# Information metric on the boundary

Kenta Suzuki  Department of Physics, Rikkyo University, Toshima, Tokyo
171-8501, Japan Center for Gravitational Physics, Yukawa Institute for
Theoretical Physics,Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan Yu-ki
<EMAIL_ADDRESS>Center for Gravitational Physics,
Yukawa Institute for Theoretical Physics,Kyoto University, Sakyo-ku, Kyoto
606-8502, Japan Takashi Tsuda  Center for Gravitational Physics, Yukawa
Institute for Theoretical Physics,Kyoto University, Sakyo-ku, Kyoto 606-8502,
Japan Masataka Watanabe  Center for Gravitational Physics, Yukawa Institute
for Theoretical Physics,Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan

###### Abstract

The information metric on the space of boundary coupling constants in two-
dimensional conformal field theories is studied. Such a metric is related to
the Casimir energy difference of the theory defined on an interval. We
concretely compute the information metric on the boundary conformal manifold
of free boson CFT as well as $SU(2)_{k}$ WZW theory, obtaining the result
expected from the symmetry of the systems. We also compute the information
metric on the space of non-conformal boundary states produced by boundary mass
perturbations in the theory of a real free scalar. The holographic dual of the
boundary information metric in the context of AdS3/BCFT2 is also discussed. We
argue that it corresponds to the area of the minimal cross section of the end-
of-the-world brane connecting two boundaries of the asymptotic BCFTs.

RUP-22-28

YITP-22-156

###### Contents

  1. 1 Introduction
  2. 2 Definitions
  3. 3 Examples
    1. 3.1 Free Scalar CFT2
    2. 3.2 SU(2) WZW Model
  4. 4 Boundary Mass Perturbations
    1. 4.1 Bosonic Case
    2. 4.2 Supersymmetric Case
  5. 5 Gravity Dual of the Boundary Information Metric
  6. 6 Conclusion
  7. A Boundary Information Metric in Superstring

##  1 Introduction

As no actual experiments can be performed on a strictly infinite volume space,
understanding of boundary conditions is a natural subject in physics. One of
the most well-studied classes of theories with boundaries is the boundary
conformal field theory (BCFT) [1, 2]. BCFT is a conformal field theory (CFT)
defined on a space-time with boundaries, which are invariant under the maximal
subgroup of the bulk conformal symmetry. BCFTs are naturally useful in
analyzing the long-wavelength limit of spin chains with a boundary or a defect
[3], and in describing the worldsheet theory of the open string [4, 5, 6].
Less trivially, they give a natural tensor factorization structure of the
Hilbert space used to compute various quantum information theoretic quantities
[7, 8, 9], or a description of the initial state in the global quantum quench
[10].

Another interesting aspect of BCFTs is their holographic dual. According to
the AdS/BCFT correspondence proposed in [11, 12, 13], the boundary condition
in BCFT is dual to what is called the End-of-the-World (ETW) brane, on which
the bulk fields satisfy the Neumann boundary conditions, terminating the bulk
space-time.222There are also explicit stringy constructions of AdS/BCFT with
boundary conditions [14, 15, 16, 17, 18, 19].  ETW branes are also useful in
analyzing the quantum nature of Black Holes, as some of the Black Hole
microstates can be constructed from them [20]. The correspondence between
Black Hole microstates and the boundary conditions (and hence their gravity
dual, the ETW brane) has been useful in devising a toy model for the unitary
Black Hole evaporation, qualitatively reproducing the Page curve obtained from
the island formula in the gravity theory [21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34].

Motivated by the fact that the boundary conditions could partially correspond
to Black Hole microstates, a natural question would be how much we can
distinguish one boundary condition from the other. In particular, one wishes
to find a physical quantity which distinguishes between boundary conditions.
Particularly in 2D CFTs, where there is a modular invariance, a natural
candidate for such a quantity is the information metric defined on the space
of boundary states. The information metric in general characterizes the
distance between normalizable states, which expresses the probability of
mistaking one state for the other. Motivated by these, in this paper, we
evaluate the information metric on the space of boundary states for various 2D
CFTs.

Let us review the definition of the information metric briefly. (More details
will be given in section 2.) The quantum information theoretic distance
between two normalisable pure states $d(\ket{\phi_{1}},\ket{\phi_{2}})$ are
usually given as

$\displaystyle
d(\ket{\phi_{1}},\ket{\phi_{2}})=1-\left|\braket{\phi_{1}}{\phi_{2}}\right|.$
(1.1)

The information metric $G_{ij}$ (usually called the Bures metric [35, 36]) is
then defined by

$\displaystyle
d(\ket{\lambda_{k}-\delta\lambda_{k}/2},\ket{\lambda_{k}+\delta\lambda_{k}/2})=G_{ij}\delta\lambda_{i}\delta\lambda_{j}+O(\delta\lambda^{4}),$
(1.2)

for a family of parameterised states $\ket{\lambda_{i}}$. As the boundary
state is not normalisable in general, we will regularize it using the heat
kernel

$\displaystyle\ket{B}\mapsto\ket{B}_{\ell}\equiv\frac{e^{-\ell
H/2}\ket{B}}{\sqrt{\braket{B}{e^{-\ell H}}{B}}}\,,$ (1.3)

so that we can use the above definition of the information metric. Here
$\ket{B}$ is a boundary state and $\ket{B}_{\ell}$ is the regularized version
with unit norm, where one usually takes $\ell\ll 1$. The information metric
computed from these states will tell us how different one boundary state
compared to the other, in the quantum information theoretic sense. Note that
the boundary state $\ket{B}$ does not have to be conformal for the above
definition to work, and we will study such a case in section 4 as well.

There are several physical interpretations of the information metric of
boundary states besides the quantum information theoretic one. Imagine a
conformal boundary condition preserving a continuous symmetry group $H$, which
is a subgroup of the larger symmetry in the bulk, $G$. It is then immediate
that there is a family of conformal boundary states on the coset $G/H$. In
other words, the coset is nothing but the boundary conformal manifold. Let us
parametrize the coset by $\lambda$, and denote the corresponding boundary
state as $\ket{B(\lambda)}$. First of all, as we will check in the main body
of the text, it is expected that the information metric on $G/H$ is
proportional to the natural metric on the coset $G/H$ for symmetry reasons. In
the context of string theory, if we have a target space description of the 2D
worldsheet CFT at hand, the information metric on the space of D-branes will
probe the target space metric.

Second, this metric is nothing but the $O(\lambda^{2})$ piece of the cylinder
partition function $\braket{B(\lambda-\delta\lambda/2)}{e^{-\ell
H}}{B(\lambda+\delta\lambda/2)}$ modulo multiplications by a constant. (Note
that the denominator of (1.3) is independent of $\lambda$ for this specific
case where the boundary conformal manifold is a coset.) Now, by virtue of
modular invariance,

$\displaystyle\braket{B(\lambda_{1})}{e^{-\ell
H}}{B(\lambda_{2})}={\mathrm{Tr}}(e^{-\pi H_{\rm open}/\ell})=e^{-\pi
E_{0}(\lambda_{1},\lambda_{2})/\ell}+O(e^{-E_{1}/\ell})$ (1.4)

where $H_{\rm open}$ is the open string Hamiltonian defined on an interval
with boundary conditions on the two ends and $E_{0}(\lambda_{1},\lambda_{2})$
($E_{1}(\lambda_{1},\lambda_{2})$) is the system’s ground state (first excited
state) energy.333When the theory is fermionic, one will have to insert
$(-)^{F}$ to the closed string amplitude. See [37, 38, 39, 40]. For general
fermionic CFTs and their boundary conditions see [41, 42, 43, 44, 45, 46, 47,
48, 49]. $E_{0}(\lambda_{1},\lambda_{2})$ is obviously symmetric, so the
$O(\delta\lambda^{2})$ piece of
$E_{0}(\lambda-\delta\lambda/2,\lambda+\delta\lambda/2)$ gives a change in the
Casimir energy when we change the boundary conditions on an interval slightly.
To summarize the information metric on $G/H$ reflects the change in the
Casimir energy on an interval when $\ell\ll 1$.

We can also extend the above to non-conformal boundary conditions. Assuming
non-conformal boundary states corresponding to such boundary conditions can be
defined, as in [50, 51], we can naturally extend the definition of the
boundary information metric to non-conformal boundary states (but of CFTs, for
simplicity). In the main body of the text, we compute the information metric
on the space of a boundary mass perturbation in free scalar theory in two
dimensions. This could potentially be interesting in the construction of the
$g$-function governing the boundary RG flow. The irreversibility of RG flows
triggered by boundary relevant perturbations is an interesting issue, and the
existence of the $g$-function was conjectured in [52] and later proven in [53,
54] (See also [55, 56, 53, 57, 11, 58, 59, 60, 54, 61, 62, 63, 64, 65, 66, 67,
68] for subsequent developments). First of all, when there is only one
relevant boundary parameter in the system, it is immediate to construct the
restricted version of the $g$-function by integrating the metric times the
beta function. More interestingly, in section 4, using the example above we
numerically find that the boundary information metric computed (which is a
scalar as we only considered only one relevant direction) is monotonic along
the RG flow. It would be interesting to study this further, but we leave it
for future work.

Another interesting aspect of the boundary information metric is its
holographic dual. In [69, 70] the gravity dual of the information metric on
the space of ordinary states was proposed. They argued that it corresponds to
the volume of the maximal time slice of the AdS spacetime under crude
approximations. Noting that the gravity dual of the boundary condition is the
ETW brane [11, 12, 13], a similar argument would result in the conclusion that
it corresponds to the area of the extremal slice of the ETW brane connecting
two boundaries of the asymptotic BCFT. In our case we should consider the
minimal one not the maximal one because it is the point where the physically
interesting things happen. The only problem of this proposal is, however, that
such a configuration for the ETW brane is absent when the two boundary
conditions are different [71], which is exactly our case. This conclusion is
too naive. The underlying assumption of [71], noticed recently, is that there
is no degrees of freedom on the ETW brane [72, 73]. This can be understood as
follows: Imagine a boundary state $\ket{B}$ of a theory with a $U(1)$ charge
$Q$, and assume that $\ket{B}\neq e^{i\theta Q}\ket{B}$. The proposal in [71]
indicates that the cylinder partition function discontinuously jumps at
$\theta=0$, computed from its gravity dual. This could be amended by realising
that the $U(1)$ global symmetry on the CFT becomes the $U(1)$ gauge symmetry
in the bulk dual, and that it effectively becomes a scalar field on the ETW
brane. Following this, we place a Klein-Gordon scalar field localized on the
ETW brane, and argue that the action of the scalar field is the gravity dual
of the boundary information metric. We furthermore solve the EOM on the ETW
brane and show that the profile of the scalar field is kink like, therefore
claiming that indeed the boundary information metric is the area of the
minimal slice of the ETW brane connecting two boundaries of the asymptotic
BCFT in a certain limit explained in the main body of the text.

The rest of the paper is organized as follows. We start in section 2 by giving
the definition of distance and the metric in the space of (in particular
boundary) states in QFTs.

In section 3 we show some examples of the calculation of the boundary
information metric. In section 3.1 we start with a real free scalar theory
with Neumann and Dirichlet boundary conditions in two-dimensional CFT and in
section 3.2 we compute the same quantity for spin 0 boundary states of
$SU(2)_{1}$ Wess-Zumino-Witten (WZW) model and extend the result to the
arbitrary level $k$ and spin $I$ case. We can recover the metric of the group
manifold $SU(2)$.

