Electromagnetic instantons and asymmetric Hawking radiation of black holes

Abstract: We argue that the topological structure of Abelian gauge theories, such as Maxwell electrodynamics, in the background of a Euclidean Schwarzschild black hole manifests itself through an asymmetry in Hawking radiation. In particular, the topology of the black hole manifold, characterised by a non-contractible 2-sphere and Euler characteristic $χ= 2$, admits non-trivial gauge-field configurations. These take the form of 2-form field strengths that are closed but not exact. From a topological perspective, such configurations are classified by the second cohomology group, which is isomorphic to $\mathbb{Z} \oplus \mathbb{Z}$, and are labelled by integer electric ($n$) and magnetic ($m$) charges, $(n,m)$. Self-dual ($n = m$) and anti-self-dual ($n = -m$) dyonic configurations carry vanishing Euclidean energy and are fully compatible with the Euclidean Schwarzschild geometry. More general dyonic configurations, by contrast, are interpreted as off-shell Euclidean field configurations. Nevertheless, both classes contribute to the thermal equilibrium vacuum and to finite-temperature correlation functions in the corresponding Lorentzian framework. Furthermore, because of the non-trivial topology, the electromagnetic $θ_{\rm EM}$-term contributes to the physical observables. In particular, it sources $CP$-asymmetric Hawking radiation, observable as an imbalance between left- and right-polarised photons in the emission spectrum. We briefly discuss some implications of this phenomenon.

• The paper demonstrates that non-trivial electromagnetic instantons in Euclidean Schwarzschild black holes yield quantized dyon sectors characterized by integer electric and magnetic charges.
• It employs semiclassical methods to show that the electromagnetic theta term induces CP-violating effects, resulting in an asymmetric chiral photon flux in Hawking radiation.
• The analysis links topological gauge configurations to black hole thermodynamics, offering insights with potential implications for quantum gravity and cosmology.

Electromagnetic Instantons and Asymmetric Hawking Radiation of Black Holes

Topological Structure of Maxwell Theory in Black Hole Backgrounds

The paper "Electromagnetic instantons and asymmetric Hawking radiation of black holes" (2604.02841) analyzes the interplay between the topological properties of Abelian gauge fields and the thermodynamic behavior of black holes within the Euclidean formalism. Focusing on a static, non-rotating, neutral Schwarzschild black hole, it establishes the presence of non-trivial $L^2$-normalizable harmonic 2-form field strengths, classified by integer-valued electric and magnetic charges. The physical realization of such configurations manifests as a statistical ensemble of dyons labeled by $(n,m)\in \mathbb{Z}\oplus \mathbb{Z}$, with electric ($n$) and magnetic ($m$) quantum numbers dual under electromagnetic duality. These states correspond to the non-trivial structure of the second cohomology group $H^2(S^2\times S^2; \mathbb{Z})$, attributed to the Euler characteristic $\chi=2$ of the compactified Euclidean Schwarzschild manifold.

A salient aspect is that the requirement of finite action, necessary for the validity of the semiclassical expansion, necessitates compactifying the manifold to $S^2_{(\tau,r)}\times S^2_{(\theta,\phi)}$. This procedure enables the existence of globally non-exact, closed harmonic forms, thus inducing quantization of both electric and magnetic charge (Dirac and dual Dirac quantization via patch overlap conditions). Explicit gauge field representatives are constructed for each $(n,m)$ sector, showing physical quantization follows from single-valuedness of gauge transition functions across patches.

Semiclassical Contributions and Thermodynamic Ensemble Interpretation

The Maxwell action evaluated on these $(n,m)$ field strengths yields $S_\mathrm{EM} = \frac{2\pi^2}{e^2}(n^2 + m^2)$, while the Euclidean energy-momentum tensor admits vanishing expectation value only for self-dual or anti-self-dual configurations ($(n,m)\in \mathbb{Z}\oplus \mathbb{Z}$0). These constitute valid classical saddles in the partition function. More general off-diagonal dyons, although not instantons in the strict sense, contribute as off-shell field configurations within the semiclassical path integral. They are physically interpreted as finite-temperature analogs of caloron constituents, with the effective dyon scale set by $(n,m)\in \mathbb{Z}\oplus \mathbb{Z}$1 and the local Hawking temperature, ensuring the dominance of thermal fluctuations near the horizon.

