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

This paper investigates the non-Abelian extension of ModMax electrodynamics by demonstrating the existence of self-dual instantons in various curved backgrounds, constructing multi-instanton solutions through perturbative methods, and deriving new gravitating configurations such as Euclidean wormholes when coupled to gravity.

The Gist

Imagine the universe as a giant, complex machine. For a long time, physicists have used a set of blueprints called Maxwell's equations to describe how electricity and magnetism work. These blueprints are perfect for most things, but they have a flaw: if you try to zoom in too close to a single electron, the math breaks down and gives you an infinite energy (a "singularity").

To fix this, scientists invented "non-linear" theories. Think of these as adding shock absorbers to the machine so it doesn't break when things get too intense. One of the newest and most interesting blueprints is called ModMax. It's special because it keeps the universe looking the same no matter how much you zoom in or out (scale invariance) and keeps the laws of physics balanced between electricity and magnetism (duality).

This paper takes that ModMax blueprint and asks: "What happens if we apply it to the strong nuclear force?"

Here is a breakdown of their discoveries, using everyday analogies:

1. The "Perfect Knots" (Instantons)

In the world of quantum physics, there are solutions called instantons. Imagine a piece of string. If you tie a knot in it, that knot is a stable, localized structure. In physics, these "knots" are temporary, self-contained bubbles of energy that pop into existence and then vanish. They are crucial for understanding how the universe tunnels between different states.

• The Old Way: In standard physics (Yang-Mills theory), these knots are well understood.
• The New Discovery: The authors found that even with the complex "shock absorbers" of ModMax theory, these knots still exist! They proved that you can tie these "perfect knots" in the ModMax fabric of the universe.
• The Twist: They found that on a curved background (like the shape of the universe in Anti-de Sitter space), the knot doesn't just disappear at the edges. Instead, it leaves a "tail" that changes the knot's identity. It's like tying a knot in a rope that stretches to infinity; the size of the knot actually changes how the rope looks far away. This means the "knot count" (a topological number) isn't always a whole number like 1 or 2; it can be a fraction depending on the knot's size.

2. The "Swiss Army Knife" of Knots (Multi-Instantons)

Usually, making multiple knots at once is a nightmare because they interact and mess each other up. However, because ModMax theory has no fixed "ruler" (scale invariance), the authors realized you can pack many of these knots together.

• The Analogy: Imagine a set of Russian nesting dolls. In standard physics, they fit together perfectly. In ModMax, the dolls are made of a stretchy, magical material. The authors figured out how to stretch and shrink these dolls to fit $N$ of them together, solving the math step-by-step. They showed that even with the complex non-linear rules, you can still build a stable cluster of these energy knots.

3. The "Ghostly Echo" (Dirac Spectrum)

When these knots exist, they affect the "ghosts" of the universe—mathematical particles called fermions (like electrons). The authors calculated how these ghosts behave on the edge of the universe (the boundary).

• The Metaphor: Imagine a bell. When you hit it, it rings at a specific pitch. The "knots" in the ModMax theory change the shape of the bell, altering the pitch of the ring. The authors calculated this new pitch. This is important because the "pitch" tells us if the universe has a hidden imbalance (an anomaly) that could explain why matter exists over antimatter.

4. Building a "Cosmic Tunnel" (Wormholes)

The most exciting part is what happens when you add Gravity to the mix.

• The Setup: The authors took their ModMax knots and asked, "What if these knots are heavy enough to bend space?"
• The Result: They found that these knots can act as the "pillars" holding open a wormhole.
• The Analogy: Usually, a wormhole is like a tunnel through a mountain. If you try to build one, it collapses instantly. But here, the "ModMax knots" act like a magical, self-supporting scaffolding. They are made of a fluid that behaves like light (ultra-relativistic), and they hold the tunnel open without collapsing.
• Secondary Hair: They also found that these wormholes can have "secondary hair." In physics, "hair" usually means extra features a black hole or wormhole can have. Think of a bald head (a standard black hole) vs. a head with a stylish hat and scarf (this new wormhole). The "hat" is a scalar field (a type of energy field) that clings to the wormhole, making it unique and stable.

