On deformations and extensions of $\text{Diff}(S^2)$

We investigate the algebra of vector fields on the sphere. First, we find that linear deformations of this algebra are obstructed under reasonable conditions. In particular, we show that $hs[\lambda]$, the one-parameter deformation of the algebra of area-preserving vector fields, does not extend to the entire algebra. Next, we study some non-central extensions through the embedding of $\mathfrak{v}\mathfrak{e}\mathfrak{c}\mathfrak{t}(S^2)$ into $\mathfrak{v}\mathfrak{e}\mathfrak{c}\mathfrak{t}(\mathbb{C}^*)$. For the latter, we discuss a three parameter family of non-central extensions which contains the symmetry algebra of asymptotically flat and asymptotically Friedmann spacetimes at future null infinity, admitting a simple free field realization.


Introduction
The algebra of vector fields on the circle vect(S 1 ) and its Virasoro central extension have played a major role in quantum gravity and theoretical high energy physics for the last half century. It is the symmetry algebra of two dimensional conformal field theories (CFT) [1], plays a fundamental role in string theory and appears constantly in a wide range of applications, among which we would like to highlight black hole microstate counting [2,3], fluid-gravity duality [4,5], and asymptotic symmetries in three dimensions [6][7][8].
Nonetheless, it is the algebra of vector fields on the sphere that naturally arises when trying to describe two dimensional membranes, instead of strings, and to approach black hole microstate counting, fluid-gravity duality and asymptotic symmetries in four dimensions.
In fact, it was shown that the spherical two dimensional membrane in light-cone gauge is invariant under area preserving diffeomorphisms on the sphere SDiff(S 2 ), whose algebra of smooth vector fields is denoted by svect(S 2 ), and that their large N SU (N ) discretization is presently the most viable path to membrane quantization [9][10][11]. More precisely, a one parameter family of algebras, known as hs[λ], reduces to SU (N ) for integer λ and becomes svect(S 2 ) in the limit λ → ∞ [12] 1 .
Furthermore, non-central extensions of the algebra of vector fields on the sphere have been recently considered to correspond to asymptotic symmetry algebras of asymptotically flat, gbms [16][17][18][19][20][21], and asymptotically decelerating spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW), gbms s [22], spacetimes at future null infinity, and asymptotically (anti) de-Sitter [23] in four spacetime dimensions. Non-central extensions of this algebra have also been discussed in the context of asymptotic symmetries in null hypersurfaces (including event horizons) [24,25]. Moreover, by means of the membrane paradigm [26,27], a connection between these asymptotic symmetries at null hypersurfaces and fluids on the sphere has been elucidated in [28,29].
Thus, the study of the algebra of vector fields on the sphere is in order to deepen into the aforementioned physical research fields. Rather surprisingly, few studies have been performed trying to investigate the structure and properties of this algebra, as far as we are aware. In [30], it has been shown that svect(S 2 ) does not admit central extensions, while [31] studied generalized Kac-Moody algebras as loop algebras for vect(S 2 ), and [32] investigated harmonic distributions on the sphere and related them to Diff(S 2 ). More recently, the representation theory of SDiff(S 2 ) has been explored using the method of coadjoint orbits in [33].
In this paper, we further investigate this algebra following two main paths. First, we analyze the structure and deformations of the algebra of globally defined vector fields on the sphere, vect(S 2 ) as well as its "chiral" subalgebra generated by holomorphic and anti-holomorphic vector fields respectively. Next, we embed vect(S 2 ) in the algebra of vector fields on the two-punctured sphere, or punctured complex plane vect(C * ), in order to investigate some of its physically relevant noncentral extensions and devising simple free field realizations for them. Our main findings are: 1. The chiral subalgebras contain half-Witt subalgebras generated by smooth vector fields on S 2 . Furthermore, these chiral subalgebras can be reconstructed from a half-Witt subalgebra and horizontal operators as described in figure 1.
2. We present an argument for the absence of linear deformations of the algebra of smooth diffeomorphisms on the sphere, vect(S 2 ). In particular, we show that the deformation family hs[λ] [12] does not extend from svect(S 2 ) to vect(S 2 ).
3. In terms of the locally defined vector fields on the two-punctured sphere, we describe a three parameter family of non-central extensions gW (a, b,ā) which contains gbms [16,17] and gbms s [22]. It generalizes and includes the W (a, b;ā,b) family of deformations for bms [34,35], and can be realized by a simple free field realization. In addition, following the fact that W (a, b;ā,b) admits a central extension,Ŵ (a, b;ā,b), we obtain an equivalent extension for gW (a, b,ā), which we nameĝ W (a, b,ā).
This paper is organized as follows: in section 2, we describe vect(S 2 ) in two different bases that are adapted to the deformation problem of vect(S 2 ) and its chiral subalgebras. We then investigate its linear deformations with the help of an attached Mathematica file vectS2deformations.nb [36]. In section 3, we discuss the structure of vect(C * ) and study some of its non-central extensions present in the context of asymptotic symmetries. We conclude with a summary of results and future endeavours in section 4.

