dS$^4$ Metamorphosis

Abstract: We study the Euclidean path integral of higher spin gravity on $S<sup>4$. Based on a one-loop analysis, we are led to a gluing formula expressing the $S<sup>4$ path integral in terms of an underlying $S<sup>3$ path integral. We view the three-sphere as a boundary hypersurface splitting the four-sphere into two halves. For a higher spin spectrum containing even spins only, the resulting boundary theory living on the $S<sup>3$ cut is the $\mathrm{Sp}(N)$ invariant sector of $N\in \mathbb{Z}<sup>+$ anti-commuting, conformally coupled free scalars, with conformal higher spin sources mediating the gluing. This boundary $\mathrm{Sp}(N)$ theory was previously shown to compute the Hartle-Hawking wavefunction at $\mathcal{I}<sup>+$ in the higher spin dS$_4$/CFT$_3$ correspondence. In contrast to the infinite spatial volume of $\mathcal{I}<sup>+$, here the conformal fields populate a finite size $S<sup>3$ hypersurface of $S<sup>4$. For theories with both bosonic and fermionic higher spin fields, the gluing formula is instead built from an $\mathcal{N}=2$ superconformal boundary field theory coupled to $U(N)$ invariant superconformal sources. Under this assumption, the leading contribution to the four-sphere partition function is $2^N$, and we observe exact cancellations at one-loop.

• The paper shows that the minimal dS₄ higher spin one-loop partition function reorganizes into a three-dimensional conformal higher spin contribution plus twice the free conformal scalar partition function on S³.
• The authors conjecture that the interacting S⁴ path integral glues two hemispheres through an Sp(N) boundary theory of anticommuting scalars, reproducing higher spin determinants and interpreting the result as a Hartle–Hawking wavefunction norm.
• For the N=2 supersymmetric model, boson–fermion cancellations yield a compact one-loop form proportional to 2^N N^{-1/16}, while higher-loop validation, group-volume definitions, and entropy interpretation remain unresolved.

The problem and the setting

The paper studies the Euclidean path integral of four-dimensional higher spin gravity with positive cosmological constant on the round $S^4$, asking what microphysical completion underlies the sphere partition function. Higher spin theories in de Sitter space are among the few models of $\Lambda>0$ quantum gravity amenable to explicit computation, since the entire perturbative spectrum — a conformally coupled scalar plus totally massless Fronsdal fields of all spins (or even spins only, in the minimal model) — is known exactly. Each Fronsdal field is a highest-depth partially massless field transforming in the discrete series representation $D^{\pm}_{s,s-1}$ of $\mathrm{SO}(1,4)$, carrying two local degrees of freedom.

The key technical input is the relation between Lorentzian Harish-Chandra characters of $\mathfrak{so}(1,4)$ representations and Euclidean one-loop sphere partition functions established in earlier work (Anninos et al., 2020): the one-loop contribution of any field in the spectrum can be written as an integral over static-patch time $\mathfrak{t}$ of its character, weighted by $(1+e^{-\mathfrak{t}})/(1-e^{-\mathfrak{t}})$. For gauge fields this involves both a bulk character diverging as $\mathfrak{t}^{-3}$ and a codimension-two edge character diverging as $\mathfrak{t}^{-1}$.

The one-loop metamorphosis

Summing the characters over the minimal (even-spin) spectrum yields a striking factorization:

$$\log \mathcal{Z}^{(1)}_{\mathrm{h.s.}} = \log \mathcal{Z}_{\mathrm{HS}} + \log \mathcal{Z}_{\mathrm{free}},$$

where the first term is the one-loop partition function of a three-dimensional conformal higher spin gauge theory on $\Lambda>0$0, equal to $\Lambda>0$1 after removing local counterterms, and the second is twice the partition function of a free conformally coupled scalar on $\Lambda>0$2, equal to $\Lambda>0$3 plus power-law divergences. This resummation is possible only for dS$\Lambda>0$4 (and dS$\Lambda>0$5); it cannot be performed explicitly in general dimensions.

