Anyonic membranes and Pontryagin statistics

TL;DR

This work generalizes the notion of anyonic statistics to higher-dimensional membrane excitations by introducing a 56-step unitary invariant $\mu_{56}$ that detects membrane statistics for $\mathbb{Z}$ and $\mathbb{Z}_N$ fusion in $d\ge4$ dimensions. The authors connect this lattice invariant to the boundary physics of higher-form ${\mathbb Z}_N$ SPT phases and show that the resulting anomalies are captured by cohomological operations, notably the first Pontryagin class modulo 3 (Pontryagin statistics) and Steenrod reduced powers, yielding a $\mathbb{Z}_3$-type structure that stabilizes in higher dimensions. They provide a rigorous proof that $\mu_{56}$ is a statistical process through local cancellation arguments, and they develop a computer-assisted framework to derive nontrivial instances of $\mu_{56}$ from cocycles, including explicit calculations up to $(7+1)$D boundaries. The results offer a unified perspective on generalized membrane statistics, higher-form anomalies, and their implications for boundary dynamics, cobordism classifications, and potential quantum-information applications in higher-dimensional topological codes.

Abstract

Anyons, unique to two spatial dimensions, underlie extraordinary phenomena such as the fractional quantum Hall effect, but their generalization to higher dimensions has remained elusive. The topology of Eilenberg-MacLane spaces constrains the loop statistics to be only bosonic or fermionic in any dimension. In this work, we introduce the novel anyonic statistics for membrane excitations in four dimensions. Analogous to the $\mathbb{Z}_N$-particle exhibiting $\mathbb{Z}_{N\times \gcd(2,N)}$ anyonic statistics in two dimensions, we show that the $\mathbb{Z}_N$-membrane possesses $\mathbb{Z}_{N\times \gcd(3,N)}$ anyonic statistics in four dimensions. Given unitary volume operators that create membrane excitations on the boundary, we propose an explicit 56-step unitary sequence that detects the membrane statistics. We further analyze the boundary theory of $(5{+}1)$D 1-form $\mathbb{Z}_N$ symmetry-protected topological phases and demonstrate that their domain walls realize all possible anyonic membrane statistics. We then show that the $\mathbb{Z}_3$ subgroup persists in all higher dimensions. In addition to the standard fermionic $\mathbb{Z}_2$ membrane statistics arising from Stiefel-Whitney classes, membranes also exhibit $\mathbb{Z}_3$ statistics associated with Pontryagin classes. We explicitly verify that the 56-step process detects the nontrivial $\mathbb{Z}_3$ statistics in 5, 6, and 7 spatial dimensions. Moreover, in 7 and higher dimensions, the statistics of membrane excitations stabilize to $\mathbb{Z}_{2} \times \mathbb{Z}_{3}$, with the $\mathbb{Z}_3$ sector consistently captured by this process.

Figures (2)

• Figure 1: Relevant cohomology groups of $K({\mathbb Z},d-2)$ and $K({\mathbb Z}_3,d-2)$ and the maps between them. Generators are indicated above or below each group in terms of cohomology operations. We take the generator $1$ for ${\mathbb Z}_n$, and $(1,0);(0,1)$ for ${\mathbb Z}_m\times {\mathbb Z}_n$. Since ${\mathbb R}/{\mathbb Z}$ is not finitely generated, its elements are written with a parameter $\lambda \in {\mathbb R}/{\mathbb Z}$. Here $B_n$ denotes an $n$-cocycle, $P^1$ is the first Steenrod reduced power for $p=3$, $Sq^i$ denotes the Steenrod square operation, and "Pontryagin" refers to the higher Pontryagin power defined in Eq. \ref{['eq:higher Pontryagin power']}. Horizontal arrows represent homomorphisms induced by the spectrum map, and vertical arrows those induced by the change of fusion group. Each arrow is annotated with the images of the corresponding generators. For instance, the label $3;0$ indicates that $\tfrac{1}{3}P^1(B_3)$ maps to three times the Pontryagin operation while $\tfrac{1}{9}B_3\cup\delta B_3$ maps to zero; the label $(1,0);(0,\tfrac{1}{2})$ indicates that $\tfrac{1}{3}P^1(B_5)$ maps to $\tfrac{1}{3}P^1(B_4)$, while $\tfrac{1}{2}B_5\cup_1 B_5$ maps to $\tfrac{1}{2}B_4^2$. For any cohomology class in this commutative diagram, the value of $\mu_{56}$ is obtained from its image in the upper-right ${\mathbb R}/{\mathbb Z}$.
• Figure 2: Analogous to Fig. \ref{['fig:Z3_Z_cohomology_mapping']}, this diagram shows the relevant cohomology groups of $K({\mathbb Z}, d-2)$ and $K({\mathbb Z}_p, d-2)$ for $p=2$ and $p \geq 5$. In all cases with $d \geq 5$, the groups $H^{d+2}(B^{d-2}{\mathbb Z}_p,{\mathbb R}/{\mathbb Z})$ map trivially to the upper-right ${\mathbb R}/{\mathbb Z}$. Consequently, $\mu_{56}$ detects nontrivial statistics in dimensions $\geq 5$ only for $G={\mathbb Z}$ or ${\mathbb Z}_{3N}$.

Theorems & Definitions (8)

• Theorem 1
• Corollary 1.1
• Theorem 2
• Lemma 3
• proof
• Definition 4
• Definition 5
• Definition 6