In section 4, we consider nonconformal boundary conditions of a CFT.
Concretely, we compute the information metric on the space of boundary mass
perturbations in free scalar theory.

In section 5 we conjecture, with supporting evidence, that the gravity dual of
the boundary information metric is given by the action of the ETW with a
scalar field localized on top.

In Appedix A we rephrase the result in Sec. 3 in terms of string theory and
D-branes. We also comment on the behavior of the information metric in the
long cylinder limit.

##  2 Definitions

We define the notion of distance and metric on the state space of Quantum
Field Theories (QFTs). Let’s say we are interested in measuring how close to
each other two (normalised) density matrices, $\rho$ and $\rho^{\prime}$, are.
One measure of such a quantity is called the fidelity (For reviews and the
information theoretic meaning of fidelity see [74].) and is defined as
follows,

$F=\operatorname{Tr}\left(\sqrt{\sqrt{\rho}\rho^{\prime}\sqrt{\rho}}\right).$
(2.1)

Similarly, the affinity $A$ and the trace norm $I$ can be defined as in [75,
76] by

$\displaystyle A$
$\displaystyle=\operatorname{Tr}[\sqrt{\rho}\sqrt{\rho^{\prime}}]\,,$ (2.2)
$\displaystyle I$
$\displaystyle=\frac{\operatorname{Tr}\rho\rho^{\prime}}{\sqrt{(\operatorname{Tr}\rho^{2})(\operatorname{Tr}\rho^{{}^{\prime}2})}}\,.$
(2.3)

For pure states, we can write the density matrices as
$\rho=\ket{\phi}\bra{\phi}$ and
$\rho^{\prime}=\ket{\phi^{\prime}}\bra{\phi^{\prime}}$, and the above
quantities become

$\displaystyle F\,=\,A\,=\,\left|\braket{\phi}{\phi^{\prime}}\right|\,,\qquad
I\,=\,\left|\braket{\phi}{\phi^{\prime}}\right|^{2}\,.$ (2.4)

The distance corresponding to affinity and trace norm is called the Bures and
the Hellinger distance,

$\displaystyle
D_{B}^{2}(\rho(\lambda+\delta\lambda/2),\rho(\lambda-\delta\lambda/2))$
$\displaystyle=2(1-F)=G^{B}_{ij}d\lambda^{i}d\lambda^{j}+\cdots,$
$\displaystyle
D_{H}^{2}(\rho(\lambda+\delta\lambda/2),\rho(\lambda-\delta\lambda/2))$
$\displaystyle=2(1-A)=G^{H}_{ij}d\lambda^{i}d\lambda^{j}+\cdots.$ (2.5)

We will call $G^{B}_{\lambda\lambda}$ and $G^{H}_{\lambda\lambda}$ the Bures
and the Hellinger metric, respectively. Note that this expansion series in
$\delta\lambda$ starts at the quadratic order.

Now, we consider the information metric on the space of boundary states. We
will hereafter only concern ourselves with the pure boundary states, although
it is perfectly possible to consider mixed boundary states.444Mixed boundary
states have recently been considered in holographic contexts in [77, 78]. The
obstacle in defining the boundary information metric is that the boundary
states are never normalizable, but we can normalise it by using the heat
kernel regularization,

$\displaystyle\ket{B}\mapsto\ket{B}_{\ell}\equiv\frac{q^{-H/2}\ket{B}}{\sqrt{\braket{B}{q^{-H}}{B}}}\,,$
(2.6)

which is nothing but (1.3) with $q\equiv e^{\ell}$. Writing the cylinder
partition function with boundary conditions $B(\lambda)$ and
$B(\lambda^{\prime})$ as

$Z_{\lambda\lambda}=\bra{B(\lambda)}q^{-H_{closed}}\Ket{B(\lambda)}\,,$ (2.7)

the Bures and the Hellinger distance (and consequently metric) can be defined
as

$\displaystyle
D_{B}^{2}=D_{H}^{2}=2-2\left|\frac{Z_{\lambda^{\prime\prime}\lambda^{\prime}}}{\sqrt{Z_{\lambda^{\prime\prime}\lambda^{\prime\prime}}}\sqrt{Z_{\lambda^{\prime}\lambda^{\prime}}}}\right|_{\lambda^{\prime}=\lambda+d\lambda,\lambda^{\prime\prime}=\lambda-d\lambda}=G^{B}_{ij}d\lambda^{i}d\lambda^{j}+\cdots,$
(2.8)

where we take the second order in the expansion series. Note that the Bures
and the Hellinger metric coincides with each other when we consider pure
states.

##  3 Examples

###  3.1 Free Scalar CFT2

We study two-dimensional free scalar theory defined by

$\displaystyle
I\,=\,\frac{1}{2}\int_{0}^{\infty}dw\int_{-\infty}^{\infty}d\tau\Big{[}(\partial_{\tau}X)^{2}+(\partial_{w}X)^{2}\Big{]}\,.$
(3.1)

We have a straight-line boundary at $w=0$ and we use string theory notation as
follows [4, 79]. The variation of the action leads to

$\displaystyle\delta
I\,=\,-\int_{0}^{\infty}dw\int_{-\infty}^{\infty}d\tau\,\delta
X(\partial_{\tau}^{2}+\partial_{w}^{2})X\,+\,\int_{w=0}d\tau\,\delta
X\partial_{w}X\,.$ (3.2)

Therefore, to make the variation principle well defined, the allowed boundary
conditions at $w=0$ are

$\displaystyle\textrm{Neumann}:$
$\displaystyle\qquad\partial_{w}X\big{|}_{w=0}\,=\,0\,,$ (3.3)
$\displaystyle\textrm{Dirichlet}:$ $\displaystyle\qquad\delta
X\big{|}_{w=0}\,=\,\partial_{\tau}X\big{|}_{w=0}\,=\,0\,.$ (3.4)

Since we have $U(1)$ currents for this theory:

$\displaystyle j(z)\,=\,i\,\partial
X\,=\,\sum_{n\in\mathbb{Z}}j_{n}\,e^{-nz}\,,\qquad\bar{j}(\bar{z})\,=\,i\,\bar{\partial}X\,=\,\sum_{n\in\mathbb{Z}}\bar{j}_{n}\,e^{-n\bar{z}}\,,$
(3.5)

it’s convenient to use the current algebra:

$\displaystyle\big{[}j_{m},j_{n}\big{]}\,=\,m\,\delta_{m,-n}\,,$ (3.6)

where we defined $z=w+i\tau$ and
$\partial=\frac{1}{2}(\partial_{w}-i\partial_{\tau})$. Therefore, we have

$\displaystyle i\,\partial_{w}X\big{|}_{w=0}\,$
$\displaystyle=\,\sum_{n\in\mathbb{Z}}\Big{(}j_{n}e^{-in\tau}+\bar{j}_{n}e^{in\tau}\Big{)}\,,$
(3.7) $\displaystyle\partial_{\tau}X\big{|}_{w=0}\,$
$\displaystyle=\,\sum_{n\in\mathbb{Z}}\Big{(}j_{n}e^{-in\tau}-\bar{j}_{n}e^{in\tau}\Big{)}\,.$
(3.8)

The corresponding boundary states $|B\rangle$ are defined by

$\displaystyle\textrm{Neumann}:$
$\displaystyle\qquad(j_{n}+\bar{j}_{-n})|B_{\textrm{N}}\rangle\,=\,0\,,$ (3.9)
$\displaystyle\textrm{Dirichlet}:$
$\displaystyle\qquad(j_{n}-\bar{j}_{-n})|B_{\textrm{D}}\rangle\,=\,0\,.$
(3.10)

These conditions are solved by the coherent states [4] as 555Boundary states
in free fermion theory was obtained in [6]. For a review, see [80].

$\displaystyle\textrm{Neumann}:$
$\displaystyle\qquad|B_{\textrm{N}}\rangle\,=\,\frac{1}{\mathcal{N}_{\textrm{N}}}\,\exp\left(-\sum_{k=1}^{\infty}\frac{1}{k}\,j_{-k}\bar{j}_{-k}\right)|0\rangle\,,$
(3.11) $\displaystyle\textrm{Dirichlet}:$
$\displaystyle\qquad|B_{\textrm{D}}\rangle\,=\,\frac{1}{\mathcal{N}_{\textrm{D}}}\,\exp\left(+\sum_{k=1}^{\infty}\frac{1}{k}\,j_{-k}\bar{j}_{-k}\right)|0\rangle\,,$
(3.12)

where $\mathcal{N}_{\textrm{N}}=\sqrt{2}$ and $\mathcal{N}_{\textrm{D}}=1$.

By using

$\displaystyle L_{n}\,$
$\displaystyle=\,\frac{1}{2}\sum_{k>-1}j_{n-k}\,j_{k}\,+\,\frac{1}{2}\sum_{k\leq-1}j_{k}\,j_{n-k}\,,$
(3.13) $\displaystyle\overline{L}_{n}\,$
$\displaystyle=\,\frac{1}{2}\sum_{k>-1}\bar{j}_{n-k}\,\bar{j}_{k}\,+\,\frac{1}{2}\sum_{k\leq-1}\bar{j}_{k}\,\bar{j}_{n-k}\,,$
(3.14)

and also noting that

$\displaystyle
j_{0}|B_{\textrm{N,D}}\rangle\,=\,\bar{j}_{0}|B_{\textrm{N,D}}\rangle\,=\,0\,,$
(3.15)

we can explicitly check that the Cardy condition is satisfied

$\displaystyle\big{(}L_{n}-\overline{L}_{-n}\big{)}|B_{\textrm{N,D}}\rangle\,=\,0\,.$
(3.16)

The Neumann-Neumann cylinder amplitude with cylinder length $l$ is given by

$\displaystyle
Z_{(N,N)}(l)\,=\,\frac{1}{\mathcal{N}_{N}^{2}}\,\frac{1}{\eta(2il)}\,,$ (3.17)

while the Dirichlet-Dirichlet cylinder amplitude is given by

$\displaystyle
Z_{(D,D)}(l)\,=\,\frac{1}{\sqrt{2l}\mathcal{N}_{D}^{2}}\,\frac{1}{\eta(2il)}\,\exp\left(-\frac{(x_{a}-x_{b})^{2}}{8\pi
l}\right)\,,$ (3.18)

where $x_{a}$ and $x_{b}$ are the boundary values of the scalar field $X$. As
usual, we defined $\eta(2il)=q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^{n})$
with $q=e^{-4\pi l}$.

In the $x_{a}\to x_{b}$ limit, regarding $\delta\lambda=x_{a}-x_{b}$, we find

$\displaystyle D_{B}^{2}\,=\,\frac{\delta\lambda^{2}}{4\pi
l}+O(\delta\lambda^{4})\,.$ (3.19)

As we will see in section A, if we regard this theory as a worldsheet theory
of an open string stretched between two D-branes, this metric coincides with
the flat target space metric on the D-brane.

###  3.2 SU(2) WZW Model

As a second example, in this section we demonstrate our idea that the boundary
information metric probes the metric of the target space. Here, we consider
SU(2)-level 1 Wess-Zumino-Witten (WZW) model and show that the boundary
information metric matches with that of SU(2). We firstly restrict to the spin
zero case and generalize our discussion to the arbitrary level $k$ and spin
$I$ case at the end of this section.

