An infinite family of $w_{1+\infty}$ invariant theories on the celestial sphere

In this note we determine the graviton-graviton OPE and the null states in any $w_{1+\infty}$ symmetric theory on the celestial sphere. Our analysis shows that there exists a discrete \textit{infinite} family of such theories. The MHV-sector and the quantum self dual gravity are two members of this infinite family. Although the Bulk Lagrangian description of this family of theories is not currently known to us, the graviton scattering amplitudes in these theories are heavily constrained due to the existence of null states. Presumably they are exactly solvable in the same way as the minimal models of $2$-D CFT.

Celestial holography [1] is an attempt to formulate a dual theory of quantum gravity in asymptotically flat space time. When the bulk is (3 + 1) dimensional, the dual theory is conjectured to be two dimensional and it lives on the celestial sphere. The correlation functions of this theory are given by the Mellin transform of the bulk S-matrix elements [2][3][4] and an important question is how to compute them in the dual formulation. Now in the absence of a Lagrangian description one can resort to the Bootstrap technique and the job is sometimes simplified if the theory has a large number of symmetries. For example, one can solve the minimal models in 2-D CFT by using the representation theory of Virasoro and other current algebras even when the Lagrangian description is not known. In celestial holography the two dimensional dual theory also has various (infinite dimensional) current algebra symmetries  and it is expected that the bootstrap approach should simplify considerably in this case 1 .
In this paper we consider dual theories which are invariant under the w 1+∞ symmetry 2 [19-22, 25, 26]. We determine the graviton-graviton OPE and the null states in such theories. So far two examples of such theories were known. One of them is the MHV-sector [16,17] and the other one is the quantum self dual gravity [22]. However, we find that there is a discrete infinite family of such theories and the MHV-sector and the quantum self dual gravity are two members of this family. Bulk description of this family of theories is not known to us and we leave this to future work.

II. GENERAL STRUCTURE OF w-INVARIANT OPE
The general structure of the w-invariant OPE between two positive helicity outgoing graviton primaries with weights ∆ 1 and ∆ 2 can be written as, where the first line contains the universal terms [19] having simple pole in z 12 . The second line contains the non-singular terms in z 12 andz 12 and they are linear combinations of the w descendants of a positive helicity graviton primary, denoted byÕ p,q i . The positive integer n p,q may be finite or infinite. Our goal is to determine the operatorsÕ p,q i and the OPE 1 Please see [29] for a discussion of conformal bootstrap approach to celestial holography. Also recently celestial holography has been used to shed light on S-matrix bootstrap in [30]. 2 To be more precise the wedge subalgebra of w 1+∞ . coefficientsC i p,q (∆ 1 , ∆ 2 ) which can appear in a w-invariant theory. Now before we proceed let us briefly describe the symmetry algebra that follows from the universal singular terms of the OPE (2.1).
So we start by defining the tower of conformally soft [31] gravitons [19] . It follows from the structure of the singular terms in (2.1) that we can introduce the following truncated mode expansion This is called the Holographic Symmetry Algebra (HSA). Now if we make the following redefinition (or discrete light transformation) [20] w p α,m = For our purpose it is more convenient to work with the HSA (2.5) rather than the w 1+∞ algebra. However, we continue to refer to the HSA as the w algebra. 3 Here we are assuming that κ = √ 32πG N = 2. 4 This is the wedge subalgebra of w 1+∞ . Now suppose we have two different w invariant theories which we call A and B. A and B have identical symmetry algebras (2.5) because there are no central charges and moreover, their operator contents are also the same. Therefore if the OPE is completely determined by w covariance then there must exist a universal form of the OPE which holds in both theory A and theory B. But how is this possible if the graviton scattering amplitudes in theory A and theory B are different ? This is still possible if the following holds: Let us denote the universal OPE by OPE w and the OPEs obtained from the graviton scattering amplitudes of theory A and theory B by OPE A and OPE B , respectively. Note that OPE w holds in both theory A and theory B. Therefore it must be true that This is a major simplification if we know the OPE and null states of either theory A or theory B.

A. What is theory A?
Our task will be simplified if we can find a theory A such that the null states of theory B can be written as a linear combination of the null states of theory A. This does not mean that the theory A and theory B have the same null states but, this implies that the null states of theory A form a basis in terms of which the null states of theory B can be expanded. Once we find such a theory A we can write, as a consequence of (2.8), Now in our case we take the theory A to be the MHV-sector. This is the simplest theory in the sense that although this is w invariant, the graviton-graviton OPE can be written as [16] a linear combination of supertranslation and sl 2 current algebra 5 descendants only.
Therefore MHV sector has the maximum number of null states, i.e, maximally constrained among all the w invariant theories. Now the OPE and the null states in the MHV-sector are known [16,17] and we will use them to construct the general w invariant OPE. 5 Supertranslation and sl 2 current algebras are subalgebras of the w 1+∞ algebra. They are generated by the soft gravitons H 1 (z,z) and H 0 (z,z).

III. NULL STATES IN THE MHV SECTOR
According to our discussion in the previous section, we can write ). This can be done at any order however, for the sake of simplicity, we focus on these two orders only. Moreover, the O(z 0 12z 1 12 ) terms in the OPE B give rise to interesting constraints on the graviton scattering amplitudes [16] in theory B. So it will be interesting to determine them.
The MHV null states that can appear at O(z 0 12z 0 12 ) are given by [16,17] where k = 1, 2, 3, · · · , ∞. However, it is more convenient to work with the new basis defined by The physical significance of this basis will become clear in the next section.
The MHV null states that can appear at O(z 0 12z 1 12 ) are given by [16,17], which is more convenient for our purpose.
There is a second set of null states at this order which involve the descendant L −1 G + ∆ . These are the null states of the Knizhnik-Zamolodchikov(KZ) type and their decoupling gives rise to differential equations for the scattering amplitudes [16,27] which are the generalization of the standard KZ equation to Celestial holography. However, we do not need the KZ-type null states to write down the OPE B because the generator L −1 7 does not belong to the w algebra. We will give the explicit form of these null states at a later part of the paper.

