The phase of de Sitter higher spin gravity

Abstract: The one-loop Euclidean partition function on the sphere is known to exhibit a nontrivial phase for massless fields of spin greater than one. Such a phase appears to be in tension with a state counting interpretation of the partition function and its relation to the de Sitter entropy. It has been recently argued that the phase associated with the gravitational path integral can be cancelled by including the contribution of an observer. In this note, we compute the total phase of Vasiliev higher spin gravity on the sphere by summing over the contributions of all spins. We evaluate the resulting infinite sum using two different regularization schemes, obtaining consistent results. We find that for the non-minimal Vasiliev theory, which includes massless fields of all integer spins, the total phase vanishes in all dimensions. This result suggests that the sphere partition function of these theories may be consistent with a counting interpretation, without explicitly including an observer.

• The paper shows that non-minimal Vasiliev higher spin gravity on S^{d+1} carries a total one-loop phase of zero, implying its sphere partition function may be interpreted statistically without invoking an observer, through summation over the negative modes for each spin $i^P_s$
• Giombi and Sun employ two regularization schemes, dimensional regularization and character regularization, both of which yield consistent results across all dimensions, solving the singularities of the dimensionless sum required for obtaining $P$ by the Hilbert function $D^{d+2}_{s-1,s-1}$
• The observed vanishing total phase represents a crucial milestone in reconciling the tension between the phase of the gravitational path integral and the statistical interpretation of de Sitter entropy, excluding an observer to previous models

The one-loop Euclidean partition function of gravity on a sphere carries a nontrivial phase arising from negative modes of the conformal factor, a fact that sits uneasily with the interpretation of $Z_{S^{d+1}}$ as encoding the de Sitter entropy $S = \log Z$. This note by Giombi and Sun extends the analysis of such phases to Vasiliev higher spin gravity, computing the "total phase" obtained by summing over all spins in the theory. The central result is that for the non-minimal Vasiliev theory — containing one massless field of every integer spin — the total phase vanishes identically in all dimensions, suggesting that its sphere partition function may admit a statistical (counting) interpretation without invoking an observer.

Background: phases of the de Sitter path integral

At the round-sphere saddle, Polchinski showed that the one-loop gravitational partition function acquires a phase $i^{d+3}$, where $i^1$ comes from the constant scalar mode and $i^{d+2}$ from the non-isometric conformal Killing vectors (CKVs). Maldacena recently argued that this phase can be cancelled by including an observer modeled as a heavy particle: $d$ of the $d+2$ CKVs that move the observer's worldline are compensated by the observer's partition function, and the residual factor is removed by imposing the Hamiltonian constraint $H_{\rm tot}=0$.

Crucially, the phase phenomenon is not exclusive to gravity. For a massless spin-$s\ge 2$ field on $S^{d+1}$, the partition function carries a phase $S = \log Z$0, where

$S = \log Z$1

with $S = \log Z$2 the dimension of the irreducible SO$S = \log Z$3 representation of highest weight $S = \log Z$4. In general $S = \log Z$5 is a degree-$S = \log Z$6 polynomial in $S = \log Z$7; on $S = \log Z$8, $S = \log Z$9, reproducing Polchinski's result at $i^{d+3}$0. The phase originates from negative modes of the Fronsdal action's trace sector after contour rotation, in direct analogy to the pure-trace graviton modes.

Setup: the higher spin sphere partition function

The paper works with the universal one-loop formula for parity-invariant effective field theories on spheres:

$i^{d+3}$1

where $i^{d+3}$2 is the total phase, $i^{d+3}$3 is the infinite-dimensional higher spin group whose dimension $i^{d+3}$4 controls the $i^{d+3}$5 dependence via $i^{d+3}$6, and $i^{d+3}$7 is fixed by the physical spectrum. The tree-level entropy is inferred from AdS holography by analytic continuation, giving $i^{d+3}$8 for the non-minimal theory and $i^{d+3}$9 for the minimal one, consistent with the dS$i^1$0/CFT$i^1$1 relation $i^1$2.

