\mathcal{C}$ is a (super) fusion $2$-category with $\Omega\mathcal{C} \cong \mathbf{SVec}$, then every indecomposable object of $\mathcal{C}$ is invertible. The equivalence classes of indecomposable objects in $\mathcal{C}$ form a finite group, which is a central double cover of the group $\pi_0

Figures (2)

  • Figure 1: The two domed cylinders in red and blue represent two objects $X,Y \in \mathcal{C}$ respectively, living in four dimensions. The purple coloured regions show the domes of the objects. Initially, we can think of one object being above the other. The dashed lines indicate places where the two sheets pass over each other in the fourth dimension, with the colour indicating which is above. The two marked points show where one of the surfaces crosses over the other in the fifth dimension, changing the order of which surface is above and below. The change in color of the dotted circle represents the fact that after the syllepsis, the object which was initially on top, is now on the bottom.
  • Figure 2: The wall is braided fusion 2-category with objects in $L^\perp$, separating the original theory $\mathcal{A}$ from the vacuum. Similar to the case of quantum Hamiltonian reductions, the wall is a bimodule for the two categories on either side.

Theorems & Definitions (31)

  • Remark
  • Definition 2.1: johnsonfreyd2020fusion
  • Theorem 2.2: johnsonfreyd2020fusion
  • Proposition 2.3
  • proof
  • Remark
  • Theorem 2.4
  • proof
  • Remark
  • Definition 3.1
  • ...and 21 more