# Higher-Dimensional Symmetry of AdS${}_{2}\times$S2 Correlators

Theresa Abl, Paul Heslop and Arthur E. Lipstein
Department of Mathematical Sciences
Durham University, Durham, DH1 3LE, United Kingdom

###### Abstract

It was recently shown that IIB supergravity on AdS${}_{5}\times$S5 enjoys 10d
conformal symmetry and that superstring theory on this background can be
described using a 10d scalar effective field theory. In this paper we adapt
these two complementary approaches to correlators of hypermultiplets in
AdS${}_{2}\times$S2. In particular, we show that 4-point correlators of
$1/2$-BPS operators in the 1d boundary can be computed using 4d conformal
symmetry and a 4d effective action in the bulk. The 4d conformal symmetry is
realised by acting with Casimirs of $SU(1,1|2)$, and is generically broken by
higher derivative corrections. We point out similar structure underlying
$\alpha^{\prime}$ corrections to IIB supergravity in AdS${}_{5}\times$S5. In
particular, while the $\alpha^{\prime 3}$ corrections can be written in terms
of a sixth order Casimir acting on a 10d conformal block, similar structure
does not appear in higher-order corrections. We note however that a specific
combination of higher derivative corrections can give rise to Witten diagrams
with higher dimensional symmetry at the integrand level, with breaking then
arising from the measure.

###### Contents

  1. 1 Introduction
  2. 2 Overview
    1. 2.1 1d Correlators
    2. 2.2 Conformal Blocks
    3. 2.3 4d Effective Action
    4. 2.4 AdS$\times$S Witten Diagrams
  3. 3 Free theory
    1. 3.1 4d Conformal Symmetry
  4. 4 2-derivative sector
    1. 4.1 Lowest-Charge Correlator
    2. 4.2 4d Conformal Symmetry
    3. 4.3 Effective Action
    4. 4.4 Physical Interpretation
  5. 5 Higher-derivative corrections
    1. 5.1 Lowest-Charge Correlators
    2. 5.2 Effective Action
    3. 5.3 4d Conformal Symmetry
  6. 6 Anomalous Dimensions from Unmixing
    1. 6.1 Degenerate Operators
    2. 6.2 Unmixing
    3. 6.3 Anomalous Dimensions
  7. 7 Comments on $\mathcal{N}=4$ SYM
    1. 7.1  10d conformal symmetry at $O(\alpha^{\prime 3})$
    2. 7.2  10d conformal symmetry at $O(\alpha^{\prime 5})$
  8. 8 Conclusion
  9. A Casimirs
    1. A.1 The quadratic Casimir in $SU(1,1|2)$
    2. A.2 Higher order Casimirs in $SU(m,m|2n)$ CFTs
  10. B Correlators of descendants
  11. C Further Unmixing Examples
    1. C.1 2-derivative sector
    2. C.2 6-derivative sector

##  1 Introduction

Understanding quantum field theory and quantum gravity in curved backgrounds
such as black holes and cosmological spacetimes is one of the most pressing
questions in theoretical physics. The AdS/CFT correspondence is a very
powerful tool for this purpose since it relates observables in curved
backgrounds to correlation functions of a CFT in the boundary [1]. Moreover,
AdS is a useful toy model since AdS4 is related by Wick rotation to dS4 which
is relevant for cosmology [2], and AdS${}_{2}\times$S2 describes the near-
horizon geometry of extremal black holes in four dimensions [3, 4]. While a
great deal of technology has been developed to compute CFT correlators, we
would also like to be able to compute things directly in the bulk in order to
have a complete grasp of the physics. Carrying out calculations directly in
the bulk is very challenging however because of the complexity of Witten
diagrams and worldsheet calculations. Nevertheless, much progress has been
made for IIB string theory in AdS${}_{5}\times$ S5. For example, it was
discovered that supergravity in this background enjoys a 10d conformal
symmetry, which is realised using 4-point correlators of protected operators
in the boundary CFT, $\mathcal{N}=4$ SYM [5]111All 4-point correlators of
protected operators in $\mathcal{N}=4$ SYM in the supergravity approximation
were first computed in [6].. This property is inherited from scale-invariance
of the corresponding flat space scattering amplitude after a conformal
transformation to AdS${}_{5}\times$ S5 and has very non-trivial consequences
for $\mathcal{N}=4$ SYM correlators. Similar results were also found in
AdS${}_{3}\times$ S3 [7, 8, 9, 10] and AdS${}_{5}\times$ S3 [11]. Subsequent
work showed that 10d conformal symmetry is broken by string corrections [12].
In another development, a simple 10d scalar effective action was conjectured
to describe all 4-point correlators of $1/2$-BPS operators in $\mathcal{N}=4$
SYM dual to tree-level string theory in AdS${}_{5}\times$ S5 to all orders in
$\alpha^{\prime}$ [13]. This action successfully reproduced previous results
obtained using different methods [14, 15, 16, 12, 17] and makes an infinite
number of new predictions.

In this paper we will develop these methods in the context of
AdS${}_{2}\times$S2. One motivation for doing so is to expand the toolbox for
studying black holes in the real world. Another is that the correlators in
this background have very simple mathematical structure, which can provide a
toy model for better understanding 10d conformal symmetry in
AdS${}_{5}\times$S5 and its breaking. On the other hand, much less is known
about the AdS2/CFT1 correspondence due to various subtleties such as the
absence of a stress tensor in a conventional 1d CFT [18] and a large
backreaction when matter is introduced in the bulk [19, 20, 21]. We will
therefore consider a non-gravitational theory in the bulk. In particular,
reducing $\mathcal{N}=8$ supergravity on AdS${}_{2}\times$S2 gives
$\mathcal{N}=2$ graviton multiplets and hypermultiplets [22, 23] and we will
restrict our attention to 4-point tree-level correlators of the latter. We
also study correlators of higher-derivative corrections to the interactions of
hypermultiplets in AdS${}_{2}\times$S2 analogous to $\alpha^{\prime}$
corrections in AdS${}_{5}\times$S5. From the point of view of the effective
action, the two-derivative interactions that arise from reducing supergravity
on AdS${}_{2}\times$S2 correspond to a $\phi^{4}$ interaction and we refer to
them as the 2-derivative sector. Higher-derivative interactions are then
described by applying derivatives to $\phi^{4}$. In addition there is a
generalised free theory described by non-interacting Witten diagrams in the
bulk.

We can formally define correlators on the boundary which have $SU(1,1|2)$
superconformal and crossing symmetry. Using these symmetries, we can then
deduce the structure of the lowest-charge correlators in the free,
2-derivative, and higher-derivative sectors. We then use 4d conformal symmetry
to obtain all higher-charge correlators in the free and 2-derivative sectors,
which correspond to Kaluza-Klein modes on the S2, and match them with the
predictions of the effective action. A key ingredient in the realisation of 4d
conformal symmetry is the action of Casimirs of the superconformal group. On
the other hand, we find that only a special infinite class of correlators
derived from the effective action with higher derivative corrections can be
reproduced by acting with Casimirs on 4d conformal blocks, suggesting that 4d
conformal symmetry is generically broken by higher derivative corrections.
This is further confirmed by the structure of anomalous dimensions of double
trace operators which we compute using a procedure for unmixing operators
which are degenerate at leading order in the central charge expansion
reminiscent of [24]. Whereas the anomalous dimensions are rational numbers in
the 2-derivative sector and correspond to eigenvalues of a Casimir operator,
they are generically irrational in higher derivative sectors.

We find similar structure in 4-point correlators corresponding to
$\alpha^{\prime}$ corrections to supergravity in AdS${}_{5}\times$S5. In
particular, $\alpha^{\prime 3}$ corrections are analogous to the 2-derivative
sector in AdS${}_{2}\times$S2 in that they are both described by a $\phi^{4}$
effective field theory and enjoy higher-d conformal symmetry. Similarly,
$\alpha^{\prime 5}$ corrections are analogous to 6-derivative corrections in
AdS${}_{2}\times$S2 and are shown to generically break 10d conformal symmetry,
in the sense that the predictions of the 10d effective action cannot be
matched by acting with Casimirs on 10d conformal blocks. We do find a
6-derivative effective action for which the integrands of the Witten diagrams
exhibit 10d conformal symmetry, but the resulting correlators are in
disagreement with ${\mathcal{N}}=4$ SYM, specifically the coefficients fixed
by localisation in [25]. This provides a new point of view on the 10d
conformal symmetry breaking.

This paper is organised as follows. In section 2 we review the formalism for
1d CFT correlators and the AdS${}_{2}\times$S2 effective action and in section
3 we compute free theory correlators and demonstrate their 4d conformal
symmetry using Casimirs. In section 4, we compute correlators in the
2-derivative sector and show agreement between the predictions of 4d conformal
symmetry and the effective action. We also provide an intuitive explanation
for the origin of this symmetry. Next, in section 5 we compute correlators in
higher derivative sectors and show that the 4d conformal symmetry is
generically incompatible with the predictions of the effective action, except
for a special infinite class of correlators. In section 6 we use an unmixing
procedure to compute the anomalous dimensions of double trace operators in the
2- and 6-derivative sectors, finding further evidence for the breaking of 4d
conformal symmetry by higher-derivative corrections. In section 7, we apply
the lessons we have learned in AdS${}_{2}\times$S2 to AdS${}_{5}\times$S5,
realising 10d conformal symmetry for $\alpha^{\prime 3}$ corrections using
Casimirs and providing a new perspective on the breaking of this symmetry by
$\alpha^{\prime 5}$ corrections. Finally, in section 8 we present our
conclusions. We also have a number of Appendices providing more details about
superconformal Casimirs, correlators of descendent operators, and further
examples of unmixing.

##  2 Overview

In this section, we will describe some aspects of superconformal correlators
and $SU(1,1|2)$ blocks in 1d. After that, we will review the effective action
in AdS$\times$S and how to compute generalised Witten diagrams from this
action.

###  2.1 1d Correlators

Note that it is not possible to define a stress tensor in 1d since any
conserved current would simply be a constant. As a result, we only consider
non-gravitational theories in the bulk whose correlators have superconformal
symmetry arising from the isometries of the background, and can be formally
thought of as correlators of a boundary CFT. It is also possible to recast the
expansion of bulk correlators in terms of Newton’s constant $G_{N}$ as an
expansion of boundary correlators in terms of a formally defined central
charge $c$. More precisely, we will associate $G_{N}\sim l_{P}^{2}$ with
$1/c$, where $l_{P}$ is the Planck length. Free theory correlators are then
$\mathcal{O}(c^{0})$, while correlators in the 2-derivative sector correspond
to $\mathcal{O}(1/c)$. In addition to the lack of a stress tensor in the
boundary, another subtlety is that AdS2 has two disconnected timelike
boundaries. In this paper, we only consider correlators on a single boundary,
but in principle one could consider correlators between the two boundaries
[26]. While the former have singularities when points in the boundary collide,
such singularities are absent in the latter.

In 1d, 4-point conformal correlators can be expressed in terms of conformal
cross ratios which take the following form:

$z\bar{z}=u=\left(\frac{x_{12}x_{34}}{x_{13}x_{24}}\right)^{2},\,\,\,(1-z)(1-\bar{z})=v=\left(\frac{x_{23}x_{14}}{x_{13}x_{24}}\right)^{2}$
(1)

where $x_{ij}=x_{i}-x_{j}$. Solving for $z,\bar{z}$ in terms of $u,v$ shows
that

$z=\bar{z}=\frac{1}{2}(1+u-v)$ (2)

which follows from the fact that the discriminant $(1+u-v)^{2}-4u$ vanishes.
Hence, conformal cross ratios in 1d correspond to a holomorphic limit of the
standard cross ratios in higher dimensions. It is convenient to define
$z=\bar{z}=x$ in terms of which cross ratios are given by

$u=x^{2},\qquad v=(1-x)^{2}.$ (3)

To describe superconformal correlators, we follow the formalism of [27] (with
$m=n=1$ in the notation of that paper) arising from analytic superspace [28,
29] and consider the super Grassmannian $Gr(1|1,2|2)$ of $(1|1)$ planes in
$(2|2)$ dimensions. Coordinates on this Grassmannian can be given as

$\displaystyle X_{i}=\left(\begin{array}[]{cc}x_{i}&\theta_{i}\\\
\bar{\theta}_{i}&y_{i}\end{array}\right)\ $ (6)

where $x_{i}$ is the (1d) space-time coordinate, $y_{i}$ is a (complex)
internal coordinate used to deal with the $SU(2)$ structure and
$\theta_{i},\bar{\theta}_{i}$ are Grassmann odd coordinates. Since we will be
dealing with correlators of four operators on this space, we added a subscript
$i=1,2,3,4$ to denote the particle number.

The 4d $\mathcal{N}=2$ hypermultiplet is the simplest to understand from a
higher dimensional perspective. This multiplet on an AdS${}_{2}\times$S2
background is dual to an infinite tower of fermionic 1d $1/2$-BPS multiplets
[22, 23]. These are fermionic superfields of scaling dimension and $SU(2)$ rep
$\Delta$. They can be written on the above super Grassmannian and decompose
into the following component fields

$\displaystyle{\Psi}_{\Delta}(x_{i},\theta_{i},\bar{\theta}_{i},y_{i})=\psi_{\Delta}+\theta_{i}\phi_{\Delta+\frac{1}{2}}+\bar{\theta}_{i}\bar{\phi}_{\Delta+\frac{1}{2}}+\theta_{i}\bar{\theta}_{i}\lambda_{\Delta+1}\qquad\Delta=\tfrac{1}{2},\tfrac{3}{2},\tfrac{5}{2},\dots$
(7)

where in each case the subscript denotes the dimension of the field. The field
$\psi_{\Delta}$ has $SU(2)$ rep $\Delta$, $\phi_{\Delta}$ has $SU(2)$ rep
$\Delta-1$, $\lambda_{\Delta}$ has $SU(2)$ rep $\Delta-2$. For the special
multiplet $\Psi_{1/2}$ the descendant $\lambda_{3/2}$ is absent (it would have
negative $SU(2)$ weight). Expanding in the $y_{i}$ coordinates manifests the
$SU(2)$ indices for these reps

$\displaystyle\psi_{\Delta}(x_{i},y_{i})$ $\displaystyle=\psi_{I_{1}\dots
I_{2\Delta}}(x_{i})y_{i}^{I_{1}}\dots y_{i}^{I_{2\Delta}}\qquad$
$\displaystyle\Delta$ $\displaystyle=\tfrac{1}{2},\,\tfrac{3}{2},\,\dots$
$\displaystyle\phi_{\Delta}(x_{i},y_{i})$ $\displaystyle=\phi_{I_{1}\dots
I_{2\Delta-2}}(x_{i})y_{i}^{I_{1}}\dots y_{i}^{I_{2\Delta-2}}\qquad$
$\displaystyle\Delta$ $\displaystyle=1,2,\dots$
$\displaystyle\lambda_{\Delta}(x_{i},y_{i})$
$\displaystyle=\lambda_{I_{1}\dots I_{2\Delta-4}}(x_{i})y_{i}^{I_{1}}\dots
y_{i}^{I_{2\Delta-4}}\qquad$ $\displaystyle\Delta$
$\displaystyle=\\{\\},\tfrac{5}{2},...$

where $y_{i}^{I}=(1,y_{i})$.

We will be interested in the 4-point functions of four $1/2$-BPS
superconformal primaries:

$\langle\psi_{\Delta_{1}}(x_{1},y_{1})\psi_{\Delta_{2}}(x_{2},y_{2})\psi_{\Delta_{3}}(x_{3},y_{3})\psi_{\Delta_{4}}(x_{4},y_{4})\rangle=\hat{P}_{\Delta_{i}}G_{\psi_{\Delta_{i}}}(x,y),$
(9)

where

$\displaystyle\hat{P}_{\Delta_{i}}$
$\displaystyle=g_{12}^{\Delta_{1}+\Delta_{2}}g_{34}^{{\Delta_{3}+\Delta_{4}}}\left(\frac{g_{24}}{g_{14}}\right)^{{\Delta_{21}}}\left(\frac{g_{14}}{g_{13}}\right)^{{\Delta_{43}}}$
(10) $\displaystyle g_{ij}$ $\displaystyle=\frac{y_{ij}}{x_{ij}}\ .$ (11)

It will also be useful shortly to define the 2-point function of $\psi_{1/2}$
operators

$\tilde{g}_{ij}=\frac{y_{ij}}{\left|x_{ij}\right|},$ (12)

which is antisymmetric under $i\leftrightarrow j$, reflecting the fermionic
statistics. The prefactor in (10) by itself transforms correctly as a
$\langle\psi_{\Delta_{1}}\psi_{\Delta_{2}}\psi_{\Delta_{3}}\psi_{\Delta_{4}}\rangle$
correlator under the bosonic subgroup, leaving a remaining function
$G_{\psi}(x,y)$ which is invariant, and thus a function of $x,y$, the space-
time and internal cross-ratios:

$\displaystyle x=\frac{x_{12}x_{34}}{x_{13}x_{24}}\qquad
y=\frac{y_{12}y_{34}}{y_{13}y_{24}}\ .$ (13)

Throughout this paper we will consider 4-point functions of $1/2$-BPS
operators and in some cases also the correlators of descendants. The $1/2$-BPS
operators are fermionic primaries $\psi_{\Delta}$ with dimensions and $SU(2)$
charges $\Delta=\tfrac{1}{2},\tfrac{3}{2},\dots$ However, it will turn out to
be most useful to label the correlators in terms of the bosonic descendant
operators $\phi_{\Delta+1/2}$ with dimensions $1,2,\dots$ Therefore, we will
define the $1/2$-BPS operators as

${\mathcal{O}}_{p}:=\left(-1\right)^{\frac{p}{2}}\frac{\psi_{p-\frac{1}{2}}}{\sqrt{2p-1}},$
(14)

where $p=1,2,\dots$ In this convention the half-BPS operators have dimension
and $SU(2)$ representation $p-\frac{1}{2}$. Furthermore, it is useful to
introduce the normalisation $\left(-1\right)^{p/2}(2p-1)^{-1/2}$ inspired by
the higher-dimensional conformal symmetry [5]. It will become clear why this
is useful in the discussions below (54) and (60). We then also label the
descendants similarly, thus defining

$L_{p}:=\left(-1\right)^{\frac{p}{2}}\frac{\phi_{p}}{\sqrt{2p-1}},$ (15)

so the superfield (7) becomes

$\displaystyle{\bf{O}}_{p}(x,y)={\mathcal{O}}_{p}(x,y)+\theta
L_{p}(x,y)+\bar{\theta}\bar{L}_{p}(x,y)+..\ ,$ (16)

where $L_{p}$ has dimension $p$ and $SU(2)$ charge $p-1$.

The correlator of primaries (9) is written:

$\displaystyle\langle{\mathcal{O}}_{p_{1}}(x_{1},y_{1}){\mathcal{O}}_{p_{2}}(x_{2},y_{2}){\mathcal{O}}_{p_{3}}(x_{3},y_{3}){\mathcal{O}}_{p_{4}}(x_{4},y_{4})\rangle=P_{p_{i}}\times
G_{p_{1}p_{2}p_{3}p_{4}}(x,y)\ ,$ (17)

where the prefactor (10) becomes

$\displaystyle
P_{p_{i}}=\hat{P}_{(2p_{i}-1)/2}=g_{12}^{p_{1}+p_{2}-1}g_{34}^{p_{3}+p_{4}-1}\left(\frac{g_{24}}{g_{14}}\right)^{{p_{21}}}\left(\frac{g_{14}}{g_{13}}\right)^{{p_{43}}}\
.$ (18)

In this form, the superconformal Ward identities are simply
$\partial_{x}G(x,y)|_{x=y}=0$, i.e. $G(x,x)$ is independent of $x$ [27]. This
implies that we can write

$\displaystyle G_{p_{i}}(x,y)=k_{p_{i}}(x,y)+\frac{x-y}{xy}H_{p_{i}}(x,y)\ ,$
$\displaystyle\text{where}\ \
k_{p_{i}}(x,y)=\kappa\left(\frac{x}{y}\right)^{p_{43}}\ ,$ (19)

with $H(x,x)$ finite and with $k_{p_{i}}(x,y)$ defined such that it is
annihilated by the super Casimir (defined later in (3.1)),
$\mathcal{C}^{SU(1,1|2)}_{1,2}\left(P_{p_{i}}k_{p_{i}}\right)=0$ with a
constant $\kappa$.