The technical obstacle is that both $i^1$3 and $i^1$4 are divergent sums over polynomials of degree up to $i^1$5 in $i^1$6. Zeta-function regularization is rejected because the answer depends on an arbitrary shift parameter with no preferred value. Instead, two independent schemes are employed.

Dimensional regularization

For the non-minimal theory, computing $i^1$7 reduces to evaluating $i^1$8. Since $i^1$9 at large $i^{d+2}$0, the sum converges for $i^{d+2}$1. Using an integral representation of $i^{d+2}$2 valid in the strip $i^{d+2}$3, the sum can be performed inside the integral, yielding $i^{d+2}$4. Adding the exactly known $i^{d+2}$5 and $i^{d+2}$6 terms gives zero throughout the strip, hence by analytic continuation:

$i^{d+2}$7

Two consequences follow immediately. First, the regularized dimension of the non-minimal higher spin algebra vanishes, so the one-loop partition function has no $i^{d+2}$8 dependence. Second, since $i^{d+2}$9, the total phase also vanishes. This is the paper's main claim: the non-minimal Vasiliev sphere partition function may be consistent with a counting interpretation without explicitly including an observer.

For the minimal theory (even spins only), the analogous computation gives

$d$0

which vanishes for even $d$1 due to the pole of $d$2 but is nonzero for odd $d$3: e.g., $d$4 on $d$5 ($d$6), $d$7 on $d$8 ($d$9), and $d+2$0 on $d+2$1. The identity $d+2$2 persists in all cases.

Character regularization

As an independent check, the authors construct a regulator adapted to the character-theoretic derivation of $d+2$3. The negative-mode content of each spin-$d+2$4 sector is encoded in a Laurent polynomial $d+2$5 whose value at $d+2$6 equals $d+2$7. Dropping terms singular at $d+2$8 defines a function $d+2$9 that decays exponentially for $H_{\rm tot}=0$0, making $H_{\rm tot}=0$1 convergent; the total phase is then read off as the constant term of the Laurent expansion around $H_{\rm tot}=0$2. At $H_{\rm tot}=0$3, the non-minimal sum yields $H_{\rm tot}=0$4, which has no constant term — total phase zero, matching dimensional regularization. The minimal sum gives constant term $H_{\rm tot}=0$5, again matching. A simpler regulator $H_{\rm tot}=0$6 with $H_{\rm tot}=0$7 produces identical results. The scheme generalizes verbatim to all $H_{\rm tot}=0$8.

Relation to dS/CFT and open questions

Under the conjectural dS/CFT correspondence, the non-minimal Vasiliev theory is dual to a complex vector model of anticommuting scalars in the $H_{\rm tot}=0$9 singlet sector, while the minimal theory is dual to an $s\ge 2$0 model. The vanishing phase and vanishing $s\ge 2$1 dependence in the non-minimal case are potentially consistent with this duality. The non-vanishing minimal-theory phase, however, lacks a clear boundary interpretation — unlike the AdS case, where minimal-theory answers are explained by the shift $s\ge 2$2. The authors speculate it may relate to the non-compactness of $s\ge 2$3 but leave this unresolved. It also remains unexplained why the odd-spin sector is essential to the cancellation mechanism, and why the regularized algebra dimension equals minus the total phase in both theories.

Conclusion

By evaluating the infinite spin sum with two independent regularization schemes that agree in every dimension, the paper establishes that the total one-loop phase of non-minimal Vasiliev higher spin gravity on $s\ge 2$4 vanishes, along with the $s\ge 2$5 dependence of the partition function. This removes, for these theories, the tension between the phase of the gravitational path integral and a statistical interpretation of the de Sitter entropy — without needing to invoke an observer. The corresponding non-zero result for the minimal theory, and its meaning under dS/CFT, remain open questions.