Supersymmetric Casimir energy on
${\cal N}=1$ conformal supergravity backgrounds

Pantelis Panopoulosa , Ioannis Papadimitrioub

a Asia Pacific Center for Theoretical Physics, Postech, Pohang 37673, Korea

bDivision of Nuclear and Particle Physics, Department of Physics, National and
Kapodistrian University of Athens, GR-157 84 Athens, Greece

<EMAIL_ADDRESS><EMAIL_ADDRESS>
We provide a first principles derivation of the supersymmetric Casimir energy
of ${\cal N}=1$ SCFTs in four dimensions using the supercharge algebra on
general conformal supergravity backgrounds that admit Killing spinors. The
superconformal Ward identities imply that there exists a continuous family of
conserved R-currents on supersymmetric backgrounds, as well as a continuous
family of conserved currents for each conformal Killing vector. These
continuous families interpolate between the consistent and covariant R-current
and energy-momentum tensor. The resulting Casimir energy, therefore, depends
on two continuous parameters corresponding to the choice of conserved currents
used to define the energy and R-charge. This ambiguity is in addition to any
possible scheme dependence due to local terms in the effective action. As an
application, we evaluate the general expression for the supersymmetric Casimir
energy we obtain on a family of backgrounds with the cylinder topology
$\mathbb{R}\times S^{3}$ and admitting two supercharges of opposite R-charge.
Our result is a direct consequence of the supersymmetry algebra, yet it
resembles more known expressions for the non-supersymmetric Casimir energy on
such backgrounds and differs from the supersymmetric Casimir energy obtained
from the zero temperature limit of supersymmetric partition functions. We
defer a thorough analysis of the relation between these results to future
work.

###### Contents

  1. 1 Introduction and summary of results
  2. 2 ${\cal N}=1$ superconformal Ward identities
  3. 3 Bardeen-Zumino terms and covariant currents
  4. 4 Conserved charges
  5. 5 Supersymmetric Casimir energy
    1. 5.1 Supercharge algebra on curved backgrounds
    2. 5.2 BPS relation and the supersymmetric Casimir energy
  6. 6 Casimir energy on $\mathbb{R}\times S^{3}$
    1. 6.1 Supersymmetric backgrounds with $\mathbb{R}\times S^{3}$ topology
    2. 6.2 Round $S^{3}$ with arbitrary $b_{1},b_{2}$
  7. A Conventions and spinor identities

##  1 Introduction and summary of results

The Casimir energy of a conformal field theory (CFT) is defined as its ground
state energy. Placing the theory on the cylinder $\mathbb{R}\times S^{d-1}$
and in the absence of other background fields it is given by

${\cal E}_{\text{Casimir}}=\int_{S^{d-1}}d^{d-1}x\sqrt{-g}\,\langle{\cal
T}_{tt}\rangle_{\text{g.s.}},$ (1.1)

where ${\cal T}_{\mu\nu}$ is the energy-momentum tensor and
$\langle...\rangle_{\text{g.s.}}$ stands for the expectation value in the
ground state of the CFT on the cylinder. This is determined by the conformal
anomaly

$\langle{\cal T}^{\mu}_{\mu}\rangle=\frac{1}{(4\pi)^{2}}(aE-cW^{2}+b\square
R)\,,$ (1.2)

where $E$ is the Euler density, $W^{2}$ the square of the Weyl tensor and the
anomaly coefficients $a$ and $c$ depend on the specific CFT. The $b$-term is
scheme-dependent and corresponds to the addition of a local counterterm
proportional to $R^{4}$ in the effective action.

To evaluate the Casimir energy, one may integrate the conformal anomaly (1.2)
in order to determine its contribution to the effective action, known as the
Riegert action [1, 2]. Its derivative with respect to the background metric
computes the expectation value of the energy-momentum tensor in the ground
state, and hence the Casimir energy (1.1). Concequently, the Casimir energy of
a CFT is in general determined by the conformal anomaly coefficients $a$ and
$c$, as well as any scheme-dependent terms, such as $b$. For example, the
Casimir energy on the round unit sphere is given by (see e.g. [3])

${\cal E}_{\text{Casimir}}=\frac{3}{4}\Big{(}a-\frac{b}{2}\Big{)}\,.$ (1.3)

The Casimir energy, therefore, is in general a scheme-dependent quantity.

The procedure outlined above applies to any CFT, including supersymmetric
conformal field theories (SCFTs). However, supersymmetry allows several
alternative approaches to computing the supersymmetric Casimir energy [4, 5,
3, 6, 7, 8, 9], while requiring that the renormalization scheme preserves
supersymmetry reduces considerably the allowed scheme dependence. One common
definition of the Casimir energy on $S^{1}_{\beta}\times M_{3}$ with $M_{3}$ a
three-manifold admitting supersymmetry is given by the “zero temperature”
limit, $\beta\to\infty$, of the supersymmetric partition function. Namely,

${\cal
E}^{\text{susy}}_{\text{Casimir}}=-\lim_{\beta\to\infty}\frac{d}{d\beta}\log
Z^{\text{susy}}_{S^{1}_{\beta}\times M_{3}}\,,$ (1.4)

since the $\beta\to\infty$ limit projects the ground state energy.

For ${\cal N}=1$ SCFTs on $S_{\beta}^{1}\times S^{3}_{b_{1},b_{2}}$ with
squashing parameters $b_{1}$ and $b_{2}$ it has been shown using
supersymmetric localization that [5]

$Z^{\text{susy}}_{S^{1}_{\beta}\times S^{3}}=e^{-\beta{\cal
E}^{\text{susy}}_{\text{Casimir}}}{\cal I}^{\text{susy}}_{S^{1}_{\beta}\times
S^{3}}\,,$ (1.5)

where ${\cal I}_{S^{1}_{\beta}\times S^{3}}$ is the superconformal index and
the exponential factor determines the Casimir energy through the definition
(1.4), namely

${\cal
E}^{\text{susy}}_{\text{Casimir}}\left(b_{1},b_{2}\right)=\frac{4\pi}{3}(|b_{1}|+|b_{2}|)(\textbf{a}-\textbf{c})+\frac{4\pi}{27}\frac{(|b_{1}|+|b_{2}|)^{3}}{|b_{1}||b_{2}|}(3\textbf{c}-2\textbf{a})\,.$
(1.6)

For $b_{1}=b_{2}=\frac{\beta}{2\pi}$, (1.6) gives the supersymmetric Casimir
energy on the round sphere

${\cal
E}_{\text{Casimir}}^{\text{susy}}\left(\frac{\beta}{2\pi},\frac{\beta}{2\pi}\right)=\frac{4}{27}\left(\textbf{a}+3\textbf{c}\right)\,.$
(1.7)

Another approach to computing the supersymmetric Casimir energy was undertaken
in [3], by reducing an ${\cal N}=1$ theory on $S_{\beta}^{1}\times M_{3}$ to
quantum mechanics. The $1d$ Hamiltonian is the sum of the Hamiltonian for
chiral and Fermi multiplets and, therefore, the ground state energy can be
computed through the expression

$\langle H_{\text{susy}}\rangle=\sum_{\text{chiral}}\langle
H_{\text{chiral}}\rangle+\sum_{\text{Fermi}}\langle
H_{\text{Fermi}}\rangle\,.$ (1.8)

The result for the Casimir energy ${\cal E}_{\text{Casimir}}=\langle
H_{\text{susy}}\rangle$ so obtained agrees with (1.6) and (1.7). Yet another
approach was considered in[6], where it was conjectured that the
supersymmetric Casimir energy in $d$ even dimensions is given by an
equivariant integral of the anomaly polynomial in two higher dimensions.

The approach we follow in the preset work, however, is based on the
supersymmetry algebra. On a curved background, the supercharges satisfy
(schematically) the algebra

$\\{Q^{\dagger},Q\\}=H-Q_{R}-{\cal E}_{\text{Casimir}}\,,$ (1.9)

where $H$ is the Hamiltonian and $Q_{R}$ is the R-charge operator. Since in a
supersymmetric vacuum $\langle\\{Q^{\dagger},Q\\}\rangle=0$, it follows that

$\langle H-Q_{R}\rangle_{\rm g.s.}={\cal E}_{\text{Casimir}}\,.$ (1.10)

We determine the general form of the Casimir energy ${\cal
E}_{\text{Casimir}}$ in (1.9) by coupling a generic ${\cal N}=1$ SCFT to
background conformal supergravity and using the operator algebra obtained in
[19]. This leads to the general expression (6) for the supersymmetric Casimir
energy in terms of local curvatures and Killing spinor bilinears. Evaluating
this general expression on backgrounds with the cylinder topology
$\mathbb{R}\times S^{3}$ and admitting two supercharges of opposite R-charge
we arrive at the expression (6.29). This result is consistent with (1.3), but
differs significantly from (1.6). We defer a thorough analysis of the relation
between these two results to a future publication.

The rest paper is organized as follows. In section 2 we review the ${\cal
N}=1$ superconformal Ward identities of conformal supergravity, including the
superconformal anomalies. In section 3 we introduce the relevant Bardeen-
Zumino terms for the R-current and energy-momentum tensor, which allow us to
rewrite the superconformal Ward identities in terms of the covariant
$R$-current and the energy-momentum tensor. In section 4 we consider bosonic
backgrounds with numerically zero anomalies, such as those admiting Killing
spinors, and show the existence of continuous families of conserved R-charges
and charges associated with conformal Killing vectors. Section 5 is devoted to
the detailed evaluation of the Casimir energy on such backgrounds using the
anomalous transformation of the supercurrent obtained in [19]. This leads to
the general BPS relation (5.43) and the “local charge” (5.44). These are the
main results of this work. As an illustrative example, in section 6, we
evaluate the Casimir energy on $\mathbb{R}\times S^{3}$, obtaining the result
(6).

##  2 ${\cal N}=1$ superconformal Ward identities

The superconformal Ward identities that four-dimensional ${\cal N}=1$ SCFTs
satisfy can be determined by coupling the theory to ${\cal N}=1$ off-shell
conformal supergravity [10, 11, 12, 13], whose field content consists of the
vielbein $e^{a}_{\mu}$, an Abelian gauge field $A_{\mu}$, and a Majorana
gravitino $\psi_{\mu}$, comprising 5+3 bosonic and 8 fermionic off-shell
degrees of freedom. Treating the supergravity fields as external background
fields, the SCFT partition function takes the form

${\cal Z}[e,A,\psi]=\int[{\cal D}\Phi]e^{iS[\Phi,e,A,\psi]}\,,$ (2.1)

where $\Phi$ denotes collectively all the microscopic degrees of
freedom.111Evaluating the path integral over the microscopic degrees of
freedom $\Phi$ in general requires the introduction of a regulator that
explicitly breaks conformal symmetry. Although the regulated theory cannot
couple consistently to conformal supergravity, the renormalized theory can,
even in the presence of superconformal anomalies. Throughout this article we
refer exclusively to the renormalized observables, which can consistently
couple to conformal supergravity.

The logarithm of the partition function on an arbitrary supergravity
background

$\mathcal{W}[e,A,\psi]=-i\log{\cal Z}[e,A,\psi]\,,$ (2.2)

is often referred to as the ‘quantum effective action’ and amounts to the
generating function of connected (renormalized) correlation functions of the
current operators

$\langle{\cal T}^{\mu}_{a}\rangle=e^{-1}\frac{\delta\mathcal{W}}{\delta
e^{a}_{\mu}},\qquad\langle{\cal
J}^{\mu}_{a}\rangle=e^{-1}\frac{\delta\mathcal{W}}{\delta
A_{\mu}},\qquad\langle{\cal
S}^{\mu}_{a}\rangle=e^{-1}\frac{\delta\mathcal{W}}{\delta\bar{\psi}_{\mu}}\,,$
(2.3)

where $e\equiv\det(e^{a}_{\mu})$ and the notation $\langle\cdot\rangle$
denotes one-point functions in the presence of arbitrary sources, i.e. on an
arbitrary supergravity background. These operators comprise the conformal
multiplet of currents and satisfy a set of superconformal Ward identities.

${\cal N}=1$ conformal supergravity is a gauge theory of the ${\cal N}=1$
superconformal algebra [10, 11, 12, 13] (see [14, 15, 16, 17] and chapter 16
of [18] for pedagogical reviews). Its local symmetries consist of
diffeomorphisms $\xi^{\mu}(x)$, Weyl transformations $\sigma(x)$, local frame
rotations $\lambda^{ab}(x)$, $U(1)$ R-symmetry transformations $\theta(x)$, as
well as ${\cal Q}$\- and ${\cal S}$-supersymmetry transformations,
parameterized respectively by the local spinors $\varepsilon(x)$ and
$\eta(x)$. Under these, the fields of ${\cal N}=1$ conformal supergravity
transform as

$\displaystyle\delta e^{a}_{\mu}=$
$\displaystyle\;\xi^{\lambda}\partial_{\lambda}e^{a}_{\mu}+e^{a}_{\lambda}\partial_{\mu}\xi^{\lambda}-\lambda^{a}{}_{b}e^{b}_{\mu}+\sigma
e^{a}_{\mu}-\frac{1}{2}\overline{\psi}_{\mu}\gamma^{a}\varepsilon\,,$
$\displaystyle\delta\psi_{\mu}=$
$\displaystyle\;\xi^{\lambda}\partial_{\lambda}\psi_{\mu}+\psi_{\lambda}\partial_{\mu}\xi^{\lambda}-\frac{1}{4}\lambda_{ab}\gamma^{ab}\psi_{\mu}+\frac{1}{2}\sigma\psi_{\mu}+{\cal
D}_{\mu}\varepsilon-\gamma_{\mu}\eta-i\gamma^{5}\theta\psi_{\mu}\,,$
$\displaystyle\delta A_{\mu}=$
$\displaystyle\;\xi^{\lambda}\partial_{\lambda}A_{\mu}+A_{\lambda}\partial_{\mu}\xi^{\lambda}+\frac{3i}{4}\overline{\phi}_{\mu}\gamma^{5}\varepsilon-\frac{3i}{4}\overline{\psi}_{\mu}\gamma^{5}\eta+\partial_{\mu}\theta\,,$
(2.4)

where

$\phi_{\mu}\equiv-\frac{1}{6}\big{(}4\delta^{[\rho}_{\mu}\delta^{\sigma]}_{\nu}+i\gamma^{5}\epsilon_{\mu\nu}{}^{\rho\sigma}\big{)}\gamma^{\nu}{\cal
D}_{\rho}\psi_{\sigma}\,,$ (2.5)

and the spinor covariant derivatives are given by

$\displaystyle{\cal D}_{\mu}\psi_{\nu}\equiv$
$\displaystyle\;\Big{(}\partial_{\mu}+\frac{1}{4}\omega_{\mu}{}^{ab}(e,\psi)\gamma_{ab}+i\gamma^{5}A_{\mu}\Big{)}\psi_{\nu}-\Gamma^{\rho}_{\mu\nu}\psi_{\rho}\equiv\big{(}\mathscr{D}_{\mu}+i\gamma^{5}A_{\mu}\big{)}\psi_{\nu}\,,$
$\displaystyle{\cal D}_{\mu}\varepsilon\equiv$
$\displaystyle\;\Big{(}\partial_{\mu}+\frac{1}{4}\omega_{\mu}{}^{ab}(e,\psi)\gamma_{ab}+i\gamma^{5}A_{\mu}\Big{)}\varepsilon\equiv\big{(}\mathscr{D}_{\mu}+i\gamma^{5}A_{\mu}\big{)}\varepsilon\,.$
(2.6)

In these expressions $\omega_{\mu}{}^{ab}(e,\psi)$ denotes the torsion-full
spin connection

$\omega_{\mu}{}^{ab}(e,\psi)\equiv\omega_{\mu}{}^{ab}(e)+\frac{1}{4}\big{(}\overline{\psi}_{a}\gamma_{\mu}\psi_{b}+\overline{\psi}_{\mu}\gamma_{a}\psi_{b}-\overline{\psi}_{\mu}\gamma_{b}\psi_{a}\big{)}\,,$
(2.7)

where $\omega_{\mu}{}^{ab}(e)$ is the torsion-free metric compatible
connection.

