(super)Schwarzian mechanics

In this paper we revisit the construction of supersymmetric Schwarzians using nonlinear realizations. We show that ${\cal N}=0,1,2,3,4$ supersymmetric Schwarzians can be systematically obtained as certain projections of Maurer-Cartan forms of superconformal groups after imposing simple conditions on them. We also present the supersymmetric Schwarzian actions defined as the integrals of products of Cartan forms. In contrast with the previous attempts to obtain the super-Schwarzians within nonlinear realizations technique, our set of constraints does not impose any restriction on the super-Schwarzians.


Introduction
The holographic relationship between the Sachdev-Ye-Kitaev model [1,2] and Jackiw-Teitelboim gravity [3,4] gives rise to numerous research activities to the Schwarzian action (see e.g. [5]), which provides the boundary description of the bulk JT gravity. The Schwarzian derivative {t, τ } defined as itself appears in seemingly unrelated fields of physics and mathematics (see e.g. [6]). The action of the bosonic Schwarzian mechanics reads (see e.g. [5]) The presence of m 2ṫ2 term in the action (1.2) modifies the realization of SL(2, R) symmetry.
The simplest way to understand the modification is to notice that the action (1.2) can be represented as [5] S (1.5) and, therefore, the action (1.2) possesses the SL(2, R) invariance via with F [τ ] defined in (1.5).
Being invariant under d = 1 conformal transformation, the Schwarzian derivative naturally appears in the transformations of the conformal stress tensor T (z) [7] T (z) = dz dz 2T (z) + {z, z} . (1.7) The N =1, 2, 3, 4 supersymmetric generalization of the Schwarzian derivative are present in the transformation properties of the current superfield J (N ) (Z) generating N -extended superconformal transformations [8]. Thus, we have complete zoo of the supersymmetric Schwarzians.
The treatment of the supersymmetric Schwarzians as the anomalous terms in the transformations of the currents superfield J (N ) (Z) [8] leads to the conclusion that the structure of the (super)Schwarzians is completely defined by the conformal symmetry and, therefore, should exist a different, probably purely algebraic, way to define the (super)Schwarzians. The main property of the (super)Schwarzians, which defines their structure, is their invariance with respect to (super)conformal transformations. The suitable way to construct (super)conformal invariants is the method of nonlinear realizations [9][10][11][12] equipped with the inverse Higgs phenomenon [13]. Such approach demonstrated how the Schwarzians can be obtained via the non-linear realizations approach, was initiated in [14] and then it was applied to different superconformal algebras in [15][16][17][18]. Later on, this approach has been extended to the cases of non-relativistic Schwarzians and Carroll algebra [19].
The reason to prefer the non-linear realizations approach to construction of supersymmetric Schwarzians to the approach related to the superconformal transformations is much wider area of its applications. Indeed, the non-linear realization method works perfectly for any (super)algebra and the set of invariant Cartan forms can be easily obtained. Thus, the main questions in such approach are • What is the role and source of the "boundary" time τ and its supersymmetric partners?
• What constraints have to be imposed on the Cartan forms? What forms nullified and how to construct the action from the surviving forms?
• What additional technique can be used to simplify the calculations?
Of course, these questions were partially analyzed and answered in the papers [14][15][16][17][18]. However, some important properties and statements were missing. Moreover, the constraints proposed in these papers looks like the results of illuminating guess. The main puzzle is the fact that the constraints were imposed on the fermionic projections of the forms, but not on the forms themselves. Thus, the questions why it is so and what happens with the full Cartan forms after imposing of such constraints have been not fully analyzed. Finally, in the cases of more complicated superconformal groups the calculations quickly become a rather cumbersome and the standard technique does not help.
In this paper we try to answer these questions. Firstly, we introduce the "boundary super-space", where supersymmetry is realized on even and odd coordinates in standard way. Secondly, the constraints will be imposed on the full Cartan forms by either nullifying them or identifying with the "boundary" forms. Finally, in the complicated situations we will invoke into game the Maurer-Cartan equations and will demonstrate their usefulness. In particular, we will show that there is only one invariant, N =3 super Schwarzian, in the case of N =3 superconformal symmetry.
