operator will enter the $SL(2,\mathbb{R})$ block decomposition of the four-
point correlation function of $\psi$ as

$\displaystyle\hat{C}_{\psi\psi}{}^{\text{D}}\hat{C}_{\psi\psi\text{D}}=\frac{\hat{\Delta}^{2}}{\hat{C}_{\text{D}}}=\frac{2\hat{\Delta}^{2}}{c}~{}.$
(A.63)

The four-point correlation function $\langle\text{D}\text{D}\psi\psi\rangle$
in this ordering can be obtained from the result of eq. (A.44) and reads
($\eta$ is defined in eq. (A.4))

$\displaystyle\langle$
$\displaystyle\text{D}(x_{1})\text{D}(x_{2})\psi(x_{3}){\psi}(x_{4})\rangle_{{\mathbb{H}^{+}}}=\frac{\mathcal{G}^{\text{D}\text{D}\psi\psi}(\eta)}{(x_{12})^{4}(x_{34})^{2\hat{\Delta}}}~{},$
$\displaystyle\mathcal{G}^{\text{D}\text{D}\psi\psi}(\eta)=\left(\frac{c}{2}+\frac{\hat{\Delta}\eta^{2}(\eta(\hat{\Delta}\eta-2)+2)}{(\eta-1)^{2}}\right)~{}.$
(A.64)

The $SL(2,\mathbb{R})$ s-channel block decomposition reads

$\displaystyle\mathcal{G}^{\text{D}\text{D}\psi\psi}(\eta)=\frac{c}{2}G(0,0,0,\eta)+\sum_{n=2,4,\dots}\hat{C}_{\text{D}\text{D}}{}^{n}\hat{C}_{\psi\psi
n}G(0,0,n,\eta)~{},$ (A.65)

with blocks given in eq. (A.3.3) and coefficients

$\displaystyle\hat{C}_{\text{D}\text{D}}{}^{n}\hat{C}_{\psi\psi
n}=\frac{\sqrt{\pi}\hat{\Delta}}{2^{2n-3}\Gamma\left(n-\frac{1}{2}\right)}(\hat{\Delta}(n-2)(n+1)+2)\Gamma(n)~{}.$
(A.66)

In particular for the first few coefficients we have

$\displaystyle\hat{C}_{\text{D}\text{D}}{}^{\text{D}}\hat{C}_{\psi\psi\text{D}}=2\hat{\Delta}~{},$
$\displaystyle\quad\hat{C}_{\text{D}\text{D}}{}^{\text{D}^{2}}\hat{C}_{\psi\psi\text{D}^{2}}=\frac{\hat{\Delta}}{5}(5\hat{\Delta}+1)~{},$
(A.67)

and so, once we combine with eq. (A.59) and eq. (A.60), we find

$\displaystyle\hat{C}_{\psi\psi\text{D}^{2}}=\frac{\hat{\Delta}}{5}(5\hat{\Delta}+1).~{}$
(A.68)

We can also consider a different ordering, e.g.

$\displaystyle\langle\text{D}(x_{1})\psi(x_{2})$
$\displaystyle\text{D}(x_{3}){\psi}(x_{4})\rangle_{{\mathbb{H}^{+}}}=\left(\frac{x_{14}}{x_{24}}\right)^{\hat{\Delta}-2}\left(\frac{x_{14}}{x_{13}}\right)^{2-\hat{\Delta}}\frac{\mathcal{G}^{\text{D}\psi\text{D}\psi}(\eta)}{(x_{12})^{2+\hat{\Delta}}(x_{34})^{2+\hat{\Delta}}}~{},\quad
x_{i}>x_{i+1}$
$\displaystyle\mathcal{G}^{\text{D}\psi\text{D}\psi}(\eta)=\eta^{\hat{\Delta}+2}\left(\frac{c}{2}+\frac{\hat{\Delta}(2(\eta-1)\eta+\hat{\Delta})}{\eta^{2}(\eta-1)^{2}}\right)~{},$
(A.69)

with $\eta$ defined as in eq. (A.4). The $s$-channel $SL(2,\mathbb{R})$ block
decomposition reads

$\displaystyle\mathcal{G}^{\text{D}\psi\text{D}\psi}(\eta)=\hat{\Delta}^{2}G(\hat{\Delta}-2,2-\hat{\Delta},\hat{\Delta},\eta)+\sum_{n=2,3,4,\dots}\hat{C}_{\text{D}\psi}{}^{n}{\hat{C}}_{\text{D}\psi
n}G(\hat{\Delta}-2,2-\hat{\Delta},\hat{\Delta}+n,\eta)~{},$ (A.70)

and the first non-zero coefficients are

$\displaystyle\hat{C}_{\text{D}\psi}{}^{2}\hat{C}_{\text{D}\psi 2}$
$\displaystyle=\frac{c}{2}+\hat{\Delta}\left(4-\frac{9}{2\hat{\Delta}+1}\right)~{},$
$\displaystyle\hat{C}_{\text{D}\psi}{}^{3}\hat{C}_{\text{D}\psi 3}$
$\displaystyle=-\frac{2\hat{\Delta}}{(\hat{\Delta}+1)(\hat{\Delta}+2)}\left((c-7)\hat{\Delta}+c+3\hat{\Delta}^{2}+2\right)~{},$
$\displaystyle\hat{C}_{\text{D}\psi}{}^{4}\hat{C}_{\text{D}\psi 4}$
$\displaystyle=\frac{\hat{\Delta}(5c(4\hat{\Delta}(\hat{\Delta}+2)+3)+4\hat{\Delta}(\hat{\Delta}(8\hat{\Delta}-19)+26)-15)}{(\hat{\Delta}+3)(2\hat{\Delta}+3)(2\hat{\Delta}+5)}~{}.$
(A.71)

Note that the negative term on the second line is due to a parity-odd operator
at level 3, see for example appendix K of [43].

####  A.3.5 Correlation functions of the displacement with a bulk operator

Consider a scalar bulk Virasoro primary $\phi_{h,h}(z,\bar{z})$ with scaling
dimension $\Delta=2h$. We want to compute its two-point correlation function
with the displacement

$\displaystyle\langle\text{D}(0)\phi_{h,h}(x+iy,x-iy)\rangle_{{\mathbb{H}^{+}}}~{}.$
(A.72)

As we have discussed, D can be traded for an insertion of $T$ on
${\mathbb{H}^{+}}$. Since $T$ is holomorphic, it does not need to be doubled,
and by the method of images

$\displaystyle\langle\text{D}(0)\phi_{h,h}(z,\bar{z})\rangle_{{\mathbb{H}^{+}}}\propto\langle
T(0)\phi_{h}(z)\phi_{h}(z^{*})\rangle~{}.$ (A.73)

Upon using eq. (A.43) we learn that

$\displaystyle\langle\text{D}(0)\phi_{h,h}(x+iy,x-iy)\rangle_{{\mathbb{H}^{+}}}=\frac{B_{\phi\text{D}}}{(2y)^{\Delta-2}(x^{2}+y^{2})^{2}}~{}.$
(A.74)

The bulk-to-boundary coefficient $B_{\phi\text{D}}$ is further related to the
one-point function in eq. (A.34) by means of a special Ward identity [33, 34]

$\displaystyle B_{\phi\text{D}}=-B_{\phi}\Delta/2~{}.$ (A.75)

The same logic applies to higher-point correlators, e.g.

$\displaystyle\langle\text{D}(x_{1})\text{D}(x_{2})\phi_{h,h}(z,\bar{z})\rangle_{{\mathbb{H}^{+}}}$
$\displaystyle\propto\langle
T(x_{1})T(x_{2})\phi_{h}(z)\phi_{h}(z^{*})\rangle~{},$ (A.76)

and from eq. (A.44) we find

$\displaystyle\langle\text{D}(x_{1})\text{D}(x_{2})\phi_{h,h}(x+iy,x-iy)\rangle_{{\mathbb{H}^{+}}}=\frac{B_{\phi}}{(2y)^{\Delta}(x_{12})^{4}}\,\left(\frac{c}{2}-\Delta\tilde{\eta}\left(1-\frac{\Delta\tilde{\eta}}{4}\right)\right)~{}.$
(A.77)

The cross-ratio $\tilde{\eta}$ is defined in eq. (B.23). The normalization of
eq. (A.77) is chosen such that the $y\rightarrow 0$ expansion of the
correlator above gives the identity contribution in the
$\text{D}\times\text{D}$ OPE, with the correct normalization. As a check of
this result, we can decompose eq. (A.77) into the bulk-boundary-boundary
conformal blocks of eq. (A.12). The second addend in eq. (A.77) can be
decomposed as follows

$\displaystyle B_{\phi}\sum_{n=2,4,6\dots}\frac{\sqrt{\pi}\Delta e^{\frac{i\pi
n}{2}}(\Delta(n-2)(n+1)+4)\Gamma(n)}{2^{2n-1}\Gamma\left(n-\frac{1}{2}\right)}\,f(0,n,\tilde{\eta}/4)~{}.$
(A.78)

In particular, we learn that

$\displaystyle\hat{C}_{\text{D}\text{D}}{}^{\text{D}}B_{\phi\text{D}}=-B_{\phi}~{}\Delta,\quad\hat{C}_{\text{D}\text{D}}{}^{\text{D}^{2}}B_{\phi\text{D}^{2}}=\frac{1}{20}B_{\phi}\Delta(5\Delta+2)~{}.$
(A.79)

As a check of this result, we note that the leftmost piece of the equation
above reproduces eq. (A.75) after using eq. (A.3.3). From the rightmost piece,
upon combining with eq. (A.60) we find

$\displaystyle B_{\phi\text{D}^{2}}=\frac{1}{20}B_{\phi}\Delta(5\Delta+2)~{}.$
(A.80)

####  A.3.6 Correlation functions with $\text{D}^{2}$

The operator $\text{D}^{2}$ is the unique global primary at level four in the
boundary identity module. It is defined upon restricting the $T\times T$ OPE
in eq. (A.40) to the real line

$\displaystyle\text{D}(x)\text{D}(0)=\frac{c/2}{x^{4}}{\hat{\mathds{1}}}+\frac{2}{x^{2}}\text{D}(0)+\frac{1}{x}\text{D}^{\prime}(0)+\frac{3}{10}\text{D}^{\prime\prime}(0)+\text{D}^{2}(0)+O(x^{2})~{},$
(A.81)

where primed operators are derivatives with respect to $x$, $O(x^{2})$ denotes
higher order contributions and $\text{D}^{2}$ is the restriction of $T^{2}$ to
the boundary.

To compute correlation functions with $\text{D}^{2}$ on the upper half-plane
we apply the following strategy. Starting from correlation functions with two
insertions of $T$ on ${\mathbb{H}^{+}}$, each $T^{2}$ insertion is obtained as

$\displaystyle\langle T^{2}(z)\dots\rangle\equiv\lim_{z\rightarrow
0}\langle\left(T(z)T(0)-\frac{c/2}{z^{4}}{\mathds{1}}-\frac{2}{z^{2}}T(0)-\frac{1}{z}T^{\prime}(0)-\frac{3}{10}T^{\prime\prime}(0)\right)\dots\rangle~{}.$
(A.82)

Finally, each $T^{2}(z)$ insertion defines a $\text{D}^{2}(x)$ insertion upon
taking the boundary limit on $z$, the latter being regular, as dictated by
holomorphy.

Our first target is

$\displaystyle\langle\text{D}^{2}(x_{1})\text{D}^{2}(x_{2})\phi(x+iy,x-iy)\rangle_{{\mathbb{H}^{+}}}~{},$
(A.83)

with $\phi$ a bulk Virasoro primary of dimension $\Delta$. For this we should
consider

$\displaystyle\langle
T(z_{0})T(z_{1})T(z_{2})T(z_{3})\phi(x+iy,x-iy)\rangle_{{\mathbb{H}^{+}}}~{},$
(A.84)

which is determined by the conformal Ward identity in terms of the correlator
of $\phi$ with zero, one, two, and three $T$ insertions, as explained above.
Note that in order to take the limit (A.82), we need to combine this five-
point correlator with the correlator of $\phi$ with one, two, and three $T$
insertions. Taking the limit (A.82) of coincident points (say, between the
first two and the second two $T$ insertions), and finally the boundary limit
of each $T^{2}$ insertion, we find

$\displaystyle\langle\text{D}^{2}(x_{1})\text{D}^{2}(x_{2})\phi(x+iy,x-iy)\rangle_{{\mathbb{H}^{+}}}=\frac{\mathcal{G}^{\text{D}^{2}\text{D}^{2}\phi}(\tilde{\eta})}{(2y)^{\Delta}(x_{12})^{8}}~{},$
(A.85)

where the cross-ratio is that of eq. (A.13) and

$\displaystyle\mathcal{G}^{\text{D}^{2}\text{D}^{2}\phi}(\tilde{\eta})={B_{\phi}}$
$\displaystyle\left(\frac{c}{10}(5c+22)-\frac{2\Delta}{5}(5c+22)\tilde{\eta}+\frac{\Delta}{50}(5(5c+64)\Delta+84)\tilde{\eta}^{2}\right.$
$\displaystyle~{}~{}~{}~{}\left.-\frac{\Delta}{25}(5\Delta+2)^{2}\tilde{\eta}^{3}+\frac{\Delta^{2}}{400}(5\Delta+2)^{2}\tilde{\eta}^{4}\right)~{}.$
(A.86)

The $B_{\phi}$ is that of eq. (A.34). This result can also be expanded into
the bulk-boundary-boundary blocks of eq. (A.12) to find

