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To formulate the universal constraints of quantum statistics data of generic long-range entangled quantum systems, we introduce the geometric-topology surgery theory on spacetime manifolds where quantum systems reside, cutting and gluing the associated quantum amplitudes, specifically in 2+1 and 3+1 spacetime dimensions. First, we introduce the fusion data for worldline and worldsheet operators capable of creating anyonic 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, 3-string, 4-string, or multi-string adiabatic loop braiding process, encoded by submanifold linkings, in the closed spacetime 3-manifolds and 4-manifolds. Third, we derive new `quantum surgery' formulas and constraints, analogous to Verlinde formula associating fusion and braiding statistics data via spacetime surgery, essential for defining the theory of topological orders, 3d and 4d TQFTs and potentially correlated to bootstrap boundary physics such as gapless modes, extended defects, 2d and 3d conformal field theories or quantum anomalies. This article is meant to be an extended and further detailed elaboration of our previous work [arXiv:1602.05951] and Chapter 6 of [arXiv:1602.05569]. Our theory applies to general quantum theories and quantum mechanical systems, also applicable to, but not necessarily requiring the quantum field theory description.

Consider $N\times N$ hermitian or symmetric random matrices $H$ with independent entries, where the distribution of the $(i,j)$ matrix element is given by the probability measure $\nu_{ij}$ with zero expectation and with variance $\sigma_{ij}^2$. We assume that the variances satisfy the normalization condition $\sum_{i} \sigma^2_{ij} = 1$ for all $j$ and that there is a positive constant $c$ such that $c\le N \sigma_{ij}^2 \le c^{-1}$. We further assume that the probability distributions $\nu_{ij}$ have a uniform subexponential decay. We prove that the Stieltjes transform of the empirical eigenvalue distribution of $H$ is given by the Wigner semicircle law uniformly up to the edges of the spectrum with an error of order $ (N \eta)^{-1}$ where $\eta$ is the imaginary part of the spectral parameter in the Stieltjes transform. There are three corollaries to this strong local semicircle law: (1) Rigidity of eigenvalues: If $\gamma_j =\gamma_{j,N}$ denotes the {\it classical location} of the $j$-th eigenvalue under the semicircle law ordered in increasing order, then the $j$-th eigenvalue $\lambda_j$ is close to $\gamma_j$ in the sense that for any $\xi>1$ there is a constant $L$ such that \[\mathbb P \Big (\exists \, j : \; |\lambda_j-\gamma_j| \ge (\log N)^L \Big [ \min \big (\, j, N-j+1 \, \big) \Big ]^{-1/3} N^{-2/3} \Big) \le C\exp{\big[-c(\log N)^{\xi} \big]} \] for $N$ large enough. (2) The proof of the {\it Dyson's conjecture} \cite{Dy} which states that the time scale of the Dyson Brownian motion to reach local equilibrium is of order $N^{-1}$. (3) The edge universality holds in the sense that the probability distributions of the largest (and the smallest) eigenvalues of two generalized Wigner ensembles are the same in the large $N$ limit provided that the second moments of the two ensembles are identical.

In the first paper of this series, “Electrodynamics and the Gauss Linking Integral on the 3-sphere and in Hyperbolic 3-space,” we developed a steady-state version of classical electrodynamics in these two spaces, including explicit formulas for the vector-valued Green’s operator, explicit formulas of Biot-Savart type for the magnetic field, and a corresponding Amp`ere’s Law contained in Maxwell’s equations, and then used these to obtain explicit inte- gral formulas for the linking number of two disjoint closed curves. In this second paper, we obtain integral formulas for twisting, writhing, and helicity, and prove that link = twist + writhe on the 3-sphere and in hyperbolic 3-space. We then use these results to derive upper bounds for the helicity of vector fields and lower bounds for the first eigenvalue of the curl operator on subdomains of these two spaces. An announcement of these results, and a hint of their proofs, can be found in the Math ArXiv, math.GT/0406276, while an expanded version of the first paper, with full proofs, can be found at math.GT/0510388.