2 Three steps towards Schwarzian. N =0 case In this Section we will repeat the construction of bosonic N =0 Schwarzian within nonlinear realization approach [14]. Mainly, while repeating the steps discussed in [14], we will point the reader's attention at the differences between our approach and those one presented in [14].

2.1
Step one: the sl(2, R) algebra The bosonic conformal group in d = 1 is infinite-dimensional. Its finite dimensional sl(2, R) subalgebra spanned by the Hermitian generators of translation P , dilatation D and conformal boost K, can be fixed by the following relations If we parameterized the SL(2, R) -group element g as then the Cartan forms would read [20] ω P = e −u dt, ω D = du − 2zdt, ω K = e u dz + z 2 dt + m 2 dt .
The infinitesimal sl(2, R) transformations g → g ′ = e i aP e i bD e i cK g (2.5) leaving the forms (2.4) invariant read At this step our consideration differs from those in [14] only by the presence of the term m 2 K in the group element (2.2). This additional term generates the m-dependent terms in the Cartan forms (2.4) and in the transformations (2.6). As we already discussed in the Introduction this modification is not important.

Step two: invariant inverse Higgs conditions
All Cartan forms in (2.4) are invariant with respect to sl(2, R) transformations (2.6). Notice, within the nonlinear realization approach we implicitly mean that the "coordinates" u and z are functions depending on time t. However, neither "time" t, nether its differentials dt are invariant under sl(2, R) transformations (2.6). Thus, to get the invariants one has to introduce the "boundary time" τ 1 and parameterize the form ω P as Let us stress again that the τ is a new "boundary time" which completely inert under sl(2, R) transformations. Correspondingly, the rest sl(2, R) forms now read Now, nullifying the form ω D we will express the field z(τ ) in terms of dilaton u(τ ) and then, using (2.7), in terms of "old time" t: This is particular case of the Inverse Higgs phenomenon [13].

Step three: the Schwarzian action
After the Second step only one field, "old time" t(τ ), and only one invariant, form ω K , remain. Form ω K now reads 2 Thus, the Schwarzian action (1.2) can be re-obtained within our approach as It proves useful to rewrite the form ω K and, therefore, the Schwarzian action (2.11) in terms of dilaton u(t) and "old time" variable t Thus, formally speaking, the action of Schwarzian mechanics is just the action of one dimensional harmonic oscillator rewritten in terms of time variable t depending on new inert time variable τ . Note, one may always change the variables t, z, u to the new onest,z,ũ by passing from the parametrization (2.2) to m = 0 one e i t(P +m 2 K) e i zK e i uD = e itP e izK e iũD (2.13) It is easy to check that these two parametrizations related as mt = tan(m t). (2.14) Clearly, this is just the transformation (1.5). Until now our consideration, being purely bosonic one, coincided with those presented in [14]. However, the generalization to the supersymmetric case will contain some new features. To obtain supersymmetric Schwarzians, one has to consider the proper superalgebra, which differs from (2.1) by the presence of supercharges Q i , superconformal charges S i and, possibly, internal symmetry generators J ij . The new ingredients include the introduction of the superconformally inert "boundary" superspace with coordinates τ and θ i using the relations where the forms △τ and dθ i are invariant with respect to standard superspace transformations δτ ∼ ǫθ, δθ ∼ ǫ. The crucial property of the conditions (2.15) is that they include the Cartan forms ω P and ω i Q themselves. Therefore, their invariance under superconformal transformations is manifest. After imposing condition ω D = 0 also, one should obtain that the remaining forms are composed of supersymmetric Schwarzians and their derivatives.
Note that the forms at the right hand side of equations (2.15) are not arbitrary but constrained by the Maurer-Cartan equations of the respective superconformal algebra. If △τ is assumed to be generalization of dτ , these equations imply that the forms at the right hand side of equations (2.15) should be standard invariant forms on superspace. This question is studied in detail in Appendices A, B and C.

