# Superconformal invariants and spinning correlators in $3d$ ${\cal N}=2$
SCFTs

Aditya Jain and Amin A. Nizami email<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics, Ashoka University,
Rajiv Gandhi Education City, NCR, India, 131029

###### Abstract

###### Contents

  1. 1 Introduction
  2. 2 Superconformal covariant structures in superspace
    1. 2.1 Polarisation spinors
    2. 2.2 Supercovariant derivative and chiral co-ordinates
  3. 3 The parity-even invariants and relations
    1. 3.1 Fermionic counterparts of the $P,Q,S$ invariants
    2. 3.2 Relations for even invariants
  4. 4 The parity-odd invariants and relations
    1. 4.1 Relations for odd invariants
  5. 5 Three-point function examples
    1. 5.1 Examples
  6. 6 Correlators of conserved currents
    1. 6.1 Free field computation of conserved current correlators
    2. 6.2 General constraints of conservation for 3-point correlators
  7. 7 Discussion
  8. A Conventions and useful relations
    1. A.1 Superconformal generators and algebra
  9. B More relations between invariants
    1. B.1 Parity even relations
    2. B.2 Parity odd relations
    3. B.3 A few more relations
  10. C Calculational details for free theory correlators
    1. C.1 Computation of $\langle J_{0}J_{0}J_{0}\rangle$
    2. C.2 Computation of $\langle J_{1}J_{1}J_{0}\rangle$

###### Contents

  1. 1 Introduction
  2. 2 Superconformal covariant structures in superspace
    1. 2.1 Polarisation spinors
    2. 2.2 Supercovariant derivative and chiral co-ordinates
  3. 3 The parity-even invariants and relations
    1. 3.1 Fermionic counterparts of the $P,Q,S$ invariants
    2. 3.2 Relations for even invariants
  4. 4 The parity-odd invariants and relations
    1. 4.1 Relations for odd invariants
  5. 5 Three-point function examples
    1. 5.1 Examples
  6. 6 Correlators of conserved currents
    1. 6.1 Free field computation of conserved current correlators
    2. 6.2 General constraints of conservation for 3-point correlators
  7. 7 Discussion
  8. A Conventions and useful relations
    1. A.1 Superconformal generators and algebra
  9. B More relations between invariants
    1. B.1 Parity even relations
    2. B.2 Parity odd relations
    3. B.3 A few more relations
  10. C Calculational details for free theory correlators
    1. C.1 Computation of $\langle J_{0}J_{0}J_{0}\rangle$
    2. C.2 Computation of $\langle J_{1}J_{1}J_{0}\rangle$

##  1 Introduction

Superconformal invariance is the largest spacetime symmetry possible in a
local relativistic quantum field theory. Consequently, it is extremely
constraining. Conformal algebras exist in all space-time dimensions but the
supersymmetry algebra can be extended to a closed superconformal algebra only
for $d\leq 6$ [1]. Also, as is well known, two and three point functions of
operators are completely fixed by (super)conformal symmetry alone [2, 3]. For
the case of 3-point functions of spinning operators, the form is still exactly
fixed, but a number of tensor structures are possible [4, 5], each coming with
an OPE coefficient encoding the dynamics of the theory. Since the number of
these tensor structures, as well as the complexity in the form of each,
increases rapidly with increasing spin of the correlator, it is useful to have
an efficient and tractable way of keeping track of the tensorial nature of
such correlators. For non-supersymmetric theories, one such approach was
provided by [6] (see also [7] which utilises the embedding space formalism).

In this paper, we will study the constraints of $\mathcal{N}=2$ superconformal
invariance on 3-point correlators of spinning primary operators in three
dimensions. We will extensively use the superspace formalism [8] and work
throughout with correlators of superfields. In particular, we will construct a
multitude of 3-point superconformal invariants and express 3-point correlators
as multinomials in these invariants. Using the polarisation spinor formalism
[6] and these invariants, we will find that the proliferating complexity of
tensor structures with increasing spin becomes tractable, and one only needs
to deal with multinomials built out of a handful of these invariants to encode
the tensor structure of general spinning correlators. Furthermore, we will
study the constraints that arise when one or more operators are conserved
currents.

Related studies of various aspects of 3-point correlators in 3d SCFTs include
[9, 10, 11, 12, 13, 14, 15, 16]. For an overview of the general constraints of
superconformal invariance in various dimensions, see [17, 18].

This paper is structured as follows. We begin by reviewing the essential
details of 3d superspace in Section 2. Here we also construct the
superconformal covariant structures which are used in later sections to build
3-point function invariants. In Sections 3, 4 we construct the primary 3-point
invariants, both parity-even and parity-odd, out of which all 3-point
functions of spinning operators are built. The relations that exist between
these invariants are also discussed in these sections. In Section 5 we give
examples of various 3-point correlators expressed as a linear combination of
linearly independent structures built out of the invariants. We move on to
discuss correlators of conserved currents in section 6. We first fix the form
of the free-theory 3-point correlators in terms of parity-even invariants.
Subsequently, using the appropriate conservation (multiplet shortening)
condition we give evidence through examples that the general 3-point conserved
current correlator in an $\mathcal{N}=2$ 3d SCFT is the sum of a parity-even
(free theory) part and a parity odd part, with two undetermined OPE
coefficients. We conclude with a discussion and future directions in Section
7.

Some technical details are relegated to the appendices. Appendix A contains
details of the various conventions used and lists various useful relations.
Appendix B lists various relations where invariants constructed out of the
covariant structures are all expressed in terms of our minimal set of
invariants. Appendix C details, through two examples, the computation of
3-point conserved current correlators in a free theory.

##  2 Superconformal covariant structures in superspace

In this section, we provide an overview of the required basic features of
superspace and construct the various covariant structures out of which 3-point
functions are built. This will help set the stage for constructing 3-point
invariants and correlators in the following sections.

A foundational work on 3d SCFTs in superspace is [9] to which the reader is
referred to for additional details. We have included the necessary details in
this section to make the paper self-contained for the most part. Note however
that our conventions are different - they are those of [10] and are summarised
in Appendix A. We will also use polarisation spinors to encode spin, in a
manner similar to [6], as explained in more detail below.

Superspace is parametrised by coordinates $(x^{\mu}_{i},\theta^{a}_{i\alpha})$
where $x$ denotes the usual bosonic coordinate, $\theta$ denotes the fermionic
(grassmanian) coordinate, $\mu=0,1,2$ is a Lorentz index, $\alpha=1,2$ is a
spinor index, $a=1,2,....,\mathcal{N}$ is the R-symmetry index and $i=1,2,3$
will label a particular superspace point.111Throughout the paper, repeated
spinor and R-symmetry indices ($\alpha,a$) will be summed over whereas
repeated superspace point indices $i,j$ etc. will never be summed unless
explicitly mentioned. We will work with Majorana spinors, the R-symmetry group
being $SO(2)$ for the $\mathcal{N}=2$ case we will be focus on. Since
$\delta_{ab}$ is the only invariant tensor for $SO(2)$, all R-symmetry
singlets will be built by contracting R-symmetry charged objects with
$\delta_{ab}$’s.

Following [6] we will use spinor notation to write an $SO(2,1)$ vector as a
bi-spinor -
$X_{\alpha}^{\,\,\beta}=x^{\mu}(\gamma_{\mu})_{\alpha}^{\,\,\beta}$, where
$\gamma_{\mu}$ are the 3d ‘Pauli’ matrices. Their form along with various
other conventions used in the paper are given in Appendix A.

The action of (super)inversion on the superspace coordinates is given by

$\begin{split}I(x^{\mu})&=\frac{x^{\mu}}{x^{2}+\frac{(\theta^{a}\theta^{a})^{2}}{16}}\,\,\,,\\\
\end{split}$ (1)

and

$\begin{gathered}I(\theta^{a}_{\alpha})=(X_{+}^{-1}\theta^{a})_{\alpha},\\\
I(\theta^{a\beta})=-(\theta^{a}X_{-}^{-1})^{\beta},\end{gathered}$ (2)

where we have defined

$X_{\pm}=X\pm\frac{i}{4}(\theta^{a}\theta^{a})\mathds{1},$ (3)

which transforms under super-inversion as

$I(X_{\pm})=X_{\pm}^{-1}.$

Now the following two structures, constructed out of 2 superspace points, are
annihilated by the supersymmetry generators:

$\displaystyle\tilde{x}_{12}^{\mu}=x_{12}^{\mu}+\frac{i}{2}\theta_{1}^{a\alpha}(\gamma^{\mu})_{\alpha}^{\,\,\beta}\theta^{a}_{2\beta},$
(4)
$\displaystyle(\tilde{X}_{12})_{\alpha}^{\,\,\beta}=(X_{12})_{\alpha}^{\,\,\beta}+i\theta^{a}_{1\alpha}\theta_{2}^{a\beta}+\frac{i}{2}(\theta^{a}_{1}\theta^{a}_{2})\delta_{\alpha}^{\leavevmode\nobreak\
\beta}.$ (5)

Super-Poincare invariant structure can thus be easily constructed out of
these. However, as may easily be checked, they do not transform covariantly
under superconformal transformations generated by $S^{a}_{\alpha}$. The
alternative definition $S^{a}_{\alpha}=IQ^{a}_{\alpha}I$ will be useful for
us.

In [9], the following structures were defined and determined to transform
homogeneously under super-inversion

$(X_{ij+})_{\alpha}^{\,\,\beta}=(X_{i+})_{\alpha}^{\,\,\beta}-(X_{j-})_{\alpha}^{\,\,\beta}+i\theta_{i\alpha}^{a}\theta_{j}^{\,\beta
a},$ (6)
$(X_{ij-})_{\alpha}^{\,\,\beta}=(X_{i-})_{\alpha}^{\,\,\beta}-(X_{j+})_{\alpha}^{\,\,\beta}-i\theta_{j\alpha}^{a}\theta_{i}^{\,\beta
a},$ (7)

with the transformation

$\displaystyle
I\left(X_{ij-}\right)_{\alpha}^{\,\,\beta}=-(X^{-1}_{j+})_{\alpha}^{\leavevmode\nobreak\
\gamma}(X_{ij-})_{\gamma}^{\leavevmode\nobreak\
\delta}(X^{-1}_{i-})_{\delta}^{\leavevmode\nobreak\ \beta},$ (8)
$\displaystyle
I\left(X_{ij-}\right)_{\alpha}^{\,\,\beta}=-(X^{-1}_{i+})_{\alpha}^{\leavevmode\nobreak\
\gamma}(X_{ij+})_{\gamma}^{\leavevmode\nobreak\
\delta}(X^{-1}_{j-})_{\delta}^{\leavevmode\nobreak\ \beta}.$ (9)

These can be written in the alternative form

$X_{ij\pm}=\tilde{X}_{ij}\pm\frac{i}{4}\theta_{ij}^{2}\mathds{1},$ (10)

which makes it clear that a particular combination of the above two objects
annihilated by the supersymmetry generators has good super-inversion
properties.

We also have the relations

$\displaystyle X_{i+}X_{i-}=\bar{x}_{i}^{2}\mathds{1},$ (11) $\displaystyle
X_{ij+}X_{ij-}=\bar{x}_{ij}^{2}\mathds{1},$ (12)

where we have defined

$\bar{x}_{i}^{2}=x_{i}^{2}+\frac{1}{16}(\theta^{a}_{i}\theta^{a}_{i})^{2},\quad\bar{x}_{ij}^{2}=\tilde{x}_{ij}^{2}+\frac{1}{16}(\theta_{ij}^{a}\theta_{ij}^{a})^{2},$
(13)

which will be used repeatedly below. We have the following homogeneous
transformation law under super-inversion:

$\bar{x}_{ij}^{2}\xrightarrow{\phantom{\leavevmode\nobreak\
\leavevmode\nobreak\ }i_{s}\phantom{\leavevmode\nobreak\ \leavevmode\nobreak\
}}\frac{\bar{x}_{ij}^{2}}{\bar{x}_{i}^{2}\bar{x}_{j}^{2}}$ (14)

With three superspace points, one can also construct

$\mathfrak{X}_{1+}=X_{12-}^{-1}X_{23+}X_{31-}^{-1},$ (15)

with again a homogeneous transformation under super-inversion,

$\mathfrak{X}_{1+}=X_{12-}^{-1}X_{23+}X_{31-}^{-1}\xrightarrow{\phantom{\leavevmode\nobreak\
\leavevmode\nobreak\ }i_{s}\phantom{\leavevmode\nobreak\ \leavevmode\nobreak\
}}-X_{1-}\mathfrak{X}_{1+}X_{1+},$ (16)

and similarly for $\mathfrak{X}_{2+},\mathfrak{X}_{3+}$.

Superspace also admits purely fermionic 3-point superconformal covariant
structures [9] defined as

$\Theta_{1\alpha}^{a}=\left((X_{21+}^{-1}\theta_{21}^{a})_{\alpha}-(X_{31+}^{-1}\theta_{31}^{a})_{\alpha}\right),$
(17)

and transforming under super-inversion as

$\Theta_{i\alpha}^{a}\xrightarrow{\phantom{\leavevmode\nobreak\
\leavevmode\nobreak\ }i_{s}\phantom{\leavevmode\nobreak\ \leavevmode\nobreak\
}}-(X_{i-})_{\alpha}^{\,\,\beta}\Theta_{i\beta}^{b}V_{ib}^{\,\,\,a}\,\,\,\,\,\,\,\,\Theta_{i}^{a\alpha}\xrightarrow{\phantom{\leavevmode\nobreak\
\leavevmode\nobreak\ }i_{s}\phantom{\leavevmode\nobreak\ \leavevmode\nobreak\
}}V^{Ta}_{i\,\,b}\,\Theta_{i}^{\beta\,b}(X_{i+})_{\beta}^{\,\,\alpha},$ (18)

with $\Theta_{2},\,\Theta_{3}$ defined similarly. Here $V_{j},\,\,j=1,2,3$ is
a matrix acting on the R-symmetry indices and has the form [9]

$V_{jb}^{\,\,\,\,a}\equiv\delta_{b}^{a}+i\theta_{jb}X_{j+}^{-1}\theta^{a}_{j}.$
(19)

It satisfies $V_{j}^{T}V_{j}=I$. As we will see in the next section, the
existence of $V$ makes it non-trivial to construct super-inversion invariants
which are also R-symmetry singlets.222Note that $V$ is trivial for
$\mathcal{N}=1$ as there is no R-symmetry in this case.

