aainstitutetext: Department of Physics, Indian Institute of Technology Ropar,
Rupnagar, Punjab 140001, Indiabbinstitutetext: Theory Division, Saha Institute
of Nuclear Physics, HBNI, 1/AF Bidhan Nagar, Kolkata, India -
700064ccinstitutetext: School of Basic Sciences, Indian Institute of
Technology Bhubaneswar, Bhubaneswar 752050, India

# Supersymmetric Graphene on Squashed Hemisphere

Rajesh Kumar Gupta, b Augniva Ray a,c and Karunava Sil
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We compute the partition function of $\mathcal{N}=2$ supersymmetric mixed
dimensional QED on a squashed hemisphere using localization. Mixed dimensional
QED is an abelian gauge theory coupled to charged matter fields at the
boundary. The partition function is a function of the complex gauge coupling
$\tau$, the choice of R-symmetry and the squashing deformation. The
superconformal R-symmetry is determined using the 3-dimensional
F-maximization. The free energy as a function of squashing deformation allows
computing correlation functions that contain the insertion of the energy-
momentum tensor. We compute the 2-point correlation function of the boundary
energy-momentum tensor by differentiating the free energy with respect to the
squashing parameter. We comment on the behaviour of the 2-point function as we
change the complex coupling $\tau$.

##  1 Introduction

Quantum field theory on a manifold with boundary finds many applications
ranging from string theory to condensed matter physics such as D-branes,
topological insulators and graphene. The fixed point in the renormalization
group flow in quantum field theory is particularly interesting since a
conformal field theory describes it. These theories find a useful application
in condensed matter system, for example, in describing the second-order phase
transition. Our focus here will be on 4-dimensional conformal field theories
in the presence of a boundary with conformally invariant boundary conditions.
One such example of 4-dimensional boundary conformal field theories (bCFT) is
mixed dimensional quantum electrodynamics where 4-dimensional electromagnetic
field interacts with charged matter fields on the 3-dimensional boundary.
These theories exhibit many interesting properties, see for example
Hsiao:2017lch ; Hsiao:2018fsc ; DiPietro:2019hqe ; Grignani:2019zxc .

One of the characteristic features of a conformal field theory is the presence
of a quantitative measure of the number of degrees of freedom that decreases
along the RG flow connecting two CFTs. In 2 and 4 dimensions, it coincides
with the central charge $c$ and $a$, respectively, while in 3-dimensional
CFTs, the free energy of the theory computed on S3 plays the similar role.
Monotonicity theorem also exists in $d$-dimensional bCFTs PhysRevLett.67.161 ;
Friedan:2003yc ; Jensen:2015swa ; Casini:2016fgb ; Casini:2018nym . The
boundary free energy defined from the hemisphere partition function as

$\frac{|Z_{HS^{d}}|^{2}}{Z_{S^{d}}}=e^{\text{div.}-2F_{\partial}}\,,$ (1)

where terms in “div.” are divergent terms which have $(d-1)$-dimensional
origin, conjectured to decrease along the RG flow triggered by the boundary
relevant operator. Our goal would be to compute the boundary free energy in
the 4-dimensional supersymmetric bCFTs.

The localization computation of the partition function for 4-dimensional
$\mathcal{N}=2$ supersymmetric theories with Neumann and Dirichlet boundary
conditions was first performed in Gava:2016oep . Subsequently, the partition
function of $\mathcal{N}=2$ supersymmetric graphene-like theories with general
boundary conditions appeared in Gupta:2019qlg . In the paper Gupta:2019qlg ,
the authors computed the partition function as a function of trial R-charge of
the charged matter fields at the boundary. The partition function takes the
form of real one-dimensional integral with integrand given in terms of
Jafferis $\ell$-function. One of the novel features of the partition function
is that it depends on the complexified gauge coupling

$\tau=\frac{\theta}{2\pi}+\frac{2\pi i}{g^{2}}\,,$ (2)

which is exactly marginal. This gives rise to a boundary conformal field
theory for every value of the complex coupling $\tau$. Thus, the partition
function computed using the method of localization is a function of the
complex coupling $\tau$ and the choice of R-symmetry. The R-symmetry is
determined using the 3-dimensional F-maximization Jafferis:2010un . Moreover,
the partition function, as a function of the background sources for gauge and
flavor currents, was used to compute boundary transport coefficients that
appear in the 2-point function of the corresponding currents.

In the present article, we will generalize the above computation to include
the metric background deformations. More specifically, we will consider
$\mathcal{N}=2$ supersymmetric mixed dimensional QED on a squashed hemisphere.
Supersymmetric theories on a squashed sphere have been studied in various
dimensions. Provided the squashing deformations preserve some supersymmetry,
the partition function as the function of the squashing parameters can be
computed using the localization technique, see for example Hama:2011ea ;
Imamura:2011wg ; Hama:2012bg ; Imamura:2012xg . The free energy as a function
of squashing deformations allows us to compute correlation functions that
contain the insertion of the energy-momentum tensor. We will be interested
here to compute the transport coefficient that appears in the 2-point function
of the energy-momentum tensor in 3-dimensional boundary.

Conformal symmetry and the conservation law fix the 2-point function of
energy-momentum tensor in 3-dimensional flat space up to a constant. It is
given by

$\displaystyle<T_{\mu\nu}(x)T_{\rho\sigma}(0)>=-\frac{\tau_{R}}{64\pi^{2}}(\delta_{\mu\nu}\partial^{2}-\partial_{\mu}\partial_{\nu})(\delta_{\rho\sigma}\partial^{2}-\partial_{\rho}\partial_{\sigma})\frac{1}{x^{2}}$
$\displaystyle+\frac{\tau_{R}}{64\pi^{2}}\Big{(}(\delta_{\mu\rho}\partial^{2}-\partial_{\mu}\partial_{\rho})(\delta_{\nu\sigma}\partial^{2}-\partial_{\nu}\partial_{\sigma})+(\mu\leftrightarrow\nu)\Big{)}\frac{1}{x^{2}}\,.$
(3)

The quantity of interest $\tau_{R}$ can be computed by placing the conformal
field theory on S${}_{b}^{3}$. It is given by the second derivative of the
squashed free energy with respect to the squashing parameter $b$
Closset:2012ru

$\tau_{R}=\frac{2}{\pi^{2}}\text{Re}\frac{\partial^{2}F_{b}}{\partial
b^{2}}\Big{|}_{b=1},\qquad\text{where}\quad F_{b}=-\ln Z_{b}\,,$ (4)

where the free energy is evaluated using the superconformal R-charge.

We extend the above computation to 2-point function in bCFT. In particular, we
compute the 2-point function of the boundary energy-momentum tensor by
differentiating the boundary-free energy as

$\tau_{R}=\frac{2}{\pi^{2}}\frac{\partial^{2}F_{\partial,b}}{\partial
b^{2}}\Big{|}_{b=1}\,.$ (5)

Basically, the idea is that after integrating over the bulk degrees of freedom
for a given conformal boundary condition (and dividing by the sphere partition
function), we can think of the $F_{\partial}$ as the free energy of some
effective CFT at the boundary. At the perturbative level, the effect of bulk
degrees of freedom can be mimicked by introducing interactions in the boundary
theory involving auxiliary fields. These were the original arguments in
Gaiotto:2014gha  that provided evidence in support of the decrease of
boundary-free energy along the boundary RG flow.

To compute $\tau_{R}$, we follow the following strategy Nishioka:2013gza : 1)
Evaluate the partition function on S3 and determine the R-charge that
maximizes the real part of the free energy, 2) Use the R-charge thus obtained
to evaluate the free energy on S${}_{b}^{3}$, and 3) Compute the second
derivative of the free energy, which is the function of squashing parameter
$b$, to obtain the expression for $\tau_{R}$.

The outline of the rest of the paper is as follows. In section 2, we review
two different ways of squashing a 4-dimensional sphere. We then discuss
supersymmetry on the squashed sphere. After that, we discuss the condition the
background fields need to satisfy to have supersymmetry on the squashed
hemisphere. In section 3, we find the partition function of mixed dimensional
QED coupled to charged matter fields at the boundary. In Section 4, we compute
the 2-point function of the energy-momentum tensor. We then conclude with a
summary in section 5. In Appendix A, we present our conventions for gamma
matrices and reality condition on fermions. In appendix B, we discuss the
supersymmetry on the squashed hemisphere. Here we also discuss the requirement
on the supergravity background fields. In appendix C and D, we present the
supersymmetric action and background fields on the squashed sphere. In
appendix E, we give explicit expressions for functions that appear in the
subleading computations of $\tau_{R}$.

##  2 Supersymmetry on squashed hemisphere

In this section, we will introduce the supersymmetric background on a squashed
hemisphere. The squashed hemisphere is obtained by considering the squashed
sphere and placing the boundary at the equator. We will be considering here
two different kinds of squashing depending on the isometry preserved by the
deformation of S4. The first kind of squashing preserves $SU(2)\times U(1)$
isometry. The metric is given by

$ds_{\textup{I}}^{2}=dr^{2}+\frac{1}{4}\sin^{2}r\Big{(}d\theta^{2}+\sin^{2}\theta\,d\phi^{2}+h(r)^{2}(d\psi+\cos\theta\,d\phi)^{2}\Big{)}$
(6)

where $0\leq r\leq\pi$, $0\leq\theta\leq\pi$, $0\leq\phi\leq 2\pi$ and
$0\leq\psi\leq 4\pi$ and $h(r)$ is an arbitrary smooth function. The
regularity of the metric near the north pole (i.e. at $r=0$) and south pole
(i.e. at $r=\pi$) requires the following behaviour the function

$\displaystyle\text{North pole:}\qquad
h(r)=1+c_{2}r^{2}+\mathcal{O}(r^{3})\,,$ $\displaystyle\text{South
pole:}\qquad h(r)=1+d_{2}(\pi-r)^{2}+\mathcal{O}((\pi-r)^{3})\,.$ (7)

The squashed hemisphere is obtained by the requirement that $0\leq
r\leq\frac{\pi}{2}$. The induced metric at the boundary is given by

$ds_{\partial\textup{I}}^{2}=\frac{1}{4}\Big{(}d\theta^{2}+\sin^{2}\theta\,d\phi^{2}+h\Big{(}\frac{\pi}{2}\Big{)}^{2}(d\psi+\cos\theta\,d\phi)^{2}\Big{)}$
(8)

The above metric on the squashed S3 preserves $SU(2)\times U(1)$ symmetry.
Thus, any arbitrary smooth function satisfying the regularity condition (2)
gives rise to a smooth deformation of the HS4. The deformation can then be
used to compute the 2-point function of energy momentum tensor of the boundary
conformal field theory. For the convenience of the later computations, will
choose the following form of the function

$h(r)=1+\frac{2\alpha}{\pi^{2}}\sin^{3}r\,.$ (9)

In the above $\alpha$ is a positive real parameter with $\alpha=0$ correspond
to unsquashed background. The above form of $h(r)$ satisfies the smoothness
criteria given in (2).

The second kind of the deformation we will consider preserves $U(1)\times
U(1)$ isometry. The squashed metric is given by

$ds_{\textup{II}}^{2}=\delta_{ab}E^{a}E^{b}\,,$ (10)

where the vielbeins are

$\displaystyle E^{1}=\ell\,\sin r\,\cos\theta\,d\phi,\quad
E^{2}=\tilde{\ell}\,\sin r\,\sin\theta\,d\chi,\quad E^{3}=f(\theta)\,\sin
r\,d\theta+h(r,\theta)dr,$ $\displaystyle E^{4}=g(r,\theta)\,dr\,.$ (11)

In the above, the range of the coordinates are $0\leq r\leq\pi$,
$0\leq\theta\leq\frac{\pi}{2}$ and $0\leq\phi,\chi\leq 2\pi$. The parameters
$\ell$ and $\tilde{\ell}$ are constants and the functions appearing in the
metric are given by

$\displaystyle
f(\theta)=\sqrt{\ell^{2}\sin^{2}\theta+\tilde{\ell}^{2}\cos^{2}\theta},\quad
g(r,\theta)=\sqrt{\rho^{2}\sin^{2}r+\frac{\ell^{2}\tilde{\ell}^{2}}{f(\theta)^{2}}\cos^{2}r}\,,$
$\displaystyle h(r,\theta)=\frac{\tilde{\ell}^{2}-\ell^{2}}{f(\theta)}\cos
r\,\sin\theta\,\cos\theta\,.$ (12)

The induced metric at the boundary of the squashed hemisphere i.e. at
$r=\frac{\pi}{2}$ is

$ds_{\partial\textup{II}}^{2}=\ell^{2}\cos^{2}\theta\,d\phi^{2}+\tilde{\ell}^{2}\sin^{2}\theta\,d\chi^{2}+(\tilde{\ell}^{2}\cos^{2}\theta+\ell^{2}\sin^{2}\theta)\,d\theta^{2}\,.$
(13)

Next, we will discuss the background which needs to be turned on in order to
preserve supersymmetry on the squashed hemisphere.

