Rotating Topological Stars in 5D Supergravity

• Rotating topological stars are horizonless, smooth solutions in five-dimensional supergravity that implement quantized angular momentum and complex charge structures via modifications of Kerr–Taub–bolt geometries.
• They exhibit standard Kaluza–Klein asymptotics and string theory embeddings, enabling microstate counting and offering a viable alternative to classical black hole models.
• Their separable geodesic and field dynamics facilitate precise studies of gravitational multipoles, stability analyses, and the extraction of potential observational signatures.

Rotating topological stars are a class of smooth, horizonless solutions in higher-dimensional gravity—particularly five-dimensional minimal supergravity—that realize angular momentum, nontrivial topology, and a rich charge structure while eschewing the conventional black hole horizon. These solutions, built from modifications of the Kerr–Taub–bolt (or related) seed geometries, possess standard Kaluza–Klein asymptotics and constitute coherent, regular microstate geometries that can mimic astrophysical black hole properties arbitrarily closely but remain fundamentally distinct due to their regular, capped interiors.

1. Geometric Construction and Parameter Space

Rotating topological stars are generated by applying hidden symmetry (sigma-model) transformations to the five-dimensional Kerr–Taub–bolt geometry within minimal supergravity. The seed metric,

$$ds^2 = -dt^2 + \frac{\Delta + a^2 \sin^2\theta}{\Sigma}(d\psi+\omega_\psi\,d\phi)^2 + \frac{\Sigma}{\Delta + a^2\sin^2\theta}\,ds^2_3,$$

with $\Delta = r^2 - 2mr + n^2 - a^2$, $\Sigma = \Delta + a^2\sin^2\theta$, and $\omega_\psi$ dependent on parameters and coordinates, features a bolt (surface where a Killing circle shrinks smoothly) at $r=r_+$. Upon transformation, the new metric includes four independent conserved charges—two from the Kaluza–Klein sector and two from the gauge field—along with quantized angular momentum determined by the dipole parameter $a$. Smoothness and global regularity at the cap enforce quantization: $$|a| = \frac{k(r_+ - r_-)}{2}, \quad k \in \mathbb{Z}_{\geq 0},$$ with identifications on the circles ensuring an asymptotic structure of $\mathbb{R}^{1,3} \times S^1$.

The parameter $k$ serves as a primary quantum number labeling a discrete tower of rotating topological star states, while further integers $\ell$, $N$ enter through identifications and the charge assignments. The solutions interpolate—depending on parameter choices—between the static topological star (for $k=0,\,a=0$), an extremal BPS limit ($k=1$), and genuinely non-supersymmetric, high-spin configurations ($k\geq2$) (Heidmann et al., 6 Oct 2025).

2. Physical Properties and Multipole Structure

Rotating topological stars differ from black holes in several key respects:

• Horizonless and Regular Interior: The geometry is globally smooth with the bolt acting as a cap, in contrast to horizons of black holes.
• Standard Kaluza–Klein Asymptotics: At large radii, the solution approaches $\mathbb{R}^{1,3} \times S^1$, suitable for embedding in higher-dimensional or string-theoretic frameworks.
• Charge Content: The solutions exhibit four charges—two Kaluza–Klein (magnetic $P_0$ and electric “momentum” $Q_0$), and two gauge field charges ($P, Q$).
• Angular Momentum Quantization: Only discrete angular momenta, linked to $k$, ensure global smoothness and periodicity.
• Gravitational Multipoles: The multipole expansion reveals distinctive features. While the raw coefficients resemble Kerr (e.g., $\widetilde{M}_{2p} = M\,a^{2p}$), after ACMC transformation the physical multipoles can differ significantly. Notably, the mass quadrupole can be prolate ($M_2>0$), unlike the oblate signature of Kerr, altering observable properties such as gravitational lensing and ringdown (Heidmann et al., 6 Oct 2025).

3. Comparison with Black Holes and Microstate Interpretation

Rotating topological stars are distinguished from black holes mainly through:

• Location in Phase Space: Although they can approach arbitrarily close (in mass, spin, and KK charges) to a highly boosted Kerr black string in conserved quantities, they remain outside the black hole extremality bound in five-dimensional STU supergravity—particularly for $k\geq2$ (Heidmann et al., 6 Oct 2025).
• Horizon vs. Cap: Black holes possess an event horizon; topological stars terminate on a smooth cap at the bolt, preventing singularity or horizon formation and avoiding associated pathologies such as the information paradox.
• Ergoregion Structure: An ergoregion is present in the five-dimensional geometry but vanishes upon reduction to four dimensions, implying that only certain Kaluza–Klein modes are subject to possible superradiant instabilities—a distinct difference with most rotating black holes.
• Microstate Role: These objects epitomize the fuzzball or microstate geometry paradigm, serving as horizonless, distinguishable states that can be counted in the statistical mechanics of black hole entropy and may constitute the building blocks of semiclassical black holes in string theory (Bianchi et al., 16 Apr 2025, Heidmann et al., 6 Oct 2025).