The quantum effective action (2.2) of any ${\cal N}=1$ SCFT is invariant under
the local symmetry transformations (2), up to local expressions in the
background supergravity fields that comprise the multiplet of superconformal
anomalies. In particular, there exists a renormalization scheme such that,
under an infinitesimal local symmetry transformation with parameters
$\Omega=(\xi,\sigma,\lambda,\theta,\varepsilon,\eta)$, the quantum effective
action transforms as

$\delta_{\Omega}\mathcal{W}[e,A,\psi]=\int
d^{4}x\,e\big{(}\sigma\mathcal{A}_{W}-\theta\mathcal{A}_{R}-\bar{\varepsilon}\mathcal{A}_{Q}+\bar{\eta}\mathcal{A}_{S}\big{)}\,,$
(2.8)

where the superconformal anomalies ${\cal A}_{W}$, ${\cal A}_{R}$, ${\cal
A}_{Q}$ and ${\cal A}_{S}$ take the form [19]

$\displaystyle{\cal A}_{W}=$
$\displaystyle\;\frac{c}{16\pi^{2}}\Big{(}W^{2}-\frac{8}{3}F^{2}\Big{)}-\frac{a}{16\pi^{2}}E+{\cal
O}(\psi^{2})\,,$ $\displaystyle{\cal A}_{R}=$
$\displaystyle\;\frac{(5a-3c)}{27\pi^{2}}\;\widetilde{F}F+\frac{(c-a)}{24\pi^{2}}{\cal
P}\,,$ $\displaystyle{\cal A}_{Q}=$
$\displaystyle\;-\frac{(5a-3c)i}{9\pi^{2}}\widetilde{F}^{\mu\nu}A_{\mu}\gamma^{5}\phi_{\nu}+\frac{(a-c)}{6\pi^{2}}\nabla_{\mu}\big{(}A_{\rho}\widetilde{R}^{\rho\sigma\mu\nu}\big{)}\gamma_{(\nu}\psi_{\sigma)}-\frac{(a-c)}{24\pi^{2}}F_{\mu\nu}\widetilde{R}^{\mu\nu\rho\sigma}\gamma_{\rho}\psi_{\sigma}+{\cal
O}(\psi^{3})\,,$ $\displaystyle{\cal A}_{S}=$
$\displaystyle\;\frac{(5a-3c)}{6\pi^{2}}\widetilde{F}^{\mu\nu}\Big{(}{\cal
D}_{\mu}-\frac{2i}{3}A_{\mu}\gamma^{5}\Big{)}\psi_{\nu}+\frac{ic}{6\pi^{2}}F^{\mu\nu}\big{(}\gamma_{\mu}{}^{[\sigma}\delta_{\nu}^{\rho]}-\delta_{\mu}^{[\sigma}\delta_{\nu}^{\rho]}\big{)}\gamma^{5}{\cal
D}_{\rho}\psi_{\sigma}$ (2.9)
$\displaystyle+\frac{3(2a-c)}{4\pi^{2}}P_{\mu\nu}g^{\mu[\nu}\gamma^{\rho\sigma]}{\cal
D}_{\rho}\psi_{\sigma}+\frac{(a-c)}{8\pi^{2}}\Big{(}R^{\mu\nu\rho\sigma}\gamma_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}g^{\mu[\nu}\gamma^{\rho\sigma]}\Big{)}{\cal
D}_{\rho}\psi_{\sigma}+{\cal O}(\psi^{3})\,.$

The anomaly coefficients $a$ and $c$ in these expressions depend on the
specific SCFT and are normalized such that for $N_{\chi}$ free chiral and
$N_{v}$ free vector multiplets [20]

$a=\frac{1}{48}(N_{\chi}+9N_{v})\,,\qquad c=\frac{1}{24}(N_{\chi}+3N_{v})\,.$
(2.10)

$W^{2}$ in (2) denotes the square of the Weyl tensor, while $E$ and ${\cal P}$
are respectively the Euler and Pontryagin densities. Namely,

$\displaystyle W^{2}\equiv$
$\displaystyle\;W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma}=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-2R_{\mu\nu}R^{\mu\nu}+\frac{1}{3}R^{2}\,,$
$\displaystyle E=$
$\displaystyle\;R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}\,,$
$\displaystyle{\cal P}\equiv$
$\displaystyle\;\frac{1}{2}\epsilon^{\kappa\lambda\mu\nu}R_{\kappa\lambda\rho\sigma}R_{\mu\nu}{}^{\rho\sigma}=\widetilde{R}^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}\,,$
(2.11)

where

$\widetilde{R}_{\mu\nu\rho\sigma}\equiv\frac{1}{2}\epsilon_{\mu\nu}{}^{\kappa\lambda}R_{\kappa\lambda\rho\sigma}\,.$
(2.12)

Finally, $P_{\mu\nu}$ in (2) denotes the Schouten tensor

$P_{\mu\nu}=\frac{1}{2}\Big{(}R_{\mu\nu}-\frac{1}{6}Rg_{\mu\nu}\Big{)}\,,$
(2.13)

and we have defined

$F^{2}\equiv F_{\mu\nu}F^{\mu\nu}\,,\qquad
F\widetilde{F}\equiv\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}\,,\qquad\widetilde{F}_{\mu\nu}\equiv\frac{1}{2}\epsilon_{\mu\nu}{}^{\rho\sigma}F_{\rho\sigma}\,.$
(2.14)

The anomalous transformation (2.8) of the quantum effective action, together
with the definition of the currents (2.3), leads to the superconformal Ward
identities [19]

$\displaystyle e^{a}_{\mu}\nabla_{\nu}\langle{\cal
T}^{\nu}_{a}\rangle+\nabla_{\nu}(\overline{\psi}_{\mu}\langle{\cal
S}^{\nu}\rangle)-\overline{\psi}_{\nu}\overleftarrow{{\cal
D}}_{\mu}\langle{\cal S}^{\nu}\rangle-F_{\mu\nu}\langle{\cal J}^{\nu}\rangle$
$\displaystyle\hskip 56.9055pt+A_{\mu}\big{(}\nabla_{\nu}\langle{\cal
J}^{\nu}\rangle+i\overline{\psi}_{\nu}\gamma^{5}\langle{\cal
S}^{\nu}\rangle\big{)}-\omega_{\mu}{}^{ab}\Big{(}e_{\nu[a}\langle{\cal
T}^{\nu}_{b]}\rangle+\frac{1}{4}\overline{\psi}_{\nu}\gamma_{ab}\langle{\cal
S}^{\nu}\rangle\Big{)}=0\,,$ $\displaystyle e^{a}_{\mu}\langle{\cal
T}^{\mu}_{a}\rangle+\frac{1}{2}\overline{\psi}_{\mu}\langle{\cal
S}^{\mu}\rangle={\cal A}_{W}\,,$ $\displaystyle e_{\mu[a}\langle{\cal
T}^{\mu}_{b]}\rangle+\frac{1}{4}\overline{\psi}_{\mu}\gamma_{ab}\langle{\cal
S}^{\mu}\rangle=0\,,$ $\displaystyle\nabla_{\mu}\langle{\cal
J}^{\mu}\rangle+i\overline{\psi}_{\mu}\gamma^{5}\langle{\cal
S}^{\mu}\rangle={\cal A}_{R}\,,$ $\displaystyle{\cal D}_{\mu}\langle{\cal
S}^{\mu}\rangle-\frac{1}{2}\gamma^{a}\psi_{\mu}\langle{\cal
T}^{\mu}_{a}\rangle-\frac{3i}{4}\gamma^{5}\phi_{\mu}\langle{\cal
J}^{\mu}\rangle={\cal A}_{Q}\,,$ $\displaystyle\gamma_{\mu}\langle{\cal
S}^{\mu}\rangle-\frac{3i}{4}\gamma^{5}\psi_{\mu}\langle{\cal
J}^{\mu}\rangle={\cal A}_{S}\,.$ (2.15)

The transformation (2.8) of the quantum effective action, and hence the local
terms in the Ward identities (2), may be modified by adding local terms to the
effective action ${\cal W}[e,A,\psi]$. For example, adding the term

$\int A\wedge{\rm Tr\,}\Big{(}\Gamma\wedge
d\Gamma+\frac{2}{3}\Gamma\wedge\Gamma\wedge\Gamma\Big{)}\,,$ (2.16)

with $\Gamma^{\mu}{}_{\nu}\equiv\Gamma^{\mu}_{\nu\rho}dx^{\rho}$ the
Christoffel connection, breaks the diffeomorphism invariance of the effective
action and modifies the form of its anomalous transformation under the rest of
the local symmetries. In particular, the addition of this term with a specific
coefficient eliminates the Pontryagin term from the R-symmetry anomaly ${\cal
A}_{R}$, as is reviewed e.g. in [21]. Although the fermionic anomalies ${\cal
A}_{Q}$ and ${\cal A}_{S}$ can be modified by the addition of such local
terms, there exists no local term that sets them to zero within conformal
supergravity. It is these fermionic anomalies ${\cal A}_{Q}$ and ${\cal
A}_{S}$ that determine the supersymmetric Casimir energy [9].

##  3 Bardeen-Zumino terms and covariant currents

The anomalies (2) are solutions of the Wess-Zumino (WZ) consistency conditions
for the local symmetry transformations (2) of ${\cal N}=1$ conformal
supergravity [19]. As such, they correspond to the so called consistent
anomalies, while the operators defined in (2.3) are known as the consistent
currents. As a consequence of the R-symmetry anomaly, the consistent currents
are not gauge invariant, while if the local term (2.16) is added to the
effective action, the consistent currents are not diffeomorphism covariant
either.

The gauge and diffeomorphism covariance of the R-current ${\cal J}^{\mu}$ and
of the stress tensor ${\cal T}^{\mu}_{a}$ can be restored by adding local
Bardeen-Zumino (BZ) terms to the currents [22]. These terms are not related to
the choice of renormalization scheme discussed above, since they cannot be
expressed as derivatives of a local term in the effective action. Instead,
they arise from a Chern-Simons action in five dimensions that cancels the
R-symmetry/mixed anomaly through the mechanism of anomaly inflow (see e.g.
[21] for an extensive discussion).

However, the BZ terms for the R-current and the energy-momentum tensor are
already encoded in the form of the anomalies (2) [9, 19]. This observation was
understood in [23] as a direct consequence of the WZ consistency conditions.
As an example, let us consider the WZ condition222We adopt the convention of
[19], where the transformations act of the transformation parameters as well.
This is analogous to the BRST treatment of the WZ consistency conditions,
where the transformation parameters are replaced by ghost fields that
themselves transform. The same conclusions are reached in the convention that
the transformation parameters do not transform – see section 3.2 of [23].

$(\delta_{\xi}\delta_{\theta}-\delta_{\theta}\delta_{\xi}){\cal W}=0\,.$ (3.1)

In the absence of the local term (2.16), $\delta_{\xi}{\cal W}=0$ and so
$\delta_{\xi}\delta_{\theta}{\cal W}=0$ and $\delta_{\theta}\delta_{\xi}{\cal
W}=0$ separately. These imply respectively that $\delta_{\theta}{\cal W}$ is
diffeomorphism invariant and $\delta_{\xi}{\cal W}$ is gauge invariant.
Focusing on $\delta_{\xi}{\cal W}$, it can be expressed in terms of the
consistent currents (2.3) as [19]

$\delta_{\xi}{\cal W}=-\int
d^{4}x\,e\,\xi^{\mu}\big{(}e^{a}_{\mu}\nabla_{\nu}\langle{\cal
T}^{\nu}_{a}\rangle+\nabla_{\nu}(\overline{\psi}_{\mu}\langle{\cal
S}^{\nu}\rangle)-\overline{\psi}_{\nu}\overleftarrow{{\cal
D}}_{\mu}\langle{\cal S}^{\nu}\rangle-F_{\mu\nu}\langle{\cal
J}^{\nu}\rangle+A_{\mu}{\cal A}_{R}\big{)}\,,$ (3.2)

whose bosonic part is not manifestly gauge invariant. As we will now
demonstrate, the form of the R-symmetry anomaly ${\cal A}_{R}$ is such that
this expression can be written in manifestly gauge invariant form in terms of
the covariant R-current and energy-momentum tensor.

Writing the R-symmetry anomaly explicitly, the bosonic terms in (3.2) take the
form

$\displaystyle\;e^{a}_{\mu}\nabla_{\nu}\langle{\cal T}^{\nu}_{a}\rangle-
F_{\mu\nu}\langle{\cal J}^{\nu}\rangle+A_{\mu}{\cal A}_{R}$ $\displaystyle\ =$
$\displaystyle\;e^{a}_{\mu}\nabla_{\nu}\langle{\cal T}^{\nu}_{a}\rangle-
F_{\mu\nu}\langle{\cal
J}^{\nu}\rangle+\frac{(5a-3c)}{54\pi^{2}}\;\epsilon^{\kappa\lambda\rho\sigma}F_{\kappa\lambda}F_{\rho\sigma}A_{\mu}+\frac{(c-a)}{48\pi^{2}}\epsilon^{\kappa\lambda\rho\sigma}R_{\kappa\lambda\alpha\beta}R_{\rho\sigma}{}^{\alpha\beta}A_{\mu}\,.$
(3.3)

Antisymmetrizing five indices in four dimensions gives zero. Hence,
$F_{[\mu\kappa}F_{\lambda\rho}A_{\sigma]}=0$ and
$R_{[\kappa\lambda\alpha\beta}R_{\rho\sigma}{}^{\alpha\beta}A_{\mu]}=0$, which
imply respectively the identities

$\epsilon^{\kappa\lambda\rho\sigma}F_{\kappa\lambda}F_{\rho\sigma}A_{\mu}=-4\epsilon^{\kappa\lambda\rho\sigma}F_{\mu\kappa}F_{\lambda\rho}A_{\sigma}\,,$
(3.4)

and

$\epsilon^{\kappa\lambda\rho\sigma}R_{\kappa\lambda\alpha\beta}R_{\rho\sigma}{}^{\alpha\beta}A_{\mu}=-4\epsilon^{\kappa\lambda\rho\sigma}R_{\mu\kappa\alpha\beta}R_{\lambda\rho}{}^{\alpha\beta}A_{\sigma}\,.$
(3.5)

Therefore, the bosonic part of (3.2) becomes

$\displaystyle\;e^{a}_{\mu}\nabla_{\nu}\langle{\cal T}^{\nu}_{a}\rangle-
F_{\mu\nu}\langle{\cal J}^{\nu}\rangle+A_{\mu}{\cal A}_{R}$ $\displaystyle\ =$
$\displaystyle\;e^{a}_{\mu}\nabla_{\nu}\langle{\cal T}^{\nu}_{a}\rangle-
F_{\mu\nu}\langle{\cal J}^{\nu}_{\rm
cov}\rangle-\frac{(c-a)}{12\pi^{2}}\epsilon^{\kappa\lambda\rho\sigma}R_{\mu\kappa\alpha\beta}R_{\lambda\rho}{}^{\alpha\beta}A_{\sigma}\,,$
(3.6)

where the covariant R-current is given by

$\langle{\cal J}^{\mu}_{\rm cov}\rangle=\langle{\cal
J}^{\mu}\rangle+P_{BZ}^{\mu}\,,$ (3.7)

with the Bardeen-Zumino term

$P_{BZ}^{\mu}=\frac{2(5a-3c)}{27\pi^{2}}\;\epsilon^{\mu\lambda\rho\sigma}F_{\lambda\rho}A_{\sigma}\,.$
(3.8)

Since the local term (2.16) is absent, the consistent R-current ${\cal
J}^{\mu}$ transforms as tensor under diffeomorphisms and so the relevant BZ
term need only restore gauge invariance.