N =1 Schwarzian
The N =1 super-Schwarzian was firstly introduced in [21]. Then it has been reproduced in [8]. Within the nonlinear realization of the supergroup OSp(1|2) it was re-constructed in [15]. Thus, in this purely illustrative Section, we will show that the our constraints (2.15) work perfectly, resulting in the proper N =1 super-Schwarzian. The crucial property of our construction is that nullifying of the N =1 super-Schwarzian, similarly to the purely bosonic case, implies nullifying of all Cartan forms, besides ω P and ω i Q . In addition, we will find the m 2 -modification of the N =1 super-Schwarzian.
The superconformal algebra osp(1|2) contains, in addition to the generators D, P , K, Hermitian fermionic supercharge Q and superconformal charge S. Their (anti)commutation relations are: The general element of the N =1 superconformal group OSp(1|2) can be parameterized in analogy with the bosonic case: The Cartan forms explicitly read The parameters of the group element (3.3) are assumed to be fields that depend on τ and θ, the coordinates of the N =1 superspace. Supersymmetry is realized on these coordinates in standard way: The inert covariant derivatives defined with respect to △τ and dθ 3 , have the form 3 These forms and transformation laws, if needed, can be obtained using the coset space techniques, for example, considering cosetg = e i τ P e θQ , with P and Q forming N =1 Poincare superalgebra. However, they are simple and standard enough to be treated even without reference to nonlinear realizations techniques.
Treating all the group parameters in (3.2) as fields that depend on τ , θ, we impose the following conditions on the forms ω P , ω Q : Finally, one has to nullify the form ω D to express the superfield z in terms of t(τ, θ) and ξ(τ, θ) Note, only first equation in (3.9) is new, the second one just follows from the equations (3.8). Altogether, equations (3.7),(3.8), (3.9) allow to express superfields u, z, ψ in terms of the Goldstone superfield ξ(τ, θ): (3.10) Calculating the remaining Cartan forms ω S and ω K with relations (3.10) and their consequences taken into account, one finds that Now, all our forms are expressed in terms of N =1 Schwarzian [21] (3.12) If we compare these expressions (3.12) with those ones from the paper [15], we conclude that the constant parameters in [15] should be chosen as g = 1, p = 0. Thus, in the case of N = 1 super-Schwarzian our constraints (2.15) are equivalent (modulo unessential "p"-freedom) to the constraints introduced in the paper [15]. However, already in the case of N =2 super-Schwarzian we will consider in the next Section the preference of our constraints (2.15) becomes evident. It is clear that the simplest invariant superfield action can be constructed as In component form, it reads where t and ξ are the first components of respective superfields. One should stress that in contrast with the bosonic case, the N =1 Schwarzian (3.12) can not be expressed in terms of the super-dilaton u only due to the presence of the last term which explicitly depends on the fermionic superfield ξ, without derivatives.
The infinitesimal Q and S transformations, generated by the element e ǫQ+εS , read All these expressions (3.11), (3.12), (3.13) are invariant with respect to these transformations and, therefore, they are invariant with respect to all OSp(1|2) transformations. It is completely clear now that if we nullify the super-Schwarzian then the all Cartan forms in (3.3) will be equal to zero, besides the forms ω P and ω Q which will coincide with the "boundary" forms △τ and dθ (3.5), as expected.

