Nested Holography

TL;DR

This paper introduces a holographic duality called nested holography, linking flat spacetime theories to conformal boundaries and AdS spaces, revealing new symmetries and connections between fermions, monopoles, and conformal structures.

Contribution

It establishes a novel nested holographic framework connecting flat space theories, conformal boundaries, and AdS spaces, with implications for symmetries and fermionic dualities.

Findings

The spin-orbit duality realizes the Hopf map.

Bulk Poincaré group transforms into the conformal group on the cylinder.

Matching of path integrals in flat space, conformal boundary, and AdS$_4$.

Abstract

Recently, we introduced a symmetry on the structure of angular momentum which 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 $\mathbb{S}^2\times\mathbb{R}$ null infinity. This cylinder has radius $R\sim1/m$ 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. Secondly, the bulk Poincar\`e group transforms into the conformal group on the cylinder. In fact, the general version of the duality yields that the dual symmetries include the BMS group, as is appropriate at null infinity. As an example, the Landau levels in $\mathbb{R}^3$ are shown to match those of a Dirac monopole on the dual $\mathbb{S}^2$, in the thermodynamic limit. This dual system is actually identified with a three-dimensional critical Ising model. The map is then realized on $N_f$ massive fermions in flat space which, indeed, are the hologram of $2N_f$ massless fermions on the cylinder. However, the dual space is really the conformal class of $\mathbb{S}^2\times\mathbb{R}$, naturally enclosing the universal cover of a conformally compactified AdS$_4$ spacetime. We argue that, in the absence of interactions, the massless fermions on the conformal boundary are in turn dual to $N_f$ massive fermions in AdS$_4$. For free fermions, all path integrals $-$the ones in $\mathbb{R}^4$ and $\mathbb{S}^2\times\mathbb{R}$ and AdS$_4-$ 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.