Comments on the Casimir energy in supersymmetric field theories

We study the Casimir energy of four-dimensional supersymmetric gauge theories in the context of the rigid limit of new minimal supergravity. Firstly, revisiting the computation of the localized partition function on $S^1\times S^3$, we recover the supersymmetric Casimir energy. Secondly, we consider the same theories in the Hamiltonian formalism on $\mathbb{R}\times S^3$, focussing on the free limit and including a one-parameter family of background gauge fields along $\mathbb{R}$. We compute the vacuum expectation value of the canonical Hamiltonian using zeta function regularization, and show that this interpolates between the supersymmetric Casimir energy and the ordinary Casimir energy of a supersymmetric free field theory.


Introduction
It is remarkable that in certain situations the path integral of an interacting supersymmetric field theory can be reduced to the computation of one-loop determinants, using the idea of localization [1]. Employing this method, the partition function Z of N = 1 supersymmetric gauge theories with an R-symmetry, defined on a Hopf surface S 1 ×M 3 , with periodic boundary conditions for the fermions on S 1 , has been computed in [2]. This is proportional [3] to the supersymmetric index [4] I(β) = Tr (−1) F e −βHsusy , (1.1) where F is the fermion number and H susy is the BPS Hamiltonian generating translations on S 1 , and commuting with at least one supercharge Q, which exists on the manifold S 1 × M 3 . Here the trace is over the Hilbert space of states on M 3 . Although this index can be refined introducing additional fugacities, associated to other conserved charges commuting with Q, below we consider the simplest case with only one fugacity β, proportional to the radius of the S 1 .
In analogy with the standard path integral definition of the vacuum energy of a field theory, from the localized partition function Z, one can define a quantity dubbed supersymmetric Casimir energy [5,2] as Interestingly, in [2] this was shown to be given by a linear combination of the central charges a and c of the supersymmetric theory, which in the simplest case reads E susy = 4 27 (a + 3c) .
(1. 3) In two dimensions, the fact that the Casimir energy is proportional to the unique central charge c has been known for long time [6], and attempts to generalize this result to higher dimensions have been discussed in several places [7,8,9]. The usual strategy consists in relating the expectation value T tt of the energy-momentum tensor to its anomalous trace T µ µ . However, this approach has some limitations. First of all, the resulting Casimir energy is ambiguous, with the ambiguities being related to the ambiguities in the trace anomaly. Moreover, previous results focus on the case of conformally flat geometries 1 . Finally, as we shall illustrate below, in a generic background with non-dynamical fields, the energy-momentum tensor is not conserved, even classically.
One may be concerned that the definition (1.2) leads to an ambiguous result, due to the existence of local counterterms, that can shift arbitrarily the value of log Z. However, it has been shown in [10] that in the context of new minimal supergravity [11], which is the framework utilized in [2] to derive (1.3), on S 1 × M 3 all the possible supersymmetric counterterms vanish, strongly suggesting that the supersymmetric Casimir energy is not ambiguous. Thus, for supersymmetric theories, we regard (1.3) as a suitable generalization of the results [6] in two dimensions. Notice that this is an exact result, thus valid for any value of the coupling constants. For example, it is insensitive to the superpotential of the theory.
The linear combination of central charges a, c in (1.3) had previously appeared in [12], in the context of studies of the supersymmetric index. Recent papers exploring the relations of the central charges a, c with the supersymmetric index include [13,14,15].
In this note we investigate further the supersymmetric Casimir energy. In particular, we show that it coincides with the vacuum expectation value (vev) of the BPS Hamiltonian H susy , appearing in the definition of the supersymmetric index H susy = E susy . (1.4) Since the results (1.3) and (1.4) are also valid at weak (or zero) coupling, one may compare these with the Casimir energy in free supersymmetric field theories, computed using zeta function techniques. In particular, for a free theory of N v vector multiplets and N χ chiral multiplets, this reads [16,7] E free = H free = 1 192 (21N v + 5N χ ) . (1.5) This is the free field value of the Casimir energy of N χ conformally coupled complex scalar fields, N χ +N v Weyl spinors, and N v Abelian gauge fields, and does not agree with (1.4), as noted in [5,2]. See e.g. [17] for a concise derivation. Working in the framework of rigid new minimal supergravity, we will consider the canonical Hamiltonian H of a multi-parameter family of supersymmetric theories defined on the round R × S 3 or its compactification to S 1 × S 3 . We will show that this interpolates continuously between H susy and H free , thus resolving the apparent tension between (1.4) and (1.5).
Generically, the Hamiltonian is not a BPS quantity, and its vev, computed using zeta function regularization, cannot be expressed as a linear combination of the anomaly coefficients a, c.
The rest of this note is organized as follows. In section 2 we introduce the background geometries and the field theories. In section 3 we reconsider the computation of the supersymmetric Casimir energy from its the path integral definition. Section 4 contains the main results of this note. We examine the field theories in the canonical formalism and derive (1.4). Our conclusions are presented in section 5. Three appendices are included. In Appendix A we collect details of the relevant spherical harmonics on the three-sphere. Appendix B contains a few details about the Hurwitz zeta function. In Appendix C we write expressions for the energy-momentum tensor and other useful formulas.