###  2.2 Conformal Blocks

In this subsection we discuss the conformal blocks in 1d. 4-point functions
can be expanded in superconformal blocks $B_{\Delta,p}(x,y)$ as follows:

$\displaystyle\langle{\mathcal{O}}_{p_{1}}(x_{1},y_{1}){\mathcal{O}}_{p_{2}}(x_{2},y_{2}){\mathcal{O}}_{p_{3}}(x_{3},y_{3}){\mathcal{O}}_{p_{4}}(x_{4},y_{4})\rangle$
$\displaystyle=\sum_{\Delta=1}^{\infty}\sum_{p=0}^{p_{1}+p_{2}-2}A^{p_{1}p_{2}p_{3}p_{4}}_{\Delta,p}\,g_{12}^{p_{1}+p_{2}-1}g_{34}^{p_{3}+p_{4}-1}\left(\frac{g_{24}}{g_{14}}\right)^{{p_{21}}}\left(\frac{g_{14}}{g_{13}}\right)^{{p_{43}}}\,B_{\Delta,p,p_{12},p_{34}}(x,y)\
,$ (20)

where $p_{ij}=p_{i}-p_{j}$, $g_{ij}$ is defined in (11), and the coefficient
$A_{\Delta,p}$ is given in terms of a sum of squares of OPE coefficients

$\displaystyle
A^{p_{1}p_{2}p_{3}p_{4}}_{\Delta,p}=\sum_{{\mathcal{O}}^{\Delta,p},\tilde{\mathcal{O}}^{\Delta,p}}C_{p_{1}p_{2}}^{{\mathcal{O}}}C_{p_{3}p_{4}}^{\tilde{\mathcal{O}}}C_{{\mathcal{O}}\tilde{\mathcal{O}}}\
.$ (21)

The superconformal blocks were derived for a general conformal group
$SU(m,m|2n)$ in [27]. Specialising to $m=n=1$, the blocks for long multiplets
are given by

$\displaystyle B^{\text{long}}_{\Delta,p,p_{12},p_{34}}(x,y)=$
$\displaystyle-(x-y)(-x)^{\Delta}{}_{2}F_{1}(\Delta+1-p_{12},\Delta+1+p_{34};2\Delta+2;x)$
$\displaystyle\times y^{-p-1}{}_{2}F_{1}(-p+p_{12},-p-p_{34};-2p;y)\ .$ (22)

For half-BPS multiplets $\Delta=p$, and the blocks are

$\displaystyle B^{\text{half-BPS}}_{p,p_{12},p_{34}}(x,y)=$
$\displaystyle\left(-\frac{x}{y}\right)^{p}\Big{(}1+(x-y)\sum_{i=1}^{k}[x^{-i}\,_{2}F_{1}(p+1-i-p_{12},p+1-i+p_{34};2p+2-2i;x)]$
$\displaystyle\times y^{i-1}\,_{2}F_{1}(i-p+p_{12},i-p-p_{34};2i-2p;y)\Big{)}\
,$ (23)

where $k=\min(p-p_{12},p+p_{34})$ and the square bracket means we take the
regular piece as $x\rightarrow 0$.

It is useful to note that the block of an operator with dimension $\Delta$ and
$SU(2)$ representation $p$ contributes as follows to the 4-point function

$\displaystyle\langle{\mathcal{O}}_{p_{1}}(x_{1},y_{1}){\mathcal{O}}_{p_{2}}(x_{2},y_{2}){\mathcal{O}}_{p_{3}}(x_{3},y_{3}){\mathcal{O}}_{p_{4}}(x_{4},y_{4})\rangle$
$\displaystyle\sim\sum_{\Delta,p}A^{p_{1}p_{2}p_{3}p_{4}}_{\Delta,p}\,g_{12}^{p_{1}+p_{2}-1}g_{34}^{p_{3}+p_{4}-1}\left(\frac{g_{24}}{g_{14}}\right)^{{p_{21}}}\left(\frac{g_{14}}{g_{13}}\right)^{{p_{43}}}(-x)^{\Delta}y^{-p}\left(1+{\mathcal{O}}(x,y)\right)\
,$ (24)

where the higher orders in $x,y$ correspond to spacetime or $SU(2)$
descendants.

###  2.3 4d Effective Action

In [13] a 10d scalar effective action was introduced which generates all half-
BPS 4-point correlators in ${\mathcal{N}}=4$ SYM described by tree-level
string theory in AdS${}_{5}\times$S5. In AdS$\times$S the effective action
takes the form

$\displaystyle V^{\text{AdS$\times$S}}(\phi)=\frac{G_{N}}{4!}\Big{[}$
$\displaystyle
A\phi^{4}+l_{s}^{4}\Big{(}3B(\nabla\phi.\nabla\phi)^{2}+6C\nabla^{2}\nabla_{\mu}\phi\nabla^{\mu}\phi\phi^{2}\Big{)}$
$\displaystyle+l_{s}^{6}\Big{(}6D(\nabla\phi.\nabla\phi)(\nabla_{\mu}\nabla_{\nu}\phi\nabla^{\mu}\nabla^{\nu}\phi)+6E\nabla_{\mu}\nabla^{2}\nabla_{\nu}\phi\nabla^{\mu}\nabla^{\nu}\phi\phi^{2}\Big{)}+\dots\Big{]}\,$
(25)

where $G_{N}$ is Newton’s constant and $l_{s}$ is the analogue of the string
length. These parameters can be written in terms of the AdS radius $R$ as
follows:

$G_{N}=R^{2}/c,\,\,\,l_{s}^{4}=aR^{4},$ (26)

where $R$ is the AdS radius, and $c$ and $a$ are analogous to the central
charge and string coupling, respectively. Some of the terms in the effective
action can be uplifted directly from the flat space action. It is important to
note however that this uplift is not unique: the action contains ambiguities
because the covariant derivatives no longer commute with each other leading to
curvature corrections which vanish in the flat space limit. Furthermore the
coefficients $B,C,..$ themselves can have an expansion in the dimensionless
parameter $l_{P}/R^{2}$:

$\displaystyle B(a)=B_{0}+B_{1}\tfrac{l_{s}^{2}}{R^{2}}+\dots$ $\displaystyle
C(a)=C_{0}+C_{1}\tfrac{l_{s}^{2}}{R^{2}}+\dots$ $\displaystyle\dots$ (27)

For simplicity, we will set $R=1$ from now on throughout this paper, but it
will be understood that terms suppressed by $1/R$ vanish in the flat space
limit. Whereas in AdS${}_{5}\times$S5 case the $\phi^{4}$ interaction
describes $\alpha^{\prime 3}$ corrections to supergravity, in
AdS${}_{2}\times$S2 it describes the 2-derivative sector obtained from
dimensional reduction of supergravity, as we will see in section 4.2. Hence,
the effective action in AdS${}_{2}\times$S2 treats the 2-derivative sector on
equal footing with higher-derivative corrections.

###  2.4 AdS$\times$S Witten Diagrams

Using the effective action, we can compute generalised Witten diagrams that
treat AdS and the sphere on equal footing. This is explained in detail in
[13], so we will just state the main formulae. The generalised bulk-to-
boundary propagator in AdS${}_{d+1}\times$Sd+1 is

$\displaystyle
G(\hat{X},\hat{Y};X,Y)=\left(-2\hat{X}.X-2\hat{Y}.Y\right)^{-\Delta}\ ,$ (28)

where $\left\\{\hat{X},\hat{Y}\right\\}$ are embedding coordinates for AdS and
sphere, respectively, written in terms of $(d+2)$-dimensional Minkowski and
Euclidean space, respectively, and which satisfy
$\hat{X}^{2}=-\hat{Y}^{2}=-1$. Similarly, $\left\\{X,Y\right\\}$ are boundary
coordinates satisfying $X^{2}=Y^{2}=0$. These are related to $d$-dimensional
boundary coordinates via $-2X_{i}\cdot X_{j}=x_{ij}^{2}$ and $-2Y_{i}\cdot
Y_{j}=y_{ij}^{2}$, where $\left\\{i,j\right\\}$ label a pair of points. Note
that the sphere does not actually have a boundary since it is compact, but the
$Y$ coordinates naturally describe $1/2$-BPS operators in the boundary of AdS.
When $\Delta=d$, the generalised bulk-to-boundary propagator satisfies the
equations of motion for a massless scalar field:

$\nabla^{2}G=0.$ (29)

Since the effective action in AdS$\times S$ describes a massless 4d scalar
field, we will therefore set $\Delta=d$ in practice. Note that the bulk-to-
boundary propagators have a normalisation

$\mathcal{C}_{\Delta}=\frac{\Gamma\left(\Delta\right)}{2\pi^{d/2}\Gamma\left(\Delta-d/2+1\right)}.$
(30)

We will dress the generalised Witten diagrams with this normalisation to
obtain correlators.

A generalised contact Witten diagram is then defined as

$\displaystyle
D^{\text{AdS}_{d+1}\times\text{S}^{d+1}}_{\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}}(X_{i},Y_{i})=\frac{1}{(-2)^{2\Sigma_{\Delta}}}\int_{\text{AdS}\times\text{S}}\frac{d^{d+1}\hat{X}d^{d+1}\hat{Y}}{(P_{1}+Q_{1})^{\Delta_{1}}(P_{2}+Q_{2})^{\Delta_{2}}(P_{3}+Q_{3})^{\Delta_{3}}(P_{4}+Q_{4})^{\Delta_{4}}}\
,$ (31)

where we introduce the shorthand

$\displaystyle P_{i}=\hat{X}.X_{i}\ ,\qquad Q_{i}=\hat{Y}.Y_{i}\ .$ (32)

It is now straightforward to expand this AdS$\times$S contact diagram into an
infinite number of standard AdS contact diagrams multiplied by spherical
analogues. In particular, using

$\displaystyle\frac{1}{(P+Q)^{\Delta}}=\sum_{p=0}^{\infty}(-1)^{p}\frac{(p+1)_{\Delta-1}}{\Gamma(\Delta)}\frac{Q^{p}}{P^{p+\Delta}}$
(33)

four times gives the expansion:

$\displaystyle
D^{\text{AdS}\times\text{S}}_{\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}}(X_{i},Y_{i})=$
$\displaystyle\sum_{p_{i}=0}^{\infty}\prod_{i=1}^{4}(-1)^{p_{i}}\frac{(p_{i}+1)_{\Delta_{i}-1}}{\Gamma(\Delta_{i})}$
$\displaystyle\times
D^{(d)}_{p_{1}+\Delta_{1},p_{2}+\Delta_{2},p_{3}+\Delta_{3},p_{4}+\Delta_{4}}(X_{i})B^{(d)}_{p_{1}p_{2}p_{3}p_{4}}(Y_{i})\
,$ (34)

where

$\displaystyle
D^{(d)}_{\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}}(X_{i})=\frac{1}{(-2)^{2\Sigma_{\Delta}}}\int_{\text{AdS}}\frac{d^{d+1}\hat{X}}{(\hat{X}.X_{1})^{\Delta_{1}}(\hat{X}.X_{2})^{\Delta_{2}}(\hat{X}.X_{3})^{\Delta_{3}}(\hat{X}.X_{4})^{\Delta_{4}}}\
,$ (35) $\displaystyle
B^{(d)}_{p_{1}p_{2}p_{3}p_{4}}(Y_{1},Y_{2},Y_{3},Y_{4})=(-2)^{2\Sigma_{p}}\int_{\text{S}}{d^{d+1}\hat{Y}}(\hat{Y}.Y_{1})^{p_{1}}(\hat{Y}.Y_{2})^{p_{2}}(\hat{Y}.Y_{3})^{p_{3}}(\hat{Y}.Y_{4})^{p_{4}}\
,$ (36)

with $\Sigma_{\Delta}=(\Delta_{1}{+}\Delta_{2}{+}\Delta_{3}{+}\Delta_{4})/2$
and $\Sigma_{p}=(p_{1}{+}p_{2}{+}p_{3}{+}p_{4})/2$. For more details, see
section 2.3 of [13]. It is convenient to define normalised functions

$\displaystyle
D^{(d)}_{\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}}(X_{i})={\mathcal{N}}^{\text{AdS}_{d+1}}_{\Delta_{i}}D_{\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}}(X_{i})\
,\,\,\,B^{(d)}_{p_{1}p_{2}p_{3}p_{4}}(Y_{i})={\mathcal{N}}^{\text{S}^{d+1}}_{p_{i}}B_{p_{1}p_{2}p_{3}p_{4}}(Y_{i}),$
(37)

where

$\displaystyle{\mathcal{N}}^{\text{AdS}_{d+1}}_{\Delta_{i}}=\frac{\tfrac{1}{2}\pi^{d/2}\Gamma(\Sigma_{\Delta}-d/2)}{(-2)^{\Sigma_{\Delta}}\prod_{i}\Gamma(\Delta_{i})}\
,\,\,{\mathcal{N}}^{\text{S}^{d+1}}_{p_{i}}={2\times
2^{\Sigma_{p}}}\frac{\pi^{d/2+1}\prod_{i}\Gamma(p_{i}{+}1)}{\Gamma(\Sigma_{p}{+}d/2{+}1)}\
.$ (38)

It is also conventional to write Witten diagrams in terms of $\bar{D}$
functions which are closely related to the D-functions described above. For
more details, see for example Appendix D of [30].

Note that the contact diagrams in (31) come from bulk interaction vertices
without derivatives. More generally, contact diagrams with derivatives take
the form

$\displaystyle\frac{1}{4!}\frac{1}{(-2)^{2\Sigma_{\Delta}}}\int_{\text{AdS}\times\text{S}}{d^{d+1}\hat{X}}{d^{d+1}\hat{Y}}\times\left(\prod_{i}\frac{Q_{i}^{n^{Q}_{i}}P_{i}^{n^{P}_{i}}\times(\Delta_{i})_{n_{i}}}{(P_{i}+Q_{i})^{\Delta_{i}+n_{i}}}\right)\times\left(\prod_{i<j}(X_{i}.X_{j})^{n^{X}_{ij}}(Y_{i}.Y_{j})^{n^{Y}_{ij}}\right)\
,$ (39)

with $n_{i}=n_{i}^{P}+n_{i}^{Q}+\sum_{j}n^{X}_{ij}+\sum_{j}n^{Y}_{ij}$.
Expanding them as above then gives

$\displaystyle(-2)^{2\Sigma_{X}+2\Sigma_{Y}}\sum_{p_{i}=0}^{\infty}\left(\prod_{i=1}^{4}(-1)^{p_{i}}\frac{(p_{i}+1)_{\Delta_{i}+n_{i}-1}}{\Gamma(\Delta_{i})}D^{(d)}_{p_{i}+\Delta_{i}+n_{i}-n^{P}_{i}}(X_{i})B_{p_{i}+n^{Q}_{i}}(Y_{i})\right)\left(\prod_{i<j}(X_{i}.X_{j})^{n^{X}_{ij}}(Y_{i}.Y_{j})^{n^{Y}_{ij}}\right)\
,$ (40)

where $\Sigma_{X},\Sigma_{Y}$ represent the sum of all the
$n^{X}_{ij},n^{Y}_{ij}$, respectively.

##  3 Free theory

In this section we will derive 1d free theory correlators and show that they
are consistent with 4d conformal symmetry. The free theory correlators can be
computed from Wick contractions. For equal charge operators we obtain

$\displaystyle\langle{\mathcal{O}}_{p}(x_{1},y_{1}){\mathcal{O}}_{p}(x_{2},y_{2}){\mathcal{O}}_{p}(x_{3},y_{3}){\mathcal{O}}_{p}(x_{4},y_{4})\rangle|_{c^{0}}$
$\displaystyle=\frac{1}{{\mathcal{N}}^{(0)}_{pppp}}\left[(\tilde{g}_{12}\tilde{g}_{34})^{2p-1}-(\tilde{g}_{13}\tilde{g}_{24})^{2p-1}+(\tilde{g}_{14}\tilde{g}_{23})^{2p-1}\right]$
$\displaystyle=\frac{1}{{\mathcal{N}}^{(0)}_{pppp}}\left(\tilde{g}_{12}\tilde{g}_{34}\right)^{2p-1}\left[1+\left(\frac{x}{y}\right)^{2p-1}\left(\left(-{\rm
sgn}x\right)+\left(\frac{1-y}{1-x}\right)^{2p-1}{\rm
sgn}\left[x\left(1-x\right)\right]\right)\right]\ ,$ (41)

where $\tilde{g}$ is defined in (12),

$\displaystyle{\mathcal{N}}^{(0)}_{p_{i}}=\left(-1\right)^{\Sigma_{p}}\sqrt{(2p_{1}-1)(2p_{2}-1)(2p_{3}-1)(2p_{4}-1)}$
(42)

comes from the normalisation in (14) and
$\Sigma_{p}=\tfrac{1}{2}\left(p_{1}+p_{2}+p_{3}+p_{4}\right)$. Choosing our
external points such that $x_{1}<x_{2}<x_{3}<x_{4}$, then
$\tilde{g}_{12}=g_{12},\tilde{g}_{34}=g_{34}$ and $0\leq x\leq 1$ and so
comparing to (17), we find that

$\displaystyle
G^{(0)}_{pppp}(x,y)=\frac{1}{{\mathcal{N}}^{(0)}_{pppp}}\left[1+\left(\frac{x}{y}\right)^{2p-1}\left(-1+\left(\frac{1-y}{1-x}\right)^{2p-1}\right)\right]\,$
(43)

where the superscript indicates free theory correlators. According to (2.1)
this decomposes into

$\displaystyle k^{(0)}_{pppp}(x,y)=\frac{1}{{\mathcal{N}}^{(0)}_{pppp}}\,,\,\
H^{(0)}_{pppp}(x,y)=\frac{1}{{\mathcal{N}}^{(0)}_{pppp}}\frac{x\,y}{x-y}\left(\frac{x}{y}\right)^{2p-1}\left[-1+\left(\frac{1-y}{1-x}\right)^{2p-1}\right]\
.$ (44)

Similarly, for unequal charges $\\{p_{i}\\}=pqpq$, where $p>q$ we obtain

$\displaystyle\langle{\mathcal{O}}_{p}(x_{1},y_{1}){\mathcal{O}}_{q}(x_{2},y_{2}){\mathcal{O}}_{p}(x_{3},y_{3}){\mathcal{O}}_{q}(x_{4},y_{4})\rangle_{c^{0}}=-\,\frac{1}{{\mathcal{N}}^{(0)}_{pqpq}}\,\tilde{g}^{2p-1}_{13}\tilde{g}^{2q-1}_{24}\
,$ (45) $\displaystyle\text{and}\
G^{(0)}_{pqpq}(x,y)=-\,\frac{1}{{\mathcal{N}}^{(0)}_{pqpq}}\left(\frac{x}{y}\right)^{p+q-1}\
,$ (46)

where the operators have dimensions and charges $(2p-1)/2,\,(2q-1)/2$ and we
decompose $G^{(0)}_{pqpq}$ into

$k^{(0)}_{pqpq}(x,y)=-\frac{1}{{\mathcal{N}}^{(0)}_{pqpq}}\left(\frac{x}{y}\right)^{q-p}\
,\ \
H^{(0)}_{pqpq}(x,y)=\frac{1}{{\mathcal{N}}^{(0)}_{pqpq}}\frac{x\,y}{x-y}\left(\frac{x}{y}\right)^{q-p}\left[1-\left(\frac{x}{y}\right)^{2p-1}\right]\
.$ (47)

###  3.1 4d Conformal Symmetry

To realise 4d conformal symmetry in the free theory, we need to consider the
correlators of descendants $L_{p},\bar{L}_{p}$ rather than the super primaries
${\mathcal{O}}_{p}$. Remarkably, as we show in appendix B, the two correlators
are related by a simple second order differential operator which is closely
related to the quadratic Casimir of the superconformal group acting on points
1 and 2:

$\displaystyle\langle
L_{p_{1}}L_{p_{2}}\bar{L}_{p_{3}}\bar{L}_{p_{4}}\rangle={\mathcal{I}}^{-1}\mathcal{C}^{SU(1,1|2)}_{1,2}\langle{\mathcal{O}}_{p_{1}}{\mathcal{O}}_{p_{2}}{\mathcal{O}}_{p_{3}}{\mathcal{O}}_{p_{4}}\rangle\
,$ (48)

where

$\displaystyle{\mathcal{I}}=x_{12}x_{34}y_{13}y_{24}-y_{12}y_{34}x_{13}x_{24}=(x-y)x_{13}x_{24}y_{13}y_{24},$
(49)

and ${\mathcal{C}}_{1,2}^{SU(1,1|2)}$ denotes the superconformal quadratic
Casimir:

$\displaystyle{\mathcal{C}}_{1,2}^{SU(1,1|2)}$
$\displaystyle=P_{p_{i}}\times\frac{x-y}{xy}\Delta^{(2)}\frac{xy}{x-y}P_{p_{i}}^{-1}\
,$ $\displaystyle\Delta^{(2)}$
$\displaystyle={\mathcal{D}}^{+}_{x}-{\mathcal{D}}^{-}_{y}\ ,$
$\displaystyle{\mathcal{D}}^{\pm}_{z}$
$\displaystyle=z^{2}\partial_{z}(1-z)\partial_{z}\pm(p_{12}+p_{43})z^{2}\partial_{z}-p_{12}p_{43}z\
.$ (50)

For a derivation of the quadratic Casimir, see Appendix A.

