On higher spin partition functions

Let us consider the standard 2-derivative free MHS field in flat $\rm{d}$-dimensional space. The corresponding partition function can be written as

$$\begin{eqnarray}&&{Z}_{\mathrm{MHS},s}={\left[{\displaystyle \frac{\mathrm{det}\;{\Delta }_{s-1\;\perp }}{\mathrm{det}\;{\Delta }_{s\;\perp }}}\right]}^{1/2}={\left[{\displaystyle \frac{{(\mathrm{det}\;{\Delta }_{s-1})}^{2}}{\mathrm{det}\;{\Delta }_{s}\mathrm{det}\;{\Delta }_{s-2}}}\right]}^{1/2},\end{eqnarray} \tag{ 2.1 }$$

where ${\Delta }_{s}$ is flat Laplacian $-{\partial }^{2}$ defined on symmetric rank s traceless tensors, and ${\Delta }_{s\perp }$ is its restriction to transverse fields. We shall assume that $\mathrm{det}\;{\Delta }_{k}$ with $k\lt 0$ is replaced by 1, i.e. ${Z}_{\mathrm{MHS},0}={(\mathrm{det}\;{\Delta }_{0})}^{-1/2}$. Let us consider a theory where each massless field with s = 0, 1, 2,... appears just once. This is field content of Vasiliev theory linearized near ${{AdS}}_{\rm{d}}$ vacuum which is dual to a large N free complex scalar theory in $d=\rm{d}-1$ dimensions. While the expansion of interacting Vasiliev theory near flat space is singular, we may formally view the free MHS partition function in flat space as a formal zero curvature limit of its one-loop counterpart in ${{AdS}}_{\rm{d}}$. Then one finds that the total partition function is trivial:

$$\begin{eqnarray}&&{({Z}_{\mathrm{MHS}})}_{\mathrm{tot}}=\underset{s=0}{\overset{\infty }{{\displaystyle \prod }}}{Z}_{\mathrm{MHS},s}={\left[{\displaystyle \frac{1}{\mathrm{det}\;{\Delta }_{0}}}\right]}^{1/2}{\left[{\displaystyle \frac{\mathrm{det}\;{\Delta }_{0}}{\mathrm{det}\;{\Delta }_{1\;\perp }}}\right]}^{1/2}{\left[{\displaystyle \frac{\mathrm{det}\;{\Delta }_{1\;\perp }}{\mathrm{det}\;{\Delta }_{2\;\perp }}}\right]}^{1/2}{\left[{\displaystyle \frac{\mathrm{det}\;{\Delta }_{2\;\perp }}{\mathrm{det}\;{\Delta }_{3\;\perp }}}\right]}^{1/2}...=1.\end{eqnarray} \tag{ 2.2 }$$

This remarkable property reminds of a supersymmetric theory where the bosonic contribution to the vacuum partition function is cancelled against the fermionic one (implying the vanishing of the vacuum energy). Here the cancellation is between the contribution of the physical spin s field determinant and the ghost determinant for spin $s+1$ field, i.e. it reflects a large gauge symmetry of the theory⁵ .

The cancellation of an infinite number of factors in (2.2) is formal (see 1 −1 + 1 −1 +... = 0) as it depends on how one groups terms together: in general, an infinite product requires a regularization and depends on its choice. The choice of regularization should be consistent with an underlying symmetry of the theory (in the present case—higher spin gauge symmetry). Let us first consider the case of $\rm{d}=4$. Observing that each spin $s\gt 0$ field has 2 dynamical degrees of freedom (see (2.1))

$$\begin{eqnarray}&&{Z}_{\mathrm{MHS},s}={({Z}_{0})}^{{\nu }_{s}},\qquad {Z}_{0}={\left[{\displaystyle \frac{1}{\mathrm{det}\;{\Delta }_{0}}}\right]}^{1/2},\qquad {\nu }_{s}={(s+1)}^{2}+{(s-1)}^{2}-2{s}^{2}=2,\end{eqnarray} \tag{ 2.3 }$$

we get

$$\begin{eqnarray}&&{Z}_{\mathrm{tot}}={({Z}_{0})}^{{\nu }_{\mathrm{tot}}},\qquad \qquad \qquad {\nu }_{\mathrm{tot}}=1+\underset{s=1}{\overset{\infty }{\sum }}{\nu }_{s}=1+2\underset{s=1}{\overset{\infty }{\sum }}\ 1=0,\end{eqnarray} \tag{ 2.4 }$$

where we used the standard Riemann zeta-function as a regularization of the sum over s: ${\sum }_{s=1}^{\infty }1={\zeta }_{R}(0)=-\frac{1}{2}$. Remarkably, in $\rm{d}=4$ the use of the simple zeta-function regularization is thus equivalent to the formal cancellation of factors in (2.2)⁶ . Note that this flat-space property ${({Z}_{\mathrm{MHS}})}_{\mathrm{tot}}=1$ is non-trivial, i.e. it holds for a flat torus, implying, in particular, the vanishing of vacuum energy or finite temperature partition function on ${S}^{1}\times {R}^{3}$. The partition function may be non-trivial if one considers an orbifold of flat space.

For a massless spin s field in d flat dimensions we get

$$\begin{eqnarray}\begin{array}{l}\mathrm{det}\;{\Delta }_{s} & = & {(\mathrm{det}\;{\Delta }_{0})}^{{N}_{s}},\quad \qquad \mathrm{det}\;{\Delta }_{\perp \;s}={(\mathrm{det}\;{\Delta }_{0})}^{{N}_{s}^{\perp }},\\ {N}_{s} & = & {\displaystyle \left({\displaystyle \genfrac{}{}{0em}{}{s+\rm{d}-1}{s}}\right)}-{\displaystyle \left({\displaystyle \genfrac{}{}{0em}{}{s+\rm{d}-3}{s-2}}\right)},\quad {N}_{s}^{\perp }={N}_{s}-{N}_{s-1},\ \ \ {\nu }_{s}={N}_{s}^{\perp }-{N}_{s-1}^{\perp }\\ & = & 2\left[s+\frac{1}{2}(\rm{d}-4)\right]{\displaystyle \frac{(s+\rm{d}-5)!}{s!(\rm{d}-4)!}}.\end{array}\end{eqnarray} \tag{ 2.5 }$$

Then ${Z}_{\mathrm{tot}}=1$ or ${\nu }_{\mathrm{tot}}=0$ is true, e.g., for general even $\rm{d}$ if one uses the following regularization:

$$\begin{eqnarray}&&{\nu }_{\mathrm{tot}}={\left.1+\underset{s=1}{\overset{\infty }{\sum }}{\nu }_{s}\ {\rm{e}}^{-\epsilon [s+\frac{1}{2}(\rm{d}-4)]}\right|}_{\mathrm{fin}.}=0.\end{eqnarray} \tag{ 2.6 }$$

Here one performs the sum for fixed $\epsilon$, then takes $\epsilon \to 0$ and finally drops all singular $\frac{1}{{\epsilon }^{n}}$ terms (in $\rm{d}=4$ this is equivalent to the standard zeta-function prescription)⁷ .

