Relation between parity-even and parity-odd CFT correlation functions in three dimensions

In this paper we relate the parity-odd part of two and three point correlation functions in theories with exactly conserved or weakly broken higher spin symmetries to the parity-even part which can be computed from free theories. We also comment on higher point functions. The well known connection of CFT correlation functions with de-Sitter amplitudes in one higher dimension implies a relation between parity-even and parity-odd amplitudes calculated using non-minimal interactions such as ${\mathcal W}^3$ and ${\mathcal W}^2 {\widetilde {\mathcal W}}$. In the flat-space limit this implies a relation between parity-even and parity-odd parts of flat-space scattering amplitudes.


Introduction
Three dimensional conformal field theory finds diverse applications in topics from cosmology to condensed matter physics [1][2][3][4][5][6][7]. They also provide interesting examples of duality. For example, the ABJM super-conformal field theory is dual to M -theory on AdS 4 × S 7 the parity-even and the parity-odd correlators in position and momentum spaces as well as in spinor-helicity variables.
We will see that for correlation functions of exactly conserved currents with spins satisfying triangle inequality, the parity-odd contribution can be obtained from the homogeneous part of the parity-even contribution. One can think of this relation as a map from parity-even correlators to parity-odd correlators. For the non-homogeneous part, there is no analogous parity-odd contribution. When the triangle inequality is violated, there is no parity-odd contribution to the correlation function of exactly conserved currents. However, for weakly broken higher-spin theories, it turns out that such correlation functions can also have a parity-odd contribution. Interestingly, when the triangle inequality is violated the only contribution to the parity-odd and the parity-even parts is the non-homogeneous contribution. In this case as well, one can find out a relation between the parity-even and the parity-odd parts of the correlation function.
Rest of the paper is structured as follows. In Section 2 we discuss the parity-even and the parity-odd projection operators that we use in this paper and the relation between them. In Section 3 we discuss the relation between the parity-even and the parity-odd two-point functions of conserved currents of arbitrary spin s. In Section 4 we describe the homogeneous and non-homogeneous contributions to 3-point functions comprising spinning operators in position space, momentum space and in spinor-helicity variables. We also describe these contributions to dS 4 amplitudes. In Section 5 we describe the relation between the parity-even and the parity-odd homogeneous parts of three-point correlation functions of conserved currents, in spinor-helicity variables and in position and momentum spaces. To describe the relation in momentum space, we rewrite momentum space expressions using a few invariants such that the relation becomes manifest. We also extend our analysis to four-point functions. We then look at similar relations for three-point dS 4 amplitudes. In Section 6 we use higher spin equations associated to weakly broken higher spin symmetry to derive the relation between parity-odd and parity-even correlators. We conclude with a summary and future directions of study in Section 7. In Appendix A we express three-point functions of conserved currents in terms of certain fundamental building blocks in momentum space. In Appendix B we collect some details of higher spin equations that relate the parity-odd and the parity-even parts of correlators comprising higher spin currents.

A relation between even and odd projectors
We observe the following relations between the parity-even and the parity-odd projectors defined above. For the spin-1 case, we have 1 k µαk π ν α (k) = χ µν (k), Similarly, for spin-2 projectors we have where (µ, ν) indicates symmetrisation with respect to µ and ν with a factor of 1/2. It is also convenient to state these relations in terms of transverse, null polarization vectors which satisfy k i · z i = 0 and z 2 i = 0. To do this, let us first write down the form of the projectors after contracting with these vectors z 1µ z 2ν π µν (k) = z 1 · z 2 z 1µ z 2ν χ µν (k) = 1 k z 1 z 2 k z 1µ z 1ν z 2ρ z 2σ Π µνρσ (k) = (z 1 · z 2 ) 2 z 1µ z 1ν z 2ρ z 2σ ∆ µνρσ (k) = z 1 z 2 k (z 1 · z 2 ) (2.7) For the spin-s case we have P even s = (z 1 · z 2 ) s , P odd We notice that the following substitution takes the parity-even projector to the parity-odd projector. More formally, the following operation takes the parity-even projector to the parity-odd projector.
3 While these are explicitly orthogonal to their momentum arguments they are different from usual projectors in the sense that they do not square to themselves. However, a contraction of three of these 'projectors' gives back the 'projector'.