In the decomposition given by (2.1) the Casimir only sees $H_{p_{i}}$:

$\displaystyle\mathcal{C}^{SU(1,1|2)}_{1,2}\langle{\mathcal{O}}_{p_{1}}{\mathcal{O}}_{p_{2}}{\mathcal{O}}_{p_{3}}{\mathcal{O}}_{p_{4}}\rangle=P_{p_{i}}\frac{x-y}{xy}\Delta^{(2)}H_{p_{i}}(x,y)=P_{p_{i}}\frac{x-y}{xy}\tilde{H}_{p_{i}}(x,y)\
,$ (51)

where we define

$\displaystyle\tilde{H}_{p_{i}}(x,y)=\Delta^{(2)}H_{p_{i}}(x,y)\ .$ (52)

The correlator in (48) is then given by

$\displaystyle\langle
L_{p_{1}}L_{p_{2}}\bar{L}_{p_{3}}\bar{L}_{p_{4}}\rangle=\frac{P_{p_{i}}}{x_{12}x_{34}y_{12}y_{34}}\tilde{H}_{p_{i}}(x,y)\
.$ (53)

Let us first consider the case of equal charges given in (44). The action of
$\Delta^{(2)}$ then yields

$\displaystyle\tilde{H}_{pppp}=\frac{x^{2p}}{y^{2p-2}}\left(1+\frac{(1-y)^{2p-2}}{(1-x)^{2p}}\right)\
.$ (54)

Note that acting with $\Delta^{(2)}$ on $H$ gives a factor of $(2p-1)^{2}$
which gets cancelled by the normalisation $1/{\mathcal{N}}^{(0)}_{pppp}$. This
is how the normalisation in (14) was chosen, up to the sign which will become
clear shortly. From (53) and (54) we get

$\langle
L_{p}L_{p}\bar{L}_{p}\bar{L}_{p}\rangle=g_{14}^{2p-2}g_{23}^{2p-2}\frac{1}{x^{2}_{14}x^{2}_{23}}+g_{13}^{2p-2}g_{24}^{2p-2}\frac{1}{x_{13}^{2}x_{24}^{2}}\
.$ (55)

Note that the correlator of bosonic descendants is given in terms of bosonic
2-point functions $g_{ij}$ (11). For the lowest case $p=1$ we have

$\displaystyle\langle
L_{1}L_{1}\bar{L}_{1}\bar{L}_{1}\rangle=\frac{1}{x_{14}^{2}x_{23}^{2}}+\frac{1}{x_{13}^{2}x_{24}^{2}}\
.$ (56)

To see the 4d conformal symmetry, now lift this to four dimensions by
replacing $x_{ij}^{2}\rightarrow
x_{ij}^{2}+y_{ij}^{2}=x_{ij}^{2}(1+g_{ij}^{2})$ which gives

$\displaystyle\langle LL\bar{L}\bar{L}\rangle_{c^{0}}$
$\displaystyle=\left[\frac{1}{x_{14}^{2}x_{23}^{2}}+\frac{1}{x_{13}^{2}x_{24}^{2}}\right]_{4d}$
$\displaystyle=\frac{1}{x_{14}^{2}x_{23}^{2}}\frac{1}{(1+g_{14}^{2})(1+g_{23}^{2})}+\frac{1}{x_{13}^{2}x_{24}^{2}}\frac{1}{(1+g_{13}^{2})(1+g_{24}^{2})}\
,$ (57)

where we have defined the generator of all operators $L_{p}$

$\displaystyle L=\sum_{p=1}^{\infty}L_{p}\ .$ (58)

This correlator is thus a 4d object which contains all 1d free theory
correlators. In particular, if we expand
$(1+g^{2})^{-1}=1-g^{2}+g^{4}-g^{6}+...$ and keep the two terms proportional
to $g_{14}^{2p-2}g_{23}^{2p-2}$ and $g_{13}^{2p-2}g_{24}^{2p-2}$, this indeed
reproduces the prediction in (55).

As another example, let us consider correlators with unequal charges in (47).
Acting on this with $\Delta^{(2)}$ gives

$\tilde{H}_{pqpq}(x,y)=\left(-1\right)^{p+q}\frac{x^{p+q}}{y^{p+q-2}}\ .$ (59)

Using (53) then yields

$\langle
L_{p}L_{q}\bar{L}_{p}\bar{L}_{q}\rangle=\left(-1\right)^{p+q}g_{13}^{2p-2}g_{24}^{2q-2}\frac{1}{x^{2}_{13}x^{2}_{24}}\
,$ (60)

which agrees with the term proportional to $g_{13}^{2p-2}g_{24}^{2q-2}$ in the
expansion of the generating function (3.1). Thus the choice of signs in our
normalisation of the half-BPS operators (14) was inspired by the realisation
of the 4d conformal symmetry in free theory, similarly to the rest of the
normalisation.

##  4 2-derivative sector

In this section, we will derive all 4-point correlators corresponding to two-
derivative interactions of bulk hypermultiplets. There is an infinite tower of
such correlators corresponding to Kaluza-Klein modes on the $S^{2}$. The basic
strategy will be to first deduce the lowest charge correlator from
superconformal symmetry and crossing, and then uplift it to arbitrary charges
using the assumption of 4d conformal symmetry. This result will then be
confirmed using Witten diagrams derived from the bulk effective action.
Finally, we will show that the 4d uplift of the lowest charge correlator
corresponds to a 4d conformal block, and provide a physical interpretation in
terms of the partial wave expansion of the 4d bulk scattering amplitude
arising in the flat space limit. This will have implications for anomalous
dimensions computed in section 6.

###  4.1 Lowest-Charge Correlator

In this subsection we determine the correlator for $p_{i}=1$ using crossing
symmetry and various other constraints. Note that crossing symmetry in 1d is
subtle. For previous treatments see [31, 32]. We make the following ansatz in
terms of bosonic 2-point functions $g_{ij}$ (11)222Although we are using
bosonic 2-point functions, the 4-point function is still a valid solution to
the Ward identities with the correct crossing symmetry properties.:

$\displaystyle\langle{\mathcal{O}}_{1}(x_{1},y_{1}){\mathcal{O}}_{1}(x_{2},y_{2}){\mathcal{O}}_{1}(x_{3},y_{3}){\mathcal{O}}_{1}(x_{4},y_{4})\rangle|_{1/c}=g_{12}g_{34}\frac{x}{y}(x-y)a(x)\
.$ (61)

If we exchange positions $1$ and $3$, which takes $x\rightarrow 1-x$ and
$y\rightarrow 1-y$, and set the result equal to minus the original expression
(by Fermi statistics) we find the crossing constraint

$a(1-x)=a(x)\ .$ (62)

Let us now consider exchanging $2$ with $3$, which takes $x\rightarrow 1/x$
and $y\rightarrow 1/y$:

$\langle{\mathcal{O}}_{1}(x_{1},y_{1}){\mathcal{O}}_{1}(x_{3},y_{3}){\mathcal{O}}_{1}(x_{2},y_{2}){\mathcal{O}}_{1}(x_{4},y_{4})\rangle|_{1/c}=g_{12}g_{34}\frac{x}{y}(x-y)\left(-\frac{1}{x^{2}}a(1/x)\right)\
.$ (63)

Equating this with minus the original expression then implies

$a\left(\frac{1}{x}\right)=x^{2}a(x)\ .$ (64)

Finally, for the third crossing condition, consider exchanging $1$ with $3$ in
(63):

$\displaystyle\langle{\mathcal{O}}_{1}(x_{3},y_{3}){\mathcal{O}}_{1}(x_{1},y_{1}){\mathcal{O}}_{1}(x_{2},y_{2}){\mathcal{O}}_{1}(x_{4},y_{4})\rangle|_{1/c}=g_{12}g_{34}\frac{x}{y}(x-y)\frac{1}{(1-x)^{2}}a\left(\frac{1}{1-x}\right)\
.$

Equating this with the expression in (61) then gives the condition

$a\left(\frac{1}{1-x}\right)=(1-x)^{2}a(x)\ ,$ (65)

which follows from (62) and (64).

Finally we consider the asymptotic structure of the correlator as
$x\rightarrow 0$ arising from knowledge of the OPE. Since the operators are
fermionic there is no exchange of the identity operator and the lowest weight
operator appearing has $\Delta=1$. More precisely, consider the conformal
block expansion in (2.2) and (2.2)

$\displaystyle\langle{\mathcal{O}}_{1}(x_{1},y_{1}){\mathcal{O}}_{1}(x_{2},y_{2}){\mathcal{O}}_{1}(x_{3},y_{3}){\mathcal{O}}_{1}(x_{4},y_{4})\rangle$
$\displaystyle=\sum_{\Delta=1}^{\infty}A^{1111}_{\Delta,0}\,g_{12}g_{34}\,B^{\text{long}}_{\Delta,0}(x,y)$
$\displaystyle\sim\sum_{\Delta=1}^{\infty}A^{1111}_{\Delta,0}\,g_{12}g_{34}\,(-x)^{\Delta}\left(1+{\mathcal{O}}(x)\right),$
(66)

and compare to (61) in the $x\rightarrow 0$ limit. This implies that $a(x)$
must satisfy the additional constraint

$a(x)=\mathcal{O}(1)\,$ (67)

as $x\rightarrow 0$ (neglecting terms containing $\log(x)$ arising from the
perturbative expansion).

We now make the following ansatz for $a(x)$ based on the perturbative OPE
structure

$\displaystyle a(x)=p(x)\log x^{2}+p(1-x){\log(1-x)^{2}}+r(x)\ ,$ (68)

where

$p(x)=\frac{1}{x-1}\frac{1}{x^{k}(1-x)^{k}}\sum_{i=0}^{m}b_{i}x^{i},\,\,\,r(x)=\frac{1}{x-1}\frac{1}{x^{k}(1-x)^{k}}\sum_{i=0}^{m}c_{i}x^{i},\,\,\,r(x)=r(1-x),$
(69)

and $k$ and $m$ are integers. This ansatz is natural since the holomorphic
limit of $\bar{D}$ functions takes this form. We then find the minimal
solution to the constraints in (62), (64) and (67):

$a(x)=\frac{\log
x^{2}}{1-x}+\frac{\log(1-x)^{2}}{x}=-\bar{D}^{\text{hol}}_{1111}(x)\ ,$ (70)

where $\bar{D}^{\text{hol}}$ is the holomorphic limit of the $\bar{D}$
function. We will obtain all higher-charge correlators in the 2-derivative
sector from 4d conformal symmetry in the next subsection.

###  4.2 4d Conformal Symmetry

Decomposing the lowest-charge correlator in (61) and (70) according to (2.1)
gives

$H_{1111}^{(2)}=-u\bar{D}^{\text{hol}}_{1111},\qquad k_{1111}^{(2)}=0,$ (71)

where the superscript ${(2)}$ indicates that we are considering 2-derivative
interactions in the bulk. We can obtain all higher charge correlators in this
sector using 4d conformal symmetry. First lift the 1d cross-ratios and
holomorphic $\bar{D}$-function to the usual cross-ratios and
$\bar{D}$-functions, using the relations (3) for the cross-ratios. To get an
object that transforms as a 4d conformal correlator, divide by
${x_{12}^{2}x_{34}^{2}}$ and view all Lorentz invariants as living in 4d. We
then obtain

$\displaystyle\langle{\mathcal{O}}{\mathcal{O}}{\mathcal{O}}{\mathcal{O}}\rangle_{1/c}={\mathcal{I}}\times\left[\frac{H^{(2)}_{1111}}{x_{12}^{2}x_{34}^{2}}\right]_{4d},\
$ (72)

where ${\mathcal{O}}$ is the generator of all half BPS operators

$\displaystyle{\mathcal{O}}=\sum_{p=1}^{\infty}{\mathcal{O}}_{p}\ .$ (73)

By the subscript $4d$ we mean that all Lorentz invariants live in 4d, so
replace $x_{ij}^{2}\rightarrow x_{ij}^{2}(1+g_{ij}^{2})$:

$\left[\frac{H^{(2)}_{1111}}{x_{12}^{2}x_{34}^{2}}\right]_{4d}=\left[\frac{-u\bar{D}_{1111}(u,v)}{x_{12}^{2}x_{34}^{2}}\right]_{4d}\
=\
\frac{-u_{4d}\bar{D}_{1111}(u_{4d},v_{4d})}{x_{12}^{2}x_{34}^{2}}\frac{1}{(1+g_{12}^{2})(1+g_{34}^{2})}\
,$ (74)

where

$\displaystyle
u_{4d}=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}}\frac{(1+g_{12}^{2})(1+g_{34}^{2})}{(1+g_{13}^{2})(1+g_{24}^{2})}\
,\ \
v_{4d}=\frac{x_{14}^{2}x_{23}^{2}}{x_{13}^{2}x_{24}^{2}}\frac{(1+g_{14}^{2})(1+g_{23}^{2})}{(1+g_{13}^{2})(1+g_{24}^{2})}\
.$ (75)

This 4d object in (72) then encodes all higher-charge correlators. To obtain
correlators with specific charges, expand in small $g_{ij}$ and take the
coefficients of the appropriate powers of $g_{ij}^{2}$. Before deriving the
general formula for any correlator, let us consider the explicit expansion for
a few examples. For the first few cases with charges $\\{p_{i}\\}=ppqq$ this
looks like:

$\displaystyle\langle{\mathcal{O}}_{1}{\mathcal{O}}_{1}{\mathcal{O}}_{1}{\mathcal{O}}_{1}\rangle_{1/c}=$
$\displaystyle{\mathcal{I}}\times g_{12}g_{34}\times$
$\displaystyle\left(-\bar{D}_{1111}\right)\ ,$
$\displaystyle\langle{\mathcal{O}}_{1}{\mathcal{O}}_{1}{\mathcal{O}}_{2}{\mathcal{O}}_{2}\rangle_{1/c}=$
$\displaystyle{\mathcal{I}}\times g_{12}g_{34}\times$ $\displaystyle
g_{34}^{2}\left(u\bar{D}_{1122}\right)\ ,$
$\displaystyle\langle{\mathcal{O}}_{1}{\mathcal{O}}_{1}{\mathcal{O}}_{3}{\mathcal{O}}_{3}\rangle_{1/c}=$
$\displaystyle{\mathcal{I}}\times g_{12}g_{34}\times$
$\displaystyle\tfrac{1}{2}g_{34}^{4}\left(-u\,\bar{D}_{1133}\right)\ ,$
$\displaystyle\langle{\mathcal{O}}_{2}{\mathcal{O}}_{2}{\mathcal{O}}_{2}{\mathcal{O}}_{2}\rangle_{1/c}=$
$\displaystyle{\mathcal{I}}\times g_{12}g_{34}\times$
$\displaystyle\left(u\,g_{12}^{2}g_{34}^{2}+g_{13}^{2}g_{24}^{2}+v\,g_{14}^{2}g_{23}^{2}\right)\left(-u\,\bar{D}_{2222}\right)\
,$
$\displaystyle\langle{\mathcal{O}}_{2}{\mathcal{O}}_{2}{\mathcal{O}}_{3}{\mathcal{O}}_{3}\rangle_{1/c}=$
$\displaystyle{\mathcal{I}}\times g_{12}g_{34}\times$
$\displaystyle\left(\tfrac{1}{2}u\,g_{12}^{2}g_{34}^{4}+g_{13}^{2}g_{24}^{2}g_{34}^{2}+v\,g_{14}^{2}g_{23}^{2}g_{34}^{2}\right)\left(u\,\bar{D}_{2233}\right)\
,$
$\displaystyle\langle{\mathcal{O}}_{3}{\mathcal{O}}_{3}{\mathcal{O}}_{3}{\mathcal{O}}_{3}\rangle_{1/c}=$
$\displaystyle{\mathcal{I}}\times g_{12}g_{34}\times$
$\displaystyle\left(\tfrac{1}{4}u^{2}g_{12}^{4}g_{34}^{4}+\tfrac{1}{4}g_{13}^{4}g_{24}^{4}+\tfrac{1}{4}v^{2}g_{14}^{4}g_{23}^{4}+u\,g_{12}^{2}g_{34}^{2}g_{13}^{2}g_{24}^{2}\right.$
$\displaystyle\left.+u\,v\,g_{12}^{2}g_{34}^{2}g_{14}^{2}g_{23}^{2}+v\,g_{13}^{2}g_{24}^{2}g_{14}^{2}g_{23}^{2}\right)\left(-u\,\bar{D}_{3333}\right)\
.$ (76)

Note that after Taylor expanding, the right hand side of these equations is
written directly in terms of 1d kinematics and so the $\bar{D}$ functions are
automatically taken in the holomorphic limit. Using the decomposition in (2.1)
we find:

$\displaystyle H^{(2)}_{1122}$ $\displaystyle=u\bar{D}^{\text{hol}}_{1122}\
,\quad H^{(2)}_{1133}=-\tfrac{1}{2}u\bar{D}^{\text{hol}}_{1133}\ ,\quad
H^{(2)}_{2222}=-2u^{2}\frac{1-y+y^{2}}{y^{2}}\bar{D}^{\text{hol}}_{2222}\ ,$
$\displaystyle H^{(2)}_{2233}$
$\displaystyle=\tfrac{1}{2}u^{2}\frac{4-4y+3y^{2}}{y^{2}}\bar{D}^{\text{hol}}_{2233}\
,\quad
H^{(2)}_{3333}=-\tfrac{3}{2}u^{3}\frac{(1-y+y^{2})^{2}}{y^{4}}\bar{D}^{\text{hol}}_{3333}\
.$ (77)

There is a simple formula for all charges which is easiest to see if one first
writes the uplifted 4-point correlator (74) in terms of normalised D-functions
in (37):

$\left[\frac{H^{(2)}_{1111}}{x_{12}^{2}x_{34}^{2}}\right]_{4d}=\left[\frac{-u\bar{D}_{1111}}{x_{12}^{2}x_{34}^{2}}\right]_{4d}=\left[-D_{1111}\right]_{4d}\
.$ (78)

The 4d $D$-function is a function of 4d
$[x_{ij}^{2}]_{4d}=x_{ij}^{2}(1+g_{ij}^{2})$ and thus, for example, the
coefficient of $\left(g_{12}^{2}\right)^{n}$ is

$\frac{\left(x_{12}^{2}\right)^{n}}{n!}\frac{d^{n}}{d\left(x_{12}^{2}\right)^{n}}D_{\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}}=\frac{\left(-x_{12}^{2}\right)^{n}}{n!}D_{\Delta_{1}+n\,\Delta_{2}+n\,\Delta_{3}\Delta_{4}}$
(79)

where we used the relation [30]

$\frac{d}{d\,x_{12}^{2}}D_{\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}}=-D_{\Delta_{1}+1\,\Delta_{2}+1\,\Delta_{3}\Delta_{4}}\
.$ (80)

Hence a general correlator with charges $p_{i}$ is given by

$\displaystyle\left[\frac{H^{(2)}_{p_{1}p_{2}p_{3}p_{4}}}{x_{12}^{2}x_{34}^{2}}\right]_{4d}$
$\displaystyle=\sum_{\\{d_{ij}\\}}\prod_{i<j}\left(\left(-1\right)^{d_{ij}}\left(g_{ij}^{2}\right)^{d_{ij}}\frac{\left(x_{ij}^{2}\right)^{d_{ij}}}{d_{ij}!}\right)(-D_{p_{1}p_{2}p_{3}p_{4}})$
$\displaystyle=\sum_{\\{d_{ij}\\}}\prod_{i<j}\left(\frac{\left(y_{ij}^{2}\right)^{d_{ij}}}{d_{ij}!}\right)\left(-1\right)^{\Sigma_{p}+1}D_{p_{1}p_{2}p_{3}p_{4}}$
$\displaystyle=(-1)^{\Sigma_{p}+1}D_{p_{1}p_{2}p_{3}p_{4}}\times
B_{p_{1}-1\,p_{2}-1\,p_{3}-1\,p_{4}-1}\ ,$ (81)

where the normalised B-function is defined in (37) and

$\displaystyle\sum_{i<j}d_{ij}=p_{i}-1\,,\ 0\leq d_{ij}=d_{ji}\,,\ d_{ii}=0\
.$ (82)

Hence, expanding the 4d formula (72), we find that correlators with arbitrary
charge in the 2-derivative sector are given by

$\displaystyle\langle{\mathcal{O}}_{p_{1}}{\mathcal{O}}_{p_{2}}{\mathcal{O}}_{p_{3}}{\mathcal{O}}_{p_{4}}\rangle_{1/c}={\mathcal{I}}\times(-1)^{\Sigma_{p}+1}D_{p_{1}p_{2}p_{3}p_{4}}\times
B_{p_{1}-1\,p_{2}-1\,p_{3}-1\,p_{4}-1}\ .$ (83)

We will derive this formula from the 4d effective action in the next
subsection.