###  2.1 Supersymmetric background on squashed hemisphere

We will be interested in $\mathcal{N}=2$ supersymmetric theory on the squashed
hemisphere backgrounds (8) and (10). The squashed metric background does not
admit any rigid supersymmetry itself. However, the supersymmetric theory can
be put on the squashed hemisphere if we turn on some non-dynamical
supergravity background fields. The theory is invariant under the rigid
supersymmetry transformations that are generated by the solution of the
Killing spinor equation

$\mathcal{D}_{\mu}\xi^{i}+T^{ab}\gamma_{ab}\gamma_{\mu}\xi^{i}=\gamma_{\mu}\xi^{\prime
i}\,.$ (14)

In addition to above, the Killing spinor also satisfies an auxiliary equation

$\not{\mathcal{D}}\not{\mathcal{D}}\xi^{i}+4\mathcal{D}_{c}T_{ab}\gamma^{ab}\gamma^{c}\xi^{i}=M\xi^{i}\,.$
(15)

The fields $T^{ab}$ and $M$ are auxiliary background fields. Also, the
$SU(2)_{R}$ connection appears in the Killing spinor equation through the
covariant derivative given by

$\mathcal{D}_{\mu}\xi^{i}=(\partial_{\mu}+\frac{1}{4}\omega_{\mu
ab}\gamma^{ab})\xi^{i}+\mathcal{V}^{i}_{\mu\,j}\xi^{j}\,.$ (16)

These Killing spinors also satisfy the symplectic Majorana condition

$\overline{\xi}_{i}\equiv(\xi^{i})^{\dagger}=\epsilon_{ij}\xi^{j\,T}C\,,$ (17)

where $C$ is the charge conjugation matrix.
Now, the presence of the boundary at $r=\frac{\pi}{2}$ breaks half of the
supersymmetry transformations i.e the boundary in our case preserves 4 out of
8 supercharges. Given a solution to the Killing spinor equations (14) and
(15), $\xi^{i}$, we define the following projected spinors Gupta:2019qlg

$\xi^{i}_{\pm}=\Pi^{i}_{\pm\,j}\xi^{j}\,,$ (18)

where the projector is given by

$\Pi^{i}_{\pm\,j}=\frac{1}{2}(\delta^{i}_{j}\pm
i\tau^{i}_{3\,j}\gamma_{5}\gamma_{n})\,.$ (19)

Here, $\gamma_{n}$ is the flat space gamma matrix corresponding to the
direction perpendicular to the boundary i.e. $n$. We require that at
$r=\frac{\pi}{2}$, $\Pi^{i}_{+\,j}\xi^{j}\Big{|}_{r=\frac{\pi}{2}}=0$, and
$\xi^{i}_{-}\Big{|}_{r=\frac{\pi}{2}}$ generates the supersymmetry on the
boundary. It is important to note that $\xi^{i}_{\pm}$ are not the solution of
the 4-dimensional Killing spinor equations.
The spinor $\xi^{i}_{-}\Big{|}_{r=\frac{\pi}{2}}$ being the Killing spinor on
the boundary, i.e. a solution to the boundary Killing spinor equation,
requires the background to satisfy certain conditions. Using the fact that the
components of spin connections corresponding to the metrics (8) and (10)
satisfy

$\omega^{Bn}_{A}=0,\qquad\text{for}\quad A,B=1,2,3\,,$ (20)

it turns out that $\xi^{i}_{-}\Big{|}_{r=\frac{\pi}{2}}$ solves 3-dimensional
Killing spinor equation if the background satisfies the following condition
(see the Appendix (B) for more details)

$T_{Bn}|_{r=\frac{\pi}{2}}=0\,.$ (21)

Note that the above condition is the requirement for having $\mathcal{N}=2$
supersymmetry with $U(1)_{R}$ symmetry at the boundary. Thus, the spinor
$\xi_{-}^{i}$ at the boundary satisfies the Killing spinor equation

$\mathcal{D}^{3d}_{A}\xi_{-}^{i}+T_{BC}(\gamma_{BC}\gamma_{A}+\frac{1}{3}\gamma_{A}\gamma_{BC})\xi_{-}^{i}=\frac{1}{3}\gamma_{A}\gamma^{B}\mathcal{D}^{3d}_{B}\xi_{-}^{i}\,.$
(22)

In particular, one can write the above as

$\nabla^{3d}_{A}\psi_{-}^{i}=\frac{i}{2}H\gamma_{A}\psi_{-}^{i}+V_{A}\gamma_{3}\psi_{-}^{i}\,,$
(23)

for some choice of $H$ and $V_{A}$ and
$\psi^{i}_{-}=(\xi_{-},\xi^{\prime}_{-})$ (see the appendix A for the
convention). This is the 3-dimensional Killing spinor equation with $H$ and
$V_{A}$ being the auxiliary fields in 3-dimensional supergravity.
In the following, we will try to find a possible solution for the supergravity
background fields $T$, $V$ and $M$ that solves the sets of Killing spinor
equation as given in Eqs. (14),(15) and also at the same time satisfies the
condition as given in (21) at the boundary. For this we have considered two
different kinds of squashing of the four dimensional manifold as already
mentioned, namely the one which preserve $SU(2)\times U(1)$ isometry and other
with $U(1)\times U(1)$ isometry. The supersymmetric backgrounds for the two
cases are separately discussed below.

####  2.1.1 Squashed sphere with $SU(2)\times U(1)$ isometry

In this section, we will consider the supersymmetry on the squashed hemisphere
described by the metric (8). The squashing was first discussed in Cabo-
Bizet:2014nia  in the context of supersymmetric localization. The presentation
below follows their analysis closely. Our choice of vielbeins are

$\displaystyle e^{1}=-\dfrac{\sin r}{2}\big{(}\cos\psi
d\theta+\sin\psi\sin\theta d\phi\big{)}\,,\qquad e^{2}=\dfrac{\sin
r}{2}\big{(}\sin\psi d\theta-\cos\psi\sin\theta d\phi\big{)}\,,$
$\displaystyle e^{3}=-\dfrac{\sin r}{2}h(r)\big{(}d\psi+\cos\theta
d\phi\big{)}\,,\qquad e^{4}=dr\,.$ (24)

Following Cabo-Bizet:2014nia , we choose an ansatz for the Killing spinors
that is compatible with the symplectic Majorana reality condition, given in
(17). The ansatz is

$\xi^{1}=\begin{pmatrix}s(r)\\\ 0\\\ ic\frac{h(r)}{s(r)}\sin{r}\\\
0\end{pmatrix},\quad\xi^{2}=\begin{pmatrix}0\\\ s(r)\\\ 0\\\
-ic\frac{h(r)}{s(r)}\sin{r}\end{pmatrix}\quad$ (25)

with $c$ being a real constant. Solving the Killing spinor equation with the
above ansatz, we find

$\begin{split}s(r)&=h(r)\cos{\frac{r}{2}}\,.\end{split}$ (26)

Note that in the limit of vanishing squashing parameter ($\alpha=0$), the
above ansatz for the Killing spinor satisfies Killing spinor equation on the
round sphere. However for non zero $\alpha$, the above spinors solve the
Killing spinor equation provided one turns on appropriate background
supergravity fields. These background fields, the antisymmetric matrix
$T^{ab}$ and the $SU(2)_{R}$ gauge field $\mathcal{V}_{\mu}$, defined in (14)
have the following form,

$T^{ab}\gamma_{ab}=\begin{pmatrix}it^{+}_{3}&&i(t^{+}_{1}-it^{+}_{2})&&0&&0\\\
i(t^{+}_{1}+it^{+}_{2})&&-it^{+}_{3}&&0&&0\\\
0&&0&&it^{-}_{3}&&i(t^{-}_{1}-it^{-}_{2})\\\
0&&0&&i(t^{-}_{1}+it^{-}_{2})&&-it^{-}_{3}\\\ \end{pmatrix}\,,$ (27)

and

$\displaystyle V_{a}\equiv\begin{pmatrix}iv_{3,a}&&i(v_{1,a}+iv_{2,a})\\\
i(v_{1,a}-iv_{2,a})&&-iv_{3,a}\end{pmatrix}\,,$ (28)

with, $\mathcal{V}_{\mu}=e^{a}_{\mu}V_{a}$. The only non zero components of
the background fields, $v_{3,3}$, $t^{+}_{3}$, $t^{-}_{3}$ and also $M$ are
given in Appendix (C). In presence of the boundary at $r=\pi/2$, we also
require to impose the boundary condition as given in (21) on the antisymmetric
tensor. Given the explicit results for the background field components in Eq.
(LABEL:vt3t3b), the above condition on the antisymmetric tensor at the
boundary translates into the following relation between $t^{+}_{3}$ and
$t^{-}_{3}$,

$\left(t^{+}_{3}-t^{-}_{3}\right)|_{r=\pi/2}=0\Rightarrow
c=\pm\left(\frac{1}{2}+\frac{\alpha}{\pi^{2}}\right).$ (29)

In other words, the real constant $c$ is fixed by considering the above
boundary condition. Moreover, due to the boundary, we expect only half of the
supersymmetry to be present compared to the case with no boundary. We define a
projected spinor $\xi^{i}_{\pm}$ as

$\xi^{i}_{\pm}=\Pi^{i}_{\pm j}\xi^{j}.$ (30)

Imposing the condition that $\xi^{i}_{+}|_{r=\pi/2}=0$ sets the value
$c=+\left(\frac{1}{2}+\frac{\alpha}{\pi^{2}}\right)=\frac{1}{2}h(\frac{\pi}{2})$.
The boundary supersymmetry is generated by the Killing spinor
$\xi^{i}_{-}\Big{|}_{r=\frac{\pi}{2}}$.
The corresponding Killing vector is given by

$K=-4h(\frac{\pi}{2})\frac{\partial}{\partial\psi}\,.$ (31)

The boundary Killing spinor equation is given by (23) and the supergravity
fields are

$H=h(\frac{\pi}{2}),\quad
V_{A}=-i\delta_{A3}\Big{(}h(\frac{\pi}{2})-\frac{1}{h(\frac{\pi}{2})}\Big{)}\,.$
(32)

####  2.1.2 Squashed sphere with $U(1)\times U(1)$ isometry

In this section, we will discuss the squashing described by the metric (10).
This particular kind of squashed geometry for S4 was considered in Hama:2012bg
. The solution of Killing spinor equations and the exact results for the
background fields were obtained in Hama:2012bg . The Killing spinors are given
by,

$\xi^{1}=\begin{pmatrix}\frac{1}{2}e^{\frac{i}{2}\left(-\theta+\phi+\chi\right)}\sin{\frac{r}{2}}\\\
-\frac{1}{2}e^{\frac{i}{2}\left(\theta+\phi+\chi\right)}\sin{\frac{r}{2}}\\\
\frac{i}{2}e^{\frac{i}{2}\left(-\theta+\phi+\chi\right)}\cos{\frac{r}{2}}\\\
-\frac{i}{2}e^{\frac{i}{2}\left(\theta+\phi+\chi\right)}\cos{\frac{r}{2}}\end{pmatrix},\quad\xi^{2}=\begin{pmatrix}\frac{1}{2}e^{\frac{i}{2}\left(-\theta-\phi-\chi\right)}\sin{\frac{r}{2}}\\\
\frac{1}{2}e^{\frac{i}{2}\left(\theta-\phi-\chi\right)}\sin{\frac{r}{2}}\\\
-\frac{i}{2}e^{\frac{i}{2}\left(-\theta-\phi-\chi\right)}\cos{\frac{r}{2}}\\\
-\frac{i}{2}e^{\frac{i}{2}\left(\theta-\phi-\chi\right)}\cos{\frac{r}{2}}\end{pmatrix}\quad$
(33)