N =2 Schwarzian
The N =2 super-Schwarzian has been introduced in [22] and then it was re-obtained in [8]. The treatment of the N =2 super-Schwarzian within the nonlinear realization of the su(1, 1|1) supergroup was initiated in [15]. The consideration performed in [15] correctly reproduced N =2 super-Schwarzian but unfortunately the constraints used there imposed the further constraint on the super-Schwarzian to be a constant. In this Section we will demonstrate that our variant of the constraints (2.15) correctly reproduce N =2 super-Schwarzian, expressed all su(1, 1|1) Cartan forms in terms of this super-Schwarzian and its derivatives. Finally, we will show that imposing the constraints on the full Cartan forms makes possible to utilize the Maurer-Cartan equations which drastically simplify all calculations. In the case of N =2 supersymmetry we are dealing with the N =2 superconformal algebra su(1, 1|1) defined by the following relations Let us remind the conjugation properties of the generators The superfields are assumed to depend on the coordinates of the "boundary" N =2 superspace τ , θ,θ. The supersymmetry transformations and differential forms, invariant with respect to them, 4 are δτ = i ǫθ +ǭθ , δθ = ǫ, δθ =ǭ, Using the invariant forms (4.3), one may easily construct the covariant derivatives Similarly to the previously considered cases, we choose the following parametrization of the general element of the N =2 superconformal group SU (1, 1|1) g = e i t(P +m 2 K) e ξQ+ξQ e ψS+ψS e i zK e i uD e φJ (4.5) where the parameters t, ξ,ξ, ψ,ψ, z, u and φ are, as we stated above, the superfunctions depending on {τ, θ,θ}. The Cartan forms Now, imposing the constraints (2.15), i.e. identifying the forms ω P , ω Q ,ω Q (4.7) with △τ , dθ, dθ (4.3) respectively, will result in the following equations (4.10) 4 If needed, they can be obtained by considering cosetg = e i τ P e θQ+θQ Finally, one has to nullify the form ω D : (4.11) From these relations one may obtain several important consequences. In particular, we have (4.14) Now, one may check that the form ω J reads Thus we see, that N =2 Schwarzian S N =2 appears automatically. One may check that the other Cartan forms, ω S ,ω S and ω K can be also expressed in terms of the N =2 Schwarzian only The transformation laws of the basic superfields t, ξ,ξ, are induced by left multiplication g ′ = g 0 g. In the case of superconformal transformations g 0 = e ǫQ+ǭQ e εS+εS the transformation laws of t and ξ,ξ read is invariant with respect to these transformations. Thus one can expect that the proper superfield Schwarzian action reads Evaluating the integral, one can find the component action where t, ξ,ξ and φ are the first components of respective superfields.
The calculations leading to the expressions (4.16) are rather involved. They become more and more complicated while passing to higher supersymmetries. However, the fact that our constraints (2.15) are imposed on the Cartan forms opens the way to use the Maurer-Cartan equations which drastically simplify the calculations. The N =2 case provides a nice possibility to demonstrate on the simplest example how everything is working on. We put this consideration in the Appendix A.
Comparing our expressions for the final Cartan forms (4.16) with the constraints which were used in the paper [15] we conclude that the constraint immediately restricts N =2 Schwarzian to be a constant. Clearly, this condition is unreasonably strong. Thus, the N =2 case is the first one in which our set of constraints (2.15) and ω D = 0 becomes preferable with respect to those ones formulated in [15].

