Supersymmetrization of deformed BMS algebras

$W(a,b)$ and $W(a,b;\bar{a},\bar{b})$ algebras are deformations of ${\mathfrak{bms}_3}$ and ${\mathfrak{bms}_4}$ algebra respectively. We present an $\mathcal{N}=2$ supersymmetric extension of $W(a,b)$ and $W(a,b;\bar{a},\bar{b})$ algebra in presence of $R-$symmetry generators that rotate the two supercharges. For $W(a,b)$ our construction includes most generic central extensions of the algebra. In particular we find that $\mathcal{N}=2$ ${\mathfrak{bms}_3}$ algebra admits a new central extension that has so far not been reported in the literature. For $W(a,b;\bar{a},\bar{b})$, we find that an infinite $U(1)_V \times U(1)_A$ extension of the algebra is not possible with linear and quadratic structure constants for generic values of the deformation parameters. This implies a similar constraint for $U(1)_V \times U(1)_A$ extension of $\mathcal{N}=2$ ${\mathfrak{bms}_4}$ algebra.


Introduction and Summary
For any theory its asymptotic symmetries are of immense physical significance. The symmetries at the asymptotic boundary of a theory depends on the boundary fall off of its constituents fields. In most examples, the asymptotic symmetry is usually enlarged compared to the bulk symmetry of the theory. However, the bulk symmetry must be contained in the asymptotic symmetry group algebra as a subalgebra. For a theory in asymptotically flat spacetimes, if one recedes from sources towards null infinity, at any finite radial distance from the source one expects the symmetry algebra to be just Poincaré. However at null infinity, the asymptotic symmetry algebra in the Bondi gauge is enhanced to the Bondi-Metzner-Sachs or the bms algebra generated by infinite number of generators known as supertranslations [1,2]. One can further extend the bms algebra by including superrotations [3], which manifests itself as a double copy of Virasoro algebra. The finite dimensional Poincaré algebra is a subalgebra of the extended bms algebra. These infinite dimensional bms algebras have gained a renewed importance due to recent developments on the relations between soft theorems and asymptotic symmetries in analyzing the vacuua of gauge theories and gravity [4][5][6][7][8][9]. It is well understood that at any null boundary in two or three dimensional spacetime one obtain infinite dimensional algebra by constructing the conserved charges without imposing any specific boundary conditions [10]. Recently, this is also realized in general dimensions [11].
It is interesting to understand how these bms algebras get modified in presence of extended supersymmetries in a theory of gravity. Furthermore in the presence of internal gauge fields, namely the R− symmetry fields, the supercharges non trivially rotate among themselves. This brings interesting dynamics to the system such as the BPS conditions gets modified in presence of the R−charges [12,13]. The effects of extended supersymmetries and R−symmetries have been extensively studied in the context of asymptotic symmetries of three dimensional supergravity theories. The supersymmetric deformations of bms 3 algebras and their consequences have been detailed in [13][14][15][16][17][18][19][20][21][22]. In particular it has been shown that the R−charges also get an infinite extensions at the null infinity and the space of the asymptotically flat cosmological solutions gets hugely extended [13,19]. A similar study in the context of four dimensional asymptotically flat extended supergravity theory has not been performed yet. This brings us to look for deformations of bms 4 algebra.
There are two distinct possible ways of generating new algebras starting from one, namely deformation and contraction of an algebra. Deformation of a Lie algebra can be viewed as an inverse procedure of Inönü-Wigner [23] contraction. While physicists have tackled more with contraction of Lie algebras, deformations of various well-known Lie algebras in physics have been recently considered in literature [24][25][26][27][28][29]. In contraction prescription one tries to obtain a new non isomorphic algebra through specific limits of a known algebra, where as in deformation prescription one deforms a Lie algebra to get new (more stable) algebras by turning on structure constants in some commutators [30,31]. For instance, one may take the limit of the Poincaré algebra by sending the speed of light to infinity (or to zero) to get Galilean (or Carroll) algebra and conversely the Galilean (or the Carroll) algebra may be deformed into the Poincaré algebra [24,32]. In recent works [33,34], it has been proven that the three and four dimensional pure bms algebras can be deformed, in their non-ideal part, into two families of new non-isomorphic infinite dimensional algebras called W algebras. In the context of three space time dimensions, these are known as W (a, b) algebras, where bms 3 corresponds to W (0, −1) and in the context of four space time dimensions these are knows as W (a, b;ā,b) algebras, where bms 4 corresponds to W (−1/2.− 1/2; −1/2, −1/2). It has been shown that by imposing appropriate boundary conditions W (0, b) as well as W (b, b; b, b) algebras are obtained as near horizon symmetry algebras of 3-and 4-dimensional black-holes [35]. Also, W (b, b; b, b) has been obtained as asymptotic symmetry algebra of flat Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes [36]. On the other hand, W (0, 1), W (0, 0) and W (0, −1) have appeared as asymptotic symmetry algebras in various gravitational theories [37][38][39].