Supersymmetric field theories
In this section we present the background geometry, that we view as a solution to the rigid limit of new supergravity and then introduce the relevant supersymmetric Lagrangians. We follow verbatim the notation of [2], to which we refer for more details.

Background geometry
We begin with the background in Euclidean signature, discussing the differences in Lorentzian signature later. We consider a background comprising the following metric where τ is a coordinate on S 1 and θ, ϕ, ς with 0 ≤ θ < π , ϕ ∼ ϕ + 2π , ς ∼ ς + 4π are coordinates on a round three-sphere 2 . We note the Ricci scalar of this metric is given by R = 6/r 2 3 , which is the same as the Ricci scalar of the 3d metric. We introduce the following orthonormal frame 3 e 1 = r 3 2 (cos ςdθ + sin θ sin ςdϕ) , e 2 = r 3 2 (− sin ςdθ + sin θ cos ςdϕ) , e 3 = r 3 2 (dς + cos θdϕ) , which corresponds to a left-invariant frame {e 1 , e 2 , e 3 } on S 3 . We will consider a class of backgrounds admitting a solution to the new minimal supersymmetry equation with the metric (2.1). Let us set r 1 = 1 and r 3 = 2 below. In our coordinates the supersymmetric complex Killing vector K reads

4)
2 For r 1 = 1 and r 3 = 2 this metric and the other background fields can be obtained specializing the background discussed in Appendix C of [2] to v = 1, b 1 = −b 2 = 1/2. Below we will set r 1 = 1 and r 3 = 2, but these can be easily restored by dimensional analysis. 3 Note that this frame is different from the frame used in [2]. and the dual one-form is We define the following "reference" values of the background fields where we have included a constant q, which can be obtained performing a (large) gauge transformation A → A + i 2 qdτ starting from the gauge choice adopted in [2]. Although in Euclidean signature this yields an ill-defined spinor, this is not true in Lorentzian signature and later in the paper this parameter will play a role. Assuming that U = κK, where κ is a constant, we can write the background fields as We also note the value of the combination where the normalization is chosen such that for q = 0 one has |ζ| 2 = 1/2 as in [2]. Indeed, because τ is a periodic coordinate, this does not make sense unless 4 q = 0.
For generic values of κ this background preserves only a SU(2) × U(1) subgroup of the isometry group SO(4) of the round three-sphere. In the following, we will be interested in two special choices of κ. In particular, the choice κ = κ ACM ≡ −1/3 corresponds (for q = 0) to the values in [2], namely where notice that A ACM is real. Another distinguished choice is κ = κ st ≡ −1, where the superscript stands for "standard", giving In particular, for this choice, the full SO(4) symmetry of the three-sphere is restored and (2.3) has the more general solution with ζ 0 any constant spinor [18,4].
Notice that, in addition to q = 0 [2], there are two other special values of the parameter q. Namely, for q = −1 the background field A st in (2.11) vanishes, while for q = 1/2 we have A cs = 0. The significance of these three values will become clearer in later sections.

