N=1,2,3 l-conformal Galilei superalgebras

The issue of constructing N=1,2,3 supersymmetric extensions of the l-conformal Galilei algebra is reconsidered following the approach in [JHEP 1709 (2017) 131]. Drawing a parallel between acceleration generators entering the superalgebra and irreducible supermultiplets of d=1, N-extended superconformal group, a new N=1 l-conformal Galilei superalgebra, two new N=2 variants, and two new N=3 versions are built. Realisations in terms of differential operators in superspace are given.


Introduction
The exploration of the non-relativistic version of the AdS/CFT-correspondence initiated in [1,2] generated a great deal of interest in non-relativistic conformal (super)groups and field theories enjoying such symmetries (earlier developments are reviewed in [3], for more recent applications see [4,5,6] and references therein). In particular, the holographic dictionary was extended to encompass strongly coupled condensed matter systems (see [7] and references therein).
A peculiar feature of non-relativistic conformal transformations is that temporal and spatial coordinates scale differently under dilatations. If one is concerned with maintaining the full Galilei algebra, the most general finite-dimensional conformal extension is given by the so called -conformal Galilei algebra [8,9] [L m , L n ] = −i(m − n)L m+n , where n = −1, 0, 1, p = − , . . . , and is a non-negative (half)integer parameter. L −1 , L 0 , L 1 generate time translations, dilatations, and special conformal transformations, respectively, which all together form sl(2, R) subalgebra. U − describes spatial translations, U − +1 is linked to Galilei boosts, while higher values of p are commonly associated with constant accelerations. 1 In accord with (1), L n and U p have conformal weights 1 and , respectively. In modern literature, the reciprocal of is called the rational dynamical exponent and (1) is sometimes referred to as the conformal Galilei algebra with rational dynamical exponent. The instances = 1 2 and = 1, known as the Schrodinger algebra and the conformal Galilei algebra, have received the utmost attention (for a review see [10]).
If the tower of acceleration generators is reduced to a single spatial translation generator, the special conformal transformations are discarded, and the dynamical exponent is regarded arbitrary, one recovers the so called Lifshitz algebra. The Lifshitz holography has interesting peculiarities and it has been extensively studied in the past (see [4,5,11,12] and references therein).
When constructing a specific dynamical realisation of a non-relativistic conformal group, generators of the corresponding Lie algebra are linked to constants of the motion. Because the number of functionally independent integrals of motion needed to integrate a differential equation correlates with its order, dynamical realisations of an > 1 2 conformal Galilei algebra in mechanics in general involve higher derivative terms. In particular, symmetries of a higher order free particle were studied in [13] while [14] established the -conformal Galilei symmetry of the Pais-Uhlenbeck oscillator. Field theories with > 1 2 conformal Galilei symmetry remain almost completely unexplored (see however the recent studies in [15,16]).
Turning to supersymmetric extensions of the -conformal Galilei algebra, there are several competing approaches to be mentioned. One can either consider a relativistic superconformal algebra in a suitable dimension and analyse its subalgebras, or implement the non-relativistic contraction, or extend d = 1, N -extended superconformal algebra by acceleration generators in a proper way (see e.g. [17,18,19,20,21,22] for = 1 2 , 1 and [23,24,25,26,27] for an arbitrary value of ).
Given a supersymmetric generalisation of (1), sl(2, R) subalgebra formed by L n is enlarged to a proper d = 1, N -extended superconformal algebra, while the acceleration generators U p are accompanied by extra superpartners. In general, the acceleration generators and their superpartners combine together to form an irreducible supermultiplet of the d = 1, N -extended superconformal algebra. It is customary to denote such supermultiplets by the triplet (N − A, N , A), in which N − A designates the number of physical real bosons, N counts the number of physical real fermions, and A quantifies auxiliary (non-propagating) real bosons. The full number of bosons equals that of fermions. In particular, the N = 1 -conformal Galilei superalgebra in [24] introduces one fermionic partner of U p , which together form a (1, 1, 0) supermultiplet. The N = 2 superalgebras in [23,24,25] link to (1,2,1) and (2,2,0) supermultiplets. N = 4 superalgebras associated with various d = 1, N = 4 supermultiplets, including the most general case of the exceptional supergroup D(2, 1; α), were investigated in [26,27]. To the best of our knowledge, the N = 3 case has not yet been explored for an arbitrary value of the parameter .
A particularly efficient method of building an N -extended -conformal Galilei superalgebra, which encompasses the previous studies in [23,24,25,26], was proposed in [27]. Given an (N − A, N , A) supermultiplet of the d = 1, N -extended superconformal algebra, one identifies the first member of the triplet with the bosonic acceleration generator U. In general, it carries an extra index which controls the number N − A chosen. Computing the commutator of U with the supersymmetry generators entering the d = 1, N -extended superconformal algebra, one obtains N superpartners of U, say S. The anticommutator of S with the supersymmetry generators then yields a combination of a new auxiliary bosonic acceleration generator, say A, and the original U. Finally, the commutator of A and the supersymmetry generators may only result in S. The number coefficients accompanying the generators on the right hand side of the structure relations are fixed by demanding that the ensuing superalgebra is finite-dimensional and that the super Jacobi identities are satisfied.
Apart from the indices which keep the balance between the components of an (N − A, N , A) supermultiplet, the acceleration generators carry extra indices specifying the sl(2, R) transformation rules. In general, given the triplet (U, S, A), one can either assign to its members conformal weights ( , + 1 2 , + 1), or alternatively choose the descending sequence ( , − 1 2 , − 1) [27]. Thus, for each value of the parameter one can construct two variants of an N -extended -conformal Galilei superalgebra, which in general are not isomorphic.
The goal of this work is to reconsider the issue of constructing N = 1, 2, 3 -conformal Galilei superalgebras following the method in [27].
In the next Section, two N = 1 -conformal Galilei superalgebras are built by enlarging the osp(1|2) superconformal algebra with a bosonic acceleration generator of conformal weight and its superpartner which either has conformal weight − 1 2 or + 1 2 . The former case corresponds to the superalgebra in [24], while the latter is new.
Sect. 3 is devoted to N = 2 -conformal Galilei superalgebras which build upon su(1, 1|1). Variants associated with (1, 2, 1) and (2, 2, 0) irreducible supermultiplets are considered. In the former case, a new N = 2 superalgebra is obtained which, in addition to , involves an arbitrary real parameter a. The latter measures the U (1) charge of the bosonic acceleration generators. It is shown that the N = 2 superalgebra introduced in [23] corresponds to setting the parameter a to zero and choosing the descending sequence of conformal weights for the acceleration generators. For a (2, 2, 0) supermultiplet, we reproduce the superalgebra in [25] and construct its twin copy.
In Sect. 4, N = 3 -conformal Galilei superalgebras are studied. It is argued that the variants associated with irreducible supermultiplets of the type (1, 3, 2) and (3, 3, 0) (other options follow by redefinition of ) are incompatible with the super Jacobi identities. Yet, introducing extra acceleration generators, such that the full set fits a component content of a real bosonic scalar superfield in R 1|3 superspace, one can circumvent the problem. Two new N = 3 -conformal Galilei superalgebras of such a kind are built, which link to a reducible supermultiplet of OSp(3|2).
In Appendix, realisations of the N = 1, 2, 3 -conformal Galilei superalgebras in terms of differential operators in superspace are given.
Throughout the paper, summation over repeated indices is understood.

