Universality of squashed-sphere partition functions

# Universality of squashed-sphere partition functions

Pablo Bueno Instituut voor Theoretische Fysica, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium    Pablo A. Cano Instituto de Física Teórica UAM/CSIC, C/ Nicolás Cabrera,13-15, C.U. Cantoblanco, 28049 Madrid, Spain    Robie A. Hennigar Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1    Robert B. Mann Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1
###### Abstract

We present several results concerning the free energy of odd-dimensional conformal field theories (CFTs) on squashed spheres. First, we propose a formula which computes this quantity for holographic CFTs dual to higher-curvature gravities with second-order linearized equations of motion. As opposed to standard on-shell action methods for Taub geometries, our formula is automatically UV-finite and only involves a simple evaluation of the corresponding bulk Lagrangian on an auxiliary pure-AdS 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 free-field numerical calculations, we conjecture that the subleading term in the squashing-parameter free-energy expansion is universally controlled by the stress-tensor three-point 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 non-supersymmetric 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 renormalization-group monotones for general odd-dimensional 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 111In particular, (1) preserves a SUU subgroup of the usual SO preserved by the usual round-sphere metric in -dimensions.. In particular, they are characterized by being Hopf fibrations over the complex projective space (), namely, . The metric on these squashed-spheres is given by

 ds2Sdε=ds2CPk(d+1)+(1+ε)(dψ+ACPk(d+1))2, (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 round-sphere 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 well-known AdS-Taub-NUT/bolt family. Our first main result — see (9) — is a universal formula for the free-energy of a broad class of holographic CFTs on squashed-spheres. The formula is automatically UV-finite and, in fact, does not require knowing the corresponding NUT solutions explicitly. It holds for higher-curvature bulk theories with second-order 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 squashing-parameter expansion of which, based on holographic and free field calculations we conjecture to be controlled by the stress-tensor three-point 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 stress-tensor, 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).

Higher-order gravities and holography on squashed-spheres: 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 field-theoretical calculations into manageable ones involving gravity techniques allows for the identification of universal properties valid for completely general CFTs. In this context, higher-curvature gravities turn out to be very useful, as they define holographic toy models for which many explicit calculations — otherwise practically inaccessible using field-theoretical 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 higher-curvature interactions generically appear as stringy corrections to the effective actions of top-down 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

 L=116πG[d(d−1)L2+R+∑n=2μnL2(n−1)R(n)], (2)

where is some length scale, is Newton’s constant, the are dimensionless couplings, and the stand for the higher-curvature 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)

 h(f∞)≡16πGL2d(d−1)[L(f∞)−2f∞(d+1)L′(f∞)]=0, (3)

where is the on-shell 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 form222The special case must be excluded from the sum, as no invariant of that order contributes to the vacua equation.

 h(f∞)=1−f∞+∑nμnfn∞. (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 second-order 333Namely, we restrict to those for which the linearized equations take the form , where is the linearized Einstein tensor, is some possible matter stress-tensor, and is the effective Newton constant. This subclass — which we shall refer to as Einstein-like 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), Quasi-topological gravity Oliva and Ray (2010); Myers and Robinson (2010) and its higher-curvature extensions Dehghani et al. (2012); Cisterna et al. (2017), Einsteinian cubic gravity in general dimensions Bueno and Cano (2016), and Generalized Quasi-topological 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 non-trivial 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 Einstein-like 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 stress-tensor two-point function444Conformal invariance completely constrains the correlator up to a theory-dependent quantity, customarily denoted , as where is a fixed dimensionless tensor structure Osborn and Petkou (1994).555Let us mention that (5) was previously proven in the particular case of Lovelock theories in Camanho et al. (2011, 2014).

 CT=−h′(f∞)CET, (5)

where stands for the Einstein gravity result666Observe 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 higher-order theories.

 CET=Γ[d+2](L/√f∞)d−18πd+22(d−1)Γ[d2]G. (6)

In AdS/CFT, the semiclassical partition function is exponentially dominated by the bulk geometry with the smallest on-shell action satisfying the appropriate boundary conditions. This means that the free energy of the holographic CFT can be accessed from the regularized on-shell action of the bulk theory evaluated on the corresponding gravity solution Aharony et al. (2000). When the boundary geometry is a squashed-sphere of the form (1), the relevant bulk solutions are of the so-called Euclidean Taub-NUT/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

 n2L2=(1+ε)(d+1)f∞. (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 Einstein-like 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)

 LECG=116πG[6L2+R−μL48P], (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 Einstein-Gauss-Bonnet 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 Einstein-like 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 on-shell 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 on-shell 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 squashed-spheres 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 Einstein-like higher-order gravity theory on a squashed can be obtained by evaluating the on-shell 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

 FSdε=(−1)(d−1)2π(d+2)2Γ[d+22]L[f∞/(1+ε)]Ld+1[f∞/(1+ε)](d+1)2. (9)

Observe that this expression is drastically simpler than the standard on-shell action approach, which relies on various theory-dependent (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 theory-independent analysis of the free energy of holographic CFTs on squashed-spheres.

