Super-Schwarzians via nonlinear realizations

The N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 1 and N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2 super-Schwarzian derivatives were originally introduced by physicists when computing a finite superconformal transformation of the super stress-energy tensor underlying a superconformal field theory. Mathematicians like to think of them as the cocycles describing central extensions of Lie superalgebras. In this work, a third possibility is discussed which consists in applying the method of nonlinear realizations to osp(1|2) and su(1, 1|1) superconformal algebras. It is demonstrated that the super-Schwarzians arise quite naturally, if one decides to keep the number of independent Goldstone superfields to a minimum.


Introduction
A recent study of supersymmetric extensions of the Sachdev-Ye-Kitaev model [1] generated renewed interest in the N = 1 and N = 2 super-Schwarzian derivatives. 1 Such derivatives were originally introduced by physicists when computing a (finite) superconformal transformation of the super stress-energy tensor underlying a superconformal field theory [3][4][5]. Mathematicians used to regard them as the cocycles describing central extensions of Lie superalgebras (see, e.g., [6] and references therein).
A remarkable property of the N = 1 and N = 2 super-Schwarzian derivatives is that they hold invariant under (finite) transformations forming OSp(1|2) and SU(1, 1|1) superconformal groups, respectively. 2 It is then natural to wonder whether the logic can be turned around so as to derive the super-Schwarzians by analysing invariants of the supergroups alone.
Given a Lie (super)algebra, a conventional means of building invariants associated with the corresponding (super)group is to apply the method of nonlinear realizations [7]. Within this framework, one starts with a coset space elementg, on which a (super)group representative g acts by the left multiplicationg = g ·g, and then constructs the Maurer-Cartan one-formsg −1 dg, which are automatically invariant under the transformation. These invariants can be used to impose constraints enabling one to express some of the (super)fields parametrizing the coset elementg in terms of the other [8]. If the algebra at hand is such that all but one (super)fields can be linked to a single unconstrained (super)field, then the last remaining Maurer-Cartan invariant describes a derivative of the latter, which holds invariant under the action of the (super)group one started with. 1 The literature on the subject is rather extensive. For a good recent account and further references see ref. [2]. 2 In modern literature, the superconformal groups are sometimes designated by the number of spacetime dimensions in which they are realised and the number of real supersymmetry charges at hand. In this nomenclature OSp(1|2) and SU(1, 1|1) are identified with the d = 1, N = 1 and d = 1, N = 2 superconformal groups, respectively.

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As an illustration, let us consider the coset space elementg = e iρ(t)P e is(t)K e iu(t)D , which builds upon the generators P , D, K forming sl(2, R) algebra 3 and three real functions ρ(t), s(t), u(t) of a temporal variable t, and compute the Maurer-Cartan invariantsg −1 dg = i (ω P P + ω K K + ω D D) (see ref. [9] for more details) The first two forms can be used to impose constraintsρe −u = 1 2 ,u − 2sρ = 0, which link u and s to a single unconstrained ρ. Substituting the result into the last remaining invariant e u ṡ + s 2ρ , one gets ...
This is the celebrated Schwarzian derivative. Note that within this framework the SL(2, R)invariance of the latter is obvious as it is built from the Maurer-Cartan invariants. The goal of this paper is to provide a similar derivation of the N = 1 and N = 2 super-Schwarzian derivatives.
The work is organized as follows. In the next section, the N = 1 super-Schwarzian derivative is obtained by applying the method of nonlinear realizations to osp(1|2) superconformal algebra. Five superfield invariants are constructed which enter the decomposition ofg −1 Dg, where D is the covariant derivative, into a linear combination of the generators of osp(1|2). Note that the Grassmann parity ofg −1 Dg is opposite to that of the conventional Maurer-Cartan invariantg −1 dg because D is an odd operator. Imposing four constraints so as to express four superfields parametrizing a coset space elementg in terms of one unconstrained fermionic superfield and substituting the result into the last remaining invariant, one reproduces the N = 1 super-Schwarzian derivative. A similar group-theoretic derivation of the N = 2 super-Schwarzian based upon su(1, 1|1) superalgebra is given in section 3. In contrast to the previous case, the N = 2 super-Schwarzian derivative comes about when one analyses the reality condition for the superfield associated with the generator of special conformal transformations. We summarise our results and discuss possible further developments in the concluding section 4. Some useful identities relevant for computation of the superconformal invariants in section 2 and section 3 are gathered in appendix.

