# Action Growth of Dyonic Black Holes and Electromagnetic Duality

## TL;DR

This paper demonstrates how to restore electromagnetic duality in the action of various Maxwell-related theories by adding a universal boundary term, ensuring the duality persists in the context of holographic complexity calculations.

## Contribution

The authors introduce a universal boundary term in both first- and second-order formalisms that restores electromagnetic duality in the action for a broad class of theories.

## Key findings

- Duality is preserved in on-shell actions across multiple theories.
- The boundary term works in both first- and second-order formalisms.
- Electromagnetic duality remains intact in the complexity=action framework.

## Abstract

Electromagnetic duality of Maxwell theory is a symmetry of equations but not of the action. The usual application of the `complexity=action' conjecture would thus loose this duality. It was recently proposed in arxiv:1901.00014 that the duality can be restored by adding some appropriate boundary term, at the price of introducing the mixed boundary condition in the variation principle. We present universal such a term in both first-order and second-order formalism for a general theory of a minimally-coupled Maxwell field. The first-order formalism has the advantage that the variation principle involves only the Dirichlet boundary condition. Including this term, we compute the on-shell actions in the Wheeler-De Witt patch and find that the duality persists in these actions for a variety of theories, including Einstein-Maxwell, Einstein-Maxwell-Dilaton, Einstein-Born-Infeld and Einstein-Horndeski-Maxwell theories.