P_{2}-\gamma V)\cdot(P_{1}+\beta P_{2}-\gamma V)\Big{)}^{\frac{a+b+c}{2}}}$
(167)
$\displaystyle=\mathcal{P}\int\frac{d\beta}{\beta}\frac{d\gamma}{\gamma}\frac{\beta^{b}\gamma^{c}}{\Big{(}\beta(P_{12}+\gamma
2P_{2}\cdot V)+\gamma(2P_{1}\cdot V-\gamma)\Big{)}^{\frac{a+b+c}{2}}}\ .$

After rescaling $\beta$ and $\gamma$, we can isolate an integral depending
only on the cross-ratio $\xi$

$\displaystyle\mathcal{F}_{(a,b,c)}(P_{1},P_{2})$
$\displaystyle=\mathcal{P}\frac{1}{(2P_{1}\cdot
V)^{(a-b)}(P_{12})^{b}}\int\frac{d\beta}{\beta}\frac{d\gamma}{\gamma}\frac{\beta^{b}\gamma^{c}}{\Big{(}\beta(1+\frac{\gamma}{\xi})+\gamma(1-\gamma)\Big{)}^{\frac{a+b+c}{2}}}$
$\displaystyle=\frac{1}{(2P_{1}\cdot V)^{a}(2P_{2}\cdot
V)^{b}}\frac{\pi}{\sin(\frac{a-b+c}{2}\pi)}\frac{\Gamma(a)\Gamma(b)(-1)^{\frac{1}{2}(-a+b-c)}}{\Gamma\left(1+\frac{a+b-c}{2}\right)\Gamma\left(\frac{a+b+c}{2}\right)}$
$\displaystyle\times\,_{2}F_{1}\left(a,b;1+\frac{a+b-c}{2};-\xi\right)$ (168)

where in the last line we used the monodromy projection to discard one of the
hypergeometric functions from the integral in the line above. In particular,
we need a function which scales as a constant in the $\xi\to 0$ limit. This
integral (B.2) is key to the derivation of the bulk CPWs (76) and (77).

###  B.3 Evaluation of the boundary standard integral

We look at two boundary points contracted with the reduced dot product
$\bullet$. Neither of the points have vanishing square under $\bullet$,

$\displaystyle\mathcal{F}_{(a,b)}$
$\displaystyle=\mathcal{P}\int\frac{d\beta}{\beta}\frac{\beta^{b}}{\Big{(}-(P_{1}+\beta
P_{2})\bullet(P_{1}+\beta P_{2})\Big{)}^{\frac{a+b}{2}}}$
$\displaystyle=\frac{\mathcal{P}}{(P_{1}\cdot V)^{a}(P_{2}\cdot
V)^{b}}\int\frac{d\beta}{\beta}\frac{\beta^{b}}{\Big{(}(1+\beta)^{2}+4\beta\xi\Big{)}^{\frac{a+b}{2}}}\
,$ (169)

where we used the definition of the cross-ratio $\xi$. Now, we can evaluate
the integral by replacing the series expansion

$\displaystyle\mathcal{F}_{(a,b)}$
$\displaystyle=\frac{\mathcal{P}}{(2P_{1}\cdot V)^{a}(2P_{2}\cdot
V)^{b}}2^{a+b}\sum_{n=0}^{\infty}\frac{\left(\frac{a+b}{2}\right)_{n}}{n!}\left(-4\xi\right)^{n}\int\frac{d\beta}{\beta}\frac{\beta^{b+n}}{(1+\beta)^{a+b+2n}}$
$\displaystyle=\frac{\Gamma(a)\Gamma\left(\frac{b-a}{2}\right)}{\Gamma\left(\frac{a+b}{2}\right)}\frac{2^{b-a}}{(P_{1}\cdot
V)^{a}(P_{2}\cdot
V)^{b}}\frac{1}{\xi^{a}}\,_{2}F_{1}\left(a,\frac{1+a-b}{2};1+a-b;-\frac{1}{\xi}\right)\
,$ (170)

where in the last line we used the hypergeometric identity

${}_{2}F_{1}\left(\alpha,\beta;\gamma;z\right)$
$\displaystyle=\frac{\Gamma(\gamma)\Gamma(\beta-\alpha)}{\Gamma(\beta)\Gamma(\gamma-\alpha)}(-z)^{-\alpha}\,_{2}F_{1}\left(\alpha,\alpha+1-\gamma;\alpha+1-\beta;\frac{1}{z}\right)$
$\displaystyle+\frac{\Gamma(\gamma)\Gamma(\alpha-\beta)}{\Gamma(\alpha)\Gamma(\gamma-\beta)}(-z)^{-\beta}\,_{2}F_{1}\left(\beta,\beta+1-\gamma;\beta+1-\alpha;\frac{1}{z}\right)\
.$

and applied the monodromy projection $\mathcal{P}$ which throws out the second
term in the hypergeometric identity. In a nutshell, the monodromy projection
should enforce that the CPW behaves as $\xi^{-a}$ in the boundary
$\xi\to\infty$ limit, matching onto the behavior expected from the OPE. The
integral (B.3) is an essential ingredient in deriving (70).

##  Appendix C Conformal Block through OPE resummation

###  C.1 Fixing the boundary OPE

This section focuses on the construction of the operator $\mathfrak{D}$ which
encodes the boundary OPE of spinor operators. Our starting point is the
definition (65) given previously. The operator $\mathfrak{D}$ is fixed by
requiring the matching of the OPE with the form of correlators involving
$\psi$ and $\rho$. Concretely, consider

$\displaystyle\expectationvalue{\psi_{\Delta_{1}}(x,z)\overline{\rho}_{\Delta,\alpha}(y)}$
$\displaystyle\propto\frac{\left(\gamma\cdot(x-y)+\gamma_{n}z\right)\mathbb{P}_{-\alpha}}{(2z)^{\Delta_{1}-\Delta}\left(\absolutevalue{x-y}^{2}+z^{2}\right)^{\Delta+\frac{1}{2}}}\
,$
$\displaystyle\expectationvalue{\rho_{\Delta,\alpha}(x)\overline{\rho}_{\Delta,\alpha}(y)}$
$\displaystyle=\frac{\mathbb{P}_{\alpha}\gamma\cdot(x-y)}{\absolutevalue{x-y}^{2\Delta+1}}\
.$

In this appendix, $\gamma\cdot x\equiv\gamma^{i}x_{i}$, and does not include
the normal components. Likewise, $\Box=\partial_{i}\partial^{i}$. We have to
match the expansion in $z$ in terms of a sum over descendants of a given
primary $\rho$. Without loss of generality, we can bring $y\rightarrow 0$ to
simplify expressions

$\displaystyle\expectationvalue{\psi_{\Delta_{1}}(x,z)\overline{\rho}_{\Delta,\alpha}(0)}$
$\displaystyle\propto(2z)^{\Delta-\Delta_{1}}\sum_{n=0}^{\infty}\frac{(-1)^{n}(\Delta+\frac{1}{2})_{n}}{n!}\times$
$\displaystyle\times\left(z^{2n}\mathbb{P}_{\alpha}\frac{\gamma\cdot
x}{\absolutevalue{x}^{2\Delta+1+2n}}+z^{2n+1}\gamma_{n}\mathbb{P}_{-\alpha}\frac{1}{\absolutevalue{x}^{2\Delta+1+2n}}\right)\
.$

The overall power is that expected by the OPE, so we can ignore it. We rewrite
each term as some derivatives acting on
$\expectationvalue{\rho\overline{\rho}}$. Note the following useful identities

$\displaystyle\Box^{n}\expectationvalue{\rho_{\Delta,\alpha}\overline{\rho}_{\Delta,\alpha}}$
$\displaystyle=4^{n}\left(\Delta+\frac{1}{2}\right)_{n}\left(\Delta-\frac{p-1}{2}\right)_{n}\frac{\mathbb{P}_{\alpha}\gamma\cdot
x}{\absolutevalue{x}^{2n+2\Delta+1}}\ ,$
$\displaystyle\Box^{n}\gamma\cdot\partial\expectationvalue{\rho_{\Delta,\alpha}\overline{\rho}_{\Delta,\alpha}}$
$\displaystyle=-2^{2n+1}\left(\Delta+\frac{1}{2}\right)_{n}\left(\Delta-\frac{p-1}{2}\right)_{n+1}\mathbb{P}_{-\alpha}\frac{1}{\absolutevalue{x}^{2n+2\Delta+1}}\
.$ (171)