$\displaystyle\frac{B_{\phi}}{10}c(5c+22)+B_{\phi}\sum_{n=2,4,6\dots}\left(c_{n}^{(1)}\Delta+c_{n}^{(2)}\Delta^{2}+c_{n}^{(3)}\Delta^{3}+c_{n}^{(4)}\Delta^{4}\right)\,f(0,n,\tilde{\eta}/4)~{},$
(A.87)

where

$\displaystyle c_{n}^{(1)}$ $\displaystyle=e^{\frac{i\pi
n}{2}}\frac{\sqrt{\pi}2^{3-2n}(50c+((n-1)n+8)((n-1)n+20))\Gamma(n)}{25\Gamma\left(n-\frac{1}{2}\right)}~{},$
$\displaystyle c_{n}^{(2)}$ $\displaystyle=e^{\frac{i\pi
n}{2}}\frac{\sqrt{\pi}2^{-2(n+1)}(n-2)(2n-1)(1800c+(n-1)n((n-1)n+678)+14760)\Gamma(n+2)}{225n\Gamma\left(n+\frac{1}{2}\right)}~{},$
$\displaystyle c_{n}^{(3)}$ $\displaystyle=e^{\frac{i\pi
n}{2}}\frac{\sqrt{\pi}2^{-2n-1}(n-4)(n-2)(n+3)((n-1)n+150)\Gamma(n+2)}{45n\Gamma\left(n-\frac{1}{2}\right)}~{},$
$\displaystyle c_{n}^{(4)}$ $\displaystyle=e^{\frac{i\pi
n}{2}}\frac{\sqrt{\pi}2^{-2n-3}(n-6)(n-4)(n-2)(n+3)(n+5)\Gamma(n+2)}{9n\Gamma\left(n-\frac{1}{2}\right)}~{}.$
(A.88)

###  A.4 Correlation functions with $T\bar{T}$

Let us now discuss how we compute correlation functions with insertions of
$T\bar{T}$ on ${\mathbb{H}^{+}}$. The first observation is that the
$T\times\bar{T}$ bulk OPE is regular, as dictated by holomorphy, and the limit
of coincident points defines a unique scalar bulk global primary of scaling
dimension equal to four. Therefore

$\displaystyle\langle\dots T\bar{T}(z)\rangle_{{\mathbb{H}^{+}}}$
$\displaystyle\equiv\lim_{z^{\prime}\rightarrow z}\langle\dots
T(z^{\prime})\bar{T}(z)\rangle_{{\mathbb{H}^{+}}}~{}$
$\displaystyle=\lim_{z^{\prime}\rightarrow z}\langle\dots
T(z^{\prime})T(z^{*})\rangle_{{\mathbb{H}^{+}}}=\langle\dots
T(z)T(z^{*})\rangle_{{\mathbb{H}^{+}}}~{}.$ (A.89)

As an immediate application of the prescription above, we learn that

$\displaystyle\langle
T\bar{T}(x+iy,x-iy)\rangle_{{\mathbb{H}^{+}}}\equiv\frac{B_{T\bar{T}}}{(2y)^{4}}=\langle
T(x+iy)T(x-iy)\rangle_{{\mathbb{H}^{+}}}=\frac{c/2}{(2y)^{4}}~{},$ (A.90)

and so $B_{T\bar{T}}=c/2$.

Let us now compute the three-point correlation function between $T\bar{T}$ and
a boundary Virasoro primary $\psi$ of scaling dimension $\hat{\Delta}$.
Starting from eq. (A.44) (with $\phi\rightarrow\psi$), we apply the
prescription in eq. (A.4) to obtain

$\displaystyle\langle\psi(x_{1})\psi(x_{2})T\bar{T}(x+iy,x-iy)\rangle_{{\mathbb{H}^{+}}}=\frac{\mathcal{G}^{\psi\psi
T\bar{T}}(\tilde{\eta})}{(2y)^{4}(x_{12})^{2\hat{\Delta}}}~{},\quad
x_{1}>x_{2}~{}.$ (A.91)

where

$\displaystyle\mathcal{G}^{\psi\psi
T\bar{T}}(\tilde{\eta})=\frac{c}{2}-2\hat{\Delta}\tilde{\eta}+\hat{\Delta}^{2}\tilde{\eta}^{2}~{}.$
(A.92)

The cross-ratio $\tilde{\eta}$ is that of eq. (A.13). The normalization is
fixed by the $\psi$ self-OPE limit, and by eq. (A.90).

Expanding this result into the bulk-boundary-boundary blocks of eq. (A.12) we
find

$\displaystyle\frac{c}{2}+\sum_{n=2,4,6\dots}(\alpha^{(1)}_{n}\hat{\Delta}+\alpha^{(2)}_{n}\hat{\Delta}^{2})~{}\,f(0,n,\tilde{\eta}/4)~{},$
(A.93)

where

$\displaystyle\alpha^{(0)}_{n}=\frac{\sqrt{\pi}e^{\frac{i\pi
n}{2}}\Gamma(n)}{4^{n-2}\Gamma\left(n-\frac{1}{2}\right)}~{},\quad\alpha^{(1)}_{n}=\frac{\sqrt{\pi}e^{\frac{i\pi
n}{2}}(n-2)(n+1)\Gamma(n)}{2^{2n-3}\Gamma\left(n-\frac{1}{2}\right)}~{}.$
(A.94)

Another interesting correlator is the three-point correlation function between
two insertions of the displacement and $T\bar{T}$ on ${\mathbb{H}^{+}}$. This
can be obtained from the four-point function of $T$ in eq. (A.3.1), upon
applying the prescription in eq. (A.4) (e.g. for the last two insertions of
$T$) and taking the boundary limit on the first two $T$ insertions. The
normalization is again fixed by the bulk-boundary limit, so we find

$\displaystyle\langle\text{D}(x_{1})\text{D}(x_{2})T\bar{T}(x+iy,x-iy)\rangle_{{\mathbb{H}^{+}}}=\frac{\mathcal{G}^{\text{D}\text{D}T\bar{T}}(\tilde{\eta})}{(2y)^{4}(x_{12})^{4}}~{},$
(A.95)

where again $\tilde{\eta}$ is that of eq. (A.13) and

$\displaystyle\mathcal{G}^{\text{D}\text{D}T\bar{T}}(\tilde{\eta})$
$\displaystyle=\frac{c^{2}}{4}\left(1-\frac{8\tilde{\eta}}{c}+\left(\frac{8}{c}+2\right)\tilde{\eta}^{2}-4\tilde{\eta}^{3}+\tilde{\eta}^{4}\right)~{}.$
(A.96)

Expanding this result into the bulk-boundary-boundary blocks of eq. (A.12) we
find

$\displaystyle\frac{c^{2}}{4}+2c\sum_{n=2,4,6\dots}(\alpha^{(0)}_{n}+\alpha^{(1)}_{n}c)~{}\,f(0,n,\tilde{\eta}/4)~{},$
(A.97)

where

$\displaystyle\alpha^{(0)}_{n}=\frac{\sqrt{\pi}e^{\frac{i\pi
n}{2}}((n-1)n-1)\Gamma(n)}{2^{2n-3}\Gamma\left(n-\frac{1}{2}\right)}~{},\quad\alpha^{(1)}_{n}=\frac{\sqrt{\pi}e^{\frac{i\pi
n}{2}}(n-3)(n-2)(n-1)\Gamma(n+3)}{9\times
4^{1+n}\Gamma\left(n-\frac{1}{2}\right)}~{}.$ (A.98)

###  A.5 A correlation function with $\text{D}^{2}$ and $T\bar{T}$

We conclude with the three-point correlation function between two insertions
of $\text{D}^{2}$ and $T\bar{T}$. The starting point is the six-point
correlation function of $T$ on ${\mathbb{H}^{+}}$, which however is completely
fixed by the conformal Ward identities. Taking the various limits, so eq.
(A.82) in the first and second $T\times T$ insertions, and eq. (A.4) in the
last $T\times T$ insertion we find ($\tilde{\eta}$ in eq. (A.13).)

$\displaystyle\langle\text{D}^{2}(x_{1})\text{D}^{2}(x_{2})T\bar{T}(x+iy,x-iy)\rangle_{{\mathbb{H}^{+}}}=\frac{\mathcal{G}^{\text{D}^{2}\text{D}^{2}T\bar{T}}(\tilde{\eta})}{(2y)^{4}(x_{12})^{8}}~{},$
(A.99)

$\displaystyle\mathcal{G}^{\text{D}^{2}\text{D}^{2}T\bar{T}}(\tilde{\eta})$
$\displaystyle=\frac{c^{2}}{20}(5c+22)\left(1-\frac{16\tilde{\eta}}{c}+\left(\frac{248}{5c}+4\right)\tilde{\eta}^{2}-\left(\frac{176}{5c}+8\right)\tilde{\eta}^{3}+\left(\frac{44}{5c}+2\right)\tilde{\eta}^{4}\right)~{}.$
(A.100)

We can expand this result into the bulk-boundary-boundary blocks of eq. (A.12)
to find

$\displaystyle\frac{c^{2}}{20}(22+5c)-\frac{4}{5}c(5c+22)\sum_{n=2,4,6\dots}(\alpha^{(0)}_{n}+\alpha^{(1)}_{n}c)~{}\,f(0,n,\tilde{\eta}/4)~{},$
(A.101)

where

$\displaystyle\alpha^{(0)}_{n}$
$\displaystyle=-\frac{\sqrt{\pi}2^{-2n-1}e^{\frac{i\pi
n}{2}}((n-1)n(11(n-1)n((n-1)n-8)+1572)-2160)\Gamma(n)}{45\Gamma\left(n-\frac{1}{2}\right)}~{},$
$\displaystyle\alpha^{(1)}_{n}$
$\displaystyle=-\frac{\sqrt{\pi}4^{-n-1}e^{\frac{i\pi
n}{2}}(n-3)(n-2)(n-1)\Gamma(n+3)}{9\Gamma\left(n-\frac{1}{2}\right)}~{}.$
(A.102)

##  Appendix B Review of minimal model BCFTs

We will consider unitary and diagonal minimal models $\mathcal{M}_{m+1,m}$
with central charge

$c=1-\frac{6}{m(m+1)}\,,\qquad{m=3,\,4,\,5,\ldots\,}$ (B.1)

The holomorphic Virasoro primaries are labelled by a pair of positive integers
$(r,s)$ as

$\displaystyle\phi_{(r,s)}(z)~{},\quad 1\leq r\leq m-1~{},\quad 1\leq s\leq
m~{},\quad(r,s)\cong(m-r,m+1-s)~{},$ (B.2)

and have the following scaling dimensions

$h_{r,s}=\frac{\bigl{(}(m+1)r-ms\bigr{)}^{2}-1}{4m(m+1)}~{}.$ (B.3)

The (holomorphic) fusion rules read

$\displaystyle\phi_{(r,s)}\times\phi_{(r^{\prime},s^{\prime})}=\sum_{\begin{subarray}{c}r^{\prime\prime}=|r-r^{\prime}|+1\\\
r+r^{\prime}+r^{\prime\prime}{\rm odd}\end{subarray}}^{r_{\rm
max}}\,\,\sum_{\begin{subarray}{c}s^{\prime\prime}=|s-s^{\prime}|+1\\\
s+s^{\prime}+s^{\prime\prime}{\rm odd}\end{subarray}}^{s_{\rm max}}\
\phi_{(r^{\prime\prime},s^{\prime\prime})}~{},$ (B.4)

where $r_{\rm max}={\rm min}(r+r^{\prime}-1,\;2m-r-r^{\prime}-1)$ and $s_{\rm
max}={\rm min}(s+s^{\prime}-1,\;2m-s-s^{\prime}+1)$.

The diagonal minimal models are obtained by gluing a copy of the holomorphic
sector with its anti-holomorphic counterpart, so that the physical spectrum of
the theory contains only scalar Virasoro primaries. The so-obtained diagonal
minimal models enjoy a $\mathbb{Z}_{2}$ symmetry. The charge of a Virasoro
primary with labels $(r,s)$ in $\mathcal{M}_{m+1,m}$ under such symmetry can
be chosen to be [44, 45] (see also [46] for a generic analysis on unitary
minimal models)

$\displaystyle\epsilon^{(m)}_{(r,s)}=(-1)^{(m+1)r+ms+1}~{}.$ (B.5)

###  B.1 Conformal boundary conditions

Let us consider placing a unitary and diagonal minimal model
$\mathcal{M}_{m+1,m}$ on the upper half-plane. The ‘elementary’ conformal
boundary conditions for $\mathcal{M}_{m+1,m}$ are parameterised by the highest
weight representations of the Virasoro algebra [35, 37, 36]

$\displaystyle{\bf a}=(a_{1},a_{2})_{m}~{},\quad 1\leq a_{1}\leq m-1~{},\quad
1\leq a_{2}\leq m~{},\quad(a_{1},a_{2})\cong(m-a_{1},m+1-a_{2})~{}.$ (B.6)

In radial quantization, the elementary conformal boundary condition (B.6)
defines a Cardy state [36]

$|{{{\bf
a}}}\rangle=\sum_{(r,s)}\frac{S_{(a_{1},a_{2})}^{\quad(r,s)}}{\sqrt{S_{(1,1)}^{\;(r,s)}}}\,|r,s\rangle\\!\rangle\,~{}.$
(B.7)

In the equation above, $|r,s\rangle\\!\rangle$ is the Ishibashi state in the
$(r,s)$ sector [47] and

$S_{(a_{1},a_{2})}^{\quad(r,s)}=\sqrt{\frac{8}{m(m+1)}}(-1)^{1+a_{1}s+a_{2}r}\,\sin\left(\tfrac{m+1}{m}\pi
a_{1}r\right)\sin\left(\tfrac{m}{m+1}\pi a_{2}s\right)\,,$ (B.8)

are the entries of the modular $S$-matrix. The Ishibashi state
$|r,s\rangle\\!\rangle$ carries the same parity as $(r,s)$ under the bulk
$\mathbb{Z}_{2}$ symmetry, see eq. (B.5), so an elementary conformal boundary
condition preserves this $\mathbb{Z}_{2}$ if the corresponding Cardy state
contains only $\mathbb{Z}_{2}$-even Ishibashi states.

Along with the Virasoro primaries that characterize the spectrum of bulk local
operators, in the presence of a boundary we can have local excitations located
at the boundary. These excitations are captured by the boundary Virasoro
primaries, which are highest-weight representations of a single copy of the
Virasoro algebra preserved by the boundary. Boundary Virasoro primaries
$\psi_{(r,s)}$ are labelled by a pair of integers

$\displaystyle\psi_{(r,s)}(x)~{},\quad 1\leq r\leq m-1~{},\quad 1\leq s\leq
m~{},\quad(r,s)\cong(m-r,m+1-s)~{},$ (B.9)

and they have scaling dimensions

$\displaystyle\psi_{(r,s)}(x):~{}\quad\hat{\Delta}_{r,s}=h_{r,s}~{}.$ (B.10)

Their fusion rules are the same as the holomorphic fusion rules of eq. (B.4).

