Higher-Spin and Vector-Model Holography

Vector-model holography provides a large-N bulk containing infinitely many light higher-spin fields. This chapter builds the dictionary from singlet operators and boundary conditions to correlation functions, then explains why this calculable regime has different locality requirements from an Einstein-like bulk.

Enter this chapter

Begin with Higher-Spin and Vector-Model Dualities to match the singlet spectrum to the proposed bulk field content.

Helpful background. Vector Models, Auxiliary Fields, and Large-N Saddles and Boundary Conditions, Alternate Quantization, and Deformations supply large-N vector saddles and alternate scalar quantization.

Topics

1. Higher-Spin and Vector-Model Dualities — States the operator, coupling, and boundary-condition map.
2. Vasiliev Higher-Spin Equations, Interactions, and Locality Obstructions — Explains the master-field equations and the problem of extracting local interactions.
3. Vector-Model Bilocals as Higher-Spin Bulk Data — Reorganizes singlet degrees of freedom as collective two-point fields.
4. Free and Critical Vector Models at the Higher-Spin Dictionary Interface — Compares scalar operator data in free and critical theories.
5. Boundary Conditions, Double-Trace Flows, and Alternate Quantization — Relates mixed scalar boundary conditions to double-trace renormalization-group flow.
6. Higher-Spin Dictionaries for Chern–Simons Matter and Parity Phases — Matches parity-violating planar data while tracking discrete gauge information.
7. Higher-Spin Correlator Tests and Ward Identities — Uses normalized correlators and Ward identities as independent dictionary tests.
8. Tensionless String Limits and Higher-Spin Enhancement — Examines which extra states appear when the string tension becomes small.
9. Higher-Spin Symmetry Breaking and Einstein-Regime Limits — Distinguishes weak symmetry breaking from decoupling the higher-spin tower.

How the ideas fit together

Free and critical vector models can share the leading conserved-current tower while differing in their scalar dimension. Changing the bulk scalar boundary condition accounts for this distinction; matching the spin list alone misses it. Once two-point normalization fixes the overall coupling, three-point functions become sharper tests Giombi and Yin 2010, §§ 4–6. Small anomalous dimensions then describe weakly broken higher-spin symmetry, not a large mass gap. Recovering an Einstein regime would require decoupling the tower and controlling its interactions, beyond perturbing a few correlators.

Claim-domain comparison

Compare the assumptions and strongest supported conclusion for each case.

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Representative claim domains and validity boundaries for Higher-Spin and Vector-Model Holography

Claim object	State, ensemble, and conventions	Approximation, status, and evidence timing	Uncertainty and counterevidence	Falsifier	Failure condition	Licensed conclusion
higher-spin dictionary	Declare boundary model, parity phase, and global data; use the volume conventions unless the page states a local replacement.	Dictionary entry or correspondence claim. Control chain: vector-model large-N data → higher-spin algebra and fields → boundary conditions and correlators → breaking and tensionless tests → non-Einstein bulk regime. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “three-point and Ward-identity tests” check is counterevidence to the promoted claim.	With operator normalization and parity data matched, the separated-point three-point function or Ward identity disagrees beyond the retained large-N order.	Inferring local Einstein gravity from a higher-spin correlator match.	a controlled correlator match
alternate boundary condition	Declare scalar branch and double-trace flow; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: vector-model large-N data → higher-spin algebra and fields → boundary conditions and correlators → breaking and tensionless tests → non-Einstein bulk regime. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “critical/free endpoint comparison” check is counterevidence to the promoted claim.	Within the allowed scalar quantization window, the proposed flow fails to reproduce the declared endpoint dimensions or normalized two-point functions.	Assuming a boundary-condition change leaves the bulk theory unchanged at every order.	a map between stated models
symmetry breaking	Declare coupling and anomalous dimensions; use the volume conventions unless the page states a local replacement.	Conditional theorem or structural result. Control chain: vector-model large-N data → higher-spin algebra and fields → boundary conditions and correlators → breaking and tensionless tests → non-Einstein bulk regime. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “gap and locality diagnostics” check is counterevidence to the promoted claim.	The claimed current nonconservation or anomalous dimension disagrees with the specified deformation at the retained order; a surviving light higher-spin tower separately blocks an Einstein limit.	Inferring an automatic string completion from higher-spin symmetry breaking.	a deformation away from the symmetric point

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Where to go next

Continue to de Sitter and Cosmological Holography to see how higher-spin examples enter a different boundary and probability problem.

References

• Giombi, S., and Yin, X. (2010). “Higher Spin Gauge Theory and Holography: The Three-Point Functions.” Journal of High Energy Physics 2010(9), 115. DOI.