Given this Killing spinor one can easily obtain the projected Killing spinors
$\xi^{i}_{+}$ and $\xi^{i}_{-}$ on the three dimensional boundary. It turns
out that $\xi^{i}_{+}$ being proportional to
$\left(\cos{\tfrac{r}{2}}-\sin{\tfrac{r}{2}}\right)$ goes to zero at
$r=\pi/2$. Moreover, for the Killing spinor equation (14) to be satisfied at
the boundary with the remaining spinor $\xi^{i}_{-}$, the antisymmetric tensor
must obey equation (21). The components of $T_{Bn}$ which are nonzero turn out
to be $T_{\phi r}$ and $T_{\chi r}$, given by,

$T_{\phi
r}=\frac{h(r,\theta)\sin{\theta}}{8f(\theta)g(r,\theta)},~{}~{}~{}~{}T_{\chi
r}=-\frac{h(r,\theta)\cos{\theta}}{8f(\theta)g(r,\theta)}\cdot$ (34)

These two components vanish at $r=\pi/2$, since $h(\frac{\pi}{2},\theta)=0$.
The corresponding Killing vector is given by

$K=-\frac{1}{\ell}\frac{\partial}{\partial\phi}-\frac{1}{\tilde{\ell}}\frac{\partial}{\partial\chi}\,.$
(35)

At the boundary, the Killing spinors (33) satisfy the 3-dimensional Killing
spinor equation (23). The background supergravity fields are given by

$H=-\frac{1}{f(\theta)},\quad
V_{1}=-\frac{i}{2\cos\theta}\Big{(}\frac{1}{f(\theta)}-\frac{1}{\ell}\Big{)},\quad
V_{2}=-\frac{i}{2\sin\theta}\Big{(}\frac{1}{f(\theta)}-\frac{1}{\tilde{\ell}}\Big{)},\quad
V_{3}=0\,.$ (36)

##  3 Supersymmetric partition function

In this section, we will compute the partition function of a supersymmetric
theory of interest on the squashed hemisphere using localization. In
particular, we will be interested in the supersymmetric theory of $n$ number
of chiral multiplets on the boundary of the squashed hemisphere coupled to an
abelian vector multiplet having propagating degrees of freedom in bulk. We
will further denote by $n_{+}$ and $n_{-}$, with $n=n_{+}+n_{-}$, the number
of positively and negatively charged chiral multiplets, respectively. The
Lagrangian of the theory on a squashed background is available in the
literature Hama:2011ea ; Hama:2012bg , however, for the convenience of a
reader, we have given the Lagrangian for vector multiplet in the appendix D.

###  3.1 Squashing preserving $SU(2)\times U(1)$

We begin with the partition function computation on the squashed hemisphere
(8). The supersymmetric locus is obtained by solving the bosonic equations
$\delta\lambda^{i}=0$. To state the solution explicitly, we choose the
deformation to be (9). With this we have a Killing spinor with

$s(r)=\cos\frac{r}{2}(1+\frac{2\alpha}{\pi^{2}}\sin^{3}r)\,.$ (37)

The solution to the variation $\delta\lambda^{i}=0$ is given by (in
perturbative expansion in $\alpha$)

$\displaystyle S=s_{0}-\frac{s_{0}\alpha}{\pi^{2}}\sin^{2}\frac{r}{2}(4+5\sin
r+4\sin 2r+\sin 3r)+...$ $\displaystyle
D_{3}=s_{0}+\frac{s_{0}\alpha}{4\pi^{2}}(-8+16\cos r+8\sin r+2\sin 2r+8\sin
3r+5\sin 4r)+...$ (38)

In the above $s_{0}$ is constant and all other fields are set to zero.
Evaluating the classical action (see appendix D) on the localization
background we obtain (to order $\alpha^{2}$)111Note that the partition
function depends on $\overline{\tau}$. This reflects the choice of the Killing
spinor. It is possible to find the Killing spinor which gives rise to the
classical action on the localization background depending on $\tau$. However,
we will not be using the partition function for the future computations, and
hence we will not proceed to find such Killing spinor.

$I=-is_{0}^{2}\Big{(}\pi-\frac{4\alpha}{\pi}+\frac{12\alpha^{2}}{\pi^{3}}+...\Big{)}\overline{\tau}\,,$
(39)

where

$\overline{\tau}=\frac{\theta}{2\pi}-\frac{2\pi i}{g^{2}_{YM}}\,.$ (40)

The $\alpha$ expansion of the action agrees with the closed form expression of
the action given as (we have checked it for a quite a few order)

$I=-is_{0}^{2}\frac{\pi}{(1+\frac{2\alpha}{\pi^{2}})^{2}}\overline{\tau}=-is_{0}^{2}\frac{\pi}{h(\frac{\pi}{2})^{2}}\overline{\tau}.$
(41)

Thus, the partition function is given by

$Z^{{\partial\textup{I}}}=\int\,d\sigma\,e^{-i\frac{\pi\sigma^{2}}{h(\frac{\pi}{2})^{2}}\overline{\tau}}Z^{{\textup{I}}}_{1-\text{loop}}(\sigma,\alpha)\,,$
(42)

where the integration contour is chosen along the imaginary direction of
$s_{0}$ i.e. $s_{0}=i\sigma$ with $\sigma\in\mathbb{R}$. The one loop
determinant for each chiral multiplet is given by Hama:2011ea 222Note that our
Killing spinor is normalized differently than Hama:2011ea . In particular, at
$r=\frac{\pi}{2}$ it is given by
$\overline{\xi}_{i}\xi^{i}=2h(\frac{\pi}{2})^{2}$. Also, the localization
background at $r=\frac{\pi}{2}$ is $S=\frac{s_{0}}{h(\frac{\pi}{2})^{2}}$ and
$D_{3}=\frac{s_{0}}{h(\frac{\pi}{2})}$.

$Z^{{\textup{I}}}_{1-\text{loop}}(\sigma,\alpha)=\prod_{n>0}\frac{n+1-q+i\frac{\sigma}{h(\frac{\pi}{2})}}{n-1+q-i\frac{\sigma}{h(\frac{\pi}{2})}}\,.$
(43)

Thus, we see that the partition function depends trivially on
$h(\frac{\pi}{2})$.

###  3.2 Squashing preserving $U(1)\times U(1)$

Next, we consider the deformation given by (10). The localization background
is obtained by solving the bosonic equations $\delta\lambda^{i}=0$ and is
given by Hama:2012bg

$D_{1}=-\frac{h(r,\theta)s_{0}}{f(\theta)g(r,\theta)}\sin(\phi+\chi),\quad
D_{2}=-\frac{h(r,\theta)s_{0}}{f(\theta)g(r,\theta)}\cos(\phi+\chi),\quad
D_{3}=-\frac{s_{0}}{f(\theta)}$ (44)

and $s_{0}$ is the constant value of the scalar field $S$. Rest all other
fields are zero. The bulk action (see appendix D) evaluated on the
localization background is given by

$I_{g}=-\frac{2\pi^{2}}{g^{2}_{YM}}\ell\tilde{\ell}\,s^{2}_{0}\,.$ (45)

Similarly, the boundary action evaluated on the localization background is

$I_{\partial g}=\frac{i\theta}{2}s^{2}_{0}\,\ell\tilde{\ell}\,.$ (46)

Thus, the complete action on the localization background is given by

$I=I_{g}+I_{\partial
g}=i\ell\tilde{\ell}s^{2}_{0}\pi\Big{(}\frac{\theta}{2\pi}+\frac{2\pi
i}{g^{2}_{YM}}\Big{)}=i\ell\tilde{\ell}\pi\tau s^{2}_{0}\,.$ (47)

The partition function is given by

$Z^{{\partial\textup{II}}}=\int
d\sigma\,e^{i\ell\tilde{\ell}\pi\tau\sigma^{2}}Z^{{\textup{II}}}_{1-\text{loop}}\,,$
(48)

where the integration contour is chosen along the imaginary direction of
$s_{0}$ i.e. $s_{0}=i\sigma$ with $\sigma\in\mathbb{R}$ and
$Z^{{\textup{II}}}_{1-\text{loop}}$ is the one loop contribution from the
matter multiplets at the boundary. The one-loop contribution from the $n_{+}$
and $n_{-}$ chiral multiplets at the boundary with R-charge $q_{+}$ and
$q_{-}$, respectively is given in terms of hyperbolic Gamma function
Hama:2011ea ; Imamura:2011wg

$Z^{{\textup{II}}}_{1-\text{loop}}=\Big{(}\Gamma_{h}(\ell\tilde{\ell}\sigma+i\omega
q_{+};i\omega_{1},i\omega_{2})\Big{)}^{n_{+}}\Big{(}\Gamma_{h}(-\ell\tilde{\ell}\sigma+i\omega
q_{-};i\omega_{1},i\omega_{2})\Big{)}^{n_{-}}\,,$ (49)

where $\omega_{1}=b,\omega_{2}=\frac{1}{b}$ and
$\omega=\frac{1}{2}(\omega_{1}+\omega_{2})$. The parameter $b$ is the
squashing parameter given by $b=\sqrt{\frac{\tilde{\ell}}{\ell}}$. For our
purposes, it is convenient to use the integral representation of the
hyperbolic Gamma function. It is given by

$\Gamma_{h}(z;w_{1},w_{2})=e^{i\int^{\infty}_{0}\frac{dy}{y}\Big{(}\frac{z-w}{w_{1}w_{2}y}-\frac{\sin
2y(z-w)}{2\sin w_{1}y\,\sin w_{2}y}\Big{)}}\,.$ (50)

The above integral is well defined for $0<\text{Im}\,z<2\text{Im}\,w$, where
$w=\frac{1}{2}(w_{1}+w_{2})$. In the case when there is no squashing, the
hyperbolic Gamma function reduces to Jafferis $\ell$-function. The precise
relation is

$\Gamma_{h}(z;i,i)=e^{\ell(1+iz)}\,.$ (51)

Next, we will use the above partition function to compute the 2-point
correlation function of boundary energy-momentum tensor.

