The Casimir Energy in Curved Space and its Supersymmetric Counterpart

We study $d$-dimensional Conformal Field Theories (CFTs) on the cylinder, $S^{d-1}\times \mathbb{R}$, and its deformations. In $d=2$ the Casimir energy (i.e. the vacuum energy) is universal and is related to the central charge $c$. In $d=4$ the vacuum energy depends on the regularization scheme and has no intrinsic value. We show that this property extends to infinitesimally deformed cylinders and support this conclusion with a holographic check. However, for $\mathcal{N}=1$ supersymmetric CFTs, a natural analog of the Casimir energy turns out to be scheme independent and thus intrinsic. We give two proofs of this result. We compute the Casimir energy for such theories by reducing to a problem in supersymmetric quantum mechanics. For the round cylinder the vacuum energy is proportional to $a+3c$. We also compute the dependence of the Casimir energy on the squashing parameter of the cylinder. Finally, we revisit the problem of supersymmetric regularization of the path integral on Hopf surfaces.


Introduction and summary
The quantum anomalies appearing in the trace of the energy-momentum tensor encode universal information about Conformal Field Theories (CFTs) in even space-time dimensions. In d = 2 the conformal anomaly c characterizes conformal field theories [1], and constrains the renormalization group (RG) flows between them [2]. In four-dimensional CFTs, there are two trace anomaly coefficients, c and a. They appear in many applications. In particular, Cardy conjectured [3] that there exists a decreasing function along RG flows, reducing to a at the fixed points [4,5,6].
Here we will be interested in the following question. Given a conformal field theory in R d , d = 2n, we can use a Weyl transformation and study the theory on S d−1 × R. This is often referred to as "radial quantization." Denoting the noncompact coordinate by τ , we can ask about the energy of the ground state E 0 , defined as The expectation value of the energy-momentum tensor is taken in the ground state of the CFT on the cylinder. We refer to E 0 as the Casimir energy.
It is crucial for the analysis below to understand whether E 0 is well defined, namely, scheme independent. In d = 2 the only dimensionless counterterm is where R denotes the Ricci scalar. This vanishes on the cylinder and thus does not shift the vacuum energy. 1 In d = 4 there are several dimensionless counterterms. A basis is given by Euler density E (4) , the square of the Weyl tensor W 2 , the Pontryagin density Tr(R ∧ R), and R 2 . Of those four dimensionless counterterms only d 4 x √ gR 2 does not vanish on S 3 × R. We could thus add this counterterm to the action with some scheme-dependent coefficient b (1. 3) The curvature of the three-sphere of radius r 3 is R = 6 r 2 3 in our conventions. As a consequence, integrating the counterterm above we get δS = − 3b 8r 3 dτ . Interpreting the coefficient of dτ as the ground state energy we see that E 0 is ambiguous. An equivalent way to understand this ambiguity is to note that (1.3) leads to a schemedependent term in the trace of the energy-momentum tensor (1.4) Since T µ µ is modified, this affects the other components of T µν and in particular the vacuum energy. By contrast, in d = 2 no dimension 2 term can be added to the right-hand side of T µ µ = − c 24π R. 1 More precisely, since there is the cosmological constant counterterm Λ 2 UV d 2 x √ g, the vacuum energy would generally have power divergent pieces E 0 ∼ Λ 2 UV r 1 (with r 1 the radius of the S 1 and Λ UV the UV cut-off) which are non-universal. So when we discuss the scheme-independence of the vacuum energy, we always have in mind the piece that remains finite when the cut-off is removed. In d = 2n dimensions, only counterterms with d-derivatives of the metric are thus relevant to us.
We see that, without additional assumptions, the ground state energy on S 3 × R is not an intrinsic property of the CFT. It depends on the regularization scheme, i.e. the precise way in which the CFT is defined (different definitions lead to different values of b). On the other hand, the Casimir energy on S 1 × R is an intrinsic observable that does not depend on the ultraviolet completion.
For completeness, let us quote the results for E 0 in d = 2 and in d = 4. In d = 4 below we use the general scheme with some b. We present a self-contained derivation of these results in Appendix A. The Casimir energy on S 1 × R is where r 1 is the radius of the circle. The Casimir energy on S 3 × R is where r 3 is the radius of the three-sphere. In fact, as we show in Appendix A, the result in d = 4 remains ambiguous even if one allows for an infinitesimal deformation of the three-sphere. By this we mean that the linear response to an infinitesimal deformation is still proportional to a − b 2 . We also provide a holographic test of this claim, by checking it against the linearized holographic Casimir energy of the supergravity solution of [7].
The three main points of the present paper are • There is a natural generalization of the notion of Casimir energy for N = 1 supersymmetric field theories on S 3 × R.
• It turns out that this Casimir energy is physical, i.e. scheme independent. We will give two proofs of this assertion and then we will evaluate the Casimir energy exactly. One proof is based on [8] and the other is based on ideas related to Chern-Simons terms in quantum mechanics. The Casimir energy comes out proportional to the linear combination a + 3c of the conformal anomalies.
• When one computes supersymmetric partition functions on manifolds with S 3 × S 1 topology, the answer is given in terms of a product of the Casimir factor and the usual supersymmetric index. This leads to consistent results in various limits of the partition function.
Let us now briefly explain why it might be useful to understand the Casimir factor for four-dimensional theories. Recall that in d = 2, in order for the torus partition function to be modular invariant, it is necessary to include the Casimir prefactor e cβ/12l (where β is the length of the thermal circle). The Casimir prefactor in four dimensions might be similarly important to manifest various relations between different path integrals such as those analogous to [9] and e.g. [10]. The Casimir prefactor might be also relevant for a better understanding of the role of the modular-like transformations in four dimensions discussed in [11,12] (and see references therein). So let us begin by discussing how we put supersymmetric N = 1 theories on S 3 × R. For theories with an R-symmetry (such as any SCFT) one can follow the procedure of [13] whereby one couples the theory to background new minimal supergravity. New minimal supergravity [14] contains the bosonic fields g µν , A µ , B µν , where A µ is the R-gauge field and B µν is a two-form. 2 It turns out that for every complex manifold with a Hermitian metric one can find appropriate A µ and B µν such that one preserves at least one rigid supercharge [16,17]. This can be viewed as a generalization of the notion of twisting [18]. Furthermore, the partition function is independent of the Hermitian metric; it only depends on the complex structure of the underlying complex manifold [19]. A particularly interesting class of complex manifolds are primary Hopf surfaces, which are topologically equivalent to S 3 × S 1 . In this case, the relevant branch of the moduli space of complex structures is two-complex dimensional, parameterized by complex numbers p, q.
For the special case of the round metric on S 3 ×S 1 , the complex structure parameters are given in terms of the S 1 -length β and the S 3 -radius r 3 as p = q = e − β r 3 . This geometry preserves in fact four supercharges and in addition to the round metric we need to activate the background fields A τ = V τ = i r 3 , where V µ = 1 4 ǫ µνρσ ∇ ν B ρσ . The two-form B µν couples to a redundant operator in conformal field theories and is not so important. However, the flat gauge field A along the time direction implies that the Hamiltonian, H susy , is shifted with respect to what we would get from radial quantization according to Above ∆ is the time translation operator that is obtained by mapping the dilatation operator in flat space to the cylinder (equivalently, ∆ is the canonical Hamiltonian for a theory in the conformally-flat background A τ = 3i 2r 3 ). The Hamiltonian (1.7) commutes with the four supercharges on the round S 3 × S 1 and so the Hilbert space is organized in representations of the appropriate superalgebra.
The Casimir energy, E susy , is thus naturally defined from the large β limit of the partition function [20] Z susy  9) in agreement with [21]. The partition function is therefore given by Z susy with I S 3 ×S 1 β being the usual supersymmetric index [22,23,24,25]. The results of [26,27,28,29] are consistent with (1.10).
Since the Casimir energy is not well defined without supersymmetry, one has to develop a supersymmetric method to regularize the a priori divergent Casimir energy. As we will explain in the following subsection, the main idea is that the expectation value of (1.7) can be related to the expectation value of the R-charge in the vacuum.
The latter can be thought of as a Chern-Simons term in quantum mechanics and this leads to a determination of the ordering ambiguities (see also [30]).
Let us mention that recovering (1.9) from a holographic computation remains an open problem. Recall that for four-dimensional SCFTs admitting a weakly coupled gravity dual, at leading order in the large N limit, one has a = c and thus we predict . Therefore this should be reproduced by the on-shell action of an appropriate supergravity solution. It was noted in [7] that an obvious candidate for such a solution is AdS 5 , with the addition of an appropriate flat background gauge field; however, its renormalised on-shell action is the same as that of pure AdS 5 , which agrees with (1.6) (with b = 0) [31], and does not agree with (1.9). It is clearly worthwhile to revisit this problem.
In the next subsection we briefly summarize the derivation of (1.9). Section 2 contains the detailed derivation of (1.9) and Section 3 contains the generalization to squashed cylinders. Appendix A contains a study (which is mostly a review) of the non-supersymmetric Casimir energy. In Appendix B we include some details about the identification of E susy with the VEV of the Hamiltonian. Finally, in Appendix C we reconsider the full partition function on S 3 × S 1 . We revisit the regularization of one-loop determinants and determine the relation of the partition function and the supersymmetric index for all p, q, a study initiated in [20] (see also [32]), thereby generalizing (1.10). The correct regularization of the one-loop determinants leads to results consistent with the Cardy-like formula [26] (see also [27]), namely Z susy as advertised in [26,30].