Lagrangians
We consider an N = 1 supersymmetric field theory with a vector multiplet transforming in the adjoint representation of a gauge group G, and a chiral multiplet transforming in a representation R, with Lagrangians given in section 2.2 of [2], evaluated in the background described in the previous section. These Lagrangians then depend on the two constant parameters q and κ, as well as on the R-charges r I of the scalar fields in the chiral multiplet, that below will be simply denoted r. We will restrict attention to Lagrangians expanded up to quadratic order in the fluctuations around a configuration where all fields vanish.
Adopting the notation of [2], we will therefore consider the following Lagrangian of a chiral multiplet where D µ = ∇ µ −iq R A µ and q R denotes the R-charges of the fields [2]. This is therefore the Lagrangian of |R| ≡ N χ free chiral multiplets, each with R-charge r I , where we will denote as N χ the dimension of the representation |R|.
The three terms in the first line of (2.13) are separately δ ζ -exact, and the parameter ǫ allows us to continuously interpolate between the localizing Lagrangian in [2], obtained for ǫ = 0, and the usual chiral multiplet Lagrangian [11], obtained for ǫ = 1. Notice that at quadratic order the term δ ζ V 3 [2] vanishes. Inserting the values of the background fields, and writing L chiral (q, κ, ǫ, r) = L chiral bos (q, κ, ǫ, r) + L chiral fer (q, κ, ǫ, r) , (2.14) the bosonic part of the Lagrangian reads where ∇ i , is the covariant derivative on the three-sphere, and we have omitted a total derivative. The fermionic part of the Lagrangian reads where a = 1, 2, 3 are frame indices on the three-sphere and γ a denote the Pauli matrices, generating the three-dimensional Clifford algebra. This expression is frame-dependent, and we used the left-invariant frame (2.2), which is useful for applying the angular momentum formalism. In particular, we used the identity Notice that the Lagrangians used in [4,19] correspond to the values ǫ = 1, κ = −1, and q = 1/2. Notice also that for r = 2/3 and ǫ = 1 the total chiral multiplet Lagrangian does not depend on κ.
Let us introduce a compact notation, writing the Lagrangians above in terms of differential operators. Denoting ℓ a the Killing vectors dual to the left-invariant frame e a , and defining the"orbital" angular momentum operators as L a = i 2 ℓ a , one finds these satisfy the SU(2) commutation relations and we have 5 −∇ i ∇ i = L 2 and ∇ ς = −iL 3 . Similarly, we identify the Pauli matrices with the the spin operator as S a = 1 2 γ a , satisfying the same SU(2) algebra. Thus the 5 Recall that here we have set r 3 = 2. In general, the three-dimensional Laplace operator is Lagrangians can be written as with the constants taking the values α b = 1 2 , and respectively.
For the vector multiplet the quadratic Lagrangian is F is the linearized field strength of the gauge field A and λ is the gaugino, both transforming in the adjoint representation of the gauge group G. This is therefore the Lagrangian of |G| ≡ N v free vector multiplets, where we will denote as N v the dimension of the gauge group G.
The fermionic part of this Lagrangian can be put in the same form as the fermionic part of the chiral multiplet Lagrangian, namely where Notice that for q = 1/2, corresponding to A cs = 0 this reduces to the standard massless Dirac operator on the three-sphere.

Supersymmetric Casimir energy
In this section we will recover in our set-up the supersymmetric Casimir energy defined in [2] as where Z is the supersymmetric partition function, namely the path integral on S 1 × S 3 with periodic boundary conditions for the fermions on S 1 , computed using localization.
Restoring the radii of S 1 and S 3 , the dimensionless parameter β in [2] is given by Differently from [2], here we will not fix the value of κ, showing that owing to the pairing of bosonic and fermionic eigenvalues in the one-loop determinant, the final result will be independent of κ. Although the computation in Euclidean signature requires to fix q = 0, we will start presenting the explicit eigenvalues for generic values of q. We will then demonstrate that the pairing occurs if and only if q = 0.