N = 1 super-Schwarzian derivative via nonlinear realizations
The N = 1 super-Schwarzian derivative where ψ(t, θ) is a real fermionic superfield and D is the covariant derivative, was first introduced in [3] by computing a finite superconformal transformation of the super stress-energy tensor underlying an N = 1 superconformal field theory. 4 Our objective in this section  4 To be more precise, in ref. [3] a complexified version of (2.1) was considered.

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is to demonstrate that (2.1) comes about naturally if one applies the method of nonlinear realizations to OSp(1|2) supergroup and keeps the number of independent Goldstone superfields to a minimum. Consider a real superspace R 1|1 parametrized by a bosonic coordinate t and a fermionic coordinate θ, θ 2 = 0. The supersymmetry transformations where a and are even and odd real supernumbers, respectively, realise the action of the d = 1, N = 1 supersymmetry algebra in the superspace. Within the method of nonlinear realizations, R 1|1 is represented by the supergroup elementg = e ith e θq , (2.4) while the left action of the supergroup on itself,g = e iah e q ·g, reproduces (2.2). The covariant derivative, which anticommutes with the supersymmetry generator, reads where ∂ t = ∂ ∂t , ∂ θ = ∂ ∂θ . Real bosonic and fermionic superfields are power series in θ which involve the bosonic components (b(t), B(t)) and their fermionic partners (f (t), F (t)). The covariant derivative (2.5) and a real fermionic superfield ψ(t, θ) are the building blocks entering eq. (2.1) above.
In order to derive the N = 1 super-Schwarzian derivative within the method of nonlinear realizations, let us consider osp(1|2) superconformal algebra where (P, D, K) are the bosonic generators of translations, dilatations and special conformal transformations, respectively. Q and S are the fermionic generators of supersymmetry transformations and superconformal boosts.

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As the next step, each generator in the superalgebra (2.7) is accompanied by a real Goldstone superfield of the same Grassmann parity and both R 1|1 and the superfields on it are represented by the element g = e ith e θq e iρ(t,θ)P e ψ(t,θ)Q e φ(t,θ)S e iµ(t,θ)K e iν(t,θ)D . (2.8) It is assumed that (h, q) (anti)commute with (P, D, K, Q, S).
Left multiplication by a group elementg = g·g, where g = e iaP e Q e σS e icK e ibD involves real bosonic parameters (a, b, c) and real fermionic parameters ( , σ), determines the action of the superconformal group OSp(1|2) on the superfields. Focusing on the infinitesimal transformations and making use of the Baker-Campbell-Hausdorff formula Note that both the original and transformed superfields depend on the same arguments (t, θ) such that the transformations affect the form of the superfields only, e.g. δρ = ρ (t, θ)− ρ(t, θ). Computing the algebra of the infinitesimal transformations (2.10), one can verify that it does reproduce the structure relations (2.7). 5 As the next step, one computes the invariant superfield combinations 5 In order to verify the structure relations (2.7), one first computes the commutators [δ1, δ2] = δ3 acting upon (ρ, µ, ν, ψ, φ) for all the transformations entering (2.10). Then one represents each δ as the product of a parameter and the corresponding generator, e.g. δa = a · P . Finally, one substitutes these into [δ1, δ2] = δ3 and discards the parameters on both the left and right hand sides of the equality. For the case at hand, this yields (2.7) after the rescaling (P, K, D) → (iP, iK, iD).

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(2.11) which originate from Note that the Grassmann parities of the invariants (2.11) are opposite to those associated with the conventional Maurer-Cartan one-formsg −1 dg because D is an odd operator. When obtaining (2.11), the identities exposed in appendix were heavily used. At this stage, one can use the invariants (2.11) so as to impose constraints enabling one to eliminate some of the Goldstone superfields. By analogy with our study of the Schwarzian derivative in [9], let us decide to keep the number of independent Goldstone superfields to a minimum and impose four conditions where g and p are even supernumbers. It seems quite natural to set the fermionic invariants to zero and to demand the bosonic invariants to take constant values as only the c-numbers are observable. The leftmost equation in (2.13) gives (2.15) Substituting these relations into the last remaining invariant ω K , one gets Up to an irrelevant constant factor, this coincides with the N = 1 super-Schwarzian derivative in (2.1). We conclude this section with a discussion of symmetries of (2.1). If one is interested in infinitesimal transformations, it suffices to consider (2.10) and focus on the transformation laws of ρ and ψ. A straightforward computation shows that both (2.1) and the supplementary condition (2.14) hold invariant.
If one is concerned with finite transformations, then, following ref. [3], one has to consider a generic super-diffeomorphism of R 1|1 which can be used to fix ρ provided ψ is known. Note that within the method of nonlinear realizations the supplementary condition (2.19) comes about as the constraint ω P = 0. Given ρ(t, θ) and ψ(t, θ) obeying (2.19), consider a coordinate transformation (2.17) and a new real fermionic superfield ψ (t , θ ) = ψ (ρ, ψ). Taking into account eq. (2.20), one gets the formula
In what follows, we shall assume that the conditions (3.10) and (3.11) hold. As a matter of fact, the method of nonlinear realizations allows one to reproduce the equation for ρ, which comes about as the constraint ω P =0, but not the chirality condition for the fermionic superfield ψ.