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 three-dimensions Klebanov et al. (2011); Casini and Huerta (2012); Pufu (2017). Indeed, this quantity has been argued to satisfy for general higher-curvature 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 round-spheres 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 two-point function charge for general odd-dimensional CFTs Bobev et al. (2017). In particular, for and , it was found (in our conventions) that

 F′′S3ε(0)=−π43CT,F′′S5ε(0)=+π615CT. (10)

Now, using (3) and (5) we find, after some manipulations,

 F′′Sdε(0)=(−1)(d−1)2πd+1(d−1)22d!CT. (11)

This expression reduces to the general results in (10), which is another highly non-trivial check of (9). Interestingly, it provides a generalization of the universal connection between and which must hold for general odd-dimensional 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 stress-tensor two-point 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 three-point function charges. A general purely field-theoretical 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 Taub-NUT 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 free-field numerics.

For general parity-even three-dimensional CFTs, the three-point function of the stress tensor is completely fixed by conformal symmetry up to two theory-dependent 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 squashed-sphere free energy of the corresponding dual theory for small values of as

 FS3ε=FS30−π4CT6ε2[1−t4630ε+O(ε2)], (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 round-sphere 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 theory-dependent quantities on general grounds. Luckily, we can use the numerical free-field 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

 T(ε)≡630ε⎡⎣1+6(FS3ε−FS30)π4CTε2⎤⎦ (13)

for the conformally-coupled 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 free-field results make us think that this property extends to arbitrary CFTs.

• Conjecture: for general three-dimensional CFTs, the subleading term in the squashing-parameter expansion of the free energy is universally controlled by the coefficient in the three-point 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 free-field and holographic higher-order 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 almost-smooth corner regions and the charge , which was eventually proven for general CFTs in Faulkner et al. (2016)777In contrast to (12), however, the subleading term in the smooth-limit expansion of the corner entanglement entropy (quartic in the deformation), was later shown not to be generically controlled by the stress tensor three-point function charges in Bueno and Witczak-Krempa (2016)..

One would expect that if our conjecture is true, an analogous expression should hold for the free energy of higher odd-dimensional squashed spheres. In that case, one would expect the term to be controlled by some combination of , and the additional stress-tensor three-point function charge, , which is nonvanishing for . In order to guess the exact relation, say, in , one could compute and holographically for some of the six-dimensional bulk theories for which Taub-NUT 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 odd-dimensional CFTs dual to higher-curvature gravities with second-order linearized equations of motion. (9) is expected to hold in the region of parameter space for which Taub-NUT geometries dominate the corresponding semiclassical partition function, something that generically occurs for small enough values of . Our formula is automatically UV-finite 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 on-shell action approaches — to the extent that it does not even require knowing the corresponding Taub-NUT 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 on-shell action of Taub-NUT 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 free-energy squashing-parameter expansion is universally controlled by the stress-tensor three-point 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 cross-checked it with the numerical results corresponding to a conformally-coupled 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 higher-order theories constructed in Bueno and Cano (2017). More ambitiously (and challenging), one could try to prove (12) in general using field-theoretical 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

 F(3)Sdε(0)= (−1)(d+1)2πd2(d2−1)Ld−116Γ[d2]fd−12∞G ⋅[(d−3)h′(f∞)−f∞h′′(f∞)]. (14)

Then, if (12) holds for general theories, it follows that for any holographic higher-order gravity of the Einstein-like class888It is immediate to check that this expression yields the right for Einsteinian cubic gravity (8), for which .,

 t4=210f∞h′′(f∞)h′(f∞). (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 higher-dimensions. In that case, we expect an expression of the form

 a(d)t2+b(d)t4=f∞h′′(f∞)h′(f∞), (16)

to hold for general Einstein-like theories in arbitrary dimensions, for some dimension-dependent constants and . Using the available results for and in Quasi-topological gravity Myers et al. (2010) and Gauss-Bonnet 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 Einstein-like 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 stress-tensor: controls the one-point function (and hence it vanishes); is proportional to the two-point function charge , see (5); we expect to control the three-point function charges and through (16). Interestingly, the “zero-point function” corresponding to the regularized round-sphere free energy also satisfies this pattern, as it can be extracted from an integral involving , namely999In even-dimensional CFTs, this expression yields — up to a factor — the coefficient of the universal logarithmic contribution to the corresponding round-sphere free energy, given by , where is proportional to one of the trace-anomaly charges ( in ), e.g., Myers and Sinha (2011).

