Holographic description of 4d Maxwell theories and their code-based ensembles

TL;DR

The paper develops a precise holographic duality between an ensemble of 4d ${ m U}(1)^g$ Maxwell theories on a spin 4-manifold ${M}_4$ and a 5d Abelian BF-type TQFT of level ${N}$ summed over 5-manifolds with boundary ${M}_4$. Boundary theories are classified by self-dual symplectic codes over ${\mathbb Z}_N$ and bulk topology sectors by orthogonal self-dual codes, recast as quantum stabilizer (CSS) codes; holomorphic quantization ties 5d/3d wavefunctions to Narain-type theta functions. For ${N}>1$ the holography becomes an ensemble duality, with the ensemble average over codes corresponding to a sum over 5d topologies and, in the large-${N}$ limit, yielding Eisenstein-series (Siegel-Weil) structures that couple bulk handlebodies to boundary moduli. The approach clarifies the relationship between 4d Maxwell theory and 4d ${\cal N}=4$ SYM boundary conditions, via SymTFT boundary data, and extends the correspondence to correlation functions by incorporating Wilson-line insertions and conformal blocks. Overall, the work provides a concrete, exactly solvable framework for ensemble holography in higher dimensions and links coding-theoretic data to topological boundary conditions and bulk topologies, with potential implications for gravity duals and gauge-group realizations in holography.

Abstract

We formulate a precise holographic duality between an ensemble of 4d $U(1)^g$ Maxwell theories living on a spin four-manifold $M_4$ and an Abelian BF-type 2-form gauge theory of level $N$, summed over all five-manifolds with boundary $M_4$. The elements of the boundary ensemble are Abelian gauge theories specified by self-dual symplectic codes over $Z_N$, that parameterize topological boundary conditions in the 5d TQFT. Similarly, the equivalence classes of topologies distinguished by the 5d theory are parameterized by orthogonal self-dual codes. Hence the holographic duality can be reformulated in the language of quantum stabilizer codes. This duality is closely related to the holographic relationship between ensembles of Narain conformal field theories in 2d and level-$N$ Abelian Chern-Simons theories in 3d. In both contexts, the duality extends to correlation functions. In the large-$N$ limit, we find that the boundary ensemble average converges to an integral over the moduli space of the gauge couplings and, when finite, is equal to an Eisenstein series of the orthogonal group, a version of the Siegel-Weil formula that appears in the 2d/3d context. As a spinoff, we clarify the holographic relationship between the gauge group of the 4d N=4 super Yang-Mills theory and the boundary conditions of the singleton sector in the bulk.

Figures (4)

• Figure 1: Left image: a $2+1$ topological system with a defect. A particular combination of defects gives rise to topological boundary conditions at the defect worldline and a CFT at the cylinder boundary. Right image: the same, after the defect worldline is fattened into a cylindrical shell (shown in red). The result is the sandwich construction: topological boundary condition at the boundary shown in red gives rise to a CFT at the other boundary.
• Figure 2: A schematic illustration of the sum over 3d topologies being equal to the sum over topological boundary conditions. The sum includes all 3d topologies, smooth topologies (handlebodies) as well as singular ones, obtained via genus reduction. Possible weights on both sides of the equality are omitted for visual simplicity.
• Figure 3: 2d CFT conformal block with the operator insertions, given for Abelian theory by \ref{['conformalblockA']}, is given by TQFT on $\Sigma\times [0,1]$ with the line operators (Wilson lines) ending at the conformal boundary (depicted in blue).
• Figure 4: Gauging the center ${\mathbb{Z}}_N$ of ${\rm U} (1)\times {\rm SU} (N)$ theory in terms of SymTFT construction.