Finally it will be useful to understand the implications of the 4d symmetry
for the correlator of descendants $\langle LL\bar{L}\bar{L}\rangle$. Notice
that (72) with (48) implies that

$\displaystyle\langle
LL\bar{L}\bar{L}\rangle_{1/c}={\mathcal{I}}^{-1}\mathcal{C}^{SU(1,1|2)}_{1,2}\langle{\mathcal{O}}{\mathcal{O}}{\mathcal{O}}{\mathcal{O}}\rangle_{1/c}={\mathcal{I}}^{-1}\mathcal{C}^{SU(1,1|2)}_{1,2}{\mathcal{I}}\times\left[\frac{H^{(2)}_{1111}}{x_{12}^{2}x_{34}^{2}}\right]_{4d}\
.$ (84)

Now note that ${\mathcal{I}}^{-1}\mathcal{C}^{SU(1,1|2)}_{1,2}{\mathcal{I}}$
is simply equal to the Casimir for the maximal bosonic subgroup $SO(1,2)\times
SO(3)$ of $SU(1,1|2)$ (see appendix A and in particular (A.1)). So we see that
the 4d symmetry manifests itself for the correlator $\langle
LL\bar{L}\bar{L}\rangle_{1/c}$ as

$\displaystyle\langle LL\bar{L}\bar{L}\rangle_{1/c}=\mathcal{C}^{SO(1,2)\times
SO(3)}_{1,2}\left[\frac{H^{(2)}_{1111}}{x_{12}^{2}x_{34}^{2}}\right]_{4d}\ .$
(85)

In other words the generator of the correlator of descendants is a
$SO(1,2)\times SO(3)$ Casimir acting on a 4d object.

###  4.3 Effective Action

All 4-point half-BPS correlators in the 2-derivative sector can be computed
from the first term in the effective action (2.3), which is simply a
$\phi^{4}$ interaction:

$\displaystyle
S_{2}=\frac{1}{4!}\,A\times\int_{\text{AdS}\times\text{S}}{d^{2}{\hat{X}}}{d^{2}{\hat{Y}}}\phi(\hat{X},\hat{Y})^{4}\
.$ (86)

Using the generalised bulk-to-boundary propagators in (28) we obtain the
AdS${}_{2}\times$S2 Witten diagram for this $\phi^{4}$ contact interaction,
leading to the following correlators (where recall ${\mathcal{O}}$ generates
all the $1/2$-BPS operators (73)):

$\displaystyle\langle{\mathcal{O}}{\mathcal{O}}{\mathcal{O}}{\mathcal{O}}\rangle^{(2)}_{\text{int}}$
$\displaystyle=\frac{1}{4!}\,A\,\frac{(\mathcal{C}_{1})^{4}}{(-2)^{4}}\int_{\text{AdS}\times\text{S}}\frac{{d^{2}{\hat{X}}}{d^{2}{\hat{Y}}}}{(P_{1}+Q_{1})(P_{2}+Q_{2})(P_{3}+Q_{3})(P_{4}+Q_{4})}$
$\displaystyle=\frac{1}{4!}\,A\,(\mathcal{C}_{1})^{4}\times
D_{1111}^{\text{AdS}_{2}\times\text{S}^{2}}\ .$ (87)

Recall that $P_{i}=\hat{X}.X_{i}$, $Q_{i}=\hat{Y}.Y_{i}$,
$\mathcal{C}_{\Delta}$ is defined in (30), and the AdS$\times$S $D$-functions
are defined in (31).

To extract correlators with specific charges one expands in the appropriate
powers in $Y_{i}$ using the Taylor expansion of the 4d bulk-to-boundary
propagators

$\displaystyle(P_{i}+Q_{i})^{-1}=\sum_{p=1}^{\infty}(-1)^{p-1}(P_{i})^{-p}(Q_{i})^{p-1}\
.$ (88)

We then obtain333This is (2.4) with $\Delta_{i}=d=1$ and with
$p_{i}\rightarrow p_{i}-1$ to account for the fact that the lowest correlator
is labelled with $p_{i}=1$ rather than $p_{i}=0$. We do not need to worry
about the minus signs in the factors $(-1)^{p}$ in (88) since
$B_{p_{1}p_{2}p_{3}p_{4}}=0$ if $p_{1}+p_{2}+p_{3}+p_{4}$ is odd.:

$\displaystyle\langle{\mathcal{O}}_{p_{1}}{\mathcal{O}}_{p_{2}}{\mathcal{O}}_{p_{3}}{\mathcal{O}}_{p_{4}}\rangle_{\text{int}}^{(2)}$
$\displaystyle=\frac{1}{4!}\,A\,\frac{(\mathcal{C}_{1})^{4}}{(-2)^{4}}\int_{\text{AdS}_{2}}{d^{2}\hat{X}}\prod_{i}\frac{1}{(P_{i})^{p_{i}}}\times\int_{\text{S}^{2}}{d^{2}\hat{Y}}\prod_{i}(Q_{i})^{p_{i}-1}$
$\displaystyle=A^{\prime}\,\left(-1\right)^{\Sigma_{p}+1}D_{p_{1}p_{2}p_{3}p_{4}}\times
B_{p_{1}-1\,p_{2}-1\,p_{3}-1\,p_{4}-1}\ ,$ (89)

which matches the result we obtained using 4d conformal symmetry in (4.2).

###  4.4 Physical Interpretation

In this section we will clarify the origin of 4d conformal symmetry. So far we
found that both the free theory (3.1) and 2-derivative sector (72) and (85)
have a four-dimensional origin. Both cases are in fact written in terms of 4d
conformal 4-point functions for operators with dimension 1. Any such conformal
4-point function has a simple expansion in terms of 4d bosonic conformal
blocks of the form [33]:

$\langle
LL\bar{L}\bar{L}\rangle=\frac{1}{x_{12}^{2}x_{34}^{2}}\sum_{\Delta,l}A_{\Delta,l}\,G_{\Delta,l}(u,v)\
,$ (90)

where

$\displaystyle G_{\Delta,l}(z,\bar{z})=$
$\displaystyle\frac{1}{z-\bar{z}}u^{\frac{1}{2}(\Delta-l)}\big{(}\left(-\tfrac{1}{2}z\right)^{l}z\,_{2}F_{1}\left(\tfrac{1}{2}(\Delta+l),\tfrac{1}{2}(\Delta+l);\Delta+l;z\right)$
$\displaystyle\times\,_{2}F_{1}\left(\tfrac{1}{2}(\Delta-l-2),\tfrac{1}{2}(\Delta-l-2);\Delta-l-2;\bar{z}\right)-\left(z\leftrightarrow\bar{z}\right)\big{)}\
.$ (91)

For the free theory correlator in (3.1), the 4d conformal block expansion is
then given by

$u+\frac{u}{v}=\sum_{n,l}A^{(0)}_{\Delta_{0},l}\,G_{\Delta_{0},l}(u,v)\ $ (92)

and we only find non-zero coefficients if $\Delta_{0}=l+2$:

$A^{(0)}_{l+2,l}=\frac{2^{1+l}(l!)^{2}}{(2\,l)!}.$ (93)

This can be understood by recalling that this is the block expansion of a 4d
scalar field of dimension 1, which is therefore massless. So the only
operators one can construct out of its OPE are the spin $l$ currents
$L\partial^{l}L$.

Next let’s do a 4d conformal block expansion of the $\log u$ part of the 4d
correlator describing the 2-derivative sector (72),(85):

$-u\bar{D}_{1111}|_{\log
u}=\sum_{\Delta_{0},l}A_{\Delta_{0},l}^{(0)}\gamma_{\Delta_{0},l}\,G_{\Delta_{0},l}(u,v).$
(94)

We find that only the 4d block with spin-0 contributes, with $\Delta_{0}=2$
and anomalous dimension $\gamma_{2,0}=1$. In other words the $\log u$ part of
the 2-derivative sector is simply a single spin-0 block $G_{2,0}$ so (using
(72))

$\displaystyle\langle{\mathcal{O}}{\mathcal{O}}{\mathcal{O}}{\mathcal{O}}\rangle_{1/c}={\mathcal{I}}\times\left[\frac{G_{2,0}(u,v)\log
u+..}{x_{12}^{2}x_{34}^{2}}\right]_{4d}\ .$ (95)

This agrees with the partial wave coefficient of the corresponding flat space
scattering amplitude, as we will now explain.

Let us first review how this works in $\mathcal{N}=4$ SYM and then adapt it to
the present case. In [5], it was observed that in the flat space limit
supergravity correlators in AdS${}_{5}\times$S5 reduce to the following
scattering amplitude:

$A_{10}=\frac{G_{N}\delta^{16}(Q)}{stu}\rightarrow G_{N}\frac{s^{3}}{tu},$
(96)

where we have taken the dilaton component. The 10d conformal symmetry of the
corresponding $\mathcal{N}=4$ SYM correlators is then inherited from the flat
space scattering amplitude after noting that $G_{N}\delta^{16}(Q)$ is
dimensionless and AdS${}_{5}\times$S5 is conformally flat. It is then possible
to relate the anomalous dimensions deduced from these correlators to partial
wave coefficients of the scattering amplitude.

In more detail, the partial wave expansion is given by

$A_{d}(s,\cos\theta)=\frac{1}{s^{(d-4)/2}}\sum_{l}(l+1)_{d-4}C_{l}(\cos\theta)\mathcal{A}_{l}^{d}(s)$
(97)

where $\cos\theta=1+\frac{2t}{s}$. Writing (96) in terms of $s$ and $\theta$,
we find

$A_{10}(s,\cos\theta)=\frac{4G_{N}s}{\sin^{2}\theta}.$ (98)

The single power of $s$ in the numerator indicates 2-derivative interactions,
as expected for supergravity. Comparing this to (97) implies that the partial
wave coefficients are schematically

$\displaystyle\mathcal{A}_{l}^{10}(s)\sim
1+\frac{R^{8}}{c}\frac{s^{4}}{\left(l+1\right)_{6}}\ ,$ (99)

where Newton’s constant $G_{N}\sim R^{8}/c$ in 10d, $R$ is the AdS radius, and
$c$ is the central charge. Remarkably, this has the same form as the
$\mathcal{N}=4$ SYM anomalous dimensions first derived in [24] after unmixing
operators whose scaling dimensions are degenerate at leading order in the
$1/c$ expansion:

$\displaystyle e^{\frac{1}{c}\gamma^{\text{(2)}}_{4d}}\sim
1+\frac{1}{c}\frac{\delta^{(8)}}{\left(l+1\right)_{6}},\,$ (100)

where $\delta^{(8)}$ is a certain factor corresponding to the eigenvalues of
an eighth order differential operator which corresponds to a Casimir of the
superconformal group $SU(2,2|4)$.

Let us now adapt this discussion to AdS${}_{2}\times$S2. In this case, the
flat space amplitude for hypermultiplets with 2-derivative interactions is
given by [7]

$A_{4}=G_{N}\delta^{4}(Q)\rightarrow G_{N}s,$ (101)

where we have taken the scalar component. Note that $G_{N}\delta^{4}(Q)$ is
dimensionless in 4d, so this amplitude has 4d conformal symmetry. The factor
of $s$ indicates 2-derivative interactions. In contrast to the 10d amplitude,
it has no $\theta$ dependence which implies that only $l=0$ contributes to the
partial wave expansion, in agreement with the conformal block analysis above.
Hence, the partial wave coefficients are given by

$\mathcal{A}_{l=0}^{4}(s)\sim 1+\frac{R^{2}}{c}\,s\ ,$ (102)

where $G_{N}\sim R^{2}/c$ in 4d. This argument suggests that the corresponding
anomalous dimensions obtained after unmixing degenerate operators in the 1d
CFT should be given by

$e^{\tfrac{1}{c}\gamma_{1d}}\sim 1+\frac{1}{c}\,\delta^{(2)}\ ,$ (103)

where $\delta^{(2)}$ is a second-order superconformal Casimir. We will verify
this in section 6.

##  5 Higher-derivative corrections

In this section, we will generalise the analysis of the previous section to
4-point correlators describing higher-derivative interactions of
hypermultiplets in AdS${}_{2}\times$S2. First we deduce the lowest-charge
correlators from crossing and relate them to the holomorphic limit of
analogous correlators in AdSd>2 found in [34]. Then we compute higher-charge
correlators using the effective action and explore the extent to which they
exhibit 4d conformal symmetry by trying to reproduce them by acting with
superconformal Casimirs on 4d conformal blocks. We find that this is not
generically possible except for a special class of correlators, suggesting
that 4d conformal symmetry is broken by higher-derivative interactions in the
bulk.

###  5.1 Lowest-Charge Correlators

In this section we will generalise the
$\left\langle\mathcal{O}_{1}\mathcal{O}_{1}\mathcal{O}_{1}\mathcal{O}_{1}\right\rangle$
correlator in the two-derivative sector in (71) to higher-derivative
interactions in the bulk. Following the method of section 4.1, we make the
ansatz

$\left\langle\mathcal{O}_{1}\mathcal{O}_{1}\mathcal{O}_{1}\mathcal{O}_{1}\right\rangle=g_{12}g_{34}(x/y)(x-y)a(x)$
(104)

where

$a(1-x)=a(x),\,\,\,a(1/x)=x^{2}a(x),\,\,\,a(x)=\mathcal{O}(1),$ (105)

as required by crossing symmetry and the constraint that the correlator
doesn’t encode the exchange of the identity operator (for more details, see
section 4.1). Taking $a(x)$ of the form in (68) and (69), we then find that
(105) only has solutions for $k=2q$ where $q$ is a non-negative integer, and
$m=3k$. The $k=0$ solution is the one previously found in the 2-derivative
sector in (71).

Further insight into these solutions can be obtained by computing the average
anomalous dimensions of operators contributing to their conformal block
expansion. The average anomalous dimensions are obtained from the conformal
block expansion of the part of the correlator proportional to $\ln u$:

$\left.a(x)\right|_{\ln
u}=\sum_{\Delta=1}A_{\Delta}^{(0)}\gamma_{\Delta}F_{\Delta}(x),\,\,\,F_{\Delta}(x)=x^{\Delta-1}\,_{2}F_{1}(\Delta+1,\Delta+1,2\Delta+2,x),$
(106)

where we expand in terms of standard non-supersymmetric 1d conformal blocks
which follow from setting $p_{i}=1$ in (2.2). The OPE coefficients are
obtained by expanding the free theory correlator in (43) with $p=1$ in terms
of these blocks and are given by

$A_{\Delta}^{(0)}=2\left(\Delta!\right)^{2}/\left(2\Delta\right)!,\,\,\,\Delta\in{\rm
odd.}$ (107)

Plugging (107) into (106), we then find that the anomalous dimensions
$\gamma_{\Delta}$ scale like $\Delta^{2k}=\Delta^{4q}$ as
$\Delta\rightarrow\infty$ for the solutions found in the previous paragraph.
In [34] it was shown that the large-$\Delta$ scaling of anomalous dimensions
is tied to the number of derivatives in the corresponding bulk interactions.
Since the $k=0$ solution corresponds to 2-derivative bulk interactions, it
follows that solutions with general $k$ correspond to interactions with
$2k+2=4q+2$ derivatives. Furthermore, using the decomposition in (2.1),
solutions corresponding to $(4q+2)$-derivative interactions in the bulk can be
written as

$H^{(4q+2)}_{1111}=x^{2}a(x)=-u\,(1+u^{q}+v^{q})\bar{D}^{\text{hol}}_{q+1\,q+1\,q+1\,q+1},$
(108)

modulo lower-derivative solutions.

Let us explain the relation to higher-derivative 4-point interactions in
AdSd>2. In that case, the solutions to the crossing equations for the dual CFT
correlators are labelled by spin $l=0,2,4,...$, and for spin $l$ there are
$l/2+1$ solutions corresponding to interaction vertices of a bulk effective
action with $2l,2l+2,...,3l$ derivatives [34]. Reducing these solution to AdS2
by taking the holomorphic limit, one finds that the $l/2+1$ solutions collapse
to a single solution, modulo lower-spin solutions [35]. This solution
corresponds to a $2l$ derivative interaction i.e. lowest number of derivatives
in the spin-$l$ tower of solutions, and is consistent with the result obtained
in the previous paragraph. For example in AdSd>2 the solutions for $l=0,2$ are
given by

$\displaystyle\left\langle 1111\right\rangle_{l=0,\partial^{0}}=$
$\displaystyle\bar{D}_{1111},$ $\displaystyle\left\langle
1111\right\rangle_{l=2,\partial^{4}}=$ $\displaystyle(1+u+v)\bar{D}_{2222},$
$\displaystyle\left\langle 1111\right\rangle_{l=2,\partial^{6}}=$
$\displaystyle\bar{D}_{3232}+\bar{D}_{2323}+u^{2}\bar{D}_{3322}+u\bar{D}_{2233}+v^{2}\bar{D}_{2332}+v\bar{D}_{3223},$
(109)

where the subscript indicates the spin and number of derivatives in the bulk
effective action, and we have taken the scaling dimensions of the external
operators to be $\Delta=1$. Taking the holomorphic limit of the 6-derivative
solution then gives a linear combination of the 4-derivative and $l=0$
solutions:

$\left.\left\langle
1111\right\rangle_{l=2,\partial^{6}}\right|_{hol}=\frac{17}{7}(1+u+v)\left.\bar{D}_{2222}\right|_{hol}-\frac{1}{7}\left.\bar{D}_{1111}\right|_{hol}.$
(110)

Hence, in the holomorphic limit there is only a 4-derivative solution at $l=2$
plus a zero derivative solution. In the present context, these correspond to
2-derivative and 6-derivative interactions of bulk hypermultiplets.

###  5.2 Effective Action

As explained in section 4.3, two-derivative interactions of the bulk theory
are described by a $\phi^{4}$ interaction in the effective action. Higher
derivative corrections are then obtained by applying covariant derivatives to
the $\phi^{4}$ vertex. 4-point correlators can then be computed from
generalised Witten diagrams in terms of unfixed coefficients of the effective
action.

In the previous subsection, we showed that the lowest higher derivative
corrections appear at six derivatives, which correspond to four-derivative
corrections in the effective action. In this subsection we will use the
effective action to compute the associated 1d CFT correlators for arbitrary
charges of the external operators. As shown in [13], there are only two
linearly independent terms in the 4-derivative effective action (corresponding
to six derivative corrections to the Lagrangian for bulk hypermultiplets):

$S_{6}=B_{0}S_{6}^{\text{main}}+C_{0}S_{6}^{\text{amb}}\ ,$ (111)

with

$\displaystyle S_{6}^{\text{main}}$
$\displaystyle=\frac{3}{4!}\int_{\text{AdS}\times\text{S}}d^{2}\hat{X}d^{2}\hat{Y}\left(\nabla\phi.\nabla\phi\right)\left(\nabla\phi.\nabla\phi\right)\
,$ $\displaystyle S_{6}^{\text{amb}}$
$\displaystyle=\frac{6}{4!}\int_{\text{AdS}\times\text{S}}d^{2}\hat{X}d^{2}\hat{Y}\nabla^{2}\nabla_{\mu}\phi\nabla^{\mu}\phi\,\phi^{2}\
.$ (112)

We refer to the second interaction as an ambiguity because it vanishes in the
flat space limit. In this limit we can commute the $\nabla^{2}$ to the right,
and then use $\nabla^{2}\phi=0$. The Witten diagram for the main interaction
is

$\displaystyle\langle{\mathcal{O}}{\mathcal{O}}{\mathcal{O}}{\mathcal{O}}\rangle^{(6)}_{\text{main}}$
$\displaystyle=\frac{1}{4!}\frac{(\mathcal{C}_{1})^{4}}{(-2)^{4}}\int_{\text{AdS}\times\text{S}}d^{2}\hat{X}d^{2}\hat{Y}\frac{N_{12}N_{34}+N_{13}N_{24}+N_{14}N_{23}}{\left(P_{1}+Q_{1}\right)^{2}\left(P_{2}+Q_{2}\right)^{2}\left(P_{3}+Q_{3}\right)^{2}\left(P_{4}+Q_{4}\right)^{2}}\
,$ (113)

where

$\displaystyle N_{ij}$
$\displaystyle=X_{i}.X_{j}+Y_{i}.Y_{j}+P_{i}P_{j}-Q_{i}Q_{j}\ .$ (114)

Furthermore, the Witten diagram for the ambiguity is

$\displaystyle\langle{\mathcal{O}}{\mathcal{O}}{\mathcal{O}}{\mathcal{O}}\rangle^{(6)}_{\text{amb}}$
$\displaystyle=-\frac{1}{4!}\frac{(\mathcal{C}_{1})^{4}}{(-2)^{4}}\int_{\text{AdS}\times\text{S}}\frac{d^{2}\hat{X}d^{2}\hat{Y}}{\prod_{i}\left(P_{i}+Q_{i}\right)}\sum_{i<j}\frac{L_{ij}}{\left(P_{i}+Q_{i}\right)\left(P_{j}+Q_{j}\right)}\
,$ (115)

where

$\displaystyle L_{ij}$
$\displaystyle=X_{i}.X_{j}-Y_{i}.Y_{j}+P_{i}P_{j}+Q_{i}Q_{j}\ .$ (116)

Using (39) and (40), we can expand (5.2) and (5.2) in terms of $D$\- and
$B$-functions, and then take the holomorphic limit of the $\bar{D}$-functions.
Using the decomposition in (2.1), the explicit expressions for some of the
correlators are:

