General two-dimensional supergravity from Poisson superalgebras

Abstract

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 theory or of any theory with an action being an (essentially) arbitrary function of curvature and torsion.Technically we proceed as follows: The bosonic part of any of these theories may be characterized by a generically nonlinear Poisson bracket on a three-dimensional target space. In analogy to a given ordinary Lie algebra, we derive all possible N = 1 extensions of any of the given Poisson (or W -) algebras. Using the concept of graded Poisson Sigma Models, any extension of the algebra yields a possible supergravity extension of the original theory, local Lorentz and super-diffeomorphism invariance follow by construction. Our procedure automatically restricts the fermionic 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 well-known dilaton theories different supergravity extensions are derived. In generic cases their field equations are solved explicitly.

Graded Poisson-sigma models and dilaton-deformed 2D supergravity algebra

“…In the gPSM formulation rigid supersymmetry is produced by the specific choice of the Poisson tensor [38] …”

“…[47] within a different approach, a derivation in terms of graded Poisson Sigma Models has been given in ref. [38]. As the bosonic potential is restricted here to be a function of the dilaton field φ only, these models have vanishing bosonic torsion (Z = 0 in (1.6)).…”

“…The Jackiw-Teitelboim model [32][33][34][35][36] corresponds to 8) and the bosonic part of the simplest non-trivial 2d supergravity model of Howe [37] -after the correct identification of the dilaton with one of the components of the superfield [38]becomes…”

“…to a gPSM with a graded Poisson tensor P IJ = − (−1) IJ P JI [38,41]. Thanks to our conventions (cf.…”

“…Attempts to simply require the absence of new singularities in the fermionic part of the action as a means to determine "allowed" extensions turned out to be unsuccessful: In a counter-example [38] two different supergravities with the same (bosonic) v of spherical reduction ((1.6) with (1.7)) were found to be free from such singularities.…”