Universality of squashedsphere partition functions
Abstract
We present several results concerning the free energy of odddimensional conformal field theories (CFTs) on squashed spheres. First, we propose a formula which computes this quantity for holographic CFTs dual to highercurvature gravities with secondorder linearized equations of motion. As opposed to standard onshell action methods for Taub geometries, our formula is automatically UVfinite and only involves a simple evaluation of the corresponding bulk Lagrangian on an auxiliary pureAdS space. The expression is closely related to the function determining the possible AdS vacua of the bulk theory in question, which we argue to act as a generating functional from which correlation functions of the boundary stress tensor can be easily characterized. Finally, based on holographic results and freefield numerical calculations, we conjecture that the subleading term in the squashingparameter freeenergy expansion is universally controlled by the stresstensor threepoint function charge for general dimensional CFTs.
Euclidean conformal field theories (CFTs) coupled to background fields can be used to learn important lessons about the dynamics of the theory in question. A prototypical example corresponds to supersymmetric CFTs, where localization techniques have allowed for notable progress — see e.g., Pestun et al. (2017). For nonsupersymmetric theories, a natural possibility consists in coupling the theory to curved background metrics. This approach has produced some exact and universal results valid for general CFTs Bobev et al. (2017); Fischetti and Wiseman (2017) and has found various applications, e.g., in holographic cosmology Anninos et al. (2013); Conti et al. (2017); Hertog et al. (2018); Hawking and Hertog (2018). Particularly interesting is the case of spherical backgrounds, whose partition functions — equivalently, free energies: — have been conjectured to be renormalizationgroup monotones for general odddimensional QFTs Klebanov et al. (2011); Casini and Huerta (2012); Pufu (2017).
In this letter, we will consider CFTs on deformed spheres and study the effect that such deformations have on . The focus will be on a particular class of squashed spheres, , which preserve a large subgroup of isometries of the round ones ^{1}^{1}1In particular, (1) preserves a SUU subgroup of the usual SO preserved by the usual roundsphere metric in dimensions.. In particular, they are characterized by being Hopf fibrations over the complex projective space (), namely, . The metric on these squashedspheres is given by
(1) 
where is a periodic coordinate which parametrizes the , is the Einstein metric on normalized so that , and is the Kähler form on . The parameter measures the degree of squashing of the sphere and, in principle, it can take values in the domain , the roundsphere corresponding to . In , which is the simplest case, , and we can write , in standard spherical coordinates.
This class of squashed spheres can be easily studied holographically Hawking et al. (1999); Dowker (1999); Chamblin et al. (1999); Emparan et al. (1999); Hartnoll and Kumar (2005); Bobev et al. (2016), as the relevant bulk geometries belong to the wellknown AdSTaubNUT/bolt family. Our first main result — see (9) — is a universal formula for the freeenergy of a broad class of holographic CFTs on squashedspheres. The formula is automatically UVfinite and, in fact, does not require knowing the corresponding NUT solutions explicitly. It holds for highercurvature bulk theories with secondorder linearized equations of motion, correctly reproducing all known results available for such theories, and passes several consistency checks coming from field theory considerations. Our second result — see (12) — is an expression for the subleading term in the small squashingparameter expansion of which, based on holographic and free field calculations we conjecture to be controlled by the stresstensor threepoint function coefficient for general CFTs. As an additional consequence of our results in the holographic context, we observe that, for the class of bulk theories just described, the function that determines the possible AdS vacua of the theory — see (4) — acts as a generating functional for the boundary stresstensor, in the sense that we can easily characterize its correlators by taking trivial derivatives of such function, drastically simplifying the standard holographic calculations — see (5), (16), (17) and (18).