One may wonder if the ${({Z}_{\mathrm{MHS}})}_{\mathrm{tot}}=1$ property may generalize to curved spaces, e.g., Ricci-flat ones. As is well known, MHS theories are not consistent in ${R}_{\mu \nu }=0$ background (do not have flat-space gauge symmetries surviving) for $s\gt 2$. However, one may formally assume that there exists a consistent theory of all higher spin fields where proper gauge symmetry is present off-shell at interacting level. Vasiliev theory does not have ${R}_{\mu \nu }=0$ as a classical vacuum solution since the limit of vanishing cosmological constant appears to be singular in the interaction terms, but one may consider formally expanding the (hopefully existing) action of Vasiliev theory near an off-shell ${R}_{\mu \nu }=0$ background and computing the resulting one-loop partition function. If ${R}_{\mu \nu }=0$ is not a classical solution this partition function will be gauge-dependent, but otherwise it may be well-defined and of interest being a direct generalization of the flat-space one (2.1), (2.2). A natural spin s counterpart of spin 2 Lichnerowicz operator on Ricci flat background may be chosen as in [17]⁸

$$\begin{eqnarray}&&{\Delta }_{\rm{L}\;s}=-{\nabla }_{s}^{2}+{X}_{s},\qquad \qquad {({X}_{s}\;\varphi )}^{{\mu }_{1}\cdots {\mu }_{s}}=-s(s-1){R}_{\nu }^{{({\mu }_{1}}_{\lambda }^{{\mu }_{2}}{\varphi }^{{\mu }_{3}\cdots {\mu }_{s})}\nu \lambda }.\end{eqnarray} \tag{ 2.7 }$$

Then one may formally consider the following generalization of the well-known spin $s=1,2$ partition functions on Ricci-flat background to any spin s [7]

$$\begin{eqnarray}&&{Z}_{\mathrm{MHS},s}={\left[{\displaystyle \frac{{(\mathrm{det}\;{\Delta }_{\rm{L}\;s-1})}^{2}}{\mathrm{det}\;{\Delta }_{\rm{L}\;s}\mathrm{det}\;{\Delta }_{\rm{L}\;s-2}}}\right]}^{1/2}.\end{eqnarray} \tag{ 2.8 }$$

Taking the product of (2.8) over s we again get that ${({Z}_{\mathrm{MHS}})}_{\mathrm{tot}}=1$ as in the flat space case (2.2):

$$\begin{eqnarray}\begin{array}{l}{({Z}_{\mathrm{MHS}})}_{\mathrm{tot}} & = & \underset{s=0}{\overset{\infty }{{\displaystyle \prod }}}{Z}_{\mathrm{MHS},s}\\ & & ={\left[{\displaystyle \frac{1}{\mathrm{det}\;{\Delta }_{\rm{L}\;0}}}\right]}^{1/2}\ {\left[{\displaystyle \frac{{(\mathrm{det}\;{\Delta }_{\rm{L}\;0})}^{2}}{\mathrm{det}\;{\Delta }_{\rm{L}\;1}}}\right]}^{1/2}\\ & & \times {\left[{\displaystyle \frac{{(\mathrm{det}\;{\Delta }_{\rm{L}\;1})}^{2}}{\mathrm{det}\;{\Delta }_{\rm{L}\;2}\mathrm{det}\;{\Delta }_{\rm{L}\;0}}}\right]}^{1/2}{\left[{\displaystyle \frac{{(\mathrm{det}\;{\Delta }_{\rm{L}\;2})}^{2}}{\mathrm{det}\;{\Delta }_{\rm{L}\;3}\mathrm{det}\;{\Delta }_{\rm{L}\;1}}}\right]}^{1/2}....=1.\end{array}\end{eqnarray} \tag{ 2.9 }$$

Here the cancellation follows from the fact that each spin s operator appears exactly twice in both the numerator and the denominator.

The ${Z}_{\mathrm{tot}}=1$ property is expected to hold also in the proper vacuum of the Vasiliev theory –${{AdS}}_{\rm{d}}$ space, and should be true to all orders in the coupling expansion. As was pointed out in [6, 8], this is the requirement of consistency of the vectorial AdS/CFT: the boundary theory of free U(N) scalar has log of its partition function scaling as N which should match the classical action of the Vasiliev theory in ${{AdS}}_{\rm{d}}$, while the 1-loop (and all higher-loop) corrections to $\mathrm{ln}{Z}_{\mathrm{tot}}$ of MHS theory should vanish (in a proper regularization). To demonstrate this even at the one-loop order (free MHS theory in ${{AdS}}_{\rm{d}}$ background) is, however, much less trivial than in the flat space background considered above. Let us introduce the operator (k = 0, 1, ...., s − 1)

$$\begin{eqnarray}&&{\Delta }_{s}({M}_{s,k}^{2})\equiv -{\nabla }_{s}^{2}+{M}_{s,k}^{2}\sigma ,\qquad \ \ \qquad {M}_{s,k}^{2}=s-(k-1)(k+\rm{d}-2),\end{eqnarray} \tag{ 2.10 }$$

where $\sigma =\pm {a}^{-2}=\pm 1$ for unit-radius ${S}^{\rm{d}}$ or euclidean ${{AdS}}_{\rm{d}}$ space ($\sigma =0$ in flat space). Let us then define the partition function of a 'partially-massless' [20] spin s field (with gauge invariance with rank k tensor parameter) [7]

$$\begin{eqnarray}&&{Z}_{s,k}={\left[{\displaystyle \frac{\mathrm{det}\;{\Delta }_{k\;\perp }({M}_{k,s}^{2})}{\mathrm{det}\;{\Delta }_{s\;\perp }({M}_{s,k}^{2})}}\right]}^{1/2}.\end{eqnarray} \tag{ 2.11 }$$

Then for the massless spin s field (having maximal gauge invariance with rank $k=s-1$ parameter) on a homogeneous conformally flat space we get the following counterpart of (2.1) (see [4, 7, 21, 22])

$$\begin{eqnarray}\begin{array}{l}{Z}_{\mathrm{MHS},s} & \equiv & {Z}_{s,s-1}={\left[{\displaystyle \frac{\mathrm{det}\;{\Delta }_{s-1\;\perp }({M}_{s-1,s}^{2})}{\mathrm{det}\;{\Delta }_{s\;\perp }({M}_{s,s-1}^{2})}}\right]}^{1/2}\\ & = & {\left[{\displaystyle \frac{{\left(\mathrm{det}\;{\Delta }_{s-1}({M}_{s-1,s}^{2})\right)}^{2}}{\mathrm{det}\;{\Delta }_{s}({M}_{s,s-1}^{2})\mathrm{det}\;{\Delta }_{s-2}({M}_{s+2,s+1}^{2})}}\right]}^{1/2},\end{array}\end{eqnarray} \tag{ 2.12 }$$

where $s\gt 0$ and for s = 0 we have ${Z}_{\mathrm{MHS},0}={[\mathrm{det}\;(-{\nabla }^{2}+{M}_{0}^{2})]}^{-1/2},\ {M}_{0}^{2}={M}_{0,-1}^{2}\sigma =2(\rm{d}-3)\sigma$.

