Bockstein Braiding Statistics Versus Three-Loop Braiding

Abstract

Braiding statistics of $p$- and $q$-dimensional topological excitations is conventionally defined in $p+q+2$ spatial dimensions. We find a novel statistical process $W_N(X,Y)=(Y^{-1}{X}^{-1}{)}^N(YX{)}^N$ for two order-$N$ excitations in $p+q+1$ dimensions, detecting the Bockstein response $A⌣{\hat{I}}^2(B)$. This new statistics and fermionic loop statistics exhaust all loop statistics in three dimensions whose fusion rules form an Abelian group $G$, classified by $H^5(B^{2}G,U(1))$. Surprisingly, conventional three-loop braiding goes beyond this classification, so it must have non-Abelian fusion rules. We suggest viewing three-loop braiding as particle-loop braiding together with exotic fusion rules between loops and point-like defects. We also try to clarify the relationship between statistics and symmetry anomaly.

20 pages, 2 figures