In this paper, our goal is to extend the W (a, b) and W (a, b;ā,b) algebras with two (fermionic) supercharges. We further consider that the two supercharges rotate among themselves due to an internal R-symmetry. Our construction is purely group theoretic with only two inputs: a) we demand the consistency of the possible extended algebra with the Jacobi identities and b) we demand that the extended algebra contains Super-Poincaré algebra as its subalgebra, for particular values of deformation parameters. The explicit construction goes as follows: • In three space time dimensions, we first introduce a set of infinite fermionic generators to grade the known W (a, b) algebras and ensure that the resulting superalgebra satisfies graded Jacobi identities. Next we perform the similar construction with infinite bosonic R−charge generators. We have further extended our analysis to include the central charges in the algebras. As we have stated above W (0, −1) gives the usual bms 3 and various Supersymmetric extensions of bms 3 algebras are well investigated [19], [20]. Our construction of supersymmetric centrally extended W (a, b) algebras in this paper reproduces the known results for a = 0, b = −1, although we get a new possible central extension.
• So far in four space time dimensions we know generic bosonic W (a, b;ā,b) algebra with central extensions [34]. As stated above bms 4 is a special case of these for a = b =ā =b = −1/2. Furthermore in [40] minimal supersymmetric generalization of bms 4 with 1 supercharge has been obtained. In this paper, we first extend bosonic W (a, b;ā,b) with a set of infinite supercharges. Next we perform further extension with infinite R−charges. In this case, the resulting algebra has not been centrally extended. Interestingly we find that for bms 4 with two supercharges, one can not infinitely extend the R charge sectors. We have shown this rigidity for both linear and quadratic dependence of the structure constants. The Jacobi identities only gets satisfied within the global sector, i.e. for N = 2 Super-Poincaré algebra with global R−charges. This is one of the prime results of this paper.
Here we must mention that both the three and four dimensional algebras constructed in this paper are purely mathematical. In both cases, the corresponding Super-Poincaré algebras are embedded in them as subalgebras for appropriate values of the deformation parameters. Thus, in principle, these algebras might show up as the asymptotic symmetry algebras for three and four dimensional asymptotically flat theories. In particular, The N = 2 extension of bms 4 is a probable candidate of four dimensional N = 2 Supergravity theories with R−charges.
Let us summarize the organisation and results of the paper below : • In section 2, we begin with a brief review of the basic properties of W (a, b) algebra .
Next we present new analysis on a N = 2 supersymmetric extension in the presence of R charge which rotate the supercharges among themselves in the rest of this section. We have completed this section with the central extension of the supersymmetric W (a, b) algebra. Equations (2.69) represent the N = 2, W (a, b) algebra, where as equations (2.105) along with the table below them represent its most generic central extensions.
• Section 3 discusses the basic properties of W (a, b,ā,b) algebra and may be skipped by the experts. We have added it for establishing the notations used in the later sections.
• In section 4, we extend the W (a, b,ā,b) algebra to include two supercharges, albeit any internal symmetry. This section forms the base of the main results of the paper that have been presented in section 5. Equations (4.20) presents the prime results of this section, which is an infinite extension of bms 4 in presence of two supercharges.
• Section 5 contains the most important results of this paper. In this section we have introduced two sets of R charges along with other N = 2 W (a, b,ā,b) generators and studied the possibility to find an infinite extension of the algebra. Here we have considered both the cases for structure constants being linear and non-linear in its arguments and performed a detailed analysis. We find a non-affirmative result (unlike the case of section 4) as discussed in the end of the section.
• In section 6, we will conclude with a discussion on main results and possible future directions.
Earlier works, such as [33,34] discussed aspects of deformation and stability of bms 3 and bms 4 algebras. In this section, we briefly state their results and observations for bms 3 . The centerless bms 3 algebra can be written as [P m , P n ] = 0 .
Physically, the J m s are identified with superrotations while the P n s are supertranslations. This algebra can be deformed into two parameters family algebra called W (a, b) where a, b are arbitrary real parameters [34]. Explicitly, the W (a, b) algebra is given by [J m , P n ] = −(n + bm + a)P m+n , It is straightforward to see that W (0, −1) corresponds to bms 3 .

Supersymmetric W (a, b) algebra
In this section, we write down a supersymmetric version of the W (a, b) algebra. Subsequently, we will introduce R and S charges and also determine the central extension to the algebra. Our approach will be somewhat operational : we start by introducing fermionic generators G s in the W (a, b) algebra where s runs over half-integers. Our goal would be to write down an extended algebra starting with the centerless W (a, b) algebra as given above by demanding consistency of Jacobi identities. For the time being, unlike bms 3 , we do not search for the realization of the algebra as the asymptotic symmetry algebra of a supersymmetric theory at null infinity in 3 spacetime dimensions.
Along with the usual W (a, b) algebra as given in (2.4), we introduce the following three commutators The above extension is motivated by various super-bms 3 algebras written in [18][19][20]. We choose to normalize the super-current generators G s in a way such that the structure constant appearing in (2.5) is unity. It is expected that any deformation of the bms 3 algebra by the parameters a and b will not change the index structure appearing on the RHS of (2.6). For bms 3 , it is known that [P m , G r ] = 0. However, it is possible that a deformation gives a non-trivial commutator between the supercurrents and supertranslation which vanishes when a = 0, b = −1 1 . This motivates us to propose (2.7) where l(m, s) is a linear function in m and s. The structure constants α and β appearing above are also assumed to be linear functions of its arguments. Our strategy will be to fix these structure constants and l(m, s) by demanding consistency of certain relevant Jacobi identities.