$\begin{array}[]{|c|c|c|l|}\hline\cr\hline\cr
m&(a_{1},a_{2})_{m}&\mathbb{Z}_{2}\text{ - preserving}&\text{Boundary
spectrum}\\\ \hline\cr\hline\cr
3&(1,2)_{3}&\checkmark&{\hat{\mathds{1}}},\psi_{(1,3)}\\\ \hline\cr
4&(1,2)_{4}&&{\hat{\mathds{1}}},\psi_{(1,3)}\\\
&(1,3)_{4}&&{\hat{\mathds{1}}},\psi_{(1,3)}\\\
&(2,1)_{4}&\checkmark&{\hat{\mathds{1}}},\psi_{(3,1)}\\\
&(2,2)_{4}&\checkmark&{\hat{\mathds{1}}},\psi_{(1,3)},\psi_{(3,1)},\psi_{(3,3)}\\\
\hline\cr 5&(1,2)_{5}&&{\hat{\mathds{1}}},\psi_{(1,3)}\\\
&(1,3)_{5}&\checkmark&{\hat{\mathds{1}}},\psi_{(1,3)},\psi_{(1,5)}\\\
&(1,4)_{5}&&{\hat{\mathds{1}}},\psi_{(1,3)}\\\
&(2,1)_{5}&&{\hat{\mathds{1}}},\psi_{(3,1)}\\\
&(2,2)_{5}&&{\hat{\mathds{1}}},\psi_{(1,3)},\psi_{(3,1)},\psi_{(3,3)}\\\
&(2,3)_{5}&\checkmark&{\hat{\mathds{1}}},\psi_{(1,3)},\psi_{(1,5)},\psi_{(3,1)},\psi_{(3,3)},\psi_{(3,5)}\\\
&(2,4)_{5}&&{\hat{\mathds{1}}},\psi_{(1,3)},\psi_{(3,1)},\psi_{(3,3)}\\\
&(2,5)_{5}&&{\hat{\mathds{1}}},\psi_{(3,1)}\\\ \hline\cr
6&(1,2)_{6}&&{\hat{\mathds{1}}},\psi_{(1,3)}\\\
&(1,3)_{6}&&{\hat{\mathds{1}}},\psi_{(1,3)},\psi_{(1,5)}\\\
&(1,4)_{6}&&{\hat{\mathds{1}}},\psi_{(1,3)},\psi_{(1,5)}\\\
&(1,5)_{6}&&{\hat{\mathds{1}}},\psi_{(1,3)}\\\
&(2,1)_{6}&&{\hat{\mathds{1}}},\psi_{(3,1)}\\\
&(2,2)_{6}&&{\hat{\mathds{1}}},\psi_{(1,3)},\psi_{(3,1)},\psi_{(3,3)}\\\
&(2,3)_{6}&&{\hat{\mathds{1}}},\psi_{(1,3)},\psi_{(1,5)},\psi_{(3,1)},\psi_{(3,3)},\psi_{(3,5)}\\\
&(2,4)_{6}&&{\hat{\mathds{1}}},\psi_{(1,3)},\psi_{(1,5)},\psi_{(3,1)},\psi_{(3,3)},\psi_{(3,5)}\\\
&(2,5)_{6}&&{\hat{\mathds{1}}},\psi_{(1,3)},\psi_{(3,1)},\psi_{(3,3)}\\\
&(2,6)_{6}&&{\hat{\mathds{1}}},\psi_{(3,1)}\\\
&(3,1)_{6}&\checkmark&{\hat{\mathds{1}}},\psi_{(3,1)},\psi_{(5,1)}\\\
&(3,2)_{6}&\checkmark&{\hat{\mathds{1}}},\psi_{(1,3)},\psi_{(3,1)},\psi_{(3,3)},\psi_{(5,1)},\psi_{(5,3)}\\\
&(3,3)_{6}&\checkmark&{\hat{\mathds{1}}},\psi_{(1,3)},\psi_{(1,5)},\psi_{(3,1)},\psi_{(3,3)},\psi_{(3,5)},\psi_{(5,1)},\psi_{(5,3)},\psi_{(5,5)}\\\
\hline\cr\hline\cr\end{array}$

Table 1: Elementary conformal boundary conditions for diagonal and unitary
minimal models with $m\leq 6$. $\mathbb{Z}_{2}$-preserving boundary conditions
are those corresponding to $\mathbb{Z}_{2}$-even Cardy states. We have not
included the conformal b.c. labeled by the identity, nor its
$\mathbb{Z}_{2}$-conjugate $(1,m)_{m}$: they are always possible and allow
only for ${\hat{\mathds{1}}}$ at the boundary.

The spectrum of allowed boundary Virasoro primaries in a given conformal b.c.
is constrained by modular invariance. With the elementary conformal boundary
condition ${\bf a}=(a_{1},a_{2})$, the allowed boundary Virasoro primaries are
those that can appear in the self-OPE of a bulk Virasoro primary labelled by
${\bf a}$. This is expressed in terms of the annulus partition function with
boundary condition $\bf a$ as follows:

$\displaystyle Z_{\bf aa}(\delta)=\sum_{\phi_{(r,s)}\in\phi_{\bf
a}\times\phi_{\bf a}}N^{(r,s)}_{{\bf a}{\bf a}}\chi_{(r,s)}(q)~{},\quad
q\equiv e^{-\pi\delta}~{},$ (B.11)

where $N^{(r,s)}_{{\bf a}{\bf a}}$ is the fusion coefficient of $\phi_{\bf
a}\times\phi_{\bf a}$ into $\phi_{(r,s)}$, $\chi_{(r,s)}(q)$ is the
(holomorphic) Virasoro character of the $(r,s)$ module and $\delta$ is the
width of the annulus. A table with a survey of elementary conformal b.c. for
diagonal minimal models with $m\leq 6$ can be found in Table 1.

Let us now briefly discuss some aspects of the OPE data. The coefficients $B$,
$\hat{C}$ and $C$ are determined via the ‘sewing’ constraints of Lewellen
[48]. For unitary and diagonal minimal models with elementary conformal b.c.,
these were solved by Runkel in ref. [49] (see [50] for the extension to the
D-series of minimal models), who has found explicit expressions for such $B$,
$\hat{C}$ and $C$ in terms of the so-called ‘fusion matrices’, or F-matrices.
Of course the $C$ coefficients are the same as those of the homogeneous
minimal model, and these can be efficiently computed via the Coulomb gas
formalism of refs. [51, 52, 53].141414In their supplementary material, the
authors of ref. [54] provided a very useful Mathematica notebook that
implements these formulae.

Note that the bulk-boundary coefficient of $\phi_{(r,s)}$ with the boundary
identity in the second line of eq. (A.1) is determined by the Cardy state to
be [55, 48]

$\displaystyle B_{(r,s)}^{{\bf
a}\,(1,1)}=\frac{S_{(a_{1},a_{2})}^{(r,s)}\sqrt{S_{(1,1)}^{(1,1)}}}{S_{(a_{1},a_{2})}^{(1,1)}\sqrt{S_{(1,1)}^{(r,s)}}}~{}.$
(B.12)

###  B.2 The $\phi_{(1,3)}$ and $\psi_{(1,3)}$ operators

In the main text we study the deformation of the minimal models by the
$\phi_{(1,3)}$ operator. In the remainder of this review we will therefore
study its properties and explicitly compute several correlation functions
involving this operator. We are also interested in the near-boundary operator
$\psi_{(1,3)}$. We note first of all that

$\Delta_{1,3}=2h_{1,3}=2\frac{m-1}{m+1}~{}.$ (B.13)

##### Bulk OPE

The bulk self-OPE of $\phi_{(1,3)}$ reads (see eq. (B.4))

$\displaystyle\phi_{(1,3)}(z,\bar{z})\phi_{(1,3)}(0,0)$
$\displaystyle=\frac{{\hat{\mathds{1}}}}{|z|^{2\Delta_{1,3}}}+\text{desc.}$
$\displaystyle m=3$ $\displaystyle\phi_{(1,3)}(z,\bar{z})\phi_{(1,3)}(0,0)$
$\displaystyle=\frac{{\mathds{1}}}{|z|^{2\Delta_{1,3}}}+\frac{C_{(1,3)(1,3)}{}^{(1,3)}}{|z|^{\Delta_{1,3}}}\phi_{(1,3)}(0,0)+\text{desc.}$
$\displaystyle m=4$ $\displaystyle\phi_{(1,3)}(z,\bar{z})\phi_{(1,3)}(0,0)$
$\displaystyle=\frac{{\mathds{1}}}{|z|^{2\Delta_{1,3}}}+\frac{C_{(1,3)(1,3)}{}^{(1,3)}}{|z|^{\Delta_{1,3}}}\phi_{(1,3)}(0,0)+\frac{C_{(1,3)(1,3)}{}^{(1,5)}}{|z|^{2\Delta_{1,3}-\Delta_{1,5}}}\phi_{(1,5)}(0,0)+\text{desc.}$
$\displaystyle m\geq 5~{},$ (B.14)

up to Virasoro descendants. The OPE coefficients are found to be

$\displaystyle C_{(1,3)(1,3)(1,3)}$
$\displaystyle=-\frac{\Gamma\left(\frac{2}{m+1}\right)\Gamma\left(\frac{1-m}{m+1}\right)\Gamma\left(\frac{3m-1}{m+1}\right)}{\Gamma\left(\frac{2-2m}{m+1}\right)\Gamma\left(\frac{2m}{m+1}\right)^{2}}\sqrt{-\frac{\Gamma\left(\frac{2-m}{m+1}\right)\Gamma\left(\frac{m}{m+1}\right)^{3}}{\Gamma\left(\frac{1}{m+1}\right)^{3}\Gamma\left(\frac{2m-1}{m+1}\right)}},$
$\displaystyle\text{for }m>3~{}$
$\displaystyle=\frac{4}{\sqrt{3}}-\frac{4\sqrt{3}}{m}-\frac{8}{\sqrt{3}m^{2}}+\frac{32\sqrt{3}\zeta(3)}{m^{3}}+O(m^{-4})~{},$
$\displaystyle C_{(1,3)(1,3)(1,5)}$
$\displaystyle=\frac{m-1}{3m-1}\sqrt{\frac{\Gamma\left(\frac{1-2m}{m+1}\right)\Gamma\left(\frac{2-m}{m+1}\right)\Gamma\left(\frac{m}{m+1}\right)\Gamma\left(\frac{4m-1}{m+1}\right)}{\Gamma\left(\frac{1}{m+1}\right)\Gamma\left(\frac{2-3m}{m+1}\right)\Gamma\left(\frac{3m}{m+1}\right)\Gamma\left(\frac{2m-1}{m+1}\right)}},$
$\displaystyle\text{for }m>4~{}$
$\displaystyle=\frac{\sqrt{5}}{3}-\frac{7\sqrt{5}}{9m}-\frac{53\sqrt{5}}{54m^{2}}+\frac{\sqrt{5}(2592\zeta(3)-691)}{324m^{3}}+O(m^{-4}).$
(B.15)

##### Bulk-boundary OPE

The bulk-boundary OPE of $\phi_{(1,3)}$ reads

$\displaystyle\phi_{(1,3)}(x+iy,x-iy)$ $\displaystyle=\frac{B_{(1,3)}^{{\bf
a}\,(1,1)}}{(2y)^{\Delta_{1,3}}}{\hat{\mathds{1}}}+\text{desc.}$
$\displaystyle m=3$ $\displaystyle\phi_{(1,3)}(x+iy,x-iy)$
$\displaystyle=\frac{B_{(1,3)}^{{\bf
a}\,(1,1)}}{(2y)^{\Delta_{1,3}}}{\hat{\mathds{1}}}+\frac{B_{(1,3)}^{{\bf
a}\,(1,3)}}{(2y)^{\Delta_{1,3}-h_{1,3}}}\psi_{(1,3)}(x)+\text{desc.}$
$\displaystyle m=4$ $\displaystyle\phi_{(1,3)}(x+iy,x-iy)$
$\displaystyle=\frac{B_{(1,3)}^{{\bf
a}\,(1,1)}}{(2y)^{\Delta_{1,3}}}{\hat{\mathds{1}}}+\frac{B_{(1,3)}^{{\bf
a}\,(1,3)}}{(2y)^{\Delta_{1,3}-h_{1,3}}}\psi_{(1,3)}(x)+\frac{B_{(1,3)}^{{\bf
a}\,(1,5)}}{(2y)^{\Delta_{1,3}-h_{1,5}}}\psi_{(1,5)}(x)+\text{desc.}$
$\displaystyle m\geq 5$ (B.16)

up to boundary Virasoro descendants. The bulk-boundary OPE coefficient
$B_{(1,3)}^{{\bf a}\,(1,1)}$ is given in eq. (B.3.1). The squared coefficients
$(B_{(1,3)}^{{\bf a}\,(1,3)})^{2}$ and $(B_{(1,3)}^{{\bf a}\,(1,5)})^{2}$ are
determined by the bulk-boundary crossing symmetry and are reported in section
B.3.1.