$\displaystyle H_{1111}^{\text{(6)}}=$
$\displaystyle\,3u\left(\bar{D}^{\text{hol}}_{1111}-5\left(1+u+v\right)\bar{D}^{\text{hol}}_{2222}\right)\
,$ $\displaystyle H_{pp11}^{\text{(6)}}=$
$\displaystyle\frac{\left(-1\right)^{p+1}u^{p}}{(p-1)!}\left(f_{1}(p)\bar{D}^{\text{hol}}_{pp11}+f_{2}(p)\,u\,\bar{D}^{\text{hol}}_{p+1\,p+1\,11}+f_{3}(p)\left(1+u+v\right)\bar{D}^{\text{hol}}_{p+1\,p+1\,22}\right)\
,$ $\displaystyle H_{p1p1}^{\text{(6)}}=$
$\displaystyle\frac{\left(-1\right)^{p+1}u^{\frac{p+1}{2}}}{(p-1)!\,y^{p-1}}\left(f_{1}(p)\bar{D}^{\text{hol}}_{p1p1}+f_{2}(p)\,\bar{D}^{\text{hol}}_{p+1\,1\,p+1\,1}+f_{3}(p)\left(1+u+v\right)\bar{D}^{\text{hol}}_{p+1\,2\,p+1\,2}\right)\
,$ $\displaystyle H_{p11p}^{\text{(6)}}=$
$\displaystyle\frac{\left(-1\right)^{p+1}u^{\frac{p+1}{2}}}{(p-1)!\,y^{p-1}}\left(f_{1}(p)\bar{D}^{\text{hol}}_{p11p}+f_{2}(p)\,\bar{D}^{\text{hol}}_{p+1\,11\,p+1}+f_{3}(p)\left(1+u+v\right)\bar{D}^{\text{hol}}_{p+1\,22\,p+1}\right)\
,$ $\displaystyle H_{2222}^{\text{(6)}}=$
$\displaystyle\,2\frac{u^{2}}{y^{2}}\,\Big{(}\left[-8\,C_{0}\,\left(1-y+y^{2}\right)+2\left(41-6y+6y^{2}\right)\right]\bar{D}^{\text{hol}}_{2222}+35\,u\left(y^{2}-1\right)\bar{D}^{\text{hol}}_{3322}$
$\displaystyle+35\left(y-2\right)y\bar{D}^{\text{hol}}_{3223}-63\left(1+u+v\right)\left(1-y+y^{2}\right)\bar{D}^{\text{hol}}_{3333}\Big{)}\
,$ (117)

where

$\displaystyle f_{1}(p)$
$\displaystyle=2\,C_{0}\,p(p-1)+p^{2}\,(4\,p^{2}-8\,p+1)\ ,\
f_{2}(p)=-(1+2p)(2\,p^{2}-3\,p+1)\ ,$ $\displaystyle f_{3}(p)$
$\displaystyle=(1+2p)(3+2p)\ .$ (118)

In the above expressions, we have set $B_{0}=4$ for convenience. Note that
$H^{\text{(6)}}_{1111}$ agrees with the prediction (108) for $q=1$, modulo the
4-point correlator in the 2-derivative sector given in (71). In the next
subsection, we will explore the extent to which 4d conformal symmetry is
realised for higher-derivative corrections.

###  5.3 4d Conformal Symmetry

In the free theory, 4d conformal symmetry was realised by acting on the
correlators
$\langle{\mathcal{O}}_{p_{1}}{\mathcal{O}}_{p_{2}}{\mathcal{O}}_{p_{3}}{\mathcal{O}}_{p_{4}}\rangle$
with a second-order Casimir, while in the 2-derivative sector it was realised
directly on the correlators without acting with a Casimir. In section 4.4, it
was further observed that the infinite tower of 2-derivative correlators can
be assembled into a 4d spin-0 block, which reduces to a scale-invariant
amplitude in the flat space limit. Since higher-derivative corrections will
give scattering amplitudes that are no longer scale-invariant, this suggests
that they can be obtained by acting with Casimirs on 4d conformal blocks. More
precisely, we expect that a correlator corresponding to a $2N$-derivative
interaction in the effective action should come from acting with a Casimir of
order $2N$. In this subsection, we find that this can indeed be realised for a
subset of higher-derivative correlators.

Let us consider correlators in the six-derivative sector computed in the
previous subsection. Since they are obtained from four-derivative interactions
in the effective action (which can have 4d spin-0 or spin-2 as explained in
section 5.1), we expect that they can be obtained by acting with a fourth-
order Casimir on the holomorphic limit of 4d spin-0 and spin-2 blocks:

$H_{p_{1}p_{2}p_{3}p_{4}}^{\text{(6)}}|_{\log
u}=\Delta_{0}^{(4)}\left(\text{4d spin-0
block}\right)_{p_{1}p_{2}p_{3}p_{4}}^{\text{hol}}+\Delta_{2}^{(4)}\left(\text{4d
spin-2 block}\right)_{p_{1}p_{2}p_{3}p_{4}}^{\text{hol}}\ ,$ (119)

where the left-hand-side is computed from the effective action in (111) with
$B_{0}=4$ for convenience, and the Casimirs on the right-hand-side are given
by

$\Delta_{i}^{(4)}=a_{i}+b_{i}\Delta^{(2)}+c_{i}\Delta^{(2)}\mathcal{D}_{x}+d_{i}\Delta^{(2)}\mathcal{D}_{y}\
,$ (120)

where $\Delta^{(2)},\,\mathcal{D}_{x},\,\mathcal{D}_{y}$ were defined in (3.1)
and $\left\\{a_{i},b_{i},c_{i},d_{i}\right\\}$ are unfixed coefficients. See
section A.2 for a discussion of general higher order Casimirs. We restrict the
left-hand-side of (119) to the $\ln u$ part of the correlator since this
encodes the scattering amplitude in the flat space limit and is therefore the
part of the correlator that is expected to have 4d conformal symmetry. In the
right-hand-side, we first Taylor expand the 4d conformal blocks in $g_{ij}$
and choose coefficients corresponding to correlators with the desired charges
as explained in section 4.2.

Remarkably, if the coefficient of the ambiguity $C_{0}=0$ then we find a
solution to (119) for $p_{i}=pp11$ and the other correlators related by
crossing:

$\Delta_{0}^{(4)}=\frac{1}{12}\left(2\,\Delta^{(2)}+5\,\left(\Delta^{(2)}\right)^{2}\right)\
,\quad\Delta_{2}^{(4)}=\frac{1}{90}\left(-2\,\Delta^{(2)}+\left(\Delta^{(2)}\right)^{2}\right)\
.$ (121)

Using these Casimir operators, 4d conformal symmetry can be realised by
correlators in the 6-derivative sector with $\\{p_{i}\\}=pp11$ and crossing
versions. This is highly non-trivial, since neither the Casimir operators nor
the 4d blocks preserve crossing symmetry beyond $1\leftrightarrow 2$ exchange,
but acting with the former on the latter produces crossing-symmetric
correlators. On the other hand, we do not find crossing-symmetric solutions to
(119) for other charge configurations. This suggests that 4d conformal
symmetry is generically broken by higher-derivative corrections. We will see
further evidence of this when we compute anomalous dimensions in the next
section, and will make further comments about this in the context of
$\mathcal{N}=4$ SYM in section 7.

##  6 Anomalous Dimensions from Unmixing

In this section we will classify double-trace operators appearing in the
conformal partial wave expansion of 4-point correlators which have degenerate
quantum numbers at leading order in $1/c$. We then review a procedure for
resolving this degeneracy and computing the anomalous dimensions of double-
trace operators after unmixing. For correlators in the 2-derivative sector we
find that the anomalous dimensions are eigenvalues of a second order conformal
Casimir, as anticipated in section 4.4, but for correlators in the
6-derivative sector some anomalous dimensions contain square roots suggesting
that 4d conformal symmetry is generically broken in agreement with the
analysis of section 5.3.

###  6.1 Degenerate Operators

Let us start by considering all long double-trace operators in the OPE of two
half-BPS operators, ${\mathcal{O}}_{p_{1}},{\mathcal{O}}_{p_{2}}$. In the free
theory the space of operators appearing in the OPE will take the form

$\mathcal{O}_{q_{1}q_{2}}=[\mathcal{O}_{q_{1}}\partial^{\Delta+1-q_{1}-q_{2}}\mathcal{O}_{q_{2}}]_{p}\
,\quad\ q_{1}\leq q_{2}\ ,$ (122)

where $p_{1}-q_{1}=p_{2}-q_{2}$ and where $[\\_]_{p}$ denotes projection onto
the $p+1$ dimensional $SU(2)$ representation. Recall the operators
$\mathcal{O}_{q_{1}}$ and $\mathcal{O}_{q_{2}}$ have half-integer scaling
dimensions $(2q_{1}-1)/2$ and $(2q_{2}-1)/2$, labelled by integers $q_{1}$ and
$q_{2}$, according to the conventions introduced in (14). Thus the scaling
dimension of the exchanged operators $\mathcal{O}_{q_{1}q_{2}}$ is $\Delta$
and the $SU(2)$ charge is $p$ ($p$ can take any value from the set
$p=q_{1}+q_{2},q_{1}+q_{2}-2,..,q_{2}-q_{1}$). There are many operators with
the same weight $(\Delta,p)$ which we now enumerate similarly to those in
${\mathcal{N}}=4$ SYM in [36].

There are two classes of operators in the spectrum, we call them class $A$ and
class $B$. We will also distinguish between the cases where $t=\Delta-p$ odd
or $t=\Delta-p$ even. We label the allowed values of pairs of $q_{1}$ and
$q_{2}$ for class $A$ and class $B$ operators with odd or even $t$ with
$(q_{1}^{A},q_{2}^{A})$, $(q_{1}^{B^{\text{to}}},q_{2}^{B^{\text{to}}})$ and
$(q_{1}^{B^{\text{te}}},q_{2}^{B^{\text{te}}})$ respectively. There are
$d=d_{A}+d_{B^{\text{to}}}+d_{B^{\text{te}}}$ operators
$\mathcal{O}_{q_{1}q_{2}}$ labelled by sets of pairs $(q_{1},q_{2})$, where
$d_{A}$ counts class $A$ and $d_{B^{\text{to}}}$, $d_{B^{\text{te}}}$ class
$B$ operators with odd and even $t$ respectively. The operators for odd and
even $t$ are summarised in Figures 1(a) and 1(b), respectively.

For $t$ odd we have both class $A$ and class $B$ operators:

$\displaystyle q_{1}^{A}=1+i_{A}+r_{A}\ ,\quad$ $\displaystyle
q_{1}^{B^{\text{to}}}=1+i_{B^{\text{to}}}+r_{B^{\text{to}}}\ ,$ $\displaystyle
q_{2}^{A}=1+i_{A}+p-r_{A}\ ,\quad$ $\displaystyle
q_{2}^{B^{\text{to}}}=1+i_{B^{\text{to}}}+(p-1)-r_{B^{\text{to}}}\ ,$
$\displaystyle i_{A}=0,\dots,\frac{t-1}{2}\ ,\quad$ $\displaystyle
i_{B^{\text{to}}}=1,\dots,\frac{t-1}{2}\ ,$ $\displaystyle
r_{A}=0,\dots,\frac{\mu_{A}-1}{2}\ ,\quad$ $\displaystyle
r_{B^{\text{to}}}=0,\dots,\frac{\mu_{B^{\text{to}}}-1}{2}\ ,$ (123)

where

$\mu_{A}=\begin{cases}p+1\ &p\ \text{even}\\\ p\ &p\ \text{odd}\\\
\end{cases}\ ,\qquad\mu_{B^{\text{to}}}=\begin{cases}p-1\ &p\ \text{even}\\\
p\ &p\ \text{odd}\end{cases}\ ,$ (124)

with $d_{A}=\tfrac{1}{4}(\mu_{A}+1)(t+1)$ and
$d_{B^{\text{to}}}=\tfrac{1}{4}(\mu_{B^{\text{to}}}+1)(t-1)$ . Class $A$
operators have even numbers of derivatives while class $B$ operators have odd
numbers of derivatives at $t$ odd.

For even $t$, there are only class B operators. These operators are
parametrised as

$\displaystyle
q_{1}^{B^{\text{te}}}=i_{B^{\text{te}}}+\lfloor{\tfrac{r_{B^{\text{te}}}+1}{2}\rfloor}\
,$ $\displaystyle
q_{2}^{B^{\text{te}}}=i_{B^{\text{te}}}+p-\lfloor{\tfrac{r_{B^{\text{te}}}}{2}\rfloor}\
,$ $\displaystyle i_{B^{\text{te}}}=1,\dots,\frac{t}{2}\ ,$ $\displaystyle
r_{B^{\text{te}}}=0,\dots,p-1\ ,$ (125)

with $d_{B^{\text{te}}}=\tfrac{1}{2}\,t\,p$ .

$q_{1}^{A}$, $q_{1}^{B^{\text{to}}}$$q_{2}^{A}$,
$q_{2}^{B^{\text{to}}}$$r_{A}$, $r_{B^{\text{to}}}$$i_{A}$,
$i_{B^{\text{to}}}$ABCDEFGH $A=\left(1,1+p\right)$
---
$B=\left(\tfrac{1}{2}\left(\mu_{A}+1\right),\tfrac{1}{2}\left(-\mu_{A}+2p+3\right)\right)$
$C=\left(\tfrac{1}{2}\left(\mu_{A}+t\right),\tfrac{1}{2}\left(-\mu_{A}+t+2p+2\right)\right)$
$D=\left(\tfrac{1}{2}\left(t+1\right),\tfrac{1}{2}\left(t+2p+1\right)\right)$
$E=\left(2,1+p\right)$
---
$F=\left(\tfrac{1}{2}\left(\mu_{B^{\text{to}}}+3\right),\tfrac{1}{2}\left(-\mu_{B^{\text{to}}}+2p+3\right)\right)$
$G=\left(\tfrac{1}{2}\left(\mu_{B^{\text{to}}}+t\right),\tfrac{1}{2}\left(-\mu_{B^{\text{to}}}+t+2p\right)\right)$
$H=\left(\tfrac{1}{2}\left(t+1\right),\tfrac{1}{2}\left(t+2p-1\right)\right)$

(a) Exchanged operators $\mathcal{O}_{q_{1}q_{2}}$ for odd $t$ parametrised
according to (6.1). The black and white nodes denote class $A$ and class
$B^{\text{to}}$ operators, respectively. The only operator acquiring non-zero
anomalous dimension in the 2-derivative sector is the one denoted by $A$. The
nodes in the grey rectangle correspond to operators which acquire non-zero
anomalous dimensions in the six-derivative sector.

$q_{1}^{B^{\text{te}}}$$q_{2}^{B^{\text{te}}}$$r_{B^{\text{te}}}$$i_{B^{\text{te}}}$IJKLMNOP
$I=\left(1,1+p\right)$
---
$J=\left(\lfloor{\frac{p+1}{2}\rfloor},p-\lfloor{\frac{p+1}{2}\rfloor}+2\right)$
$K=\left(\lfloor{\frac{p+1}{2}\rfloor}+\frac{t-2}{2},p-\lfloor{\frac{p+1}{2}\rfloor}+\frac{t+2}{2}\right)$
$L=\left(\frac{t}{2},\frac{t}{2}+p\right)$
$M=\left(2,1+p\right)$
---
$N=\left(\lfloor{\frac{p+2}{2}\rfloor},p-\lfloor{\frac{p+2}{2}\rfloor}+3\right)$
$O=\left(\lfloor{\frac{p+2}{2}\rfloor}+\frac{t-2}{2},p-\lfloor{\frac{p+2}{2}\rfloor}+\frac{t+4}{2}\right)$
$P=\left(1+\frac{t}{2},\frac{t}{2}+p\right)$

(b) The exchanged operators $\mathcal{O}_{q_{1}q_{2}}$ for even $t$
parametrised according to (6.1). All operators contributing at even $t$ belong
to class B and are split into two groups which do not mix with each other. The
black and white nodes denote operators with even and odd $r_{B^{\text{te}}}$,
respectively. The nodes in the grey rectangle correspond to operators which
acquire non-zero anomalous dimensions in the six-derivative sector.

###  6.2 Unmixing

We have seen above that there can be many different double-trace operators
with the same quantum numbers $(\Delta,p)$ at leading order in $1/c$. To solve
this mixing problem we follow the method developed for $\mathcal{N}=4$ SYM in
[24, 12].

First we expand the correlator in terms of long blocks as follows:

$\displaystyle
H(x,y)=\frac{x\,y}{x-y}\sum_{\Delta,p}A^{p_{i}}_{\Delta,p}B^{\text{long}}_{\Delta,p,p_{12},p_{34}}(x,y)\
,$ (126)

where the superconformal blocks are given in (2.2). The coefficients of the
decomposition are given as a sum of squares of OPE coefficients as follows

$A^{p_{i}}_{\Delta,p}=\sum_{{\mathcal{O}}^{\Delta,p}}C_{p_{1}p_{2}\mathcal{O}}C_{p_{3}p_{4}\mathcal{O}}\
,$ (127)

where the sum goes over the degenerate operators. Expanding the OPE data in
$1/c$ and $a$ (which is the analogue of a string coupling introduced in
section 2.3), gives

$\displaystyle\Delta_{{\mathcal{O}}}$
$\displaystyle=\Delta^{(0)}+\frac{1}{c}\left(\gamma^{\text{(2)}}+a\,\gamma^{\text{(6)}}+\dots\right)+\mathcal{O}(c^{-2})\
,$ $\displaystyle C_{pp\mathcal{O}}$
$\displaystyle=C_{pp\mathcal{O}}^{(0)}+\left(a\,C_{pp\mathcal{O}}^{\text{(6)}}+\dots\right)+{\mathcal{O}}(c^{-1})\
,$ (128)

where the anomalous dimensions $\gamma$ depend on
$\left\\{\Delta,p,i,r\right\\}$, and the superscripts $\left\\{0,2,6\right\\}$
label the free, 2-derivative, and 6-derivative sectors, respectively. Note
that we allow for $\mathcal{O}(a)$ corrections to the OPE coefficients which
are not suppressed in $1/c$. Such corrections could in principle arise and
were shown to be non-zero in $\mathcal{N}=4$ SYM [12].

Plugging (6.2) into (126) then gives

$\displaystyle H(x,y)=$
$\displaystyle\frac{x\,y}{x-y}\sum_{\Delta^{(0)},\,p}\big{[}A^{{(0)}}_{\Delta^{(0)},\,p}B^{\text{long}}_{\Delta^{(0)},\,p,p_{12},p_{34}}(x,y)$
$\displaystyle+\frac{1}{c}\log
u\sum_{\Delta^{(0)},\,p}\left(M^{\text{(2)}}_{\Delta^{(0)},\,p}+a\,M^{\text{(6)}}_{\Delta^{(0)},\,p}+\dots\right)B^{\text{long}}_{\Delta^{(0)},\,p,p_{12},p_{34}}(x,y)+\dots\big{]}\
,$ (129)

where

$\displaystyle A^{(0)}_{\Delta,p}$
$\displaystyle=\sum_{{\mathcal{O}}^{\Delta,p}}C^{(0)}_{p_{1}p_{2}{\mathcal{O}}}C^{(0)}_{p_{3}p_{4}{\mathcal{O}}}\
,\quad
M^{\text{(2)}}_{\Delta,p}=\sum_{{\mathcal{O}}^{\Delta,p}}\gamma^{\text{(2)}}C^{(0)}_{p_{1}p_{2}{\mathcal{O}}}C^{(0)}_{p_{3}p_{4}{\mathcal{O}}}\
,$ $\displaystyle M^{\text{(6)}}_{\Delta,p}$
$\displaystyle=\sum_{{\mathcal{O}}^{\Delta,p}}\left(\gamma^{\text{(6)}}C^{(0)}_{p_{1}p_{2}{\mathcal{O}}}C^{(0)}_{p_{3}p_{4}{\mathcal{O}}}+\gamma^{\text{(2)}}C^{(0)}_{p_{1}p_{2}{\mathcal{O}}}C^{\text{(6)}}_{p_{3}p_{4}{\mathcal{O}}}+\gamma^{\text{(2)}}C^{\text{6}}_{p_{1}p_{2}{\mathcal{O}}}C^{(0)}_{p_{3}p_{4}{\mathcal{O}}}\right)\
,$ $\displaystyle 0$
$\displaystyle=\quad\sum_{{\mathcal{O}}^{\Delta,p}}C_{p_{1}p_{2}{\mathcal{O}}}^{(0)}C_{p_{3}p_{4}{\mathcal{O}}}^{(6)}+C_{p_{1}p_{2}{\mathcal{O}}}^{(6)}C_{p_{3}p_{4}{\mathcal{O}}}^{(0)}\
,$ (130)

and we denote $\Delta^{(0)}$ by $\Delta$ for simplicity. The last line follows
from the fact that there are no $\mathcal{O}(a)$ corrections to the correlator
since in the bulk stringy states only appear on internal lines and are
therefore accompanied by Newton’s constant to give $\mathcal{O}(a/c)$
corrections. Note that $A^{(0)}$, $M^{\text{(2)}}$, $M^{\text{(6)}}$ are
determined from the conformal partial wave expansions of correlators in the
free, 2-derivative, and 6-derivative sectors, respectively. Given this data,
one then solves the above equations for the anomalous dimensions and OPE
coefficients $\gamma$ and $C$, respectively. These are referred to as the
unmixing equations.