The Jacobi identity involving the generators G r , G s and P m is given by Next, we use Jacobi identities on the operators J m , G s and G r to determine α(m, s). The corresponding Jacobi identity is A similar exercise on the above Jacobi identity followed by equating the coefficient of G m+n+s yields α(m, s)α(n, m + s) + (m − n)α(m + s, s) = α(n, s)α(m, n + s) . (2.14) It can be easily seen that the structure constant α(m, s) as determined in (2.12) indeed satisfies the above equality. We thus end up with a possible N = 1 supersymmetric extension of W (a, b) algebra given by The supercurrent generators G s and H s can be used to define the following linear combinations The newly defined generators Q 1 r and Q 2 r satisfy Our next aim is to write the generalized algebra in the presence of R-charges. Physically speaking, the R-charge generators rotate the supercharge generators and thus brings in an additional non trivialities in the algebra. It is known that introduction of R charge generators necessitates the introduction of S-charge generators [41] in the context of bms 3 . Motivated by the N = 2 bms 3 algebra as discussed in [20] we begin our analysis by proposing the following relations involving the R-charge and S-charge generators In writing the above ansatz, we have assumed that an R−deformation of the N = 2 super-W (a, b) algebra will not affect the index structure of the undeformed algebra. The Jacobi identity for the operators Q 1 m , Q 2 n and R s is given by, Further, noting that [S m , R n ] = 0 implies that the LHS of (2.29) is independent of the translation generator P m . In order to make this Jacobi identity consistent the coefficient of the translation generator on RHS must vanish identically implying β (m, s) = β(m, r) . (2.30) Since the above has to be true for arbitrary half-integer values of r, s and integer values of m and also both β and β must be linear, we conclude that both β (m, s) and β(m, r) depend only on m. For simplicity, we denote the structure constant appearing in (2.25) as β(m) since, β (m) = β(m) . (2.31) Demanding further consistency of (2.29) yields, σ(r + s, m) = β(m)(η(r, s + m) − η(r + m, s)) . (2.32) The two Jacobi identities involving Q 1 r , J n and R m and, R l , J m and J n , Assuming the structure constants γ(m, n) and β(m) to be linear in m and n we explicitly take them to be of the form, The above ansatz for γ(m, n) along with (2.36) ensures Clearly the above set of equations are over constrained but admits the following consistent solution while β 0 is a non-zero constant that cannot be fixed further. One can easily check that this is also consistent with the Jacobi identity for R m , R n and Q i r . Thus, we can write the following commutation relations Similar to the ansatz for γ(m, n), we assume the following anstaz for κ(m, n) κ(m, n) = κ 0 + κ 1 m + κ 2 n. (2.47) Plugging the above ansatz into (2.46) we get The Jacobi identity for the operators J l , P m and R n , σ(m, n) = σ 0 + σ 1 m + σ 2 n , (2.51) one can substitute it back into (2.50) to get the set of seven relations: (2.55) The above system of equations have two consistent solution sets (detailed in appendix A): In light of the above, we can rewrite (2.26) and (2.27) as Finally, we need to find the structure constant η(r, s) appearing in we use (2.32) to see that we must have a relation of the form It is quite evident that a choice of parameters as defined in Case II in our preceding analysis is inconsistent with the above equation since the LHS is a constant and independent of m. However, from the structure constants of Case I, we arrive at the relation Finally, the Jacobi identity for Q 1 r , Q 2 s and J m can be written as Using (2.26) and (2.27), the above gives rise to, Assuming, κ(m, n) = a + m + (b + 1)n, the above equation will be satisfied for an ansatz of the form (2.57) provided, Thus, the R and S-charge sector algebra under a deformation reads as 2 : Redefining S n → S n /η 1 and R n → β 0 R n , we arrive at, The full W (a, b) algebra including R and S charge generators takes the form where indices {m, n, p, q} ∈ Z while {r, s} ∈ Z + 1 2 and i ∈ {1, 2}. All other commutators vanish. The conformal weight of the generators P m and S m are −b + 1 and −b respectively, while the weight of Q 1 , Q 2 is − b 2 + 1. For the specific case a = 0 and b = −1 which corresponds to supersymmetric-bms 3 , we recover the same algebra as given in [20] 3 .

