Higher-Spin de Sitter Examples

Higher-spin dS/CFT is an unusually explicit cosmological holography proposal: analytically continue the AdS$_4$ Vasiliev/$O(N)$ dictionary to de Sitter and replace the commuting vector model by an $Sp(N)$ model of anticommuting scalars. The boundary partition function can then compute candidate Hartle–Hawking wavefunction coefficients. Exact calculability is a strength; nonunitarity, contour dependence, and the absence of a general static-patch inner product remain real limits.

Required background. Higher-Spin and Vector-Model Dualities supplies the AdS dictionary; dS/CFT Dictionaries and Analytic Continuation fixes what continuation can establish.

Helpful background. Vasiliev Higher-Spin Equations, Interactions, and Locality Obstructions supplies the bulk interaction caveat; Conjugation and Reflection Positivity supplies the boundary test.

The Sp(N) proposal

Let $\chi^a$ be anticommuting scalars with symplectic form $\Omega_{ab}$. The singlet action is schematically

$$S_{Sp(N)}=\frac12\int d^3x\,\Omega_{ab}\,\partial_\mu\chi^a\partial^\mu\chi^b.$$

Closed vector loops acquire a minus sign relative to commuting $O(N)$ scalars, implementing an $N\to-N$ continuation of singlet correlators. The conjecture identifies the $Sp(N)$ generating functional with the future Hartle–Hawking wavefunction of a parity-even higher-spin theory in dS$_4$, with the free and critical models corresponding to the two scalar boundary conditions Anninos, Hartman, and Strominger 2017, §§ 2–4.

The model has the desired Euclidean conformal symmetry and a computable current tower. But anticommuting scalars violate the positivity properties of an ordinary unitary CFT. Its partition function is therefore candidate wavefunction data, not a conventional positive-norm boundary probability theory.

First application: continue a normalized correlator

Begin with the AdS relation for a scalar singlet $J_0$,

$$\langle J_0(\mathbf k)J_0(-\mathbf k)\rangle_{O(N)}=N\,C(k).$$

Continue $L_{\mathrm{AdS}}\to i/H$ and $N\to-N$, together with the regular AdS contour to the Bunch–Davies contour. The nonlocal kernel maps to a dS wavefunction coefficient

$$\psi_2^{\mathrm{dS}}(k)=e^{i\vartheta_2}N\,C(k),$$

where the phase $e^{i\vartheta_2}$ is fixed by the radius, field, and contour conventions. At three points the same continuation predicts every tensor structure and a correlated phase; local polynomial terms remain counterterm dependent. This is a real quantitative test only when two-point normalizations and the scalar boundary condition are held fixed.

Functional-determinant calculations evaluate the $Sp(N)$ partition function on $S^3$ with metric and scalar-source deformations. They find the pure de Sitter configuration as a local maximum but also nontrivial behavior for large deformations, directly probing wavefunction normalizability rather than merely matching symmetries Anninos et al. 2014, §§ 3–5.

Reality, contour, and observer limits

Bulk fields may be real while their late-time wavefunction coefficients are complex. The correct conjugation pairs the forward and backward contours; it is not reflection positivity of the single Euclidean $Sp(N)$ functional. Double-trace changes correspond to boundary-condition or basis transforms and must be implemented on the wavefunction, not treated as unitary RG evolution.

The construction is semiclassical at large $N$, with $G_NH^2\sim1/N$. It has no higher-spin gap and no Einstein truncation. Vasiliev pseudo-locality persists after continuation. No generic $g_s$, $\alpha'H^2$, Kaluza–Klein, or metastable de Sitter embedding is supplied.

Adversarial control: request positivity and a patch observable

Compute the norm matrix of simple $Sp(N)$ states under ordinary radial conjugation; negative or indefinite entries show that the boundary model is not a unitary Euclidean CFT. Then request a normalized finite-time static-detector response from the partition function. Without a Lorentzian inner product and reconstruction map, the late-time coefficient does not supply it. These failures limit the interpretation without erasing the successful wavefunction calculations.

The evidence ceiling is an explicit, highly testable late-time higher-spin wavefunction proposal with spectrum, correlator, and determinant checks. It is not a universal dS dual, a positive boundary Hilbert space, a static-patch completion, or an Einstein/string construction. Those missing structures must remain visible in the final status comparison.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Anninos, D., Denef, F., Konstantinidis, G., and Shaghoulian, E. (2014). “Higher Spin de Sitter Holography from Functional Determinants.” Journal of High Energy Physics 2014(2), 007. DOI.
• Anninos, D., Hartman, T., and Strominger, A. (2017). “Higher Spin Realization of the dS/CFT Correspondence.” Classical and Quantum Gravity 34, 015009. DOI.