The BZ term for the energy-momentum tensor ${\cal T}^{\mu\nu}={\cal T}^{\mu
a}e^{\nu}_{a}$ takes the form [21]

$\displaystyle P^{\mu\nu}_{BZ}=$
$\displaystyle\;-\frac{1}{2}\nabla_{\lambda}\big{(}X^{\lambda\mu\nu}+X^{\lambda\nu\mu}-X^{\mu\nu\lambda}\big{)}\,,$
$\displaystyle X^{\mu\lambda}{}_{\nu}=$
$\displaystyle\;-\frac{(c-a)}{12\pi^{2}}\big{(}\epsilon^{\mu\rho\kappa\sigma}R^{\lambda}{}_{\nu\kappa\sigma}+\epsilon^{\lambda\rho\kappa\sigma}R^{\mu}{}_{\nu\kappa\sigma}\big{)}A_{\rho}\,.$
(3.9)

With a bit of algebra this can be simplified to

$P^{\mu\nu}_{BZ}=\frac{(c-a)}{6\pi^{2}}\nabla_{\lambda}\big{(}\epsilon^{\alpha\rho\kappa\sigma}R^{\lambda\beta}{}_{\kappa\sigma}A_{\rho}\big{)}\delta^{\mu}_{(\alpha}\delta^{\nu}_{\beta)}\,.$
(3.10)

Evaluating its covariant divergence we find

$\displaystyle\nabla_{\mu}P_{BZ}^{\mu\nu}=$
$\displaystyle\;\frac{(c-a)}{12\pi^{2}}\nabla_{\mu}\nabla_{\lambda}\big{(}\epsilon^{\mu\rho\kappa\sigma}R^{\lambda\nu}{}_{\kappa\sigma}A_{\rho}+\epsilon^{\nu\rho\kappa\sigma}R^{\lambda\mu}{}_{\kappa\sigma}A_{\rho}\big{)}$
(3.11) $\displaystyle=$
$\displaystyle\;\frac{(c-a)}{12\pi^{2}}\Big{(}\big{(}[\nabla_{\mu},\nabla_{\lambda}]+\nabla_{\lambda}\nabla_{\mu}\big{)}(\epsilon^{\mu\rho\kappa\sigma}R^{\lambda\nu}{}_{\kappa\sigma}A_{\rho})+\frac{1}{2}[\nabla_{\mu},\nabla_{\lambda}](\epsilon^{\nu\rho\kappa\sigma}R^{\lambda\mu}{}_{\kappa\sigma}A_{\rho})\Big{)}$
$\displaystyle=$
$\displaystyle\;\frac{(c-a)}{24\pi^{2}}\Big{(}[\nabla_{\mu},\nabla_{\lambda}](2\epsilon^{\mu\rho\kappa\sigma}R^{\lambda\nu}{}_{\kappa\sigma}A_{\rho}-\epsilon^{\nu\rho\kappa\sigma}R^{\mu\lambda}{}_{\kappa\sigma}A_{\rho})+\nabla_{\lambda}(\epsilon^{\mu\rho\kappa\sigma}R^{\lambda\nu}{}_{\kappa\sigma}F_{\mu\rho})\Big{)}\,,$

where we made use of the second Bianchi identity
$\nabla_{[\mu}R^{\kappa\lambda}{}_{\nu\rho]}=0$.

Moreover, using the first Bianchi identity,
$R^{\kappa}{}_{[\lambda\nu\rho]}=0$, we obtain

$[\nabla_{\mu},\nabla_{\lambda}](2\epsilon^{\mu\rho\kappa\sigma}R^{\lambda\nu}{}_{\kappa\sigma}A_{\rho}-\epsilon^{\nu\rho\kappa\sigma}R^{\mu\lambda}{}_{\kappa\sigma}A_{\rho})=2\epsilon^{\rho\alpha\kappa\sigma}R^{\nu}{}_{\alpha\mu\lambda}R^{\mu\lambda}{}_{\kappa\sigma}A_{\rho}\,,$
(3.12)

and hence

$\nabla_{\mu}P_{BZ}^{\mu\nu}=\frac{(c-a)}{12\pi^{2}}\epsilon^{\rho\alpha\kappa\sigma}R^{\nu}{}_{\alpha\mu\lambda}R^{\mu\lambda}{}_{\kappa\sigma}A_{\rho}+\nabla_{\mu}L^{\mu\nu}\,,$
(3.13)

where

$L^{\mu\nu}=\frac{(c-a)}{12\pi^{2}}F_{\rho\sigma}\widetilde{R}^{\rho\sigma\mu\nu}\,.$
(3.14)

It follows that the bosonic part of (3.2) takes the form

$e^{a}_{\mu}\nabla_{\nu}\langle{\cal T}^{\nu}_{a}\rangle-
F_{\mu\nu}\langle{\cal J}^{\nu}\rangle+A_{\mu}{\cal
A}_{R}=\nabla_{\nu}\langle{\cal T}^{\nu}_{\text{cov}\,\mu}\rangle-
F_{\mu\nu}\langle{\cal J}^{\nu}_{\rm
cov}\rangle-\nabla_{\nu}L^{\nu}{}_{\mu}\,,$ (3.15)

where

$\langle{\cal T}_{\text{cov}}^{\mu\nu}\rangle=\langle{\cal
T}^{\mu\nu}\rangle+P^{\mu\nu}_{BZ}\,.$ (3.16)

Since $L_{\mu\nu}$ is gauge invariant, we have demonstrated that all terms in
(3.2) can be made manifestly gauge invariant in terms of the covariant
R-current and energy-momentum tensor.

The covariant R-current and energy-momentum tensor help simplify not only the
diffeomorphism and R-symmetry Ward identities, but also those for ${\cal Q}$\-
and ${\cal S}$-supersymmetry. In particular, in terms of the covariant
currents, the Ward identities (2) take the form

$\displaystyle e^{a}_{\mu}\nabla_{\nu}\langle{\cal
T}^{\nu}_{\text{cov}\,a}\rangle+\nabla_{\nu}(\overline{\psi}_{\mu}\langle{\cal
S}^{\nu}\rangle)-\overline{\psi}_{\nu}\overleftarrow{{\cal
D}}_{\mu}\langle{\cal S}^{\nu}\rangle-F_{\mu\nu}\langle{\cal
J}_{\text{cov}}^{\nu}\rangle={\cal A}_{D\,\mu}^{\rm cov}\,,$ $\displaystyle
e^{a}_{\mu}\langle{\cal
T}^{\mu}_{\text{cov}\,a}\rangle+\frac{1}{2}\overline{\psi}_{\mu}\langle{\cal
S}^{\mu}\rangle={\cal A}^{\rm cov}_{W}\,,$ $\displaystyle
e_{\mu[a}\langle{\cal
T}^{\mu}_{\text{cov}\,b]}\rangle+\frac{1}{4}\overline{\psi}_{\mu}\gamma_{ab}\langle{\cal
S}^{\mu}\rangle=0\,,$ $\displaystyle\nabla_{\mu}\langle{\cal J}_{\rm
cov}^{\mu}\rangle+i\overline{\psi}_{\mu}\gamma^{5}\langle{\cal
S}^{\mu}\rangle={\cal A}_{R}^{\rm cov}\,,$ $\displaystyle{\cal
D}_{\mu}\langle{\cal
S}^{\mu}\rangle-\frac{1}{2}\gamma^{a}\psi_{\mu}\langle{\cal
T}^{\mu}_{a\,\text{cov}}\rangle-\frac{3i}{4}\gamma^{5}\phi_{\mu}\langle{\cal
J}^{\mu}_{\rm cov}\rangle={\cal A}_{Q}^{\rm cov}\,,$
$\displaystyle\gamma_{\mu}\langle{\cal
S}^{\mu}\rangle-\frac{3i}{4}\gamma^{5}\psi_{\mu}\langle{\cal J}^{\mu}_{\rm
cov}\rangle={\cal A}_{S}^{\rm cov}\,.$ (3.17)

with the covariant anomalies given by the simple expressions

$\displaystyle{\cal A}_{D\,\mu}^{\rm cov}=$
$\displaystyle\;\nabla_{\nu}L^{\nu}{}_{\mu}=\frac{(a-c)}{12\pi^{2}}\nabla^{\nu}(F_{\rho\sigma}\widetilde{R}^{\rho\sigma}{}_{\mu\nu})\,,$
$\displaystyle{\cal A}^{\rm cov}_{W}=$ $\displaystyle\;{\cal
A}_{W}=\frac{c}{16\pi^{2}}\Big{(}W^{2}-\frac{8}{3}F^{2}\Big{)}-\frac{a}{16\pi^{2}}E+{\cal
O}(\psi^{2})\,,$ $\displaystyle{\cal A}_{R}^{\rm cov}=$
$\displaystyle\;\frac{(5a-3c)}{9\pi^{2}}\;\widetilde{F}F+\frac{(c-a)}{24\pi^{2}}{\cal
P}\,,$ $\displaystyle{\cal A}_{Q}^{\rm cov}=$
$\displaystyle\;\frac{(c-a)}{24\pi^{2}}F_{\mu\nu}\widetilde{R}^{\mu\nu\rho\sigma}\gamma_{\rho}\psi_{\sigma}+{\cal
O}(\psi^{3})\,,$ $\displaystyle{\cal A}_{S}^{\rm cov}=$
$\displaystyle\;\Big{[}\frac{(5a-3c)}{6\pi^{2}}\widetilde{F}^{\rho\sigma}+\frac{ic}{6\pi^{2}}F^{\mu\nu}\big{(}\gamma_{\mu}{}^{[\sigma}\delta_{\nu}^{\rho]}-\delta_{\mu}^{[\sigma}\delta_{\nu}^{\rho]}\big{)}\gamma^{5}+\frac{3(2a-c)}{4\pi^{2}}P_{\mu\nu}g^{\mu[\nu}\gamma^{\rho\sigma]}$
$\displaystyle+\frac{(a-c)}{8\pi^{2}}\Big{(}R^{\mu\nu\rho\sigma}\gamma_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}g^{\mu[\nu}\gamma^{\rho\sigma]}\Big{)}\Big{]}{\cal
D}_{\rho}\psi_{\sigma}+{\cal O}(\psi^{3})\,.$ (3.18)

##  4 Conserved charges

The superconformal Ward identities determine the conserved charges and their
algebra. In preparation for the derivation of the supersymmetric Casimir
energy in the subsequent sections, we consider the R-charge and the charges
associated with conformal Killing vectors and spinors on supergravity
backgrounds that admit Killing spinors. A key property of such backgrounds is
that the supercoformal anomalies are numerically zero, resulting in continuous
families of conserved bosonic charges.

Starting with the R-charge, we define the one-parameter family of R-currents
[9]

$\langle{\cal J}^{\mu}_{\omega_{\cal J}}\rangle\equiv\langle{\cal
J}^{\mu}\rangle+\omega_{{\cal J}}P_{BZ}^{\mu}\,,$ (4.1)

where $P_{BZ}^{\mu}$ is the BZ term given in (3.8). Clearly, $\langle{\cal
J}^{\mu}_{0}\rangle$ is the consistent R-current, while $\langle{\cal
J}^{\mu}_{1}\rangle$ is the covariant one. On a bosonic background the
divergence of this current is

$\nabla_{\mu}\langle{\cal J}^{\mu}_{\omega_{\cal J}}\rangle=(1+2\omega_{{\cal
J}})\frac{5a-3c}{27\pi^{2}}\widetilde{F}F+\frac{c-a}{24\pi^{2}}{\cal P}\,,$
(4.2)

which is nonzero on a generic supergravity background. However, for conformal
supergravity backgrounds that admit Killing spinors both terms in the
R-symmetry anomaly are numerically zero, i.e. $\widetilde{F}F=0$ and ${\cal
P}=0$. As a result, on supersymmetric backgrounds there exists a continuous
family of conserved R-charges defined as

$Q_{R}^{\omega_{\cal J}}=\int_{\cal C}d\sigma_{\mu}\langle{\cal
J}^{\mu}_{\omega_{\cal J}}\rangle\,,$ (4.3)

where ${\cal C}$ is a Cauchy surface and $d\sigma_{\mu}$ is the corresponding
infinitesimal area element. Notice that although the anomalies are numerically
zero on supersymmetric backgrounds, the BZ terms are not necessarily
vanishing, and hence the value of the R-charge (4.3) in general depends on the
parameter $\omega_{\cal J}$.

Let us now consider the conserved charges associated with conformal Killing
vectors of supersymmetric backgrounds. A conformal Killing vector ${\cal
K}^{\mu}$ satisfies the relations

$\mathcal{L}_{\cal K}\,g_{\mu\nu}=\nabla_{\mu}{\cal K}_{\nu}+\nabla_{\nu}{\cal
K}_{\mu}=\frac{1}{2}(\nabla_{\rho}{\cal
K}^{\rho})g_{\mu\nu}\,,\qquad\mathcal{L}_{\cal
K}A_{\mu}=\partial_{\mu}\Lambda_{\cal K}\,,$ (4.4)

where ${\cal L}_{\cal K}$ denotes the Lie derivative with respect to ${\cal
K}^{\mu}$ and $\Lambda_{\cal K}$ is an arbitrary R-symmetry gauge parameter.
The Killing condition on the R-symmetry gauge field $A_{\mu}$ is equivalent to
the gauge-invariant condition ${\cal L}_{\cal K}F_{\mu\nu}=0$.

As with the R-current, we define the continuous family of energy-momentum
tensors

$\langle{\cal T}^{\mu\nu}_{\omega_{\cal T}}\rangle\equiv\langle{\cal
T}^{\mu\nu}\rangle+\omega_{{\cal T}}P_{BZ}^{\mu\nu}\,,$ (4.5)

where $P_{BZ}^{\mu\nu}$ is the BZ term (3.10). Notice again that $\langle{\cal
T}^{\mu\nu}_{0}\rangle$ corresponds to the consistent energy-momentum tensor,
while $\langle{\cal T}^{\mu\nu}_{1}\rangle$ is the covariant one. The
divergence of this current on a bosonic background is given by

$\nabla_{\mu}\langle{\cal T}^{\mu}_{\omega_{\cal
T}\,\nu}\rangle=F_{\nu\mu}\langle{\cal J}^{\mu}_{\rm cov}\rangle+\omega_{{\cal
T}}\frac{(a-c)}{12\pi^{2}}\nabla^{\mu}(F_{\rho\sigma}\widetilde{R}^{\rho\sigma}{}_{\nu\mu})\,.$
(4.6)

In order to determine the conserved charge associated with the conformal
Killing vector ${\cal K}^{\mu}$ we evaluate the divergence

$\displaystyle\;\nabla_{\mu}\big{(}(\langle{\cal T}^{\mu}_{\omega_{\cal
T}\,\nu}\rangle+A_{\nu}\langle{\cal J}^{\mu}_{\omega_{\cal J}}\rangle){\cal
K}^{\nu}\big{)}$ $\displaystyle=$ $\displaystyle\;\nabla_{\mu}\langle{\cal
T}^{\mu}_{\omega_{\cal T}\,\nu}\rangle{\cal K}^{\nu}+\frac{1}{4}\langle{\cal
T}^{\mu}_{\omega_{\cal T}\,\mu}\rangle\nabla_{\rho}{\cal
K}^{\rho}+\partial_{\mu}(A_{\nu}{\cal K}^{\nu})\langle{\cal
J}^{\mu}_{\omega_{\cal J}}\rangle+A_{\nu}{\cal
K}^{\nu}\nabla_{\mu}\langle{\cal J}^{\mu}_{\omega_{\cal J}}\rangle$
$\displaystyle=$ $\displaystyle\;{\cal K}^{\nu}F_{\nu\mu}\langle{\cal
J}^{\mu}_{\rm cov}\rangle+(F_{\mu\nu}{\cal
K}^{\nu}+\partial_{\mu}\Lambda_{\cal K})\langle{\cal J}^{\mu}_{\omega_{\cal
J}}\rangle$ $\displaystyle\;+\omega_{{\cal T}}\frac{(a-c)}{12\pi^{2}}{\cal
K}^{\nu}\nabla^{\mu}(F_{\rho\sigma}\widetilde{R}^{\rho\sigma}{}_{\nu\mu})+\frac{1}{4}{\cal
A}_{W}\nabla_{\rho}{\cal K}^{\rho}+A_{\nu}{\cal
K}^{\nu}\nabla_{\mu}\langle{\cal J}^{\mu}_{\omega_{\cal J}}\rangle$
$\displaystyle=$ $\displaystyle\;(\omega_{{\cal J}}-1)F_{\mu\nu}{\cal
K}^{\nu}P_{BZ}^{\mu}+\nabla_{\mu}(\Lambda_{\cal K}\langle{\cal
J}^{\mu}_{\omega_{\cal J}}\rangle)$ $\displaystyle\;+\omega_{{\cal
T}}\frac{(a-c)}{12\pi^{2}}{\cal
K}^{\nu}\nabla^{\mu}(F_{\rho\sigma}\widetilde{R}^{\rho\sigma}{}_{\nu\mu})+\frac{1}{4}{\cal
A}_{W}\nabla_{\rho}{\cal K}^{\rho}+(A_{\nu}{\cal K}^{\nu}-\Lambda_{\cal
K})\nabla_{\mu}\langle{\cal J}^{\mu}_{\omega_{\cal J}}\rangle\,.$ (4.7)