Outline of the derivation
Let us now summarize the main points in the derivation of the supersymmetric Casimir energy for SCFTs.
In section 2 we consider an SCFT on the conformally flat background S 3 × R, with round metric on S 3 . Preserving supersymmetry requires to turn on background fields of the new minimal supergravity multiplet A t = V t = 1 r 3 . Focusing on two supercharges of opposite R-charge (out of the four preserved by the background), the supersymmetry algebra on S 3 × R takes the form Here R is the generator of U(1) R symmetry while J 3 is a Cartan generator of the isometries of the three-sphere. The Casimir energy is given by the VEV of the Hamiltonian appearing in the algebra, E susy = H susy . Here, the expectation value is evaluated in the vacuum of the theory coupled to the various background fields. Since the supercharge is time-independent, we can then Wick rotate and compactify the time direction on a circle of length β, and E susy gives the dominant contribution to the supersymmetric partition function for β → ∞ as in (1.8).
Because of the separation of scales between the radius of the three-sphere and the radius of the circle, it is natural to study the reduction of the theory on the threesphere. The result of this reduction is a supersymmetric quantum mechanics with infinitely many degrees of freedom. We are thus led to consider a 1d system with Hamiltonian H susy , a global symmetry with charge Σ and a supersymmetry algebra of the form Unbroken supersymmetry implies that r 3 H susy = Σ . Therefore, when we reduce to 1d, the supersymmetric term in the effective action that computes H susy contains (1.14) where A Σ t is the background gauge field associated to the symmetry generated by Σ. The "cosmological constant" term dt |g tt | is associated to the VEV of H susy , while the Chern-Simons (CS) term, dt A Σ t , is associated to the VEV of Σ, i.e. the charge of the vacuum.
In quantum mechanics (1.14) is an allowed counterterm, and this reflect the fact that H susy and Σ can be shifted by an arbitrary amount, because the algebra (1.13) is not affected as long as r 3 H susy = Σ . By contrast, if the quantum mechanical model arises from a local higher-dimensional model, the allowed counterterms must descend from local higher-dimensional counterterms. It easy to see that the quantummechanical CS term (1.14) cannot descend from a higher-dimensional counterterm and thus it is scheme independent. As a consequence, since the vacuum energy is fixed by supersymmetry to be the same as the 1d CS term, the vacuum energy is physical. Furthermore, since the CS term cannot depend on continuous coupling constants, the vacuum energy is also independent of continuous coupling constants. If we further assume the existence of a weakly coupled point, we can reliably compute the Casimir energy using a free field theory.
We can then consider a free chiral multiplet. Supersymmetry implies that upon reduction on the S 3 the resulting Lagrangian and supersymmetry transformations must be organized in terms of 1d multiplets and Lagrangians. In particular, there are two types of multiplets, that we will call "short." These are the chiral (φ, ψ) and the Fermi (λ, f ) multiplets. A key point is that the combination of the two multiplets can form a reducible but indecomposable representation of supersymmetry. Thus they can join to form a "long" multiplet, with the coupling between the short components controlled by a parameter p. When p = 0, the short multiplets are decoupled.
The scalar fields of the 1d chiral multiplets arise from "Fourier modes" of the scalar fields in the 4d chiral multiplet. For example, on the round sphere we can use the spherical harmonics and write φ = φ ℓ,m,n Y ℓ,m,n . Here ℓ, m, n are the usual SU(2) l × SU(2) r quantum numbers. Similarly, one can expand the two-component fermion of the 4d chiral multiplet in spinorial harmonics, and the 1d fermions ψ and λ may be identified with linear combinations of its components. The deformation parameter p that governs the shortening of the multiplets is related to the 4d quantum number as p 2 = (ℓ − 2m)(ℓ + 2 + 2m), hence the shortening condition p = 0 is satisfied if and only if m = ℓ/2 or m = −ℓ/2 − 1.
We will see that in the correct renormalization scheme (whose existence we established above), the Hamiltonian of a long multiplet exactly vanishes on the vacuum, while for the short multiplets we find where r is the R-charge of the 4d chiral multiplet. Thus, given that the long multiplets do not contribute, the expectation value of the total Hamiltonian is The two infinite sums can be regularized using different methods, for example using the Hurwitz zeta function, or a cut-off method, (see also [20] and [30]) yielding in any case the result 4 H susy = E susy = 4 27r 3 (a + 3c) . (1.17) Using a similar strategy, in section 3 we compute the supersymmetric Casimir energy for an N = 1 theory (with an R-symmetry) on a class of Hopf surfaces with U(1) 3 symmetry, considered in [20]. Let us describe this briefly. Again we start from the free chiral multiplet Lagrangian in four dimensions, and reduce this to an infinite set of fields in 1d, comprising long and short multiplets. Since the quantized Hamiltonian of long multiplets vanishes on the vacuum, we can focus on the short multiplets. The shortening conditions are expressed as first-order differential equations for the eigenstates of the Hamiltonians for chiral and Fermi multiplets, which are labeled by two non-negative integers n 1 , n 2 . This can be seen using the method of [33] or, equivalently, utilising the variables of [32]. Each chiral and Fermi multiplet contributes as chiral : H chiral = 1 2r 3 |b|n 1 + |b −1 |n 2 + r 2 (|b| + |b −1 |) , Fermi : where b is a parameter deforming the complex structure of the Hopf surface. Again the two sums in (1.16) must be regulated separately. This can be done using the Barnes double zeta function, which is a two-parameter generalization of the Hurwitz zeta function. In the end, we obtain which is the expression (5.10) in [20] (with the redefinition b 1 = βb 2πr 3 and b 2 = βb −1 2πr 3 ).

Supersymmetric Casimir energy
In this section we perform a manifestly supersymmetric analysis of the Casimir energy problem on S 3 ×R. Our strategy is to reduce to a one-dimensional quantum mechanical problem. Using the same idea, in section 3 we will discuss the more general case leading to (1.19).

Definition
Consider an N = 1 theory with an R-symmetry. The energy-momentum tensor can be embedded in the so-called R-multiplet (see, for instance, [34]), hence the theory can be coupled canonically to new minimal supergravity. The bosonic part of the supergravity multiplet consists of the metric g µν and two auxiliary fields, the R-gauge field A µ and a conserved one-form V µ , ∇ µ V µ = 0, dual to the field strength of a two-form gauge field B µν . Gravity can then be made non-dynamical by taking a rigid limit, so that one is left with an ordinary field theory on a four-manifold M 4 . In this limit generically all the supergravity fields above are turned on, and play the role of background fields. For particular choices of background fields and manifolds M 4 one can show that the theory on M 4 preserves some rigid supersymmetry. The condition is that there exists a nontrivial solution to the generalized Killing spinor equation [13] ( An N = 1 theory on a curved manifold M 4 presents the same short-distance behavior as in flat space. So in general there will be divergences in the computation of e.g. the partition function on M 4 , which need to be renormalized. Assuming that the theory is regularized in a way which respects both supersymmetry and the R-symmetry, we infer that two different regularization schemes differ by some local counterterm, namely some local action constructed using the background new minimal supergravity fields, d 4 x √ g L(g µν , A µ , V µ ). This has to respect supersymmetry and gauge invariance. Local terms of dimension four affect the finite part of the partition function, as they remain finite when the UV cut-off is removed. In [8] it was shown that when there exist two Killing spinors of opposite chirality, then all local counterterms of dimension four evaluate to zero. This means that such partition functions are in fact scheme-independent. 5 The results of [19,18] tell us that Z susy is independent of coupling constants and only depends on complex structure parameters of the underlying complex four-manifold M 3 × S 1 . The findings of this work are consistent with that.
The partition function can be interpreted as usual as a trace over the Hilbert space Z susy where H susy generates the time evolution along S 1 . Then the Casimir energy is extracted from the large circle limit as The scheme-independence of the partition function implies that the vacuum energy is universal (below we will also give a new proof of this fact). This should be contrasted with the non-universality of the ordinary Casimir energy E 0 discussed in the introduction.

