Remarks on D(2,1;a) super-Schwarzian derivative

It was recently demonstrated that N=1,2,3,4 super-Schwarzian derivatives can be constructed by applying the method of nonlinear realisations to finite-dimensional superconformal groups OSp(1|2), SU(1,1|1), OSp(3|2), SU(1,1|2), respectively, thus avoiding the use of superconformal field theory techniques. In this work, a similar construction is applied to the exceptional supergroup D(2,1;a), which describes the most general N=4 supersymmetric extension of SL(2,R), with the aim to study possible candidates for a D(2,1;a) super-Schwarzian derivative.


Introduction
Over the past few decades, extensive studies of the AdS/CF T -correspondence brought into focus a plethora of interesting field theories, which hold invariant under conformal transformations. Most recent in the chain of dualities is the Schwarzian theory which connects to two-dimensional models of gravity. 1 In this context, the Schwarzian derivative and its supersymmetric counterparts play the central role.
The super-Schwarzian derivatives were introduced within the framework of N -extended superconformal field theory in the mid-eighties [3,4,5,6]. They showed up when computing the anomalous term in a finite superconformal transformation of the super stress-energy tensor. Mathematicians link them to central extensions of infinite-dimensional Lie superalgebras (see [7] and references therein), which also revealed the bound N ≤ 4 on an admissible number of supersymmetry parameters [8].
As is well known [14], the most general N = 4 supersymmetric extension of SL(2, R) is given by the exceptional supergroup D(2, 1; a). Structure relations of the corresponding Lie superalgebra (see Appendix B) involve an arbitrary real parameter a. To the best of our knowledge, despite the fact that D(2, 1; a) admits an infinite-dimensional extension [15], no attempt was made to build a D(2, 1; a) super-Schwarzian derivative by applying superconformal field theory methods.
The goal of this work is to extend the recent group-theoretic analysis of the SU (1, 1|2) super-Schwarzian derivative in [11], which corresponds to a = −1, to the case of arbitrary 1 The literature on the subject is rather extensive. For good recent accounts and further references see [1,2].
2 A more precise definition of a superconformal diffeomorphism is given below in Sect. 2. Note also that, because conformal transformations in R 1|N involve the inversion t → 1 t , SL(2, R) does not act globally on R 1|N , but rather on S 1|N . values of a. Surprisingly enough, the construction of a D(2, 1; a) super-Schwarzian derivative obeying the two defining properties mentioned above turns out to be problematic. Yet, a reasonable alternative is proposed and its properties are established.
The paper is organised as follows. In Sect. 2, a superconformal diffeomorphism t = ρ(t, θ,θ), θ α = ψ α (t, θ,θ), where t is a real Grassmann-even variable and θ α are complex Grassmann-odd coordinates with α = 1, 2, is considered. It is argued that the conventional chirality constraint on the fermionic superfieldD α ψ β = 0, which normally follows from the condition that the covariant derivative transforms homogeneously under the superconformal isomorphisms, i.e. D α = (D α ψ β )D β , turns out to be incompatible with D(2, 1; a) symmetry. A weaker condition is proposed, The requirement that the covariant derivatives algebra is preserved under the superconformal diffeomorphism gives rise to extra quadratic constraints on ψ α andψ α .
In Sect. 3, infinitesimal D(2, 1; a) transformations acting upon the form of the superfields ρ and ψ α are constructed by applying the method of nonlinear realisations. They are subsequently used for verifying the invariance of constraints imposed upon ψ α as well as for analysing the infinitesimal limit of finite D(2, 1; a) transformations, which result from setting a generalised D(2, 1; a) super-Schwarzian derivative to zero.