##  4 $\tau_{R}$ computation

In this section, we will compute the 2-point function of boundary energy-
momentum tensor for various matter degrees of freedom at the boundary
interacting with a bulk photon. The 2-point function of energy momentum tensor
in flat space is given by

$\displaystyle<T_{\mu\nu}(x)T_{\rho\sigma}(0)>$ $\displaystyle=$
$\displaystyle-\frac{\tau_{R}}{64\pi^{2}}(\delta_{\mu\nu}\partial^{2}-\partial_{\mu}\partial_{\nu})(\delta_{\rho\sigma}\partial^{2}-\partial_{\rho}\partial_{\sigma})\frac{1}{x^{2}}$
(52)
$\displaystyle+\frac{\tau_{R}}{64\pi^{2}}\Big{(}(\delta_{\mu\rho}\partial^{2}-\partial_{\mu}\partial_{\rho})(\delta_{\nu\sigma}\partial^{2}-\partial_{\nu}\partial_{\sigma})+(\mu\leftrightarrow\nu)\Big{)}\frac{1}{x^{2}}\,.$

As discussed previously, the quantity of interest, $\tau_{R}$ can be computed
by placing the theory on a squashed sphere. It is given by

$\tau_{R}=\frac{2}{\pi^{2}}\text{Re}\frac{\partial^{2}F}{\partial
b^{2}}\Big{|}_{b=1},\qquad F=-\ln Z^{\partial}\,.$ (53)

###  4.1 Free matter at the boundary

We begin with the case of the free conformal matter at the boundary i.e. there
is no interaction with the bulk photon. This case will be important for our
future discussion when we will consider the interaction to be very small. For
a conformal chiral matter, we have $q=\frac{1}{2}$. Thus, the one-loop
determinant is given by

$Z^{{}^{\prime}\text{chiral}}_{1-\text{loop}}=e^{-F}=\Gamma_{h}(\omega(iq);i\omega_{1},i\omega_{2})=\exp\Big{(}i\int^{\infty}_{0}\frac{dx}{x}\Big{[}-\frac{\omega(iq)-i\omega}{\omega_{1}\omega_{2}x}+\frac{\sin(2x(iq\omega-\omega))}{2\sinh\omega_{1}x\,\sinh\omega_{2}x}\Big{]}\Big{)}\,.$
(54)

In the above $\omega_{1}=b,\omega_{2}=\frac{1}{b}$ and
$\omega=\frac{1}{2}(\omega_{1}+\omega_{2})$. Thus, the free energy is given as

$F=\int^{\infty}_{0}\frac{dx}{x}\Big{[}-\frac{(b+b^{-1})(q-1)}{2x}+\frac{\sinh(x(b+b^{-1})(q-1))}{2\sinh
bx\,\sinh b^{-1}x}\Big{]}\,.$ (55)

Substituting the R-charge $q=\frac{1}{2}$ in the above, we get the free energy

$F=-\int^{\infty}_{0}\frac{dx}{x}\Big{[}-\frac{(b+b^{-1})}{4x}+\frac{\sinh(\frac{x}{2}(b+b^{-1}))}{2\sinh
bx\,\sinh b^{-1}x}\Big{]}\,.$ (56)

Calculating the second derivative w.r.t $b$ and then setting $b=1$, we obtain

$\frac{\partial^{2}F}{\partial
b^{2}}\Big{|}_{b=1}=-\int^{\infty}_{0}\frac{dx}{x}\Big{(}-\frac{1}{2x}-\frac{x}{2}\frac{\cosh
x}{\sinh^{2}x}+\frac{x^{2}}{\sinh^{3}x}\Big{)}=\frac{\pi^{2}}{8}\,.$ (57)

Thus, the coefficient $\tau_{R}$ for a free chiral multiplet is

$\tau_{R}=\frac{1}{4}\,.$ (58)

###  4.2 Interacting case: Non chiral theory

Next, we will consider the partition function of non-chiral matter at the
boundary interacting with the bulk photon. It consists of an equal number of
chiral multiplets of positively and negatively charged coupled to a single
$U(1)$ gauge field. In this case, the squashing dependent partition function
is given by333Here onwards we will consider only the squashing dependent part
of the partition function, ignoring the overall factor of $\ell\tilde{\ell}$.

$Z^{\partial}=\int\,d\sigma\,e^{i\sigma^{2}\pi\tau}\,(\Gamma_{h}(\sigma+i\omega
q_{+};i\omega_{1},i\omega_{2}))^{n}(\Gamma_{h}(-\sigma+i\omega
q_{-};i\omega_{1},i\omega_{2}))^{n}\,.$ (59)

In the above expression, we have not included the monopole charge $q_{t}$
since it vanishes at the extremum in the present case.

We first consider the case of $n=1$ i.e. of two oppositely charged chiral
multiplets. Using the exponential form of the hyperbolic gamma function, the
partition function is given by

$\displaystyle Z^{\partial}$ $\displaystyle=$
$\displaystyle\int\,d\sigma\,e^{i\sigma^{2}\pi\tau+G(\sigma)}\,,$ (60)

where

$G(\sigma)=\int^{\infty}_{0}\frac{dy}{y}\Big{[}\frac{2\omega(q_{f}-1)}{y}\\\
+\frac{i\sin(2y(\sigma+i\omega q_{+}-i\omega))+i\sin(2y(-\sigma+i\omega
q_{-}-i\omega))}{2\sinh by\,\sinh b^{-1}y}\Big{]}\,.$

We will further write $q_{\pm}$ as $q_{\pm}=q_{f}\pm q_{g}$.
Large $\tau$ analysis: We will start with saddle point calculation in the weak
coupling limit i.e. $|\tau|>>1$. In this limit, the matter degrees of freedom
at the boundary interact weakly with the bulk photon, and as a result, one
would expect $\tau_{R}\sim\frac{2}{4}=\frac{1}{2}$ to leading order in large
$|\tau|$ expansion. We will see this explicitly below.
In the large $|\tau|$ limit, the partition function on S3 is extremum at the
value $q_{\pm}$ given by Gupta:2019qlg

$q_{g}=0,\quad\text{and}\quad
q_{f}=\frac{1}{2}-\frac{\sin\alpha}{\pi|\tau|}-\frac{\pi^{2}-4+(4+(1+2n)\pi^{2})\cos
2\alpha}{4\pi^{2}|\tau|^{2}}+\mathcal{O}(|\tau|^{-3})\,,$ (61)

where $\tau=|\tau|e^{i\alpha}$ . We will do saddle point calculation with the
above choice of $q_{f}$ and $q_{g}$. We see that $\sigma=0$ is the saddle
point, since the function $G(\sigma)$ satisfies

$\frac{\partial G(\sigma)}{\partial\sigma}\Big{|}_{\sigma=0}=0\,.$ (62)

Let the function has the following Taylor series expansion about $\sigma=0$

$G(\sigma)=G_{0}+\frac{1}{2}G_{2}\sigma^{2}+\mathcal{O}(\sigma^{3})$ (63)

The value of the above functions are

$G_{0}=\int^{\infty}_{0}\frac{dy}{y}\Big{[}\frac{2\omega(q_{f}-1)}{y}\\\
-\frac{\sinh(2y\omega(q_{f}-1))}{\sinh by\,\sinh b^{-1}y}\Big{]}\,,$

and

$G_{2}=\int^{\infty}_{0}dy\Big{[}\frac{4y\sinh(2y\omega(q_{f}-1))}{\sinh
by\,\sinh b^{-1}y}\Big{]}\,.$ (64)

Note that both the integrals are entirely convergent since $q_{f}<2$. Thus the
partition function to a leading order is given by

$Z^{\partial}=e^{G_{0}}\Big{(}\frac{1}{\sqrt{-i\tau}}+\frac{1}{2}G_{2}\frac{1}{2\pi(-i\tau)^{3/2}}\Big{)}+\mathcal{O}(|\tau|^{-5/2})\,.$
(65)

Thus the real part of free energy is

$|Z^{\partial}|^{2}=e^{-2\text{Re}F_{\partial}}=\frac{e^{\overline{G}_{0}+G_{0}}}{|\tau|}\Big{(}1+\frac{i}{4\pi|\tau|^{2}}(G_{2}\overline{\tau}-\overline{G}_{2}\tau)\Big{)}+\mathcal{O}(|\tau|^{-3})\,.$
(66)

Thus the expression for $\tau_{R}$ is

$\tau_{R}=\frac{2}{\pi^{2}}\text{Re}\frac{\partial^{2}F_{\partial}}{\partial
b^{2}}\Big{|}_{b=1}=-\frac{1}{\pi^{2}}\frac{\partial^{2}}{\partial
b^{2}}(\overline{G}_{0}+G_{0}+\frac{i}{4\pi|\tau|^{2}}(\overline{\tau}G_{2}-\tau\overline{G}_{2}))\Big{|}_{b=1}$
(67)

We will calculate the above one by one

$\frac{\partial^{2}}{\partial
b^{2}}(\overline{G}_{0}+G_{0})\Big{|}_{b=1}=-\frac{\pi^{2}}{2}-\frac{\pi}{3}\frac{\sin\alpha}{|\tau|}-\frac{\pi}{3}\frac{\sin\alpha}{|\tau|}+\mathcal{O}(\tau^{-2})=-\frac{\pi^{2}}{2}-\frac{2\pi}{3}\frac{\sin\alpha}{|\tau|}+\mathcal{O}(\tau^{-2})$
(68)

Next we calculate

$\frac{\partial^{2}}{\partial
b^{2}}(\frac{i}{4\pi|\tau|^{2}}(\overline{\tau}G_{2}-\tau\overline{G}_{2}))\Big{|}_{b=1}=\frac{\text{Im}\tau}{2\pi|\tau|^{2}}\frac{\partial^{2}}{\partial
b^{2}}G_{2}\Big{|}_{b=1}=\frac{\text{Im}\tau}{2\pi|\tau|^{2}}\frac{\pi^{2}}{2}(-8+\pi^{2})=\frac{\pi\text{Im}\tau}{4|\tau|^{2}}(-8+\pi^{2})$
(69)

Thus we have

$\tau_{R}=\frac{1}{2}+\frac{\sin\alpha}{12\pi|\tau|}(32-3\pi^{2})+\mathcal{O}(\tau^{-2})\,,$
(70)

where $\tau=|\tau|e^{i\alpha}$.
We can easily extend the above analysis for the case of $n_{\pm}=n$. In this
case, the partition function is given by

$Z^{\partial}=\int\,d\sigma\,e^{i\sigma^{2}\pi\tau+\tilde{F}(\sigma)}\,,$ (71)

where

$\tilde{F}(\sigma)=n\int^{\infty}_{0}\frac{dy}{y}\Big{[}\frac{2\omega(q_{f}-1)}{y}+\frac{i\sin(2y(\sigma+i\omega
q_{+}-i\omega))+i\sin(2y(-\sigma+i\omega q_{-}-i\omega))}{2\sinh by\,\sinh
b^{-1}y}\Big{]}$ (72)

Since $n$ is an overall factor, one, therefore, has

$\tau_{R}=\frac{n}{2}+\frac{n\sin\alpha}{12\pi|\tau|}(32-3\pi^{2})+\mathcal{O}(\tau^{-2})\,.$
(73)

Small $\tau$ analysis: Next, we consider the large coupling expansion, i.e.
$|\tau|<<1$, for the coefficient $\tau_{R}$. We will focus here the case of
$n_{+}=n_{-}=1$. The partition function is

$Z^{\partial}=\int\,d\sigma\,e^{i\sigma^{2}\pi\tau}\,\Gamma_{h}(\sigma+i\omega
q_{+};i\omega_{1},i\omega_{2})\Gamma_{h}(-\sigma+i\omega
q_{-};i\omega_{1},i\omega_{2})\,.$ (74)

We will evaluate the above partition function for the value of $q_{f}$ and
$q_{g}$ which extremizes the corresponding $S^{3}$ partition function in the
limit of $|\tau|<<1$. To the leading order, the extremum occurs at $q_{g}=0$
and

$q_{f}=\frac{1}{3}+\frac{54\sqrt{3}-90\pi+8\sqrt{3}\pi^{2}}{9\Big{(}8\pi(3\sqrt{3}-2\pi)-27\Big{)}}\,|\tau|\sin\alpha+\mathcal{O}(\tau^{2})\,.$
(75)

Thus, we have

$\tau_{R}=\frac{2}{\pi^{2}}\text{Re}\frac{\partial^{2}F}{\partial
b^{2}}\Big{|}_{b=1}=-\frac{2}{\pi^{2}}\text{Re}\Big{(}\frac{1}{Z^{\partial}}\partial^{2}_{b}Z^{\partial}\Big{|}_{b=1}\Big{)}\,,$
(76)

where the RHS is evaluated at (75).
In the above we have used the relation

$\partial_{b}\Gamma_{h}(\pm\sigma+iq_{\pm}\omega;ib,\frac{i}{b})\Big{|}_{b=1}=0\,.$
(77)