Vector fields on S 2
We begin by reviewing some basic properties of the algebra of smooth vector fields on the sphere vect(S 2 ). In the following subsection, we then address the problem of linear deformations of this algebra.

Description of the classical algebra
The generators of vect(S 2 ) fall naturally into two classes 2 • Area-preserving • Non area-preserving where l > 0 and −l ≤ m ≤ l denote the orbital and magnetic quantum numbers respectively, while θϕ = 1 sin(θ) and g ab are the inverse volume form and metric of the round sphere respectively. Let us now summarize some features of this algebra that will be useful in the sequel: 1. The area preserving vector fields T l m form a closed subalgebra called svect(S 2 ) and {T l 0 } form an abelian closed subalgebra of the latter 3 . On the other hand, the non-area preserving vector fields S l m do not close on themselves.
2. The generators with l = 1 form a subalgebra. In particular, generate the so(3) subalgebra of rotations. Together with and the commutation relations they generate the subalgebra so(1, 3) of conformal diffeomorphisms on the sphere.
3. The Lie algebra isomorphism so(1, 3) sl(2, R) ⊕ sl(2, R) is made manifest by the complex linear combinations, 4. The generators with > 1 transform as vectors under the so(3) subalgebra of rotations, that is (B l m ∈ {T l m , S l m } ) In addition, they transform in a representation of so(1, 3) but, since the latter is infinite dimensional, this will not be of use here.

Chiral subalgebras
The decomposition of the algebra as in (2.7) does not generalise to > 1. However, there are subalgebras A ± for > 1. This is more easily seen in stereographic coordinates z = e iϕ cot(θ/2) ,z = z * . (2.9) In these coordinates we have which makes the decomposition of vect(S 2 ) into holomorphic-and anti holomorphic vector fields manifest, that is This reveals further subalgebras. Here we list some of their features: 1. The l = 1 subalgebra (2.7) is recovered with Furthermore, (A l m ) ± transform as vectors under the so(3) subalgebra of rotations

The chiral algebras
The subalgebras generated by {(A ± ) + } (and similarly by {(A ± ) − }) are isomorphic to the subalgebra of the Witt algebra, [L n , L m ] = (m − n)L m+n , generated by L n with n > 0, usually called half-Witt algebra. This can be seen by a change of normalization, for instance These half-Witt subalgebras miss a lowering operator and, contrary to the usual two dimensional conformal field theory on the sphere, the corresponding vector fields are regular everywhere. 5 3. More generally, the generators of the chiral subalgebra A + (and similarly A − ) can be constructed from a single generator (A 2 −2 ) + with a raising operator, A 1 −1 and a horizontal operator, T 1 ±1 , as described in fig. 1. Figure 1: Generation of chiral generators starting from A 2 ∓2 and then first acting repeatedly with A ∓ = A 1 ∓1 to obtain A ∓ and then acting with Schematically we can understand the algebra as vect(S 2 ) (so(3) → hW ) ∪ (so(3) → hW ), where so(3) → hW denotes the action of so(3) on half-Witt and the bar on top signals parity conjugation. This structure resembles and contains so(1, 3) sl(2, R) ⊕ sl(2, R).