Higherorder gravities and holography on squashedspheres: AdS/CFT Maldacena (1999); Witten (1998); Gubser et al. (1998) provides a powerful playground for exploring the physics of strongly coupled CFTs. In some cases, the possibility of mapping intractable fieldtheoretical calculations into manageable ones involving gravity techniques allows for the identification of universal properties valid for completely general CFTs. In this context, highercurvature gravities turn out to be very useful, as they define holographic toy models for which many explicit calculations — otherwise practically inaccessible using fieldtheoretical techniques — can be performed explicitly. The idea is that, if a certain property is valid for general theories, it should also hold for these models. This approach has been successfully used before, e.g., in the identification of monotonicity theorems in various dimensions Myers and Sinha (2010, 2011), or in the characterization of entanglement entropy universal terms Bueno et al. (2015); Bueno and Myers (2015); Mezei (2015); Chu and Miao (2016). Naturally, particular highercurvature interactions generically appear as stringy corrections to the effective actions of topdown models admitting holographic duals Gross and Sloan (1987). For the purposes just described, however, it is more useful to consider bulk models which are particularly amenable to holographic calculations — see e.g., Camanho and Edelstein (2010); Buchel et al. (2010); Myers et al. (2010); de Boer et al. (2010); Camanho et al. (2014); Bueno et al. (2018a).
The Lagrangian of such kind of models can be generally written, in bulk dimensions, as
(2) 
where is some length scale, is Newton’s constant, the are dimensionless couplings, and the stand for the highercurvature terms, constructed from linear combinations of order curvature invariants. The AdS vacua of any theory of the form (2) can be obtained by solving Bueno et al. (2017)
(3) 
where is the onshell Lagrangian on pure AdS with radius . This can be easily obtained evaluating all Riemann tensors in (2) as . Also, . It is easy to argue that the can always be chosen so that the function in (3) reduces to the form^{2}^{2}2The special case must be excluded from the sum, as no invariant of that order contributes to the vacua equation.
(4) 
Naturally, for Einstein gravity one just finds , and the action scale coincides with the AdS radius.
Let us further restrict (2) to the particular subclass of theories whose linearized equations on maximally symmetric backgrounds are secondorder ^{3}^{3}3Namely, we restrict to those for which the linearized equations take the form , where is the linearized Einstein tensor, is some possible matter stresstensor, and is the effective Newton constant. This subclass — which we shall refer to as Einsteinlike Bueno et al. (2017) — contains infinitely many theories and includes, among others: all Lovelock Lovelock (1970, 1971) and some Lovelock theories Bueno et al. (2016), Quasitopological gravity Oliva and Ray (2010); Myers and Robinson (2010) and its highercurvature extensions Dehghani et al. (2012); Cisterna et al. (2017), Einsteinian cubic gravity in general dimensions Bueno and Cano (2016), and Generalized Quasitopological gravity Hennigar et al. (2017), among others Karasu et al. (2016); Li et al. (2018a, b). The vast majority of all known theories of the form (2) admitting nontrivial black hole and Taub solutions belong to this class.
As we show here, the function contains a surprisingly great deal of additional nontrivial information for Einsteinlike theories. Firstly, given one such theory, it determines the effective gravitational constant through — a detailed proof can be found in appendix A. From the dual CFT point of view, this translates into the following relation with the charge , which fully characterizes the CFT stresstensor twopoint function^{4}^{4}4Conformal invariance completely constrains the correlator up to a theorydependent quantity, customarily denoted , as where is a fixed dimensionless tensor structure Osborn and Petkou (1994).^{5}^{5}5Let us mention that (5) was previously proven in the particular case of Lovelock theories in Camanho et al. (2011, 2014).
(5) 
where stands for the Einstein gravity result^{6}^{6}6Observe that our convention for differs from that in Bobev et al. (2017) by a factor . It agrees, however, with the convention in Myers and Sinha (2011); Bueno and Myers (2015); Buchel et al. (2010); Myers et al. (2010). Note also that it is customary to write Einstein gravity results in terms of , instead of alone. This is irrelevant for Einstein gravity itself, for which , but needs to be kept in mind for higherorder theories.