###  B.3 The bulk two-point function of $\phi_{(1,3)}$

We start from the bulk two-point function of $\phi_{(1,3)}$ on the upper half-
plane:

$\displaystyle\langle\phi_{(1,3)}(x_{1}+iy_{1},x_{1}-iy_{1})\phi_{(1,3)}(x_{2}+iy_{2},x_{2}-iy_{2})\rangle_{{\mathbb{H}^{+}}}~{}.$
(B.17)

To compute such a correlator, we employ the method of images and solve

$\displaystyle\left({\mathcal{L}}_{-3}^{(z_{4})}-\frac{2}{h_{1,3}+2}{\mathcal{L}}_{-1}{\mathcal{L}}_{-2}^{(z_{4})}+\frac{1}{(h_{1,3}+1)(h_{1,3}+2)}{\mathcal{L}}_{-1}^{3}\right)\langle\phi_{(1,3)}(z_{1})\phi_{(1,3)}(z_{2})\phi_{(1,3)}(z_{3})\phi_{(1,3)}(z_{4})\rangle=0~{},$
(B.18)

where ${\cal L}_{-n}^{(\cdot)}$ is the differential operator defined in eq.
(A.29) and ${\cal L}_{-1}=\partial_{z_{4}}$. This holomorphic correlator takes
the following form

$\displaystyle\langle\phi_{(1,3)}(z_{1})\phi_{(1,3)}(z_{2})\phi_{(1,3)}(z_{3})\phi_{(1,3)}(z_{4})\rangle=\frac{\mathcal{G}(\eta)}{(z_{12}z_{34})^{2h_{1,3}}}~{}.$
(B.19)

The cross-ratio $\eta$ is defined as ($z_{ij}\equiv z_{i}-z_{j}$)

$\displaystyle\eta$
$\displaystyle=\frac{z_{12}z_{34}}{z_{13}z_{24}}=-\frac{4y_{1}y_{2}}{(y_{1}-y_{2})^{2}+(x_{1}-x_{2})^{2}}~{}.$
(B.20)

The differential equation for $\mathcal{G}(\eta)$ reads

$\begin{split}0&=\eta^{2}(\eta-1)^{3}(m+1)^{3}\mathcal{G}^{\prime\prime\prime}(\eta)+2\eta(\eta-1)^{2}(m+1)^{2}(\eta(m+3)+m-3)\mathcal{G}^{\prime\prime}(\eta)\\\
&+2(\eta-1)(m+1)\left(\eta\left(m^{2}+m-6\right)+\eta^{2}(m(3-2m)+3)-3m+3\right)\mathcal{G}^{\prime}(\eta)\\\
&+4(\eta-2)\eta(m-1)^{2}m\,\mathcal{G}(\eta)~{}.\end{split}$ (B.21)

In order to solve this equation, it is convenient to define another function

$\displaystyle\mathcal{G}(\eta)=\tilde{\mathcal{G}}\left(\tilde{\eta}\equiv\frac{\eta^{2}}{1-\eta}\right)~{}.$
(B.22)

The cross-ratio $\tilde{\eta}$ is real and positive in the physical region. In
fact, in terms of the cross-ratio $\xi$ defined in eq. (A.8) we have that

$\displaystyle\tilde{\eta}=\xi^{-1}(1+\xi)^{-1}~{}=\frac{16y_{1}^{2}y_{2}^{2}}{\left((y_{1}-y_{2})^{2}+(x_{1}-x_{2})^{2}\right)\left((y_{1}+y_{2})^{2}+(x_{1}-x_{2})^{2}\right)}.$
(B.23)

There are three independent solutions to this equation, corresponding to the
boundary operators ${\hat{\mathds{1}}},\psi_{(1,3)}$ and $\psi_{(1,5)}$, see
e.g. eq. (B.2). The three Virasoro blocks at finite $m$ read

$\displaystyle V_{(1,1)}(\tilde{\eta})$
$\displaystyle=\,_{3}F_{2}\left(\frac{1-m}{m+1},\frac{2m}{m+1},-\frac{2m-2}{m+1};\frac{3+m}{2m+2},\frac{2-m}{m+1};-\frac{\tilde{\eta}}{4}\right)~{},$
$\displaystyle V_{(1,3)}(\tilde{\eta})$
$\displaystyle=\tilde{\eta}^{{h_{1,3}}/{2}}\,_{3}F_{2}\left(\frac{5m-1}{2m+2},\frac{1-m}{2m+2},-\frac{3m-3}{2m+2};\frac{3-m}{2m+2},\frac{3m+1}{2m+2};-\frac{\tilde{\eta}}{4}\right),$
$\displaystyle V_{(1,5)}(\tilde{\eta})$
$\displaystyle=\tilde{\eta}^{{h_{1,5}}/{2}}\,_{3}F_{2}\left(\frac{1}{m+1},\frac{m}{m+1},\frac{4m-1}{m+1};\frac{3m}{m+1},\frac{5m+1}{2m+2};-\frac{\tilde{\eta}}{4}\right)~{}.$
(B.24)

The final solution is

$\displaystyle\langle$
$\displaystyle\phi_{(1,3)}(x_{1}+iy_{1},x_{1}-iy_{1})\phi_{(1,3)}(x_{2}+iy_{2},x_{2}-iy_{2})\rangle_{{\mathbb{H}^{+}}}=\frac{\tilde{\mathcal{G}}(\tilde{\eta})}{(4y_{1}y_{2})^{2h_{1,3}}}~{},$
$\displaystyle\tilde{\mathcal{G}}(\tilde{\eta})=(B_{(1,3)}^{{\bf
a}\,(1,1)})^{2}V_{(1,1)}(\tilde{\eta})+(B_{(1,3)}^{{\bf
a}\,(1,3)})^{2}V_{(1,3)}(\tilde{\eta})+(B_{(1,3)}^{{\bf
a}\,(1,5)})^{2}V_{(1,5)}(\tilde{\eta})~{}.$ (B.25)

We will fix the coefficients using bulk-boundary crossing symmetry.

####  B.3.1 Bulk-boundary crossing

The correlator in eq. (B.3) is manifestly written in terms of the boundary
data. By expanding it around $\tilde{\eta}=\infty$ we can read off the bulk-
channel data. For a systematic expansion around $\tilde{\eta}=\infty$ it is
convenient to define the following basis of bulk-channel blocks151515These can
be obtained upon ‘inverting’ the blocks of eq. (B.3) using standard
hypergeometric transformations and picking up the correct linear combination
of ‘inverted blocks’ consistently with the bulk OPE limit.

$\displaystyle V^{\text{bulk}}_{(1,1)}(\tilde{\eta})$
$\displaystyle=\tilde{\eta}^{2h_{1,3}}\,_{3}F_{2}\left(\frac{1}{m+1},\frac{2-2m}{m+1},\frac{3-3m}{2m+2};\frac{2}{m+1},\frac{3-3m}{m+1};-\frac{4}{\tilde{\eta}}\right)~{},$
$\displaystyle V^{\text{bulk}}_{(1,3)}(\tilde{\eta})$
$\displaystyle=\tilde{\eta}^{h_{1,3}}\,_{3}F_{2}\left(\frac{m}{m+1},\frac{1-m}{m+1},\frac{1-m}{2m+2};\frac{2m}{m+1},\frac{2-2m}{m+1};-\frac{4}{\tilde{\eta}}\right)~{},$
$\displaystyle V^{\text{bulk}}_{(1,5)}(\tilde{\eta})$
$\displaystyle=\tilde{\eta}^{2h_{1,3}-h_{1,5}}\,_{3}F_{2}\left(\frac{2m}{m+1},\frac{4m-1}{m+1},\frac{5m-1}{2m+2};\frac{4m}{m+1},\frac{5m-1}{m+1};-\frac{4}{\tilde{\eta}}\right)~{}.$
(B.26)

The bulk-boundary crossing is then simply the statement that

$\displaystyle(B_{(1,3)}^{{\bf
a}\,(1,1)})^{2}V_{(1,1)}(\tilde{\eta})+(B_{(1,3)}^{{\bf
a}\,(1,3)})^{2}V_{(1,3)}(\tilde{\eta})+(B_{(1,3)}^{{\bf
a}\,(1,5)})^{2}V_{(1,5)}(\tilde{\eta})=$ $\displaystyle
V^{\text{bulk}}_{(1,1)}(\tilde{\eta})+C_{(1,3)(1,3)(1,3)}B_{(1,3)}^{{\bf
a}\,(1,1)}V^{\text{bulk}}_{(1,3)}(\tilde{\eta})+C_{(1,3)(1,3)(1,5)}B_{(1,5)}^{{\bf
a}\,(1,1)}V^{\text{bulk}}_{(1,5)}(\tilde{\eta})~{}.$ (B.27)

The bulk coefficients are given in eq. (B.2), while from eq. (B.12) we have
that

$\displaystyle B_{(1,3)}^{{\bf a}\,(1,1)}$
$\displaystyle=\left(1+2\cos\left(\frac{2\pi
a_{2}m}{m+1}\right)\right)\sqrt{\frac{{\sin\left(\frac{\pi}{m}\right)\sin\left(\frac{\pi
m}{m+1}\right)}}{{\sin\left(\frac{\pi}{m}\right)\sin\left(\frac{3\pi
m}{m+1}\right)}}}$
$\displaystyle=\sqrt{3}-\frac{2\pi^{2}\left(2a_{2}^{2}-1\right)}{\sqrt{3}m^{2}}+\frac{4\pi^{2}\left(2a_{2}^{2}-1\right)}{\sqrt{3}m^{3}}+O(m^{-4})~{},$
$\displaystyle B_{(1,5)}^{{\bf a}\,(1,1)}$ $\displaystyle=\sin\left(\frac{5\pi
a_{2}m}{m+1}\right)\csc\left(\frac{\pi
a_{2}m}{m+1}\right)\sqrt{\frac{{\sin\left(\frac{\pi}{m}\right)\sin\left(\frac{\pi
m}{m+1}\right)}}{{\sin\left(\frac{\pi}{m}\right)\sin\left(\frac{5\pi
m}{m+1}\right)}}}$
$\displaystyle=\sqrt{5}+\frac{2\pi^{2}\sqrt{5}\left(1-2a_{2}^{2}\right)}{m^{2}}+\frac{4\pi^{2}\sqrt{5}\left(2a_{2}^{2}-1\right)}{m^{3}}+O(m^{-4}),$
(B.28)

and the large-$m$ limit is taken assuming a generic conformal b.c. ${\bf a}$,
with $a_{1},a_{2}\ll m$. The coefficients $(B_{(1,3)}^{{\bf a}\,(1,3)})^{2}$
and $(B_{(1,3)}^{{\bf a}\,(1,5)})^{2}$ are determined by the bulk-boundary
crossing symmetry for any finite $m$.

##### Solution to bulk-boundary crossing for $m=3$ and $m=4$

Let us start from the simplest case, $m=3$, where only the identity is present
in both the bulk and the boundary channels. Crossing imposes that
$(B_{(1,3)}^{{\bf a}\,(1,1)})^{2}=1$ which, upon comparing to the first of eq.
(B.3.1), is true if $a_{2}=1,2,3$, as it should.

For the case with $m=4$ we have only two operators in both channels:
${\hat{\mathds{1}}}$ and $\psi_{(1,3)}$ in the boundary channel,
${\mathds{1}}$ and $\phi_{(1,3)}$ in the bulk channel. The only remaining
coefficient is $(B_{(1,3)}^{{\bf a}\,(1,3)})^{2}$, which can be non-zero only
if $a_{2}=2,3$ and is fixed by crossing as follows

$\displaystyle(B_{(1,3)}^{{\bf
a}\,(1,3)})^{2}=-\frac{24\sqrt{2-\frac{2}{\sqrt{5}}}\left(25\
2^{2/5}\sqrt{\pi\left(9-4\sqrt{5}\right)}\Gamma\left(\frac{3}{5}\right)\Gamma\left(\frac{7}{10}\right)-12\pi\Gamma\left(-\frac{6}{5}\right)\right)}{3125\Gamma\left(\frac{3}{5}\right)\Gamma\left(\frac{8}{5}\right)\Gamma\left(\frac{13}{5}\right)}\simeq
0.663053~{}.$ (B.29)

##### Solution to bulk-boundary crossing for general $m>4$

We find

$\displaystyle(B_{(1,3)}^{{\bf a}\,(1,3)})^{2}$
$\displaystyle=\frac{\sqrt{\pi}\sec\left(\frac{2\pi}{m+1}\right)\Gamma\left(\frac{m}{m+1}\right)\Gamma\left(\frac{1-m}{2m+2}\right)\Gamma\left(\frac{3m-1}{m+1}\right)}{2\Gamma\left(\frac{m-1}{m+1}\right)\Gamma\left(\frac{2m-1}{m+1}\right)\Gamma\left(\frac{3-m}{2m+2}\right)\Gamma\left(\frac{3m+1}{2m+2}\right)}$
$\displaystyle-\frac{1}{\pi}{\sin\left(\frac{2\pi}{m+1}\right)\Gamma\left(\frac{2m}{m+1}\right)\Gamma\left(\frac{2}{m+1}-1\right)}C_{(1,3)(1,3)(1,3)}B_{(1,3)}^{{\bf
a}\,(1,1)}$
$\displaystyle+\frac{2^{\frac{6}{m+1}-4}\Gamma\left(\frac{2-m}{m+1}\right)\Gamma\left(\frac{3-3m}{2m+2}\right)\Gamma\left(\frac{1-m}{2m+2}\right)\Gamma\left(\frac{m+3}{2m+2}\right)\Gamma\left(\frac{5m-1}{2(m+1)}\right)}{\pi^{3/2}\left(\sec\left(\frac{\pi}{m+1}\right)+\sec\left(\frac{3\pi}{m+1}\right)\right)\Gamma\left(\frac{2-2m}{m+1}\right)^{2}\Gamma\left(\frac{2m}{m+1}\right)\Gamma\left(\frac{3m+1}{2m+2}\right)}~{}$
$\displaystyle~{}~{}~{}~{}\times\left(\Gamma\left(\frac{1-m}{m+1}\right)\Gamma\left(\frac{2m}{m+1}\right)-\pi\csc\left(\frac{4\pi}{m+1}\right)\right)(B_{(1,3)}^{{\bf
a}\,(1,1)})^{2}~{},$ $\displaystyle(B_{(1,3)}^{{\bf a}\,(1,5)})^{2}$
$\displaystyle=\frac{(m-1)\left(\sec\left(\frac{2\pi}{m+1}\right)-2\right)\Gamma\left(-\frac{1}{m+1}\right)\Gamma\left(4-\frac{5}{m+1}\right)}{2(m-2)(3m-1)\Gamma\left(\frac{m-2}{m+1}\right)\Gamma\left(\frac{3m}{m+1}\right)}~{}$
$\displaystyle+\frac{\sqrt{\pi}\sec\left(\frac{2\pi}{m+1}\right)\Gamma\left(\frac{2m}{m+1}\right)\Gamma\left(4-\frac{5}{m+1}\right)\Gamma\left(\frac{3}{2}-\frac{1}{m+1}\right)}{2\Gamma\left(\frac{3m}{m+1}\right)\Gamma\left(3-\frac{4}{m+1}\right)\Gamma\left(\frac{5}{2}-\frac{2}{m+1}\right)\Gamma\left(\frac{1}{m+1}-\frac{1}{2}\right)}C_{(1,3)(1,3)(1,3)}B_{(1,3)}^{{\bf
a}\,(1,1)}$
$\displaystyle-\frac{2^{\frac{6}{m+1}-4}\Gamma\left(\frac{2-m}{m+1}\right)\Gamma\left(\frac{m+3}{2m+2}\right)\Gamma\left(\frac{4m-1}{m+1}\right)}{\Gamma\left(\frac{2-2m}{m+1}\right)\Gamma\left(\frac{1-m}{m+1}\right)\Gamma\left(\frac{2m}{m+1}\right)\Gamma\left(\frac{3m}{m+1}\right)\Gamma\left(\frac{5m+1}{2m+2}\right)}~{}$
$\displaystyle~{}~{}~{}~{}\times\frac{\left(\pi\csc\left(\frac{4\pi}{m+1}\right)\sec\left(\frac{\pi}{m+1}\right)+\sec\left(\frac{3\pi}{m+1}\right)\Gamma\left(\frac{1-m}{m+1}\right)\Gamma\left(\frac{2m}{m+1}\right)\right)}{\left(\sec\left(\frac{\pi}{m+1}\right)+\sec\left(\frac{3\pi}{m+1}\right)\right)}(B_{(1,3)}^{{\bf
a}\,(1,1)})^{2}~{},$ (B.30)

with $C_{(1,3)(1,3)(1,3)}$ and $B_{(1,3)}^{{\bf a}\,(1,1)}$ given in equations
(B.2) and (B.3.1) (respectively). These results are consistent with those
obtained using F-matrices in ref. [49].