Relation between parity-even and parity-odd two-point functions
In this section we establish the relation between the parity-odd and the parity-even parts of two-point correlation functions in a generic 3d CFT. We will concern ourselves with only the momentum dependent structures and ignore constants. We start with the 2-point function of the spin-1 current.

JJ
The parity-even and parity-odd parts of the two-point function of the spin-1 current are given by One can easily check that the following relations hold These relations can also be interpreted as the ones between parity-odd and parity-even projectors discussed in (2.5).

T T
The above relation can be extended to the T T 2-point function. The parity-odd and the parity-even 2-point functions are given by In this case, we need to symmetrize as follows This is because the right hand side is symmetric under (µ ↔ ν) by definition. Therefore, the left hand side must also be symmetric under this exchange. To generalize this to higher spin currents, it is convenient to use polarization vectors. The parity-even and the parity-odd two-point functions of spin-2 currents after contracting with polarization vectors are given by More generally where z s 1 is shorthand notation for z µ 1 1 · · · z µs . We see that the map in (2.10) transforms the parity-even two-point function to the parity-odd two-point function.

Relation between parity-even and parity-odd in spinor-helicity variables
All the statements made above can be seen manifestly in spinor-helicity variables in terms of which the 2-point function has a unique structure and the parity-even and parity-odd correlators differ only by a factor of i 4 . For the spin-s current we have We now turn our attention to three and higher point functions. To understand them, let us first explain a few important aspects of correlation functions. We also discuss the relation between parity-even and the parity-odd two-point functions in position space in a later section.

Generality : Homogeneous (h) vs Non-homogeneous (nh) parts of CFT correlators and amplitudes
In this section we discuss the distinction between the homogeneous and the non-homogeneous parts of correlation functions and amplitudes. This distinction will be useful in relating the parity-odd and parity-even parts of correlation functions.

h vs nh in CFT correlation functions
Any CFT correlation function can be separated into homogeneous and non-homogeneous parts, see [1] for a discussion in momentum space. In what follows we do this identification in momentum space, spinor-helicity variables and in position space.

Momentum space
The non-homogeneous part of the correlation function saturates the Ward-Takahashi (WT) identity, i.e. k 1µ 1 J µ 1 ···µs 1 (k 1 )J s 2 J s 3 · · · J sn h = 0 k 1µ 1 J µ 1 ···µs 1 (k 1 )J s 2 J s 3 · · · J sn nh = terms from the WT Identity (4.1) To be more concrete, let us consider the example of J µ J ν T ρσ . The correlator can be written as When dotted with external momentum, the local piece reproduces the WT identity. The transverse piece can as well be split into homogeneous and non-homogeneous pieces It is easiest to distinguish between homogeneous and non-homogeneous contributions in spinor-helicity variables. In spinor-helicity variables, the action of the special conformal generator on the homogeneous piece gives zero whereas on the non-homogeneous piece it gives the terms that appear in the WT identity. It is also the case that at the level of three-point functions, homogeneous and non-homogeneous contributions have different pole structures in E = k 1 + k 2 + k 3 . Generically the homogeneous contribution is always more singular. As an example, let us consider the parity-even part of the stress-tensor three-point function [30,49] in spinor-helicity variables where c 123 = k 1 k 2 k 3 and b 123 = (k 1 k 2 + k 2 k 3 + k 3 k 1 ). Note that c T comes from the parity-even two-point function of the stress tensor (3.8). The term proportional to c 1 is the homogeneous contribution whereas the term proportional to c T is the non-homogeneous contribution. It is clear that the pole structure in E of the homogeneous and non-homogeneous pieces are different. Let us also emphasize that the homogeneous piece contributes only to the − − − and + + + helicity components whereas the non-homogeneous piece contributes to all helicity components.

