Symmetries in quantum field theory and quantum gravity

TL;DR

The paper uses AdS/CFT to justify longstanding swampland constraints: bulk theories cannot host exact global symmetries, and any bulk gauge theory must be accompanied by charged states transforming under all finite representations of the gauge group. It formalizes the notions of global, gauge, and higher-form symmetries, develops a lattice-gauge-theory perspective, and proves no-global-symmetry theorems in holographic quantum gravity via entanglement-wedge technology. It then establishes a precise bulk-boundary dictionary: boundary splittable global symmetries correspond to bulk long-range gauge symmetries, with completeness of charged bulk states ensuring boundary operators realize all irreps. The work extends the framework to $p$-form symmetries and closes with a weak gravity conjecture argument that emergent bulk gauge fields satisfy a convex-hull condition for multi-$U(1)$ scenarios, highlighting a deep UV/IR connection in holography with potential phenomenological implications.

Abstract

In this paper we use the AdS/CFT correspondence to refine and then establish a set of old conjectures about symmetries in quantum gravity. We first show that any global symmetry, discrete or continuous, in a bulk quantum gravity theory with a CFT dual would lead to an inconsistency in that CFT, and thus that there are no bulk global symmetries in AdS/CFT. We then argue that any "long-range" bulk gauge symmetry leads to a global symmetry in the boundary CFT, whose consistency requires the existence of bulk dynamical objects which transform in all finite-dimensional irreducible representations of the bulk gauge group. We mostly assume that all internal symmetry groups are compact, but we also give a general condition on CFTs, which we expect to be true quite broadly, which implies this. We extend all of these results to the case of higher-form symmetries. Finally we extend a recently proposed new motivation for the weak gravity conjecture to more general gauge groups, reproducing the "convex hull condition" of Cheung and Remmen. An essential point, which we dwell on at length, is precisely defining what we mean by gauge and global symmetries in the bulk and boundary. Quantum field theory results we meet while assembling the necessary tools include continuous global symmetries without Noether currents, new perspectives on spontaneous symmetry-breaking and 't Hooft anomalies, a new order parameter for confinement which works in the presence of fundamental quarks, a Hamiltonian lattice formulation of gauge theories with arbitrary discrete gauge groups, an extension of the Coleman-Mandula theorem to discrete symmetries, and an improved explanation of the decay $π^0\toγγ$ in the standard model of particle physics. We also describe new black hole solutions of the Einstein equation in $d+1$ dimensions with horizon topology $\mathbb{T}^p\times \mathbb{S}^{d-p-1}$.

Figures (29)

• Figure 1: A bulk time slice viewed from above, with the boundary timeslice $\Sigma$ split up into disjoint spatial regions $R_i$. We've shaded the entanglement wedge of each $R_i$ grey, and the point in the center lies in none of these entanglement wedges.
• Figure 2: Constructing a symmetry insertion on a torus in the path integral of a QFT on a spacetime that is topologically $\mathbb{R}^3$: the "upper" operator on the left hand side is a deformation of $U^\dagger(g,\mathbb{R}^2)$, while the "lower" operator is a deformation of $U(g,\mathbb{R}^2)$. If we bring them together the blue sections cancel, leaving the green torus. Since the $U(g,\mathbb{R}^2)$ commute with $T_{\mu\nu}$ they are topological, so it does not matter where we join them. If there are no charged insertions inside the torus then we can further collapse it to nothing, while if a charged operator is inserted inside the torus, say an operator $\mathcal{O}$ at the black dot in the figure, then the joint insertion amounts to inserting $U^\dagger(g,\mathbb{R}^{2})\mathcal{O} U(g,\mathbb{R}^2)=D(g)\mathcal{O}$ into the path integral.
• Figure 3: Splittability of any global symmetry for a lattice theory. Here each dot is a spin, so a spatial region $R$, shaded blue, corresponds to a subset of the spins, shaded red. To produce a localized symmetry operator we take the product over the $U_i(g)$ associated to the red spins.
• Figure 4: A counterexample to the split property: electrodynamics on a spatial torus. The flux operator through $S$ is equal to the flux operator through $S'$, but they live in spacelike-separated regions $R$ and $\hat{R}$.
• Figure 5: Re-routing unbreakable lines. Here we have a symmetry exchanging blue and red lines, and we can arrange for it to act locally in the shaded region by rerouting the blue line around the boundary of the region. This is not possible however when the region has multiple boundary components which are not contractible, for example as in figure \ref{['torusfig']}.

Theorems & Definitions (57)

• Conjecture 1
• Conjecture 2
• Conjecture 3
• Definition 2.1
• Definition 2.2
• Conjecture 4
• Definition 2.3
• Theorem 2.1
• Definition 2.4
• Definition 2.5