From these we can write

$\displaystyle\expectationvalue{\psi_{\Delta_{1}}(x,z)\overline{\rho}_{\Delta,\alpha}(0)}$
$\displaystyle\propto(2z)^{\Delta-\Delta_{1}}\sum_{n=0}^{\infty}\frac{1}{n!}\left(-\frac{z^{2}}{4}\Box\right)^{n}\left(\frac{1}{\left(\Delta-\frac{p-1}{2}\right)_{n}}\right.$
$\displaystyle-\left.\frac{z\gamma_{n}\gamma\cdot\partial}{2}\frac{1}{\left(\Delta-\frac{p-1}{2}\right)_{n+1}}\right)\expectationvalue{\rho_{\Delta,\alpha}\overline{\rho}_{\Delta,\alpha}}\
,$ (172)

which fully specifies the form of the differential operator:

$\mathfrak{D}_{(\Delta)}(z,x,\partial_{x})=\sum_{n=0}^{\infty}\frac{1}{n!}\left(-\frac{z^{2}}{4}\Box_{x}\right)^{n}\left(\frac{1}{\left(\Delta-\frac{p-1}{2}\right)_{n}}-\gamma_{n}\frac{z\gamma\cdot\partial_{x}}{2\left(\Delta-\frac{p-1}{2}\right)_{n+1}}\right)\
.$ (173)

To obtain the equivalent expression for the boundary OPE of $\overline{\psi}$,
one should take the complex conjugate of the $\psi$ OPE:

$\displaystyle\overline{\psi}(y,w)$
$\displaystyle=\sum_{\Delta,\alpha}\frac{\mu_{\Delta,\alpha}}{(2w)^{\Delta_{1}-\Delta}}\overline{\rho}_{\Delta,\alpha}(y)\overline{\mathfrak{D}}_{(\Delta)}(w,y,\overset{\leftarrow}{\partial}_{y})$
(174)
$\displaystyle\overline{\mathfrak{D}}_{(\Delta)}(w,y,\overset{\leftarrow}{\partial}_{y})$
$\displaystyle=\sum_{k=0}^{\infty}\frac{1}{k!}\left(-\frac{w^{2}}{4}\overset{\leftarrow}{\Box}_{y}\right)^{k}\left(\frac{1}{\left(\Delta-\frac{p-1}{2}\right)_{n}}+\gamma_{n}\frac{w\gamma\cdot\overset{\leftarrow}{\partial}_{y}}{2\left(\Delta-\frac{p-1}{2}\right)_{n+1}}\right)\
,$

where the sign change in the second term comes from anticommuting the
$\gamma_{n}$ and $\gamma_{i}$ after complex conjugation.

We now turn to the computation of the conformal blocks. These are given by
inserting the boundary OPE expansion for a single primary field into both
operators inside $\expectationvalue{\psi(x,z)\overline{\psi}(y,w)}$.

We obtain the following expression for the sum of blocks times spinor
structures

$\displaystyle\widehat{\mathcal{W}}_{\Delta,\alpha}(u)$
$\displaystyle=\frac{1}{(2z)^{\Delta_{1}-\Delta}(2w)^{\Delta_{2}-\Delta}}\mathfrak{D}_{(\Delta)}(z,x,\partial_{x})\expectationvalue{\rho_{\Delta,\alpha}(x)\overline{\rho}_{\Delta,\alpha}(y)}\overline{\mathfrak{D}}_{(\Delta)}(w,y,\overset{\leftarrow}{\partial}_{y})\
.$ (175)

This expression is complicated, and we will satisfy ourselves with only a
partial check of the boundary conformal blocks, keeping only the terms
proprtional to $\not{\partial}$. To further simplify the expression, we set
$y=0$ by translation symmetry.

The check proceeds by brute force, evaluating the derivatives and making use
of various Pochhammer identities. The goal is to try to organize the sum to
match the sum (B.8) from Dolan:2001wg 666For an alternative reference see
gradshtein . This double sum defines an $F_{4}$ hypergeometric of 2 variables,
formula (9.180). This reduction to a single Gauss hypergeometric then follows
by formula (9.182), sub-equation 7.:

$\displaystyle\sum_{n,k}\frac{A^{n}B^{k}}{(A+B+C)^{\lambda+n+k}}\frac{(\lambda)_{n+k}(\rho)_{n+k}}{n!k!(\rho)_{n}(\rho)_{k}}$
$\displaystyle=\frac{1}{C^{\lambda}}\sum_{n}\frac{(\lambda)_{2n}}{n!(\rho)_{n}}\left(\frac{AB}{C}\right)^{n}$
(176)
$\displaystyle=\frac{1}{C^{\lambda}}\,_{2}F_{1}\left(\frac{\lambda}{2},\frac{\lambda+1}{2};\rho;\frac{4AB}{C^{2}}\right)\
.$

In particular, one is able to identify $|x|^{2}=A+B+C$, $A=-z^{2}$, and
$B=-w^{2}$. One finds hypergeometric functions which depend on
$(1+2\xi)^{-2}$, which after a quadratic transformation can be written in
terms of hypergeometrics which depend on $-\frac{1}{\xi}$, and further
simplify to the form (70) previously given.

###  C.2 Fixing the bulk OPE

The goal of this section is to fully specify the differential operator
encoding the bulk OPE of the two spinor fields, as given through the equation
(64). To do so, we will take as a starting point the differential operator
generating the bulk OPE of scalar fields

$\displaystyle\phi_{1}(x)\phi_{2}(y)=\sum_{\Delta}\frac{C_{123}}{\absolutevalue{x-y}^{\Delta_{1}+\Delta_{2}-\Delta_{3}}}C_{(\Delta_{1},\Delta_{2},\Delta_{3})}(s,\partial_{y})\phi_{3}(y)\bigg{\rvert}_{s_{\mu}=(x-y)_{\mu}}+(\ell>0)\
.$ (177)

The differential operator $C$ is given by Ferrara:1973yt ; Billo:2016vm .

$\displaystyle C^{(\Delta_{1},\Delta_{2},\Delta_{3})}(s,\partial_{y})=$
$\displaystyle\int_{0}^{1}\frac{du}{B(a,b)}u^{a-1}(1-u)^{b-1}e^{us\cdot\partial_{y}}\sum_{m=0}^{\infty}\left(-\frac{s^{2}u(1-u)}{4}\Box_{y}\right)^{m}\times$
$\displaystyle\frac{1}{m!\left(\Delta_{3}+1-\frac{d}{2}\right)_{m}}\ .$ (178)

We leave $s$ free so that $[s,\partial_{y}]=0$, meaning one should not act
with derivatives on any factor involving $s$. This is convenient to obtain
this simple, factorised expression. The parameters $a$ and $b$ are given by
$a=\frac{\Delta_{3}+\Delta_{1}-\Delta_{2}}{2}$ and
$b=\frac{\Delta_{3}+\Delta_{2}-\Delta_{1}}{2}$.

