On the SuperConformal Quantum Mechanics in the Nonlinear Realizations Approach

In the framework of nonlinear realizations we rederive the action of the N=2 SuperConformal Quantum Mechanics (SCQM). We propose also the WZNW -- like construction of the interaction term in the lagrangian with the help of Cartan's Omega forms.

In general the coordinate x n in (2.2) transforms through the infinitesimal transformation function ε(τ ) and coordinates x k , k ≤ n. In addition the transformation law for parameter x n contains the term with n + 1-st derivative of the parameter ε(τ ).
The simplest transformation law have the dimension-one coordinate τ which transforms as the coordinate of the one-dimensional space under the reparametrization. The coordinates x 0 and x 1 transform correspondingly as the dilaton and one-dimensional Cristoffel symbol.
At this stage it is natural to consider all parameters as the fields in one-dimensional space parametrized by the coordinate τ . However, in general all fields x m (τ ) can depend on some coordinate σ, which plays the role of additional parameter. Even more, such dependence on the additional parameter σ can take place only for fields x m with m ≥ M with fixed M. This will not lead to any contradictions with the transformation laws (2.4)-(2.7). As we will see, the introduction of such additional coordinates gives the possibility to construct some nontrivial invariants of the transformations (2.3).
As we already mentioned, the conformal group in one dimension is a subgroup of (2.2), namely the ones generated by L −1 , L 0 and L 1 As was shown in [11] the one-dimensional conformal mechanics introduced in [1] can be described on the language of invariant differential Cartan's forms connected with the parametrization (2.8) of the conformal group. Moreover, by the linear change of basis of the conformal algebra one can describe [8] on the same footing the "new" conformal mechanics of [12].
In this Section we reproduce the results of [11] using the natural matrix realization for the generators of SL(2, R) group: translation H = L −1 , dilatation D = L 0 and conformal Such representation of the generators of the conformal group can be easily generalized to the superconformal case [16], including the extended ones. So, in the purely bosonic case the element of the conformal group in one dimension can be parametrized as a product of three matrix multipliers The parameters in (2.8) and (2.10) are connected by the relations t = −τ, x = e x 0 /2 and p = −x 1 x. The conformal group transformation of these new variables are where parameters of the transformation are constrained by the unimodularity condition ad − bc = 1. Using the representation (2.2) one can calculate also the transformations of functions x(τ ) and p(τ ) under the most general (finite) reparametrization: (2.14) The invariant differential Cartan's form, calculated with the help of (2.10) is All matrix elements in (2.15) are invariant under the transformations (2.11). One can recognize among them the einbein differential form ω H .

The action integral for Conformal Mechanics
All these differential forms can be used for construction of an invariant action. The simplest one is the linear combination The first term in this expression is appropriately normalized to get the correct kinetic term. The parameter Λ plays the role of cosmological constant. One can find p by solving its equation of motion, insert it back in the lagrangian and get the action of De Alfaro, Fubini and Furlan [1] with the coupling constant γ = Λ + α 2 /2. So, the parameter α simply renormalizes the cosmological constant Λ. At this point we should note that the integrand in the action S (??) is invariant only up to the total derivative, though it was deduced from the expression (2.16) in which the integrand is strictly invariant because it was constructed out of invariant Cartan's Omega forms ω K , ω D and ω H . The reason of this lies in the utilization of some equations of motion and partial integration when (2.16) is transformed into (2.17). The same situation will take place also in the more complicated case of N = 2 SCQM.
The additional invariant actions can be constructed as [17] with arbitrary function F of two invariant variables which are the coefficients in the expressions of invariant differential one-forms ω K and ω D in terms of only one (in dimension one) independent invariant one-form ω H . Some of these actions will have form (2.17), but in general the actions (2.18) will include the higher degrees of the velocityẋ. The another sort of invariants in the action can be constructed by introducing the dependence of the group element (2.8) or (2.10) on some parameter σ. Indeed, one can consider the special dependence of the group element (2.10) on some new parameter σ such that t does not depend on it, whereas functions x 0 (t, σ) and x 1 (t, σ) are subject to the following boundary conditions So, the boundary group elements in (2.10) are where G 0 is the identity element of the group. The parameters ǫ n (in our case n = 0, 1, 2) of transformation (2.3) are assumed to be independent of σ. It means that condition (2.21) will be the same for transformed quantities. On the other hand the group element G 0 in the boundary condition (2.20) will vary. But this variation is very simple, as one can see from the transformation laws of x 0 and x 1 . Moreover, both of them are total derivatives. All this can serve as an argumentation of the following construction, leading in general to some integral invariants on the group. In our case there are two independent invariant differential one-forms -ω H and dσ. So, the coefficients in expanding of all other Cartan's forms in terms of these two forms are invariant as well. For example, such one is In the presence of new coordinate σ the invariant integration measure is The result of integration over σ is the difference of two terms at points σ = 1 and σ = 0. The last one is invariant by virtue of the transformation laws (2.5)-(2.6) of x 0 and x 1 near the identity element which corresponds to x 0 = 0 and x 1 = 0. Indeed, they transform as total derivatives. Because by construction S 1 is invariant, the term at the point σ = 1 should be also invariant, though the integrand in this term transforms as a total derivative. In the case under consideration it is not wondering, because this invariant simply reproduces the already known third term in (2.16). Nevertheless, as we will see later, such procedure of constructing can lead to new invariants in more complicated cases.