N = 1 -conformal Galilei superalgebras
Before we proceed to the construction of the N = 1 -conformal Galilei superalgebras, let us fix the structure relations of osp(1|2). They are most easily obtained by focusing on the Hamiltonian formulation of a free superparticle in R 1|1 superspace. Introducing a real bosonic canonical pair (x, p) and a self-conjugate real fermion θ, which obey the Poisson brackets {x, p} = 1, {θ, θ} = −i, one can readily verify that the functions form a closed superalgebra. Promoting the Poisson brackets to (anti)commutators, one arrives at the structure relation of osp(1|2) where L n , with n = −1, 0, 1, form sl(2, R) subalgebra and Q r , with r = − 1 2 , 1 2 , encode the supersymmetry generator and its superconformal partner. (L n , Q r ) are assumed to be Hermitian operators. In accord with (3), they have conformal weights 1 and 1 2 , respectively. In order to construct an N = 1 supersymmetric extension of the -conformal Galilei superalgebra, one extends osp(1|2) with the bosonic acceleration generator U m introduced in Eq. (1) above. Because the commutator of U m and Q r is Grassmann-odd, it should be regarded as a new fermionic generator, say S p . In general, the range of values of the lower index p correlates with conformal weight of S p . Two options are available [27]. One can either assign the conformal weight + 1 2 to S p , or alternatively choose − 1 2 . In the former case, one has with p = − − 1 2 , . . . , + 1 2 . The anticommutator of Q r and S p is then uniquely determined by the range of values of the indices and the super Jacobi identities It is assumed that U m and S p (anti)commute with each other. To the best of our knowledge, this N = 1 -conformal Galilei superalgebra is new. A twin copy of the superalgebra arises if one assigns S p with the conformal weight − 1 2 . Three structure relations are modified accordingly , while the rest remains unchanged. Written in different notations, this variant was proposed in [24].
For reader's convenience, realisations of the N = 1 -conformal Galilei superalgebras in terms of differential operators in superspace are given in Appendix.
Note that the two versions above are not entirely identical. In particular, the second option involves two less fermionic generators S p . Curiously enough, the latter follows from the former if one implements the formal substitution and interchanges commutators with anticommutators where appropriate. As the Grassmann parities of the generators get altered, the change should be regarded as a kind of duality transformation. Worth mentioning also is that, discarding the lower indices attributed to sl(2, R), the set of the acceleration generators (U, S) fits a component content of a real scalar superfield The latter links to a (1, 1, 0) supermultiplet of the OSp(1|2) superconformal group. An N = 2 -conformal Galilei superalgebra builds upon su(1, 1|1). In order to establish structure relations of the latter, it suffices to consider the Hamiltonian formulation of a free superparticle in R 1|2 superspace where (x, p) and (θ,θ), with θ † =θ, form bosonic and fermionic canonical pairs obeying the Poisson brackets {x, p} = 1, {θ,θ} = −i. Computing the brackets between the functions in (7) and quantising the result, one obtains the structure relations of su(1, 1|1) (Hermitian conjugates are omitted) It is assumed that the bosonic generators L n and J are Hermitian, while (Q r ) † =Q r . As above, L n , with n = −1, 0, 1, form sl(2, R) subalgebra, Q r , with r = − 1 2 , 1 2 , involve the supersymmetry operator and its superconformal partner, while J generates u(1) R-symmetry subalgebra. (L n , Q r ,Q r , J) have conformal weights (1, 1 2 , 1 2 , 0), respectively.

