Abstract
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.
Highlights
We apply the tools of quantum mechanics in physics and surgery theory in mathematics [9, 10]
We explore the constraints between the 2+1 dimensional (2+1D) and 3+1D topological orders and the geometric-topology properties of 3- and 4-manifolds
(2) We provide the braiding statistics data of particles and strings encoded by submanifold linking, in the 3- and 4-dimensional closed spacetime manifolds
Summary
It is known that the quantum statistics of particles in 2+1D begets anyons, beyond the familiar statistics of bosons and fermions, while Verlinde formula [12] plays a key role to dictate the consistent anyon statistics. We derive a set of quantum surgery formulas analogous to Verlinde’s constraining the fusion and braiding quantum statistics of anyon excitations of particle and string in 3+1D. A further advancement of our work, comparing to the pioneer work Ref.[3] on 2+1D Chern-Simons gauge theory, is that we apply the surgery idea to generic 2+1D and 3+1D topological orders without assuming quantum field theory (QFT) or gauge theory description. JW gratefully acknowledges the Schmidt Fellowship at IAS supported by Eric and Wendy Schmidt and the NSF Grant PHY1314311.
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