This operator is fixed by requiring that its action on a scalar two point
function reproduces the parts needed to generate the three point function

$\displaystyle\left.C_{(\Delta_{1},\Delta_{2},\Delta_{3})}(s,\partial_{y})\left[\frac{1}{\absolutevalue{y-z}^{2\Delta_{3}}}\right]\right\rvert_{s=(x-y)}=\frac{1}{\absolutevalue{x-z}^{\Delta_{3}+\Delta_{1}-\Delta_{2}}\absolutevalue{y-z}^{\Delta_{3}+\Delta_{2}-\Delta_{1}}}\
.$ (179)

The general tactic is now to write the correlator involving spinors using
derivatives and sums of correlators involving only scalars. Then, linear
combinations of $C$’s with shifted dimension and derivatives can be used to
construct the spinor $\mathfrak{C}$. Consider the first structure in the three
point function

$\displaystyle\expectationvalue{\psi(x)\bar{\psi}(y)\phi(z)}_{\circ}^{(1)}$
$\displaystyle=\gamma\cdot(x-y)\expectationvalue{\phi_{\Delta_{1}+\frac{1}{2}}(x)\phi_{\Delta_{2}+\frac{1}{2}}(y)\phi_{\Delta_{3}}}$
(180)
$\displaystyle=\left.\frac{\gamma\cdot(x-y)C^{(\Delta_{1}+\frac{1}{2},\Delta_{2}+\frac{1}{2},\Delta_{3})}(s,\partial_{y})}{\absolutevalue{x-y}^{\Delta_{1}+\Delta_{2}+1-\Delta_{3}}}\bigg{[}\expectationvalue{\phi_{\Delta_{3}}(y)\phi_{\Delta_{3}}(z)}\bigg{]}\right\lvert_{s=x-y}\
,$

from which we obtain that the bulk OPE of the first structure can be
encapsulated efficiently using this $C$ differential operator, i.e. we find

$\displaystyle\mathfrak{C}_{(\Delta_{1},\Delta_{2},\Delta)}^{(1)}(x-y,\partial_{y})[\bullet]=\left.\gamma\cdot(x-y)C_{(\Delta_{1}+\frac{1}{2},\Delta_{2}+\frac{1}{2},\Delta)}(s,\partial_{y})[\bullet]\right\lvert_{s=x-y}\
.$ (181)

For the second structure, we have to be more creative. First note we can
rewrite it as

$\displaystyle(\not{x}-\not{z})(\not{y}-\not{z})$
$\displaystyle=\not{s}(\not{y}-\not{z})+\absolutevalue{y-z}^{2}\ .$ (182)

Clearly, the second term can be reabsorbed into the power-laws to yield a
scalar three point function with shifted weights. Meanwhile, the first term
should be rewritten using factors of $s_{\mu}$ and $\partial_{y}$ acting on
the three point function given out by the action of $C$. One can check the
following identity

$\displaystyle\not{s}(\not{y}-\not{z})\expectationvalue{\phi_{\Delta_{1}}(x)\phi_{\Delta_{2}}(y)\phi_{\Delta_{3}}(z)}$
$\displaystyle=\frac{\left(\Delta_{3}-\Delta_{2}-\Delta_{1}+\not{s}\not{\partial}_{y}\right)}{2+\Delta_{1}-\Delta_{2}-\Delta_{3}}\expectationvalue{\phi_{\Delta_{1}}(x)\phi_{\Delta_{2}-1}(y)\phi_{\Delta_{3}-1}(z)}$
(183)
$\displaystyle=\frac{1}{\absolutevalue{x-y}^{\Delta_{1}+\Delta_{2}-\Delta_{3}}}\left.\frac{\not{s}\not{\partial}_{y}C_{(\Delta_{1},\Delta_{2}-1,\Delta_{3}-1)}(s,\partial_{y})}{2+\Delta_{1}-\Delta_{2}-\Delta_{3}}\frac{1}{|y-z|^{2\Delta_{3}-2}}\right\lvert_{s=x-y}\
.$

Meanwhile, the second piece can be incorporated using the same shifted
dimension for $\Delta_{2}$ and $\Delta_{3}$. Taking into account the explicit
form of the correlator $\expectationvalue{\psi\bar{\psi}\phi}_{\circ}^{(2)}$,
and the powers entering it, we end up with the final expression

$\displaystyle\expectationvalue{\psi(x)\bar{\psi}(y)\phi(z)}_{\circ}^{(2)}=(\not{x}-\not{z})(\not{z}-\not{y})\expectationvalue{\phi_{\Delta_{1}+\frac{1}{2}}(x)\phi_{\Delta_{2}+\frac{1}{2}}(y)\phi_{\Delta_{3}+1}}$
(184)
$\displaystyle=\left.\frac{1+\frac{\not{s}\not{\partial}_{y}}{1+\Delta_{1}-\Delta_{2}-\Delta_{3}}}{\absolutevalue{x-y}^{\Delta_{1}+\Delta_{2}-\Delta_{3}}}C_{(\Delta_{1}+\frac{1}{2},\Delta_{2}-\frac{1}{2},\Delta_{3})}(s,\partial_{y})\frac{1}{|y-z|^{2\Delta_{3}}}\right\lvert_{s=x-y}$

from which we gather that

$\displaystyle\mathfrak{C}_{(\Delta_{1},\Delta_{2},\Delta)}^{(2)}(x-y,\partial_{y})[\bullet]=\left.\absolutevalue{s}\left(1+\frac{\not{s}\not{\partial}_{y}}{1+\Delta_{1}-\Delta_{2}-\Delta}\right)C_{(\Delta_{1}+\frac{1}{2},\Delta_{2}-\frac{1}{2},\Delta)}(s,\partial_{y})[\bullet]\right\lvert_{s=x-y}\
.$ (185)

Using these results, we can compute the conformal blocks for the exchange of
bulk scalars.

###  C.3 Bulk OPE Resummation

The bulk conformal partial waves are given by the action of
$\mathfrak{C}^{(i)}_{(\Delta_{1},\Delta_{2},\Delta)}$ on the one point
function of a scalar operator

$\displaystyle\mathcal{W}_{\Delta,i}(x,y)=\frac{\mathfrak{C}^{(i)}_{(\Delta_{1},\Delta_{2},\Delta)}[\expectationvalue{\phi_{\Delta}}]}{\absolutevalue{x-y}^{\Delta_{1}+\Delta_{2}+1-\Delta_{3}}}\
.$ (186)

Since the $\mathfrak{C}^{(i)}_{(\Delta_{1},\Delta_{2},\Delta)}$ are built from
the $C$’s, one can first compute the result for $C$ to build the result for
the spinors. The result for $C$ is of course the bulk block of a scalar field
McAvity:1995tm :

$\displaystyle C_{(\Delta_{1},\Delta_{2},\Delta_{3})}(s,\partial_{y})$
$\displaystyle\frac{1}{(2y\cdot n)^{\Delta_{3}}}=$
$\displaystyle\frac{1}{2^{\Delta_{3}}}\sum_{m=0}^{\infty}\int_{0}^{1}\frac{du}{B(a,b)}\frac{u^{a+m-1}(1-u)^{b+m-1}}{(y\cdot
n+us\cdot
n)^{\Delta_{3}+2m}}\left(-\frac{s^{2}}{4}\right)^{m}\frac{(\Delta_{3})_{2m}}{m!\left(\Delta_{3}+1-\frac{d}{2}\right)_{m}}\
.$

We used that the exponential $e^{us\cdot\partial_{y}}$ generates a translation
in the $y$ direction. The $u$ integral is an Euler parametrisation of a
hypergeometric function with parameters including a shift by $m$. Inputting
the explicit values for $a$ and $b$, the hypergeometric function becomes a
rational function of $x\cdot n$ and $y\cdot n$. The sum can then be done and
gives yet another hypergeometric function

$\displaystyle\left.C_{(\Delta_{1},\Delta_{2},\Delta_{3})}(s,\partial_{y})[\expectationvalue{\phi_{\Delta_{3}}(y)}]\right\lvert_{s=x-y}$
$\displaystyle=\frac{\,{}_{2}F_{1}\left(\frac{\Delta_{3}+\Delta_{1}-\Delta_{2}}{2},\frac{\Delta_{3}+\Delta_{2}-\Delta_{1}}{2};\Delta_{3}+1-\frac{d}{2};-\xi\right)}{(2y\cdot
n)^{\frac{\Delta_{2}+\Delta_{3}-\Delta_{1}}{2}}(2x\cdot
n)^{\frac{\Delta_{1}+\Delta_{3}-\Delta_{2}}{2}}}\ ,$ (187)

and we made the cross-ratio explicit. Computing the block is now
straightforward by acting with derivatives. One finds back the expressions
(76) and (77) coming from the conformal integrals. In checking
$\mathcal{W}_{\Delta,2}(x,y)$, the hypergeometric identity

$\displaystyle\left(1-\frac{2\xi}{1+\Delta_{12}-\Delta_{3}}\partial_{\xi}\right){}_{2}F_{1}$
$\displaystyle\left(\frac{\Delta_{3}+\Delta_{12}+1}{2},\frac{\Delta_{3}-\Delta_{12}-1}{2},\Delta_{3}-\frac{d}{2}+1,-\xi\right)=$
$\displaystyle{}_{2}F_{1}\left(\frac{\Delta_{3}+\Delta_{12}+1}{2},\frac{\Delta_{3}-\Delta_{12}+1}{2},\Delta_{3}-\frac{d}{2}+1,-\xi\right)$
(188)

is useful.

