Symmetry spans enforce gaplessness when a symmetry E embedded into two larger symmetries C and D has no compatible gapped phase that restricts from both.
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Graph-state inner products are governed by the F2-rank of the adjacency matrix and the Arf invariant, yielding a nonlocal Bell-pair factorization of the Hilbert space.
Defect 't Hooft anomalies trap charges at symmetry-line junctions and thereby drive categorical scattering into twist operators.
Z_k symmetries from Pythagorean triples in two free Weyl fermions yield non-invertible defects that generate all U(1)^2-preserving boundaries for two Dirac fermions.
Classification of 2D fermionic systems with Z2 flavor symmetry yields 16 consistent superfusion categories labeled by anomaly invariants (ν_W, ν_Z, ν_WZ).
Quantization of axions on dS_D yields Hilbert space H = L^2(S^1) ⊗ F with zero-mode U(1) charge, producing non-dS-invariant charged sectors and Hadamard Wightman functions that become asymptotically invariant.
Proposes a bosonized lattice construction of anomaly-free 2D non-Abelian chiral gauge theories in which left and right bulk contributions cancel at finite spacing when quadratic indices match.
Abelian 2d gauge theories show rich phase structure with c=1 and c=1/2 critical lines; chiral versions realize symmetric mass generation for fermions.
Constructs Z_N extended fusion rings and modular partition functions for nonanomalous subgroups, extending to multicomponent systems and orbifoldings in CFTs.
A survey of non-invertible symmetries with constructions in the Ising model and applications to neutral pion decay and other systems.