Taking the product of (2.12) over s we conclude that there is no straightforward cancellations of determinant factors that happened in flat space in (2.2) or in (2.9): the same-spin operators that appear in the numerator and denominator of ${\prod }_{s=0}^{\infty }{Z}_{\mathrm{MHS},s}$ are different for non-zero curvature, i.e. $\sigma \;\ne\;0$: they have different mass-like terms in (2.10). For example, in $\rm{d}=4$ case we get

$$\begin{eqnarray}&&{({Z}_{\mathrm{MHS}})}_{\mathrm{tot}}={\left[{\displaystyle \frac{1}{\mathrm{det}\;{\Delta }_{0}(2)}}\right]}^{1/2}{\left[{\displaystyle \frac{\mathrm{det}\;{\Delta }_{0}(0)}{\mathrm{det}\;{\Delta }_{1\;\perp }(3)}}\right]}^{1/2}{\left[{\displaystyle \frac{\mathrm{det}\;{\Delta }_{1\;\perp }(-3)}{\mathrm{det}\;{\Delta }_{2\;\perp }(2)}}\right]}^{1/2}{\left[{\displaystyle \frac{\mathrm{det}\;{\Delta }_{2\;\perp }(-8)}{\mathrm{det}\;{\Delta }_{3\;\perp }(-1)}}\right]}^{1/2}....\end{eqnarray} \tag{ 2.13 }$$

Using spectral zeta-function regularization one has, in a homogeneous space background,

$$\begin{eqnarray}&&\mathrm{ln}\;\mathrm{det}\;{\Delta }_{s}=-{\zeta }_{{\Delta }_{s}}(0)\mathrm{ln}{({La})}^{2}-\zeta {\prime }_{{\Delta }_{s}}(0),\qquad \ \ \ \ \ \ \ \ \ \ \ {\zeta }_{{\Delta }_{s}}(z)={\bar{\zeta }}_{{\Delta }_{s}}(z)\;{V}_{\rm{d}}.\end{eqnarray} \tag{ 2.14 }$$

Here L is UV cutoff, a is the curvature radius and ${V}_{\rm{d}}$ is the volume. In a non-compact space like ${{AdS}}_{\rm{d}}$ where the volume is formally divergent this relation requires an IR regularization (see, e.g., [14] and references there). Dropping power IR divergences, regularized ${V}_{\rm{d}}$ is then finite for even $\rm{d}$ and log divergent for odd $\rm{d}$ ($\rm{R}$ is an IR cutoff)

$$\begin{eqnarray}&&{V}_{\rm{d}=\mathrm{even}}={k}_{1},\qquad \qquad {V}_{\rm{d}=\mathrm{odd}}={k}_{2}\mathrm{lnR}+{k}_{3}.\end{eqnarray} \tag{ 2.15 }$$

As was shown in [8] using the explicit form of higher spin heat kernel for ${{AdS}}_{\rm{d}}$ [23], keeping the argument z of spectral $\zeta$-function non-zero, summing over spins and then taking $z\to 0$ one finds that ${\bar{\zeta }}_{\mathrm{tot}}(z)=O({z}^{2})$, i.e. that ${\bar{\zeta }}_{\mathrm{tot}}(0)=0,\ \ \bar{\zeta }{\prime }_{\mathrm{tot}}(0)=0$. Thus for any ${{AdS}}_{\rm{d}}$ with $\rm{d}\gt 3$ one has⁹

$$\begin{eqnarray}&&{({Z}_{\mathrm{MHS}})}_{\mathrm{tot}}=\underset{s=0}{\overset{\infty }{{\displaystyle \prod }}}{Z}_{\mathrm{MHS},s}=1.\end{eqnarray} \tag{ 2.16 }$$

The same conclusion should be true also in dimensional regularization used in [14] (see also appendix A) where ${V}_{\rm{d}}={\pi }^{{\displaystyle \frac{\rm{d}-1}{2}}}\Gamma (-{\displaystyle \frac{\rm{d}-1}{2}})$ contains a pole for odd $\rm{d}\to \rm{d}-\varepsilon$ (i.e. $\frac{1}{\varepsilon }\sim \mathrm{lnR}$). Here the product ${\bar{\zeta }}_{{\Delta }_{s}}(z)\;{V}_{\rm{d}}$ in (2.14) has also a non-trivial finite part whose s-dependent coefficient need not be the same as the coefficient of the $\frac{1}{\varepsilon }$ pole part. However, since ${\sum }_{s}\bar{\zeta }{\prime }_{\mathrm{eff},s}(0)=0$ property was shown in [8] to be true for any $\rm{d}$, the multiplication by s-independent factor V_d sj=hould not change this conclusion.

In general, if one uses dimensional (e.g. proper-time) UV cutoff L, then $\mathrm{lndet}\;{\Delta }_{s}$ in (2.14) will contain also power divergences. The coefficients of such ${L}^{d-2n}$ terms in $\rm{d}$ dimensions are controlled by the Seeley coefficients or ${\zeta }_{{\Delta }_{s}}(0)$ functions in $\rm{d}\prime =\rm{d}-2n$ dimensions. Since the sum over s of the corresponding combination of $\zeta (0)$ vanishes in any $\rm{d}\gt 2$ [8], this suggests that all power divergences should thus be absent too, demonstrating ${Z}_{\mathrm{tot}}=1$ in the proper-time cutoff regularization. This cancellation of power divergences is again analogous to what happens in supersymmetric theories.

For example, in d = 4 there will be L⁴ divergence with the coefficient of the total number of degrees of freedom (which vanishes according to (2.4), (2.6)) and also L² divergence proportional to $\mathrm{tr}(\frac{1}{6}R-{M}^{2})$ for an operator $-{\nabla }^{2}+{M}^{2}$ (here $R=12\sigma {a}^{-2}$ is the scalar curvature of the background metric). The coefficient of the R-term is again the total number of degrees of freedom, while the contribution of the M² terms is found to be (we suppress the overall sign factor $\sigma$)

$$\begin{eqnarray}&&2+\underset{s=1}{\overset{\infty }{\sum }}({M}_{s,s-1}^{2}{N}_{s}^{\perp }-{M}_{s-1,s}^{2}{N}_{s-1}^{\perp })=2+4\underset{s=1}{\overset{\infty }{\sum }}(1+2{s}^{2})=0,\end{eqnarray} \tag{ 2.17 }$$

where in the last step we used the standard zeta-function regularization equivalent to ${\rm{e}}^{-\epsilon s}$ cutoff. In any even $\rm{d}$ one finds the same vanishing result using the regularization factor $f(s,\epsilon )=\mathrm{exp}[-\epsilon (s+\frac{\rm{d}-4}{2})]$ as in [8].