Why Does This Matter?

This paper is like finding a new type of Lego brick.

1. It works: It proves that these complex, non-linear theories can still support the fundamental structures (knots) we rely on for quantum mechanics.
2. It's flexible: It shows that these structures can exist in curved spaces and even hold open wormholes.
3. It connects: It bridges the gap between abstract math (duality and conformal symmetry) and physical reality (gravity and wormholes).

In short, the authors took a fancy new theory of electricity, showed it can tie the same complex knots as the old theory, and then used those knots to build a stable, non-singular tunnel through the fabric of space-time. It's a step toward understanding how the universe might be stitched together at its most fundamental level.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the extension of ModMax electrodynamics—a recently discovered, conformally invariant, and duality-invariant nonlinear electrodynamics theory characterized by a dimensionless coupling constant $\gamma$—to the non-Abelian (Yang-Mills) sector.

• Context: While ModMax is well-studied in the Abelian (Maxwell) limit, its non-Abelian generalization remains largely unexplored.
• Key Question: Do (anti-)self-dual instanton solutions exist in non-Abelian ModMax theory? If so, how do they deform compared to standard Yang-Mills (YM) instantons, and what are their implications for topological invariants and gravity?
• Challenge: Standard YM instantons rely on the linearity of the field strength in the self-duality condition ($F = \pm \tilde{F}$). In ModMax, the constitutive relations are highly nonlinear, potentially destroying the existence of regular, finite-action instanton solutions.

2. Methodology

The authors employ a combination of analytical construction, perturbation theory, and numerical analysis within a Euclidean framework.

• Theoretical Framework:

  • They define the Non-Abelian ModMax (NAM) action for an $SU(2)$ gauge group, parameterized by $\gamma$. The action depends on two gauge invariants, $X = \frac{1}{4}\text{Tr}(F_{\mu\nu}F^{\mu\nu})$ and $Y = \frac{1}{4}\text{Tr}(\tilde{F}_{\mu\nu}F^{\mu\nu})$.
  • They derive the field equations and the constitutive tensor $P_{\mu\nu}$.
  • They establish a generalized (anti-)self-duality condition: $P_{\mu\nu} = \pm \tilde{F}_{\mu\nu}$. They prove that configurations satisfying this condition have a vanishing energy-momentum tensor and automatically solve the full field equations due to the Bianchi identity.

• Analytical Solutions (Constant Curvature):

  • They construct solutions on constant curvature spaces: Flat space ($\mathbb{R}^4$), Euclidean de Sitter ($S^4$), and Euclidean anti-de Sitter ($H^4$).
  • Using an ansatz aligned with $SU(2)$ left-invariant forms, they reduce the system to a single first-order differential equation for the profile function $a(r)$.

• Multi-Instanton Construction:

  • To address the difficulty of solving the nonlinear equations for multiple instantons, they utilize the 't Hooft ansatz ($A_\mu = -\bar{\eta}_{\mu\nu}\partial_\nu \log \Phi$).
  • Since an exact analytic solution for $\Phi(x)$ is intractable, they employ perturbation theory in the parameter $\gamma$. They expand $\Phi = \Phi^{(0)} + \gamma \Phi^{(1)} + \dots$, where $\Phi^{(0)}$ is the standard YM solution.

• Gravitational Coupling:

  • The theory is coupled to Einstein gravity with a cosmological constant and a conformally coupled scalar field.
  • They search for self-gravitating solutions (dyons and wormholes) by relaxing the self-duality condition, allowing for a non-vanishing energy-momentum tensor.