The relation (3.4) implies that

$F_{\mu\nu}{\cal
K}^{\nu}P_{BZ}^{\mu}=\frac{(5a-3c)}{27\pi^{2}}F\widetilde{F}A_{\mu}{\cal
K}^{\mu}\,,$ (4.8)

and so we conclude that

$\displaystyle\;\nabla_{\mu}\big{(}\langle{\cal T}^{\mu}_{\omega_{\cal
T}\,\nu}\rangle{\cal K}^{\nu}+(A_{\nu}{\cal K}^{\nu}-\Lambda_{\cal
K})\langle{\cal J}^{\mu}_{\omega_{\cal J}}\rangle\big{)}=\frac{1}{4}{\cal
A}_{W}\nabla_{\rho}{\cal K}^{\rho}+\omega_{{\cal
T}}\frac{(a-c)}{12\pi^{2}}{\cal
K}^{\nu}\nabla^{\mu}(F_{\rho\sigma}\widetilde{R}^{\rho\sigma}{}_{\nu\mu})$
$\displaystyle\;+(\omega_{{\cal J}}-1)\frac{(5a-3c)}{27\pi^{2}}A_{\nu}{\cal
K}^{\nu}F\widetilde{F}+(A_{\nu}{\cal K}^{\nu}-\Lambda_{\cal
K})\Big{(}(1+2\omega_{{\cal
J}})\frac{5a-3c}{27\pi^{2}}\widetilde{F}F+\frac{c-a}{24\pi^{2}}{\cal
P}\Big{)}\,.$ (4.9)

It follows that if the anomalies $\widetilde{F}F$, ${\cal P}$, ${\cal A}_{W}$
as well as $\nabla_{\mu}L^{\mu\nu}$ vanish numerically, as is the case for
supersymmetric backgrounds, then there exists a two-parameter family of
conserved currents for any conformal Killing vector ${\cal K}^{\mu}$,

$\nabla_{\mu}\big{(}\langle{\cal T}^{\mu}_{\omega_{\cal T}\,\nu}\rangle{\cal
K}^{\nu}+(A_{\nu}{\cal K}^{\nu}-\Lambda_{\cal K})\langle{\cal
J}^{\mu}_{\omega_{\cal J}}\rangle\big{)}=0\,,$ (4.10)

and hence a two-parameter family of conserved charges

$Q^{\omega_{\cal T},\omega_{\cal J}}[{\cal K}]=\int_{\cal
C}d\sigma_{\mu}\big{(}\langle{\cal T}^{\mu}_{\omega_{\cal T}\,\nu}\rangle{\cal
K}^{\nu}+(A_{\nu}{\cal K}^{\nu}-\Lambda_{\cal K})\langle{\cal
J}^{\mu}_{\omega_{\cal J}}\rangle\big{)}\,.$ (4.11)

Finally, we consider the conserved charges associated with (conformal) Killing
spinors of ${\cal N}=1$ conformal supergravity. These are solutions of the
Killing spinor equation obtained by setting the local symmetry transformation
of the gravitino in (2) to zero. On a bosonic background this leads to the
Killing spinor equation

${\cal D}_{\mu}\varepsilon_{0}=\gamma_{\mu}\eta_{0}\,.$ (4.12)

In the following we will also need the Majorana conjugate equation

$\bar{\varepsilon}_{0}\overleftarrow{{\cal
D}}_{\mu}=-\bar{\eta}_{0}\gamma_{\mu}\,.$ (4.13)

Expressing $\eta_{0}$ in terms of $\varepsilon_{0}$ through the algebraic
relation

$\eta_{0}=\frac{1}{4}\gamma^{\nu}{\cal D}_{\nu}\varepsilon_{0}\,,$ (4.14)

the Killing spinor equation can be written in the form

${\cal D}_{\mu}\varepsilon_{0}=\frac{1}{4}\gamma_{\mu}\gamma^{\nu}{\cal
D}_{\nu}\varepsilon_{0}\,.$ (4.15)

On a bosonic background the Ward identities (2) imply that the supercurrent is
covariantly conserved and has zero $\gamma$-trace, i.e.

${\cal D}_{\mu}\langle{\cal S}^{\mu}\rangle=0\,,\qquad\gamma_{\mu}\langle{\cal
S}^{\mu}\rangle=0\,.$ (4.16)

These in turn imply that

$\displaystyle\nabla_{\mu}(\bar{\varepsilon}_{0}\langle{\cal
S}^{\mu}\rangle)=\bar{\varepsilon}_{0}\overleftarrow{{\cal
D}}_{\mu}\langle{\cal S}^{\mu}\rangle=-\bar{\eta}\gamma_{\mu}\langle{\cal
S}^{\mu}\rangle=0\,,$ (4.17)

and, hence, the quantity

$Q[\varepsilon_{0}]=\int_{\cal
C}d\sigma_{\mu}\bar{\varepsilon}_{0}\langle{\cal S}^{\mu}\rangle\,,$ (4.18)

corresponds to the conserved supercharge associated with the (conformal)
Killing spinor $\varepsilon_{0}$. Note that the Killing spinor
$\varepsilon_{0}$ must be commuting in order for the supercharge (4.18) to be
Grassmann-valued.

##  5 Supersymmetric Casimir energy

We are now in a position to obtain the general form of the supersymmetric
Casimir energy on ${\cal N}=1$ conformal supergravity backgrounds,
generalizing the result of [9] to generic SCFTs with $a\neq c$. Our starting
point is the anomalous transformation of the supercurrent ${\cal S}^{\mu}$
under local ${\cal Q}$\- and ${\cal S}$-supersymmetries. These transformations
follow directly from the anomalous superconformal Ward identities (2) and are
given by [19]

$\displaystyle\delta_{\varepsilon}\langle{\cal S}^{\mu}\rangle=$
$\displaystyle\;\frac{1}{2}\gamma^{a}\varepsilon\langle{\cal
T}^{\mu}_{\text{cov}\,a}\rangle+\frac{i}{8}\big{(}4\delta^{[\mu}_{\nu}\delta^{\rho]}_{\sigma}+i\gamma^{5}\epsilon^{\mu}{}_{\nu}{}^{\rho}{}_{\sigma}\big{)}\gamma^{\nu}\gamma^{5}{\cal
D}_{\rho}\big{(}\varepsilon\langle{\cal
J}^{\sigma}_{\text{cov}}\rangle\big{)}+\frac{(a-c)}{24\pi^{2}}F_{\rho\sigma}\widetilde{R}^{\rho\sigma\mu\nu}\gamma_{\nu}\varepsilon\,,$
$\displaystyle\rule{0.0pt}{28.45274pt}\delta_{\eta}\langle{\cal
S}^{\mu}\rangle=$ $\displaystyle\;\frac{3i}{4}\gamma^{5}\eta\langle{\cal
J}^{\mu}_{\text{cov}}\rangle+\frac{(5a-3c)}{6\pi^{2}}{\cal
D}_{\nu}(\widetilde{F}^{\mu\nu}\eta)-\frac{ic}{6\pi^{2}}\big{(}\gamma^{[\mu}{}_{\rho}\delta^{\nu]}_{\sigma}-\delta^{[\mu}_{\rho}\delta^{\nu]}_{\sigma}\big{)}\gamma^{5}{\cal
D}_{\nu}(F^{\rho\sigma}\eta)$ (5.1)
$\displaystyle-\frac{3(2a-c)}{4\pi^{2}}{\cal
D}_{\nu}\big{(}P_{\rho\sigma}g^{\rho[\sigma}\gamma^{\mu\nu]}\eta\big{)}-\frac{(a-c)}{8\pi^{2}}{\cal
D}_{\nu}\Big{[}\Big{(}R^{\mu\nu\rho\sigma}\gamma_{\rho\sigma}-\frac{1}{2}Rg_{\rho\sigma}g^{\rho[\sigma}\gamma^{\mu\nu]}\Big{)}\eta\Big{]}\,.$

###  5.1 Supercharge algebra on curved backgrounds

Taking the transformation parameters $\varepsilon$, $\eta$ to be the
(commuting) components of a conformal Killing spinor, $\varepsilon_{0}$,
$\eta_{0}=\frac{1}{4}\gamma^{\nu}{\cal D}_{\nu}\varepsilon_{0}$, the
transformations (5) compute the algebra of the corresponding supercharges
through the relation

$\langle\\{Q[\varepsilon_{0}],Q[\varepsilon_{0}]\\}\rangle=\int_{\cal
C}d\sigma_{\mu}\bar{\varepsilon}_{0}(\delta_{\varepsilon_{0}}+\delta_{\eta_{0}})\langle{\cal
S}^{\mu}\rangle\,.$ (5.2)

Since the anticommutator of the supercharges vanishes on BPS states, this
relation determines the BPS relation among the bosonic conserved charges on an
arbitrary supersymmetric background, and hence the general form of the
supersymmetric Casimir energy.

Our task, therefore, is to evaluate the r.h.s. of (5.2) using th supercurrent
transformations (5). The term
$\bar{\varepsilon}_{0}\delta_{\varepsilon_{0}}\langle{\cal S}^{\mu}\rangle$
takes the form

$\frac{1}{2}\bar{\varepsilon}_{0}\gamma_{\nu}\varepsilon_{0}\Big{(}\langle{\cal
T}^{\mu\nu}_{\text{cov}}\rangle+\frac{(a-c)}{24\pi^{2}}F_{\rho\sigma}\widetilde{R}^{\rho\sigma\mu\nu}\Big{)}+\frac{i}{8}\bar{\varepsilon}_{0}\big{(}4\delta^{[\mu}_{\nu}\delta^{\rho]}_{\sigma}+i\gamma^{5}\epsilon^{\mu}{}_{\nu}{}^{\rho}{}_{\sigma}\big{)}\gamma^{\nu}\gamma^{5}{\cal
D}_{\rho}\big{(}\varepsilon_{0}\langle{\cal
J}^{\sigma}_{\text{cov}}\rangle\big{)}\,,$ (5.3)

Integrating by parts, the term proportional to the covariant R-current becomes

$\nabla_{\rho}{\cal V}_{{\cal
Q}}^{\mu\rho}-\frac{i}{8}\bar{\varepsilon}_{0}\overleftarrow{{\cal
D}}_{\rho}\big{(}4\delta^{[\mu}_{\nu}\delta^{\rho]}_{\sigma}+i\gamma^{5}\epsilon^{\mu}{}_{\nu}{}^{\rho}{}_{\sigma}\big{)}\gamma^{\nu}\gamma^{5}\varepsilon_{0}\langle{\cal
J}^{\sigma}_{\text{cov}}\rangle\,,$ (5.4)

where

${\cal V}_{{\cal
Q}}^{\mu\rho}=\frac{i}{8}\bar{\varepsilon}_{0}\big{(}4\delta^{[\mu}_{\nu}\delta^{\rho]}_{\sigma}+i\gamma^{5}\epsilon^{\mu}{}_{\nu}{}^{\rho}{}_{\sigma}\big{)}\gamma^{\nu}\gamma^{5}\varepsilon_{0}\langle{\cal
J}^{\sigma}_{\text{cov}}\rangle\,.$ (5.5)

The Killing spinor equation (4.13) and the identity (A.4) in Appendix A, allow
us to simplify the second term in (5.4) to

$-\frac{3i}{4}(\bar{\eta}_{0}\gamma^{5}\varepsilon_{0})\langle{\cal
J}^{\mu}_{\text{cov}}\rangle\,.$ (5.6)

Therefore, we conclude that the term
$\bar{\varepsilon}_{0}\delta_{\varepsilon_{0}}\langle{\cal S}^{\mu}\rangle$
takes the form

$\bar{\varepsilon}_{0}\delta_{\varepsilon_{0}}\langle{\cal
S}^{\mu}\rangle=\frac{1}{2}\bar{\varepsilon}_{0}\gamma_{\nu}\varepsilon_{0}\Big{(}\langle{\cal
T}^{\mu\nu}_{\text{cov}}\rangle+\frac{(a-c)}{24\pi^{2}}F_{\rho\sigma}\widetilde{R}^{\rho\sigma\mu\nu}\Big{)}-\frac{3i}{4}(\bar{\eta}_{0}\gamma^{5}\varepsilon_{0})\langle{\cal
J}^{\mu}_{\text{cov}}\rangle+\nabla_{\rho}{\cal V}_{\cal Q}^{\rho\mu}\,.$
(5.7)

Let us next focus on the term

$\displaystyle\bar{\varepsilon}_{0}\delta_{\eta_{0}}\langle{\cal
S}^{\mu}\rangle=$
$\displaystyle\;\frac{3i}{4}\bar{\varepsilon}_{0}\gamma^{5}\eta_{0}\langle{\cal
J}^{\mu}_{\text{cov}}\rangle+\frac{(5a-3c)}{6\pi^{2}}\bar{\varepsilon}_{0}{\cal
D}_{\nu}(\widetilde{F}^{\mu\nu}\eta_{0})-\frac{ic}{6\pi^{2}}\bar{\varepsilon}_{0}\big{(}\gamma^{[\mu}{}_{\rho}\delta^{\nu]}_{\sigma}-\delta^{[\mu}_{\rho}\delta^{\nu]}_{\sigma}\big{)}\gamma^{5}{\cal
D}_{\nu}(F^{\rho\sigma}\eta_{0})$
$\displaystyle-\frac{3(2a-c)}{4\pi^{2}}\bar{\varepsilon}_{0}{\cal
D}_{\nu}\big{(}P_{\rho\sigma}g^{\rho[\sigma}\gamma^{\mu\nu]}\eta_{0}\big{)}-\frac{(a-c)}{8\pi^{2}}\bar{\varepsilon}_{0}{\cal
D}_{\nu}\Big{[}\Big{(}R^{\mu\nu\rho\sigma}\gamma_{\rho\sigma}-\frac{1}{2}Rg_{\rho\sigma}g^{\rho[\sigma}\gamma^{\mu\nu]}\Big{)}\eta_{0}\Big{]}\,,$
(5.8)

and consider in turn all local terms.

