The Parisi-Sourlas Uplift and Infinitely Many Solvable $4d$ CFTs

Emilio Trevisani

Laboratoire de Physique Théorique et Hautes Énergies, CNRS & Sorbonne
Université,
4 Place Jussieu, 75252 Paris, France
Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique,
91191, Gif-sur-Yvette, France

###### Abstract

Parisi-Sourlas (PS) supersymmetry is known to emerge in some models with
random field type of disorder. When PS SUSY is present the $d$-dimensional
theory allows for a $d-2$-dimensional description. In this paper we
investigate the reversed question and we provide new indications that any
given CFTd-2 can be uplifted to a PS SUSY CFTd. We show that any scalar four-
point function of a CFTd-2 is mapped to a set of 43 four-point functions of
the uplifted CFTd which are related to each other by SUSY and satisfy all
necessary bootstrap axioms. As a byproduct we find 43 non trivial relations
between conformal blocks across dimensions.

We test the uplift in generalized free field theory (GFF) and find that PS
SUSY is a powerful tool to bootstrap an infinite class of previously unknown
GFF observables. Some of this power is shown to persist in perturbation theory
around GFF.

We explain why all diagonal minimal models admit an uplift and we show exact
results for correlators and CFT data of the $4d$ uplift of the Ising model.
Despite being strongly coupled $4d$ CFTs, the uplifted minimal models contain
infinitely many conserved currents and are expected to be integrable.

###### Contents

  1. 1 Introduction
  2. 2 Background and review of Parisi-Sourlas CFTs
    1. 2.1 CFT basics
    2. 2.2 Conformal blocks
    3. 2.3 Parisi-Sourlas supersymmetry
    4. 2.4 Dimensional reduction
    5. 2.5 Reduction and uplift of Lagrangians
    6. 2.6 Uplift of CFTs
  3. 3 Extracting primary components of correlation functions in superspace
    1. 3.1 Two-point functions
    2. 3.2 Three-point functions
    3. 3.3 Four-point functions
  4. 4 Conformal block expansion in PS CFTs
    1. 4.1 43 relations between conformal blocks across dimensions
    2. 4.2 Examples
    3. 4.3 Poles in the conformal blocks and SUSY
  5. 5 Bootstrapping GFF using PS SUSY
    1. 5.1 Bootstrapping GFF using PS supersymmetry
    2. 5.2 Bootstrapping OPE coefficients
    3. 5.3 Bootstrapping correlators and extension to perturbation theory
  6. 6 The $4d$ Parisi-Sourlas uplift of diagonal minimal models
    1. 6.1 Example: Parisi-Sourlas uplift of the Ising minimal model
    2. 6.2 Conformal block decomposition of the uplifted correlators
    3. 6.3 Comments on the operator spectrum
  7. 7  Comments on the loss of information in the reduction
    1. 7.1 On the operators that dimensionally reduce to zero
    2. 7.2 On the uplift of one-dimensional GFF
  8. 8 Conclusion and outlook
  9. A Coefficients of the relations between conformal blocks
  10. B Resolving the singularities of the conformal blocks
  11. C Global multiplets inside Virasoro multiplets
    1. C.1 Global conformal block decomposition of minimal model correlators
  12. D More conformal block decompositions in the uplifted Ising model

##  1 Introduction

Parisi-Sourlas supersymmetry was first introduced in [1] in order to explain a
very peculiar behaviour of critical theories with quenched random field type
of disorder. The story started a few years prior when Aharony, Imry, and Ma
conjectured that the IR fixed point of random field models is described by
pure models in two less dimensions [2]. This was motivated by the behaviour of
the Feynman diagrams of the random field Ising model in $d$ dimensions, which
in the IR limit match the diagrammatic computations of the pure Ising model in
$d-2$ dimensions. Parisi and Sourlas provided an explanation of such behaviour
by conjecturing that random field models have emergent supersymmetry in the
IR, which they further showed (through a perturbative argument) to be
responsible for the dimensional reduction [1]. Namely, following picture 1,
the relation A of Aharony, Imry, and Ma, was explained by Parisi and Sourlas
by a combination of B and C.

RFd f.p.PS CFTdCFTd-2BADC Figure 1:  Relations between theories: the fixed
point of a random field theories in $d$ dimensions (RFd f.p.), a Parisi-
Sourlas SUSY CFTd (PS CFTd) and a pure CFT in two lower dimensions (CFTd-2).

The explanation by Parisi and Sourlas is very appealing since the emergence of
symmetries at a fixed point is not uncommon. Moreover the original
perturbative argument for C was later made non-perturbative and more rigorous
[3, 4, 5, 6]. A CFT argument was also recently provided in [7].

Unfortunately the beautiful conjecture by Parisi-Sourlas was quickly found to
have flaws. In particular the arrow A for the Ising model clearly cannot be
fully correct. Indeed the RF Ising model has a critical point in $3\leq d\leq
5$ [8], while the pure $\hat{d}\equiv d-2$-dimensional Ising model only has a
critical point in $2\leq\hat{d}\leq 3$. Thus A is certainly violated in $d=3$.
Furthermore in later numerical studies it was found that both A and B fail in
$d=4$ [9]. It came as a surprise that numerical simulations in $d=5$ [10, 11]
gave instead strong indications for the both the emergence of supersymmetry B
and the dimensional reduction A. The Parisi-Sourlas conjecture can also be
studied in a different random field model, RF $\phi^{3}$, which has a phase
transition between $d\leq 2<8$. In this case numerical simulations indicate
that the fixed point is always compatible with A [12].

While many studies were produced in the past decades, no fully conclusive
explanation of when/why the conjecture works was found. This motivated a
revised study of the emergence of SUSY in RF models (arrow A) using a
perturbative RG setup [13, 14, 15, 16]. This series of works showed in epsilon
expansion that the PS CFT for the Ising case is reached in the vicinity of the
upper critical dimension, namely at $d=6-\epsilon$ for small $\epsilon$.
Conversely, when $\epsilon$ is of order $1$, some SUSY breaking deformations
develop a large negative anomalous dimension becoming relevant for $d$ less
than a critical value $d_{c}$. A two-loop analysis predicts that $d_{c}\approx
4.2-4.7$, which is in perfect agreement with the numerical simulation
described above. The same setup was also applied to the RF $\phi^{3}$ model,
showing that in this case all SUSY breaking deformations remain irrelevant in
all dimensions, again in agreement with the numerical results. In summary it
was found that the SUSY fixed point sometimes is not reached because of new
relevant SUSY breaking perturbations. However, whenever it is reached,
dimensional reduction is expected to always occur. This thus provides an
explanation for when/why the Parisi-Sourlas conjecture works.

While this could seem the conclusion of a story, by revisiting the topic new
interesting directions emerged. In particular, as we mentioned, the arrow C of
figure 1 was also studied in an axiomatic CFT context in [7]. In this paper it
was first defined what is a PS CFTd. Then it was shown in which sense such a
theory has a description in terms of a CFTd-2. It was shown that a huge part
of the spectrum and OPE coefficients of (also non protected) superprimaries of
the PS CFTd exactly matches the CFT data of the CFTd-2. E.g. the spectrum and
OPE coefficients of _all_ the scalar operators would match in the two theories
(in $d\geq 3$). This result was obtained by using arguments only based on
symmetry and therefore applies to any possible PS CFTd. The work of [7] thus
motivated a new question: given any CFTd-2 is it possible to define a PS CFTd
which dimensionally reduces to it? This is what we refer to as the Parisi-
Sourlas dimensional uplift, represented as the arrow D of figure 1, and it is
the focus of this paper. In particular the results of this paper are in
support of the conjecture that, independently on the properties of the CFTd-2,
the uplift always exists. This is mostly independent from random field models,
meaning that the uplifted models may or may not emerge at the fixed point of a
disordered system. One may thus ask why should we study the uplift. There are
various motivations.

One important motivation is to understand the space of possible consistent
CFTs, which is the ultimate goal of the conformal bootstrap. If the uplift
always exists, then it can also be iterated. Therefore given a CFTd one could
in principle define an infinite tower of PS CFTd+2n for $n=1,2,\dots$ as shown
in figure 2.

CFTdCFTd+2CFTd+4$\dots$ Figure 2:  Iterated uplift.

There is an important catch: PS CFTs are non-unitary. Therefore figure 2 shows
that the space of non-unitary CFTs is infinitely much larger than the one of
unitary ones: for every unitary CFT there exists an infinite tower of non
unitary CFTs in higher dimension. The tower in figure 2 has increasing amount
of SUSY. It is interesting to speculate what could be the fixed point of this
sequence (indeed this would live in infinite dimensions and would enjoy
infinite SUSY).

Another very interesting motivation is to find the first examples of fully
solvable non-trivial CFTs in $d>2$. Indeed in $d>2$ there is no example of
interacting CFT where the full CFT data is accessible exactly. The situation
is much better in $2d$ where in the minimal models it is possible to compute
spectrum, OPE coefficients and even correlation functions exactly. The idea is
thus to uplift all minimal models obtaining infinitely many towers of
infinitely many solvable CFTs in all even dimensions. Notice also that such
models might be physically relevant. For example the uplift of the Yang-Lee
minimal model describes the $4d$ universality class of RF $\phi^{3}$ model
(which also describes systems like branched polymers and lattice animals
[12]). In this universality class there exists also a model defined by Brydges
and Imbrie [17] (see also [18]) which has SUSY at the microscopic level. The
microscopic SUSY ensures the model to dimensionally reduce all along the RG
flow and not only at the IR fixed point. In the same spirit, in [19] Cardy
recently proposed a supersymmetric microscopic model that flows to the
uplifted Ising model. In $d=4$ this thus flows to the uplifted Ising minimal
model. It would be very interesting to see whether new supersymmetric
microscopic theories can be defined such that they flow to other minimal
models.

Another motivation is to use the supersymmetry of the uplifted theory as a
tool for solving problems of the original theory. This might sound surprising
since we argued that the two theories are somewhat equivalent. However the
information is packaged very differently in the two formulations and it can
happen that the SUSY formulation is more suitable to answer some questions, as
we shall exemplify in the following.

Other applications will appear as we will enter the details of the paper. For
example the SUSY structure of the uplifted theory provides some powerful
kinematical constraint on CFTs. In particular we will see that PS SUSY implies
a set of $43$ relations between conformal blocks in $d-2$ and in $d$
dimensions, which are interesting in their own rights. For example one of such
equation was already obtained by Dolan and Osborn to compute conformal blocks
in higher dimensions from the knowledge of conformal blocks in lower
dimensions. With this work we find that such equation actually has a very
clear physical interpretation in the SUSY theory.

The plan of the paper is as follows. We start section 2 with a basic
introduction on CFTs, which will be followed by a review on PS SUSY and
dimensional reduction/uplift.

In section (3) we explicitly show how to extract different primary components
from superspace correlators with two, three and four scalar insertions. We
show that all of them are (as expected) compatible with conformal symmetry
which is a further check that PS CFTs are well defined. In the four-point
function case we show that there are 43 independent primary components in a
generic four-point function which are obtained by acting with 43 differential
operators (in the conformal cross ratios) on the lowest component.

In section 4 we show that these differential operators on $d-2$-dimensional
conformal blocks can be written in terms of a linear combination of at most
five $d$-dimensional conformal blocks. We further analyze these relation and
explain how the linear combinations are tuned to cancel possible poles of the
conformal blocks.

In section 5 we provide some examples of how to use the supersymmetry of the
uplifted theory as a tool to compute observables of the original theory. We
find that in the uplift of generalized free field theory (GFF) there exists an
infinite class of non trivial correlators which have some vanishing primary
components. In terms of four-point functions this means that some of the
differential operators of section 3 annihilate the correlator. We then show
that this constraint can be used in two ways. Firstly to obtain a recurrence
relation for the coefficients in the conformal block decomposition, which we
also show how to solve in the case of
$\langle\phi\phi^{n}\phi^{m}\phi^{m}\rangle$ in closed form for any $n,m$
(which is a new result in GFF). Secondly we explain how to use the
differential operators to fix the correlator itself (up to some constants).
Then we also show how this type of logic can be generalized to perturbative
computations around GFF, giving rise to differential equations for the
perturbative correlators.

In section 6 we start by briefly introducing the minimal models. We give an RG
argument of why diagonal minimal models should have a well defined uplift. We
then exemplify how to uplift all four-point functions of Virasoro primaries in
the Ising minimal model. We then proceed at computing the first few terms of
their conformal block decomposition. We end the section by commenting on the
properties of the spectrum which must contain an infinite number of conserved
higher spin currents.

Section 7 is devoted to a general property of all uplifted theories: they are
expected to have a larger set of operators than the reduced ones. These extra
operators are projected to zero under dimensional reduction. In section 7 we
exemplify a class of such operators. We then consider the uplift of a $1d$ GFF
four-point function as a toy model to show how the larger kinematic space of
the uplifted theory (in $3d$ there are two independent cross ratios instead of
the single cross ratio of one-dimensional CFTs) is exactly reconstructed by
the extra operators of the uplifted theory, which in this case are the spin
$\ell\geq 1$ traceless and symmetric operators.

In section 8 we conclude the paper and mention future directions. More details
are deferred to the appendices. Finally, a Mathematica code with the
definitions of the differential operators (and some extra checks) is attached
to the publication.

##  2 Background and review of Parisi-Sourlas CFTs

###  2.1 CFT basics

To set our conventions let us start by reviewing some basic facts on
$d$-dimensional CFTs (see e.g. [20] for a more detailed review). Correlation
functions up to three insertions are fixed by symmetry. One-point functions
vanish besides the one of the identity which equals one. The two-point
function of a scalar primary $O$ with dimension $\Delta$ takes the form

$\langle O(x_{1})O(x_{2})\rangle=\frac{1}{(x_{12}^{2})^{\Delta}}\,,$ (2.1)

where $x^{\mu}_{ij}\equiv x^{\mu}_{i}-x^{\mu}_{j}$ and
$x_{i}\in\mathbb{R}^{d}$. In a unitary theory the basis of primary operators
can be diagonalized such that the two-point functions of different primaries
vanish. Three-point functions of scalar primaries $O_{i}$ with dimensions
$\Delta_{i}$ take the form

$\langle
O_{1}(x_{1})O_{2}(x_{2})O_{3}(x_{3})\rangle=\frac{\lambda_{123}}{(x_{12}^{2})^{\frac{\Delta_{1}+\Delta_{2}-\Delta_{3}}{2}}(x_{13}^{2})^{\frac{\Delta_{1}+\Delta_{3}-\Delta_{2}}{2}}(x_{23}^{2})^{\frac{\Delta_{2}+\Delta_{3}-\Delta_{1}}{2}}}\,,$
(2.2)

where $\lambda_{123}$ is a theory-dependent quantity called OPE coefficient.
With four insertions we can build two independent conformal invariant cross
ratios,

$u\equiv\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}}\,,\qquad
v\equiv\frac{x_{14}^{2}x_{23}^{2}}{x_{13}^{2}x_{24}^{2}}\,,$ (2.3)

therefore a four-point functions of scalars can be fixed as

$\langle
O_{1}(x_{1})O_{2}(x_{2})O_{3}(x_{3})O_{4}(x_{4})\rangle=K_{\Delta_{i}}(x_{i})f(u,v)\,,$
(2.4)

where $f(u,v)$ is a theory-dependent function and $K$ is a kinematic factor
defined as

$K_{\Delta_{i}}(x_{i})\equiv\frac{1}{(x_{12}^{2})^{\frac{1}{2}(\Delta_{1}+\Delta_{2})}(x_{34}^{2})^{\frac{1}{2}(\Delta_{3}+\Delta_{4})}}\left(\frac{x_{14}^{2}}{x_{24}^{2}}\right)^{-\frac{\Delta_{12}}{2}}\left(\frac{x_{14}^{2}}{x_{13}^{2}}\right)^{\frac{\Delta_{34}}{2}}\,,$
(2.5)

where $\Delta_{ij}\equiv\Delta_{i}-\Delta_{j}$. Sometimes we shall use
different cross ratios as $z,\bar{z}$ and $\rho,\bar{\rho}$ [21]. They can be
related to $u,v$ using

$u=z\bar{z}\,,\qquad v=(1-z)(1-\bar{z})\,,\qquad
z=\frac{4\rho}{(1+\rho)^{2}}\,,\qquad\bar{z}=\frac{4\bar{\rho}}{(1+\bar{\rho})^{2}}\,.$
(2.6)

A crucial property of CFTs is the operator product expansion (OPE) which
allows to replace two operator insertions by a sum of single operators
insertions,

$O_{1}(x)O_{2}(0)=\sum_{\Delta\,\ell}\lambda_{12O}\left[\frac{x_{\mu_{1}}\cdots
x_{\mu_{\ell}}}{|x|^{\Delta_{1}+\Delta_{2}-\Delta+\ell}}O^{\mu_{1}\dots\mu_{\ell}}(x)+\dots\right]\,,$
(2.7)

where the sum runs over all operators $O$ with dimension $\Delta$ and spin
$\ell$ and the dots stand for the contribution of the descendants operators.
Using the OPE of operators $O_{1}$ and $O_{2}$ inside the four-point function
(2.4), we obtain

$f(u,v)=\sum_{\Delta\,\ell}a_{\Delta\ell}\,g_{\Delta\ell}(u,v)\,,$ (2.8)

where $a_{\Delta\ell}\equiv\lambda_{12O}\lambda_{O34}$. This formula
represents the decomposition of a four-point function in conformal blocks
$g_{\Delta\ell}$ which are functions fixed by symmetry that encode the
exchange of a primary $O$ with dimension $\Delta$ and spin $\ell$ along with
all its descendants. The blocks $g_{\Delta\ell}$ are function of the variables
$\Delta_{12}$, $\Delta_{34}$ and $d$, but we will often suppress these
dependencies in order to streamline the notation.

###  2.2 Conformal blocks

Conformal blocks are fixed by symmetry. In even dimensions $d$ they can be
expressed in closed form [22]. For example in $d=2,4$ they read [23]

$\displaystyle
g^{(d=2)}_{\Delta\ell}(z,\bar{z})=\frac{k_{\Delta-\ell}(z)k_{\Delta+\ell}(\bar{z})+k_{\Delta+\ell}(z)k_{\Delta-\ell}(\bar{z})}{2^{\ell+\delta_{\ell,0}}}\,,$
(2.9) $\displaystyle
g^{(d=4)}_{\Delta\ell}(z,\bar{z})=\frac{z\bar{z}}{\bar{z}-z}\frac{\big{[}k_{\Delta-\ell-2}(z)k_{\Delta+\ell}(\bar{z})-k_{\Delta+\ell}(z)k_{\Delta-\ell-2}(\bar{z})\big{]}}{2^{\ell}}\,,$
(2.10)

where $k_{\eta}$ is defined as follows

$\displaystyle k_{\eta}(z)\equiv
z^{\eta/2}\,{}_{2}F_{1}(\tfrac{\eta-\Delta_{12}}{2},\tfrac{\eta+\Delta_{34}}{2},\eta,z)\,.$
(2.11)

For generic dimensions $d$, the blocks are typically not known in a closed
form but nevertheless they can be determined in several ways (see [20] for a
review of various techniques). For this paper it will be useful to keep in
mind a recursion relation [24, 25, 26, 27] that defines the blocks from their
analytic structure in $\Delta$. Indeed conformal blocks have poles in $\Delta$
with residue proportional to other blocks,111In even dimensions higher order
poles are allowed to appear, but here we work in arbitrary $d$ as an analytic
continuation around odd $d$, where only single poles appear.

$g^{(d)}_{\Delta\ell}\sim\frac{R_{A}}{\Delta-\Delta^{*}_{A}}\;g^{(d)}_{\Delta_{A}\ell_{A}}\,.$
(2.12)

The label $A\equiv\mbox{type},n$ is specified by a type between
$\textrm{I},\textrm{II},\textrm{III}$ and an integer $n$ in some range, while
all other labels appearing in (2.12) are defined in table 1.