##### Large-$m$ expansion

In the large-$m$ expansion (we have in mind a sufficiently generic conformal
b.c. ${\bf a}$, with $a_{1},a_{2}\ll m$), the solutions of eq. (B.3.1) read

$\displaystyle(B_{(1,3)}^{{\bf a}\,(1,3)})^{2}$
$\displaystyle=\frac{32\pi^{2}\left(a_{2}^{2}-1\right)}{3m^{2}}-\frac{160\pi^{2}\left(a_{2}^{2}-1\right)}{3m^{3}}-\frac{32\pi^{2}\left(a_{2}^{2}-1\right)\left(\pi^{2}\left(4a_{2}^{2}-7\right)-21\right)}{9m^{4}}+O(m^{-5})~{},$
$\displaystyle(B_{(1,3)}^{{\bf a}\,(1,5)})^{2}$
$\displaystyle=\frac{8\pi^{4}\left(a_{2}^{2}-1\right)\left(a_{2}^{2}-4\right)}{9m^{4}}+O(m^{-5})~{}.$
(B.31)

###  B.4 Correlator between four $\psi_{(1,3)}$

Consider the four-point correlation function of $\psi_{(1,3)}$. The associated
differential equation and its solutions where given in section B.3. The fusion
rules read

$\displaystyle\psi_{(1,3)}\times\psi_{(1,3)}$
$\displaystyle={\hat{\mathds{1}}}$ $\displaystyle m=3$
$\displaystyle\psi_{(1,3)}\times\psi_{(1,3)}$
$\displaystyle={\hat{\mathds{1}}}+\psi_{(1,3)}$ $\displaystyle m=4$
$\displaystyle\psi_{(1,3)}\times\psi_{(1,3)}$
$\displaystyle={\hat{\mathds{1}}}+\psi_{(1,3)}+\psi_{(1,5)}$ $\displaystyle
m\geq 5~{}.$ (B.32)

The solution is

$\displaystyle\langle\psi_{(1,3)}(x_{1})\psi_{(1,3)}(x_{2})\psi_{(1,3)}(x_{3})\psi_{(1,3)}(x_{4})\rangle_{{\mathbb{H}^{+}}}=\frac{\tilde{\mathcal{G}}(\tilde{\eta})}{(x_{12}x_{34})^{2h_{1,3}}}~{},$
$\displaystyle\tilde{\mathcal{G}}$
$\displaystyle(\tilde{\eta})=V_{(1,1)}(\tilde{\eta})+({\hat{C}}_{(1,3)(1,3)}^{{\bf
a}(1,3)})^{2}V_{(1,3)}(\tilde{\eta})+({\hat{C}}_{(1,3)(1,3)}^{{\bf
a}(1,5)})^{2}V_{(1,5)}(\tilde{\eta})~{}.$ (B.33)

In the equation above we defined

$\displaystyle\mathcal{G}(\eta)=\tilde{\mathcal{G}}\left(\tilde{\eta}\equiv\frac{\eta^{2}}{1-\eta}\right)~{}.$
(B.34)

The form of the $V_{(r,s)}(\tilde{\eta})$ is given in eq. (B.3), while
$\tilde{\eta}$ is defined in terms of the standard cross ratio $\eta$ of eq.
(A.4) as

$\displaystyle\tilde{\eta}=\frac{\eta^{2}}{1-\eta}~{}=\frac{x_{12}^{2}x_{34}^{2}}{x_{12}x_{14}x_{23}x_{24}}~{},$
(B.35)

which is positive if $x_{i}>x_{i+1}$.

####  B.4.1 Boundary crossing

The squared boundary OPE coefficients in eq. (B.4) are fixed by crossing
symmetry, which is the statement that (recall the definition in eq. (B.22))

$\displaystyle\mathcal{G}(\eta)=\left(\frac{1-\eta}{\eta}\right)^{-2h_{1,3}}\mathcal{G}(1-\eta)~{}.$
(B.36)

For $m=3$ we have that ${\hat{C}}_{(1,3)(1,3)(1,3)}^{{\bf
a}}={\hat{C}}_{(1,3)(1,3)(1,5)}^{{\bf a}}=0$ and boundary crossing symmetry is
ensured by the identity block alone. For $m=4$, or for those conformal b.c.
that do not allow $\psi_{(1,5)}$ we have that
${\hat{C}}_{(1,3)(1,3)(1,5)}^{{\bf a}}=0$ and crossing is satisfied if

$\displaystyle({\hat{C}}_{(1,3)(1,3)(1,3)}^{{\bf
a}})^{2}=-\frac{\sqrt{\pi}\,\Gamma\left(3-\frac{4}{m+1}\right)\Gamma\left(\frac{1}{m+1}-\frac{1}{2}\right)\left(\pi\csc\left(\frac{3\pi}{m+1}\right)-\Gamma\left(\frac{1}{m+1}\right)\Gamma\left(\frac{m}{m+1}\right)\right)}{\Gamma\left(\frac{1}{m+1}\right)\Gamma\left(\frac{m-1}{m+1}\right)\Gamma\left(2-\frac{3}{m+1}\right)\Gamma\left(\frac{3}{2}-\frac{1}{m+1}\right)\Gamma\left(\frac{2}{m+1}-\frac{1}{2}\right)}~{},$
(B.37)

which vanishes identically for $m=3$, but is positive otherwise. This result
is consistent with the known results from the literature, see e.g. ref. [49].
For $m>4$ we find generically a one-parameter family of solutions to the
crossing equation, so that additional correlators (and crossing constraints)
are needed in order to compute ${\hat{C}}_{(1,3)(1,3)(1,3)}^{{\bf a}}$ and
${\hat{C}}_{(1,3)(1,3)(1,5)}^{{\bf a}}$.

###  B.5 Correlator between two $\psi_{(r,s)}$ and one $\phi_{(1,3)}$

Next, we compute the boundary-boundary-bulk correlation function between a
boundary Virasoro primary $\psi_{(r,s)}$ (assuming it exists) and
$\phi_{(1,3)}$,

$\displaystyle\langle\psi_{(r,s)}(x_{1})\psi_{(r,s)}(x_{2})\phi_{(1,3)}(x+iy,x-iy)\rangle_{{\mathbb{H}^{+}}}~{},\quad
x_{1}>x_{2}~{}.$ (B.38)

We recall that the scaling dimension of $\psi_{(r,s)}$ is
$\Delta_{r,s}=h_{r,s}$. The correlator of eq. (B.38) satisfies a third order
differential equation. To compute such a correlator, we employ the method of
images and solve

$\displaystyle\left({\mathcal{L}}_{-3}^{(z_{4})}-\frac{2}{h_{1,3}+2}{\mathcal{L}}_{-1}{\mathcal{L}}_{-2}^{(z_{4})}+\frac{1}{(h_{1,3}+1)(h_{1,3}+2)}{\mathcal{L}}_{-1}^{3}\right)\langle\psi_{(r,s)}(z_{1})\psi_{(r,s)}(z_{2})\phi_{(1,3)}(z_{3})\phi_{(1,3)}(z_{4})\rangle=0~{},$
(B.39)

where ${\cal L}_{-n}^{(\cdot)}$ is the differential operator defined in eq.
(A.29) and ${\cal L}_{-1}=\partial_{z_{4}}$. This holomorphic correlator takes
the following form:

$\displaystyle\langle\psi_{(r,s)}(z_{1})\psi_{(r,s)}(z_{2})\phi_{(1,3)}(z_{3})\phi_{(1,3)}(z_{4})\rangle=\frac{\mathcal{G}(\eta)}{(z_{12})^{2h_{r,s}}(z_{34})^{2h_{1,3}}}~{}.$
(B.40)

The cross-ratio $\eta$ is defined as

$\displaystyle\eta$
$\displaystyle=\frac{z_{12}z_{34}}{z_{13}z_{24}}=\frac{2iyx_{12}}{(x_{1}-z)(x_{2}-z^{*})}~{}.$
(B.41)

The differential equation for $\mathcal{G}(\eta)$ reads

$\begin{split}0&=(\eta-1)^{3}\eta^{2}(m+1)^{3}\mathcal{G}^{\prime\prime\prime}(\eta)+2(\eta-1)^{2}\eta(m+1)^{2}(\eta(m+3)+m-3)\mathcal{G}^{\prime\prime}(\eta)\\\
&-(\eta-1)(m+1)\left(12\eta-2\eta
m(m+1)+6m-6\right)\mathcal{G}^{\prime}(\eta)\\\
&-(\eta-1)\eta^{2}(m+1)\left(m((r-s)((m+2)r-ms)-2)+r^{2}-7\right)\mathcal{G}^{\prime}(\eta)\\\
&+(\eta-2)\eta(m-1)(mr-ms+r-1)(mr-ms+r+1)\mathcal{G}(\eta)~{}.\end{split}$
(B.42)

In order to solve this equation, it is convenient to define another function

$\displaystyle\mathcal{G}(\eta)=\tilde{\mathcal{G}}\left(\tilde{\eta}\equiv\frac{\eta^{2}}{\eta-1}\right)~{}.$
(B.43)

The cross-ratio $\tilde{\eta}$ so defined is real and $\tilde{\eta}\in[0,4]$
(its explicit expression in terms of the coordinates is given in eq. (A.13)).
There are three independent solutions to this equation corresponding to the
boundary operators ${\hat{\mathds{1}}},\psi_{(1,3)}$ and $\psi_{(1,5)}$ (see
the fusion rules in eq. (B.4)). These Virasoro blocks at finite $m$ are found
to be

$\displaystyle V_{(1,1)}(\tilde{\eta})$
$\displaystyle=\,_{3}F_{2}\left(\frac{1-m}{m+1},\frac{1+mq-r}{m+1},\frac{1-mq+r}{m+1};\frac{2-m}{m+1},\frac{m+3}{2m+2};\frac{\tilde{\eta}}{4}\right)~{},$
$\displaystyle V_{(1,3)}(\tilde{\eta})$
$\displaystyle=\tilde{\eta}^{h_{1,3}/2}\,_{3}F_{2}\left(\frac{1-m}{2m+2},\frac{m+2mq-2r+1}{2m+2},\frac{m-2mq+2r+1}{2m+2};\frac{3-m}{2m+2},\frac{3m+1}{2m+2};\frac{\tilde{\eta}}{4}\right),$
$\displaystyle V_{(1,5)}(\tilde{\eta})$
$\displaystyle=\tilde{\eta}^{h_{1,5}/2}\,_{3}F_{2}\left(\frac{m}{m+1},\frac{2m+mq-r}{m+1},\frac{2m-mq+r}{m+1};\frac{3m}{m+1},\frac{5m+1}{2m+2};\frac{\tilde{\eta}}{4}\right)~{},$
(B.44)

in terms of the parameter $q$ defined as

$\displaystyle q\equiv s-r\in\mathbb{Z}~{}.$ (B.45)

The final solution is then

$\displaystyle\langle\psi_{(r,s)}$
$\displaystyle(x_{1})\psi_{(r,s)}(x_{2})\phi_{(1,3)}(x+iy,x-iy)\rangle_{{\mathbb{H}^{+}}}=\frac{\tilde{\mathcal{G}}(\tilde{\eta})}{(x_{12})^{2h_{r,s}}(2y)^{2h_{1,3}}}~{},$
$\displaystyle\tilde{\mathcal{G}}(\tilde{\eta})=B_{(1,3)}^{{\bf
a}\,(1,1)}V_{(1,1)}(\tilde{\eta})+\alpha_{(1,3)}V_{(1,3)}(\tilde{\eta})+\alpha_{(1,5)}V_{(1,5)}(\tilde{\eta})~{},$
(B.46)

where $B_{(1,3)}^{{\bf a}\,(1,1)}$ is the one-point function coefficient of
$\phi_{(1,3)}$ computed in eq. (B.3.1) and we denoted

$\displaystyle\alpha_{(1,3)}\equiv{\hat{C}}_{(r,s)(r,s)(1,3)}^{{\bf
a}}B_{(1,3)}^{{\bf
a}\,(1,3)}~{},\quad\alpha_{(1,5)}\equiv{\hat{C}}_{(r,s)(r,s)(1,5)}^{{\bf
a}}B_{(1,3)}^{{\bf a}\,(1,5)}~{}.$ (B.47)

Of course some of the three-point function coefficients
${\hat{C}}_{(r,s)(r,s)(1,3)}^{{\bf a}},~{}{\hat{C}}_{(r,s)(r,s)(1,5)}^{{\bf
a}}$ can be zero in a given conformal b.c. Instead, $B_{(1,3)}^{{\bf
a}\,(1,3)},~{}B_{(1,3)}^{{\bf a}\,(1,5)}$ are generically non-zero (see e.g.
eq. (B.3.1)).