$\frac{(5a-3c)}{6\pi^{2}}\bar{\varepsilon}_{0}{\cal
D}_{\rho}(\widetilde{F}^{\mu\rho}\eta_{0}):$

Integrating by parts and using the Killing spinor equation (4.13) this term
becomes

$\nabla_{\rho}{\cal V}_{{\cal
S}_{1}}^{\mu\rho}+\frac{(5a-3c)}{6\pi^{2}}\widetilde{F}^{\mu\rho}(\bar{\eta}_{0}\gamma_{\rho}\eta_{0})\,,$
(5.9)

where

${\cal V}_{{\cal
S}_{1}}^{\mu\rho}=\frac{(5a-3c)}{6\pi^{2}}\widetilde{F}^{\mu\rho}\bar{\varepsilon}_{0}\eta_{0}\,.$
(5.10)

$-\frac{ic}{6\pi^{2}}\bar{\varepsilon}_{0}(\gamma^{[\mu}{}_{\rho}\delta^{\nu]}_{\sigma}-\delta^{[\mu}_{\rho}\delta^{\nu]}_{\sigma})\gamma^{5}{\cal
D}_{\nu}(F^{\rho\sigma}\eta_{0}):$

Similarly, this term can be written in the form

$\nabla_{\rho}{\cal V}^{\mu\rho}_{{\cal
S}_{2}}-\frac{ic}{6\pi^{2}}\bar{\eta}_{0}\gamma_{\rho}\big{(}\gamma^{[\mu}{}_{\nu}\delta^{\rho]}_{\sigma}-\delta^{[\mu}_{\nu}\delta^{\rho]}_{\sigma}\big{)}\gamma^{5}\eta_{0}\,F^{\nu\sigma}\,,$
(5.11)

with

${\cal V}^{\mu\rho}_{{\cal
S}_{2}}=-\frac{ic}{6\pi^{2}}\bar{\varepsilon}_{0}\big{(}\gamma^{[\mu}{}_{\nu}\delta^{\rho]}_{\sigma}-\delta^{[\mu}_{\nu}\delta^{\rho]}_{\sigma}\big{)}\gamma^{5}\eta_{0}\,F^{\nu\sigma}\,.$
(5.12)

Moreover, the second term in (5.11) gives

$\displaystyle-\frac{ic}{6\pi^{2}}\bar{\eta}_{0}\gamma_{\rho}\big{(}\gamma^{[\mu}{}_{\nu}\delta^{\rho]}_{\sigma}-\delta^{[\mu}_{\nu}\delta^{\rho]}_{\sigma}\big{)}\gamma^{5}\eta_{0}\,F^{\nu\sigma}$
$\displaystyle=-\frac{ic}{6\pi^{2}}\bar{\eta}_{0}(\gamma_{\sigma}\gamma^{\mu}{}_{\nu}-\delta^{\mu}_{\nu}\gamma_{\sigma})\gamma^{5}\eta_{0}F^{\nu\sigma}$
$\displaystyle=-\frac{ic}{6\pi^{2}}\big{(}\bar{\eta}_{0}\gamma^{\sigma}\gamma^{\mu\nu}\gamma^{5}\eta_{0}\,F_{\nu\sigma}-\bar{\eta}_{0}\gamma_{\sigma}\gamma^{5}\eta_{0}F^{\mu\sigma}\big{)}\,.$
(5.13)

The second term vanishes, since
$\bar{\eta}_{0}\gamma_{\sigma}\gamma^{5}\eta_{0}=0$ for commuting spinors,
while

$\bar{\eta}_{0}\gamma^{\sigma}\gamma^{\mu\nu}\gamma^{5}\eta_{0}=\bar{\eta}_{0}\gamma^{\mu\nu}\gamma^{\sigma}\gamma^{5}\eta_{0}=\bar{\eta}_{0}(\gamma^{\mu\nu\sigma}+\gamma^{\mu}g^{\nu\sigma}-\gamma^{\nu}g^{\mu\sigma})\gamma^{5}\eta_{0}=i\epsilon^{\mu\nu\sigma\rho}\bar{\eta}_{0}\gamma_{\rho}\eta_{0}\,.$
(5.14)

Hence,

$-\frac{ic}{6\pi^{2}}\bar{\eta}_{0}\gamma_{\rho}\big{(}\gamma^{[\mu}{}_{\nu}\delta^{\rho]}_{\sigma}-\delta^{[\mu}_{\nu}\delta^{\rho]}_{\sigma}\big{)}\gamma^{5}\eta_{0}\,F^{\nu\sigma}=\frac{c}{6\pi^{2}}\epsilon^{\mu\nu\sigma\rho}F_{\nu\sigma}\bar{\eta}_{0}\gamma_{\rho}\eta_{0}=\frac{c}{3\pi^{2}}\widetilde{F}^{\mu\rho}\,\bar{\eta}_{0}\gamma_{\rho}\eta_{0}\,,$
(5.15)

and therefore (5.11) can be simplified to

$\nabla_{\rho}{\cal V}^{\mu\rho}_{{\cal
S}_{2}}+\frac{2c}{6\pi^{2}}\widetilde{F}^{\mu\rho}\,\bar{\eta}_{0}\gamma_{\rho}\eta_{0}\,.$
(5.16)

$-\frac{3(2a-c)}{4\pi^{2}}\bar{\varepsilon}_{0}{\cal
D}_{\nu}(P_{\rho\sigma}g^{\rho[\sigma}\gamma^{\mu\nu]}\eta_{0}):$

Using integration by parts and the Killing spinor equation (4.13) this term
becomes

$\nabla_{\rho}{\cal V}_{{\cal
S}_{3}}^{\mu\rho}-\frac{3(2a-c)}{4\pi^{2}}P_{\nu\sigma}\,\bar{\eta}_{0}\gamma_{\rho}\,g^{\nu[\sigma}\gamma^{\mu\rho]}\eta_{0}\,,$
(5.17)

where

${\cal V}_{{\cal
S}_{3}}^{\mu\rho}=-\frac{3(2a-c)}{4\pi^{2}}\bar{\varepsilon}_{0}(P_{\nu\sigma}g^{\nu[\sigma}\gamma^{\mu\rho]}\eta_{0})\,.$
(5.18)

The second term in (5.17) can be simplified as

$\displaystyle-\frac{3(2a-c)}{4\pi^{2}}P_{\nu\sigma}\,\bar{\eta}_{0}\gamma_{\rho}\,g^{\nu[\sigma}\gamma^{\mu\rho]}\eta_{0}$
$\displaystyle=-\frac{3(2a-c)}{4\pi^{2}}\frac{2}{3}P_{\nu\sigma}(-g^{\nu\sigma}\bar{\eta}_{0}\gamma^{\mu}\eta_{0}+g^{\mu\nu}\bar{\eta}_{0}\gamma^{\sigma}\eta_{0})$
$\displaystyle=-\frac{(2a-c)}{2\pi^{2}}\left(-\frac{1}{6}R\,\delta^{\mu}_{\sigma}+P^{\mu}_{\,\,\sigma}\right)\bar{\eta}_{0}\gamma^{\sigma}\eta_{0}\,,$
(5.19)

where in the last step we used that $P\equiv P_{\nu\sigma}g^{\nu\sigma}=R/6$.
Observing that

$-\frac{1}{6}R\,\delta^{\mu}_{\sigma}+P^{\mu}_{\,\,\sigma}=\frac{1}{2}\left(R^{\mu}_{\,\,\,\nu}-\frac{1}{2}R\,\delta^{\mu}_{\nu}\right)$
(5.20)

we arrive at the final form of the expression (5.17):

$\nabla_{\rho}{\cal V}_{{\cal
S}_{3}}^{\mu\rho}-\,\frac{(2a-c)}{4\pi^{2}}\left(R^{\mu}_{\,\,\,\nu}-\frac{1}{2}R\,\delta^{\mu}_{\nu}\right)\bar{\eta}_{0}\gamma^{\nu}\eta_{0}\,.$
(5.21)

$-\frac{(a-c)}{8\pi^{2}}\bar{\varepsilon}_{0}\,{\cal
D}_{\nu}\big{[}\big{(}R^{\mu\nu\rho\sigma}\gamma_{\rho\sigma}-\frac{1}{2}Rg_{\rho\sigma}g^{\rho[\sigma}\gamma^{\mu\nu]}\big{)}\eta_{0}\big{]}:$

This term is similarly written as

$\nabla_{\rho}{\cal V}_{{\cal
S}_{4}}^{\mu\rho}-\frac{(a-c)}{8\pi^{2}}\bar{\eta}_{0}\gamma_{\rho}\Big{(}R^{\mu\rho\nu\sigma}\gamma_{\nu\sigma}-\frac{1}{2}Rg_{\nu\sigma}g^{\nu[\sigma}\gamma^{\mu\rho]}\Big{)}\eta_{0}\,,$
(5.22)

with

${\cal V}_{{\cal
S}_{4}}^{\mu\rho}=-\frac{(a-c)}{8\pi^{2}}\bar{\varepsilon}_{0}\Big{(}R^{\mu\rho\nu\sigma}\gamma_{\nu\sigma}-\frac{1}{2}Rg_{\nu\sigma}g^{\nu[\sigma}\gamma^{\mu\rho]}\Big{)}\eta_{0}\,.$
(5.23)

The first term in the square bracket in (5.22) gives

$\displaystyle\bar{\eta}_{0}\gamma_{\rho}\left(R^{\mu\rho\nu\sigma}\gamma_{\nu\sigma}\right)\eta_{0}$
$\displaystyle=-R^{\mu}_{\,\,\,\,\rho\nu\sigma}\,\bar{\eta}_{0}\left(\gamma^{\nu\sigma\rho}+\gamma^{\nu}g^{\sigma\rho}-\gamma^{\sigma}g^{\nu\rho}\right)\eta_{0}$
$\displaystyle=-2R^{\mu}_{\,\,\,\nu}\,\bar{\eta}_{0}\gamma^{\nu}\eta_{0}\,,$
(5.24)

where we used the property
$\bar{\eta}_{0}\gamma_{\rho}\gamma_{\nu\sigma}\eta_{0}=-\bar{\eta}_{0}\gamma_{\nu\sigma}\gamma_{\rho}\eta_{0}$
for commuting spinors. Moreover, the second term in the square bracket in
(5.22) becomes

$\frac{1}{2}R\,g_{\nu\sigma}(\bar{\eta}_{0}\gamma_{\rho}g^{\nu[\sigma}\gamma^{\mu\rho]}\eta_{0})=\frac{1}{2}R\,g_{\nu\sigma}\left(-\frac{2}{3}\,g^{\nu\sigma}\bar{\eta}_{0}\gamma^{\mu}\eta_{0}+\frac{2}{3}\,g^{\mu\nu}\bar{\eta}_{0}\gamma^{\sigma}\eta_{0}\right)=-R\,\bar{\eta}_{0}\gamma^{\mu}\eta_{0}\,,$
(5.25)

where we used the identity (A.11) in Appendix A with $\varepsilon=\eta$.

We therefore conclude that

$-\frac{(a-c)}{8\pi^{2}}\bar{\eta}_{0}\gamma_{\rho}\Big{(}R^{\mu\rho\nu\sigma}\gamma_{\nu\sigma}-\frac{1}{2}Rg_{\nu\sigma}g^{\nu[\sigma}\gamma^{\mu\rho]}\Big{)}\eta_{0}=\frac{(a-c)}{4\pi^{2}}\left(R^{\mu}_{\,\,\,\nu}-\frac{1}{2}R\,\delta^{\mu}_{\nu}\right)\bar{\eta}_{0}\gamma^{\nu}\eta_{0}\,,$
(5.26)

and hence (5.22) reduces to

$\nabla_{\rho}{\cal V}_{{\cal
S}_{4}}^{\mu\rho}+\,\frac{(a-c)}{4\pi^{2}}\left(R^{\mu}_{\,\,\,\nu}-\frac{1}{2}R\,\delta^{\mu}_{\nu}\right)\bar{\eta}_{0}\gamma^{\nu}\eta_{0}\,.$
(5.27)

Returning to the overall variation (5.1), we observe that (5.9) and (5.16)
combine to

$\frac{(5a-c)}{6\pi^{2}}\widetilde{F}^{\mu\rho}(\bar{\eta}_{0}\gamma_{\rho}\eta_{0})\,,$
(5.28)

while (5.21) and (5.27) reduce to

$-\frac{a}{4\pi^{2}}\Big{(}R^{\mu}_{\,\,\,\nu}-\frac{1}{2}R\,\delta^{\mu}_{\nu}\Big{)}\bar{\eta}_{0}\gamma^{\nu}\eta_{0}\,.$
(5.29)

Therefore, gathering all terms, the variation (5.1) is written as

$\bar{\varepsilon}_{0}\delta_{\eta_{0}}\langle{\cal
S}^{\mu}\rangle=\frac{3i}{4}\bar{\varepsilon}_{0}\gamma^{5}\eta_{0}\langle{\cal
J}^{\mu}_{\text{cov}}\rangle+\Big{[}\frac{(5a-c)}{6\pi^{2}}\widetilde{F}^{\mu}{}_{\nu}-\frac{a}{4\pi^{2}}\Big{(}R^{\mu}_{\,\,\,\nu}-\frac{1}{2}R\,\delta^{\mu}_{\nu}\Big{)}\Big{]}\bar{\eta}_{0}\gamma^{\nu}\eta_{0}+\nabla_{\rho}\mathcal{V}_{\cal
S}^{\mu\rho}\,,$ (5.30)

with

$\displaystyle{\cal V}_{\cal S}^{\mu\rho}=$
$\displaystyle\;\frac{(5a-3c)}{6\pi^{2}}\bar{\varepsilon}_{0}(\widetilde{F}^{\mu\rho}\eta_{0})+\frac{ic}{6\pi^{2}}\varepsilon_{0}\big{(}\gamma^{[\mu}{}_{\nu}\delta^{\rho]}_{\sigma}-\delta^{[\mu}_{\nu}\delta^{\rho]}_{\sigma}\big{)}\gamma^{5}(F^{\nu\sigma}\eta_{0})$
$\displaystyle\;-\frac{3(2a-c)}{4\pi^{2}}\bar{\varepsilon}_{0}\big{(}P_{\nu\sigma}g^{\nu[\sigma}\gamma^{\mu\rho]}\eta\big{)}-\frac{(a-c)}{8\pi^{2}}\varepsilon_{0}\Big{(}R^{\mu\rho\nu\sigma}\gamma_{\nu\sigma}-\frac{1}{2}Rg_{\nu\sigma}g^{\nu[\sigma}\gamma^{\mu\rho]}\Big{)}\eta_{0}\,.$
(5.31)

Combining (5.7) and (5.30), and using the commuting spinor identity
$\bar{\eta}_{0}\gamma^{5}\varepsilon_{0}=-\bar{\varepsilon}_{0}\gamma^{5}\eta_{0}$,
we arrive at the result

$\displaystyle\bar{\varepsilon}_{0}(\delta_{\varepsilon_{0}}+\delta_{\eta_{0}})\langle{\cal
S}^{\mu}\rangle=$
$\displaystyle\;\frac{1}{2}\bar{\varepsilon}_{0}\gamma_{\nu}\varepsilon_{0}\Big{(}\langle{\cal
T}^{\mu\nu}_{\text{cov}}\rangle+\frac{(a-c)}{24\pi^{2}}F_{\rho\sigma}\widetilde{R}^{\rho\sigma\mu\nu}\Big{)}+\frac{3i}{2}(\bar{\varepsilon}_{0}\gamma^{5}\eta_{0})\langle{\cal
J}^{\mu}_{\text{cov}}\rangle$ (5.32)
$\displaystyle\;+\Big{[}\frac{(5a-c)}{6\pi^{2}}\widetilde{F}^{\mu}{}_{\nu}-\frac{a}{4\pi^{2}}\Big{(}R^{\mu}_{\,\,\,\nu}-\frac{1}{2}R\,\delta^{\mu}_{\nu}\Big{)}\Big{]}\bar{\eta}_{0}\gamma^{\nu}\eta_{0}+\nabla_{\rho}({\cal
V}_{\cal Q}^{\rho\mu}+{\cal V}_{\cal S}^{\mu\rho})\,.$

Inserting this in (5.2) and dropping the total derivative terms we conclude
that BPS states satisfy the integral constraint

$\displaystyle 0=$ $\displaystyle\;\int_{\cal
C}d\sigma_{\mu}\bar{\varepsilon}_{0}(\delta_{\varepsilon_{0}}+\delta_{\eta_{0}})\langle{\cal
S}^{\mu}\rangle$ $\displaystyle=$ $\displaystyle\;\int_{\cal
C}d\sigma_{\mu}\Big{\\{}\frac{1}{2}\bar{\varepsilon}_{0}\gamma_{\nu}\varepsilon_{0}\Big{(}\langle{\cal
T}^{\mu\nu}_{\text{cov}}\rangle+\frac{(a-c)}{24\pi^{2}}F_{\rho\sigma}\widetilde{R}^{\rho\sigma\mu\nu}\Big{)}+\frac{3i}{2}(\bar{\varepsilon}_{0}\gamma^{5}\eta_{0})\langle{\cal
J}^{\mu}_{\text{cov}}\rangle$ $\displaystyle\;\hskip
128.0374pt+\Big{[}\frac{(5a-c)}{6\pi^{2}}\widetilde{F}^{\mu}{}_{\nu}-\frac{a}{4\pi^{2}}\Big{(}R^{\mu}_{\,\,\,\nu}-\frac{1}{2}R\,\delta^{\mu}_{\nu}\Big{)}\Big{]}\bar{\eta}_{0}\gamma^{\nu}\eta_{0}\Big{\\}}\,.$
(5.33)

We will now show that this constraint corresponds to the BPS relation among
the bosonic charges of supersymmetric states and determines the supersymmetric
Casimir energy.