Sect. 4 is devoted to the construction of D(2, 1; a) invariants along the lines in [11]. First, each generator in the corresponding Lie superalgebra is accompanied by a Goldstone superfield of the same Grassmann parity and a conventional group-theoretic elementg is introduced. Then the odd analogues of the Maurer-Cartan invariantsg −1 D αg are computed. They are subsequently used to link some of the Goldstone superfields enteringg to a single fermionic superfield ψ α as well as to construct a D(2, 1; a) analogue of the N = 4 super-Schwarzian derivative in [6,11]. It is demonstrated that such a candidate lacks the conventional composition law unless extra nonlinear constraints are imposed upon ψ α .
In Sect. 5, a covariant projection method is used to solve the nonlinear constraints explicitly. A finite form of a D(2, 1; a) transformation acting in the Grassmann-odd sector of R 1|4 is obtained. It is argued that a natural generalisation of the N = 4 super-Schwarzian derivative to the D(2, 1; a) case turns out to be trivial. As a matter of fact, it is superseded by one of the nonlinear constrains, which was imposed upon ψ α when securing the composition law in Sect. 4.
An alternative definition of a D(2, 1; a) super-Schwarzian derivative and its peculiar features are discussed in Sect. 6. The composition law and the change under D(2, 1; a) transformations are established, which resemble the way in which the covariant derivative is transformed under the generalised superconformal diffeomorphism. It is suggested that the difficulty in defining a D(2, 1; a) super-Schwarzian derivative with conventional properties is related to the fact that the chirality condition on the fermionic superfield ψ α is incompatible with D(2, 1; a) symmetry.
In Sect. 7, we draw a parallel with the N = 3 case, which was recently studied in [12] within a similar framework. Like for D(2, 1; a), the basic fermionic superfield does not obey the chirality condition. A generalised N = 3 super-Schwarzian derivative is introduced and its properties are established. It is shown that the conventional N = 3 super-Schwarzian derivative [5] can be constructed in terms of the generalised object. A new OSp(3|2) invariant is proposed.
In the concluding Sect. 8, we summarise our results and discuss possible further developments.
Appendix A contains our spinor conventions. Structure relations of the Lie superalgebra associated with the exceptional superconformal group D(2, 1; a) are gathered in Appendix B.
The D(2, 1; a) Maurer-Cartan invariants are exposed in Appendix C. Throughout the paper, summation over repeated indices is understood.

Generalised superconformal diffeomorphisms
An N -extended super-Schwarzian derivative [3,4,5,6] is intimately connected with finitedimensional superconformal transformations acting in R 1|N superspace parametrized by (t, θ i ), i = 1, . . . , N . In order to obtain the latter from the former, one considers a generic superdiffeomorphism t = ρ(t, θ), which is specified by a real bosonic superfield ρ and a set of real fermionic superfields ψ i , and then confines oneself to a subgroup of superconformal diffeomorphisms, under which the covariant derivative D i transforms homogeneously [3] Eq.
(2) yields constraints on ρ and ψ i (see the discussion in [5]). In particular, ρ is fixed provided ψ i is known. Note that for even N one usually introduces complex Grassmann-odd coordinates θ α , α = 1, . . . , N 2 , in which case (2) implies the chirality conditionD α ψ β = 0. Having solved the constraints, which follow from (2), one then identifies ψ i with the argument of an N -extended super-Schwarzian derivative and sets the latter to vanish. The resulting fermionic superfield coincides with a finite-dimensional superconformal transformation acting in the Grassmann-odd sector of R 1|N superspace, while ρ is fixed from (2) provided ψ i is known.
A natural way out is to admit the weaker conditions where D α ,D α are the covariant derivatives in R 1|4 superspace which obey where ∂ t = ∂ ∂t . Note that, after implementing the generalised superconformal diffeomorphism (3), the covariant derivatives D β andD β get mixed up. This fact will have an impact on the properties of a generalised D(2, 1; a) super-Schwarzian derivative to be introduced below.
From Eq. (3) one gets the constraints which allow one to fix ρ provided ψ α is known. Here and below we use the abbreviations Our convention for other contractions similar to (8) is that spinor indices entering the first factor on the right hand side stand in their natural position, i.e. ψ α ,ψ α , D α ,D α . The algebra of the covariant derivatives (5) along with the generalised homogeneity conditions (3) yield further quadratic restrictions where Eq. (7) and the identity The explicit solution to these equations will be discussed in Sect. 5.
