Multiplet calculus and compensating fields in D=2 conformal supergravity

The construction of conformal (1,1) and (1,0) supergravity is discussed. The representations of the local supersymmetry algebra on scalar, spinor and vector multiplets are given, and a procedure for constructing invariant actions is established. Several previously known and unknown Lagrange densities for these multiplets are constructed. A compensating multiplet of fields with non-linear transformation law is found. It can be used to construct conformal actions for the (super)gravity fields, such as the dilaton term in string theories. It can also be used to pass to Poincare supergravity by classical gauge fixing. It has a dual multiplet compensating for local Lorentz transformations. From this multiplet one can construct a new kind of Poincare-like supergravity with local dilatation invariance, but broken d=2 Lorentz invariance. (orig.)