2025/09/17 by Feng, Yitao, Xue, Hanyu, Li, Yuyang +4 · 1 citation
#FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Quantum Physics (quant-ph) #Strongly Correlated Electrons (cond-mat.str-el)
paper · doi:10.48550/arxiv.2509.14314
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 ℤN-particle exhibiting ℤN× gcd(2,N) anyonic statistics in two dimensions, we show that the ℤN-membrane possesses ℤN× 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 ℤN symmetry-protected topological phases and demonstrate that their domain walls realize all possible anyonic membrane statistics. We then show that the ℤ3 subgroup persists in all higher dimensions. In addition to the standard fermionic ℤ2 membrane statistics arising from Stiefel-Whitney classes, membranes also exhibit ℤ3 statistics associated with Pontryagin classes. We explicitly verify that the 56-step process detects the nontrivial ℤ3 statistics in 5, 6, and 7 spatial dimensions. Moreover, in 7 and higher dimensions, the statistics of membrane excitations stabilize to ℤ2 × ℤ3, with the ℤ3 sector consistently captured by this process.