Linear deformations of vect(S 2 )
In this section, we investigate the linear deformations of vect(S 2 ) and its chiral subalgebras A ± . By linear we mean that the commutation relations do not generate higher powers of the generators. This problem is hard to tackle analytically because of the complicated form of the structure constants. Here, we reformulate the problem in a way that can be analyzed level by level with the help of Mathematica.
To study deformations, we first have to specify the conditions that we impose on any consistent deformation. Concretely we impose: 1. The Jacobi identities have to be satisfied.
2. The generators (T j m , S j m ) have to transform as spherical tensors under T 1 m (i.e. the isometry group of S 2 is not deformed).
3. The possible deformations have to include the classical algebra vect(S 2 ) as a limit in the deformation parameter.
4. The generators are required to have a definite transformation under parity.
The general ansatz for the commutators, imposing covariance under the rotation group (condition 2) reads as follows: with the m-dependence of the commutators completely fixed by the so(3) subalgebra (second condition) together with the Wigner-Eckart theorem such that, using the conventions of [12]: where the the combinatorial factor [a] n = a(a − 1)...(a − n + 1) is a Pochhammer symbol.
There are infinitely many free coefficients in the above commutators. These will be reduced by imposing parity invariance and the Jacobi identity, while taking care at the same time that the solutions remain in the classical branch. Furthermore, for each j there is a rescaling freedom of the generators (T j m , S j m ) which we fix by choosing coefficients that do not vanish in vect(S 2 ) and assign them a value. The practical way to perform this analysis at the computational level is to solve the Jacobi identities order by order in j max 6 , replacing the coefficients of lower j in terms of the ones with higher j and analyze the resultant algebra at every level to observe if there are free parameters left.
In vectS2deformations.nb, attached to this note, we carry out this analysis up to j max = 7 and we observe that the algebra of commutators with j ≤ 3 is completely determined. In fact, it is clear from the computations that the number of independent equations grows faster than the number of free coefficients. It is not possible for us to continue this analysis to large j max due to lack of computational power. Nevertheless, from the computational perspective it is rather evident that the algebra will not admit deformations at higher j. On this ground, we arrive at the following claim: The algebra of smooth diffeomorphisms on the sphere, vect(S 2 ), does not admit linear deformations satisfying the Jacobi identities, parity and vector representation of the generators under rotations 7 .
Let us contrast this result to the well known one parameter linear deformation of svect(S 2 ), which is also known as higher spin algebra or hs[λ] [12]. The latter is obtained from the so-called lone-star product of area-preserving generators [12]: where the generalized hypergeometric function 4 F 3 is evaluated at z = 1. Moreover, if |λ| ∈ N + the lone-star product corresponds to associative matrix multiplication compatible with SU (N ), being N = |λ|, and the limit λ → ∞ leads to svect(S 2 ). Using this product, one obtains the deformed commutator as m represents both T j m and S j m . 7 Note that parity seems to follow naturally from the Jacobi identities, so it might actually not be a necessary requirement.

Hs[λ] is not compatible with vect(S 2 ).
The one parameter deformation of the algebra of area-preserving diffeomorphisms on the sphere, hs[λ], does not extend to the full algebra of smooth diffeomorphisms on the sphere, vect(S 2 ), if the Jacobi identities, parity and vector representation of the generators under rotations are satisfied 7 .
This result agrees and provides further support for the previous proposal on the absence of linear deformations of vect(S 2 ).

Deformations of the chiral subalgebra
While vect(S 2 ) appears to admit no linear deformations, this might still leave room to the possibility that, in addition to svect(S 2 ), other subalgebras admit linear deformations. With this in mind, we now consider possible linear deformations of the chiral subalgebras A ± . In this case, the assumptions we adopt are: 1. The Jacobi identities have to be satisfied.
2. The generators (A j m ) ± have to transform as spherical tensors under T 1 m .
3. The possible deformations have to include the non-deformed classical chiral subalgebras.
The second requirement leads to the ansatz Performing an analysis identical to the previous cases up to j max = 7, we observe that the algebra of commutators with j ≤ 3 is completely determined. It is again clearly noticeable that the number of independent equations grows faster than the number of free coefficients, and from the computational perspective it seems clear that the algebra will not admit deformations at higher j. Thus, we collect strong evidence in favor of: No linear deformations of chiral A ± subalgebras. The chiral subalgebras of smooth diffeomorphisms on the sphere, A ± , do not admit linear deformations satisfying the Jacobi identities, and vector representation of the generators under rotations.