Central extensions of supersymmetric W (a, b)
One can show that the W (a, b) algebra for generic values of its parameters just admits one central term in its Witt part but for certain specific values of a and b, it admits various central extensions which were classified in [42]. The most general centrally extended 2 η1 = 0 or β0 = 0 are also viable choices of parameters that satisfy the Jacobi identities. But for interpreting R0 as the R−symmetry generator we must consider non-zero values of those parameters. 3 There is a typo in Eq. 3.19 and Eq. 8.59 of [20]. The structure constant in [Mn, Rm] commutator will be −2m instead of −4m otherwise the (G 1 r , G 2 s , Rm) will not be satisfied.
supersymmetric W (a, b) algebra can be written as, where the Jacobi identity between the generators are expected put constraints on un- One can fix the central term v(m, n) of the [J m , P n ] commutator in the following way. The Jacobi identity between J m , J n and P l leads to Specific values of a and b, yield even more non-trivial solutions. We systematically tabulate all the cases below jp mδ m+n,0 , 5. a = 0 and b is arbitrary where, v(m, n) = C (8) Here the subscript jp denotes the central extension in [J, P ] commutator. Out of the 8 central terms appearing in the above five scenarios only C jp and C (5) jp are the non trivial ones. Other central terms can be absorbed by a simple redefinition of P m → P m + Bδ m,0 and choosing the constant B subsequently in an appropriate manner. Thus we drop the remaining central terms C (2) jp , C (6) jp , C (7) jp and C (8) jp for the remaining analysis.
The above analysis demonstrates that there may be certain values for the parameters a and b for which certain central terms will be allowed in the algebra. This opens up a host of possibilities in the central extension. We will focus on the most general extension that is admissible for arbitrary values of a and b.
The commutator [J m , Q 1 r ] may admit a central term given by, is an arbitrary function. The Jacobi identity between J m , J n and Q 1 r gives us, The x 1 central term appearing in the [J m , , Q 1 r ] commutator is identically zero since a central term proportional to δ m+r,0 is identically zero as m is an integer and r is a halfinteger. An identically similar argument is true for x 2 (m, r) which is the central extension in the [J m , Q 2 r ] commutator as given in (2.73).
The central term in [J m , S n ] commutator is denoted by z(m, n) and the full commutator is written as [J m , S n ] = −(a + n + (b + 1)m)S n+m + z(m, n) .
Again, performing the shift S m → S m + Sδ m,0 will remove some of the constants appearing above with an appropriate choice of S. A detailed analysis reveals that we can drop C (0) js and C (4) js .
The anticommutator {Q 1 r , Q 2 s } may have the possible central term f (r, s) and is given by, The Jacobi identity of Q 1 r , Q 2 s and J m gives,  6. When a = 0 and b is arbitrary, we recover we get f (r, s) = 0 identically. g 1 (m, n) is an arbitrary symmetric function which denotes the central term in the [R m , Q 1 r ] commutator and is given by, For g 1 (m, r) ∝ δ m+r,0 , we can easily conclude that this will be zero identically since m is an integer and r is a half-integer.
The commutator of [P m , R n ] may admit a central term h(m, n) which appears as follows, The Jacobi identity of J m , R n and P l leads to (bm + a + l)h(l + m, n) + nh(l, m + n) = 2nz(m, n + l) . 6. For a = 0 and arbitrary b, we must have k(r, s) = 0.
A quick glance at (2.70)-(2.81) tells us that the central extension to the commutator [R m , R n ] will affect the Jacobi identity between J m , R n and R p . Denoting the central extension in this case as the J m , R n , R p Jacobi identity leads to pw(n, p + m) = nw(p, m + n) . The Jacobi identity between P m , R n and S l gives which naturally implies s(m, n) = 0 identically. The reader can easily verify that the Jacobi identities of (R m , S n , Q i r ), (R m , P n , Q i r ) and (Q 1 r , Q 2 s , P m ) implies f i (m, n) = h i (m, n) = t 2 (m, n) = 0. The supertranslation commutator [P m , P n ] also does not admit any central term, as discussed in further details in an earlier work [33] by one of the authors.
Finally, we consider the anticommutator {Q 1 r , Q 1 s } which may admit a central term as It is clear that w 1 (r, s) must be symmetric in its arguments. The Jacobi identity between Q 1 r , Q 1 s and R m gives For m = 0, we see that w 1 (m, n) should be antisymmetric which is clearly a contradiction. Thus, w 1 (m, n) = 0 identically and a similar argument involving Q 2 r yields w 2 (m, n) = 0. This completes a full description of the central extension for the W (a, b) algebra which clearly depends on the values of the parameters a and b.
Here, we tabulate the W (a, b) for a = 0 and b = −1, which is the same as the super-bms 3   Having established a realization of the extended W (a, b) algebra, we now move on to generalize the above analysis in four spacetime dimensions. Like before, our starting point will be the bms 4 algebra which can be thought of as a special case of the more general W (a, b;ā,b) algebra [34,43]. In the early 1960s, [1,2] attempted to understand and study the radiation that will be detected by a distant observer. Interestingly, they found that the full set of symmetries for an asymptotically flat spacetime 4 is an infinite dimensional group spanned by the so-called supertranslation and superrotations generators which is dubbed the bms 4 group.
The infinite dimensional centerless asymptotic symmetry algebra of four dimensional flat spacetime, conventionally known as the bms 4 algebra [34,46], is given by where the indices m, n, p, q, k, l ∈ Z. The generators L m andL m forming two independent copies of the Witt algebra are known to correspond to superrotations while the generators T p,q are known to correspond to supertranslations. Following [34,47], we briefly state the map between the global sector of bms 4 algebra and Poincaré algebra.