The action of the WZW model on Riemann surface $\Sigma$ is given by

$\displaystyle
I=\frac{k}{8\pi}\int_{\Sigma}d^{2}x\mathrm{Tr}(\partial_{\alpha}g\partial_{\alpha}g^{-1})-\frac{ik}{12\pi}\int_{B}\mathrm{Tr}(g^{-1}dg\wedge
g^{-1}dg\wedge g^{-1}dg),$ (3.20)

where $g$ is a map from $\Sigma$ to the Lie group $G$, and $B$ is an arbitrary
3-dimensional manifold with boundary $\Sigma$. This action is well-defined,
independent to the choice of $B$ if $k$ is integer. $k$ is restricted to
positive to make the action bounded below. The WZW model is a 2-dimensional
CFT, with the symmetry of current algebra. The currents are giben by

$\displaystyle J=$ $\displaystyle~{}J^{a}T_{a}=-k\partial gg^{-1}\,,$ (3.21)
$\displaystyle\bar{J}=$
$\displaystyle~{}\bar{J}^{a}T_{a}=kg^{-1}\bar{\partial}g\,,$ (3.22)

where $T_{a}$ is the generator of $G$. Taking the Laurent expansion of these
currents

$\displaystyle J^{a}(z)=$
$\displaystyle~{}\sum_{n\in\mathbb{Z}}\frac{J^{a}_{n}}{z^{n+1}},$ (3.23a)
$\displaystyle\bar{J}^{a}(\bar{z})=$
$\displaystyle~{}\sum_{n\in\mathbb{Z}}\frac{\bar{J}^{a}_{n}}{\bar{z}^{n+1}},$
(3.23b)

their modes, $J_{n}^{a}$ and $\bar{J}_{n}^{a}$, satisfy current algebras:

$\displaystyle\left[J_{n}^{a},J_{m}^{b}\right]\,$
$\displaystyle=\,if^{ab}{}_{c}J^{c}_{n+m}+kn\,\delta^{ab}\delta_{n+m,0},$
(3.24a) $\displaystyle\left[\bar{J}_{n}^{a},\bar{J}_{m}^{b}\right]\,$
$\displaystyle=\,if^{ab}{}_{c}\bar{J}^{c}_{n+m}+kn\,\delta^{ab}\delta_{n+m,0}.$
(3.24b)

In our setup we set $G=SU(2)$ and we consider $k=1$ case for a while.666We
consider the diagonal model (A-series in ADE classification).

In the CFT with current algebra symmetry, boundary states must satisfy the
gluing conditions

$\displaystyle(L_{n}-\bar{L}_{-n})|B\rangle=0,$ (3.25)
$\displaystyle(J^{a}_{n}+\Omega\bar{J}^{a}_{-n})|B\rangle=0,$ (3.26)

where $\Omega$ is an automorphism of the algebra. In the case of SU(2) WZW
model, we can rotate $\bar{J}^{a}_{-n}$ by $g\in SU(2)$. The choice of $g$ is
arbitrary, thus gives the moduli space of boundary states. In [81], a spin 0
boundary state in the level-$1$ SU(2) WZW model with gluing condition $g$ is
given by

$\Ket{B,g}=\frac{1}{2^{\frac{1}{4}}}\sum_{(j,m,n)}D^{j}_{m,n}(g)|j,m,n\rangle\rangle,$
(3.27)

where $|j,m,n\rangle\rangle$ is Virasoro Ishibashi state, $m,n=-j,\cdots,j$
and $j$ labels eigenvalues of $J_{z}$. The explicit form of Klebsch-Gordan
coefficients are

$D^{j}_{m,n}(g)=\sum_{l=max(0,n-m)}^{min(j-m,j+n)}\frac{\sqrt{(j+m)!(j-m)!(j+n)!(j-n)!}}{(j-m-l)!(j+n-l)!l!(m-n+l)!}a^{j+n-l}d^{j-m-l}b^{l}c^{m-n+l},$
(3.28)

where we represent $g\in SU(2)$ as

$g\,=\,\begin{pmatrix}a&b\\\ c&d\\\ \end{pmatrix}.$ (3.29)

We originally consider a cylinder amplitude between $g_{1}$ and $g_{2}$ with
circumference $R$ and length $L$. By a conformal mapping, it can be mapped to
an annulus with an inner radius $\exp(-2\pi L/R)$ and an outer radius 1. The
annulus partition function can be represented in terms of Klebsch-Gordan
coefficients and characters

$\displaystyle\mathcal{A}=$
$\displaystyle\bra{B,g_{1}}q^{\frac{1}{2}(L_{0}+\bar{L_{0}}-\frac{c}{12})}\Ket{B,g_{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\sum_{j\in\frac{\mathbb{Z}}{2}}\sum_{m,n}(-1)^{-m,-n}D^{j}_{m,n}(g_{1})D^{j}_{m,n}(g_{2})\chi_{j^{2}}(q),$
(3.30)

where

$\displaystyle q$ $\displaystyle=e^{-4\pi L/R},$ (3.31)
$\displaystyle\chi_{h}(q)$
$\displaystyle=\theta_{\sqrt{2}j}(q)-\theta_{\sqrt{2}(j+1)}(q),$ (3.32)
$\displaystyle\theta_{s}(q)$
$\displaystyle=\frac{q^{\frac{s^{2}}{2}}}{q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^{n})}.$
(3.33)

Figure 1: The amplitude between two boundary states on $SU(2)$.

The sum of the representation matrices gives

$\sum_{m,n}D^{j}_{m,n}(g_{1}^{-1})D^{j}_{m,n}(g_{2})=\sum_{n}D^{j}_{n,n}(g_{1}^{-1}g_{2}).$
(3.34)

If we take the Jordan normal form of $\hat{g}=g_{1}^{-1}g_{2}$, then we obtain

$D^{j}_{n,n}(\hat{g})=\hat{a}^{2n},$ (3.35)

where $\hat{g}=\begin{pmatrix}\hat{a}&0\\\ 0&1/\hat{a}\\\ \end{pmatrix}\in
SU(2)$. After a simple algebra we obtain

$\mathcal{A}=\sum_{n}\frac{\tilde{q}^{n^{2}+\frac{in\alpha}{\pi}-\frac{\alpha^{2}}{4\pi^{2}}}}{\eta(\tilde{q})},$
(3.36)

where $\hat{a}=e^{\alpha}$ and $\tilde{q}=e^{-\frac{\pi R}{L}}$.

Let us now calculate the information metric. We should evaluate the amplitude
with the same position in the group manifold, where we set $\alpha=0$

$Z_{gg}=\sum_{n}\frac{\tilde{q}^{n^{2}}}{\eta(\tilde{q})}.$ (3.37)

We can also calculate a perturbed amplitude for an infinitesimal $\alpha$

$Z_{gg^{\prime}}=\sum_{n}\frac{\tilde{q}^{n^{2}+\frac{in\alpha}{\pi}-\frac{\alpha^{2}}{4\pi^{2}}}}{\eta(\tilde{q})}.$
(3.38)

Then, the Bures distance is given by

$D_{B}^{2}=2\left(1-\left|\frac{Z_{gg^{\prime}}}{Z_{gg}}\right|\right)=2(1-e^{\frac{R\alpha^{2}}{4\pi
L}})\sim\frac{R\beta^{2}}{2\pi L},$ (3.39)

where $\beta=i\alpha$.

To derive a metric, we denote the $g$ and $g^{\prime}$ as
$g=\begin{pmatrix}e^{i\xi_{1}}\sin\eta&e^{i\xi_{2}}\cos\eta\\\
-e^{-i\xi_{2}}\cos\eta&e^{-i\xi_{1}}\sin\eta\\\ \end{pmatrix}$ and
$g^{\prime}=\begin{pmatrix}e^{i\xi^{\prime}_{1}}\sin\eta^{\prime}&e^{i\xi^{\prime}_{2}}\cos\eta^{\prime}\\\
-e^{-i\xi^{\prime}_{2}}\cos\eta^{\prime}&e^{-i\xi^{\prime}_{1}}\sin\eta^{\prime}\\\
\end{pmatrix}$, respectively. This is called Hopf coordinates of $SU(2)$.
After we diagonalize the matrix $g^{-1}g^{\prime}$ we obtain

$\beta\sim\sqrt{(\sin^{2}\eta(\Delta\xi_{1}^{2})+\cos^{2}\eta(\Delta\xi_{2})^{2}+(\Delta\eta)^{2})},$
(3.40)

where we define $\Delta\xi_{1}=\xi^{\prime}_{1}-\xi_{1}$,
$\Delta\xi_{2}=\xi^{\prime}_{2}-\xi_{2}$ and $\Delta\eta=\eta^{\prime}-\eta$.
Here we also ignored higher-order terms in $\Delta\xi_{i}$. From this the
information metric becomes

$D_{B}^{2}=\frac{R}{2\pi L}(d\eta^{2}+\sin^{2}\eta d\xi_{1}^{2}+\cos^{2}\eta
d\xi_{2}^{2}),$ (3.41)

which reproduces the metric of SU(2) in Hopf coordinates. We note that in the
closed string picture the modulus is $L/R$ and the boundary information metric
is proportional to an inverse of the modulus.

As one can see, in the short cylinder limit $|Z_{gg^{\prime}}/Z_{gg}|$ picks
up $\tilde{q}^{h}$, where $h$ is a conformal weight of the open-string vacuum
with boundary conditions $g$ and $g^{\prime}$. This means that when we
calculate the information metric in the short cylinder limit we do not need to
explicitly construct the set of boundary states if we know $h$. In the case of
SU(2)-level $k$ WZW model, $h$ corresponding to the spin 0 representation of
$g$ and $g^{\prime}$ is determined in [82]:

$\displaystyle h=k\left(\frac{\beta}{2\pi}\right)^{2}.$ (3.42)

The Bures distance is

$\displaystyle D^{2}_{B}\sim\frac{kR\beta^{2}}{2\pi L},$ (3.43)

here $k=1$ corresponds to (3.39). More generically, $Z_{gg^{\prime}}$ for
generic spin (labelled by $I,J$) representation is found in [83]:

$\displaystyle
Z_{(g,I)(g^{\prime},J)}=\sum_{j}N^{j}_{IJ}{\mathrm{Tr}}_{\mathcal{H}_{j}}\tilde{q}^{L_{0}-\frac{c}{24}-\frac{2\beta}{2\pi}J^{3}_{0}+k\left(\frac{\beta}{2\pi}\right)^{2}}.$
(3.44)

We can read off the open-string vacuum contribution $h$ for general spin $I=J$
to be (3.42).

##  4 Boundary Mass Perturbations

So far in this paper, we have been considering BCFTs which keep maximal
subgroup of the dulk conformal symmetry at its boundary. Such maximal subgroup
on the boundary is called boundary conformal symmetry. For these cases, the
parameter $\lambda$ is the coordinate of the boundary conformal manifold. Now,
we study the case when we break this boundary conformal symmetry and take
$\lambda$ as the breaking parameter. In particular in this section, we
consider an example of such breaking by adding boundary mass perturbations. In
the context of string theory, such boundary mass perturbations were studied as
the so called boundary string field theory [50, 84, 85, 51, 86].

###  4.1 Bosonic Case

In this subsection, we consider a cylinder amplitude of scalar fields with a
modified Neumann condition. In the context of string field theory this was
already studied in [50, 51]. In this section we will follow the formalism of
[51].