Discussion
Let us briefly analyze the main result obtained in this section and its implications.
We have gathered strong evidence in favor of a no-go theorem for deformations of vect(S 2 ) under the following assumptions: the deformation is linear, the Jacobi identities have to be satisfied and the generators have to transform as spherical tensors under T 1 m . The last two conditions are necessary if we aim to obtain deformations which are Lie algebras and whose isometry group is still that of the sphere, allowing us to use the Wigner-Eckart theorem in the ansatz (2.15)-(2.17). We cannot exclude non-linear deformations akin to W -algebras [39] in the case of vect(S 1 ), the Witt algebra. To our knowledge, such non-linear deformations have not been explored even for the area preserving subalgebra svect(S 2 ). Linearity could be relaxed at the cost of the ansatz (2.15)-(2.17) having to be modified in order to include terms non-linear in the generators in the RHS. This analysis would be certainly more involved, as we would have to figure out how to efficiently use the Wigner-Eckart theorem and the conservation of angular momentum to constrain the allowed non-linearities. Besides, the computational power required to perform such an analysis is beyond our present capacities. Nevertheless, such an exploration is definitely worth to be pursued in future studies.
Most of the implications of this result are surely yet to be unveiled, even though we can already notice some important consequences. Firstly, the rigidity under linear deformations of vect(S 2 ) is in sharp contrast with the well-known hs[λ] deformation of svect(S 2 ). The latter reduces to SU (N ) for integer λ and defines a large N discretization of svect(S 2 ) [12], which has been linked to membrane quantization [9][10][11]. The possibility of a similar discretization for vect(S 2 ) is ruled out by our analysis, at least at a linear level, which points towards a fundamental difference between algebras of diffeomorphism and their area preserving subalgebras. Furthermore, we expect the rigidity of vect(S 2 ) to play a fundamental role in the understanding of its potential representations and (quantum) deformations. For instance, our result might well pose constraints to generalize the quantum deformations of the bms algebra, studied in [40], to gbms [16,17], gbms s [22] and bmsw [20], which are non-central extensions of vect(S 2 ) arising in the study of asymptotically flat and FLRW spacetimes.
It would certainly be interesting to explore whether the rigidity of vect(S 2 ) extends to other two-dimensional surfaces like the plane or the torus. As far as we are aware, such analysis have not been performed so far in the literature. Unfortunately, our algorithm does not straightforwardly extend to theses spaces. The main obstacle is the lack of spherical symmetry organizing the generators and the subsequent loss of the Wigner-Eckart theorem, which severely constrains the free coefficients to be determined in (2.15)-(2.17). As a consequence, the number of free coefficients grows substantially making it very challenging to constrain them efficiently.
As a final comment, let us note that, while vect(S 1 ) admits a central extension, the Virasoro algebra, it was known for a long time [30] that vect(S 2 ) does not admit central extensions 8 . On the other hand, non-central extensions do exist but their description is cumbersome due to the complicated form of the structure constants of vect(S 2 ). In the next section we will discuss some extensions by embedding vect(S 2 ) in vect(C * ).

