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We study the symmetries of closed Majorana chains in 1+1d, including thetranslation, fermion parity, spatial parity, and time-reversal symmetries. Thealgebra of the symmetry operators is realized projectively on the Hilbertspace, signaling anomalies on the lattice, and constraining the long-distancebehavior. In the special case of the free Hamiltonian (and small deformationsthereof), the continuum limit is the 1+1d free Majorana CFT. Its continuumchiral fermion parity $(-1)^{F_\text{L}}$ emanates from the lattice translationsymmetry. We find a lattice precursor of its mod 8 't Hooft anomaly. Using aJordan-Wigner transformation, we sum over the spin structures of the latticemodel (a procedure known as the GSO projection), while carefully tracking theglobal symmetries. In the resulting bosonic model of Ising spins, the Majoranatranslation operator leads to a non-invertible lattice translation symmetry atthe critical point. The non-invertible Kramers-Wannier duality operator of thecontinuum Ising CFT emanates from this non-invertible lattice translation ofthe transverse-field Ising model.

Majorana chain and Ising model -- (non-invertible) translations, anomalies, and emanant symmetries