$\mbox{Type},n\phantom{\Big{(}}\qquad\qquad\qquad$ | $\Delta^{\star}_{A}\qquad\qquad\qquad$ | $n_{A}\qquad\quad$ | $\ell_{A}\qquad$
---|---|---|---
I, $n\in[1,\infty]$ | $1-\ell-n$ | $n$ | $\ell+n$
II, $n\in[1,\ell]$ | $\ell+d-1-n$ | $n$ | $\ell-n$
III, $n\in[1,\infty]$ | $\frac{d}{2}-n$ | $2n$ | $\ell$
Table 1:  Position of the poles and labels of the blocks at the residue in
(2.12), where $\Delta_{A}\equiv\Delta^{*}_{A}+n_{A}$.

The coefficients $R_{A}$ at the residue are known in a closed form as follows

$\begin{array}[]{ccl}R_{\textrm{I},n}&\\!\\!\\!=&\\!\\!\frac{-n(-2)^{n}}{(n!)^{2}}\
{\prod\limits_{\delta=\Delta_{12},\Delta_{34}}}\left(\frac{\delta+1-n}{2}\right)_{n}\vspace{0.1cm}\\\
R_{\textrm{II},n}&\\!\\!\\!=&\\!\\!\frac{-n\,\ell!}{(-2)^{n}(n!)^{2}(\ell-n)!}\frac{(d+\ell-n-2)_{n}}{\left(\frac{d}{2}+\ell-n\right)_{n}\left(\frac{d}{2}+\ell-n-1\right)_{n}}\
{\prod\limits_{\delta=\Delta_{12},\Delta_{34}}}\left(\frac{\delta+1-n}{2}\right)_{n}\
,\vspace{0.1cm}\\\
R_{\textrm{III},n}&\\!\\!\\!=&\\!\\!\frac{-n(-1)^{n}\left(\frac{d}{2}-n-1\right)_{2n}}{(n!)^{2}\left(\frac{d}{2}+\ell-n-1\right)_{2n}\left(\frac{d}{2}+\ell-n\right)_{2n}}\
\prod\limits_{\delta=\Delta_{12},\Delta_{34}}\ \prod\limits_{\sigma=\pm 1}\
\left(\frac{\delta-\sigma\frac{d}{2}-\sigma\ell-n+1+\sigma}{2}\right)_{n}\
.\end{array}$ (2.13)

One can thus reconstruct the full conformal blocks by summing all the poles
and an entire function in $\Delta$. This gives rise to the following
recurrence relation of the conformal blocks as functions of the radial
coordinates $r\equiv|\rho|$ and $\eta\equiv\cos\arg\rho$,

$\displaystyle h_{\Delta\ell}(r,\eta)$
$\displaystyle=h_{\infty\ell}(r,\eta)+\sum_{A}\frac{R_{A}}{\Delta-\Delta^{\star}_{A}}(4r)^{n_{A}}h_{\Delta_{A}\,\ell_{A}}(r,\eta)\
,$ (2.14)

where $g^{(d)}_{\Delta\ell}(r,\eta)\equiv(4r)^{\Delta}h_{\Delta\ell}(r,\eta)$
and the sum over $A$ is a sum over the three types and over $n$. Finally the
regular part in $\Delta$ of $h_{\Delta\ell}$ is defined by

$h_{\infty\ell}(r,\eta)\equiv\frac{\ell!\left(1-r^{2}\right)^{1-\frac{d}{2}}}{(-2)^{\ell}\,(\frac{d}{2}-1)_{\ell}}\frac{\left(r^{2}-2\eta
r+1\right)^{\frac{\Delta_{12}-\Delta_{34}-1}{2}}}{\left(r^{2}+2\eta
r+1\right)^{\frac{\Delta_{12}-\Delta_{34}+1}{2}}}\
C_{\ell}^{\frac{d}{2}-1}(\eta)\ ,$ (2.15)

where $C_{\ell}^{\nu}(x)$ is a Gegenbauer polynomial. The normalization of the
blocks is chosen to match the one of the closed form solutions in $d=2,4$
written above.

###  2.3 Parisi-Sourlas supersymmetry

In this section we present a short introduction to PS SUSY based on the more
complete results of [7].

In order to introduce PS supersymmetry it is convenient to work in a
superspace $\mathbb{R}^{d|2}$, parametrized by coordinates
$y^{a}=\\{x^{\alpha},\theta,\bar{\theta}\\}$, where the index $a$ takes $d+2$
possible values $a=1,\dots d,\theta,\bar{\theta}$. A QFT with PS SUSY is
invariant under superrotations and supertranslations which are the
transformations that preserve the superspace distance

$y^{2}\equiv y^{a}y^{b}(g_{d|2})_{ab}\,,\qquad\mbox{with
}g_{d|2}=\left(\begin{array}[]{cc}g_{d}&0\\\
0&g_{\textrm{sp}(2)}\end{array}\right)\,,$ (2.16)

where $g_{d}$ is just the rank-$d$ identity matrix which defines the metric of
a Euclidean $d$-dimensional space, while
$g_{\textrm{sp}(2)}=\left(\begin{smallmatrix}0&-1\\\
1&0\end{smallmatrix}\right)$ is the symplectic metric. We name the generator
of supertranslations and superrotations respectively as $P^{a}$ and $M^{ab}$.
They can be represented as

$P^{a}=\partial^{a}\,,\qquad
M^{ab}=y^{a}\partial^{b}-(-1)^{[a][b]}y^{b}\partial^{a}\,,$ (2.17)

where the derivative in superspace is defined as
$\partial_{a}\equiv\frac{\partial}{\partial
y^{a}}\equiv(\partial_{\alpha},\partial_{\theta},\partial_{\bar{\theta}})$ and
the bracket in $[a]$ measures if the index $a$ is bosonic or fermionic, namely
$[a]=0$ for $a=1,\dots,d$, while $[\theta]=1=[\bar{\theta}]$. A PS CFT is
further invariant under superdilation $D$ and special superconformal
transformation $K^{a}$, which are represented as

$D=-y^{a}\partial_{a}\,,\qquad
K^{a}=2y^{a}y^{b}\partial_{b}-y^{b}y_{b}\partial_{a}\,.$ (2.18)

The full set of generators of a PS CFT satisfies the $osp(d+1,1|2)$ algebra.

It is convenient to consider local (super) operators ${\mathcal{O}}(y)$
inserted at points $y^{a}$ in superspace. We will consider operators that have
a well defined superconformal dimension $\Delta$ under the action of the
superdilation generator $D$ and that transform in irreducible representations
of $OSp(d|2)$, such that tensor indices will be of the form
$a_{i}=1,\dots,d,\theta,\bar{\theta}$. Standard $OSp(d|2)$ representations are
labelled by Young tableaux with $[d/2]$ rows of length
$\ell_{1}\geq\ell_{2}\geq\dots\geq\ell_{[d/2]}\geq 0$ and an arbitrary long
column, as follows boxsize=1.4em

& ⋯ ⋯ [ ℓ_1]
[ $\vdots$] [ $\vdots$] [ $\vdots$] [ $\iddots$]
⋯ [ ℓ_[d/2]]

$\vdots$

(2.19)

where rows are graded symmetric, columns graded antisymmetric and all traces
are removed. For example the spin $\ell$ graded symmetric and traceless
representation is labelled by a single row with $\ell$ boxes. We can write a
spin $\ell$ operator in components as follows222The following notation
slightly differs from the one of [7] in order to make R-symmetry properties of
the operators more readable.

${\mathcal{O}}^{a_{1}\dots a_{\ell}}(y)={\mathcal{O}}_{0}^{a_{1}\dots
a_{\ell}}(x)+\theta{\mathcal{O}}_{\bar{\theta}}^{a_{1}\dots
a_{\ell}}(x)+\bar{\theta}{\mathcal{O}}_{\theta}^{a_{1}\dots
a_{\ell}}(x)+\theta\bar{\theta}{\mathcal{O}}_{\theta\bar{\theta}}^{a_{1}\dots
a_{\ell}}(x)\,.$ (2.20)

When $\ell=0$ the components ${\mathcal{O}}_{0}$ and
${\mathcal{O}}_{\theta\bar{\theta}}$ are bosonic while
${\mathcal{O}}_{\theta}$ and ${\mathcal{O}}_{\bar{\theta}}$ are fermionic. As
a convention, throughout the paper we will use upper indices of operators to
denote graded tensor indices, while the subscript of the operator will be
reserved to denote the components. In terms of the usual dilation generator,
the component operators in (2.20) have dimensions $\Delta$, $\Delta+1$,
$\Delta+2$ where each theta in the expansion is responsible for a one unit
shift in the conformal dimensions. Since $\theta$ and $\bar{\theta}$ do not
carry indices, all components of ${\mathcal{O}}$ transform in the same
$OSp(d|2)$ representation. However every $OSp(d|2)$ representation can be
decomposed in $SO(d)$ ones, so each component of (2.20) can be further
decomposed in four different operators with $SO(d)$-spin equal to $\ell$,
$\ell-1$, $\ell-2$, e.g.

${\mathcal{O}}_{0}^{\alpha_{1}\dots\alpha_{\ell}}(x)\,,\quad{\mathcal{O}}_{0}^{\alpha_{1}\dots\alpha_{\ell-1}\theta}(x)\,,\quad{\mathcal{O}}_{0}^{\alpha_{1}\dots\alpha_{\ell-1}\bar{\theta}}(x)\,,\quad{\mathcal{O}}_{0}^{\alpha_{1}\dots\alpha_{\ell-2}\theta\bar{\theta}}(x)\,.$
(2.21)

Every time an index takes the value $a=\theta,\bar{\theta}$, then the operator
changes grading. E.g. ${\mathcal{O}}_{0}^{\alpha_{1}\dots\alpha_{\ell}}$ and
${\mathcal{O}}_{0}^{\alpha_{1}\dots\alpha_{\ell-2}\theta\bar{\theta}}$ are
bosonic while ${\mathcal{O}}_{0}^{\alpha_{1}\dots\alpha_{\ell-1}\theta}$ and
${\mathcal{O}}_{0}^{\alpha_{1}\dots\alpha_{\ell-1}\bar{\theta}}$ are
fermionic. It is also worth commenting on the R-symmetry of the model, which
is $Sp(2)$ and it is generated by $M^{\theta\theta}$,
$M^{\bar{\theta}\bar{\theta}}$ and
$M^{\theta\bar{\theta}}=M^{\bar{\theta}\theta}$. It is easy to see that, given
a scalar ${\mathcal{O}}$, the component ${\mathcal{O}}_{\bar{\theta}}$ has
charge $-1$ under $M^{\theta\bar{\theta}}$, while ${\mathcal{O}}_{\theta}$ has
charge $+1$. The bosonic components are charged zero. Similarly by taking
indices in the direction $\theta,\bar{\theta}$ we select operators with
different R-symmetry charge. E.g. given a vector ${\mathcal{O}}^{a}$,
${\mathcal{O}}_{0}^{\theta}$ has charge $+1$,
${\mathcal{O}}_{\theta}^{\theta}$ has charge $+2$, while
${\mathcal{O}}_{\bar{\theta}}^{\theta}$ has charge 0. As usual, R-symmetry is
a global symmetry, thus all non-vanishing correlation functions must be
invariant under $Sp(2)$.

Since the theory has superconformal symmetry, it is convenient to classify the
operators as superconformal primaries and descendants. The super primaries are
operators ${\mathcal{O}}(0)$ defined such that they are annihilated by the
special superconformal generators, $[K^{a},{\mathcal{O}}(0)]=0$.
Superdescendants are instead obtained by acting with derivatives
$\partial_{a}$ on the superprimaries.

Correlation functions of local operators are restricted by superconformal
symmetry. One-point functions take the form
$\langle{\mathcal{O}}(y)\rangle=\delta_{{\mathcal{O}}\mathbb{I}}$, while two-
and three-point functions are fixed as

$\displaystyle\langle{\mathcal{O}}(y_{1}){\mathcal{O}}(y_{2})\rangle=\frac{1}{(y_{12}^{2})^{\Delta}}\,.$
(2.22)
$\displaystyle\langle{\mathcal{O}}_{1}(y_{1}){\mathcal{O}}_{2}(y_{2}){\mathcal{O}}_{3}(y_{3})\rangle=\frac{\lambda_{123}}{(y_{12}^{2})^{\frac{\Delta_{1}+\Delta_{2}-\Delta_{3}}{2}}(y_{13}^{2})^{\frac{\Delta_{1}+\Delta_{3}-\Delta_{2}}{2}}(y_{23}^{2})^{\frac{\Delta_{2}+\Delta_{3}-\Delta_{1}}{2}}}\,,$
(2.23)

where $y^{a}_{ij}\equiv y^{a}_{i}-y^{a}_{j}$. Four-point functions can be
written as

$\langle{\mathcal{O}}_{1}(y_{1}){\mathcal{O}}_{2}(y_{2}){\mathcal{O}}_{3}(y_{3}){\mathcal{O}}_{4}(y_{4})\rangle=K_{\Delta_{i}}(y_{i})f(U,V)$
(2.24)

where the superspace cross ratios are defined as

$U\equiv\frac{y_{12}^{2}y_{34}^{2}}{y_{13}^{2}y_{24}^{2}}\,,\qquad
V\equiv\frac{y_{14}^{2}y_{23}^{2}}{y_{13}^{2}y_{24}^{2}}\,,$ (2.25)

and $K_{\Delta_{i}}$ is the same as in (2.5), where we replace $x$ by $y$,
namely

$K_{\Delta_{i}}(y_{i})\equiv\frac{1}{(y_{12}^{2})^{\frac{1}{2}(\Delta_{1}+\Delta_{2})}(y_{34}^{2})^{\frac{1}{2}(\Delta_{3}+\Delta_{4})}}\left(\frac{y_{14}^{2}}{y_{24}^{2}}\right)^{-\frac{\Delta_{12}}{2}}\left(\frac{y_{14}^{2}}{y_{13}^{2}}\right)^{\frac{\Delta_{34}}{2}}\,.$
(2.26)

One can also introduce other cross ratios like the ones of (2.6), e.g.
$U=Z\bar{Z},V=(1-Z)(1-\bar{Z})$ and similarly for $\rho$ and $\bar{\rho}$. The
OPE in superspace takes the form

${\mathcal{O}}_{1}(y){\mathcal{O}}_{2}(0)=\sum_{\Delta\,\ell}\lambda_{12{\mathcal{O}}}\left[\frac{y_{a_{1}}\cdots
y_{a_{\ell}}}{|y|^{\Delta_{1}+\Delta_{2}-\Delta+\ell}}{\mathcal{O}}^{a_{1}\dots
a_{\ell}}(x)+\dots\right]\,,$ (2.27)

where because of the graded commuting properties of $y^{a}$, the spin $\ell$
operators ${\mathcal{O}}^{a_{1}\dots a_{\ell}}$ transform in the graded
symmetric and traceless representation of $OSp(d|2)$. The dots contain
contributions from the superdescendants. We can apply the OPE in the four-
point function and get

$f(U,V)=\sum_{\Delta\,\ell}a_{\Delta\ell}\,G^{(d|2)}_{\Delta\ell}(U,V)\,,$
(2.28)

where $a_{\Delta\ell}\equiv\lambda_{12O}\lambda_{O34}$ and
$G^{(d|2)}_{\Delta\ell}$ is the superspace superconformal block, which is
equal to a usual conformal blocks $g^{(d-2)}_{\Delta\ell}$ in two less
dimensions [7]

$G^{(d|2)}_{\Delta\ell}=g^{(d-2)}_{\Delta\ell}\,.$ (2.29)

The equality between superspace superblocks and lower-dimensional blocks was
proven in [7] to hold also for correlators of generic tensor operators.

###  2.4 Dimensional reduction

Any given PS CFT can be dimensionally reduced to a CFT in two less dimension
$\hat{d}\equiv d-2$. Let us explain how this works. The main idea is that,
given a correlator in superspace, if one restricts all the insertions to
$\mathcal{M}_{\hat{d}}\equiv\\{y\in\mathbb{R}^{d|2}:y=(\hat{x},0,0,0,0),\hat{x}\in\mathbb{R}^{\hat{d}}\\}$
as in figure 3, then one defines the dimensionally reduced theory. Namely all
reduced correlators are obtained as

$\langle{\mathcal{O}}_{1}(y_{1})\dots\mathcal{O}_{n}(y_{n})\rangle|_{\mathcal{M}_{\hat{d}}}=\langle\hat{O}_{1}(\hat{x}_{1})\dots\hat{O}_{n}(\hat{x}_{n})\rangle\,,$
(2.30)

where $\hat{O}_{i}$ are operators of a CFT${}_{{\hat{d}}}$. Here we considered
${\mathcal{O}}_{i}$ and $\hat{O}_{i}$ to be scalars, but the generalization to
operators with spin is straightforward (see comments below and in [7]).

$\hat{x}_{1}$$\hat{x}_{n}$$y_{1}$$y_{n}$$\dots$$\dots$$\mathbb{R}^{d\mid
2}$$\mathcal{M}_{\hat{d}}$ Figure 3:  Dimensional reduction: all points
$y_{i}\in\mathbb{R}^{d|2}$ are projected to $\mathcal{M}_{\hat{d}}$ .

It is well known that any set of correlators which arises from any type of
reduction automatically satisfies the axioms of crossing and conformal block
decomposition as a usual CFT${}_{{\hat{d}}}$. What is special to this kind of
reduction is that the resulting CFT${}_{{\hat{d}}}$ can be shown to be
local333The precise statement is that, whenever the PS CFT contains a
conserved super stress tensor, then the reduced theory contains a conserved
stress tensor (as long as $\hat{d}>1$). When $\hat{d}=1$, as expected, the
reduced stress tensor does not exist as we shall discuss in section 7.  and
with a very small operator content, which is smaller (or equal) than the
superprimary content of the original PS CFTd [7]. Both properties are very
atypical since: 1) the stress tensor is not expected to be conserved on
$\mathcal{M}_{\hat{d}}$ since in principle the energy is allowed to leak
orthogonally to $\mathcal{M}_{\hat{d}}$ and 2) usually for every operator in a
higher dimensional theory we expect infinitely many operators in the
dimensionally reduced theory, and therefore a much larger spectrum. These
remarkable properties are due to a decoupling of infinitely many operators
because of SUSY. Let us explain how this decoupling arises.

Restricting operator insertions to a hyperplane in the CFT literature is often
called “trivial defect”. In our case the trivial defect breaks the
$OSp(d+1,1|2)$ symmetry to $SO(d-1,1)\times OSp(2|2)$ where $SO(d-1,1)$ is
understood as the conformal group in ${\hat{d}}$ dimensions and $OSp(2|2)$ is
a global symmetry. The prescription to get dimensional reduction (2.30)
focuses on the singlet sector of the $OSp(2|2)$ symmetry. For any given
primary operator ${\mathcal{O}}$ there is an infinite set of $OSp(2|2)$
singlets in the trivial defect. Indeed, in addition to the $OSp(2|2)$-singlet
primary ${\mathcal{O}}(\hat{x})$ considered in (2.30), one may also consider
$OSp(2|2)$-singlet primary operators obtained as
$(\partial_{\perp}^{2})^{n}{\mathcal{O}}(\hat{x})$, where $\partial_{\perp}$
is the derivative in the direction perpendicular to the hyperplane
$\mathcal{M}_{{\hat{d}}}$, namely
$\partial_{\perp}^{a}=g_{2|2}^{ab}\partial_{b}$. The statement is that all
these operators decouple from the theory if they are inserted in correlators
of only $OSp(2|2)$-singlets. This can be easily understood, since the
$g_{2|2}$ metric orthogonal to the plane has two free indices that cannot be
contracted with anything to build an $OSp(2|2)$-singlet. Indeed

$g_{2|2}^{ab}\,\hat{x}_{i\,b}=0\,,\qquad\qquad g_{2|2}^{ab}\ g_{2|2\,ba}=0\,,$
(2.31)

the former relation due to the orthogonality of the vectors $\hat{x}_{i}$ and
the perpendicular metric, while the latter due to the properties of the
supertrace of the orthosymplectic metric which satisfies
$\textrm{str}\,g_{d|2}=d-2$. A similar decoupling works at the level of
operators with spin. In this case to define operators on the trivial defect we
must project the tensor indices to directions either orthogonal or parallel to
$\mathcal{M}_{{\hat{d}}}$. The only $OSp(2|2)$-singlets which do not decouple
have all indices along the trivial defect $\mathcal{M}_{{\hat{d}}}$. So in
practice given a superfield ${\mathcal{O}}_{Y}$ that transforms in a
representation of $OSp(d|2)$ labelled by a Young tableau $Y$ as in (2.19), the
reduced field transforms in a representation of $SO(\hat{d})$ labelled by the
same Young tableau $Y$. It can happen that the dimension of the representation
$Y$ in $SO(\hat{d})$ is zero. When this happens, the dimensionally reduced
field vanishes, so the superfield ${\mathcal{O}}_{Y}$ is projected to zero
upon dimensional reduction. We will expand on this point in section 7.

