N=4 Super-Schwarzian Theory on the Coadoint Orbit and PSU(1,1|2)

An N=4 super-Schwarzian theory is formulated by the coadjoint orbit method. It is discovered that the action has symmetry under PSU(1,1|2).


Introduction
The Schwarzian theory has been drawing a lot of attention giving a hope to be a theoretical gateway between the SYK model and the D = 2 effective gravity. The Liouville gravity, which is the simplest one for the D = 2 effective gravity, was extensively studied by various methods after the pioneering Polyakov's work. (See [1,2] for an overview of the studies at the early stage.) Among them the coadjoint orbit method proposed by Alekseev and Shatashvili is the most geometrical [3]. In a recent paper [4] this coadjoint orbit method was revisited to study the Schwarzian theory. The Hamilton structure of the Schwarzian theory was clarified through the formulation by the coadjoint orbit method.
The first aim of this paper is to formulate an N = 4 super-Schwarzian theory by means of the coadjoint orbit method. The lower symmetric cases have been discussed in the past few years. (See [5,6] for instance.) But a proper account on the differential geometry has been given only recently in [4], but for the non-supersymmetric case.
As for supersymmetrization of the coadjoint orbit method for the Liouville gravity it is opportune to give a brief summary of the studies. After the work [3] the coadjoint orbit method was generalized to formulate the (1, 0) and (2, 0) supersymmetric theories in [7] and [9,8] respectively. The left-moving sector was extended so as to admit the (1, 0) and (2, 0) superconformal symmetry. In the right-moving sector the conformal symmetry remained non-supersymmetric, but the symmetry SL(2) got promoted to OSp(2|1) and OSp(2|2) for the respective supersymmetric theories [10,11]. A further extension of the coadjoint orbit method to the N = (4, 0) supersymmetric case has not been discussed, although the N = 4 superconformal algebra has been known since a long time ago [12].
Once formulated an N = 4 super-Schwarzian theory the second aim of this paper is to examine if it has a symmetry further generalizing OSp(2|2). We show that the theory indeed has symmetry under PSU (1,1|2).
The paper is organized as follows. In Section 2 we give a short summary of the coadjoint orbit method. In Section 3 we discuss the N = 4 superconformal diffeomorphism and give the N = 4 super-Schwarzian derivative, which is a key element for the paper. Following these arguments the coadjoint orbit method is worked out to construct the N = 4 super-Schwarzian theory in Section 4. In Section 5 we show that the theory admits symmetry under PSU (1,1|2). Three Appendices are devoted to help the calculations in the main body of the paper.

A short summary of the coadjoint orbit mothod
We shall start with a brief review about the general construction of the Kirillov-Kostant 2-form on the coadjoint orbit of a Lie-group G [3]. Let g to be a Lie-algebra of G and g * the dual space of g. An element g ∈ G acts on an element a ∈ g by Ad(g)a = gag −1 , while the group action on an element X of the dual space g * is defined by means of an invariant quadratic form < Ad * (g)X, a >=< X, Ad(g −1 )a > . (2.1) Elements a ∈ g and X ∈ g * are called adjoint and coadjoint vectors respectively. We define the Kirillov-Kostant 2-form as where Y is a g-valued 1-form related with X by Here ad * (Y ) is the infinitesimal coadjoint action on X determined according to (2.1), i.e.,

