General Two-Dimensional Supergravity from Poisson Superalgebras

TL;DR

This paper develops a geometric framework for constructing the most general N=1 supergravity theories in two dimensions, extending to arbitrary N, by utilizing graded Poisson Sigma Models and Poisson superalgebras.

Contribution

It introduces a method to derive supergravity extensions of generalized dilaton gravities using Poisson superalgebras, avoiding superfield constraints and providing explicit solutions.

Findings

Constructed supergravity actions for various dilaton theories.

Derived all possible N=1 extensions of Poisson (W-) algebras.

Explicitly solved field equations for many models.

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.