###  5.2 BPS relation and the supersymmetric Casimir energy

In order to relate the constraint (5.1) to the conserved R-charge and
conformal Killing charges, respectively (4.3) and (4.11), we begin by showing
that the spinor bilinear $\bar{\varepsilon}_{0}\gamma^{\mu}\varepsilon_{0}$ is
a conformal Killing vector. In fact, for later use we show more generally that
if $\varepsilon_{0}$ and $\varepsilon_{0}^{\prime}$ are conformal Killing
spinors, then the spinor bilinear

${\cal
K}^{\mu}(\varepsilon_{0},\varepsilon_{0}^{\prime})\equiv\bar{\varepsilon}_{0}^{\prime}\gamma^{\mu}\varepsilon_{0}\,,$
(5.34)

is a conformal Killing vector.

We have,

$\displaystyle\nabla_{\mu}{\cal
K}_{\nu}(\varepsilon_{0},\varepsilon_{0}^{\prime})=$
$\displaystyle\;\bar{\varepsilon}_{0}^{\prime}\overleftarrow{{\cal
D}}_{\mu}\gamma_{\nu}\varepsilon_{0}+\bar{\varepsilon}_{0}^{\prime}\gamma_{\nu}{\cal
D}_{\mu}\varepsilon_{0}$ $\displaystyle=$
$\displaystyle\;-\bar{\eta}_{0}^{\prime}\gamma_{\mu}\gamma_{\nu}\varepsilon_{0}+\bar{\varepsilon}_{0}^{\prime}\gamma_{\nu}\gamma_{\mu}\eta_{0}$
$\displaystyle=$
$\displaystyle\;\bar{\varepsilon}_{0}\gamma_{\nu}\gamma_{\mu}\eta_{0}^{\prime}+\bar{\varepsilon}_{0}^{\prime}\gamma_{\nu}\gamma_{\mu}\eta_{0}\,,$
(5.35)

where we used the Killing spinor equations (4.12) and (4.13) and the fact that
for commuting spinors
$\bar{\eta}_{0}^{\prime}\gamma_{\mu}\gamma_{\nu}\varepsilon_{0}=-\bar{\varepsilon}_{0}\gamma_{\nu}\gamma_{\mu}\eta_{0}^{\prime}$.
Hence,

$\nabla_{\mu}{\cal
K}_{\nu}(\varepsilon_{0},\varepsilon_{0}^{\prime})+\nabla_{\nu}{\cal
K}_{\mu}(\varepsilon_{0},\varepsilon_{0}^{\prime})=g_{\mu\nu}(\bar{\varepsilon}_{0}\eta_{0}^{\prime}+\bar{\varepsilon}_{0}^{\prime}\eta_{0})=\frac{1}{2}g_{\mu\nu}\nabla_{\rho}{\cal
K}^{\rho}(\varepsilon_{0},\varepsilon_{0}^{\prime})\,,$ (5.36)

which confirms that ${\cal K}^{\mu}(\varepsilon_{0},\varepsilon_{0}^{\prime})$
is a conformal Killing vector.

It follows that

${\cal K}_{0}^{\mu}\equiv\bar{\varepsilon}_{0}\gamma^{\mu}\varepsilon_{0}\,,$
(5.37)

is a conformal Killing vector and so (5.1) can be written in the form

$\displaystyle 0=$ $\displaystyle\;Q_{\omega_{\cal T},\omega_{\cal J}}[{\cal
K}_{0}]+\int_{\cal
C}d\sigma_{\mu}\Big{\\{}\big{(}3i(\bar{\varepsilon}_{0}\gamma^{5}\eta_{0})-A_{\nu}{\cal
K}_{0}^{\nu}+\Lambda_{{\cal K}_{0}}\big{)}\langle{\cal J}^{\mu}_{\omega_{\cal
J}}\rangle$ $\displaystyle+\Big{(}(1-\omega_{\cal
T})P_{BZ}^{\mu\nu}+\frac{(a-c)}{24\pi^{2}}F_{\rho\sigma}\widetilde{R}^{\rho\sigma\mu\nu}\Big{)}{\cal
K}_{0\nu}+3i(\bar{\varepsilon}_{0}\gamma^{5}\eta_{0})(1-\omega_{\cal
J})P_{BZ}^{\mu}$
$\displaystyle\;+\Big{[}\frac{(5a-c)}{3\pi^{2}}\widetilde{F}^{\mu}{}_{\nu}-\frac{a}{2\pi^{2}}\Big{(}R^{\mu}_{\,\,\,\nu}-\frac{1}{2}R\,\delta^{\mu}_{\nu}\Big{)}\Big{]}\bar{\eta}_{0}\gamma^{\nu}\eta_{0}\Big{\\}}\,.$
(5.38)

However, the coefficient multiplying the R-current $\langle{\cal
J}^{\mu}_{\omega_{\cal J}}\rangle$ is a constant, since

$\partial_{\mu}\big{(}3i(\bar{\varepsilon}_{0}\gamma^{5}\eta_{0})-A_{\nu}{\cal
K}_{0}^{\nu}+\Lambda_{{\cal
K}_{0}}\big{)}=3i\partial_{\mu}(\bar{\varepsilon}_{0}\gamma^{5}\eta_{0})-F_{\mu\nu}{\cal
K}_{0}^{\nu}\,,$ (5.39)

and

$\displaystyle 3i\partial_{\mu}(\bar{\varepsilon}_{0}\gamma^{5}\eta_{0})=$
$\displaystyle\;3i\bar{\varepsilon}_{0}\overleftarrow{{\cal
D}}_{\mu}\gamma^{5}\eta_{0}+3i\bar{\varepsilon}_{0}\gamma^{5}{\cal
D}_{\mu}\eta_{0}$ $\displaystyle=$
$\displaystyle\;-3i\bar{\eta}_{0}\gamma_{\mu}\gamma^{5}\eta_{0}-\frac{3i}{2}\bar{\varepsilon}_{0}\gamma^{5}\Big{(}P_{\mu\nu}+\frac{2i}{3}F_{\mu\nu}\gamma^{5}-\frac{1}{3}\widetilde{F}_{\mu\nu}\Big{)}\gamma^{\nu}\varepsilon_{0}$
$\displaystyle=$ $\displaystyle\;F_{\mu\nu}{\cal K}_{0}^{\nu}\,,$ (5.40)

where we used the fact that
$\bar{\eta}_{0}\gamma_{\mu}\gamma^{5}\eta_{0}=\bar{\varepsilon}_{0}\gamma_{\mu}\gamma^{5}\varepsilon_{0}=0$
for commuting spinors and [19]

${\cal
D}_{\mu}\eta_{0}=-\frac{1}{2}\Big{(}P_{\mu\nu}+\frac{2i}{3}F_{\mu\nu}\gamma^{5}-\frac{1}{3}\widetilde{F}_{\mu\nu}\Big{)}\gamma^{\nu}\varepsilon_{0}\,.$
(5.41)

$\Phi_{{\cal K}_{0}}\equiv
3i(\bar{\varepsilon}_{0}\gamma^{5}\eta_{0})-A_{\nu}{\cal
K}_{0}^{\nu}+\Lambda_{{\cal K}_{0}}=\text{const.}\,.$ (5.42)

We conclude that (5.1) can be further simplified to

$Q^{\omega_{\cal T},\omega_{\cal J}}[{\cal K}_{0}]+\Phi_{{\cal
K}_{0}}Q_{R}^{\omega_{\cal J}}+Q_{\rm local}^{\omega_{\cal T},\omega_{\cal
J}}[{\cal K}_{0}]=0\,,$ (5.43)

where

$\displaystyle Q_{\rm local}^{\omega_{\cal T},\omega_{\cal J}}[{\cal
K}_{0}]\equiv$ $\displaystyle\;\int_{\cal
C}d\sigma_{\mu}\Big{\\{}\Big{(}(1-\omega_{\cal
T})P_{BZ}^{\mu\nu}+\frac{(a-c)}{24\pi^{2}}F_{\rho\sigma}\widetilde{R}^{\rho\sigma\mu\nu}\Big{)}{\cal
K}_{0\nu}$ (5.44) $\displaystyle\;\hskip-56.9055pt+(1-\omega_{\cal
J})(\Phi_{{\cal K}_{0}}+A_{\nu}{\cal K}_{0}^{\nu}-\Lambda_{{\cal
K}_{0}})P_{BZ}^{\mu}+\Big{[}\frac{(5a-c)}{3\pi^{2}}\widetilde{F}^{\mu}{}_{\nu}-\frac{a}{2\pi^{2}}\Big{(}R^{\mu}_{\,\,\,\nu}-\frac{1}{2}R\,\delta^{\mu}_{\nu}\Big{)}\Big{]}\bar{\eta}_{0}\gamma^{\nu}\eta_{0}\Big{\\}}\,,$

is a local charge that depends only on the supergravity background. As we
demonstrate in the next section, when applied to the global timelike Killing
vector, the relation (5.43) determines the energy of a BPS state in terms of
the conserved R-charge, with (5.44) corresponding to the supersymmetric
Casimir energy.

##  6 Casimir energy on $\mathbb{R}\times S^{3}$

The BPS relation (5.43) and the local charge (5.44) hold for any
supersymmetric background of ${\cal N}=1$ conformal supergravity with
numerically vanishing superonformal anomalies. Such backgrounds have been
studied extensively [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34] (see also
[35, 36] for earlier work). In this section we will focus on a concrete
background for which the supersymmetric Casimir energy has been computed in
the literature by other means. In particular, we will apply our general result
to an example in the class of backgrounds with topology $\mathbb{R}\times
S^{3}$ (or $S^{1}\times S^{3}$ in Euclidean signature) that admit two
supercharges with opposite R-charge.

###  6.1 Supersymmetric backgrounds with $\mathbb{R}\times S^{3}$ topology

Following [5], we consider four-dimensional backgrounds of topology
$\mathbb{R}\times S^{3}$ that admit a non-singular complete direct product
metric of the form333The corresponding Euclidean backgrounds are obtained by
setting $t=i\tau$, see e.g. [32, 37].

$ds^{2}=-\Omega^{2}(\rho)dt^{2}+f^{2}(\rho)d\rho^{2}+m_{IJ}(\rho)\,d\varphi_{I}\,d\varphi_{J},\quad
I,J=1,2\,,$ (6.1)

where $\varphi_{I}\in[0,2\pi]$, $\rho\in[0,1]$, while the functions
$\Omega(\rho)$, $f(\rho)$ are positive definite, as is the symmetric matrix
$m_{IJ}(\rho)$. This metric possesses an $\mathbb{R}\times U(1)^{2}$ isometry
($U(1)^{3}$ in its Euclidean form) corresponding to the commuting Killing
vectors $\partial_{t}$, $\partial_{\varphi_{1}}$ and $\partial_{\varphi_{2}}$.

Demanding that the background admits two supercharges with opposite R-charge
requires the existence of a globally defined null Killing vector of the form

${\cal
K}=\frac{1}{2}(\partial_{t}+b_{I}\partial_{\varphi_{I}})=\frac{1}{2}(\partial_{t}+b_{1}\partial_{\varphi_{1}}+b_{2}\partial_{\varphi_{2}})\,,$
(6.2)

where $b_{1}$, $b_{2}$ are real parameters. The requirement that this is null
fixes

$\Omega^{2}=b^{I}m_{IJ}b^{J}\,.$ (6.3)

However, note that the metric on $S^{3}$, which is parameterized by $f(\rho)$
and $m_{IJ}(\rho)$, is not constrained by supersymmetry.

Finally, the globally well defined R-symmetry gauge field takes the form
[5]444This differs by an overall minus sign compared to [5] due to different
conventions.

$A=-\frac{\Omega}{8fc}\partial_{\rho}\Big{[}(c^{2}+a_{\chi}^{2})\Big{(}\frac{d\varphi_{1}}{b_{1}}-\frac{d\varphi_{2}}{b_{2}}\Big{)}+2a_{\chi}\Big{(}\frac{d\varphi_{1}}{b_{1}}+\frac{d\varphi_{2}}{b_{2}}+dt\Big{)}\Big{]}-\frac{1}{2}d\omega\,,$
(6.4)

where

$a_{\chi}=\frac{1}{\Omega^{2}}(b_{1}^{2}m_{11}-b_{2}^{2}m_{22})\,,\quad
c=\frac{2|b_{1}b_{2}|}{\Omega^{2}}\sqrt{\det(m_{IJ})}\,,\quad\omega={\rm
sgn}(b_{1})\varphi_{1}+{\rm sgn}(b_{2})\varphi_{2}\,.$ (6.5)

These expressions describe a family of non-singular backgrounds that admit two
supercharges of opposite R-charge. They are parameterized by the arbitrary
non-singular metric on $S^{3}$ and the real parameters $b_{1}$, $b_{2}$. The
squashed (Berger) three-sphere is a special case of these backgrounds [5].
However, for simplicity we will illustrate our results by considering the
background with arbitrary $b_{1}$, $b_{2}$ and the round metric on $S^{3}$.