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As the next step, let us impose constraints similar to those in the preceding section where g and p are complex even supernumbers. They allow one to express (µ, ν, λ, φ) in terms of ψ which, in their turn, ensure φ to be a chiral superfield,Dφ = 0, and force the remaining invariants to vanish, ω K = ω J = ωS = 0. Finally, taking into account that µ is a real superfield,μ = µ, one gets the equation the left hand side of which reproduces the N = 2 super-Schwarzain derivative (3.1). As compared to the N = 1 case, the constraints (3.15) turn out to be more stringent and result in a variant of N = 2 super-Schwarzain mechanics in which the super-Schwarzian derivative is equal to a (coupling) constantpḡ −pg gḡ . The latter is an N = 2 analogue of the model studied recently in [11]. As was mentioned in the Introduction, our primarily concern in this work is to understand how the super-Schwarzian derivatives may be obtained within the method of nonlinear realizations. The dynamics of the specific model (3.17) will not be studied any further.

Discussion
To summarize, in this work we have demonstrated that the N = 1 and N = 2 super-Schwarzian derivatives can be obtained within the method of nonlinear realizations applied to OSp(1|2) and SU(1, 1|1) superconformal groups, thus providing an alternative to the existing approaches. Let us discuss possible further developments. Although it is not quite clear whether the construction in this work might tell us something new about superconformal field theory, it can definitely be used to generate super-Schwarzians invariant under a given supergroup. Such objects are indispensable for constructing N > 2 supersymmetric extensions of the Sachdev-Ye-Kitaev model. In this regard, the most pressing issue is to generalise 7 As in the N = 1, after computing the algebra one has to rescale the bosonic generators (P, D, K, J) → (iP, iD, iK, iJ ) so as to fit the notation in (3.7). 8 The standard form of SL(2, R) transformations with ad − cb = 1 is recovered by rescaling ψ → 1
In the literature there is some controversy on the latter point. In ref. [5] it is stated that an N = 4 super-Schwarzian is a non-local expression and only the covariant derivative of it is given in explicit form. However, because the R-symmetry subalgebra in [5] is so(4), the case seems to correspond to Osp(4|2) rather than SU(1, 1|2) (see also a related work [12]). Mathematicians report an obstruction to obtain a projective cocycle for N ≥ 3 (see, e.g., the discussion in [6]). An N = 4 super-Schwarzian proposed in [13] does not seem to be invariant under finite SU(1, 1|2) transformations (any super-Schwarzian should be a homogeneous function of degree zero under the rescaling ψ → bψ).
A preliminary consideration shows that an N = 4 super-Schwarzian generated by the method of nonlinear realizations might read where ψ α is a fermionic chiral superfield on R 1|4 superspace carrying an SU(2) spinor index α = 1, 2,ψ α is its complex conjugate (ψ α ) * =ψ α , and D α ,D α are the covariant derivatives. Above we abbreviated DψDψ = D α ψ βDαψ β . Yet, it turns out that along with (4.1) extra quadratic constraints on ψ α appear, which still need to be understood. We hope to report on the progress as well as to describe a more general case of the D(2, 1; α) super-Schwarzian elsewhere. An elegant derivation of the N = 1 and N = 2 super-Schwarzian derivatives within the context of a one-dimensional Osp(N |2M ) pseudoparticle mechanics was proposed in [14]. It would be interesting to see if the analysis in [14] can be generalised to the N = 4 case, which should link to the Osp(4|2) super-Schwarzian in [5].
A connection between the conventional second order conformal mechanics and the Schwarzian mechanics was discussed in a very recent work [15]. It would be interesting to explore whether the analysis in [15] can be extended to produce the super-Schwarzians. The construction of higher derivative superconformal mechanics of the Schwarzian type along the lines in [16] is of interest as well.
Note that these relations are also valid for the su(1, 1|1) superconformal algebra, in which case Q and S are regarded complex. In that case the identities involvingQ,S follow by the Hermitian conjugation. When computing the superconformal invariants (3.14), the following identities: Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.