Duality between Electric and Magnetic Black Holes

TL;DR

The authors investigate the quantum implications of electric–magnetic duality in Einstein–Maxwell gravity, showing that the Maxwell action’s sign flip under duality does not spoil duality at the semiclassical level when electric charge is fixed via a charge-projection operator. They demonstrate that the semiclassical Euclidean partition functions for dual electric and magnetic black holes, including in cosmological de Sitter backgrounds and in backgrounds with external fields (Ernst solutions), are identical, implying duality-invariant pair production rates. The work also clarifies entropy definitions for charged black holes in both asymptotically flat and cosmological spacetimes, finding the familiar area law $S = \mathcal{A}/4$ in these contexts, including for the collective horizon area in de Sitter instantons. Overall, the results provide strong evidence that electric–magnetic duality persists as a quantum symmetry in gravity in a nuanced, boundary-condition–dependent way, rather than as a naive invariance of the action alone.

Abstract

A number of attempts have recently been made to extend the conjectured $S$ duality of Yang Mills theory to gravity. Central to these speculations has been the belief that electrically and magnetically charged black holes, the solitons of quantum gravity, have identical quantum properties. This is not obvious, because although duality is a symmetry of the classical equations of motion, it changes the sign of the Maxwell action. Nevertheless, we show that the chemical potential and charge projection that one has to introduce for electric but not magnetic black holes exactly compensate for the difference in action in the semi-classical approximation. In particular, we show that the pair production of electric black holes is not a runaway process, as one might think if one just went by the action of the relevant instanton. We also comment on the definition of the entropy in cosmological situations, and show that we need to be more careful when defining the entropy than we are in an asymptotically-flat case.