In what follows, two corollaries of the last equation in (9) will be extensively used. The complex conjugation rules where ψ α is a complex fermionic superfield and ρ is a real bosonic superfield, will prove helpful as well.
Note that for a = −1 the supergroup D(2, 1; a) simplifies to SU (1, 1|2) × SU (2), which is easily seen by inspecting the structure relations of the corresponding Lie superalgebra (see Appendix B). In this particular case, the chirality conditionD α ψ β = 0 proves to be compatible with SU (1, 1|2) symmetry and the homogeneous transformation law for the covariant derivative D α = (D α ψ β )D β . The constraints (9) then reduce to (D α ψ γ ) D βψ γ = 1 2 δ β α DψDψ , which, in their turn, are equivalent to the linear equation D α ψ β +D αψ β = 0 [11]. For generic values of a, D(2, 1; a) involves transformations which interchange ψ α and ψ α . For this reason, the chirality condition would be too restrictive and one is led to deal with the whole set of nonlinear constraints (9).

Infinitesimal D(2, 1; a) transformations
In this section, we construct infinitesimal D(2, 1; a) transformations acting upon the form of the superfields ρ and ψ α introduced in the preceding section. They will prove useful when verifying the invariance of constraints, to be imposed upon ψ α later, as well as for analysing the infinitesimal limit of finite D(2, 1; a) transformations, which result from setting a generalised D(2, 1; a) super-Schwarzian derivative to zero. Technically, it suffices to apply the method of nonlinear realisations [13] to D(2, 1; a) and treat ρ and ψ α as Goldstone superfields associated with the generators of translation and supersymmetry transformation, respectively.
As the first step, one considers the structure relations of Lie superalgebra associated with D(2, 1; a) (see Appendix B) as well as the d = 1, N = 4 supersymmetry algebra {q α ,q β } = 2hδ α β . Then each generator of the former is accompanied by a Goldstone superfield of the same Grassmann parity, while coordinates of R 1|4 superspace are linked to h and q α . Afterwards, the group-theoretic element is introduced where (P, D, K) are bosonic generators of translations, dilatations, and special conformal transformations, respectively. J l , with l = 1, 2, 3, generate the R-symmetry subalgebra su(2). One more su(2) is realised by I ± , I 3 , for which the Cartan basis is chosen. Q α and S α are fermionic generators of supersymmetry transformations and superconformal boosts,Q α , S α being their Hermitian conjugates. Accordingly, (ρ, µ, ν, λ l , k 3 ) are real bosonic superfields, k − and k + are complex conjugates of each other, while (ψ α ,ψ α ) and (φ α ,φ α ) form complex conjugate fermionic pairs. In what follows, ρ and ψ α are identified with those in the preceding section.

A group-theoretic analysis
In this section, we extend a recent analysis of SU (1, 1|2) [11] to the case of D(2, 1; a). Our primary concern is to construct the analogues of the Maurer-Cartan invariants, which gave rise to the SU (1, 1|2) super-Schwarzian derivative in [11].
In the previous section, the group-theoretic element (12) was introduced. By making use of the covariant derivative D α , one can build the Grassmann-odd analogues of the Maurer-Cartan one-forms The superfields (ω P ) α , . . . , (ωS) β α turn out to be rather bulky and are displayed in Appendix C. By construction,g −1 D αg hold invariant under the transformation (13) and, hence, (ω P ) α , . . . , (ωS) β α provide D(2, 1; a) invariants. They can be used to impose constraints which allow one to eliminate some of the superfields entering (12) from the consideration as well as to study candidates for a D(2, 1; a) super-Schwarzian derivative. Note thatg −1D αg results in the complex conjugate invariants.
