A generalization of the DHR theorem for higher form symmetries

Abstract

The Doplicher-Haag-Roberts (DHR) reconstruction theorem shows that standard ($0$-form) internal symmetries are associated to groups in relativistic quantum field theory in spacetime dimension $D>2$. In particular, non-invertible symmetry structures in $D>2$ correspond to the choice of a subtheory of a unique parent one, where the symmetry is a compact group. We present a theorem that generalizes this result to higher form symmetries. We first re-formulate the DHR theorem in terms of Haag duality violations (HDV) for regions with non-trivial homotopy group $\pi_0$ in the finite index case. In this light, the theorem states that the category associated with such HDV is the dual of a group, and it can be extended to spontaneous symmetry breaking scenarios. Then, after eliminating $\pi_0$ sectors via DHR reconstruction, we show that the HDV corresponding to regions with non-trivial $\pi_i$, $0<i<D-2$, are associated with abelian groups. Physically, the result shows that generalized order/disorder parameters in $D>2$ are labeled by such groups, in agreement with the case of confinement order parameters in Yang-Mills theories (Wilson and 't Hooft loops). For the special case of $D=4n$ and loops of dimension $k=2 n-1$, the group is further required to have a Hermitian character table. This does not rule out the possibility of an extra $\mathbb{Z}_2$ factor that is not achievable by Lagrangian gauge models. In the way we find a new proof of the group-like origin of internal symmetries, and analyze the sectors for more general regions, e.g., direct sums, knots, and links. In particular, we find that generalized knot order parameters are classified by the unknot order parameters, and the commutator of knot non-local operators is determined by the linking number.