Acceleration generators versus (1, 2, 1) supermultiplet
Continuing to draw a parallel between the acceleration generators entering an N -extended -conformal Galilei superalgebra and irreducible supermultiplets of the d = 1, N -extended superconformal group, in this subsection we construct an N = 2 -conformal Galilei superalgebra inspired by a (1, 2, 1) supermultiplet. An analogy with a real bosonic superfield in where the range of values of the indices carried by the generators is prescribed by In order to specify (anti)commutation relations between (U, S,S, A) and (Q r ,Q r , J), one builds the most general expressions compatible with the conformal weights, Grassmann parities, and index ranges chosen, and analyses the super Jacobi identities. After a straightforward computation one gets along with their Hermitian conjugates. Here a is an arbitrary real parameter. Note that for an arbitrary value of a both the bosonic and fermionic acceleration generators do not commute with J. This is to be contrasted with the su(1, 1|1) subalgebra, in which only the fermions are sensitive to the R-symmetry transformation. It is assumed that (U, S,S, A) (anti)commute with each other. To the best of our knowledge, this N = 2 -conformal Galilei superalgebra is new. The descending sequence of conformal weights , − 1 2 , − 1 2 , − 1 can be considered likewise and the result reads (Hermitian conjugates are omitted) where U m : m = − , . . . , , S p : p = − + 1 2 , . . . , − 1 2 , A q : q = − + 1, . . . , − 1. In contrast to the previous case, the superalgebra (11) is defined for ≥ 1. Note that at a = 0 one recovers the N = 2 superalgebra introduced in [23].
At first glance the variants in (9), (10) and (11) are not related. Given a value of , the latter contains two less fermionic operators S p and four less bosonic elements A q . Yet, the replacement → + 1 in the descending chain of conformal weights , − 1 2 , − 1 2 , − 1 yields the ascending one read in the reverse order + 1, + 1 2 , + 1 2 , . To put it in other words, the N = 2 -conformal Galilei superalgebra defined by Eqs. (9), (10) contains the same number of generators as the N = 2 ( + 1)-conformal Galilei superalgebra in (11). The following change establishes the isomorphism of the twin copies.