Denoting the Lorentz generators as M µν and the translations as P µ , we known that in four spacetime dimensions, they satisfy the algebra where the indices µ, ν, ρ, σ ∈ {0, 1, 2, 3} and η µν ≡ diag(−1, +1, +1, +1) is the flat Minkowski metric. We can define the generator of rotations and boosts as There are various equivalent ways in which one can specify asymptotic behaviour of spacetimes. For a detailed exposition, the reader is urged to consult [44,45] and references therein.
respectively, where ijk is the Levi-Civita tensor and the indices i, j, k ∈ {1, 2, 3}. Further, we define the quantities The translation generators P µ can be mapped to linear combinations of T p,q where (p, q) ∈ {0, 1} as follows: This demonstrates that appropriate combinations of the global part of bms 4 algebra consisting of the operators {L ±1 , L 0 ,L ±1 ,L 0 , T 1,0 , T 0,1 , T 0,0 , T 1,1 } can be suitably repackaged to give the Poincaré algebra. It is noteworthy that the T 0,1 , T 1,0 , T 0,0 and T 1,1 gives rise to the translation generators while a certain combination of (L m ,L m ) where (m, n) ∈ {±1, 0} gives rise to the Lorentz generators. In some sense, this gives us further intuition to associate T p,q with supertranslations while associating L m andL m with superrotations.
Similarly, for four spacetime dimensions it has been proved that bms 4 algebra is not rigid and can be deformed into four parameters family algebra called W where a, b,ā andb are arbitrary real parameters. In this way, bms 4 algebra (3.1) can be viewed as W (− 1 2 , − 1 2 ; − 1 2 , − 1 2 ). This algebra can be viewed as two copies of W (a, b) and W (ā,b) with the identification of T p,q as a product of supertranslation generators of both the algebras. However, as we see in the next section, this structure does not extend in the supersymmetric extensions of the algebra. Another interesting case is W (0, 0; 0, 0), which represents an infinite dimensional algebra of the symmetries of the near horizon geometry of nonextremal black holes [48]. The algebra with a = b =ā =b = − 1+s 2 for 0 < s < 1 describes the asymptotic symmetry algebra of decelerating FLRW spacetime [36].
4 Supersymmetric W (a, b;ā,b) algebra from N = 2 super-bms 4 In this section we write down a supersymmetric extension of W (a, b;ā,b) algebra with two supercharges. To get to this, like the bms 3 algebra, our first goal is to write the supersymmetrized bms 4 algebra. The first attempt towards the construction of a super-bms algebra was carried out in [49], which however did not consider superrotation generators. [50], further explored asymptotic fermionic charges in N = 1 supergravity on four dimensional asymptotically flat background 5 . [40,53] have derived such an algebra by analysing OPEs of appropriate operators of Einstein-Yang-Mills theory at the celestial sphere. However, their convention for indices on the supertranslation generators is different from ours. This changes the index structure that appears in the commutators. We fix the index structure of the supersymmetrized bms 4 algebra by demanding consistency between its global part and the four dimensional super-Poincaré algebra which along with (3.2)-(3.4) now also contains Our starting point in the current context is the algebra stated at (3.1). However, as mentioned earlier, we need to determine the index structure once we include the supercurrent generators which we denote by Q i r andQ i r where i = 1, 2 while r ∈ Z + 1 2 . We begin with the global algebra described in the earlier section 3 to determine the indices. Hence, we propose the following ansatz involving the supertranslation and superrotation generators with the fermionic supercurrent generators 2) and all other elements in the super-algebra are zero. We must note here that a priori, it is not necessary that the other possible (anti)-commutators are zero for a four dimensional asymptotically flat supergravity theory, however we consider this simplified deformation for the purpose of this paper. Our approach is pragmatic-we simply want to write a possible supersymmetric extension of bms 4 algebra, such that its global part coincides with the super-Poincaré algebra, without bothering about the fact if it can be realized from the asymptotic symmetry analysis of a physical supergravity theory. The OPE analysis of [40,53] found that the only non-zero commutators in super-bms 4 algebra are the ones mentioned above in (4.2). This motivates us to propose the ansatz as written above.
We further make the simplifying assumption that the functions parametrizing the indices f, g, h,h are all linear in their arguments 6 .
We demand the map between supercurrent modes and the fermionic generators of the super-Poincaré algebra to be Linearity of the indices implies f (r, s) = f 0 + f 1 r + f 2 s , g(r, s) = g 0 + g 1 r + g 2 s . (4.5) where f i and g i are constants. Using (4.1), along with (3.14)-(3.17), we must have Thus, the mapping (4.3) requires the functions f (r, s) and g(r, s) to satisfy Thus, we need to solve for the six unknowns f i , g i (i = 0, 1, 2) appearing in (4.5) from the above eight equations. As it turns out, (detailed in appendix B) there does exist a consistent solution to the above system given by Thus, we eventually recover The exact map (4.3) can be seen to be (4.10) Using the above map, along with the map described in Sec. 3, we get, (4.11) 6 This structure does not hold for certain symmetry algebras such as the one discussed in [54].