Explicitly, higher Seeley coefficients for an operator $-{\nabla }^{2}+{M}^{2}$ on a constant curvature space are given by ${\sum }_{k,n}\mathrm{tr}({R}^{k}{M}^{2n})$ and thus are expressed in terms of terms proportional to $\mathrm{tr}({M}^{2n})$. equation (2.17) is interpreted as $\mathrm{tr}{M}^{2}=0$; one can show also the validity of its higher power analogs or mass sum rules $\mathrm{tr}({M}^{2n})=0$, where

$$\begin{eqnarray}\begin{array}{l}\mathrm{tr}({M}^{2n}) & = & {[2(\rm{d}-3)]}^{n}+\underset{s=1}{\overset{\infty }{{\displaystyle \sum }}}{\rm{e}}^{-\epsilon (s+\frac{\rm{d}-4}{2})}\\ & & \times \left[{M}_{s,s-1}^{2n}\;{N}_{s}+{M}_{s+2,s+1}^{2n}\;{N}_{s-2}-2{M}_{s-1,s}^{2n}\;{N}_{s-1}\right].\end{array}\end{eqnarray} \tag{ 2.18 }$$

Here the first term is the contribution from the spin 0 field¹⁰ .

It is natural to conjecture that the ${Z}_{\mathrm{tot}}=1$ property (2.16) should be true not only for the 1-loop partition function, but also for the exact ${{AdS}}_{\rm{d}}$ vacuum partition function of the Vasiliev theory (i.e. at any quantum loop order in the $1/N$ coupling expansion). As already mentioned, this the requirement of the vectorial AdS/CFT duality: the logarithm of the partition function of the dual free U(N) scalar theory has only the order N term (that should match the vacuum value of the classical action of the Vasiliev theory). This further strengthens the analogy with a supersymmetric or topological quantum field theory.

Finally, let us note that the property (2.16) need not apply to quotients of the ${{AdS}}_{\rm{d}}$ space—for example, the MHS partition function on thermal quotient of ${{AdS}}_{\rm{d}}$ is non-trivial (see [9] and references there).

Let us now consider the free partition function for CHS theory [7, 12]. The flat space action for a free CHS field in d dimensions is $\int {d}^{d}x\ {\varphi }_{s}{P}_{s}{\partial }^{2s+d-4}{\varphi }_{s}$, where P_s is projector to transverse traceless totally symmetric rank s field. Here s = 0 is a non-dynamical scalar, s = 1 is the Maxwell vector, s = 2 is the Weyl graviton, etc. The corresponding partition function in d = 4 is [7]

$$\begin{eqnarray}&&{Z}_{\mathrm{CHS},s}={\left[{\displaystyle \frac{{(\mathrm{det}\;{\Delta }_{s-1})}^{s+1}}{{(\mathrm{det}\;{\Delta }_{s})}^{s}}}\right]}^{1/2}=\underset{k=0}{\overset{s-1}{{\displaystyle \prod }}}\ {\left[{\displaystyle \frac{\mathrm{det}\;{\Delta }_{k\;\perp }}{\mathrm{det}\;{\Delta }_{s\;\perp }}}\right]}^{1/2},\end{eqnarray} \tag{ 2.19 }$$

where as in (2.1) the operator ${\Delta }_{s}=-{\partial }^{2}$ is defined on symmetric traceless tensors.

CHS fields having dimension $2-s$ are sources or 'shadow fields' for spin s conserved bilinear currents ${J}_{s}(\phi )$ built out of a free U(N) scalar field; they are also boundary values for the corresponding dual MHS theory in AdS_d+1. An interacting CHS theory may be defined as an induced one [24–26], obtained by integrating out $\phi$ in the path integral defined by the action $\int {d}^{4}x[\partial {\phi }^{*}\partial \phi +{\sum }_{s}{J}_{s}(\phi ){\varphi }_{s}]$. The resulting interacting CHS theory contains all fields with spins s = 0, 1, 2,.... The corresponding free partition function in flat background is given by

$$\begin{eqnarray}&&{({Z}_{\mathrm{CHS}})}_{\mathrm{tot}}=\underset{s=1}{\overset{\infty }{{\displaystyle \prod }}}{Z}_{\mathrm{CHS},s}={\left[{\displaystyle \frac{\mathrm{det}\;{\Delta }_{0}}{\mathrm{det}\;{\Delta }_{1}}}\right]}^{1/2}{\left[{\displaystyle \frac{{(\mathrm{det}\;{\Delta }_{1})}^{3}}{{(\mathrm{det}\;{\Delta }_{2})}^{2}}}\right]}^{1/2}{\left[{\displaystyle \frac{{(\mathrm{det}\;{\Delta }_{2})}^{4}}{{(\mathrm{det}\;{\Delta }_{3})}^{3}}}\right]}^{1/2}....\end{eqnarray} \tag{ 2.20 }$$

Formally cancelling similar factors in the numerator and the denominator of (2.20) leads, in contrast to (2.2), to a non-trivial result

$$\begin{eqnarray}&&{({Z}_{\mathrm{CHS}})}_{\mathrm{tot}}\ \to \ ({Z}_{\mathrm{CHS}}){\prime }_{\mathrm{tot}}=\underset{s=0}{\overset{\infty }{{\displaystyle \prod }}}\mathrm{det}\;{\Delta }_{s}.\end{eqnarray} \tag{ 2.21 }$$

This rearrangement of an infinite product in (2.20) effectively corresponds to its particular regularization.

Alternatively, we may use that each ${Z}_{\mathrm{CHS},s}$ factor (2.19) may be written as in (2.3), i.e. as ${({Z}_{0})}^{{\nu }_{s}}={[\mathrm{det}\;{\Delta }_{0}]}^{-{\nu }_{s}/2}$ where here ${\nu }_{s}=s(s+1)$ is the number of dynamical degrees of freedom of a CHS field in d = 4. Then

$$\begin{eqnarray}&&{({Z}_{\mathrm{CHS}})}_{\mathrm{tot}}=\underset{s=0}{\overset{\infty }{{\displaystyle \prod }}}{({Z}_{0})}^{{\nu }_{s}}={({Z}_{0})}^{{\nu }_{\mathrm{tot}}},\ \ \qquad \ \ \ {\nu }_{\mathrm{tot}}=\underset{s=0}{\overset{\infty }{\sum }}{\nu }_{s},\ \ \ \ \ \ \ {\nu }_{s}=s(s+1).\end{eqnarray} \tag{ 2.22 }$$

