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A general theory of topological classification of defects is introduced. We illustrate the application of tools from algebraic topology, including homotopy and cohomology groups, to classify defects including several explicit calculations for crystals in ℝ^2, S^2, 2-dimensional cylinder, 2-dimensional annulus, and 2-tori. A set of physically motivated assumptions is formulated in order to justify the classification process and also to expose certain inherent inconsistencies in the considered methodology, particularly for crystal lattices.

We compute the spectral form factor of two integrable quantum-critical many-body systems in one spatial dimension. The spectral form factor of the quantum Ising chain is periodic in time in the scaling limit described by a conformal field theory; we also compute corrections from lattice effects and deviation from criticality. Criticality in the random Ising chain is described by rare regions associated with a strong randomness fixed point, and these control the long-time limit of the spectral form factor.