General two-dimensional supergravity from Poisson superalgebras

We provide the geometric actions for most general N = 1 supergravity in two spacetime dimensions. Our construction implies an extension to arbitrary N . This provides a supersymmetrization of any generalized dilaton gravity th eory or of any theory with an action being an (essentially) arbitrary funct ion of curvature and torsion. Technically we proceed as follows: The bosoni c part of any of these theories may be characterized by a generically nonli near Poisson bracket on a three-dimensional target space. In analogy to a g iven ordinary Lie algebra, we derive all possible N = 1 extensions of any o f the given Poisson (or W-) algebras. Using the concept of graded Poisson s igma models, any extension of the algebra yields a possible supergravity ex tension of the original theory, local Lorentz and super-diffeomorphism inva riance follow by construction. Our procedure automatically restricts the fe rmionic extension to the minimal one; thus local supersymmetry is realized on-shell. By avoiding a superfield approach we are also able to circumvent in this way the introduction of constraints and their solution. For many we ll-known dilaton theories different supergravity extensions are derived. In generic cases their field equations are solved explicitly.