Further assuming linearity of the structure constant α(m, r) and h(m, r) appearing in (4.2), we see that the above global sector is consistent provided one has An identical exercise on the "barred" sector first leads us to the relations This shows that we must have (4.14) Thus, a particular realization of the N = 2 super-bms 4 algebra can be written as while the other (anti)-commutators are identically zero. One can easily check that all the Jacobi identities are satisfied for the above written algebra. Now, that we have fixed the indices in the super-bms 4 algebra, we assume that deformations do not change that and thus will carry over to the W (a, b;ā,b) algebra. Thus, we now propose the following supersymmetrized W (a, b;ā,b) algebra The Jacobi identities of Q i r ,Q j s andL m similarly gives Thus, a N = 2 supersymmetric W (a, b;ā,b) algebra is given by 20) and all other commutators are zero. It satisfies all the Jacobi identities. The above algebra is a simple extension of bosonic W (a, b;ā,b) algebra with two supercurrent generators, as it does not contain any rotation among the supercurrent generators. In the context of super-Poincaré algebra, the R-charge rotates the SUSY generators amongst themselves. To be mathematically precise, the R−extension is the largest subgroup of the automorphism group of the supersymmetry algebra which commutes with the Lorentz group. In the next section we perform the R-extension of the above N = 2 supersymmetric W (a, b;ā,b) algebra, by further introducing the R-charge generators to rotate the two supercurrent generators.

R-extended supersymmetric W (a, b;ā,b) algebra
Before going to R-extension of the W (a, b;ā,b) algebra, let us briefly recall some important points about super-Poincare algebras in presence of R−symmetry. As discussed in [55] generic N = 2 super-Poincare algebra contains two species of R-symmetry generatorsvectorial and axial which act on the supercharges as As a matter of fact theories with extended supersymmetries are extremely rich precisely due to the presence of these two kinds of supercharges. These R-symmetries can in fact be used as a powerful tool to define various twistings in a theory (A-type or B-type) resulting in what is known as Topological Field Theories whose correlators happen to be independent of the background metric. Although conventionally theories with either one kind of R-symmetry are considered, in general the full theory does contain both kinds of R-symmetries. In the current context since we are specifically interested in N = 2 SUSY, the R-symmetry generator in fact generates the group U (1) V × U (1) A . Following [55], we will associate R 0 with the vectorial R-symmetry whileR 0 will be associated as the axial R-symmetry. Another important aspect of super-Poincare algebras is the fact that the R−charges commute with the bosonic Poincare generators. We utilize these facts in our constructions. To be precise, in the following we demand that our R-extended W (a, b;ā,b) algebra must have vanishing commutators of the R−charge generators with other bosonic generators in the global sector for the deformation W (−1/2, −1/2; −1/2, −1/2) 7 In order to make the extension of R-charges in the context of W (a, b;ā,b) algebra, we will start with a general ansatz,

(5.3)
It must be noted that in the above ansatz, although i is a repeated index on the RHS, there is no sum over i. Although we will be primarily interested in N = 2 but the discussion in this section is valid for arbitrary value of N . Thus we consider i = 1, 2, .., N . Furthermore, since we are working in a very general setting, we assume that β i is different for i = 1, 2, .., N . This is of course not the most general extension one can think of but rather a simpler starting point which is also consistent with the global sub-sector (5.1)-(5.2).
In the following analysis, we will assume that the indices appearing in the above ansatz are linear in its arguments, which is a basic feature of most algebras. Consistency with the global subsector i.e. (5.1)-(5.2) fixes the form of the indices as where c,c, k andk are integers and the structure constants are non-zero at least in the global subsector i.e. when r = ± 1 2 and n = 0. We have to find their form away from the global sector. Subsequently, we will concentrate on the commutators [R n , T p,q ] and [R n , T p,q ]. Consider the Jacobi identity for the operators Q i r ,Q j s and R n which gives us Using (5.4), in the above, we get, In the above equation i, j are free indices. In particular for N = 2 SUSY , we see that, which also implies β (1) (r, n) = β (2) (r, n) andβ (1) (r, n) =β (2) (r, n). An identical exercise with Q i r ,Q j s andR n gives leading to the condition that κ (1) (r, n) = κ (2) (r, n) andκ (1) (r, n) =κ (2) (r, n). Given the relation between the structure constants for i = 1, 2, we see that the index is extraneous and hence we will simply be dropping it from our notation subsequently. Thus, we may write more simply Note that the above commutation relations also ensure that the Jacobi identities between T p,q , T m,n , R l as well as T p,q , Q i r , R n (and it's corresponding counterparts with Q i r replaced withQ j s and R n replaced withR n ) are also satisfied. Now, that we have identified R n and R n to the vectorial and axial R-supercurrents, we make a further assumption that [R n ,R m ] = 0 . (5.9) The above assumptions applied to the Jacobi identity of R n ,R m and Q i r leads to a relation between the structure constants κ(cn + r, m)β(r, n) = κ(r, m)β(km + r, n) , (5.10) and analogously the Jacobi identity of R n ,R m andQ i r gives rise to, κ(cn + r, m)β(r, n) =κ(r, m)β(km + r, n) . (5.11) The above two relations seems to put certain constraints on the free parameters c,c, k and k. However, we will explore this subsequently.