$X_{ij\pm},\Theta_{i\alpha}^{a}$ are the primary building blocks, together
with the polarisation spinors that we discuss next, out of which 3-point
invariants for spinning correlators will be constructed.

###  2.1 Polarisation spinors

We will adopt the formalism of [6] to encode the tensor structure of the
correlators and use polarisation spinors ($\lambda_{\alpha}$) as trackers of
the spin of the operators. Our $\lambda$’s are 2-component, real, bosonic
objects that transform in the spinor representation of the 3d Lorentz group.
We define

$O_{\alpha_{1}\alpha_{2}.....\alpha_{2s}}\equiv(\gamma^{\mu_{1}})_{\alpha_{1}\alpha_{2}}(\gamma^{\mu_{2}})_{\alpha_{3}\alpha_{4}}....(\gamma^{\mu_{s}})_{\alpha_{2s-1}\alpha_{2s}}O_{\mu_{1}\mu_{2}.....\mu_{s}},$
(20)

and

$O_{s}\equiv
O_{\alpha_{1}\alpha_{2}.....\alpha_{2s}}\lambda^{\alpha_{1}}\lambda^{\alpha_{2}}.....\lambda^{\alpha_{2s}},$
(21)

as a scalar object (with the spin encoded in the $\lambda$’s) representing a
spin $s$ symmetric superconformal primary operator. We will be interested in
3-point functions of the form

$\langle
O_{s_{1}}(x_{1},\theta_{1},\lambda_{1})O_{s_{2}}(x_{2},\theta_{2},\lambda_{2})O_{s_{3}}(x_{3},\theta_{3},\lambda_{3})\rangle$

with a homogeneity in $\lambda$ of
$\lambda_{1}^{2s_{1}}\lambda_{2}^{2s_{2}}\lambda_{3}^{2s_{3}}$. This
correlator will thus be a function of
$(x_{i},\theta_{i},\lambda_{i}),\,\,(i=1,2,3)$, which are co-ordinates in
(extended) superspace.333Note that $O_{s}(x,\theta)$ represents a local
operator superfield, i.e. it is an operator super-multiplet with both bosonic
and fermionic operators as components.

Their transformation under super-inversions is the same as that of $\theta$’s
:

$\lambda_{\alpha}\rightarrow(X_{+}^{-1}\lambda)_{\alpha}\,\,\,\,\,,\,\,\,\,\,\,\lambda^{\beta}\rightarrow-(\lambda
X_{-}^{-1})^{\beta}.$ (22)

In higher dimensions, where we have mixed symmetry tensor operators, one has
to include fermionic polarisation spinors as well.

###  2.2 Supercovariant derivative and chiral co-ordinates

In studying conservation constraints in Section 6 we will make much use of the
supercovariant derivative $D^{a}_{\alpha}$ which as a differential operator in
superspace is defined as

$\displaystyle
D^{a}_{\alpha}\equiv\frac{\partial}{\partial\theta^{a\alpha}}+\frac{i}{2}\theta^{a\beta}\partial_{\beta\alpha},$
(23)

where $\partial_{\beta\alpha}=(\gamma^{\mu})_{\beta\alpha}\partial_{\mu}$.

As is well-known, the operator $D^{a}_{\alpha}$ anti-commutes with all the
supersymmetry generators

$\displaystyle\\{D^{a}_{\alpha},Q^{b}_{\beta}\\}=0.$ (24)

Also,

$\displaystyle\\{D^{a}_{\alpha},D^{b}_{\beta}\\}=-P_{\alpha\beta}\delta^{ab}.$
(25)

where $P_{\alpha\beta}=P^{\mu}(\gamma_{\mu})_{\alpha\beta}$.

#### Chiral basis for ${\cal N}=2$

The $R$-symmetry group for ${\cal N}=2$ is $SO(2)$, which is isomorphic to
$U(1)$. Thus, instead of working with real coordinates
$\theta^{a\alpha}=(\theta^{1\alpha},\theta^{2\alpha})$, one can introduce the
following chiral basis,

$\displaystyle\theta^{\alpha}\equiv\frac{1}{\sqrt{2}}(\theta^{1\alpha}+i\theta^{2\alpha})\
,\qquad\bar{\theta}^{\alpha}\equiv\frac{1}{\sqrt{2}}(\theta^{1\alpha}-i\theta^{2\alpha}).$
(26)

The corresponding supercovariant derivatives can then be written as

$\displaystyle D_{\alpha}=\frac{1}{\sqrt{2}}(D^{1}_{\alpha}-iD^{2}_{\alpha})\
,\qquad\bar{D}_{\alpha}=-\frac{1}{\sqrt{2}}(D^{1}_{\alpha}+iD^{2}_{\alpha}).$
(27)

Expanding, one gets

$\displaystyle
D_{\alpha}=\frac{\partial}{\partial\theta^{\alpha}}+\frac{i}{2}\bar{\theta}^{\beta}\partial_{\beta\alpha}\
,\qquad\bar{D}_{\alpha}=\frac{\partial}{\partial\bar{\theta}^{\alpha}}+\frac{i}{2}\theta^{\beta}\partial_{\beta\alpha}.$
(28)

Also, the (anti-)chiral supercovariant derivatives satisfy the algebra

$\displaystyle\\{D_{\alpha},\bar{D}_{\beta}\\}=-P_{\alpha\beta},\qquad\\{D_{\alpha},D_{\beta}\\}=\\{\bar{D}_{\alpha},\bar{D}_{\beta}\\}=0.$
(29)

##  3 The parity-even invariants and relations

We will now determine 3-point invariant structures which will be used for
writing down expressions for spinning correlators in $\mathcal{N}=2$ 3d SCFTs.
We will find new invariant structures, both parity-even and parity-odd. These
new fermionic structures do not, of course, exist in the non-supersymmetric
case [6], but they are also different from the $\mathcal{N}=1$ case considered
in [10], as we will discover 444One may wonder how there are additional
invariants for 3-point function structures given that $\mathcal{N}=2$ susy is
more constraining than $\mathcal{N}=1$. Firstly, with more grassmanian
variables there can be a higher non-vanishing degree of $\Theta$ in an
invariant. Specifically, for ${\cal N}=1$, two factors of $\Theta$ is the
highest possible in a non-vanishing term, while for ${\cal N}=2$, invariants
with $\mathcal{O}(\Theta^{4})$ (like $R^{\prime}$ defined below, which do not
have a counterpart in $\mathcal{N}=1$) are also possible. Also, note that
since the size of the supermultiplet increases with $\mathcal{N}$, the
superspace 3-point function will, with increasing $\mathcal{N}$, contain more
and more ‘elementary’ 3-point functions in its component expansion. Since more
elementary correlators are contained within a single superspace correlator for
larger $\mathcal{N}$, it is expected that a larger number of invariants is
required to encapsulate the structure..

In this section, we will construct all the parity even 3-point superconformal
invariants using the covariant structures of the previous section, and also
the polarisation spinors encoding spin information. The bosonic555By bosonic
we mean those invariants which are non-vanishing when all the grassmanian
coordinates are put to zero. Likewise, fermionic invariants are defined as
those that vanish identically when the grassmanian coordinates are set to
zero, and thus exist only in supersymmetric theories. Note that one could have
called “fermionic” the invariants which are used to build up the 3-point
function structures of free/critical fermion CFTs (for example) with no susy -
however this is not our usage. For us, fermionic invariants exist only in
superspace and vanish in ordinary (non-supersymmetric) CFTs. We thank the
referee for a comment regarding this matter. invariants are direct
generalisations of the non-supersymmetric ($\mathcal{N}=0$) ones in [6]

$P_{1}\equiv\lambda_{2}X_{23-}^{-1}\lambda_{3}\,,\,\,\,\,\,P_{2}\equiv\lambda_{3}X_{31-}^{-1}\lambda_{1}\,,\,\,\,\,\,P_{3}\equiv\lambda_{1}X_{12-}^{-1}\lambda_{2}$
(30)
$Q_{1}\equiv\lambda_{1}\mathfrak{X}_{1+}\lambda_{1}\,,\,\,\,\,\,Q_{2}\equiv\lambda_{2}\mathfrak{X}_{2+}\lambda_{2}\,\,\,\,\,\,Q_{3}\equiv\lambda_{3}\mathfrak{X}_{3+}\lambda_{3}.$
(31)

where $\mathfrak{X}_{i+}$ is defined in Eq. (15). Using the transformations of
the covariant structures under superinversion listed in Section 2, it is
straightforward to check that these are parity-even superconformal invariants.

The fermionic parity-even invariants in the $\mathcal{N}=1$ case (with no
R-symmetry) are [10] $R_{i}=\lambda_{i}{\Theta}_{i},\,\,i=1,2,3$. For the
$\mathcal{N}=2$ case these objects will be charged under R-symmetry:
$R_{i}^{a}=\lambda_{i}{\Theta}_{i}^{a}$. However, in this case R-symmetry
singlets like $R_{i}^{2}$ are vanishing, whereas those like $R_{1}\cdot
R_{2}=\delta_{ab}R_{1}^{a}R_{2}^{b}$ are not super-inversion invariant because
of the non-trivial $V$ factors in the transformation of ${\Theta}$’s. Thus we
can not straightforwardly build on the $\mathcal{N}=1$ fermionic invariants to
get the $\mathcal{N}=2$ ones and must undertake an ab-initio construction.

###  3.1 Fermionic counterparts of the $P,Q,S$ invariants

As a first step towards constructing the fermionic invariants, we will first
define a number of covariant fermionic counterparts to the $P,Q,S$ invariants
above by replacing one or more $\lambda$’s by $\Theta$. The requisite
definitions, together with the transformation under super-inversion ($i_{s}$),
are

$\displaystyle\pi_{ij}^{a}$
$\displaystyle=\lambda_{i}X_{ij+}\Theta_{j}^{a}\xrightarrow{\phantom{\leavevmode\nobreak\
\leavevmode\nobreak\ }i_{s}\phantom{\leavevmode\nobreak\ \leavevmode\nobreak\
}}-\dfrac{1}{\bar{x}_{i}^{2}}\pi_{ij}^{b}(V_{j})_{b}^{\ a},$ (32)
$\displaystyle\vspace{1em}\Pi_{ij}^{ab}$
$\displaystyle=\Theta_{i}^{a}X_{ij+}\Theta^{b}_{j}\xrightarrow{\phantom{\leavevmode\nobreak\
\leavevmode\nobreak\ }i_{s}\phantom{\leavevmode\nobreak\ \leavevmode\nobreak\
}}(V^{T}_{i})^{a}_{\ c}\Pi_{ij}^{cd}(V_{j})_{d}^{\ b},$ (33)
$\displaystyle\vspace{1em}\omega^{a}_{i}$
$\displaystyle=\lambda_{i}\mathfrak{X}_{i+}\Theta_{i}^{a}\xrightarrow{\phantom{\leavevmode\nobreak\
\leavevmode\nobreak\ }i_{s}\phantom{\leavevmode\nobreak\ \leavevmode\nobreak\
}}-\bar{x}_{i}^{2}\omega_{i}^{b}(V_{i})_{b}^{\ a},$ (34)
$\displaystyle\vspace{1em}\Omega_{i}^{ab}$
$\displaystyle=\Theta_{i}^{a}\mathfrak{X}_{i+}\Theta_{i}^{b}\xrightarrow{\phantom{\leavevmode\nobreak\
\leavevmode\nobreak\ }i_{s}\phantom{\leavevmode\nobreak\ \leavevmode\nobreak\
}}\bar{x}_{i}^{4}(V^{T}_{i})^{a}_{\ c}\Omega_{i}^{cd}(V_{i})_{d}^{\ b},$ (35)
$\displaystyle\vspace{1em}\sigma_{13}^{a}$
$\displaystyle=\dfrac{\lambda_{1}X_{12+}X_{23+}\Theta_{3}^{a}}{\sqrt{\bar{x}_{12}^{2}\bar{x}_{23}^{2}\bar{x}_{31}^{2}}}\xrightarrow{\phantom{\leavevmode\nobreak\
\leavevmode\nobreak\ }i_{s}\phantom{\leavevmode\nobreak\ \leavevmode\nobreak\
}}\bar{x}_{3}^{2}\sigma_{13}^{b}(V_{3})_{b}^{\ a},$ (36)
$\displaystyle\vspace{1em}\Sigma_{13}^{ab}$
$\displaystyle=\dfrac{\Theta^{a}_{1}X_{12+}X_{23+}\Theta^{b}_{3}}{\sqrt{\bar{x}_{12}^{2}\bar{x}_{23}^{2}\bar{x}_{31}^{2}}}\xrightarrow{\phantom{\leavevmode\nobreak\
\leavevmode\nobreak\ }i_{s}\phantom{\leavevmode\nobreak\ \leavevmode\nobreak\
}}-\bar{x}_{1}^{2}\bar{x}_{3}^{2}(V^{T}_{1})^{a}_{\
c}\Sigma^{cd}_{13}(V_{3})_{d}^{\ b}.$ (37)

These objects are charged under R-symmetry and are superconformal covariant -
they transform homogeneously under super-inversion.666Cyclic permutations of
the last two structures above can be written down straightforwardly.

We can now build the required invariants by suitable contraction of the
various covariant fermionic structures constructed above. However, not all
R-symmetry singlets constructed out of the above covariant structures will be
invariant under super-inversion. This subtlety is due to the existence of the
$V$ matrix in R-symmetry space. Let us take a simple example to explain this
feature. A natural choice for a fermionic parity-even invariant777Note that
for $\mathcal{N}=1$, both $R_{1},R_{2}$ are separately invariant and thus so
is $R_{1}\cdot R_{2}$. would be $R_{1}\cdot R_{2}\equiv
R_{1}^{a}R_{2}^{b}\delta_{ab}$. However, under superinversion we get the $V$
matrix at points 1 and 2 and thus superconformal invariance is not maintained.
On the other hand, consider the R-symmetry singlet
$\frac{1}{\bar{x}_{12}^{2}}\pi_{13}^{a}\pi_{23}^{a}$ which, like $R_{1}\cdot
R_{2}$, is also of homogeneity $\lambda_{1}\lambda_{2}$. In this case, one can
check that super-inversion invariance is maintained.

