Towards Bulk Locality: A Systematic Construction of Contact Interactions from Chord Diagrams

Abstract: Chord diagrams encode boundary correlators in the double-scaled holographic Sachdev-Ye-Kitaev model, but currently capture only a limited class of bulk interactions that yield pure power-law correlators. In this article, we investigate a general construction based on Fock-space flux models with arbitrary periodic lattice size, clarifies how lattice dimensions control probe configurations and bulk contact vertices. Developing a systematic matching scheme and using the chord path integral formalism, we compute three- to six-point contact correlators and reproduce a broad class of AdS$_2$ scalar contact Witten diagrams, including those with logarithmic singularities. The results demonstrate that chord diagrams, in full generality, provide a microscopic description of bulk contact interactions and thereby establish a principled framework for reconstructing bulk locality from boundary data.

• The paper develops a systematic method for constructing bulk contact interactions using chord diagram combinatorics and flux models in AdS2/CFT1.
• It employs Fock-space flux models on periodic hypercubic lattices where the lattice size L enforces a closure condition that selects allowed probe configurations.
• By tuning microscopic flux parameters, the framework reproduces conformal correlators exhibiting both power-law behavior and logarithmic singularities, thus bridging boundary data to bulk locality.

Systematic Construction of Bulk Contact Interactions from Chord Diagrams

Introduction and Motivation

The paper "Towards Bulk Locality: A Systematic Construction of Contact Interactions from Chord Diagrams" (2605.30970) develops a general framework for reconstructing bulk contact interactions in AdS$_2$/CFT$_1$ holography, using combinatorially defined chord diagrams arising naturally in the double-scaled SYK (DSSYK) landscape and its Fock-space flux model generalizations. The primary objective is to establish how boundary data, via chord combinatorics, encodes a broad class of bulk local contact couplings—including those leading to logarithmic singularities and rational functions in correlators—thereby bridging a key gap in the program of emergent bulk locality in low-dimensional holography.

Fock-Space Flux Models and Periodic Lattices

The starting point is a generalization of the DSSYK model through the construction of Fock-space flux models, extending the Hilbert space to a periodic hypercubic lattice of arbitrary size $L$ in each direction. The microscopic Hamiltonian describes particles hopping between $L^N$ Fock-space sites, with each hop accumulating a random magnetic flux (subject to quantization enforced by periodicity). Both Hamiltonian and probe operators are constructed using these fluxed transport operators.

A crucial insight is the direct correspondence between the lattice parameter $L$ and the set of allowed probe operator configurations, each labeled by a sequence of $\pm1$ signs. The lattice closure condition, $\sum_i \epsilon_i \equiv 0 \mod L$ for probe signs $\epsilon_i$, dictates which bulk contact vertices are accessible in the boundary microscopic theory.

Chord Diagram Rules and Chord Path Integral Formalism

The computation of multi-point correlation functions is encoded diagrammatically through chord diagrams—a graphical representation of the pairwise contraction patterns of fermions in the trace over Hamiltonian and probe operator products. The paper develops detailed rules for associating microscopic weights to these diagrams, with intersection factors $q^{\Delta_{S}}$ assigned according to the number and arrangement of chord-probe intersections. The exponents $\Delta_S$ are determined by the covariance structure of the underlying fluxes.

Correlation functions, systematically evaluated in the large-$_1$0 limit, are represented as functionals within the chord path integral framework. The dynamical variable is the (bilocal) density $_1$1 of $_1$2-chords connecting boundary time intervals, with the corresponding path integral action mapping directly to power-law correlators in the conformal ($_1$3) limit.

Explicit Construction of Contact $_1$4-Point Correlators

For three- through six-point contact correlators, the analysis demonstrates that the chord path integral produces conformally covariant, factorized power-law expressions parametrized by the allowed probe sign configurations and the geometry of the chord intersections. The closure condition for the lattice enforces selection rules on the contributing sign configurations for each $_1$5 and lattice size $_1$6.

