Self-dual instantons and gravitating dyons in non-Abelian ModMax theory

TL;DR

This paper explores non-Abelian ModMax electrodynamics, demonstrating the existence of self-dual instantons, constructing explicit solutions including multi-instantons, and coupling the theory to gravity to find novel Euclidean wormhole solutions.

Contribution

It introduces the non-Abelian extension of ModMax theory, constructs explicit instanton solutions, and couples the theory to gravity to find new Euclidean wormhole configurations.

Findings

Existence of (anti-)self-dual instantons in non-Abelian ModMax theory

Construction of the generalization of the BPST instanton for SU(2)

Coupling to gravity yields Euclidean wormholes and smooth solutions with secondary hair.

Abstract

Motivated by the recent interest in conformal and duality invariant nonlinear electrodynamics, we study the non-Abelian extension of ModMax electrodynamics. The theory is parameterized by a single dimensionless constant, and it is continuously connected to Yang-Mills theory in its vanishing limit. We show that the theory admits (anti-)self-dual instantons, despite the additional nonlinearities that characterize the non-Abelian ModMax theory. For $SU(2)$, we construct the generalization of the BPST instanton and extend this solution to Euclidean de Sitter and anti-de Sitter backgrounds. In the latter case, the Chern-Pontryagin index depends on the instanton size since the configuration is not a pure gauge at infinity; a property already pointed out in Yang-Mills on negative-curvature backgrounds by Callan and Wilczek. We compute the contribution of the latter to the spectrum of the Dirac operator at the boundary, which is crucial for determining the non-local contributions to the Dirac index. Then, we show that the ansatz constructed with 't Hooft symbols accommodates multi-instantons in the non-Abelian ModMax theory. The system of (anti-)self-dual equations reduces to a single nonlinear equation, which can be perturbatively solved order by order in the parameter that controls the nonlinearity. Following such a strategy, we provide a formal solution for the $N$-instanton configuration to first order in the expansion. Then, we couple non-Abelian ModMax theory to gravity with a conformally coupled scalar field and construct new gravitating solutions that describe Euclidean wormholes and other smooth configurations with secondary hair.