The N = 4 superconformal diffeomorphism
The N = 4 superconformal group is a group of which elements are superdiffeomorphism in the N = 4 superspace. A general account of the N-extended superconformal group was given in [13,14]. The case of N = 4 was studied in [15]. Here we elaborate their arguments. The N = 4 superspace is described by the supercoordinates Here x is a real coordinate. θ a , a = 1, 2, are fermionic ones, while θ a , a = 1, 2, their complex conjugates. The supercovariant derivatives are defined by so as to satisfy We consider N = 4 superdiffeomorphisms The supercovariant derivatives change as by the chain rule. Impose the chirality conditions (3.6) and the superconformal conditions Then (3.4) and (3.5) become supercovariant derivatives as When the supercovariant derivatives satisfy these transformation properties, the transformations in (3.3) are called superconformal diffeomorphisms. Elements of the superconformal group consist of such diffeomorphisms. A superfield with weight 0, denoted by Ψ 0 (x, θ), transforms as by the superconformal diffeomorphisms. Infinitesimally it reads which may be put in the form by using the supercovariant derivatives (3.1) and an infinitesimal parameter v = v(x, θ) When the superconformal conditions (3.7) are imposed, the infinitesimally small parameters δx, δθ and δθ are constrained as which become respectively by using the chirality condition (3.6) in the infinitesimal form Using this we write the transformation δΨ 0 , given by (3.9), in a supercovariant form as Hereinafter we do not write the arguments of superfields explicitly if they are simply (x, θ) as here. This transformation law can be generalized to define a superfield having arbitrary weight w and charge q as However the charge part of this transformation drops out, since [D θc , D c θ ]v = 0 as discussed in Appendix B.
A posteriori we recognize that the superconformal diffeomorphisms (3.3) may be given by superfields with weight 0, but the fermionic ones are constrained by the chirality condition (3.6). That is, 14) We now propose that the N = 4 super-Schwarzian derivative 1 with the above superdiffeomorphisms. When expanded in components by using the formulae in Appendix A, the purely bosonic part takes the form in which h is the lowest component of the superfield f and O(η) indicates contributions of fermion fields. The top component does not coincides with the non-supersymmetric Schwarzian derivative. However using this S(f, ϕ; x, θ) we will find an N = 4 super-Schwarzian action in the next Section, of which purely bosonic part is the usual nonsupersymmetric Schwarzian one. (See (4.14) and the argument thereafter.) Or without going through such an argument we may convince ourselves that S(f, ϕ; x, θ) in this form is indeed the N = 4 super-Schwarzian derivative. Namely it obeys the anomalous superdiffeomorphism with weight 0 The N = 4 super-Schwarzian derivative of this form appeared in [14].
which can be easily checked by the chain rule. Infinitesimally it reads The last term is the conformal anomaly. From (3.18) it follows that the quantity det[D θa ϕ b (x, θ)] is a superfield transforming as Ψ 1 , given by (3.12). On the other hand we can easily show that the quantity is also a superfield obeying the same transformation law as Ψ 1 . Hence showing

N = super-Schwarzian theory
Now we are in a position to discuss the coadjoint orbit method to apply for the N = 4 superconformal algebra. The superconformal algebra g and the dual space g * are centrally extended. Their elements are given by Here k and c are central elements. u and b are bosonic superfields, obeying the superconformal transformations of Ψ −1 and Ψ 0 given by (3.12) respectively. The transformation of the latter may be centrally extended. The volume element of the N = 4 superspace, dxd 4 θ, has weight 1, so that the invariant quadratic form is defined by The centrally extended superconformal algebra g is given by the infinitesimal adjoint Then using the relation (2.4) yields the corresponding coadjoint action ad * (v, l) on (b, c) ∈ g * is given by We think of a coadjoint orbit O (b,c) , whose initial point is (b, c) ∈ g * . The finite form of (4.3) is generated on the coadjoint orbit by the superdiffeomorphism (3.3) as 2 Here S(f, ϕ; x, θ) is the super-Schwarzian derivative given by (3.16). Now the Kirillov-Kostant 2-form (2.2) can be given by on the coadjoint orbit O (b,c) . Here the commutator was given by (4.2). (y, 0) is a centrally extended g-valued 1-form in a space parameterizing the coadjoint orbit. Namely we think of the superfields f, ϕ c , ϕ c in a fictitious space beyond the D = 1, N = 4 superspace as f (x, θ, t 1 , t 2 , · · ·) etc. The 1-form y is a function of them. We should have written it as y(f, ϕ) according to our convention. But we would not like to do it for simplicity hereinafter too. y is determined so that the exterior derivative of the quantity (4.4), which is an element of g * , is induced by the infinitesimal coadjoint action (4.3) on it along the Keep in mind that the exterior derivative acts only on the coordinates t 1 , t 2 , · · ·. It is the most important step in our arguments to find an explicit form of y by solving this equation. It turns out that the solution is given by with ∆ defined by (3.19). Once found y as a solution to (4.6), the centrally extended commutator in (4.5) becomes from (4.2). We shall verify that y given by (4.7) indeed solves the equation (4.6). Using (4.3) we may rewrite (4.6) by a pair of the equations Compare these with the respective superdiffeomorphisms and δ v S(f, ϕ; x, θ) given by (3.18). The former one can be verified by the infinitesimal vari- The equations (4.9) and (4.10) require that the exterior derivatives of b(f, ϕ) and S(f, ϕ; x, θ) coincide with their superconfomal diffeomorphisms, if the infinitesimal parameter v is replaced by y. In a mathematical language we can put it as Here i v is the anti-derivative of the differential form, implying the operation by (4.1) with (4.4) and (4.8). Take the anti-derivative and use (4.13). By integration by part we get owing to by (4.9) and (4.10). Thus (4.14) has been shown. It is worth knowing about non-supersymmetric approximation of S(f, ϕ; x, θ). By using (3.17) and the expanding formula of v, given in Appendix A, we find the top component of the integrand as vS(f, ϕ; x, θ) = · · · · · · + 1 2 Here α is the lowest component of v and dv = 0. Upon integrating the first term by part the top component of the integrand becomes the ordinary Schwarzian derivative multiplied by −2. So there is nothing wrong to have claimed that S(f, ϕ; x, θ) given by (3.16) is the N = 4 super-Schwarzian derivative of which purely bosonic part is given by (3.17). Finally putting v = 1 in (4.14) leads us to the desired super-Schwarzian action.

