A generalization of the DHR theorem for higher form symmetries
Horacio Casini, Javier M. Magan
TL;DR
This paper extends the DHR reconstruction framework to higher-form and generalized symmetries in relativistic QFT by recasting Haag duality violations (HDV) through a π0-completion procedure. It proves that, after curing π0-related HDV, higher-form symmetry sectors in D>2 are abelian and labeled by group data, with the D=4n case enforcing Hermitian character tables and linking-number-controlled commutators for knot-like regions. A dimensionally-reduced, HDV-based argument shows that confinement-type order parameters and knotted/linked ring operators inherit group-like structures, generalizing the familiar Wilson/'t Hooft loop story to a broader algebraic setting. The results imply a group-origin for internal and generalized symmetries in higher dimensions and provide a rigorous framework for classifying order/disorder parameters in terms of abelian groups and their duals, with explicit constraints from topology and dimension.
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 $π_0$ 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 $π_0$ sectors via DHR reconstruction, we show that the HDV corresponding to regions with non-trivial $π_i$, $0<i<D-2$, 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.