Calculating the second derivative w.r.t $b$, we obtain

$\partial^{2}_{b}Z^{\partial}\Big{|}_{b=1}=\int\,d\sigma\,e^{i\sigma^{2}\pi\tau}\,e^{\ell(1-i\sigma-
q_{-})+\ell(1+i\sigma-
q_{+})}\Big{(}X^{+}(\sigma,q_{+})+X^{-}(\sigma,q_{-})\Big{)}\,,$ (78)

where

$\displaystyle X^{\pm}(\sigma,q_{\pm})$ $\displaystyle=$
$\displaystyle\partial^{2}_{b}\Big{(}i\int^{\infty}_{0}\frac{dx}{x}\Big{[}\frac{\sin(2x(\pm\sigma+iq_{\pm}\omega-i\omega))}{2\sinh
bx\,\sinh\frac{x}{b}}-\frac{\sigma+i\omega
q_{\pm}-i\omega}{x}\Big{]}\Big{)}\Big{|}_{b=1}\,,$ $\displaystyle=$
$\displaystyle\int^{\infty}_{0}dx\,\Big{[}(q_{\pm}-1)\Big{(}\frac{1}{x^{2}}-\frac{\cosh(2x(1-q_{\pm}\mp
i\sigma))}{\sinh^{2}x}\Big{)}-\frac{\sinh(2x)-2x}{2\sinh^{4}x}\sinh(2x(1-q_{\pm}+\pm
2ix\sigma))\Big{]}$

In the above we have used the relation

$\Gamma_{h}(z;i,i)=e^{\ell(1+iz)}\,,$ (80)

where $\ell(z)$ is the Jafferis $\ell$-function. To evaluate (78) at strong
coupling, we go to the dual frame where the computation reduces to the weak
coupling computation. The dual frame is obtained by using the Fourier
transform Gupta:2019qlg

$\int d\sigma\,e^{\ell(1-q_{+}+i\sigma)+\ell(1-q_{-}-i\sigma)+2\pi
i\kappa\sigma}=e^{\ell(1-q_{+}-q_{-})+\ell(\frac{q_{+}+q_{-}}{2}+i\kappa)+\ell(\frac{q_{+}+q_{-}}{2}-i\kappa)+\pi\kappa(q_{+}-q_{-})}\,.$
(81)

Using the above identity, we obtain

$\displaystyle\partial^{2}_{b}Z^{\partial}\Big{|}_{b=1}$ $\displaystyle=$
$\displaystyle\int\,d\sigma\,\,d\kappa\,e^{-2\pi
i\kappa\sigma}e^{i\sigma^{2}\pi\tau}\,e^{\ell(1-q_{+}-q_{-})+\ell(i\kappa+\frac{q_{-}+q_{+}}{2})+\ell(-i\kappa+\frac{q_{-}+q_{+}}{2})}e^{\pi\kappa(q_{+}-q_{-})}$
$\displaystyle\times\Big{(}X^{+}(\sigma,q_{+})+X^{-}(\sigma,q_{-})\Big{)}\,,$
$\displaystyle=$
$\displaystyle\int\,d\kappa\,e^{-\frac{i\pi\kappa^{2}}{\tau}}\,e^{\ell(1-q_{+}-q_{-})+\ell(i\kappa+\frac{q_{-}+q_{+}}{2})+\ell(-i\kappa+\frac{q_{-}+q_{+}}{2})}e^{\pi\kappa(q_{+}-q_{-})}$
$\displaystyle\times\int_{\text{Im}\tilde{\sigma}=\frac{\kappa\text{Im}\tau}{|\tau|^{2}}}\,d\tilde{\sigma}\,e^{i\tilde{\sigma}^{2}\pi\tau}\Big{(}X^{+}(\tilde{\sigma}+\frac{\kappa}{\tau},q_{+})+X^{-}(\tilde{\sigma}+\frac{\kappa}{\tau},q_{-})\Big{)}\,,$
$\displaystyle=$
$\displaystyle\int\,d\kappa\,e^{-\frac{i\pi\kappa^{2}}{\tau}}\,e^{\ell(1-q_{+}-q_{-})+\ell(i\kappa+\frac{q_{-}+q_{+}}{2})+\ell(-i\kappa+\frac{q_{-}+q_{+}}{2})}e^{\pi\kappa(q_{+}-q_{-})}$
$\displaystyle\times\int^{\infty}_{-\infty}\,d\tilde{\sigma}\,e^{i\tilde{\sigma}^{2}\pi\tau}\Big{(}X^{+}(\tilde{\sigma}+\frac{p}{\tau},q_{+})+X^{-}(\tilde{\sigma}+\frac{p}{\tau},q_{-})\Big{)}\,,$
$\displaystyle=$
$\displaystyle\int\,d\kappa\,e^{-\frac{i\pi\kappa^{2}}{\tau}}\,e^{\ell(1-2q_{f})+\ell(i\kappa+q_{f})+\ell(-i\kappa+q_{f})}$
$\displaystyle\times\int^{\infty}_{0}\,\frac{dx}{\sqrt{-i\tau}}e^{-\frac{ix^{2}}{\pi\tau}}\Big{[}e^{\frac{ix^{2}}{\pi\tau}}\frac{2(q_{f}-1)}{x^{2}}-\frac{2(q_{f}-1)}{\sinh^{2}x}\cos\frac{2px}{\tau}\cosh(2x(q_{f}-1))$
$\displaystyle-\frac{\sinh(2x)-2x}{\sinh^{4}x}\cos\frac{2px}{\tau}\sinh(2x(1-q_{f}))\Big{]}\,.$

In the above we have deformed the contour back to the real axis and also
substituted $q_{\pm}=q_{f}$, since $q_{g}=0$. In the limit $|\tau|\sim 0$, we
evaluate the above integral in the saddle point approximation. In this
approximation, we get

$\displaystyle\frac{1}{Z^{\partial}}\partial^{2}_{b}Z^{\partial}\Big{|}_{b=1}$
$\displaystyle=$
$\displaystyle\int^{\infty}_{0}\,dx\Big{[}\frac{2(q_{f}-1)}{x^{2}}+\frac{1}{2}e^{-2\ell(\frac{1}{3})+\ell(i\frac{x}{\pi}+\frac{1}{3})+\ell(-i\frac{x}{\pi}+\frac{1}{3})}(2+|\tau|f_{1}(x)+|\tau|f_{1}(-x)-2|\tau|f_{1}(0))\times$
(83)
$\displaystyle\times\Big{(}-\frac{2(q_{f}-1)}{\sinh^{2}x}\cosh(2x(q_{f}-1))-\frac{\sinh(2x)-2x}{\sinh^{4}x}\sinh(2x(1-q_{f}))\Big{)}\Big{]}+\mathcal{O}(|\tau|^{2})\,,$

where the function $f_{1}(x)$ is given in the appendix E. Thus, we have

$\displaystyle\tau_{R}$ $\displaystyle=$
$\displaystyle-\frac{2}{\pi^{2}}\text{Re}\int^{\infty}_{0}\,dx\Big{[}\frac{2(q_{f}-1)}{x^{2}}+\frac{1}{2}e^{-2\ell(\frac{1}{3})+\ell(i\frac{x}{\pi}+\frac{1}{3})+\ell(-i\frac{x}{\pi}+\frac{1}{3})}(2+|\tau|f_{1}(x)+|\tau|f_{1}(-x)-2|\tau|f_{1}(0))\times$
(84)
$\displaystyle\times\Big{(}-\frac{2(q_{f}-1)}{\sinh^{2}x}\cosh(2x(q_{f}-1))-\frac{\sinh(2x)-2x}{\sinh^{4}x}\sinh(2x(1-q_{f}))\Big{)}\Big{]}+\mathcal{O}(|\tau|^{2})\,,$

with the R-charge $q_{f}$ to the first order in $|\tau|$ given in (75). We
could not evaluate the above integral explicitly; however, numerical
evaluation is possible for the various values of the angle $\alpha$. In
particular, for $\alpha=\frac{\pi}{2}$, we obtain

$\tau_{R}=0.545-0.07035|\tau|+\mathcal{O}(|\tau|^{2})\,.$ (85)

Figure 1: The plot of $\tau_{R}$ vs. $\text{Im}\tau$ for the $n_{+}=n_{-}=1$
theory. The dots represent the numerical computations and the solid lines are
the saddle point approximation.

The full behaviour of $\tau_{R}$ as a function of $\text{Im}\tau$ can be seen
in the figure 1.
Large n-analysis: We can also find the expression for $\tau_{R}$ in the large
$n$-limit. In this case, we consider the partition function given by

$Z^{\partial}=\int\,d\sigma\,e^{i\sigma^{2}\pi\tau}\,(\Gamma_{h}(\sigma+i\omega
q_{+};i\omega_{1},i\omega_{2}))^{n}(\Gamma_{h}(-\sigma+i\omega
q_{-};i\omega_{1},i\omega_{2}))^{n}\,.$ (86)

The computation of $\tau_{R}$ in $\frac{1}{n}$-expansion proceeds in a similar
manner as in previous cases and we will not repeat here. To compute
$\tau_{R}$, we need to know the R-charge which maximize the free energy on
$S^{3}$. In the $\frac{1}{n}$-expansion it is given by Gupta:2019qlg

$q_{f}=\frac{1}{2}-\frac{2}{\pi^{2}n}+\mathcal{O}(n^{-2}),\quad\text{and}\quad
q_{g}=0\,.$ (87)

Using the above expression for the R-charge, the explicit computation then
gives

$\tau_{R}=\frac{n}{2}-2\Big{(}1-\frac{8}{3\pi^{2}}\Big{)}+\mathcal{O}(n^{-1})\,.$
(88)

###  4.3 Interacting case: Chiral theory

Next, we consider the chiral theory with a positively charged matter field at
the boundary i.e. $n_{+}=1,n_{-}=0$. The squashing dependent partition
function is given by

$Z_{\text{chiral}}^{\partial}=\int\,d\sigma\,e^{i\sigma^{2}\pi\tau}\,\Gamma_{h}(\sigma+i\omega
q_{g};i\omega_{1},i\omega_{2})e^{2\pi\omega q_{t}\sigma}\,,$ (89)

where $q_{t}$ is the monopole charge. Note that since
$\partial_{b}\omega\Big{|}_{b=1}=0$, we have a vanishing one point function of
energy momentum tensor

$\partial_{b}Z_{\text{chiral}}^{\partial}\Big{|}_{b=1}=0\,.$ (90)

For the computation of $\tau_{R}$, we need the second derivative of the
partition function with respect to the squashing parameter and is given by

$\partial^{2}_{b}Z_{\text{chiral}}^{\partial}\Big{|}_{b=1}=\int\,d\sigma\,e^{i\pi\tau\sigma^{2}+\ell(1-q_{g}+i\sigma)+2\pi
q_{t}\sigma}X(\sigma)\,,$ (91)

where

$\displaystyle X(\sigma)$ $\displaystyle=$ $\displaystyle 2\pi\sigma
q_{t}+\int^{\infty}_{0}dx\,\Big{[}\frac{q_{g}-1}{x^{2}}+\frac{(1-q_{g})\cosh(2x(q_{g}-1)-2ix\sigma)}{\sinh^{2}x}$
(92)
$\displaystyle-\frac{1}{2\sinh^{4}x}(\sinh(2x)-2x)\sinh(2x(1-q_{g})+2ix\sigma)\Big{]}$

We first calculate the $\tau_{R}$ in the weak coupling limit i.e.
$|\tau|\rightarrow\infty$. The superconformal R-symmetry is determined by the
value of $q_{g}$ and $q_{t}$ that extremize the partition function on S3. In
the saddle point approximation, these are given by Gupta:2019qlg

$q_{g}=\frac{1}{2}-\frac{\text{Im}\tau}{\pi|\tau|^{2}}+\mathcal{O}(|\tau|^{-2}),\quad
q_{t}=-\frac{\text{Re}\tau}{4|\tau|^{2}}+\mathcal{O}(|\tau|^{-2})\,.$ (93)

The computation of $\tau_{R}$ proceeds similarly as in the non-chiral case and
we will not repeat here. The saddle point approximation in the limit
$|\tau|\rightarrow\infty$ gives

$\tau_{R}=\frac{1}{4}+\frac{\sin\alpha}{24\pi|\tau|}(32-3\pi^{2})+\mathcal{O}(\tau^{-2})\,.$
(94)

Note that to the order $\mathcal{O}(\frac{1}{|\tau|})$, the result of
$\tau_{R}$ is exactly half of the non chiral case (70).