Embedding in vect(C * )
We can embed vect(S 2 ) in vect(C) simply by replacing (2.12) by arbitrary smooth holomorphic and anti-holomorphic vector fields on C. More generally, if we allow the vector fields to be singular at the origin, we can choose the following basis of vect(C * ) L m,n = −z m+1zn ∂ z ,L m,n = −z mzn+1 ∂z , [L m,n ,L r,s ] = −rL m+r,n+s + nL m+r,n+s .
In fact, (3.2) makes it clear that (3.1) is isomorphic to vect(C * ) and to vect(T 2 ) (see [41][42][43][44][45][46][47] for a detailed analysis and some representations). This is not surprising since both can be obtained from the two-punctured sphere with suitable identifications. Thus we actually have which is compatible with the geometric picture of the cylinder being an open subset either to S 2 or T 2 . Unlike (A l m ) ± , the basis (3.1) does not diagonalize the so(3) Casimir 9 but, instead, simultaneously diagonalizes As already mentioned, these vector fields are generally singular on S 2 , for z,z → 0 and z,z → ∞. In fact, they form an over-complete basis for the global vector fields in vect(S 2 ). This can be seen by noticing that the global vector fields on S 2 have the form 1 (1 + zz) l−1 P (z azb )∂ z and 1 (1 + zz) l−1 P (z azb )∂z , (3.5) where P (z azb ) is a polynomial. Thus expanding around the south pole (zz → 0) or, around the north pole (zz → ∞) they are clearly infinite linear combinations of (3.1). We see that this gives two different ways of representing a smooth vector field on S 2 as an infinite linear combination of the elements (3.1). This is analogous to the fact that the Taylor expansion of a rational function of two variables z and w in two different regions (|z| |w| and |w| |z|) leads to different formal power series representing the function [48].

Extensions of vect(C * )
As shown in [42,44], vect(C * ) and vect(T 2 ) do not admit central extensions either. However, there are non-central extensions which reduce to the Virasoro central extension when viewed as a subalgebra [42,44]. They can be described as Consequently, we find that, unless n = 0 and/or r = 0, which correspond to the Witt subalgebras, we are forced to to set b =b. Therefore, the family of algebras is actually gW (a, b;ā). Some examples of algebras in this family are given by gbms gW (− 1 2 , − 1 2 ; − 1 2 ) and gbms s gW (− 1+s 2 , − 1+s 2 ; − 1+s 2 ). We note in passing that if the superrotation-like vector fields appearing in the near horizon symmetry algebras described in [51,52] and [53] are not constrained to satisfy the conformal Killing equation, the latter are described by gW (0, 0; 0) and gW (a, a; a) respectively.

Free field realization
It might seem difficult to find a representation of these complicated algebras. Nevertheless, it turns out that there exists a Heisenberg-like construction which provides us with a free field realization for the family gW (a, b;ā). This is given by (3.14) with [a α,β ,ā γ,δ ] = δ α+γ,0 δ β+δ,0 (3.17) and b =b. This free field realization helps us to visualize the physical meaning of the uniparametric family of deformations gbms s gW (− 1+s 2 , − 1+s 2 ; − 1+s 2 ), being s related to the weight in the lattice. Besides, it is evident that this representation describes also the subfamily W (a, b;ā,b) with n = 0 in (3.14) and m = 0 in (3.15) and sheds light on the symmetries of the coefficients a,ā, b,b described in section 5.3 of [35]. vect(S 2 ) since, as mentioned above, c 1 = c 2 = 0 for the latter [30]. The same happens to the centrally extendedŴ (a, b;ā,b) algebras, although one can define them abstractly using the punctured complex plane as suggested by the works of [54,37,34,35,55]. In fact,Ŵ (a, b;ā,b) is a subfamily of (3.18) after using the conditions mS m,n + nŜ m,n = 0 , S m,0 = S 0,0 δ m0 ,Ŝ 0,n =Ŝ 0,0 δ n0 , which allow to recover Virasoro central extensions in the one-dimensional limit.

