Quantum Statistics and Spacetime Topology: Quantum Surgery Formulas

Abstract

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.

arXiv comment35 pages, 3d and 4d figures, 3 tables. An extended sequel and further detailed elaboration of [arXiv:1602.05951] and Chapter 6 of Thesis [arXiv:1602.05569] in 2015

References in corpus (21)

• Non-Abelian Anyons and Topological Quantum Computation
• Anyons in an exactly solved model and beyond
• Generalized Global Symmetries
• Topological Defect Lines and Renormalization Group Flows in Two Dimensions
• Duality and defects in rational conformal field theory
• Generalised twisted partition functions
• Boundary conformal field theory approach to the two-dimensional critical Ising model with a defect line
• Kramers-Wannier duality from conformal defects
• Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy
• Twisted Gauge Theory Model of Topological Phases in Three Dimensions
• Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions
• Ground State Degeneracy of Topological Phases on Open Surfaces
• From gauge to higher gauge models of topological phases
• Ground-state degeneracy for abelian anyons in the presence of gapped boundaries
• Universal Topological Data for Gapped Quantum Liquids in Three Dimensions and Fusion Algebra for Non-Abelian String Excitations
• Multi-kink topological terms and charge-binding domain-wall condensation induced symmetry-protected topological states: Beyond Chern-Simons/BF theory
• Invertible defects and isomorphisms of rational CFTs
• Tunneling Topological Vacua via Extended Operators: (Spin-)TQFT Spectra and Boundary Deconfinement in Various Dimensions
• Universal Wave Function Overlap and Universal Topological Data from Generic Gapped Ground States
• Anyon and Loop Braiding Statistics in Field Theories with a Topological $\Theta -$term
• Modular transformations through sequences of topological charge projections

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