(6) 
In AdS/CFT, the semiclassical partition function is exponentially dominated by the bulk geometry with the smallest onshell action satisfying the appropriate boundary conditions. This means that the free energy of the holographic CFT can be accessed from the regularized onshell action of the bulk theory evaluated on the corresponding gravity solution Aharony et al. (2000). When the boundary geometry is a squashedsphere of the form (1), the relevant bulk solutions are of the socalled Euclidean TaubNUT/bolt class Hawking et al. (1999); Chamblin et al. (1999); Emparan et al. (1999) — see appendix B for details. Such solutions are characterized by the NUT charge which, on general grounds, holography maps to the squashing parameter of the boundary geometry through
(7) 
Naturally, constructing Taub solutions is a more challenging task than classifying the vacua of the theory and, in fact, only a few examples of such solutions have been constructed for Einsteinlike Lagrangians of the form (2). The simplest instances in correspond to Einsteinian cubic gravity Bueno et al. (2018b), whose Lagrangian is given by Bueno and Cano (2016)
(8) 
where is a new cubic invariant and is a dimensionless coupling. In , analytic Taub solutions have been constructed for Einstein Awad and Chamblin (2002) and EinsteinGaussBonnet gravity Dehghani and Mann (2005); Dehghani and Hendi (2006); Hendi and Dehghani (2008) and there have been a number of holographic applications of these solutions Astefanesei et al. (2005); Clarkson et al. (2004); Lee (2008); Shaghoulian (2017). Very recently, additional solutions have been discovered for other Einsteinlike theories (both in and ) in Bueno et al. (2018b).
In all these cases, the thermodynamic properties of the solutions can be accessed analytically. In particular, the computation of regularized onshell actions can be performed after the introduction of various boundary terms and counterterms which account for the various UV divergences Chamblin et al. (1999); Balasubramanian and Kraus (1999); Brihaye and Radu (2008); Teitelboim and Zanelli (1987); Dehghani and Vahidinia (2011); Bueno et al. (2018a). As long as the solution is the dominant saddle, the resulting onshell action computes the free energy of the dual theory on a squashed sphere . For sufficiently small , the relevant saddle is generically of the NUT type.
A universal formula for holographic squashedspheres free energy: Rather strikingly, we observe that the following simple pattern holds in all cases: the free energy of a holographic CFT dual to an Einsteinlike higherorder gravity theory on a squashed can be obtained by evaluating the onshell Lagrangian of the corresponding theory on pure AdS. The dependence on the squashing parameter appears encoded in the AdS radius of this auxiliary geometry, which is given by . Explicitly, we claim that the following formula holds
(9) 
Observe that this expression is drastically simpler than the standard onshell action approach, which relies on various theorydependent (boundary and counter) terms to yield a finite result for each theory. Instead, (9) is automatically free of UV divergences, and allows us to perform a general theoryindependent analysis of the free energy of holographic CFTs on squashedspheres.
First, note that if we set , we recover the result for the free energy of the theory on a round , which plays a crucial role in establishing monotonicity theorems, particularly in threedimensions Klebanov et al. (2011); Casini and Huerta (2012); Pufu (2017). Indeed, this quantity has been argued to satisfy for general highercurvature bulk theories, with the proportionality coefficient precisely agreeing with the one predicted by (9) — see e.g., Myers and Sinha (2011); Bueno et al. (2018a).
In addition, we know that the round sphere is a local extremum for the function Bobev et al. (2017), namely, for general theories. This property is also nicely implemented in (9). Indeed, comparing with (3), it is straightforward to show that, according to (9), which of course vanishes by definition, as is nothing but the embedding condition of AdS on the corresponding theory. It is remarkable how holography ties the CFT fact that roundspheres are local extrema of the free energy as a function of the squashing parameter, to the requirement that the AdS geometry solves the bulk field equations.
Furthermore, we know that is fully determined by the stress tensor twopoint function charge for general odddimensional CFTs Bobev et al. (2017). In particular, for and , it was found (in our conventions) that
(10) 
Now, using (3) and (5) we find, after some manipulations,
(11) 
This expression reduces to the general results in (10), which is another highly nontrivial check of (9). Interestingly, it provides a generalization of the universal connection between and which must hold for general odddimensional CFTs (holographic or not).
