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

TL;DR

The paper introduces a one-parameter non-Abelian ModMax theory for SU(2) that deforms Yang–Mills while preserving conformal and duality invariance, and proves the existence of (anti-)self-dual instantons on flat and constant-curvature spaces. It generalizes BPST and ILP instantons and develops a perturbative multi-instanton construction via a 't Hooft ansatz, with the YM limit recovered as $\gamma \to 0$. The work further couples the theory to gravity with a conformally coupled scalar, yielding regular gravitating configurations including Euclidean wormholes and gravitational instantons with secondary hair, along with gauge-invariant charges and a boundary Dirac spectrum analysis. It opens avenues for holographic inquiries and parity/anomaly studies at AdS boundaries, linking nonlinear conformal gauge dynamics to rich Euclidean geometries and topological features.

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.

Figures (2)

• Figure 1: Plot of the potential $V(a)$ as a function of the generalized coordinate $a$. The shaded area represents the allowed region for solutions, where $Q\geq V(a)$.
• Figure 2: Profile of the oscillatory solution of the ModMax field equations. Here, we choose the initial conditions $a(0)=0.25$, $a'(0)=0$, and we have used $\gamma = \{ \ln 2\,, \ln 3\,, \ln 5\}$ for the blue, orange, and green plots, respectively. Notice that the role of the ModMax parameter $\gamma$ is directly related to the frequency of the oscillatory solution: the larger its value, the higher the frequency.