On multipoint Ward identities for
superconformal line defects

Gabriel Bliarda
aLaboratoire de Physique, ,
École Normale Supérieure,
Université PSL, CNRS, Sorbonne Université, Université Paris Cité,
24 rue Lhomond, F-75005 Paris, France

<EMAIL_ADDRESS>
###### Abstract

Superconformal Ward identities are revisited in the context of superconformal
line defects. Multipoint correlators of topological operators inserted on
superconformal lines are studied. In particular, it is known that protected
operators preserving enough of the supersymmetry become topological after
performing a topological twist. By definition, such a correlator is constant
in the topological limit. By analysing the topological constraint on the OPE
of such operators, the correlator is further constrained away from this limit.
The constraints on multipoint correlators match the known superconformal Ward
identities in the case of 4-point functions. This allows for an simple and
universal derivation of the superconformal Ward identities governing the
multipoint correlation functions of such operators. This concept is
illustrated by 1/2-BPS operators with an $su(2)$ R-symmetry and further
explored in the case of the displacement multiplet on the 1/2-BPS Wilson line
in 4d $\mathcal{N}=4$ super Yang-Mills theory supporting the conjectured
multipoint Ward identities in the literature.

###### Contents

  1. 1 Introduction
  2. 2 The simplest multipoint Ward identities
    1. 2.1 Topological sector
    2. 2.2 Topological OPE and quadratic Casimir
    3. 2.3 Multipoint Ward identities
  3. 3 An application: multipoint correlators on the 1/2 BPS Wilson line in $\mathcal{N}=4$ SYM
  4. 4 Conclusion
  5. A Simple Lie algebras
  6. B Linearity of the Casimir operator
  7. C Higher-points
  8. D $su(N)$

##  1 Introduction

Superconformal field theories (SCFTs) play a central role in modern
theoretical physics, appearing both in the context of string theory through
the AdS/CFT correspondence and as supersymmetric extensions to critical
phenomena and condensed matter systems. While the additional symmetries makes
the systems less rich in complexity, there is still much to be known and we
are far from a complete description of these theories. On the other hand,
these symmetries allow us to delve into more complex problems. One such
problem is that of extended operators such as conformal defects. These arise
naturally in any CFT [1] and their understanding is essential to a full
description of the theory. In fact, part of the additional data which is
introduced by these non-local operators can be studied as an independent
defect-SCFT where the broken symmetries of the embedding theory play a special
role as operator insertions. In this context, much work has been done, mainly
on four-point functions.

Superconformal theories are extremely constrained; on one hand, the conformal
theory reduces the study of all correlators to a countable set, the conformal
data $\\{\Delta,c_{123}\\}$, on the other, the supersymmetry protects the
dimension of special operators and provides tools such as localisation [2] to
evaluate some of the conformal data at finite coupling. Such operators
preserving half the supersymmetries -1/2 BPS- generically depend on the
position of insertion and their polarisation under the R-symmetry. The
constraints coming from the supersymmetry for four-point correlators of such
operators in [3, 4, 5, 6] leads to relations between the position and the
polarisation cross-ratios; the superconformal Ward identities (SCWI).

These SCWI constrain the form of correlators in the theory and as such are a
very useful tool which reduces the number of variables and the difficulty in
the study of a 4-point correlator. This has been used in [7, 8, 9, 10, 11, 12]
in this context to study the four-point correlators with more ease. The recent
interest in multipoint correlators justifies the search for more general Ward
identities. For example, in [13], the superconformal Ward identities could be
used to extract the CFT data from the single R-symmetry channel computed via
the conformal gauge. The resulting understanding can help extend the numerical
bootstrap of 6-point correlators[14] from that of scalars to that of
constrained supermultiplets. The R-symmetry dependence of correlators is then
reduced and the space-time dependent part can then effectively be studied
using other methods such as Witten diagrams [15, 16, 17, 18] and Mellin
amplitudes[19, 20].

In the context of superconformal line defect correlators, the case is even
clearer as there is only one spacetime cross-ratio for four external
insertions. In this case, the SCWIs reduce the problem of a three-variable
function to that of a single function of a single cross-ratio [21]. This was
then used to constrain the CFT data through the numerical bootstrap [22], to
fix the perturbative form of the correlator at weak [23, 24] and strong
coupling [25]. Being a devil’s advocate, one could ask why bother with higher-
point functions since the OPE relates these all to 3-point functions and can
therefore be found by looking at the 4-point functions. However, just as
4-point correlators contain an infinite number of 3-point functions, 6-point
functions contain an infinite number of 4-point functions and so on. On one
hand, this means that certain sectors of the CFT data corresponding to
operators which are not exchanged in the known 4-point functions can be
accessed[14, 26, 27]. Furthermore, there is a technical aspect wherein it is
easier to look at a single higher-point function than to consider an infinite
number of correlators. On the other hand, there is the technical difficulty of
the mixing of correlators. While this is a feature of perturbations in CFTs so
does not concern methods such as the numerical bootstrap, it certainly is a
roadblocks when considering the analytic bootstrap such as in [28, 29]. In
this case, resolving the mixing problem requires the authors to analyse
correlators with generic long multiplets as external operators and requires a
detailed construction of both the anomalous dimension matrix and the long
operators themselves at strong coupling. The way that higher-point functions
avoid such an issue is two-fold. First, the additional operators allow for the
analysis of long operators through the OPE without the need for an explicit
construction of an operator in terms of the elementary fields. Furthermore,
when organising the exchanged operators in terms of their ’length’, the free
theory already contains non-trivial information. Finally, there is an upper
limit to the highest-point function which contains all the relevant
information in a theory[14].

On another hand, protected operators such as the displacement multiplet in
supersymmetric theories play a special role in describing deformations of the
line [30, 31, 32, 33]. In this context, understanding higher-point functions
can help describe deformations and setups different from the usual cusp
solution [34]. As such, the analysis of multipoint correlators in the context
of supersymmetric defect theories is essential in the understanding of the CFT
data and the deformations of the defect. Despite the many symmetries at play,
understanding these correlators away from the topological limit is
challenging. The many R-symmetry channels transform the bosonic case of a
function depending on $n-3$ variables to as many functions as there are
channels. In the case of 4 external operators, these are related by the SCWI
which reduces the problem to a single function of a single variable. This note
explores the generalisation of this using a different method as that used in
[7, 21]. This method not only allows for the derivation of the SCWI in a
highly adaptable manner, but also explains why the number of constraints from
the SCWI increases as the number of insertions increases despite the symmetry
group staying the same. Such a method is analogous to the deriving of
superconformal blocks but uses the vanishing of the eigenvalue to constrain
the full correlator instead of individual blocks.

The note proceeds as follows; the first section 2 will be focused on the
derivation of the SCWI for multipoint insertions with an R-symmetry group
$SU(2)$. Since the R-symmetry and the conformal group act in a very similar
way, this leads to the simplest multipoint Ward identities. The section
includes the three key elements to the derivation, in the first, the concept
of the topological sector and the topological twist such as is [35] are
reviewed and the relevant topologically twisted conformal algebra given
explicitly. In the second, the operator product expansion between two
topological operators is studied and the constraints from the quadratic
Casimir of the twisted conformal algebra are shown. In the final section, the
linearisation of the quadratic Casimir as a differential equation gives the
generalisation of the superconformal Ward identities.

$\displaystyle(\partial_{\chi_{i}}+\partial_{\rho_{i}})f(\chi_{i},\rho_{i})|_{\rho_{i}\rightarrow\chi_{i}}=0$
(1.1)

where $\chi_{i}$ are the space-time cross ratios, $\rho_{i}$ are the
R-symmetry cross ratios, and the topological limit
$\rho_{i}\rightarrow\chi_{i}$ is taken. In the second section, the multipoint
correlators of 1/2-BPS insertions on the 1/2-BPS Wilson line defect in
$\mathcal{N}=4$ super Yang-Mills (SYM) are considered. As expected the
equivalent generalisation of the known SCWIs match the 4-point results and
support the conjecture for multipoint Ward identities made in [36] in a
certain limit.

$\displaystyle(\frac{1}{2}\partial_{\chi_{i^{*}}}+\partial_{\zeta_{i^{*}}})f(\chi_{i},\zeta_{i}\bar{\zeta}_{i},(1-\zeta_{i})(1-\bar{\zeta}_{i}),(\zeta_{i}-\zeta_{j})\bar{\eta_{ij}})\raisebox{-5.69046pt}{$\Biggr{|}$}_{\raisebox{5.69046pt}{$\begin{subarray}{l}\zeta_{i}\rightarrow\chi_{i}\\\
\eta_{ij}\rightarrow(\chi_{i}-\chi_{j})\\\ \bar{\zeta}_{i^{*}}\rightarrow
1\end{subarray}$}}=0.$ (1.2)

where $\chi_{i}$ are the $n-3$ space-time cross-ratios,
$\zeta_{i},\bar{\zeta}_{i},\eta_{ij}$ are the $2(n-3)+\frac{1}{2}(n-3)(n-4)$
R-symmetry cross-ratios. These Ward identities explains why the number of
constraints increases with the number of external insertions and agrees with
the conjectured Ward identities in [36]. A number of technical appendices
complete this note.

##  2 The simplest multipoint Ward identities

The case of an $\mathcal{N}=2$ supersymmetry with $su(2)$ R-symmetry is
particularly enlightening because it will appear as a subgroup of all the
other cases considered. In this case, the Ward identities were found over 20
years ago in [3] and the superconformal blocs derived recently in [37]. In
order to generalise these results to a larger number of insertions, the
relevant Ward identities must be found. The method above consists of fixing
the superconformal cross-ratios using the symmetries of the correlator. The
variation under a generic supersymmetric transformation then generates terms
with spurious poles whose vanishing denominator gives the Ward identities.
This powerful formalism [3] is universal for four-point correlators and has
been used to derive many of the supersymmetric Ward identities present in the
literature [21, 37]. The question then arises in how to extend this to higher-
point correlators. The first problem arising is that the supersymmetry is no
longer enough to fully fix the fermionic coordinates to 0. This in general
leads to nilpotent (or fermionic) cross-ratios. The resulting independent
functions can then be fixed by the OPE and considering the superconformal
Casimir such as in [38, 39, 40, 41, 42]. Given these additional terms, it is
unclear whether the same amount of supersymmetry will generate more
constraints for an increasing number of external operators. The results from
this note suggests that this is indeed the case and provides an intuitive
physical understanding to why that is the case. In thius sense, one would
expect that extending the work of [3] to higher-point functions will indeed
lead to many spurious poles in the coincident limit of space-time and
R-symmetry variables and correspondingly, many superconformal Ward identities.