We thus have the following set of parity-even fermionic invariants

$\bar{R}_{1}\equiv\frac{1}{\bar{x}_{23}^{2}}\pi_{21}^{a}\pi_{31}^{a},\,\,\,\,\bar{R}_{2}\equiv\frac{1}{\bar{x}_{31}^{2}}\pi_{32}^{a}\pi_{12}^{a},\,\,\,\,\bar{R}_{3}\equiv\frac{1}{\bar{x}_{12}^{2}}\pi_{13}^{a}\pi_{23}^{a}.$
(38)

We also have a fermionic parity-even invariant without any $\lambda$’s (such
an object does not exist in the $\mathcal{N}=1$ case)

$R^{\prime}\equiv\Pi_{ij}^{ab}\Pi_{ij}^{ab}$ (39)

Here we have a sum over $a,b$ but no sum over $i,j$. It is clearly neutral
under R-symmetry and it can be checked that it is super-inversion invariant as
well.888Note that $i\neq j$. It turns out that the three different pairs of
$(i,j)$ are all equivalent and thus there is only one $R^{\prime}$ invariant
which needs to be defined here. The choice of $\bar{R}_{i}$ and $R^{\prime}$
may seem arbitrary. For example, we can also build the following object by
contracting the above covariant structures:
$\dfrac{\bar{x}_{23}^{2}\bar{x}_{31}^{2}}{\bar{x}_{12}^{2}}\sigma_{13}^{a}\sigma_{23}^{a}$
. This R-symmetry neutral object (or its cyclic permutations) can be checked
to be parity-even and super-inversion invariant, and one may wonder if it
defines a new parity-even fermionic invariant. Likewise, one could ask the
same question for the parity-even, R-symmetry singlet, superconformally
invariant structure
$\frac{\bar{x}_{12}^{2}\bar{x}_{31}^{2}}{\bar{x}_{23}^{2}}\Omega^{aa}_{1}$.
However, one can show that

$\frac{\bar{x}_{12}^{2}\bar{x}_{31}^{2}}{\bar{x}_{23}^{2}}\Omega^{aa}_{1}=\frac{i}{2}R^{\prime},\,\quad\dfrac{\bar{x}_{23}^{2}\bar{x}_{31}^{2}}{\bar{x}_{12}^{2}}\sigma_{13}^{a}\sigma_{23}^{a}=\bar{R}_{3}-\dfrac{i}{2}R^{\prime}P_{3},$
(40)

so that both these cases can be expressed in terms of the existing set of
parity-even invariants. There are many more examples like these. A systematic
analysis (see Appendix B) shows that we can work with the set of invariants
$P_{i},Q_{i},\bar{R}_{i}$ and $R^{\prime}$, and all the parity-even structures
that are constructed from the above covariant building blocks can be expressed
in terms of these.

###  3.2 Relations for even invariants

To summarise, the parity-even invariants are $P_{i},Q_{i},\bar{R}_{i}$ and
$R^{\prime}$. With the action on three superspace points of the
$\mathcal{N}=2$ superconformal group in 3d, with 19 generators (11 bosonic and
8 fermionic), we expect to be able to construct $9\times 3-19=8$ such
independent invariants. Thus there should be two relations between the 10
constructed invariants. Indeed we find,999These extend the corresponding
relations in [6, 10].

$\displaystyle\sum_{i=1}^{3}P_{i}^{2}Q_{i}-2P_{1}P_{2}P_{3}-Q_{1}Q_{2}Q_{3}+\frac{i}{2}\sum_{i=1}^{3}\bar{R}_{i}P_{i}Q_{i}+\frac{1}{4}R^{\prime}P_{1}P_{2}P_{3}=0,$
(41)

at ${\cal O}(\lambda_{1}^{2}\lambda_{2}^{2}\lambda_{3}^{2})$, and the
(cyclically related) triplet of relations at ${\cal
O}(\lambda_{i}^{2}\lambda_{j}\lambda_{k})$:

$\displaystyle
P_{1}\bar{R}_{2}+P_{2}\bar{R}_{1}+Q_{3}\bar{R}_{3}-\frac{i}{2}R^{\prime}P_{1}P_{2}$
$\displaystyle=0,$ $\displaystyle
P_{2}\bar{R}_{3}+P_{3}\bar{R}_{2}+Q_{1}\bar{R}_{1}-\frac{i}{2}R^{\prime}P_{2}P_{3}$
$\displaystyle=0,$ (42) $\displaystyle
P_{3}\bar{R}_{1}+P_{1}\bar{R}_{3}+Q_{2}\bar{R}_{2}-\frac{i}{2}R^{\prime}P_{3}P_{1}$
$\displaystyle=0$

Together, these make the number of independent parity-even invariants to be
eight, as required.

Other relations between the parity-even invariants, which are higher order in
the $\lambda$’s, can be generated from the above relations. Some of these are
listed at the end of Appendix B.

##  4 The parity-odd invariants and relations

In this section we turn to the construction of 3-point function invariants
which are odd under parity. These pick up a minus sign under a super-inversion
transformation. As previously, there are some of these which are direct
extensions of the $\mathcal{N}=0$ case [6] :

$S_{1}=\frac{\lambda_{3}X_{31+}X_{12+}\lambda_{2}}{\sqrt{\bar{x}^{2}_{12}\bar{x}^{2}_{23}\bar{x}^{2}_{31}}}\,,\,\,\,\,\,S_{2}=\frac{\lambda_{1}X_{12+}X_{23+}\lambda_{3}}{\sqrt{\bar{x}^{2}_{12}\bar{x}^{2}_{23}\bar{x}^{2}_{31}}}\,,\,\,\,\,\,S_{3}=\frac{\lambda_{2}X_{23+}X_{31+}\lambda_{1}}{\sqrt{\bar{x}^{2}_{12}\bar{x}^{2}_{23}\bar{x}^{2}_{31}}}.$
(43)

Given the transformation laws of the building blocks in Section 2, it is easy
to check that these transform as $S_{i}\rightarrow-S_{i}$ under super-
inversion and hence are parity-odd.

Also, there now exists a unique parity-odd fermionic invariant:

$T^{\prime}=\sqrt{\dfrac{\bar{x}_{12}^{2}\bar{x}_{31}^{2}}{\bar{x}_{23}^{2}}}\displaystyle\Theta^{a}_{1\alpha}\Theta^{a\alpha}_{1}$
(44)

Under super-inversion: $T^{\prime}\rightarrow-T^{\prime}$.101010It can be
checked that cyclic permutations of $1,2,3$ give the same invariant.

Like in the parity-even case, there are other parity-odd invariant structures
that can be built from the covariant building blocks of Section 2 but they can
all be expressed in terms of $S_{i}$ and $T^{\prime}$(see Appendix B).

###  4.1 Relations for odd invariants

Since the product of two odd invariants is an even one, we must be able to
express it in terms of the minimal set of even invariants. For example, we
have

$\displaystyle T^{\prime}S_{i}=-2\bar{R}_{i}.$ (45)

We further have relations containing products of $S_{i}$’s which is expected
to be an even invariant. Indeed, we find

$\displaystyle S_{1}^{2}=P_{1}^{2}-Q_{2}Q_{3}+i\bar{R}_{1}P_{1},\quad
S_{1}S_{2}=-P_{1}P_{2}+P_{3}Q_{3},$ (46)

and cyclic permutations.

Similarly for $T^{\prime}$, we get

$\displaystyle T^{\prime 2}=2R^{\prime}.$ (47)

We also have various linear relations between products of even and odd
invariants.

  * •

${\cal O}(\lambda_{1}\lambda_{2})$:

$\displaystyle\bar{R}_{i}T^{\prime}=-S_{i}R^{\prime}$ (48)

  * •

${\cal O}(\lambda_{1}^{2}\lambda_{2}^{2})$:

$\displaystyle
P_{3}^{2}T^{\prime}-Q_{1}Q_{2}T^{\prime}+2S_{3}\bar{R}_{3}-iP_{3}S_{3}R^{\prime}=0,$
(49)

and cyclic permutations.

  * •

${\cal O}(\lambda_{1}^{2}\lambda_{2}\lambda_{3})$:

$\displaystyle
Q_{1}S_{1}+P_{2}S_{3}+P_{3}S_{2}+\frac{i}{2}T^{\prime}P_{2}P_{3}=0,$ (50)
$\displaystyle S_{2}\bar{R}_{3}-S_{3}\bar{R}_{2}=0,$ (51) $\displaystyle
S_{2}\bar{R}_{3}+S_{3}\bar{R}_{2}+T^{\prime}(P_{1}Q_{1}-P_{2}P_{3})=0,$ (52)

and cyclic permutations.

  * •

${\cal O}(\lambda_{1}^{2}\lambda_{2}^{2}\lambda_{3}^{2})$:

$\displaystyle
P_{1}Q_{1}S_{1}+P_{2}Q_{2}S_{2}-P_{3}Q_{3}S_{3}+2P_{1}P_{2}S_{3}+\frac{i}{2}T^{\prime}P_{1}P_{2}P_{3}=0,$
(53)
$\displaystyle\bar{R}_{1}(S_{2}P_{3}+\frac{1}{2}Q_{1}S_{1})+\bar{R}_{2}(S_{3}P_{1}+\frac{1}{2}Q_{2}S_{2})+\bar{R}_{3}(S_{1}P_{2}+\frac{1}{2}Q_{3}S_{3})+\frac{i}{4}R^{\prime}S_{1}S_{2}S_{3}=0,$
(54)

and cyclic permutations.

  * •

${\cal O}(\lambda_{1}^{3}\lambda_{2}^{3}\lambda_{3}^{2})$:

$\displaystyle(P_{1}^{2}Q_{1}-P_{2}^{2}Q_{2})P_{3}S_{3}+(P_{3}^{2}-Q_{1}Q_{2}+iP_{3}\bar{R}_{3})(P_{1}Q_{1}S_{1}-P_{2}Q_{2}S_{2})=0,$
(55)

and cyclic permutations.

Like in the parity-even case, these relations between the various parity-odd
structures have to be taken into account when enumerating all the independent
structures for the 3-point correlators in terms of our minimal set of
invariants.

##  5 Three-point function examples

The purpose of constructing the parity even and odd invariants is to make
simpler the enumeration of tensor structures for particular 3-point
correlators, a task we turn to in this section. Before turning to specific
examples of 3-point functions of spinning operators, let us make some general
observations.

The general 3-point function of spinning operators has the following form in
terms of the invariants constructed in the previous sections:

$\langle
O_{s_{1}}(x_{1},\theta_{1},\lambda_{1})O_{s_{2}}(x_{2},\theta_{2},\lambda_{2})O_{s_{3}}(x_{3},\theta_{3},\lambda_{3})\rangle=\frac{1}{\bar{x}_{12}^{m_{123}}\bar{x}_{23}^{m_{231}}\bar{x}_{31}^{m_{312}}}\sum_{n}\mathcal{G}_{n}(P_{i},Q_{i},\bar{R}_{i},R^{\prime},S_{i},T^{\prime}),$
(56)

with
$m_{ijk}\equiv(\Delta_{i}-s_{i})+(\Delta_{j}-s_{j})-(\Delta_{k}-s_{k})$,111111$\Delta_{i}$
and $s_{i}$ denote, respectively, the scaling dimension and spin of the
operator $O_{s}$ the sum being over all the linearly independent invariant
structures $\mathcal{G}_{n}$, each with homogeneity
$\lambda_{1}^{2s_{1}}\lambda_{2}^{2s_{2}}\lambda_{3}^{2s_{3}}$. Now the
3-point function has to be linear in the parity-odd invariants and in
$R^{\prime}$ or $\bar{R}_{i}$’s. We thus have the following general structure
for $\mathcal{G}_{n}$:

$\displaystyle\mathcal{G}_{n}=G_{n}^{(0)}(P_{i},Q_{i})+a_{n}^{(1)}G_{n}^{(0)}(P_{i},Q_{i})R^{\prime}+a_{n}^{(2)}G_{n}^{(1)}(P_{i},Q_{i})\bar{R}_{j}+b_{n}^{(1)}G_{n}^{(0)}(P_{i},Q_{i})T^{\prime}$
$\displaystyle+b_{n}^{(2)}G_{n}^{(1)}(P_{i},Q_{i})S_{k}+b_{n}^{(3)}G_{n}^{(1)}(P_{i},Q_{i})R^{\prime}S_{k}+b_{n}^{(4)}G_{n}^{(2)}(P_{i},Q_{i})\bar{R}_{j}S_{k},$

where $G_{n}^{(a)}(P_{i},Q_{i})$ is a monomial in $P$’s and $Q$’s with every
term in the r.h.s above having homogeneity
$\lambda_{1}^{2s_{1}}\lambda_{2}^{2s_{2}}\lambda_{3}^{2s_{3}}$.

The coefficients of the monomials can be further constrained when one or more
operators are conserved currents, as we will see in a later section. Besides
this, constraints also arise when two or more operators are identical, giving
rise to permutation invariance in the correlator. Our invariants have the
following transformation under a swap of superspace points 2 and 3:

$\displaystyle A_{1}\rightarrow-A_{1}\qquad A_{2}\rightarrow-A_{3},\qquad
A_{3}\rightarrow-A_{2},$ $\displaystyle R^{\prime}\rightarrow
R^{\prime},\qquad T^{\prime}\rightarrow T^{\prime},$

where $A_{i}$ stands for any of $P_{i},Q_{i},\bar{R}_{i},S_{i}$. This will be
useful when enumerating correlator structures below.