Position space
Let us now distinguish between the homogeneous and non-homogeneous contributions in position space. We consider two examples to illustrate the distinction.
The WT identity for correlation functions of the form J( which implies that the correlator has only a homogeneous part. In terms of certain conformal structures the parity-even and the parity-odd parts of the correlator can be expressed as follows [21] (4.9) For details of the notation see [21]. Let us consider another example T (x 1 )J(x 2 )J(x 3 ) . The correlator is given by [21] T ( where c j is the coefficient of the two-point function of J µ fixed by the WT identity. The term proportional to c 1 is homogeneous and the term proportional to c j is the non-homogeneous contribution. One can always add a homogeneous piece to the non-homogeneous piece of the correlator. Doing this, one obtains We observe that in the representation (4.11), the homogeneous and the non-homogeneous pieces can be obtained from each other by taking P 3 → −P 3 . One can write down a general polynomial for the parity-even homogeneous and nonhomogeneous contributions to correlation functions involving general conserved currents as follows As is clear we have It would be interesting to find out if there exists a similar relation between homogeneous and non-homogeneous contributions in momentum space. It is useful to note the following representation of three-point correlation functions in the free bosonic and free fermionic theories (4.14) which precisely matches the representation given in equations 15 and 16 of [50]. Let us note that (4.14) implies We can express the homogeneous and the non-homogeneous pieces in terms of the free boson and free fermion answers as follows A simple way to understand (4.16) is the following where in the last step we have identified the WT identity for the bosonic and fermionic theories. This identification requires us to identify the two-point function of boson and fermion. This also fixes the relative normalization. We emphasise that in this section we have considered correlators that satisfy triangle inequality (4.25). Outside the triangle inequality all the contributions are non-homogeneous and we discuss them in the later sections.

h vs nh in Amplitudes
We will now distinguish between the homogeneous and non-homogeneous contributions to amplitudes. As we shall see, the gravity amplitude automatically comes in a way which separates out the homogeneous and non-homogeneous contributions. For this purpose, let us consider the n-point amplitude M µ 1 ,µ 2 ···µ s 1 +s 2 ···sn (k 1 , k 2 · · · , k n ) of spinning particles with spins s 1 , s 2 · · · s n . The homogeneous and non-homogeneous parts of the amplitude can be defined as As an example, let us consider the three graviton dS 4 amplitude. It has three contributions. Two parity-even contributions come from W 3 and the Einstein gravity part, and a parityodd contribution comes from W 2 W. It can be easily checked that the contributions from W 3 and W 2 W are homogeneous whereas the Einstein gravity part is non-homogeneous. Let us check this explicitly. The contribution from W 3 is given by [21] where i are transverse polarization tensors. To analyse (4.18) we have to replace one of the polarization tensors with the momentum which implies that the W 3 contribution is homogeneous. It is easy to check that a similar conclusion holds for W 2 W. Let us now consider the contribution from Einstein gravity (EG) The analogue of (4.18) is Thus this contribution is non-homogeneous. It is interesting to point out that in gravity, there exists a natural distinction between the homogeneous and non-homogeneous parts.
In CFT, the homogeneous and non-homogeneous parts come together as we saw in the case of the free bosonic and free fermionic theories (4.14).
A useful gauge where amplitudes take a simple form is given by In this gauge the M W 3 structure reduces to The other interaction which gives rise to non-homogeneous contributions is the Yang-Mills term 5 whereas terms such as F 3 , F 2 F , φF 2 and φF F contribute to homogeneous amplitudes.

Summary of 3-point functions for exactly conserved currents
Let us consider correlation functions comprising conserved currents with spins s 1 , s 2 , s 3 such that they satisfy triangle inequality where i, j, k can be any of 1, 2, 3. The most general three-point function inside the triangle can be written as the sum of parity-even free boson and free fermion contributions and a parity-odd contribution which can also be written as where we used from (4.6) Correlation functions involving a scalar operator of scaling dimension ∆ can be written as Note that there is no homogeneous contribution to J s O ∆ O ∆ and no non- Let us now consider the case when the spins violate triangle inequality, i.e. say where for simplicity we have assumed s 1 > s 2 and s 2 ≥ s 3 . In such cases when the currents are exactly conserved there is no parity-odd contribution to the correlation function and the only contributions are non-homogeneous [52] 5 Relating parity-even and odd correlation function for exactly conserved currents In this section we work out the relation between the parity-even and the parity-odd parts of correlation functions in spinor-helicity variables, momentum space and in position space. The homogeneous part of correlation functions in spinor-helicity variables was discussed in [41]. In position space it is given by (4.16). In [41], momentum space expressions were also written down. However, to construct a map from parity-even to parity-odd, it is convenient to rewrite the expressions in a slightly different way which makes the map manifest. We will start our analysis using spinor-helicity variables.