The approach this note takes compared to the method described above is rather
a detour on the superconformal constraints. The goal is to develop a set of
purely bosonic identities satisfied by the correlator of the superconformal
primary of the multiplet. The supersymmetry then works in the background. The
main advantage of this formalism is purely technical: Once the system is set
up, one can forget entirely about supersymmetry and consider only bosonic
quantities which greatly reduces the difficulty of the problem. As such, these
Ward identities have been derived for up to 12 external operators. While this
may seem reductive, many of the results and much of the CFT data in
superconformal defect setups [43, 22, 44] can be obtained by considering
purely bosonic functions.

The concept will be illustrated in the context of $su(2)$ R-symmetry and 1d
conformal defects. However, this only relies on a $Q-$exact conformal algebra
which also exists in other setups [45, 46]. The first subsection will
establish the topological sector and the twisted conformal algebra which is
$Q$-exact under supercharges which annihilate the superprimary. One can then
act on the operators with the twisted translations therefore annihilating the
correlator; it no longer depends on position. Then the second subsection 2.2
will look at the OPE between two such operators. In such a scenario, the
exchanged operator is annihilated by the same supercharges as the external
operators and will therefore also be topological. Just as in a usual CFT, the
quadratic Casimir can be used to construct blocs in this twisted conformal
algebra with the difference that the exchanged operators all have vanishing
eigenvalue. The quadratic Casimir acting on the external legs will therefore
annihilate the correlator and give a constraint much like the Ward identities.
The final subsection 2.3 will look at the explicit form of these equations and
form linear differential constraints which match the superconformal Ward
identities for 4 points and extend it to multipoint superconformal Ward
identities.

###  2.1 Topological sector

Let us consider 1-dimensional $\mathcal{N}=2$ superconformal theory with
supercharges

$\displaystyle Q_{a}$ $\displaystyle\bar{Q}^{a}$ $\displaystyle S_{a}$
$\displaystyle\bar{S}^{a}$ . (2.1)

satisfying

$\displaystyle(Q_{a})^{\dagger}$ $\displaystyle=\bar{S}^{a}$
$\displaystyle(\bar{Q}^{a})^{\dagger}$ $\displaystyle=S_{a},$ (2.2)

and the 1/2-BPS supermultiplet $\Phi_{\Delta,\lambda}$ annihilated by $Q_{1}$
and $\bar{Q}^{2}$. The labels $\\{\Delta,\lambda\\}$ are the conformal and
$su(2)$ weights respectively. The superprimary of this multiplet then
satisfies

$\displaystyle Q_{1}\phi=\bar{Q}\phi=S_{a}\phi=\bar{S}^{a}\phi=0.$ (2.3)

We can form the following fermionic operators

$\displaystyle\mathbb{Q}$ $\displaystyle=Q_{1}+S_{2}$
$\displaystyle\mathbb{Q}^{\dagger}$ $\displaystyle=\bar{Q}^{2}+\bar{S}^{1}$
(2.4)

such that the following twisted 1d conformal algebra is
$\mathbb{Q}/\mathbb{Q}^{\dagger}$-exact:

$\displaystyle\tilde{P}$
$\displaystyle=P+R^{1}_{2}=\\{\mathbb{Q},\bar{Q}^{1}\\}$ (2.5)
$\displaystyle\tilde{K}$
$\displaystyle=K-R^{2}_{1}=(\tilde{P}^{\dagger})=-\\{\mathbb{Q}^{\dagger},S_{1}\\}$
(2.6) $\displaystyle\tilde{D}$
$\displaystyle=D-\frac{1}{2}(R^{1}_{1}-R^{2}_{2})=\frac{1}{2}\\{\mathbb{Q},\mathbb{Q}^{\dagger}\\}$
(2.7)

Where the details of the $su(2)$ and $sl(2)$ algebras are reminded in appendix
A. The particular choice of $\tilde{K}=(\tilde{P})^{\dagger}$ seems natural to
preserve some of the properties of the conformal group.
$\Phi_{\Delta,\lambda}$ is therefore a topological operator with vanishing
eigenvalue under $\tilde{D}$

$\displaystyle\tilde{\Delta}=\Delta-\frac{1}{2}\lambda=0.$ (2.8)

The topological operators satisfy the following translation

$\displaystyle\mathcal{O}_{top}(\tau)=\text{e}^{-\tau\tilde{P}}\mathcal{O}_{top}(0)\text{e}^{\tau\tilde{P}}$
(2.9)

So that when parametrising the translations in space and in the R-symmetry
variable as :

$\displaystyle\mathcal{O}_{\Delta,\lambda}(t,y)=\text{e}^{-Pt-
E_{+}y}\mathcal{O}_{\Delta,\lambda}(0,0)\text{e}^{Pt+E_{+}y}$ (2.10)

The topological limit corresponds to

$\displaystyle y_{i}\rightarrow t_{i}$ (2.11)

###  2.2 Topological OPE and quadratic Casimir

The operators above can be made topological in the limit 2.11, we therefore
have the constraint that the multipoint correlator of the 1/2-BPS operators
considered above is a constant in this topological limit

$\displaystyle\langle\mathcal{O}_{\Delta,2\Delta}(t_{1},y_{1})...\mathcal{O}_{\Delta,2\Delta}(t_{n},y_{n})\rangle\underset{y_{i}\rightarrow
t_{i}}{=}\mathbb{F}_{0,n}$ (2.12)

However, considering this topological twist of $sl(2)$ and $su(2)$ contains
additional constraints from considering the OPE expansion in this topological
sector. The topological operators have a natural OPE expansion in terms of
other topological operators:

$\displaystyle\hat{O}_{1}\times\hat{O}_{2}$
$\displaystyle\sim\sum_{i}c_{12i}O_{i}=0$ (2.13)
$\displaystyle\mathbb{Q}(\hat{O}_{1}\times\hat{O}_{2})$
$\displaystyle\sim\sum_{i}c_{12i}\mathbb{Q}O_{i}=0$ (2.14) $\displaystyle
O_{i}$ $\displaystyle=\hat{O}_{i}$ (2.15)

Where we have used the fact that $\mathbb{Q}$ commutes with
$P=\\{\mathbb{Q},\bar{Q}_{1}\\}$. Just as in the usual conformal case,the
twisted conformal algebra is naturally expressed in terms of conformal blocks
which are eigenvalues under the quadratic Casimir[47]. Let us consider the
insertion of a resolution of the identity in a conformal correlator of
operators with a topological limit

$\displaystyle\langle O_{1}....O_{n}\rangle$ $\displaystyle=\langle
O_{1}..O_{j}\sum_{i}\ket{O_{i}}\bra{O_{i}}..O_{n}\rangle$ (2.16)

We consider the conformal Casimir of the twisted conformal algebra

$\displaystyle\tilde{C}^{(2)}_{a_{1}...a_{j}}=\frac{1}{2}\\{\sum_{i}\tilde{D}_{a_{i}},\sum_{i}\tilde{D}_{a_{i}}\\}-\frac{1}{2}\\{\sum_{i}\tilde{P}_{a_{i}},\sum_{i}\tilde{K}_{a_{i}}\\}$
(2.17)

The Casimir acting on the points $\\{1,....,j\\}$ of the correlator above
gives:

$\displaystyle(\tilde{C}_{1...j}^{(2)}\langle
O_{1}..O_{j}\sum_{i}\ket{O_{i}}\bra{O_{i}}..O_{n}\rangle)_{y_{j}\rightarrow
t_{j}}$ $\displaystyle=\sum_{i}\langle
O_{1}..O_{j}\hat{C}_{\Delta_{i}}O_{i}\rangle\langle
O_{i}..O_{n}\rangle_{y_{j}\rightarrow t_{j}}$ (2.18)
$\displaystyle=\sum_{i}\langle\hat{O}_{1}..\hat{O}_{j}\hat{C}_{\Delta_{i}}\hat{O}_{i}\rangle\langle\hat{O}_{i}..\hat{O}_{n}\rangle$
(2.19)
$\displaystyle=\sum_{i}\tilde{\Delta}_{i}(\tilde{\Delta}_{i}-1)\langle\hat{O}_{1}..\hat{O}_{j}\hat{O}_{i}\rangle\langle\hat{O}_{i}..\hat{O}_{n}\rangle$
(2.20) $\displaystyle=0$ (2.21)

Where we have used the fact that in the topological limit, only topological
operators are exchanged, and they satisfy 2.8. The quadratic Casimir equation
therefore not only constrains the blocs, but the full correlator. Therefore,
we know that the twisted quadratic Casimir acting on any number of points of a
multipoint function will annihilate the correlator.111The attentice reader
will worry about the order of limits, this is addressed in appendix B and is
the reason why the linearity of the differential operator is important. This
set of quadratic differential equations for the correlator is naturally
expressed in terms of spacetime and R-symmetry variables and is reminiscent of
superconformal Ward identities. These set of equations will lead to the
superconformal Ward identities for the $n$-point correlator presented in the
next subsection.

###  2.3 Multipoint Ward identities

Let us consider the four-point correlator of 1/2-BPS operators of conformal
dimension $\Delta$ considered above

$\displaystyle\mathcal{A}_{4}=\langle\mathcal{O}_{\Delta,2\Delta}(t_{1},y_{1})\mathcal{O}_{\Delta,2\Delta}(t_{2},y_{2})\mathcal{O}_{\Delta,2\Delta}(t_{3},y_{3})\mathcal{O}_{\Delta,2\Delta}(t_{4},y_{4})\rangle$
$\displaystyle=(\frac{y_{13}y_{24}}{t_{13}t_{24}})^{2\Delta}f_{4}(\chi,\rho)$
(2.22)

where

$\displaystyle\chi$ $\displaystyle=\frac{t_{12}t_{34}}{t_{13}t_{24}}$
$\displaystyle\rho$ $\displaystyle=\frac{y_{12}y_{34}}{y_{13}y_{24}}.$ (2.23)

The multipoint quadratic Casimir acts on the correlator as

$\displaystyle(C^{(2)}_{i}\mathcal{A}_{4})|_{y_{i}=t_{i}}$
$\displaystyle=\tilde{\Delta}(\tilde{\Delta}-1)f_{4}(\chi_{i},\chi_{i})$
(2.24) $\displaystyle(C^{(2)}_{1,2}\mathcal{A}_{4})|_{y_{i}=t_{i}}$
$\displaystyle=\chi^{2}\left(\partial_{\rho}+(\chi-1)\partial_{\rho}^{2}+\partial_{\chi}+(\chi-1)\left(2\partial_{\rho}\partial_{\chi}+\partial_{\chi}^{2}\right)\right)f_{4}(\chi,\rho)|_{\rho=\chi}$
(2.25) $\displaystyle(C^{(2)}_{2,3}\mathcal{A}_{4})|_{y_{i}=t_{i}}$
$\displaystyle=-(\chi-1)^{2}\left(\partial_{\rho}+\partial_{\chi}+\chi\left(\partial_{\rho}^{2}+2\partial_{\rho}\partial_{\chi}+\partial_{\chi}^{2}\right)\right)f_{4}(\chi,\rho)|_{\rho=\chi}$
(2.26)