###  5.1 Examples

###  $\langle O_{s}O_{0}O_{0}\rangle$

This correlator of one spin-$s$ and two scalar operators has homogeneity
$\lambda_{1}^{2s}$ and thus the possible structures are

$\displaystyle Q_{1}^{s},\,\,\,Q_{1}^{s}R^{\prime},\,\,\,Q_{1}^{s}T^{\prime}.$
(57)

In this case all the 3 structures are independent, so the general form of this
correlator is

$\displaystyle\langle
O_{s}O_{0}O_{0}\rangle=\frac{1}{\bar{x}_{12}^{m_{123}}\bar{x}_{23}^{m_{231}}\bar{x}_{31}^{m_{312}}}\Big{(}a_{1}Q_{1}^{s}+a_{2}Q_{1}^{s}R^{\prime}+b_{1}Q_{1}^{s}T^{\prime}\Big{)},\quad
s=0,2,4,\ldots.$ (58)

Further we note that when $s$ is an odd integer, each of the 3 structures
above pick up a minus sign under $2\leftrightarrow 3$ whereas the correlator
is symmetric under this swap. Thus, we can conclude that $\langle
O_{s}O_{0}O_{0}\rangle$ vanishes for $s$ odd.

### $\langle O_{1}O_{1}O_{0}\rangle$

The possible structures for the correlator of two identical spin one operators
and one scalar operator (homogeneity $\lambda_{1}^{2}\lambda_{2}^{2}$) are:

$\displaystyle\begin{aligned}
&P_{3}^{2},\,\,\,\,Q_{1}Q_{2},\,\,\,P_{3}^{2}R^{\prime},\,\,\,Q_{1}Q_{2}R^{\prime},\,\,\,P_{3}\bar{R}_{3}\,\,\,\text{(parity
even)}\\\
&Q_{1}Q_{2}T^{\prime},\,\,\,P_{3}S_{3},\,\,\,P_{3}S_{3}R^{\prime},\,\,\,\bar{R}_{3}S_{3}\,\,\,\,\text{(parity
odd)}\end{aligned}$ (59)

The structure $P_{3}^{2}T^{\prime}$ is also possible but can be eliminated
using Eq. (49).

### $\langle O_{s}O_{1}O_{0}\rangle$

The possible structures for this correlator are:

$\displaystyle\begin{aligned} &Q_{1}^{s}Q_{2},\ Q_{1}^{s}Q_{2}R^{\prime},\
Q_{1}^{s-1}P_{3}^{2},\ Q_{1}^{s-1}P_{3}^{2}R^{\prime},\
Q_{1}^{s-1}P_{3}\bar{R}_{3}\quad\text{(parity even)}\\\\[4.5pt]
&Q_{1}^{s}Q_{2}T^{\prime},\ Q_{1}^{s-1}P_{3}S_{3},\
Q_{1}^{s-1}P_{3}S_{3}R^{\prime},Q_{1}^{s-1}\bar{R}_{3}S_{3}\quad\text{(parity
odd)}\end{aligned}$ (60)

We thus note that the possible structures for $\langle O_{s}O_{1}O_{0}\rangle$
are just $Q_{1}^{s-1}$ times the possible structures for $\langle
O_{1}O_{1}O_{0}\rangle$.

### $\langle O_{s}O_{\frac{1}{2}}O_{0}\rangle$

Only half-integral values of $s$ are allowed, since there is no invariant in
the ${\cal N}=2$ theory with homogeneity ${\cal O}(\lambda)$.The allowed
structures are:

$\displaystyle\begin{aligned} &Q_{1}^{s-\frac{1}{2}}P_{3},\
Q_{1}^{s-\frac{1}{2}}P_{3}R^{\prime},\
Q_{1}^{s-\frac{1}{2}}\bar{R}_{3}\quad\text{(parity even)}\\\
&Q_{1}^{s-\frac{1}{2}}P_{3}T^{\prime},Q_{1}^{s-\frac{1}{2}}S_{3},\
Q_{1}^{s-\frac{1}{2}}S_{3}R^{\prime},\quad\text{(parity odd)}\end{aligned}$
(61)

### $\langle O_{2}O_{2}O_{0}\rangle$

The possible independent structures for this correlator are:

$\displaystyle\begin{aligned} &Q_{1}^{2}Q_{2}^{2},\
Q_{1}^{2}Q_{2}^{2}R^{\prime},\ Q_{1}Q_{2}P_{3}^{2},\
Q_{1}Q_{2}P_{3}^{2}R^{\prime},\ Q_{1}Q_{2}P_{3}\bar{R}_{3},\ \\\ &P_{3}^{4},\
P_{3}^{4}R^{\prime},\ P_{3}^{3}\bar{R}_{3}\quad\text{(parity even)}\\\\[4.5pt]
&Q_{1}^{2}Q_{2}^{2}T^{\prime},\ Q_{1}Q_{2}P_{3}^{2}T^{\prime},\
Q_{1}Q_{2}S_{3}P_{3},\ Q_{1}Q_{2}S_{3}P_{3}R^{\prime},\
Q_{1}Q_{2}S_{3}\bar{R}_{3}\\\ &P_{3}^{3}S_{3},\ P_{3}^{3}S_{3}R^{\prime},\
P_{3}^{2}S_{3}\bar{R}_{3}\quad\text{(parity odd)}\end{aligned}$ (62)

Again, some possible structures have been eliminated using the relations.

### $\langle O_{s}O_{2}O_{0}\rangle$

Once again, the possible structures for $\langle O_{s}O_{2}O_{0}\rangle$ are
just $Q_{1}^{s-2}$ times the above possible structures for $\langle
O_{2}O_{2}O_{0}\rangle$.

### $\langle O_{1}O_{1/2}O_{1/2}\rangle$

The only possible structures with the required symmetry under
$2\leftrightarrow 3$ are parity odd

$P_{2}S_{3}-P_{3}S_{2}$ (63)

The structure $\bar{R}_{2}S_{3}-\bar{R}_{3}S_{2}$ is also possible but
vanishes by Eq. (51).

### $\langle O_{2}O_{1/2}O_{1/2}\rangle$

The independent structures are

$\displaystyle\begin{aligned} &Q_{1}P_{2}P_{3},\ Q_{1}P_{2}P_{3}R^{\prime},\
Q_{1}^{2}P_{1},\ Q_{1}P_{1}^{2}R^{\prime},\
Q_{1}^{2}\bar{R}_{1}\qquad\text{(parity even)}\\\\[4.5pt]
&Q_{1}P_{2}P_{3}T^{\prime},\ Q_{1}^{2}P_{1}T^{\prime},\ Q_{1}^{2}S_{1},\
Q_{1}^{2}S_{1}R^{\prime}\qquad\text{(parity odd)}\end{aligned}$ (64)

Under $2\leftrightarrow 3$ this correlator picks up a minus sign because of
the anti-commuting fermionic operators. Other possible structures include
$Q_{1}(P_{2}\bar{R}_{3}+P_{3}\bar{R}_{2}),\,\,Q_{1}(P_{2}S_{3}+P_{3}S_{2}),\,\,Q_{1}(P_{2}S_{3}+P_{3}S_{2})R^{\prime}$
but can be expressed in terms of the above structures by using the relations
in Sections 3 and 4.

### $\langle O_{s}O_{1/2}O_{1/2}\rangle$

The possible structures for this correlator, for $s$ even, are $Q_{1}^{s-2}$
times the allowed structures for $\langle O_{2}O_{1/2}O_{1/2}\rangle$. For $s$
odd, they are $Q_{1}^{s-1}(P_{2}S_{3}-P_{3}S_{2})$.

### $\langle O_{2}O_{1}O_{1}\rangle$

The independent structures are

$\displaystyle\begin{aligned} &Q_{1}^{2}Q_{2}Q_{3},\
Q_{1}^{2}Q_{2}Q_{3}R^{\prime},\ Q_{1}^{2}P_{1}^{2},\
Q_{1}^{2}P_{1}^{2}R^{\prime},\ Q_{1}P_{1}P_{2}P_{3},\
Q_{1}P_{1}P_{2}P_{3}R^{\prime},\ P_{2}^{2}P_{3}^{2},\\\
&P_{2}^{2}P_{3}^{2}R^{\prime},\ Q_{1}^{2}P_{1}\bar{R}_{1},\
Q_{1}P_{2}P_{3}\bar{R}_{1}\qquad\text{(parity even)}\\\\[4.5pt]
&Q_{1}^{2}Q_{2}Q_{3}T^{\prime},\ Q_{1}P_{1}P_{2}P_{3}T^{\prime},\
Q_{1}^{2}P_{1}S_{1},\ Q_{1}^{2}P_{1}S_{1}R^{\prime},\\\
&Q_{1}P_{2}P_{3}S_{1},\ Q_{1}P_{2}P_{3}S_{1}R^{\prime},\
Q_{1}^{2}S_{1}\bar{R}_{1},\
P_{2}^{2}S_{3}\bar{R}_{3}+P_{3}^{2}S_{2}\bar{R}_{2}\qquad\text{(parity
odd)}\end{aligned}$ (65)

These have the requisite symmetry under a $2\leftrightarrow 3$ swap. Other
structures, which are possible on grounds of homogeneity and symmetry can be
expressed in terms of the above list by using the relations in Sections 3 and
4.

### $\langle O_{s}O_{1}O_{1}\rangle$

The possible independent structures for this correlator are simply
$Q_{1}^{s-2}$ times the structures for $\langle O_{2}O_{1}O_{1}\rangle$ above,
for $s=2,3,...$.

### $\langle O_{2}O_{2}O_{2}\rangle$

This correlator is fully symmetric under $1\leftrightarrow 2\leftrightarrow
3$, and the independent structures are

$\displaystyle\begin{aligned} &Q_{1}^{2}Q_{2}^{2}Q_{3}^{2},\
Q_{1}^{2}Q_{2}^{2}Q_{3}^{2}R^{\prime},\ P_{1}^{2}P_{2}^{2}P_{3}^{2},\
P_{1}^{2}P_{2}^{2}P_{3}^{2}R^{\prime},\ Q_{1}Q_{2}Q_{3}P_{1}P_{2}P_{3},\
Q_{1}Q_{2}Q_{3}P_{1}P_{2}P_{3}R^{\prime}\\\\[4.5pt]
&\sum_{i}P_{i}^{4}Q_{i}^{2},\ R^{\prime}\sum_{i}P_{i}^{4}Q_{i}^{2},\
P_{1}P_{2}P_{3}\sum_{\text{cyc}}P_{1}P_{2}\bar{R}_{3},\
Q_{1}Q_{2}Q_{3}\sum_{\text{cyc}}P_{1}P_{2}\bar{R}_{3}\quad\text{(parity
even)}\\\\[9.00002pt] &Q_{1}^{2}Q_{2}^{2}Q_{3}^{2}T^{\prime},\
P_{1}^{2}P_{2}^{2}P_{3}^{2}T^{\prime},\
Q_{1}Q_{2}Q_{3}P_{1}P_{2}P_{3}T^{\prime},\
T^{\prime}\sum_{i}P_{i}^{4}Q_{i}^{2},\ \sum_{i}P_{i}^{3}Q_{i}^{2}S_{i},\\\
&R^{\prime}\sum_{i}P_{i}^{3}Q_{i}^{2}S_{i},\
P_{1}P_{2}P_{3}\sum_{\text{cyc}}P_{1}P_{2}S_{3},\
R^{\prime}P_{1}P_{2}P_{3}\sum_{\text{cyc}}P_{1}P_{2}S_{3}\qquad\text{(parity
odd)}\end{aligned}$ (66)

Several other linearly dependent structures that are possible have been
removed using various relations between the invariants.

##  6 Correlators of conserved currents

Our analysis so far has been general and the only constraint on the
operators/correlators has been $\mathcal{N}=2$ superconformal symmetry. In
this section we specialise to the case when one or more operators $O_{s}$ in
the 3-point function is a conserved current. We will denote a conserved
current by $J_{s}$ which will obey the shortening condition
$D^{a}_{\alpha_{1}}J^{\alpha_{1}\alpha_{2}.....\alpha_{2s}}=0$ or
equivalently,

$D^{a\alpha}\frac{\partial}{\partial\lambda^{\alpha}}J_{s}=0$ (67)

This is of interest because the stress-energy tensor is always an exactly
conserved current in any SCFT, one can have (one or more) conserved spin one
currents in the presence of global symmetries, and furthermore Chern-Simons-
Matter theories have higher-spin currents for weakly broken higher-spin
symmetries (when these symmetries are exact, the CFT is free [19]).The
presence of conserved currents imposes further constraints on the structure
correlators when the above conservation/shortening condition is imposed - some
of the coefficients in the 3-point function are determined in terms of others
as we will see below. For non-supersymmetric CFTs in various dimensions, [5]
had shown the existence of a fixed number (two in 3 dimensions) of (parity-
even) tensor structures for the 3-point correlator of conserved currents. In
[6] it was shown that for 3d CFTs, conserved current correlators also admit a
third structure which is parity-odd. For $\mathcal{N}=1$ SCFTs, evidence was
provided in [10] (see also [11, 15, 16]) that 3-point conserved current
correlators have one parity-even and one parity-odd structure.

###  6.1 Free field computation of conserved current correlators

We will first focus on the special case of 3-point correlators in a free field
$\mathcal{N}=2$ SCFT. Such a theory possesses an infinite number of exactly
conserved higher spin currents. The form of the currents in terms of the
elementary (anti-) chiral superfield is [10]

$\displaystyle J_{s}=\sum_{r=0}^{s}\left\\{(-1)^{r(2r+1)}\begin{pmatrix}2s\\\
2r\end{pmatrix}\partial^{r}\bar{\Phi}\,\partial^{s-r}\Phi+(-1)^{(r+1)(2r+1)}\begin{pmatrix}2s\\\
2r+1\end{pmatrix}\partial^{r}\bar{D}\bar{\Phi}\,\partial^{s-r-1}D\Phi\right\\}$
(68)

where
$\partial=i\lambda^{\alpha}(\gamma^{\mu})_{\alpha}\beta\lambda^{\beta}\partial_{\mu},\
D=\lambda^{\alpha}D_{\alpha}$, and $s=0,1,2\ldots$.

It is straightforward to directly compute 3-point correlators of such
conserved currents by using Wick’s theorem. The basic non-zero Wick
contractions are between the chiral superfield and its conjugate, and given by

$\displaystyle\langle\bar{\Phi}_{1}\Phi_{2}\rangle=\frac{1}{y_{12}}$ (69)

where $\Phi_{i}$ denotes the superfield at superspace point $x_{i}$ and
$y_{12}$ is defined [9, 11] as

$\displaystyle y_{12}^{2}=\bar{x}_{12}^{2}v_{12},\quad
v_{12}=\frac{1}{2}\left((V_{12})^{aa}+i\epsilon_{ab}(V_{12})^{ab}\right)$ (70)

where
$V_{12}^{ab}=\delta^{ab}+i\theta_{12}^{a\alpha}(X_{12+}^{-1})_{\alpha}^{\
\beta}\theta_{12\beta}^{b}$.