Figure 2: The plot of $\tau_{R}$ vs. $\text{Im}\tau$ for the $n_{+}=1,n_{-}=0$
theory. The dots represent the numerical computations and the solid lines are
the saddle point approximation.

For the computation of $\tau_{R}$ at the strong coupling $|\tau|<1$, we will
use the following identity Gupta:2019qlg

$\int d\sigma\,e^{\frac{i\pi}{2}\sigma^{2}+2\pi
i\sigma\kappa+\ell(1-q_{g}+i\sigma)}=e^{-2\pi
i\kappa^{2}+\ell(\frac{q_{g}+1}{2}+i\kappa)-\frac{3\pi
i}{8}(q_{g}+2i\kappa-\frac{1}{3})^{2}+\frac{i\pi}{12}}\,.$ (95)

Using the above identity, the derivative of the partition function can be
written as

$\displaystyle\partial^{2}_{b}Z_{\text{chiral}}^{\partial}\Big{|}_{b=1}$
$\displaystyle=$ $\displaystyle\int\,d\kappa\,dp\,e^{-2\pi ip\kappa}e^{2\pi
q_{t}\kappa}e^{i\pi(\tau-\frac{1}{2})\kappa^{2}}e^{-2\pi
ip^{2}}e^{\ell(\frac{\tilde{q}+1}{2})-\frac{3\pi
i}{8}(\tilde{q}-\frac{1}{3})^{2}+\frac{i\pi}{12}}X(\kappa)\,,$ (96)

where the function $X(\kappa)$ is given in (92). After some simplifications
and performing the integration over $\kappa$, the integral on the RHS can be
written as

$\displaystyle\partial^{2}_{b}Z_{\text{chiral}}^{\partial}\Big{|}_{b=1}=e^{\frac{\pi
i}{2}q^{2}_{t}-\frac{3\pi i}{8}(q_{g}-\frac{1}{3})^{2}-\frac{3\pi
i}{2}q_{t}(q_{g}-\frac{1}{3})+\frac{i\pi}{12}}\int\,dp\,e^{-\frac{\pi
i}{2}p^{2}}e^{\ell(\frac{q_{g}+1}{2}+q_{t}+ik)+2\pi
p(\frac{3q_{g}-1}{4}-\frac{q_{t}}{2})}\tilde{Y}(p)\,.$

with

$\displaystyle\tilde{Y}(k)$ $\displaystyle=$ $\displaystyle 2\pi
q_{t}\frac{e^{i\frac{\pi}{4}}}{\sqrt{-i(\tau-\frac{1}{2})^{3}}}e^{-\frac{i\pi}{\tau-\frac{1}{2}}k^{2}}k+\int^{\infty}_{-\infty}dx\,\frac{1}{2\sqrt{-i(\tau-\frac{1}{2})}}\Big{[}\frac{q_{g}-1}{x^{2}}e^{-\frac{i\pi}{\tau-\frac{1}{2}}k^{2}}$
$\displaystyle+\frac{(1-q_{g})}{\sinh^{2}x}e^{2x(q_{g}-1)}e^{-\frac{i}{\pi(\tau-\frac{1}{2})}(\pi
k+x)^{2}}+\frac{(\sinh(2x)-2x)}{2\sinh^{4}x}e^{2x(q_{g}-1)}e^{-\frac{i}{\pi(\tau-\frac{1}{2})}(\pi
k+x)^{2}}\Big{]}\,.$

Next, we compute the integral (4.3) in saddle point approximation when the
coupling $\tau\sim\frac{1}{2}$. Let us define a complex coupling
$\tau^{\prime}$ by

$\tau^{\prime}=-\frac{1}{2}-\frac{1}{\tau-\frac{1}{2}}\,.$ (99)

In the limit $|\tau^{\prime}|>>1$, the superconformal R-symmetry is given by

$q_{g}=\frac{1}{4}+\mathcal{O}(|\tau^{\prime}|^{-2}),\quad
q_{g}+2q_{t}=\frac{2\sin\alpha^{\prime}}{\pi|\tau^{\prime}|}+\frac{1}{|\tau^{\prime}|^{2}}\Big{(}\frac{3}{4}-\frac{3}{\pi^{2}}+(\frac{5}{4}+\frac{3}{\pi^{2}})\cos
2\alpha^{\prime}\Big{)}+\mathcal{O}(|\tau^{\prime}|^{-3})\,,$ (100)

where $\tau^{\prime}=|\tau^{\prime}|e^{i\alpha^{\prime}}$. Evaluating the
integral (4.3) in the saddle point approximation, using the above R-charge, we
find to the first subleading order

$\tau_{R}=\frac{1}{4}-\frac{2}{\pi^{2}}\text{Re}\frac{1}{\tau^{\prime}}\int_{-\infty}^{\infty}dx\,g(x,\alpha^{\prime})+\mathcal{O}(\tau^{\prime-2})\,,$
(101)

where the explicit form the function $g(x,\alpha^{\prime})$ is given in the
appendix E. For $\alpha^{\prime}=\frac{\pi}{2}$, we have

$\tau_{R}=\frac{1}{4}+\frac{0.0570}{|\tau^{\prime}|}+\mathcal{O}(|\tau^{\prime}|^{-2})\,.$
(102)

The full behaviour of $\tau_{R}$ as a function of $\text{Im}\tau$ can be seen
in the figure 2.

##  5 Conclusion

Our original motivation for the present work is to compute the correlation
function of the energy-momentum tensor for the 4-dimensional bCFT. The most
general form of the 2-point function of the energy-momentum tensor requiring
the conservation law and conformal invariance is known McAvity:1993ue ;
McAvity:1995zd ; Herzog:2017xha . It is given in terms of 3 arbitrary
functions that encode the dynamics of the bCFT. Our goal was to compute these
coefficients in a supersymmetric bCFT. In this direction, we computed the
partition function of the mixed dimensional supersymmetric dimensional QED on
squashed hemispheres. We considered two different squashings that preserve
either $SU(2)\times U(1)$ or $U(1)\times U(1)$ isometry of the original
sphere. The boundary free energy depends on the deformation parameter. In the
case of the squashing that preserves $SU(2)\times U(1)$ isometry of the
original sphere, the free energy is trivial as a function of the deformation.
In contrast, the boundary-free energy depends on the squashing parameter in
the case of the squashing that preserves $U(1)\times U(1)$ isometry. We then
computed the coefficient $\tau_{R}$ by differentiating the free energy twice
with respect to the squashing parameter. On important feature of $\tau_{R}$ is
that it depends on the bulk marginal coupling $\tau$. We then find the
behaviour of $\tau_{R}$ as we change the coupling. For the non-chiral case, we
find that the coefficient $\tau_{R}$ decreases from strong coupling to weak
coupling. In contrast, the computation in the chiral case reveals that it
first increases and then decreases as we change the coupling. It would be
interesting to understand the physical reason behind it.

As a future direction, it would be interesting to extend the above computation
for the case when the bulk involves non-abelian gauge fields, in particular
for the $\mathcal{N}=4$ SYM. See VanRaamsdonk:2020djx  for a recent work in
this direction. Another direction would be to compute the anomalous
contributions to the trace of the energy-momentum tensor. These contributions
depend on the extrinsic curvature of the boundary. For the computation of
these contributions, we need to consider the manifold where the boundary has
non zero extrinsic curvature. The squashed hemisphere has a vanishing
extrinsic curvature. It would be interesting to see if it is possible to find
a squashing deformation that preserves some supersymmetry so that the
localization computation can be done, and also has non zero extrinsic
curvature. For example, one way to obtain non zero extrinsic curvature is to
start with a 4-dimensional sphere and put the boundary at
$r\neq\frac{\pi}{2}$.

## Acknowledgments

We thank Imtak Jeon for useful discussion. AR thanks Arnab Kundu, Dharmesh
Jain and Rudranil Basu for various discussions on supersymmetric field
theories. He is also grateful for the hospitality extended by IIT, Ropar
during Jan., 2020 where parts of the project were discussed. He is supported
by “Senior Research Fellowship” under Government of India.

##  Appendix A Charge Conjugation and gamma matrices

In Euclidean signature, the Clifford algebra is given by

$\\{\gamma^{a},\gamma^{b}\\}=2\,\delta^{ab}\ ,$ (103)

where the gamma matrices are Hermitian $(\gamma^{a})^{\dagger}=\gamma^{a}$.
Our convention for the gamma matrices is the same that Narain et al.
Gava:2016oep  use

$\gamma^{a}=\left(\begin{array}[]{cc}0&-\mathrm{i}\sigma^{a}\\\
\mathrm{i}\sigma^{a}&0\end{array}\right)\
;\;\;\;\gamma^{4}=\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)\
,\;\;\;\gamma^{5}=\gamma^{1}\gamma^{2}\gamma^{3}\gamma^{4}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\ .$ (104)

The 4-dimensional charge conjugation matrix is $C=\gamma_{1}\gamma_{3}={\rm
diag}(-\mathrm{i}\sigma_{2},-\mathrm{i}\sigma_{2})$. We choose $C$ to define
symplectic Majorana fermions

$\overline{\xi}_{i}\equiv(\xi_{i})^{\dagger}=\epsilon_{ij}(\xi^{j})^{T}C\ .$
(105)

In particular, $\xi_{1}^{\dagger}=\xi^{2T}C$ and
$\xi_{2}^{\dagger}=-\xi^{1T}C$. Furthermore, $\xi^{2}=C\xi_{1}^{*}$. The
matrix $C$ satisfies the further conjugation properties

$\gamma_{a}^{*}=\gamma_{a}^{T}=C\gamma_{a}C^{-1}\
,\;\;\;C^{\dagger}=C^{T}=C^{-1}=-C\ ,\;\;\;C^{*}=C\ .$ (106)

Let us denote the component of the projector as

$\Pi^{1}_{\pm 1}=\pi_{\pm},\qquad\Pi^{2}_{\pm 2}=\pi_{\mp}$ (107)

where

$\pi_{\pm}=\frac{1}{2}(1\pm i\gamma_{5}\gamma_{n})\,.$ (108)

These satisfy

$C\pi_{\pm}=\pi^{T}_{\mp}C,\qquad\gamma_{A}\pi_{\pm}=\pi_{\pm}\gamma_{A}\,.=\psi_{\mp}\gamma_{n}$
(109)

Next, we define the relation between the 4 dimensional gamma matrices,
$\gamma_{A}$, and 3 dimensional gamma matrices, $\Gamma_{A}$. It is given by

$\Gamma_{A}=\pi_{+}\gamma_{A}$ (110)

With this we see that

$\Gamma^{T}_{A}=\gamma^{T}\pi^{T}_{+}=C\gamma_{A}C^{-1}$ (111)

The above gamma matrices satisfy 3 dimensional transpose relation with the
charge conjugation matrix

$\tilde{C}=C\gamma_{n},\quad\Rightarrow\quad\Gamma^{T}_{A}=-\tilde{C}\Gamma_{A}\tilde{C}^{-1}\,.$
(112)

Given a spinor $\psi^{i}$, the projected spinor is given by
$\psi^{i}_{\pm}=\Pi^{i}_{\pm j}\psi^{j}$. Furthermore, we define

$\psi^{1}_{\pm}=\psi_{\pm},\quad\psi^{2}_{\pm}=\gamma_{n}\psi^{\prime}_{\pm},\qquad\text{where}\quad\pi_{\pm}\psi^{\prime}_{\pm}=\psi^{\prime}_{\pm}\,.$
(113)

We find the reality condition

$\psi^{\dagger}_{\pm}=\psi^{\prime
T}_{\pm}\tilde{C},\quad\psi^{\prime\dagger}_{\pm}=-\psi^{T}_{\pm}\tilde{C}$
(114)

Thus, the pairs $(\psi_{\pm},\psi^{\prime}_{\pm})$ satisfy the condition of
3-dimensional sympletctic Majorana condition.