PSU(1,1|2) symmetry
In this section we show symmetry of the acion (4.14) under PSU (1,1|2). The action depends on the initial point b of the coadjoint orbit. We discuss the issue dividing the dependence into two cases. For each case we are involved in different realization of the PSU(1,1|2) symmetry.
We expect it to be realized on a supermanifold whose local coordinates are the superdiffeomorphism f, ϕ a , ϕ a discussed in Section 3 and their complex conjugate f , ϕ a , ϕ a . Such a supermanifold is given by the coset space PSU(1,1|2)/{SU(2)⊗U(1)} for which the generators of PSU(1,1|2) are decomposed as The coset generators L, F a , F a correspond to the coordinates f, ϕ a , ϕ a . The fermionic coordinates ϕ a and ϕ a are doublets of the subgroup SU(2). It is well-known that PSU(1,1|2) can be embedded in the larger supergroup D(2,1;γ). We may write the fermionic generators and the corresponding coordinates by using the notation of D(2,1;γ) as with the identifications See [16] for the more precise relation between the generators of PSU(1,1|2) and D(2,1;γ).

(5.5)
Here R A and R A aα are the Killing vectors satisfying the Lie algebra of PSU(1,1|2). ǫ A are infinitesimal parameters of the transformation corresponding to the generators of PSU(1,1|2), given by (5.1).
It is not guaranteed at all that the PSU(1,1|2) transformations generated by these Killing vectors (5.4) and (5.5) respect the chirality conditions (3.6) as well as the superconformal conditions (3.7). So we claim that This claim will be verified in Appendix C. Therefore it makes perfect sense to study the transformation property of the N = 4 super-Schwarzian action by the Killing vectors (5.4) and (5.5). Remarkably we find the quantity ∆, given by (3.19), to obey a fairly simple transformation as This follows by a straightforward calculation with the use of ∆ written in the notation of D(2, 1; γ) as As the result the action (4.14) with b(f, ϕ) = 0 transforms as δ ǫ H| v=1,b=0 = c dxd 4 θδ ǫ log ∆ = c dxd 4 θ ǫ L 0 + 2f ǫ L + ϕ cγ ǫF dδ ǫ cd ǫ˙γδ , (5.9) in which S(f, ϕ; x, θ) = log ∆ owing to (3.16) and (3.20). We find that the top component of the integrand is of the form ∂ x (· · ·), when the superfields f and ϕ cγ are expanded in components as in Appendix A and use is made of the second equation in (A.1). Therefore the Schwarzian action H| v=1,b=0 is invariant under the PSU(1,1|2) transformations generated by the Killing vectors (5.4) and (5.5).
The infinitesimal parameter v of the N = 4 superdiffeomorphism is expanded in components in Appendix A. The modes of the components α = e ±inx α ±n , α 0 , (5.10) β a = e ± 1 2 inx β a± 1 2 n , (5.11) β a = e ± 1 2 inx β a ± 1 2 n , (5.12) (5.13) span the N = 4 superconformal algebra [12]. The PSU(1,1|2) symmetry is realized also by the modes of the diffeomorphisms with n odd. They sequentially correspond to the generators It is wise to write the Schwarzian action (4.14) as In the second line we have abused the definition (4.4) since the c-dependence of the initial point of the coadjoint orbit O (b,c) is implicit. But as for the arguments of the initial point b we have made it explicit as b(x, θ) against the convention employed below (3.10). Now the question is if there exists a certain configuration of b(x, θ) with which the Schwarzian action is invariant by the superdiffeomorphism given by (5.10)∼(5.13) with n odd. It may be examined at the initial point of the coadjoint orbit O b,c , i.e., Here use was made of (3.18) and (4.11). We may proceed the argument quite analogously to the non-supersymmetric case, but in a much simpler way. The Schwarzian action is found as Having conformal weight 2 the field b(x) gets scaled with a factor (∂ x h) 2 by the coadjoint action. Assuming b(x) = 0 we require that under the diffeomorphism with an infinitesimal parameter α. It is important to observe that this is a third-order equation for α. If b(x) is constant, then it is solved by any constant α. It implies that the action is invariant under U(1) symmetry generated by L 0 . If b(x) is fixed to be 1 2 cn 2 , then (5.17) admits three independent solutions of the form (5.10). The symmetry of the action is enhanced to SL(2). This result is well-known in [19,20,3,8] as well as [4]. For the case of b(x) = 0 refer to a comment in the end of the paper.
Let us turn to the N = 4 Schwarzian action (5.14). The superfield b(x, θ) is expanded in components as the superfield f was done in Appendix A, i.e., b(x, θ) = a + θ · γ + γ · θ + θ · θi + (θσ i θ)s i Here the arguments of the component fields have been omitted according to our convention. Put this expansion as well as that of v, also given in Appendix A, into the second line of (5.15). Calculating its top component we have 5 We find that it is vanishing by the diffeomorphism (5.10)∼(5.13) when the initial point b has a configuration such as with Thus the Schwarzian action is invariant under PS(1,1|2). But it is worth remarking that we do not encounter boundary terms at all in examining the symmetry of the integrand. It is also worth recognize that the solution contains the non-supersymmetric one in the previous paragraph by setting 2d = −b.
The reader may ask about symmetry for the density of the Schwarzian action (5.14). Then the variation (5.15) is required to vanish at lower orders of θ as well. The resulting differential equations are too stringent to be satisfied by the above solution. For instance at the lowest order of θ it reads More stringent equations come out at higher orders. Nonetheless it is not hard to see that all the equations are satisfied by the subset of the modes in (5.10)∼(5.13), when b has a configuration such as Therefore the subgroup SU(2)⊗U(1) is also a symmetry of the density of the Schwarzian action (5.14).
We content ourselves with these solutions, although our analysis of the differential equations is not exhaustive at all. In summary, the partition function of the N = 4 super-Schwarzian theory is given by   1|2). For that we have found a configuration of b such as given by (5.19). We comment the case of b = 0 for the non-supersymmetric Schwarzian action (5.16), which we have not discussed in Section 5. The non-supersymmetric Schwarzian derivative S(h; x) is invariant under SL(2) realized by the Killing vectors for the coset space SL(2)/U(1). However (5.9) implies that the Schwarzian derivative S(f, ϕ; x, θ) for the N = 4 case is invariant only modulo boundary terms ∂ x (· · ·) by the same transformation. This discrepancy is not a problem because the purely bosonic part of S(f, ϕ; x, θ) is given by (3.17) and the top component giving the action reads The additional term is invariant modulo the boundary term 2ǫL∂ 2 x h under SL(2) realized by the Killing vectors. It is consistent with (5.9).
It is desirable to study quantum dynamics of the N = 4 super-Schwarzian action. Our study on this is in progress. It is also desirable to extend the D = 2 Liouville gravity to the N = 4 supersymmetric one. It will be reported in [21].

