Living on the Edge: A Toy Model for Holographic Reconstruction of Algebras with Centers

TL;DR

This work extends the HaPPY holographic code by adding edge degrees of freedom and interconnecting six copies via copying tensors to model bulk gauge fields or linearized gravitons. The resulting edge-mode code defines an isometry from bulk to boundary and yields bulk algebras with nontrivial centers localized on interior entanglement-wedge boundaries, which can be reconstructed from complementary boundary regions. The authors derive an FLM-like entropy structure, with RT-like surface terms and state-dependent bulk/auxiliary contributions, and they discuss interpretations in extended bulk Hilbert spaces that relate to graviton or gauge-field entropies. Overall, the model provides a concrete, tractable framework to study subregion duality, center algebras, and quantum corrections in holography, with connections to lattice gauge theory and potential extensions to Yang–Mills and gravity.

Abstract

We generalize the Pastawski-Yoshida-Harlow-Preskill (HaPPY) holographic quantum error-correcting code to provide a toy model for bulk gauge fields or linearized gravitons. The key new elements are the introduction of degrees of freedom on the links (edges) of the associated tensor network and their connection to further copies of the HaPPY code by an appropriate isometry. The result is a model in which boundary regions allow the reconstruction of bulk algebras with central elements living on the interior edges of the (greedy) entanglement wedge, and where these central elements can also be reconstructed from complementary boundary regions. In addition, the entropy of boundary regions receives both Ryu-Takayanagi-like contributions and further corrections that model the $\frac{δ\text{Area}}{4G_N}$ term of Faulkner, Lewkowycz, and Maldacena. Comparison with Yang-Mills theory then suggests that this $\frac{δ\text{Area}}{4G_N}$ term can be reinterpreted as a part of the bulk entropy of gravitons under an appropriate extension of the physical bulk Hilbert space.

Figures (11)

• Figure 1: (a) The fundamental tensor $T$ of the pentagon code showing the bulk leg (dashed line, red in color version) and the network legs (solid lines). (b) These units are contracted along their networks legs to form a pentagonal tiling of the hyperbolic plane.
• Figure 2: An unsuccessful first attempt to add edge degrees of freedom. A copy of the tensor $G$ has been attached to each of the 5 network legs of the tensor $T$ from figure \ref{['pentagoncode']}. The bulk input leg of $G$ is drawn in small dashes. This attempt does not succeed, as the tensor annihilates bulk states lacking particular correlations among the 6 bulk inputs.
• Figure 3: Our code is built from 6 copies of the code from happy by contracting the tensor $G$ with a pair of neighboring bulk inputs. The relevant two $T$-tenors are shown here, where we have chosen them both to be part of the same copy of the pentagon code.
• Figure 4: (a) The structure near each vertex of our edge-mode code. The thick black legs carry 5 indices. The central input (long dashes, red in color version) corresponds to a bulk matter field as in happy while the inputs on each edge (short dashes, green in color version) are to be interpreted as degrees of freedom of a bulk gauge field. (b) A sketch of the full edge-mode code.
• Figure 5: Pushing $\sigma_z$ through $G$