Summary and conclusions
The object of study of this paper is the algebra of vector fields on the sphere. Besides being mathematically interesting per se and scarcely studied, it pops up ubiquitously in physics literature. Although Diff(S 2 ) plays a major role in membrane theory [9][10][11] and fluid-gravity duality [28,29], our main motivation emerges from recent investigations in asymptotically flat [16,17,[19][20][21] and asymptotically spatially flat FLRW [49,50,22] spacetimes, where the asymptotic symmetry algebras contain as superrotation subalgebra that of vector fields on S 2 .
In section 2, we restricted to smooth vector fields, which form the algebra vect(S 2 ). Firstly, we described the structure of this algebra in the conventional area preserving (T l m ) and non-area preserving (S l m ) vector fields, which we used to investigate possible linear deformations of vect(S 2 ) with the help of an attached Mathematica file vectS2deformations.nb [36], where explicit details on the computations can be found. Next, with the help of stereographic coordinates, we found a more illuminating chiral basis that splits into vector fields with purely holomorphic (A l m ) + and antiholomorphic components (A l m ) − . In section 3, we loosened the smoothness condition for the vector fields and embedded vect(S 2 ) in vect(C * ), allowing for two punctures. In terms of it, we examined physically relevant non-central extensions and came up with some simple free field realizations. Remarkably, the two-punctured Riemann sphere, where the conformal subalgebra of (3.2) and of (3.6) is consistently realized, has been argued to be the relevant one for celestial scattering amplitudes and soft theorems in the context of bms [56,54,37,55]. Analogously, we expect the complete (3.2) and (3.6) to play the equivalent role for gbms.
Let us recall our most important results: • By means of the chiral basis, we observed that (A l ± ) + and (A l ± ) − describe half-Witt subalgebras generated by smooth vector fields on S 2 . Moreover, both chiral subalgebras A ± can be reconstructed from a half-Witt subalgebra and the action of rotation operators as described in the picture 1. We find the chiral basis especially illuminating and hope that it will help in future studies of this important algebra.
• We found that the Jacobi identities fix the structure constants for small values of j completely, which suggests that vect(S 2 ) does not admit linear deformations satisfying Jacobi identities, being compatible with parity and transforming in given representations of the rotation group. In particular, we showed that the higher-spin one parameter deformation of svect(S 2 ), hs[λ] [12], does not extend to vect(S 2 ) under these requirements. For the chiral subalgebras of vect(S 2 ), A ± we find by the same method that it should not admit linear deformations satisfying Jacobi identities and vector representation of the generators under rotations.
• In terms of the locally defined vector fields on the two-punctured sphere, we uncovered a three parameter family of non-central extensions gW (a, b,ā) which contains the asymptotic symmetry algebra of asymptotically flat (gbms [16,17]) and asymptotically decelerating spatially flat FLRW (gbms s [22]) spacetimes at future null infinity. It generalizes and contains the W (a, b;ā,b) family of deformations for bms [34,35] and admits a simple free field realization compatible with the ones described in [41,43]. In addition, guided by the fact that W (a, b;ā,b) admits a central extension,Ŵ (a, b;ā,b), obtained by centrally extending both Witt algebras, we found an equivalent extension for gW (a, b,ā), which we denoted byĝ W (a, b,ā).
Finally, we briefly list some open questions and especially interesting research directions.
• It was shown in [30] that svect(S 2 ) does not admit any central extension and our results strongly suggest that the complete algebra vect(S 2 ) does not admit linear deformations. It would be certainly interesting to explore whether or not vect(S 2 ) admits non-central extensions and/or non-linear deformations [39].
• Locally, Witt and Virasoro algebras have been shown to not admit linear deformations [57,58], although they allow for non-linear ones [59,60]. Taking into account that vect(C * ) ← vect(T 2 ) is its more direct two dimensional generalization, it would be desirable to investigate its possible linear and non-linear deformations. In particular, svect(T 2 ) admits a one parameter deformation similar to hs[λ] [61] which might or not extend to vect(T 2 ).
• We did not dig into field realizations of vect(S 2 ) andĝ W (a, b,ā) but it would be appealing to explore them in detail.
• Recently, a double non-central extension of vect(S 2 ), called Weyl bms (bmsw), has been proposed in [20] to be the most general extension of the bms algebra 12 . It would be very interesting to generalize this construction to asymptotically FLRW spacetimes, to study its family of deformations and to analyze possible field realizations in a similar way we described in this paper for the other non-central extensions. Similar considerations apply for the so-called corner symmetry and extended corner symmetry algebras [19,21], which also non-centrally extend vect(S 2 ).