##  Appendix D Renormalisation Group Analysis of Interacting Fermions in
$d=3-\epsilon$

We present the renormalisation group analysis of the system of fermions
coupled to scalars in $d=3-\epsilon$. For completeness sake, we also consider
a potential for the scalars. To the best of our knowledge, these results are
new.

###  D.1 Lagrangian and Feynman Rules

The Lagrangian is given in eq. (3), and the conventions for the spinors are as
before. We take $a,b,c,\ldots=1\ldots N_{f}$ or $1\ldots N_{s}$, depending on
context. We will work in $d=3-\epsilon$, where both interactions are
marginally relevant. We write the renormalized fields using brackets,
$[\bullet]$. The various $Z$-factors from which we extract anomalous
dimensions and that we use to renormalize the fields are defined as follows:

$\displaystyle\psi$ $\displaystyle=\sqrt{Z_{\psi}}[\psi]\ ,$
$\displaystyle\phi$ $\displaystyle=\sqrt{Z_{\phi}}[\phi]\ ,$ $\displaystyle
Z_{\psi}[\overline{\psi}][\psi]=Z_{\sigma}[\overline{\psi}\psi]\ ,$ (189)
$\displaystyle\gamma_{\psi}$
$\displaystyle=\frac{\mu}{2}\partialderivative{\mu}\log(Z_{\psi})\ ,$
$\displaystyle\gamma_{\phi}$
$\displaystyle=\frac{\mu}{2}\partialderivative{\mu}\log(Z_{\phi})\ ,$
$\displaystyle\gamma_{\sigma}=\mu\partialderivative{\mu}\log(\frac{Z_{\psi}}{Z_{\sigma}})\
.$

The couplings are also renormalized, giving rise to beta functions

$\displaystyle g$
$\displaystyle=\frac{Z_{g}}{Z_{\phi}Z_{\psi}}\mu^{\epsilon}[g]\ ,$
$\displaystyle\beta_{g}$
$\displaystyle=g\left(2(\gamma_{\psi}+\gamma_{\phi})-\mu\partialderivative{\mu}\log(Z_{g})-\epsilon\right)\
,$ (190) $\displaystyle h$
$\displaystyle=\frac{Z_{h}}{Z_{\psi}^{3}}\mu^{2\epsilon}[h]\ ,$
$\displaystyle\beta_{h}$
$\displaystyle=h\left(3\gamma_{\phi}-\mu\partialderivative{\mu}\log(Z_{h})-2\epsilon\right)\
.$ (191)

For convenience later, we write $Z_{A}=1+\delta_{A}$. The anomalous dimensions
$\gamma_{\psi}$ and $\gamma_{\phi}$ have to be positive by unitarity. We will
use a renormalisation condition such that the loop corrections to the
correlators vanish when all momenta (except one) are equal to some $p_{\mu}$
such that $p\cdot p=\mu^{2}$. The Feynman rules777The diagrams have all been
drawn using TikZ-Feynman Ellis:2016jkw . are

$\displaystyle=-\frac{\not{p}}{p^{2}}$ $\displaystyle=-\not{p}\delta_{\psi}$
(192) $\displaystyle=\frac{-i}{p^{2}}$ $\displaystyle=ip^{2}\delta_{\phi}$
(193) $\displaystyle=-ig$ $\displaystyle=-ig\delta_{g}$ (194)
$\displaystyle=-ih$ $\displaystyle=-ih\delta_{\lambda}$ (195)

We omitted the $O(N_{s})\times O(N_{f})$ tensor structure of the vertices.
When computing loop diagrams, one has to analytically continue each momentum
integral to Euclidean signature. Hence for each loop we have to add an $i$. We
will also abbreviate the momentum integral

$\displaystyle\int dk\equiv\int\frac{d^{d}k}{(2\pi)^{d}}\ .$ (196)

For the renormalisation group analysis, it is convenient to use a QFT
normalisation of the coupling, as we just gave. In the rest of this paper, we
used a CFT adapted normalisation for $g$:
$\kappa_{s}g=2\lambda\sqrt{\epsilon}$ where $\kappa_{s}^{-1}=4\pi$. To derive
the epsilon scaling of the coupling, let us instead write $g=8\pi g_{*}$ and
use $g_{*}$ for now instead of $\lambda$. We find it convenient to introduce
here also $h=1920\pi^{2}h_{*}$.

###  D.2 Field Strength Renormalisation

The renormalisation of the fermion propagator is computed through

$\displaystyle 0$ $\displaystyle=\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[small,baseline=(a)]
\vertex(a); \vertex[ right=of a](b) ; \vertex[right= of b] (c) ;
\vertex[right= of c] (d); \diagram* { (a) --[fermion] (b) --[fermion] (c)
--[fermion] (d), (b) --[scalar,half right, looseness=1.5] (c) --[scalar, half
right, looseness=1.5] (b) }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\feynmandiagram[layeredlayout,inline=-0.0pt,horizontal=atob]{a--[fermion]b[crosseddot]--[fermion]c,};$
(197) $\displaystyle=-\frac{g^{2}N_{s}}{2}\int
dk\,dl\frac{\not{k}}{k^{2}l^{2}(l-p+k)^{2}}-\not{p}\delta_{\psi}\ .$ (198)

The integral can be evaluated using Feynman parameters and dimensional
regularization. Asking that the equality be achieved at a scale $p\cdot
p=\mu^{2}$ extracting only the logarithmic piece, one finds

$\displaystyle\delta_{\psi}$
$\displaystyle=\frac{g^{2}N_{s}}{96\pi^{2}}\log(\mu)+\ldots$ (199)

which translates into an anomalous dimension

$\displaystyle\gamma_{\psi}=\frac{g_{*}^{2}N_{s}}{3}+\ldots$ (200)

This result matches (141) that we obtained from crossing symmetry.

To find the beta function, one needs also the renormalisation of the scalar
sector. We look at two contributions, one from a $g^{2}$ interaction and one
from an $h^{2}$ diagram:

$\displaystyle\feynmandiagram[layeredlayout,horizontal=atob,inline=-0.0pt]{a--[scalar]b[crosseddot]--[scalar]c,};$
(201)

The renormalisation condition is then

$\displaystyle\delta_{\phi}=$
$\displaystyle\overbrace{-\frac{h^{2}(N_{s}+2)(N_{s}+4)}{15(5!)p^{2}}\int\frac{dk_{1}\ldots
dk_{4}}{k_{1}^{2}\ldots
k_{4}^{2}(p-k_{1}-k_{2}-k_{3}-k_{4})^{2}}}^{\Sigma_{h}}$ (202)
$\displaystyle\underbrace{-\frac{2g^{2}N_{f}}{p^{2}}\int dk\,dl\frac{k\cdot
l}{k^{2}l^{2}(k-p+l)^{2}}}_{\Sigma_{g}}\ .$

The computation of $\Sigma_{g}$ is quite similar to the fermion case

$\displaystyle\Sigma_{g}$
$\displaystyle=\frac{4}{3}g_{*}^{2}N_{f}\log(\mu)+\ldots$ (203)

The second piece, $\Sigma_{h}$ can be computed by chaining the loop integrals.
There is no particular difficulty beyond keeping track of the different
overall coefficients. We find

$\displaystyle\Sigma_{h}$
$\displaystyle=\frac{(N_{s}+2)(N_{s}+4)}{6}h_{*}^{2}\log(\mu)+\ldots$ (204)