It is convenient to write (6.2) in matrix form as follows:

$\displaystyle\mathcal{O}(1):$ $\displaystyle\ \hat{A}_{\Delta,p}^{(0)}$
$\displaystyle=\mathbb{C}^{(0)}\left(\mathbb{C}^{(0)}\right)^{T}\ ,$
$\displaystyle\mathcal{O}\left(1/c\right):$ $\displaystyle\
\hat{M}_{\Delta,p}^{\text{(2)}}$
$\displaystyle=\mathbb{C}^{(0)}\hat{\gamma}^{\text{(2)}}\left(\mathbb{C}^{(0)}\right)^{T}\
,$ $\displaystyle\mathcal{O}(a):$ $\displaystyle\ 0$
$\displaystyle=\mathbb{C}^{(0)}\left(\mathbb{C}^{\text{(6)}}\right)^{T}+\mathbb{C}^{\text{(6)}}\left(\mathbb{C}^{(0)}\right)^{T}\
,$ $\displaystyle\mathcal{O}\left(a/c\right):$ $\displaystyle\
\hat{M}^{\text{(6)}}_{\Delta,p}$
$\displaystyle=\mathbb{C}^{(0)}\hat{\gamma}^{\text{(6)}}\left(\mathbb{C}^{(0)}\right)^{T}+\mathbb{C}^{(0)}\hat{\gamma}^{\text{(2)}}\left(\mathbb{C}^{\text{(6)}}\right)^{T}+\mathbb{C}^{\text{(6)}}\hat{\gamma}^{\text{(2)}}\left(\mathbb{C}^{(0)}\right)^{T}\
.$ (131)

In more detail, the matrices are
$\left((d_{A}+d_{B^{\text{to}}})\times(d_{A}+d_{B^{\text{to}}})\right)$ and
$(d_{B^{\text{te}}}\times d_{B^{\text{te}}})$ for odd and even $t$,
respectively. Note that $\hat{\gamma}$ is a diagonal matrix of anomalous
dimensions, $\hat{A}^{(0)}$ is a diagonal matrix, and $\hat{M}^{\text{(2)}}$
and $\hat{M}^{\text{(6)}}$ are symmetric matrices. When $t$ is odd, the
matrices are block-diagonal, so class $A$ and $B$ operators can be treated
independently, and similarly when $t$ is even operators with
$r_{B^{\text{te}}}$ even or odd can be treated independently.

In practice, it is useful to define a matrix $\tilde{c}$ such that

$\displaystyle\tilde{c}\tilde{c}^{T}=\text{Id}\
,\quad\mathbb{C}^{(0)}=\left(\hat{A}^{(0)}\right)^{\tfrac{1}{2}}\cdot\tilde{c}\
.$ (132)

The unmixing equations in the 2-derivative sector then become:

$\displaystyle\tilde{c}\cdot\hat{\gamma}^{\text{(2)}}\cdot\tilde{c}^{T}=\left(\hat{A}^{(0)}\right)^{-\frac{1}{2}}\cdot\hat{M}^{\text{(2)}}\cdot\left(\hat{A}^{(0)}\right)^{-\frac{1}{2}}\
.$ (133)

From this we see that the columns of $\tilde{c}$ are eigenvectors of the
matrix
$\left(\hat{A}^{(0)}\right)^{-\frac{1}{2}}\cdot\hat{M}^{\text{(2)}}\cdot\left(\hat{A}^{(0)}\right)^{-\frac{1}{2}}$
and the corresponding eigenvalues are the anomalous dimensions. The anomalous
dimensions can also be computed directly as the eigenvalues of the matrix
$\hat{M}^{\text{(2)}}\cdot\left(\hat{A}^{(0)}\right)^{-1}$. Similarly, the
anomalous dimensions in the 6-derivative sector are the ${\mathcal{O}}(a)$
eigenvalues of
$\left(\hat{M}^{\text{(2)}}+a\hat{M}^{\text{6-deriv}}\right)\cdot\left(\hat{A}^{(0)}\right)^{-1}$.

Let us illustrate the method with a few simple examples in the 2-derivative
sector. For the simplest case $\left(\Delta,p\right)=\left(1,0\right)$ there
is only one exchanged operator of the form (122):
${\mathcal{O}}_{1}{\mathcal{O}}_{1}$. Performing the conformal block expansion
of the correlator $H^{\text{(2)}}_{1111}$ in (71) and comparing to (6.2), we
find that the coefficients of the expansion for
$\left(\Delta,p\right)=\left(1,0\right)$ are

$\displaystyle A^{(0)}_{1111}(1,0)=1\ ,\quad M^{\text{(2)}}_{1111}(1,0)=2.$
(134)

The unmixing equations in (6.2) are then simply

$A^{(0)}_{1111}(1,0)=\left(C^{(0)}_{1,0}\right)^{2}\ ,\quad
M^{\text{(2)}}_{1111}(1,0)=\gamma^{\text{(2)}}_{1,0}\left(C^{(0)}_{1,0}\right)^{2}\
.$ (135)

Solving these equations then gives

$\gamma^{\text{(2)}}_{1,0}=2\ ,\quad C^{(0)}_{1,0}=1\ .$ (136)

The next simplest example is $\left(\Delta,p\right)=\left(3,0\right)$. In this
case, there are two possible exchanged operators,
${\mathcal{O}}_{2}{\mathcal{O}}_{2}$ and
${\mathcal{O}}_{1}\partial^{2}{\mathcal{O}}_{1}$, and we have

$\displaystyle\hat{A}^{(0)}_{3,0}=\begin{pmatrix}A^{(0)}_{1111}&0\\\
0&A^{(0)}_{2222}\end{pmatrix}_{(3,0)}=\begin{pmatrix}\frac{1}{10}&0\\\
0&\frac{1}{18}\end{pmatrix}\ ,\
\hat{M}^{\text{(2)}}_{3,0}=\begin{pmatrix}M^{\text{(2)}}_{1111}&M^{\text{(2)}}_{1122}\\\
M^{\text{(2)}}_{1122}&M^{\text{(2)}}_{2222}\end{pmatrix}_{(3,0)}=\begin{pmatrix}\frac{1}{5}&\frac{1}{3}\\\
\frac{1}{3}&\frac{5}{9}\end{pmatrix}\ .$ (137)

Solving the unmixing equations in (6.2):

$\displaystyle\hat{\gamma}^{\text{(2)}}_{3,0}=\begin{pmatrix}12&0\\\
0&0\end{pmatrix}\
,\quad\mathbb{C}^{(0)}_{3,0}=\begin{pmatrix}\frac{1}{2\sqrt{15}}&\frac{1}{2\sqrt{3}}\\\
\frac{\sqrt{5}}{6\sqrt{3}}&-\frac{1}{6\sqrt{3}}\end{pmatrix}\ .$ (138)

Note that there is only one non-zero anomalous dimension. This structure
extends to general $\Delta$ and $p$ in the 2-derivative sector: at each weight
there is only one non-zero anomalous dimension, which is a rational number. In
the 6-derivative sector all the anomalous dimensions will generically be non-
zero and contain square roots. We will provide an explanation for this in the
next subsection. See Appendix C for further examples of unmixing.

###  6.3 Anomalous Dimensions

We will now apply the unmixing procedure described in the previous subsection
to compute anomalous dimensions of double trace operators contributing to the
conformal partial wave expansion of 4-point correlators in the 2-derivative
and 6-derivative sectors. We obtain general formulas for the anomalous
dimensions by working out many examples.

Let us start with the 2-derivative sector. In that case we find that only
class $A$ operators acquire anomalous dimensions while class $B$ operators
decouple. In more detail, we find that only one operator acquires a non-zero
anomalous dimension for each $\left(\Delta,p\right)$, and the value is given
by

$\gamma_{\Delta,p}^{\text{(2)}}=\delta^{(2)}=(\Delta-p)(\Delta+1+p).$ (139)

We recognize this to be the eigenvalue of the Casimir $\Delta^{(2)}$ in (3.1)
acting on superconformal blocks:

$\Delta^{(2)}\left(\frac{x\,y}{x-y}B^{\text{long}}_{\Delta,p,p_{12},p_{34}}\right)=\delta^{(2)}\left(\frac{x\,y}{x-y}B^{\text{long}}_{\Delta,p,p_{12},p_{34}}\right)\
.$ (140)

This was anticipated from 4d conformal symmetry in (103). The fact that only
one operator acquires a non-zero anomalous dimension for each
$\left(\Delta,p\right)$ can be understood from the following formula relating
4d spin to 1d quantum numbers:

$\displaystyle
l_{4d}=2\left(i_{A}+r_{A}+i_{B^{\text{to}}}+i_{B^{\text{te}}}+r_{B^{\text{te}}}-\frac{1+(-1)^{r_{B^{\text{te}}}+1}}{2}\right)\
,$ (141)

which has an analogous form in $\mathcal{N}=4$ SYM [12]. In particular,
recalling that correlators in the 2-derivative sector come from the 4d spin-0
block (as explained in section 4.4) and setting $l_{4d}=0$, we see that class
$B$ operators cannot contribute since $i_{B}\geq 1$ and only the class $A$
operator with $i_{A}=r_{A}=0$ can contribute.

We can similarly see that both class $A$ and class $B$ operators acquire
anomalous dimensions in the six derivative sector, since the corresponding
correlators can be expanded in terms of 4d blocks with spin-0 and spin-2, as
explained in section 5.1. It is useful to separately consider the cases where
$t=\Delta-p$ is odd or even. In the results below we have set the coefficient
$B_{0}=4$ in (111) for notational simplicity. One can obtain anomalous
dimensions for general $B_{0}$ by inserting a factor of $B_{0}/4$ in all terms
that are not multiplied by $C_{0}$.

#### odd $t$

As explained above, there are two possibilities: $l_{4d}=0,2$. The condition
$l_{4d}=0$ can only be satisfied by class $A$ operators with quantum numbers
$i_{A}=r_{A}=0$ and in this case the anomalous dimensions are

$\left(\gamma^{A}\right)_{0,0}^{\Delta,p}=\delta^{(2)}\left(\frac{2}{9}\,(1-6\,C_{0})+\frac{1}{18}\,(1+12\,C_{0})\,\delta^{(2)}+\frac{5}{12}\,\left(\delta^{(2)}\right)^{2}+\frac{1}{9}\,(1+12\,C_{0})\,\delta^{(y)}\right)\
,$ (142)

where $\delta^{(2)}$ and $\delta^{(y)}$ are the eigenvalues of $\Delta^{(2)}$
and $\mathcal{D}_{y}$ in (3.1), respectively:

$\delta^{(2)}=(\Delta-p)(\Delta+p+1)\ ,\quad\delta^{(y)}=p(p+1)\ .$ (143)

For $l_{4d}=2$, there are three possible non-zero anomalous dimensions:
$\left(i_{A},r_{A}\right)\in\left\\{\left(1,0\right),\left(0,1\right)\right\\}$
for class $A$, and $i_{B^{\text{to}}}=1$ for class $B$, and the corresponding
anomalous dimensions are

$\displaystyle\left(\gamma^{A}\right)_{1,0}^{\Delta,p}=$
$\displaystyle\frac{1}{180}\,\delta^{(2)}\Big{(}4-2\,\delta^{(2)}+3\,\left(\delta^{(2)}\right)^{2}-4\,\delta^{(y)}$
$\displaystyle-4\,\sqrt{8\,p^{3}+4\,p^{4}-4\,p^{2}\Delta(1+\Delta)-4\,p\,(1+\Delta+\Delta^{2})+(-1+2\,\Delta+2\,\Delta^{2})^{2}}\Big{)}\
,$ (144) $\displaystyle\left(\gamma^{A}\right)_{0,1}^{\Delta,p}=$
$\displaystyle\frac{1}{180}\,\delta^{(2)}\Big{(}4-2\,\delta^{(2)}+3\,\left(\delta^{(2)}\right)^{2}-4\,\delta^{(y)}$
$\displaystyle+4\,\sqrt{8\,p^{3}+4\,p^{4}-4\,p^{2}\Delta(1+\Delta)-4\,p\,(1+\Delta+\Delta^{2})+(-1+2\,\Delta+2\,\Delta^{2})^{2}}\Big{)}\
,$ (145) $\displaystyle\left(\gamma^{B^{\text{to}}}\right)_{1}^{\Delta,p}=$
$\displaystyle\frac{1}{60}\,\delta^{(2)}\left(-2\,\delta^{(2)}+\left(\delta^{(2)}\right)^{2}-4\,\delta^{(y)}\right)\
.$ (146)

Unlike in the 2-derivative sector, all operators have non-zero anomalous
dimensions and the latter contain square roots, which signals the breaking of
4d conformal symmetry. This is consistent with the conclusions in section 5.3.

#### even $t$

For even $t$, operators only start to contribute for $l_{4d}=2$ and we get two
different non-zero anomalous dimensions for quantum numbers
$\left(i_{B^{\text{te}}},r_{B^{\text{te}}}\right)\in\left\\{\left(1,0\right),(1,1)\right\\}$:

$\left(\gamma^{B^{\text{te}}}\right)_{1,0}^{\Delta,p}=\frac{1}{60}\left(\delta^{(2)}\right)^{2}\left(\delta^{(2)}-2\right)\
,$ (147)
$\left(\gamma^{B^{\text{te}}}\right)_{1,1}^{\Delta,p}=\frac{1}{60}\left(\delta^{(2)}\right)^{2}\left(\delta^{(2)}+2\right)\
.$ (148)

Note that (147) is proportional to the eigenvalue of $\Delta_{\text{2}}^{(4)}$
in (121).

##  7 Comments on $\mathcal{N}=4$ SYM

The main focus of this paper has been on 1d ${\mathcal{N}}=4$ SCFT, its
relation to a bulk theory on AdS${}_{2}\times$S2 and in particular the
investigation of a hidden 4d symmetry. The primary motivation for this came
from the 4d ${\mathcal{N}}=4$ SCFT context, related to string theory in
AdS${}_{5}\times$S5, which was shown to possess a hidden 10d symmetry [5].
This 10d symmetry has been further investigated in a number of works (see e.g.
[17, 13, 37, 38]). It was also generalised to AdS${}_{3}\times$S3 in [7, 8, 9,
10]. Thus we can view the AdS${}_{2}\times$S2 setting here as a toy model for
the AdS${}_{5}\times$S5 case. In this section we then draw out some of the
lessons learned in AdS${}_{2}\times$S2 and apply them to ${\mathcal{N}}=4$
SYM.

We will only very briefly review the ${\mathcal{N}}=4$, AdS${}_{5}\times$S5
story here, and will do so mainly by comparison to the 1d story detailed in
earlier parts of the paper. Indeed the entire formalism we have used makes the
generalisation particularly natural, needing minimal modification. The SCFT
(${\mathcal{N}}=4$ SYM) naturally lives on analytic superspace [28, 29, 27]
which has the same matrix form as in the 1d case (6) but where each term
$x,y,\theta,\bar{\theta}$ becomes itself a $2\times 2$ matrix. The half BPS
superfield of (16) thus has more component fields in its expansion – since
there are now four $\theta$s and four $\bar{\theta}$s – but we will still
focus on just two important component fields: the highest weight state
${\mathcal{O}}_{p}(x,y)$ and the highest term in the $\theta$ (only)
expansion, $L_{p}(x,y)$:

$\displaystyle{\bf{O}}_{p}(x,y)={\mathcal{O}}_{p}(x,y)+..+\theta^{4}L_{p}(x,y)+..+\bar{\theta}^{4}\bar{L}_{p}(x,y)+..\
.$ (149)

Here ${\mathcal{O}}_{p}$ has scaling dimension $p$ and $SU(4)$ rep $[0,p,0]$
whereas $L_{p}$ has dimension $p+2$ and $SU(4)$ rep $[0,p-2,0]$.

As in the 1d case (see (58), (73)) we define a single object which generates
all (appropriately normalised) half BPS operators ${\mathcal{O}}_{p}$ and
their descendants $L_{p}$:

$\displaystyle{\mathcal{O}}=\sum_{p=2}^{\infty}{\mathcal{O}}_{p}\qquad\qquad
L=\sum_{p=2}^{\infty}L_{p}\ .$ (150)

Then the 10d symmetry of free and $1/c$ (tree level supergravity) discovered
in [5] takes the form (compare this with the 1d SCFT equivalent cases with
manifest 4d symmetry (3.1),(72))

$\displaystyle\langle LL\bar{L}\bar{L}\rangle_{c^{0}}$
$\displaystyle=\left[\frac{1}{x_{14}^{8}x_{23}^{8}}+\frac{1}{x_{13}^{8}x_{24}^{8}}\right]_{10d},$
(151)
$\displaystyle\langle{\mathcal{O}}{\mathcal{O}}{\mathcal{O}}{\mathcal{O}}\rangle_{1/c}$
$\displaystyle={\mathcal{I}}_{{\mathcal{N}}=4}\times\left[\frac{\bar{D}_{2422}}{x_{13}^{8}x_{24}^{8}}\right]_{10d},\
$ (152)

where we are neglecting overall numerical coefficients,

$\displaystyle{\mathcal{I}}_{{\mathcal{N}}=4}$
$\displaystyle=(x-y)(x-\bar{y})(\bar{x}-y)(\bar{x}-\bar{y})x_{13}^{4}x_{24}^{4}y_{13}^{4}y_{24}^{4},$
(153)

and

$\displaystyle\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}}$
$\displaystyle=x\bar{x}\qquad\frac{x_{14}^{2}x_{23}^{2}}{x_{13}^{2}x_{24}^{2}}=(1-x)(1-\bar{x})$
$\displaystyle\frac{y_{12}^{2}y_{34}^{2}}{y_{13}^{2}y_{24}^{2}}$
$\displaystyle=y\bar{y}\qquad\frac{y_{14}^{2}y_{23}^{2}}{y_{13}^{2}y_{24}^{2}}=(1-y)(1-\bar{y})\
.$ (154)

Now similarly to the 1d case (see (48)), the correlator of descendants is
related to the correlator of primaries by a Casimir acting at points 1 and 2.
This time the Casimir is eighth order [39, 5]

$\displaystyle\langle
L_{p_{1}}L_{p_{2}}\bar{L}_{p_{3}}\bar{L}_{p_{4}}\rangle={\mathcal{I}}_{{\mathcal{N}}=4}^{-1}\,\mathcal{C}^{(8);SU(2,2|4)}_{1,2}\langle{\mathcal{O}}_{p_{1}}{\mathcal{O}}_{p_{2}}{\mathcal{O}}_{p_{3}}{\mathcal{O}}_{p_{4}}\rangle\
,$ (155)

where ${\mathcal{C}}_{1,2}^{(8);SU(2,2|4)}$ denotes an eighth order
superconformal Casimir defined explicitly as:

$\displaystyle{\mathcal{C}}_{1,2}^{(8);SU(2,2|4)}$
$\displaystyle=\hat{P}_{p_{i}}\left(\frac{(x{-}y)(x{-}\bar{y})(\bar{x}{-}y)(\bar{x}{-}\bar{y})}{x\bar{x}y\bar{y}(x{-}\bar{x})(y{-}\bar{y})}\right)\Delta^{(8)}\left(\frac{x\bar{x}y\bar{y}(x{-}\bar{x})(y{-}\bar{y})}{(x{-}y)(x{-}\bar{y})(\bar{x}{-}y)(\bar{x}{-}\bar{y})}\right)\hat{P}_{p_{i}}^{-1}\
,$ $\displaystyle\Delta^{(8)}$
$\displaystyle=\left({\mathcal{D}}^{+}_{x}-{\mathcal{D}}^{-}_{y}\right)\left({\mathcal{D}}^{+}_{\bar{x}}-{\mathcal{D}}^{-}_{y}\right)\left({\mathcal{D}}^{+}_{x}-{\mathcal{D}}^{-}_{\bar{y}}\right)\left({\mathcal{D}}^{+}_{\bar{x}}-{\mathcal{D}}^{-}_{\bar{y}}\right)\
,$ (156)

with ${\mathcal{D}}^{\pm}_{z}$ defined in (3.1c) and $\hat{P}_{p_{i}}$ in (10)
(but with now $g_{ij}=y_{ij}^{2}/x_{ij}^{2}$). The superconformal Casimirs are
related to Casimirs for the maximal bosonic subgroup via conjugation with
${\mathcal{I}}_{{\mathcal{N}}=4}$ as discussed in (191):

$\displaystyle\mathcal{C}^{SU(2,2|4)}_{1,2}=\left(\frac{(x{-}y)(x{-}\bar{y})(\bar{x}{-}y)(\bar{x}{-}\bar{y})}{(x\bar{x}y\bar{y})^{2}}\right)\,\mathcal{C}^{SO(1,5)\times
SO(6)}_{1,2}\,\left(\frac{(x\bar{x}y\bar{y})^{2}}{(x{-}y)(x{-}\bar{y})(\bar{x}{-}y)(\bar{x}{-}\bar{y})}\right)\
.$ (157)

Putting all this together we see that for the descendant correlator at order
$1/c$ (i.e. the supergravity approximation) the 10d symmetry manifests as a
$SO(1,5)\times SO(6)$ Casimir acting on a 10d symmetric function (again
similarly to in 1d - compare with (85))

$\displaystyle\langle
LL\bar{L}\bar{L}\rangle_{1/c}=\mathcal{C}^{(8);SO(1,5)\times
SO(6)}_{1,2}\,\left[\frac{\bar{D}_{2422}}{x_{13}^{8}x_{24}^{8}}\right]_{10d}\
.$ (158)

We now examine whether higher-derivative corrections can be written as higher-
order Casimirs acting on 10d functions. We find that the answer is yes for the
$\alpha^{\prime 3}$ correction, but for higher corrections things are not so
simple.

