Non-supersymmetric type II asymmetric orbifolds with Z_k x Z_k Abelian point groups (k=2,3,4) admit vanishing one-loop vacuum energy via sector-wise conservation of a supercharge-like operator.
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Defects for discrete symmetries encoded in bordism groups Ω^ξ_2(BG) and H_2(BG;Z) are described as brane networks rather than isolated objects, extending the Cobordism Conjecture and demonstrated in 4d supergravity from string/M-theory.
Family unification models from E7 cosets have no global sigma model anomalies because the torsion parts of their bordism groups vanish, as computed via the Atiyah-Hirzebruch spectral sequence, including when isotropy subgroups are gauged.
Dai-Freed anomalies for chiral fermion actions in heterotic asymmetric orbifolds recover the level-matching constraints plus mod-2 conditions for even m.
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