Hairy AdS Soliton in 3D Gravity

• Hairy AdS soliton is a smooth, horizon-free solution featuring a self-interacting scalar field that uniquely determines its negative mass ground state.
• Its relaxed AdS boundary conditions and asymptotic structure preserve Virasoro symmetry, enabling Cardy formula microstate counting for associated hairy black holes.
• The soliton’s fixed parameters serve as a robust holographic dual for confining and symmetry-broken phases, underpinning advanced studies in three-dimensional gravity.

A hairy AdS soliton is a smooth, horizonless, globally regular solution to Einstein gravity (often with negative cosmological constant) coupled to one or more matter fields—typically a (charged or neutral) scalar—with “hair” denoting the presence of nontrivial scalar field profiles that backreact on the geometry. Unlike standard AdS solitons, these background configurations possess additional scalar degrees of freedom, spontaneously breaking certain symmetries and giving rise to distinctive phase structure—most notably, the appearance of a negative, fixed energy that is interpreted as a Casimir effect in the dual field theory. Hairy AdS solitons play a central role in holography as the ground states of “hairy sectors” and can serve as endpoints of instabilities or as duals to confining, gapped, or symmetry-broken states.

1. Exact Solutions and Defining Properties

Hairy AdS solitons in 2+1 dimensions are constructed as exact solutions to Einstein gravity coupled to self-interacting scalar fields. The defining metric and scalar profiles for the canonical case (for self-interaction parameter ν = 0) are:

• Metric:

$$ds^2 = l^2 \left\{ -\frac{4(1+\rho^2)^4}{(3+2\rho^2)^2} d\tau^2 + \frac{64(1+\rho^2)^3}{(3+2\rho^2)^4} d\rho^2 + \frac{64}{81} \rho^2(1+\rho^2) d\phi^2 \right\}$$

• Scalar:

$$\phi(\rho) = \operatorname{arctanh} \sqrt{\frac{1}{3+2\rho^2}}$$

Generalizations for ν > –1 yield metrics and scalar profiles dependent on ν, with fixed couplings αν and cν uniquely determined by ν and an auxiliary function Θ_ν. These solitons are globally regular, free of horizons and curvature singularities, and devoid of integration constants; their physical properties are fixed by the couplings and the scalar self-interaction, not by arbitrary parameters. The absence of integration constants means the mass of the soliton is uniquely determined by the underlying parameters of the model.

2. Asymptotic Structure and Relaxed AdS Boundary Conditions

The soliton solution is asymptotically AdS but with “relaxed” boundary conditions compared to the Brown-Henneaux class, owing to the slow fall-off of the scalar field: $$\phi = \frac{\chi}{r^{1/2}} + \alpha \frac{\chi^3}{r^{3/2}} + O(r^{-5/2})$$ Correspondingly, the asymptotic metric components deviate from standard AdS fall-offs; however, the conserved charges and asymptotic symmetry algebra remain well-defined. Despite the lesser decay rate, the spacetime remains asymptotically AdS in a generalized sense and supports the standard conformal structure at infinity. This slow decay ensures that both the metric and the scalar profile contribute non-trivially to the conserved charges and the algebra of canonical generators. The resulting configurations can be consistently endowed with charges and central extensions, as expected for AdS holography.

3. Asymptotic Symmetry Algebra: Virasoro and Central Extension

Imposing the relaxed boundary conditions, the asymptotic symmetry group continues to be two copies of the Virasoro algebra—that is, the two-dimensional conformal group. The canonical realization (by the Regge-Teitelboim method) gives the standard central charge: $c^+ = c^- = c = \frac{3l}{2G}$ This is essential for the application of CFT methods (notably Cardy's formula) in counting microstates. The contributions from the scalar and the gravitational sector combine so that the overall algebra structure is unaltered, allowing Cardy-based entropy calculations even in the presence of nontrivial scalar profiles.

4. Hairy Black Holes, Disconnected Sectors, and the Ground State

The theory also admits “hairy” black holes, i.e., solutions with horizons and scalar hair. These are not continuously connected to the standard “bald” BTZ black holes: the scalar hair cannot be turned off continuously without changing the mass, so each family corresponds to a distinct sector. While the BTZ black hole entropy can be microscopically accounted for via Cardy’s formula assuming global AdS as the ground state, this fails for the hairy black holes since the sector is disconnected. The corresponding ground state must be taken as the soliton described above, with fixed negative mass (M_sol) and no integration constants. This role of the soliton as the vacuum is further substantiated by its bounded negative mass, determined by the coupling constants and bounded from below by AdS spacetime mass (e.g., for certain potentials –1/(8G) < M_sol < 0).

5. Microscopic Entropy and Cardy Formula

With the soliton as the ground state, the entropy of the corresponding hairy black holes is obtained exactly from the asymptotic state counting via Cardy’s formula. In terms of shifted Virasoro operators: $\tilde{L}_0^\pm = L_0^\pm - c^\pm/24$ and the soliton fixing the ground state energy: $\tilde{\Delta}_0^\pm = \frac{l}{2} M_{\text{sol}}$ The entropy in the microcanonical ensemble is

$$S = 4\pi [\sqrt{-\tilde{\Delta}_0^+ \tilde{\Delta}^+} + \sqrt{-\tilde{\Delta}_0^- \tilde{\Delta}^-}]$$

where, for nonrotating hairy black holes, the identification $\tilde{\Delta}^\pm = (Ml \pm J)/2$ holds. This reproduces the semiclassical Bekenstein-Hawking entropy S = A/(4G) for hairy black holes. The key feature is that the central charge itself need not appear explicitly in the statistical counting; the calculation can be written solely in terms of the shifted Virasoro spectrum, with the ground state set by the soliton mass.

6. The Role of the Soliton and Virasoro Operator Spectrum

Notably, the Cardy entropy can be equivalently written as an explicit function of the spectrum of shifted Virasoro operators: $$\rho(\tilde{\Delta}^\pm) \propto \exp[4\pi \sqrt{-\tilde{\Delta}_0^\pm \tilde{\Delta}^\pm}]$$ The density of states (and thus entropy) is thus controlled directly by the ground state eigenvalue, itself fixed by the soliton properties. This mechanism provides evidence for the more general principle that, when the sector ground state is not AdS, matching the lowest eigenvalue—rather than the central charge—is the crucial ingredient for entropy counting in the “hairy” sector.

7. Summary, Implications, and Significance

Hairy AdS solitons thus serve as the unique regular, horizon-free, and integration-constant–free ground states of sectors with scalar “hair,” possessing fixed negative masses determined by the parameters of the theory. They are asymptotically AdS in a relaxed sense, preserve the conformal symmetry and central extension of the pure AdS theory, and resolve the entropy puzzle for the corresponding hairy black holes by enabling a correct microscopic counting via Cardy’s formula, using the soliton’s negative mass to set the lowest Virasoro eigenvalue. This integrated approach (including explicit analytic geometry, canonical methods, relaxed boundary conditions, and CFT counting) underpins a robust understanding of the entropy and microstate structure of hairy black holes in three dimensions (Correa et al., 2010, Correa et al., 2011). The methodology generalizes to other constructions and dimensions, providing a framework for analyzing disconnected ground states and their holographic implications in AdS gravity and beyond.