The differential equations above can be recast in a linear form

$\displaystyle\left(\frac{C^{(2)}_{1,2}}{\chi}-\frac{C^{(2)}_{2,3}}{1-\chi}\right)f(\chi_{i},\rho_{i}))|_{\rho=\chi}=\left(\partial_{\chi}+\partial_{\rho}\right)f(\chi_{i},\rho_{i}))|_{\rho=\chi}$
(2.27)

which gives the Ward identity for the four-point function:

$\displaystyle\left(\partial_{\chi}+\partial_{\rho}\right)f(\chi_{i},\rho_{i}))|_{\rho=\chi}=0$
(2.28)

For higher point functions, we consider the correlator

$\displaystyle\mathcal{A}_{n}=\langle\mathcal{O}_{\Delta,2\Delta}(t_{1},y_{1})...\mathcal{O}_{\Delta,2\Delta}(t_{n},y_{n})\rangle=K_{n,\Delta,\lambda}(t_{i},y_{i})f_{n}(\chi_{i,n},\rho_{i,n})$
(2.29)

where

$\displaystyle K_{n,\Delta,\lambda}(t_{i},y_{i})$
$\displaystyle=\left(\left(\frac{t_{1,n}t_{n-1,n}}{t_{1,n-1}}\right)^{n-2}\prod_{k=1}^{n-1}\frac{1}{t_{k,n}^{2}}\right)^{\Delta}\left(\left(\frac{y_{1,n}y_{n-1,n}}{y_{1,n-1}}\right)^{2-n}\prod_{k=1}^{n-1}y_{k,n}^{2}\right)^{\frac{\lambda}{2}}$
(2.30)

and

$\displaystyle\chi_{i,n}$
$\displaystyle=\frac{t_{1,i+1}t_{n-1,n}}{t_{i+1,n}t_{1,n-1}}$
$\displaystyle\rho_{i,n}$
$\displaystyle=\frac{y_{1,i+1}y_{n-1,n}}{y_{i+1,n}y_{1,n-1}}.$ (2.31)

There are now non-trivial constraints from $\frac{n(n-3)}{2}$ quadratic
Casimirs which can be recast into $n-3$ linear differential equations.

For $n=5$, for example, we have:

$\displaystyle\sum
b^{j}_{a_{i}}\mathcal{C}^{(2)}_{a_{i}}=\left(\partial_{\chi_{j}}+\partial_{\rho_{j}}\right)f_{5}$
(2.32)

where

$\displaystyle b^{1}_{1,2}$ $\displaystyle=\frac{1}{\chi_{1}}$ $\displaystyle
b^{1}_{2,3}$
$\displaystyle=\frac{\chi_{2}-1}{(\chi_{1}-1)(\chi_{1}-\chi_{2})}$
$\displaystyle b^{1}_{3,4}$ $\displaystyle=\frac{1}{1-\chi_{1}}$
$\displaystyle b^{1}_{1,2,3}$ $\displaystyle=0$ $\displaystyle b^{1}_{2,3,4}$
$\displaystyle=\frac{1}{\chi_{1}-1}$ (2.33) $\displaystyle b^{2}_{1,2}$
$\displaystyle=-\frac{1}{\chi_{2}}$ $\displaystyle b^{2}_{2,3}$
$\displaystyle=\frac{\chi_{1}}{\chi_{2}(\chi_{2}-\chi_{1})}$ $\displaystyle
b^{2}_{3,4}$ $\displaystyle=\frac{1}{\chi_{2}-1}$ $\displaystyle
b^{2}_{1,2,3}$ $\displaystyle=\frac{1}{\chi_{2}}$ $\displaystyle
b^{2}_{2,3,4}$ $\displaystyle=0.$ (2.34)

The action of a twisted quadratic Casimir on the correlator can be seen as the
vanishing of the eigenvalue of the exchange operator in one of the OPE
channels depicted in figure 1.

$\mathcal{C}_{12}^{(2)}$$1$$2$$3$$4$$5$

$\mathcal{C}_{123}^{(2)}$$1$$2$$3$$4$$5$

$\mathcal{C}_{23}^{(2)}$$1$$2$$3$$4$$5$

$\mathcal{C}_{34}^{(2)}$$1$$3$$4$$2$$5$

$\mathcal{C}_{234}^{(2)}$$1$$2$$3$$5$$4$

Figure 1: OPE diagrams of the different configurations on which the twisted
quadratic casimir acts on in the case of the 5-point function. The light grey
lines represent the exchange of a topological operator with eigenvalue 0 under
the twisted quadratic casimir leading to the vanishing of the correlator. For
identical external operators, some of these are related by crossing, but the
resulting equations are not dependent on that fact.

This gives us the Ward identities:

$\displaystyle\left(\partial_{\chi_{1}}+\partial_{\rho_{1}}\right)f_{5}(\chi_{i},\rho_{i})|_{\rho_{i}=\chi_{i}}$
$\displaystyle=0$
$\displaystyle\left(\partial_{\chi_{2}}+\partial_{\rho_{2}}\right)f_{5}(\chi_{i},\rho_{i})|_{\rho_{i}=\chi_{i}}$
$\displaystyle=0$ (2.35)

The cases for higher points correlators are shown in the appendix C.

This gives the general multipoint Ward identities for the operators with a
topological sector with an $su(2)$ R-symmetry:

$\displaystyle\left(\partial_{\chi_{j}}+\partial_{\rho_{j}}\right)f_{n}(\chi_{i},\rho_{i})|_{\rho_{i}=\chi_{i}}=0\qquad\forall
j\in\\{1,....,n-3\\}$ (2.36)

##  3 An application: multipoint correlators on the 1/2 BPS Wilson line in
$\mathcal{N}=4$ SYM

In the case of the 1/2 BPS operators on the 1/2-BPS Wilson line defect in
$\mathcal{N}=4$ SYM, the R-symmetry is $sp(4)\sim so(5)$ and we can define two
pairs of nilpotent supercharges.222See, for example, the supercharges in
equation (2.8) of [10, 48] and the footnote 13. This explains the similarity
between the $su(2)$ case in the previous section and this one for $sp(4)$.
Using the notation in [28, 29], the 1/2-BPS operators are annihilated by the
supercharges $\tilde{Q}_{\alpha}^{a}$ and the superconformal descendents are
generated by the action of $Q_{\alpha a}$. As a consequence, there are two
topological sectors. The four-point correlator can be conveniently written
using the embedding formalism [3]:

$\displaystyle\langle\Phi_{\Delta,[k,0]}(t_{1},Y_{1})\Phi_{\Delta,[k,0]}(t_{2},Y_{2})\Phi_{\Delta,[k,0]}(t_{3},Y_{3})\Phi_{\Delta,[k,0]}(t_{4},Y_{4})\rangle=\frac{(Y_{12}Y_{34})^{k}}{(t_{12}t_{34})^{2\Delta}}f(\chi,\zeta\bar{\zeta},(1-\zeta)(1-\bar{\zeta}))$
(3.1)

where the space time cross ratio is the same

$\displaystyle\chi=\frac{t_{12}t_{34}}{t_{13}t_{24}}$ (3.2)

and the R-symmetry cross ratios are

$\displaystyle U$
$\displaystyle=\zeta\bar{\zeta}=\frac{Y_{12}Y_{34}}{Y_{13}Y_{24}}$ (3.3)
$\displaystyle V$
$\displaystyle=(1-\zeta)(1-\bar{\zeta})=\frac{Y_{14}Y_{23}}{Y_{13}Y_{24}}$
(3.4)

Which can be achieved by setting ($Y_{ij}=|\vec{y}_{i}-\vec{y}_{j}|^{2}$)

$\displaystyle y_{1}$ $\displaystyle=y_{4}^{-1}=0$ $\displaystyle
y_{3}=(1,0,...)$ $\displaystyle
y_{2}=(\frac{\zeta+\bar{\zeta}}{2},\frac{1}{2}i(\zeta-\bar{\zeta}),0...).$
(3.5)

In this notation, the shortening condition satisfied for both topological
charges is333Note that between the $so(5)$ and $sp(4)$ notation, the two
dynkin labels are swapped[28].

$\displaystyle\tilde{D}_{i}\Phi_{\Delta,[a,b]}=0$ $\displaystyle\Delta=a$
$\displaystyle b=0$ (3.6)

which is satisfied by the 1/2-BPS multiplets related to the displacement
multiplet (i.e those of the form $(\mathcal{D}_{1})^{k}$). The two topological
limits are

$\displaystyle\zeta\rightarrow\chi$ $\displaystyle\bar{\zeta}\rightarrow\chi$
(3.7)

and there is a symmetry between the barred and unbarred variables. The Ward
identities were derived for this case in [21] and read

$\displaystyle(\frac{1}{2}\partial_{\chi}+\partial_{\zeta})f|_{\zeta\rightarrow\chi}=0.$
(3.8)

or equivalently as[36]

$\displaystyle(\frac{1}{2}\partial_{\chi}+\alpha\partial_{r}-(1-\alpha)\partial_{s})f(\chi,r,s)|_{r\rightarrow\alpha\chi,s\rightarrow(1-\alpha)(1-\chi)}=0$
(3.9)

where the function is explicitly symmetric in $\zeta,\bar{\zeta}$. In this
formalism, it is clear that the $sp(4)$ algebra can be seen as two $su(2)$
algebras, one rotating and the other projecting [49] while the symmetry
between $\zeta$ and $\bar{\zeta}$ is the automorphism between the two copies
of $su(2)$.