 FSd=(−1)(d+1)2πd2(d+1)(d−1)Ld−116Γ[d2]G∫f∞h(x)x(d+3)2dx. (17)

Integrating by parts in this expression, and using (5) and (6), it is possible to find the suggestive relation

 CT=(−1)(d−1)2Γ[d+2]πd+1(d−1)2f∞[∂FSd∂f∞], (18)

which is equivalent to the one recently found in Li et al. (2018c), and which connects two seemingly unrelated quantities, such as and 101010In 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 stress-tensors. The situation for those is a bit more complicated, as conformal symmetry no longer fixes them completely up to a few theory-dependent 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 pre-doctoral grant and partially by the MINECO/FEDER, UE grant FPA2015-66793-P, and the “Centro de Excelencia Severo Ochoa” Program grant SEV-2016-0597. RBM and RAH were supported in part by the Natural Sciences and Engineering Research Council of Canada.

Appendices

## Appendix A Effective Newton constant in Einstein-like theories

The spectrum of general theories on maximally symmetric backgrounds (m.s.b.) was characterized in Bueno et al. (2017). For any given higher-order theory of this kind, the linearized field equations around a m.s.b. with curvature scale , are given by

 12ELab= +[e−2Λ(a(D−1)+c)+(2a+c)¯□]GLab+[a+2b+c][¯gab¯□−¯∇a¯∇b]RL −Λ[a(D−3)−2b(D−1)−c]¯gabRL, (19)

where and stand for the linearized Einstein tensor and the Ricci scalar, respectively. As we can see, there is a fixed theory-independent tensorial structure which is weighted by linear combinations of four theory-dependent parameters, which were denoted in Bueno et al. (2017). These parameters determine the physical quantities of the theory — namely, the masses of the ghost-like 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

 m2s=e(D−2)−4Λ(a+bD(D−1)+c(D−1))2a+Dc+4b(D−1),m2g=−e+2Λ(D−3)a2a+c,8πGeff=14e−8aΛ(D−3). (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

 L2L′(f∞) = −2eD(D−1), (21) L4L′′(f∞) = 4D(D−1)(a+bD(D−1)+c(D−1)). (22)

As we can see from (20), theories with Einstein-like 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 Taub-NUT solutions in higher-order gravities

On general grounds, the metric of Euclidean Taub-NUT/bolt solutions with base space for generic higher-curvature gravities takes the form

 ds2=V(r)(dτ+nACPk)2+dr2W(r)+(r2−n2)ds2CPk. (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 Dirac-Misner string Misner (1963) associated with the potential , its period must be fixed to . All known examples of this kind of solution for higher-curvature gravities additionally satisfy . We restrict to this case in the following. The condition of being asymptotically locally AdS implies that the function behaves as

 V(r)=f∞r2L2+O(1),whenr→∞. (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 :

 V′(rH)=4πβτ. (25)

This is the broad picture, but of course constructing actual solutions for a particular higher-order gravity is a difficult problem. As we mentioned, in the general case the solution is determined by two functions that satisfy a highly non-linear system of equations, including higher-order derivatives. However, for the class of theories that we are considering here – namely Einstein and Lovelock gravities, Einsteinian cubic gravity, Quasi-topological gravity, or in general, those of the Generalized Quasi-topological class – the problem of finding solutions is drastically simplified Bueno et al. (2018b). As we have mentioned, for these , and the higher-curvature equations of motion reduce, in each case, to a single third-order differential equation for . Interestingly, this allows for an integrable factor that effectively turns this into a second-order equation. Schematically we have

 ddrE[V(r),V′(r),V′′(r),r]=0⇒E[V(r),V′(r),V′′(r),r]=C, (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 second-order 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 Taub-NUT solutions known in the literature, computed using the standard on-shell action approach. This includes Einstein gravity and Gauss-Bonnet in general dimensions as well as the recently constructed solutions of Einsteinian cubic gravity and Quartic Generalized Quasi-topological gravities in and respectively.

In the case of -dimensional Gauss-Bonnet, the complete Euclidean action, including the generalized Gibbons-Hawking boundary term Gibbons and Hawking (1977); Myers (1987); Teitelboim and Zanelli (1987) and counterterms Balasubramanian and Kraus (1999); Brihaye and Radu (2008) reads

 IGBE= −∫dd+1x√g16πG[d(d−1)L2+R+λGBL2X4(d−2)(d−3)]−18πG∫∂ddy√h[K+2L2λGB(d−2)(d−3)[J−2GijKij]], −18πG∫∂ddy√h{−(d−1)(f∞+2)3Lf1/2∞−L(3f∞−2)Θ[d−3]2f3/2∞(d−2)R (27)

where is the Gauss-Bonnet density, is the extrinsic curvature of the boundary with its trace, with

 Jij=13(2KKikKkj+KklKklKij−2KikKklKlj−K2Kij), (28)

and is the Einstein tensor of the boundary metric . We have also explicitly included the counterterms that ensure a finite on-shell action for . The dots stand for additional contributions that are required in higher-dimensions. Computing the on-shell action of Taub-NUT solutions in this theory yields

 FEGBSdε=(−1)(d−1)2πd2(1+ε)(d+1)2d(d−1)Ld−116Γ[d+22]f(d+1)2∞G[1−f∞(d+1)(d−1)(1+ε)+(f∞−1)(d+1)(d−3)(1+ε)2], (29)

which is in precise agreement with the result obtained using the conjectured relationship (9).