Finally, we need to fix the algebra between the R-supercurrents and the superrotations L m andL m . For this purpose we consider a set of Jacobi identities and fixing the form of the commutators between the R-charge supercurrents and the superrotation generators.
1. Jacobi identity for L m , Q i r , R n and L m ,Q j s , R n We start with the Jacobi identity for L m , Q i r and R n which is given by Now, if we consider the [R n , L m ] to be of the form as (5.14), we easily see that the part [R u(n,m) ,Q j s ] and [R v(n,m) ,Q j s ] will be generically non-zero individually for arbitrary values of n, m and s however a linear combination with specific forms of h 1 andh 1 might presumably ensure that the expression vanishes. Note that (5.14) has certain features which puts it in stark contrast to its 3 dimensional W (a, b) analog. Firstly, the first term appearing on the RHS in the above equation has no analog for the W (a, b) algebra as stated explicitly in (2.69). We can also see, commutators similar to the W (a, b) algebra i.e. 3. Jacobi identity for L m , Q i r ,R n and L m ,Q j s ,R n The Jacobi identity for L m , Q i r andR n leads to 5. Jacobi identity for L m , T p,q , R n andL m , T p,q , R n The Jacobi identity for the operators L m , T p,q and R n is given by 28) which is consistent with the ansatz (5.19).
6. Jacobi identity for L m , T p,q ,R n andL m , T p,q ,R n The Jacobi identity for L m , T p,q ,R n leads to , n (ā +bm + q) T p,−kn+m+q = 0 .
(5.30) One can easily see that the above two equations are indeed consistent with the proposed ansatze (5.19) and (5.25) respectively.
Armed with a series of ansatze and a number of Jacobi identities, we enumerate below the index structure and which equations they follows from.
• Plugging ansatz (5.14) in (5.13), we get, (5.31) The above equation must be satisfied for arbitrary permissible values of m, n and r. Also, by definition, β and κ cannot be identically zero. This implies, Since clearly the indices must be integers, we see both k and c must divide every integer which naturally implies that both can takes values ±1. Using (5.15), we further get,c where ξ is some real number. Further looking at (5.31), we can have either w 1 to be identically zero or t 1 = cn + m. Further, (5.27) gives (5.33) In the above expression, each of the coefficients of the T cn+m+p,q and T p,−cn−ξm+q has to vanish individually.

(5.47)
Finally, we have another family of Jacobi identities involving two superrotation generator and one R-supercurrent generator. The Jacobi identities for L m ,L n , R n and L m ,L n ,R n are trivially satisfied. We list down the equations that follow from the non-trivial Jacobi identities systematically 1. Jacobi identity for L m , L n and R p implies 3. Jacobi identity forL m ,L n and R p implies (5.59)

Algebra with linear structure constants
In our analysis so far, the structure constants have been kept arbitrary. However the above equations are severely constraining in determining the form of the structure constants appearing above and solving them generally seems quite difficult. Hence, at this point, we make two powerful simplifying assumptions, which will somewhat reduce the complexity of the equations above. The assumptions are • The structure constants appearing in the proposed algebra are linear in its arguments.
One immediate consequence of the above is that (5.10) and (5.11) is now trivially satisfied. With all of this conclusions, we have a more simplified algebra, given by, Now, we will focus on (5.33) specifically which yields two equations given by (5.68) The first equation written above must be true for arbitrary integral values of m, n and p. However, it contains a term of the form w 1 (n)p which must vanish, implying that w 1 (n) = 0 identically. A similar argument can be applied to the Jacobi identities (5.35), (5.37) and (5.39) to reach the conclusion that w 2 = w 3 = w 4 = 0 too. Plugging in the linear forms of h 1 (n),h 1 (n) and β(n) in the equations (5.67) and (5.68), we get, which leads us to conclude that the structure constant β must be identically zero which is clearly in contradiction with (5.63). (5.35), (5.37). Similarly (5.39) leads to the conclusion thatβ, κ andκ must also be vanishing. Thus we essentially find that an infinite dimensional extension of R-charges in N = 2 bms 4 algebra is impossible with linear structure constants. Therefore we conclude that a generic N = 2 W (a, b;ā,b) algebra (4.20) of section 4 can not have infinite R−extension with linear structure constants .