Relation in spinor-helicity variables
Consider a general correlator of the form J s 1 J s 2 J s 3 where s 1 ≥ s 2 ≥ s 3 and s 2 + s 3 > s 1 .
In spinor-helicity variables, the homogeneous part of the correlator is given by The only other non-zero contribution to the homogeneous piece comes from the + + + helicity component which can be obtained by complex conjugating(5.1). The other helicity components have only non-homogeneous contribution [41]. It is interesting to note that the parity-even and the parity-odd parts of the homogeneous part of the correlation function are the same up to factors of i. To avoid clutter, we introduce the following notation

Relation in momentum space
The homogeneous part of any 3-point function of the form J s 1 J s 2 J s 3 can be written in terms of a finite number of building blocks as was observed in [41]. In [41] we had introduced Let us split correlators into two families, one satisfying s 1 + s 2 + s 3 = even and the other satisfying s 1 + s 2 + s 3 = odd.
The parity-even and the parity-odd homogeneous pieces of correlators involving a scalar operator are given by Correlation functions involving all spinning operators are given by Let us note that the results presented in (5.4) and (5.5) are significantly simpler than what appears in [41]. To reach these results we have made repeated use of degeneracy and Schouten identities. The details are worked out in Appendix A. Results in (5.4) and (5.5) can be symmetrized appropriately. However, it can be shown that all such symmetrized terms are related by degeneracy and Schouten identities to the expressions in (5.4) and (5.5).
Let us now consider the second family of correlators in which s 1 + s 2 + s 3 = odd.
When s 1 + s 2 + s 3 = odd for general spins s 1 , s 2 , s 3 , the correlator can be written as Relating parity-even to parity-odd From their expressions in (5.3) one can check that Furthermore R 123 and P 123 are related by We can see that (5.8) and (5.9) map parity-even correlators to parity-odd correlators. We see the following simple relation between the parity-even and the parity-odd results : even h → odd. (5.10) Let us note that for s 1 = 0, s 2 = 0, s 3 = 0 we could have chosen any z i · z j and replaced it with either 1 k i z i z j k i or 1 k j z i z j k j and it would have given us the map (5.10). We also note that the map in (5.10) is exactly the same as the one observed at the level of two-point functions in (2.10).
Using Todorov operator one can rewrite the relation in (5.10) as This maps the parity-even homogeneous contribution in momentum space to the parity-odd contribution One can check that this reduces to the relation one has in spinor-helicity variables.

Summary of map
Using all the results discussed above one can show the following in general 7 In the above we used the first spin for mapping. However note that the mapping works out exactly the same way for the second and the third spins as well.

Relation in position space
The relation between the parity-even and the parity-odd parts of the correlation function in momentum space can easily be Fourier transformed to get the relations in position space. One can define a kernel which maps the parity-even contribution to the parity-odd contribution. The inverse kernel also exists which maps the parity-odd contribution to the parity-even contribution. Let us start with the two-point function of the spin-1 current The parity-odd piece is a contact term. This is true for the parity-odd part of the two-point function of any spin-s conserved current.
For the spin-1 current it is easy to check that the parity-odd and the parity-even parts are related as

Three-point function
We now turn our attention to three-point functions. A Fourier transform of the relation in momentum space gives where we have suppressed indices of the second and third operators. There are other useful representations as well such as It is possible to invert the relation and express the parity-even result in terms of the parityodd result as follows which can also be rewritten as (5.20) Using (4.12), we can symbolically write Let us denote the kernel by K. We then have We shall derive this relation in section 6. As a concrete example one can consider J µ J ν O ∆ given in (4.9). Using the star triangle relation 8 , it is straightforward to show that