The matrix representation of the N = 2 SuperConformal Group
The N = 2 SuperConformal group in one dimensional space is an eight-parameter subgroup of the infinitedimensional N = 2 Super Virasoro group with the following algebra of its generators Our parametrization for coset space of N = 2 SuperConformal group over the U(1) subgroup generated by U is: The transformation laws of the coset space parameters under the infinitesimal left shift are: One can see that in the point (x = 1, x 1 = ψ = 0) the variables (x, x 1 , ψ) transform as a total derivatives. The differential Cartan's form, calculated with the help of (3.7) is where Due to the fact that we consider the coset space G SC /U(1) instead of the whole group G SC /U(1), not all of Cartan's forms are invariant. Namely, ω Q , ωQ, ω S and ωS transform homogeneously as linear representations of U(1). As one can easily see the forms ω Q , ωS carry the same charge under the U(1) transformations, whereas ωQ and ω S carry the opposite equal charge. In turn, ω U transform as a total differential.

The action integral for N = 2 SuperConformal Mechanics
All Cartan's forms can be expanded in terms of three independent ones ω H , ω Q , ωQ using the formula df (t, θ,θ) = xω Q Df + xωQDf + x 2 ω H (ḟ − iψDf + iψDf ), (3.24) where D andD are flat covariant derivatives The coefficients in such expansions of ω H , ω D , ω S and ω K are invariant under the transformations (3.9)-(3.13) or transform as some linear representation of U(1), Two of these expansions which are useful in the construction of invariants are Since ω D and ω H are invariant, the coefficient is invariant as well. At the same time the coefficients are mutually conjugated and inert under the transformations (3.9)-(3.13). So, their sum L K = I S/Q + IS /Q gives one more possible term in the Lagrangian 3 . Indeed, it describes the kinetic term of the SCQM in the superfield formulation.
Using the invariant measure dv = dtdθdθ one can construct the invariant action in the form With the help of the equations of motion for ψ andψ the action S K can be rewritten in the form S K = 4 dtdθdθDxDx, (3.36) in which the integrand is invariant only up to the total derivatives (see the remark after the eq. (2.17)). Note, that the left hand sides of the equations (3.34)-(3.35) coincide with the coefficients (3.29)-(3.30). So, they vanish some part of the Cartan's Omega form ω D , playing the role of inverse Higgs effect [13]. It means that there is no need to use the eq's -(3.34)-(3.35) as independent ones arising from the inverse Higgs effect as in [6].
One can construct some other invariant terms for the action, for example but this term leads simply to redefinition of overall coefficient in (3.36). The additional invariants in the action can be constructed by the procedure described in the previous Section. We again introduce the dependence of the group element (3.7) on some parameter σ such that t, θ,θ do not depend on it, whereas functions x 0 (t, σ), x 1 (t, σ) andψ(t, σ) are subject to the boundary conditions ln x(t, 0) = x 1 (t, 0) = ψ(t, 0) = 0, (3.38) ln x(t, 1) = ln x(t), x 1 (t, 1) = x 1 (t), ψ(t, 1) = ψ(t). Using the arguments of the previous Section one can show that the expressioñ is invariant and leads to additional invariant term in the action So, the total action S = S 0 +S 1 reproduces the action of N = 2 SuperConformal Quantum mechanics [2]- [3] including both the kinetic and potential terms.

Conclusions
In this paper, we applied the methods of nonlinear realizations approach for construction of the actions of Conformal and N = 2 SuperConformal Quantum Mechanics. We have shown that both the kinetic and interaction terms of these models can be constructed by using the invariant Cartan's Omega-forms. The interaction part of the action looks like the well known WZNW term. We have shown also that the Inverse Higgs Effect in both cases is a consequence of the equations of motion for some variables. It would be interesting to analyze the possibility of such duality between the Inverse Higgs Effect and equation of motions for auxiliary variables in more complicated theories, like N = 4 SuperConformal Quantum Mechanics. Besides, it is interesting to investigate in this model the possibility getting with the help of the equations of motion of some irreducibility conditions for the basic superfield, which were obtained originally by the inverse Higgs effect [5].