The (unrefined) partition function takes the form [2]
where I(β) is the supersymmetric index, and the pre-factor arises from the regularization of one-loop determinants in the chiral multiplets and vector multiplets, respectively [2] (see also [20]). Since the index I(β) does not contribute to (3.1), in order to compute E susy we can restrict attention to Ψ (0) chi and Ψ (0) vec , and thus effectively set the constant gauge field A 0 = 0 in the one-loop determinants around the localization locus in [2]. In particular, as the vector multiplet Lagrangian does not depend on κ and ǫ, its contribution to E susy can be simply borrowed from [2].
For example, by setting |b 1 | = |b 2 | = r 1 /r 3 in eq. (4.33) of [2], one obtains For the chiral multiplet, we revisit the computation of the one-loop determinant (with A 0 = 0) by working out the explicit eigenvalues, for an arbitrary choice of the parameters κ and ǫ. The eigenvalues of the operators O b and O f can be obtained with elementary methods from the theory of angular momentum in quantum mechanics [21].
We distinguish two types of eigenvalues of O f . For any ℓ = 1, 2, 3, ... we have where here the quantum number m takes the values m = − ℓ 2 , . . . , ℓ 2 − 1. Furthermore, for any ℓ = 0, 1, 2, . . . , we have the two special eigenvalues Again, each eigenvalue has degeneracy (ℓ+1), due to the SU(2) R symmetry. Expanding the fields in Kaluza-Klein modes on S 1 as and similarly for ψ, we obtain the following eigenvalues for each mode For generic values of the quantum numbers ℓ, m, we say that the eigenvalues of the Inserting the values of the parameters given in (2.21) and (2.22) we find that this is satisfied if and only if q = 0, and for any value of κ, ǫ, r. Let us then set q = 0 in the rest of this section. Restoring generic values of the radius r 3 of the S 3 , the one-loop determinant for a fixed k is where the products are over all the bosonic and fermionic eigenvalues, including the special ones. However, using the condition (3.11) all the paired eigenvalues cancel out 6 .
For m = ℓ/2 the generic fermionic eigenvalues do not exist, thus there are unpaired bosonic eigenvalues, obtained setting m = ℓ/2 in (3.6), which read 13) and remain in the denominator of (3.12). Therefore, taking into account the contribution of the special fermionic eigenvalues in the numerator, and including the degeneracies, we obtain where we defined n 0 = ℓ + 1 and used that α f = −1 and Upon obvious identifications, this coincides with the one-loop determinant of an N = 2 chiral multiplet on the round three-sphere, originally derived in [22] and [23], although our operators O b and O f are slightly more general and interpolate between those used in these two references. In particular, the Lagrangians used in [22] correspond to κ = −1/3 and ǫ = 0, precisely as in [2], while those used in [23] correspond to κ = −1 and ǫ = 1. Recall that in all cases we have q = 0. Defining where r 1 can be restored simply rescaling the coordinate τ → r 1 τ , one finds Z [23]. Alternatively, (3.14) can be written in terms of special functions by integrating the differential equation where the Hurwitz zeta function has been used to regularize the infinite sum [22] (see Appendix B).
In order to take the limit β → ∞ it is more convenient to write (3.14) as an infinite product over two integers, namely Regularizing the infinite product over the Kaluza-Klein modes as in [2], one obtains Z 1−loop in terms of Barnes triple-gamma functions, which eventually can be written as where Γ e is elliptic gamma function [2] and 7 From this, one finds the contribution of a chiral multiplet to (3.1) to be where notice that the radii r 3 and r 1 have dropped out. This is exactly the contribution of a chiral multiplet with R-charge r to the total supersymmetric Casimir energy computed in [2], although we emphasize that here this has been derived for arbitrary values of the parameters κ and ǫ.
Combining the contributions of the chiral multiplets and the vector multiplets we recover the result with the anomaly coefficients are defined as where R denotes the R-symmetry charge, and "tr" runs over the fermionic fields in the multiplets. 7 Actually, the terms proportional to r 3 /r 1 in (3.20) and (3.4) are not present if one uses a slightly different regularization, consistent with the results of [14]. However, this does not affect the terms proportional to r 1 /r 3 , which are relevant for the computation of E susy . We thank B. Assel, D. Cassani, L. Di Pietro, and Z. Komargodski for discussions on this issue.
In the remainder of the paper we will show that (3.22) is also equal to the expectation value of the BPS Hamiltonian H susy , appearing in supersymmetric index (3.24) Therefore we now turn to the Hamiltonian formalism, working in a background with a non-compact time direction, thus with β → ∞ from the outset.