Universal expansion on the squashing parameter:
As we have seen, the leading term in the expansion of is quadratic in the deformation, and proportional to the stresstensor twopoint function charge for general CFTs. A question left open in Bobev et al. (2017) was the possibility that the subleading term, cubic in , could present an analogous universal behavior, in the sense of being fully characterized by the corresponding threepoint function charges. A general purely fieldtheoretical approach looked extremely challenging, even in , and the available partial results — numerical for a free scalar and a free fermion, and analytic for holographic Einstein gravity — did not suffice to provide a conclusive answer. In particular, the exact result for the free energy in holographic Einstein gravity is a polynomial of order in , namely, , which means that its Taylor expansion around is trivial, and precisely ends with the quadratic piece — which is of course controlled by in agreement with (10), as can be readily verified using (6).
Happily, the new TaubNUT solutions constructed in Bueno et al. (2018b) for Einsteinian cubic gravity provide us with an additional family of holographic models for which we can access the cubic contribution, and explore its possible universality by testing it against the freefield numerics.
For general parityeven threedimensional CFTs, the threepoint function of the stress tensor is completely fixed by conformal symmetry up to two theorydependent quantities Osborn and Petkou (1994). These can be chosen to be , plus an additional dimensionless quantity, customarily denoted Hofman and Maldacena (2008). Using the result obtained in Bueno et al. (2018a) for in holographic Einsteinian cubic gravity, we can express the squashedsphere free energy of the corresponding dual theory for small values of as
(12) 
where the holographic mapping between boundary and bulk quantities is given by: , and , which naturally reduce to the Einstein gravity results in the limit.
As we can see, the leading correction to the roundsphere result agrees with the general result (10), as it should. But now we have a nontrivial subleading piece, cubic in , and proportional to . In principle, it is far from obvious that the cubic term should not depend on additional theorydependent quantities on general grounds. Luckily, we can use the numerical freefield results in Bobev et al. (2017) to perform two highly nontrivial tests of the possible validity of (12) beyond holography. In order to do so, we study the function
(13) 
for the conformallycoupled scalar (s) and the free Dirac fermion (f) free energies near — for details on the numerical method utilized in the computation of and see appendix D. Naturally, if (12) held for these theories, we should obtain which, for the scalar and the fermion are respectively given by and Osborn and Petkou (1994); Buchel et al. (2010). The result of this analysis is shown in Fig. 1, where it is manifest that this is precisely satisfied in both cases. The extremely different nature of the theories and techniques used in deriving the holographic and freefield results make us think that this property extends to arbitrary CFTs.

Conjecture: for general threedimensional CFTs, the subleading term in the squashingparameter expansion of the free energy is universally controlled by the coefficient in the threepoint function of the stress tensor. In particular, we conjecture that (12) holds for general theories.
The level of evidence provided here in favor of (12) — involving freefield and holographic higherorder gravity calculations — is very similar to the one initially presented in Bueno et al. (2015); Bueno and Myers (2015) concerning the universal relation between the entanglement entropy of almostsmooth corner regions and the charge , which was eventually proven for general CFTs in Faulkner et al. (2016)^{7}^{7}7In contrast to (12), however, the subleading term in the smoothlimit expansion of the corner entanglement entropy (quartic in the deformation), was later shown not to be generically controlled by the stress tensor threepoint function charges in Bueno and WitczakKrempa (2016)..
One would expect that if our conjecture is true, an analogous expression should hold for the free energy of higher odddimensional squashed spheres. In that case, one would expect the term to be controlled by some combination of , and the additional stresstensor threepoint function charge, , which is nonvanishing for . In order to guess the exact relation, say, in , one could compute and holographically for some of the sixdimensional bulk theories for which TaubNUT solutions have recently been constructed Bueno et al. (2018b), and follow the same steps taken here for Einsteinian cubic gravity.