###  6.2 Round $S^{3}$ with arbitrary $b_{1},b_{2}$

Defining the angular coordinate $\theta=\pi\rho$, the metric (6.1)
corresponding to the round $S^{3}$ is

$ds^{2}=-\Omega^{2}dt^{2}+d\theta^{2}+m_{IJ}d\varphi_{I}d\varphi_{J}\,,$ (6.6)

with

$m_{11}=4\cos^{2}\frac{\theta}{2}\,,\qquad
m_{22}=4\sin^{2}\frac{\theta}{2}\,,\qquad m_{12}=0\,,\qquad f=\pi\,.$ (6.7)

These expressions for $m_{IJ}$ completely determine the background. In
particular,

$\Omega^{2}=b^{I}m_{IJ}b^{J}=2\big{(}b_{1}^{2}+b_{2}^{2}+(b_{1}^{2}-b_{2}^{2})\cos\theta\big{)}\,,$
(6.8)
$a_{\chi}=\frac{1}{\Omega^{2}}(b_{1}^{2}m_{11}-b_{2}^{2}m_{22})=\frac{b_{1}^{2}-b_{2}^{2}+(b_{1}^{2}+b_{2}^{2})\cos\theta}{b_{1}^{2}+b_{2}^{2}+(b_{1}^{2}-b_{2}^{2})\cos\theta}\,,$
(6.9)

and

$c=\frac{2|b_{1}b_{2}|}{\Omega^{2}}\sqrt{\det(m_{IJ})}=\frac{2|b_{1}b_{2}|\sin\theta}{b_{1}^{2}+b_{2}^{2}+(b_{1}^{2}-b_{2}^{2})\cos\theta}\,.$
(6.10)

Inserting these in (6.4) we obtain the R-symmetry gauge field

$A=\frac{{\rm
sgn}(b_{1}b_{2})}{\Omega}(b_{2}d\varphi_{1}+b_{1}d\varphi_{2})+\frac{|b_{1}b_{2}|}{\Omega}\,dt-\frac{1}{2}d\omega\,.$
(6.11)

### Killing spinors

The Killing spinors are solutions of the conformal Killing equation (4.15). In
order to solve this equation we introduce a suitable local frame for the
background metric (6.6), namely

$\displaystyle e^{0}=$ $\displaystyle\;\Omega dt\,,$ $\displaystyle e^{1}=$
$\displaystyle\;\frac{\Omega}{2b_{1}b_{2}}\big{(}(1+a_{\chi})b_{2}d\varphi_{1}+(1-a_{\chi})b_{1}d\varphi_{2}\big{)}\,,$
$\displaystyle e^{2}=$ $\displaystyle\;d\theta\,,$ $\displaystyle e^{3}=$
$\displaystyle\;\frac{\Omega}{2b_{1}b_{2}}(b_{2}d\varphi_{1}-b_{1}d\varphi_{2})\,,$
(6.12)

so that

$ds^{2}=-(e^{0})^{2}+(e^{1})^{2}+(e^{2})^{2}+(e^{3})^{2}\,.$ (6.13)

We also use the Weyl representation of the gamma matrices so that
$\gamma^{\mu}=e^{\mu}_{a}\gamma^{a}$ with

$\gamma^{a}=\left(\begin{array}[]{cc}0&\sigma^{a}\\\ \bar{\sigma}^{a}&0\\\
\end{array}\right),\qquad\sigma_{a}=(-\mathbb{1},\sigma_{i}),\quad\bar{\sigma}_{a}=(\mathbb{1},\sigma_{i})\,,$
(6.14)

and

$\gamma^{ab}=\left(\begin{array}[]{cc}\sigma^{ab}&0\\\ 0&\bar{\sigma}^{ab}\\\
\end{array}\right),\qquad\sigma_{ab}=\frac{1}{2}[\sigma_{a},\bar{\sigma}_{b}],\quad\bar{\sigma}_{ab}=\frac{1}{2}[\bar{\sigma}_{a},\sigma_{b}]\,.$
(6.15)

Moreover, the chirality matrix takes the form

$\gamma^{5}=\left(\begin{array}[]{cc}\mathbb{1}&0\\\ 0&-\mathbb{1}\\\
\end{array}\right)\,.$ (6.16)

Th commuting Majorana spinors $\varepsilon_{0}$ and $\eta_{0}$ can then be
expressed as

$\varepsilon_{0}=\left(\begin{array}[]{c}\epsilon\\\ \tilde{\epsilon}\\\
\end{array}\right)\,,\qquad\eta_{0}=\left(\begin{array}[]{c}\zeta\\\
\tilde{\zeta}\\\ \end{array}\right)\,,$ (6.17)

where $\epsilon$, $\zeta$ are left-handed two-component Weyl spinors and
$\tilde{\epsilon}\equiv i\sigma_{2}\epsilon^{*}$, $\tilde{\zeta}\equiv
i\sigma_{2}\zeta^{*}$. Moreover, the Killing spinor equation (4.15) is
equivalent to the two-component spinor equations

${\cal
D}^{L}_{\mu}\epsilon=\sigma_{\mu}\tilde{\zeta}\,,\quad\tilde{\zeta}=\frac{1}{4}\bar{\sigma}^{\mu}\,{\cal
D}^{L}_{\mu}\epsilon\,,\qquad{\cal
D}^{R}_{\mu}\tilde{\epsilon}=\bar{\sigma}_{\mu}\zeta\,\,,\quad\zeta=\frac{1}{4}\sigma^{\mu}\,{\cal
D}^{R}_{\mu}\epsilon\,,$ (6.18)

where the chiral derivatives are ${\cal
D}^{L}_{\mu}=\partial_{\mu}+\frac{1}{4}\omega^{ab}_{\mu}\sigma_{ab}+iA_{\mu}$
and ${\cal
D}^{R}_{\mu}=\partial_{\mu}+\frac{1}{4}\omega^{ab}_{\mu}\bar{\sigma}_{ab}-iA_{\mu}$.

For generic $b_{1}$ and $b_{2}$ the Killing spinor equation (4.15) admits the
unique solution

$\varepsilon_{0}=e^{\frac{i}{2}\omega}\sqrt{\frac{\Omega}{2}}\left(\begin{matrix}-1\\\
1\\\ 0\\\
0\end{matrix}\right)+e^{-\frac{i}{2}\omega}\sqrt{\frac{\Omega}{2}}\left(\begin{matrix}0\\\
0\\\ 1\\\ 1\end{matrix}\right)\,,$ (6.19)

while $\eta_{0}=\frac{1}{4}\gamma^{\nu}{\cal D}_{\nu}\varepsilon_{0}$ takes
the form

$\eta_{0}=-\frac{i}{(2\Omega)^{3/2}}e^{\frac{i}{2}\omega}\left(\begin{matrix}0\\\
0\\\ -2|b_{1}b_{2}|+\Omega\Omega^{\prime}\\\
2|b_{1}b_{2}|+\Omega\Omega^{\prime}\end{matrix}\right)-\frac{i}{(2\Omega)^{3/2}}e^{-\frac{i}{2}\omega}\left(\begin{matrix}2|b_{1}b_{2}|+\Omega\Omega^{\prime}\\\
2|b_{1}b_{2}|-\Omega\Omega^{\prime}\\\ 0\\\ 0\end{matrix}\right)\,.$ (6.20)

Having determined the conformal Killing spinor, we can evaluate the three
spinor bilinears that enter in the BPS relation (5.43) and the local charge
(5.44). We find

$\displaystyle\bar{\varepsilon}_{0}\gamma^{\mu}\varepsilon_{0}=-4{\cal
K}^{\mu}\,,$ $\displaystyle
3i\bar{\varepsilon}_{0}\gamma^{\mu}\eta_{0}-A_{\mu}\bar{\varepsilon}_{0}\gamma^{\mu}\varepsilon_{0}=-(|b_{1}|+|b_{2}|)\,,$
$\displaystyle\bar{\eta}_{0}\gamma^{\mu}\eta_{0}=\frac{1}{4}{\cal
K}^{\mu}-\frac{1}{2\Omega^{2}}\big{(}b_{1}^{2}+b_{2}^{2},0,(b_{1}^{2}-b_{2}^{2})b_{1},-(b_{1}^{2}-b_{2}^{2})b_{2}\big{)}\,,$
(6.21)

where ${\cal K}$ is the globally defined null Killing vector (6.2). These
expressions reaffirm the results we obtained earlier. In particular,
$\bar{\varepsilon}_{0}\gamma^{\mu}\varepsilon_{0}$ is a Killing vector, as it
should, while the R-charge potential (5.42) is indeed constant (note that
$\Lambda_{{\cal K}_{0}}=0$ in this case since ${\cal L}_{{\cal
K}_{0}}A_{\mu}=0$).

### Casimir energy

We now have all ingredients in order to evaluate the local charge (5.44) on
this supersymmetric background. Since
$\bar{\varepsilon}_{0}\gamma^{\mu}\varepsilon_{0}=-4{\cal
K}^{\mu}\sim-2\partial_{t}$, the Casimir energy corresponds to

${\cal E}_{\rm Casimir}^{\omega_{\cal T},\omega_{\cal
J}}\equiv-\frac{1}{2}Q_{\rm local}^{\omega_{\cal T},\omega_{\cal J}}[{\cal
K}_{0}]=Q_{\rm local}^{\omega_{\cal T},\omega_{\cal J}}[2{\cal K}]\,,$ (6.22)

and, therefore, it is given by

$\displaystyle{\cal E}_{\rm Casimir}^{\omega_{\cal T},\omega_{\cal J}}\equiv$
$\displaystyle\;\int_{\cal C}d\sigma_{\mu}\Big{\\{}\big{(}2(1-\omega_{\cal
T})P_{BZ}^{\mu\nu}-L^{\mu\nu}\big{)}{\cal K}_{\nu}-(1-\omega_{\cal
J})\frac{3i}{2}(\bar{\varepsilon}_{0}\gamma^{5}\eta_{0})P_{BZ}^{\mu}$
$\displaystyle\;\hskip
85.35826pt+\Big{[}\frac{a}{4\pi^{2}}\Big{(}R^{\mu}_{\,\,\,\nu}-\frac{1}{2}R\,\delta^{\mu}_{\nu}\Big{)}-\frac{(5a-c)}{6\pi^{2}}\widetilde{F}^{\mu}{}_{\nu}\Big{]}\bar{\eta}_{0}\gamma^{\nu}\eta_{0}\Big{\\}}\,,$
(6.23)

where ${\cal K}^{\mu}$ is the globally defined Killing vector in (6.2),
$L^{\mu\nu}$ is given in (3.14), while the BZ terms are given in (3.8) and
(3.10).

Evaluating the R-current BZ term on the background specified by the metric
(6.6) and R-symmetry gauge field (6.11) we find

$P_{BZ}^{\mu}=\frac{2(5a-3c)}{27\pi^{2}}\frac{(b_{1}^{2}-b_{2}^{2})}{2\Omega^{4}}\big{(}-{\rm
sgn}(b_{1}b_{2})(|b_{1}|-|b_{2}|),0,-|b_{1}|b_{2},|b_{2}|b_{1}\big{)}\,.$
(6.24)

Similarly, the only nonzero components of the stress tensor BZ term (3.10)
become

$\displaystyle P^{t\varphi_{1}}_{BZ}=$
$\displaystyle\;\frac{(c-a)}{3\pi^{2}}\frac{(b_{1}^{2}-b_{2}^{2})}{\Omega^{8}}(2\Omega-5|b_{1}|)\,b_{1}^{2}b_{2}^{2}\,{\rm
sgn}(b_{2})\,,$ $\displaystyle P^{t\varphi_{2}}_{BZ}=$
$\displaystyle\;-\frac{(c-a)}{3\pi^{2}}\frac{(b_{1}^{2}-b_{2}^{2})}{\Omega^{8}}(2\Omega-5|b_{2}|)\,b_{1}^{2}b_{2}^{2}\,{\rm
sgn}(b_{1})\,.$ (6.25)

Moreover, the nonzero components of the tensors $L^{\mu\nu}$ and
$\widetilde{F}_{\mu\nu}$ are respectively

$\displaystyle L^{t\varphi_{1}}=$
$\displaystyle\;\frac{(a-c)}{24\pi^{2}}\frac{(b_{1}^{2}-b_{2}^{2})^{2}}{\Omega^{6}}|b_{1}|{\rm
sgn}(b_{2})(1-\cos\theta)\,,$ $\displaystyle L^{t\varphi_{2}}=$
$\displaystyle\;\frac{(a-c)}{24\pi^{2}}\frac{(b_{1}^{2}-b_{2}^{2})^{2}}{\Omega^{6}}|b_{2}|{\rm
sgn}(b_{1})(1+\cos\theta)\,,$ $\displaystyle L^{\varphi_{1}\varphi_{2}}=$
$\displaystyle\;-\frac{(a-c)}{48\pi^{2}}\frac{(b_{1}^{2}-b_{2}^{2})}{\Omega^{4}}|b_{1}b_{2}|\,,$
(6.26)

and

$\displaystyle\widetilde{F}_{t\varphi_{1}}=$
$\displaystyle\;-\frac{(b_{1}^{2}-b_{2}^{2})}{\Omega^{2}}|b_{1}|{\rm
sgn}(b_{2})(1+\cos\theta)\,,$ $\displaystyle\widetilde{F}_{t\varphi_{2}}=$
$\displaystyle\;\frac{(b_{1}^{2}-b_{2}^{2})}{\Omega^{2}}|b_{2}|{\rm
sgn}(b_{1})(1-\cos\theta)\,,$
$\displaystyle\widetilde{F}_{\varphi_{1}\varphi_{2}}=$
$\displaystyle\;\frac{2(b_{1}^{2}-b_{2}^{2})}{\Omega^{4}}|b_{1}b_{2}|\sin^{2}\theta\,,$
(6.27)

while the Einstein tensor on the supersymmetric background (6.6) takes the
form

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}={\rm
diag}\Big{(}0,\frac{b_{1}^{2}+b_{2}^{2}}{\Omega^{2}},\frac{b_{2}^{2}m_{11}}{\Omega^{2}}\Big{(}\frac{4b_{1}^{2}}{\Omega^{2}}+1\Big{)},\frac{b_{1}^{2}m_{22}}{\Omega^{2}}\Big{(}\frac{4b_{2}^{2}}{\Omega^{2}}+1\Big{)}\Big{)}-\frac{3}{4}g_{\mu\nu}\,.$
(6.28)

Putting everything together we can evaluate the integral (6) on the Cauchy
surface defined by the constant time slices. The result is

$\displaystyle{\cal E}_{\rm Casimir}^{\omega_{\cal T},\omega_{\cal J}}=$
$\displaystyle\;\frac{3a}{2}\frac{|b_{1}||b_{2}|}{(|b_{1}|+|b_{2}|)}+\frac{(|b_{1}|-|b_{2}|)^{2}}{9b_{1}b_{2}(|b_{1}|+|b_{2}|)}\big{(}c_{1}(|b_{1}|^{2}+|b_{2}|^{2})+c_{2}|b_{1}||b_{2}|\big{)}\,,$
(6.29)

where the coefficients $c_{1}$ and $c_{2}$ are given by

$\displaystyle c_{1}=$ $\displaystyle\;(5a-3c)(\omega_{\cal
J}-1)-(a-c)(2\omega_{\cal T}-1)+\frac{3}{2}(5a-c)\,,$ $\displaystyle c_{2}=$
$\displaystyle\;9a\,{\rm sgn}(b_{1}b_{2})+2(5a-3c)(\omega_{\cal
J}-1)-3(a-c)(\omega_{\cal T}-1)+\frac{3}{2}(5a-c)\,.$ (6.30)

For $|b_{1}|=|b_{2}|$, in which case the metric (6.6) is conformally flat, (6)
reduces to

${\cal E}^{\rm Casimir}=\frac{3a}{4}|b|\,,\qquad|b_{1}|=|b_{2}|=|b|\,,$ (6.31)

independently of the values of $\omega_{\cal T}$ and $\omega_{\cal J}$. This
result is in agreement with the expression (1.3) in the scheme where the
coefficient of the $R^{4}$ counterterm is set to zero.

## Acknowledgments

The work of PP was supported by the National Research Foundation of Korea
(NRF) grant funded by the Korea government (MSIT) (No. 2023R1A2C1006975) and
from the JRG Program at the APCTP through the Science and Technology Promotion
Fund and Lottery Fund of the Korean Government. IP would like to thank the
Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and
hospitality during the programme “Black holes: bridges between number theory
and holographic quantum information” where work on this paper was undertaken.
This work was supported by EPSRC grant no EP/R014604/1.