For the higher-point functions, the correlator will in general depend on $n-3$
space-time cross ratios and $2(n-3)+\frac{(n-4)(n-3)}{2}$ R-symmetry cross-
ratios. These map conveniently to the $\chi_{i}$ and
$\zeta_{i},\bar{\zeta}_{i},\eta_{ij}$ variables where we keep the automorphism
by imposing the constraint
$\bar{\eta}_{ij}(\zeta_{i}-\zeta_{j})=\eta_{ij}(\bar{\zeta}_{i}-\bar{\zeta}_{j})$
for the dummy variable $\bar{\eta}_{ij}$. For higher points, some of these
R-symmetry cross ratios become degenerate which reduces them to $3(n-3)-1$
variables. In terms of the Mandelstam variables $r_{i},s_{i},t_{ij}$, this
amounts to keeping the definitions for $r$ and $s$ and having $n-4$
independent $t_{ij}$ variables described by $\frac{(n-4)(n-3)}{2}$ partially
redundant variables. This works since each Ward identity depends on $n-4$ of
these variables.444One way to see this is in analogy to the Mellin variables
$\gamma_{ij}$ as explained in [19], the redundant description allows for a
simpler formulation. If needed, one could fix one variable in term of the
others. Explicitly, we consider the $n-$point function

$\displaystyle\langle\Phi_{k,[k,0]}(t_{1},Y_{1})...\Phi_{k,[k,0]}(t_{n},Y_{n})\rangle=K_{n,k,k}(t_{i},Y_{i})f(\chi_{i},r_{i},s_{i},t_{ij})$
(3.10)

where the prefactor is defined in 2.30 and the cross ratios are given by

$\displaystyle\chi_{i,n}$
$\displaystyle=\frac{t_{1,i+1}t_{n-1,n}}{t_{i+1,n}t_{1,n-1}}$ $\displaystyle
t_{i,j,n}$
$\displaystyle=\eta_{ij}(\bar{\zeta}_{i}-\bar{\zeta}_{j})=\bar{\eta}_{ij}(\zeta_{i}-\zeta_{j})=\frac{Y_{1,n}Y_{n-1,n}Y_{i+1,j+1}}{Y_{1,n-1}Y_{i+1,n}Y_{j+1,6}}$
(3.11) $\displaystyle r_{i,n}$
$\displaystyle=\zeta_{i}\bar{\zeta}_{i}=\frac{Y_{1,i+1}Y_{n-1,n}}{Y_{i+1,n}Y_{1,n-1}}$
$\displaystyle s_{i,n}$
$\displaystyle=(1-\zeta_{i})(1-\bar{\zeta}_{i})=\frac{Y_{1,n}Y_{i+1,n-1}}{Y_{i+1,n}Y_{1,n-1}}$
(3.12)

The generalisation of the Ward identity 3.8 of the same type as 2.36 would
then be

$\displaystyle(\frac{1}{2}\partial_{\chi_{i^{*}}}+\partial_{\zeta_{i^{*}}})f(\chi_{i},\zeta_{i}\bar{\zeta_{i}},(1-\zeta_{i})(1-\bar{\zeta_{i}}),\bar{\eta}_{ij}(\zeta_{i}-\zeta_{j})))|_{\zeta_{i}\rightarrow\chi_{i},\eta_{ij}\rightarrow\chi_{i}-\chi_{j}}=0$
(3.13)

where the constraint
$\bar{\eta}_{ij}(\zeta_{i}-\zeta_{j})=\eta_{ij}(\bar{\zeta}_{i}-\bar{\zeta}_{j})$
restores the symmetry between the barred and unbarred variables. However, when
recasting the Casimir in its linear form, this can only be achieved by setting
$\bar{\zeta}_{i^{*}}\rightarrow 1$. Rewriting this in terms of the $r_{i}$,
$s_{i}$ and $t_{ij}$ variables used in [36] is then easy which gives $n-3$
linear superconformal Ward identities

$\displaystyle\lim_{\alpha_{i^{*}}\rightarrow
1}(\frac{1}{2}\partial_{\chi_{i^{*}}}+\alpha_{i^{*}}\partial_{r_{i^{*}}}-(1-\alpha_{i^{*}})\partial_{s_{i^{*}}}+\sum_{j}(\alpha_{i^{*}}-\alpha_{j})\partial_{t_{{i^{*}}j}})f(\chi_{i},r_{i},s_{i},t_{ij}))\raisebox{-5.69046pt}{$\Biggr{|}$}_{\raisebox{5.69046pt}{$\begin{subarray}{l}r_{i}\rightarrow\alpha_{i}\chi_{i}\\\
s_{i}\rightarrow(1-\alpha_{i})(1-\chi_{i})\\\
t_{ij}\rightarrow(\alpha_{i}-\alpha_{j})(\chi_{i}-\chi_{j})\end{subarray}$}}=0.$
(3.14)

where the sum of these terms then gives a weaker form of the superconformal
Ward identity for multipoints conjectured in [36]:

$\displaystyle\lim_{\alpha_{i}\rightarrow
1}\sum_{k=1}^{n-3}\left(\frac{1}{2}\partial_{\chi_{k}}+\alpha_{k}\partial_{r_{k}}-(1-\alpha_{k})\partial_{s_{k}}\right)\mathcal{A}_{\Delta_{1}...\Delta_{n}}\raisebox{-5.69046pt}{$\Biggr{|}$}_{\raisebox{5.69046pt}{$\begin{subarray}{l}r_{i}\rightarrow\alpha_{i}\chi_{i}\\\
s_{i}\rightarrow(1-\alpha_{i})(1-\chi_{i})\\\
t_{ij}\rightarrow(\alpha_{i}-\alpha_{j})(\chi_{i}-\chi_{j})\end{subarray}$}}=0$
(3.15)

where we have used the fact that

$\displaystyle\sum_{ij}(\alpha_{i}-\alpha_{j})\partial_{t_{ij}}=0.$ (3.16)

It would be interesting to see if the linearising of the quadratic Casimir can
directly give the sum conjectured in [36]. These annihilate the 5- and 6-point
superblocks derived in [26, 27, 50] as well as the strong coupling results in
[27]. These identities can only be summed by having a coincident limit so give
a weaker version of the Ward Identity in [36]. However, these still contain
$n-3$ constraints so can probably be rewritten in a more convenient form. The
increase on the number of constraints is clear from the structure as well as
the derivation of the Ward identity. There are $n-3$ Ward identities
corresponding to the different linearisations of the quadratic Casimir
equations. For each of these, there are $n-4$ variables ($\bar{\zeta}$ in the
first formulation and $\alpha$ in the second) that are unfixed from the
topological limit. This being said, it is unclear how these combine and if how
many constraints these Ward identities give for a generic $n-$point correlator
though it was conjectured that the number of independent constraints should
grow as $C\frac{n}{2}$[36] at large $n$ where $C$ is the Catalan number.555The
author thanks Julien Barrat for pointing this out.

##  4 Conclusion

In this note, the $n$-point correlators of supersymmetric operators with a
topological limit were studied. These are known to satisfy topological
constraints, that is, the correlator is constant in the topological limit. To
see this, one can construct a twisted conformal algebra which is exact under
the supercharges annihilating the superprimary. Therefore, when acting on the
correlator with the twisted translations, one is effectively acting with a
supercharges giving 0. However, one can further constrain these correlators by
considering the operator product expansion (OPE) between two such operators.
In particular, the topological condition satisfied by the primary field $\Phi$
will also be satisfied by these exchanged operators. As a consequence, when
acting with the generators of the twisted conformal algebra on the external
legs of an OPE diagram, one annihilates the whole multipoint function. The
corresponding twisted quadratic Casimir acting on one point only gives the
shortening condition of $\Phi$. However, the remaining quadratic Casimirs
acting on the correlator lead to non-trivial linear differential equations
involving both the R-symmetry and the spacetime coordinates of the correlator.
Recast in its linear form for the simplest case of $su(2)$ R-symmetry, this
gives the Ward identities

$\displaystyle(\partial_{\chi_{i}}+\partial_{\rho_{i}})f(\chi_{i},\rho_{i})|_{\rho_{i}\rightarrow\chi_{i}}=0.$
(4.1)

One of the questions one could ask in the usual derivation of multipoint Ward
identities such as in [36] was why there should be an increasing number of
constraints for an increasing number of external points while the number of
supersymmetries remained fixed. In particular, when deriving them, one would
first need to fix the superconformal invariants before analysing the spurious
poles from the variations in supersymmetry.

Rather, in this note, the source of these constraints are clear; when acting
on an $n-$point function with the twisted conformal Casimir, one isolates the
eigenvalue of the corresponding exchanged topological operator which, by
definition, must be zero. For an increasing number of external legs, there is
an increasing number of internal exchanged operators which in turn leads to an
increasing number of constraints from the Ward identities.

This analysis finds its applications in the plethora of superconformal
theories and superconformal defects, such as [21, 43, 51, 48]. In particular,
this provides a method to prove the multipoint Ward identities conjectured in
[36] which are crucial to the construction of the superconformal blocks needed
for the analysis of analytic or numerical bootstrap such as in [52, 53]. In
this light, the note explores the equivalent multipoint Ward identities for
correlators of insertions on the 1/2 BPS Wilson line in $\mathcal{N}=4$ SYM
which are

$\displaystyle(\frac{1}{2}\partial_{\chi_{i^{*}}}+\partial_{\zeta_{i^{*}}})f(\chi_{i},\zeta_{i}\bar{\zeta}_{i},(1-\zeta_{i})(1-\bar{\zeta}_{i}),(\zeta_{i}-\zeta_{j})\bar{\eta_{ij}})\raisebox{-5.69046pt}{$\Biggr{|}$}_{\raisebox{5.69046pt}{$\begin{subarray}{l}\zeta_{i}\rightarrow\chi_{i}\\\
\eta_{ij}\rightarrow(\chi_{i}-\chi_{j})\\\ \bar{\zeta}_{i^{*}}\rightarrow
1\end{subarray}$}}=0.$ (4.2)

These can be rewritten as

$\displaystyle\lim_{\alpha_{i}\rightarrow
1}(\frac{1}{2}\partial_{\chi_{i}}+\alpha_{i}\partial_{r_{i}}-(1-\alpha_{i})\partial_{s_{i}}+\sum_{j}(\alpha_{i}-\alpha_{j})\partial_{t_{ij}})f(\chi_{i},r_{i},s_{i},t_{ij}))\raisebox{-5.69046pt}{$\Biggr{|}$}_{\raisebox{5.69046pt}{$\begin{subarray}{l}r_{i}\rightarrow\alpha_{i}\chi_{i}\\\
s_{i}\rightarrow(1-\alpha_{i})(1-\chi_{i})\\\
t_{ij}\rightarrow(\alpha_{i}-\alpha_{j})(\chi_{i}-\chi_{j})\end{subarray}$}}=0$
(4.3)

and can be seen either as a rewriting or a limit of the multipoint Ward
identities in [36]. This paves the way for a derivation using the superspace
formalism and indicates that there would indeed be poles at the coincident
point $\zeta_{i}=\chi_{i}$ While being technically more challenging, this
would be extremely interesting since the resulting construction then gives
crucial information for the descendents of the operators which is not given by
the bosonic correlator above.666At the time of publication, work is being done
in this direction[54].

