Supersymmetry and Positive Energy in Classical and Quantum Two-Dimensional Dilaton Gravity

TL;DR

This paper demonstrates that in a supersymmetric two-dimensional dilaton gravity model, the ADM energy is non-negative under certain conditions, and extends positive energy theorems to quantum and black hole contexts, revealing deep links between supersymmetry and energy positivity.

Contribution

It introduces a supersymmetric version of 2D dilaton gravity, proves a positive energy theorem in both classical and quantum regimes, and relates supersymmetry to energy bounds in black hole spacetimes.

Findings

The linear dilaton vacuum breaks half the supersymmetries.

The ADM energy is non-negative for smooth initial data obeying the dominant energy condition.

A quantum positive energy theorem holds despite quantum stress tensor indefiniteness.

Abstract

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 ADM 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 {\it 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 $e^{- 2 \phi_H}$, where $\phi_H$ 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.