The total number of CHS degrees of freedom vanishes if one uses the regularization suggested by the relation to the MHS theory in ${{AdS}}_{\rm{d}}$ with $\rm{d}=d+1$ (in which also the total conformal anomaly vanishes [5, 7, 8]). Indeed, doing the sum in (2.22) with the $\mathrm{exp}[-\epsilon (s+\frac{d-3}{2})]$ cutoff as in (2.6) we get in the d = 4 case

$$\begin{eqnarray}&&{\nu }_{\mathrm{tot}}={\left.\underset{s=0}{\overset{\infty }{\sum }}s(s+1)\ {\rm{e}}^{-\epsilon (s+\frac{1}{2})}\right|}_{\mathrm{fin}.}=0,\ \ \ \ \qquad \rm{i}.\rm{e}.\qquad \ \ \ \ \ \ \ {({Z}_{\mathrm{CHS}})}_{\mathrm{tot}}=1,\end{eqnarray} \tag{ 2.23 }$$

as in the MHS case in (2.2), (2.6). This conclusion generalizes to the case of the CHS theory in d even dimensions where the partition function is [7]

$$\begin{eqnarray}&&{Z}_{\mathrm{CHS},s}={\left[{\left({\displaystyle \frac{1}{\mathrm{det}\;{\Delta }_{s\perp }}}\right)}^{\frac{d-4}{2}}\;\underset{k=0}{\overset{s-1}{{\displaystyle \prod }}}\ {\displaystyle \frac{\mathrm{det}\;{\Delta }_{k\perp }}{\mathrm{det}\;{\Delta }_{s\perp }}}\right]}^{1/2}={({Z}_{0})}^{{\nu }_{s}},\end{eqnarray} \tag{ 2.24 }$$

$$\begin{eqnarray}&&{\nu }_{s}={\displaystyle \frac{(d-3)(2s+d-4)(2s+d-2)(s+d-4)!}{2(d-2)!\ s!}}.\end{eqnarray} \tag{ 2.25 }$$

Then ${\nu }_{\mathrm{tot}}={\left.{\sum }_{s=0}^{\infty }{\nu }_{s}\ {\rm{e}}^{-\epsilon (s+\frac{d-3}{2})}\right|}_{\mathrm{fin}.}=0$ and thus ${({Z}_{\mathrm{CHS}})}_{\mathrm{tot}}=1$. Notice that here d is the dimension of the boundary where CHS theory is defined; the related MHS theory defined in d = d + 1 has equivalent regularization used, e.g., in (2.6). This regularization should be the one that is consistent with the symmetries of the CHS theory.

At the same time, if we start with the rearranged product (2.21) that can be written as

$$\begin{eqnarray}&&({Z}_{\mathrm{CHS}}){\prime }_{\mathrm{tot}}={({Z}_{0})}^{-2{N}_{\mathrm{tot}}},\ \ \ \ \ \ \ \ \ \ {N}_{\mathrm{tot}}=\underset{s=0}{\overset{\infty }{\sum }}{N}_{s},\ \ \ \ \ \ \ \ \ \ \ {N}_{s}={(s+1)}^{2},\end{eqnarray} \tag{ 2.26 }$$

we find that ${N}_{\mathrm{tot}}=\frac{1}{24}$ if one uses the same regularization as in (2.23)¹¹ . This illustrates an ambiguity associated with formal rearrangements of an infinite product: the result depends on a regularization.

In contrast to the 2-derivative MHS theory the CHS theory is expected to admit a Ricci-flat (or, more generally, Bach) background as its classical solution and each CHS field should have proper gauge invariance in such background. Thus the free CHS partition function in ${R}_{{mn}}=0$ should be well-defined (gauge-independent). If one makes a bold conjecture (known to be true for s = 1, 2)¹² that the $2s$ derivative covariant CHS operator can be factorized into a product of standard 2-derivative spin s Lichnerowicz operators (2.7) [7]:

$$\begin{eqnarray}&&{Z}_{\mathrm{CHS},s}={\left[{\displaystyle \frac{{(\mathrm{det}\;{\Delta }_{\rm{L}s-1})}^{s+1}}{{(\mathrm{det}\;{\Delta }_{\rm{L}s})}^{s}}}\right]}^{1/2}.\end{eqnarray} \tag{ 2.27 }$$

The same rearrangement of the infinite product as in (2.20) then gives the following Ricci-flat space generalization of (2.21)

$$\begin{eqnarray}&&{({Z}_{\mathrm{CHS}})}_{\mathrm{tot}}=\underset{s=1}{\overset{\infty }{{\displaystyle \prod }}}{Z}_{\mathrm{CHS},s}\ \ \to \ \ ({Z}_{\mathrm{CHS}}){\prime }_{\mathrm{tot}}=\underset{s=0}{\overset{\infty }{{\displaystyle \prod }}}\mathrm{det}\;{\Delta }_{\rm{L}\;s}.\end{eqnarray} \tag{ 2.28 }$$

From (2.27) one finds, in particular, the following expression for the ${\beta }_{1}=\rm{c}-\rm{a}$ conformal anomaly coefficient (coefficient of the ${R}^{*}{R}^{*}$ term in trace anomaly on Ricci-flat background) [7]:

$$\begin{eqnarray}&&{\rm{c}}_{s}-{\rm{a}}_{s}=\frac{1}{720}{\nu }_{s}(4-45{\nu }_{s}+15{\nu }_{s}^{2}),\qquad \ \ \ \ \ \ {\nu }_{s}=s(s+1).\end{eqnarray} \tag{ 2.29 }$$

Then ${\sum }_{s=1}^{\infty }({\rm{c}}_{s}-{\rm{a}}_{s})=0$ in the same regularization [8] as in (2.23).

As was shown in [5, 7, 8], the sum of the conformal anomaly ${\rm{a}}_{s}$-coefficients over all s vanishes (this is true, in particular, in the same regularization as used in (2.23)). One might then expect that the total partition function of the CHS theory on a conformally-flat space should be simply related to the one in flat space. For example, ${Z}_{\mathrm{tot}}({S}^{4})$ with the product over s computed with the regularization in (2.23) may again be equal to 1.

This would be consistent with the relation between the spin s MHS partition function in AdS₅ and the corresponding CHS partition function on S⁴ [5, 7, 11] (see also [28, 29])

$$\begin{eqnarray}&&{Z}_{\mathrm{CHS},s}({S}^{4})={\displaystyle \frac{{Z}_{\mathrm{MHS},s}^{-}({\mathrm{AdS}}_{5})}{{Z}_{\mathrm{MHS},s}^{+}({\mathrm{AdS}}_{5})}}.\end{eqnarray} \tag{ 2.30 }$$

Here ${Z}_{\mathrm{MHS},s}^{+}$ is the MHS partition function with the standard (Dirichlet) b.c. (as assumed in (2.12)) while ${Z}_{\mathrm{MHS},s}^{-}$ is its alternative (Neumann) b.c. counterpart. The relation (2.30) should be true for any s = 0, 1, 2,... and should thus also apply also to the total products over s. The computation of the spectral zeta-function in [8] (in which the infinite AdS₅ volume was assumed to factorize uniformly) implies (2.16), i.e. ${({Z}_{\mathrm{MHS}}^{+}({\mathrm{AdS}}_{5}))}_{\mathrm{tot}}=1$ and also ${({Z}_{\mathrm{MHS}}^{-}({\mathrm{AdS}}_{5}))}_{\mathrm{tot}}=1$ ¹³ . Equivalently, these properties hold in the same regularization of the sum over spins as used in (2.23). Assuming the validity of (2.30) one should then expect to find that in this summation prescription

$$\begin{eqnarray}&&{\left({Z}_{\mathrm{CHS}}\right)}_{\mathrm{tot}}({S}^{4})=\underset{s=1}{\overset{\infty }{{\displaystyle \prod }}}{Z}_{\mathrm{CHS},s}({S}^{4})=1.\end{eqnarray} \tag{ 2.31 }$$