This analysis provides an efficient way to derive superconformal Ward
identities, avoiding the complications of grassmann coordinates while focusing
only on the space-time and R-symmetry coordinates. Furthemore, this formalism
seems widely applicable to systems with a topological sector and a
topologically twisted algebra. Beyond the other line defect cases listed in
the classification of [55], there are several higher-dimensional examples that
could use this same method. In particular, there is a full two-dimensional
$Q-$exact conformal algebra in the case of $\mathcal{N}=4$ SYM so it should
follow that the same method can be used to derive the SCWI for the 1/2-BPS
surface defect in this theory. In $d=3$, the configuration of topological
operators in ABJM [56] lying in a circle around the Wilson line could be
studied in this setting. Additionally, there is the symmetry between the
1-dimensional and codimension-1 defect in $\mathcal{N}=4$ SYM[21] which
indicates that there should be a similar derivation for this $p=3$ defect. The
only uncertain aspect when extending this analysis to higher dimensions is
whether the Casimir equations can be recast into a linear form. One example in
which this is clearly not the case is the four point function with two 1/2-BPS
topological operators and two general longs such as those studied in [29].
Beyond the absence of a topological limit, the condition from the twisted
Casimir acting on points 1 and 2 is quadratic and cannot be linearised. As a
consequence the order of limits discussed in the main text plays a role and
there are no constraints akin to the SCWI. The application to higher-
dimensions is a logical extension to this work and it will be interesting to
see if the same simplifications occur.

In a system where this method can be applied, the resulting SCWI can be used
to fix the form of the superconformal blocs in the OPE expansion of the
$n-$point correlator of the superconformal primary of the topological
multiplet, such as is [7, 43, 22, 40]. For 5 and 6-point functions of the
displacement multiplet in the 1/2-BPS Wilson line defect in $\mathcal{N}=4$
SYM, this was done in [27] in anticipation using the conjecture of [36]. The
form of these superconformal blocs as well as that of the correlator
constrained by the SCWI is then enough to study the system using many other
techniques; If the correlator is known through explicit computations, the
unmixed CFT data can then be obtained. Likewise, the numerical bootstrap can
be used with these blocs to obtain bounds on the CFT data [22, 14].
Perturbatively, the analytical bootstrap can be used to find the form of the
correlator from this expansion, an ansatz and the crossing equation. The case-
by-case use of these SCWI to fix the form of the superconformal blocs in
different theories is a rich area which supports the recent interest in
higher-point correlators.

This analysis facilitates the study of multipoint correlators, particularly in
the context of line defects. The analysis of these multipoint correlators can
then shed light on several aspects including finite deformations of defects,
resolving the mixing problem and determining the complete CFT data of the
defect theory.

## Acknowlegements

The author is grateful to Aleix Gimenei-Grau for sharing unpublished notes
which led to the initial inspiration, and to Miguel Paulos, Philine Van Vliet,
Adam Chalabi and Maxime Trepanier for very useful discussions. The author
acknowledges an ongoing project [54] whose scope is complementary to this work
and thanks the authors for discussions. This work was funded by the European
Union (ERC, FUNBOOTS, project number 101043588). Views and opinions expressed
are however those of the author(s) only and do not necessarily reflect those
of the European Union or the European Research Council. Neither the European
Union nor the granting authority can be held responsible for them.

##  Appendix A Simple Lie algebras

### $SL(2,\mathbb{C})$

The Conformal algebra of the 1d-defect is $SL(2,\mathbb{C})$, generated by
$\\{P,K,D\\}$ satisfying the commutation relations

$\displaystyle[D,P]$ $\displaystyle=P$ $\displaystyle[D,K]$ $\displaystyle=-K$
$\displaystyle[K,P]$ $\displaystyle=2D$ (A.1)

We can write these as differential operators acting on a field

$\displaystyle\Phi_{\Delta}(t)$ $\displaystyle=\text{e}^{-Pt}\Phi_{\Delta}(0)$
(A.2) $\displaystyle P$ $\displaystyle=-\partial_{t}$ (A.3) $\displaystyle K$
$\displaystyle=-t^{2}\partial_{t}-2\Delta t$ (A.4) $\displaystyle D$
$\displaystyle=t\partial_{t}+\Delta$ (A.5)

The quadratic Casimir is then written as :

$\displaystyle\mathcal{C}^{(2)}_{a_{1},...,a_{n}}=\frac{1}{2}\\{\sum_{i}D_{i},\sum_{i}D_{i}\\}-\frac{1}{2}\\{\sum_{i}P_{i},\sum_{i}K_{i}\\}$
(A.6)

Explicitly, this gives the Casimir eigenvalue and 2-point differential
operator

$\displaystyle\mathcal{C}^{(1)}_{a_{1}}$ $\displaystyle=\Delta(\Delta-1)$
(A.7) $\displaystyle\mathcal{C}^{(2)}_{1,2}$
$\displaystyle=(\Delta_{1}+\Delta_{2})(\Delta_{1}+\Delta_{2}-1)+(t_{1}-t_{2})\left(2\Delta_{2}\partial_{t_{1}}-2\Delta_{1}\partial_{t_{2}}+(t_{1}-t_{2})\partial_{t_{1}}\partial_{t_{2}}\right)$
(A.8)

Which give the usual differential equation for the 4-point conformal bloc:

$\displaystyle K(t_{i})^{-1}\mathcal{C}^{(2)}_{1,2}K(t_{i})G_{\Delta}(\chi)$
$\displaystyle=\Delta(\Delta-1)G_{\Delta}(\chi)$ (A.9) $\displaystyle
G_{\Delta}(\chi)$
$\displaystyle=\chi^{\Delta}{}_{2}F_{1}(\Delta,\Delta,2\Delta,\chi)$ (A.10)

where we factor use the prefactor

$\displaystyle K(t_{i})=\frac{1}{(t_{12}t_{23})^{2\Delta}}$ (A.11)

for two identical operators $\Delta_{1}=\Delta_{2}=\Delta$

### $SU(2)$

In the case of SU(2) in the Chevalley basis, we have the commutation
relations:

$\displaystyle[H_{0},E_{\pm}]$ $\displaystyle=\pm E_{\pm}$
$\displaystyle[E_{+},E_{-}]$ $\displaystyle=2H_{0}$ (A.12)

For an operator of spin J, we have the following differential action of the
elements of the algebra

$\displaystyle\Phi_{\lambda}(y)$
$\displaystyle=\text{e}^{-yE_{-}}\Phi_{\lambda}(0)$ (A.13) $\displaystyle
E_{m}$ $\displaystyle=-\partial_{y}$ (A.14) $\displaystyle E_{+}$
$\displaystyle=y^{2}\partial_{y}-\lambda y$ (A.15) $\displaystyle H_{0}$
$\displaystyle=y\partial_{y}+\frac{\lambda}{2}$ (A.16)

The quadratic Casimir in this basis is777Note that the Cartan is scaled by 2
compared to the Chevalley basis, and that the Casimir differs from the sl(2)
case because of the rescaling of the Dilation operator $D\rightarrow-iD$ which
is standard in $1d$.

$\displaystyle\mathcal{C}^{(2)}_{su(2),a_{i}}=\frac{1}{2}\\{\sum_{i}H^{i}_{0},\sum_{i}H^{i}_{0}\\}+\frac{1}{2}\\{\sum_{i}E^{i}_{-},\sum_{i}E^{i}_{+}\\}$
(A.17)

The eigenfunctions of the 2-point Casimir, the su(2) blocs for 4-point
functions, are then found to be (once the prefactor 2.30 is factored out):

$\displaystyle\mathcal{C}^{(2)}_{su(2),1,2}H_{\lambda}$
$\displaystyle=\frac{\lambda}{2}(\frac{\lambda}{2}+1)H_{\lambda}$ (A.18)
$\displaystyle H_{\lambda}$
$\displaystyle=\rho^{J}{}_{2}F_{1}(-\frac{\lambda}{2},-\frac{\lambda}{2},-\lambda,\rho)$
(A.19)

### Twisted conformal algebra

Combining these two previous sections, we can write an operator of spin
$J=\frac{\lambda}{2}$ and weight $\Delta$ as:

$\displaystyle\Phi_{\Delta,\lambda}(t,y)=\text{e}^{-tP-
yE_{-}}\Phi_{\Delta,\lambda}(0,0)$ (A.20)

Where $\Phi_{\Delta,\lambda}$ is a primary of both the $sl(2)$ and the $su(2)$
and therefore satisfies:

$\displaystyle K\Phi_{\Delta,\lambda}(0,0)$
$\displaystyle=(D-\Delta)\Phi_{\Delta,\lambda}(0,0)=0$ (A.21) $\displaystyle
E_{+}\Phi_{\Delta,\lambda}(0,0)$
$\displaystyle=(H_{0}-\frac{\lambda}{2})\Phi_{\Delta,J}(0,0)=0$ (A.22)

The twisted conformal algebra can then be written as :

$\displaystyle\tilde{P}$ $\displaystyle=P+E_{-}$ (A.23)
$\displaystyle\tilde{K}$ $\displaystyle=K-E_{+}$ (A.24)
$\displaystyle\tilde{D}$ $\displaystyle=D-H_{0}$ (A.25)

Since the $su(2)$ and the $sl(2)$ commute, we can just add the operators to
form the differential operators of the twisted algebra and the twisted
quadratic Casimir

$\displaystyle\tilde{\mathcal{C}}_{a_{i}}^{(2)}$
$\displaystyle=\frac{1}{2}\\{\sum_{i}\tilde{D}_{i},\sum_{i}\tilde{D}_{i}\\}-\frac{1}{2}\\{\sum_{i}\tilde{P}_{i},\sum_{i}\tilde{K}_{i}\\}$
(A.26)

this gives the Casimir Eigenvalue

$\displaystyle\mathcal{C}^{(2)}_{a_{1}}$
$\displaystyle=\tilde{\Delta}(\tilde{\Delta}-1)$ (A.27)
$\displaystyle\tilde{\Delta}$ $\displaystyle=\Delta-\frac{\lambda}{2}$ (A.28)

and the higher-point Casimir acting on multipoint correlators of topological
operators give the corresponding Ward identities in the main text.