We consider a free scalar with a following boundary term

$S=-\frac{1}{4\pi\alpha^{\prime}}\int_{0}^{2\pi}d\sigma\int_{-\frac{\pi}{t}}^{0}d\tau\partial_{\mu}X\partial^{\mu}X-\frac{\lambda\delta(\tau=0)}{4\pi\alpha^{\prime}}\int_{0}^{2\pi}d\sigma
X^{2}-\frac{\lambda\delta(\tau=-\frac{\pi}{t})}{4\pi\alpha^{\prime}}\int_{0}^{2\pi}d\sigma
X^{2}.$ (4.1)

The boundary condition at $\tau=0,-\pi/t$ becomes

$(\partial_{\tau^{\prime}}X+t\lambda X)|_{\tau=0,-\frac{\pi}{t}}=0,$ (4.2)

which we call a modified Neumann condition.

The mode expansion in complexified coordinates $z=e^{\tau+i\sigma}$ becomes

$X=X_{0}-\frac{ip\alpha^{\prime}}{2}\log\left|z\right|^{2}+i\sqrt{\frac{\alpha^{\prime}}{2}}\sum_{m}\frac{1}{\sqrt{m}}\left(\alpha_{m}e^{-m(\tau+i\sigma)}+\tilde{\alpha}_{m}e^{-m(\tau-i\sigma)}\right).$
(4.3)

The commutation relation is given by

$\displaystyle[X_{0},p]$ $\displaystyle=i$
$\displaystyle[\alpha_{m},\alpha_{-n}]$ $\displaystyle=\delta_{mn}.$ (4.4)

Plugging (4.3) into (4.2), we obtain the boundary conditions on the left and
right creation and annihilation operators

$\left.\left(\alpha_{m}-\frac{m-t\lambda}{m+t\lambda}e^{2m\tau}\tilde{\alpha}_{-m}\right)\right|_{\tau=-\frac{\pi}{t},0}=0.$
(4.5)

Foe general $\tau$ the boundary states can be described by

$\displaystyle\alpha_{m}\Ket{\lambda}_{\tau}$
$\displaystyle=\frac{m-t\lambda}{m+t\lambda}e^{2m\tau}\tilde{\alpha}_{-m}\Ket{\lambda}_{\tau}$
$\displaystyle\tilde{\alpha}_{m}\Ket{\lambda}_{\tau}$
$\displaystyle=\frac{m-t\lambda}{m+t\lambda}e^{2m\tau}\alpha_{-m}\Ket{\lambda}_{\tau}$
$\displaystyle i\hat{p}\Ket{\lambda}_{\tau}$
$\displaystyle=-\frac{t\lambda\hat{X}_{0}}{\alpha^{\prime}(1-t\lambda\tau)}\Ket{\lambda}_{\tau},$
(4.6)

where $\Ket{\lambda}_{\tau}$ represents the boundary state at general $\tau$.
We can find the boundary state, which satisfies the above conditions

$\Ket{\lambda}_{\tau}=\mathcal{N}_{\lambda}\sqrt{\frac{2t\lambda}{\alpha^{\prime}(1-t\lambda\tau)}}\exp\left(\sum_{m=1}^{\infty}(\frac{m-t\lambda}{m+t\lambda})\alpha_{-m}\tilde{\alpha}_{-m}e^{2m\tau}\right)\exp{\left(-\frac{t\lambda\hat{X_{0}^{2}}}{2\alpha^{\prime}(1-t\tau\lambda)}\right)}\Ket{0}.$
(4.7)

We can fix the normalization as follows. The tree-level partition function
factor, which corresponds to the disk amplitude, was found in [50] to be

$Z_{0}=\left(\sqrt{t\lambda}e^{\gamma
t\lambda}\Gamma(t\lambda)\right)^{\frac{2}{\alpha^{\prime}}}.$ (4.8)

Then, we have

$\mathcal{N}_{\lambda}=N_{0}Z_{0},$ (4.9)

where $N_{0}$ can be fixed by comparing to the open string channel and
$N_{0}=\frac{1}{\sqrt{2}}$.

The cylinder amplitude between boundary states at $\tau=-\pi/t$ and $\tau=0$
becomes

$\displaystyle Z$
$\displaystyle=\bra{\lambda}_{\tau=0}\Ket{\lambda}_{\tau=-\pi/t}$
$\displaystyle=N_{0}^{2}Z_{0}^{2}\sqrt{\frac{2t\lambda}{\pi\alpha^{\prime}(2+\pi\lambda)}}\prod_{m=1}^{\infty}\frac{1}{1-\left(\frac{m-t\lambda}{m+t\lambda}\right)^{2}e^{-\frac{2\pi
m}{t}}}$
$\displaystyle=\frac{1}{2}\left(\sqrt{t\lambda}\Gamma(t\lambda)e^{t\lambda\gamma}\right)^{\frac{4}{\alpha^{\prime}}}\sqrt{\frac{2t\lambda}{\pi\alpha^{\prime}(2+\pi\lambda)}}\prod_{m=1}^{\infty}\frac{1}{1-\left(\frac{m-t\lambda}{m+t\lambda}\right)^{2}e^{-\frac{2\pi
m}{t}}},$ (4.10)

where $\gamma$ is the Euler’s constant. We note that in the $\lambda\to 0$
limit, $Z$ diverges as $Z\propto\lambda^{-1/2}$. This is because the Neumann
boundary condition (i.e. strictly at $\lambda=0$) is distinct from the
$\lambda\to 0$ limit. For the Neumann boundary condition, the boundary state
is independent of the zero mode $\hat{X}_{0}$, so the corresponding amplitude
excludes the zero mode integral. On the other hand, for the modified Neumann
case ($\lambda>0$) we have the zero mode integral

$\displaystyle\int_{-\infty}^{\infty}\frac{d\hat{X}_{0}}{2\pi}\,\exp\left(-\frac{t\lambda}{2\alpha^{\prime}}\hat{X}_{0}^{2}\right)\exp\left(-\frac{t\lambda}{2\alpha^{\prime}(1-t\lambda)}\hat{X}_{0}^{2}\right)\,=\,\sqrt{\frac{\alpha^{\prime}(1-t\lambda)}{2\pi
t\lambda(2-t\lambda)}}\,,$ (4.11)

in the amplitude (4.10). This Gaussian integral is responsible for the
$Z\propto\lambda^{-1/2}$ divergence.

Figure 2: Log plot of the information metric $G$ with $t=1,4,9$, where we
chose $\alpha^{\prime}=2$ and $m_{\textrm{max}}=10$. As we explain around
table 1, $m_{\textrm{max}}=10$ is sufficient for the case when
$t=\mathcal{O}(1)$ and $\lambda=\mathcal{O}(1)$. For larger $t$, higher
$m_{\textrm{max}}$ we need.

For the calculation of the information metric, we calculate the amplitude from
the boundary condition $\lambda$ to $\lambda^{\prime}$, which is slightly
perturbed from $\lambda$. The partition function becomes

$\displaystyle Z$
$\displaystyle=\bra{\lambda^{\prime}}_{\tau=0}\Ket{\lambda}_{\tau=-\pi/t}$
$\displaystyle=\frac{1}{2}\\!\left(\sqrt{t\lambda}\Gamma(t\lambda)e^{t\lambda\gamma}\right)^{\frac{2}{\alpha^{\prime}}}\\!\left(\sqrt{t\lambda^{\prime}}\Gamma(t\lambda^{\prime})e^{t\lambda^{\prime}\gamma}\right)^{\frac{2}{\alpha^{\prime}}}\\!\sqrt{\frac{2t\lambda^{2}}{\alpha^{\prime}\pi(\lambda+\lambda^{\prime}+\lambda\lambda^{\prime}\pi)}}$
$\displaystyle\cdot\prod_{m=1}^{\infty}\frac{1}{1-\left(\frac{m-t\lambda}{m+t\lambda}\right)\left(\frac{m-t\lambda^{\prime}}{m+t\lambda^{\prime}}\right)e^{-\frac{2\pi
m}{t}}}.$ (4.12)

From this amplitude, the information metric is then computed by

$\displaystyle G(\lambda)\,=\,\lim_{\epsilon\to
0}\frac{1-P(\lambda+\epsilon,\lambda-\epsilon)}{\epsilon^{2}}\,,\qquad
P(\lambda_{1},\lambda_{2})\,=\,\frac{Z(\lambda_{1},\lambda_{2})}{\sqrt{Z(\lambda_{1},\lambda_{1})Z(\lambda_{2},\lambda_{2})}}\,.$
(4.13)

We can obtain an analytical expression, but the result is lengthy, so here we
instead show numerical plots in Figure 2, as well as the disk amplitude
behavior. These plots show that the information metric monotonically decreases
as $\lambda$ grows.

For the numerical evaluation of the information metric, we need to put an
upper cutoff $m_{\textrm{max}}$ on the infinite product in the amplitude

$\displaystyle
Z(\lambda_{1},\lambda_{2})\,=\,A(\lambda_{1},\lambda_{2})\prod_{m=1}^{m_{\textrm{max}}}B_{m}(\lambda_{1},\lambda_{2})\,,$
(4.14)

where

$\displaystyle A(\lambda_{1},\lambda_{2})\,$
$\displaystyle=\,\frac{1}{2}\\!\left(\sqrt{t\lambda_{1}}\Gamma(t\lambda_{1})e^{t\lambda_{1}\gamma}\right)^{\frac{2}{\alpha^{\prime}}}\\!\left(\sqrt{t\lambda_{2}}\Gamma(t\lambda_{2})e^{t\lambda_{2}\gamma}\right)^{\frac{2}{\alpha^{\prime}}}\\!\sqrt{\frac{2t\lambda_{1}\lambda_{2}}{\alpha^{\prime}\pi(\lambda_{1}+\lambda_{2}+\lambda_{1}\lambda_{2}\pi)}}\,,$
$\displaystyle B_{m}(\lambda_{1},\lambda_{2})\,$
$\displaystyle=\,\left[1-\left(\frac{m-t\lambda}{m+t\lambda}\right)\left(\frac{m-t\lambda^{\prime}}{m+t\lambda^{\prime}}\right)e^{-\frac{2\pi
m}{t}}\right]^{-1}\,.$ (4.15)

In table 1, we show that $B_{m}(\lambda_{1},\lambda_{2})$ is very quickly
suppressed as $m$ grows, at least for $t=\mathcal{O}(1)$ and
$\lambda=\mathcal{O}(1)$.

$m$ | $B_{m}|^{t=2.2}_{\lambda_{1}=\lambda_{2}=2.3}-1$ | $B_{m}|^{t=4.6}_{\lambda_{1}=\lambda_{2}=2.3}-1$ | $B_{m}|^{t=2.2}_{\lambda_{1}=\lambda_{2}=8.7}-1$ | $B_{m}|^{t=4.6}_{\lambda_{1}=\lambda_{2}=8.7}-1$
---|---|---|---|---
1 | 2.649 $\times$ $10^{-2}$ | 2.116 $\times$ $10^{-1}$ | 4.893 $\times$ $10^{-2}$ | 3.002 $\times$ $10^{-1}$
2 | 6.215 $\times$ $10^{-4}$ | 3.123 $\times$ $10^{-2}$ | 2.178 $\times$ $10^{-3}$ | 5.630 $\times$ $10^{-2}$
3 | 1.242 $\times$ $10^{-5}$ | 5.202 $\times$ $10^{-3}$ | 1.010 $\times$ $10^{-4}$ | 1.245 $\times$ $10^{-2}$
4 | 1.496 $\times$ $10^{-7}$ | 8.639 $\times$ $10^{-4}$ | 4.679 $\times$ $10^{-6}$ | 2.846 $\times$ $10^{-3}$
|  |  |  |
Table 1: Suppression in $m$ for $B_{m}-1$. We can see for $t=\mathcal{O}(1)$
and $\lambda=\mathcal{O}(1)$, the suppression is very fast.