As expected, these coefficients are positive, since
$\Delta_{\phi}=\frac{d-2}{2}+\gamma_{\phi}\geq\frac{d-2}{2}$, by unitarity.
The end result is that

$\displaystyle\delta_{\phi}=\frac{1}{6}\left(8N_{f}g_{*}^{2}+(N_{s}+2)(N_{s}+4)h_{*}^{2}\right)\log(\mu)+\ldots$
(205)

###  D.3 Scalar Vertex Renormalisation

The diagrams that renormalise the $\phi^{6}$ vertex are at leading order

The multiplicity count of those diagram can be found by using double-line
notation. As previously, we split the computation in two

$\displaystyle-i\delta_{h}h+ih(\Sigma_{h}^{{}^{\prime}}+\Sigma_{g}^{{}^{\prime}})=0\
,$ (207)

with obvious form of the contributions from each diagram. The first piece is
given by

$\displaystyle\Sigma_{h}^{{}^{\prime}}$
$\displaystyle=h\frac{22+3N_{s}}{15}\int\frac{dk_{1}dk_{2}}{k_{1}^{2}k_{2}^{2}(p-k_{1}-k_{2})^{2}}$
(208) $\displaystyle=-8(22+3N_{s})h_{*}\log(\mu)+\ldots$

while the second is

$\displaystyle\Sigma_{h}$ $\displaystyle=-30g^{2}N_{f}\int
dkdl\frac{k\cdot(k+l-p)}{k^{2}l^{2}(l+k-p)^{2}(l-2p)^{2}}$ (209)
$\displaystyle=-60N_{f}g_{*}^{2}\log(\mu)+\ldots$

The end result is then

$\displaystyle\delta_{h}=-\left(8(3N_{s}+22)h_{*}+60N_{f}g_{*}^{2}\right)\log(\mu)+\ldots$
(210)

which implies a beta function

$\displaystyle\beta_{h}\propto
h_{*}\left(8N_{f}g_{*}^{2}+(3N_{s}+22)h_{*}-\frac{\epsilon}{4}+\ldots\right)$
(211)

As anticipated $g_{*}$ scales as $\sqrt{\epsilon}$, while $h_{*}$ scales as
$\epsilon$. Thus the $h_{*}^{2}$ contribution to the anomalous dimension of
$\phi$ will be higher order.

###  D.4 Yukawa Vertex Renormalisation

The diagrams at order $g^{2}$ are all convergent; hence one must go to order
$g^{3}$ to find a non-zero contribution. Indeed, $O(g^{3})$ is consistent with
$g^{2}\sim\epsilon$ scaling at the fixed point. There are three families of
divergent diagrams, which we consider separately

$\displaystyle-ig\delta_{g}+ig(\Sigma_{1}+\Sigma_{2}+\Sigma_{3})=0.$ (212)

The first family is given by bug type diagrams,

We will impose the renormalisation condition at the point
$p_{1}=p_{2}=p_{3}=p$, such that $p^{2}=\mu^{2}$. The sum of the four diagrams
drawn evaluates in this simplified configuration to

$\displaystyle\Sigma_{1}=2g^{2}\int\frac{dk\,dl}{k^{2}l^{2}(k-2p)^{2}}\left(\frac{\not{k}\not{l}}{(l-k-p)^{2}}+\frac{\not{l}\not{k}}{(l-k+p)^{2}}\right)\
.$ (215)

Standard methods yield

$\displaystyle\Sigma_{1}$ $\displaystyle=8g_{*}^{2}\log(\mu)+\ldots.$ (216)

We still have two diagrams to consider. The first one is made up of an
internal fermion loop, the second of an internal scalar loop:

The diagram with the internal fermion loop gives

$\displaystyle\Sigma_{2}$ $\displaystyle=2N_{f}g^{2}\int
dk\,dl\frac{k\cdot(k+p-l)}{k^{2}l^{2}(k+p-l)^{2}(l-2p)^{2}}$ (218)

Extracting the log we find

$\displaystyle\Sigma_{2}$ $\displaystyle=-4N_{f}g^{2}_{*}\log(\mu)+\ldots$
(219)

For the diagram with the internal scalar loop, we have instead

$\displaystyle\Sigma_{3}$ $\displaystyle=\frac{g^{2}N_{s}}{2}\int
dk\,dl\frac{\not{k}(\not{k}-2\not{p})}{l^{2}k^{2}(k-2p)^{2}(l-k+3p)^{2}}$
(220) $\displaystyle=-2N_{s}g_{*}^{2}\log(\mu)+\ldots$

The end result of this computation is

$\displaystyle\delta_{g}=(8-4N_{f}-2N_{s})g_{*}^{2}\log(\mu)+\ldots$ (221)

from which we can compute the beta function

$\displaystyle\beta_{g}\propto
g_{*}\left(g_{*}^{2}(N_{s}+2N_{f}-3)-\frac{3}{8}\epsilon+\ldots\right)$ (222)

As long as $N_{f}$ and $N_{s}>1$ there is a unitary fixed point in
$3-\epsilon$ dimensions. It is tempting to speculate that the beta function
vanishes in $d=3$ for $N_{f}=N_{s}=1$ because of supersymmetry.

###  D.5 Fermion Bilinear Renormalisation

We are interested in the anomalous dimension of
${:}\overline{\psi}\psi{:}\equiv\sigma$. We must renormalise the divergences
of the $\psi$ fields on their own as well as the new divergences that show up
because of compositeness. We computed $Z_{\psi}$ to the requisite perturbative
order already. The new divergences will be rescaled away by the $Z_{\sigma}$
factor we introduced at the beginning of this appendix. To fix $Z_{\sigma}$,
we need to regularise diagrams with $\sigma$ insertions. The simplest
situation is to consider a process in which
$\sigma\rightarrow\psi\overline{\psi}$. We find possible contributions from
two diagrams at $O(g^{2})$:

It turns out that the second diagram is convergent; hence we can safely drop
it. We then regularise the first diagram at a momentum scale $p^{2}=\mu^{2}$
by adding a counterterm $\delta_{\sigma}$:

$\displaystyle\delta_{\sigma}$ $\displaystyle=-\frac{g^{2}}{2}N_{s}\int
dk\,dl\frac{(\not{k}-\not{p})\not{k}}{k^{2}l^{2}(k-p)^{2}(l-k+p)^{2}}\ .$
(224)

These integrals are straightforward. Isolating the logarithm, we extract

$\displaystyle\delta_{\sigma}$
$\displaystyle=-2g_{*}^{2}N_{s}\log(\mu)+\ldots$ (225)

The anomalous dimension of the composite operator is determined as usual

$\displaystyle\gamma_{\sigma}$
$\displaystyle=\mu\partialderivative{\mu}\log(\frac{Z_{\psi}}{Z_{\sigma}})=\frac{8}{3}g_{*}^{2}N_{s}\
.$ (226)

This result matches (146) we computed using crossing symmetry and the
equations of motion.

##  Appendix E Relation to EAdSp+1 Amplitude

Intuitively it is understood that one can consider a bCFT and perform a Weyl
transformation to obtain a QFT in AdS Paulos:2016fap . It is useful to give an
explicit form to this mapping in the embedding space picture. Embedding space
offers a simple geometric and coordinate invariant view of the procedure. The
goal of this section is to bridge the gap between our work, which we view
along the bCFT perspective, and the various results coming from the study of
QFT in AdS Carmi:2018qzm ; Giombi:2021cnr .