In a series of recent works [10,11,12], it was suggested to choose the constraints so as to link all the superfields entering the group-theoretic element (12) to ρ and ψ α . Unfortunately, for the case at hand it proved difficult to handle the triplet (k ± , k 3 ). Below, we follow a broader road and impose just as many constraints as is needed to build an analogue of the SU (1, 1|2) super-Schwarzian derivative in [6,11].
All the reservations made, let us discuss the Maurer-Cartan invariants. Comparing (ω P ) α in Appendix C with Eq. (6) above, one concludes that it vanishes. Had we not chosen to impose (6) earlier, it might have been obtained here by setting (ω P ) α = 0.
A study of the D(2, 1; −1) case in [11] showed that the fermionic superfield ψ α was to be chiral. Within the method of nonlinear realisations, a suitable constraint was provided by ωQ β α = 0. Imposing a similar condition for generic a, one gets where k = k − k + . These equations allow one to express k tan k k ± in terms of ψ α , but more importantly, they imply an extra (complex) quadratic constraint on the fermionic superfield which, in its turn, has an impact on the restrictions (9) revealed above. The first two conditions in (9) are now satisfied identically, while the last equation simplifies to 3 which means that, up to a scalar superfield factor, D α ψ γ is a unitary matrix. Given the infinitesimal D(2, 1; a) transformations (15), it is straightforward to verify that (18) and (19) do hold invariant.
Recall that for a = −1 the superfield ψ α can be chosen chiral. Hence, (18) is irrelevant in that case, while (19) guarantees that the covariant derivatives algebra {D α ,D β } = 2iδ β α ∂ t is preserved under the superconformal diffeomorphism D α = (D α ψ β )D β . As was mentioned above, the chirality condition turns out to be incompatible with D(2, 1; a) symmetry and the meaning of the novel restriction (18) is that D αψ β is proportional to D α ψ β . A group-theoretic derivation of the SU (1, 1|2) super-Schwarzian derivative in [11] revealed that it is constructed in terms of the dilaton superfield ν, which, in its turn, is linked to ψ α by means of constraints. Let us carry out a similar analysis for an arbitrary value of a.
Focusing on the candidate (25) and taking into account the identity where (3) and and are imposed. 5 Taking into account that the infinitesimal transformations (15) leave DψDψ + DψDψ invariant but for the dilatation, special conformal transformation, and superconformal boost 4 Our presentation here is schematic. External indices carried by a super-Schwarzian derivative and Grassmann-odd coordinates are omitted. 5 Note that (30) and (31) could be obtained from a more general D(2, 1; a)-invariant constraint one can verify that (29), (30), and (31) do hold invariant. Note that Eq. (29) implies that, up to a scalar superfield factor, D α ψ β is the same asD αψ β . In particular, the quadratic constraint (19) is now satisfied identically.

Consider a component decomposition of a generic fermionic superfield defined on
Here Latin letters stand for bosonic components, Greek letters designate fermionic ones, and (σ a ) α β , a = 1, 2, 3, are the Pauli matrices (see Appendix A). The complex conjugate partner readsψ where In order to solve the superfield constraints (18), (29), (30), we use the covariant projection method, in which components of a superfield are linked to its covariant derivatives evaluated at θ α =θ α = 0. For the case at hand, one finds 6 where Computing the covariant derivatives of (18), (29), (30) and taking into account (37), after rather tedious calculation, one gets Here w a is a real vector parameter generating SU (2) group. Constant parameters (v, q,q), of which v is real while q andq are complex conjugate to each other, give rise to another SU (2). Note that in obtaining Eq. (38), the following identities were repeatedly used. For a real bosonic function u(t) entering (38) one reveals the differential equation where c 0 and c 1 are constants of integration. Along with an additive constant of integration, which occurs when solving (7), c 0 and c 1 generate SL(2, R) transformations. The complex fermionic function α σ (t) in (38) is found to obey two differential equations where the Grassmann-odd parameters σ and κ σ are associated with the global supersymmetry transformations and superconformal boosts, respectively. The resulting superfield (34) determines a finite D(2, 1; a) transformation acting in the Grassmann-odd sector of R 1|4 superspace, which correctly reduces to (15) in the infinitesimal limit. In particular, in order to reproduce the infinitesimal form of the superconformal boosts entering (15), one sets c 0 = 1, considers c 1 to be small, such that 1 1+c 1 t ≈ 1−c 1 t, and identifies c 1 (κκ)κ γ with the infinitesimal κ γ in (15). The resulting transformation is a superposition of the supersymmetry transformation, special conformal transformation parametrized by c 1 and the superconformal boost associated with c 1 (κκ)κ γ . A finite D(2, 1; a) transformation acting in the Grassmann-even sector of R 1|4 can be found by integrating (7).