The verification of (2.30) by directly computing the determinants appearing on the both sides of the equality turns out to be non-trivial. The expression for ${Z}_{\mathrm{CHS},s}({S}^{4})$ depends on a choice of UV regularization, while ${Z}_{\mathrm{MHS},s}^{\pm }({\mathrm{AdS}}_{5})$ depends on a choice of IR regularization, and these regularizations should be properly coordinated for (2.30) to hold. The question of how to do this was previously addressed only in the spin 0 analog of the relation (2.30) in [14, 30] where ${Z}_{\mathrm{CHS},s}$ is replaced by the partition function of the order $2r$ GJMS operator [31] and ${Z}_{\mathrm{MHS},s}$—by the AdS₅ partition function of the massive scalar with ${m}^{2}=\Delta (\Delta -4)={r}^{2}-4$. We shall discuss this case in detail appendix A.1 below. For $s\gt 0$ the relation (2.30) was verified for the leading singular (logarithmically divergent) parts only: the spin-dependent coefficient [5] of the IR divergent $\mathrm{lnR}$ term (see (2.15)) in $\mathrm{ln}\frac{{Z}_{\mathrm{MHS},s}^{-}({\mathrm{AdS}}_{5})}{{Z}_{\mathrm{MHS},s}^{+}({\mathrm{AdS}}_{5})}$ indeed matches the (conformal anomaly ${\rm{a}}_{s}$) coefficient [7] of the UV divergent $\mathrm{ln}\Lambda$ term in $\mathrm{ln}{Z}_{\mathrm{CHS},s}({S}^{4})$ ¹⁴ .

If one uses an IR regularization in AdS₅ in which the volume universally factorizes as in (2.14) then the coefficients of the $\mathrm{lnR}$ and finite parts in the MHS side of (2.30) appear to have the same spin dependence. This is certainly not so a priori for the coefficients of the UV divergent and finite parts on the CHS side of (2.30)¹⁵ . As was pointed out in the s = 0 case in [14], to be able to systematically match the finite parts of the partition functions on S^d and AdS_d+1 one may use dimensional regularization, i.e. $d\to d-\varepsilon$. Then for even d the regularized AdS_d+1 volume ${\pi }^{\frac{d}{2}}\Gamma (-\frac{d}{2})$ will have $\frac{1}{\varepsilon }$ pole term that may hit an order $\varepsilon$ term in ${\bar{\zeta }}_{{\Delta }_{s}}$ in (2.14) to produce a non-trivial finite contribution that may match the finite term present in the S^d partition function. Generalizing the dimensional regularization approach of [14] to spin $s\gt 0$ case on AdS₅ side, in appendix A.1 we shall demonstrate the matching of the most non-trivial transcendental finite parts of the CHS and MHS sides of (2.30). We shall then provide a check of the validity of the ${({Z}_{\mathrm{CHS}})}_{\mathrm{tot}}=1$ property (B.1) in this regularization consistent with (2.30) using the same summation over spins prescription as in (2.23).

Explicitly, the CHS partition function on S⁴ is given by the following generalization of (2.19) (see (2.10), (2.11), (2.12)) [7]

$$\begin{eqnarray}\begin{array}{l}{Z}_{\mathrm{CHS},s}({S}^{4}) & = & \underset{k=0}{\overset{s-1}{{\displaystyle \prod }}}{Z}_{s,k},\ \ \ {Z}_{s,k}={\left[{\displaystyle \frac{\mathrm{det}\;{\Delta }_{k\;\perp }({M}_{k,s}^{2})}{\mathrm{det}\;{\Delta }_{s\;\perp }({M}_{s,k}^{2})}}\right]}^{1/2},\\ {\left.{M}_{s,k}^{2}\right|}_{d=4} & = & s-(k-1)(k+2).\end{array}\end{eqnarray} \tag{ 2.32 }$$

For the corresponding conformal anomaly $\rm{a}$-coefficient one finds

$$\begin{eqnarray}&&{\rm{a}}_{s}={\displaystyle \frac{1}{720}}{\nu }_{s}(3{\nu }_{s}+14{\nu }_{s}^{2}),\qquad \qquad \ \ \ \ \ {\nu }_{s}=s(s+1),\end{eqnarray} \tag{ 2.33 }$$

and then ${\sum }_{s=1}^{\infty }{\rm{a}}_{s}=0$ in the same regularization as in (2.23)¹⁶ . Using the known spectra of the second-order Laplace operators on S⁴ one may also compute explicitly the finite part of the determinants in (2.32)¹⁷ and match the CHS expression with the MHS one in (2.30) (see appendix A.1).

An analog of (B.1) may be expected to be true for any conformally-flat space, e.g., for $\mathbb{R}\times {S}^{3}$ (indeed, the vanishing of the CHS Casimir energy on S³ was demonstrated in [10]). At the same time, this need not apply to orbifolds of conformally-flat space: for example, the finite temperature CHS partition function on ${S}_{\beta }^{1}\times {S}^{3}$ is non-trivial [10] (and is equal to the ratio (2.30) of the non-trivial MHS partition functions on thermal quotient of AdS₅)¹⁸ .

An important observation made in [20] is that restricted to a conformally flat homogeneous background (AdS₄ or S⁴) the Lagrangian (3.6) admits an analog of the scalar gauge invariance in (3.9),

$$\begin{eqnarray}&&\delta {\varphi }_{\mu \nu }=\left({\nabla }_{\mu }\;{\nabla }_{\nu }-{\displaystyle \frac{1}{4}}\;{g}_{\mu \nu }\;{\nabla }^{2}\right)\sigma .\end{eqnarray} \tag{ 3.12 }$$

This curved-space generalization of the scalar gauge invariance (3.9) is an important consistency requirement. As we shall explain in appendix A, this invariance generalizes also to the Einstein-space backgrounds provided the coefficient $\omega$ in (3.6) is chosen to be

$$\begin{eqnarray}&&\omega =-2,\end{eqnarray} \tag{ 3.13 }$$

i.e. is the same as in the expansion of the Einstein action (3.7) or in the Lichnerowicz operator²⁵ . The presence of the scalar gauge invariance in a gravitational background indicates that one may be able to couple this conformal rank 2 tensor field to Einstein gravity in a consistent way (see [47]).

This choice is special also for another (related) reason: in this case the differential operator in (3.6) factorizes on an Einstein background: the spin 2 and spin 1 transverse parts in the curved space analog of the decomposition (3.10) decouple in the action. This makes the ${\nabla }^{2}$ kinetic operator effectively diagonal and allows one to write the curved space generalization of the partition function (3.11) in a simple form in terms of determinants of the standard Lichnerowicz spin s = 2, 1, 0 operators.