###  7.1  10d conformal symmetry at $O(\alpha^{\prime 3})$

Recently the 4-point correlators of all $1/2$-BPS operators at
$O(\alpha^{\prime 3}/c)$ have been obtained in Mellin space [14, 15, 16] and
translated to position space [13]. We now claim that these $\alpha^{\prime 3}$
corrections can be written as a fourteenth order Casimir of the maximal
bosonic subgroup acting on a 10 dimensional function:

$\displaystyle\left\langle LL\bar{L}\bar{L}\right\rangle|_{\alpha^{\prime
3}/c}$ $\displaystyle=\mathcal{C}^{(8+6);SO(1,5)\times
SO(6)}_{1,2}\left[\frac{\bar{D}_{4411}(u,v)}{x_{13}^{8}x_{24}^{8}}\right]_{10d}\
.$ (159)

where the fourteenth order Casimir factors
$\mathcal{C}^{(8+6)}=\mathcal{C}^{(8)}\mathcal{C}^{(6)}$ into the above eighth
order Casimir $\mathcal{C}^{(8)}$ times a sixth order Casimir
$\mathcal{C}^{(6)}$ which is defined similarly to (7) but in terms of

$\displaystyle\Delta^{(6)}$
$\displaystyle=-\tfrac{1}{8}\left({\mathcal{D}}_{x}^{+}{+}{\mathcal{D}}_{\bar{x}}^{+}{-}{\mathcal{D}}_{y}^{-}{-}{\mathcal{D}}_{\bar{y}}^{-}\right)^{3}+\left(({\mathcal{D}}_{x}^{+})^{2}{+}({\mathcal{D}}_{\bar{x}}^{+})^{2}{-}({\mathcal{D}}_{y}^{-})^{2}{-}({\mathcal{D}}_{\bar{y}}^{-})^{2}\right)-\tfrac{3}{2}\left({\mathcal{D}}_{x}^{+}{+}{\mathcal{D}}_{\bar{x}}^{+}{-}{\mathcal{D}}_{y}^{-}{-}{\mathcal{D}}_{\bar{y}}^{-}\right)\
.$ (160)

The form (159) follows naturally from the 10d structure of the free theory
(151) and the supergravity correction (158) together with the hint that
dimensional analysis of the corresponding term in the effective action
translates into a Casimir of a certain order (here sixth order beyond
supergravity due to the dimension six coefficient $\alpha^{\prime 3}$).
Further just as the supergravity correction can be expanded in 10d blocks [5]
so can the $\alpha^{\prime 3}$ correction. On the other hand, the
$\alpha^{\prime 3}$ correction arises from an effective 10d
AdS${}_{5}\times$S5 scalar contact Witten diagram [13], and thus the classic
arguments of [34] lifted to the 10d situation imply that only a 10d spin-0
block should appear. Indeed the $\log u$ part of $\bar{D}_{4411}$ is precisely
the 10d spin-0 block, just as the 1d CFT correlators in the two-derivative
sector arise from a 4d spin-0 block (95).

The precise form of the sixth order Casimir (160) was found by taking an
arbitrary linear combination of the Casimirs of order six and below (or more
accurately Casimirs of order 14 down to 8 containing a $\Delta^{(8)}$ factor
using the basis in (188)). We then evaluated the action of these operators on
the $\log u$ part of the 10d function $\bar{D}_{4411}$ expanded into 4d
superblocks. In fact, since the 4d blocks are eigenfunctions of the Casimirs
and the eigenvalues are explicitly known [48] this becomes simple algebra. The
resulting anomalous dimensions were then compared with the results for these
anomalous dimensions found in [12]. Matching the results gave the above unique
Casimir.

The result (159) can instead be written in terms of the correlator of
primaries via (155)

$\displaystyle\langle{\mathcal{O}}{\mathcal{O}}{\mathcal{O}}{\mathcal{O}}\rangle|_{\alpha^{\prime
3}/c}$
$\displaystyle={\mathcal{I}}_{{\mathcal{N}}=4}\times\mathcal{C}^{(6);SO(1,5)\times
SO(6)}_{1,2}\left[\frac{\bar{D}_{4411}(u,v)}{x_{13}^{8}x_{24}^{8}}\right]_{10d}\
.$ (161)

One can also check explicitly that the component correlators agree with known
results, for example for the lowest component term one can check that

$\displaystyle\frac{x\bar{x}y\bar{y}}{(x-\bar{x})(y-\bar{y})}\Delta^{(6)}\frac{(x-\bar{x})(y-\bar{y})}{x\bar{x}y\bar{y}}\left(u^{4}\bar{D}_{4411}\right)=60\bar{D}_{4444}\
.$ (162)

This then reproduces the result of [14] for this correction. One can also
verify higher charge correlators by comparing with the results in Mellin space
derived in [15, 12] or in position space derived in [13].

###  7.2  10d conformal symmetry at $O(\alpha^{\prime 5})$

We now wish to see if the above structure generalises to $O(\alpha^{\prime
5})$. Similarly to the 6-derivative sector in AdS${}_{2}\times$S2, we will
find that it is not possible to reproduce the predictions of the effective
action by acting with Casimirs on 10d conformal blocks except for a certain
infinite class of operators. We then consider an alternative approach by
constructing an integrand with 10d conformal symmetry from the generalised
Witten diagrams of the effective action. Remarkably, here we do find a unique
candidate possessing 10d symmetry at the level of the integrand. The
combination of terms in the effective action which possess this integrand-
level symmetry do not agree with the predictions arising from localisation
results [25] in ${\mathcal{N}}=4$ SYM however. These results do provide an
alternative point of view on the breaking of 10d conformal symmetry by
$\alpha^{\prime 5}$ corrections, which was first suggested by the structure of
anomalous dimensions obtained after unmixing [12], and will hopefully lead to
a more systematic understanding of how this symmetry is broken.

#### Conformal Functions from 10d Casimirs

Let us then see if all $1/2$-BPS correlators packaged together can be written
as Casimirs acting on 10d conformal functions at $O(\alpha^{\prime 5})$. More
concretely, since the effective action at $O(\alpha^{\prime 5})$ contains four
derivatives (see [13]), following the general arguments of [34] we might
expect the 10d function to be related to 10d spin-0 and spin-2 blocks (just as
the $O(\alpha^{\prime 3})$ correction was related to a 10d spin-0 block).
Furthermore, since the $\alpha^{\prime 5}$ corrections correspond to
10-derivative corrections to supergravity, we expect to act on these blocks
with Casimirs of order $10+8=18$. We therefore propose the following structure
(similar to the one proposed in 1d (119)):

$\displaystyle\left\langle LL\bar{L}\bar{L}\right\rangle|_{\alpha^{\prime
5}/c}|_{\log u}$
$\displaystyle=\mathcal{C}^{(8+10)}_{1,2}\left[\frac{\text{block}_{2}^{d=10}}{X_{12}^{4}X_{34}^{4}}\right]_{10d}+\mathcal{C}^{\prime(8+10)}_{1,2}\left[\frac{\text{block}_{0}^{d=10}}{X_{12}^{4}X_{34}^{4}}\right]_{10d}\
,$ (163)

where the 10d blocks can be found in [5].

Interestingly we find a unique and remarkably simple solution which correctly
reproduces all the $1/2$-BPS correlators of the form $\langle 22pp\rangle$
arising from the simplest four-derivative effective action $\int
d^{10}x(\nabla\phi.\nabla\phi)^{2}$ (called $S^{\text{main}}$ in (5.2a)). The
solution is

$\displaystyle\left\langle LL\bar{L}\bar{L}\right\rangle|_{\text{main}}|_{\log
u}$
$\displaystyle=\mathcal{C}^{(8)}_{1,2}\mathcal{C}^{(8)}_{1,2}(2-\tfrac{1}{2}\mathcal{C}^{(2)}_{1,2})\left(\frac{\text{block}_{2}^{d=10}+28\text{block}_{0}^{d=10}}{X_{12}^{4}X_{34}^{4}}\right)-72\,\mathcal{C}^{(8)}_{1,2}\mathcal{C}^{(8)}_{1,2}\left(\frac{\text{block}_{0}^{d=10}}{X_{12}^{4}X_{34}^{4}}\right)\
,$ (164)

and we note that all the high order Casimirs reduce to products of the eighth
order Casimir (7) together with the quadratic Casimir
$\mathcal{C}^{(2)}_{1,2}$.444The quadratic Casimir is defined in a similar way
to (7) in terms of
$\Delta^{(2)}={\mathcal{D}}^{+}_{x}+{\mathcal{D}}^{+}_{\bar{x}}-{\mathcal{D}}^{-}_{y}-{\mathcal{D}}_{\bar{y}}^{-}$.
However it does not correctly give the more general half BPS operators
$\langle p_{1}p_{2}p_{3}p_{4}\rangle$ resulting from $S_{\text{main}}$. This
is in complete analogy with what happens in the 1d case (see section 5.3 in
particular the last paragraph) where as soon as derivatives first appear in
the effective action, the 4d conformal symmetry is broken. Furthermore there
is no solution of the form (163) which yields the $\alpha^{\prime 5}/c$
correlators predicted by localisation in ${\mathcal{N}}=4$ SYM [25], even for
the case of $\langle 22pp\rangle$ correlators. It therefore appears that
breaking of 10d conformal symmetry is related to the presence of derivatives
in the effective action.

#### A 10d Conformal Integrand

This leads one to ask whether instead the higher conformal symmetry might
remain at the level of the integrand of the sum of generalised Witten diagrams
derived from the effective action rather than explicitly at the functional
level via Casimir operators.555We thank Simon Caron-Huot and Frank Coronado
for suggesting this possibility.

It turns out that this is the case for a specific four-derivative effective
action, although this action does not apparently coincide with the actual one
predicted by the localisation results of [25]. We will write the result for a
SCFT with general dimension $d$, corresponding to higher dimensional conformal
group $SO(2,2d+2)$. For ${\mathcal{N}}=4$ SYM we have $d=4$ whereas for the 1d
CFT we have $d=1$. In detail, we find that the following higher dimensional
effective Lagrangian:

$\displaystyle L_{\text{conf}}=$
$\displaystyle\,\Big{(}2d^{2}\nabla_{\mu}\nabla^{2}\nabla_{\mu}\phi\phi^{3}-2d(1+d)\nabla^{2}\nabla_{\mu}\phi\nabla_{\mu}\phi\phi^{2}-2d^{2}\nabla_{\nu}\nabla_{\mu}\phi\nabla_{\nu}\nabla_{\mu}\phi\phi^{2}$
$\displaystyle+4d(1+d)\nabla_{\nu}\nabla_{\mu}\phi\nabla_{\mu}\phi\nabla_{\nu}\phi\phi-(1+d)(1+2d)\nabla_{\mu}\phi\nabla_{\mu}\phi\nabla_{\nu}\phi\nabla_{\nu}\phi\Big{)}\
,$ (165)

has a corresponding Witten diagram expression which has integrand-level 10d
conformal symmetry.

Replacing each scalar in (7.2) by the higher dimensional bulk to boundary
propagator (28) and summing over permutations yields the Witten diagram
integrand

$\displaystyle\tfrac{1}{6}d^{2}(1+d)\sum_{S_{4}}\left(\frac{\frac{1+d}{4}{\bm{P}}_{1}^{2}{\bm{P}}_{2}^{2}{\bm{X}}_{34}^{2}-\frac{1+d}{2}{\bm{P}}_{1}^{2}{\bm{P}}_{2}{\bm{P}}_{3}{\bm{X}}_{34}{\bm{X}}_{24}+\frac{1+2d}{8}{\bm{P}}_{1}{\bm{P}}_{2}{\bm{P}}_{3}{\bm{P}}_{4}{\bm{X}}_{34}{\bm{X}}_{12}-\frac{1}{8}{\bm{P}}_{1}^{2}{\bm{P}}_{2}^{2}{\bm{P}}_{3}^{2}{\bm{P}}_{4}^{2}}{{\bm{P}}_{1}^{6}{\bm{P}}_{2}^{6}{\bm{P}}_{3}^{6}{\bm{P}}_{4}^{6}}\right),$
(166)

where ${\bm{X}}_{ij}:=X_{i}.X_{j}+Y_{i}.Y_{j}$,
${\bm{P}}_{i}:=P_{i}+Q_{i}=\hat{X}.X_{i}+\hat{Y}.Y_{i}$ . We see that this can
be written entirely in terms of higher dimensional propagators
$P+Q=\hat{X}.X_{i}+\hat{Y}.Y_{i}$ and $SO(2,2d+2)$ invariants
$X_{i}.X_{j}+Y_{i}.Y_{j}$. Furthermore it has uniform weight -4 in each of the
projective coordinates $(X_{i},Y_{i})$. Thus the integrand possesses 10d
conformal symmetry. Compare this for example with the Witten diagrams arising
from the simplest forms of the four-derivative Lagrangians in (5.2) and (5.2)
where the numerators can be seen to destroy the higher dimensional conformal
symmetry even at the integrand level. This is reminiscent of the integrand
level 10d conformal symmetry of perturbative half BPS 4-point functions [40]
noticed in [38].

Now the integration of this integrand is over AdS${}_{5}\times$S5 rather than
the Weyl equivalent 10d flat space, and so the breaking of the symmetry must
arise from this measure. In the case of $\alpha^{\prime 3}$ given by a
$\phi^{4}$ interaction we found that the corresponding integral possesses 10d
conformal symmetry, but only after acting with a sixth order Casimir (163).
This Casimir then presumably arises as a consequence of performing the Weyl
transformation from $AdS_{5}\times S^{5}$ to flat space. In the current case
the Weyl transformation is no longer so simple however (indeed notice that the
weights in the integration variables $\hat{X},\hat{Y}$ is non uniform in
(166)) and a Casimir acting on a 10d function no longer works.

Note that to obtain the Lagrangian in (7.2) we wrote down an ansatz involving
all possible 4-derivative Lagrangians (so all possible ways of acting with 4
non commuting derivatives on 4 scalars) and imposed that the result be a
function of $P_{i},Q_{i}$ through the combinations ${\bm{P}}_{i}:=P_{i}+Q_{i}$
only. Interestingly, the result was then automatically a function of
$X_{i}.X_{j},Y_{i}.Y_{j}$ through the combination
${\bm{X}}_{ij}:=X_{i}.X_{j}+Y_{i}.Y_{j}$ only, without us having to impose
this condition directly. Using integration-by-parts the effective Lagrangian
possessing the integrand level symmetry (7.2) can be written as a specific
linear combination of $L_{\text{main}}$ and $L_{\text{amb}}$ (5.2) as

$\displaystyle
L_{\text{conf}}\,\sim\,(1{+}3d)\Big{(}-(1{+}2d)\nabla_{\mu}\phi\nabla_{\mu}\phi\nabla_{\nu}\phi\nabla_{\nu}\phi-2d\nabla^{2}\nabla_{\mu}\phi\nabla_{\mu}\phi\phi\phi\Big{)}\
.$ (167)

Putting $d=4$ this combination is however not the one predicted by the
localisation results of [25] (compare with the combination in [13] eq(71)).
Thus we see that although there is a unique four derivative effective action
which implies an integrand with 10d conformal symmetry, it is apparently not
the one chosen by ${\mathcal{N}}=4$ SYM.

##  8 Conclusion

In this paper we explored two complementary approaches for computing
correlators of hypermultiplets in AdS${}_{2}\times$S2. The first makes use of
4d conformal symmetry in the free and 2-derivative sectors of the theory,
which makes it possible to deduce 4-point correlators of protected operators
in the dual 1d CFT with arbitrary charges by lifting the lowest charge
correlator to 4d. Note that the dual CFT is defined only formally since we are
not considering gravity in the bulk. In order to realise 4d conformal
symmetry, a crucial role is played by Casimirs of the superconformal group
$SU(1,1|2)$. For example, we must act with these Casimirs on free theory
correlators before lifting to 4d, and after unmixing correlators in the
2-derivative sector, the resulting anomalous dimensions turn out to be
eigenvalues of the Casimirs.

The second approach we develop is based on a 4d scalar effective action
analogous to the 10d effective action for IIB string theory in
AdS${}_{5}\times$S5 [13]. Using the effective action, one can compute
generalised Witten diagrams which treat AdS and S on equal footing, giving a
whole tower of 4-point correlators corresponding to harmonics on the sphere.
Whereas the 10d effective action describes $\alpha^{\prime}$ corrections to
supergravity, the 4d effective action treats the 2-derivative sector (which
arises from dimensional reduction of supergravity) on equal footing with
higher derivative corrections. We find that the predictions of the 4d
effective action are in perfect agreement with those of 4d conformal symmetry
in the 2-derivative sector, but are generally incompatible with 4d conformal
symmetry for higher derivative corrections except for a special infinite class
of correlators. Moreover, after unmixing the correlators obtained from the
higher-derivative effective action, we find that the resulting anomalous
dimensions are no longer rational which further suggests that 4d conformal
symmetry is broken.

We then apply the lessons we learned in 1d to $\mathcal{N}=4$ SYM, for which
there is a lot more data to compare to. Using superconformal Casimirs of
$SU(2,2|4)$, we find that $\alpha^{\prime 3}$ corrections enjoy 10d conformal
symmetry, similarly to 2-derivative interactions in AdS${}_{2}\times$S2, while
$\alpha^{\prime 5}$ corrections generically break the 10d conformal symmetry,
similarly to 6-derivative corrections in AdS${}_{2}\times$S2. So it is not in
general possible to reproduce the predictions of the 10d effective action by
acting with Casimirs on 10d conformal blocks beyond $\alpha^{\prime 3}$,
except for a special class of correlators. We do on the other hand find that
there exists a 6-derivative action (corresponding to a four-derivative scalar
effective action) for which the integrand of the corresponding Witten diagrams
has 10d conformal symmetry, although this combination does not appear to be
consistent with the results of localisation in $\mathcal{N}=4$ SYM at
$O(\alpha^{\prime 5})$.

There are a number of directions for future work. First of all it would be
interesting to have a more systematic understanding of the (breaking of) 10d
conformal symmetry by stringy corrections in AdS${}_{5}\times$S5. Since
$\alpha^{\prime 3}$ corrections enjoy 10d conformal symmetry and are described
by a $\phi^{4}$ interaction in the effective action, the breaking seems to be
associated with derivative interactions in the effective action. It would be
interesting to derive the appearance of the sixth order Casimir at
$O(\alpha^{\prime 3})$ from first principles using a Weyl transformation from
AdS${}_{5}\times$S5 to flat space, and to use similar reasoning to understand
the breaking of 10d conformal symmetry at $O(\alpha^{\prime 5})$ more
systematically. In [5], 10d conformal symmetry was also used to deduce the
leading logarithmic part of all 4-point correlators of $1/2$-BPS operators in
$\mathcal{N}=4$ SYM to all loops in the supergravity approximation, so it
would be interesting to use 4d conformal symmetry to do the analogue of this
for 1d correlators, which we expect to have a simpler structure. For example,
whereas 1-loop correlators in $\mathcal{N}=4$ SYM have transcendentality four,
recent results [35] suggest that they should have transcendentality two in 1d.

Moreover since AdS${}_{2}\times$S2 describes the near-horizon geometry of
extremal 4d black holes, we hope this work will provide new tools for studying
realistic black holes. As a first step, it would be interesting to study
correlators of graviton multiplets AdS${}_{2}\times$S2 and their relation to
$\mathcal{N}=4$ SYK models [41, 42], $\mathcal{N}=4$ JT supergravity and the
associated $\mathcal{N}=4$ Schwarzian theory deduced in [43]. Another
important aspect of AdS2 is that it has two disconnected timelike boundaries.
In this paper, we have only considered correlators on a single boundary, but
it would be interesting to generalise our results to correlators connecting
the two boundaries and explore the effects of higher-derivative corrections in
this context, which may have implications for black hole entropy [26, 44] or
the weak gravity conjecture [45, 46].

