Quantum state space geometry. The Fubini-Study metric measures distinguishability of pure states on CP¹. Berry curvature = field strength of Berry connection.
Fubini-Study metric on CP¹ (Bloch sphere):
ds² = (dψ|dψ) − |(ψ|dψ)|² = ¼(dθ²+sin²θ dφ²)
Berry connection: A = i⟨ψ|∇_k|ψ⟩
Berry curvature: Ω = ∇×A (antisymmetric 2-form)
Chern number: C = (1/2π)∫ Ω d²k
Left: Bloch sphere with state trajectory.
Right: Berry curvature Ω(k_x,k_y) in BZ.