####  B.5.1 Removing unphysical singularities

The r.h.s. of eq. (B.5) features unphysical singularities when the
configuration is such that $\tilde{\eta}\in[4,\infty]$ ends up on the branch
cut of the Virasoro blocks of eq. (B.5). In order to remove these
singularities we must require that $\text{Disc}~{}\tilde{\mathcal{G}}=0$
across that cut. This is one of the ‘sewing constraints’ of Lewellen [48] for
the open-open-closed string amplitude on the upper half-plane. The condition
that unphysical singularities in defect-defect-bulk blocks must disappear from
the physical correlation functions must hold in higher dimensions as well, and
it has been exploited in refs. [28, 56] to prove ‘triviality’ of certain free
theory conformal defects and in refs. [57, 58] to constrain the space of
conformal boundary conditions for a theory of a free massless scalar field.
See also refs. [59, 29] for applications of this idea for $O(N)$ models with
boundaries of defects and [30] for applications in QFTs in AdS.

##### Case with $m=3$

The only non-identity boundary primary allowed to appear in any conformal b.c
is $\psi_{(1,3)}$ with self-OPE

$\displaystyle\psi_{(1,3)}\times\psi_{(1,3)}$
$\displaystyle={\hat{\mathds{1}}}.$ (B.48)

For this special case, eq. (B.5) contains only the identity block $V_{(1,1)}$,
which becomes a polynomial in $\tilde{\eta}$. Hence
$\text{Disc}~{}\tilde{\mathcal{G}}=0$ and all is well.

##### Case with $m=4$

This is the first non-trivial case. The non-identity boundary primaries
allowed to appear in any conformal b.c are
$\psi_{(1,3)},\psi_{(3,1)},\psi_{(3,3)}$ (see Table 1) with self-OPEs

$\displaystyle\psi_{(1,3)}\times\psi_{(1,3)}$
$\displaystyle={\hat{\mathds{1}}}+\psi_{(1,3)}~{},$
$\displaystyle\psi_{(3,1)}\times\psi_{(3,1)}$
$\displaystyle={\hat{\mathds{1}}}~{},$
$\displaystyle\psi_{(3,3)}\times\psi_{(3,3)}$
$\displaystyle={\hat{\mathds{1}}}+\psi_{(1,3)}~{}.$ (B.49)

We have to inspect the singularity structure of the correlator (B.5) for each
of the possible allowed boundary operators. When $\psi_{(r,s)}=\psi_{(3,1)}$
there is only $V_{(1,1)}$, and it is easy to check that
$\text{Disc}~{}\tilde{\mathcal{G}}=0$ automatically. For $\psi_{(1,3)}$ and
$\psi_{(3,3)}$ this condition implies that

$\displaystyle\alpha_{(1,3)}=B_{(1,3)}^{{\bf
a}\,(1,1)}\frac{\Gamma\left(-\frac{3}{10}\right)\Gamma\left(\frac{7}{10}\right)\sec\left(\pi\left(r-\frac{4s}{5}\right)\right)}{4\sqrt[5]{2}\Gamma\left(r-\frac{4s}{5}+\frac{1}{5}\right)\Gamma\left(-r+\frac{4s}{5}+\frac{1}{5}\right)}~{}.$
(B.50)

As a check, the case of $r=1$, $s=3$ in the equation above is consistent with
the (squared) OPE coefficient obtained from bulk-boundary crossing in eq.
(B.3.1) times the (squared) OPE coefficient obtained from boundary crossing in
eq. (B.37), as it should. One can also check that eq. (B.50) is consistent
with the known results obtained from F-matrices in ref. [49].

##### General $m$, only $V_{(1,1)}$ and $V_{(1,3)}$

Consider a conformal b.c. ${\bf a}=(a_{1},a_{2})_{m}$ for which the following
two properties are satisfied:

  1. 1.

The boundary primary $\psi_{(r,s)}$ is allowed to exist;

  2. 2.

$\alpha_{(1,3)}\equiv{\hat{C}}_{(r,s)(r,s)(1,3)}^{{\bf a}}B_{(1,3)}^{{\bf
a}\,(1,3)}\neq 0$, but $\alpha_{(1,5)}\equiv{\hat{C}}_{(r,s)(r,s)(1,5)}^{{\bf
a}}B_{(1,3)}^{{\bf a}\,(1,5)}=0$ (either because $\psi_{(1,5)}$ does not
exist, or ${\hat{C}}_{(r,s)(r,s)(1,5)}^{{\bf a}}=0$) or it is subleading with
respect to $\alpha_{(1,3)}$ in the large-$m$ limit,

then the condition $\text{Disc}~{}\tilde{\mathcal{G}}=0$ requires that

$\displaystyle\alpha_{(1,3)}=-B_{(1,3)}^{{\bf
a}\,(1,1)}\frac{\Gamma\left(\frac{2-m}{m+1}\right)\Gamma\left(\frac{1-m}{2m+2}\right)\Gamma\left(\frac{m+3}{2m+2}\right)\Gamma\left(\frac{2rm-2sm+m+2r+1}{2m+2}\right)\Gamma\left(\frac{-2rm+2sm+m-2r+1}{2m+2}\right)}{2^{\frac{m-1}{m+1}}\Gamma\left(\frac{1-m}{m+1}\right)\Gamma\left(\frac{3-m}{2m+2}\right)\Gamma\left(\frac{3m+1}{2m+2}\right)\Gamma\left(\frac{mr+r-ms+1}{m+1}\right)\Gamma\left(\frac{ms+1-mr-r}{m+1}\right)}~{}.$
(B.51)

This is consistent with earlier results in the literature computed from
F-matrices, see [49]. For finite $(r,s)$, the large-$m$ expansion of the
former result gives (in taking this limit we take $a_{1},a_{2}\ll m$)

$\displaystyle\alpha_{(1,3)}=\frac{2\pi(r-s)(s-\text{sgn}(r-s))}{\sqrt{3}m}+\frac{2\pi(s+1)\left(s+2(s-r)H_{s-r}-1\right)}{\sqrt{3}m^{2}}+O(m^{-3})~{},$
(B.52)

with $H_{n}$ the $n^{\text{th}}$ harmonic number.

##### General $m$, all blocks

For a generic choice of conformal b.c. and of the external field
$\psi_{(r,s)}$ we find a one-parameter family of solutions to the condition
that $\text{Disc}~{}\tilde{\mathcal{G}}=0$, so that only a linear combination
of $\alpha_{(1,3)}$ and $\alpha_{(1,5)}$ gets fixed. In order to determine
$\alpha_{(1,3)}$ and $\alpha_{(1,5)}$ completely, one has to then consider the
full set of ‘sewing’ constraints.

###  B.6 The $\phi_{(3,1)}$ and $\psi_{(3,1)}$ operators

We are also interested in the irrelevant deformation of the minimal models
generated by the $\phi_{(3,1)}$ operator with dimension

$\Delta_{3,1}=2h_{3,1}=2+4/m~{}.$ (B.53)

##### Bulk OPE

The bulk self-OPE of $\phi_{(3,1)}$ reads (see eq. (B.4))

$\displaystyle\phi_{(3,1)}(z,\bar{z})\phi_{(3,1)}(0,0)$
$\displaystyle=\frac{{\mathds{1}}}{|z|^{2\Delta_{3,1}}}+\text{desc.}$
$\displaystyle m=4$ $\displaystyle\phi_{(3,1)}(z,\bar{z})\phi_{(3,1)}(0,0)$
$\displaystyle=\frac{{\mathds{1}}}{|z|^{2\Delta_{3,1}}}+\frac{C_{(3,1)(3,1)}{}^{(3,1)}}{|z|^{\Delta_{3,1}}}\phi_{(3,1)}(0,0)+\text{desc.}$
$\displaystyle m=5$ $\displaystyle\phi_{(3,1)}(z,\bar{z})\phi_{(3,1)}(0,0)$
$\displaystyle=\frac{{\mathds{1}}}{|z|^{2\Delta_{3,1}}}+\frac{C_{(3,1)(3,1)}{}^{(3,1)}}{|z|^{\Delta_{3,1}}}\phi_{(3,1)}(0,0)+\frac{C_{(3,1)(3,1)}{}^{(5,1)}}{|z|^{2\Delta_{3,1}-\Delta_{5,1}}}\phi_{(5,1)}(0,0)+\text{desc.}$
$\displaystyle m\geq 6.$ (B.54)

The bulk OPE coefficients are found to be [51, 52, 53]

$\displaystyle C_{(3,1)(3,1)(3,1)}$
$\displaystyle=\frac{2^{\frac{4}{m}+2}(m+2)\Gamma\left(\frac{3}{2}+\frac{2}{m}\right)\Gamma\left(-\frac{m+2}{2m}\right)}{m^{2}\Gamma\left(\frac{3}{2}+\frac{1}{m}\right)\Gamma\left(-\frac{m+4}{2m}\right)}\sqrt{\frac{\Gamma\left(\frac{1}{m}\right)\Gamma\left(-\frac{m+3}{m}\right)}{\Gamma\left(2+\frac{3}{m}\right)\Gamma\left(\frac{m-1}{m}\right)}},$
$\displaystyle\text{for }m>4~{}$
$\displaystyle=\frac{4}{\sqrt{3}}+\frac{4\sqrt{3}}{m}-\frac{20}{\sqrt{3}m^{2}}+\frac{28-96\zeta(3)}{m^{3}}+O(m^{-4})~{},$
$\displaystyle C_{(3,1)(3,1)(5,1)}$
$\displaystyle=-\frac{m(m+2)\Gamma\left(\frac{1}{m}\right)\Gamma\left(-\frac{m+3}{m}\right)}{(2m+3)(3m+4)\Gamma\left(2+\frac{3}{m}\right)\Gamma\left(-\frac{1}{m}\right)}\sqrt{\frac{\Gamma\left(4+\frac{5}{m}\right)\Gamma\left(\frac{m-1}{m}\right)}{\Gamma\left(-3-\frac{5}{m}\right)\Gamma\left(\frac{1}{m}\right)}},$
$\displaystyle\text{for }m>5~{}$
$\displaystyle=\frac{\sqrt{5}}{3}+\frac{7\sqrt{5}}{9m}-\frac{95\sqrt{5}}{54m^{2}}+\frac{\sqrt{5}(1579-2592\zeta(3))}{324m^{3}}+O(m^{-4})~{}.$
(B.55)

##### Bulk-boundary OPE

The bulk-boundary OPE of $\phi_{(3,1)}$ reads

$\displaystyle\phi_{(3,1)}(x+iy,x-iy)$ $\displaystyle=\frac{B_{(3,1)}^{{\bf
a}\,(1,1)}}{(2y)^{\Delta_{3,1}}}{\hat{\mathds{1}}}+\text{desc.}$
$\displaystyle m=4$ $\displaystyle\phi_{(3,1)}(x+iy,x-iy)$
$\displaystyle=\frac{B_{(3,1)}^{{\bf
a}\,(1,1)}}{(2y)^{\Delta_{3,1}}}{\hat{\mathds{1}}}+\frac{B_{(3,1)}^{{\bf
a}\,(3,1)}}{(2y)^{\Delta_{3,1}-h_{3,1}}}\psi_{(3,1)}(x)+\text{desc.}$
$\displaystyle m=5$ $\displaystyle\phi_{(3,1)}(x+iy,x-iy)$
$\displaystyle=\frac{B_{(3,1)}^{{\bf
a}\,(1,1)}}{(2y)^{\Delta_{3,1}}}{\hat{\mathds{1}}}+\frac{B_{(3,1)}^{{\bf
a}\,(3,1)}}{(2y)^{\Delta_{3,1}-h_{3,1}}}\psi_{(3,1)}(x)+\frac{B_{(3,1)}^{{\bf
a}\,(5,1)}}{(2y)^{\Delta_{3,1}-h_{5,1}}}\psi_{(5,1)}(x)+\text{desc.}$
$\displaystyle m\geq 6.$ (B.56)

The bulk-boundary OPE coefficient $B_{(3,1)}^{{\bf a}\,(1,1)}$ is given in eq.
(B.7). The squared coefficients $(B_{(3,1)}^{{\bf a}\,(3,1)})^{2}$ and
$(B_{(3,1)}^{{\bf a}\,(5,1)})^{2}$ are determined by the bulk-boundary
crossing symmetry, as we show in the next section.

###  B.7 The bulk two-point function of $\phi_{(3,1)}$

We start from the bulk two-point function of $\phi_{(3,1)}$ on the upper half-
plane. To compute such a correlator, we employ the method of images and solve

$\displaystyle\left({\mathcal{L}}_{-3}^{(z_{4})}-\frac{2}{h_{3,1}+2}{\mathcal{L}}_{-1}{\mathcal{L}}_{-2}^{(z_{4})}+\frac{1}{(h_{3,1}+1)(h_{3,1}+2)}{\mathcal{L}}_{-1}^{3}\right)\langle\phi_{(3,1)}(z_{1})\phi_{(3,1)}(z_{2})\phi_{(3,1)}(z_{3})\phi_{(3,1)}(z_{4})\rangle=0~{},$
(B.57)

where ${\cal L}_{-n}^{(\cdot)}$ is the differential operator defined in eq.
(A.29) and ${\cal L}_{-1}=\partial_{z_{4}}$. This holomorphic correlator takes
the following form

$\displaystyle\langle\phi_{(3,1)}(z_{1})\phi_{(3,1)}(z_{2})\phi_{(3,1)}(z_{3})\phi_{(3,1)}(z_{4})\rangle=\frac{\mathcal{G}(\eta)}{(z_{12}z_{34})^{2h_{3,1}}}~{}.$
(B.58)

The cross-ratio $\eta$ is defined as in eq. (B.20). We have three independent
solutions to this equation, corresponding to the boundary operators
${\hat{\mathds{1}}},\psi_{(3,1)}$ and $\psi_{(5,1)}$. In terms of the function
$\tilde{\mathcal{G}}$ defined as in eq. (B.22) with cross-ratio (B.23), the
three Virasoro blocks at finite $m$ read

