It is known that quantum statistics of quasiparticles in 2+1 spacetime dimensions beget anyons, beyond the familiar statistics of bosons and fermions, while Verlinde formula dictates the consistent anyon statistics. In 3+1 dimensions, although quasiparticles cannot be anyonic, extended quasi-excitations like strings can have anyonic string statistics. In this work, we first introduce the fusion and braiding data of particle and string excitations creatable from the operators of (world-) lines/sheets, and all are well-defined in gapped states of matter with intrinsic topological orders. We then apply the geometric-topology surgery theory to derive a set of quantum surgery formulas analogous to Verlinde's constraining the fusion and braiding quantum statistics of anyonic particle and anyonic string excitations in 2+1 and 3+1 dimensions, essential for the theory of bulk topological orders and potentially correlated to bootstrap boundary physics.
Abstract The relation between braid and exclusion statistics is examined in one-dimensional systems, within the framework of Chern–Simons statistical transmutation in gauge invariant form with an appropriate dimensional reduction. If the matter action is anomalous, as for chiral fermions, a relation between braid and exclusion statistics can be established explicitly for both mutual and nonmutual cases. However, if it is not anomalous, the exclusion statistics of emergent low energy excitations is not necessarily connected to the braid statistics of the physical charged fields of the system. Finally, we also discuss the bosonization of one-dimensional anyonic systems through T-duality.
Quasiparticle excitations in
3+1 dimensions can be either bosons or fermions. In this work, we introduce the notion of fermionic loop excitations in
(3+1)-dimensional
(3+1d) topological phases. Specifically, we construct a new many-body lattice invariant of gapped Hamiltonians, the loop self-statistics $\ensuremath{\mu}=\ifmmode\pm\else\textpm\fi{}1$, that distinguishes two bosonic topological orders that both superficially resemble
3+1dZ2 gauge theory coupled to fermionic charged matter. The first has fermionic charges and bosonic
Z2 gauge flux loops (FcBl) and is just the ordinary fermionic toric code. The second has fermionic charges and fermionic loops (FcFl) and, as we argue, can only exist at the boundary of a nontrivial
(4+1)-dimensional
(4+1d) invertible phase, stable without any symmetries, i.e., it possesses a gravitational anomaly. We substantiate these claims by constructing an explicit exactly solvable
4+1d model using a method that bootstraps a boundary theory into a bulk Hamiltonian, analogous to that of Walker and Wang, and computing the loop self-statistics in the fermionic
Z2 gauge theory hosted at its boundary. We also show that the FcFl phase has the same gravitational anomaly as all-fermion quantum electrodynamics. Our results are in agreement with the recent classification of nondegenerate braided fusion 2-categories by Johnson-Freyd, and with the cobordism prediction of a nontrivial
Z2-classified
4+1d invertible phase with action $S=\frac{1}{2}\ensuremath{\int}{w}_{2}{w}_{3}$.