Our next example is ECG plus a quartic generalized quasi-topological term in . The Euclidean action with generalized boundary and counterterms reads 111111Note 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 Einstein-like higher-order gravities a finite on-shell action for asymptotically AdS spaces is obtained by using the Gibbons-Hawking-York boundary term along with the counterterms for Einstein gravity all weighted by — c.f. Eq. (4.19) of that work.

 IE=−∫d4x√g16πG[6L2+R−μL48P−ξL616Q]−(1+3μf2∞+2ξf3∞)8πG∫∂Md3x√h[K−2√f∞L−L2√f∞R], (30)

where

 P= 12R c da bR e fc dR a be f+RcdabRefcdRabef−12RabcdRacRbd+8RbaRcbRac, Q= −44RabcdR  efabR g hc eRdgfh−5RabcdR  efabR  ghceRdfgh+5RabcdR    eabcRfghdRfgh    e+24RabRcdefR gc eaRdgfb. (31)

Evaluating the on-shell action for Taub-NUT solutions we find  Bueno et al. (2018b)

 FS3ε=−πL2(1+ε)2Gf2∞[12−f∞(1+ε)−μf3∞(1+ε)3−ξf4∞(1+ε)4], (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 quasi-topological theories in is given by Bueno et al. (2018b)

 IE= −∫d6x√g16πG[20L2+R+λGBL26X4−ξL6216S−ζL6144Z] −1−4λGBf∞+8(ξ+ζ)f3∞8πG∫d5x√h[K−4√f∞L−L6√f∞R−L318f3/2∞(RabRab−516R2)] (33)

where is the Gauss-Bonnet density and

 S =992RacRabRbdRcd+28RabRabRcdRcd−192RacRabRbcR−108RabRabR2 +1008RabRcdRRacbd+36R2RabcdRabcd−2752RacRabRdeRbdce+336RRaecfRabcdRbedf −168RRabefRabcdRcdef−1920RabRacdeRbfdhRcfeh+152RabRabRcdefRcdef +960RabRacdeRbcfhRdefh−1504RabRacbdRcefhRdefh+352RabefRabcdRcehiRdfhi −2384RaecfRabcdRbheiRdhfi+4336RabefRabcdRcheiRdhfi−143RabefRabcdRcdhiRefhi −436RabceRabcdRdfhiRefhi+2216RaecfRabcdRbhdiRehfi−56RabcdRabcdRefhiRefhi, (34) Z =−112RacRabRbdRcd−36RabRabRcdRcd+18RabRabR2−144RabRcdRRacbd −9R2RabcdRabcd+72RabRRacdeRbcde+576RacRabRdeRbdce−400RabRcdRacefRbdef +48RRaecfRabcdRbedf+160RacRabRbdefRcdef−992RabRacdeRbfdhRcfeh +18RabRabRcdefRcdef−8RabRacdeRbcfhRdefh+238RabefRabcdRcehiRdfhi −376RaecfRabcdRbheiRdhfi+1792RabefRabcdRcheiRdhfi−4RabefRabcdRcdhiRefhi −284RabceRabcdRdfhiRefhi+320RaecfRabcdRbhdiRehfi, (35)

are two densities belonging to the quartic generalized quasi-topological family of theories Ahmed et al. (2017). Computing the on-shell Euclidean action for Taub-NUT solutions of this theory yields

 FS5ε=π2L4(1+ε)3Gf3∞[23−f∞1+ε+2λGBf2∞(1+ε)2−2(ξ+ζ)f4∞(1+ε)4] (36)

which, again, precisely matches the result obtained using the conjectured relationship (9).

Finally, let us note that on-shell Euclidean actions for Taub solutions in Einstein gravity and Gauss-Bonnet gravity have been previously computed in Clarkson et al. (2003); Khodam-Mohammadi 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

 2kk!(k+1)k (37)

in dimensions. These factors are precisely the ratio of the volume of a product of 2-spheres 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); Khodam-Mohammadi and Monshizadeh (2009), gives us confidence that the results presented here are correct.

## Appendix D Free-field calculations

The numerical results for a free (conformally-coupled) 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

 Zs=∫Dϕe−12∫d3x√g[(∂ϕ)2+Rϕ28],Zf