Consider fields in a bCFT viewed in the embedding space, with points specified
by $P^{A}P_{A}=0$. Given a constant vector $V^{A}$, there is an AdS foliation
of the lightcone along the direction $V$, i.e. there is a map from insertions
on the lightcone in $\mathbb{R}^{1,d+1}$ to insertions on an AdS slice of
$\mathbb{R}^{1,d}$, immersed in the higher space. The explicit map is given by
a projection and rescaling

$\displaystyle X^{A}$
$\displaystyle=\left(\eta^{AB}-V^{A}V^{B}\right)\frac{P_{B}}{P\cdot
V}\rightarrow X^{A}X_{A}=-1\ .$ (227)

The orthogonal projection kills off one component, while the rescaling brings
us to a canonical AdS slice. Because the vector is transverse, the inner
product on the AdS slice is given by the reduced inner product $\bullet$.
Intrinsic coordinates on the manifold spanned by the $X^{A}$ see an
$EAdS_{p+1}$ metric. Viewed from the bCFT embedding, this is simply a change
of section, whose effect is to rescale the fields by their Weyl factors. This
parametrisation is singular at boundary points, $P\cdot V\rightarrow 0$, which
are sent to infinity. Effectively, this maps insertions of fields in a bCFT to
those of fields in AdS. This map is invariant under the residual conformal
symmetry of the boundary, which is kept manifest. The case of $dS_{p+1}$
follows through, at the kinematic level, by considering $V^{2}=-1$ and working
on the lightcone of the space $\mathbb{R}^{2,d}$.

Let us now see how generic fields behave under this mapping. First, from the
scaling property

$\displaystyle\phi_{AdS}(X^{A})=2^{\Delta}\phi_{bCFT}\left(\frac{P^{A}}{P\cdot
V}\right)=(2P\cdot V)^{\Delta}\phi_{bCFT}(P)\ .$ (228)

Hence for bulk insertions, the equivalent expression in $AdS$ follows
straightforwardly by taking power laws out of correlation functions. For
boundary insertions, this rescaling is not well defined since $P\cdot V=0$,
and one must retain the scaling law. We find back that bulk insertions at
$P^{A}$ are mapped to bulk fields at coordinate $X^{A}$ on the AdS
hyperboloid, $X^{A}X_{A}=-1$. Boundary insertions are mapped to insertions on
the lightcone of $R^{1,d}$, the conformal boundary of AdS. The fate of
tensorial objects is similarly dealt with. In the bCFT embedding space,
tensors satisfy

$\displaystyle P^{A}T_{A\ldots}$ $\displaystyle=0\ ,$ $\displaystyle
T_{A\ldots}$ $\displaystyle\sim T_{A\ldots}+\alpha P_{A}\ T^{\prime}_{\ldots}\
.$ (229)

Both conditions are needed to kill two components for each index. Meanwhile,
tensors in the embedding space of AdS have one component killed by the
transversality relation, $X^{A}T_{A\ldots}=0$. This last relation is not true
of general bulk bCFT tensors. However, one is free to use the gauge freedom to
select the representative tensor out of the gauge-equivalence class which is
transverse to both $P^{A}$ and $X^{A}$, equivalently, to both $P^{A}$ and
$V^{A}$. In the index-free picture, the argument follows through by applying
this logic to the polarisation vector $Z^{A}$. The bulk bCFT polarisation
$Z^{A}$ is gauge-fixed to $W^{A}=Z^{A}-\frac{Z\cdot V}{P\cdot V}P^{A}$.
$X^{A}W_{A}=0=V^{A}W_{A}$. The renaming into $W$ is to match with both the
usual notation in AdS, and the notation for the boundary polarisation vectors
which satisfy generally $W^{A}V_{A}=0$.

The rule to translate tensorial insertions from $bCFT$ to $AdS$ is then to
take the correlation function and perform the rewriting:

$\displaystyle Z_{i}^{A}$ $\displaystyle\rightarrow W_{i}^{A}\ ,$
$\displaystyle W_{i}^{A}V_{A}$ $\displaystyle=0=W^{A}W_{A}\ ,$ (230)
$\displaystyle P_{i}^{A}$ $\displaystyle\rightarrow P_{i}\cdot
V(X_{i}^{A}+V^{A})\ ,$ $\displaystyle X_{i}^{A}V_{A}$ $\displaystyle=0\ ,$
$\displaystyle A\cdot B$ $\displaystyle\rightarrow A\bullet B+(A\cdot
V)(B\cdot V)\ .$

One can also take an expression written using the AdS variables and rewrite
them using bCFT variables. One should simply give arbitrary distance
$P_{i}\cdot V$ to each individual insertion. As an example, consider the bCFT
two-point function cross-ratio. Under this transformation, it becomes the
usual AdS geodesic distance

$\displaystyle\xi$ $\displaystyle=-\frac{P_{1}\cdot P_{2}}{4P_{1}\cdot
VP_{2}\cdot V}=-\frac{\left(-X_{1}\bullet
X_{2}-1\right)}{4}=\frac{(X_{1}-X_{2})\bullet(X_{1}-X_{2})}{2}\ .$ (231)

We can also consider the tensor structures entering a bCFT bulk-bulk
correlation function of symmetric traceless operators. These are usually
constructed starting from the basis

$\displaystyle{\mathcal{S}}_{1}$ $\displaystyle=\frac{P_{2}\cdot
P_{1}Z_{1}\cdot Z_{2}-P_{2}\cdot Z_{1}P_{1}\cdot Z_{2}}{P_{1}\cdot
P_{2}}\rightarrow W_{1}\bullet W_{2}+\frac{4}{\xi}W_{1}\bullet
X_{2}W_{2}\bullet X_{1}\ ,$ (232) $\displaystyle{\mathcal{S}}_{2}$
$\displaystyle=\left(Z_{1}\cdot V-\frac{P_{2}\cdot Z_{1}P_{1}\cdot
V}{P_{1}\cdot P_{2}}\right)\left(Z_{2}\cdot V-\frac{P_{1}\cdot Z_{2}P_{2}\cdot
V}{P_{1}\cdot P_{2}}\right)\rightarrow\frac{16}{\xi^{2}}W_{1}\bullet
X_{2}W_{2}\bullet X_{1}\ ,$

where we performed the substitution described and rewrote the final results
using the AdS variables. Looking now at the whole correlator, we have found a
mapping between the correlator in the bCFT and in AdS

$\displaystyle\expectationvalue{F_{L}(P_{1},Z_{1})F_{L}(P_{2},Z_{2})}$
$\displaystyle=\sum_{k=0}^{l}\frac{({\mathcal{S}}_{1})^{k}({\mathcal{S}}_{2})^{L-k}}{(2P_{1}\cdot
V)^{\Delta_{1}}(2P_{2}\cdot V)^{\Delta_{2}}}g_{k}(\xi)\ ,$ (233)
$\displaystyle\expectationvalue{F_{L}(X_{1},W_{1})F_{L}(X_{2},W_{2})}_{AdS}$
$\displaystyle=\sum_{k=0}^{L}(W_{1}\bullet W_{2})^{k}(W_{1}\bullet
X_{2}W_{2}\bullet X_{1})^{L-k}f_{k}(z)\ ,$ $\displaystyle f_{k}(z)$
$\displaystyle=\sum_{n=k}^{L}\binom{n}{k}\left(\frac{\xi}{4}\right)^{n+k-2L}g_{n}\left(\xi\right)\
.$

The functions of the cross-ratio are the same up to a change of basis. The
physical content is that the blocks are left unchanged, as is expected by
conformal invariance. The functions $g_{k}$ or $f_{k}$ admit an expansion in
terms of bulk and boundary conformal blocks. On the bCFT side, the bulk blocks
are all known. They can all be extrapolated from the knowledge of the seed
block for a scalar and by acting with weight shifting operators
Karateev:2018uk . The seed conformal blocks Costa:2011vf  consist of the
exchange of a boundary spin-$L$ operator between two spin-$L$ bulk operators,
and configurations with higher bulk spin can be reached by acting with another
weight shifting operator Billo:2016vm ; Herzog:2021spv .

The bulk expansion is convergent whenever the bulk is conformal, which
corresponds to a CFT in AdS. The boundary expansion however is always present.
The seed boundary conformal blocks can be found by solving the conformal
Casimir equation, which in the AdS picture is equivalent to considering the
propagator for massive tensorial fields with spin-$L$ and some specific mass-
squared. This has been done and the boundary blocks are known Costa:2014kfa .
It follows that the seed blocks for tensorial operators in a bCFT two point
function are all known.