###  2.5 Reduction and uplift of Lagrangians

There are explicit realizations of PS QFTs. An important class is defined
through the following scalar action in superspace [1],

$S_{d|2}=\int d^{d}xd\theta d\bar{\theta}\
\frac{1}{2}\partial^{a}\Phi(y)\partial_{a}\Phi(y)+V(\Phi(y))\,,$ (2.32)

where $V$ is a given polynomial potential and the scalar superfield $\Phi$ can
be decomposed in components as

$\Phi(x,\theta,\bar{\theta})=\varphi(x)+\theta\bar{\psi}(x)+\bar{\theta}\psi(x)+\theta\bar{\theta}\omega(x)\,.$
(2.33)

We can integrate out the Grassmann variables obtaining the action

$S_{d|2}=\int d^{d}x\
\partial^{\mu}\varphi\partial_{\mu}\omega-\omega^{2}+\partial^{\mu}\psi\partial_{\mu}\bar{\psi}+\omega
V^{\prime}(\varphi)+\psi\bar{\psi}V^{\prime}(\varphi)\,,$ (2.34)

where we notice that the classical dimensions of the fields is

$[\varphi]=\frac{d}{2}-2\,,\qquad[\psi]=[\bar{\psi}]=\frac{d}{2}-1\,,\qquad[\omega]=\frac{d}{2}\,.$
(2.35)

This action can be dimensionally reduced to [1]

$S_{{\hat{d}}}=\int
d^{{\hat{d}}}\hat{x}\frac{1}{2}(\partial_{\mu}\hat{\phi}(\hat{x}))^{2}+V(\hat{\phi}(\hat{x}))\,.$
(2.36)

In this case the dimensional reduction is more powerful than in (2.30). Indeed
here we know that correlators on the _lhs_ of (2.30) which are computed using
the action (2.32) can be equivalently calculated by the prescription in the
_rhs_ defined using the action (2.36), e.g.

$\langle\Phi(y_{1})\dots\Phi(y_{n})\rangle_{S_{d|2}}\
\bigg{|}_{\mathcal{M}_{{\hat{d}}}}=\langle\hat{\phi}(\hat{x}_{1})\dots\hat{\phi}(\hat{x}_{n})\rangle_{S_{{\hat{d}}}}\,.$
(2.37)

Therefore there exist two separate computations which give the same result.
Because of this we can conclude that when a PS SUSY Lagrangian is available,
not only dimensional reduction is available, but also the uplift. Indeed the
equivalence (2.37) can be also read backwards, implying that every scalar
action of the form (2.36) can be uplifted to (2.32) which can be used to
compute the same observables. The prescription to uplift Lagrangians is thus
trivial and it amounts to uplifting fields to superfields, and derivatives to
superderivatives,

$\hat{\phi}(\hat{x})\to\Phi(y)\,,\qquad\partial_{\mu}\to\partial_{a}\,.$
(2.38)

This can be further generalized to any number or scalar fields and also to
theories where the fundamental fields are not scalars, see for example the
realizations for spinors and gauge fields presented in [28].

###  2.6 Uplift of CFTs

While we showed that given a Lagrangian, its uplift is somewhat
straightforward, this result does not help much in defining the uplift of CFTs
which typically do not have useful Lagrangian descriptions. However in CFT we
can use the power of symmetry to uplift some observables for free. This is
indeed a basic idea in CFT where it is known that one can use conformal
symmetry to set the operator insertions to special locations. E.g. by knowing
two and three-point functions of scalar operators on a line one can recover
their full dependence on coordinates, and similarly four-point functions of
scalar operators are completely determined by their expression on a plane. The
same idea works for PS CFTs. For these theories the dimensionally reduced two
and three-point functions of scalar operators determine the uplifted ones as
long as $\hat{d}\geq 1$, while for four-point functions of scalar operators
the uplifted correlator is fully reconstruct if $\hat{d}\geq 2$. The
reconstruction in these cases is trivial and it amounts to replace

$\hat{x}\to y\,,$ (2.39)

in correlation functions. For example a two-point function
$\langle\hat{O}_{1}(\hat{x}_{1})\hat{O}_{2}(\hat{x}_{2})\rangle=|\hat{x}_{12}|^{-2\Delta}$
is simply uplifted to
$\langle{\mathcal{O}}_{1}(y_{1}){\mathcal{O}}_{2}(y_{2})\rangle=|y_{12}|^{-2\Delta}$,
where now the distance is computed using the $g_{d|2}$ metric instead of a
$g_{d-2}$ metric. Similarly for any scalar four-point function
$\langle\hat{O}_{1}(\hat{x}_{1})\hat{O}_{2}(\hat{x}_{2})\hat{O}_{3}(\hat{x}_{3})\hat{O}_{4}(\hat{x}_{4})\rangle=F(\hat{x}_{i}\cdot\hat{x}_{j})$,
as long as $\hat{d}\geq 2$, then its uplift is just given by $F(y_{i}\cdot
y_{j})$ where all distances are replaced by superspace distances.

Of course this prescription cannot be used to fix all possible observables. In
general for any fixed dimension $\hat{d}$, when the number of insertion is
high enough and/or the operators have spin large enough, then a piece of the
information of the correlation function cannot be reconstructed. One can
easily see when this happens for any given observable by counting the number
of cross ratios and invariant tensor structures [29]. E.g. in $\hat{d}=1$ a
four-point function is fixed in terms of a single cross ratio and therefore
one cannot simply reconstruct a full PS CFT four-point function in $d=3$, but
only its restriction to a line (we expand on this point in section 7). This is
also related to the fact that some operators of the PS CFT are projected to
zero after dimensional reduction. This is easy to see, indeed in the PS CFT
there can exist operators with an arbitrary number of graded antisymmetric
indices [7], but in $\hat{d}$ dimensions one cannot antisymmetrize more that
$\hat{d}$ indices otherwise the result is trivially zero. This will be further
discussed in section 7.

In this paper we will mostly focus on scalar correlators of $n\leq 4$
insertions in dimensions $\hat{d}\geq 2$ so for these cases the uplift will
trivially work. It is interesting to stress that this is a huge number of
observables which know a lot about the theory. Indeed we are saying that all
possible four-point functions of scalar operators will be exactly uplifted.
The latter contain in their conformal block decomposition a lot of information
about operators with spin. It is an interesting open question whether the
information contained in this infinite set of correlators is enough to fix the
rest of the observables by e.g. applying bootstrap techniques. We leave this
question for future investigations.

##  3 Extracting primary components of correlation functions in superspace

One of the results of this paper is to put on a firm footing the uplift of
scalar $n$-point functions for $n\leq 4$. For the uplift to make sense all
components of the superspace correlator must behave appropriately as
correlators of a CFTd. In the following we show how to obtain the different
primary components from the superspace correlators for the cases of two, three
and four-point functions. We explicitly show the form of all components and
that they indeed have the correct covariance properties of correlators of
primary operators. Moreover we will show that all the primary components can
be fully reconstructed by the knowledge of the lowest component, which is
fixed by the dimensionally reduced theory. This result will be important for
the rest of the paper where we will input a four-point function of a
CFT${}_{\hat{d}}$ to obtain all the primary components of the uplifted four-
point functions. Since this philosophy works for any CFT${}_{\hat{d}}$ (at
least when $\hat{d}>1$) this result also supports the conjecture that the
dimensional uplift works for any theory.

Given a scalar superprimary ${\mathcal{O}}(y_{i})$ with dimension $\Delta_{i}$
inserted at a point $y_{i}$ in superspace, the correspondent primaries are
defined as follows444One could change the normalization of the primaries and
require that their two-point function is unit normalized. We did not do it
here since we privileged having a simpler form for the differential operators.
Moreover in our normalization $\Delta$ never appears at the denominator thus
avoiding singularities in the differential operators at special values of
$\Delta$.

$\displaystyle{\mathcal{O}}(y_{i})|_{0}$
$\displaystyle={\mathcal{O}}_{0}(x_{i})\,,$ (3.1)
$\displaystyle\partial_{\theta_{i}}{\mathcal{O}}(y_{i})|_{0}$
$\displaystyle={\mathcal{O}}_{\bar{\theta}}(x_{i})\,,$ (3.2)
$\displaystyle\partial_{\bar{\theta}_{i}}{\mathcal{O}}(y_{i})|_{0}$
$\displaystyle={\mathcal{O}}_{\theta}(x_{i})\,,$ (3.3) $\displaystyle{\bf
D}_{[{\bf i}]}{\mathcal{O}}(y_{i})|_{0}$
$\displaystyle=\tilde{\mathcal{O}}_{\theta\bar{\theta}}(x_{i})\,,$ (3.4)

where throughout the paper the notation $|_{0}$ will mean that we set to zero
all $\theta_{i}$ and $\bar{\theta}_{i}$. The differential operator in (3.4) is
defined as

${\bf D}_{[{\bf
i}]}\equiv\partial_{x_{i}}^{2}-(2\Delta_{i}-d+2)\partial_{\bar{\theta}_{i}}\partial_{\theta_{i}}\,.$
(3.5)

Notice that, while
${\mathcal{O}}_{0},{\mathcal{O}}_{\theta},{\mathcal{O}}_{\bar{\theta}}$ are
the same as the components defined in (2.20), the tilded operator
$\tilde{\mathcal{O}}_{\theta\bar{\theta}}$ differs from the component
${\mathcal{O}}_{\theta\bar{\theta}}$. The two operators are related since
$\tilde{\mathcal{O}}_{\theta\bar{\theta}}$ is a linear combination of
${\mathcal{O}}_{\theta\bar{\theta}}$ with a descendant of ${\mathcal{O}}_{0}$.
In practice we can think of $\tilde{\mathcal{O}}_{\theta\bar{\theta}}$ as an
improved version of ${\mathcal{O}}_{\theta\bar{\theta}}$ which is primary;
indeed ${\mathcal{O}}_{\theta\bar{\theta}}$ by itself is not a primary
operator.

Correlation functions of a PS CFTd can be written in terms of superspace
distances $y_{ij}^{2}$. By opportunely acting with the operators
$\partial_{\theta_{i}},\partial_{\bar{\theta}_{i}},{\bf D}_{[{\bf i}]}$ on
superspace correlators we can extract the correlators of CFTd primaries. The
generic form is as follows

${\bf
D}_{[{\vec{S}}]}\langle{\mathcal{O}}_{1}(y_{1})\dots\mathcal{O}_{n}(y_{n})\rangle|_{0}\,,$
(3.6)

where ${\bf D}_{[{\vec{S}}]}$ is a differential operator in
$x_{i},\theta_{i},\bar{\theta}_{i}$ defined as follows

${\bf D}_{[\vec{S}]}\equiv{\bf D}_{[S_{1}\dots S_{m}]}\equiv{\bf
D}_{[S_{1}]}{\cdots}{\bf D}_{[S_{m}]}\,,$ (3.7)

where each $S_{k}$ can be either ${\bf i}$ or ${i{\bar{j}}}$ and we define

${\bf
D}_{[{i\bar{j}}]}\equiv\partial_{\bar{\theta}_{j}}\partial_{\theta_{i}}\,,\quad(i\neq
j)\,.$ (3.8)

We find it convenient to use ${\bf D}_{[{i\bar{j}}]}$ instead of the building
blocks $\partial_{\theta_{i}}$, $\partial_{\bar{\theta}_{j}}$, because all
observables must be $Sp(2)$ invariants, so if $\partial_{\theta_{i}}$ is used
then we must also use $\partial_{\bar{\theta}_{j}}$ for some $j$. As an
example the operators will be of the form ${\bf D}_{[1\bar{2}3\bar{4}]}={\bf
D}_{[1\bar{2}]}{\bf D}_{[3\bar{4}]}$ (of course this could be equivalently
written in terms of ${\bf D}_{[1\bar{4}]}{\bf D}_{[3\bar{2}]}$), ${\bf
D}_{[\bf{124}]}={\bf D}_{[{\bf 1}]}{\bf D}_{[{\bf 2}]}{\bf D}_{[{\bf 4}]}$,
${\bf D}_{[{\bf{2}}1\bar{4}]}={\bf D}_{[{\bf{2}}]}{\bf D}_{[1\bar{4}]}$.

It will also be useful to introduce some shift operators
$\Sigma_{[{\vec{S}}]}$, which take into account that the dimension of the
operator $\Delta_{i}$ is shifted by one unit if ${\bf D}_{[{\vec{S}}]}$
contains $\partial_{\bar{\theta}_{i}}$ or $\partial_{\theta_{i}}$ and by two
units if ${\bf D}_{[{\vec{S}}]}$ contains
$\partial_{x_{i}}^{2}-(2\Delta_{i}-d+2)\partial_{\bar{\theta}_{i}}\partial_{\theta_{i}}$.
To be precise

$\Sigma_{[{\vec{S}}]}\equiv\Sigma_{[S_{1}\dots
S_{m}]}\equiv\Sigma_{[S_{1}]}{\cdots}\Sigma_{[S_{m}]}\,,$ (3.9)

where $\Sigma_{[S_{i}]}$ implement the following shifts

$\Sigma_{[{\bf
i}]}g(\Delta_{k})=g(\Delta_{k}+2\delta_{ik}),\qquad\Sigma_{[{i}{\bar{j}}]}g(\Delta_{k})=g(\Delta_{k}+\delta_{ik}+\delta_{jk}),$
(3.10)

for any function of $g$ that depends on the conformal dimensions $\Delta_{k}$,
where $k=1,\dots,n$ in formula (3.6).

###  3.1 Two-point functions

As we explained in section 2.3, the scalar two-point function in superspace is
fixed as (2.22). Using the differential operators defined above, we can easily
compute all the associated primary components as follows

$\displaystyle\langle{\mathcal{O}}_{0}(x_{1}){\mathcal{O}}_{0}(x_{2})\rangle$
$\displaystyle\equiv\langle{\mathcal{O}}(y_{1}){\mathcal{O}}(y_{2})\rangle\big{|}_{0}=\frac{1}{(x_{12}^{2})^{\Delta}}\,,$
(3.11)
$\displaystyle\langle{\mathcal{O}}_{\theta}(x_{2}){\mathcal{O}}_{\bar{\theta}}(x_{1})\rangle$
$\displaystyle\equiv{\bf
D}_{[1\bar{2}]}\langle{\mathcal{O}}(y_{1}){\mathcal{O}}(y_{2})\rangle\big{|}_{0}=\frac{2\Delta}{(x_{12}^{2})^{\Delta+1}}\,,$
(3.12)
$\displaystyle\langle\tilde{\mathcal{O}}_{\theta\bar{\theta}}(x_{1})\tilde{\mathcal{O}}_{\theta\bar{\theta}}(x_{2})\rangle$
$\displaystyle\equiv{\bf D}_{[{\bf
12}]}\langle{\mathcal{O}}(y_{1}){\mathcal{O}}(y_{2})\rangle\big{|}_{0}=\frac{-4\Delta(\Delta+1)(-d+2\Delta+2)(-d+2\Delta+4)}{(x_{12}^{2})^{\Delta+2}}\,.$
(3.13)

We notice that all these two-point functions have the correct properties to be
two-point functions of primary operators. Moreover we further checked that the
two-point functions of different primaries are diagonal, namely
$\langle\tilde{\mathcal{O}}_{\theta\bar{\theta}}{\mathcal{O}}_{0}\rangle=0$.
Indeed one of the ways to fix the differential operator ${\bf D}_{[\bf{i}]}$
is to require that the latter is true. We notice that the normalization of the
two-point functions above may be zero. This happens when $\Delta=0$, which is
the trivial case of the identity, but also at the free field dimension
$\Delta=\frac{d-2}{2}$ and at $\Delta=\frac{d-4}{2}$ which is the dimension of
a free field in the dimensionally reduced theory. In section (6) we will see
that indeed a field with dimensions $\Delta=1$ in $d=4$ will appear, in that
circumstance we will explain how the vanishing norm (3.13) will have important
implications.

###  3.2 Three-point functions

We now focus on three-point functions of scalar operators, which in superspace
are defined as (2.23). This observable contains 14 inequivalent three-point
functions of primary operators corresponding to the different primary
components of the supermultiplets. To capture the three-point functions of all
components at once we use the following compact formula

${\bf
D}_{[\vec{S}]}\langle{\mathcal{O}}_{1}(y_{1}){\mathcal{O}}_{2}(y_{2}){\mathcal{O}}_{3}(y_{3})\rangle\bigg{|}_{0}=c_{[\vec{S}]}\lambda_{123}\Sigma_{[\vec{S}]}\frac{1}{(x_{12}^{2})^{\frac{\Delta_{1}+\Delta_{2}-\Delta_{3}}{2}}(x_{13}^{2})^{\frac{\Delta_{1}+\Delta_{3}-\Delta_{2}}{2}}(x_{23}^{2})^{\frac{\Delta_{2}+\Delta_{3}-\Delta_{1}}{2}}}\,,$
(3.14)

where ${\bf D}_{[\vec{S}]}$ are the differential operators defined in (3.7)
and $\Sigma_{[\vec{S}]}$ implement the shifts in $\Delta_{i}$ according to
(3.9). In this notation we only need to define what the coefficients
$c_{[\vec{S}]}$ are. They encode the relation between the OPE coefficient of
the superprimary and the ones of its superdescendants. When $\vec{S}$ is the
empty list, which corresponds to the lowest component, the coefficient equals
one, namely $c_{[\ ]}=1$. All other cases are below

$\displaystyle\small\begin{array}[]{lll}c_{[{\bf
1}]}=-\alpha_{13,2}\alpha_{12,3}\,,&c_{[{\bf
2}]}=-\alpha_{23,1}\alpha_{21,3}\,,&c_{[{\bf
3}]}=-\alpha_{31,2}\alpha_{32,1}\,,\\\
c_{[{1\bar{2}}]}=-\alpha_{12,3}\,,&c_{[{2\bar{3}}]}=-\alpha_{23,1}\,,&c_{[{3\bar{1}}]}=-\alpha_{31,2}\,,\\\
c_{[{\bf 1}{2\bar{3}}]}=\alpha_{13,2}\alpha_{12,3}(\beta-d+2)\,,&c_{[{\bf
2}3{\bar{1}}]}=\alpha_{23,1}\alpha_{21,3}(\beta-d+2)\,,&c_{[{\bf
3}{1\bar{2}}]}=\alpha_{31,2}\alpha_{32,1}(\beta-d+2)\,,\end{array}$ (3.18)
$\displaystyle\small\begin{array}[]{ll}c_{[{\bf
12}]}=\alpha_{12,3}\left(\alpha_{12,3}+2\right)(\beta-d+2)(d-4-\alpha_{12,3})\,,&c_{[{\bf
23}]}=\alpha_{23,1}\left(\alpha_{23,1}+2\right)(\beta-d+2)(d-4-\alpha_{23,1})\,,\\\
c_{[{\bf
13}]}=\alpha_{13,2}\left(\alpha_{13,2}+2\right)(\beta-d+2)(d-4-\alpha_{13,2})\,,&c_{[{\bf
123}]}=\alpha_{23,1}\alpha_{12,3}\alpha_{13,2}(\beta-2d+6)(\beta-d+2)(\beta-d+4)\,,\end{array}$
(3.21)

where $\alpha_{ij,k}\equiv\Delta_{i}+\Delta_{j}-\Delta_{k}$ and
$\beta\equiv\Delta_{1}+\Delta_{2}+\Delta_{3}$. For the sake of clarity we show
a couple of three-point functions in a more explicit form,

$\displaystyle\langle{\mathcal{O}}_{0\,1}(x_{1}){\mathcal{O}}_{0\,2}(x_{2})\tilde{\mathcal{O}}_{\theta\bar{\theta}\,3}(x_{3})\rangle$
$\displaystyle=\frac{\lambda_{123}\left(\Delta_{1}-\Delta_{2}-\Delta_{3}\right)\left(\Delta_{1}-\Delta_{2}+\Delta_{3}\right)}{(x_{12}^{2})^{\frac{\Delta_{1}+\Delta_{2}-\Delta_{3}-2}{2}}(x_{13}^{2})^{\frac{\Delta_{1}+\Delta_{3}+2-\Delta_{2}}{2}}(x_{23}^{2})^{\frac{\Delta_{2}+\Delta_{3}+2-\Delta_{1}}{2}}}\,,$
(3.22)
$\displaystyle\langle{\mathcal{O}}_{0\,1}(x_{1}){\mathcal{O}}_{\bar{\theta}\,2}(x_{2}){\mathcal{O}}_{\theta\,3}(x_{3})\rangle$
$\displaystyle=\frac{\lambda_{123}(\Delta_{2}+\Delta_{3}-\Delta_{1})}{(x_{12}^{2})^{\frac{\Delta_{1}+\Delta_{2}-\Delta_{3}}{2}}(x_{13}^{2})^{\frac{\Delta_{1}+\Delta_{3}-\Delta_{2}}{2}}(x_{23}^{2})^{\frac{\Delta_{2}+\Delta_{3}-\Delta_{1}+2}{2}}}\,.$
(3.23)

As a comment, let us mention that it is a non-trivial consistency check that
in all 14 cases these take the form of three-point functions of primary
operators. The shifts $\Sigma_{[\vec{S}]}$ keep track of the correct scaling
of the operators and thus the theory behaves correctly as a CFTd. The OPE
coefficients of the (primary) superdescendants are just related to the ones of
the superprimary by some coefficients $c_{[\vec{S}]}$ which are fixed by SUSY.
Therefore by knowing the lowest component one can reconstruct all the other
ones.