##  Appendix B Linearity of the Casimir operator

Near the topological limit, the bloc decomposition can be separated into 2
parts

$\displaystyle\sum_{\Delta}G_{\Delta}(\chi,\rho)\underset{\rho\rightarrow\chi}{=}\sum_{\text{topo}}G_{\tilde{\Delta}}(\chi,\chi)+\sum_{\text{not
topo}}(\rho-\chi)G_{\tilde{\Delta}}(\chi,\chi)+O((\rho-\chi)^{2})$ (B.1)

In the topological limit, the second term trivially disappears. When acting on
the OPE with a linear differential operator made from the twisted conformal
Casimirs

$\displaystyle\mathcal{L}=\sum_{a_{i}}\mathcal{C}^{(2)}_{a_{i}}$ (B.2)

we can write it as:

$\displaystyle\mathcal{L}\sum_{\Delta}G_{\Delta}(\chi,\rho)\underset{\rho\rightarrow\chi}{=}$
(B.3)
$\displaystyle\quad\sum_{\text{topo}}\mathcal{L}G_{\tilde{\Delta}}(\chi,\chi)+\sum_{\text{not
topo}}\mathcal{L}(\rho-\chi)G_{\tilde{\Delta}}(\chi,\chi)+\sum_{\text{not
topo}}(\rho-\chi)\mathcal{L}G_{\tilde{\Delta}}(\chi,\chi)+O((\rho-\chi)^{2})$

The first term has 0-eigenvalue under the quadratic Casimir (2.8), the
$(\rho-\chi)$ term is annihilated by the quadratic Casimir (by definition, the
quadratic Casimir commutes with the twisted translations.) so that in the
topological limit, all three terms vanish for a linear differential operator.
Thankfully, such an operator can be constructed as a linear combination
quadratic Casimirs. We therefore have the equation:

$\displaystyle\sum_{a_{i}}b_{a_{i}}\tilde{C}^{(2)}_{a_{i}}\langle
O_{1}....O_{n}\rangle\underset{\rho\rightarrow\chi}{=}0$ (B.4)

In particular, this means that the topological limit of the linearised
quadratic Casimir acting on a multipoint correlator of operators with a
topological limit vanishes. This gives the superconformal Ward identity
constraints on the $n-$point correlators of the superprimary in 2.3.

##  Appendix C Higher-points

For higher points, the expressions quickly become overwhelming. It is
convenient to define the distance:

$\displaystyle\chi_{i}(j,k)=\frac{1}{\chi_{i}-\chi_{j}}-\frac{1}{\chi_{i}-\chi_{k}}$
(C.1)

where we extend the definition of $\chi$ to

$\displaystyle\chi_{0}$ $\displaystyle=0$ $\displaystyle\chi_{n-2}$
$\displaystyle=1$ $\displaystyle\chi_{n-1}$ $\displaystyle=\infty$ (C.2)

so that we have, for example:

$\displaystyle\chi_{i}(0,n-1)=\frac{1}{\chi_{i}}$ (C.3)

In this language we have a list of the coefficients leading to the linear
combination of quadratic Casimirs resented in the main text

###  $n=6$ points

$\displaystyle b^{1}_{1,2}=-\chi_{1}(5,0)$ $\displaystyle
b^{1}_{2,3}=\chi_{1}(2,3)$ $\displaystyle b^{1}_{3,4}=-\chi_{1}(3,4)$
$\displaystyle b^{1}_{4,5}=0$ $\displaystyle b^{1}_{1,2,3}=0$ $\displaystyle
b^{1}_{2,3,4}=\chi_{1}(3,4)$ $\displaystyle b^{1}_{3,4,5}=\chi_{1}(4,5)$
$\displaystyle b^{1}_{1,2,3,4}=0$ $\displaystyle
b^{1}_{2,3,4,5}=-\chi_{1}(4,5)$ (C.4) $\displaystyle
b^{2}_{1,2}=\chi_{2}(5,0)$ $\displaystyle b^{2}_{2,3}=-\chi_{2}(0,1)$
$\displaystyle b^{2}_{3,4}=\chi_{2}(3,4)$ $\displaystyle
b^{2}_{4,5}=-\chi_{2}(4,5)$ $\displaystyle b^{1}_{1,2,3}=0$ $\displaystyle
b^{2}_{2,3,4}=0$ $\displaystyle b^{2}_{3,4,5}=\chi_{1}(4,5)$ $\displaystyle
b^{1}_{1,2,3,4}=0$ $\displaystyle b^{1}_{2,3,4,5}=0$ (C.5) $\displaystyle
b^{3}_{1,2}=0$ $\displaystyle b^{3}_{2,3}=\chi_{3}(0,1)$ $\displaystyle
b^{3}_{3,4}=\chi_{3}(1,2)$ $\displaystyle b^{3}_{4,5}=\chi_{3}(4,5)$
$\displaystyle b^{3}_{1,2,3}=-\chi_{3}(0,5)$ $\displaystyle
b^{3}_{2,3,4}=-\chi_{3}(0,1)$ $\displaystyle b^{3}_{3,4,5}=0$ $\displaystyle
b^{3}_{1,2,3,4}=\chi_{3}(0,5)$ $\displaystyle b^{3}_{2,3,4,5}=0$ (C.6)

###  $n=7$ points

$\displaystyle b^{1}_{1,2}=-\chi_{1}(6,0)$ $\displaystyle
b^{1}_{2,3}=\chi_{1}(2,3)$ $\displaystyle b^{1}_{3,4}=-\chi_{1}(3,4)$
$\displaystyle b^{1}_{4,5}=0$ $\displaystyle b^{1}_{5,6}=0$ $\displaystyle
b^{1}_{1,2,3}=0$ $\displaystyle b^{1}_{2,3,4}=\chi_{1}(3,4)$ $\displaystyle
b^{1}_{3,4,5}=-\chi_{1}(4,5)$ $\displaystyle b^{1}_{4,5,6}=0$ $\displaystyle
b^{1}_{1,2,3,4}=0$ $\displaystyle b^{1}_{2,3,4,5}=\chi_{1}(4,5)$
$\displaystyle b^{1}_{3,4,5,6}=-\chi_{1}(5,6)$ $\displaystyle
b^{1}_{1,2,3,4,5}=0$ $\displaystyle b^{1}_{2,3,4,5,6}=\chi_{1}(5,6)$ (C.7)
$\displaystyle b^{2}_{1,2}=\chi_{2}(6,0)$ $\displaystyle
b^{2}_{2,3}=\chi_{2}(0,1)$ $\displaystyle b^{2}_{3,4}=\chi_{1}(3,4)$
$\displaystyle b^{2}_{4,5}=\chi_{2}(4,5)$ $\displaystyle b^{1}_{5,6}=0$
$\displaystyle b^{2}_{1,2,3}=-\chi_{2}(6,0)$ $\displaystyle b^{2}_{2,3,4}=0$
$\displaystyle b^{2}_{3,4,5}=-\chi_{2}(4,5)$ $\displaystyle
b^{2}_{4,5,6}=-\chi_{2}(5,6)$ $\displaystyle b^{2}_{1,2,3,4}=0$ $\displaystyle
b^{2}_{2,3,4,5}=0$ $\displaystyle b^{2}_{3,4,5,6}=\chi_{2}(5,6)$
$\displaystyle b^{2}_{1,2,3,4,5}=0$ $\displaystyle b^{2}_{2,3,4,5,6}=0$ (C.8)
$\displaystyle b^{3}_{1,2}=0$ $\displaystyle b^{3}_{2,3}=\chi_{3}(0,1)$
$\displaystyle b^{3}_{3,4}=-\chi_{3}(1,2)$ $\displaystyle
b^{3}_{4,5}=\chi_{3}(4,5)$ $\displaystyle b^{3}_{5,6}=\chi_{3}(5,6)$
$\displaystyle b^{3}_{1,2,3}=\chi_{3}(6,0)$ $\displaystyle
b^{3}_{2,3,4}=\chi_{3}(0,1)$ $\displaystyle b^{3}_{3,4,5}=0$ $\displaystyle
b^{3}_{4,5,6}=-\chi_{3}(5,6)$ $\displaystyle b^{3}_{1,2,3,4}=-\chi_{3}(6,0)$
$\displaystyle b^{3}_{2,3,4,5}=0$ $\displaystyle b^{3}_{3,4,5,6}=0$
$\displaystyle b^{3}_{1,2,3,4,5}=0$ $\displaystyle b^{3}_{2,3,4,5,6}=0$ (C.9)
$\displaystyle b^{4}_{1,2}=0$ $\displaystyle b^{4}_{2,3}=0$ $\displaystyle
b^{4}_{3,4}=\chi_{4}(1,2)$ $\displaystyle b^{4}_{4,5}=-\chi_{4}(2,3)$
$\displaystyle b^{4}_{5,6}=\chi_{4}(5,6)$ $\displaystyle b^{4}_{1,2,3}=0$
$\displaystyle b^{4}_{2,3,4}=\chi_{4}(0,1)$ $\displaystyle
b^{4}_{3,4,5}=-\chi_{4}(1,2)$ $\displaystyle b^{4}_{4,5,6}=0$ $\displaystyle
b^{4}_{1,2,3,4}=\chi_{4}(6,0)$ $\displaystyle b^{4}_{2,3,4,5}=\chi_{4}(0,1)$
$\displaystyle b^{4}_{3,4,5,6}=0$ $\displaystyle
b^{4}_{1,2,3,4,5}=-\chi_{4}(6,0)$ $\displaystyle b^{4}_{2,3,4,5,6}=0$ (C.10)

###  $n\geq 8$ points

Until $n=12$, the same Ward identities are found. The relevant coefficients
still have a simple expression in terms of $\chi_{i}(j,k)$ but are not
particularly enlightening so have not been included in this note.

##  Appendix D $su(N)$

Following the classification of [55], the requirements are met by the
representations of type $A_{1}\bar{A}_{1}$ for the supergroup
$su(1,1|\frac{1}{2}\mathcal{N})$. In the case of extended supersymmetry, there
are some choices to be made in terms of the topological charges used.

  * •

$\mathcal{N}=2$

For $\mathcal{N}=2$, there are no R-symmetry indices and correspondingly not
enough supercharges to form a topological operator.

