# Xiao-Gang Wen, Chern-Simons, Nonabelian Topological Order 

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Goal: classify all possible phases of matter.
Can model *spin liquids*, someone wrote a wave function for it.
Quantum Hall states: 2d electron gas.
Fractional quantum Hall states have different phases even when there is no symmetry (and thus no symmetry to break).
See energy gaps, literally show up as gaps in the spectrum of Hamiltonians.
Topological order: having a specific type of ground state degeneracy.
"Topological" usually means robust against small perturbations, particularly ones that break symmetries.
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Conjecture: chiral spin states and quantum Hall states are described at low energies by a path integral over an action which includes a Chern-Simons term corresponding to a $\U_1$ gauge.
Theme: things locally look the same, but can differ globally.
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Cut electron into partons, write wave function for each and take product (gluing).
Degeneracies: locally the same wave functions but globally different wave functions.
See nonabelian statistics: modular tensor categories.
Monopoles: defects or punctures in spacetime.
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Looking for a mathematical language to describe long-range entanglement.
Current directions: category theory, Chern-Simons theory.
Framing anomaly: path integral depends on the framing of spacetime.
Believe the path integral in Chern-Simons leads to a gravitational Chern-Simons term $\omega_{\CS}^{\text{grav}}$.
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