Consequences of the supersymmetry algebra
In this section we make a few observations based on the supersymmetry algebra. Let us consider the simple case M 4 = S 3 ×R, where S 3 is a round three-sphere of radius r 3 and R is the time direction. By fixing the other background fields to A t = V t = 1 r 3 , with all other components vanishing, we can preserve four supercharges 6 for any N = 1 theory with an R-symmetry [13]. In fact one could take any constant A t for the flat gauge field. However, only in the case that A t = 1 r 3 one gets time-independent supercharges. So we will make this choice throughout.
The superalgebra preserved by the background is [22] where H susy is the generator of translations along the circle, R is the R-charge, the J i l , i = 1, 2, 3, are the generators of the SU(2) l ⊂ SU(2) l × SU(2) r isometry of the sphere and σ i are the Pauli matrices. The supercharges Q α , α = 1, 2, form a doublet of SU(2) l , while the SU(2) r subgroup does not appear in the superalgebra.
A first remark is that we assume the vacuum does not break supersymmetry. Suppose the vacuum were not supersymmetric, in which case either Q 1 , or Q 2 (or both) would not annihilate the vacuum. Then, Q|VAC is a new state with the same value of H susy , but contributing with an opposite sign to the index or partition function. Therefore if supersymmetry were broken, the index on S 3 × R would not receive a contribution from the unit operator. In the case of SCFTs, the fact that supersymmetry is unbroken on S 3 × R is a simple consequence of radial quantization. Indeed, in this case, the spectrum of H susy is identical to the spectrum of ∆ − 1 2r 3 R, which has a gap above the unit operator.
Another simple observation is that J 3 l annihilates the vacuum, J 3 l |VAC = 0. Indeed, J 3 l appears with different signs on the right hand side of {Q 1 , Q † 1 } and {Q 2 , Q † 2 }. Hence, it must vanish or one of the Q's does not annihilate the vacuum. (Also, if J 3 l were nonzero on the vacuum, the vacuum would not be unique.) It is useful to focus on the algebra of one specific supercharge, say Q 1 . We will also rescale R and J 3 l to reabsorb the radius r 3 of S 3 . Then the Q 1 -algebra reads From this we conclude that H susy = R . We will approach the problem of determining H susy by reducing the theory on the three-sphere. In this way we get a quantum mechanics (QM) theory with infinitely many degrees of freedom. The theory has four supercharges, Q 1 , Q 2 and their Hermitian conjugates. The R-symmetry group is SU(2) l × U(1) and the supercharges furnish the (2, 1) representation. SU(2) r is a global symmetry in the quantum mechanics theory.
From the fact that H susy = R we deduce that the generating functional of connected correlators W contains at zero momentum the terms where we restored the three-sphere radius r 3 .
The first term in parenthesis is a "cosmological constant" term in one dimension, while the second is a Chern-Simons term. The former determines the ground state energy and the latter fixes the R-charge of the vacuum, in such a way that H susy = R is preserved. Since the relation H susy = R is a consequence of supersymmetry, the one-dimensional local term (2.7) must be the bosonic part of a supersymmetric action in a one-dimensional version of new minimal supergravity. We leave the systematic study of such terms to the future. Equation (2.7) implies that H susy is physical (i.e. scheme independent) without relying on the classification of four-derivatives counterterms of [8]. The idea is that if we just studied a quantum mechanics theory of finitely many degrees of freedom, then there would be no meaningful way to compute H susy itself. Indeed, we could always add a counterterm of the form (2.7) with an arbitrary coefficient, shifting the values of H susy and R at will, while preserving H susy = R . 7 So without additional assumptions, there is no possible way to fix H susy itself at the level of quantum mechanics. However, it is crucial that in the present study the quantum mechanics theory derives from a four-dimensional local quantum field theory. Then, the counterterm (2.7) would be admissible only if it came from a local term in four dimensions. It is easy to convince oneself that there is no way to derive (2.7) -with its specific normalization -by dimensional reduction of a four-dimensional counterterm on S 3 × R with radius r 3 . Indeed to get the normalization right for the Chern-Simons term, it needs to come from a term in four dimensions of mass dimension four. But then it is straightforward to see that one cannot write anything, regardless of supersymmetry, that would look like dtA t after integrating over the sphere.
This argument can be rephrased as follows: If we have a quantum mechanics theory with finitely many degrees of freedom, the charge of the vacuum (under some global symmetry) is ambiguous because we can add the Chern-Simons counterterm at will.
But if it comes from a local four-dimensional QFT, then the Chern-Simons term has to come from some local term in four dimensions. In the present context, it is easy to convince oneself that such a term (of dimension 4) does not exist. Therefore, the R-charge of the vacuum becomes physical and so does the ground state energy by the relation H susy = R .
Another observation that follows directly from (2.7) is that the Casimir energy cannot depend on continuous coupling constants (and hence on the RG scale). This follows from the fact that H susy = R and that to compute R we need to evaluate the coefficient of dtA t . If the coefficient of A t had depended on continuous coupling constants, we could have promoted them to time-dependent fields and lose the gauge invariance under small R-symmetry gauge transformations (and this cannot be accounted for by any anomaly). This is similar to the arguments in [26] (and references therein). This conclusion is consistent with the arguments of [19,18].
An important consequence of this observation is that it is sufficient to calculate the Casimir energy starting from a free field theory in 4d. We henceforth assume that such a free point exists in the space of continuous coupling constants. It would be very interesting to generalize our considerations to non-Lagrangian theories. (It is not currently clear to us how to do so.) To summarize, by considering the supersymmetry algebra and reducing to a quantum mechanical problem, we established that what we need to compute is the coefficient of the generating functional (2.7) in quantum mechanics. We perform this computation below, after having introduced some notions of supersymmetric quantum mechanics.
Before coming to these issues, let us comment about the case in which two supercharges are preserved instead of four. This is in fact the generic supersymmetric case, pertinent to various deformations of S 3 (these will be discussed in section 3) as well as to other topologies. In this case, some of the claims above are valid while some others are not necessarily true. The most crucial difference is that there is no SU(2) l algebra in general, rather, there is just some U(1) isometry of M 3 , generated by J 3 . Thus, a simple argument that J 3 vanishes in the vacuum does not exist. So now in the vacuum we have H susy = R + 2J 3 . A quantum mechanical term like (2.7) would still exist, so the claims made below eq. (2.7) still apply. In particular, the vacuum energy is still independent of the renormalization scheme and of the coupling constants. We will use this to compute E susy for SCFTs on M 3 × S 1 , where M 3 has S 3 topology.