Discussion & outlook: In the first part of this letter, we have presented a new formula for the free energy of odddimensional CFTs dual to highercurvature gravities with secondorder linearized equations of motion. (9) is expected to hold in the region of parameter space for which TaubNUT geometries dominate the corresponding semiclassical partition function, something that generically occurs for small enough values of . Our formula is automatically UVfinite and only involves the evaluation of the Lagrangian of the corresponding theory on an auxiliary AdS geometry, which represents a drastic simplification with respect to the usual onshell action approaches — to the extent that it does not even require knowing the corresponding TaubNUT bulk geometry. We have argued that (9) satisfies various highly nontrivial properties expected from general CFT considerations Bobev et al. (2017), which AdS/CFT elegantly connects to bulk statements. Additionally, our formula is also satisfied in all known cases in which the corresponding holographic calculation involving the onshell action of TaubNUT geometries has been performed — see appendix C. Additional checks for other holographic theories or, preferably, a general proof of (9) would be very desirable.
In the second part, we have conjectured that the subleading term in the freeenergy squashingparameter expansion is universally controlled by the stresstensor threepoint function coefficient , as given in (12), for general dimensional CFTs (holographic or not). In deriving (12), we have made use of the free energy result for holographic Einsteinian cubic gravity, and then we have crosschecked it with the numerical results corresponding to a conformallycoupled scalar and a free Dirac fermion, finding perfect agreement. Naturally, it would be convenient to gather additional evidence. For this, one could consider the holographic duals of the set of higherorder theories constructed in Bueno and Cano (2017). More ambitiously (and challenging), one could try to prove (12) in general using fieldtheoretical techniques.
The validity of (12) would have additional consequences in the holographic context. As one can easily check, can always be written in terms of th (and lower) derivatives of . For example, one finds
(14) 
Then, if (12) holds for general theories, it follows that for any holographic higherorder gravity of the Einsteinlike class^{8}^{8}8It is immediate to check that this expression yields the right for Einsteinian cubic gravity (8), for which .,
(15) 
Hence, one would be able to obtain the coefficient by taking a couple of trivial derivatives of . This represents a dramatic simplification with respect to the standard holographic calculations involving energy fluxes — see e.g., Hofman and Maldacena (2008); Buchel et al. (2010); Myers et al. (2010). It is natural to expect that this formula generalizes to higherdimensions. In that case, we expect an expression of the form
(16) 
to hold for general Einsteinlike theories in arbitrary dimensions, for some dimensiondependent constants and . Using the available results for and in Quasitopological gravity Myers et al. (2010) and GaussBonnet Buchel et al. (2010), it is straightforward to set: and . In fact, a formula equivalent to (16) valid in the particular case of Lovelock theories — for which — was shown to be true in Camanho et al. (2011, 2014) for the same value of . This provides additional support for the validity of (16) for general Einsteinlike theories. It would be interesting to test the validity for such additional theories in various dimensions and, if correct in general, to determine the value of .
Paying some attention to the results gathered so far, we find that acts as a sort of generating functional from which one can extract correlators of the boundary stresstensor: controls the onepoint function (and hence it vanishes); is proportional to the twopoint function charge , see (5); we expect to control the threepoint function charges and through (16). Interestingly, the “zeropoint function” corresponding to the regularized roundsphere free energy also satisfies this pattern, as it can be extracted from an integral involving , namely^{9}^{9}9In evendimensional CFTs, this expression yields — up to a factor — the coefficient of the universal logarithmic contribution to the corresponding roundsphere free energy, given by , where is proportional to one of the traceanomaly charges ( in ), e.g., Myers and Sinha (2011).
(17) 
Integrating by parts in this expression, and using (5) and (6), it is possible to find the suggestive relation
(18) 
which is equivalent to the one recently found in Li et al. (2018c), and which connects two seemingly unrelated quantities, such as and ^{10}^{10}10In terms of , the relation reads , which is valid in general (even and odd) dimensions..