N =3 Schwarzian
The N = 3 super Schwarzian has been introduced in [8]. Then it was re-obtained within the nonlinear realization of the supergroup OSp(3|2) in [17]. However, the constraints introduced in [17] lead, besides the N = 3 super Schwarzian S N =3 , to some new OSp(3|2) invariants with unclear geometric meaning. In this Section we will demonstrate that our constraints (2.15) and ω D = 0 being applied to superalgebra osp(3|2) results in the Cartan forms expressed in terms of S N =3 and its derivatives only. The osp(3|2) superalgebra contains 6 bosonic (P, D, K, J i ) and 6 fermionic generators Q i , S j obeying the following (anti)commutators: Here, all generators are chosen to be hermitean, i, j, k . . . = 1, 2, 3 and ǫ ijk is completely antisymmetric symbol, ǫ 123 = 1. We parameterize the general element of the N =3 superconformal group as 5 with the invariant Cartan forms defined as The forms of P , D, K generators read The forms of fermionic generators and J i , unlike (5.5), include rotations, induced by the exponent e i φ k J k , which can be parameterized with SO(3) matrix M ij : The hatted forms here are We treat the parameters of the OSp(3|2) group element as superfields that depend on the coordinates of the N =3 superspace, τ and θ i . The N =3 supersymmetry is realized on these coordinates in standard way, 6 Correspondingly, the N =3 covariant derivatives read Just as before, we enforce the following invariant constraints on the Cartan forms As was suggested by the results of the previous Section, much information about the Schwarzian can be obtained by analyzing structure of the Cartan forms with the help of Maurer-Cartan equations Leaving detailed description of this calculation to the Appendix B, we present here only the result: Therefore, all the Cartan forms (5.4) can be written in terms of just one fermionic superfield S and its derivatives, with no constraints on S coming from Maurer-Cartan equations. It is natural to identify this fermionic superfield with the N =3 super-Schwarzian: To relate S N =3 to the group superfield parameters, one should study the conditions (5.10) explicitly, writing all their projections with respect to △τ and dθ i : The condition D i t + i D i ξ j ξ j = 0 can be considered as primary one. From it, one can obtain and D i ξ j is proportional to the orthogonal matrix. This way one can also obtain the derivative of u, D i e u = −2i D i ξ jξj . Condition ω D = 0 then just expresses z in terms of u as z = 1 2 e −uu . With these conditions taken into account, one can write down dθ p projection of ω J i as To proceed further, one should note that, as a consequence of (5.13), the fermionic superfield ξ i satisfies a quadratic relation To obtain it, one should take the relation D m ǫ ipq D p D q t = −2i ǫ imk D kṫ + 1 3 δ im ǫ pqr D p D q D r t , which follows just from anticommutation relations of D i , and substitute D i t = −i D i ξ j ξ j (5.13). Also taking into account that the form ω J i (5.15) reduces to Obtained S N =3 is just the already known N =3 Schwarzian [8,17]. The obvious candidate for Schwarzian action in N =3 case is This is further substantiated by the fact that dτ projection of ω K , which defines the component Schwarzian action, contains ǫ pqr D p D q D r S N =3 . Using this property to calculate component form of (5.19), one obtains Here, t, ξ i and M ij are the first components of respective superfields, and s is the first (fermionic) component of the N =3 Schwarzian (5.18). It should be taken as independent, as calculating D i , D j }S N =3 using D i S N =3 extracted from △τ projection of ω J i Cartan form, one arrives just to an identity. In terms of Cartan forms, the integral (5.19) could be written as Let us, finally, consider the construction of the N =4 Schwarzian. In this paper, we do not make an attempt to consider the N =4 superconformal algebra D(2, 1, α) and concentrate on its particular limit, the su(1, 1|2) superalgebra. The corresponding N =4 Schwarzian has been constructed in [8,23] and then it was re-obtained within the nonlinear realization su(1, 1|2) superalgebra in [16]. In this Section we are going to use our set of constraints (2.15) and ω D = 0 to demonstrate that all the Cartan forms in this case can be expressed in terms of N =4 Schwarzian and its derivatives. The su(1, 1|2) superalgebra contains usual conformal generators P , D, K, supersymmetric and superconformal charges Q α , Q α = Q α † , S α , S α = S α † and generators of the su(2) subalgebra T α β = − T β α † , T α α = 0 : The generators D, K, P commute with su(2); the commutators of su(2) with themselves and fermionic generators read Here, indices α, β, . . . = 1, 2 can be raised and lowered with help of antisymmetric tensors ǫ αβ , ǫ αβ , ǫ αβ ǫ βγ = δ γ α , ǫ 12 = ǫ 21 = 1. The SU (1, 1|2) group element can be parameterized as g = e i tP e ξ α Qα+ξαQ α e ψ α Sα+ψαS α e i zK e λ β α Tα β e i uD . (6. 