Figure 3: Left: Log plot of the disk amplitude $Z_{0}$ with $t=1,2,3$. Right:
Log plot of the action $S$ defined in (4.16) for $t=1,2,3$.

The result that the information metric (which is a scalar in our case) is
monotonic along the RG flow in our case reminds us of the $c$-function [87].
The boundary version of such an object is given by the action $S$ discussed in
[85, 52]. This action is related to the disk partition function (4.8) by

$\displaystyle
S\,=\,\left(1-\lambda\frac{\partial}{\partial\lambda}\right)Z_{0}\,.$ (4.16)

We plot $S$ and $Z_{0}$ for our case in figure 3. The apparent monotonicity in
our example seems to indicate that it is easier to distinguish boundary
conditions with larger degrees of freedom. Unfortunately though, we could not
find any concrete relationship between the two. Nevertheless, we have an upper
bound of the information metric which monotonically decreases along the RG
trajectories [88]. This does not prove the information metric itself
monotonically decreases 777In particular, the information metric obtained from
the quantum Renyi relative entropies in [88] is defined by ordinary states,
while we are discussing the information metric defined from boundary states.
We will discuss difference and similarity between these two in our future
work. , but it’s not surprise to have such behaviour in many other models. It
would also be interesting to understand the relation of the boundary
information metric to the quantum information theoretic proof of the
$g$-theorem as in [54, 88, 89].

###  4.2 Supersymmetric Case

Next, we consider a supersymmetric theory:

$\displaystyle S_{\textrm{bulk}}\,=\,\frac{1}{2}\int dzd\bar{z}\Big{(}\partial
X\bar{\partial}X+\psi\bar{\partial}\psi+\tilde{\psi}\partial\tilde{\psi}\Big{)}\,,$
(4.17)

with the boundary mass perturbation

$\displaystyle\partial_{\tau}X+uX\,$ $\displaystyle=\,0\,,$ (4.18)
$\displaystyle\partial_{\tau}\psi+u\psi\,$
$\displaystyle=\,i\eta(\partial_{\tau}\tilde{\psi}-u\tilde{\psi})\,,$ (4.19)

on the boundary, where $\eta=\pm 1$.

The cylinder amplitudes are computed in [86] as

$\displaystyle Z_{\textrm{NS-NS}}(u,v)\,$
$\displaystyle=\,\mathcal{N}_{\textrm{NS-
NS}}\,4^{u+v}uvB(u,u)B(v,v)(u+v+luv)^{-1/2}$
$\displaystyle\qquad\times\frac{f_{3}^{7}(q)f_{3}^{(u,v)}(q)-f_{4}^{7}(q)f_{4}^{(u,v)}(q)}{f_{1}^{7}(q)f_{1}^{(u,v)}(q)}\,,$
(4.20)

for the NS-NS sector, and

$\displaystyle
Z_{\textrm{R-R}}(u,v)\,=\mathcal{N}_{\textrm{R-R}}\,\sqrt{\frac{uvq}{u+v+luv}}\frac{f_{2}^{7}(q)f_{2}^{(u,v)}(q)}{f_{1}^{7}(q)f_{1}^{(u,v)}(q)}\,,$
(4.21)

for the R-R sector. The numerical constants $\mathcal{N}_{\textrm{NS-NS}}$ and
$\mathcal{N}_{\textrm{R-R}}$ are independent of the boundary mass $u$ or $v$.
Here we defined $q=e^{-l}$ and

$\displaystyle f_{1}^{(u,v)}(q)\,$
$\displaystyle=\,q^{\frac{1}{12}}\prod_{n=1}^{\infty}\left(1-\frac{n-u}{n+u}\frac{n-v}{n+v}\,q^{2n}\right)\,,$
(4.22) $\displaystyle f_{2}^{(u,v)}(q)\,$
$\displaystyle=\,\sqrt{2}q^{\frac{1}{12}}\prod_{n=1}^{\infty}\left(1+\frac{n-u}{n+u}\frac{n-v}{n+v}\,q^{2n}\right)\,,$
(4.23) $\displaystyle f_{3}^{(u,v)}(q)\,$
$\displaystyle=\,q^{-\frac{1}{24}}\prod_{r=\frac{1}{2}}^{\infty}\left(1+\frac{r-u}{r+u}\frac{r-v}{r+v}\,q^{2r}\right)\,,$
(4.24) $\displaystyle f_{4}^{(u,v)}(q)\,$
$\displaystyle=\,q^{-\frac{1}{24}}\prod_{r=\frac{1}{2}}^{\infty}\left(1-\frac{r-u}{r+u}\frac{r-v}{r+v}\,q^{2r}\right)\,,$
(4.25)

with $f_{i}(q)=f_{i}^{(0,0)}(q)$. We show numerical plots for the information
metrics computed from these cylinder amplitudes in Figure 4.

Figure 4: Left: Log plot of the information metric $G$ with $l=1,2,3$ for the
NS-NS sector (left) and the R-R sector (right), where we chose
$n_{\textrm{max}}=10$ and $r_{\textrm{max}}=19/2$.

##  5 Gravity Dual of the Boundary Information Metric

In this section, we conjecture a gravity dual of the boundary information
metric. Later, we will see that the gravity dual of the boundary information
metric appears as a geometrical object, which we call an “area of the minimal
cross section.”

To clarify the problem, let us consider AdS3/BCFT2 at finite temperature.
Here, we consider a connected end-of-the-world brane (ETW brane) in the
thermal AdS3 background. The metric of an Euclidean thermal AdS3 is given by

$ds^{2}=R^{2}\left(\frac{d\tau^{2}}{z^{2}}+\frac{dz^{2}}{h(z)z^{2}}+\frac{h(z)dx^{2}}{z^{2}}\right),$
(5.1)

where $h(z)=1-(z/z_{0})^{2}$. Note that here we compactify the $x$ and $\tau$
as

$\displaystyle x$ $\displaystyle\sim x+2\pi z_{0},$ $\displaystyle\tau$
$\displaystyle\sim\tau+2\pi z_{H}.$ (5.2)

The asymptotic AdS region is given by $z=0$ and we take the region of BCFT as

$0\leq x\leq\pi z_{0}.$ (5.3)

Next we determine the position of a brane. We can solve the equation of motion
for a brane and the profile is solved as [71, 12]

$x(z)=z_{0}\arctan\left(\frac{RTz}{z_{0}\sqrt{h(z)-R^{2}T^{2}}}\right)$ (5.4)

for $0\leq x\leq\pi z_{0}/2$ and

$x(z)=\pi
z_{0}-z_{0}\arctan\left(\frac{RTz}{z_{0}\sqrt{h(z)-R^{2}T^{2}}}\right)$ (5.5)

for $\pi z_{0}/2\leq x\leq\pi z_{0}$.

The gravity dual of an information metric between the overlap of states in
2-dim CFT has already been considered in [70]. In that case, the gravity dual
of an information metric is expected to represent the area of the time-slice
hypersurface in the Janus solution [90]. Here, we consider the information
metric of a boundary counterpart. The overlap of boundary states is always
divergent and so is the information metric. Thus, we expect that the gravity
dual also contains divergence. In 2-dim CFT the boundary information metric is
proportional to an inverse power of modulus in the short length cylinder
limit. This can be expected as follows. Let us consider an overlap of boundary
states on a cylinder with modulus $s$

$Z_{ab}=\bra{a}e^{-sH}\Ket{b},$ (5.6)

where we consider the $s\ll 1$ case. In the open string picture, we can write
the partition function as

$Z_{ab}=\operatorname{Tr}e^{-\frac{1}{s}H_{open}}=e^{-\frac{1}{s}(E_{0}(\lambda_{a},\lambda_{b})+\cdots)},$
(5.7)

where we expand the energy spectrum in the open string channel and consider
the vacuum energy $E_{0}$. Here, the $\lambda$ is a modulus of the marginal
deformation. In general the energy spectrum $E$ depends on the modulus $s$,
but in the short cylinder limit $E_{0}$ is independent of $s$. In the
$s\rightarrow 0$ limit, the $E_{0}$ takes a finite value and its expansion
series start with the constant (as a function of $s$). Therefore, $E_{0}$ is a
function of $\lambda_{a}$ and $\lambda_{b}$. By considering the Taylor
expansion with respect to the $\lambda_{b}-\lambda_{a}$ the information metric
reads

$ds^{2}=\frac{1}{s}d\lambda_{i}^{2}+O(s^{0}),$ (5.8)

where $\lambda_{i}$ are the coordinates of the moduli space. From this, we can
argue that the gravity dual of this boundary information metric also behaves
as $1/s$. In the thermal AdS3 a natural candidate of this gravity dual is a
minimal cross section of the ETW brane. We note that in the BTZ black hole
background there is no gravity dual because the ETW branes are disconnected.
This matches with the CFT calculation (5.8), where the BTZ black hole
corresponds to an $s\rightarrow 0$ or a long cylinder limit and the metric
shrinks. Taking into the above facts we conjecture that

the gravity dual of the boundary information metric is an area of a cross
section of a connected end-of-the-world brane, which minimizes its area.

Figure 5: A schematic figure of a minimal cross section in the thermal AdS3,
which is expected to be dual of the boundary information metric. The two BCFTs
live on each of the boundaries of the cylinder and boundary states live on the
two corners of the cylinder. The minimal cross section $\Gamma_{A}$ is
determined by minimizing the area of the whole cross sections on the ETW brane
$Q$.

We will call this gravity dual as a “minimal cross section” of the ETW brane.
Notice that the short length cylinder limit corresponds to the $\beta=2\pi
z_{0}\rightarrow 0$ limit. In this limit the minimal cross section of an ETW
brane approaches the asymptotic boundary and it indeed behaves as

$G_{\lambda\lambda}\sim\textrm{Area}[\Gamma_{A}]=\frac{2\pi
z_{H}R}{z_{0}\sqrt{1-T^{2}R^{2}}},$ (5.9)

which indeed diverges in a inverse power of a modulus $z_{0}$. Note that if we
make a quess that the gravity dual is another cross section such as
$z=const\neq z_{0}\sqrt{1-R^{2}T^{2}}$, then the area of this cross section
behaves $1/z$. Of course, we can manipulate the constant slice as $z=az_{0}$,
but we expect that this is not the case. Firstly, as we notice later we have
$\mathbf{Z}_{2}$ symmetry $x\rightarrow\pi R/2-x$ and it is natural to expect
that the gravity dual should correspond to the geometrical object at the
turning point $x=\pi R/2$. Secondly, as this can also be described just below,
the action of the scalar field on the ETW brane has a divergence at the
turning point, which means that the integrand at the turning point effectively
contributes.

Here, we denote the abstract coordinate of moduli as $\lambda$ and we call the
minimal cross section of the ETW brane as $\Gamma_{A}$. Note that this is the
first order approximation of the kink solution as we revisit in the end of
this section. We also note that this conjecture is an explicit example of our
main statement:

the information metric of boundary states corresponds to the metric of moduli
space of a target space.