##  Appendix A Conventions and spinor identities

Our spacetime and spinor conventions are those of [18]. The tangent space
metric is $\eta={\rm diag\,}(-1,1,1,1)$ and the Levi-Civita symbol
$\varepsilon_{\mu\nu\rho\sigma}=\pm 1$ satisfies $\varepsilon_{0123}=1$.
Moreover, the Levi-Civita tensor is defined as
$\epsilon_{\mu\nu\rho\sigma}=\sqrt{-g}\;\varepsilon_{\mu\nu\rho\sigma}=e\;\varepsilon_{\mu\nu\rho\sigma}$.
Finally, the chirality matrix is

$\gamma^{5}=i\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}\,.$ (A.1)

Several gamma matrix and spinor identities we use in this manuscript are given
in Appendix A of [19]. Here we quote only the gamma matrix identities

$\gamma^{\mu\nu\rho\sigma}=i\epsilon^{\mu\nu\rho\sigma}\gamma^{5},\qquad\gamma^{\mu\nu\rho}=i\epsilon^{\mu\nu\rho\sigma}\gamma_{\sigma}\gamma^{5},\qquad\gamma^{\mu\nu}=\frac{i}{2}\epsilon^{\mu\nu}{}_{\rho\sigma}\gamma^{\rho\sigma}\gamma^{5}$
(A.2)

and the anticommuting spinor flip relations under Majorana conjugation

$\displaystyle\bar{\varepsilon}\,\gamma^{5}\eta=\bar{\eta}\gamma^{5}\varepsilon\,,$
$\displaystyle\bar{\varepsilon}\,\gamma_{\mu}\eta=-\bar{\eta}\gamma_{\mu}\varepsilon\,,$
$\displaystyle\bar{\varepsilon}\,\gamma_{\mu}\gamma^{5}\eta=\bar{\eta}\,\gamma_{\mu}\gamma^{5}\,,\varepsilon$
$\displaystyle\varepsilon\,\gamma^{\sigma}\gamma^{\mu\nu}\gamma^{5}\eta=-\bar{\eta}\,\gamma^{\mu\nu}\gamma^{\sigma}\gamma^{5}\varepsilon\,,$
$\displaystyle\varepsilon\,\gamma^{\sigma}\gamma^{\mu\nu}\eta=\bar{\eta}\,\gamma^{\mu\nu}\gamma^{\sigma}\varepsilon\,.$
(A.3)

However, the evaluation of the Casimir energy involves bilinears of commuting
spinors, for which the flip relation (A) hold with an additional minus sign.
Using these we now prove two identities commuting spinor identities that we
use extensively in the our analysis.

$\bar{\eta}\gamma_{\rho}\big{(}4\delta^{[\mu}_{\nu}\delta^{\rho]}_{\sigma}+i\gamma^{5}\epsilon^{\mu}{}_{\nu}{}^{\rho}{}_{\sigma}\big{)}\gamma^{\nu}\gamma^{5}\varepsilon=-6\delta^{\mu}_{\sigma}\,\bar{\eta}\gamma^{5}\varepsilon\,.$
(A.4)

Proof

The first term can rewritten as follows:

$\begin{split}\bar{\eta}\gamma_{\rho}(4\delta^{[\mu}_{\nu}\delta^{\rho]}_{\sigma})\gamma^{\nu}\gamma^{5}\varepsilon&=2\,\bar{\eta}\gamma_{\rho}(\delta^{\mu}_{\nu}\delta^{\rho}_{\sigma}-\delta^{\rho}_{\nu}\delta^{\mu}_{\sigma})\gamma^{\nu}\gamma^{5}\epsilon\\\
&=2\,\bar{\eta}\gamma_{\sigma}\gamma^{\mu}\gamma^{5}\epsilon-2\delta^{\mu}_{\sigma}\,\bar{\eta}\gamma_{\nu}\gamma^{\nu}\gamma^{5}\varepsilon\\\
&=2g_{\kappa\sigma}\bar{\eta}\gamma^{\kappa}\gamma^{\mu}\gamma^{5}\varepsilon-8\,\delta^{\mu}_{\sigma}\,\bar{\eta}\gamma^{5}\varepsilon\,,\end{split}$
(A.5)

where in the second step we used that $\gamma_{\nu}\gamma^{\nu}=4$. The first
term of eq. (A.5) becomes

$\begin{split}2g_{\kappa\sigma}\bar{\eta}\gamma^{\kappa}\gamma^{\mu}\gamma^{5}\varepsilon&=2g_{\kappa\sigma}\bar{\eta}(\gamma^{\kappa\mu}+g^{\kappa\mu})\gamma^{5}\varepsilon\\\
&=2g_{\kappa\sigma}\,\bar{\eta}\gamma^{\kappa\mu}\gamma^{5}\varepsilon+2\delta^{\mu}_{\sigma}\,\bar{\eta}\gamma^{5}\varepsilon\,,\end{split}$
(A.6)

and expressing $\gamma^{\mu\nu}$ in terms of the Levi-Civita symbol using eq.
(A.2), we get

$\begin{split}2g_{\kappa\sigma}\,\bar{\eta}\gamma^{\kappa\mu}\gamma^{5}\varepsilon&=ig_{\kappa\sigma}\,\epsilon^{\kappa\mu}{}_{\nu\rho}\,\bar{\eta}\gamma^{\nu\rho}\varepsilon=-i\epsilon^{\mu}{}_{\sigma\nu\rho}\,\bar{\eta}\gamma^{\nu\rho}\varepsilon\,.\end{split}$
(A.7)

Inserting eq. (A.6) we obtain

$2g_{\kappa\sigma}\bar{\eta}\gamma^{\kappa}\gamma^{\mu}\gamma^{5}\varepsilon=-i\epsilon^{\mu}{}_{\sigma\nu\rho}\,\bar{\eta}\gamma^{\nu\rho}\varepsilon+2\delta^{\mu}_{\sigma}\,\bar{\eta}\gamma^{5}\varepsilon\,,$
(A.8)

so that eq. (A.5) is written as

$\bar{\eta}\gamma_{\rho}(4\delta^{[\mu}_{\nu}\delta^{\rho]}_{\sigma})\gamma^{\nu}\gamma^{5}\varepsilon=-i\epsilon^{\mu}{}_{\sigma\nu\rho}\,\bar{\eta}\gamma^{\nu\rho}\varepsilon-6\delta^{\mu}_{\sigma}\,\bar{\eta}\gamma^{5}\varepsilon\,\,.$
(A.9)

The second term of the spinor part in eq. (5.4) is easily treated and gives

$i\bar{\eta}\gamma_{\rho}\gamma^{5}\epsilon^{\mu}{}_{\nu}{}^{\rho}{}_{\sigma}\gamma^{\nu}\gamma^{5}\varepsilon=i\epsilon^{\mu}{}_{\sigma\rho\nu}\,\bar{\eta}\gamma^{\rho}\gamma^{\nu}\varepsilon_{0}=i\epsilon^{\mu}{}_{\sigma\nu\rho}\,\bar{\eta}\gamma^{\nu\rho}\varepsilon_{0}\,\,.$
(A.10)

Summing eq. (A.9) and eq. (A.10) the Levi-Civita part cancels out and get eq.
(A.4). $\blacksquare$

$\bar{\varepsilon}\gamma_{\rho}\,g^{\nu[\sigma}\gamma^{\mu\rho]}\eta=-\frac{i}{3}\epsilon^{\sigma\mu\nu\rho}\,\bar{\eta}\gamma_{\rho}\gamma^{5}\varepsilon-\frac{2}{3}\,g^{\nu\sigma}\bar{\varepsilon}\gamma^{\mu}\eta+\frac{2}{3}\,g^{\mu\nu}\bar{\varepsilon}\gamma^{\sigma}\eta\,.$
(A.11)

Proof

Let us first consider the $\gamma$-terms. We have

$\gamma_{\rho}\,g^{\nu[\sigma}\gamma^{\mu\rho]}=\frac{1}{3}\gamma_{\rho}\left(g^{\nu\sigma}\gamma^{\mu\rho}+g^{\nu\rho}\gamma^{\sigma\mu}+g^{\nu\mu}\gamma^{\rho\sigma}\right)=\frac{1}{3}(-3g^{\nu\sigma}\gamma^{\mu}+\gamma^{\nu}\gamma^{\sigma\mu}+3\,g^{\mu\nu}\gamma^{\sigma})\,,$
(A.12)

where we used that $\gamma_{\rho}\gamma^{\rho\mu}=3\gamma^{\mu}$. For
commuting spinors
$\bar{\varepsilon}\gamma^{\nu}\gamma^{\mu\rho}\eta=-\bar{\eta}\gamma^{\mu\rho}\gamma^{\nu}\varepsilon$
and hence

$\bar{\varepsilon}\gamma_{\rho}\,g^{\nu[\sigma}\gamma^{\mu\rho]}\eta=-g^{\nu\sigma}\bar{\varepsilon}\gamma^{\mu}\eta+g^{\mu\nu}\bar{\varepsilon}\gamma^{\sigma}\eta-\frac{1}{3}\,\bar{\eta}\gamma^{\sigma\mu}\gamma^{\nu}\varepsilon\,.$
(A.13)

Note that the last term may be written as

$\bar{\eta}\gamma^{\sigma\mu}\gamma^{\nu}\varepsilon=\bar{\eta}\left(\gamma^{\sigma\mu\nu}+\gamma^{\sigma}g^{\mu\nu}-\gamma^{\mu}g^{\sigma\nu}\right)\varepsilon\,.$
(A.14)

The first term in this expression gives
$\bar{\eta}\gamma^{\sigma\mu\nu}\varepsilon=i\epsilon^{\sigma\mu\nu\rho}\,\bar{\eta}\gamma_{\rho}\gamma^{5}\varepsilon$.
Inserting eq. (A.14) in eq. (A.13) we obtain eq. (A.11). $\blacksquare$

Note that for $\varepsilon=\eta$, the first term on the r.h.s of (A.14)
vanishes and we get

$\bar{\eta}\gamma_{\rho}\,g^{\nu[\sigma}\gamma^{\mu\rho]}\eta=-\frac{2}{3}\,g^{\nu\sigma}\bar{\eta}\gamma^{\mu}\eta+\frac{2}{3}\,g^{\mu\nu}\bar{\eta}\gamma^{\sigma}\eta\,.$
(A.15)

## References

  * [1] R. J. Riegert, _A Nonlocal Action for the Trace Anomaly_ , _Phys. Lett. B_ 134 (1984) 56.
  * [2] E. S. Fradkin and A. A. Tseytlin, _Conformal Anomaly in Weyl Theory and Anomaly Free Superconformal Theories_ , _Phys. Lett. B_ 134 (1984) 187.
  * [3] B. Assel, D. Cassani, L. Di Pietro, Z. Komargodski, J. Lorenzen and D. Martelli, _The Casimir Energy in Curved Space and its Supersymmetric Counterpart_ , _JHEP_ 07 (2015) 043 [1503.05537].
  * [4] H.-C. Kim and S. Kim, _M5-branes from gauge theories on the 5-sphere_ , _JHEP_ 05 (2013) 144 [1206.6339].
  * [5] B. Assel, D. Cassani and D. Martelli, _Localization on Hopf surfaces_ , _JHEP_ 08 (2014) 123 [1405.5144].
  * [6] N. Bobev, M. Bullimore and H.-C. Kim, _Supersymmetric Casimir Energy and the Anomaly Polynomial_ , _JHEP_ 09 (2015) 142 [1507.08553].
  * [7] F. Brünner, D. Regalado and V. P. Spiridonov, _Supersymmetric Casimir energy and $SL(3,\mathbb{Z})$ transformations_, _JHEP_ 07 (2017) 041 [1611.03831].
  * [8] P. Benetti Genolini, D. Cassani, D. Martelli and J. Sparks, _The holographic supersymmetric Casimir energy_ , _Phys. Rev. D_ 95 (2017) 021902 [1606.02724].
  * [9] I. Papadimitriou, _Supercurrent anomalies in 4d SCFTs_ , _JHEP_ 07 (2017) 038 [1703.04299].
  * [10] M. Kaku, P. K. Townsend and P. van Nieuwenhuizen, _Gauge Theory of the Conformal and Superconformal Group_ , _Phys. Lett._ 69B (1977) 304.
  * [11] M. Kaku, P. K. Townsend and P. van Nieuwenhuizen, _Superconformal Unified Field Theory_ , _Phys. Rev. Lett._ 39 (1977) 1109.
  * [12] M. Kaku, P. K. Townsend and P. van Nieuwenhuizen, _Properties of Conformal Supergravity_ , _Phys. Rev._ D17 (1978) 3179.
  * [13] P. K. Townsend and P. van Nieuwenhuizen, _Simplifications of Conformal Supergravity_ , _Phys. Rev._ D19 (1979) 3166.
  * [14] P. Van Nieuwenhuizen, _Supergravity_ , _Phys. Rept._ 68 (1981) 189.
  * [15] B. de Wit, _CONFORMAL INVARIANCE IN EXTENDED SUPERGRAVITY_ , in _First School on Supergravity Trieste, Italy, April 22-May 6, 1981_ , p. 0267, 1981.
  * [16] B. de Wit, _MULTIPLET CALCULUS_ , in _September School on Supergravity and Supersymmetry Trieste, Italy, September 6-18, 1982_ , 1983.
  * [17] E. S. Fradkin and A. A. Tseytlin, _CONFORMAL SUPERGRAVITY_ , _Phys. Rept._ 119 (1985) 233.
  * [18] D. Z. Freedman and A. Van Proeyen, _Supergravity_. Cambridge Univ. Press, Cambridge, UK, 2012.
  * [19] I. Papadimitriou, _Supersymmetry anomalies in $\mathcal{N}=1$ conformal supergravity_, _JHEP_ 04 (2019) 040 [1902.06717].
  * [20] D. Anselmi, D. Z. Freedman, M. T. Grisaru and A. A. Johansen, _Nonperturbative formulas for central functions of supersymmetric gauge theories_ , _Nucl. Phys._ B526 (1998) 543 [hep-th/9708042].
  * [21] K. Jensen, R. Loganayagam and A. Yarom, _Thermodynamics, gravitational anomalies and cones_ , _JHEP_ 02 (2013) 088 [1207.5824].
  * [22] W. A. Bardeen and B. Zumino, _Consistent and Covariant Anomalies in Gauge and Gravitational Theories_ , _Nucl. Phys._ B244 (1984) 421.
  * [23] R. Minasian, I. Papadimitriou and P. Yi, _Anomalies and supersymmetry_ , _Phys. Rev. D_ 105 (2022) 065005 [2104.13391].
  * [24] H. Samtleben and D. Tsimpis, _Rigid supersymmetric theories in 4d Riemannian space_ , _JHEP_ 05 (2012) 132 [1203.3420].
  * [25] C. Klare, A. Tomasiello and A. Zaffaroni, _Supersymmetry on Curved Spaces and Holography_ , _JHEP_ 08 (2012) 061 [1205.1062].
  * [26] T. T. Dumitrescu, G. Festuccia and N. Seiberg, _Exploring Curved Superspace_ , _JHEP_ 08 (2012) 141 [1205.1115].
  * [27] J. T. Liu, L. A. Pando Zayas and D. Reichmann, _Rigid Supersymmetric Backgrounds of Minimal Off-Shell Supergravity_ , _JHEP_ 10 (2012) 034 [1207.2785].
  * [28] T. T. Dumitrescu and G. Festuccia, _Exploring Curved Superspace (II)_ , _JHEP_ 01 (2013) 072 [1209.5408].
  * [29] A. Kehagias and J. G. Russo, _Global Supersymmetry on Curved Spaces in Various Dimensions_ , _Nucl. Phys._ B873 (2013) 116 [1211.1367].
  * [30] C. Closset, T. T. Dumitrescu, G. Festuccia and Z. Komargodski, _Supersymmetric Field Theories on Three-Manifolds_ , _JHEP_ 05 (2013) 017 [1212.3388].
  * [31] H. Samtleben, E. Sezgin and D. Tsimpis, _Rigid 6D supersymmetry and localization_ , _JHEP_ 03 (2013) 137 [1212.4706].
  * [32] D. Cassani, C. Klare, D. Martelli, A. Tomasiello and A. Zaffaroni, _Supersymmetry in Lorentzian Curved Spaces and Holography_ , _Commun. Math. Phys._ 327 (2014) 577 [1207.2181].
  * [33] P. de Medeiros, _Rigid supersymmetry, conformal coupling and twistor spinors_ , _JHEP_ 09 (2014) 032 [1209.4043].
  * [34] K. Hristov, A. Tomasiello and A. Zaffaroni, _Supersymmetry on Three-dimensional Lorentzian Curved Spaces and Black Hole Holography_ , _JHEP_ 05 (2013) 057 [1302.5228].
  * [35] M. Blau, _Killing spinors and SYM on curved spaces_ , _JHEP_ 11 (2000) 023 [hep-th/0005098].
  * [36] S. M. Kuzenko, _Symmetries of curved superspace_ , _JHEP_ 03 (2013) 024 [1212.6179].
  * [37] D. Cassani and D. Martelli, _Supersymmetry on curved spaces and superconformal anomalies_ , _JHEP_ 10 (2013) 025 [1307.6567].