##  Appendix B Projector and Killing Spinor

In this section, we discuss in details the supersymmetry on a squashed
hemisphere. Now, the presence of boundary breaks half of the supersymmetry
transformation, i.e. the boundary in our case preserves only 4 out of 8
supercharges. For the squashing deformation to preserve the supersymmetry on
the hemisphere, the background needs to satisfy certain conditions. We begin
with the projector, which was discussed in the case of supersymmetric field
theory on the hemisphere Gupta:2019qlg . It is given by

$\Pi^{i}_{\pm\,j}=\frac{1}{2}(\delta^{i}_{j}\pm
i\tau^{i}_{3\,j}\gamma_{5}\gamma_{n})\,.$ (115)

Here, $\gamma_{n}$ is the flat space gamma matrix corresponding to the
direction perpendicular to the boundary. The projector satisfies the following
conditions

$\gamma_{A}\Pi^{i}_{\pm\,j}=\Pi^{i}_{\pm\,j}\gamma_{A},\qquad\gamma_{n}\Pi^{i}_{\pm\,j}=\Pi^{i}_{\mp\,j}\gamma_{n}\,,$
(116)

where $A$ is the boundary index.
Given a Killing spinor $\xi^{i}$ on the squashed sphere, we define the
following

$\xi^{i}_{\pm}=\Pi^{i}_{\pm\,j}\xi^{j}\,.$ (117)

We require the projector so that half of the supersymmetry transformation
parameter vanishes at the boundary. In other words, we require that if $\xi$
be a Killing spinor, then at $r=\frac{\pi}{2}$,
$\Pi^{i}_{+\,j}\xi^{j}\Big{|}_{r=\frac{\pi}{2}}=0$ and
$\xi^{i}_{-}\Big{|}_{r=\frac{\pi}{2}}$ generates the supersymmetry at the
boundary.
We start with the Killing spinor equation on the squashed S4

$\mathcal{D}_{\mu}\xi^{i}+\gamma^{ab}T_{ab}\gamma_{\mu}\xi=\frac{1}{4}\gamma_{\mu}(\gamma^{\nu}\mathcal{D}_{\nu}\xi^{i})\,,$
(118)

where the auxiliary field $T_{ab}$ is non zero for the squashing. Let
$A,B,C..$ be the boundary flat indices and $n$ be the normal direction flat
index. Then

$\mathcal{D}_{A}\xi^{i}+\gamma^{ab}T_{ab}\gamma_{A}\xi=\frac{1}{4}\gamma_{A}(\gamma^{\nu}\mathcal{D}_{\nu}\xi^{i})\,,$
(119)

which can be written in terms of 3-dimensional covariant derivative as

$\displaystyle\nabla^{3d}_{A}\xi^{i}+\mathcal{V}^{i}_{A\,j}\xi^{j}+\frac{1}{2}\omega^{Bn}_{A}\gamma_{Bn}\xi^{i}+(T_{BC}\gamma_{BC}+2T_{Bn}\gamma_{Bn})\gamma_{A}\xi^{i}$
$\displaystyle=\frac{1}{4}\gamma_{A}(\gamma^{B}\nabla^{3d}_{B}\xi^{i}+\gamma^{B}\mathcal{V}^{i}_{B\,j}\xi^{j}+\gamma^{n}\mathcal{D}_{n}\xi^{i}+\frac{1}{2}\gamma^{B}\omega^{Cn}_{B}\gamma_{Cn}\xi^{i})\,.$

Normal direction Killing spinor equation is

$\mathcal{D}_{n}\xi^{i}+\gamma^{ab}T_{ab}\gamma_{n}\xi=\frac{1}{4}\gamma_{n}(\gamma^{\nu}\mathcal{D}_{\nu}\xi^{i})\,,$
(121)

which can be further written as

$\gamma^{n}\mathcal{D}_{n}\xi^{i}+\gamma^{n}\gamma^{ab}T_{ab}\gamma_{n}\xi=\frac{1}{4}(\gamma^{A}\nabla^{3d}_{A}\xi^{i}+\gamma^{B}\mathcal{V}^{i}_{B\,j}\xi^{j}+\frac{1}{2}\gamma^{B}\omega^{Cn}_{B}\gamma_{Cn}\xi^{i}+\gamma^{n}\mathcal{D}_{n}\xi^{i})\,.$
(122)

Solving the above we get

$\gamma^{n}\mathcal{D}_{n}\xi^{i}=\frac{1}{3}(\gamma^{A}\nabla^{3d}_{A}\xi^{i}+\gamma^{B}\mathcal{V}^{i}_{B\,j}\xi^{j}+\frac{1}{2}\gamma^{B}\omega^{Cn}_{B}\gamma_{Cn}\xi^{i}-4\gamma^{n}\gamma^{ab}T_{ab}\gamma_{n}\xi^{i})\,.$
(123)

Substituting the above in (B), we get

$\displaystyle\nabla^{3d}_{A}\xi^{i}+\mathcal{V}^{i}_{A\,j}\xi^{j}+\frac{1}{2}\omega^{Bn}_{A}\gamma_{Bn}\xi^{i}+(T_{BC}\gamma_{BC}+2T_{Bn}\gamma_{Bn})\gamma_{A}\xi^{i}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\gamma_{A}\Big{(}\frac{4}{3}\gamma^{B}\nabla^{3d}_{B}\xi^{i}+\frac{4}{3}\gamma^{B}\mathcal{V}^{i}_{B\,j}\xi^{j}$
$\displaystyle-4\gamma^{n}\gamma^{ab}T_{ab}\gamma_{n}\xi\,+\,\frac{2}{3}\gamma^{B}\omega^{Cn}_{B}\gamma_{Cn}\xi^{i}\Big{)}\,.$

Simplifying further, we obtain

$\displaystyle\nabla^{3d}_{A}\xi^{i}+\mathcal{V}^{i}_{A\,j}\xi^{j}+\frac{1}{2}\omega^{Bn}_{A}\gamma_{Bn}\xi^{i}+T_{BC}(\gamma_{BC}\gamma_{A}+\frac{1}{3}\gamma_{A}\gamma_{BC})\xi^{i}-2T_{Bn}(\gamma_{B}\gamma_{A}\gamma_{n}+\frac{1}{3}\gamma_{A}\gamma_{B}\gamma_{n})\xi^{i}$
$\displaystyle\qquad\qquad\qquad=\frac{1}{3}\gamma_{A}\gamma^{B}\mathcal{D}^{3d}_{B}\xi^{i}+\frac{1}{3}\gamma_{A}\gamma^{B}\mathcal{V}^{i}_{B\,j}\xi^{j}+\frac{1}{6}\gamma_{A}\gamma^{B}\omega^{Cn}_{A}\gamma_{Cn}\xi^{i}\,.$
(125)

In the above $\nabla^{3d}_{A}$ is the 3-dimensional spacetime covariant
derivative. Now, we apply the projector $\Pi^{i}_{\pm j}$ and evaluate the
above at $r=\frac{\pi}{2}$. We get with $\Pi^{i}_{-j}$

$\displaystyle\mathcal{D}^{3d}_{A}\xi_{-}^{i}+V^{i}_{A\,j}\xi_{+}^{j}+\frac{1}{2}\omega^{Bn}_{A}\gamma_{Bn}\xi_{+}^{i}+T_{BC}(\gamma_{BC}\gamma_{A}+\frac{1}{3}\gamma_{A}\gamma_{BC})\xi_{-}^{i}-2T_{Bn}(\gamma_{B}\gamma_{A}\gamma_{n}+\frac{1}{3}\gamma_{A}\gamma_{B}\gamma_{n})\xi_{+}^{i}$
$\displaystyle\qquad\qquad\qquad=\frac{1}{3}\gamma_{A}\gamma^{B}\mathcal{D}^{3d}_{B}\xi_{-}^{i}+\frac{1}{3}\gamma_{A}\gamma^{B}V^{i}_{B\,j}\xi_{+}^{j}+\frac{1}{6}\gamma_{A}\gamma^{B}\omega^{Cn}_{A}\gamma_{Cn}\xi_{+}^{i}\,,$
(126)

and with $\Pi^{i}_{+j}$

$\displaystyle\mathcal{D}^{3d}_{A}\xi_{+}^{i}+V^{i}_{A\,j}\xi_{-}^{j}+\frac{1}{2}\omega^{Bn}_{A}\gamma_{Bn}\xi_{-}^{i}+T_{BC}(\gamma_{BC}\gamma_{A}+\frac{1}{3}\gamma_{A}\gamma_{BC})\xi_{+}^{i}-2T_{Bn}(\gamma_{B}\gamma_{A}\gamma_{n}+\frac{1}{3}\gamma_{A}\gamma_{B}\gamma_{n})\xi_{-}^{i}$
$\displaystyle\qquad\qquad\qquad=\frac{1}{3}\gamma_{A}\gamma^{B}\mathcal{D}^{3d}_{B}\xi_{+}^{i}+\frac{1}{3}\gamma_{A}\gamma^{B}V^{i}_{B\,j}\xi_{-}^{j}+\frac{1}{6}\gamma_{A}\gamma^{B}\omega^{Cn}_{A}\gamma_{Cn}\xi_{-}^{i}\,,$
(127)

In the above, we have defined

$\mathcal{D}^{3d}_{A}\xi_{+}^{i}=\nabla^{3d}_{A}\xi_{+}^{i}+V^{(3)}_{A}\sigma^{i}_{3\,j}\xi_{+}^{j},\quad\text{and}\quad
V^{i}_{B\,j}=V^{(1)}_{B}\sigma^{i}_{1\,j}+V^{(2)}_{B}\sigma^{i}_{2\,j}\,.$
(128)

Now, we require that the Killing spinor is such that

$\xi_{+}^{i}\Big{|}_{r=\frac{\pi}{2}}=0$ (129)

Using the fact that our squashed metric background is such that

$\omega^{Bn}_{A}=0,\qquad\text{for}\quad A,B=1,2,3\,,$ (130)

we get at $r=\frac{\pi}{2}$

$\displaystyle\mathcal{D}^{3d}_{A}\xi_{-}^{i}+T_{BC}(\gamma_{BC}\gamma_{A}+\frac{1}{3}\gamma_{A}\gamma_{BC})\xi_{-}^{i}=\frac{1}{3}\gamma_{A}\gamma^{B}\mathcal{D}^{3d}_{B}\xi_{-}^{i}\,.$
(131)

This is the 3-dimensional Killing spinor equation. Next, we consider the case
of (B). We get at $r=\frac{\pi}{2}$

$\displaystyle\frac{1}{2}\omega^{Bn}_{A}\gamma_{Bn}\xi_{-}^{i}-2T_{Bn}(\gamma_{B}\gamma_{A}\gamma_{n}+\frac{1}{3}\gamma_{A}\gamma_{B}\gamma_{n})\xi_{-}^{i}=\frac{1}{6}\gamma_{A}\gamma^{B}\omega^{Cn}_{A}\gamma_{Cn}\xi_{-}^{i}$
(132)

The above is the consistency condition the Killing spinor $\xi^{i}_{-}$ need
to satisfy for the existence of supersymmetry on a manifold with boundary.
Using the fact that our squashed metric backgrounds are such that

$\omega^{Bn}_{A}=0,\qquad\text{for}\quad A,B=1,2,3\,,$ (133)

we obtain at $r=\frac{\pi}{2}$

$T_{Bn}(\gamma_{B}\gamma_{A}\gamma_{n}+\frac{1}{3}\gamma_{A}\gamma_{B}\gamma_{n})\xi_{-}^{i}=0$
(134)

This should be true for every $A$. In particular, one possible solution of the
above can be that the background is such that $T_{Bn}|_{r=\frac{\pi}{2}}=0$.
This will not impose any condition of the Killing spinor.