4. Dynamics, Separation of Variables, and Integrability

Both test-particle (geodesic) motion and field perturbations in the rotating topological star background demonstrate complete separability:

• Hamilton–Jacobi Separability: The geodesic equations admit a separation ansatz, yielding Carter-like constants and enabling analysis of photon rings and orbits—a property inherited from the underlying Kerr–NUT sector.
• Klein–Gordon Equation: Scalar perturbations separate into ordinary differential equations in $r$ and $\theta$, with both the radial and angular equations of confluent Heun type. This structure permits full spectral and stability studies—mapping quasinormal modes and scattering processes—as well as explicit evaluation of wave propagation.
• Implications: The existence of separability and integrability is instrumental for both analytic studies of field dynamics and for the potential extraction of ringdown or lensing observables, facilitating detailed comparison to black hole signals (Heidmann et al., 6 Oct 2025, Bianchi et al., 16 Apr 2025).

5. String Theory Embedding and Quantum Numbers

Rotating topological stars admit a string-theoretic interpretation via embedding into five-dimensional minimal supergravity as the dimensional reduction of three-charge brane condensates:

• Brane Origin: In M-theory, configurations stem from the (non)-extremal intersection of three orthogonal M5-branes; in type IIB, dualities map the objects to KK-monopoles, D1-, and D5-branes.
• Charge Quantization: The stringy context explains the observed charge and spin quantization, relating them to wrapped brane numbers and the geometry of the compactification circles.
• Microstate Counting: The discrete tower of solutions (labeled by quantum number $k$ and other integers) underpins the perspective of counting horizonless microstates associated to a black hole of given charge and angular momentum—the essence of the fuzzball scenario (Bianchi et al., 16 Apr 2025, Bah et al., 2020).

6. Astrophysical and Theoretical Implications

Rotating topological stars function as prototypes for coherent black hole microstates and as models for exotic, horizonless replacements of classical black holes in gravitational and astrophysical contexts:

• Supergravity Solutions and Phenomenology: In the near-vacuum limit, the solutions can approach the mass, spin, and charge properties of observed black holes (e.g., highly boosted Kerr black strings), suggesting that distinguishing between horizonless topological stars and classical black holes may challenge gravitational wave and electromagnetic observations.
• Ergoregion Instabilities and Observational Signatures: The unique ergoregion structure—present only in 5D—affects the possible emission of superradiant instabilities and constraints from ringdown signals.
• Future Research Directions: Open directions include detailed quasinormal mode analysis, (in)stability with respect to various perturbations, further generalization to include more charges or rotation parameters, and comparative studies to other microstate constructions such as the JMaRT or multi-center Weyl solutions (Heidmann et al., 6 Oct 2025, Bianchi et al., 16 Apr 2025, Bah et al., 2020).

7. Quantization, Smoothness, and Multipole Towers

A defining attribute of rotating topological stars is the quantization of physical parameters dictated by regularity at the cap and asymptotic consistency:

Quantum Number	Geometric Role	Physical Interpretation
k	Spin quantization	Discrete angular momentum
N, ℓ	Charge/periodicity conditions	Magnetic/electric quant.

The tower of solutions generated through this quantization impacts both the allowed multipole moments of the stars and the set of astrophysically relevant states that can be constructed. The adjustment of these quantum numbers underlies the selection rules and spectrum of microstates in both gravitational and quantum string theory frameworks (Heidmann et al., 6 Oct 2025).

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In summary, rotating topological stars are quantized, smooth, horizonless solitons constructed in higher-dimensional gravity and supported by nontrivial topology, angular momentum, and charge. Their theoretical framework integrates developments from classical general relativity, supergravity, and string theory, and they stand as critical models for the paper of microstate geometry, black hole structure, and observable deviations from classical horizon-based compact objects (Heidmann et al., 6 Oct 2025, Bianchi et al., 16 Apr 2025).