Supersymmetric quantum mechanics
Let us model the situation above, governed by the two supercharges algebra (2.4), with where H generates time translations, while Σ is some Hermitian conserved charge. At the formal level, we can just redefine H by Σ. However, in order to be able to connect more easily to the reduction over S 3 , we will keep the algebra in the form (2.4).
We can define two types of multiplets: a chiral multiplet (φ, ψ), and a Fermi multiplet (λ, f ), where φ, f are complex and commuting while ψ, λ are complex and anticommuting. These two multiplets have the following supersymmetry transformations chiral : where on all the fields we define D t = ∂ t − iσ, with σ the charge of the field under Σ. The complex parameter ζ is independent of time and uncharged under Σ. In the variations of the Fermi multiplet there appears a parameter p. When p = 0, the chiral and Fermi multiplets are independent of each other. We will refer to each of the decoupled multiplets as "short." When instead p = 0 the two multiplets form one reducible but indecomposable representation of supersymmetry. Thus, for p = 0 we call the combined chiral and Fermi multiplets a "long" multiplet.
On each component of a multiplet with charge σ, the transformations (2.9) give The supersymmetric Lagrangian of a long multiplet takes the form where µ is an additional free parameter, giving a mass to ψ. For p = 0, the first and the second lines are the Lagrangians of a free chiral and free Fermi multiplet, respectively, and are separately supersymmetric. 8 We now pass to Hamiltonian formalism and quantize the theory. The canonical momenta are The canonical (anti-)commutation relations are together with their Hermitian conjugates.
The Hamiltonian reads where again when p = 0 the first line gives the Hamiltonian of a chiral multiplet, while the second line is the Hamiltonian of a Fermi multiplet. The field f has been set to zero by its equation of motion. Note that we have introduced a constant α, parameterizing the usual ordering ambiguity.
In terms of canonical variables, the charge Σ reads where α parameterizes the ordering ambiguity in this operator. The supercharge is 16) and is free of ordering ambiguities. Evaluating {Q, Q † } we find that (2.8) is upheld provided we take Hence supersymmetry fixes the ordering ambiguity in H − Σ. Of course, after having solved for α we still have the freedom to shift H and Σ by an equal amount, corresponding to the remaining parameter α. Without additional assumptions, this freedom would have remained in the framework of ordinary quantum mechanics in one dimension.
In order to explain how to fix the ordering ambiguity that is left, it is useful to recall that we are computing the coefficient of a CS term in the low-energy 1d effective action. This term takes the form where A Σ t is the background gauge field associated to the charge Σ. A single fermion of mass M and charge q shifts the coefficient of the Chern-Simons term by q 2 sgn(M) [36]. We can think about that as if we are starting from some theory in the UV with Chern-Simons coefficient k uv and then we integrate out the massive fermion leading to a Chern-Simons coefficient in the infrared k ir (this interpretation was elaborated upon in [37] A simple way to derive (2.21) is as follows. First, from dimensional analysis and the fact that M and k are odd under charge conjugation we infer where x is a coefficient, independent of M . To fix x we can consider a free fermion with mass M and charge q with a constant background gauge field A Σ t . This has Hamiltonian H = (M + qA Σ t )(ψψ † +α), From the point of view of the quantum mechanics, the arbitrariness in the charge of the vacuum corresponds to the arbitrariness in the UV coefficient k uv . However, our theory arises from a higher-dimensional model. As already observed, it is easy to convince oneself that a term like (2.18) cannot be generated by dimensional reduction of a four-dimensional local term. So we must take i.e. no Chern-Simons contact term in the UV generating functional. This key requirement fixes the ordering ambiguity in H. Together with (2.21), this implies that multiplets containing pairs of fermions with masses of opposite sign do not contribute to the Casimir energy. We will see below that as long as the Hamiltonian is bounded from below, a long multiplet necessarily contains fermions with masses of opposite sign.
As a result, the choice of the ordering coefficient must be such that H and Σ vanish in the ground state of a long multiplet. This leads to the conclusion that the correct choice of the ordering constant is We will use this choice in the following and one can verify that in all cases the results are consistent with (2.22). Incidentally, it turns out that (2.23) also corresponds to Weyl ordering for the Hamiltonian. 10

Spectrum of the Hamiltonian
We now study the spectrum of the Hamiltonian and determine the vacuum state.

Long multiplet
Let us start from the bosonic sector of (2.14):

24)
whereα is an arbitrary ordering constant. The partition function is given by . (2.20) The idea now is that we can keep the ultraviolet fixed and consider two different RG flows, one with positive M and one with negative M . By subtracting the resulting Chern-Simons terms in the infrared (which we will read out from the charge of the vacuum), we will find 2x. If M > 0 then taking M → ∞ we can read off the CS term (i.e. charge) in the IR to be qα dtA Σ t . On the other hand, if M < 0 we read out the CS term in the IR by taking the limit M → −∞ and we find q(α+1) dtA Σ t . Subtracting these yields 2x = −q. 10 This explains why our final result is identical to that of [30] for the VEV of H. But, unlike [30], our result for the VEV of Σ in the vacuum manifestly respects the BPS condition H = Σ.
where we have included half of the ordering constant appearing there (the other half will enter in the fermionic sector). This ensures Weyl ordering. We can introduce creation operators a † , b † and annihilation operators a, b via The canonical commutation relations between φ and Π φ (and their Hermitian conjugates) imply that these satisfy [a, Then the bosonic Hamiltonian can be written as where in the second line we have emphasized that H bosonic is Weyl ordered. The state annihilated by a and b has energy µ 2 + p 2 . Acting on this with (a † ) m (b † ) n (with m, n positive integers) we obtain a state with energy We see that in order for the Hamiltonian to have a spectrum that is bounded from below we need to assume µ 2 + p 2 > |µ + σ|. 11 Hence the state of minimum energy in the bosonic sector is the one with m = n = 0.
Next we address the fermionic sector. The Hamiltonian reads where we have kept the ordering constant that ensures Weyl ordering. We can make a unitary U(2) rotation to diagonalize the above matrix. This preserves the anticommutation relations. The eigenvalues are Denoting the eigenvectors u + , u − , u † + , u † − , the Hamiltonian is thus with {u ± , u † ± } = 1. The charge operator Σ takes the form Starting with the state |0 which is annihilated by both u † ± , we can act with u − , u + or u − u + . The spectrum therefore consists of four states with the following energy and charge: state Since we assumed µ 2 + p 2 > |µ + σ|, the state of lowest energy is u − |0 .
We now combine the information obtained studying the bosonic and fermionic sectors of the Hamiltonian and identify a state with minimum energy that respects supersymmetry. Adding H bosonic and H fermionic , the complete Hamiltonian is One can also check that the full charge operator reads From the discussion above, the state with minimum energy is clearly where m = 0, n = 0 indicates that no bosonic oscillators are excited, and by x − we mean that we excite one fermionic oscillator with eigenvalue x − . Its total energy is and thus vanishes due to an exact cancellation between the bosonic and the fermionic contributions. Since we have just one fermionic oscillator the charge is Σ = 0 , hence the relation (H − Σ)|VAC = 0 is satisfied and supersymmetry is unbroken in the vacuum, as expected.
We conclude that the long multiplets yield a vanishing contribution to the vacuum energy and charge: Note that this is a consequence of our choice of ordering constant, and as argued at the end of the previous subsection this is the correct choice for a quantum mechanics arising from a higher-dimensional theory.
If we had a theory of long multiplets only, the vacuum energy would just be zero. However, if short multiplets are also present, this is not the case, as we now show.

Fermi multiplet
Consider the Fermi multiplet. Then the supercharge identically vanishes. The Hamiltonian and the charge generator take the same form, The only two states have energy − 1 2 σ and + 1 2 σ. The contribution of a Fermi multiplet to the vacuum energy and charge is thus (2.39)

Chiral multiplet
The bosonic sector of the chiral multiplet can be treated as we did for the long multiplet, setting p = 0. The full Hamiltonian and charge operator can thus be written as Since p = 0, the condition for the Hamiltonian to be bound from below becomes |µ| > |µ + σ| . Which state has minimum energy depends on the values of µ and σ. Note that (2.42) requires µ and σ to have opposite signs. If µ > 0, σ < 0, then (2.42) implies −2µ < σ < 0, and the state number 1 has minimum energy H = − 1 2 σ; since H = Σ, this state is supersymmetric, while the state 2 is non-supersymmetric. Conversely, if µ < 0 and σ > 0, then from (2.42) we deduce 0 < σ < −2µ, hence the state number 1 now has higher energy and the state 2 is the supersymmetric vacuum, with H = 1 2 σ. Thus, a chiral multiplet contributes to the vacuum energy and charge as (2.43) In conclusion, the analysis in supersymmetric quantum mechanics establishes that a long multiplet yields a vanishing contribution to the vacuum energy and charge, that a Fermi multiplet contributes as in (2.39) while a chiral multiplet contributes as in (2.43).