IV. ORGANIZATION OF THE OPE AT EVERY ORDER
We have argued that the OPE in the theory B can be written as, where N p,q k are the MHV null states. Now, an infinite number of null states can potentially appear at every order of the OPE B . So we need some guiding principle which tells us which null states can appear at a given order and hopefully, we have only a finite number of them.
The guiding principle is nothing but the w-covariance of the OPE. In fact, if we want to classify the states which can appear at a given order of the OPE B , then we can work with the sl 2 (R) subalgebra of the w algebra generated by the operators We denote this by sl 2 (R) V 8 .

7L
−1 is a generator of the w algebra. 8 Here V stands for vertical. Please see Fig.1 for an explanation.

Now using the relations
and the w-covariance of the OPE, i.e, the fact that both sides of the OPE transform in the same way under w transformations, one can easily check that: The MHV null states that can appear at O(z pzq ) of the OPE, transform in a representation of the sl 2 (R) V subalgebra. Moreover, this is the unique sl 2 subalgebra with this property.
We now verify this explicitly.
A. Action of sl 2 (R) V on the null states Let us consider the action of the sl 2 (R) V on the MHV null states Ω k (∆) that can appear at O(z 0z0 ). It is given by, and H 0 0,0 = 2L 0 is obviously diagonal on these states. So we can see that the null states Ω k (∆) indeed form a representation of sl 2 (R) V . However, this representation is reducible. For let us consider the subspace spanned by the (infinite) set of states {Ω n+1 (∆), Ω n+2 (∆), Ω n+3 (∆), · · · } where n ≥ 0. It is easy to see from (4.4) and (4.5) that this set is closed under the action of the sl 2 (R) V . Therefore the representation carried by the states {Ω k (∆)} k=1,2,... is reducible. As a consequence we can set the states Ω k+1 (∆) = 0, k ≥ n ≥ 0 (4.6) without violating the sl 2 (R) V symmetry. After doing this we are left with n null states which transform in a representation of the sl 2 (R) V as a consequence of (4.6). We can also set these states to zero without violating the sl 2 (R) V symmetry.
The action of the sl 2 (R) V on the MHV null states Π k (∆) which can appear at O(z 0z1 ) is given by, and Therefore we can see the states Π k (∆) form a representation of the sl 2 (R) V .
This representation is reducible because the subspace spanned by the states , · · · } form a representation of the sl 2 (R) V . We can get a smaller representation spanned by the states There is no other nontrivial representation of sl 2 (R) V in the subspace spanned by the states  weights. These currents transform in an irreducible highest weight representation of the sl 2 (R) V . This can be seen from the following commutation relations following from (2.5), Therefore, starting from the current H 1 1 2 (z) we can generate any other w current by the combined action of the sl 2 (R) and sl 2 (R) V (Fig.1). Now, the way it is reflected in the structure of the OPE is the following. Suppose we consider the OPE of two positive helicity gravitons. The OPE is w invariant if the singular terms have the following universal structure Now it has been shown in [28] that the leading term inz is uniquely determined by the sl 2 (R) V invariance 11 . Once we know the leading term, the subleading terms inz of O(z q z ), q ≥ 2 are determined by the sl 2 (R) invariance 12 . Therefore if we make sure that the sl 2 (R) V and the sl 2 (R) symmetries are not broken then we are bound to have w invariance. We now discuss the consequences of this.
In section-(IV) we have shown that the equations are sl 2 (R) V invariant. Now one can easily check that where H 0 0,1 is, upto normalization,L 1 . So the MHV null states Ω k (∆) are sl 2 (R) primaries and as a result the equations (5.3) are sl 2 (R) invariant. This implies that (5.3) is also w invariant. 11 It is assumed that the leading term is of O(z z ). This follows from the structure of the universal leading term in the collinear limit of two positive helicity gravitons. 12 Here the assumption is that the Vir is not part of the symmetry algebra.This is further discussed in [15].
So if we want to impose (5.6) without breaking the sl 2 (R) invariance then we also have to impose Ω k+1 (∆) = 0, k ≥ n ≥ 0 (5.8) Therefore, (5.6) is w invariant if (5.8) holds and we know from our previous analysis that {Π 1 (∆), · · · , Π n (∆)}, respectively. The crucial point here is that the value of the integer n, is not fixed by w invariance. So we get different w invariant OPEs for different choices of the integer n. For example, n = 0 gives the MHV-sector. Similarly, direct calculation of graviton-graviton OPE using the scattering amplitudes shows that n = 4 gives the OPE of the quantum self-dual gravity theory which is known to be w invariant [22]. The existence of this discrete infinite family of w invariant OPEs is the main outcome of our analysis.
As we have already discussed the states {Ω 1 (∆), ..., Ω n (∆)} are not all independent because of the existence of the additional null state relations (4.8) We can further simplify (7.1) by using these null state relations. However, we choose to leave it in this form because the OPE coefficients are particularly nice when written in terms of these linearly dependent states. Now different values of n give OPEs in different w invariant theories. We now discuss the KZ type null states of these theories.

Similarly, we have
Decoupling of null states gives rise to differential equations which the graviton scattering amplitudes in this theory have to satisfy. So in principle we know a lot about these theories however, solving these equations may not be easy in practice. We hope to return to this in near future.