##  Appendix C Background fields for the squashed geometry with $SU(2)\times
U(1)$ isometry

$\begin{split}&v_{3,3}=-\frac{1}{h(r)}\left(1+\frac{h^{\prime}(r)}{2}\right)+\frac{h(r)}{2\sin^{2}{r}}+\frac{1}{2}\cot{\frac{r}{2}},\\\
&t^{+}_{3}=\frac{1}{16c\sin^{2}{\frac{r}{2}}}\left(2h^{\prime}(r)+\left(\frac{h(r)}{\sin{r}}\right)^{2}-\left(\cot{r}+\cot{\frac{r}{2}}\right)h(r)\right),\\\
&t^{-}_{3}=\frac{c}{4h(r)\cos^{2}{\frac{r}{2}}}\biggl{(}h(r)-\sin{r}\biggr{)},\\\
&M=2\left(\frac{h(r)}{\sin^{2}{r}}+\cot{\frac{r}{2}}\right)\frac{h^{\prime}(r)}{h(r)}+\left(\frac{h(r)}{\sin^{2}{r}}-2\cot{\frac{r}{2}}\right)\frac{h(r)}{\sin^{2}{r}}+\left(\cot^{2}{r}-\cot^{2}{\dfrac{r}{2}}-2\right).\end{split}$
(135)

##  Appendix D Supersymmetry transformations and vector multiplet Lagrangian

First we list the SUSY transformations for the fields in the vector multiplet

$\displaystyle\delta S=\overline{\xi}_{i}\lambda^{i}\,,$ $\displaystyle\delta
P=\overline{\xi}_{i}\gamma_{5}\lambda^{i}\,,$ $\displaystyle\delta
A_{\mu}=\overline{\xi}_{i}\gamma_{\mu}\lambda^{i}\,,$
$\displaystyle\delta\vec{D}=i\vec{\tau}^{i}_{j}\overline{\xi}_{i}\gamma^{a}\mathcal{D}_{a}\lambda^{j}\,,$
$\displaystyle\delta\lambda^{i}=\gamma^{a}\partial_{a}S\xi^{i}-\gamma^{a}\partial_{a}P\gamma_{5}\xi^{i}-\frac{1}{2}F_{ab}\gamma^{ab}\xi^{i}+2(ST_{ab}+\frac{1}{2}P\epsilon_{abcd}T_{cd})\gamma^{ab}\xi^{i}-i\vec{D}\cdot\vec{\tau}^{i}{}_{j}\xi^{j}$
$\displaystyle\qquad+\frac{1}{2}S\not{\mathcal{D}}\xi^{i}-\frac{1}{2}P\not{\mathcal{D}}\gamma_{5}\xi^{i}$
$\displaystyle\delta\overline{\lambda}_{i}=\overline{\xi}_{i}\gamma^{a}\partial_{a}S-\overline{\xi}_{i}\gamma_{5}\gamma^{a}\partial_{a}P+\frac{1}{2}\overline{\xi}_{i}F_{ab}\gamma^{ab}-2\overline{\xi}_{i}(ST_{ab}+\frac{1}{2}P\epsilon_{abcd}T_{cd})\gamma^{ab}+i\vec{D}\cdot\vec{\tau}^{j}{}_{i}\overline{\xi}_{j}$
$\displaystyle+\frac{1}{2}S\mathcal{\mathcal{D}}_{a}\overline{\xi}_{i}\gamma^{a}+\frac{1}{2}P\mathcal{\mathcal{D}}_{a}\overline{\xi}_{i}\gamma^{a}\gamma_{5}$
(136)

In the above $T_{ab}$ is a 2 form and not self or antiself dual i.e.
$T_{ab}=T^{+}_{ab}+T^{-}_{ab}$ and

$\displaystyle\mathcal{D}_{\mu}\lambda^{i}=(\partial_{\mu}+\frac{1}{4}\omega_{\mu
ab}\gamma^{ab})\lambda^{i}+\frac{1}{2}\mathcal{V}^{i}_{\mu\,j}\lambda^{j}$
$\displaystyle\mathcal{D}_{\mu}\xi^{i}=(\partial_{\mu}+\frac{1}{4}\omega_{\mu
ab}\gamma^{ab})\xi^{i}+\frac{1}{2}\mathcal{V}^{i}_{\mu\,j}\xi^{j}$ (137)

We will mention here the Lagrangian of the abelian theory. The action for the
theory is

$I_{V}=I_{g}+I_{\partial g}+I_{\theta}$ (138) $\displaystyle I_{g}$
$\displaystyle=$
$\displaystyle\frac{1}{2g^{2}}\int_{\mathcal{M}}d^{4}x\Big{(}\frac{1}{2}F_{\mu\nu}F^{\mu\nu}-8F^{\mu\nu}(ST_{\mu\nu}+\frac{1}{2}P\epsilon_{\mu\nu\rho\eta}T^{\rho\eta})+\overline{\lambda_{i}}\not{\mathcal{D}}\lambda^{i}$
$\displaystyle+(\partial_{\mu}P)^{2}-(\partial_{\mu}S)^{2}-\vec{D}^{2}-\frac{M}{2}(P^{2}-S^{2})+16(S^{2}+P^{2})T_{ab}T^{ab}+16SP\epsilon_{abcd}T^{ab}T^{cd}\Big{)},$
$\displaystyle I_{\partial g}$ $\displaystyle=$
$\displaystyle\frac{1}{g^{2}}\int_{\partial\mathcal{M}}d^{3}x\left(\frac{i}{4}\tau_{3}{}^{i}{}_{j}\,\overline{\lambda_{i}}\gamma_{5}\lambda^{j}+P(D_{3}-\partial_{n}P)\right)\,,$
(140) $\displaystyle I_{\theta}$ $\displaystyle=$
$\displaystyle\frac{\theta}{4\pi^{2}}\left[\frac{i}{4}\int_{\mathcal{M}}d^{4}xF_{\mu\nu}\widetilde{F}^{\mu\nu}+\int_{\partial\mathcal{M}}d^{3}x\left(\frac{1}{2}\tau_{3}{}^{i}{}_{j}\,\overline{\lambda_{-}}_{i}\lambda^{j}_{-}-iS(D_{3}-\partial_{n}P)\right)\right]\,.$

##  Appendix E Functions

The explicit form of the function that appears in the expression for
$\tau_{R}$ (84) is

$\displaystyle f_{1}(x)$ $\displaystyle=$
$\displaystyle\frac{e^{\frac{i}{2}(3\alpha+\frac{\pi}{2})}}{18\pi(1+2\cosh
2x)^{2}}\Big{[}4\pi^{2}(\cosh 4x-3\cosh 2x-7)-3\sqrt{3}\pi((1+2\cosh
2x)(-3+2a\sinh\alpha)$ (142) $\displaystyle+20x\sinh
2x)+18x(3x-2a\sinh\alpha(\sinh 2x+4\sinh 4x))\Big{]}\,,$

where
$a=\frac{54\sqrt{3}-90\pi+8\sqrt{3}\pi^{2}}{9\Big{(}8\pi(3\sqrt{3}-2\pi)-27\Big{)}}$.

The explicit form of the function that appears in the expression for
$\tau_{R}$ (101) is

$\displaystyle g(x,\alpha^{\prime})$ $\displaystyle=$
$\displaystyle\frac{e^{i\frac{\pi}{24}}}{64\sqrt{2}\pi\tau^{\prime}}e^{2x}(1+e^{2x})^{-\frac{5}{2}+\frac{ix}{\pi}}\frac{1}{\sinh^{3}x}\Big{(}-8(\pi+ix)\frac{x}{\sinh
x}$ (143) $\displaystyle-8\cosh x(\pi+\pi
x-2ix^{2}+2e^{i\alpha^{\prime}}(i\pi+2x-2x\coth x)\sin\alpha^{\prime}$
$\displaystyle+2(3i\pi^{2}+4\pi x-8ix^{2})\sinh x\Big{)}(3\cosh 2x-3-8x+4\sinh
2x)\,.$

## References

  * (1) W.-H. Hsiao and D. T. Son, Duality and universal transport in mixed-dimension electrodynamics, Phys. Rev. B96 (2017), no. 7 075127, [arXiv:1705.0110].
  * (2) W.-H. Hsiao and D. T. Son, Self-dual $\nu$ = 1 bosonic quantum Hall state in mixed-dimensional QED, Phys. Rev. B 100 (2019), no. 23 235150, [arXiv:1809.0688].
  * (3) L. Di Pietro, D. Gaiotto, E. Lauria, and J. Wu, 3d Abelian Gauge Theories at the Boundary, JHEP 05 (2019) 091, [arXiv:1902.0956].
  * (4) G. Grignani and G. W. Semenoff, Defect QED: Dielectric without a Dielectric, Monopole without a Monopole, JHEP 11 (2019) 114, [arXiv:1909.0327].
  * (5) I. Affleck and A. W. W. Ludwig, Universal noninteger “ground-state degeneracy” in critical quantum systems, Phys. Rev. Lett. 67 (Jul, 1991) 161–164.
  * (6) D. Friedan and A. Konechny, On the boundary entropy of one-dimensional quantum systems at low temperature, Phys. Rev. Lett. 93 (2004) 030402, [hep-th/0312197].
  * (7) K. Jensen and A. O’Bannon, Constraint on Defect and Boundary Renormalization Group Flows, Phys. Rev. Lett. 116 (2016), no. 9 091601, [arXiv:1509.0216].
  * (8) H. Casini, I. Salazar Landea, and G. Torroba, The g-theorem and quantum information theory, JHEP 10 (2016) 140, [arXiv:1607.0039].
  * (9) H. Casini, I. Salazar Landea, and G. Torroba, Irreversibility in quantum field theories with boundaries, JHEP 04 (2019) 166, [arXiv:1812.0818].
  * (10) E. Gava, K. S. Narain, M. N. Muteeb, and V. I. Giraldo-Rivera, $N=2$ gauge theories on the hemisphere $HS^{4}$, Nucl. Phys. B920 (2017) 256–297, [arXiv:1611.0480].
  * (11) R. Kumar Gupta, C. P. Herzog, and I. Jeon, Duality and Transport for Supersymmetric Graphene from the Hemisphere Partition Function, JHEP 05 (2020) 023, [arXiv:1912.0922].
  * (12) D. L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, JHEP 05 (2012) 159, [arXiv:1012.3210].
  * (13) N. Hama, K. Hosomichi, and S. Lee, SUSY Gauge Theories on Squashed Three-Spheres, JHEP 05 (2011) 014, [arXiv:1102.4716].
  * (14) Y. Imamura and D. Yokoyama, N=2 supersymmetric theories on squashed three-sphere, Phys. Rev. D 85 (2012) 025015, [arXiv:1109.4734].
  * (15) N. Hama and K. Hosomichi, Seiberg-Witten Theories on Ellipsoids, JHEP 09 (2012) 033, [arXiv:1206.6359]. [Addendum: JHEP10,051(2012)].
  * (16) Y. Imamura, Supersymmetric theories on squashed five-sphere, PTEP 2013 (2013) 013B04, [arXiv:1209.0561].
  * (17) C. Closset, T. T. Dumitrescu, G. Festuccia, and Z. Komargodski, Supersymmetric Field Theories on Three-Manifolds, JHEP 05 (2013) 017, [arXiv:1212.3388].
  * (18) D. Gaiotto, Boundary F-maximization, arXiv:1403.8052.
  * (19) T. Nishioka and K. Yonekura, On RG Flow of $\tau_{RR}$ for Supersymmetric Field Theories in Three-Dimensions, JHEP 05 (2013) 165, [arXiv:1303.1522].
  * (20) A. Cabo-Bizet, E. Gava, V. I. Giraldo-Rivera, M. N. Muteeb, and K. Narain, Partition Function of $N=2$ Gauge Theories on a Squashed $S^{4}$ with $SU(2)\times U(1)$ Isometry, Nucl. Phys. B 899 (2015) 149–164, [arXiv:1412.6826].
  * (21) D. McAvity and H. Osborn, Energy momentum tensor in conformal field theories near a boundary, Nucl. Phys. B 406 (1993) 655–680, [hep-th/9302068].
  * (22) D. McAvity and H. Osborn, Conformal field theories near a boundary in general dimensions, Nucl. Phys. B 455 (1995) 522–576, [cond-mat/9505127].
  * (23) C. P. Herzog and K.-W. Huang, Boundary Conformal Field Theory and a Boundary Central Charge, JHEP 10 (2017) 189, [arXiv:1707.0622].
  * (24) M. Van Raamsdonk and C. Waddell, Holographic and Localization Calculations of Boundary F for ${\cal N}=4$ SUSY Yang-Mills Theory, arXiv:2010.1452.