  * •

$\mathcal{N}=4$

For $\mathcal{N}=4$, we have the operators of type $A_{1}\bar{A}_{1}$
annihilated by

$\displaystyle Q_{1}$ $\displaystyle\bar{Q}^{2}$ (D.1)

with the unitarity bound

$\displaystyle\Delta=\frac{1}{2}R$ (D.2)

These are precisely the conditions specified above, so we have the Ward
identity on the $n$-point function

$\displaystyle\langle[0]^{(R)}_{\frac{R}{2}}(t_{1},y_{1})....[0]^{(R)}_{\frac{R}{2}}(t_{n},y_{n})\rangle$
$\displaystyle=A_{n,R,\frac{R}{2}}(t_{i},y_{i})f_{n,A_{1}\bar{A}_{1}}(\chi_{i},\rho_{i})$
(D.3)
$\displaystyle\left(\partial_{\chi_{j}}+\partial_{\rho_{j}}\right)f_{n,A_{1}\bar{A}_{1}}(\chi_{i},\rho_{i})$
$\displaystyle=0$ (D.4)

  * •

$\mathcal{N}=6$ Let us now consider the Ward identities constraining
correlators of operators charged under an $su(2)$ subgroup of the R-symmetry

$\displaystyle\Phi_{\Delta,(R_{1},0,0...)}$
$\displaystyle\bar{\Phi}_{\Delta,(0,...,0,R_{N-1})}$ (D.5)

are annihilated by the supercharges

$\displaystyle\\{Q_{1},\bar{Q}^{N}\\}$ (D.6)

We can form the topological charges:

$\displaystyle\mathbb{Q}=Q_{1}+S_{N}$
$\displaystyle\mathbb{Q}^{\dagger}=\bar{Q}^{N}+\bar{S}^{1}$ (D.7)

that generate the twisted conformal algebra

$\displaystyle\tilde{P}$ $\displaystyle=P+E_{-}$ $\displaystyle\tilde{K}$
$\displaystyle=P-E_{-}$ $\displaystyle\tilde{D}$ $\displaystyle=D-$ (D.8)

such that we recover the $su(2)$ Ward identities above for the field

$\displaystyle\Phi_{\Delta,(\lambda_{1},0,...)}(t,y)$
$\displaystyle=\text{e}^{-tP-yE_{-}}\Phi_{\Delta,(\lambda_{1},0,...)}(0,0)$
(D.9)

where:

$\displaystyle E_{-}$ $\displaystyle=R^{1}_{N}$ $\displaystyle E_{+}$
$\displaystyle=R^{N}_{1}$ $\displaystyle
H_{0}=\frac{1}{2}(R_{1}^{1}-R_{N}^{N})=\frac{1}{2}\sum_{i=1}^{N-1}\lambda_{i}$
(D.10)

Since the differential operators can only act on one point each, we can
separatly define

$\displaystyle\bar{\Phi}_{\Delta,(0,...,0,\lambda)}(t,\bar{y})$
$\displaystyle=\text{e}^{-tP-\bar{y}E_{-}}\Phi_{\Delta,(0,...0,,\lambda)}(0,0)$
(D.11)

So that the $u(1)$-invariant $su(2)$ distance is naturally defined:

$\displaystyle(1\bar{2})$ $\displaystyle=y_{1}-\bar{y}_{2}$ (D.12)

This gives the same Ward identities as before:

$\displaystyle\langle\Phi_{\Delta,(\Delta,0,...)}\bar{\Phi}_{\Delta,(0,...,0,\Delta)}\Phi_{\Delta,(\Delta,0,...)}\bar{\Phi}_{\Delta,(0,...,0,\Delta)}\rangle$
$\displaystyle=\left(\frac{(1\bar{2})(3\bar{4})}{t_{12}t_{34}}\right)^{\Delta}f(\chi_{i},\rho_{i})$
(D.13)
$\displaystyle\left(\partial_{\chi_{i}}+\partial_{\rho_{i}}\right)f(\chi_{i},\rho_{i})|_{\rho_{i}\rightarrow\chi_{i}}$
$\displaystyle=0$ (D.14)

where we use the interdependence of the R-symmetry variables to find
u(1)-invariant terms, for example for the 4-point function:

$\displaystyle\rho$
$\displaystyle=\frac{(1\bar{2})(3\bar{4})}{(13)(\bar{2}\bar{4})}$
$\displaystyle 1-\rho$
$\displaystyle=-\frac{(1\bar{4})(3\bar{2})}{(13)(\bar{2}\bar{4})}$ (D.15)

So that:

$\displaystyle\frac{\rho}{\rho-1}=\frac{(1\bar{2})(3\bar{4})}{(1\bar{4})(3\bar{2})}$
(D.16)

Equivalently, one can find an embedding complex vector $Y_{a},\bar{Y}^{a}$ as
in [7] such that

$\displaystyle Y_{i}\cdot\bar{Y}_{j}=y_{ij}.$ (D.17)

In this setting, it is possible to express $\\{E_{-},E_{+},H_{0}\\}$ as
differential operators acting only on $y$ such that the $su(2)$ results hold.

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Emulator. Shaded regions show $1\sigma$ error estimated with $204$ jackknife
resampling. Bottom panels show fractional differences of $\xi_{+/-}$
measurements and theory. The shaded region shows $1\sigma$ error. The vertical
dashed line shows ten times the ray-tracing resolution.

##  Appendix F Conditional luminosity function

We compare the conditional luminosity function in Cardinal, Buzzard v2.0, and
DES-Y3 data. The measurements follow the prescriptions detailed in
2020ApJ...897...15T. Figure 19 shows the result. We find that for the lower
two redshift bins, the central galaxy luminosity distributions are consistent
between Buzzard v2.0 and Cardinal. For the highest redshift bin, the central
galaxy luminosity in Cardinal is somewhat fainter than Buzzard v2.0. However,
the relative brightness of centrals and satellites in Cardinal is more
consistent with the DES-Y3 data than Buzzard v2.0. This is an important
feature for the performance of cluster finders, as indicated in
2022OJAp....5E...1K and 4x2pt1. In terms of the satellites, we find that the
satellite luminosity distributions in Cardinal are more consistent with the
DES-Y3 data than Buzzard v2.0 on the faint end. On the bright end, Cardinal
and Buzzard’s satellite luminosity distributions are consistent but are
fainter than the DES-Y3 data.

Figure 19: The conditional luminosity function for central galaxies (solid
lines) and satellite galaxies (dashed lines) for redMaPPer clusters in Buzzard
v2.0 and Cardinal (blue and orange lines, respectively) and the DES-Y3 data
(black lines). Different panels show the different richness and redshift bins,
as indicated in the legend in each panel. Richness increases from top to
bottom, and redshift increases from left to right. Error bars and shaded
regions show the $1\sigma$ errors estimated from $50$ jackknife resampling.

##  Appendix G Algorithm of avoiding widening red sequence with conditional
abundance matching

Here, we present an algorithm that uses redMaPPer to avoid widening the width
of the red sequence in the conditional abundance matching step. First, we run
the redMaPPer cluster finder in the mocks generated before applying the
conditional abundance matching. For each galaxy in the mock, we obtain a
$\chi^{2}$, quantifying the consistency of the multidimensional colors with
the empirically constructed red-sequence model, and $z_{\rm{red}}$, the
redshift that minimizes the $\chi^{2}$ (See details in Redmapper1). For the
galaxies consistent with the red sequence (i.e., small $\chi^{2}$),
$z_{\rm{red}}$ provides a good estimator of galaxies’ redshift, with a typical
uncertainty $\sim 0.02$. Second, we abundance-match this $\chi^{2}$ measured
in mocks to $\chi^{2}$ measured in the data by enforcing the equality of
$p(<\chi^{2}|m_{z})$. In this way, we can avoid the possibility that
$\chi^{2}$ distributions in the mocks and data might differ. Third, we select
red galaxies in the data and mocks using the abundance-matched $\chi^{2}$.
According to redmapper4, we select red galaxies as $\chi^{2}<20$. Third, we
perform abundance matching separately for blue and red galaxies. For blue
galaxies with $\chi^{2}>20$, we enforce the equality of equation 25 between
mocks and data. For red galaxies with $\chi^{2}<20$, we use the $z_{\rm{red}}$
information. To avoid the possibility that $z_{\rm{red}}$ distributions in
mocks and data differ, we first compute $z_{\rm{a,red}}$, the abundance
matched $z_{\rm{red}}$ in Cardinal by matching $p(<z_{\rm{red}}|m_{z})$
between Cardinal and data. We then match the overall color distributions of
mocks and data by enforcing the equality of the following equation,

$P(<c_{i}|m_{z},c_{j<i},z_{\rm{a,red}}).$ (G1)

##  Appendix H Algorithm of matching red sequence in mocks and data

In this appendix, we describe an algorithm that matches red sequences in mocks
and the data. We first run the redMaPPer algorithm on mocks and data to obtain
a red-sequence model. This model has the following functional form,

$\displaystyle P(\bm{c}|z,m_{z})\propto$ $\displaystyle\exp(-0.5\chi^{2}),$
(H1) $\displaystyle\chi^{2}=$
$\displaystyle\left(\bm{c}-\langle\bm{c}|z,m_{z}\rangle\right)\left(\bm{C}_{\rm{int}}(z)+\bm{C}_{\rm{err}}(z)\right)^{-1}$
$\displaystyle\left(\bm{c}-\langle\bm{c}|z,m_{z}\rangle\right)^{T},$

where $\bm{c}=[g-r,r-i,i-z]$, $\bm{C}_{\rm{int}}(z)$ is the intrinsic scatter
of red sequence, and $\bm{C}_{\rm{err}}(z)$ is the observational noise. The
mean color $\langle\bm{c}|z,m_{z}\rangle$ is modeled as a simple power law
$c(z)+s(z)(m_{z}-m_{p}(z))$, where $c(z)$, $s(z)$, and $m_{p}(z)$ are free
parameters. With $P(\bm{c}|z,m_{z})$ models for mocks and data, we can enforce
the consistency of red sequence using the following algorithm,

for galaxies with redshift $z$ and observed magnitudes $m_{g,r,i,z}$ do

$m^{o}_{g,r,i,z}\leftarrow$ noisy realizations of $m_{g,r,i,z}$ using equation
20.

Calculate $\chi^{2}$ using $m^{o}_{g,r,i,z}$ and $P_{m}(\bm{c}|z,m_{z})$

if $\chi^{2}<100$ and $R_{n}<R(z)$ and $M_{n}>M(z)$ then

for $c\in\\{g-r,r-i,i-z\\}$ do

$c+=\frac{F(1-\frac{\chi^{2}}{100})}{G(z)}$
$\left(\left(c-\langle\bm{c}|z,m_{z}\rangle_{m}\right)\left(\frac{s_{d}}{s_{m}}\right)^{2}+\langle\bm{c}|z,m_{z}\rangle_{d}-c\right)$

end for

$m_{g}\leftarrow c_{g-r}+c_{r-i}+c_{i-z}+m_{z}$

$m_{r}\leftarrow c_{r-i}+c_{i-z}+m_{z}$

$m_{i}\leftarrow c_{i-z}+m_{z}$

end if

end for

In the above algorithm, $x_{d}$ represents the model evaluated using DES-Y3
data, $x_{m}$ represents the model evaluated using mocks, $R_{n}$ is the
distance to the nearest resolved halo whose mass is $M_{n}$, and $s$ is the
square root of the diagonal terms of $\bm{C}_{\rm{int}}(z)$. The
$(1-\frac{\chi^{2}}{100})$ term is to ensure a smooth transition between red
galaxies and blue galaxies. F controls the non-linear dependence of the color
shift and $(1-\frac{\chi^{2}}{100})$, which is given by
$F(x)=0.5+0.5{\rm{erf}}((x-0.1)/0.05)$, where ${\rm{erf}}$ is the error
function. $(s_{d}/s_{m})^{2}$ term ensures consistency of the scatter in the
observed colors, which are proportional to the square root of the scatter of
$c$ due to the Poisson draws. The $G(z)$ removes additional noise in the red
sequence introduced by the CAM method. This additional noise becomes large
when the galaxies’ magnitude approaches the survey depth limits. Therefore,
G(z) must be larger at higher redshifts when galaxies are fainter than their
low redshift counterparts. We empirically find that
$G(z)=2(0.5+0.5{\rm{erf}}((z-0.8)/0.3)+1.2$ produces reasonable galaxy
properties. Further, shifting all galaxies with $\chi^{2}<100$ will produce
too many red galaxies. We, therefore, control the number of red galaxies and
their clustering with two additional cuts $M_{n}>M(z)$ and $R_{n}<R(z)$.
Again, because the additional noise introduced by the CAM model becomes larger
at higher redshifts, $M(z)$ should decrease, and $R(z)$ should increase with
redshifts. If $M(z)$ becomes too small, the clustering of red-sequence
galaxies will become too small. We increase R(z) to ensure enough red galaxies
and strong enough clustering. Empirically, we find

$\displaystyle R(z)$
$\displaystyle=6\left(0.5+0.5{\rm{erf}}\left(\left(z-0.54\right)/0.1\right)\right)+1.2\,h^{-1}{\rm{Mpc}}$
(H2) $\displaystyle M(z)$
$\displaystyle=10^{-0.25\left(0.5+0.5{\rm{erf}}\left((z-0.65)/0.05\right)\right)+12.90}\,h^{-1}M_{\odot}$

give reasonable red galaxy properties.