A Superfields in components
In the body of the paper the N = 4 super-Schwarzian derivative S was needed to be expanded in components. We give here only the expansion for the basic ones. The superfields f, ϕ c , ϕ c which describe the N = 4 superdiffeomorphism are expanded as with θ · ψ ≡ θ a ψ a , ψ · θ ≡ ψ a θ a etc. Note that the component fields of ϕ c and ϕ c got the argument x shifted so that the chirality conditions (3.6) are satisfied. By imposing the superconformal conditions (3.7) they become with the remaining constraints Now the component fields have the argument x, which has been omitted for simplicity. It is important to note that all of their top components are of the form ∂ x (· · ·). Use the second equation of (A.1) in order to see this for the one of f . In the end of Section 4 we also need to expand the infinitesimal parameter v(x, θ) of the N = 4 superdiffeomorphism. By (3.10) and (3.11) it satisfies in which α, β a , β a , t i are independent parameters of the superdiffeomorphisms.

B Proofs of some formulae in Sections 3 and 4
We prove the various formulae required for the arguments in Sections 3 and 4. We begin by the following formulae They were studied in [15]. (B.2) and (B.3) follow from (B.1). (B.1) can be shown by taking the supercovariant derivative of (3.7) and using the algebra (3.2) and the chirality C Proof of (5.6) and (5.7) in Section 5 We show the formulae (5.6) and (5.7) following from the chirality and superconformal conditions respectively. To this end it is convenient to write the Killing vectors in the doublet notation Here we have used the same doublet notation also for ǫ F aα , ǫ F aα and R A aα as given for ϕ aα by (5.2). Then it is immediate to see that (5.6) holds owing to (3.6) and (3.7). (5.7) can be also checked by a few of calculations. We do it explicitly for the first equation of (5.7) as an example. From (C.3) it follows that by using (3.6) and (3.7). For the case of ǫ = ǫ F we have By the same calculation we have also from (C.2) and (C.1) respectively. It is now clear that the first relation of (5.7) is satisfied for the case of ǫ = ǫ F . It can be checked similarly for other cases than ǫ = ǫ F .