Relation between parity-even and parity-odd three-point dS 4 amplitude
It is known that CFT correlation functions and dS 4 amplitudes are related to each other [1]. This readily implies that the parity-even and the parity-odd parts of the dS 4 amplitudes are related just as they are in CFT correlators. To be explicit in the computations, we will consider the three-point graviton amplitude. The parity-even parts of the correlator are given by (4.19) and (4.21). The parity-odd part of the amplitude is We choose a gauge in which k µ = (| k|, k i ) and z µ = (0, z). It is easy to check that the contributions from W 3 and W 2 W are related to each other by Similar relations hold between contributions from φW W and φW 2 , φF F and φF 2 , and F 2 F and F 3 . Similar statements can be made for interactions involving higher-spins. We also notice that there is no such relation involving the amplitude given by Einstein gravity part (4.21).
In this section we first argue abstractly that if there is a parity-even homogeneous solution to the conformal Ward identity then there also exists a parity-odd homogeneous solution. To show this let us consider the simplest example of T OOO . 8 The star triangle relation is given by

T OOO even
The ansatz for the transverse-traceless part of the parity-even correlator is given by The WT identity for the correlator is [36] We separate the correlation function into homogeneous and non-homogeneous parts T OOO = T OOO h,even + T OOO nh,even . (5.29) Let us now convert the ansatz in (5.27) to spinor-helicity variables The positive helicity component is obtained by complex conjugation. The conformal Ward identity in spinor-helicity variables is given by . Plugging (5.29) in (5.31) we see that the homogeneous piece satisfies In general it is not known how to solve this equation in terms of an arbitrary function of conformal cross-ratios. However, for our purpose, it is sufficient to know just the equation (5.32).

T OOO odd
Let us now write down ansatz for the parity-odd contribution. It can be written as The WT identity for the parity-odd part is trivial Thus we see that the parity-odd part has only homogeneous contribution Transforming the ansatz (5.33) to spinor-helicity variables we obtain T − OOO odd = B 1 12 2 2 1 2 + B 2 13 2 3 1 2 + B 3 14 2 4 1 2 (5.36) Let us note that even though the parity-odd ansatz (5.33) looks completely different from the parity-even ansatz(5.27), in spinor-helicity variables they become identical. The conformal Ward identity is again given by It is easy to show that if {A 1 , A 2 , A 3 } is a solution to (5.32), one of the solutions of (5.37) As in the case of three-point functions, we find a map between the parity-even and the parity-odd homogeneous parts of the correlator as follows One can easily generalize this discussion to correlation functions involving more than one spinning operator as well as to arbitrary n-point functions. We conclude that for an arbitrary n-point function, we can construct a parity-odd homogeneous solution using a parityeven homogeneous solution by doing the epsilon-transformation we discussed above.

A derivation using weakly broken higher spin symmetry
In this section we make use of weakly broken higher spin symmetry to derive relations presented in the previous sections between parity-even and parity-odd correlation functions. Until now we considered only exactly conserved currents. We will now consider theories with weakly broken higher-spin symmetry at large N . We show that for two-point functions of such currents the relation between even and odd correlators continues to hold. For three-point functions when the spins satisfy triangle inequality, the same relation holds between the even and odd correlators. For weakly broken higher spin currents, parity-odd contribution to correlators exist even when triangle inequality is violated. We show that the relation between even and odd continues to hold in this case.

