Nested holography

Abstract

Recently, we introduced a symmetry on the structure of angular momentum that interchanges internal and external degrees of freedom. The spin-orbit duality is a holographic map that projects a massive theory in four-dimensional flat spacetime onto the three-dimensional S2×R null infinity. This cylinder has radius R∼1/m and, quantum mechanically, its vacuum state is a fuzzy sphere. Progress shows that, first, this duality realizes the Hopf map, a fact manifest on the superparticle. Second, the bulk Poincarè group transforms into the conformal group on the cylinder. In fact, the general version of the duality yields that the dual symmetries include the Bondi-Metzner-Sachs group, as is appropriate at null infinity. As an example, the Landau levels in R3 are shown to match those of a Dirac monopole on the dual S2, in the thermodynamic limit. This dual system is actually identified with a three-dimensional critical Ising model. The map is then realized on Nf massive fermions in flat space which, indeed, are the hologram of 2Nf massless fermions on the cylinder. However, the dual space is really the conformal class of S2×R, naturally enclosing the universal cover of a conformally compactified four-dimensional anti–de Sitter (AdS4) spacetime. We argue that, in the absence of interactions, the massless fermions on the conformal boundary are, in turn, dual to Nf massive fermions in AdS4. For free fermions, all path integrals—the ones in R4 and S2×R and AdS4—are shown to match. Hence, AdS/CFT duality emerges into a larger context, where one holography nests inside the other, suggesting a complete holographic bridge between fields in flat space and the AdS superstring. Published by the American Physical Society 2025