Hamiltonian formalism
In this section we will study the theories defined in section 2.2 in a background R × S 3 in Lorentzian signature, obtained from the geometry in section 2.1 by a simple analytic continuation. In particular, we take the metric where t denotes the time coordinate on R, and ds 2 (S 3 ) is the metric on S 3 , given in equation (2.1). Below we continue to set r 3 = 2. The background fields are obtained Moreover, the dynamical fields obey φ = φ † and ψ = ψ † from the start. The σ-matrices generating the appropriate Clifford algebra are obtained setting σ 0 αα = iσ 4 αα = 1 1 αα and σ 0 αα = i σ 4 αα = 1 1 αα , with the remaining components unchanged, such that with η ab = diag(−1, 1, 1, 1). The Lorentzian spinor ζ solving equation (2.3) for generic κ is then again with a more general solution for the special value κ = κ st = −1 [18,4].

Conserved charges
The Hamiltonian 8 density H = H bos + H fer , associated to the chiral multiplet Lagrangian (2.13), is obtained as usual, by defining the canonical momenta and its bosonic and fermionic parts read The Hamiltonian is then obtained by integrating 9 over the spatial S 3 , The R-symmetry current J µ R can be derived either from the Noether procedure or as the functional derivative of the action with respect to A µ , namely and it reads This is conserved, i.e. ∇ µ J µ R = 0, and the corresponding conserved charge R is obtained by contracting it with the time-like Killing vector ∂ t , and integrating on the S 3 , which yields Rotational symmetry along the Killing vector ∂ ς gives rise to a conserved current with the corresponding conserved angular momentum Finally, supersymmetry gives rise to the conserved supercurrent Using the equations of motion for the dynamical fields, after some calculation, one can verify that ∇ µ (ζJ µ susy ) = 0 . (4.13) Note that ∇ µ ζ = 0, and therefore J µ susy is not conserved by itself, as is the case in the standard flat-space computation. Contracting ζJ µ susy with the time-like Killing vector ∂ t , we obtain the conserved supercharge where we defined In summary, applying the Noether procedure to the Lagrangian (2.13), we have derived expressions for the Hamiltonian H, R-charge R, angular momentum J 3 , and supercharge Q. These will provide the relevant operators in the quantized theory.
Let us discuss briefly other currents that can be considered, which however are not conserved generically. In particular, the usual energy-momentum tensor, defined as is not conserved in the presence of non-dynamical fields. This remains true even if T µν is contracted with a vector field that generates a symmetry of the metric and the other background fields. Thus, for example, T tt does not define a conserved quantity, and in particular it does not coincide with the canonical Hamiltonian. Denoting the non-dynamical vector fields as A I µ , with F I = dA I , and the associated currents as J µ I , in general the energy-momentum tensor (4.17) obeys the Ward identity In the present case, after a tedious computation, one finds that the energy-momentum satisfies where J µ FZ is the Ferrara-Zumino current 20) and Neither J µ FZ nor J µ K are conserved. Explicit expressions for T µν , J µ FZ , and J µ K are given in Appendix C. Note that in this context, we must formally treat K µ as a background field, although it was introduced in the Lagrangian as a shift of the original fields A µ and V µ . For the usual chiral multiplet Lagrangian with ǫ = 1, however, one has J µ K = 0. For a generic Killing vector ξ, that is also a symmetry of the background fields, L ξ A = L ξ V = 0, we can define a conserved current as One can show that indeed ∇ µ Y µ ξ = 0. In particular, for ξ = ∂ t , one finds that the conserved charge is the Hamiltonian density up to a total derivative on the S 3 .

