Abstract
In many gauge theories, the existence of particles in every representation of the gauge group (also known as completeness of the spectrum) is equivalent to the absence of one-form global symmetries. However, this relation does not hold, for example, in the gauge theory of non-abelian finite groups. We refine this statement by considering topological operators that are not necessarily associated with any global symmetry. For discrete gauge theory in three spacetime dimensions, we show that completeness of the spectrum is equivalent to the absence of certain Gukov-Witten topological operators. We further extend our analysis to four and higher spacetime dimensions. Since topological operators are natural generalizations of global symmetries, we discuss evidence for their absence in a consistent theory of quantum gravity.
Highlights
D − q − 1 and are labeled by elements g of the group G
For discrete gauge theory in three spacetime dimensions, we show that completeness of the spectrum is equivalent to the absence of certain Gukov-Witten topological operators
Likewise in AdSd/CFTd−1, under certain assumptions, it has been argued that q-form global symmetries with q ≤ d − 3 of the boundary conformal field theory (CFT) are necessarily gauged in the bulk [10]
Summary
An intrinsic description of the global symmetry is given in terms of its symmetry operators. It is independent of the explicit Lagrangian presentation (if it exists) of the underlying system. A q-form global symmetry transformation is implemented by a symmetry operator Ug(M (d−q−1)) that has support on a codimension-(q + 1) closed manifold M (d−q−1). When G = U(1) and there is a conserved (q +1)-form Noether current j, the symmetry operator is Uθ(M (d−q−1)) = exp[iθ M(d−q−1) j], with θ ∈ [0, 2π) the U(1) group element. The current conservation equation d( j) = 0 implies that the symmetry operator Uθ(M (d−q−1)) is independent of small deformations of M (d−q−1). When M (d−q−1) is taken to be the whole space at a given time, the topological nature of Uθ(M (d−q−1)) means that the U(1) charge is conserved in time
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