Three-point function with slightly broken higher spin current
In theories with weakly broken higher-spin symmetry even when triangle inequality is violated (4.30), there is one parity-odd contribution in (4.26), i.e.
J s 1 J s 2 J s 3 odd = 0 even outside the triangle.
We also have It turns out that all the contribution to the correlator when the spins violate triangle inequality is non-homogeneous [52]. Let us also note that when the spins satisfy triangle inequality, there is no change in the structure of the correlation function.
A derivation of the parity-even-odd relation In this section we use slightly broken higher symmetry [25] to derive the relation between the parity-even and the parity-odd part of correlation functions 9 . Let us start with correlation functions of the form T µν J s 1 J s 2 where spins s 1 and s 2 are arbitrary and the currents need not be conserved. In the following we also use the notation where z i are polarization vectors such that Let us consider the spin-4 current J 4 that obeys the following non-conservation equation in the quasi-fermionic theory [25] ∂ σ J σ z 1 z 1 z 1 = 80 7 where we have used the notation z µ ∂ µ = ∂ z . In (6.5), T denotes the trace of the stress tensor 10 , O denotes the scalar operator with scaling dimension ∆ = 2 + O( 1 N ) and λ is the coupling constant introduced in [25].
The charge associated to J 4 labelled Q 4 has the following action on O Let us now consider the action of Q 4 on the three-point correlator OJ s 1 J s 2 . It leads to the following higher spin equation in position space Upon utilizing the algebra in (6.6) and the current equation (6.5) we get In this section we closely follow Appendix D of [25]. 10 Even though the stress-tensor is traceless, it can lead to a non-trivial trace WT identity.
where by standard terms we mean correlators with a single insertion of the scalar operator O. After a large N factorisation we obtain the following We now express the three-point functions in the quasi-fermionic theory as [25] OO Let us now look at terms with the pole structure 1 1+ λ 2 . This gives us the following relation Let us now look at terms with the pole structure λ 1+ λ 2 . This gives us the following relation Now we make use of the Todorov operator [61] along with (4.17) and the fact that the trace WT identity is also identical for the free boson and the free fermion. We obtain the following two equations from the above two In Fourier space, these relations take the following form and We now use Π νρ αβ (k 1 ) to project the RHS and the LHS of the above equations. Using the following we obtain Similarly we also obtain This relation translates to the following in position space In (6.19) one can as well take the k 1 on the LHS to the RHS and upon converting the resulting equation to position space one obtains This is precisely the relation between odd and even correlators that we derived in (5.21). Note that we could have argued this relation directly from (6.14) by writing down the explicit structure in position space and realizing that (6.21) indeed holds. As a simple example we could have s 1 = s 2 = 2 which corresponds to the T T T case. Even though we have used large-N techniques, here we have derived results that are perfectly valid even at finite N as was discussed in previous sections. Let us now consider the case when the spins of the operators in T J s 1 J s 2 are such that they violate triangle inequality, i.e. s 1 > s 2 + 2. In this case the correlators that appear in the higher spin equation are purely non-homogeneous. However, the above relation between parity-odd and parity-even correlators will continue to hold. Our analysis can be easily extended to operators with higher spin. See Appendix B for details.
Note that one can obtain similar relations by working in the quasi-bosonic theory.

J s T O
Let us consider a correlator of the kind J s T O such that the spins violate triangle inequality, i.e. s > 2. We will now derive the relation between the parity-odd part of J s T O and the same correlator in free theories where it is parity-even. To do so we consider the action of Q 4 on the three-point correlator J s OO . It leads to the following higher spin equation : Following similar steps as above we obtain In position space this relation takes the form This is precisely the relation we obtained in Section 5.3. The analysis can also be generalised to J s 1 J s 2 O of which a special case is J s J s O .

T OOO
In this section we obtain the relation between the parity-odd and the parity-even parts of four-point functions. Let us consider T OOO . To obtain this correlator we consider the action of Q 4 on the four-point function of scalar operators OOOO . Following the steps in the previous sections and making use of the following [62] (a similar identification of the quasi-fermionic correlator has been made in the position and Mellin spaces in [53]) and [55] respectively) We note that T OOO FF is parity-odd and T OOO CB is parity-even and they are homogeneous 11 : k µ 1 T µν OOO F F,CB = 0 (6.26) From the higher spin equations we obtain In Fourier space these equations take the following form and Thus we obtain a relation between the parity-even and the parity-odd parts of T OOO 12 .