It is also natural to wonder about correlators involving four or more stresstensors. The situation for those is a bit more complicated, as conformal symmetry no longer fixes them completely up to a few theorydependent constants. Hence, the connection between , and such correlators is less evident to us.
Acknowledgements. We would like to thank Nikolay Bobev, Alejandro Ruipérez and Yannick Vreys for useful discussions and comments. The work of PB was supported by a postdoctoral fellowship from the National Science Foundation of Belgium (FWO). The work of PAC is funded by Fundación la Caixa through a “la Caixa  Severo Ochoa” International predoctoral grant and partially by the MINECO/FEDER, UE grant FPA201566793P, and the “Centro de Excelencia Severo Ochoa” Program grant SEV20160597. RBM and RAH were supported in part by the Natural Sciences and Engineering Research Council of Canada.
Appendices
Contents
Appendix A Effective Newton constant in Einsteinlike theories
The spectrum of general theories on maximally symmetric backgrounds (m.s.b.) was characterized in Bueno et al. (2017). For any given higherorder theory of this kind, the linearized field equations around a m.s.b. with curvature scale , are given by
(19) 
where and stand for the linearized Einstein tensor and the Ricci scalar, respectively. As we can see, there is a fixed theoryindependent tensorial structure which is weighted by linear combinations of four theorydependent parameters, which were denoted in Bueno et al. (2017). These parameters determine the physical quantities of the theory — namely, the masses of the ghostlike graviton () and the scalar mode (), and the effective Newton constant — and can be straightforwardly computed for a given theory following the procedure presented in Bueno et al. (2017). The relations between and read
(20) 
As we mentioned before, is the curvature scale of the background, which in the context of the present paper we write as . Now, a simple adaptation of Eqs. (2.22) and (2.23) in Bueno et al. (2017) yields the following general relations
(21)  
(22) 
As we can see from (20), theories with Einsteinlike spectrum — i.e., those for which and hence (19) reduces to — satisfy the constraints , . Taking these relations into account, as well as the definition of in (3), it follows that for this class of theories, as anticipated in the main text.
Appendix B TaubNUT solutions in higherorder gravities
On general grounds, the metric of Euclidean TaubNUT/bolt solutions with base space for generic highercurvature gravities takes the form
(23) 
Here, and are functions to be determined by the field equations, and is the NUT charge. The coordinate is periodic, and in order to remove the DiracMisner string Misner (1963) associated with the potential , its period must be fixed to . All known examples of this kind of solution for highercurvature gravities additionally satisfy . We restrict to this case in the following. The condition of being asymptotically locally AdS implies that the function behaves as
(24) 
Then, we see that the boundary metric at is conformally equivalent to the squashed sphere metric (1), with and with the squashing parameter given by (7). Hence, in the holographic context, these metrics have the correct boundary geometries so as to describe the dual theories on squashed spheres. On the other hand, one has to impose regularity of the solution in the bulk. In general, there is a value of such that , and we distinguish two qualitatively different cases. If the solution is of the NUT type, whereas if it is a bolt. In both cases, absence of a conical singularity imposes the following condition on the derivative of :
(25) 
This is the broad picture, but of course constructing actual solutions for a particular higherorder gravity is a difficult problem. As we mentioned, in the general case the solution is determined by two functions that satisfy a highly nonlinear system of equations, including higherorder derivatives. However, for the class of theories that we are considering here – namely Einstein and Lovelock gravities, Einsteinian cubic gravity, Quasitopological gravity, or in general, those of the Generalized Quasitopological class – the problem of finding solutions is drastically simplified Bueno et al. (2018b). As we have mentioned, for these , and the highercurvature equations of motion reduce, in each case, to a single thirdorder differential equation for . Interestingly, this allows for an integrable factor that effectively turns this into a secondorder equation. Schematically we have
(26) 
where is an integration constant that in all cases is proportional to the total mass of the solution. In the case of Einstein and Lovelock gravities, the last equation is actually algebraic Dehghani and Mann (2005) and the resolution is trivial. In general, it is a secondorder differential equation and two conditions are needed in order to specify a solution. It turns out the asymptotic condition (24) together with the regularity condition (25) suffice to determine it. Expanding the equation (26) near the cap , we find constraints that completely determine the allowed values of and . In the NUT case the radius is already fixed and the regularity condition fixes the mass as a function of the NUT charge, . In the bolt case there can be several values of the radius for the same value of , and for all of them we obtain as well . For each one of these sets of parameters , the function can be constructed from to infinity by using numerical methods, and we always find that for a given it is unique. A remarkable feature of these theories is that the thermodynamics of the solutions can be characterized fully analytically. Indeed, it is also possible to compute exactly the free energy by evaluating the corresponding regularized Euclidean actions. This is illustrated in appendix C.