The computational procedure is explained in more detail through examples in
Appendix C. Below we simply quote the final results.121212The $f$ subscript
denotes a free theory correlator. We have removed constant multiplicative
overall factors. As expected, we find that the result of the calculation can
be expressed in terms of only the parity-even invariants,

$\displaystyle\langle
J_{0}J_{0}J_{0}\rangle_{f}=\frac{1}{\bar{x}_{12}\bar{x}_{23}\bar{x}_{31}}\left(1+\frac{1}{16}R^{\prime}\right)$
(71)

$\displaystyle\langle
J_{2}J_{0}J_{0}\rangle_{f}=\frac{Q_{1}^{2}}{\bar{x}_{12}\bar{x}_{23}\bar{x}_{31}}\left(1+\frac{5}{16}R^{\prime}\right)$
(72)

$\displaystyle\langle
J_{4}J_{0}J_{0}\rangle_{f}=\frac{Q_{1}^{4}}{\bar{x}_{12}\bar{x}_{23}\bar{x}_{31}}\left(1+\frac{9}{16}R^{\prime}\right)$
(73)

$\displaystyle\langle
J_{1}J_{1}J_{0}\rangle_{f}=\frac{1}{\bar{x}_{12}\bar{x}_{23}\bar{x}_{31}}\left[Q_{1}Q_{2}\left(1+\frac{9}{16}R^{\prime}\right)+P_{3}^{2}\left(2+\frac{3}{8}R^{\prime}\right)+2iP_{3}\bar{R}_{3}\right]$
(74)

###  6.2 General constraints of conservation for 3-point correlators

We will now directly use the results of Section 5 and impose conservation (Eq.
(67)) on the correlator to obtain additional constraints. The general
procedure is as follows. Since the 3-point correlator is a multinomial in the
invariants,131313Note that the exponents $m_{ijk}$ here are all unity for the
case of conserved currents because the unitarity bound is saturated in this
case.

$\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle=\frac{1}{{\bar{x}}_{12}{\bar{x}}_{23}{\bar{x}}_{31}}\sum_{n}a_{n}\mathcal{G}_{n},$
(75)

the imposition of the conservation constraints

$D_{i}^{a\alpha}\frac{\partial}{\partial\lambda_{i}^{\alpha}}\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle=0,\quad i=1,2,3$ (76)

(no sum over $i$) will lead to linear relations between various coefficients
of the multinomial. The computation gets rather messy. We set some superspace
coordinates to special values, and worked through a number of examples on
Mathematica. The results are tabulated in Table 1. Overall $\bar{x}_{ij}$
factors (see Eq. (75)) are suppressed. On setting all fermions to zero, our
results match with the non-supersymmetric case of [6].

This analysis is independent of the free theory computation in the previous
subsection but corroborates the parity-even results obtained there, since the
parity-even part of the various correlators obtained by applying the
conservation constraint is the same as the result of the free theory
computation.

Correlator |  Parity-even part |  Parity-odd part
---|---|---
$\langle J_{0}J_{0}J_{0}\rangle$ |  $\vphantom{\bigg{.}}1+\frac{1}{16}R^{\prime}\vphantom{\bigg{.}}$ |  $T^{\prime}$
| $\langle J_{s}J_{0}J_{0}\rangle$
---
$(s\geq 2)$
$\vphantom{\bigg{.}}Q_{1}^{s}\left(1+\frac{2s+1}{16}R^{\prime}\right)\vphantom{\bigg{.}}$ |  0
$\langle J_{1}J_{1}J_{0}\rangle$ |  | $Q_{1}Q_{2}\left(1+\frac{9}{16}R^{\prime}\right)+P_{3}^{2}\left(2+\frac{3}{8}R^{\prime}\right)$
---
$+2iP_{3}\bar{R}_{3}$
| $Q_{1}Q_{2}T^{\prime}-S_{3}\bar{R}_{3}-2iS_{3}P_{3}$
---
| $\langle J_{s}J_{1}J_{0}\rangle$
---
$(s\geq 2)$
| $Q_{1}^{s}Q_{2}\left(1+\frac{3(2s+1)}{16}R^{\prime}\right)$
---
$+Q_{1}^{s-1}P_{3}^{2}\left(2s+\frac{s(2s+1)}{8}R^{\prime}\right)$
$+2isQ_{1}^{s-1}P_{3}\bar{R}_{3}$
0
$\langle J_{2}J_{2}J_{0}\rangle$ |  | $Q_{1}^{2}Q_{2}^{2}\left(1+\frac{25}{16}R^{\prime}\right)+Q_{1}Q_{2}P_{3}^{2}\left(8+\frac{22}{4}R^{\prime}\right)$
---
$+8iQ_{1}Q_{2}P_{3}\bar{R}_{3}+P_{3}^{4}\left(\frac{8}{3}-\frac{1}{2}R^{\prime}\right)$
$+\frac{16i}{3}P_{3}^{3}\bar{R}_{3}$
|
$Q_{1}^{2}Q_{2}^{2}T^{\prime}+Q_{1}Q_{2}P_{3}S_{3}\left(-\frac{2i}{3}+\frac{i}{6}R^{\prime}\right)$
---
$-\frac{5}{3}Q_{1}Q_{2}S_{3}\bar{R}_{3}+P_{3}^{3}S_{3}\left(-\frac{4i}{3}-\frac{i}{6}R^{\prime}\right)$
| $\langle J_{s}J_{2}J_{0}\rangle$
---
$(s>2)$
| $Q_{1}^{s}Q_{2}^{2}\left(1+\frac{5(2s+1)}{16}R^{\prime}\right)$
---
$+Q_{1}^{s-1}Q_{2}P_{3}^{2}\left(4s+\frac{s(6s-1)}{4}R^{\prime}\right)$
$+Q_{1}^{s-2}P_{3}^{4}\left(\frac{4s(s-1)}{3}+\frac{s(s-1)(2s-7)}{12}R^{\prime}\right)$
$+4isQ_{1}^{s-1}Q_{2}P_{3}\bar{R}_{3}+\frac{8s(s-1)}{3}iQ_{1}^{s-2}P_{3}^{3}\bar{R}_{3}$
0
$\langle J_{2}J_{1}J_{1}\rangle$ |  | $Q_{1}^{2}Q_{2}Q_{3}\left(1+\frac{21}{16}R^{\prime}\right)+Q_{1}^{2}P_{1}^{2}\left(-\frac{2}{5}-\frac{3}{8}R^{\prime}\right)$
---
$+2iQ_{1}^{2}P_{1}\bar{R}_{1}+Q_{1}P_{1}P_{2}P_{3}\left(-\frac{8}{5}-\frac{3}{2}R^{\prime}\right)$
$+P_{2}^{2}P_{3}^{2}\left(\frac{8}{5}+\frac{1}{2}R^{\prime}\right)-4iQ_{1}P_{2}P_{3}\bar{R}_{1}$
$Q_{1}^{2}Q_{2}Q_{3}T^{\prime}-Q_{1}^{2}P_{1}S_{1}\left(2i+2iR^{\prime}\right)$
$-Q_{1}P_{1}P_{2}P_{3}T^{\prime}+Q_{1}P_{2}P_{3}S_{1}\left(4i+2iR^{\prime}\right)$
$+2Q_{1}^{2}S_{1}\bar{R}_{1}-(P_{2}^{2}S_{3}\bar{R}_{3}+P_{3}^{2}S_{2}\bar{R}_{2})$
| $\langle J_{s}J_{1}J_{1}\rangle$
---
$(s>2)$
| $Q_{1}^{s}Q_{2}Q_{3}\left(1+\frac{3(2s+3)}{16}R^{\prime}\right)$
---
$+Q_{1}^{s}P_{1}^{2}\left(-\frac{2(s-1)}{2s+1}-\frac{3(s-1)}{8}R^{\prime}\right)$
$+Q_{1}^{s-1}P_{1}P_{2}P_{3}\left(-\frac{4s}{2s+1}-\frac{3s}{4}R^{\prime}\right)$
$+Q_{1}^{s-2}P_{2}^{2}P_{3}^{2}\left(\frac{4s(s-1)}{2s+1}+\frac{s(s-1)}{4}R^{\prime}\right)$
$+2iQ_{2}^{s}P_{1}\bar{R}_{1}-2isQ_{1}^{s-1}P_{2}P_{3}\bar{R}_{1}$
0
$\langle J_{2}J_{2}J_{2}\rangle$ |  | $Q_{1}^{2}Q_{2}^{2}Q_{3}^{2}\left(3+\frac{103}{16}R^{\prime}\right)$
---
$+P_{1}^{2}P_{2}^{2}P_{3}^{2}\left(\frac{128}{25}+\frac{72}{5}R^{\prime}\right)$
$+P_{1}P_{2}P_{3}Q_{1}Q_{2}Q_{3}\left(-\frac{16}{5}-11R^{\prime}\right)$
$+\sum_{i}P_{i}^{4}Q_{i}^{2}\left(-\frac{8}{5}-\frac{5}{2}R^{\prime}\right)$
$+\sum_{\text{cyc}}P_{1}P_{2}\bar{R}_{3}\left(\frac{64i}{5}P_{1}P_{2}P_{3}-8iQ_{1}Q_{2}Q_{3}\right)$
|
$9Q_{1}^{2}Q_{2}^{2}Q_{3}^{2}T^{\prime}+38P_{1}^{2}P_{2}^{2}P_{3}^{2}T^{\prime}$
---
$-8Q_{1}Q_{2}Q_{3}P_{1}P_{2}P_{3}T^{\prime}-5T^{\prime}\sum_{i}P_{i}^{4}Q_{i}^{2}$
$+\sum_{i}P_{i}^{3}Q_{i}^{2}S_{i}\left(-8i-6iR^{\prime}\right)$
$+P_{1}P_{2}P_{3}\sum_{\text{cyc}}P_{1}P_{2}S_{3}\left(-40i-46iR^{\prime}\right)$
Table 1: Results from conservation constraints. These results have been
checked case by case up to $s=6$. The current superfield
$J_{0}=\bar{\Phi}\Phi$ has $\Delta=1$.

The table of results shows clearly that for $\mathcal{N}=2$ 3d SCFTs the
3-point correlator of conserved currents of various spins always has only two
undetermined constant coefficients that come with the two parts - a parity-
even part arising from the free theory, and a parity-odd part which comes from
interactions:

$\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle=\frac{1}{{\bar{x}}_{12}{\bar{x}}_{23}{\bar{x}}_{31}}\Big{(}a\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{even}+b\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{odd}\Big{)}.$ (77)

Furthermore, we also note from the examples, that the coefficient of the
parity-odd structure vanishes when the 3-point correlator is ‘outside the
triangle’. 141414This means that the inequality between the spins
$s_{i}+s_{j}\geq s_{k}$ is violated for some spins, see [6], [19]. Inside the
triangle, conservation constraints allow a parity-odd structure, as our table
shows. However, for a correlator involving an exactly conserved higher spin
($s>2$) current, we would expect the theory to be free along the lines of [19]
and hence the parity-odd structure would not arise, although this can’t be
inferred from conservation constraints alone. We would like thank Sachin Jain
for discussions regarding this point.

##  7 Discussion

In this paper we studied 3-point correlators of spinning operators in
$\mathcal{N}=2$ 3d SCFTs in terms of superconformal invariants. Our main
results include:

  * •

Construction of the parity-even invariants and determination of relations
between them.

  * •

Construction of the parity-odd invariants and determination of relations
between them.

  * •

Expressions for several 3-point functions of spinning operators in terms of a
linearly independent basis made up of the minimal set of invariants.

  * •

Computing the 3-point conserved current correlators in the free field case and
expressing them in terms of the invariants.

  * •

Studying the constraints of conservation - and verifying through several
examples that conserved current 3-point correlators contain 2 OPE coefficients
comprising of the parity-even and odd parts of the correlator.

We would like to emphasise again that the construction of the invariants in
the $\mathcal{N}=2$ case is more involved and subtle compared to the
$\mathcal{N}=1,\,0$ cases due to the presence of R-symmetry. Although the
$P,Q,S$ invariants for $\mathcal{N}=0$ have a relatively straightforward
extension in superspace, the fermionic invariants for $\mathcal{N}=2$ are
completely different from the $\mathcal{N}=1$ case, although built out of
similar building blocks which are now charged under R-symmetry. To build our
minimal set of $\mathcal{N}=2$ invariants, we had to deal with two layers of
complexity. Firstly, as discussed in Section 3, not all R-symmetry singlets
built by suitably contracting the covariant building blocks will be super-
inversion invariant. Secondly, even when they are, they will not be
independent and we found (in Sections 3, 4 and Appendix B) relations which
give our final minimal set of invariants:
$P_{i},Q_{i},\bar{R}_{i},R^{\prime},S_{i},T^{\prime}$.

It would be interesting to take forward this work in various directions:

  * •

Higher dimensions –  The same methods should work for correlators in SCFTs in
higher than 3 dimensions. The supercovariant structures in this case are known
from the work of Park and Osborn [20, 21, 22, 23]. One new feature would be
the existence of mixed symmetry tensors not present in the 3d case, which
would engender the use of grassmanian polarisation spinors.

  * •

More supersymmetry –  It should be straightforward, though cumbersome, to
extend this analysis to higher supersymmetry. Off-shell superspace
formulations for SCFTs in three dimensions exist for $\mathcal{N}=1,2,3,4$
[24]. For $\mathcal{N}>2$, there will be other invariant tensors of
$SO(\mathcal{N})$ which may also be used to build R-symmetry singlets. For
example, for $\mathcal{N}=3$ we can have $\delta_{ab}$ as well as the anti-
symmetric $\epsilon_{abc}$.

  * •

Generating functions – Building on the various examples, it would be
interesting to work out the form of the generating function for 3-point
correlators in the $\mathcal{N}=1,2$ free theory. The form of this generating
function is known for the non-supersymmetric case [6] in terms of the parity-
even invariants $P\,\text{and}\,\,Q$, but not known for the SCFT case.