###  3.3 Four-point functions

Scalar four-point functions in superspace are written as (2.24), in terms of a
single function $f(U,V)$ of superspace cross ratios $U,V$ . Of course since
the theory is a CFT, all components of this superspace four-point function
should be written in terms of usual $d$-dimensional cross ratios $u,v$. The
lowest component is just $f(u,v)$. The other components are obtained by acting
with a differential operator in $u,v$ that acts on $f(u,v)$. The latter can be
computed by acting with some differential operators on a four-point function,
as follows

${\bf
D}_{[{\vec{S}}]}K_{\Delta_{i}}(y_{i})f(U,V)|_{\theta,\bar{\theta}=0}=(\Sigma_{[{\vec{S}}]}K_{\Delta_{i}}(x_{i}))D_{[{\vec{S}}]}f(u,v)$
(3.24)

where ${\bf D}_{[{\vec{S}}]}$ and $\Sigma_{[{\vec{S}}]}$ are defined in (3.7)
and (3.9) while $D_{[{\vec{S}}]}$ is a differential operator in $u,v$ which we
computed for all 43 components. Some cases are too lengthy to be shown in a
paper,555Indeed the maximal order of the differential operator in
$\partial_{u}$ and $\partial_{v}$ is eight, and the coefficients are
polynomials of the seven variables
$d,\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4},u,v$ Because of this, the
resulting expressions are sometimes several pages long. but they will be
collected in a Mathematica file included with the publication. Below we show
their explicit form in a few instances. First the lowest component is the
trivial differential operator $D_{[\;]}=1$. Let us now focus on the operators
$D_{[{\bf i}]}$ for $i=1,2,3,4$,

$D_{[{\bf
1}]}\,\equiv\left[-\Delta_{12}^{2}+\Delta_{34}^{2}u-\Delta_{34}\Delta_{12}(u+v-1)-2\Delta_{2}\left(\Delta_{12}+\Delta_{34}(v-1)\right)\right]\,-2u\left(\Delta_{12}(u+v-1)-\Delta_{34}(u+v-1)+2\Delta_{2}v-2v\right)\partial_{u}\,-2v\left(-2\left(\Delta_{34}+1\right)u+\Delta_{12}(u+v-1)+2\Delta_{2}(v-1)\right)\partial_{v}\,\\\
+4uv(u+v-1)\partial_{u}\partial_{v}\,+4u^{2}v\partial_{u}^{2}\,+4uv^{2}\partial_{v}^{2}\,\;,$
(3.25) $D_{[{\bf
2}]}\,\equiv\left(\Delta_{1}^{2}-\Delta_{2}^{2}\right)\,-4\left(\Delta_{1}-1\right)u\partial_{u}\,+\left(2\left(-\Delta_{1}+\Delta_{2}+2\right)u-2\left(\Delta_{1}+\Delta_{2}\right)(v-1)\right)\partial_{v}\,\\\
+4u^{2}\partial_{u}^{2}\,+4uv\partial_{v}^{2}\,+4u(u+v-1)\partial_{u}\partial_{v}\,\;,$
(3.26) $D_{[{\bf
3}]}\,\equiv\left(\Delta_{4}^{2}-\Delta_{3}^{2}\right)\,-4\left(\Delta_{4}-1\right)u\partial_{u}\,+\left(2\Delta_{3}(u-v+1)-2\Delta_{4}(u+v-1)+4u\right)\partial_{v}\,\\\
+4u^{2}\partial_{u}^{2}\,+4uv\partial_{v}^{2}\,+4u(u+v-1)\partial_{u}\partial_{v}\,\;,$
(3.27) $D_{[{\bf
4}]}\,\equiv\left[\Delta_{12}\left(\Delta_{12}-\Delta_{34}\right)u+\left(\Delta_{4}+\Delta_{3}\right)\left(\Delta_{34}+\Delta_{12}(v-1)\right)\right]\,+4uv(u+v-1)\partial_{u}\partial_{v}\,\\\
-2u\left(\Delta_{12}(u+v-1)+\Delta_{34}(-u+v+1)+2\Delta_{4}v-2v\right)\partial_{u}\,+4u^{2}v\partial_{u}^{2}\,\\\
+2v\left(-2\Delta_{12}u+\Delta_{34}(u-v+1)+2u-2\Delta_{4}(v-1)\right)\partial_{v}\,+4uv^{2}\partial_{v}^{2}\,\;.$
(3.28)

We further exemplify the form of $D_{[{i\bar{j}}]}$ for $1\leq i<j\leq 4$,

$D_{[{1\bar{2}}]}\,\equiv\left(-\Delta_{1}-\Delta_{2}\right)\,+2u\partial_{u}\,\;,$
(3.29)
$D_{[{3\bar{4}}]}\,\equiv\left(-\Delta_{3}-\Delta_{4}\right)\,+2u\partial_{u}\,\;,$
(3.30)
$D_{[{1\bar{3}}]}\,\equiv-2u^{3/2}\partial_{u}\,-\Delta_{34}\sqrt{u}\,-2\sqrt{u}v\partial_{v}\,\;,$
(3.31)
$D_{[{2\bar{4}}]}\,\equiv-2u^{3/2}\partial_{u}\,+\Delta_{12}\sqrt{u}\,-2\sqrt{u}v\partial_{v}\,\;,$
(3.32)
$D_{[{1\bar{4}}]}\,\equiv\left(-\Delta_{12}+\Delta_{34}\right)\sqrt{u}\,+2\sqrt{u}v\partial_{v}\,\;,$
(3.33) $D_{[{2\bar{3}}]}\,\equiv 2\sqrt{u}\partial_{v}\,\;.$ (3.34)

It is easy to see that $D_{[{i\bar{j}}]}=-D_{[{\bar{i}j}]}$, so these
components are redundant and are not taken into account in the counting. Let
us also show the cases of four fermionic components. In this case there are
only three independent operators given by

$D_{[{1\bar{2}3\bar{4}}]}\,\equiv\left(\Delta_{1}+\Delta_{2}\right)\left(\Delta_{3}+\Delta_{4}\right)\,+4u^{2}\partial_{u}^{2}\,+2\left(\Delta_{12}-\Delta_{34}-2\right)u\partial_{v}\,\\\
-2\left(\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4}-2\right)u\partial_{u}\,-4uv\partial_{v}^{2}\,\;,$
(3.35) $D_{[{1\bar{3}2\bar{4}}]}\,\equiv
2\left(-\Delta_{12}+\Delta_{34}+2\right)u^{2}\partial_{u}\,+4u^{3}\partial_{u}^{2}\,+8u^{2}v\partial_{u}\partial_{v}\,-\Delta_{12}\Delta_{34}u\,-2\left(\Delta_{12}-\Delta_{34}-2\right)u(v-1)\partial_{v}\,+4u(v-1)v\partial_{v}^{2}\,$
(3.36)
$D_{[{1\bar{2}4\bar{3}}]}\,\equiv\left(\left(\Delta_{1}+\Delta_{2}\right)\left(\Delta_{3}+\Delta_{4}\right)+\Delta_{12}\Delta_{34}u\right)\,+2\left(\Delta_{12}-\Delta_{34}-2\right)uv\partial_{v}\,+2\left(\left(\Delta_{12}-\Delta_{34}-2\right)u-\left(\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4}-2\right)\right)u\partial_{u}\,\\\
-8u^{2}v\partial_{u}\partial_{v}\,+4\left(1-u\right)u^{2}\partial_{u}^{2}\,-4uv^{2}\partial_{v}^{2}\,\\\
$ (3.37)

The other combinations are equivalent, e.g.
$D_{[{\bar{1}2\bar{3}4}]}=D_{[{1\bar{2}3\bar{4}}]}$. Finally we show a single
representative of the family $D_{[{{\bf i}j\bar{k}}]}$ since these are already
quite lengthy,

$D_{[{{\bf
1}2\bar{3}}]}\,\equiv+\left(\Delta_{1}+\Delta_{2}\right)\Delta_{34}\sqrt{u}\left(-d+2\Delta_{1}+2\right)\,-2\left(\Delta_{1}+\Delta_{2}-\Delta_{34}-2\right)u^{3/2}\left(d-2\left(\Delta_{1}+1\right)\right)\partial_{u}\,\\\
+\left(4u^{3/2}v\left(d-3\Delta_{1}-\Delta_{2}+\Delta_{34}+2\right)-4\left(\Delta_{12}-\Delta_{34}-2\right)(u-1)u^{3/2}\right)\partial_{u}\partial_{v}\,\\\
-2\sqrt{u}\left[\left(\Delta_{1}+\Delta_{2}\right)v\left(d-2\Delta_{1}+\Delta_{34}\right)+\left(\Delta_{12}-\Delta_{34}-2\right)\left(\Delta_{1}+\Delta_{2}+\Delta_{34}u+2u\right)\right]\partial_{v}\,\\\
+8u^{3/2}v^{2}\partial_{v}^{3}\,+8u^{3/2}v\left(v+u-1\right)\partial_{u}\partial_{v}^{2}\,+4u^{5/2}\left(d-2\left(\Delta_{1}+1\right)\right)\partial_{u}^{2}\,\\\
+8u^{5/2}v\partial_{u}^{2}\partial_{v}\,+4\sqrt{u}v\left(\left(\Delta_{1}+\Delta_{2}-\Delta_{12}u+2\Delta_{34}u+6u\right)-\left(\Delta_{1}+\Delta_{2}\right)v\right)\partial_{v}^{2}\,\;.$
(3.38)

The operator (3.38) has an explicit dependence on $d$. This feature is shared
with all the other operators which we are omitting from the text.

All differential operators are written in a form where the derivatives in $u$
and $v$ are on the right and thus act directly on the function. This
formulation is the most convenient for computations, however there may exist a
different formulation which makes the differential operators more compact. We
did not invest time in searching for compact rewritings, which we postpone for
future work.

We checked that the differential operators automatically satisfy crossing. For
example the crossing $1\leftrightarrow 3$ is written as

$D_{[\vec{S}]}f(u,v)\big{|}_{\Delta_{1}\leftrightarrow\Delta_{3},u\leftrightarrow
v}=\sigma_{\vec{S^{\prime}}}\frac{D_{[\vec{S}^{\prime}]}f(u,v)}{\Sigma_{[\vec{S}^{\prime}]}v^{\frac{-\Delta_{2}-\Delta_{3}}{2}}u^{\frac{\Delta_{1}+\Delta_{2}}{2}}}\,,$
(3.39)

where we assume that $f(u,v)$ is crossing covariant, namely
$f(u,v)=v^{\frac{-\Delta_{2}-\Delta_{3}}{2}}u^{\frac{\Delta_{1}+\Delta_{2}}{2}}f(v,u)$
(which must be the case since it comes from a well defined CFTd-2). Here we
define $[\vec{S}^{\prime}]=[\vec{S}]|_{1\leftrightarrow 3}$ and
$\sigma_{\vec{S}}=\pm 1$ is the fermionic signature of $\vec{S}$, namely it
computes how many permutations of the fermionic operators are needed to
restore the canonical order, e.g. $\sigma_{\bar{2}1{\bf 4}{\bf
3}}=\sigma_{\bar{2}1{\bf 3}{\bf 4}}=-\sigma_{1\bar{2}{\bf 3}{\bf 4}}=-1$.

We further checked the crossing $1\leftrightarrow 2$ and $1\leftrightarrow 4$,
which also work out correctly. These equations generically map $D_{[\vec{S}]}$
into a different $D_{[\vec{S^{\prime}}]}$, which is a check that all the
differential operators are correct. Moreover this implies that all primary
components of the superspace four-point functions automatically satisfy
crossing in all channels. This is an important check for the uplift, indeed we
find that given one four-point function in $d-2$ we get 43 new four-point
functions in $d$ dimensions which are crossing covariant and consistent with
supersymmetry. In the next section we explain that they can also be
automatically decomposed in $d$-dimensional conformal blocks.

##  4 Conformal block expansion in PS CFTs

In this section we want to study how the different components of a four-point
function are all compatible with the conformal block decomposition. This will
give rise to $43$ relations between conformal blocks in $d-2$ and $d$
dimensions.

As explained in section 2.3, the scalar four-point function in superspace is
determined in terms of a single function $f(U,V)$ of super cross ratios which
can be decomposed in superspace superconformal blocks
$G^{(d|2)}_{\Delta\,\ell}$ as in (2.28). On the other hand $f(u,v)$ also
defines the dimensionally reduced four-point function (which depends on the
usual cross ratios $u,v$) and thus it can be decomposed in $(d-2)$-dimensional
blocks

$f(u,v)=\sum_{\Delta\ell}\tilde{a}_{\Delta\ell}g^{(d-2)}_{\Delta\,\ell}(u,v)\,.$
(4.1)

where in principle $\tilde{a}_{\Delta\ell}$ and $a_{\Delta\ell}$ of the two
decompositions could be different. In [7] it was shown that they are actually
the same $\tilde{a}_{\Delta\ell}=a_{\Delta\ell}$, since also the conformal
blocks are the same function (2.29). This already explains that given a scalar
four-point function in $d-2$ dimensions compatible with the conformal block
decomposition (and other bootstrap axioms like crossing) one can define a
scalar four-point function in $\mathbb{R}^{d|2}$ superspace which is
compatible with the superconformal block decomposition (and the other
bootstrap axioms).

However it is also important to check that the $d$-dimensional PS CFT behaves
as a good CFTd and therefore its four-point functions are decomposed in terms
of $d$-dimensional conformal blocks $g^{(d)}_{\Delta\,\ell}$. In particular
this means that each component of $G^{(d|2)}_{\Delta\,\ell}(U,V)$ should be
decomposed in terms of a finite number of $g^{(d)}_{\Delta\,\ell}$ in a
precise linear combination fixed by supersymmetry. In [7] we showed that this
is indeed true for the lowest component. In particular since the lowest
component of $G^{(d|2)}_{\Delta\,\ell}(U,V)$ is also equal to the block
$g^{(d-2)}_{\Delta\,\ell}$ itself, we obtained the following beautiful formula
relating conformal blocks in $d-2$ and $d$ dimensions [7]

$g^{(d-2)}_{\Delta\,\ell}=g^{(d)}_{\Delta\,\ell}+c_{2,0}\,g^{(d)}_{\Delta+2\,\ell}+c_{1,-1}\,g^{(d)}_{\Delta+1\,\ell-1}+c_{0,-2}\,g^{(d)}_{\Delta\,\ell-2}+c_{2,-2}\,g^{(d)}_{\Delta+2\,\ell-2}\
,$ (4.2)

where the generic scalar block in $d-2$ dimensions is written as a linear
combination of only five blocks in $d$ dimensions. The coefficients can be
written in closed form as follows:

$\displaystyle c_{2,0}$
$\displaystyle=\textstyle-\frac{(\Delta-1)\Delta\left(\Delta-\Delta_{12}+\ell\right)\left(\Delta+\Delta_{12}+\ell\right)\left(\Delta-\Delta_{34}+\ell\right)\left(\Delta+\Delta_{34}+\ell\right)}{4(d-2\Delta-4)(d-2\Delta-2)(\Delta+\ell-1)(\Delta+\ell)^{2}(\Delta+\ell+1)}\,,$
$\displaystyle c_{1,-1}$
$\displaystyle=\textstyle-\frac{(\Delta-1)\Delta_{12}\Delta_{34}\ell}{(\Delta+\ell-2)(\Delta+\ell)(d-\Delta+\ell-4)(d-\Delta+\ell-2)}\,,$
(4.3) $\displaystyle c_{0,-2}$
$\displaystyle=\textstyle-\frac{(\ell-1)\ell}{(d+2\ell-6)(d+2\ell-4)}\,,$
$\displaystyle c_{2,-2}$
$\displaystyle=\textstyle\frac{(\Delta-1)\Delta(\ell-1)\ell\left(d-\Delta-\Delta_{12}+\ell-4\right)\left(d-\Delta+\Delta_{12}+\ell-4\right)\left(d-\Delta-\Delta_{34}+\ell-4\right)\left(d-\Delta+\Delta_{34}+\ell-4\right)}{4(d-2\Delta-4)(d-2\Delta-2)(d+2\ell-6)(d+2\ell-4)(d-\Delta+\ell-5)(d-\Delta+\ell-4)^{2}(d-\Delta+\ell-3)}\,.$

Because of this relation it is trivial to see that if $f(u,v)$ is decomposed
in $g^{(d-2)}_{\Delta\,\ell}$, then it can be also decomposed in
$g^{(d)}_{\Delta\,\ell}$ and the explicit form of the coefficients $c_{i,j}$
can be used to simply map the OPE coefficients of the $d-2$ dimensional
decomposition, to the OPE coefficients of the $d$-dimensional one.

This relation however was obtained by only considering the lowest component of
$f(U,V)$. Now we want to see how this relation generalizes for the other
components.

###  4.1 43 relations between conformal blocks across dimensions

As we showed in section 3.3, inside a single superspace four-point functions
there are 43 inequivalent four-point functions specified by the list
${\vec{S}}$. Using (3.24) we can reconstruct each primary four-point function
by acting with a given differential operator $D_{[{\vec{S}}]}$ in the
variables $u,v$ on the function $f(u,v)$ that specifies the lowest component.
This is of course also true at the level of conformal blocks if we replace
$f\to G^{(d|2)}_{\Delta\,\ell}$ in (3.24). In particular this replacement
defines what are the superconformal blocks for each component. In other words
$G^{(d|2)}_{\Delta\,\ell}(U,V)$ is the _superspace superconformal blocks_ ,
while $D_{[{\vec{S}}]}G^{(d|2)}_{\Delta\,\ell}(u,v)$ defines what is usually
called a _superconformal block_ , which is just a function of $u,v$. Each
$D_{[{\vec{S}}]}G^{(d|2)}_{\Delta\,\ell}(u,v)$ must be decomposed in a finite
number of $d$-dimensional blocks. Using again formula (2.29), we can replace
$G^{(d|2)}\to g^{d-2}$ and thus obtain $42$ extra relations between blocks in
$d-2$ and $d$ dimensions labelled by the component $\vec{S}$. We can capture
all such relations by the following compact formula

$\boxed{\
\phantom{\Bigg{|}}D_{[{\vec{S}}]}g^{(d-2)}_{\Delta\,\ell}=\sum_{(i,j)\in
P_{[{\vec{S}}]}}c^{[{\vec{S}}]}_{i,j}\
\Sigma_{[{\vec{S}}]}\,g^{(d)}_{\Delta+i\,\ell+j}\ \phantom{\Bigg{|}}}$ (4.4)

where we recall that $\Sigma_{[\vec{S}]}$ just implements some shifts in the
conformal dimensions $\Delta_{k}$ of the external operators by some units as
defined in (3.9) and (3.10), while the differential operators
$D_{[{\vec{S}}]}$ are defined in section 3.3 and explicitly given in a
Mathematica file attached to the publication. We computed in a closed form all
coefficients $c^{[{\vec{S}}]}_{i,j}$ for all 43 possible choices of $\vec{S}$
as we will show in the following section and in appendix A (for convenience in
the Mathematica file we also include the full list of the coefficients
together with a check of (4.4)).