Now that the situation for tensors is clearly laid out, we can turn to our
main interest, spinors. These are defined in the bCFT embedding space through
the eigenvalue equation

$\displaystyle P^{A}\Gamma_{A}\Psi=0\rightarrow
X^{A}\Gamma_{A}\psi=\left(-V\cdot\Gamma\right)\Psi\ .$ (234)

We can define $\Gamma_{\circ}=V\cdot\Gamma$. We then find fermions in AdS
which satisfy the eigenvalue equation outlined in Nishida:2018opl . Note, this
clearly holds in any dimension, since the embedding of spinors on the
lightcone is insensitive to $d$. Consider the representation of $\Psi$. For a
bCFT${}_{d}\sim$AdSp+1, we use embedding spinors of $\mathbb{R}^{1,d+1}$.
These always take the form of a doublet of Dirac spinors of two dimensions
lower. Consider $d$ odd; then $\Psi$ is a Dirac spinor of
$\mathbb{R}^{1,d+1}$, which is also the Dirac spinor of $\mathbb{R}^{1,d}$.
This is an irreducible representation in even total dimension; a
$\Gamma_{\circ}$ matrix is available in the embedding of $EAdS_{d}$. If $d$ is
even, $\Psi$ is a Dirac spinor of $\mathbb{R}^{1,d+1}$, and it is made up of a
symplectic doublet of Dirac spinors of $\mathbb{R}^{1,d}$. As seen from the
embedding space of $AdS_{d}$, this forms a reducible representation, whose
symplectic form gives rise to a $\Gamma_{\circ}$-type matrix. This
construction is precisely the one highlighted in Pethybridge:2021rwf , where
this structure was constructed explicitly to study the embedding of fermions
in dS. The output of this discussion is that one can use the same embedding
formalism in (A)dS in both even and odd dimensions. To further make contact
with the eigenvalue equation used there, note that one can redefine the
$\Gamma$-matrix in the AdS picture as $\Gamma^{\prime}_{A}=-i\Gamma\cdot
V(\Gamma^{B}-V^{B}\Gamma\cdot V)$. Then, one finds the eigenvalue equation:

$\displaystyle X^{A}\Gamma^{\prime}_{A}\Psi=i\Psi\ .$ (235)

The factor of $i$ is needed to preserve the signs in the Clifford Algebra.
Where $V^{A}$ timelike, one would not need the $i$, and we would find back the
constraint for dS. Note that taking $V^{A}V_{A}=-1$ yields a projection to
$EdS_{d}\equiv S_{d}$. To obtain Lorentzian $dS_{p+1}$ one needs to consider a
Lorentzian bCFTd, which lifts to an embedding space $\mathbb{R}^{2,p+1}$. It
is quite interesting to note that, although the embedding picture for AdSp+1,
$CFT_{p}$ and dSp+1 live in the same space, $\mathbb{R}^{1,p+1}$, the map from
bCFTd to these is signature dependent. Regarding the index-free formalism for
spinors, no modification is needed beyond the identification
$\not{V}\leftrightarrow\Gamma_{\circ}$. As we did for the tensor, we can
relate the bulk-bulk correlators in both pictures, by simply translating the
structures in the dictionary:

$\displaystyle\expectationvalue{\Psi(P_{1},\overline{S}_{1})\overline{\Psi}(P_{2},S_{2})}$
$\displaystyle=\frac{\overline{S}_{1}S_{2}F(\xi)+\overline{S}_{1}\not{V}S_{2}G(\xi)}{(2P_{1}\cdot
V)^{\Delta_{1}+\frac{1}{2}}(2P_{2}\cdot V)^{\Delta_{2}+\frac{1}{2}}}\ ,$ (236)
$\displaystyle\expectationvalue{\Psi(X_{1},\overline{S}_{1})\overline{\Psi}(X_{2},S_{2})}_{AdS}$
$\displaystyle=\overline{S}_{1}S_{2}F\left(\xi\right)+\overline{S}_{1}\Gamma_{\circ}S_{2}G\left(\xi\right)\
.$

Using these equations, one can easily make contact between our computations
and the ones done in AdS, in a coordinate invariant manner.

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# Bounds of MIN_NCC and MAX_NCC and filtering scheme for graph domain
variables

Dimitri Justeau-Allaire Corresponding author<EMAIL_ADDRESS>CIRAD,
UMR AMAP, Montpellier, France  Institut Agronomique néo-Calédonien (IAC),
Nouméa, New Caledonia  AMAP, Univ Montpellier, CIRAD, CNRS, INRAE, IRD,
Montpellier, France  Philippe Birnbaum  CIRAD, UMR AMAP, Montpellier, France
Institut Agronomique néo-Calédonien (IAC), Nouméa, New Caledonia  AMAP, Univ
Montpellier, CIRAD, CNRS, INRAE, IRD, Montpellier, France  Xavier Lorca
Centre de Génie Industriel, IMT Mines Albi, Albi, France

(May 2018)

###### Abstract

Graph domain variables and constraints are an extension of constraint
programming introduced by Dooms et al. [3]. This approach had been further
investigated by Fages in its PhD thesis [4]. On the other hand, Beldiceanu et
al. presented a generic filtering scheme for global constraints based on graph
properties [1]. This scheme strongly relies on the computation of graph
properties’ bounds and can be used in the context of graph domain variables
and constraints with a few adjustments [2]. Bounds of MIN_NCC and MAX_NCC had
been defined for the graph-based representation of global constraint for the
path_with_loops graph class [1]. In this note we generalize those bounds for
graph domain variables and for any graph class. We also provide a filtering
scheme for any graph class and arbitrary bounds.

##  Introduction and Notations

We use the notations introduced in [1].

  * •

$G$: The graph domain variable.

  * •

$V_{T}$: The set of mandatory vertices.

  * •

$E_{T}$: The set of mandatory arcs.

  * •

$V_{U}$: The set of potential vertices.

  * •

$E_{U}$: The set of potential arcs.

  * •

$V_{F}$: The set of removed vertices.

  * •

$E_{F}$: The set of removed arcs.

  * •

$G(V_{T},E_{T})$: The graph kernel.

  * •

$G(V_{TU},E_{TU})$: The graph envelope.

  * •

$cc(G)$: The set of connected components of $G$, $cc_{[cond]}(G)$ denotes the
set of connected components of $G$ verifying $cond$.

  * •

If $i$ is a connected component, $l_{i}$ is the length of any elementary
sequence made from all the vertices of $i$, or more simple, $l_{i}$ is the
size of $i$.

##  The MIN_NCC Constraint

Given a graph domain variable $G$ and an integer domain variable $P$, the
constraint $\text{MIN\\_NCC}(G,P)$ holds if $P$ is the size of the smallest
connected component of $G$. We denote by $\underline{\text{MIN\\_NCC}}(G)$ and
$\overline{\text{MIN\\_NCC}}(G)$ the lower and upper bounds of the MIN_NCC
graph property, computed from $G$. Sharp bounds of MIN_NCC had been defined in
[1] for graph-based representation of global constraints, in the particular
case of the path_with_loops graph class. We adjusted them for the context of
graph domain variables, and for any graph class.

$\displaystyle\underline{\text{MIN\\_NCC}}(G)=\left\\{\begin{array}[]{ll}0&\text{if
}|V_{T}|=0;\\\ 1&\text{if }|V_{T}|\geq 1\land|V_{U}|\geq 1;\\\
\min_{i\in\textit{cc}(G(V_{T},E_{T}))}l_{i}&\text{if }|V_{T}|\geq
1\land|V_{U}|=0.\end{array}\right.$ (4)
$\displaystyle\overline{\text{MIN\\_NCC}}(G)=\left\\{\begin{array}[]{ll}\min_{i\in\textit{cc}_{[|V_{T}|\geq
1]}(G(V_{TU},E_{TU}))}l_{i}&\text{if }|V_{T}|\geq 1;\\\
\max_{i\in\textit{cc}(G(V_{TU},E_{TU}))}l_{i}&\text{if
}|V_{T}|=0.\end{array}\right.$ (7)

We now suggest a filtering scheme for the $\text{MIN\\_NCC}(G,P)$ constraint
that is adapted from the filtering scheme introduced in [1] for the
path_with_loops graph class in the context of graph-based representation of
global constraints.