Acceleration generators versus (2, 2, 0) supermultiplet
Our next example is an N = 2 -conformal Galilei superalgebra which links to a (2, 2, 0) supermultiplet of SU (1, 1|1). The first number in the braces implies that one has to consider two real bosonic acceleration generators of the conformal weight , or equivalently one complex operator The Hermitian conjugateŪ m = (U m ) † , with m = − , . . . , , obeys the analogous equation. The commutator of Q r from su(1, 1|1) and U m yields a complex fermionic generator, which is to be regarded as the analogue of two propagating real fermions entering a (2, 2, 0) supermultiplet. Assigning the superpartner of U m with the conformal weight + 1 2 , one has (Hermitian conjugates are omitted) with S p : p = − − 1 2 , . . . , + 1 2 . As to the remaining (anti)commutators, one considers the most general expressions compatible with the conformal weights, Grassmann parities, and index ranges chosen, and requires the fulfilment of the super Jacobi identities. The result reads (Hermitian conjugates are omitted) It is assumed that (U,Ū, S,S) (anti)commute with each other. To the best of our knowledge, this superalgebra has not yet been presented in the literature. Its existence was envisaged by S. Krivonos in [27] (see a footnote on p. 2).
Turning to the alternative in which the superpartner of the bosonic acceleration generator has the conformal weight − 1 2 , one finds with S p : p = − + 1 2 , . . . , − 1 2 . A similar consideration of other (anti)commutation relations gives along with their Hermitian conjugates. Written in different notations, this variant of the N = 2 -conformal Galilei superalgebra was proposed in [25]. Note that the superalgebra can be derived from the variant above by applying a formal substitution (which similarly to that in Sect. 2 alters the Grassmann parity) and interchanging commutators with anticommutators where appropriate. For reader's convenience, realisations of the N = 2 -conformal Galilei superalgebras in terms of differential operators in superspace are given in Appendix.

Structure relations of osp(3|2)
N = 3 -conformal Galilei superalgebras build upon osp(3|2). The simplest way to establish structure relations of the latter is to consider the Hamiltonian formulation of a free superparticle moving in R 1|3 superspace which is parametrized by a real bosonic coordinate x and real fermions θ a , a = 1, 2, 3. Introducing the momentum p canonically conjugate to x, regarding θ a as self-conjugate variables, and imposing the brackets {x, p} = 1, {θ a , θ b } = −iδ ab , one can verify that the set of functions where abc is the Levi-Civita symbol, forms a closed superalgebra. Promoting the brackets to (anti)commutators in the usual way, one obtains the structure relations of osp(3|2) As above, L n , with n = −1, 0, 1, generate the conformal subalgebra sl(2, R), Q a r , with r = − 1 2 , 1 2 and a = 1, 2, 3, encode the supersymmetry operator and its superconformal partner, while J a , with a = 1, 2, 3, generate so(3) R-symmetry subalgebra. As follows from (20), (L n , Q a r , J a ) have conformal weights (1, 1 2 , 0), respectively. All operators are assumed to be Hermitian.
Similarly to the analysis above, one can try to construct N = 3 -conformal Galilei superalgebras which link to (1,3,2) and (3, 3, 0) supermultiplets (the options (2, 3, 1) and (0, 3, 3) follow by redefinition of ). In the former case, the starting point would be the triplet of acceleration generators (U, S a , A), where U and its superpartner S a , with a = 1, 2, 3, are real, while A is complex. Assigning them with the conformal weights ( , + 1 2 , + 1) and analysing the most general structure relations compatible with the Grassmann parities, one finds out that the super Jacobi identities do not hold in the sector involving two operators of the type Q a r and one acceleration generator from the triplet (U, S a , A). Similar problem occurs for a (3, 3, 0) supermultiplet, which would be represented by the Hermitian operators (U a , S a ), with a = 1, 2, 3.
A natural way out is to turn to reducible supermultiplets of OSp(3|2), the simplest of which is a real bosonic scalar superfield defined in R 1|3 superspace where U (x), A c (x) are Grassmann-even and S a (x), P (x) are Grassmann-odd. Below we construct two new N = 3 -conformal Galilei superalgebras, acceleration generators of which fit the reducible supermultiplet (U, S a , A a , P ).

