Quantum Statistics and Spacetime Surgery

TL;DR

This paper uses geometric-topology surgery theory on spacetime manifolds to analyze quantum statistics constraints in 2+1 and 3+1 dimensions, linking fusion and braiding data through new quantum surgery formulas.

Contribution

It introduces fusion and braiding statistics data for particles and strings, and derives new quantum surgery constraints analogous to the Verlinde formula for topological orders.

Findings

Fusion data for worldline and worldsheet operators defined in gapped topological states.

Braiding statistics characterized by geometric Berry matrices and linkings.

Quantum surgery constraints relating fusion and braiding data, aiding topological order theory.

Abstract

We apply the geometric-topology surgery theory on spacetime manifolds to study the constraints of quantum statistics data in 2+1 and 3+1 spacetime dimensions. First, we introduce the fusion data for worldline and worldsheet operators capable creating anyon excitations of particles and strings, well-defined in gapped states of matter with intrinsic topological orders. Second, we introduce the braiding statistics data of particles and strings, such as the geometric Berry matrices for particle-string Aharonov-Bohm and multi-loop adiabatic braiding process, encoded by submanifold linkings, in the closed spacetime 3-manifolds and 4-manifolds. Third, we derive new quantum surgery constraints analogous to Verlinde formula associating fusion and braiding statistics data via spacetime surgery, essential for defining the theory of topological orders, and potentially correlated to bootstrap boundary physics such as gapless modes, conformal field theories or quantum anomalies.