Indeed, let us consider first the Ricci-flat background. Using the covariant version of the decomposition (3.10)

$$\begin{eqnarray}&&{\varphi }_{\mu \nu }={\varphi }_{\mu \nu }^{\perp }+{\nabla }_{(\mu }\;{V}_{\nu )}^{\perp }+\left({\nabla }_{\mu }{\nabla }_{\nu }-{\displaystyle \frac{1}{4}}\;{g}_{\mu \nu }\;{\nabla }^{2}\right)\sigma \end{eqnarray} \tag{ 3.14 }$$

in (3.6) with ${R}_{\mu \nu }=0$ and $\omega =-2$ we get

$$\begin{eqnarray}\begin{array}{l}{\mathcal{L}}_{\mathrm{CST},2} & = & {\nabla }^{\lambda }{\varphi }^{\mu \nu }\;{\nabla }_{\lambda }{\varphi }_{\mu \nu }-{\displaystyle \frac{4}{3}}\;{({\nabla }_{\mu }{\varphi }^{\mu \nu })}^{2}-2\;{C}_{\mu \nu \rho \lambda }\;{\varphi }^{\mu \rho }\;{\varphi }^{\nu \lambda }\\ & = & {\varphi }_{\mu \nu }^{\perp }\;{\Delta }_{\rm{L}\;2}\;{\varphi }^{\perp \;\mu \nu }+{\displaystyle \frac{2}{3}}{V}_{\mu }^{\perp }\;{({\Delta }_{\rm{L}\;1})}^{2}\;{V}^{\perp \;\mu },\end{array}\end{eqnarray} \tag{ 3.15 }$$

where ${\Delta }_{\rm{L}\;s}$ are the Lichnerowicz operators (2.7) on Ricci flat background. As was mentioned above, the decoupling of σ (which is a manifestation of the scalar gauge invariance) and separation of ${\varphi }_{\mu \nu }^{\perp }$ and ${V}_{\mu }^{\perp }$ is the consequence of the choice of $\omega$ in (3.13)²⁶ .

The Jacobian of the transformation in (3.14) is $J={[\mathrm{det}\;{\Delta }_{\rm{L}\;1\;\perp }\ {(\mathrm{det}\;{\Delta }_{0})}^{2}]}^{1/2}$, so we get the following generalization of the flat-space expression (3.11)

$$\begin{eqnarray}&&{Z}_{\mathrm{CST},2}={\left[{\displaystyle \frac{{(\mathrm{det}\;{\Delta }_{0})}^{2}}{\mathrm{det}\;{\Delta }_{\rm{L}\;2\;\perp }\;\mathrm{det}\;{\Delta }_{\rm{L}\;1\;\perp }}}\right]}^{1/2}={\left[{\displaystyle \frac{{(\mathrm{det}\;{\Delta }_{0})}^{3}}{\mathrm{det}\;{\Delta }_{\rm{L}\;2}}}\right]}^{1/2},\end{eqnarray} \tag{ 3.16 }$$

where ${({\Delta }_{\rm{L}\;2})}_{\mu \nu ,\alpha \beta }=-{g}_{\mu (\alpha {g}_{\beta )}\nu }{\nabla }^{2}-2{C}_{\mu \alpha \nu \beta }$, ${({\Delta }_{\rm{L}\;1})}_{\mu \nu }=-{g}_{\mu \nu }{\nabla }^{2}$ and ${\Delta }_{0}=-{\nabla }^{2}$.

Let us now compute the partition function corresponding to the action (3.6) on conformally flat Einstein background, e.g., S⁴. Using (3.14) we find that σ decouples (manifesting scalar gauge invariance) and the dependence ${\varphi }^{\perp }$ and ${V}^{\perp }$ separates (we consider unit-radius S⁴, i.e. R = 12)

$$\begin{eqnarray}&&{\mathcal{L}}_{\mathrm{CST},2}({S}^{4})={\varphi }_{\mu \nu }^{\perp }\;{\Delta }_{2\perp }(4){\varphi }^{\perp \;\mu \nu }+{\displaystyle \frac{2}{3}}\;{V}_{\mu }^{\perp }\;{\Delta }_{1\perp }(3){\Delta }_{1\perp }(-3){V}^{\perp \;\mu }.\end{eqnarray} \tag{ 3.17 }$$

As in (2.10)–(2.12) here ${\Delta }_{s\perp }({M}^{2})=-{\nabla }^{2}+{M}^{2}$, acting in transverse symmetric traceless tensors of rank s. Taking into account the Jacobian of the transformation (3.14) (see, e.g., (A.8) in [7])

$$\begin{eqnarray}&&J={\left[\mathrm{det}\;{\Delta }_{1\perp }(-3)\mathrm{det}\;{\Delta }_{0}(-4)\mathrm{det}\;{\Delta }_{0}(0)\right]}^{1/2},\end{eqnarray} \tag{ 3.18 }$$

we end with the following partition function (generalizing (3.11) in the flat-space case where all mass terms are zero)

$$\begin{eqnarray}&&{Z}_{\mathrm{CST},2}={Z}_{\mathrm{2,0}}\ {Z}_{\mathrm{1,0}},\qquad \qquad {Z}_{\mathrm{1,0}}={\left[{\displaystyle \frac{\mathrm{det}\;{\Delta }_{0}(0)}{\mathrm{det}\;{\Delta }_{1\perp }(3)}}\right]}^{1/2}={\left[{\displaystyle \frac{{(\mathrm{det}\;{\Delta }_{0}(0))}^{2}}{\mathrm{det}\;{\Delta }_{1}(3)}}\right]}^{1/2},\end{eqnarray} \tag{ 3.19 }$$

$$\begin{eqnarray}&&{Z}_{\mathrm{2,0}}={\left[{\displaystyle \frac{\mathrm{det}\;{\Delta }_{0}(-4)}{\mathrm{det}\;{\Delta }_{2\perp }(4)}}\right]}^{1/2}={\left[{\displaystyle \frac{\mathrm{det}\;{\Delta }_{0}(-4)\mathrm{det}\;{\Delta }_{1}(-1)}{\mathrm{det}\;{\Delta }_{2}(4)}}\right]}^{1/2}.\end{eqnarray} \tag{ 3.20 }$$

Here ${Z}_{n,k}$ are defined as in (2.10), (2.11), i.e. ${Z}_{\mathrm{1,0}}$ is the standard Maxwell partition function and ${Z}_{\mathrm{2,0}}$ is the partition function corresponding to the s = 2 partially massless field [20]. ${Z}_{\mathrm{2,0}}$ appears, together with the massless spin 2 (Einstein graviton) partition function ${Z}_{\mathrm{2,1}}$, in the Weyl graviton partition function ${Z}_{\mathrm{CHS},2}={Z}_{\mathrm{2,1}}{Z}_{\mathrm{2,0}}$ which is a special case of (2.32) (see [7]). Thus we have the following relation

$$\begin{eqnarray}&&{Z}_{\mathrm{CST},2}={\displaystyle \frac{{Z}_{\mathrm{CHS},2}\ {Z}_{\mathrm{1,0}}}{{Z}_{\mathrm{2,1}}}}.\end{eqnarray} \tag{ 3.21 }$$

In appendix B we shall also present the expression for the finite temperature partition function on ${S}_{\beta }^{1}\times {S}^{3}$ and discuss its interpretation in terms of counting of conformal operators in a spin 2 CFT in ${\mathbb{R}}^{4}$ corresponding to 'shortened' representation with shadow counterpart as $(3;1,1)-(5;0,0)$.