Dimensional reduction of a 4d chiral multiplet
Consider a free four-dimensional chiral multiplet (φ, ψ α , F ) on S 3 × R. The Lagrangian and supersymmetry transformations can be found in [13]. The only parameter appearing in the Lagrangian is the charge r under the background R-symmetry gauge field. Here we will restrict to 0 < r ≤ 2. 12 This range is compatible with the inequalities mentioned in the previous subsection, ensuring that the spectrum of the Hamiltonian is bounded from below. Expanding in appropriate spherical harmonics, the chiral multiplet reduces to a one-dimensional theory with infinitely many fields. These organize in one-dimensional multiplets with different values of the parameters µ, p, σ introduced above. Some have p = 0 and are thus long multiplets, while some others have p = 0 and are thus short multiplets, either chiral or Fermi.
More explicitly, we can expand the scalars in spherical harmonics Y ℓ,m,n transforming in representations ( ℓ 2 , ℓ 2 ) of SU(2) l × SU(2) r . The quantum number ℓ is a non-negative integer. For a fixed ℓ, the quantum numbers m, n of the scalar harmonic Y ℓ,m,n range 44) and similarly for the auxiliary field F . The fermionic field ψ α can be expanded in spinorial harmonics. A review of spinor spherical harmonics on S 3 can be found in [38,30]. A single 4d fermion reduces to two infinite series of 1d fermions furnishing the representation l ( ℓ−1 2 , ℓ 2 ) ⊕ ( ℓ+1 2 , ℓ 2 ) of SU(2) l × SU(2) r . 13 Integrating over S 3 and using the orthonormality of the spherical harmonics, the action of a four-dimensional chiral multiplet gives rise to a one-dimensional action for an infinite number of fields. These arrange in multiplets of supersymmetric quantum mechanics labeled by ℓ, m, n, and one can check that the Lagrangian of each of these multiplets takes the form (2.11). Here we do not need to present all details of the reduction. All we need to know is how the R-charge r and the quantum numbers ℓ, m, n map into the parameters σ, p, µ entering in (2.11) and characterizing each multiplet in supersymmetric quantum mechanics. Actually, the discussion in subsection 2.4 shows that for the purpose of determining the vacuum energy we just need to know when a multiplet is shortened (namely when p = 0), if it is a chiral or a Fermi multiplet, and what is the value of its charge σ.
By comparing the four-dimensional algebra (2.4) with (2.8), we deduce that we must identify (restoring the S 3 radius r 3 ) Σ = 1 r 3 (R + 2J 3 l ), and therefore Moreover, reducing the four-dimensional Lagrangian to one dimension, one finds 14 hence the shortening condition p = 0 is satisfied if and only if m = ℓ/2 or m = −ℓ/2−1.
In the former case a chiral multiplet is obtained, with charge σ = 1 r 3 (ℓ+r). In the latter case a Fermi multiplet is obtained, with charge σ = − 1 r 3 (ℓ + 2 − r). Recalling (2.39), (2.43) we conclude that the respective contribution to the vacuum energy is: (2.47) The expectation value of the Hamiltonian is obtained by adding up the contributions 13 The symmetry between left and right is broken by the choice of spin bundle. 14 More generally, one could easily restore the dependence on the parameter κ. This affects only µ but not p 2 and σ. In the notation of [30] one finds that the parameter µ is related with the parameters in the four-dimensional Lagrangian as of all chiral and Fermi multiplets: where the (ℓ + 1) factor comes from the degeneracy associated with SU(2) r .
To regularize the sum, we dress the terms in the sum with some decreasing weights. Taking the small t limit and dropping the diverging term in t −2 , 15 we obtain a regularized result for the vacuum energy that, after recalling the trace anomaly coefficients [39] a = 3 32 This is the result advertised in eq. (1.9) of the introduction.
One could consider a supersymmetric regularization with a different function f (tH ℓ,m,n ) of the H ℓ,m,n operators. It can be shown, using an Euler-MacLaurin expansion (see appendix C for a related application) that for all smooth functions f such that f (0) = 1 (and such that the series converges), one obtains the same result for the finite piece in the small t expansion. This is in agreement with the fact that the supersymmetric Casimir energy is unambiguous. The regularization using the Hurwitz zeta function [30] also reproduces the same result.
It is possible to contrast our results with several previous works in which localization techniques on S 3 × S 1 were utilized. Comparing with [20] (see also [32] and [40] where similar localization techniques are used in other topologies), one finds agreement regarding the vacuum energy. However, as we will briefly discuss in appendix C, the regularization scheme of [20] in fact does not preserve supersymmetry, as it violates certain SUSY Ward identities in the small circle limit. Our result for the vacuum energy also agrees with that of [30] (this method is a Hamiltonian version of [21]), but as mentioned in footnote 10 the result of [30] does not preserve some SUSY Ward identities as well.

Supersymmetric Casimir energy on a deformed three-sphere
In this section we study the chiral multiplet on a supersymmetric S 3 × S 1 background with more general metric and complex structure. We will use results known from the computation of the partition function on these spaces, based on localization, to implement the Hamiltonian approach to the evaluation of the Casimir energy.

Shortening conditions on chiral multiplets
We start by reviewing some results of [32], where the fermionic degrees of freedom in the chiral multiplet were conveniently redefined. This made it particularly easy to show that in backgrounds preserving two supercharges of opposite R-charge, the modes contributing to the partition function solve first-order differential equations that can be interpreted as shortening conditions.
We work in Euclidean signature and follow the conventions of [20]. 16 We assume the existence of at least one positive-chirality spinor ζ α and one negative-chirality spinor ζα satisfying the Killing spinor equations These independent equations are the Euclidean version of the supersymmetry condition given in (2.1). Note that ζ has R-charge +1 while ζ has R-charge −1. In Euclidean signature the chiral multiplet is made of (φ, ψ α , F ) with R-charge (r, r − 1, r − 2) and of the independent fields ( φ, ψα, F ) with R-charge (−r, −r + 1, −r + 2). The supersymmetry transformations are where on a field of R-charge q the covariant derivative D µ is defined as with ∇ µ the Levi-Civita connection. A supersymmetric Lagrangian is where R is the Ricci scalar on the four-manifold.
Following [32] (see section 5 therein), we decompose the fermion fields ψ, ψ in anticommuting scalars as 17 Note that B has R-charge r − 2 while C has R-charge r. Similarly, B has R-charge −r + 2 while C has R-charge −r. We also introduce the complex vectors which define a complex frame. These satisfy K µ K µ = Y µ Y µ = 1 2 , with all other contractions vanishing. The vectors K, K have vanishing R-charge, while Y has Rcharge +2 and Y has R-charge −2. They satisfy hence K is a (complex) Killing vector. Finally, we define the differential operatorŝ In the new variables the supersymmetry transformations (3.2) take the form (here we distinguish between δ ≡ δ ζ and δ ≡ δ ζ ): with similar transformations for the tilded fields. The Lagrangian (3.4) can then be rewritten as Here, κ is a function describing a redundancy in the choice of the background fields [17]; it will play no important role for us as it drops from the final answer.
In [32] it was showed that in a background with two supercharges ζ, ζ, the partition function of a chiral multiplet reduces to where λ B , λ φ are eigenvalues determined by the first-order differential conditionŝ 14) The equations on the left can be read as shortening conditions for the chiral multiplet. Indeed,L Y B = 0 allows to set C (and φ) to zero consistently with its equation of motion. Similarly,L Y φ = 0 permits to set B to zero consistently with its supersymmetry transformation in (3.10). In order to set B = 0 consistently with its equation of motion one also needsL Y C = 0; this condition is also needed to set F = 0 respecting its supersymmetry variation. Similar considerations apply to the tilded fields. (In [32], the latter were related to the untilded fields via Hermitian conjugation).
These shortenings are better understood by reducing to lower dimensions, where (B, F ) and (φ, C) define irreducible representations of supersymmetry. In [32], this was done in an S 2 × T 2 background by expanding in monopole harmonics on S 2 and noting that the aforementioned equations correspond to shortenings of the (0, 2) supersymmetry multiplets on T 2 . Therefore only modes in shortened multiplets contribute to the one-loop determinant of the chiral multiplet partition function. The long multiplets lead to paired bosonic and fermionic eigenmodes and hence yield a trivial contribution.
Similarly, in the following we consider S 3 × S 1 with a general metric and reduce on S 3 to supersymmetric quantum mechanics. This allows to make contact with the approach to the computation of the Casimir energy developed in section 2.

