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

Published 3 Oct 2025 in hep-th | (2510.03392v1)

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.

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Summary

The paper formulates a rigorous holographic duality between 4d Maxwell ensembles and 5d BF-type TQFTs using self-dual stabilizer codes.
It employs quantum stabilizer codes to classify boundary conditions and bulk topologies via Construction A, bridging quantum information and topological field theory.
In the large-N limit, the ensemble average produces an Eisenstein series reflecting a semiclassical bulk geometric average over moduli spaces.

Holographic Duality for 4d Maxwell Theories and Code-Based Ensembles

Overview and Motivation

This work establishes a precise holographic duality between ensembles of four-dimensional $U(1)^g$ Maxwell theories on spin four-manifolds and five-dimensional Abelian BF-type topological quantum field theories (TQFTs) of level $N$ , summed over all five-manifolds with fixed boundary. The boundary ensemble is parameterized by self-dual symplectic codes over $\mathbb{Z}_N$ , which correspond to topological boundary conditions in the 5d TQFT. The equivalence classes of bulk topologies are similarly classified by orthogonal self-dual codes. This duality is reformulated in the language of quantum stabilizer codes, providing a bridge between quantum information theory and topological field theory.

The construction generalizes the well-known 2d/3d Narain CFT/Chern-Simons duality, extending the correspondence to correlation functions and ensemble averages. In the large- $N$ limit, the boundary ensemble average converges to an integral over the moduli space of gauge couplings, yielding an Eisenstein series of the orthogonal group—a manifestation of the Siegel-Weil formula.

Topological Boundary Conditions and Code Theory

The 5d BF-type TQFT is defined by the action

$\frac{N}{2\pi} \sum_{I=1}^g \int B_2^I \wedge dC_2^I$

on a spin four-manifold $M_4$ . The Hilbert space depends only on the 2-cohomology of $M_4$ , and is isomorphic to the Hilbert space of a 3d Chern-Simons theory compactified on a Riemann surface $\Sigma$ of genus $g$ .

Topological boundary conditions in the 5d theory are classified by maximal self-dual symplectic codes over $\mathbb{Z}_N$ of length $2g$. Each code defines a quantum stabilizer state in the Hilbert space, and the set of inequivalent bulk topologies is similarly classified by orthogonal self-dual codes. This correspondence is formalized via the Construction A mapping from codes to lattices, and the resulting states are stabilized by the generalized Pauli group generated by surface operators wrapping 2-cycles in $M_4$ .

Holographic Correspondence: Bulk/Boundary Duality

The central holographic relation is

$\langle Z_{\text{Maxwell}} \rangle = \sum_{\text{topologies}} Z_{\text{5d CS}}$

where the left side is the ensemble-averaged partition function of boundary Maxwell theories, and the right side is the sum over bulk wavefunctions for all 5d topologies with fixed boundary. For $N=1$ , the ensemble reduces to a single theory and the sum over topologies is trivial. For $N>1$ , the ensemble is nontrivial and parameterized by codes.

The duality extends to correlation functions of $U(1)$ primaries, with each partition function in the boundary ensemble invariant under the 4d modular group $O(n,n,\mathbb{Z})$ . The bulk theory inherits an additional $Sp(2g,\mathbb{Z}_N)$ symmetry, which permutes theories in the ensemble.

Figure 1: Left: A 2+1 topological system with a defect, where topological boundary conditions at the defect worldline yield a CFT at the cylinder boundary. Right: The defect worldline is fattened into a cylindrical shell, resulting in the sandwich construction.

Figure 2: Schematic illustration of the sum over 3d topologies being equivalent to the sum over topological boundary conditions, including both smooth and singular topologies.

Quantization and Wavefunction Construction

Holomorphic quantization of the 5d theory proceeds via dimensional reduction from a 7d Chern-Simons-type theory. The wavefunctions are constructed as holomorphic functions of boundary data, with explicit dependence on the modular parameters of the underlying geometry. The basis states are labeled by codewords, and the action of the modular and orthogonal groups is implemented via surface operators.

The explicit form of the wavefunctions is given by generalized theta functions, with the partition function for a Maxwell theory on $M_4$ expressed as

$Z_\tau[M_4] = \frac{\theta(\tau, E, \xi, \bar{\xi})}{\Phi_{5d}}$

where $\theta$ is a Siegel-Narain theta function and $\Phi_{5d}$ encodes the determinant of Laplacians on $M_4$ .

Large- $N$ Limit and Siegel-Weil Formula

In the large- $N$ limit, the ensemble average over boundary theories becomes an integral over the moduli space of gauge couplings, and the sum over bulk topologies approaches a sum over handlebodies. The resulting average is an Eisenstein series of the orthogonal group, a higher-dimensional analog of the Siegel-Weil formula known from the 2d/3d context.

Correlation Functions and Operator Insertions

The duality extends to correlation functions of local operators. In the 2d/3d case, conformal blocks with operator insertions correspond to bulk path integrals with line operators ending at insertion points. In the 4d/5d case, surface operators in the bulk ending on contours at the boundary compute correlators of Wilson lines in the boundary theory.

Figure 3: 2d CFT conformal block with operator insertions, realized in the bulk by TQFT on $\Sigma \times [0,1]$ with Wilson lines ending at the conformal boundary.

Gauge Group of 4d $\mathcal{N}=4$ SYM and Bulk Boundary Conditions

The holographic dictionary for 4d $\mathcal{N}=4$ SYM is clarified: the choice of gauge group ( $U(N)$ or $SU(N)$ ) is determined by the boundary conditions for the bulk TQFT fields $B_2, C_2$ . Holomorphic quantization yields the $U(N)$ theory, while topological boundary conditions correspond to $SU(N)$ theories. The construction is formalized via the SymTFT approach, and the bulk partition function is shown to evaluate the appropriate boundary theory depending on the quantization scheme.

Figure 4: Gauging the center $\mathbb{Z}_N$ of $U(1) \times SU(N)$ theory in the SymTFT construction.

Implications and Future Directions

This work demonstrates that ensemble holography is not restricted to lower dimensions or the absence of dynamical gravitons. The bulk theory is a topological theory of gravity, and both single-theory and ensemble dualities are realized depending on the properties of the bulk TQFT. The connection to quantum stabilizer codes provides a new perspective on anyon condensation and topological boundary conditions in higher-dimensional TQFTs.

In the large- $N$ limit, the bulk sum over topologies becomes semiclassical, including only handlebody geometries. The duality extends to correlators of $U(1)$ primaries and their descendants, and the formalism suggests possible generalizations to Virasoro TQFTs and more general boundary ensembles.

Conclusion

The paper formulates a rigorous holographic duality between ensembles of 4d Maxwell theories and 5d Abelian TQFTs, with the ensemble structure and bulk topology classified by self-dual codes. The duality is exact for partition functions and extends to correlation functions, with the large- $N$ limit yielding a geometric average over moduli spaces described by Eisenstein series. The connection to quantum information theory via stabilizer codes enriches the interpretation of topological boundary conditions and anyon condensation. The results suggest that ensemble holography is a general feature of topological gravity theories in any dimension, and open avenues for further exploration of holographic dualities involving more general symmetry algebras and dynamical gravity.