  * •

Bootstrap –  Since 4-point superconformal blocks can be built from the product
of 3-point functions of spinning operators, our analysis should aid in
implementations of the superconformal bootstrap for 4-point functions in 3d
$\mathcal{N}=2$ SCFTs. Bootstrap studies of 3d SCFTs include [25, 26] for
$\mathcal{N}=2$ and [27, 28, 29] for $\mathcal{N}=1$ (see also [30, 31] for
higher $\mathcal{N}$).

  * •

Momentum space –  It would be instructive to carry out our general analysis in
momentum space. Firstly, this is useful because most perturbative computations
of correlators in specific theories are carried out directly in momentum
space. Secondly, the structure of momentum space correlators in (non-
supersymmetric) CFTs has been actively investigated in the last decade,
starting with the works [32, 33, 34], with a number of interesting conceptual
and technical results and it would be natural to study this also for SCFTs. In
particular, conserved current correlators with spin have a simple form in
momentum space (and more so in terms of spinor-helicity variables) for 3d CFTs
[32, 35, 36, 37] and it would be interesting to investigate the same for SCFT
correlators.

  * •

$\mathcal{N}=2$ Chern-Simons-Matter theories –  A prime example of 3d
Lagrangian CFTs are the Chern-Simons theories coupled to matter. Such theories
are exactly conformal and have a well-controlled weakly broken higher-spin
symmetry [38]. Our analysis should be of use in computations of correlators in
$\mathcal{N}=2$ 3d Chern-Simons-Matter theories [39].

## Acknowledgements

AAN would like to thank J.H. Park and S. Jain for useful suggestions and
comments on the draft. AJ wishes to acknowledge support from the Council of
Scientific and Industrial Research (CSIR) for a Junior Research Fellowship
(NET-JRF).

## Appendices

##  Appendix A Conventions and useful relations

We use the mostly plus $\\{-1,1,1\\}$ metric convention. Spinors - which can
be taken to be Majorana - transform under the (double cover of the) 3d Lorentz
group, which is $SL(2,R)$. We thus use the fundamental representation and
impose the Majorana condition on the 2-component spinors which is the
following reality condition $\psi_{\alpha}=\psi^{*}_{\alpha}$. In this
representation, superconformal theories with $\mathcal{N}$ extended
supersymmetry possess an $SO(\mathcal{N})$ R-symmetry group which is a
subgroup of the full superconformal group. For 3 dimensions, we can make a
choice of a real basis for the $\gamma$ matrices

$(\gamma_{\mu})_{\alpha}^{\leavevmode\nobreak\
\beta}\equiv(i\sigma^{2},\sigma^{1},\sigma^{3})=\left(\left(\begin{array}[]{cc}\mbox{
}0&\mbox{ }1\\\ -1&\mbox{ }0\end{array}\right),\left(\begin{array}[]{cc}0&1\\\
\mbox{ }1&\mbox{ }0\end{array}\right),\left(\begin{array}[]{cc}1&\mbox{ }0\\\
0&-1\end{array}\right)\right)$ (78)

These matrices when both indices are up (or down) are symmetric

$\begin{split}(\gamma_{\mu})_{\alpha\beta}\equiv(\mathds{1},\sigma^{3},-\sigma^{1})&\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
(\gamma_{\mu})^{\alpha\beta}\equiv(\mathds{1},-\sigma^{3},\sigma^{1})\end{split}$
(79)

The anti-symmetric $\epsilon$ symbol is defined as
$\epsilon^{12}=-1=\epsilon_{21}$.

$\begin{split}\epsilon\gamma^{\mu}\epsilon^{-1}=-(\gamma^{\mu})^{T}\\\
\epsilon\Sigma^{\mu\nu}\epsilon^{-1}=-(\Sigma^{\mu\nu})^{T}\end{split}$ (80)

with $\Sigma^{\mu\nu}=-\frac{i}{4}[\gamma^{\mu},\gamma^{\nu}]$ as the Lorentz
group generators. The $\gamma$ matrices satisfy

$(\gamma_{\mu}\gamma_{\nu})_{\alpha}^{\leavevmode\nobreak\
\beta}=\eta_{\mu\nu}\delta_{\alpha}^{\leavevmode\nobreak\
\beta}+\epsilon_{\mu\nu\rho}(\gamma^{\rho})_{\alpha}^{\leavevmode\nobreak\
\beta}$ (81)

with $\epsilon_{\mu\nu\rho}$ as the Levi-Civita symbol, and we use
$\epsilon_{012}=1$ ($\epsilon^{012}=-1$).

The spinor Lorentz transformation is

$\psi^{\prime}_{\alpha}\rightarrow-(\Sigma_{\mu\nu})_{\alpha}^{\leavevmode\nobreak\
\beta}\psi_{\beta}.$

The conventions for raising and lowering with $\epsilon$ are

$\begin{split}\psi^{\beta}=\epsilon^{\beta\alpha}\psi_{\alpha}\\\
\psi_{\alpha}=\epsilon_{\alpha\beta}\psi^{\beta}\end{split}$ (82)

and we denote $\psi\chi\equiv\psi^{\alpha}\chi_{\alpha}$.

###  A.1 Superconformal generators and algebra

The generators of the 3d superconformal algebra in differential operator form
are:

$\begin{split}P_{\mu}&=-i\partial_{\mu},\\\
M_{\mu\nu}&=-i\left(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu}-\frac{1}{2}\epsilon_{\mu\nu\rho}(\gamma^{\rho})_{\alpha}^{\mbox{
}\beta}\theta^{a}_{\beta}\frac{\partial}{\partial\theta^{a}_{\alpha}}\right)+{\cal
M}_{\mu\nu},\\\ D&=-i\left(x^{\nu}\partial_{\nu}+\frac{1}{2}\theta^{\alpha
a}\frac{\partial}{\partial\theta^{\alpha}_{a}}\right)+\Delta,\\\
K_{\mu}&=-i\left(\left(x^{2}+\frac{(\theta^{a}\theta^{a})^{2}}{16}\right)\partial_{\mu}-2x_{\mu}\left(x\cdot\partial+\theta^{\alpha
a}\frac{\partial}{\partial\theta^{\alpha}_{a}}\right)+(\theta^{a}X_{+}\gamma_{\mu})^{\beta}\frac{\partial}{\partial\theta_{a}^{\beta}}\right)\\\
&=x^{\nu}M_{\nu\mu}-x_{\mu}D+\frac{i}{2}(\theta^{a}\gamma_{\mu}X)^{\alpha}\frac{\partial}{\partial\theta_{a}^{\alpha}}-\frac{i}{16}(\theta^{a}\theta^{a})^{2}\partial_{\mu}+\frac{(\theta^{a}\theta^{a})}{4}(\theta^{b}\gamma_{\mu})^{\alpha}\frac{\partial}{\partial\theta_{b}^{\alpha}},\\\
I^{ab}&=-i\left(\theta^{a}\frac{\partial}{\partial\theta_{b}}-\theta^{b}\frac{\partial}{\partial\theta_{a}}\right)+{\cal
I}^{ab},\\\
Q_{\alpha}^{a}&=\frac{\partial}{\partial\theta^{\alpha}_{a}}-\frac{i}{2}\theta^{\beta
a}(\gamma^{\mu})_{\beta\alpha}\partial_{\mu},\\\
S_{\alpha}^{a}&=-(X_{+})_{\alpha}^{\leavevmode\nobreak\
\beta}Q_{\beta}^{a}-i\theta^{a}\theta^{b}\frac{\partial}{\partial\theta^{\alpha}_{b}}-i\theta^{a}_{\alpha}\theta^{b\beta}\frac{\partial}{\partial\theta^{\beta}_{b}}+\frac{i}{2}(\theta^{b}\theta^{b})\frac{\partial}{\partial\theta_{a}^{\alpha}}\\\
&=-(X_{-})_{\alpha}^{\leavevmode\nobreak\
\beta}\frac{\partial}{\partial\theta_{a}^{\beta}}+\frac{\theta_{\alpha}^{a}}{2}D+\frac{1}{4}\epsilon_{\mu\nu\rho}(\gamma^{\rho}\theta^{a})_{\alpha}M^{\mu\nu}-\frac{(\theta^{b}\theta^{b})}{8}\theta^{a\beta}\partial_{\beta\alpha}-\frac{3i}{4}\left(\theta_{\alpha}^{a}\theta\frac{\partial}{\partial\theta}+\theta^{a}\theta^{b}\frac{\partial}{\partial\theta_{b}^{\alpha}}\right)\leavevmode\nobreak\
.\end{split}$ (83)

where $\cal{M},\cal{I}$ arise due to the tensor structure of the operators on
which these generators act.

The superconformal algebra generated by these is:

$\begin{split}[M_{\mu\nu},M_{\rho\lambda}]&=i\left(\eta_{\mu\rho}M_{\nu\lambda}-\eta_{\nu\rho}M_{\mu\lambda}-\eta_{\mu\lambda}M_{\nu\rho}+\eta_{\nu\lambda}M_{\mu\rho}\right),\\\
[M_{\mu\nu},P_{\lambda}]&=i(\eta_{\mu\lambda}P_{\nu}-\eta_{\nu\lambda}P_{\mu}),\\\
[M_{\mu\nu},K_{\lambda}]&=i(\eta_{\mu\lambda}K_{\nu}-\eta_{\nu\lambda}K_{\mu}),\\\
[D,P_{\mu}]&=iP_{\mu}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ [D,K_{\mu}]=-iK_{\mu},\\\
[P_{\mu},K_{\nu}]&=2i(\eta_{\mu\nu}D-M_{\mu\nu}),\\\
[I_{ab},I_{cd}]&=i\left(\delta_{ac}I_{bd}-\delta_{bc}I_{ad}-\delta_{ad}I_{bc}+\delta_{bd}I_{ac}\right),\\\
\\{Q_{\alpha}^{a},Q_{\beta}^{b}\\}&=(\gamma^{\mu})_{\alpha\beta}P_{\mu}\delta^{ab},\\\
[I_{ab},Q^{\alpha}_{c}]&=i(\delta_{ac}Q^{\alpha}_{b}-\delta_{bc}Q^{\alpha}_{a}),\\\
\\{S_{\alpha}^{a},S_{\beta}^{b}\\}&=(\gamma^{\mu})_{\alpha\beta}K_{\mu}\delta^{ab},\\\
[I_{ab},S^{\alpha}_{c}]&=i(\delta_{ac}S^{\alpha}_{b}-\delta_{bc}S^{\alpha}_{a}),\\\
[K_{\mu},Q_{\alpha}^{a}]&=i(\gamma_{\mu})_{\alpha}^{\leavevmode\nobreak\
\beta}S_{\beta}^{a},\\\
[P_{\mu},S_{\alpha}^{a}]&=i(\gamma_{\mu})_{\alpha}^{\leavevmode\nobreak\
\beta}Q_{\beta}^{a},\\\
[D,Q_{\alpha}^{a}]&=\frac{i}{2}Q_{\alpha}^{a}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
[D,S_{\alpha}^{a}]=-\frac{i}{2}S_{\alpha}^{a},\\\
[M_{\mu\nu},Q_{\alpha}^{a}]&=-(\Sigma_{\mu\nu})_{\alpha}^{\leavevmode\nobreak\
\beta}Q_{\beta}^{a},\\\
[M_{\mu\nu},S_{\alpha}^{a}]&=-(\Sigma_{\mu\nu})_{\alpha}^{\leavevmode\nobreak\
\beta}S_{\beta}^{a},\\\
\\{Q^{a}_{\alpha},S^{b}_{\beta}\\}&=\left(\epsilon_{\beta\alpha}D-\frac{1}{2}\epsilon_{\mu\nu\rho}(\gamma^{\rho})_{\alpha\beta}M^{\mu\nu}\right)\delta^{ab}+\epsilon_{\beta\alpha}I^{ab}.\end{split}$
(84)

The rest of the (anti)-commutators are zero.

##  Appendix B More relations between invariants

We had the following set of superconformal invariants which were used to
construct various 3-point correlators:

parity even: $\displaystyle P_{i},\ Q_{i},\ \bar{R}_{i},\ R^{\prime}$ (85)
parity odd: $\displaystyle S_{i},\ T^{\prime}$ (86)

We noted earlier that using the covariant structures we seem to be able to
build other invariants as well. However, all such additional invariants can be
expressed in terms of the above minimal set of invariants. We show this
explicitly below by listing manifold relations where an invariant structure is
expressed in terms of our minimal set. We have classified the relations
according to the degree in $\lambda$ and ${\Theta}$, an ${\cal
O}(\lambda^{m}\Theta^{n})$ relation will have terms, each of which has $m$
$\lambda$’s and $n$ $\Theta$’s.

###  B.1 Parity even relations

In this subsection we enumerate parity-even invariant structures built from
the covariant building blocks. These will all be expressed in terms of our
claimed minimal set of parity-even invariants:
$P_{i},Q_{i},\bar{R}_{i},R^{\prime}$.

  1. 1.

${\cal O}(\Theta^{2})$:

    1. (a)

$\displaystyle\frac{\bar{x}_{12}^{2}\bar{x}_{31}^{2}}{\bar{x}_{23}^{2}}\Omega^{aa}_{1}=\frac{i}{2}R^{\prime}$

  2. 2.

${\cal O}(\Theta^{4})$:

    1. (a)

$\bar{x}_{12}^{4}\Sigma_{12}^{ab}\Sigma_{12}^{ab}=-R^{\prime},\quad\bar{x}_{12}^{4}\Omega_{1}^{ab}\Omega_{2}^{ab}=R^{\prime}$

  3. 3.