3. Key Contributions and Results

A. Existence of Generalized Instantons

• Generalized BPST Instanton: The authors successfully construct a generalized Belavin-Polyakov-Schwartz-Tyupkin (BPST) instanton for $SU(2)$ on flat space, de Sitter, and anti-de Sitter backgrounds.
• Deformation: The solution is controlled by the parameter $\gamma$. In the limit $\gamma \to 0$, it recovers the standard YM instanton.
• Chern-Pontryagin Index:
  • On flat and de Sitter spaces, the topological charge (Chern-Pontryagin index) remains an integer ($\pm 1$).
  • Crucial Finding: On Euclidean AdS ($H^4$), the configuration is not a pure gauge at infinity. Consequently, the Chern-Pontryagin index becomes non-integer and depends on the instanton size (integration constant). This reproduces the behavior found by Callan and Wilczek in standard YM theory on negative-curvature backgrounds.

B. Dirac Spectrum and Topological Invariants

• To correctly compute topological invariants on manifolds with boundaries (like AdS), the bulk integral must be supplemented by boundary terms.
• The authors compute the spectrum of the Dirac operator on the conformal boundary of Euclidean AdS in the presence of the ModMax instanton.
• This spectral analysis is essential for determining the Atiyah-Patodi-Singer (APS) $\eta$-invariant, which accounts for the spectral asymmetry and ensures the consistency of the Dirac index in the presence of non-Abelian fields.

C. Multi-Instanton Solutions

• Using the 't Hooft ansatz and perturbative expansion, they derive a formal solution for $N$-instanton configurations to first order in $\gamma$.
• The problem reduces to solving a Poisson equation with a source term determined by the leading-order YM solution. The resulting correction $\Phi^{(1)}$ is explicitly calculated, confirming that multi-instanton configurations exist and smoothly reduce to YM solutions as $\gamma \to 0$.

D. Gravitating Dyons and Wormholes

• Non-Abelian Dyons: By coupling the theory to gravity and relaxing the self-duality condition, they find solutions describing gravitating non-Abelian dyons. The ModMax parameter $\gamma$ influences the oscillatory frequency of the gauge field profile.
• Equation of State: The energy-momentum tensor of the self-gravitating ModMax field behaves as an ultra-relativistic fluid ($p = \rho/3$).
• Euclidean Wormholes:
  • They construct regular Euclidean wormhole solutions connecting asymptotic regions.
  • With Cosmological Constant: Solutions exist for both positive (de Sitter) and negative (AdS) $\Lambda$. In the AdS case, a throat exists where the area coordinate is minimal.
  • Without Cosmological Constant: They find a wormhole solution connecting two asymptotically flat regions.
  • Regularity: Crucially, the inclusion of the conformally coupled scalar field and the nonlinear ModMax terms ensures that all curvature invariants (Kretschmann scalar) and field invariants remain regular everywhere, avoiding the singularities often found in such gravitational configurations.

4. Significance

• Theoretical Consistency: The work demonstrates that the desirable properties of ModMax (conformal invariance, duality invariance) can be successfully extended to the non-Abelian sector without destroying the existence of topological solitons (instantons).
• Topological Nuances: It highlights the subtle dependence of topological charges on the background geometry (specifically AdS) and the necessity of boundary spectral analysis (Dirac spectrum) for correct topological counting.
• Gravitational Physics: The construction of regular, self-gravitating wormholes and instantons with "secondary hair" (scalar fields) provides new exact solutions in Einstein-Maxwell-like theories. These solutions are relevant for:
  • Holography: Understanding non-perturbative effects in AdS/CFT, particularly regarding multi-boundary geometries and entanglement entropy.
  • Singularity Resolution: Showing how nonlinear electrodynamics and conformal scalars can prevent gravitational singularities.
• Future Directions: The paper opens avenues for studying holographic magnetotransport in dual CFTs and investigating parity anomalies via the computed Dirac spectrum.

In summary, this paper establishes the non-Abelian ModMax theory as a robust framework for studying nonlinear gauge-gravity interactions, providing explicit instanton solutions, multi-instanton perturbative expansions, and regular gravitational wormhole geometries.