The partition function is described as a discrete sum over these dyon sectors:

$(n,m)\in \mathbb{Z}\oplus \mathbb{Z}$2

and, crucially, observables are obtained by ensemble averages over this space. For the neutral, non-rotating Schwarzschild BH, both the mean electric and magnetic charges vanish, as does the mean energy-momentum tensor, ensuring consistency with the classical vacuum expectation. However, higher moments and composite topological observables are generally non-vanishing due to the rich structure of the ensemble.

Physical Consequences of the Electromagnetic Theta Term

A central result is the demonstration that the electromagnetic topological angle $(n,m)\in \mathbb{Z}\oplus \mathbb{Z}$3, normally unobservable in pure Maxwell theory in trivial backgrounds, becomes a physical parameter in this context. The presence of non-trivial $(n,m)\in \mathbb{Z}\oplus \mathbb{Z}$4 dyon configurations engenders a nonzero Pontryagin charge $(n,m)\in \mathbb{Z}\oplus \mathbb{Z}$5, with the ensemble-averaged value proportional to $(n,m)\in \mathbb{Z}\oplus \mathbb{Z}$6 and exponentially suppressed as $(n,m)\in \mathbb{Z}\oplus \mathbb{Z}$7 for leading sectors. Thus, $(n,m)\in \mathbb{Z}\oplus \mathbb{Z}$8 sources $(n,m)\in \mathbb{Z}\oplus \mathbb{Z}$9 violation even in the absence of matter or external monopoles.

Most strikingly, the non-trivial vacuum expectation value for the Pontryagin density translates, via the magnetic helicity current, into a non-zero net helicity flux of photons in the Hawking radiation spectrum. The paper establishes that, even for neutral and non-rotating black holes, Hawking radiation is generically $n$0-asymmetric if $n$1, producing an imbalance between left- and right-circularly polarized photons. The effect is nonperturbative and suppressed for large black holes, but it is a robust consequence of the topological sector structure and independent of the traditional sources (such as angular momentum or external fields).

Theoretical and Practical Implications

The research provides a transparent procedure for embedding electromagnetic $n$2 violation into gravitational backgrounds and elucidates the physical meaning of the $n$3 parameter. These effects are theoretically relevant for the non-perturbative structure of quantum gravity coupled to gauge fields and suggest concrete topological origins for chiral selection effects in black hole evaporation.

On the practical side, while the effect is suppressed for large black holes, the cumulative contribution from many small black holes (e.g., in virtual black hole dominated vacua or during a phase of primordial black hole domination) could be significant. The resulting chiral photon excess or anomalous charge carries potential observational signatures, with implications for cosmological polarization and (via axion couplings) for dark matter phenomenology. The analysis indicates that virtual black holes, especially when operating near the Planck scale, can substantially alter the effective axion mass and other IR observables by reducing the classical exponential suppression.

Prospects and Future Directions

The authors outline extensions including:

• Explicit realization of $n$4 via embedding into grand unified or anomalous Abelian frameworks, and radiative generation from chiral matter/axion couplings.
• Systematic analysis of the impact of fermionic zero modes and their effect on vacuum structure, condensate formation, and pseudo-Goldstone spectrum, in analogy with known index theorems for instantons.
• Quantitative study of ensembles of virtual (microscopic) or cosmological black holes and their role in generating cosmologically relevant chiral excesses or anomalous charge.
• Extension of the construction to broader classes of topological gravitational backgrounds (e.g., charged black holes, wormholes).

Conclusion

This work establishes the physical significance of the electromagnetic $n$5-angle in the context of black hole thermodynamics, showing that topologically non-trivial Abelian gauge field configurations in gravity backgrounds generate $n$6-asymmetric Hawking radiation, observable as a net chiral photon flux. The analysis provides formal tools for quantifying these topological contributions and connects instanton physics, axion phenomenology, and quantum gravity in a transparent framework. These results delineate a direct topological origin for $n$7 violation in black hole evaporation, with broad implications for gravitational, cosmological, and gauge field physics.