${\cal O}(\lambda^{2}\Theta^{2})$:

    1. (a)

$R_{1}^{a}R_{1}^{a}=\dfrac{\bar{x}_{23}^{2}}{\bar{x}_{12}^{2}\bar{x}_{31}^{2}}\pi_{12}^{a}\pi_{12}^{a}=\dfrac{\bar{x}_{23}^{2}\bar{x}_{12}^{2}}{\bar{x}_{31}^{2}}\sigma_{12}^{a}\sigma_{12}^{a}=\dfrac{\bar{x}_{12}^{2}\bar{x}_{31}^{2}}{\bar{x}_{23}^{2}}\omega_{1}^{a}\omega_{1}^{a}=0$

    2. (b)

$\sqrt{\dfrac{\bar{x}_{12}^{2}\bar{x}_{31}^{2}}{\bar{x}_{23}^{2}}}R_{1}^{a}\sigma_{21}^{a}=-\bar{R}_{3},\quad\dfrac{\bar{x}_{23}^{2}\bar{x}_{31}^{2}}{\bar{x}_{12}^{2}}\sigma_{13}^{a}\sigma_{23}^{a}=\bar{R}_{3}-\dfrac{i}{2}R^{\prime}P_{3},\quad\dfrac{\bar{x}_{23}^{2}}{\bar{x}_{13}^{2}}\pi_{12}^{a}\omega_{2}^{a}=\bar{R}_{3}$

  4. 4.

${\cal O}(\lambda^{2}\Theta^{4})$:

    1. (a)

$R_{1}^{a}\Pi_{23}^{ab}R_{2}^{b}=-\dfrac{1}{2}R^{\prime}P_{3},\quad\dfrac{\bar{x}_{12}^{2}\bar{x}_{31}^{2}}{\bar{x}_{23}^{2}}R_{1}^{a}\Omega_{1}^{ab}R_{1}^{b}=-\dfrac{1}{2}R^{\prime}Q_{1}$

    2. (b)

$\dfrac{1}{\bar{x}_{12}^{2}}\pi_{12}^{a}\Pi_{23}^{ab}\pi_{23}^{b}=\dfrac{1}{2}R^{\prime}P_{3},\quad\bar{x}_{23}^{2}\sigma_{12}^{a}\Pi_{23}^{ab}\sigma_{23}^{b}=-\dfrac{1}{2}R^{\prime}P_{3},\quad\bar{x}_{12}^{2}\omega_{1}^{a}\Pi_{12}^{ab}\omega_{2}^{b}=-\dfrac{1}{2}R^{\prime}P_{3},\\\
\pi_{12}^{a}\Pi_{21}^{ab}\omega_{1}^{b}=-\dfrac{1}{2}R^{\prime}Q_{1}$

    3. (c)

${\bar{x}_{12}^{2}}\pi_{12}^{a}\Sigma_{21}^{ab}\sigma_{31}^{b}=-\dfrac{1}{2}R^{\prime}P_{2},\quad{\bar{x}_{23}^{4}}\sigma_{12}^{a}\Sigma_{23}^{ab}\omega_{3}^{b}=\dfrac{1}{2}R^{\prime}P_{2}$

    4. (d)

$\dfrac{\bar{x}_{12}^{2}\bar{x}_{23}^{2}}{\bar{x}_{13}^{4}}\pi_{12}^{a}\Omega_{2}^{ab}\pi_{32}^{b}=-\dfrac{1}{2}R^{\prime}P_{2},\quad\dfrac{\bar{x}_{23}^{4}\bar{x}_{12}^{2}}{\bar{x}_{13}^{4}}\pi_{12}^{a}\Omega_{2}^{ab}\omega_{2}^{b}=\dfrac{1}{2}R^{\prime}P_{3},\quad\\\
\dfrac{\bar{x}_{12}^{4}\bar{x}_{31}^{4}}{\bar{x}_{23}^{4}}\omega_{1}^{a}\Omega_{1}^{ab}\omega_{1}^{b}=\dfrac{1}{2}R^{\prime}Q_{1},\quad\dfrac{\bar{x}_{12}^{4}\bar{x}_{31}^{4}}{\bar{x}_{23}^{4}}\sigma_{21}^{a}\Omega_{1}^{ab}\sigma_{31}^{b}=-\dfrac{1}{2}R^{\prime}P_{1}$

  5. 5.

${\cal O}(\lambda^{4}\Theta^{4})$:

    1. (a)

$(R_{1}^{a}R_{2}^{a})^{2}=-\dfrac{1}{2}R^{\prime}(Q_{1}Q_{2}-P_{3}^{2})$

    2. (b)

$\dfrac{1}{\bar{x}_{12}^{4}}(\pi_{12}^{a}\pi_{23}^{a})^{2}=-\dfrac{1}{2}R^{\prime}(Q_{1}Q_{2}-P_{3}^{2}),\quad\dfrac{1}{\bar{x}_{12}^{4}}(\pi_{12}^{a}\pi_{21}^{a})^{2}=-\dfrac{1}{2}R^{\prime}(Q_{1}Q_{2}-P_{3}^{2}),\\\
\dfrac{1}{\bar{x}_{13}^{4}}(\pi_{12}^{a}\pi_{31}^{a})^{2}=-\dfrac{1}{2}R^{\prime}(Q_{1}Q_{3}-P_{2}^{2})$

    3. (c)

$\bar{x}_{23}^{4}(\sigma_{12}^{a}\sigma_{23}^{a})^{2}=-\dfrac{1}{2}R^{\prime}(Q_{1}Q_{2}-P_{3}^{2}),\quad\bar{x}_{12}^{4}(\sigma_{12}^{a}\sigma_{21}^{a})^{2}=-\dfrac{1}{2}R^{\prime}(Q_{1}Q_{2}-P_{3}^{2}),\\\
\bar{x}_{12}^{4}(\sigma_{12}^{a}\sigma_{31}^{a})^{2}=-\dfrac{1}{2}R^{\prime}(Q_{1}Q_{3}-P_{2}^{2})$

    4. (d)

$\bar{x}_{12}^{4}(\omega_{1}^{a}\omega_{2}^{a})^{3}=-\dfrac{1}{2}R^{\prime}(Q_{1}Q_{2}-P_{3}^{2})$

    5. (e)

$\dfrac{\bar{x}_{31}^{2}}{\bar{x}_{23}^{2}\bar{x}_{12}^{2}}(R^{a}_{1}\pi^{a}_{21})^{2}=\dfrac{1}{2}R^{\prime}P_{3}^{2},\quad\dfrac{\bar{x}_{31}^{2}}{\bar{x}_{23}^{2}\bar{x}_{12}^{2}}(R^{a}_{1}\pi^{a}_{23})^{2}=\dfrac{1}{2}R^{\prime}P_{3}^{2},\\\
\quad\dfrac{\bar{x}_{23}^{2}}{\bar{x}_{12}^{2}\bar{x}_{31}^{2}}(R^{a}_{1}\pi^{a}_{12})^{2}=\dfrac{1}{2}R^{\prime}Q_{1}^{2},\\\
\dfrac{1}{\bar{x}_{13}^{2}}R_{1}^{a}\pi_{12}^{a}R_{1}^{b}\pi^{b}_{32}=-\dfrac{1}{2}Q_{1}P_{2},\quad\dfrac{1}{\bar{x}_{23}^{2}}R_{1}^{a}\pi_{21}^{a}R_{1}^{b}\pi^{b}_{31}=-\dfrac{1}{2}R^{\prime}P_{2}P_{3},\\\
\dfrac{1}{\bar{x}_{12}^{2}}R_{1}^{a}\pi_{23}^{a}R_{1}^{b}\pi^{b}_{13}=-\dfrac{1}{2}R^{\prime}Q_{1}P_{3}$

    6. (f)

$\dfrac{\bar{x}_{13}^{2}\bar{x}_{12}^{2}}{\bar{x}_{23}^{2}}(R^{a}_{1}\sigma^{a}_{21})^{2}=-\dfrac{1}{2}R^{\prime}(Q_{1}Q_{2}-P_{3}^{2}),\quad\dfrac{\bar{x}_{23}^{2}\bar{x}_{13}^{2}}{\bar{x}_{12}^{2}}(R^{a}_{1}\sigma^{a}_{23})^{2}=-\dfrac{1}{2}R^{\prime}(Q_{1}Q_{2}-P_{3}^{2}),\quad\\\
\dfrac{\bar{x}_{23}^{2}\bar{x}_{12}^{2}}{\bar{x}_{31}^{2}}(R^{a}_{1}\sigma^{a}_{12})^{2}=0$

    7. (g)

$\dfrac{\bar{x}_{13}^{2}\bar{x}_{12}^{2}}{\bar{x}_{23}^{2}}(R^{a}_{1}\omega^{a}_{1})^{2}=\dfrac{1}{2}R^{\prime}Q_{1}^{2},\quad\dfrac{\bar{x}_{12}^{2}\bar{x}_{23}^{2}}{\bar{x}_{13}^{2}}(R^{a}_{1}\omega^{a}_{2})^{2}=\dfrac{1}{2}R^{\prime}P_{3}^{2}$

    8. (h)

$(\pi_{12}^{a}\sigma_{21}^{a})^{2}=\dfrac{1}{2}R^{\prime}P_{3}^{2},\quad(\pi_{12}^{a}\sigma_{31}^{a})^{2}=\dfrac{1}{2}R^{\prime}P_{2}^{2},\quad\dfrac{\bar{x}_{23}^{4}}{\bar{x}_{13}^{4}}(\pi_{12}^{a}\sigma_{21}^{a})^{2}=\dfrac{1}{2}R^{\prime}P_{2}^{2},\\\
\dfrac{\bar{x}_{23}^{4}}{\bar{x}_{12}^{4}}(\pi_{12}^{a}\sigma_{21}^{a})^{2}=\dfrac{1}{2}R^{\prime}P_{3}^{2}$

Note that other relations can be generated by cyclic permutation of $1,2,3$ in
the above list.

###  B.2 Parity odd relations

In this subsection we list the various parity-odd invariant structures built
from the covariant building blocks. These will all be expressed in terms of
our minimal set of invariants.

  1. 1.

${\cal O}(\Theta^{4})$:

    1. (a)

$\bar{x}^{2}_{12}\Pi^{ab}_{12}\Sigma^{ab}_{12}=0$

  1. 1.

${\cal O}(\lambda^{2}\Theta^{2})$:

    1. (a)

$\sqrt{\dfrac{\bar{x}_{12}^{2}\bar{x}_{31}^{2}}{\bar{x}_{23}^{2}}}R_{1}^{a}\omega_{1}^{a}=\dfrac{1}{2}T^{\prime}Q_{1},\quad\sqrt{\dfrac{\bar{x}_{13}^{2}}{\bar{x}_{12}^{2}\bar{x}_{23}^{2}}}R_{1}^{a}\pi_{21}^{a}=\dfrac{1}{2}T^{\prime}P_{3}$

    2. (b)

$\dfrac{\bar{x}_{23}^{2}}{\bar{x}_{13}^{2}}\pi_{12}^{a}\sigma_{12}^{a}=\dfrac{1}{2}T^{\prime}Q_{1},\quad\dfrac{\bar{x}_{23}^{2}}{\bar{x}_{13}^{2}}\pi_{12}^{a}\sigma_{32}^{a}=\dfrac{1}{2}T^{\prime}P_{2},\quad{\dfrac{\bar{x}_{13}^{2}}{\bar{x}_{23}^{2}\bar{x}_{23}^{2}}}\sigma_{12}^{a}\omega_{2}^{a}=\dfrac{1}{2}T^{\prime}P_{3}-\dfrac{i}{2}R^{\prime}S_{3}$

  2. 2.

${\cal O}(\lambda^{2}\Theta^{4})$:

    1. (a)

$\bar{x}_{12}^{2}R_{1}^{a}\Sigma_{12}^{ab}R_{2}^{b}=-\dfrac{1}{2}R^{\prime}S_{3},\quad\bar{x}_{12}^{4}\omega_{1}^{a}\Sigma_{12}^{ab}\omega_{2}^{b}=-\dfrac{1}{2}R^{\prime}S_{3},\quad\pi_{21}^{a}\Sigma_{12}^{ab}\pi_{12}^{b}=\dfrac{1}{2}R^{\prime}S_{3},\\\
\dfrac{\bar{x}_{23}^{2}\bar{x}_{13}^{2}}{\bar{x}_{12}^{4}}\pi_{31}^{a}\Sigma_{12}^{ab}\pi_{32}^{b}=0,\quad\dfrac{\bar{x}_{12}^{2}}{\bar{x}_{23}^{4}}\pi_{21}^{a}\Sigma_{12}^{ab}\pi_{32}^{b}=-\dfrac{1}{2}R^{\prime}S_{1},\quad\dfrac{\bar{x}_{12}^{2}}{\bar{x}_{13}^{4}}\pi_{31}^{a}\Sigma_{12}^{ab}\pi_{12}^{b}=-\dfrac{1}{2}R^{\prime}S_{2}$

    2. (b)

$\sqrt{\dfrac{\bar{x}_{23}^{2}}{\bar{x}_{13}^{2}\bar{x}_{12}^{2}}}R_{1}^{a}\Pi_{12}^{ab}\pi_{12}^{b}=0,\quad\sqrt{\dfrac{\bar{x}_{12}^{2}}{\bar{x}_{13}^{2}\bar{x}_{32}^{2}}}R_{1}^{a}\Pi_{12}^{ab}\pi_{32}^{b}=\dfrac{1}{2}R^{\prime}S_{2},\\\
\sqrt{\dfrac{\bar{x}_{12}^{2}\bar{x}_{23}^{2}}{\bar{x}_{13}^{2}}}R_{1}^{a}\Pi_{12}^{ab}\omega_{2}^{b}=-\dfrac{1}{2}R^{\prime}S_{3},\quad\pi_{21}^{a}\Pi_{12}^{ab}\sigma_{12}^{b}=\dfrac{1}{2}R^{\prime}S_{3},\quad\pi_{21}^{a}\Pi_{12}^{ab}\sigma_{32}^{b}=-\dfrac{1}{2}R^{\prime}S_{1},\\\
\dfrac{\bar{x}_{12}^{2}}{\bar{x}_{13}^{2}}\pi_{31}^{a}\Pi_{12}^{ab}\sigma_{12}^{b}=\dfrac{1}{2}R^{\prime}S_{3},\quad\bar{x}_{12}^{2}\omega_{1}^{a}\Pi_{12}^{ab}\sigma_{12}^{a}=0$

    3. (c)

${\dfrac{\bar{x}_{12}^{2}\bar{x}_{13}^{4}}{\bar{x}_{23}^{4}}}\pi_{21}^{a}\Omega_{1}^{ab}\sigma_{21}=0,\quad{\dfrac{\bar{x}_{13}^{2}\bar{x}_{12}^{4}}{\bar{x}_{23}^{4}}}\pi_{31}^{a}\Omega_{1}^{ab}\sigma_{21}=-\dfrac{1}{2}R^{\prime}S_{1},\quad\dfrac{\bar{x}_{12}^{4}\bar{x}_{13}^{4}}{\bar{x}_{23}^{4}}\omega_{1}^{a}\Omega_{1}^{ab}\sigma_{21}^{a}=\dfrac{1}{2}R^{\prime}S_{3}$

  3. 3.