  1. 1.

Trivial case: if $\underline{P}>|V_{TU}|$ then fail.

  2. 2.

Trivial case: if $\underline{P}>0$ and $|E_{TU}|<\max(1,\underline{P}-1)$ then
fail.

  3. 3.

If $\overline{\text{MIN\\_NCC}}(G)<\underline{P}$ then fail. Note that a
sufficient condition is $|\textit{cc}_{[|V_{TU}|<\underline{P}\land|V_{T}|\geq
1]}(G(V_{TU},E_{TU}))|\geq 1$, that is the existence of a mandatory connected
component in the envelope strictly smaller than $\underline{P}$.

  4. 4.

If $\underline{\text{MIN\\_NCC}}(G)>\overline{P}$ then fail.

  5. 5.

If $\underline{P}<\underline{\text{MIN\\_NCC}}(G)$ then set $\underline{P}$ to
$\underline{\text{MIN\\_NCC}}(G)$.

  6. 6.

If $\overline{P}>\overline{\text{MIN\\_NCC}}(G)$ then set $\overline{P}$ to
$\overline{\text{MIN\\_NCC}}(G)$.

  7. 7.

Every $U$-vertex of any optional connected component smaller than
$\underline{P}$ (that is, in
$\textit{cc}_{[|V_{TU}|<\underline{P}\land|V_{T}|=0]}\allowbreak(G(V_{TU},E_{TU}))$)
is turned into a $F$-vertex.

  8. 8.

If $\underline{\text{MIN\\_NCC}}(G)<\underline{P}$ then:

    1. (a)

If $|V_{T}|\geq 1$, every $U$-vertex of any mandatory connected component of
size $\underline{P}$ (that is, in
$\textit{cc}_{[|V_{TU}|=\underline{P}\land|V_{T}|\geq 1]}(G(V_{TU},E_{TU}))$),
is turned into a $T$-vertex. $P$ can then be instantiated to $\underline{P}$.

    2. (b)

Consider each connected component
$i\in\textit{cc}_{[|V_{T}|<\underline{P}]}(G(V_{T},E_{T}))$, if there is
exactly one $U$-arc ($x$, $y$) outgoing from $i$ ($x\in i\land y\notin i$), it
is turned into a $T$-arc. If $y$ is a $U$-vertex, it is turned into a
$T$-vertex. Similarly, if there are several $U$-arcs outgoing from $i$ but all
of them are connected to a single $U$-vertex, then it is turned into a
$T$-vertex.

  9. 9.

If $G$ had been modified in step 8b, recompute
$\underline{\text{MIN\\_NCC}}(G)$ and repeat steps 4 and 5.

  10. 10.

If $\overline{\text{MIN\\_NCC}}(G)>\overline{P}$ then:

    1. (a)

If $\overline{P}=0$ every $U$-vertex is turned into a $F$-vertex.

    2. (b)

If
$\overline{P}=1\land|V_{U}|=1\land\min_{i\in\textit{cc}(G(V_{T},E_{T}))}l_{i}>1$
then the only $U$-vertex is turned into a $T$-vertex and every $U$-arc
outgoing from it and which is not a loop is turned into a $F$-arc.

##  The MAX_NCC Constraint

Given a graph domain variable $G$ and an integer domain variable $P$, the
constraint $\text{MAX\\_NCC}(G,P)$ holds if $P$ is the size of the largest
connected component of $G$. We denote by $\underline{\text{MAX\\_NCC}}(G)$ and
$\overline{\text{MAX\\_NCC}}(G)$ the lower and upper bounds of the MAX_NCC
graph property, computed from $G$. Sharp bounds of MAX_NCC had been defined in
[1] for graph-based representation of global constraints, in the particular
case of the path_with_loops graph class. Those bounds remain unchanged in the
context of graph domain variables, for any graph class.

$\displaystyle\underline{\textbf{MAX\\_NCC}}(G)=\max_{i\in\textit{cc}(G(X_{T},E_{T}))}l_{i};$
(8)
$\displaystyle\overline{\textbf{MAX\\_NCC}}(G)=\max_{i\in\textit{cc}(G(X_{TU},E_{TU}))}l_{i}.$
(9)

We now suggest a filtering scheme for the $\text{MAX\\_NCC}(G,P)$ constraint
that is adapted from the filtering scheme introduced in [1] for the
path_with_loops graph class in the context of graph-based representation of
global constraints.

  1. 1.

Trivial case: if $\underline{P}>|X_{TU}|$ then fail.

  2. 2.

Trivial case: if $\underline{P}>0$ and $|E_{TU}|<\max(1,\underline{P}-1)$ then
fail.

  3. 3.

If $\overline{\textbf{MAX\\_NCC}}(G)<\underline{P}$ then fail.

  4. 4.

If $\underline{\textbf{MAX\\_NCC}}(G)>\overline{P}$ then fail. Note that a
sufficient condition is
$|\textit{cc}_{[|X_{T}|>\overline{P}]}(G(X_{T},E_{T}))|\geq 1$, that is the
existence of connected component in the kernel strictly greater than
$\overline{P}$.

  5. 5.

If $\underline{P}<\underline{\text{MAX\\_NCC}}(G)$ then set $\underline{P}$ to
$\underline{\text{MAX\\_NCC}}(G)$.

  6. 6.

If $\overline{P}>\overline{\text{MAX\\_NCC}}(G)$ then set $\overline{P}$ to
$\overline{\text{MAX\\_NCC}}(G)$.

  7. 7.

If $\overline{\text{MAX\\_NCC}}(G)>\overline{P}$ then:

    1. (a)

If $\overline{P}=1$ then every $U$-arc $(x,y)$ such that $x\neq y$ is turned
into a $F$-arc.

    2. (b)

If $\overline{P}=0$ then every $U$-vertex is turned into a $F$-vertex.

    3. (c)

Every $U$-arc linked to a maximal size connected component in the kernel, that
is in $\textit{cc}_{[|X_{T}|=\overline{P}]}(G(X_{T},E_{T}))$ is turned into a
$F$-arc.

    4. (d)

Any $U$-arc linking two connected components of $G(X_{T},E_{T})$ of size
$s_{1}$ and $s_{2}$ such that $s_{1}+s_{2}>\overline{P}$ is turned into a
$F$-arc.

    5. (e)

If $G$ had been modified in step 7d, recompute
$\overline{\text{MAX\\_NCC}}(G)$ and repeat steps 3 and 6.

  8. 8.

If $|\textit{cc}_{[|V_{TU}|\geq\underline{P}]}(G(V_{TU},E_{TU}))|=1$ and the
size of this single candidate component is exactly $\underline{P}$ then any
$U$-vertex of it is turned into a $T$-vertex. $P$ can then be instantiated to
$\underline{P}$.

## References

  * Beldiceanu et al. [a] Nicolas Beldiceanu, Mats Carlsson, Sophie Demassey, and Thierry Petit.  Graph Properties Based Filtering.  In _Principles and Practice of Constraint Programming - CP 2006_ , Lecture Notes in Computer Science, pages 59–74. Springer, Berlin, Heidelberg, a.  ISBN 978-3-540-46267-5 978-3-540-46268-2.  doi: 10.1007/11889205˙7.
  * Beldiceanu et al. [b] Nicolas Beldiceanu, Thierry Petit, and Guillaume Rochart.  Bounds of graph parameters for global constraints.  40(4):327–353, b.  ISSN 0399-0559, 1290-3868.  doi: 10.1051/ro:2007001.
  * [3] Gregoire Dooms, Yves Deville, and Pierre Dupont.  CP(Graph): Introducing a Graph Computation Domain in Constraint Programming.  In _Principles and Practice of Constraint Programming - CP 2005_ , Lecture Notes in Computer Science, pages 211–225. Springer, Berlin, Heidelberg.  ISBN 978-3-540-29238-8 978-3-540-32050-0.  doi: 10.1007/11564751˙18.
  * [4] Jean-Guillaume Fages.  On the use of graphs within constraint-programming.  20(4):498–499.  ISSN 1383-7133, 1572-9354.  doi: 10.1007/s10601-015-9223-9.