The appearance of three-dimensional divergences from a closed four-manifold path integral is the central puzzle motivating the paper's proposal. Combining the one-loop sum with a postulated on-shell action $\Lambda>0$6 — inferred from the AdS$\Lambda>0$7/CFT$\Lambda>0$8 duality via the relation $\Lambda>0$9 — gives a leading contribution to the Gibbons-Hawking horizon entropy that is positive, as expected.

The gluing formula

The authors conjecture that the full interacting partition function takes the form

$D^{\pm}_{s,s-1}$0

where $D^{\pm}_{s,s-1}$1 is the partition function of $D^{\pm}_{s,s-1}$2 anti-commuting conformally coupled real scalars on an equatorial $D^{\pm}_{s,s-1}$3 cut of $D^{\pm}_{s,s-1}$4, sourced by a bilocal object $D^{\pm}_{s,s-1}$5 coupling to all $D^{\pm}_{s,s-1}$6-invariant quadratic currents (a scalar plus conserved currents at each even spin). The statistics reversal encodes the sign difference between AdS$D^{\pm}_{s,s-1}$7 and dS$D^{\pm}_{s,s-1}$8 on-shell actions. Because all currents are quadratic in the $D^{\pm}_{s,s-1}$9, the sourced partition function is an exact functional determinant; expanding around $\mathrm{SO}(1,4)$0 produces, at Gaussian order, determinants of higher spin current two-point functions that reproduce precisely the three-dimensional conformal higher spin one-loop answer. Since conformal higher spin fields in three dimensions carry no local degrees of freedom, all spin-$\mathrm{SO}(1,4)$1 character contributions vanish, leaving only the $\mathrm{SO}(1,4)$2 sector — confirming the gluing formula to one loop. The group volume contributes factors $\mathrm{SO}(1,4)$3 reflecting the normalization of generators by the coupling.

The formula remains formal in several respects: ultraviolet divergences requiring three-dimensional (not four-dimensional) counterterms, the infinite dimension of the higher spin group, the undefined volume of that group, and the unregularized Polchinski phase $\mathrm{SO}(1,4)$4, for which recent work (Giombi et al., 21 Jan 2026) proposes $\mathrm{SO}(1,4)$5 for the minimal model.

Relation to the Hartle-Hawking wavefunction

The bilinear structure invites the identification $\mathrm{SO}(1,4)$6, so that the $\mathrm{SO}(1,4)$7 partition function is the norm of the Hartle-Hawking wavefunction. Notably, the same $\mathrm{SO}(1,4)$8 boundary theory computes the Hartle-Hawking wavefunction at $\mathrm{SO}(1,4)$9 in the higher spin dS$\mathfrak{so}(1,4)$0/CFT$\mathfrak{so}(1,4)$1 correspondence (Anninos et al., 2011), but here the conformal fields live on a finite-size $\mathfrak{so}(1,4)$2 hypersurface rather than the infinite-volume future boundary. The interpretation offered is that the gluing formula implements conformal boundary conditions (fixing the conformal three-metric and $\mathfrak{so}(1,4)$3) on the finite hypersurface; pushing the hypersurface to $\mathfrak{so}(1,4)$4 corresponds to tuning $\mathfrak{so}(1,4)$5 to a special complex value. The norm interpretation carries a caveat: gravitational sphere path integrals generally carry a saddle-dependent phase from the complexified conformal-mode contour [Polchinski:1988ua], so the identification $\mathfrak{so}(1,4)$6 is marked as conjectural throughout.

The $\mathfrak{so}(1,4)$7 super-gluing formula

For the non-minimal supersymmetric higher spin model containing massless fields of every integer and half-integer spin, the completion is proposed to be built from a free $\mathfrak{so}(1,4)$8 superconformal $\mathfrak{so}(1,4)$9 vector model ($\mathfrak{t}$0 anti-commuting complex scalars and $\mathfrak{t}$1 commuting Dirac fermions) coupled to $\mathfrak{t}$2-invariant superconformal sources packaged in a chiral–anti-chiral bilocal superfield $\mathfrak{t}$3. A series of cancellations then occurs:

• The fermionic tower of AdS$\mathfrak{t}$4 one-loop characters vanishes identically on its own.
• The bosonic and fermionic functional determinants cancel their $\mathfrak{t}$5 terms and their $\mathfrak{t}$6 divergences.
• All supersymmetric conformal higher spin characters vanish except at $\mathfrak{t}$7, where the two scalar contributions (operators of weight $\mathfrak{t}$8 and $\mathfrak{t}$9) differ by sign and cancel.
• The Polchinski phase plausibly vanishes because each integer-spin field doubles, and the Dirac spectra come in complex-conjugate pairs despite imaginary fermionic "mass".
• Remarkably, the regularized dimension of the higher spin supergroup is finite, $(1+e^{-\mathfrak{t}})/(1-e^{-\mathfrak{t}})$0, unlike the strictly infinite bosonic case.

