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We derive a selection rule among the $(1+1)$-dimensional SU(2) Wess-Zumino-Witten theories, based on the global anomaly of the discrete $\mathbb{Z}_2$ symmetry found by Gepner and Witten.
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In general, the double degeneracy of the ground states in the symmetric and antisymmetric sectors with respect to the g → − g g\to-g does not necessarily mean the spontaneous symmetry breaking: it could correspond to a topological degeneracy two ground states in the presence of periodic boundary conditions are indistinguishable by any local observable. However, in one spatial dimension, which is the focus of the present Letter, such a topological order with long-range entanglement is absent Chen et al. 2011 and thus the ground-state degeneracy between the two sectors does imply a spontaneous symmetry breaking
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