Appendix C Explicit checks of formula (9)
We have verified that our conjectured formula (9) correctly reproduces the free energies of all TaubNUT solutions known in the literature, computed using the standard onshell action approach. This includes Einstein gravity and GaussBonnet in general dimensions as well as the recently constructed solutions of Einsteinian cubic gravity and Quartic Generalized Quasitopological gravities in and respectively.
In the case of dimensional GaussBonnet, the complete Euclidean action, including the generalized GibbonsHawking boundary term Gibbons and Hawking (1977); Myers (1987); Teitelboim and Zanelli (1987) and counterterms Balasubramanian and Kraus (1999); Brihaye and Radu (2008) reads
(27) 
where is the GaussBonnet density, is the extrinsic curvature of the boundary with its trace, with
(28) 
and is the Einstein tensor of the boundary metric . We have also explicitly included the counterterms that ensure a finite onshell action for . The dots stand for additional contributions that are required in higherdimensions. Computing the onshell action of TaubNUT solutions in this theory yields
(29) 
which is in precise agreement with the result obtained using the conjectured relationship (9).
Our next example is ECG plus a quartic generalized quasitopological term in . The Euclidean action with generalized boundary and counterterms reads ^{11}^{11}11Note that here we use the simple method for generating generalized boundary and counterterms introduced in Bueno et al. (2018a). There it was found that for Einsteinlike higherorder gravities a finite onshell action for asymptotically AdS spaces is obtained by using the GibbonsHawkingYork boundary term along with the counterterms for Einstein gravity all weighted by — c.f. Eq. (4.19) of that work.
(30) 
where
(31) 
Evaluating the onshell action for TaubNUT solutions we find Bueno et al. (2018b)
(32) 
which matches precisely the results from the conjectured formula (9).
As our last example, the Euclidean action with generalized boundary terms for the quartic generalized quasitopological theories in is given by Bueno et al. (2018b)
(33) 
where is the GaussBonnet density and
(34)  
(35) 
are two densities belonging to the quartic generalized quasitopological family of theories Ahmed et al. (2017). Computing the onshell Euclidean action for TaubNUT solutions of this theory yields
(36) 
which, again, precisely matches the result obtained using the conjectured relationship (9).
Finally, let us note that onshell Euclidean actions for Taub solutions in Einstein gravity and GaussBonnet gravity have been previously computed in Clarkson et al. (2003); KhodamMohammadi and Monshizadeh (2009). Our results agree with those calculations up to an overall factor of in , a factor of in , and more generally by a factor of
(37) 
in dimensions. These factors are precisely the ratio of the volume of a product of 2spheres to the volume of . This discrepancy was observed in Bobev et al. (2017) in the case . Both there and in the present work properly accounting for these factors is important for matching the general results expected from field theory considerations, e.g., the proportionality factor between and . This, combined with our careful analysis of the computations in Clarkson et al. (2003); KhodamMohammadi and Monshizadeh (2009), gives us confidence that the results presented here are correct.
Appendix D Freefield calculations
The numerical results for a free (conformallycoupled) scalar field and a free Dirac fermion used in the main text were presented in Bobev et al. (2017). We quickly summarize them here, along with some further details on the manipulations we performed to produce the curves in Fig. 1.
In each case, the corresponding partition functions are given, for a generic background metric, by