Key numerical result: For 4-point functions, the power-law exponents in the correlators depend on the choice of configuration and are given by $_1$7 and $_1$8, extracted from the underlying flux ensemble. By tuning these parameters systematically, the combinatorics of the chord diagram expansion can produce an extensive set of conformally invariant boundary correlators.

For higher points ($_1$9), additional joint parameters ($L$0) become relevant, reflecting more intricate intersection patterns. The method extends recursively, preserving the closure and positivity constraints imposed by the covariance of the flux variables.

Linear Combinations and Parameter Degeneracy

A critical advance is the realization that general scalar contact Witten diagrams in AdS$L$1 contain not only pure power-laws but also logarithmic singularities and rational functions, which cannot be constructed from a single probe configuration. The framework introduces two computational mechanisms:

1. Linear Combinations: The physical correlator is a weighted sum over all allowed probe sign configurations, with the weights $L$2 reflecting the microscopic flux distribution and probe operator statistics. These weights correspond to the amplitudes of different bulk contact vertices encoded in the boundary theory.
2. Parameter Degeneracy: Logarithmic terms are obtained by considering nearly degenerate exponents in different configurations and taking controlled singular limits (e.g., $L$3 as $L$4 with diverging weights). This mechanism allows the generation of logarithmic singularities mirroring those in AdS$L$5 contact Witten diagrams.

Systematic matching is performed by tuning flux parameters (subject to covariance positivity) and configuration weights to reproduce the full library of known AdS$L$6 scalar contact Witten diagrams, including those with non-trivial analytic structure.

Theoretical and Practical Implications

The work provides a robust and principled method for reconstructing the full set of local contact interactions in the AdS$L$7 bulk from discrete, microscopic boundary data. The explicit realization that the lattice size $L$8, via the closure condition on probe configurations, functions as a "regulator" controlling the accessible space of bulk local vertices is significant: in the large-$L$9 limit, a dense function space of derivative bulk couplings is recovered.

The linear combination structure parallels the effective field theory expectation that generic contact interactions are sums over allowed (derivative) vertices, with weights set by ultraviolet (UV) data. The degeneracy mechanism for logarithmic terms illuminates the origin of anomalous scaling (e.g., marginal couplings, critical points) from microscopic boundary physics.

Practically, these results offer a blueprint for constructing boundary models—beyond SYK itself—whose chord diagram expansion can be engineered to realize desired bulk locality properties, potentially informing new designs for holographic codes or tensor network models of emergent geometry.

Future Directions

Several concrete avenues for further investigation emerge:

• Non-Gaussian Fluxes: The current approach focuses on Gaussian-distributed fluxes, leading only to (logarithms of) power-laws. Extension to non-Gaussian flux ensembles could enable matching to more general transcendental structures (e.g., polylogarithms) in higher-loop bulk diagrams.
• Exchange and Loop Diagrams: While contact interactions are fully captured, general bulk exchange diagrams—corresponding to internal splittings in the chord diagram formalism—require systematic analysis and the development of analogous matching schemes.
• Higher-Dimensional Generalization: The combinatorial structures and closure rules may inform construction of tensor models encoding higher-dimensional holographic locality, e.g., for higher AdS$L^N$0/CFT$L^N$1 dualities.
• Connection to JT Gravity and Quantum Error Correction: Bridging the combinatorial chord description with gravitational effective actions (especially JT gravity coupled to matter) could clarify the interpretation of configuration weights, the role of $L^N$2 as a bulk UV scale, and potential connections to holographic quantum error correction.

Conclusion

By systematically extending the chord diagram formalism to general lattice sizes, enumerating all allowed boundary probe configurations, and establishing a principled linear combination and parameter degeneracy prescription, the work demonstrates that chord diagram combinatorics provide a complete microscopic encoding of local bulk contact interactions in AdS$L^N$3/CFT$L^N$4 holography. The framework captures both rational and logarithmic analytic structures in correlation functions and is constrained by physical requirements (positivity of the flux covariance matrix). The results represent a significant advance in reconstructing bulk locality from solvable boundary models and outline a roadmap for further developments toward higher-dimensional and more general emergent gravitational duals.