Canonical quantization
We now expand the dynamical fields in terms of creation and annihilation operators.
Let us first focus on the scalar field. In order for the field φ to solve its equation of motion, we expand it as where Y mn ℓ ( x) are the scalar spherical harmonics on a three-sphere of unit radius (see Appendix A for further details), and The canonical commutation relations , hold by taking the oscillators to satisfy From (4.6) it follows that the Hamiltonian of the scalar field reads Notice that we have used the Weyl ordering prescription, as this is the correct one for comparison with the path integral approach.
For the fermion, we expand the field ψ in terms of the spinor spherical harmonics S ± ℓmn . As discussed in Appendix A, these are eigenspinors of the operator O f , with the eigenvalues λ ± ℓm given in equation (3.7). In addition, there are the "special" spherical harmonics, with λ special± ℓ given in equation (3.8). We expand the field ψ as Here we included S special± in the sums by defining (4.34) Of course, by imposing the anti-commutation relations one finds the field ψ α and the conjugate momentum π α = i ψα σ 0αα satisfy the canonical The mode expansion (4.32) can now be inserted into the conserved charges of section 4.1, recalling that these have to be Weyl ordered. For example, the Hamiltonian density in (4.6) becomes Inserting now the mode expansion and integrating over the S 3 yields the quantized Hamiltonian In the next section we will turn to the computation of the expectation values of these Hamiltonians, and we will show that the infinite sums can be evaluated with (Hurwitz) zeta function regularization in two special cases. One case is obtained for q = 0, for which we can use the pairing of bosonic and fermionic eigenvalues discussed in section 3 to evaluate the vev of H = H bos + H fer . Another case is obtained for β f = β b = 0, where we will be able to evaluate the vevs of H bos and H fer separately.
Thus, for simplicity in the remainder of this section we restrict to β f = β b = 0. Using the mode expansions of the fields, and after Weyl ordering, we obtain expressions for the remaining conserved charges of section 4.1. For the R-charge, equation (4.10), this leads to For the J 3 angular momentum, equation (4.11), we get and finally the supercharge (4.14) reads By direct computation, one can now verify the following commutation relations where we restored the radius r 3 of the S 3 . Note that the Hamiltonian commutes with Q only for q = 0, which from equation (3.11) is the value required for the pairing of eigenvalues. By conjugating equation (4.41), one can further verify that which coincide for example with eq. in (7) of [19] as well as with eq. (6.11) in [24], where H| q=1/2 corresponds to ∆ in the latter reference.
Although these commutation relations are here written for the chiral multiplet, it is straightforward to verify that they hold also for the vector multiplet, and hence for the total H tot = H + H vec , and similarly for the other operators. It was noticed in [5] that these may be formally derived from the abstract supersymmetry algebra of new minimal supergravity.

