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We apply the geometric-topology surgery theory on the spacetime manifold 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.

condensed matter, geometric topology, surgery theory, quantum field theory, topological field theory, quantum anomalies, conformal field theory, topological order, fusion and braiding statistics, Verlinde formula, Berry phase, Aharonov-Bohm effect, knots and links, Borromean rings, fractional quantum Hall effect, anyons, anyonic string