Finally it is crucial that the summation over $P_{[{\vec{S}}]}$ is finite; in
particular it contains at most $5$ terms depending on the choice of
${{\vec{S}}}$. Indeed $P_{[{\vec{S}}]}$ is defined as one the following three
possible sets

$P_{[{\vec{S}}]}=\left\\{\begin{array}[]{ll}P^{(0)}\equiv\\{(0,0),(0,-2),(1,-1),(2,0),(2,-2)\\}&\mbox{if
}Q_{1}[{\vec{S}}]+Q_{2}[{\vec{S}}]=0\,,\\\
P^{(1)}\equiv\\{(0,-1),(1,0),(1,-2),(2,-1)\\}&\mbox{if
}Q_{1}[{\vec{S}}]+Q_{2}[{\vec{S}}]=\pm 1\,,\\\
P^{(2)}\equiv\\{(1,-1)\\}&\mbox{if }Q_{1}[{\vec{S}}]+Q_{2}[{\vec{S}}]=\pm
2\,,\end{array}\right.$ (4.5)

where $Q_{k}$ is the $Sp(2)$ R-symmetry charge (computed by
$M^{\theta\bar{\theta}}$) of the operator at the position $k$. Namely if the
$k$-th operator is a fermion then $Q_{k}=1$, if it is an antifermion
$Q_{k}=-1$, while if it is bosonic $Q_{k}=0$. The charge of the $k$-th
operator can be extracted from the list $\vec{S}$ in a very simple way, by
defining
$Q_{k}[{\vec{S}}]=Q_{k}[S_{1},\dots,S_{n}]=Q_{k}[S_{1}]+\dots+Q_{k}[S_{n}]$
with

$Q_{k}[i\bar{j}]=\delta_{ik}-\delta_{jk}\,,\qquad Q_{k}[{\bf i}]=0\,.$ (4.6)

In table 2 we present a table that summarizes all components $[{\vec{S}}]$ of
the four-point function and their associated sets $P_{[{\vec{S}}]}$.

$P_{[{\vec{S}}]}$ |  $\\#$ | $[{\vec{S}}]$
---|---|---
$\phantom{\bigg{|}}P^{(0)}\phantom{\bigg{|}}$ | $26$ | $\begin{array}[]{c}[\,],[1{\bar{2}}],[3{\bar{4}}],[{\bf 1}],[{\bf 2}],[{\bf 3}],[{\bf 4}],[{\bf 1}3{\bar{4}}],[{\bf 2}3{\bar{4}}],[{\bf 3}1{\bar{2}}],[{\bf 4}1{\bar{2}}],[1{\bar{2}}3{\bar{4}}],[1{\bar{2}}4{\bar{3}}],\\\ {}[{\bf 12}],[{\bf 13}],[{\bf 14}],[{\bf 23}],[{\bf 24}],[{\bf 34}],[{\bf 12}3{\bar{4}}],[{\bf 34}1{\bar{2}}],[{\bf 123}],[{\bf 124}],[{\bf 134}],[{\bf 234}],[{\bf 1234}]\end{array}$
$\phantom{\bigg{|}}P^{(1)}\phantom{\bigg{|}}$ | $16$ | $\begin{array}[]{c}[1{\bar{3}}],[1{\bar{4}}],[2{\bar{3}}],[2{\bar{4}}],[{\bf{23}}1{\bar{4}}],[{\bf{13}}2{\bar{4}}],[{\bf{24}}1{\bar{3}}],[{\bf{14}}2{\bar{3}}],\\\ {}[{\bf{1}}2{\bar{3}}],[{\bf{1}}2{\bar{4}}],[{\bf{2}}1{\bar{3}}],[{\bf{2}}1{\bar{4}}],[{\bf{3}}1{\bar{4}}],[{\bf{3}}2{\bar{4}}],[{\bf{4}}1{\bar{3}}],[{\bf{4}}2{\bar{3}}]\\\ \end{array}$
$\phantom{\bigg{|}}P^{(2)}\phantom{\bigg{|}}$ | $1$ | $\begin{array}[]{c}[1{\bar{3}}2{\bar{4}}]\end{array}$
Table 2:  For all possible components $[{\vec{S}}]$ of four-point function we
show which is their associated $P_{[{\vec{S}}]}$. The set $P^{(0)}$ appears in
26 instances, $P^{(1)}$ in 16 while $P^{(2)}$ only in a single one.

Of course it is natural that the _rhs_ of (4.4) depends on the charge of the
operators at position $1$ and $2$. Indeed, since the $Sp(2)$ charge is
conserved, the charge of these two operators specifies which operator can flow
in their OPE and thus which conformal blocks can be exchanged. Keeping this in
mind let us now explain why the specific form of (4.5) arises.

Let us first consider the case $Q_{1}+Q_{2}=0$ when bosons are exchanged. In
this case there are 5 different primaries that can be exchanged for every spin
$\ell$ super primary.666For this counting we consider $\ell$ large enough
otherwise there may be less than 5 coefficients. The coefficients
$c^{[\vec{S}]}_{i,j}$ have zeros which automatically take into account when
some contributions do not arise. Similar remarks hold for the other cases
below.  These are obtained from a single spin $\ell$ super primary
${\mathcal{O}}^{a_{1}\dots a_{\ell}}$ as follows

${\mathcal{O}}^{\alpha_{1}\dots\alpha_{\ell}}_{0}\,,\quad{\mathcal{O}}^{\alpha_{1}\dots\alpha_{\ell-2}\theta\bar{\theta}}_{0}\,,\quad{\mathcal{O}}^{\alpha_{1}\dots\alpha_{\ell}}_{\theta\bar{\theta}}\,,\quad{\mathcal{O}}^{\alpha_{1}\dots\alpha_{\ell-1}\bar{\theta}}_{\theta}({\mathcal{O}}^{\alpha_{1}\dots\alpha_{\ell-1}\theta}_{\bar{\theta}})\,,\quad{\mathcal{O}}^{\alpha_{1}\dots\alpha_{\ell-2}\theta\bar{\theta}}_{\theta\bar{\theta}}\,.$
(4.7)

These have quantum numbers $(\Delta,\ell)$, $(\Delta+2,\ell)$,
$(\Delta+1,\ell-1)$, $(\Delta+2,\ell)$, $(\Delta+2,\ell-2)$ as prescribed by
(4.5) and (4.4). To be precise the operators above may not be all primaries,
but they can be always improved to be primaries, so this argument is enough to
explain the counting and the quantum numbers of the conformal blocks in the
_rhs_ of (4.4). Similarly when $Q_{1}+Q_{2}=\pm 1$ there are 4 primaries which
are fermionic (or antifermionic), which are related to the following operators

${\mathcal{O}}^{\alpha_{1}\dots\alpha_{\ell-1}\theta}_{0}\,,\quad{\mathcal{O}}^{\alpha_{1}\dots\alpha_{\ell}}_{\theta}\,,\quad{\mathcal{O}}^{\alpha_{1}\dots\alpha_{\ell-2}\theta\bar{\theta}}_{\theta}\,,\quad{\mathcal{O}}^{\alpha_{1}\dots\alpha_{\ell-1}\theta}_{\theta\bar{\theta}}\,,$
(4.8)

for $Q_{1}+Q_{2}=1$ and similarly for the opposite charge. These corresponds
to operators with quantum numbers $(\Delta,\ell-1)$, $(\Delta+1,\ell)$,
$(\Delta+1,\ell-2)$,$(\Delta+2,\ell-1)$, as shown in (4.5). Finally when
$Q_{1}+Q_{2}=\pm 2$ there is a single charge-2 (or $-2$) fermion

${\mathcal{O}}^{\alpha_{1}\dots\alpha_{\ell-1}\theta}_{\theta}({\mathcal{O}}^{\alpha_{1}\dots\alpha_{\ell-1}\bar{\theta}}_{\bar{\theta}})\,,$
(4.9)

with quantum numbers $(\Delta+1,\ell-1)$. This simple analysis fully explains
the sets in (4.5) and thus the structure of equation (4.4). In what follows we
explicitly show the form of $c^{[{\vec{S}}]}_{i,j}$.

###  4.2 Examples

In the previous section we introduced all ingredients for the equation (4.4)
besides the coefficient $c^{[{\vec{S}}]}_{i,j}$. Here we want to show their
form in a few examples (the rest can be found in appendix A). While doing it
we will also provide a few explicit examples of equation (4.4) for some
choices of $\vec{S}$.

Let us first consider the case of $\vec{S}={\bf i}$ for $i=1,\dots 4$, namely
when there is a single level-2 superdescendant and all other operators are
lowest components. Let us exemplify in this case all the ingredients entering
(4.4). The differential operators $D_{[{\bf i}]}$ are defined in equations
(3.25)-(3.28). The charges $Q_{1}$ and $Q_{2}$ defined in (4.6) are zero, thus
$P_{[{{\bf i}}]}=P^{(0)}$ which contains 5 terms. We thus find that there are
five $d$-dimensional conformal blocks appearing in the _rhs_ of (4.4), with
$\Delta_{i}$ shifted by two units (namely $\Delta_{i}\to\Delta_{i}+2$). We
therefore obtain that (4.4) can be unpacked as follows

$D_{[{\bf i}]}g^{(d-2)}_{\Delta\,\ell}=c^{[{\bf i}]}_{0,0}\Sigma_{{\bf
i}}g^{(d)}_{\Delta\,\ell}+c^{[{\bf i}]}_{0,-2}\Sigma_{{\bf
i}}g^{(d)}_{\Delta\,\ell-2}+c^{[{\bf i}]}_{1,-1}\Sigma_{{\bf
i}}g^{(d)}_{\Delta+1\,\ell-1}+c^{[{\bf i}]}_{2,0}\Sigma_{{\bf
i}}g^{(d)}_{\Delta+2\,\ell}+c^{[{\bf i}]}_{2,-2}\Sigma_{{\bf
i}}g^{(d)}_{\Delta+2\,\ell-2}\,,$ (4.10)

where $i=1,2,3,4$. For $i=1$ the exact form of the coefficients is as follows,

$\begin{array}[]{ll}c^{\bf{[1]}}_{0,0}&\\!\\!\\!=\left(-\Delta-\Delta_{12}-\ell\right)\left(-\Delta+\beta_{12}+\ell\right)\\\
c^{\bf{[1]}}_{0,-2}&\\!\\!\\!=\frac{(\ell-1)\ell\left(d-\Delta-\Delta_{12}+\ell-2\right)\left(d+\Delta-\beta_{12}+\ell-2\right)}{(d+2\ell-4)(d+2\ell-2)}\\\
c^{\bf{[1]}}_{1,-1}&\\!\\!\\!=\frac{(1-\Delta)\Delta_{34}\ell\left(d-\beta_{12}\right)\left(\Delta+\Delta_{12}+\ell\right)\left(-d+\Delta+\Delta_{12}-\ell+2\right)}{(\Delta+\ell-2)(\Delta+\ell)(-d+\Delta-\ell)(-d+\Delta-\ell+2)}\\\
c^{\bf{[1]}}_{2,0}&\\!\\!\\!=\frac{(1-\Delta)\Delta\left(\Delta+\Delta_{12}+\ell\right)\left(\Delta+\Delta_{12}+\ell+2\right)\left(\Delta-\Delta_{34}+\ell\right)\left(\Delta+\Delta_{34}+\ell\right)\left(-d+\Delta+\Delta_{12}-\ell+2\right)\left(-d+\Delta+\beta_{12}+\ell\right)}{4(2\Delta-d)(-d+2\Delta+2)(\Delta+\ell-1)(\Delta+\ell)^{2}(\Delta+\ell+1)}\\\
c^{\bf{[1]}}_{2,-2}&\\!\\!\\!=\frac{(1-\Delta)\Delta(\ell-1)\ell\left(\Delta+\Delta_{12}+\ell\right)\left(d-\Delta-\Delta_{12}+\ell-4\right)\left(d-\Delta-\Delta_{12}+\ell-2\right)\left(2d-\Delta-\beta_{12}+\ell-2\right)\left(d-\Delta-\Delta_{34}+\ell-2\right)\left(d-\Delta+\Delta_{34}+\ell-2\right)}{4(d-2\Delta-2)(d-2\Delta)(d+2\ell-4)(d+2\ell-2)(d-\Delta+\ell-3)(d-\Delta+\ell-2)^{2}(d-\Delta+\ell-1)}\\\
\end{array}\,,$

(4.11)

where we introduced the notation $\beta_{ij}\equiv\Delta_{i}+\Delta_{j}$.

All other coefficients $c^{\bf{[k]}}_{i,j}$ (for $k=2,3,4$) are obtained by a
simple map of the coefficient $c^{\bf{[1]}}_{i,j}$, namely

$c^{\bf{[2]}}_{i,j}=\pi_{(12)(34)}c^{\bf{[1]}}_{i,j}\,,\qquad
c^{\bf{[3]}}_{i,j}=\pi_{(13)(24)}c^{\bf{[1]}}_{i,j}\,,\qquad
c^{\bf{[4]}}_{i,j}=\pi_{(23)(14)}c^{\bf{[1]}}_{i,j}\,,$ (4.12)

where $\pi$ implements permutations of the labels $\Delta_{i}$. In particular

$\pi_{(ij)}F(\Delta_{k})\equiv
F(\Delta_{k})|_{\Delta_{i}\leftrightarrow\Delta_{j}}\,,\qquad\pi_{(i_{1}j_{1})\dots(i_{n}j_{n})}F(\Delta_{k})\equiv\pi_{(i_{1}j_{1})}\cdots\pi_{(i_{n}j_{n})}F(\Delta_{k})\,,$
(4.13)

for any function $F$ of the conformal dimensions $\Delta_{k}$.

Let us mention a common feature that we will see for most of choices of
$\vec{S}$: the relations (4.4) can be though as equations for the conformal
blocks, but the latter only depend on the differences $\Delta_{12}$ and
$\Delta_{34}$, while the coefficients $c^{[\vec{S}]}_{i,j}$ (see e.g. (4.11))
as well as the differential operators $D_{[\vec{S}]}$ (see e.g. (3.25)) depend
also on $\beta_{12}$, $\beta_{34}$ (polynomially). Therefore one can expand
the equations (4.4) in $\beta_{12}$ and $\beta_{34}$ and each term of the
expansion can be understood as a valid equation for the blocks.

Let us now consider the case of $\vec{S}={i\bar{j}}$ ($i\neq j$), which
corresponds to two level-one superdescendants and two lowest components. There
are two distinct situations. First we consider $(i,j)=(1,2),(3,4)$. In this
case $P_{[i\bar{j}]}$ contains 5 terms. Therefore the action of
$D_{[{i\bar{j}}]}$ (defined in (3.29) and (3.30)) on the $d-2$-dimensional
conformal blocks can be written in terms of a sum of five conformal blocks in
$d$-dimensions,

$D_{[{i\bar{j}}]}g^{(d-2)}_{\Delta\,\ell}=c^{[{i\bar{j}}]}_{0,0}g^{(d)}_{\Delta\,\ell}+c^{[{i\bar{j}}]}_{0,-2}g^{(d)}_{\Delta\,\ell-2}+c^{[{i\bar{j}}]}_{1,-1}g^{(d)}_{\Delta+1\,\ell-1}+c^{[{i\bar{j}}]}_{2,0}g^{(d)}_{\Delta+2\,\ell}+c^{[{i\bar{j}}]}_{2,-2}g^{(d)}_{\Delta+2\,\ell-2}\,.$
(4.14)

Here the conformal blocks are not shifted since they only depend on
$\Delta_{12}$ and $\Delta_{34}$ and thus the shift act trivially. The form of
the coefficients is as follows,

$\begin{array}[]{ll}c^{{[1\bar{2}]\ }}_{0,0}=\Delta-\beta_{12}-\ell\\\
c^{{[1\bar{2}]\
}}_{0,-2}=\frac{(\ell-1)\ell\left(\beta_{12}-d-\Delta-\ell+2\right)}{(d+2\ell-4)(d+2\ell-2)}\\\
c^{{[1\bar{2}]\
}}_{1,-1}=\frac{(\Delta-1)\Delta_{12}\Delta_{34}\ell\left(\beta_{12}-d\right)}{(\Delta+\ell-2)(\Delta+\ell)(-d+\Delta-\ell)(-d+\Delta-\ell+2)}\\\
c^{{[1\bar{2}]\
}}_{2,0}=\frac{(\Delta-1)\Delta\left(\Delta-\Delta_{12}+\ell\right)\left(\Delta+\Delta_{12}+\ell\right)\left(\Delta-\Delta_{34}+\ell\right)\left(\Delta+\Delta_{34}+\ell\right)\left(-d+\Delta+\beta_{12}+\ell\right)}{4(2\Delta-d)(-d+2\Delta+2)(\Delta+\ell-1)(\Delta+\ell)^{2}(\Delta+\ell+1)}\\\
c^{{[1\bar{2}]\
}}_{2,-2}=\frac{(\Delta-1)\Delta(\ell-1)\ell\left(d-\Delta+\Delta_{12}+\ell-2\right)\left(\Delta_{12}-d+\Delta-\ell+2\right)\left(\beta_{12}-2d+\Delta-\ell+2\right)\left(\Delta-d-\Delta_{34}-\ell+2\right)\left(\Delta-d+\Delta_{34}-\ell+2\right)}{4(d-2\Delta)(d-2(\Delta+1))(d+2\ell-4)(d+2\ell-2)(d-\Delta+\ell-3)(d-\Delta+\ell-2)^{2}(d-\Delta+\ell-1)}\\\
\end{array}\,.$ (4.15)

The other case $[3\bar{4}]$ is defined by

$c^{{[3\bar{4}]}}_{i,j}\equiv\pi_{(13)(24)}c^{{[1\bar{2}]}}_{i,j}\,.$ (4.16)

In the cases $(i,j)=(1,3),(1,4),(2,3),(2,4)$ the set $P_{[i\bar{j}]}$ contains
4 terms. We thus find that the action of $D_{[{i\bar{j}}]}$ (defined in
(3.31)-(3.34)) on the $d-2$-dimensional conformal blocks can be written in
terms of a sum of only four conformal blocks in $d$-dimensions, with
$\Delta_{i}$ and $\Delta_{j}$ shifted by one unit,

$D_{[{i\bar{j}}]}g^{(d-2)}_{\Delta\,\ell}=c^{[{i\bar{j}}]}_{0,-1}\Sigma_{{i\bar{j}}}g^{(d)}_{\Delta\,\ell-1}+c^{[{i\bar{j}}]}_{1,0}\Sigma_{{i\bar{j}}}g^{(d)}_{\Delta+1\,\ell}+c^{[{i\bar{j}}]}_{1,-2}\Sigma_{{i\bar{j}}}g^{(d)}_{\Delta+1\,\ell-2}+c^{[{i\bar{j}}]}_{2,-1}\Sigma_{{i\bar{j}}}g^{(d)}_{\Delta+2\,\ell-1}\,.$
(4.17)

The coefficient for $[1\bar{3}]$ takes the following form

$\begin{array}[]{l}c^{{[1\bar{3}]\ }}_{0,-1}=-\ell\\\ c^{{[1\bar{3}]\
}}_{1,0}=-\frac{(\Delta-1)\left(\Delta+\Delta_{12}+\ell\right)\left(\Delta+\Delta_{34}+\ell\right)}{2(\Delta+\ell-1)(\Delta+\ell)}\\\
c^{{[1\bar{3}]\
}}_{1,-2}=\frac{(\Delta-1)(\ell-1)\ell\left(d-\Delta-\Delta_{12}+\ell-2\right)\left(d-\Delta-\Delta_{34}+\ell-2\right)}{2(d+2\ell-4)(d+2\ell-2)(d-\Delta+\ell-2)(d-\Delta+\ell-1)}\\\
c^{{[1\bar{3}]\
}}_{2,-1}=\frac{(\Delta-1)\Delta\ell\left(\Delta+\Delta_{12}+\ell\right)\left(\Delta+\Delta_{34}+\ell\right)\left(d-\Delta-\Delta_{12}+\ell-2\right)\left(d-\Delta-\Delta_{34}+\ell-2\right)}{4(2\Delta-d)(-d+2\Delta+2)(\Delta+\ell-1)(\Delta+\ell)(-d+\Delta-\ell+1)(-d+\Delta-\ell+2)}\\\
\end{array}\,.$ (4.18)

The other three coefficients are defined by

$c^{{[2\bar{4}]}}_{i,j}=\pi_{(14)(23)}c^{{[1\bar{3}]}}_{i,j}\,,\quad
c^{{[1\bar{4}]}}_{i,j}=(-1)^{j}\pi_{(34)}c^{{[1\bar{3}]}}_{i,j}\,,\quad
c^{{[1\bar{3}]}}_{i,j}=(-1)^{j}\pi_{(12)}c^{{[1\bar{3}]}}_{i,j}\,,$ (4.19)

where $(-1)^{j}$ changes the sign of the coefficients $c_{0,-1}$ and
$c_{2,-1}$ (notice that these correspond to operators with spin $\ell-1$, thus
they have an extra sign with respect to operators with spin $\ell$ or
$\ell-2$).