It has to be kept in mind, however, that for a = −1 and chiral ψ α , the extra condition (30) vanishes and (33) comes onto the scene as the legitimate SU (1, 1|2) super-Schwarzian derivative.
6. An alternative candidate That (33) is an unsuitable candidate for a D(2, 1; a) super-Schwarzian derivative comes as a nasty surprise, which forces us to take a step back and reconsider the material in Sect. 4.
Let ψ α be a complex fermionic superfield, which is subject to two quadratic constraints As was mentioned above, the former is the analogue of the chirality condition, while the latter guarantees that the covariant derivatives algebra is preserved under the generalised superconformal diffeomorphism. Our analysis in the preceding section shows that, although Eqs. (41) relate some of the components in (34) to each other, they do not fix u(t) and α σ (t) and, hence, leaves one with an infinite-dimensional group of transformations. Let us introduce the third-rank tensor which is symmetric in the first pair of indices, and its complex conjugate partner As follows from our analysis in the preceding section, setting S (αβ)γ to vanish one reduces the infinite-dimensional group of superconformal isomorphisms to the finite-dimensional subgroup D(2, 1; a).
Another unconventional feature of S (αβ)γ is that it does not hold exactly invariant under a finite D(2, 1; a) transformation acting upon the argument ψ α (t, θ,θ) → ψ α (t , θ ,θ ). Rather, it transforms covariantly This is an immediate consequence of (42) and the fact that S (αβ)γ vanishes when acting upon the fermionic superfield, which determines a D(2, 1; a) transformation in the Grassmann-odd sector of R 1|4 . Note that (46) resembles the transformation law of the covariant derivative under the generalised superconformal diffeomorphism (3). It is rather likely that the difficulty in defining a D(2, 1; a) super-Schwarzian derivative with conventional properties is related to the fact that the chirality condition on the fermionic superfield ψ α is incompatible with D(2, 1; a) symmetry. Above, it was this point which led us to admit the mixed transformation rule (3).

A parallel with the N = 3 case
In a recent work [12], the N = 3 super-Schwarzian derivative [5] was constructed by applying the method of nonlinear realisations to the finite-dimensional superconformal group OSp(3|2). In this section, we discuss an N = 3 analogue of the generalised super-Schwarzian derivative formulated in the preceding section.