3) The Cartan forms defined in standard way ) are rather involved. Explicitly, the forms of the scalar bosonic generators, P, D and K read The forms for su(2) generators T α β and fermionic generators are We subject the forms to the usual conditions (2.15) where △τ , dθ α , dθ α are invariant with respect to N =4 supersymmetry transformations 7 Correspondingly, the N =4 covariant derivatives read From now on we will treat all fields as the superfields depending on the coordinates of "boundary" superspace τ, θ α ,θ α . The analysis of Maurer-Cartan equation, which we leave for Appendix C, shows that all the Cartan forms, aside of constrained ω P , ω Q α , ω Q α , ω D , can be written in terms of three quantities S α β , S α α = 0: Unlike previously considered systems, S α β satisfy differential constraints D (γ S αβ) = 0, D (γ S αβ) = 0, (6.12) which imply that S αβ form an N =4, d = 1 vector multiplet. In full agreement with the previous cases we call S αβ by N =4 super-Schwarzian The last step is to express N =4 super-Schwarzian in terms of our basic superfields t, ξ α ,ξ α . The relations between the fields t, ξ α ,ξ α , ψ α ,ψ α , u, z and constraints on them can be obtained by expanding relations (6.8) in terms of △τ , dθ α , dθ α . The ω P conditions read 14) The ω Q α and ω Q α conditions read Finally, the ω D = 0 conditions are It can be shown that the conditions (6.14), (6.15) are equivalent to and define the N =4 linear multiplet, with one "physical" boson, four fermions and three "auxiliary" bosons (though in the system under discussion all are dynamical). It can be shown that the same phenomenon as in N =2 case happens: the commutator of covariant derivatives, acting on t, reduces to τ -derivative of ξ α ,ξ β : and ξ α ,ξ β can not be expressed entirely in terms of t. This again does not put the system on-shell: In comparison, calculating traceless part of D β , D α u using (6.15), (6.16), one finds: Therefore, the N =4 Schwarzian reads as expected. As S αβ satisfies the irreducibility conditions of the vector multiplet, D µ , D ν S µν transforms w.r.t. supersymmetry as an auxiliary field (gets shifted by a total derivative). Therefore, is the N =4 Schwarzian action. In terms of Cartan forms, it can be presented as Evaluating the integral in (6.23), one can obtain the component action Here, as usual, t, ξ α ,ξ α , λ α β are the first components of respective superfields.

Conclusion
In this work we re-consider the application of the method of nonlinear realizations to the N = 0, 1, 2, 3, 4 (super)conformal groups. As compared to the previous attempts to utilize the nonlinear realizations for construction of the super-Schwarzians [14][15][16][17][18], our consideration is based on the minimal set of constraints imposed on the Cartan forms. These constraints include • The constraints on the forms of N -extended super Poincare generators ω P = △τ, ω i Q = dθ i . Here, the forms △τ, dθ i depend on the coordinates of "boundary" superspace τ, θ i and they are invariant with respect to rigid N -extended supersymmetry transformations; • The final constraint reads ω D = 0. It provides some variant of the Inverse Higgs Phenomenon constraints [13].
We explicitly show that this minimal set of constraints is enough to express all Cartan forms of the N = 0, 1, 2, 3, 4 (super)conformal groups in terms of corresponding (super-)Schwarzians and their derivatives. In the cases of higher supersymmetries the calculations quickly become rather cumbersome. Having at hands the constraints written on the Cartan forms (not on their projections!) it proved useful to use the Maurer-Cartan equations which help to express all Cartan forms in terms of the single object -N super-Schwarzian. However, to find the expression of the N super-Schwarzian in terms of the basic superfields one has to again use all set of constraints.
The idea to use the "boundary" superspace to impose the proper constrains on the Cartan forms was firstly formulated in [14,15]. However, the full set of constraints used in the papers [14][15][16][17][18] seems to be unessentially strong. At least our analysis shows that these constraints unavoidably restrict super-Schwarzians.
We are planning to apply the proposed approach to N -extended superconformal group including the variant of OSp(4|2) superconformal symmetry. Another interesting problem is to obtain non-relativistic and/or Carrollian versions of the Schwarzian [19].