Let us consider in details also the three cases when all four operators are
level-one superdescendant. In the first case $1\bar{2}3\bar{4}$, the set
$P_{[1\bar{2}3\bar{4}]}$ contains 5 terms and thus

$D_{[{1\bar{2}3\bar{4}}]}g^{(d-2)}_{\Delta\,\ell}=c^{{[1\bar{2}3\bar{4}]}}_{0,0}g^{(d)}_{\Delta\,\ell}+c^{{[1\bar{2}3\bar{4}]}}_{0,-2}g^{(d)}_{\Delta\,\ell-2}+c^{{[1\bar{2}3\bar{4}]}}_{1,-1}g^{(d)}_{\Delta+1\,\ell-1}+c^{{[1\bar{2}3\bar{4}]}}_{2,0}g^{(d)}_{\Delta+2\,\ell}+c^{{[1\bar{2}3\bar{4}]}}_{2,-2}g^{(d)}_{\Delta+2\,\ell-2}\,,$
(4.20)

where $D_{[{1\bar{2}3\bar{4}}]}$ is defined in (3.35) and the coefficients
read

$\begin{array}[]{l}c^{{[1\bar{2}3\bar{4}]}}_{0,0}=\left(\beta_{12}-\Delta+\ell\right)\left(\beta_{34}-\Delta+\ell\right)\\\
c^{{[1\bar{2}3\bar{4}]}}_{0,-2}=\frac{(\ell-1)\ell\left(d+\Delta-\beta_{12}+\ell-2\right)\left(\beta_{34}-d-\Delta-\ell+2\right)}{(d+2\ell-4)(d+2\ell-2)}\\\
c^{{[1\bar{2}3\bar{4}]}}_{1,-1}=\frac{(\Delta-1)\ell\left(\Delta_{12}\Delta_{34}\left(-d+\beta_{12}\right)\left(d-\beta_{34}\right)+(\Delta+\ell-2)(\Delta+\ell)(d-\Delta+\ell-2)(d-\Delta+\ell)\right)}{(\Delta+\ell-2)(\Delta+\ell)(d-\Delta+\ell-2)(d-\Delta+\ell)}\\\
c^{{[1\bar{2}3\bar{4}]}}_{2,0}=\frac{(\Delta-1)\Delta\left(\Delta_{12}-\Delta-\ell\right)\left(\Delta+\Delta_{12}+\ell\right)\left(\Delta-\Delta_{34}+\ell\right)\left(\Delta+\Delta_{34}+\ell\right)\left(-d+\Delta+\beta_{12}+\ell\right)\left(-d+\Delta+\beta_{34}+\ell\right)}{4(2\Delta-d)(-d+2\Delta+2)(\Delta+\ell-1)(\Delta+\ell)^{2}(\Delta+\ell+1)}\\\
c^{{[1\bar{2}3\bar{4}]}}_{2,-2}=\frac{(\Delta-1)\Delta(\ell-1)\ell\left(d-\Delta-\Delta_{12}+\ell-2\right)\left(2d-\Delta-\beta_{12}+\ell-2\right)\left(d-\Delta+\Delta_{12}+\ell-2\right)\left(d-\Delta-\Delta_{34}+\ell-2\right)\left(2d-\Delta-\beta_{34}+\ell-2\right)\left(d-\Delta+\Delta_{34}+\ell-2\right)}{4(d-2\Delta)(d-2(\Delta+1))(d+2\ell-4)(d+2\ell-2)(d-\Delta+\ell-3)(d-\Delta+\ell-2)^{2}(d-\Delta+\ell-1)}\\\
\end{array}\,.$

(4.21)

The case $1\bar{2}4\bar{3}$ is similar and the coefficients are related to the
ones above by
$c^{{[1\bar{2}4\bar{3}]}}_{i,j}=(-1)^{j}\pi_{(34)}c^{{[1\bar{2}3\bar{4}]}}_{i,j}$.

We now consider the case $1\bar{3}2\bar{4}$. This is the only case where the
set $P_{[1\bar{3}2\bar{4}]}$ contains only one term. The single coefficient is
defined as $c_{1,-1}^{[1\bar{3}2\bar{4}]}=2(\Delta-1)\ell$, therefore the
final formula takes the very simple form

$D_{[{1\bar{3}2\bar{4}}]}g^{(d-2)}_{\Delta\,\ell}=2(\Delta-1)\ell
g^{(d)}_{\Delta+1\,\ell-1}\,,$ (4.22)

where $D_{[{1\bar{3}2\bar{4}}]}$ is defined in (3.36). The differential
operator $D_{[{1\bar{3}2\bar{4}}]}/(2(\Delta-1)\ell)$ is very useful since it
can be applied to define conformal blocks in higher dimensions knowing the
ones in lower dimensions.

Indeed the relation (4.22) was already found in equation (4.37) of [30]777The
normalization of the blocks is different
$g^{(d),here}_{\Delta,\ell}=\frac{(-2)^{\ell}\left(\frac{d-2}{2}\right)_{\ell}}{(d-2)_{\ell}}g^{(d),there}_{\Delta,\ell}$.
by searching for a differential operator that shifts the dimension of the
conformal blocks. Now we can give a physical interpretation to this relation
as arising from supersymmetry.

We avoid the exemplifications of all other cases, but it should be clear that
all 43 relations in (4.4) can be easily implemented using the differential
operators and the coefficients defined in the Mathematica file attached to the
publication. For completeness in appendix A we also define the remaining
coefficients.

###  4.3 Poles in the conformal blocks and SUSY

In this section we aim at studying the possible singularities in $\Delta$ of
conformal blocks. Below we warm up by considering a unitary CFT and show that
in this case the blocks are always finite even when they naively look
singular. Then we turn to a PS CFT and consider the superconformal blocks,
which by equation (4.4) are written as a linear combination of conformal
blocks. We will show that in this case it may also happen that some blocks in
the linear combination diverge, but the specific linear combination provided
by (4.4) is fine tuned to cancel all singularities and gives a finite result.
On one hand this remark can be understood as a consistency check of (4.4). On
the other hand it will also uncover some interesting features which will play
an important role in the conformal block decompositions of the following
sections.

To start let us consider the conformal block decomposition of a four-point
correlation function in usual (compact unitary) CFTs. In this case it is
expected that both the OPE coefficients and the conformal blocks appearing in
the decomposition are finite quantities and sum up to a finite result. The
fact that conformal blocks are finite is non trivial since they have poles in
$\Delta$ as shown in (2.12), which for convenience we rewrite here

$g_{\Delta\ell}\sim\frac{R_{A}}{\Delta-\Delta^{*}_{A}}g_{\Delta_{A}\ell_{A}}\,,$
(4.23)

where $\Delta^{*}_{A},\Delta_{A}=\Delta^{*}_{A}+n_{A},\ell_{A}$ are defined in
table 1 and the coefficient $R_{A}$ is defined in (2.13). So the conformal
blocks seem to diverge for the exchange of an operator with labels
$\Delta=\Delta^{*}_{A},\ell$. At a first sight this might be puzzling since
these labels include the ones of a free scalar ($\Delta=d/2-1,\ell=0$) and a
conserved tensors (with $\Delta=d+\ell-2$ for any spin $\ell=1,2,\dots$). So
naively it may seem that free scalars and conserved tensors cannot be
exchanged. How can this be compatible with the fact that these operators exist
in physical theories? The answer is simple. In unitary theories when
$\Delta,\ell$ equals the ones of a free scalar or a conserved tensor then the
associated residue $R_{A}$ vanishes to cancel the pole. This gives a non
trivial restriction on the possible set of correlation functions that exchange
such operators. For example for a free scalar exchange the conformal block
diverges as

$g_{\Delta\,\ell=0}\sim\frac{R_{\textrm{III},1}|_{\ell=0}}{\Delta-(d/2-1)}g_{\Delta+2,\ell=0}\,,\qquad\mbox{for
}\Delta\to d/2-1.$ (4.24)

The residue at $\Delta=d/2-1$ vanishes when

$R_{\textrm{III},1}|_{\ell=0}=\frac{\left(d/2-1-\Delta_{12}\right)\left(d/2-1+\Delta_{12}\right)\left(d/2-1-\Delta_{34}\right)\left(d/2-1+\Delta_{34}\right)}{4(d-2)d}=0$
(4.25)

which implies that either $\Delta_{12}=\pm(d/2-1)$ or
$\Delta_{34}=\pm(d/2-1)$. This is indeed what happens in free theory: in order
to exchange a free field in an OPE, the dimensions of the (external) operators
in the correlation function must differ by $\pm(d/2-1)$, e.g.
$\phi^{n}\times\phi^{n-1}\sim\phi$.

Similarly when we exchange a conserved tensor of spin $\ell$ the conformal
block diverges as

$g_{\Delta\,\ell}\sim\frac{R_{\textrm{II},1}}{\Delta-(d+\ell-2)}g_{\Delta+1,\ell-1}\,,\qquad\mbox{for
}\Delta\to d+\ell-2.$ (4.26)

Thus we need to require that the residue at $\Delta=d+\ell-2$ has to vanish,
namely

$R_{\textrm{II},1}=\frac{\Delta_{12}\Delta_{34}\ell!(d+\ell-3)}{2(\ell-1)!(d+2\ell-4)(d+2\ell-2)}=0\,,$
(4.27)

which is possible only for either $\Delta_{12}=0$ or $\Delta_{34}=0$. This
condition is indeed required by Ward identities of the conserved tensors
$\langle\phi_{\Delta_{1}}\phi_{\Delta_{2}}T^{\mu_{1}\dots\mu_{\ell}}\rangle\propto\delta_{\Delta_{1}\Delta_{2}}$.

We thus found that in unitary theories the singularities of the conformal
blocks are avoided because the residue $R_{A}$ is zero. Let us discuss what
this means physically. Let us consider a conformal multiplet/module labelled
by a primary with dimension $\Delta$ and spin $\ell$. Now we want to see what
happens to this multiplet when changing $\Delta$. Using the notation of
equation (4.23), when $\Delta=\Delta^{*}_{A}$ the module becomes
singular/reducible, namely it contains a descendant labelled by
$\Delta_{A},\ell_{A}$ which becomes primary. The primary descendant together
with all its descendants define the submodule. All states of the submodule
have vanishing norm. These are responsible for the pole in the conformal
block. The condition $R_{A}=0$ is equivalent to the fact that there is a
shortening condition that the primary descendant must satisfy. For example
$\square\phi=0$ for a free scalar or
$\partial_{\mu}J^{\mu\mu_{2}\dots\mu_{\ell}}=0$ for a spin $\ell$ conserved
current. These shortening conditions can be understood as a way to mod out the
primary descendant and its associated multiplet. This is necessary in a
unitary theory to avoid non-trivial states with vanishing norms. In the
following we see how this does not happen in PS CFTs.

#### Avoiding poles with Parisi-Sourlas SUSY

Above we showed that a tuning of the conformal blocks is required when
operators at the unitarity bound are exchanged. Here we exemplify how this is
not necessary in PS CFTs. This will also generalize to exchanges below the
unitarity bound which are naively singular (as we detail in appendix B).

Let us exemplify the new mechanism for the case of an exchanged scalar
superblock with $\Delta=d/2-1$. We further think of this superblock as arising
from the uplift of a $\hat{d}=d-2$-dimensional block. Since in $\hat{d}$
dimensions nothing special happens for exchanges of operators of dimensions
$\Delta=d/2-1=\hat{d}/2$, we would expect that the same is true for the
superblocks of the uplifted theory, however this is non-trivial since in
$d$-dimensions the blocks have a pole at this value of $\Delta$ (unless (4.25)
is satisfied). Let us show how this pole is erased in the case of the lowest
component of a four-point function. The lowest component of a scalar
($\ell=0$) superblock takes the form

$g_{\Delta\,\ell=0}+c^{(\ell=0)}_{2,0}g_{\Delta+2\,\ell=0}\,,\qquad\mbox{where
}c^{(\ell=0)}_{2,0}=-\frac{\left(\Delta-\Delta_{12}\right)\left(\Delta+\Delta_{12}\right)\left(\Delta-\Delta_{34}\right)\left(\Delta+\Delta_{34}\right)}{4\Delta(\Delta+1)(d-2\Delta-4)(d-2\Delta-2)}\,.$
(4.28)

We can explicitly see that the coefficient $c^{(\ell=0)}_{2,0}$ in front of
$g_{\Delta+2\,\ell=0}$ diverges as follows

$c_{2,0}\sim-\frac{R_{\textrm{III},1}}{\Delta-(d/2-1)}\,\qquad\mbox{for
}\Delta\to d/2-1.$ (4.29)

The contribution of $c_{2,0}g_{\Delta+2\,\ell=0}$ exactly cancels the pole at
$\Delta\to d/2-1$ of $g_{\Delta\,\ell=0}$ defined in formula (2.12), and thus
we get that the combination provided by (the lowest component of) the
superblock

$\tilde{g}_{d/2-1\,\ell=0}\equiv\lim_{\Delta\to
d/2-1}g_{\Delta\,\ell=0}+c_{2,0}g_{\Delta+2\,\ell=0}$ (4.30)

is finite. Therefore in a PS CFT the pole at $\Delta=d/2-1$ in the conformal
block cancels thanks to supersymmetry and the conformal block
$\tilde{g}_{d/2-1\,\ell=0}$ can always be exchanged independently on the value
of $\Delta_{12}$ and $\Delta_{34}$. This mechanism is unconventional because
the pole of the block is cancelled because of a divergent OPE coefficient (to
be precise it is only the kinematic part $c_{2,0}$ of the OPE which diverges).
In all theories with finite OPE coefficients this can never happen and the
only mechanism allowed is the one of the cancellation of the residues $R_{A}$.

The fact that $R_{A}\neq 0$ means that the module of the primary descendant is
not modded out and thus it will be part of the spectrum. Another consequence
of this is that we can consider these primary descendants as insertions in a
correlation function without getting a vanishing result, because no shortening
condition has to be satisfied (this is in contrast with the unitary cases
described above where if one considers e.g. $\square\phi$ as an insertion in
any free boson correlation function, the resulting correlator vanishes). We
will see how all these facts will play a role in the study of the uplifted
minimal models.

There are other important cases where this mechanism takes place. Indeed,
besides for exchanges of operators at the unitarity bound, the conformal
blocks diverge for infinitely more values of $\Delta=\Delta^{*}_{A}$ (the
unitary values appear only for the type $\textrm{III},n=1$ when $\ell=0$ and
for $\textrm{II},n=1$ when $\ell>0$, but in general one could consider any of
the three types $\textrm{I},\textrm{II},\textrm{III}$ for any $n$). All these
extra cases lye below the unitarity bounds, so these exchanges do not arise in
unitary theories. However these may arise in non-unitary theories, like the PS
CFTs. For such theories there should be a mechanism to cancel the
singularities whenever one of these operators is exchanged.

First it is interesting to ask which are the possible problematic exchanges
which arise in the Parisi-Sourlas uplift of a unitary $\hat{d}\equiv
d-2$-dimensional theory (these are the theories that we consider in the rest
of the paper). The answer is easy: the operators of such uplifts satisfy
bounds which arise by uplifting the unitarity-bounds of the lower dimensional
theory. Namely they satisfy $\Delta\geq\frac{d}{2}-2$ for scalar operators and
$\Delta\geq d+\ell-3$ for operators of spin $\ell\geq 1$. In particular in
these theories one can exchange scalar operators with dimensions
$\Delta=\frac{d}{2}-2,\frac{d}{2}-1$ and operators with dimensions
$\Delta=d+\ell-3,d+\ell-2,d+\ell-1$ and spin $\ell$. For all these values the
blocks have singularities. For the uplift to make sense it is crucial that all
these singularities are resolved. In appendix B we show that all these poles
are erased, sometimes because $R_{A}$ vanishes and sometimes because of the
special linear combination of the blocks provided by supersymmetry which gives
rise to new regularized blocks that exactly cancels the singularity as we
showed for (4.30). The specific form of the regularized blocks computed in
appendix B is going to play a role in the next sections, when we will
decompose correlations functions of the uplifted minimal models.

##  5 Bootstrapping GFF using PS SUSY

In the previous sections we determined a set of kinematic properties shared by
all PS CFTs. We now want to apply the PS CFT framework to the simple theory of
a generalized free boson field (GFF). Below we will start by a basic
introduction to GFF to set some conventions. We define it in ${\hat{d}}=d-2>1$
dimensions (the case $\hat{d}=1$ is studied in section 7.2) so that the
uplifted theory lives in $d>3$ dimensions (for the purposes of this section
one could also drop the hat of ${\hat{d}}$ in every formula). In the next
subsection we will define the uplifted theory. We will show that, besides
being well defined, the PS uplift of GFF can be used to infer some properties
of the original GFF, thus helping to bootstrap some observables. As an example
we will be able to compute, without doing any Wick contraction, the conformal
block decomposition of $\langle\phi\phi^{n}\phi^{m}\phi^{m}\rangle$ for any
power $n,m$, which is a new result in GFF. We will also explain how the uplift
can be used to constrain perturbative computations around GFF.

GFF can be defined through a non-local $\hat{d}$-dimensional Lagrangian
$\phi\square^{\xi}\phi$, where $\xi$ is a real parameter. The theory is
Gaussian and therefore is solvable. An equivalent way to define the theory
without introducing a Lagrangian is as follows. The defining property of GFF
is that correlation function of fields $\phi$ can be computed through Wick
contractions where the propagators are
$\langle\phi(x)\phi(0)\rangle=|x|^{-2\Delta_{\phi}}$. Since the theory enjoys
a $\mathbb{Z}_{2}$ symmetry (which acts as $\phi\to-\phi$) correlation
functions of an odd number of $\phi$ fields vanish.

When $\Delta_{\phi}={\hat{d}}/2-1$ (which corresponds to $\xi=1$ in the
Lagrangian) the theory is the usual free theory, which has local equations of
motions $\square\phi=0$ and infinitely many conserved (higher spin) currents.
However we shall study the more generic case of $\Delta_{\phi}$ being a
generic real parameter. In this case the field $\phi$ does not satisfy any
local shortening condition. Similarly no local composite of $\phi$ (built out
of an integer number of $\phi$ dressed with an integer number of derivatives)
can play the role of a conserved current/tensor, simply because it would not
have integer conformal dimensions (which is required by representation
theory). So naively we do not expect shortening conditions for any operator or
in other words we do not expect the correlation functions of GFF to satisfy
any local differential equation. In the next subsection we shall see that such
differential equations can be defined because of the existence of the PS
uplift. Before that let us continue the review of GFF by describing some
observables.