Conclusion
As mentioned earlier, symmetry algebras are powerful tools which severely constrain the dynamics and vacua of gauge and gravity theories. In this work, we have concentrated on supersymmetric W (a, b) and W (a, b;ā,b) algebras which are deformations of the asymptotic symmetry algebra of supergravity theories in three and four spacetime dimensions respectively. Earlier works [33,34] has established that generic deformations of bms 3 algebras involve two parameters a and b while generic deformations of bms 4 algebras involve a, b,ā andb. The a = 0, b = −1 centerless R-extension of supersymmetric W (a, b) algebra, given by (2.69) indeed matches with earlier results of [20] where the authors performed an asymptotic symmetry analysis to obtain the super-bms 3 algebra. We also classified and wrote down possible central extension of supersymmetric, R-extended W (a, b) algebra. We observed interesting and novel central charges appearing in the {Q 1 r , Q 2 s } anti-commutator, denoted by f (r, s) for various values of a and b. We also find that [J m , R n ] admits a quadratic central charge which has not been realized through the asymptotic symmetry analysis performed in [20] although it seems to be present for arbitrary values of a and b. From the analysis of [20], it is certain that such a central term would not arise with the usual Barnich-Compere boundary conditions. Nevertheless, the study of [56] clearly indicates that with an asymptotically Rindler like behavior will modify the asymptotic symmetry algebra with such central extensions. Thus it remains an interesting open problem to find the importance of this central term in the context of three dimensional asymptotically flat supergravity theory in more generic contexts. As mentioned earlier, the W (0, 0) and W (0, 1) algebras have also appeared as asymptotic symmetry algebras of gravity theories [37,38], so it is worth exploring appropriate boundary/falloff conditions to obtain supersymmetric W (0, 0) and W (0, 1) algebras as asymptotic symmetry algebras in some supergravity theory. To conclude, the analysis of the present work, being mathematically rigorous, provides new asymptotic algebras and hence opens up the possibilities of finding new boundary conditions for supergravity fields. We hope to report on these possibilities in future works.
The construction of the R-extended supersymmetric W (a, b;ā,b) algebra turned out to be more involved. Physically, the R-charge generators are supposed to rotate the global SUSYgenerators which motivates us to propose an ansatz of the form (5.3). To be mathematically precise, we essentially tried to extend the super-W (a, b;ā,b) algebra (written explicitly in (4.20)) by a U (1) V × U (1) A group where each sector is represented by infinitely many generators. One of the sectors of the U (1) V symmetry can be thought of as vectorial R-symmetry while the other copy of the U (1) A symmetry can be thought of as axial Rsymmetry. We considered two primary guiding principles to fix the algebra • For a = b =ā =b = − 1 2 , the global subalgebra must be identical to the R-extended super-Poincaré algebra.
• The indices appearing in all the proposed commutators involving the R-charges must be linear in its arguments.
In order to simplify our calculations, we considered linear as well as quadratic structure constants. In either cases, we realized that having infinitely many R-charges is in contradiction with one or more Jacobi identities that must be followed by such a graded Lie algebra. Essentially, it seems there is an obstruction in the u(1) × u(1) extension of super-W (a, b;ā,b) algebra-which will naturally hinder the construction of an R-extended super-bms 4 algebra. Recent work [57] has carried out u(1) and u(N ) extensions of bms 4 by analysing celestial amplitudes of Einstein-Maxwell and Einstein-Yang-Mills theories and have indeed obtained non-trivial asymptotic symmetry algebras at the boundary which does include infinitely many generators parametrizing the u(1) or u(N ) symmetry. This is however, not in contradiction with our results. Our demand on the behaviour of R-charges i.e. it must non-trivially rotate the global SUSY generators forces us to demand (5.63) which ensures the [Q i r , R m ] commutator to be non-zero for the global sector. Such a constraint need not be followed for the u(1) or u(N ) gauge groups that enter the analysis of [57]. Relaxing (5.63) in our current work does indeed recover the symmetry algebras derived in [57]. Finally, the methodology of our construction by throughly analysing all possible Jacobi identities along with imposing consistency with the global subsector is quite general. It is possible to adapt this algorithm to construct u(1) or u(N ) extension for other exotic symmetry algebras. It is worth emphasizing that although we have studied R-extended super-W (a, b) and super-W (a, b;ā,b) algebras in this papers, it is only for specific values of a and b, we are aware of physical theories of gravity or supergravity where these are realized as boundary symmetry algebras. It will be interesting to explore what kind of supergravity theories give rise to these wide range of W -algebras for more generic values of a, b,ā andb.
We would also like to mention that the construction of the present paper is not exhaustive. Cases that have not been considered here are technically difficult ones to address and there is no other clarifications/reasoning to not consider them. It will be good to find an exhaustive construction of all possible supersymmetrization of deformed bms algebras and we hope to report on them in future.
Let us conclude the paper with the importance of the study of supersymmetric extensions of the deformations of bms algebras. As is well understood, bms algebras are symmetries of asymptotically flat gravity theories at their null boundaries. Some of their deformations have also been realized as the symmetry algebra at the horizon of certain black hole backgrounds. In the context of three spacetime dimensions, the presence of extended supersymmetries and internal R−symmetries plays crucial roles in characterizing the soft hair modes (that gives non trivial cosmological solutions) and their thermodynamics [13,19,58,59]. The similar study has not been performed for four space time dimensions, where non trivial black hole and gravitational wave solutions exist. In the context of bms 4 , the soft hair modes contributes to Black Hole entropy, albeit they do not correspond to the entire microscopic degeneracy. The microscopic degeneracy for a class of four dimensional N = 2, 4, 8 supersymmetric BPS black holes are very well understood [60][61][62][63][64][65]. It would be interesting to understand how much of this entropy is contributed by the soft hairs. Our present results suggest that, for Black Holes appearing in N = 2 spergravity theory, where the internal R-symmetry (that only scales the supercharges) will not have any contributions to the soft hairs. The similar study for other supergravity theories with exotic internal symmetries remain an open problem to study in the future.