Casimir energy
We are now ready to compute the vacuum expectation value of the Hamiltonian. This yields infinite sums which we regularize using the standard zeta function method. Thus, for an operator A, we define its vacuum expectation value as where, denoting with λ A n the set of all the eigenvalues (here n is a multi-index) of A and with d A n their degeneracies, the generalised zeta function is defined as One must be careful with the regularization of the infinite sums. In particular, if A = B + C with corresponding eigenvalues denoted as λ B n and λ C n , one has This lack of additivity is related to the lack of associativity of functional determinants, det(BC) = det(B) · det(C), which is known as "multiplicative anomaly". See e.g. [25].
In the present context, we use the following prescription for dealing with the infinite sums: for each given operator, we sum independently the eigenvalues corresponding to every different field. In particular, we define the vev of each operator as the sum of the vevs of the terms containing the fields φ, ψ, A, and λ, respectively. Therefore, for example, and similarly for R, J 3 , and Q. This recipe is in accordance with [26] and, perhaps more importantly, yields to the supersymmetric Casimir energy.
The vevs of the scalar and fermion Hamiltonians, (4.29) and (4.38), are respectively. However, due to the square roots appearing in both sets of eigenvalues ω ± ℓm and λ ± ℓm , the vevs in (4.51) cannot in general be separately regularized with any 12 zeta function and written in closed form.
In the special case q = 0, we can take advantage of the pairing as discussed in section 3 to compute the vev of the Hamiltonian of the chiral multiplet, H = H bos +H fer . Thus, setting q = 0 one has The eigenvalues not included in equation (4.52) are the "special" fermion eigenvalues, which we can write as and the "unpaired" bosonic eigenvalues We used here that α f < 0 and assumed β f ≤ − Notice that the first and third term in the first line further exactly cancelled and in the last step we regularized separately the two remaining sums using the Hurwitz zeta function. See Appendix B. To summarize, since for q = 0 one has 2(β f + µ) = 1 − r, the vev of the Hamiltonian of a chiral multiplet with R-charge r is where we restored r 3 . This result is valid for any value of r, κ, ǫ. Notice that if we were to combine the two sums in the middle line of (4.55), before regularization, we would get a different (wrong) result! It is now simple to combine this with the contributions of the fields in the vector multiplet, and recover the supersymmetric Casimir energy E susy of section 3. The Casimir energy of the gauge field does not depend on any of our parameters and is simply given by the result for an Abelian gauge field H gauge = 11 120r 3 (see e.g. [7]) multiplied by the dimension of the gauge group N v . For the gaugino λ, the Casimir energy is computed as for the fermion ψ above, but using the operator O vec f in equation (2.25), giving simply H gaugino = − 1 120r 3 , again multiplied by N v . Adding everything together, we obtain Next, we consider the special case β b = β f = 0. This corresponds to setting κ = −1 and ǫ = 1, but leaving arbitrary q. In this case, both sums in (4.51) and can be separately regularized using Hurwitz zeta function, as the square roots in ω ± ℓm and λ ± ℓm are absent, namely and where we dropped the subscript m, as this quantum number becomes degenerate. Thus, regularizing the sums as described at the beginning of this subsection using the Hurwitz zeta function, we obtain the finite Casimir energies and H fer = 1 240 10(q + 1) 3 (1 + q) (r − 1) 4 +20(q + 1) 3 (r − 1) 3 − 10(q + 1)(r − 1) − 1 . Adding (4.60) and (4.61), we obtain the Casimir energy of a chiral multiplet with R-charge r This generalizes straight-forwardly to an arbitrary number of chiral multiplets. As before, we can include easily an arbitrary number N v of vector multiplets as well. In this case, for the gaugino, the Casimir energy can be obtained by formally setting r = 2q 1+q in equation (4.61), and reads Combining these results, we find that (for κ = −1, ǫ = 1) the Casimir energy of a supersymmetric gauge theory with N v vector multiplets and N χ chiral multiplets with R-charges r I is given by the following expression where we restored the radius r 3 of the three-sphere. First of all, setting q = 0 as in [2], we see that (4.64) reduce to in agreement with (4.57).
In general, however, equation (4.64) cannot be written as a linear combination of a and c. In the special case q = 1/2 and r I = 2/3, corresponding to the usual conformally coupled scalars, Weyl spinors, and gauge fields, the Casimir energy (4.64) reduces to in accordance with standard zeta function computations (see e.g. [17]). In particular, notice that for theories with N χ = 3N v (so that a = c), such as N = 4 super-Yang Mills, this becomes simply 3 4r 3 a. However, the agreement with the CFT result of [8] for the Casimir energy is accidental [7,8]. Finally, we note that for q = −1, the Casimir energy is independent of the R-charges and reads This is simply because in this case A = 0, and therefore the Lagrangian does not depend on the R-charges, as we observed at the end of section 2.1.
We can also easily compute the vev 14 of the R-charge operator R, by using the regularization described above. For a single chiral multiplet of R-charge r we get one could naively conclude that H tot is given by the sum of the vevs on the right hand side, namely H tot q=0 − 1 2 q R tot . However, this does not hold, as noted in (4.49). In particular, we have +6q(q + 1) 2 (r I − 1) 2 + 4(q + 1)q 2 (r I − 1) + q 3 , (4.72) 14 The vev of the supercharge Q is zero. For the chiral multiplet this follows from (4.14) or its mode expansion (4.41). Similarly, for the vector multiplet it follows from the explicit expression of Q vec , which is that is generically non-zero, for q = 0. The correct prescription is that of regularizing the combined contributions, rather than separately the two terms in the right hand side of (4.71). For example, for q = 1/2 and r I = 2/3 the former yields (4.66), while the latter does not. Note that had the R-charge operator R tot been normal ordered, by definition we would have had : R tot : = 0 and H tot q=0 = H tot . The results we have discussed in this note crucially rely on the fact that the operators we are using are not normal ordered. See also [27] for a similar discussion.