Similarly to the material in Sect. 2, a superconformal diffeomorphism of R 1|3 is determined by a real bosonic superfield ρ and a triplet of real fermionic superfields ψ i , i = 1, 2, 3, which give rise to a coordinate transformation t = ρ(t, θ), θ i = ψ i (t, θ), under which the covariant derivative D i = ∂ ∂θ i − iθ i ∂ ∂t transforms homogeneously D i = (D i ψ j ) D j . The latter condition and the fact that the algebra of covariant derivatives {D i , D j } = −2iδ ij ∂ ∂t is preserved, yield constraints upon ρ and ψ i (for more details see [12]) where the dot designates the derivative with respect to t and DψDψ = (D i ψ j ) (D i ψ j ). An N = 3 analogue of (42) reads which is antisymmetric in the first pair of indices. Considering a superconformal diffeomorphism t = ρ(t, θ), θ i = ψ i (t, θ) and changing the argument ψ i (t, θ) → Ω i (t , θ ), one finds the generalised composition law where the identity (DΩDΩ) = 1 3 (DψDψ) (D ΩD Ω) was used. Note that Eq. (48) can be obtained by analogy with the D(2, 1; a) case. Constructing the Maurer-Cartan invariants similar to those in Appendix C and specifying to (ω S ) ik (ω Q ) jk − (ω S ) jk (ω Q ) ik one obtains a candidate for an N = 3 super-Schwarzian derivative (for more details see [12]) Then one reveals that the latter does not obey a composition law unless the constraint It remains to analyse the equation S [ij]k [ψ(t, θ); t, θ] = 0 and verify that the resulting ψ i does specify a finite OSp(3|2) transformation acting in the Grassmann-odd sector of R 1|3 . Considering a component decomposition where (b ai , g i ) are Grassmann-even functions of t, (α i , β abi ) are their Grassmann-odd partners, and abc is the Levi-Civita symbol, one first computes the covariant projections 7 Then one solves the quadratic constraint (D j ψ k ) = 1 3 δ ij DψDψ in (47), which yields [12] b ij = u(t)exp(ξ) ij , where u(t) is an arbitrary bosonic function of t, the matrixξ ij = ξ k kij involves a real bosonic vector parameter ξ k such that exp(ξ) ij = exp(−ξ) ji , and b 2 = b ij b ij . Note that, similarly to D i ψ j , the bosonic component b ij obeys the equation b ik b jk = 1 3 δ ij b 2 , which means that the parameter ξ k represents a finite SO(3)-transformation.
Note that, guided by the N = 3 analogy, one could try to contract S (αβ)γ andS (αβ)γ in the preceding section with the covariant derivatives of ψ γ andψ γ in an attempt to form a D(2, 1; a) super-Schwarzian derivative, obeying the conventional composition law (27). A close inspection shows that, while the expression aD λψ γ + bD λψ γ S (αβ)γ + aD λ ψ γ + bD λ ψ γ S (αβ) where a, b are constants, does hold invariant under finite D(2, 1; a) transformations, its fails to produce a reasonable transition matrix M (see (27)).

Conclusion
To summarise, in this work we extended a recent group-theoretic analysis of the N = 4 super-Schwarzian derivative [11] to the case of the exceptional supergroup D(2, 1; a). The latter describes the most general N = 4 supersymmetric extension of the conformal group in one dimension SL(2, R). An analogue of the N = 4 super-Schwarzian derivative was built in terms of the Maurer-Cartan invariants. It was demonstrated that it lacked the conventional composition law unless extra nonlinear constraints were imposed upon the argument. The explicit solution of the constrains then showed that the natural candidate proved trivial and had to be superseded by one of the nonlinear constraints. An alternative candidate for a D(2, 1; a) super-Schwarzian derivative was proposed and its properties were established. A parallel with the N = 3 case was drawn and a generalised N = 3 super-Schwarzian derivative was proposed. A new OSp(3|2) invariant was constructed. Turning to possible further developments, it would be interesting to explore whether the alternative candidate for a D(2, 1; a) super-Schwarzian derivative proposed in Sect. 6 can be supported by superconformal field theory computations based upon an infinite-dimensional extension of D(2, 1; a) [15]. A possibility to use it for constructing a D(2, 1; a) supersymmetric extension of the Sachdev-Ye-Kitaev model is worth studying as well.
As is known, the supergroups SU (1, 1|n), Osp(n|2), and Osp(4 * |2n) involve SL(2, R) subgroup. It would be interesting to investigate whether consistent super-Schwarzian derivatives can be associated to them. Note that a direct solution of superfield constraints may turn out to be problematic in that context, as the component decomposition becomes more involved with n growing.
Another interesting open problem is a formulation of super-Schwarzian derivatives in superspace with universal cosmological attraction or repulsion [17]. Here a is a real parameter and (σ c ) β α are the Pauli matrices. Note that for a = −1 the superalgebra reduces to su(1, 1|2) ⊕ su (2).