In GFF it is straightforward to compute correlation functions by Wick
contractions. A simple example is

$\langle\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})\rangle=\frac{1}{x_{12}^{2\Delta_{\phi}}x_{34}^{2\Delta_{\phi}}}\left[1+u^{\Delta_{\phi}}+u^{\Delta_{\phi}}v^{-\Delta_{\phi}}\right]\,.$
(5.1)

The OPE in GFF can also be fixed by Wick contractions, for example

$\phi(x)\phi(0)=\frac{1}{|x|^{2\Delta_{\phi}}}+\sum_{n,\ell}\lambda_{n,\ell}|x|^{2n+\ell}[\phi\phi]_{n,\ell}\,.$
(5.2)

where the first term is obtained as by Wick contraction of $\phi(x)\phi(0)$
and is associated to the exchange of the identity operator, while
$[\phi\phi]_{n,\ell}$ are called double twist operators and $\lambda_{n,\ell}$
are the associated OPE coefficients. The double twists are obtained by Taylor
expansion of $:\phi(x)\phi(0):$ around $x=0$ and are defined as

$[\phi\phi]_{n,\ell}\equiv:\phi\square^{n}\partial^{\mu_{1}}\cdots\partial^{\mu_{\ell}}\phi:+\dots\,,$
(5.3)

where the dots take into account terms that make the operator symmetric and
traceless in the indices $\mu_{1}\dots\mu_{\ell}$ together with terms that
make the operator primary. By taking the OPE in a four-point function we
obtain

$\langle\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})\rangle=\frac{1}{x_{12}^{2\Delta_{\phi}}x_{34}^{2\Delta_{\phi}}}[1+\sum_{n,\ell}a_{n,\ell}g^{({\hat{d}})}_{2\Delta_{\phi}+2n+\ell\,\ell}]\,.$
(5.4)

Since we know the exact form of the four-point function we can easily read off
the coefficients $a_{n,\ell}=\lambda_{n,\ell}^{2}$ by simply expanding the
conformal blocks and the four-point functions in powers of the cross ratios
and match the coefficients. There are more sophisticated ways to compute
$a_{n,\ell}$ from the four-point function, e.g. the inversion formula of [31],
which however will not be needed for this work.

The same logic can be applied for any four-point function. In the following we
show how one can use the PS uplift as a tool to compute the conformal block
decomposition of a four-point function of
$\langle\phi\phi^{n_{2}}\phi^{n_{3}}\phi^{n_{4}}\rangle$ for generic
$n_{2},n_{3},n_{4}$. In this case the conformal block expansion takes the form

$\langle\phi(x_{1})\phi^{n_{2}}(x_{2})\phi^{n_{3}}(x_{3})\phi^{n_{4}}(x_{4})\rangle=K_{\Delta_{i}}(x_{i})[a^{n_{2},n_{3},n_{4}}g_{\Delta=-\Delta_{12}\,\,\ell=0}^{({\hat{d}})}+\sum_{n,\ell}a^{n_{2},n_{3},n_{4}}_{n,\ell}g_{\Delta=\Delta_{1}+\Delta_{2}+2n+\ell\,\,\ell}^{({\hat{d}})}]\,,$
(5.5)

where $\Delta_{1}=\Delta_{\phi}$, $\Delta_{i}=n_{i}\Delta_{\phi}$ ($i=2,3,4$)
and the contribution at $\Delta=-\Delta_{12}=(n_{2}-1)\Delta_{\phi}$ comes by
taking a Wick contraction of $\phi$ with the operator $\phi^{n_{2}}$. Very
interestingly PS supersymmetry gives a set of recurrence relations for
$a_{n,\ell}$, which we were able to solve in the case of $n_{3}=n_{4}$.
Besides the specific result it is very interesting that just kinematic
considerations of PS supersymmetry can be used to fix the dynamical data
$a_{n,\ell}$ with the only input of the ansatz (5.5).

Before explaining how this bootstrap problem works, let us present the
simplest set of GFF correlators. Because of the tower of double twist
operator, in a four-point function we typically expect the exchange of an
infinite number of operators. However there is a set of “extremal” four-point
function where only one operator is exchanged. These take the simple form

$\langle\phi^{n_{1}}(x_{1})\phi^{n_{2}}(x_{2})\phi^{n_{3}}(x_{3})\phi^{N}(x_{4})\rangle=K_{\Delta_{i}}(x_{i})u^{-\frac{\Delta_{34}}{2}}N!=K_{\Delta_{i}}(x_{i})N!g^{({\hat{d}})}_{\Delta=-\Delta_{34}\;\ell=0}(u,v)\,,$
(5.6)

where $N=n_{1}+n_{2}+n_{3}$ and $\Delta_{34}=(n_{3}-N)\Delta_{\phi}$. The
factorial is simply obtained from the combinatorics. Nicely enough this
actually corresponds to a single conformal block exchange.888Similarly if the
operator $\phi^{N}$ is at position $x_{1}$ we get
$g^{({\hat{d}})}_{\Delta=\Delta_{12}\,\ell=0}(u,v)=u^{\frac{\Delta_{12}}{2}}$.
If it is at $x_{2}$, the single block is
$g^{({\hat{d}})}_{\Delta=-\Delta_{12}\,\ell=0}(u,v)=u^{-\frac{\Delta_{12}}{2}}v^{\frac{1}{2}\left(\Delta_{12}-\Delta_{34}\right)}$.
While if it is at $x_{3}$, the single block is
$g^{({\hat{d}})}_{\Delta=\Delta_{34}\,\ell=0}(u,v)=u^{\frac{\Delta_{34}}{2}}v^{\frac{1}{2}\left(\Delta_{12}-\Delta_{34}\right)}$.
So in particular these exchanges are crossing covariant, meaning that a single
exchange in the $s$-channel is mapped to a single exchange in the $t$-channel.
Whenever $n_{1}=1$ and $1+n_{2}+n_{3}=N$ we can compare the result (5.6) with
the ansatz (5.5), which shows that all the coefficients
$a^{n_{2},n_{3},n_{4}}_{n,\ell}$ of the double twist operators vanish, leaving
as only contribution $a^{n_{2},n_{3},N}=N!$.

Below we want to study $a^{n_{2},n_{3},n_{4}}_{n,\ell}$ for generic values of
$n_{i}$.

###  5.1 Bootstrapping GFF using PS supersymmetry

Given the GFF in $d-2$ dimensions, we can obtain its relative PS CFTd by
explicitly uplifting the Lagrangian of the theory, namely
$\phi(\partial^{\mu}\partial_{\mu})^{\xi}\phi\to\Phi(\partial^{a}\partial_{a})^{\xi}\Phi$.
Like GFF, its uplifted version can be defined by saying that all correlators
are computed by Wick contractions, where now the two-point function lives in
superspace $\langle\Phi(y_{1})\Phi(y_{2})\rangle=|y_{12}|^{-2\Delta_{\phi}}$,
where $\Delta_{\phi}$ is the same as in the dimensionally reduced theory. When
$\Delta_{\phi}={d}/2-2$, the model becomes the uplift of free theory but we
will keep $\Delta_{\phi}$ generic.999For the connection of uplifted free
theory and random field models see [32]. Composite operators are mapped by
following the rules that $\phi\to\Phi$ and $\partial_{\mu}\to\partial_{a}$.

Now we show how the existence of the uplift can be used to constrain (the
dimensionally reduced) GFF. For example to constrain GFF correlation
functions, we can leverage the simple fact that

${\bf D}_{[\bf 1]}\langle\Phi(y_{1})\Phi(y_{2})\rangle|_{0}=0\,,$ (5.7)

where ${\bf D}_{[\bf
1]}\Phi(y_{1})|_{0}\equiv\tilde{\Phi}_{\theta\bar{\theta}}(x_{1})$ is the
superdescendant of $\Phi_{0}$. This two-point function vanishes simply because
it involves two different primaries, namely
$\langle\tilde{\Phi}_{\theta\bar{\theta}}(x_{1})\Phi_{0}(x_{2})\rangle=0$.101010
Notice that
$\tilde{\Phi}_{\theta\bar{\theta}}=(d-2-2\Delta_{\phi})\omega+\square\varphi$.
This primary is identically zero in the uplift of local free theory because
$\square\varphi=-2\omega$ and $\Delta_{\phi}=(d-4)/2$. On the contrary it is
non vanishing in GFF, where we do not impose the latter constraints. In fact
in GFF the operator $\tilde{\Phi}_{\theta\bar{\theta}}$ has a non vanishing
two-point function.  Indeed any correlation function of the following form
vanishes,

${\bf D}_{[\bf
1]}\langle\Phi(y_{1})\Phi^{n_{2}}(y_{2})\dots\Phi^{n_{k}}(y_{k})\rangle|_{0}=0\,,$
(5.8)

for generic $k$ and $n_{1},\dots,n_{k}$. This is true because the correlation
function in GFF is computed by Wick contraction and therefore it is equal to
as a sum of terms that contain a vanishing term ${\bf D}_{[\bf
1]}\langle\Phi(y_{1})\Phi(y_{i})\rangle|_{0}=0$. For the sake of clarity let
us show how the vanishing result appears in the simplest four-point function

$\displaystyle\begin{split}{\bf D}_{[\bf
1]}\langle\Phi(y_{1})\Phi(y_{2})\Phi(y_{3})\Phi(y_{4})\rangle|_{0}=&+\langle\Phi(y_{3})\Phi(y_{4})\rangle{\bf
D}_{[\bf 1]}\langle\Phi(y_{1})\Phi(y_{2})\rangle|_{0}\\\
&+\langle\Phi(y_{2})\Phi(y_{4})\rangle{\bf D}_{[\bf
1]}\langle\Phi(y_{1})\Phi(y_{3})\rangle|_{0}\\\
&+\langle\Phi(y_{2})\Phi(y_{3})\rangle{\bf D}_{[\bf
1]}\langle\Phi(y_{1})\Phi(y_{4})\rangle|_{0}=0\,.\end{split}$ (5.9)

Equation (5.8) can be generalized to other differential operators: all those
that do not involve ${\bf D}_{[{\bf i}]}$ at other points $i\neq 1$. Let us
exemplify this for four-point functions,

${\bf
D}_{[\vec{S}]}\langle\Phi(y_{1})\Phi^{n_{2}}(y_{2})\Phi^{n_{3}}(y_{3})\Phi^{n_{4}}(y_{4})\rangle|_{0}=0\,,\qquad[\vec{S}]={[{\bf
1}]},{[{\bf 1}3\bar{4}]},{[{\bf 1}2\bar{4}]},{[{\bf 1}2\bar{3}]}\,.$ (5.10)

Moreover for other differential operators the result is not zero but it
greatly simplifies. E.g. for $S={[{\bf 12}]},{[{\bf
12}3\bar{4}]},{[1\bar{2}]}$ the field at position $1$ must be contracted with
the one at position $2$ and no possible contraction $1-3$ and $1-4$ are
allowed: the latter are the contractions responsible to double twist
contributions, which for these correlation functions are thus absent. In this
case we obtain that the correlator is written in terms of a single conformal
block

${\bf
D}_{[\vec{S}]}\langle\Phi(y_{1})\Phi^{n_{2}}(y_{2})\Phi^{n_{3}}(y_{3})\Phi^{n_{4}}(y_{4})\rangle|_{0}=N_{[\vec{S}]}^{n_{2},n_{3},n_{4}}\left(\Sigma_{[\vec{S}]}K_{\Delta_{i}}(x_{i})\right)g_{\Delta=-\Delta_{12}\,\ell=0}^{(d)}\,,\qquad{[\vec{S}]}={[{\bf
12}]},{[{\bf 12}3\bar{4}]},{[1\bar{2}]}.$ (5.11)

where $N_{[\vec{S}]}^{n_{2},n_{3},n_{4}}$ is an overall constant and
$g_{\Delta=-\Delta_{12}\,\ell=0}^{(d)}(u,v)=u^{-\frac{\Delta_{12}}{2}}v^{\frac{\Delta_{12}-\Delta_{34}}{2}}$.
Because of the absence of double twist operators, the correlators (5.11)
behave like the extremal correlators defined in (5.6).111111E.g. in a simple
case ${\bf D}_{[{\bf
12}]}\langle\Phi\Phi\Phi^{2}\Phi^{2}\rangle|_{0}=-8\Delta_{\phi}\left(\Delta_{\phi}+1\right)\left(-d+2\Delta_{\phi}+2\right)\left(-d+2\Delta_{\phi}+4\right)\left(\Sigma_{[{\bf
12}]}K_{\Delta_{i}}(x_{i})\right)$, which is constant in $u,v$ and only
exchanges the identity.

In all cases (5.10) and (5.11) we found that the double twist contributions
are annihilated by the differential operators. Below we explain how to exploit
this fact to find recurrence relations for the OPE coefficients in the
conformal block decomposition of the correlators. In section 5.3 we also show
how to use (5.10) to find the correlators themselves.

###  5.2 Bootstrapping OPE coefficients

We now show how to use (5.10) and (5.11) to find recurrence relations for the
OPE coefficients in the conformal block decomposition of
$\langle\phi\phi^{n_{2}}\phi^{n_{3}}\phi^{n_{4}}\rangle$. We further show how
to solve these recurrence relations when $n_{3}=n_{4}$, giving rise to new
closed form expressions for these OPE coefficients.

To start we uplift the ansatz (5.5) and act with the operators ${\bf
D}_{[\vec{S}]}$ with $[\vec{S}]={[{\bf 1}]},{[{\bf 1}3\bar{4}]},{[{\bf
1}2\bar{4}]},{[{\bf 1}2\bar{3}]}$. According to (5.10) we find

$\displaystyle\begin{split}0=&\left(\Sigma_{[\vec{S}]}K_{\Delta_{i}}(x_{i})\right)^{-1}{\bf
D}_{[\vec{S}]}\langle\Phi(y_{1})\Phi^{n_{2}}(y_{2})\Phi^{n_{3}}(y_{3})\Phi^{n_{4}}(y_{4})\rangle\big{|}_{0}\\\
=&a^{\\{n_{k}\\}}D_{[\vec{S}]}g_{\Delta=-\Delta_{12}\,\ell=0}^{(d-2)}(u,v)+\sum_{n,\ell}a^{\\{n_{k}\\}}_{n,\ell}D_{[\vec{S}]}g_{\Delta=\beta_{12}+2n+\ell\,\ell}^{(d-2)}(u,v)\,,\end{split}$
(5.12)

where we use the notation
$\beta_{12}\equiv\Delta_{1}+\Delta_{2}=(1+n_{2})\Delta_{\phi}$. From equation
(4.4) it is easy to see that
$D_{[\vec{S}]}g_{\Delta=-\Delta_{12}\,\ell=0}^{(d-2)}(u,v)=0$ for
$[\vec{S}]={[{\bf 1}]},{[{\bf 1}3\bar{4}]},{[{\bf 1}2\bar{4}]},{[{\bf
1}2\bar{3}]}$. On the other hand $D_{[\vec{S}]}$ does not trivially vanish
when acting on each double twist conformal block. Miraculously there is a very
non trivial cancellation which arises in the sum. Using (4.4) we find that

$\displaystyle\sum_{n,\ell}a^{\\{n_{k}\\}}_{n,\ell}\sum_{(i,j)\in
P_{[{\vec{S}}]}}\left(c^{[{\vec{S}}]}_{i,j}\
\Sigma_{[{\vec{S}}]}\,g_{\Delta+i\,\ell+j}^{(d)}(u,v)\right)_{\Delta=\beta_{12}+2n+\ell}=0\,.$
(5.13)

We set $\Delta=\beta_{12}+2n+\ell$ because the coefficients
$c^{[{\vec{S}}]}_{i,j}$ are functions of $\Delta$, which should be evaluated
to the correct value depending on the exchanged block. Because of the shifts
in $i$ and $j$, now the blocks at different values of $n$ and $\ell$ can mix.
It is however simple to rearrange the formula above collecting the
contributions of $d$-dimensional blocks with given $\Delta$ and $\ell$. Let us
do it first for the cases $[{\vec{S}}]={[{\bf 1}]},{[{\bf 1}3\bar{4}]}$ (we
will see that this can be extended to ${[\vec{S}]}={[{\bf 12}]},{[{\bf
12}3\bar{4}]},{[1\bar{2}]}$) which have $P_{[{\vec{S}}]}=P^{(0)}$,

$\displaystyle\left.0=\sum_{n,\ell}\left(a^{\\{n_{k}\\}}_{n,\ell}\tilde{c}^{[{\vec{S}}]}_{0,0}+a^{\\{n_{k}\\}}_{n-1,\ell+2}\tilde{c}^{[{\vec{S}}]}_{0,-2}+a^{\\{n_{k}\\}}_{n-1,\ell+1}\tilde{c}^{[{\vec{S}}]}_{1,-1}+a^{\\{n_{k}\\}}_{n-1,\ell}\tilde{c}^{[{\vec{S}}]}_{2,0}+a^{\\{n_{k}\\}}_{n-2,\ell+2}\tilde{c}^{[{\vec{S}}]}_{2,-2}\right)\Sigma_{[{\vec{S}}]}\,g_{\Delta\,\ell}^{(d)}\right|_{\Delta=\beta_{12}+2n+\ell}\,,$
(5.14)

where $a^{\\{n_{k}\\}}_{n<0,\ell}\equiv 0$ and the tilded coefficients
$\tilde{c}$ are related to $c$ by simple shifts of $\Delta$ and $\ell$, namely

$\tilde{c}^{[{\vec{S}}]}_{i,j}\equiv
c^{[{\vec{S}}]}_{i,j}\big{|}_{\Delta\to\Delta-i\;\ell\to\ell-j}\,.$ (5.15)

Similarly we can write this expansion for the cases $[{\vec{S}}]={[{\bf
1}2\bar{4}]},{[{\bf 1}2\bar{3}]}$ which have $P_{[{\vec{S}}]}=P^{(1)}$,

$\displaystyle\left.0=\sum_{n,\ell}\left(a^{\\{n_{k}\\}}_{n+1,\ell}\tilde{c}^{[{\vec{S}}]}_{0,-1}+a^{\\{n_{k}\\}}_{n,\ell}\tilde{c}^{[{\vec{S}}]}_{1,0}+a^{\\{n_{k}\\}}_{n-1,\ell+2}\tilde{c}^{[{\vec{S}}]}_{1,-2}+a^{\\{n_{k}\\}}_{n-1,\ell+1}\tilde{c}^{[{\vec{S}}]}_{2,-1}\right)\Sigma_{[{\vec{S}}]}\,g_{\Delta\,\ell}^{(d)}\right|_{\Delta=\beta_{12}+2n+\ell+1}\,.$
(5.16)

Now let us discuss the cases ${[\vec{S}]}={[{\bf 12}]},{[{\bf
12}3\bar{4}]},{[1\bar{2}]}$ of equation (5.11). In practice the argument goes
the same way besides that in these cases
$D_{[\vec{S}]}g_{\Delta=-\Delta_{12}\,\ell=0}^{(d-2)}(u,v)\propto
g_{\Delta=-\Delta_{12}\,\ell=0}^{(d)}(u,v)$. But we can forget about this
contribution in (5.12) and focus on the sum of double traces that vanishes,
thus equation (5.13) still holds for these cases. Now since the blocks in
(5.14) and (5.16) are linearly independent,121212If the blocks were linearly
dependent, the conformal block decomposition would not be unique. In the case
of (5.14) and (5.16) it is easy to see that this is not the case by expanding
the blocks in powers of the cross ratios. in order to get zero the terms in
the parentheses must be vanishing. In summary with this simple argument we
have found seven recurrence relations for $a^{\\{n_{k}\\}}_{n,\ell}$ which
take the form

$\boxed{\begin{array}[]{ll}0=a^{\\{n_{k}\\}}_{n,\ell}\tilde{c}^{[{\vec{S}}]}_{0,0}+a^{\\{n_{k}\\}}_{n-1,\ell+2}\tilde{c}^{[{\vec{S}}]}_{0,-2}+a^{\\{n_{k}\\}}_{n-1,\ell+1}\tilde{c}^{[{\vec{S}}]}_{1,-1}+a^{\\{n_{k}\\}}_{n-1,\ell}\tilde{c}^{[{\vec{S}}]}_{2,0}+a^{\\{n_{k}\\}}_{n-2,\ell+2}\tilde{c}^{[{\vec{S}}]}_{2,-2}\bigg{|}_{\Delta=\beta_{12}+2n+\ell}\,,&{\scriptstyle{[\vec{S}]}={{[{\bf
1}]},{[{\bf 1}3\bar{4}]},\atop{[{\bf 12}]},{[{\bf
12}3\bar{4}]},{[1\bar{2}]}.}}\\\
0=a^{\\{n_{k}\\}}_{n+1,\ell}\tilde{c}^{[{\vec{S}}]}_{0,-1}+a^{\\{n_{k}\\}}_{n,\ell}\tilde{c}^{[{\vec{S}}]}_{1,0}+a^{\\{n_{k}\\}}_{n-1,\ell+2}\tilde{c}^{[{\vec{S}}]}_{1,-2}+a^{\\{n_{k}\\}}_{n-1,\ell+1}\tilde{c}^{[{\vec{S}}]}_{2,-1}\bigg{|}_{\Delta=\beta_{12}+2n+\ell+1}\,,&{\scriptstyle[\vec{S}]={\scriptsize[{\bf
1}2\bar{4}]},{[{\bf 1}2\bar{3}]}.}\end{array}}$ (5.17)