In d = 4 the conformal anomaly coefficients $\rm{a}$ and $\rm{c}$ in

$$\begin{eqnarray}&&{b}_{4}={\beta }_{1}{R}^{*}{R}^{*}+{\beta }_{2}\left({R}_{\mu \nu }^{2}-{\displaystyle \frac{1}{3}}{R}^{2}\right)=-\rm{a}\;{R}^{*}{R}^{*}+\rm{c}\;{C}^{2},\qquad {\beta }_{1}=\rm{c}-\rm{a},\ \ \ \ \ {\beta }_{2}=2\rm{c},\end{eqnarray} \tag{ 3.22 }$$

can be found by doing two separate computations: (i) on conformally-flat space like S⁴ (finding $\rm{a}$ coefficient) and (ii) on a Ricci-flat space (finding $\rm{c}-\rm{a}$ coefficient). The $\rm{a}$ coefficient corresponding to (3.6) thus does not depend on $\omega$ and is readily obtained from the expression for the partition function in (3.19). Using that [7]

$$\begin{eqnarray}\begin{array}{l}\rm{a}[{\Delta }_{s}({M}^{2})] & = & {\displaystyle \frac{1}{144}}{(s+1)}^{2})\left[{(s+1)}^{2}-3{M}^{4}+12{M}^{2}-{\displaystyle \frac{63}{5}}\right],\\ \rm{a}[{\Delta }_{\perp \;s}({M}^{2})] & = & {\displaystyle \frac{1}{720}}(2s+1)\left[30{s}^{3}+85{s}^{2}+10s-58\right.\\ & & -\left.30({s}^{2}-2){M}^{2}-15{M}^{4}\right],\end{array}\end{eqnarray} \tag{ 3.23 }$$

we get

$$\begin{eqnarray}&&{\rm{a}}_{\mathrm{CST},2}={\rm{a}}_{\mathrm{2,0}}+{\rm{a}}_{\mathrm{1,0}}={\displaystyle \frac{53}{45}}+{\displaystyle \frac{31}{180}}={\displaystyle \frac{27}{20}}.\end{eqnarray} \tag{ 3.24 }$$

At the same time, computing the conformal anomaly $\rm{c}$ coefficient for the action in (3.6) on a Ricci-flat background is, in general, a non-trivial problem. The reason is that the corresponding second order operator is a non-minimal one, i.e. has 'non-diagonal' highest derivative part (see [48]). This problem is, however, readily solved in the special $\omega =-2$ case (3.13) where one finds the explicit factorized expression for the partition function (3.16). Using that [7, 17]

$$\begin{eqnarray}&&{\beta }_{1}[{\Delta }_{\rm{L}\;s}]={\displaystyle \frac{1}{720}}{(s+1)}^{2}\left[21-20{(s+1)}^{2}+3{(s+1)}^{4}\right],\end{eqnarray} \tag{ 3.25 }$$

we find

$$\begin{eqnarray}&&{({\beta }_{1})}_{\mathrm{CST},2}={\displaystyle \frac{31}{30}},\ \ \ \ \ \ \ \ \ \ \rm{i}.\rm{e}.\ \ \ \ \ \ \ \ \ \ \ {\rm{c}}_{\mathrm{CST},2}={\displaystyle \frac{143}{60}}.\end{eqnarray} \tag{ 3.26 }$$

Let us now explain how these results fit into the general expressions for the conformal anomaly coefficients of massive ${SO}(2,4)$ representations discussed in [11]. Given a 4d conformal field in representation $(\Delta \prime ;{j}_{1},{j}_{2})$ the corresponding shadow field $(\Delta ;{j}_{1},{j}_{2})$ with $\Delta =4-\Delta \prime$ is associated (as a boundary value) to a massive higher spin in AdS₅ with mass ${m}^{2}={(\Delta -2)}^{2}-{s}^{2},\ \ s={j}_{1}+{j}_{2}$ [49] (the corresponding kinetic operator for ${j}_{1}\geqslant {j}_{2}$ is $-{\nabla }^{2}+\Delta (\Delta -4)-2{j}_{1}$). A non-unitary representation field in 4d corresponds to a 'partially massless' field in 5d. The CST field $(1;1,1)$ is thus related to $(3;1,1)$ spin 2 field in AdS₅. In the case of $(\Delta ;\frac{s}{2},\frac{s}{2})$ massive field one finds via AdS₅ computation [5] (as in [11] we use hat to distinguish a long representation from shortened one)

$$\begin{eqnarray}&&\hat{\rm{a}}(\Delta ;\frac{s}{2},\frac{s}{2})={\displaystyle \frac{1}{720}}{(s+1)}^{2}{(\Delta -2)}^{3}\left(-3{\Delta }^{2}+12\Delta +5{s}^{2}+10s-7\right).\end{eqnarray} \tag{ 3.27 }$$

In the present case, accounting for the scalar gauge invariance, we should get (see (3.5); here $\Delta \prime =1$ for the rank 2 field and $\Delta \prime =-1$ for the scalar gauge parameter field)

$$\begin{eqnarray}&&{\rm{a}}_{\mathrm{CST},2}=\hat{\rm{a}}(3;1,1)-\hat{\rm{a}}(5;0,0)={\displaystyle \frac{27}{20}}.\end{eqnarray} \tag{ 3.28 }$$

This is indeed in agreement with direct computation in (3.24).

The expression for ${\beta }_{1}=\rm{c}-\rm{a}$ coefficient for a conformal field associated to a representation $(\Delta ;\frac{s}{2},\frac{s}{2})$ field in AdS₅ suggested in [11] was

$$\begin{eqnarray}\begin{array}{l}{\hat{\beta }}_{1}(\Delta ;\frac{s}{2},\frac{s}{2}) & = & {\displaystyle \frac{1}{720}}{(s+1)}^{2}(\Delta -2)\left[-3{(\Delta -2)}^{4}\right.\\ & & -\left.5({s}^{2}+2s-3){(\Delta -2)}^{2}+8{s}^{3}+2{s}^{2}-12s-8\right].\end{array}\end{eqnarray} \tag{ 3.29 }$$

Then using also (3.28) we get

$$\begin{eqnarray}&&{\rm{c}}_{\mathrm{CST},2}=\hat{\rm{c}}(3;1,1)-\hat{\rm{c}}(5;0,0)={\displaystyle \frac{143}{60}},\end{eqnarray} \tag{ 3.30 }$$

in agreement with (3.26). The equivalent result is found from the expression for ${\beta }_{1}$ proposed in [50, 51]

$$\begin{eqnarray}&&{\hat{\beta }}_{1}\prime (\Delta ;\frac{s}{2},\frac{s}{2})={\displaystyle \frac{1}{180}}{(s+1)}^{2}(\Delta -2)\left(1+{\displaystyle \frac{1}{4}}s(s+2)[3s(s+2)-14]\right),\end{eqnarray} \tag{ 3.31 }$$

i.e. the present s = 2 case does not distinguish between the two expressions for ${\beta }_{1}$ or $\rm{c}$ discussed in [11].