Acknowledgements

We thank Francesco Aprile, Simon Caron-Huot, Shai Chester, Frank Coronado,
James Drummond, Matthew Hydeman, Madalena Lemos, Hynek Paul and Michele
Santagata for useful conversations. TA is supported by a Durham Doctoral
Studentship, AL by the Royal Society as a Royal Society University Research
Fellowship holder, and PH by an STFC Consolidated Grant ST/P000371/1. We also
acknowledge support from the European Union’s Horizon 2020 research and
innovation programme under the Marie Sklodowska-Curie grant agreement
No.764850 “SAGEX”.

##  Appendix A Casimirs

In this Appendix we will derive various properties of superconformal Casimir
operators in 1d and 4d.

###  A.1 The quadratic Casimir in $SU(1,1|2)$

Superblocks are eigenfunctions of the quadratic super Casimir at points 1 and
2 acting on the correlator. To consider this we develop the formalism slightly
in order to consider the coordinates $X_{i}$ explicitly as sections on the
super Grassmannian of $1|1$ planes inside a $2|2$ dimensional vector space.
Note that the formalism and super Casimir outlined below generalises naturally
from the supergroup $SU(1,1|2)$ to any supergroup of the form $SU(m,m|2n)$
[27].

Our space is the Grassmannian $Gr(1|1,2|2)$ , the space of $1|1\times 2|2$
matrices $u^{\alpha}{}_{A}$. Here the small Greek indices refer to the local
isotropy group $GL(1|1)$ whilst the big Latin indices refer to the global
group $GL(2|2)$. Explicitly, one can put coordinates on this Grassmannian as

$\displaystyle(u_{i})^{\alpha}{}_{B}=\left(\delta^{\alpha}_{\beta},(X_{i})^{\alpha}{}_{\dot{\beta}}\right)\text{
,
}\bar{u}^{B}{}_{\dot{\alpha}}=\left(\begin{array}[]{c}-(X_{i})^{\beta}{}_{\dot{\alpha}}\\\
\delta^{\dot{\beta}}_{\dot{\alpha}}\end{array}\right),\quad(X_{i})^{\alpha}{}_{\dot{\beta}}=\left(\begin{array}[]{cc}x_{i}&\theta_{i}\\\
\bar{\theta}_{i}&y_{i}\end{array}\right)\ .$ (172)

We thus have
$u^{\alpha}_{iB}\bar{u}^{B}_{j\dot{\alpha}}=X_{ij\dot{\alpha}}^{\alpha}$. Then
the generators of the superconformal group, $SU(1,1|2)$, (at point $i$) are
given simply as

$\displaystyle D^{A}_{iB}=u^{\alpha}_{iA}\frac{\partial}{\partial
u^{\alpha}_{iB}}.$ (173)

The quadratic Casimir operator acting on the 4-point function at points $1$
and $2$ is then given as

$\displaystyle{\mathcal{C}}_{1,2}^{SU(1,1|2)}=\frac{1}{2}(D_{1B}^{A}+D_{2B}^{A})(D_{1A}^{B}+D_{2A}^{B}).$
(174)

Acting with the Casimir on the supercorrelator in the form (199) and commuting
through the prefactor gives

$\displaystyle{\mathcal{C}}_{1,2}^{SU(1,1|2)}\langle\Psi_{p_{1}}\Psi_{p_{2}}\Psi_{p_{3}}\Psi_{p_{4}}\rangle=P_{p_{i}}\times\left[\left(\left(p_{12}{-}p_{34}\right)\left(x^{2}\frac{\partial}{\partial
x}+y^{2}\frac{\partial}{\partial
y}\right)+p_{34}p_{12}(x-y)\right)f+\frac{1}{2}{\mathcal{C}}_{1,2}^{SU(2|2)}f\right].$
(175)

To obtain this we used that $\text{sdet}(M)=\text{exp}(\text{str}\log(M))$ to
deal with differentiating the propagators $g_{ij}:=\text{sdet}(X_{ij}^{-1})$
and then just applied the double derivative directly using
$D_{12B}^{A}u_{iC}^{\alpha}=u_{iB}^{\alpha}\delta^{A}_{C}\text{ and
}D_{12A}^{B}\bar{u}_{i\dot{\delta}}^{C}=-\delta^{C}_{A}\bar{u}_{i\dot{\delta}}^{B}$
for $i=1$ or $2$.

We now consider the Casimir acting on $f(Z)=f(x,y)=$ the conjugation invariant
function of the cross-ratio matrix $Z$, which is thus a function of the
eigenvalues of $Z$ only as discussed above (201). By examining the action of
the Casimir on arbitrary products of traces of powers of $Z$,
$\prod_{i}\text{tr}(Z^{i})^{a_{i}}$, and the corresponding expressions as
polynomials of eigenvalues we find that

$\displaystyle{\mathcal{C}}_{1,2}^{SU(2|2)}f(x,y)$
$\displaystyle=\left[\left(x^{2}\frac{\partial}{\partial
x}-\frac{2xy}{x-y}\right)(1-x)\frac{\partial}{\partial
x}-\left(y^{2}\frac{\partial}{\partial
y}-\frac{2yx}{y-x}\right)(1-y)\frac{\partial}{\partial y}\right]f(x,y)$ (176)
$\displaystyle=\frac{x-y}{xy}\left(x^{2}\partial_{x}(1-x)\partial_{x}-y^{2}\partial_{y}(1-y)\partial_{y}\right)\frac{xyf(x,y)}{x-y}$
(177)

Putting this together, we then obtain the action of the quadratic super
Casimir on the correlator:

$\displaystyle{\mathcal{C}}_{1,2}^{SU(1,1|2)}$
$\displaystyle=P_{p_{i}}\times\frac{x-y}{xy}\Delta^{(2)}\frac{xy}{x-y}P_{p_{i}}^{-1}\
$ $\displaystyle\Delta^{(2)}$
$\displaystyle={\mathcal{D}}^{(p_{12},p_{43})}_{x}-{\mathcal{D}}^{(-p_{12},-p_{43})}_{y}$
$\displaystyle{\mathcal{D}}^{(p_{12},p_{43})}_{x}$
$\displaystyle=x^{2}\partial_{x}(1-x)\partial_{x}+(p_{12}+p_{43})x^{2}\partial_{x}-p_{12}p_{43}x\
.$ (178)

Similar (simpler) computations give us the Casimirs of the subgroups $SU(1,1)$
and $SU(2)$ acting on the correlator as

$\displaystyle{\mathcal{C}}_{1,2}^{SU(1,1)}$
$\displaystyle=P_{p_{i}}{\mathcal{D}}^{(p_{12},p_{43})}_{x}P_{p_{i}}^{-1}$
$\displaystyle{\mathcal{C}}_{1,2}^{SU(2)}$
$\displaystyle=P_{p_{i}}{\mathcal{D}}^{(-p_{12},-p_{43})}_{y}P_{p_{i}}^{-1},$
(179)

and so we can write the action of superconformal Casimir on the correlator
directly in terms of the Casimir of the subgroups:

$\displaystyle{\mathcal{C}}_{1,2}^{SU(1,1|2)}$
$\displaystyle=\frac{x-y}{xy}\left({\mathcal{C}}_{1,2}^{SU(1,1)}-{\mathcal{C}}_{1,2}^{SU(2)}\right)\frac{xy}{x-y}$
$\displaystyle={\mathcal{I}}\left({\mathcal{C}}_{1,2}^{SU(1,1)}-{\mathcal{C}}_{1,2}^{SU(2)}\right){\mathcal{I}}^{-1}\
,$ (180)

where in the second line we recall (49) and use that
$x_{12},x_{34},y_{12},y_{34}$ commute with ${\mathcal{C}}_{12}$.

###  A.2 Higher order Casimirs in $SU(m,m|2n)$ CFTs

The above formulae in fact generalise very nicely in two ways (see [48] for
more details). All higher order Casimirs for 4-point functions in an
$SU(m,m|2n)$ CFT for arbitrary $m,n$ (where we are interested in $m=n=1$ for
the 1d CFT case and $m=n=2$ for the ${\mathcal{N}}=4$ SYM case) are given in
terms of supersymmetric functions of the differential operators
${\mathcal{D}}_{z}^{\pm}$ defined in (3.1). A symmetric function
$f(x_{1},..,x_{n})$ is one which is invariant under permutations of the $n$
variables. It can equivalently be viewed as a function of an $n\times n$
matrix $M$ invariant under conjugation, $f(M)=f(G^{-1}MG)$. The equivalence of
these descriptions occurs by identifying the variables $x_{i}$ with the
eigenvalues of $M$. Similarly a supersymmetric function can be viewed as a
function of an $(m|n)\times(m|n)$ supermatrix $M$ invariant under conjugation
$f(M)=f(G^{-1}MG)$. In terms of the eigenvalues of $M$ which we label
$(x_{1},..,x_{m}|y_{1},..,y_{n})$ the space of such functions is equivalent to
the space of doubly symmetric functions (symmetric separately in the two sets
of variables $x_{i}$ and $y_{j}$) which also satisfies the additional
constraint $\partial f(t,x_{2},..|t,y_{2},..)/\partial t=0$ (see e.g. [49] and
references therein). A nice basis for the supersymmetric functions is provided
by super Schur polynomials.

Any higher order Casimir for a 4-point function in an $SU(m,m|2n)$ theory (or
more precisely its action on conformally invariant functions) is simply a
supersymmetric function $f$ of appropriate differential operators
${\mathcal{D}}_{z}^{\pm}$, conjugated with a super Vandermonde determinant
(and appropriate prefactor):

$\displaystyle f({x_{1}},..,{x_{m}}|{y_{1}},..,{y_{n}})$
$\displaystyle\rightarrow
P_{p_{i}}V^{-1}f({\mathcal{D}}_{x_{1}}^{+},..,{\mathcal{D}}_{x_{m}}^{+}|{\mathcal{D}}_{y_{1}}^{-},..,{\mathcal{D}}_{y_{n}}^{-})VP_{p_{i}}^{-1},$
(181)

where the super Vandermonde determinant is

$\displaystyle V$ $\displaystyle=\frac{\prod_{1\leq i<j\leq
m}(x_{i}^{-1}-x_{j}^{-1})\prod_{1\leq i<j\leq
n}(y_{i}^{-1}-y_{j}^{-1})}{\prod_{\begin{subarray}{c}1\leq i\leq m\\\ 1\leq
j\leq n\end{subarray}}(x_{i}^{-1}-y_{j}^{-1})}\ .$ (182)

The bosonic case ($n=0$) of this result follows from the results of Shimeno
[50] for $BC_{m}$ hypergeometric functions (see theorem 2.1) following the
identification of the higher order Casimirs for blocks with the defining
system of higher order differential equations for these hypergeometric
functions in [48]. The supersymmetric generalisation is then naturally given
by the above expression. The corresponding eigenvalues for superblocks of this
Casimir are also given in [48].

#### Higher order Casimirs in the 1d SCFT

The supersymmetric functions in the case $m=n=1$ (relevant for the 1d SCFT)
are straightforward to classify: they take the form

$\displaystyle f^{(a,b)}(x|y)$ $\displaystyle=(x-y)x^{a}y^{b}\ ,$ (183)

together with the trivial constant function. Thus we classify all higher order
Casimirs in the 1d SCFT as

$\displaystyle{\mathcal{C}}_{1,2}^{(a,b);SU(1,1|2)}$
$\displaystyle=\hat{P}_{p_{i}}\frac{x-y}{xy}\,\Delta^{(a,b)}\,\frac{xy}{x-y}\hat{P}_{p_{i}}^{-1}\
$ $\displaystyle\Delta^{(a,b)}$
$\displaystyle=\left({\mathcal{D}}^{+}_{x}-{\mathcal{D}}^{-}_{y}\right)({\mathcal{D}}^{+}_{x})^{a}({\mathcal{D}}^{-}_{y})^{b}\
.$ (184)

The case $a=b=0$ is then precisely the 1d quadratic Casimir of (3.1) and
higher order Casimirs in 1d are considered in (120) and the following.

Let us compare this with the higher order Casimirs for the maximal bosonic
subgroup $SU(1,1)\times SU(2)$. These correspond to the product of the cases
$(m,n)=(1,0)$ and $(m,n)=(0,1)$ in the general case (181) and take the form

$\displaystyle{C}_{1,2}^{(a,b);SU(1,1)\times SU(2)}$
$\displaystyle=\hat{P}_{p_{i}}\,({\mathcal{D}}^{+}_{x})^{a}({\mathcal{D}}^{-}_{y})^{b}\,\hat{P}_{p_{i}}^{-1}.\
$ (185)

We then find the relation

$\displaystyle{\mathcal{C}}_{1,2}^{(a,b)}=\frac{x-y}{xy}\left({C}_{1,2}^{(a+1,b)}-{C}_{1,2}^{(a,b+1)}\right)\frac{xy}{x-y}\
,$ (186)

of which (A.1) is a special case.

#### Higher order Casimirs in ${\mathcal{N}}=4$ SYM

The space of supersymmetric functions in the case $m=n=2$ (relevant for
${\mathcal{N}}=4$ SYM) is more intricate, but a nice basis for them is given
by the super Schur polynomials, labelled by Young diagrams. They fall into two
classes: short and long. We only discuss the long ones as they are the only
ones needed here, but all cases, both long and short, can be found in [47].
The long supersymmetric functions in the case $m=n=2$ are spanned by the
functions666This basis is equivalent to but not identical to the Schur
polynomial basis for long supersymmetric polynomials.

$\displaystyle f^{(a,\bar{a}|b,\bar{b})}(x,\bar{x}|y,\bar{y})$
$\displaystyle=(x-y)(x-\bar{y})(\bar{x}-y)(\bar{x}-\bar{y})(x^{a}\bar{x}^{\bar{a}}+x^{\bar{a}}\bar{x}^{a})(y^{b}\bar{y}^{\bar{b}}+y^{\bar{b}}\bar{y}^{b})\
.$ (187)

Noting that the first four factors translate directly to $\Delta^{(8)}$ (7)
under the prescription (181), we see that the space of corresponding higher
order Casimirs is spanned by the operators

$\displaystyle{\mathcal{C}}_{1,2}^{(a,\bar{a}|b,\bar{b});SU(2,2|4)}$
$\displaystyle=\hat{P}_{p_{i}}\left(\frac{(x{-}y)(x{-}\bar{y})(\bar{x}{-}y)(\bar{x}{-}\bar{y})}{x\bar{x}y\bar{y}(x{-}\bar{x})(y{-}\bar{y})}\right)\Delta^{(8)}\Delta^{(a,\bar{a}|b,\bar{b})}\left(\frac{x\bar{x}y\bar{y}(x{-}\bar{x})(y{-}\bar{y})}{(x{-}y)(x{-}\bar{y})(\bar{x}{-}y)(\bar{x}{-}\bar{y})}\right)\hat{P}_{p_{i}}^{{-}1}$
(188) $\displaystyle\Delta^{(a,\bar{a}|b,\bar{b})}$
$\displaystyle=\Big{(}({\mathcal{D}}_{x}^{+})^{a}({\mathcal{D}}_{\bar{x}}^{+})^{\bar{a}}+({\mathcal{D}}_{x}^{+})^{\bar{a}}({\mathcal{D}}_{\bar{x}}^{+})^{a}\Big{)}\Big{(}({\mathcal{D}}_{y}^{-})^{b}({\mathcal{D}}_{\bar{y}}^{-})^{\bar{b}}+({\mathcal{D}}_{y}^{-})^{\bar{b}}({\mathcal{D}}_{\bar{y}}^{-})^{b}\Big{)}\
.$ (189)

This is the basis of higher order Casimirs used to find the 14th order Casimir
in the $\alpha^{\prime 3}$ correction (163)-(160) as well as the higher order
Casimirs considered at higher orders in $\alpha^{\prime}$ in section 7.2.

Let us compare this super Casimir with the higher order Casimirs for the
maximal bosonic subgroup $SU(2,2)\times SU(4)$ which are obtained from the
general formula (181) by taking the product of the cases $(m,n)=(2,0)$ and
$(m,n)=(0,2)$ and take the form

$\displaystyle{C}_{1,2}^{(a,\bar{a}|b,\bar{b});SU(2,2)\times SU(4)}$
$\displaystyle=\hat{P}_{p_{i}}\left(\frac{x\bar{x}y\bar{y}}{(x{-}\bar{x})(y{-}\bar{y})}\right)\,\Delta^{(a,\bar{a}|b,\bar{b})}\,\left(\frac{(x{-}\bar{x})(y{-}\bar{y})}{x\bar{x}y\bar{y}}\right)\,\hat{P}_{p_{i}}^{{-}1},\
$ (190)

where $\Delta^{(a,\bar{a}|b,\bar{b})}$ is in (189). We find that super
Casimirs are related to the bosonic Casimirs via

$\displaystyle{\mathcal{C}}_{1,2}^{(a,\bar{a}|b,\bar{b});SU(2,2|4)}=\left(\frac{(x{-}y)(x{-}\bar{y})(\bar{x}{-}y)(\bar{x}{-}\bar{y})}{(x\bar{x}y\bar{y})^{2}}\right){C}_{1,2}^{(*);SU(2,2)\times
SU(4)}\left(\frac{(x\bar{x}y\bar{y})^{2}}{(x{-}y)(x{-}\bar{y})(\bar{x}{-}y)(\bar{x}{-}\bar{y})}\right),$
(191)

where ${C}_{1,2}^{(*)}$ is a linear combination of the bosonic Casimirs in
(190) obtained by multiplying by $\Delta^{(8)}$ and expanding, so the first
and last terms for example are

$\displaystyle{C}_{1,2}^{(*)}={C}_{1,2}^{(a+2,\bar{a}+2|b,\bar{b})}-\dots+{C}_{1,2}^{(a,\bar{a}|b+2,\bar{b}+2)}\
.$ (192)

##  Appendix B Correlators of descendants

In this Appendix we wish to compute the correlator of superconformal
descendants (see (7)):

$\displaystyle\partial_{\theta_{1}}\partial_{\theta_{2}}\partial_{\bar{\theta}_{3}}\partial_{\bar{\theta}_{4}}\langle\Psi_{p_{1}}\Psi_{p_{2}}\Psi_{p_{3}}\Psi_{p_{4}}\rangle|_{\theta_{i}=\bar{\theta}_{i}=0}$
(193)

and prove the result (48).

We must first understand the general form the supercorrelator must take
according to the superconformal Ward identities. Superconformal symmetry
$SU(1,1|2)$ fixes the correlator in terms of a conjugation invariant function
of a cross ratio matrix (see [27, 47]) The general arguments in [27, 47] show
that superconformal symmetry $SU(1,1|2)$ fixes the correlator to
$P_{p_{i}}\times f(Z)$ where $f(Z)$ is a conjugation invariant function
($f(Z)=f(G^{-1}ZG)$) of the cross-ratio matrix $Z$

$\displaystyle
Z:=X_{12}X_{24}^{-1}X_{43}X_{31}^{-1}=\left(\begin{array}[]{cc}x&\xi\\\
\bar{\xi}&y\end{array}\right),\qquad X_{ij}:=X_{i}-X_{j},\qquad
X_{i}:=\left(\begin{array}[]{cc}x_{i}&\theta_{i}\\\
\bar{\theta}_{i}&y_{i}\end{array}\right)\ .$ (198)

It thus becomes a function of the eigenvalues of $Z$ only, $\hat{x},\hat{y}$
so

$\displaystyle\langle\Psi_{p_{1}}\Psi_{p_{2}}\Psi_{p_{3}}\Psi_{p_{4}}\rangle=P_{p_{i}}\times
f(\hat{x},\hat{y})$ (199)

where

$\displaystyle\hat{x}=x+\frac{\xi\bar{\xi}}{x-y}\qquad\hat{y}=y+\frac{\xi\bar{\xi}}{x-y}$
(200)

That the eigenvalues of $Z$ are given by (200) can be easily checked by
verifying that they give the same super trace and super determinant:

$\displaystyle\text{str}(Z)=x-y=\hat{x}-\hat{y},\qquad\text{sdet}(Z)=(x-\xi\bar{\xi}/y)/y=\hat{x}/\hat{y}.$
(201)

Plugging these definitions into Mathematica with the help of the Grassmann
package [51] to deal with the Grassmann odd variables one finds that the
Grassmann odd derivatives acting on $f$ are consistent with the following
differential operator:

$\displaystyle\partial_{\theta_{1}}\partial_{\theta_{2}}\partial_{\bar{\theta}_{3}}\partial_{\bar{\theta}_{4}}f(\hat{x},\hat{y})|_{\theta_{i}=\bar{\theta}_{i}=0}=\frac{1}{x_{12}x_{34}y_{12}y_{34}}\left({\mathcal{D}}^{(0,0)}_{\hat{x}}-{\mathcal{D}}^{(0,0)}_{\hat{y}}\right)\frac{\hat{x}\hat{y}}{\hat{x}-\hat{y}}f(\hat{x},\hat{y})\
.$ (202)

Pulling this through the prefactor then gives the action on the correlator
itself: