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 bounda \try regions. In addition, the entropy of boundary regions receives both Ryu-Takayanagi-like contributions and further corrections that model the frac{delta mathrm{Area}}{4{G}_N} term of Faulkner, Lewkowycz, and Maldacena. Comparison with Yang-Mills theory then suggests that this frac{delta mathrm{Area}}{4{G}_N} term can be reinterpreted as a part of the bulk entropy of gravitons under an appropriate extension of the physical bulk Hilbert space.
Highlights
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 entanglement wedge, and where these central elements can be reconstructed from complementary boundary regions
It is natural to consider modifications of the HaPPY code inspired by lattice gauge theory and having additional degrees of freedom that live on the links of the bulk lattice
Following [2] we show that the entropy of a boundary region A can be written in a form analogous to the explicit terms in the FLM formula (1.1), but in constrast to the original HaPPY code, for general codes of the form (3.6) there is a non-trivial term playing the role of δArea 4GN
Summary
The fact that gauge theories are described canonically by a connection and a conjugate electric flux makes it natural to describe these degrees of freedom as living on the links of a discrete graph-like model, as is common in lattice gauge theory. One might add to each of the non-bulk legs of the fundamental unit a three index tensor Gijk, whose role in the network is to link two adjacent tensors to a common input modelling the electric flux of some bulk gauge field. The resulting code defines an isometry from the bulk degrees of freedom to the boundary, and has many of the same features as the code described in [2] This is because one may view this edge-mode code as six copies of the HaPPY pentagon code. This observation allows us to import all of the main technology from [2] including operator pushing, the greedy entanglement wedge construction, and the Ryu-Takayanagi formula for entanglement entropy. the additional tensors G introduce certain subtleties which we will discuss in depth
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