Quantum Statistics and Spacetime Surgery
Juven Wang, Xiao-Gang Wen, Shing-Tung Yau
TL;DR
The paper addresses constraints on quantum statistics data for excitations in $2+1D$ and $3+1D$ topological orders using spacetime surgery. It introduces fusion data for worldline and worldsheet operators and braiding data encoded by submanifold linkings, and derives Verlinde-like formulas by cutting and gluing spacetime via mapping class group transformations such as $\hat{\mathcal{S}}$, $\hat{\mathcal{T}}$, and $\mathcal{S}^{xyz}$. The main contributions are the explicit definitions of $\mathcal{F}^{S^1}$, $\mathcal{F}^{S^2}$, $\mathcal{F}^{T^2}$ and braiding data including $\mathcal{S}$, $\mathcal{T}$, $\tilde{L}^{(S^2,S^1)}$, $\tL^{Tri}$, and the generalized Verlinde-like constraints for 2+1D and 3+1D topological orders, potentially linking bulk data to boundary physics. This framework is notable for being largely QFT/gauge-field-free, relying on path-integral amplitudes of submanifold insertions in fixed-point topological orders, with potential implications for bootstrapping boundary CFTs and anomalies. However, it omits boundary degrees of freedom such as the chiral central charge $c_-$ and leaves questions of sufficiency and completeness for future work. The approach can be applied to a broad class of topological orders, including Dijkgraaf-Witten theories, and is extendable to generic spacetime dimensions.
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.