Super-Schwarzians via nonlinear realizations

Abstract

The N = 1 and N = 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.

Schwarzian mechanics via nonlinear realizations

“…Because the N = 4 super-Schwarzian derivative is expected to involve the fermionic superfield ψ α only, one is led to use the invariants (26), so as to eliminate (ν, µ, λ a , φ α ,φ α ) from the consideration. Guided by a recent analysis of the N = 1 and N = 2 super-Schwarzian derivatives in a similar setting [8], let us impose the following constraints…”

“…In a recent work [8], a third alternative was studied, which consists in applying the method of nonlinear realizations [9] to finite-dimensional superconformal groups. In this setting, a super-Schwarzian derivative is linked to the supergroup invariants.…”

“…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. In particular, in [8] the N = 1 and N = 2 super-Schwarzian derivatives were obtained by applying the procedure to OSp(1|2) and SU (1, 1|1) superconformal groups, respectively. 3 The goal of this paper is to provide a similar derivation of the N = 4 super-Schwarzian associated with SU (1, 1|2) superconformal group.…”

(Super)Schwarzian mechanics

“…The suitable way to construct (super)conformal invariants is the method of nonlinear realizations [10][11][12][13] equipped with the inverse Higgs phenomenon [14]. Such approach demonstrated how the Schwarzians can be obtained via the non-linear realizations approach, was initiated in [15] and then it was applied to different superconformal algebras in [16][17][18][19]. Later on, this approach has been extended to the cases of non-relativistic Schwarzians and Carroll algebra [20].…”

“…Of course, these questions were partially analyzed and answered in the papers [15][16][17][18][19]. However, some important properties and statements were missing.…”