SYMPLECTIC ACTIONS ON COADJOINT ORBITS

We present a compact expression for the field theoretical actions based on the symplectic analysis of coadjoint orbits of Lie groups. The final formula for the action density αc becomes a bilinear form 〈(S, 1/λ), (y, my)〈, where S is a 1-cocycle of the Lie group (a schwarzian type of derivative in conformal case), λ is a coefficient of the central element of the algebra and Y ≡ (y, my) is the generalized Maurer-Cartan form. In this way the action is fully determined in terms of the basic group theoretical objects. This result is illustrated on a number of examples, including the superconformal model with N = 2. In this case the method is applied to derive the N = 2 superspace generalization of the D=2 Polyakov (super-)gravity action in a manifest (2,0) supersymetric form. As a byproduct we also find a natural (2, 0) superspace generalization of the Beltrami equations for the (2, 0) supersymmetric world-sheet metric describing the transition from the "conformal" to the "chiral" gauge. © 1990.