Reduction on deformed three-sphere
We consider the class of supersymmetric backgrounds studied in [20]. These are complex manifolds with the topology of S 3 × S 1 known as primary Hopf surfaces. The complex structure moduli space is described by two parameters p, q. In the notation of [20], p = e −2π|b 1 | , q = e −2π|b 2 | and b 1 , b 2 are chosen real. The circle S 1 is parameterized by τ ∼ τ + 2π. The three-sphere is described as a torus fibration over an interval: the torus angular coordinates are ϕ 1 ∼ ϕ 1 + 2π and ϕ 2 ∼ ϕ 2 + 2π, while the interval coordinate is ρ ∈ [0, 1]. The metric is where I, J = 1, 2. Supersymmetry imposes the constraint which ensures Hermiticity of the metric. f and m IJ are arbitrary functions of ρ, except that m IJ must be positive definite and suitable boundary conditions need to be satisfied at the extrema of the interval. As ρ → 0, we require that where f 2 > 0 and m 11 (0) > 0 are constants. Similar boundary conditions are taken at ρ → 1. The background field A is given by where m = det m IJ , a prime denotes derivative with respect to ρ, and We will not need the expression of the remaining background field V .
The complex vectors K and Y defined in (3.7) take the form where ε JK is the antisymmetric symbol, with ε 12 = +1. In order to evaluate the differential operators (3.9) it will be useful to record that In principle, the action of the 4d chiral multiplet can be reduced to one dimension by expanding all the fields in an orthonormal basis of eigenfunctions of the relevant differential operators, and performing the integral over the three-manifold M 3 . This yields a 1d action for infinitely many 1d multiplets. In practice, carrying out this reduction on a general background such as the one above is not feasible, as the eigenfunctions are not known. However, for the purpose of computing the Casimir energy there is no need to perform the complete reduction. Indeed, comparing the supersymmetry transformations (3.10) with the one-dimensional ones in (2.9), it is clear that the modes that do not satisfy the shortening conditions in (3.13), (3.14) are going to produce long multiplets in 1d. As explained in section 2.4, these do not contribute to the Casimir energy. Therefore we can focus our attention on the subsector that does satisfy either one of the shortening conditions in (3.13), (3.14). Let us study the two cases in turn.

Reduction to 1d Fermi multiplets
We start analyzing the conditions (3.13), which lead to 1d Fermi multiplets. We Fourier expand the dependence of B on the torus coordinates ϕ 1 , ϕ 2 as where n 1 , n 2 are integer. Recalling (3.21), (3.22) and that B has R-charge r − 2, one can see that the conditionL Y B = 0 yields for each choice of n 1 , n 2 :  for some constant k. This is normalizable when n 2 sgn(b 2 ) ≤ 0. A similar analysis of the behavior near ρ = 1 leads to require n 1 sgn(b 1 ) ≤ 0. It is thus convenient to redefine n 1 → −n 1 sgn(b 1 ), n 2 → −n 2 sgn(b 2 ). After this redefinition, the integers n 1 , n 2 must take non-negative values, and the expansion (3.24) becomes Recalling (3.20), (3.23), we can also computê and λ B n 1 ,n 2 = −|b 1 |n 1 − |b 2 |n 2 + r − 2 2 (|b 1 | + |b 2 |) . (3.30) We are now ready to perform the reduction to one dimension. SettingL Y B = 0 and φ = C = 0, the chiral multiplet Lagrangian (3.11) becomes (3.31) and the supersymmetry variations simplify to We can expand F as (note that the dependence on ρ, ϕ 1 , ϕ 2 is chosen the same as that of B, so F also satisfiesL Y F = 0). Similar expansions hold for B and F . The action associated with the Lagrangian (3.31) reduces to an action for an infinite set of Fermi multiplets in one dimension: This is an infinite set of decoupled 1d multiplets, labeled by n 1 , n 2 . It is also straightforward to see that the 4d supersymmetry variations decompose into a set of 1d variations for the Fermi multiplets: δb n 1 ,n 2 = f n 1 ,n 2 , δb n 1 ,n 2 = 0 , δf n 1 ,n 2 = 0 , δf n 1 ,n 2 = D τ b n 1 ,n 2 .
From the form of the covariant derivative in (3.29), we see that λ B n 1 ,n 2 should be identified with the charge σ in section 2. More precisely, σ Fermi = 2π β λ B n 1 ,n 2 . 18 Recalling eq. (2.39), the contribution of the infinitely many Fermi multiplets to the vacuum expectation values of Σ and H is thus
The sum (3.47) can be written as where ζ 2 is the Barnes double zeta function, defined as with b 1 , b 2 , x real and positive. At s = −1, it evaluates to (see e.g. [42]) This gives for the vacuum energy This is the same expression appearing in eq. (5.10) of [20]. Redefining b 1 = βb 2πr 3 and b 2 = βb −1 2πr 3 , where β coincides with the length of S 1 , gives the result (1.19) advertised in the introduction.

Back to the round case
Let us conclude by coming back to the S 3 ×S 1 background with SU(2) l ×SU(2) r ×U(1) symmetry discussed in subsection 2.5. The round sphere is just S 3 b with b = 1. In this case |b 1 | = |b 2 | = β 2πr 3 and the result above for the Casimir energy simplifies to the result in subsection 2.5. In more detail, the sum (3.47) becomes where in the second line we have defined ℓ = n 1 + n 2 . So the regularization used above and the one used in subsection 2.5 are compatible.
It is also straightforward to see that the shortening conditions match. When expanding the chiral multiplet (φ, C, B, F ) in scalar spherical harmonics, the differential operators discussed above becomê where q R is the charge of the field the operator is acting on, and we introduced the SU(2) l generators L ± l , L 3 . Then the conditionL Y φ = 0 is nothing but L + l Y ℓ,m,n = 0, which selects the harmonics with highest quantum number m, namely Y ℓ, ℓ 2 ,n . The conditionL Y B = 0 translates into L − l Y ℓ,m,n = 0 and thus selects Y ℓ,− ℓ 2 ,n . Under an infinitesimal Weyl transformations g µν → (1 + 2σ)g µν , the transformation of W is given by the Weyl anomaly. In d = 2 and d = 4 respectively we get We have set the coefficient b that appears in (1.3) to zero. The above infinitesimal Weyl transformations can be integrated to the so-called Dilaton action, 19 see e.g. [43,44] W where S D has the following explicit expressions in d = 2 and d = 4 Taking a derivative of (A.4) with respect to the metric, we find We see that the change in the VEV of the energy-momentum tensor under Weyl rescaling is fixed by the Dilaton action.

A.1 The cylinder and its infinitesimal deformation
In the case of a conformally flat geometry with metric e 2σ δ µν (such as S d−1 × R) we can use the fact that the VEV in flat space is unambiguously fixed to vanish, i.e.
T µ ν R d = 0. One then finds for the energy-momentum tensor on this space These results were first derived in [45] using a different method.
Consider in particular the round cylinder S d−1 × R with the conformally flat metric Here dΩ 2 d−1 denotes the metric on a (d − 1)-sphere with unit radius, and r d−1 is the radius of the sphere. Evaluating the energy-momentum tensor in this case, one finds the following non-zero components of T µ ν in d = 2: and in d = 4: Note that T µ µ S 3 ×R = 0, because the Weyl anomalies evaluate to zero on the cylinder. We can easily reintroduce the b dependence of the result in d = 4 by taking a functional derivative of the associated counterterm − b This results in a shift of the coefficient a → a − b 2 . Integrating T τ τ over S 1 and S 3 respectively gives the Casimir energies quoted in the text (1.5) and (1.6).
Above we have reviewed how to compute the Casimir energy on a conformally flat background e 2σ δ µν . We will now slightly extend our considerations by taking a geometry that is a generic small perturbation of a conformally flat one, i.e. g µν = e 2σ δ µν +h µν . Let us compute the change in the ground state energy to first order in h µν . The idea is that we can infer the ground state energy from the change in the partition function W as a result of the perturbation.
This can be approached as follows: from the definition (A.1) we have so we just need to know T µν e 2σ δ , i.e. the VEV in the conformally flat original space.
For instance, consider the round cylinder S 3 × R with a time-independent perturbation h ij of the metric on the three-sphere. The result for the VEV of the energymomentum tensor on the cylinder was given in equation (A.12). Therefore we find that Since the metric perturbation is assumed time-independent, we can interpret this integral as .15) and the coefficient of dτ gives the correction to the energy of the ground state. The final result for the Casimir energy is We see that to order O(h) the Casimir energy is still scheme dependent, being proportional to the same combination 2a − b as the leading order result. However, the ratio of the leading term and the subleading term is unambiguous (in fact, the ratio is fixed by imagining a small perturbation that only changes the radius of the three-sphere).
One can speculate that going to second order in the perturbation O(h 2 ) the canomaly will also appear and that some terms will be scheme independent. Notice, however, that at this order a new counterterm, proportional to d 4 x √ g W 2 , could contribute.