##  Appendix I Galaxy radial profiles around clusters

This appendix details our analyses on constructing galaxy radial profiles
around redMaPPer clusters. We construct radial profiles for redMaPPer clusters
in simulation and data using exactly the same algorithm to ensure an apples-
to-apples comparison. Our calculation pipeline is similar to the one presented
in 2018ApJ...864...83C but has important differences in details. We emphasize
that it is necessary to recalculate the galaxy profile around redMaPPer
clusters for data because redMaPPer has been sufficiently changed.

We wish to measure galaxy density profiles around redMaPPer clusters using
photometric data. This density profile ($\Sigma_{g}$) can be related to
correlation functions via the following equation,

$\Sigma_{g}(R)=\langle\Sigma_{g}\rangle w(R),$ (I1)

where $R$ is the comoving distance to galaxy clusters,
$\langle\Sigma_{g}\rangle$ is the mean density of the galaxy samples, and
$w(R)$ is the correlation functions between galaxies and clusters. Without
knowing galaxy redshifts, we cannot evaluate equation I1 directly. For a
photometric dataset like DES, we must construct an estimator that gives
equation I1 but does not require a knowledge of galaxy redshifts. For the
simplicity of the argument, let’s first assume galaxy cluster samples all have
the same redshifts $z_{c}$. We can then define our estimator of
$\Sigma_{g}(R)$ as

$\hat{\Sigma}_{g}(R)=\left(\frac{DD_{CT}(\theta)}{DR_{CR}(\theta)}\frac{N_{R}}{N_{T}}-1\right)\frac{N_{T}}{A},$
(I2)

where $DD_{CT}$ denotes the number of pairs between clusters and galaxy
samples in photometric data, $DD_{CR}$ denotes the number of pairs between
clusters and random points describing survey footprints, $N_{T}$ is the total
number of considered galaxies, $N_{R}$ is the total number of randoms, and $A$
is the survey area in the comoving unit at redshift $z_{c}$. In the above
equation, $\theta$ is the angular separation of the pairs corresponding to a
comoving distance $R$ at redshift $z_{c}$.

We now show that $\hat{\Sigma}_{g}(R)$ is an unbiased estimator of
$\Sigma_{g}(R)$. The full galaxy sample ($T$) can be divided into two groups:
(1) galaxies physically associated with galaxy clusters ($g$) and (2) other
galaxies ($f$). Because pair counting is an additive process, $DD_{CT}$ is
then a sum of $DD_{Cg}$ (pairs between clusters and associated galaxies) and
$DD_{Cf}$ (pairs between clusters and other galaxies). Further, clusters and
unassociated galaxies are not clustered, so $DD_{Cf}$ is then
$N_{c}N_{f}\zeta(\theta)$, where $N_{f}$ is the number of unassociated
galaxies and $\zeta$ is some geometric factor that depends on angular
separation. By the same argument $DD_{CR}$ is $N_{c}N_{R}\zeta(\theta)$. With
these, we have

$\displaystyle\hat{\Sigma}_{g}(R)$ $\displaystyle=$
$\displaystyle\left(\frac{DD_{CT}(\theta)}{DR_{CR}(\theta)}\frac{N_{R}}{N_{T}}-1\right)\frac{N_{T}}{A}$
(I3) $\displaystyle=$
$\displaystyle\left(\frac{DD_{Cg}(\theta)+DD_{Cf}(\theta)}{DR_{CR}(\theta)}\frac{N_{R}}{N_{g}+N_{f}}-1\right)\frac{N_{g}+N_{f}}{A}$
$\displaystyle=$
$\displaystyle\left(\frac{DD_{Cg}(\theta)+N_{c}N_{f}\zeta(\theta)}{DR_{CR}(\theta)}\frac{N_{R}}{N_{g}+N_{f}}-1\right)\frac{N_{g}+N_{f}}{A}$
$\displaystyle=$
$\displaystyle\left(\frac{DD_{Cg}(\theta)}{DR_{CR}(\theta)}\frac{N_{R}}{N_{g}+N_{f}}+\frac{N_{c}N_{f}\zeta(\theta)}{N_{c}N_{R}\zeta(\theta)}\frac{N_{R}}{N_{g}+N_{f}}-1\right)\frac{N_{g}+N_{f}}{A}$
$\displaystyle=$
$\displaystyle\left(\frac{DD_{Cg}(\theta)}{DR_{CR}(\theta)}\frac{N_{R}}{N_{g}}-1\right)\frac{N_{g}}{A}$
$\displaystyle=$
$\displaystyle\left(\frac{DD_{Cg}(\theta)}{DR_{CR}(\theta)}\frac{N_{R}}{N_{g}}-1\right)\langle\Sigma_{g}\rangle,$

In the above equation, we see the terms in the parenthesis are the
Davis–Peebles estimator (1983ApJ...267..465D) of $w(R)$. We note that the
above derivation is only exact when considering Davis–Peebles estimator. Using
Landy–Szalay estimator (1993ApJ...412...64L) to estimate $\Sigma_{g}(R)$ like
the one described in 2018ApJ...864...83C would include additional terms and
potentially lead to biases.

In survey data, not all clusters live at the same redshifts. We, therefore,
follow the prescription described in 2018ApJ...864...83C for the full survey
data. We first divide the redMaPPer clusters with $\lambda>20$ from
$z=0.2-0.55$ into $\Delta z=0.025$ bins. We estimate galaxy density profiles
around clusters for each redshift bin ( $\hat{\Sigma}_{g}(R)_{i}$) using
equation I3. We then estimate the $\Sigma_{g}(R)$ using the weighted sum of
$\hat{\Sigma}_{g}(R)_{i}$, with the number of clusters in each redshift bin as
the weight. To avoid mixing galaxies with different luminosities across the
entire redshift range, we further make a cut on galaxy samples at each
redshift bin $M_{\rm{r}}-5\log(h)<-20$. The $M_{\rm{r}}$ is calculated
assuming galaxies are at the same redshift as galaxy clusters.

In Cardinal, we repeat all the above calculations.

Figure 20: Color distributions of mock samples with the blue histogram
representing the true color distribution. The red histogram corresponds to the
samples when colors are shifted by $0.1$ magnitude. The black histogram shows
the case when we summarize the color using SED templates. To generate the
black histogram, we apply the biases shown in the right panel of figure 8 to
the galaxies in the red histogram.

##  Appendix J Color template bias simulation

This section employs a simple simulation to offer valuable insights into the
potential impact of the color biases presented in figure 8 on galaxy color
distribution. We accomplish this by generating mock galaxy samples through a
combination of two Gaussian distributions. Specifically, we draw 100k blue
galaxies with a mean $g-r$ color of $1$ and a standard deviation of $0.3$,
followed by drawing 5k red galaxies with a mean $g-r$ of $1.65$ and a standard
deviation of $0.05$. The resulting galaxy color distribution is presented as
the blue histogram in figure 20, designed to mirror the cosmos samples’ color
distribution at $z=0.43-0.63$, as depicted in figure 5 of JoeBuzzard. Next, we
shift the color up by $0.1$ magnitude, a typical value when comparing the
$g-r$ color distribution between Buzzard and DES-Y3 data. The resulting color
distribution is illustrated as the red histogram in figure 20. Subsequently,
we incorporate the color biases displayed in the right panel of figure 8,
resulting in a galaxy sample with the color distribution depicted as the black
histogram in figure 20. Strikingly, the black histogram reproduces two key
features in figure 5 of JoeBuzzard. First, the red-sequence galaxies
corresponding to the black histogram have a similar mean color to those in the
unaltered galaxy samples. Second, the color distribution of red-sequence
galaxies is significantly more peaked than those of unaltered samples. This
simple yet compelling simulation further supports our hypothesis regarding the
color discrepancies’ origin between Buzzard and data.

##  Appendix K Differences between Cardinal and Buzzard v2.0

This section summarizes the main improvements from Buzzard v2.0 to Cardinal.
We divide these improvements into four categories, summarized below.

  1. 1.

Subhalo abundance matching:

    1. (a)

We consider a luminosity-dependent scatter.

    2. (b)

Orphan subhalos are included, whose abundances are controlled by a specially-
designed orphan model.

    3. (c)

The group–galaxy cross-correlation function is used in addition to the galaxy-
galaxy correlation function to constrain the model.

  2. 2.

Environmentally dependent galaxy color model:

    1. (a)

We consider a luminosity-dependent correlation between galaxy color and
distances to nearby massive halos at fixed galaxy luminosity.

    2. (b)

When using distances to nearby massive halos as a local environment proxy, we
normalize them by the halo size.

    3. (c)

We apply a nonlinear transform on distances to nearby massive halos to break
the degeneracy of large-scale color-dependent clustering and galaxy color
gradient around massive halos.

    4. (d)

We introduce an additional parameter that controls the color of low-mass
centrals.

  3. 3.

Observational effects:

    1. (a)

We increase the fidelity of faint galaxy photometric uncertainties by swapping
the Gaussian distribution used to generate noise of galaxy fluxes to a Poisson
distribution.

    2. (b)

We empirically correct artifacts in the raytracing shears due to limited
resolution.

  4. 4.

Afterburner:

    1. (a)

We demonstrate the limitations of kcorrect SED templates in describing red
galaxy colors.

    2. (b)

To mitigate this problem, we develop a new algorithm that employs the
conditional abundance matching technique and a photometric dataset to correct
galaxy colors empirically.

    3. (c)

We develop a new algorithm for matching the red sequence between mocks and
data that takes into account the slope of the color–magnitude relations.