Supersymmetry and positive energy in classical and quantum two-dimensional dilaton gravity

An N=1 supersymmetric version of two-dimensional dilaton gravity coupled to matter is considered. It is shown that the linear dilaton vacuum spontaneously breaks half the supersymmetries, leaving broken a linear combination of left and right supersymmetries which squares to time translations. Supersymmetry suggests a spinorial expression for the Arnowitt-Deser-Misner energy M, as found by Witten in four-dimensional general relativity. Using this expression it is proven that M is non-negative for smooth initial data asymptotic (in both directions) to the linear dilaton vacuum, provided that the (not necessarily supersymmetric) matter stress tensor obeys the dominant energy condition. A quantum positive-energy theorem is also proven for the semiclassical large-N equations, despite the indefiniteness of the quantum stress tensor. For black-hole spacetimes, it is shown that M is bounded from below by ${\mathit{e}}_{\mathit{H}}^{\mathrm{-}2\mathrm{\phi }}$, where ${\mathrm{\phi }}_{\mathit{H}}$ is the value of the dilaton at the apparent horizon, provided only that the stress tensor is positive outside the apparent horizon. This is the two-dimensional analogue of an unproven conjecture due to Penrose. Finally, supersymmetry is used to prove positive-energy theorems for a large class of generalizations of dilaton gravity which arise in consideration of the quantum theory.