Summary and future directions
In this paper we explicitly showed that for exactly conserved currents, one can relate the parity-odd part of the correlation function to the parity-even homogeneous contribution to the correlator 13 . We wrote down an explicit relation between the two in position space, momentum space and in spinor-helicity variables. The fact that, for exactly conserved currents, there does not exist a parity-odd non-homogeneous contribution can be understood by a simple analysis of the WT identity. However, for weakly broken higher spin theories, when the spins violate triangle inequality there are only non-homogeneous contributions. It turns out that in such cases, a relation exists between the even and the odd non-homogeneous pieces.
We explicitly wrote down a relation between the parity-even and the parity-odd parts of amplitudes in one higher dimension. We found that in four dimensions parity-even interactions arising from non-minimal coupling map to parity-odd interactions arising from non-minimal coupling. We recover the know fact that gravity naturally splits up the correlation into homogeneous and non-homogeneous part [21]. We also briefly discussed the case of four-point functions.
It would be interesting to generalize our results in the following ways. Firstly, it would be interesting to verify the relation between the parity-even and the parity-odd four-point homogeneous correlation functions. It would be interesting to check if similar relations hold for the non-homogeneous parts of four-point functions. One easy way would be to see if the parity-even and the parity-odd WT identities could be mapped to each other in spinorhelicity variables. It would also be interesting to verify if similar relations exist between the parity-even and the parity-odd homogeneous parts of 4 photon and 4 graviton amplitudes [63].
Another interesting direction would be to explore how the relation between even and odd correlators comes about in perturbation theory when we compute correlation functions in Chern-Simons matter theories [45,46,54,64]. The parity-odd contribution comes from odd loop orders whereas parity-even contribution comes from even loop orders. This implies that the relation between the even and odd parts of correlators gives a relation between the even and odd loop order calculations. The parity-odd contribution leads to anyonic nature exhibited by Chern-Simons matter theories [47,65,66]. A related story is that of flux 12 This relation is weaker than the one we discussed in (6.30) because this holds true only with the permutations taken into account. In position space similar relations were explicitly understood in [53]. 13 There is no analogue of parity-odd non-homogeneous correlator which can be obtained from a parityeven non-homogeneous correlator. attachment [4,5]. It would be interesting to explore these aspects of Chern-Simons matter theories in the light of the relation between the parity-odd and the parity-even correlators that we obtained in this paper. It would be interesting to find out if similar relation holds at finite temperature. This would lead to a relation between viscosity, conductivity and Hall viscosity and Hall conductivity computed in [3,67].

Acknowledgments
The work of SJ and RRJ is supported by the Ramanujan Fellowship. We acknowledge our debt to the people of India for their steady support of research in basic sciences. We thank S. D. Chowdhury, L. Janagal, S. Sinha and E. Skvortsov for useful discussions and correspondence. We thank A. Mehta and A. Suresh for collaboration at an early stage of this work.

A Three-point function in momentum space
In this appendix we give details of the three-point functions used in the main text. For our purpose, it is sufficient to concentrate on the homogeneous part of the correlation function. The homogeneous part of any 3-point function of the form J s 1 J s 2 J s 3 can be written in terms of a finite number of building blocks which are given below [41].
It was shown in [41], for this class of correlators we only require Q ij and S ij . Let us consider J s 1 J s 2 J s 3 such that s 1 ≥ s 2 ≥ s 3 , s 1 ≤ s 2 + s 3 . For this case we have where b ij = k i k j and c 123 = k 1 k 2 k 3 . For scalar operator with ∆ = 1 we just need to do a shadow transform to (A.6). For generic scalar operator dimension ∆ can also be obtained easily. Now we use following interesting identities We can easily check that where J 2 = 4(k 2 1 k 2 2 − (k 1 · k 2 ) 2 . The last line of (A.9) is zero because RHS is exactly degeneracy factor [33]. Let us emphasize the fact that This is because the way the degeneracy works is only with Q 2 ij . Using this identity we can rewrite (A.5) as follows Remarkably we can use the following identity This turns (A.11) into One can symmetrize above in 1, 2, 3 however they are just related by degeneracy 14 . One can easily check (A.13) to be correct by going to spinor-helicity variables and matching it with (5.1). For parity-odd case there exists another useful representation. For this we use following identities Using this identities we can write the parity-odd result as follows We can also write the odd part as There is a simple interpretation of the formula in (A.13). The factors Qij and Sij are exactly the same as JJO2 even and JJO2 odd respectively. In spinor-helicity variables Q −+ = Q +− = S −+ = S +− = 0 and the only non-zero components are ++ and −−. Since we want only the − − − and + + + components of the homogeneous piece to be non-zero, we require at least two factors of Q or one factor Q and one factor of S. This will ensure that any mixed helicity component is zero. We can then appropriately multiply zi · zj to account for the spin.

B Higher-spin correlators
Let us consider the correlator J 4 J s 1 J s 2 . We consider the action of the charge Q 6 associated to the spin-6 current J 6 on the correlator OJ s 1 J s 2 . To do so we make use of the following algebra and the current equation The higher spin equation that arises from the action of Q 6 on OJ s 1 J s 2 is One can make use of the higher spin equation (6.9) to get rid of correlators involving the stress-tensor from the above equation. This leaves us with the following