A.2 The Casimir energy and holography
We now summarize the status of the Casimir energy in the context of the gauge/gravity duality. We will mainly focus on four-dimensional CFTs admitting a dual description in terms of solutions of type IIB supergravity of the type AdS 5 × M 5 , and their deformations. Similar considerations can be made for six-dimensional CFTs with AdS 7 gravity duals.
The comparison of the Casimir energy of a four-dimensional CFT with the expectation value of the holographic energy-momentum tensor was first made in [46], although in this reference the authors focussed on N = 4 SYM and its AdS 5 ×S 5 dual, and some of the comments made there apply only to this example. Let us review the discussion of [46]. Using the holographic energy-momentum tensor, computed in global AdS 5 , one obtains for the Casimir energy, defined exactly as in (1.1), the expression 20 It is important to note that this result is obtained using a minimal holographic renor- is, in some sense, accidental, and may be misleading. Indeed for generic n 0 , n 1/2 , n 1 , the expression (A.18) for E free 0 is not proportional to the a anomaly: it also includes a contribution from b. In fact, the scheme leading to (A.18) generically also gives a R contribution to the trace anomaly. Once the particular matter content of N = 4 SYM is specified, the coefficient b happens 20 This was referred to as "mass of global AdS 5 " in [46]. We have used the standard formula for the Newton constant G −1 where ℓ is the AdS radius and in the specific case vol ′ (S 5 ) = π 3 .
to vanish and the scheme leading to (A.18) effectively coincides with the scheme b = 0 which was used in the holographic computation.
More generally, one can ask whether the computation of the Casimir energy in a deformed cylinder can be reproduced by a five-dimensional supergravity solution deforming AdS 5 . At least at leading order in the deformation, there exists a holographic counterpart to the formula (A.13). Consider a one-parameter family of deformations of the boundary metric, with parameter µ. Then by simply applying the chain rule to the renormalised on-shell action one obtains 21 where g µν is the finite metric on the boundary, and T µν is the holographic energymomentum tensor. Setting g = g (0) + µh, with µ infinitesimal, and expanding (A. 20) at first order in µ leads exactly to the holographic version of the formula (A.13). Thus, the holographic Casimir energy on an infinitesimally squashed cylinder is guaranteed to agree with the field theory result. In the next subsection, we verify this in an explicit example.
For the supersymmetric version of the Casimir energy the situation is different. One of the main points emphasized in this paper is that on the field theory side, computing in the free limit gives a reliable result for E susy , valid also at strongly coupled points. However, how to reproduce this in the gravity side remains an open problem. In [7] it was suggested that a supersymmetric treatment of holographic renormalization might reveal the existence of new boundary terms, that would lead to a matching of the onshell action of existing solutions with E susy . Another possibility is that there exist other (Euclidean) supersymmetric solutions, whose on-shell action would match with E susy , using the standard holographic renormalization technology. In light of the results of the present paper, it will be very interesting to revisit this problem.
Let us also mention that there exists a similar open problem in the context of SCFTs on the six-dimensional cylinder S 5 ×R. A supersymmetric version of the Casimir energy was discussed in [21], where it was also noted that it does not match the standard holographic Casimir energy in AdS 7 . It may be useful to prove that this quantity is physical in 6d SCFTs by performing an analysis like the one in the present paper, and to investigate the seven-dimensional holographic dual.

A.3 Holographic check of E 0 on a squashed cylinder
Below we will compare the result of the first-order correction to the ordinary Casimir energy due to a non-conformally flat geometry (A. 16), with a corresponding holographic result, that can be easily extracted from the gravity solution presented in [7]. Although the solution in [7] is supersymmetric, we can obtain from this an expression for the Casimir energy, that is valid independently of supersymmetry. In order to make the discussion as self-contained as possible, we will begin recalling relevant aspects of the solution in [7], referring the reader to this reference for more details.
This is a one-parameter family of supersymmetric solutions deforming AdS 5 , constructed as an asymptotically locally anti de Sitter (AlAdS) solution of five-dimensional minimal gauged supergravity.
An analytic continuation to Euclidean signature yields the gravity dual to a class of four-dimensional N = 1 supersymmetric gauge theories on a curved manifold with topology S 3 ×S 1 . In particular, the boundary metric is that of a squashed three-sphere preserving SU(2) × U(1) isometry, and there is a non-trivial background gauge field coupling to the R-symmetry current. So, this is an instance of a Hopf surface, as discussed in the main part of the text [7]. In a coordinate system, the five-dimensional metric and the graviphoton field take the following asymptotic form near to the boundary, corresponding to ρ → ∞: ds 2 5d = dρ 2 + e 2ρ ds 2 bdry + . . . , where the boundary values read and respectively. Here ℓ is the AdS radius, that can be identified with the radius of the S 3 , and the left-invariant one-forms σ a 's are defined as usual (see [7]). The full fivedimensional solution is determined in terms of the single parameter v, measuring the squashing of the boundary metric. For v 2 = 1 the solution reduces to AdS 5 and the boundary metric is conformally flat. The parameter 2a 0 is an overall scale, that can be set to any value by simply shifting the radial coordinate ρ. 22 Notice that the background gauge field in (A.23) comprises a constant part along dt, that is necessary to have well defined supercharges in the compactified geometry, as discussed around (2.3). However, in order to compare with the result above, which is valid in the absence of such term 23 (and independently of supersymmetry), we should remove this by shifting the graviphoton field as A 5d → A 5d − 1 2 √ 3ℓ dt. It is simple to check that the solution with this new gauge field is still non-singular, and its relevant properties may be extracted from [7], by following through this simple change of gauge.
This can be compared with the result of the holographic computation. The Euclidean on-shell action of five-dimensional supergravity was computed exactly as a function of the parameter v in [7], and reads where we rewrote the five-dimensional Newton constant in terms of the a anomaly of the dual CFT. This was obtained in a gauge such that A bdry τ = 0, so we should shift the gauge to A bdry τ = 0 and correspondingly the on-shell action using eq. (4.14) therein. In fact, the shift due to the change of gauge is proportional to ǫ 2 , so at linear order, this shift is immaterial. In any case, after doing this shift, we obtain exactly the expression appearing in (4.35) therein for E 0 , times dτ . The on-shell action (A.29) expanded at first order in ǫ, gives thus we get perfect agreement with the dual field theory result (A.28). Note that (A.29) has been evaluated using a holographic renormalization scheme without the d 4 x √ gR 2 counterterm, precisely as in the conformally flat background.