${\cal O}(\lambda^{4}\Theta^{4})$:

    1. (a)

$\sqrt{\dfrac{\bar{x}_{23}^{2}}{\bar{x}_{12}^{2}\bar{x}_{31}^{2}}}R_{1}^{a}R_{2}^{a}R_{1}^{b}\pi_{12}^{b}=\dfrac{1}{2}R^{\prime}S_{3}Q_{1},\quad\sqrt{\dfrac{\bar{x}_{12}^{2}}{\bar{x}_{23}^{2}\bar{x}_{31}^{2}}}R_{1}^{a}R_{2}^{a}R_{1}^{b}\pi_{32}^{b}=-\dfrac{1}{2}R^{\prime}S_{3}P_{2},\\\
\sqrt{\dfrac{\bar{x}_{12}^{2}\bar{x}_{23}^{2}}{\bar{x}_{31}^{2}}}R_{1}^{a}R_{2}^{a}R_{1}^{b}\omega_{2}^{b}=\dfrac{1}{2}R^{\prime}S_{3}P_{3}$

    2. (b)

$\dfrac{1}{\bar{x}_{12}^{2}}\pi_{21}^{a}\pi_{12}^{a}\pi_{21}^{b}\sigma_{12}^{b}=-\dfrac{1}{2}R^{\prime}S_{3}P_{3},\quad\dfrac{\bar{x}_{ik}^{4}}{\bar{x}_{23}^{4}}\pi_{21}^{a}\sigma_{21}^{a}\pi_{21}^{b}\omega_{1}^{b}=-\dfrac{1}{2}R^{\prime}Q_{2}S_{3}$

Again, we can obtain other relations by cyclically permuting $1,2,3$ above.

###  B.3 A few more relations

Using the two relations (41) and (3.2), other relations with higher
homogeneity in $\lambda^{\prime}s$ can be generated. For example, using Eq.
(3.2), one gets the ${\cal O}(\lambda_{1}^{2}\lambda_{2}^{2}\lambda_{3}^{2})$
fermionic relation

$P_{1}P_{2}\bar{R}_{3}+P_{2}P_{3}\bar{R}_{1}+P_{3}P_{1}\bar{R}_{2}+\frac{1}{2}\sum_{i}Q_{i}P_{i}\bar{R}_{i}-\frac{3i}{4}R^{\prime}P_{1}P_{2}P_{3}=0.$
(87)

More combinations of the relations yield

$\displaystyle\begin{aligned}
&\frac{i}{2}\sum_{i}P_{i}^{3}Q_{i}^{2}\bar{R}_{i}+\frac{i}{2}\sum_{\text{cyc}}P_{1}^{2}Q_{1}(P_{2}Q_{2}\bar{R}_{2}+P_{3}Q_{3}\bar{R}_{3})+2\sum_{\text{cyc}}P_{1}^{2}P_{2}^{2}Q_{1}Q_{2}+\sum_{i}P_{i}^{4}Q_{i}^{2}\\\
&\qquad\qquad-
Q_{1}Q_{2}Q_{3}\sum_{i}P_{i}^{2}Q_{i}+\frac{1}{4}R^{\prime}P_{1}P_{2}P_{3}\sum_{i}P_{i}^{2}Q_{i}-2P_{1}P_{2}P_{3}\sum_{i}P_{i}^{2}Q_{i}=0\end{aligned}$
(88) $\displaystyle\begin{aligned} &-\frac{i}{4}\sum
P_{i}^{3}Q_{i}^{2}\bar{R}_{i}-\frac{1}{2}\sum_{\text{cyc}}P_{1}^{2}P_{2}^{2}Q_{1}Q_{2}+\frac{1}{4}Q_{1}Q_{2}Q_{3}\sum_{i}P_{i}^{2}Q_{i}-\sum_{i}P_{i}^{4}Q_{i}^{2}+P_{1}^{2}P_{2}^{2}P_{3}^{2}\\\
&\qquad\qquad\qquad+\frac{1}{2}Q_{1}Q_{2}Q_{3}P_{1}P_{2}P_{3}-\frac{1}{8}P_{1}^{2}P_{2}^{2}P_{3}^{2}R^{\prime}=0\end{aligned}$
(89)

Also, we can have purely fermionic relations, since the product of
$R^{\prime}$ with any other fermionic invariant vanishes,

$\displaystyle\left(2P_{1}P_{2}P_{3}-\sum_{i}P_{i}^{2}Q_{i}+Q_{1}Q_{2}Q_{3}\right)R^{\prime}=0$
(90)
$\displaystyle\left(2\sum_{\text{cyc}}P_{1}^{2}P_{2}^{2}Q_{1}Q_{2}-Q_{1}Q_{2}Q_{3}\sum_{i}P_{i}^{2}Q_{i}-2P_{1}P_{2}P_{3}\sum_{i}P_{i}^{2}Q_{i}+\sum_{i}P_{i}^{4}Q_{i}^{2}\right)R^{\prime}=0$
(91)

and so on.

Higher order parity-odd relations can also be generated. For example, at
${\cal O}(\lambda_{1}^{4}\lambda_{2}^{4}\lambda_{3}^{4})$ we have:

$\displaystyle\left(\frac{1}{2}\sum_{\text{cyc}}Q_{1}P_{1}^{2}(P_{2}Q_{2}S_{2}+P_{3}Q_{3}S_{3})-\frac{1}{2}\sum_{i}P_{i}^{3}Q_{i}^{2}S_{i}+P_{1}P_{2}P_{3}\sum_{i}P_{i}Q_{i}S_{i}\right)R^{\prime}=0$
(92)

These relations are utilised in Section 5 where we write the linearly
independent parity-even structures for various 3-point functions.

##  Appendix C Calculational details for free theory correlators

In this appendix we provide some calculation details for results quoted in
Section 6. In particular, we will explain through two examples how one
computes current correlators in the free-field theory using Wick contractions
of the chiral scalar superfield. For ${\cal N}=2$, the basic superfields are
(anti-)chiral,

$\displaystyle\bar{D}_{\alpha}\Phi=D_{\alpha}\bar{\Phi}=0$ (93)

and obey the following free field equations of motion,

$\displaystyle
D^{\alpha}D_{\alpha}\Phi=\bar{D}^{\alpha}\bar{D}_{\alpha}\bar{\Phi}=0\,\,.$
(94)

Solving the above, we get the super-multiplet for the (anti-)chiral
superfields to have the form

$\displaystyle\Phi$
$\displaystyle=\phi+\theta\psi-\frac{i}{2}\theta\gamma^{\mu}\bar{\theta}\partial_{\mu}\phi$
(95) $\displaystyle\bar{\Phi}$
$\displaystyle=\bar{\phi}+\bar{\theta}\psi^{*}+\frac{i}{2}\theta\gamma^{\mu}\bar{\theta}\partial_{\mu}\phi\,\,.$
(96)

Here $\phi$ and $\psi$ satisfy, respectively, the free Klein-Gordon and Dirac
equations,
($\theta\gamma^{\mu}\bar{\theta}\equiv\theta^{\alpha}(\gamma^{\mu})_{\alpha}^{\
\beta}\bar{\theta}_{\beta}$).

Next, one can construct the 2-point function

$\displaystyle\langle\bar{\Phi}_{1}\Phi_{2}\rangle=\frac{1}{y_{12}}$ (97)

where $\Phi_{i}$ is the superfield at superspace location
$(x_{i},\theta_{i})$, while $y_{12}$ is built from $\bar{x}_{12}$ [11],

$\displaystyle y_{12}^{2}=\bar{x}_{12}^{2}v_{12},\quad
v_{12}=\frac{1}{2}\left((V_{12})^{aa}+i\epsilon_{ab}(V_{12})^{ab}\right)$ (98)

where we’ve used the 2-point structure
$V_{12}^{ab}=\delta^{ab}+i\theta_{12}^{a\alpha}(X_{12+}^{-1})_{\alpha}^{\
\beta}\theta_{12\beta}^{b}$.

This form of the 2-point correlator can be explicitly checked by expanding the
superfields and using the expressions

$\displaystyle\langle\bar{\phi}_{1}\phi_{2}\rangle=\langle\phi_{1}\bar{\phi}_{2}\rangle$
$\displaystyle=\frac{1}{x_{12}}$ (99)
$\displaystyle\langle\psi_{1\alpha}^{*}\psi_{2\beta}\rangle=-\langle\psi_{2\alpha}\psi_{1\beta}^{*}\rangle$
$\displaystyle=i\frac{(X_{12})_{\alpha\beta}}{x_{12}^{3}}$ (100)

###  C.1 Computation of $\langle J_{0}J_{0}J_{0}\rangle$

The spin-0 conserved supercurrent is

$\displaystyle J_{0}=\bar{\Phi}\Phi$ (101)

The correlator $\langle J_{0}J_{0}J_{0}\rangle$ can be written as

$\displaystyle\langle
J_{0}J_{0}J_{0}\rangle=\left\langle(\bar{\Phi}_{1}\Phi_{1})(\bar{\Phi}_{2}\Phi_{2})(\bar{\Phi}_{3}\Phi_{3})\right\rangle$
(102)

Since this is a free theory correlator it can be computed by Wick contractions
on the the right hand side,

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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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to0.4pt{\vbox to14.22pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{}{}{}{}}{}{{}{}}{}{{}}{}{
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{}{}{}\pgfsys@moveto{0.0pt}{6.65088pt}\pgfsys@lineto{0.0pt}{20.47318pt}\pgfsys@lineto{0.0pt}{20.23708pt}\pgfsys@lineto{0.0pt}{6.88698pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\vbox
to22.50002pt{}{}\rangle\\\
&=\left(\frac{1}{y_{13}}\right)\left(\frac{1}{y_{21}}\right)\left(\frac{1}{y_{32}}\right)+\left(\frac{1}{y_{12}}\right)\left(\frac{1}{y_{31}}\right)\left(\frac{1}{y_{23}}\right)\end{aligned}$
(103)

We find that the result of this computation is equal to a linear combination
of the parity even invariant structures. Since there is no spin in this
example, the only one allowed is $R^{\prime}$,

$\displaystyle\langle
J_{0}J_{0}J_{0}\rangle=\frac{-2}{\bar{x}_{12}\bar{x}_{23}\bar{x}_{31}}\left(1+\frac{1}{16}R^{\prime}\right)$
(104)

###  C.2 Computation of $\langle J_{1}J_{1}J_{0}\rangle$

The spin-0 and spin-1 conserved supercurrents are,

$\displaystyle J_{0}=\bar{\Phi}\Phi\ ,\qquad
J_{1}=\bar{\Phi}(\partial\Phi)-2\bar{D}\bar{\Phi}D\Phi-(\partial\bar{\Phi})\Phi$
(105)

Thus, the 3-point correlator

$\displaystyle\begin{aligned} \langle
J_{1}J_{1}J_{0}\rangle&=\left\langle\left(\bar{\Phi}_{1}(\partial_{1}\Phi_{1})-2\bar{D}_{1}\bar{\Phi}_{1}D_{1}\Phi_{1}-(\partial_{1}\bar{\Phi}_{1})\Phi_{1}\right)\right.\\\
&\left.\qquad\qquad\times\left(\bar{\Phi}_{2}(\partial_{2}\Phi_{2})-2\bar{D}_{2}\bar{\Phi}_{2}D_{2}\Phi_{2}-(\partial_{2}\bar{\Phi}_{2})\Phi_{2}\right)\left(\bar{\Phi}_{3}\Phi_{3}\right)\right\rangle\end{aligned}$
(106)

can be calculated by expanding and performing all possible Wick contractions.

As an example, one of the nine Wick contraction terms in the above expansion
is

$\displaystyle\begin{aligned}
\left\langle\bar{\Phi}_{1}(\partial_{1}\Phi_{1})\bar{\Phi}_{2}(\partial_{2}\Phi_{2})\bar{\Phi}_{3}\Phi_{3}\right\rangle&=\Big{\langle}\leavevmode\hbox
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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to0.4pt{\vbox to14.22pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{}{}{}{}}{}{{}{}}{}{{}}{}{
{}{}{}{}{}}{}{{}{}}{{}{}} {}{}{ {}{}{}{}{}}{}{{}{}}{{}{}} {}{}{
{}{}{}{}{}}{}{{}{}}
{}{}{}\pgfsys@moveto{0.0pt}{6.65088pt}\pgfsys@lineto{0.0pt}{20.47318pt}\pgfsys@lineto{0.0pt}{20.23708pt}\pgfsys@lineto{0.0pt}{6.88698pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\vbox
to22.50002pt{}{}\Big{\rangle}\\\
&=\left(\frac{1}{y_{13}}\right)\left(\partial_{1}\frac{1}{y_{21}}\right)\left(\partial_{2}\frac{1}{y_{32}}\right)+\left(\partial_{2}\frac{1}{y_{12}}\right)\left(\partial_{1}\frac{1}{y_{31}}\right)\left(\frac{1}{y_{23}}\right)\end{aligned}$
(107)

Computing and collecting all the terms, the Wick contractions give as
expected, a linear combination of all the linearly indepenedent parity-even
invariants with homogeneity $\lambda_{1}^{2}\lambda_{2}^{2}$,

$\displaystyle\langle
J_{1}J_{1}J_{0}\rangle=\frac{-2}{\bar{x}_{12}\bar{x}_{23}\bar{x}_{31}}\left[Q_{1}Q_{2}\left(1+\frac{9}{16}R^{\prime}\right)+P_{3}^{2}\left(2+\frac{3}{8}R^{\prime}\right)+2iP_{3}\bar{R}_{3}\right]$
(108)

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