With ω P = △τ and ω Q = dθ,ω Q = dθ, dω P equation (A.2) is satisfied identically, as While Ψ is yet undetermined, just one equation (A.6) is strong enough to show that the form ω S can not have a dθ -projection, and dθ and dθ projections of ω J are absent. Also it relates dθ projection of ω S and △τ projection of ω J : A = −1/2S. The analysis of dω Q equation (A.7) leads to analogous results B = 0, A = 1/2S. Most convenient next step would be to study dω J equation (A.5). As we already reduced ω J to ω J = i △τ S, taking into account that we find (A.14) Comparing this with the right hand side of (A.5), where ω S = △τ Ψ − 1/2dθS andω S = △τ Ψ + 1/2dθS, we find that dθ × dθ terms cancel from (A.5), while the rest imply 15) and the forms ω S ,ω S , ω J can be written in terms of just one quantity S: Next step further would be to check dω D equation (A.4). As the left hand side of (A.4) is zero due condition ω D = 0, we do not need to take the differential of anything. Simply putting results for the forms ω S ,ω S and the ansatz for ω K into (A.4), we find that dθ × dθ terms cancel and others imply To determine C, we should consider dω S or dω S equations (A.8), (A.9). The left hand side of (A.8) can be calculated from (A.16) as Substituting the ω S ,ω S and ω J (A.16) to the right hand side, as well as the ansatz for ω K (A.10), one obtains Thus all dθ × dθ, dθ × dθ terms cancel out and one finds As every projection of all the forms is already found in terms of S and its derivatives, one can only check by direct calculation that dω K equation (A.3) is satisfied. It is indeed so, with no constraints imposed on S.
The results (A.16), (A.20) are in full agreement with ones obtained by straightforward calculation of (4.16). Though for N =2 Schwarzians this analysis was somewhat tedious and not particularly easier than direct calculation of multiplet defining conditions, it is still important. At first, it shows that the structure of Cartan forms in N = 1 and N =2 cases is not a coincidence and reflects fundamental properties of supersymmetric Schwarzians. Secondly, in the N =3 and N =4 cases, the irreducibility conditions of multiplets become more and more important, while remaining highly nonlinear, and calculation of their consequences becomes increasingly difficult. Therefore, analysis of Maurer-Cartan equations becomes more convenient way to identify proper Schwarzians even from technical point of view.

B Maurer-Cartan equations for OSp(3|2)
The main Maurer-Cartan equation for osp(3|2) with general structure of the Cartan form given by (5.4) can be reduced to a set of relations The primary conditions are ω P = △τ , ω Q i = dθ i and ω D = 0. The remaining forms can be expanded in terms of △τ and dθ i as Analyzing equations (B.1), one can obtain that dω P equation is satisfied automatically. The part of dω Q equation, proportional to △τ ∧ dθ, implies that A ij = ǫ ijk B k , and dθ ∧ dθ part is satisfied if ǫ ijk S jm + ǫ ijm S jk = 0, which happens if and only if S ij = δ ij S.
Analyzing dω J equation one obtains from dθ ∧ dθ part that B j = i D j S, while the △τ ∧ dθ part implies that 3) The dθ ∧ dθ part of dω D equation is then satisfied automatically, and the rest implies just Σ i = i Ψ i . The dω S equation is more complicated, producing two relations: Substituting B i , Ψ i into these equations, one finds that of the first one only δ ij component survives while the second one is satisfied automatically. To show this, one should use the identity it can be straightforwardly checked that dω P equation (C.1) is satisfied identically. Substituting (C.3) into the dω Q equation, one finds Therefore, one should take B αβ = Σ γβ α = Σ γ β α = 0 and A β α = −i S β α . Considering dω Q equation in the same way, one finds also A α β = i S α β .
The dθ×dθ terms also cancel from dω S equation, and the remaining △ 1 τ d 2 θ α −△ 2 τ d 1 θ α term implies that To obtain this, one should use the vector multiplet conditions (C.8) to find Using this, it is easy to prove that dω K equation does not lead to any new conditions. Finally, all the forms are written in terms of S α β and its derivatives (6.11).