4.2.
Acceleration generators of conformal weights ( , + 1 2 , + 1, + 3 2 ) As usual, one starts with a Hermitian operator U m of the conformal weight As U m does not carry so(3) indices, it commutes with J a . The commutator of U m with the fermionic operators Q a r is Grassmann-odd and, hence, should be regarded as a new fermionic Hermitian generator of the conformal weight + 1 with S a p : p = − − 1 2 , . . . , + 1 2 . The structure of so(3) indices entering the anticommutator {Q a r , S b s } and the fact that it is Grassmann-even suggest the structure relation where A a q : q = − − 1, . . . , + 1, with a = 1, 2, 3, is a new bosonic Hermitian generator to be included into the ensuing superalgebra. The factor − ((2 + 1) r − p) entering the first term is so designed as to keep the index carried by U from leaving the range − , . . . , , as well as to guarantee the fulfilment of the super Jacobi identities.
In its turn, the extra bosonic generator A a q has the conformal weight + 1 and carries so(3) vector index, which imply while the commutator of Q a r and A b q produces a new fermionic Hermitian generator P s with P s : s = − − 3 2 , . . . , + 3 2 . Similarly to (23), the meaning of the factor i ((2 + 2) r − q) entering the second term is to balance in a proper way the range of values of indices carried by the acceleration generators on both sides of the equality (25) as well as to ensure the super Jacobi identities.
Finally, the conformal weights of Q a r and A b m add up to yield that of P s while the structure of indices entering the anticommutator {Q a r , P s } implies The factor − ((2 + 3) r − s) harmonises the range of values of indices on both sides of (27) and conforms to super Jacobi identities. It is assumed that [J a , P r ] = 0 and the acceleration generators commute with each other.

Acceleration generators of conformal weights
where it is assumed that ≥ 3 2 . The range of values of the lower indices carried by the acceleration generators is specified by It is straightforward to verify that the super Jacobi identities are satisfied. Note that the two variants of the N = 3 l-conformal Galilei superalgebra obtained in this section are related to each other by a duality transformation similar to (6) and (18) → Realisations of the N = 3 -conformal Galilei superalgebras in terms of differential operators in superspace can be found in Appendix.

Conclusion
To summarise, in this work the issue of constructing N = 1, 2, 3 supersymmetric extensions of the -conformal Galilei algebra was reconsidered along the lines in [27]. A new N = 1 version, two new N = 2 superalgebras, and two new N = 3 variants were built. Their realisations in terms of differential operators in superspace were given.
Turning to possible further developments, it would be interesting to study dynamical realisations of the superalgebras proposed above. Only a limited number of such examples are available (see e.g. [28,29,30] for = 1 2 and [31,32] for an arbitrary value of ) and a deeper understanding of peculiarities of an N -extended -conformal Galilei supersymmetry is desirable. In particular, it would be interesting to build an N = 2 model that would assign a clear physical meaning to the parameter a introduced in Sect. 3.
Above it was assumed that the acceleration generators commute with each other. Analysis of possible central extensions is an interesting task. A related open problem is the construction of Casimir elements.
In this work, only finite-dimensional superalgebras were considered. Allowing the index n of L n to carry any integer value, the index r characterising Q r to take any half-integer value and similarly for the acceleration generators, one would automatically generate infinitedimensional extensions. A possibility to use them within the context of superconformal field theory is worth studying.
As is well known, the Galilei algebra can be obtained from the Poincaré algebra by applying the non--relativistic contraction. A similar derivation of the -conformal Galilei algebra is available only for = 1 2 [34] and = 1 [35], which relies upon so(2, d + 1). It would be interesting to study in which way the full tower of acceleration generators for an arbitrary value of can be obtained from so(p, q) (with properly adjusted p and q) by applying the Inönü-Wigner contraction and whether such a consideration can be extended to encompass the supersymmetric cases studied in this work.
A variant corresponding to the ascending sequence of conformal weights is obtained via the isomorphism (12).