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We introduce a class of projected entangled pair states (PEPS) which is basedon a group symmetry twisted by a 3-cocycle of the group. This twisted symmetrygives rise to a new standard form for PEPS from which we construct a family oflocal Hamiltonians which are gapped, frustration-free and include fixed pointsof the renormalization group flow. Moreover, we advance the classification of2D gapped quantum spin systems by showing how this new standard form for PEPSdetermines the emergent topological order of these local Hamiltonians.Specifically, we identify their universality class as Dijkgraaf-Wittentopological quantum field theory (TQFT).

Twisted Injectivity in PEPS and the Classification of Quantum Phases