$\displaystyle V_{(1,1)}(\tilde{\eta})$
$\displaystyle=\,_{3}F_{2}\left(-\frac{4}{m}-2,-\frac{2}{m}-1,\frac{2}{m}+2;-\frac{3}{m}-1,\frac{1}{2}-\frac{1}{m};-\frac{\tilde{\eta}}{4}\right)~{},$
$\displaystyle V_{(3,1)}(\tilde{\eta})$
$\displaystyle=\tilde{\eta}^{{h_{3,1}}/{2}}\,_{3}F_{2}\left(-\frac{3}{m}-\frac{3}{2},-\frac{1}{m}-\frac{1}{2},\frac{3}{m}+\frac{5}{2};-\frac{2}{m}-\frac{1}{2},\frac{1}{m}+\frac{3}{2};-\frac{\tilde{\eta}}{4}\right)~{},$
$\displaystyle V_{(5,1)}(\tilde{\eta})$
$\displaystyle=\tilde{\eta}^{{h_{5,1}}/{2}}\,_{3}F_{2}\left(\frac{1}{m}+1,\frac{5}{m}+4,-\frac{1}{m};\frac{2}{m}+\frac{5}{2},\frac{3}{m}+3;-\frac{\tilde{\eta}}{4}\right)~{}.$
(B.59)

The final solution is

$\displaystyle\langle$
$\displaystyle\phi_{(3,1)}(x_{1}+iy_{1},x_{1}-iy_{1})\phi_{(3,1)}(x_{2}+iy_{2},x_{2}-iy_{2})\rangle_{{\mathbb{H}^{+}}}=\frac{\tilde{\mathcal{G}}(\tilde{\eta})}{(4y_{1}y_{2})^{2h_{3,1}}}~{},$
$\displaystyle\tilde{\mathcal{G}}(\tilde{\eta})=(B_{(3,1)}^{{\bf
a}\,(1,1)})^{2}V_{(1,1)}(\tilde{\eta})+(B_{(3,1)}^{{\bf
a}\,(3,1)})^{2}V_{(3,1)}(\tilde{\eta})+(B_{(3,1)}^{{\bf
a}\,(5,1)})^{2}V_{(5,1)}(\tilde{\eta})~{}.$ (B.60)

In the bulk channel we find

$\displaystyle V^{\text{bulk}}_{(1,1)}(\tilde{\eta})$
$\displaystyle=\tilde{\eta}^{2h_{3,1}}\,_{3}F_{2}\left(-\frac{4}{m}-2,-\frac{3}{m}-\frac{3}{2},-\frac{1}{m};-\frac{6}{m}-3,-\frac{2}{m};-\frac{4}{\tilde{\eta}}\right)~{},$
$\displaystyle V^{\text{bulk}}_{(3,1)}(\tilde{\eta})$
$\displaystyle=\tilde{\eta}^{h_{3,1}}\,_{3}F_{2}\left(\\-\frac{2}{m}-1,-\frac{1}{m}-\frac{1}{2},\frac{1}{m}+1;-\frac{4}{m}-2,\frac{2}{m}+2;-\frac{4}{\tilde{\eta}}\right)~{},$
$\displaystyle V^{\text{bulk}}_{(5,1)}(\tilde{\eta})$
$\displaystyle=\tilde{\eta}^{2h_{3,1}-h_{5,1}}\,_{3}F_{2}\left(\frac{2}{m}+2,\frac{3}{m}+\frac{5}{2},\frac{5}{m}+4;\frac{4}{m}+4,\frac{6}{m}+5;-\frac{4}{\tilde{\eta}}\right)~{}.$
(B.61)

Bulk-boundary crossing symmetry requires that

$\displaystyle(B_{(3,1)}^{{\bf
a}\,(1,1)})^{2}V_{(1,1)}(\tilde{\eta})+(B_{(3,1)}^{{\bf
a}\,(3,1)})^{2}V_{(3,1)}(\tilde{\eta})+(B_{(3,1)}^{{\bf
a}\,(5,1)})^{2}V_{(5,1)}(\tilde{\eta})=$ $\displaystyle
V^{\text{bulk}}_{(1,1)}(\tilde{\eta})+C_{(3,1)(3,1)(3,1)}B_{(3,1)}^{{\bf
a}\,(1,1)}V^{\text{bulk}}_{(3,1)}(\tilde{\eta})+C_{(3,1)(3,1)(5,1)}B_{(5,1)}^{{\bf
a}\,(1,1)}V^{\text{bulk}}_{(5,1)}(\tilde{\eta})~{}.$ (B.62)

From eq. (B.12) we have that

$\displaystyle B_{(3,1)}^{{\bf a}\,(1,1)}$
$\displaystyle=\left(1+2\cos\left(\frac{2\pi
a_{1}(m+1)}{m}\right)\right)\sqrt{\frac{\sin\left(\frac{\pi}{m}\right)\sin\left(\frac{\pi
m}{m+1}\right)}{\sin\left(\frac{3\pi}{m}\right)\sin\left(\frac{\pi
m}{m+1}\right)}}$
$\displaystyle=\sqrt{3}+\frac{2\pi^{2}\left(1-2a_{1}^{2}\right)}{\sqrt{3}m^{2}}+O(m^{-4})~{},$
$\displaystyle B_{(5,1)}^{{\bf a}\,(1,1)}$ $\displaystyle=\sin\left(\frac{5\pi
a_{1}(m+1)}{m}\right)\csc\left(\frac{\pi
a_{1}(m+1)}{m}\right)\sqrt{\frac{\sin\left(\frac{\pi}{m}\right)\sin\left(\frac{\pi
m}{m+1}\right)}{\sin\left(\frac{5\pi}{m}\right)\sin\left(\frac{\pi
m}{m+1}\right)}}$
$\displaystyle=\sqrt{5}+\frac{2\sqrt{5}\pi^{2}\left(1-2a_{1}^{2}\right)}{m^{2}}+O(m^{-4}).$
(B.63)

The coefficients $(B_{(3,1)}^{{\bf a}\,(3,1)})^{2}$ and $(B_{(3,1)}^{{\bf
a}\,(5,1)})^{2}$ are determined by the bulk-boundary crossing symmetry for any
finite $m$.

####  B.7.1 Solution to bulk-boundary crossing

In the simplest cases of $m=4$, only the identity is present in both the bulk
and the boundary channels. Crossing imposes that $(B_{(3,1)}^{{\bf
a}\,(1,1)})^{2}=1$, which upon comparing to eq. (B.7), happens if $a_{1}=1,2$,
as it should. For the case with $m=5$ we have only two operators in both
channels: ${\hat{\mathds{1}}}$ and $\psi_{(3,1)}$ in the boundary channel,
${\mathds{1}}$ and $\phi_{(3,1)}$ in the bulk channel. The only remaining
coefficient is $(B_{(3,1)}^{{\bf a}\,(3,1)})^{2}$, which can be non-zero only
if $a_{1}=2$, and it is fixed by crossing symmetry as follows:

$\displaystyle(B_{(3,1)}^{{\bf a}\,(3,1)})^{2}$
$\displaystyle=\frac{3\Gamma\left(-\frac{14}{5}\right)\Gamma\left(\frac{19}{10}\right)}{3125\
2^{7/10}\pi^{3/2}\Gamma\left(-\frac{8}{5}\right)}$ $\displaystyle\times$
$\displaystyle\left(10584\sqrt{5-\sqrt{5}}\Gamma\left(-\frac{14}{5}\right)\Gamma\left(-\frac{7}{5}\right)-15625\sqrt[10]{2}\left(3\sqrt{5}-7\right)\sqrt{\pi}\,\Gamma\left(\frac{13}{10}\right)\right)\simeq
2.28878~{}.$ (B.64)

For $m>5$ we have

$\displaystyle(B_{(3,1)}^{{\bf a}\,(3,1)})^{2}$
$\displaystyle=C_{(3,1)(3,1)(3,1)}B_{(3,1)}^{{\bf a}\,(1,1)}$
$\displaystyle+(B_{(3,1)}^{{\bf
a}\,(1,1)})^{2}\frac{\left(\csc\left(\frac{2\pi}{m}\right)+\csc\left(\frac{4\pi}{m}\right)\right)\Gamma\left(\frac{5}{2}+\frac{3}{m}\right)\Gamma\left(-\frac{m+2}{2m}\right)^{2}\Gamma\left(-\frac{2(m+3)}{m}\right)}{6\left(\sec\left(\frac{\pi}{m}\right)+\sec\left(\frac{3\pi}{m}\right)\right)\Gamma\left(\frac{3}{2}+\frac{1}{m}\right)\Gamma\left(2+\frac{2}{m}\right)\Gamma\left(-\frac{2(m+2)}{m}\right)^{2}}~{},$
$\displaystyle(B_{(3,1)}^{{\bf a}\,(5,1)})^{2}$
$\displaystyle=C_{(3,1)(3,1)(3,1)}B_{(3,1)}^{{\bf a}\,(1,1)}\frac{\pi
2^{-\frac{4}{m}-3}\sec\left(\frac{2\pi}{m}\right)\Gamma\left(\frac{3}{2}+\frac{1}{m}\right)\Gamma\left(4+\frac{5}{m}\right)}{\Gamma\left(\frac{3}{2}+\frac{2}{m}\right)\Gamma\left(\frac{5}{2}+\frac{2}{m}\right)\Gamma\left(3+\frac{3}{m}\right)\Gamma\left(-\frac{m+2}{2m}\right)}$
$\displaystyle+(B_{(3,1)}^{{\bf
a}\,(1,1)})^{2}\frac{2^{-\frac{2(m+1)}{m}}m\sec\left(\frac{2\pi}{m}\right)\Gamma\left(\frac{1}{2}-\frac{1}{m}\right)\Gamma\left(4+\frac{5}{m}\right)\Gamma\left(-\frac{m+3}{m}\right)}{\sqrt{\pi}(3m+4)\Gamma\left(3+\frac{3}{m}\right)\Gamma\left(-\frac{m+2}{m}\right)}~{},$
(B.65)

with $C_{(3,1)(3,1)(3,1)}$ and $B_{(3,1)}^{{\bf a}\,(1,1)}$ given
(respectively) in equations (B.6) and (B.7). In the large-$m$ expansion (with
$a_{1},a_{2}\ll m$)

$\displaystyle(B_{(3,1)}^{{\bf a}\,(3,1)})^{2}$
$\displaystyle=\frac{32\pi^{2}\left(a_{1}^{2}-1\right)}{3m^{2}}+\frac{32\pi^{2}\left(a_{1}^{2}-1\right)}{m^{3}}-\frac{32\pi^{2}\left(\pi^{2}\left(4a_{1}^{2}-7\right)+15\right)\left(a_{1}^{2}-1\right)}{9m^{4}}+O(m^{-5})~{},$
$\displaystyle(B_{(3,1)}^{{\bf a}\,(5,1)})^{2}$
$\displaystyle=\frac{8\pi^{4}(a_{1}^{2}-4)(a_{1}^{2}-1)}{9m^{4}}+O(m^{-5})~{}.$
(B.66)

###  B.8 Correlator between four $\psi_{(3,1)}$

Consider the four-point correlation function of $\psi_{(3,1)}$. The associated
differential equation and its solutions where given in section B.7. The fusion
rules read

$\displaystyle\psi_{(3,1)}\times\psi_{(3,1)}$
$\displaystyle={\hat{\mathds{1}}}$ $\displaystyle m=4$
$\displaystyle\psi_{(3,1)}\times\psi_{(3,1)}$
$\displaystyle={\hat{\mathds{1}}}+\psi_{(3,1)}$ $\displaystyle m=5$
$\displaystyle\psi_{(3,1)}\times\psi_{(3,1)}$
$\displaystyle={\hat{\mathds{1}}}+\psi_{(3,1)}+\psi_{(5,1)}$ $\displaystyle
m\geq 6~{}.$ (B.67)

The solution is

$\displaystyle\langle\psi_{(3,1)}(x_{1})\psi_{(3,1)}(x_{2})\psi_{(3,1)}(x_{3})\psi_{(3,1)}(x_{4})\rangle_{{\mathbb{H}^{+}}}=\frac{\tilde{\mathcal{G}}(\tilde{\eta})}{(x_{12}x_{34})^{2h_{3,1}}}~{},$
$\displaystyle\tilde{\mathcal{G}}$
$\displaystyle(\tilde{\eta})=V_{(1,1)}(\tilde{\eta})+({\hat{C}}_{(3,1)(3,1)}^{{\bf
a}(3,1)})^{2}V_{(3,1)}(\tilde{\eta})+({\hat{C}}_{(3,1)(3,1)}^{{\bf
a}(5,1)})^{2}V_{(5,1)}(\tilde{\eta})~{}.$ (B.68)

In the equation above we defined, is given by

$\displaystyle\mathcal{G}(\eta)=\tilde{\mathcal{G}}\left(\tilde{\eta}\equiv\frac{\eta^{2}}{1-\eta}\right)~{}.$
(B.69)

The form of the $V_{(r,s)}(\tilde{\eta})$ is given in eq. (B.7), while
$\tilde{\eta}$ is defined in terms of the standard cross ratio $\eta$ as

$\displaystyle\tilde{\eta}=\frac{\eta^{2}}{1-\eta}~{}=\frac{x_{12}^{2}x_{34}^{2}}{x_{12}x_{14}x_{23}x_{24}}~{},$
(B.70)

which is positive if $x_{i}>x_{i+1}$.

####  B.8.1 Boundary crossing

The squared boundary OPE coefficients in eq. (B.8) are fixed by crossing
symmetry, which in terms of the $\mathcal{G}$ previously defined, requires
that:

$\displaystyle\mathcal{G}(\eta)=\left(\frac{1-\eta}{\eta}\right)^{-2h_{3,1}}\mathcal{G}(1-\eta)~{}.$
(B.71)

For $m=4$ we have that ${\hat{C}}_{(3,1)(3,1)(3,1)}^{{\bf
a}}={\hat{C}}_{(3,1)(3,1)(5,1)}^{{\bf a}}=0$ and boundary crossing symmetry is
ensured by the identity block alone. For $m=5$, or for those conformal b.c.
that do not allow $\psi_{(5,1)}$ we have that
${\hat{C}}_{(3,1)(3,1)(5,1)}^{{\bf a}}=0$ and crossing is satisfied if

$\displaystyle({\hat{C}}_{(3,1)(3,1)(3,1)}^{{\bf
a}})^{2}=\frac{2^{\frac{m-2}{m}}\pi\cos\left(\frac{2\pi}{m}\right)\Gamma\left(3+\frac{4}{m}\right)\Gamma\left(-\frac{m+2}{2m}\right)}{\left(2\cos\left(\frac{2\pi}{m}\right)+1\right)\Gamma\left(\frac{1}{2}+\frac{1}{m}\right)\Gamma\left(\frac{3}{2}+\frac{1}{m}\right)\Gamma\left(2+\frac{3}{m}\right)\Gamma\left(-\frac{m+4}{2m}\right)}~{},$
(B.72)

which vanishes identically for $m=4$, but is positive otherwise. For $m>5$ we
find generically a one-parameter family of solutions to the crossing equation,
so that additional correlators (and crossing constraints) are needed in order
to compute ${\hat{C}}_{(3,1)(3,1)(3,1)}^{{\bf a}}$ and
${\hat{C}}_{(3,1)(3,1)(5,1)}^{{\bf a}}$, see e.g. ref. [49].

