[2607.02280] Bockstein braiding statistics

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Summary:Braiding phenomena, from the charge-flux Aharonov-Bohm impact to anyonic statistics in fractional quantum Corridor programs, are paradigmatic manifestations of topology in quantum physics. Strange mutual braiding between $p$- and $q$-dimensional excitations happens in $d=p+q+2$ spatial dimensions. On this work, we introduce a common development of mutual statistics within the adjoining dimension $d=p+q+1$, relevant to excitations obeying $mathbb Z_N$ fusion for arbitrary $N$ and all excitation dimensions $p$ and $q$. The corresponding invariant is the Berry part amassed in a easy $4N$-step microscopic unitary course of constructed from native excitation operators on lattices. This course of measures the linking of 1 excitation with the $N$-fold fusion junction of the opposite, encompassing particle-particle statistics in a single dimension, particle-loop statistics in two dimensions, and loop-loop or particle-membrane statistics in three dimensions. We set up the quantization and bilinearity of the invariant and present that its field-theory response is ruled by the Bockstein homomorphism, motivating the title Bockstein braiding statistics. Deciphering the excitation operators as open symmetry operators turns the identical invariant right into a direct microscopic diagnostic of combined anomalies between symmetries. We reveal this diagnostic in a (1+1)D spin chain, the place the nontrivial Bockstein braiding part proves the combined anomaly between the spin-flip symmetry $prod X$ and the nearest-neighbor controlled-$Z$ symmetry $prod CZ$. We assemble specific (2+1)D and (3+1)D lattice analogs, yielding new anomalous symmetry pairs, and apply the framework to strongly coupled (3+1)D continuum gauge theories. Nontrivial Bockstein braiding guidelines out a completely symmetric gapped part, obstructs simultaneous condensation of the 2 excitations, and implies fractionalization of higher-form symmetries.