Notice that the coefficients $\tilde{c}^{[{\vec{S}}]}_{i,j}$ are all known in
a closed form as shown in section 4.2. Therefore these relations are very
explicit. Let us show a couple of examples for concreteness. A simple
recursion can be obtained by summing the relation labelled by $[\vec{S}]=[{\bf
1}2\bar{4}]$ with the one of $[\vec{S}]={[{\bf 1}2\bar{3}]}$. The result is

$\begin{array}[]{l}\frac{2(\ell+1)\left(n_{3}-n_{4}\right)\Delta_{\phi}\left(-\beta_{12}+d-n-2\right)\left(\beta_{12}+\ell+2n-2\right)\left(\beta_{12}+\ell+2n-1\right)\left(2\beta_{12}-d+2\ell+2n+2\right)\left(d-2\Delta_{\phi}-2n-2\right)\left(\Delta_{\phi}+\ell+n\right)}{\left(-\beta_{12}+d-2n-2\right)\left(-\beta_{12}+d-2n-1\right)\left(\beta_{12}+2\ell+2n-1\right)\left(\beta_{12}+2\ell+2n\right)\left(2\beta_{12}-d+2\ell+4n\right)}a^{\\{n_{k}\\}}_{n-1,\ell+1}\\\
+\frac{2(\ell+1)(\ell+2)(d+2\ell+2n-2)\left(-\beta_{12}+d-n-2\right)\left(\beta_{12}+\ell+2n-1\right)\left(d-2\Delta_{\phi}-2n-2\right)}{(d+2\ell-2)(d+2\ell)\left(-\beta_{12}+d-2n-1\right)}a^{\\{n_{k}\\}}_{n-1,\ell+2}\\\
+\frac{4n\left(\beta_{12}+\ell+2n-1\right)\left(2\beta_{12}-d+2\ell+2n+2\right)\left(\Delta_{\phi}+\ell+n\right)}{\beta_{12}+2\ell+2n-1}a^{\\{n_{k}\\}}_{n,\ell}=0\,.\end{array}$
(5.18)

Another nice combination is obtained by subtracting $[\vec{S}]=[{\bf
1}2\bar{4}]$ with $[\vec{S}]={[{\bf 1}2\bar{3}]}$ (we also divide by the
factor
$2(\ell+1)\left(\beta_{12}+\ell+2n-1\right)\left(d-2\Delta_{\phi}-2(n+1)\right)$
to shorten the expression, this is normally allowed since we are taking
$\Delta_{\phi}$ to be an arbitrary real number),

$\begin{array}[]{l}\frac{\left(-\beta_{12}+d-n-2\right)\left(\beta_{12}+\ell+2n-2\right)\left(-2\beta_{12}+d-2(\ell+n+1)\right)\left(\Delta_{\phi}+\ell+n\right)\left(\left(-\beta_{12}+d-2(n+1)\right)\left(\beta_{12}+2(\ell+n)\right)-\left(n_{3}-n_{4}\right){}^{2}\Delta_{\phi}^{2}\right)}{\left(-\beta_{12}+d-2n-1\right)\left(-\beta_{12}+d-2(n+1)\right)\left(\beta_{12}+2\ell+2n-1\right)\left(\beta_{12}+2(\ell+n)\right)\left(-2\beta_{12}+d-2(\ell+2n)\right)\left(-2\beta_{12}+d-2(\ell+2n+1)\right)}a^{\\{n_{k}\\}}_{n-1,\ell+1}\\\
-\frac{(\ell+2)\left(n_{3}-n_{4}\right)\Delta_{\phi}(d+2(\ell+n-1))\left(-\beta_{12}+d-n-2\right)}{(d+2\ell-2)(d+2\ell)\left(-\beta_{12}+d-2n-1\right)\left(-\beta_{12}+d-2(n+1)\right)}a^{\\{n_{k}\\}}_{n-1,\ell+2}\\\
+\frac{2n\left(n_{3}-n_{4}\right)\Delta_{\phi}\left(-2\beta_{12}+d-2(\ell+n+1)\right)\left(\Delta_{\phi}+\ell+n\right)}{(\ell+1)\left(\beta_{12}+2\ell+2n-1\right)\left(\beta_{12}+2(\ell+n)\right)\left(-d+2\Delta_{\phi}+2n+2\right)}a^{\\{n_{k}\\}}_{n,\ell}\\\
+\frac{2n(d+2(\ell+n-1))}{\left(\beta_{12}+\ell+2n-1\right)\left(-d+2\Delta_{\phi}+2n+2\right)}a^{\\{n_{k}\\}}_{n,\ell+1}=0\,.\\\
\end{array}$ (5.19)

Similar nice combinations can be defined using the remaining five recurrence
relations, but we will omit them here to avoid clutter. One can in principle
try to solve this set of seven relations for generic $n_{i}$. We did not
invest enough time in this, however we want to point out that the solution of
the recursion is straightforward when $n_{3}=n_{4}$. Indeed in this case the
two recursions above greatly simplify (they only involve two terms) and can be
solved by131313The recurrence relations take the following form which can be
trivially solved
$A_{n}=A_{n-1}\prod_{i=1}^{j}(p_{i}+q_{i}n)^{r_{i}}\,\Longrightarrow
A_{n}=A_{0}\prod_{i=1}^{j}\left[q_{i}^{n}\left(\tfrac{p_{i}}{q_{i}}+1\right)_{\\!\\!n}\,\right]^{r_{i}}\,,$
(5.20) for any set of constants $p_{i},q_{i},r_{i}$ and any $j\in\mathbb{N}$.
The second line of (5.21) comes by solving (5.19) in $n$ while the first line
comes by solving (5.18) in $\ell$ (using the dependence in $n$ just computed).

$\boxed{\begin{array}[]{ll}a^{n_{2},n_{3},n_{3}}_{n,\ell}=&a^{n_{2},n_{3}}_{0}\frac{\left(\frac{\beta_{12}-1}{2}\right)_{\frac{\ell}{2}}\left(\frac{\beta_{12}}{2}\right)_{\frac{\ell}{2}}\left(\frac{\Delta_{\phi}}{2}\right)_{\frac{\ell}{2}}\left(\frac{\Delta_{\phi}+1}{2}\right)_{\frac{\ell}{2}}}{\ell!\left(\frac{\beta_{12}-1}{4}\right)_{\frac{\ell}{2}}\left(\frac{\beta_{12}+1}{4}\right)_{\frac{\ell}{2}}}\times\\\
&\qquad\qquad\times\frac{\left(-d+\beta_{12}+3\right)_{n}\left(-\frac{d}{2}+\Delta_{\phi}+2\right)_{n}\left(\frac{\ell+\beta_{12}-1}{2}\right)_{n}\left(\frac{\ell+\beta_{12}}{2}\right)_{n}(\ell+\Delta_{\phi})_{n}\left(-\frac{d}{2}+\ell+\beta_{12}+1\right)_{n}}{2^{4n}n!\left(\frac{d}{2}+\ell-1\right)_{n}\left(\frac{-d+\beta_{12}+3}{2}\right)_{n}\left(\ell+\frac{\beta_{12}}{2}-\frac{1}{2}\right)_{n}\left(\frac{-d+2\ell+2\beta_{12}+2}{4}\right)_{n}\left(\frac{-d+2\ell+2\beta_{12}+4}{4}\right)_{n}}\,.\end{array}}$
(5.21)

Here we solved the recursion only for $\ell$ even since the equality
$n_{3}=n_{4}$ selects only even spin operators in the OPE. Notice that $n_{2}$
enters in the definition of $\beta_{12}=(1+n_{2})\Delta_{\phi}$, while $n_{3}$
only appears in the overall constant $a^{n_{2},n_{3}}_{0}$, which is the seed
of the recursion for $\ell=0$ and $n=0$. We checked that this solution
automatically satisfies the remaining five recurrence relations.

Interestingly we found in a closed form all the OPEs of double traces
exchanges of any correlator
$\langle\phi\phi^{n_{2}}\phi^{n_{3}}\phi^{n_{3}}\rangle$ without ever
computing any correlation function. The expression is given up to an overall
constant which is the seed of the recursion and can be fixed from explicit
computations of the simplest double twist exchange, $\phi^{n_{2}+1}$. Indeed
the computation of such coefficient is a simple combinatorial exercise which
is solved by141414 The rationale is the following: we need to find
$\phi^{n_{2}+1}$ in the OPE of $\phi^{n_{3}}\times\phi^{n_{3}}$, we thus
choose $\frac{n_{2}+1}{2}$ fields out of the $n_{3}$ of both $\phi^{n_{3}}$
fields, which gives ${n_{3}\choose\frac{n_{2}+1}{2}}^{2}$ (these form the
field $\phi^{n_{2}+1}$), the remaining $n_{3}-\frac{n_{2}+1}{2}$ fields of
each $\phi^{n_{3}}$ should be contracted giving a
$(n_{3}-\frac{n_{2}+1}{2})!$. Moreover we need to introduce an extra factor of
$(n_{2}+1)!$ to account for the normalization of the operator
$\phi^{n_{2}+1}$. Putting the factors together and simplifying the expression,
we obtain (5.22).

$a^{n_{2},n_{3}}_{0}=n_{3}!{{n_{2}+1}\choose{\frac{n_{2}+1}{2}}}\left(\frac{1-n_{2}+2n_{3}}{2}\right)_{\frac{n_{2}+1}{2}}\,,$
(5.22)

where $n_{2}$ must be even, otherwise the correlator vanishes. We also notice
that the correlator is extremal when $n_{2}=1+2n_{3}$ and thus
$a^{n_{2},n_{3}}_{0}=0$ for all $n_{2}\geq 1+2n_{3}$ as it should. The other
OPE coefficient which is missing in the ansatz (5.5) is the one of the
exchange of $\phi^{n_{2}-1}$ which can also be trivially computed by
combinatorics $a^{n_{2},n_{3},n_{3}}=n_{2}a^{n_{2}-2,n_{3}}_{0}$, indeed the
combinatorics is the same as the one of $a^{n_{2},n_{3}}_{0}$ but with two
less fields $n_{2}$, and adding an extra factor of $n_{2}$ to take into
account the left OPE $\phi\times\phi^{n_{2}}\sim n_{2}\phi^{n_{2}-1}$.

As a consistency check, for the case of the four-point function of
$\langle\phi\phi\phi\phi\rangle$ these coefficients were computed in [33]; it
is easy to see that our solution matches theirs once we restrict to $n_{2}=1$
and we shift $d\to d+2$ (recall that the original GFF leaves in two less
dimension).

In summary this method solves for the OPE coefficients that are harder to
compute (the ones of double twist operators which contain derivatives), while
the remaining ones can be obtained from a one-line combinatoric computation.
Moreover, at a more abstract level, PS SUSY provides a novel way to understand
why double twist families (in GFF) are tied together: one cannot simply modify
one of the OPE coefficients in the family without violating the recursion
relations provided by supersymmetry.

###  5.3 Bootstrapping correlators and extension to perturbation theory

In the previous section we found a sophisticated way to extract information
from supersymmetry. There is also another simple way to make use these
constraints. Indeed in (5.10) we found that a differential operator was
annihilating the correlators, therefore we obtained a differential equation
that the correlators must satisfy. Below we show how to use this information
to fix the form of the GFF correlators
$\langle\phi\phi^{n_{2}}\phi^{n_{3}}\phi^{n_{4}}\rangle$ up to a few
constants. Moreover we will see that this logic can be also used to bootstrap
the form of correlation functions in perturbation theory around GFF.

Let us first see how to use (5.10) to fix the correlation function. We require
that the GFF correlators are written as finite sums of powers $u^{a}v^{b}$.
This is always true if the correlator contains a finite number of fields and
derivatives. Then we notice that the differential operators $D_{[\vec{S}]}$
acting on $u^{a}v^{b}$ produce a sum of terms with different powers, e.g.
$D_{[{\bf 1}]}u^{a}v^{b}$ is a linear combination of
$u^{a}v^{b},u^{a+1}v^{b},u^{a}v^{b+1}$. Because of this reason and the fact
that the correlator is written as a finite sum of powers, the action
$D_{[\vec{S}]}$ must annihilate separately each power. By requiring

$D_{[\vec{S}]}u^{a}v^{b}=0\,,$ (5.23)

we obtain a constraint on the possible powers $a$ and $b$. If we apply this
logic to $\langle\phi\phi^{n_{2}}\phi^{n_{3}}\phi^{n_{4}}\rangle$ using e.g.
${[\vec{S}]}=[{\bf 1}]$ we find that there are only three allowed powers and
thus the correlator must take the form

$\displaystyle\begin{split}\langle\phi(x_{1})\phi^{n_{2}}(x_{2})\phi^{n_{3}}(x_{3})\phi^{n_{4}}(x_{4})\rangle=&K_{\Delta_{i}}(x_{i})\bigg{[}k_{1}u^{\frac{n_{2}-1}{2}\Delta_{\phi}}v^{-\frac{n_{2}+n_{3}-n_{4}-1}{2}\Delta_{\phi}}\\\
&\quad+k_{2}u^{\frac{n_{2}+1}{2}\Delta_{\phi}}v^{-\frac{n_{2}+n_{3}-n_{4}-1}{2}\Delta_{\phi}}+k_{3}u^{\frac{n_{2}+1}{2}\Delta_{\phi}}v^{-\frac{n_{2}+n_{3}-n_{4}+1}{2}\Delta_{\phi}}\bigg{]}\,,\end{split}$
(5.24)

were $k_{i}$ are undetermined coefficients. In this case the computation of
the four-point function is actually quite simple, indeed by some combinatorics
it is easy to find that the four-point function indeed matches the form above
and that the coefficients are151515The OPE coefficients in the previous
section are related to these coefficients as
$a^{n_{2},n_{3}}_{0}=2n_{3}C_{n_{2},n_{3},n_{3}-1}$ and
$a^{n_{2},n_{3},n_{3}}=n_{2}C_{n_{2}-1,n_{3},n_{3}}$.

$k_{1}=n_{2}C_{n_{2}-1,n_{3},n_{4}}\,,\qquad
k_{2}=n_{3}C_{n_{2},n_{3}-1,n_{4}}\,,\qquad
k_{3}=n_{4}C_{n_{2},n_{3},n_{4}-1}\,,$ (5.25)

where we defined
$C_{m_{1},m_{2},m_{3}}\equiv\frac{m_{1}!m_{2}!m_{3}!}{\left(\frac{m_{1}+m_{2}-m_{3}}{2}\right)!\left(\frac{m_{1}-m_{2}+m_{3}}{2}\right)!\left(\frac{-m_{1}+m_{2}+m_{3}}{2}\right)!}$.

In this example we were able to obtain the ansatz (5.24) without doing any
Wick contraction. This may not seem a very impressive result since the direct
computation of the four-point function is quite straightforward. However this
idea can be applied to more generic cases, where the computation is more
involved as we show below.

Let us give an example of how to use this same logic for perturbative
computations. We consider the one-loop integral of a $\phi^{4}$ perturbation
around GFF,

$F(x_{i})\equiv\int
d^{\hat{d}}x_{0}\langle\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})\phi^{4}(x_{0})\rangle=\int
d^{\hat{d}}x_{0}\;\prod_{i=1}^{4}|x_{i0}|^{-2\Delta_{\phi}}\,.$ (5.26)

Instead of trying to compute the integral in $\mathbb{R}^{\hat{d}}$ we uplift
the correlator to $\mathbb{R}^{d|2}$,

$F(y_{i})\equiv\int
d^{d}x_{0}d\theta_{0}d\bar{\theta}_{0}\langle\Phi(y_{1})\Phi(y_{2})\Phi(y_{3})\Phi(y_{4})\Phi^{4}(y_{0})\rangle\,.$
(5.27)

The fermionic part of the integral has the effect of taking the highest
component of $\Phi^{4}$ which is written in terms of
$\Phi_{\theta}\Phi_{\bar{\theta}}\Phi_{0}^{2}$ and
$\Phi_{\theta\bar{\theta}}\Phi_{0}^{3}$. Moreover can rewrite
$\Phi_{\theta\bar{\theta}}$ in terms of the primary
$\tilde{\Phi}_{\theta\bar{\theta}}$ and $\partial_{x}^{2}\Phi_{0}$. So
$F(y_{i})$ is obtained as a linear combination of the following five-point
functions integrated over $x_{0}$,

$\displaystyle\langle\Phi\Phi\Phi\Phi\,[\tilde{\Phi}_{\theta\bar{\theta}}\Phi_{0}^{3}](x_{0})\rangle\,,\quad\langle\Phi\Phi\Phi\Phi\,[(\partial_{x}^{2}\Phi_{0})\Phi_{0}^{3}](x_{0})\rangle\,,\quad\langle\Phi\Phi\Phi\Phi\,[\Phi_{\theta}\Phi_{\bar{\theta}}\Phi_{0}^{2}](x_{0})\rangle\,,$
(5.28)

where we suppressed the position $y_{1},y_{2},y_{3},y_{4}$ of the first four
$\Phi$ fields to shorten the notation. Looking at the correlators above, just
from factorization and the fact that two-point functions of different
primaries are diagonal, we see that they are annihilated every time we act
with at least two differential operators of the form ${\bf D}_{[{\bf i}]}$ or
${\bf D}_{[{i\bar{j}}]}$, namely

${\bf D}_{[\vec{S}]}F(y_{i}){|}_{0}=0\,,\qquad[\vec{S}]=[{\bf ij}],[{\bf
i}{j\bar{k}}],[{i\bar{j}}{k\bar{l}}],[{\bf ijk}],[{\bf 1234}]\,,$ (5.29)

where $i,j,k,l$ are all different and run over $1,2,3,4$.

The same logic can be generalized to the correlator of $n$ different
generalized free bosons $\phi_{i}$ with dimension $\Delta_{i}$, coupled by the
interaction $\phi_{1}\cdots\phi_{n}$ where at one loop we find

$F_{\Delta_{1},\dots,\Delta_{n}}(x_{i})\equiv\int
d^{\hat{d}}x_{0}\langle\phi_{1}(x_{1})\dots\phi_{n}(x_{n})[\phi_{1}\cdots\phi_{n}](x_{0})\rangle=\int
d^{\hat{d}}x_{0}\;\prod_{i=1}^{n}|x_{i0}|^{-2\Delta_{\phi_{i}}}\,,$ (5.30)

Again the correlator can be uplifted to superspace and it is easy to see that
for the same reasons of above, it should satisfy a large set of differential
equations. In particular by acting with two or more differential operators of
the type ${\bf D}_{[{\bf i}]}$ or ${\bf D}_{[{j\bar{k}}]}$ the uplifted
correlator is annihilated.

Let us now focus of the case $\hat{d}=\sum_{i=1}^{n}\Delta_{i}$, when the
integral is conformal. We further consider a four-point function where by
conformal symmetry we can write

$F_{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}}(x_{i})=K_{\Delta_{i}}(x_{i})f_{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}}(u,v)\,,$
(5.31)

where $K_{\Delta_{i}}$ is the kinematic factor of (2.5). The set of equations
(5.29) can then be written directly in terms of $u$ and $v$ using the
differential operator $D_{[\vec{S}]}$,

$D_{[\vec{S}]}f_{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}}(u,v)=0\,,\qquad[\vec{S}]=[{\bf
ij}],[{\bf i}{j\bar{k}}],[{i\bar{j}}{k\bar{l}}],[{\bf ijk}],[{\bf 1234}]\,.$
(5.32)

These types of integrals are very common in the literature. Indeed $F$
computes the so called $D$-function which defines a contact Witten diagram in
AdS (while $f$ computes the $\bar{D}$-function) see for example [23]).161616
Let us match our conventions with the definition of $D$ and $\bar{D}$ in [23],