To justify the above conjecture, let us consider a more concrete a example: a
real massless scalar field on the ETW brane with the following action

$S_{scalar}=\sigma\int_{Q}d^{2}x\sqrt{h}(\partial\phi)^{2},$ (5.10)

where $\sigma$ is some constant. This corresponds to considering an exactly
marginal deformation on the coset coming from breaking $U(1)$ symmetry. Note
also that even though there is also a configuration with disconnected ETW
branes as an on-shell solution, its on-shell action is smaller than the one we
will consider in the $z_{0}/z_{H}\to 0$ limit, as was pointed out in [71]. If
we solve the equation of motion with boundary conditions $\phi(x=0)=0$ and
$\phi(x=\pi z_{0})=\phi_{0}$ under a fixed induced metric $h$, we expect that
there is a sharp jump of $\phi$ around $x=\pi z_{0}/2$. This can be explained
as follows. The induced metric on the brane profile is evaluated from (5.4)
and (5.5)

$\displaystyle h_{\tau\tau}=\frac{R^{2}}{z^{2}}\,,\qquad
h_{zz}=\frac{R^{2}}{z^{2}(h(z)-R^{2}T^{2})}.$ (5.11)

The Klein-Gordon equation for the massless field becomes

$\Box\phi=\frac{1}{\sqrt{h}}\partial_{z}(\sqrt{h}h^{zz}\partial_{z}\phi)=0,$
(5.12)

where we assumed that $\partial_{\tau}\phi=0$ for rotational invariance. Then,
the first derivative can be solved as

$\partial_{z}\phi=\frac{C_{1}}{\sqrt{h(z)-T^{2}R^{2}}},$ (5.13)

where $C_{1}$ is some integration constant. We can integrate again and obtain

$\phi=C_{1}z_{0}\arctan\left(\frac{z}{z_{0}\sqrt{h(z)-R^{2}T^{2}}}\right)+C_{2},$
(5.14)

where $C_{2}$ is another integration constant. The problem is that $z$ is not
single valued with respect to $x$ (for example, we have $z=0$ for $x=0$ and
$x=\pi z_{0}$) and this makes us hard to impose the boundary conditions at
$z=0$. From (5.13) $\phi$ monotonically increases as $0\leq x\leq\pi z_{0}$
and taking into account the symmetry $x\leftrightarrow\pi z_{0}-x$ we find
that

$\phi(x)=\begin{cases}\phi(z),&(0\leq x\leq\pi z_{0}/2)\\\
\phi_{0}-\phi(z),&(\pi z_{0}/2\leq x\leq\pi z_{0})\end{cases}$ (5.15)

where we suitably use $x$ and $z$ in each region. From this, the boundary
conditions can be rewritten as

$\displaystyle\phi(z=0)\,=\,0\,,\qquad\phi(z_{*})\,=\,\frac{\phi_{0}}{2}\,,$
(5.16)

where $z_{*}=z_{0}\sqrt{1-R^{2}T^{2}}$ represents the turning point of the ETW
brane. Substituting this into (5.15) we find that

$\displaystyle C_{1}\,=\,\frac{\phi_{0}}{\pi z_{0}}\,,\qquad C_{2}\,=\,0\,.$
(5.17)

Note that this determines uniquely the $\phi(x)$ via (5.15). Schematic plots
for the relations among $x$, $z$ and $\phi$ are shown in figure 6.

Figure 6: Schematic plots for the relations among $x$, $z$ and $\phi$. The
blue region corresponds to $0\leq x\leq\pi z_{0}/2$ and the orange region
corresponds to $\pi z_{0}/2\leq x\leq\pi z_{0}$.

With these background solutions and the action (5.10), we can evaluate the
tree-level on-shell action as

$S_{scalar}=\frac{2\sigma z_{H}\phi_{0}^{2}}{z_{0}}$ (5.18)

and we expect that this represents the gravity dual of a boundary information
metric because

$\displaystyle\braket{\phi(x=0)=0}{\phi(x=\pi z_{0})=\phi_{0}}$
$\displaystyle=e^{-S_{scalar}}$ $\displaystyle\sim 1-\frac{2\sigma
z_{H}}{z_{0}}\phi_{0}^{2}.$ (5.19)

Then, the information metric reads

$G_{\phi\phi}=\frac{4\sigma z_{H}}{z_{0}},$ (5.20)

which reproduces the expected result $G_{\lambda\lambda}\propto 1/z_{0}$ as in
(5.9).

Note that the gravity dual of the boundary information metric is ambiguous by
mutiplying some $O(1)$ coefficient. For example, in the perturbation of
boundary scalar fields the gravity dual has a smooth gradient of scalar
fields. The statement that the gravity dual of the information metric is given
by the minimal cross-section means that we approximate such a solution with a
kink-like profile. The situation is the same in [70] where the gravity dual of
the bulk information metric was studied.

Even though we only considered an exact marginal deformation in the direction
of the $U(1)$ coset, it would be possible to generalise this to other
parameter spaces. For exactly marginal boundary deformations, the computation
of the information metric corresponds to evaluating the on-shell action with
corresponding scalar fields on top of the connecting ETW brane. We expect
similar computation as what we do in this case. Generalisations to relevant
and irrelevant deformations are also possible with a suitable potential for
the scalar fields on the ETW brane [91]. It would be interesting to explicitly
carry out the computation to study if the monotonicity of the information
metric is holographically satisfied or not.

##  6 Conclusion

In this paper, we introduced a novel information metric whose distance is
defined by the inner product of boundary states, which we called a boundary
information metric. We expect that this boundary information metric describes
a metric of a moduli space in a target space under a suitable perturbation of
boundary states and confirmed this by explicit calculations. We examined the
boundary information metric in the SU(2) WZW model and a mass deformed real
free scalar and its SUSY counterpart in 2-dim CFT. In the SU(2) WZW model, we
can reproduce the metric of the target space, a metric of the SU(2). In the
mass deformed free scalar case, under a flow from the Neumann boundary
condition to the Dirichlet boundary condition, the boundary information metric
is monotonically decreasing, although the disk amplitude itself is not. In the
supersymmetric case, the information metric monotonically decreases. However,
the action (4.16) defined from the disk amplitude becomes also monotonic both
in the bosonic and supersymmetric cases.

As an application to the holography, we conjectured that the boundary
information metric corresponds to the area of a minimal cross section of an
ETW brane in the thermal AdS space, which corresponds to a short cylinder
limit. We confirmed this conjecture by calculating the area of the minimal
cross section and reproduce a modulus dependence $1/z_{0}$. To confirm more
concretely, we also considered a real free scalar field on the ETW brane and
calculated the action with perturbed boundary conditions. From this, we
evaluated the boundary information metric and reproduced the modulus
dependence $1/z_{0}$, similarly. In this calculation, we assumed that the
gradient of the fields on the ETW brane changes so sharply that it localizes
as a minimal cross section geometrically in the first order approximation.
Therefore, we expect that it is hard to reproduce the coefficients of the
boundary information metric from the calculation of the minimal cross section
on the ETW brane. Toward explicit formulation of the boundary states in
AdS/BCFT, symmetric orbifold formulation seems to be useful, which will be one
of future directions of our boundary information metric [92, 93, 94]. We can
also examine the behavior of the information metric under boundary RG-flows
including the relevant deformation. We can consider scalar field on the ETW
brane and tune the potential as in [91]. We expect that we can analyze the
information metric under the backreacted solution.

Finally, we briefly commented on the relation to the string theory in the
Appendix A. In this analysis, we can reproduce the inverse power of the
modulus dependence for the fixed modulus and the flat metric of the transverse
directions to the D-branes. We cannot interpret this metric in string theory
directly because we do not integrate over the moduli of the worldsheet, but
even in that case, we can probe the metric of the target space even though its
coefficient seems to be meaningless. It will be a future direction to study
the relation of the this boundary information metric calculated with the
modulus fixed and the target space geometry.

## Acknowledgements

We are grateful to Tadashi Takayanagi for stimulating discussions and support
throughout the project. This work is supported by MEXT KAKENHI Grant-in-Aid
for Transformative Research Areas (A) through the “Extreme Universe”
collaboration: Grant Number 21H05187. The work of KS is also supported by the
Simons Foundation through the “It from Qubit” collaboration. MW is supported
by Grant-in-Aid for JSPS Fellows (No. 22J00752).

##  Appendix A Boundary Information Metric in Superstring

In this section, we will consider boundary states in a worldsheet CFT of
superstring theory and check that the boundary information metric corresponds
to the metric in the target space, which is just a flat 10-dim Minkowski
space. Below, we will follow the argument in [95, 80].

We consider a Dp-brane extending along the $x^{0},\cdots,x^{p}$ directions.
Then, the boundary conditions of open strings can be described as

$\displaystyle\partial_{\tau}X^{\mu}\Ket{B,\eta}=(\psi^{\mu}+i\eta\tilde{\psi^{\mu}})\Ket{B,\eta}=0,$
$\displaystyle\partial_{\sigma}X^{i}\Ket{B,\eta}=(\psi^{i}-i\eta\tilde{\psi^{i}})\Ket{B,\eta}=0,$
(A.1)

where we represent the tangential and perpendicular directions to the D-brane
as $x^{\mu}$ and $x^{i}$, respectively. The $\eta$ labels the spin structure
of D-branes or the R and NS sector of open strings stretching between
D-branes. In terms of the creation and annihilation operators we can rewrite
these boundary conditions as

$\displaystyle(\alpha^{\mu}_{n}+\tilde{\alpha}^{\mu}_{-n})\Ket{B,\eta}=0,$
$\displaystyle(\alpha^{i}_{n}-\tilde{\alpha}^{i}_{-n})\Ket{B,\eta}=0,$
$\displaystyle(\psi^{\mu}_{r}+i\eta\tilde{\psi}^{\mu}_{-r})\Ket{B,\eta}=0,$
$\displaystyle(\psi^{i}_{r}-i\eta\tilde{\psi}^{i}_{-r})\Ket{B,\eta}=0.$ (A.2)

We can integrate these and the boundary states can be described by coherent
states

$\Ket{B,\eta}\sim\exp{\sum_{n=0}^{\infty}\frac{1}{n}(\alpha^{\mu}_{-n}\tilde{\alpha}^{\mu}_{-n}-\alpha^{i}_{-n}\tilde{\alpha}^{i}_{-n})-i\eta\sum_{r>0}^{\infty}}(\psi^{\mu}_{-r}\tilde{\psi}^{\mu}_{-r}-\psi^{i}_{-r}\tilde{\psi}^{i}_{-r})\Ket{B,\eta}_{(0)},$
(A.3)

where the $\Ket{B,\eta}_{(0)}$ represents a boundary state for zeromodes. To
move onto zeromodes let us firstly consider the NS sector. We will denote the
left and right worldsheet fermion number as $F$ and $\tilde{F}$, respectively.
Their actions on the boundary states are

$\displaystyle(-1)^{F}\Ket{B,\eta}_{NSNS}=-\Ket{B,-\eta}_{NSNS},$
$\displaystyle(-1)^{\tilde{F}}\Ket{B,\eta}_{NSNS}=-\Ket{B,-\eta}_{NSNS}.$
(A.4)

Then, GSO invariant boundary states are obtained as

$\Ket{B}_{NSNS}=\frac{1+(-1)^{F}}{2}\frac{1+(-1)^{\tilde{F}}}{2}\Ket{B,+}_{NSNS}=\frac{1}{2}(\Ket{B,+}_{NSNS}-\Ket{B,-}_{NSNS}).$
(A.5)

In the RR sector we can similarly obtain

$\Ket{B}_{RR}=\frac{1+(-1)^{F+p}}{2}\frac{1+(-1)^{\tilde{F}+p}}{2}\Ket{B,+}_{RR}=\frac{1}{2}(\Ket{B,+}_{RR}+\Ket{B,-}_{RR}).$
(A.6)

From this we can calculate an amplitude between two parallel Dp-branes at
$x^{i}$ and $y^{i}$ in the long length approximation ($s\rightarrow\infty$) as
in (Fig.7) [95]

$\displaystyle\mathcal{A}_{x^{i}y^{i}}$
$\displaystyle=\frac{\alpha^{\prime}}{2}\frac{T_{p}^{2}}{4}V_{p+1}\int
ds(\frac{dk}{2\pi})^{9-p}e^{ik(x-y)-\frac{s}{2}\alpha^{\prime}k^{2}}(\frac{f_{3}(q)^{8}-f_{4}(q)^{8}-f_{2}(q)^{8}}{2f_{1}(q)^{8}})$
(A.7) $\displaystyle=\frac{\alpha^{\prime}T_{p}^{2}}{16}V_{p+1}\int\frac{\pi
dt}{t^{6}}(\frac{t}{2\pi^{2}\alpha^{\prime}})^{\frac{9-p}{2}}e^{-\frac{(x-y)^{2}}{2\pi\alpha^{\prime}}t}(\frac{f_{3}(\tilde{q})^{8}-f_{4}(\tilde{q})^{8}-f_{2}(\tilde{q})^{8}}{2f_{1}(\tilde{q})^{8}})$
(A.8)
$\displaystyle\sim(1-1)V_{p+1}2\pi(4\pi^{2}\alpha^{\prime})^{3-p}G_{9-p}(|x^{i}-y^{i}|^{2}),$
(A.9)

where $G_{D}(y)=\frac{\Gamma(\frac{D}{2})}{2\pi^{\frac{D}{2}}}y^{2-D}$ is a
massless scalar Green’s function in $d$ dimensions and $q$ and $\tilde{q}$ are
given by

$\displaystyle q$ $\displaystyle=e^{-s}$ $\displaystyle\tilde{q}$
$\displaystyle=e^{-\pi t}=e^{-\frac{\pi^{2}}{s}},$ (A.10)

where $s$ is the modulus of a cylinder in the closed string channel. However,
here we need a short cylinder limit ($s\rightarrow 0$). In this limit the
amplitude can be approximated as

$\displaystyle\mathcal{A}_{x^{i}y^{i}}$
$\displaystyle=\frac{\alpha^{\prime}}{2}\frac{T_{p}^{2}}{4}V_{p+1}\int
ds\left(\frac{dk}{2\pi}\right)^{9-p}e^{ik(x-y)-\frac{s}{2}\alpha^{\prime}k^{2}}\left(\frac{f_{3}(q)^{8}-f_{4}(q)^{8}-f_{2}(q)^{8}}{2f_{1}(q)^{8}}\right)$
$\displaystyle=\frac{\alpha^{\prime}}{16}\frac{T_{p}^{2}}{(2\pi^{3}\alpha^{\prime})^{\frac{9-p}{2}}}V_{p+1}\int_{0}^{\infty}dss^{\frac{p-1}{2}}e^{-\frac{(x-y)^{2}}{2\alpha^{\prime}s}}\left(\frac{f_{3}(q)^{8}-f_{4}(q)^{8}-f_{2}(q)^{8}}{f_{1}(q)^{8}}\right)$
(A.11)

Figure 7: A schematic figure of D-branes scattering. Two parallel D-branes
exchange closed strings and generate forces coming from a RR-sector and a
NSNS-sector. Usually, we evaluate the amplitude and fix the tension of
D-branes.

In the calculation of 2-dim CFT, we usually introduce a cutoff for the
boundary state via

$\Ket{\psi}=e^{-\epsilon H}\Ket{B}.$ (A.12)

Keeping this in mind, it is natural to treat the modulus as a cutoff because
here the $\epsilon$ corresponds to the cylinder modulus $s$. Note that this
cut-off is not compatible with the calculation in the string theory because we
do not integrate over the modulus. For the fixed modulus, we can calculate the
boundary information metric in the $y\rightarrow x$ limit from (A.11)

$\displaystyle D_{B}^{2}$
$\displaystyle=(1-1)\frac{2}{\pi\alpha^{\prime}s}dx_{i}^{2},$ (A.13)

which reproduces the flat metric along the transverse direction to the
D-branes. Here, we again note that we do not integrate over the modulus of a
cylinder. In the $s\rightarrow\infty$ limit the boundary information metric
approaches to zero. If we consider $s\rightarrow 0$ limit or a short distance
limit, it indeed diverges as $1/\epsilon$. Note that if we consider a long
cylinder limit for the fixed modulus, then the information metric is the same
as (A.13).

Several comments are in order. Firstly, the cylinder amplitude indeed vanishes
due to the BPS condition (we denote as $(1-1)$ in front of (A.9)). One may
wonder if it is meaningless to take the variation of the originally vanishing
quantity. However, it does not affect our statement because we can only
consider the RR sector or NSNS sector individually. In the NSNS sector the
exchange of gravitons and dilatons generates the attraction force and in the
RR sector the exchange of the RR fields generates the repulsion force. Because
the boundary states are proportional to the D-brane tension, the magnitude of
the amplitude is also proportional to the tension and has physical meanings.

Secondly, we note that the boundary information metric depends on the modulus.
In the long cylinder limit in the closed string channel, where
$s\rightarrow\infty$, the boundary information metric becomes zero. In this
limit, only the massless modes survive. This is consistent with a physical
intuition. If we change the position of branes in the long cylinder limit, we
can only detect the massless modes and the disk amplitudes. In the long
cylinder limit, the energy spectrum always takes the vacuum. The translation
along the transverse direction does not change the tension of the branes,
hence the information metric vanishes. We can generalize this logic explicitly
below.

Now, let us consider an overlap of boundary states with boundary conditions
$a$ and $b$, which is slightly perturbed from $a$ by a relevant operator
$\mathcal{O}$

$S\rightarrow S+\lambda\int dx\mathcal{O}.$ (A.14)

Especially, here we consider an almost marginal operator for simplicity. In
the long cylinder limit, the overlap of boundary states factorizes into the
disk amplitude and the bulk propagator

$\braket{a}{b}\sim g_{a}g_{b}e^{-E_{0}L},$ (A.15)

where we define the disk amplitude $g_{a}=\braket{a}{0}$. In the above, we
also approximate the energy of intermediate particles by the vacuum one. In
this limit, the boundary information metric can be approximated as

$\displaystyle D_{B}^{2}=$ $\displaystyle
2\left(1-\left|\frac{\braket{a}{b}}{\sqrt{\braket{a}{a}}\sqrt{\braket{b}{b}}}\right|\right)$
$\displaystyle=2\left(1-\left|\frac{g_{a}g_{b}e^{-E_{0}L}}{\sqrt{g_{a}^{2}e^{-E_{0}L}}\sqrt{g_{b}^{2}e^{-E_{0}L}}}\right|\right)\cdots$
$\displaystyle=0,$ (A.16)

where we ignore the sub-leading terms since they are exponentially suppressed
$\mathcal{O}(e^{-E_{n}L})$ and $E_{n}$ is the energy of the n-th excited
state.

The inverse power of modulus dependence in the boundary information metric is
a basic feature of 2-dim CFTs. These results match with the ones in the
AdS3/BCFT2 case. Especially, the boundary information metric becomes divergent
(an inverse power of the modulus) in the short length limit of the cylinder
(thermal AdS3) and zero in the long cylinder limit (the BTZ black-hole).

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Figure S15: Two-transition factors $D_{\kappa\chi}$ vs new-boson mass $m_{\phi}$ for variable transition pairs ($\chi$,$\kappa$) coded with different colors (see legend) calculated using $D_{\chi}$ and $D_{\kappa}$ in Fig. S14. Solid, dashed, and dash-dotted lines are for ambit, GRASP2018, and Ref. [22], respectively (some of dashed and dash-dotted lines are missing as the corresponding $D_{\kappa\chi}$ are not available; see Table LABEL:tab:el_factors_2). Table S9:  Calculated and experimental values of three-transition electronic factors $f_{\eta\kappa\chi}$ and $Z_{\eta\kappa\chi}$ ($Z\in\\{K,G^{(2)},D\\}$) for $\chi,\kappa,\eta\in\\{\alpha,\beta,\gamma,\delta,\epsilon\\}$. The quantities are defined in Sec. II.6. The values are calculated from the two-transition values in Table LABEL:tab:el_factors_2. $f_{\eta\kappa\chi}$ is dimensionless. The units of $K_{\eta\kappa\chi}$, $G^{(2)}_{\eta\kappa\chi}$, $D_{\eta\kappa\chi}$, and $\upsilon_{ne}D_{\chi\kappa}$ are GHz$\cdot$u, kHz/fm4, kHz/fm4, $10^{3}$ THz, and kHz, respectively. The last three columns (“Fit”) are for data from fit of corresponding 3D King plots $\overline{\overline{\nu}}^{AA^{\prime}}_{\eta}=K_{\eta\kappa\chi}+f_{\eta\kappa\chi}\overline{\overline{\nu}}^{AA^{\prime}}_{\chi}+f_{\eta\chi\kappa}\overline{\overline{\nu}}^{AA^{\prime}}_{\kappa}$ (“Linear“), and $G^{(2)}_{\eta\kappa\chi}\overline{\overline{[\delta\langle{r^{2}}\rangle^{2}]}}^{AA^{\prime}}$ (“QFS”) or $\upsilon_{ne}D_{\eta\kappa\chi}\overline{\overline{a}}^{AA^{\prime}}$ (“New boson”) terms in addition to the relation. $\chi^{2}_{\eta\kappa\chi}$ and $s_{\eta\kappa\chi}$ are $\chi^{2}$ and the significance of fit, respectively.  | GRASP |  ambit | Ref. [22] | Fit
---|---|---|---|---
| Cal. | Exp. | Cal. | Exp. | Cal. | Exp. | Linear | QFS | New boson
$f_{\beta\gamma\alpha}$ | 0.81292 |  |  |  |  |  | 0.978(26) | 0.998(27) | 1.052(36)
$f_{\beta\delta\alpha}$ | 1.0214 |  |  |  |  |  | 1.023(13) | 1.018(13) | 0.993(16)
$f_{\beta\epsilon\alpha}$ |  |  |  |  | 1.0054 |  | 1.14(10) | 1.058(99) | 0.86(12)
$f_{\delta\gamma\alpha}$ | 19.612 |  |  |  |  |  | 1.867(41) | 1.877(41) | 1.965(51)
$f_{\epsilon\gamma\alpha}$ |  |  |  |  |  |  | 1.049(30) | 1.046(33) | 1.040(45)
$f_{\epsilon\delta\alpha}$ |  |  |  |  |  |  | 0.701(13) | 0.707(14) | 0.717(19)
$f_{\delta\gamma\beta}$ | 24.126 |  |  |  |  |  | 1.885(35) | 1.880(37) | 1.868(49)
$f_{\epsilon\gamma\beta}$ |  |  |  |  |  |  | 1.090(28) | 1.047(31) | 0.989(43)
$f_{\epsilon\delta\beta}$ |  |  |  |  |  |  | 0.673(11) | 0.695(13) | 0.722(19)
$f_{\epsilon\delta\gamma}$ |  |  |  |  |  |  | -0.2082(32) | -0.2146(38) | -0.2223(54)
$f_{\beta\alpha\gamma}$ | -0.076559 |  |  |  |  |  | -0.015(12) | -0.006(12) | 0.018(16)
$f_{\beta\alpha\delta}$ | -0.010629 |  |  |  |  |  | -0.019(21) | -0.010(21) | 0.030(26)
$f_{\beta\alpha\epsilon}$ |  |  |  |  | 0.021058 |  | -0.15(12) | -0.06(12) | 0.18(15)
$f_{\delta\alpha\gamma}$ | 7.2027 |  |  |  |  |  | 0.565(18) | 0.570(19) | 0.609(23)
$f_{\epsilon\alpha\gamma}$ |  |  |  |  |  |  | 0.104(14) | 0.103(15) | 0.100(20)
$f_{\epsilon\alpha\delta}$ |  |  |  |  |  |  | 0.191(21) | 0.180(23) | 0.165(31)
$f_{\delta\beta\gamma}$ | 9.0498 |  |  |  |  |  | 0.583(16) | 0.580(17) | 0.575(22)
$f_{\epsilon\beta\gamma}$ |  |  |  |  |  |  | 0.128(13) | 0.109(14) | 0.082(20)
$f_{\epsilon\beta\delta}$ |  |  |  |  |  |  | 0.223(19) | 0.187(22) | 0.143(32)
$f_{\epsilon\gamma\delta}$ |  |  |  |  |  |  | 0.580(12) | 0.557(14) | 0.529(20)
$K_{\beta\gamma\alpha}$ | -34.804 | 89(21) |  | -80(160) |  |  | 206(66) | 154(69) | 17(91)
$K_{\beta\delta\alpha}$ |  |  |  |  |  |  | 126.9(7.7) | 124.2(7.8) | 111.1(9.0)
$K_{\beta\epsilon\alpha}$ |  |  |  |  |  |  | 120.48(23) | 120.53(21) | 121.79(43)
$K_{\delta\gamma\alpha}$ |  |  |  |  |  |  | -2880(110) | -2900(110) | -3130(130)
$K_{\epsilon\gamma\alpha}$ |  |  |  |  |  |  | -596(79) | -587(85) | -570(120)
$K_{\epsilon\delta\alpha}$ |  |  |  |  |  |  | -67.6(7.6) | -63.8(8.3) | -59(11)
$K_{\delta\gamma\beta}$ |  |  |  |  |  |  | -3207(97) | -3190(100) | -3160(130)
$K_{\epsilon\gamma\beta}$ |  |  |  |  |  |  | -865(77) | -749(86) | -590(120)
$K_{\epsilon\delta\beta}$ |  |  |  |  |  |  | -160.3(5.5) | -150.1(6.4) | -138.8(8.8)
$K_{\epsilon\delta\gamma}$ |  |  |  |  |  |  | 986(23) | 1031(27) | 1083(38)
$G^{(2)}_{\beta\gamma\alpha}$ | 2.0139 | -13.4(2.6) |  | -77.3(7.5) |  |  |  | 57(18) |
$G^{(2)}_{\beta\delta\alpha}$ |  |  |  | -74.1(4.6) |  |  |  | 58(18) |
$G^{(2)}_{\beta\epsilon\alpha}$ |  |  |  | -72.8(3.4) | -645.43 | -636.1(8.6) |  | 57(18) |
$G^{(2)}_{\delta\gamma\alpha}$ |  |  |  | -106(11) |  |  |  | 94(24) |
$G^{(2)}_{\epsilon\gamma\alpha}$ |  |  |  | -24.4(9.6) |  |  |  | -6(22) |
$G^{(2)}_{\epsilon\delta\alpha}$ |  |  |  | -6.9(5.6) |  |  |  | -23(19) |
$G^{(2)}_{\delta\gamma\beta}$ |  |  |  | 50.2(6.8) |  |  |  | -13(27) |
$G^{(2)}_{\epsilon\gamma\beta}$ |  |  |  | 59.9(6.0) |  |  |  | -65(22) |
$G^{(2)}_{\epsilon\delta\beta}$ |  |  |  | 52.7(7.2) |  |  |  | -63(20) |
$G^{(2)}_{\epsilon\delta\gamma}$ |  |  |  | 36.1(6.1) |  |  |  | -58(18) |
$D_{\beta\gamma\alpha}$a | -43.386 | 14.8(9.8) |  | 9.2(3.9) |  |  |  |  |
$D_{\beta\delta\alpha}$a | 2.7384 | 6.0(2.1) |  | 6.9(1.8) |  |  |  |  |
$D_{\beta\epsilon\alpha}$a |  |  |  | 11.0(5.1) | 7.2462 | 12.0(4.4) |  |  |
$D_{\delta\gamma\alpha}$a | 4339.4 | 293(14) |  | 78.7(5.6) |  |  |  |  |
$D_{\epsilon\gamma\alpha}$a |  |  |  | -9.3(5.0) |  |  |  |  |
$D_{\epsilon\delta\alpha}$a |  |  |  | -22.3(2.2) |  |  |  |  |
$D_{\delta\gamma\beta}$a | 5386.1 | 264(13) |  | 60.6(5.2) |  |  |  |  |
$D_{\epsilon\gamma\beta}$a |  |  |  | -19.0(4.6) |  |  |  |  |
$D_{\epsilon\delta\beta}$a |  |  |  | -27.7(2.4) |  |  |  |  |
$D_{\epsilon\delta\gamma}$a |  |  |  | -53.2(2.7) |  |  |  |  |
$\upsilon_{ne}D_{\beta\gamma\alpha}$ |  |  |  |  |  |  |  |  | 54(17)
$\upsilon_{ne}D_{\beta\delta\alpha}$ |  |  |  |  |  |  |  |  | 51(15)
$\upsilon_{ne}D_{\beta\epsilon\alpha}$ |  |  |  |  |  |  |  |  | 55(17)
$\upsilon_{ne}D_{\delta\gamma\alpha}$ |  |  |  |  |  |  |  |  | 88(24)
$\upsilon_{ne}D_{\epsilon\gamma\alpha}$ |  |  |  |  |  |  |  |  | -5(21)
$\upsilon_{ne}D_{\epsilon\delta\alpha}$ |  |  |  |  |  |  |  |  | -20(17)
$\upsilon_{ne}D_{\delta\gamma\beta}$ |  |  |  |  |  |  |  |  | -12(23)
$\upsilon_{ne}D_{\epsilon\gamma\beta}$ |  |  |  |  |  |  |  |  | -58(19)
$\upsilon_{ne}D_{\epsilon\delta\beta}$ |  |  |  |  |  |  |  |  | -57(18)
$\upsilon_{ne}D_{\epsilon\delta\gamma}$ |  |  |  |  |  |  |  |  | -52(16)
$\chi^{2}_{\beta\gamma\alpha}$ |  |  |  |  |  |  | 10.532 |  |
$\chi^{2}_{\beta\delta\alpha}$ |  |  |  |  |  |  | 10.9 |  |
$\chi^{2}_{\beta\epsilon\alpha}$ |  |  |  |  |  |  | 8.724 |  |
$\chi^{2}_{\delta\gamma\alpha}$ |  |  |  |  |  |  | 15.221 |  |
$\chi^{2}_{\epsilon\gamma\alpha}$ |  |  |  |  |  |  | 0.065554 |  |
$\chi^{2}_{\epsilon\delta\alpha}$ |  |  |  |  |  |  | 1.4067 |  |
$\chi^{2}_{\delta\gamma\beta}$ |  |  |  |  |  |  | 0.23876 |  |
$\chi^{2}_{\epsilon\gamma\beta}$ |  |  |  |  |  |  | 8.3928 |  |
$\chi^{2}_{\epsilon\delta\beta}$ |  |  |  |  |  |  | 10.248 |  |
$\chi^{2}_{\epsilon\delta\gamma}$ |  |  |  |  |  |  | 10.481 |  |
$s_{\beta\gamma\alpha}$ |  |  |  |  |  |  | 3.2454$\sigma$ |  |
$s_{\beta\delta\alpha}$ |  |  |  |  |  |  | 3.3015$\sigma$ |  |
$s_{\beta\epsilon\alpha}$ |  |  |  |  |  |  | 2.9536$\sigma$ |  |
$s_{\delta\gamma\alpha}$ |  |  |  |  |  |  | 3.9014$\sigma$ |  |
$s_{\epsilon\gamma\alpha}$ |  |  |  |  |  |  | 0.25604$\sigma$ |  |
$s_{\epsilon\delta\alpha}$ |  |  |  |  |  |  | 1.1861$\sigma$ |  |
$s_{\delta\gamma\beta}$ |  |  |  |  |  |  | 0.48863$\sigma$ |  |
$s_{\epsilon\gamma\beta}$ |  |  |  |  |  |  | 2.897$\sigma$ |  |
$s_{\epsilon\delta\beta}$ |  |  |  |  |  |  | 3.2012$\sigma$ |  |
$s_{\epsilon\delta\gamma}$ |  |  |  |  |  |  | 3.2375$\sigma$ |  |
a At $m_{\phi}=1$ eV. Values over different $m_{\phi}$’s are shown in Fig. S16 |  |  |  |

(a) $(\alpha,\gamma,\beta)$ transitions (b) $(\alpha,\delta,\beta)$
transitions (c) $(\alpha,\epsilon,\beta)$ transitions (d)
$(\alpha,\gamma,\delta)$ transitions

Figure S16: Calculated $D_{\eta\kappa\chi}$ vs new-boson mass $m_{\phi}$ for
all different choices of three transitions $(\chi,\kappa,\eta)$ out of five
available transitions $\alpha$, $\beta$, $\gamma$, $\delta$, and $\epsilon$,
each corresponding to one of the subfigures (a – j). Solid lines correspond to
the $D_{\eta\kappa\chi}$ obtained from $D_{\kappa\chi}$ and $D_{\eta\chi}$ in
Fig. S15, and $f_{\eta\chi\kappa}=G^{(4)}_{\eta\chi}/G^{(4)}_{\kappa\chi}$
ratio from the linear fit in 3D King plot (see Table LABEL:tab:el_factors_3).
Shaded regions for $D_{\eta\kappa\chi}$ indicate 95% confidence intervals that
arise from fitted $f_{\eta\chi\kappa}$’s uncertainty. (Figures and caption
continue on the next page.)

(e) $(\alpha,\gamma,\epsilon)$ transitions (f) $(\alpha,\delta,\epsilon)$
transitions (g) $(\beta,\gamma,\delta)$ transitions (h)
$(\beta,\gamma,\epsilon)$ transitions

Figure S16: (Continued) Dashed lines show $D_{\eta\kappa\chi}$ calculated
purely from ASCs (i.e., using calculated $f_{\eta\chi\kappa}$). Blue, red, and
green colors correspond to ASCs performed using GRASP2018, ambit, and in Ref.
[22], respectively. (Figures continued on the next page.)

(i) $(\beta,\delta,\epsilon)$ transitions (j) $(\gamma,\delta,\epsilon)$
transitions

Figure S16: (Continued)