Conclusions
The main purpose of this note was to examine aspects of the supersymmetric Casimir energy of N = 1 gauge theories introduced in [2], in the simplest case where the partition function depends only on one fugacity. It has been argued recently in [10] that this quantity does not suffer from ambiguities, and therefore it should be a good physical observable.
Firstly, by revisiting the localization computation in [2], we have verified explicitly that its value does not depend on the choice of the parameter κ, characterizing the background fields A and V , as expected. Secondly, we reproduced it by evaluating the expectation value of the BPS Hamiltonian that appears in the definition of the supersymmetric index, as anticipated in [12]. Our computations also clarify the relation of the supersymmetric Casimir energy with the Casimir energy of free conformal fields theories, demonstrating that these two quantities arise as the expectation values of two different Hamiltonians, evaluated using the same zeta function regularization method.
An obvious extension of this note is to compute the vev of the BPS Hamiltonian in the more general case of supersymmetric theories on a Hopf surface with two complex structure parameters, and to check that this agrees with the general form of the supersymmetric Casimir energy [2]. It should also be rewarding to study this problem from the viewpoint of [9,28,7,8], or using the effective action approach of [14].
The discrepancy between the supersymmetric Casimir energy and the renormalized on-shell action in AdS 5 , pointed out in [5], remains unexplained. We think it is imperative to elucidate how to reproduce the supersymmetric Casimir energy from a holographic computation.

A Spherical harmonics A.1 Scalar spherical harmonics
In this appendix we give some details on the scalar and spinor spherical harmonics on the three-sphere, starting with the former, following [18,29]. We can obtain the metric on the unit three-sphere by considering a parametrization on R 4 ≃ C 2 with the metric, The three-sphere of unit radius is then defined by The isometry group is SO(4) ≃ SU(2) L × SU(2) R , with generators 15 L L a and L R a , with a = 1, 2, 3, satisfying As usual, we define raising and lowering operators, In the (u, v)-coordinates, these are represented by while the 15 In the main text, we use the operators L L a , but drop the superscript L.
In terms of these operators, the scalar Laplacian is The spherical harmonics Y mn ℓ are constructed starting from the highest weight state, which is annihilated by the raising operators L L + and L R + . The number m (n) can be lowered by and take values − ℓ 2 ≤ m, n ≤ ℓ 2 . The spherical harmonics are eigenfunctions of the operator O b , equation (2.20), and also We normalize the spherical harmonics as [29] Y ℓ 2 −a, ℓ where the sum is over all integer values of k for which the exponents are non-negative, and Now specifically taking u = i sin θ 2 e i(ϕ−ς)/2 and v = cos θ 2 e −i(ϕ+ς)/2 , one finds the metric ds 2 S 3 = 1 4 dθ 2 + sin 2 θdϕ 2 + (dς + cos θdϕ) 2 , (A.14) and With the above normalization, the spherical harmonics satisfy as well as the completeness relation,

B Hurwitz zeta function
In this appendix we include the definition of the Hurwitz zeta function and some useful properties. This is defined as the analytic continuation to complex s = 1, of the following series The following formulas used in the text are easily proved

C Energy-momentum tensor and other currents
In this appendix we provide explicit expressions for the energy-momentum tensor and other currents obtained from the (quadratic) chiral multiplet Lagrangian (2.13). Denoting with S the corresponding action, the energy-momentum tensor is defined as A straightforward but tedious computation yields where the lower parenthesis denote symmetrization of the indices. Recall that we defined D µ = ∇ µ − iq R A µ , with q R the R charges of the fields [2].