The final one-loop result is

$(1+e^{-\mathfrak{t}})/(1-e^{-\mathfrak{t}})$1

with the residual volume independent of $(1+e^{-\mathfrak{t}})/(1-e^{-\mathfrak{t}})$2, so ratios of partition functions are clean: $(1+e^{-\mathfrak{t}})/(1-e^{-\mathfrak{t}})$3, which is rational when $(1+e^{-\mathfrak{t}})/(1-e^{-\mathfrak{t}})$4 and $(1+e^{-\mathfrak{t}})/(1-e^{-\mathfrak{t}})$5 share the appropriate sixteenth-power structure. The authors suggest the $(1+e^{-\mathfrak{t}})/(1-e^{-\mathfrak{t}})$6 partition function may be amenable to localization and hence one-loop exact, which would open a quantitative route to a microscopic account of the de Sitter horizon entropy. A further obstacle is identified here: the super-volume of the candidate seed group $(1+e^{-\mathfrak{t}})/(1-e^{-\mathfrak{t}})$7 vanishes due to unsaturated Grassmann zero modes (shown explicitly in the appendix), whereas the $(1+e^{-\mathfrak{t}})/(1-e^{-\mathfrak{t}})$8 counterparts $(1+e^{-\mathfrak{t}})/(1-e^{-\mathfrak{t}})$9 and $\mathfrak{t}^{-3}$0 have non-vanishing volumes; soaking up these zero modes may require inserting operators analogous to picture-changing operators.

Limitations and open questions

Several assumptions underpin the results. The gluing formulas are verified only to one loop; higher-order terms require a definition of the measure $\mathfrak{t}^{-3}$1, for which only a proposal exists. The dS$\mathfrak{t}^{-3}$2 on-shell actions are postulated by analogy with AdS$\mathfrak{t}^{-3}$3/CFT$\mathfrak{t}^{-3}$4 rather than computed, and the precise map between $\mathfrak{t}^{-3}$5 and $\mathfrak{t}^{-3}$6 beyond leading order is undetermined. The Polchinski phase in the bosonic theory lacks a definitive regularization, and the volume of the (super-)higher spin group has no rigorous mathematical definition — exponentiation of higher spin algebras into groups is itself not on firm footing. Whether a gluing formula of this type is a defining property of $\mathfrak{t}^{-3}$7 quantum gravity, or an artifact of the special higher spin structure, is left open. Finally, the speculation that consistency conditions (integrality or rationality of partition-function ratios) could restrict the discretuum of allowed $\mathfrak{t}^{-3}$8 values remains conjectural.

Conclusion

The paper shows that the one-loop $\mathfrak{t}^{-3}$9 partition function of minimal higher spin gravity reorganizes exactly into a three-dimensional character sum, suggesting a gluing formula in which two hemispheres are sewn along an $\mathfrak{t}^{-1}$0 carrying the $\mathfrak{t}^{-1}$1 invariant sector of free anti-commuting scalars coupled to conformal higher spin sources — the same boundary data that encode the Hartle-Hawking wavefunction at $\mathfrak{t}^{-1}$2. In the supersymmetric non-minimal model, exact cancellations reduce the answer to $\mathfrak{t}^{-1}$3 times group-theoretic factors, with a finite regularized supergroup dimension and a candidate one-loop-exact structure. These results connect the Euclidean sphere path integral, the Lorentzian wavefunction norm, and the Gibbons-Hawking entropy within a single calculable framework, while leaving the status of the gluing formula beyond one loop, the definition of higher spin group volumes, and the interpretation of the $\mathfrak{t}^{-1}$4 counting as horizon entropy as concrete open problems.