###  B.9 Correlator between two $\psi_{(r,s)}$ and one $\phi_{(3,1)}$

Next, we consider the boundary-boundary-bulk correlation function

$\displaystyle\langle\psi_{(r,s)}(x_{1})\psi_{(r,s)}(x_{2})\phi_{(3,1)}(x+iy,x-iy)\rangle_{{\mathbb{H}^{+}}}~{},\quad
x_{1}>x_{2}~{}.$ (B.73)

This correlator satisfies the third order differential equation of (B.39), now
with $h_{1,3}\rightarrow h_{3,1}$. There are three independent solutions
(corresponding to the boundary operators ${\hat{\mathds{1}}},\psi_{(3,1)}$ and
$\psi_{(5,1)}$):

$\displaystyle V_{(1,1)}(\tilde{\eta})$
$\displaystyle=\,_{3}F_{2}\left(-\frac{2}{m}-1,\frac{r}{m}-\frac{1}{m}+r-s,-\frac{r}{m}-\frac{1}{m}-r+s;-\frac{3}{m}-1,\frac{1}{2}-\frac{1}{m};\frac{\tilde{\eta}}{4}\right)~{},$
$\displaystyle V_{(3,1)}(\tilde{\eta})$
$\displaystyle=\tilde{\eta}^{h_{3,1}/2}\,_{3}F_{2}\left(-\frac{1}{m}-\frac{1}{2},\frac{r}{m}+r-s+\frac{1}{2},-\frac{r}{m}-r+s+\frac{1}{2};-\frac{2}{m}-\frac{1}{2},\frac{1}{m}+\frac{3}{2};\frac{\tilde{\eta}}{4}\right),$
$\displaystyle V_{(5,1)}(\tilde{\eta})$
$\displaystyle=\tilde{\eta}^{h_{5,1}/2}\,_{3}F_{2}\left(\frac{1}{m}+1,\frac{r}{m}+\frac{2}{m}+r-s+2,-\frac{r}{m}+\frac{2}{m}-r+s+2;\frac{2}{m}+\frac{5}{2},\frac{3}{m}+3;\frac{\tilde{\eta}}{4}\right)~{}.$
(B.74)

The cross-ratio $\tilde{\eta}$ is defined as in eq. (A.13). The final solution
is then

$\displaystyle\langle\psi_{(r,s)}(x_{1})$
$\displaystyle\psi_{(r,s)}(x_{2})\phi_{(3,1)}(x+iy,x-iy)\rangle_{{\mathbb{H}^{+}}}=\frac{\tilde{\mathcal{G}}(\tilde{\eta})}{(x_{12})^{2h_{r,s}}(2y)^{2h_{3,1}}}~{},$
$\displaystyle\tilde{\mathcal{G}}(\tilde{\eta})=B_{(3,1)}^{{\bf
a}\,(1,1)}V_{(1,1)}(\tilde{\eta})+\alpha_{(3,1)}V_{(3,1)}(\tilde{\eta})+\alpha_{(5,1)}V_{(5,1)}(\tilde{\eta})~{},$
(B.75)

where $B_{(3,1)}^{{\bf a}\,(1,1)}$ is the one-point function coefficient of
$\phi_{(3,1)}$ computed in eq. (B.7) and we denoted

$\displaystyle\alpha_{(3,1)}\equiv{\hat{C}}_{(r,s)(r,s)(3,1)}^{{\bf
a}}B_{(3,1)}^{{\bf
a}\,(3,1)}~{},\quad\alpha_{(5,1)}\equiv{\hat{C}}_{(r,s)(r,s)(5,1)}^{{\bf
a}}B_{(3,1)}^{{\bf a}\,(5,1)}~{}.$ (B.76)

####  B.9.1 Removing unphysical singularities

Let us now look for possible unphysical singularities in the r.h.s. of eq.
(B.9). There is a branch cut at $\tilde{\eta}\in[4,\infty]$, so we must
require that $\text{Disc}~{}\tilde{\mathcal{G}}=0$ across that cut. For the
b.c. where $\psi_{(3,1)}$ and $\psi_{(5,1)}$ are not allowed in the boundary
spectrum (see Table 1), $V_{(1,1)}$ alone ensures that
$\text{Disc}~{}\tilde{\mathcal{G}}=0$.

For generic $m>5$, and any conformal b.c. ${\bf a}=(a_{1},a_{2})_{m}$ such
that

  1. 1.

The boundary primary $\psi_{(r,s)}$ is allowed to exist;

  2. 2.

$\alpha_{(3,1)}\neq 0$, but
$\alpha_{(5,1)}\equiv{\hat{C}}_{(r,s)(r,s)(5,1)}^{{\bf a}}B_{(3,1)}^{{\bf
a}\,(5,1)}=0$ (either because $\psi_{(5,1)}$ does not exist, or
${\hat{C}}_{(r,s)(r,s)(5,1)}^{{\bf a}}=0$) or it is subleading with respect to
$\alpha_{(3,1)}$ in the large-$m$ limit,

then the condition $\text{Disc}~{}\tilde{\mathcal{G}}=0$ requires that

$\displaystyle\alpha_{(3,1)}=-B_{(3,1)}^{{\bf
a}\,(1,1)}\frac{2^{-\frac{6}{m}-4}\sqrt{\pi}\,\Gamma\left(\frac{1}{2}-\frac{1}{m}\right)\Gamma\left(-\frac{m+2}{2m}\right)\Gamma\left(-\frac{m+3}{m}\right)\sec\left(\pi\left(\frac{r}{m}+r-s\right)\right)}{\Gamma\left(\frac{3}{2}+\frac{1}{m}\right)\Gamma\left(-\frac{2(m+2)}{m}\right)\Gamma\left(s-\frac{mr+r+1}{m}\right)\Gamma\left(\frac{mr+r-ms-1}{m}\right)}~{}.$
(B.77)

For finite $(r,s)$, the leading large-$m$ expansion (with $a_{1},a_{2}\ll m$)
of the former result gives

$\displaystyle\alpha_{(3,1)}=\frac{2\pi(r-s)(\text{sgn}(r-s)+r)}{\sqrt{3}m}+O(m^{-2})~{}.$
(B.78)

##  Appendix C Some integrals on AdSp+1

We would like to compute the following integrals:

$\displaystyle I_{p}(\alpha)\equiv\int
d^{p}\vec{x}\int_{a}^{\infty}\frac{dz}{z^{p+1}}~{}v^{\alpha}~{},\quad\hat{I}_{p}(\alpha)\equiv\int\frac{d^{p}\vec{x}}{a^{p}}~{}v^{\alpha}\bigg{\rvert}_{z=a}~{},$
(C.1)

with $a,\alpha>0$ and

$\displaystyle
v\equiv\frac{z^{2}(x_{1}-x_{2})^{2}}{(z^{2}+(x_{1}-x)^{2})(z^{2}+(x_{2}-x)^{2})}~{},$
(C.2)

being $\vec{x}_{1},\vec{x}_{2},\vec{x}$ parallel coordinates along the AdSp+1
boundary (in Poincaré coordinates) whereas $z$ is transverse, and
$x_{12}\equiv|\vec{x}_{1}-\vec{x_{2}}|$. Let us start with the first integral.
After going to polar coordinates, for the angular integral we find

$\displaystyle I_{p}(\alpha)$
$\displaystyle=S_{p-1}\int_{a}^{\infty}\frac{dz}{z^{p+1}}\int_{0}^{\infty}r^{r-1}dr\int_{0}^{2\pi}(\sin\theta)^{p-3}~{}v^{\alpha}$
$\displaystyle=S_{p}\int_{0}^{\infty}dr\int_{a}^{\infty}dz~{}{\frac{r^{p-1}x_{12}^{2\alpha}z^{2\alpha-p-1}}{[\left(r^{2}+z^{2}\right)\left(r^{2}+x_{12}^{2}+z^{2}\right)]^{\alpha}}\,_{2}F_{1}\left(\frac{\alpha}{2},\frac{\alpha+1}{2};\frac{p}{2};\frac{4r^{2}x_{12}^{2}}{\left(r^{2}+x_{12}^{2}+z^{2}\right)^{2}}\right)}{}~{}.$
(C.3)

Next we Taylor expand $\,{}_{2}F_{1}$, so that the above equation becomes an
infinite sum of elementary functions. Each term in this expansion can be
further mellinized, so that we find

$\displaystyle I_{p}(\alpha)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{2^{2n+1}\pi^{p/2}\left(\frac{\alpha}{2}\right)_{n}\left(\frac{\alpha+1}{2}\right)_{n}}{n!\left(\frac{p}{2}\right)_{n}\Gamma\left(\frac{p}{2}\right)}\int_{-i\infty}^{i\infty}\frac{dt}{2\pi
i}~{}c_{n}(\alpha,t)\,u^{4t-2\alpha-4n-p}~{},$ (C.4)

where $u\equiv a/x_{12}$ and

$\displaystyle
c_{n}(\alpha,t)\equiv\frac{\Gamma\left(n+\frac{p}{2}\right)\Gamma\left(2t-n-\frac{p}{2}\right)\Gamma(2n-2t+2\alpha)\Gamma\left(3n+\frac{p}{2}-2t+\alpha\right)}{(2\alpha+4n+p-4t)\Gamma(2n+\alpha)\Gamma\left(3n+\frac{p}{2}-2t+2\alpha\right)}~{}.$
(C.5)

We can close the contour to the left, and pick up poles at $-n-p/2+2t=-k$,
$k=0,1,2,3,\dots$. After summing over all these residues we find

$\displaystyle R_{n}(\alpha)$
$\displaystyle\equiv\int_{-i\infty}^{i\infty}\frac{dt}{2\pi
i}~{}c_{n}(\alpha,t)\,u^{4t-2\alpha-4n-p}$
$\displaystyle=\sum_{k=0}^{\infty}\frac{(-1)^{k}\Gamma\left(n+\frac{p}{2}\right)\Gamma(k+2n+\alpha)\Gamma\left(k+n+2\alpha-\frac{p}{2}\right)}{4k!(\alpha+k+n)\Gamma(2n+\alpha)\Gamma(k+2n+2\alpha)}\left(\frac{1}{u^{2}}\right)^{\alpha+k+n}.$
(C.6)

Plugging this result into the expression for $I_{p}(\alpha)$ we get

$\displaystyle I_{p}(\alpha)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{2^{2n+1}\pi^{p/2}\left(\frac{\alpha}{2}\right)_{n}\left(\frac{\alpha+1}{2}\right)_{n}}{n!\Gamma\left(\frac{p}{2}\right)\left(\frac{p}{2}\right)_{n}}R_{n}(\alpha)~{}$
$\displaystyle=\frac{2^{-2\alpha}\pi^{\frac{p+1}{2}}\Gamma\left(2\alpha-p/2\right)}{\Gamma(1+\alpha)\Gamma\left(\alpha+1/2\right)}\left(\frac{1}{u^{2}}\right)^{\alpha}\,_{3}F_{2}\left(\alpha,\alpha,2\alpha-\frac{p}{2};\alpha+\frac{1}{2},\alpha+1;-\frac{1}{4u^{2}}\right)~{}.$
(C.7)

For small $a$ and finite $x_{12}$ (namely small $u$) we find

$\displaystyle I_{p}(\alpha)$
$\displaystyle=\frac{\pi^{p/2}\Gamma\left(\alpha-\frac{p}{2}\right)}{\Gamma(\alpha)}\left(\log(1/u^{2})-\psi(\alpha)+\psi\left(\alpha-\frac{p}{2}\right)\right)$
$\displaystyle+u^{2\alpha-p}\left(\frac{\pi^{\frac{p}{2}+2}\csc^{2}\left(\frac{1}{2}\pi(p-2\alpha)\right)\Gamma\left(2\alpha-\frac{p}{2}\right)}{\Gamma(\alpha)^{2}\Gamma(p-2\alpha+1)\Gamma\left(-\frac{p}{2}+\alpha+1\right)^{2}}+O(u^{2})\right)~{},$
(C.8)

where $\psi(x)$ is the digamma function. Similar tricks can be used in order
to compute the second integral in (C.1), for which we find

$\displaystyle\hat{I}_{p}(\alpha)$
$\displaystyle=\sum_{n,k=0}^{\infty}\frac{(-1)^{k}\pi^{p/2}\Gamma(k+2n+\alpha)\Gamma\left(k+n+2\alpha-\frac{p}{2}\right)}{\Gamma(\alpha)\Gamma(k+1)\Gamma(n+1)\Gamma(k+2(n+\alpha))}\left(\frac{1}{u^{2}}\right)^{\alpha+k+n}~{}.$
(C.9)

Therefore after performing the infinite sum we obtain

$\displaystyle\hat{I}_{p}(\alpha)=\frac{2^{1-2\alpha}\pi^{\frac{p+1}{2}}\Gamma\left(2\alpha-p/2\right)}{\Gamma(\alpha)\Gamma\left(\alpha+1/2\right)}\left(\frac{1}{u^{2}}\right)^{\alpha}\,_{2}F_{1}\left(\alpha,2\alpha-\frac{p}{2};\alpha+\frac{1}{2};-\frac{1}{4u^{2}}\right)~{}.$
(C.10)

For small $u$ we have that

$\displaystyle\hat{I}_{p}(\alpha)$
$\displaystyle=\frac{2\pi^{p/2}\Gamma\left(\alpha-\frac{p}{2}\right)}{\Gamma(\alpha)}+u^{2\alpha-p}\left(\frac{\pi^{\frac{p+1}{2}}2^{2\alpha-p+1}\Gamma\left(\frac{p}{2}-\alpha\right)\Gamma\left(2\alpha-\frac{p}{2}\right)}{\Gamma(\alpha)^{2}\Gamma\left(\frac{p}{2}-\alpha+\frac{1}{2}\right)}+O(u^{2})\right)~{}.$
(C.11)

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