B E susy and the Hamiltonian
In this Appendix we show that the supersymmetric vacuum energy E susy , defined within the path integral approach as coincides with the VEV of the charge associated to the time translation symmetry. In other words, E susy is the VEV of the Hamiltonian, E susy = H susy . This is wellknown, in particular in the context of quantum field theories at non-zero temperature. Nevertheless in the presence of additional non-dynamical background fields it may be useful to spell out some details.
E susy can be expressed in terms of the energy-momentum tensor and the currents appearing in the R-multiplet in the following way. Consider a background M 3 ×S 1 β as in the main text, then perform a trivial rescaling of the S 1 coordinate τ so that τ ∼ τ + 1 and the background fields g τ τ , A τ and V τ acquire a dependence on β. Applying the chain rule to the variation of − log Z with respect to β, E susy can be written as where the energy-momentum tensor T µν , the R-current J µ R and the Ferrara-Zumino current J µ FZ are defined as These are components of the R-multiplet, which at the linear level have canonical couplings to the metric, to A and to V , respectively. However, we remark that we do not set A and V to zero after having taken the variation, so these are currents in the presence of sources.
The expression in (B.2) shows that E susy receives a contribution from the current in the R-multiplet in addition to the one from the temporal component of the energymomentum tensor. This is an alternate way to see that E susy is different from the ordinary Casimir energy E 0 defined in eq. (1.1).
We now construct the charge associated with the time translation symmetry and compare it with (B.2). Recall that in the presence of background fields other than the metric, the energy-momentum tensor T µν is in general not conserved, ∇ µ T µν = 0. This can be easily seen by revisiting the standard conservation proof (see e.g. Appendix E of [47]) allowing for fields that do not satisfy their Euler-Lagrange equation. In particular, this applies to a supersymmetric field theory defined via the rigid limit of new minimal supergravity. One can see that the non-conservation equation of the energy-momentum tensor reads where we used that the R-current is conserved, ∇ µ J µ R = 0, while generically ∇ µ J µ FZ = 0. Although T µν is not conserved, when the background admits a Killing symmetry generated by a vector ξ µ one can introduce a modified energy-momentum current that is conserved, ∇ µ Y µ ξ = 0. This is easily seen using L ξ g = L ξ A = L ξ V = 0. One can show that Y µ ξ is just the canonical Noether current associated to the symmetry generated by ξ. Thus a conserved charge can be defined in the usual way.
We recall that two such Killing vectors exist in Euclidean backgrounds preserving two supercharges of opposite R-charge [16,17]. In Lorentzian signature, at least one null Killing vector exists [35]. Our M 3 × S 1 β background admits a Killing vector generating (Euclidean) time translations, ξ = ∂ ∂τ . The associated conserved charge, to be identified with the Hamiltonian, is where u µ is the unit time-like vector, that in the second line we expressed as u = u µ dx µ = √ g τ τ dτ (in the second line we also multiplied the three-dimensional integral by 1 = 1 β S 1 dτ ; since the charge is constant, this can be rewritten as a four-dimensional integral). The VEV of the final expression in (B.6) for β → ∞ is immediately recognized as the E susy given in (B.2), that is what we wanted to show.
Finally, note that the Hamiltonian in (B.6) coincides with the supersymmetric H susy used in section 2 only for the special choice A τ = i/r 3 , which guarantees timeindependent supersymmetry.

C Regularization of one-loop determinant
In this Appendix we reconsider the regularization of the one-loop determinant for a free chiral multiplet on the Hopf surface M 3 × S 1 (with M 3 ≃ S 3 ) computed in [20].
It was found in [26] that the partition function on the Hopf surface has a universal behaviour in the small β limit, where β is the length of S 1 . In the case of the round S 3 × S 1 , this takes the form 24 log Z susy where r 3 is the radius of S 3 . The result found in [20] are incompatible with this expansion, as in [20] the order O(β −1 ) vanishes instead. We propose here an alternative method to regularize one-loop determinant which on one hand agrees with the small beta expansion of [26] and on the other hand reproduces the large β behaviour leading to the Casimir energy (1.19). Moreover, we show that this regularization method is equivalent to a cut-off regularization which manifestly preserves supersymmetry. The main difference between the procedure discussed below and that in [20] is that here all the Kaluza-Klein (KK) modes on S 1 are dealt with in a manifestly symmetric way, while in [20] these were somewhat artificially split and combined with the Fourier modes on the three-sphere, in the triple gamma functions. However, the precise reason for the discrepancy of the two methods remains unclear. While we were writing up this paper, [29] appeared, presenting a similar alternative regularization method of one-loop determinants for the round S 3 × S 1 .
One-loop determinant regularizations are based on the use of multiple gamma functions [48] and generalized zeta functions. The main mathematical tools that we will use are presented in a convenient way in appendices A and B of [49]. We will also rely on [50].
The one-loop determinant suffers from UV divergences. It is given for a free chiral multiplet 25 by the formal expression − r 2 (a 1 + a 2 ) − n 0 + (n 1 + 1)a 1 + (n 2 + 1)a 2 r 2 (a 1 + a 2 ) + n 0 + n 1 a 1 + n 2 a 2 , where a 1 = ib 1 , a 2 = ib 2 in the notations of [20] and here we consider b 1 > 0, b 2 > 0. r is the R-charge of the chiral multiplet. The analysis of this paper allows to re-interpret each factor appearing in the numerator of (C.2) as the contribution from a single Fermi multiplet to the one-loop determinant and each factor in the denominator as the contribution of a single chiral multiplet. A crucial difference with the Hamiltonian quantization analysis is that now we have a tower of KK modes on S 1 with KK level parameterized by n 0 ∈ Z.

C.2 Cut-off regularization
We present now a different method of regularization which uses a cut-off on the momentum modes. The treatment of the KK modes in this method is more transparent and the result agrees with (C.11), thus corroborating it.
A reliable way of regularizing the one-loop determinant is to introduce a cut-off on the momentum of the modes or more generally a smooth truncation of the modes above a momentum scale Λ (this means that the contributions of modes above Λ are counted with decreasing weights). Note that this preserves supersymmetry since each momentum in the product carries the contribution of a specific multiplet in supersymmetric quantum mechanics. The (logarithm of the) regularized determinant can be expanded in powers of the cut-off at large Λ. The diverging contributions are expected to be removed by local supersymmetric counterterms, while the finite piece yields the regularized result, possibly subject to ambiguities. In the case at hand we compute the partition function on a curved manifold admitting two Killing spinors of new minimal supergravity of opposite chirality and it was shown in [8] that there is no ambiguity in the finite result.
For simplicity, we focus here on the case of the round S 3 × S 1 , which amounts to setting a 1 = a 2 = a ∈ iR >0 . In this case, the chiral multiplet one-loop determinant simplifies to Z = n 0 ∈Z n 1 ,n 2 ≥0 −u − n 0 + (n 1 + n 2 + 2)a u + n 0 + (n 1 + n 2 )a = We regularize the sum by introducing a cut-off β 2π Λ on the S 1 β KK momentum level n 0 and β 2π|a| Λ on the S 3 KK momentum level m. 27 To simplify slightly the notations we simply set β = 2π in the following. The regularization is done by using any smooth decreasing function f such that f (0) = 1 and going to zero at infinity sufficiently fast to make the sum converge. The regularized sum over S 3 modes is with c 1 = 2i ∞ 0 f . The finite part of the large Λ expansion is complicated as it receives an infinite number of contributions from the Euler-MacLaurin expansion, however it is easy to see that it is independent of the function f and we can trust that it will reproduce the (logarithm of the) hyperbolic gamma function at a 1 = a 2 , since this is the well-known result for the 3d chiral multiplet one-loop determinant. We obtain F [u + n 0 ] ≃ e c 1 Λa −2 (u−a+n 0 ) Γ h (u + n 0 , a, a) , (C.17) 27 The relative factor |a| = β 2πr3 takes into account the ratio of scales between S 1 β and S 3 KK modes. 28 The Euler-MacLaurin formula for a convergent sum reads: (u ′ + n) 2 − a 2 6 + c 1 Λ(u ′ + n) f n Λ where u ′ = u − a = (r − 1)a andf is a second regulating function implementing the smooth cut-off on S 1 KK modes. Applying again the Euler-MacLaurin formula we obtain C(u, a, a) = 1 with c 2 = 4 ∞ 0 yf (y)dy + 8c 1 iπ ∞ 0f (y)dy. The finite piece reproduces the result (C.10) for a 1 = a 2 = a, so that the cut-off regularization result Z reg will match the regularization described above. The diverging piece should be removed with a dimension two supersymmetric counterterm constructed with new minimal supergravity background fields [8]. We have obtained log Z ≃ iπ 2 c 2 Λ 2 (r − 1)a −1 + log Z reg .

(C.21)
There is a single supergravity term of mass dimension two in new minimal supergravity [8]; it is the usual Einstein-Hilbert new minimal supergravity action, that we can take with coefficient Λ 2 . Its bosonic part is given by where the constant κ parametrizes a freedom in the choice of background. The choice κ = a is the most natural, because it preserves the SU(2) l × SU(2) r isometries of S 3 . This is precisely the term needed to remove the divergent piece in (C.21). We conclude that the cut-off regularization further validates the regularization method presented above.