Supersymmetric Ergoregions in String Theory

• Supersymmetric Ergoregions are spatial regions in certain gravitational backgrounds where a preferred Killing vector turns spacelike, marking the appearance of unconventional BPS excitations.
• They arise in horizonless microstate geometries and supertube backgrounds, characterized by sign changes in the metric component and formulated via null-gauged WZW models.
• These regions enable holographic mapping of BPS state transitions and illustrate how orbifold singularities and twisted sectors contribute to energy extraction and dynamic stability.

Supersymmetric ergoregions are spatial domains within certain supersymmetric gravitational backgrounds—typically involving rotation and multiple charges—where an asymptotically timelike Killing vector field becomes spacelike, despite the absence of an event horizon. These ergoregions may support atypical energetic phenomena, including the existence of BPS excitations with zero or negative energy relative to infinity, and can have significant implications for microstate geometry, supergravity solution structure, string theory quantization, and holography.

1. Geometric Definition and Construction of Supersymmetric Ergoregions

Unlike standard black hole ergoregions, which arise outside rotating horizons (e.g., Kerr solutions), supersymmetric ergoregions appear in horizonless, rotating bound states such as supersymmetric microstate geometries and supertube backgrounds (Martinec et al., 7 Aug 2025). The defining metric feature is that the norm of a preferred Killing vector—often $\partial/\partial t$, where $t$ aligns with the boundary time in AdS$_3$/CFT$_2$—changes sign inside a finite region: $g_{tt} = -\frac{f_0 - \Delta_p}{\Sigma}$ with $f_0$ a function of coordinates, $\Delta_p$ a positive parameter set by conserved charges, and $\Sigma$ another background-dependent function. The ergosurface is defined by $f_0 = \Delta_p$, and the ergoregion is $f_0 < \Delta_p$.

In these backgrounds, the six-dimensional metric can be cast generically as [(Martinec et al., 7 Aug 2025), Eq. 2.12]: $$ds^2 = \frac{1}{\Sigma}\left( -f_0\,du\,dv + ... \right) + d\cdot d$$ with $u = t - y$, $v = t + y$ for the null directions, and where absence of a horizon is ensured by global regularity and smoothness conditions. The base space may include orbifold singularities, which affect the local operator spectrum.

2. Physical Properties and BPS Excitations

A distinct haLLMark of supersymmetric ergoregions is the existence of localized BPS excitations—geodesics and string states—with zero or negative energy as measured from asymptotic infinity:

• The asymptotic energy, as computed from charge quantization (momentum along $y$), may be negative for states inside the ergoregion: $E_{\text{asymptotic}} \propto -n_y$ where $n_y$ is the quantized momentum along the string direction. For specific values, certain excitations exactly saturate $E_{\text{asymptotic}} = 0$.

Classical analysis of null geodesics in the ergoregion reveals that these can maintain zero “cap” energy (local energy in the co-rotating frame) and negative or vanishing global energy [(Martinec et al., 7 Aug 2025) Section 2.2]. These negative-energy BPS modes manifest as coherent collective excitations, reducing both total energy and longitudinal momentum charge, while preserving supersymmetry.

3. Worldsheet Description and Gauged WZW Model Quantization

Supersymmetric ergoregions admit a worldsheet string description via null-gauged Wess–Zumino–Witten (WZW) coset models: $$\frac{SL(2,\mathbb{R}) \times SU(2) \times \mathbb{R}_t \times S^1_y}{U(1)_L \times U(1)_R}$$ [(Martinec et al., 7 Aug 2025) Eq. 3.1] Here, the gauging takes place along null isometries, resulting in the physical time coordinate mapped to the CFT boundary time, and relating cap (local) energy to asymptotic energy via gauge constraints.

Vertex operators are constructed from WZW primaries and include contributions from twisted sectors localized at orbifold singularities. These operators are subject to BRST and Virasoro constraints, leading to quantization conditions that directly mirror properties of BPS excitations in the bulk. For instance, winding sectors (fractional spectral flow) correspond to distinct charge quantization, with conserved charges invariant under large gauge transformations.

4. Holographic Mapping and CFT Transitions

In the AdS$_3$/CFT$_2$ holographic dual, the presence of a supersymmetric ergoregion corresponds to transitions among heavy BPS states in the dual two-dimensional CFT. Specifically, vertex operator transitions in the bulk map to strand recombination processes, such as the joining of multiple small-winding strands into longer fractionally wound configurations. These transitions conserve overall charge but can lower the energy by shifting momentum and angular momentum quantum numbers.

Table: Correspondence Between Bulk Excitations and CFT State Transitions

Bulk Feature	Worldsheet Operator	CFT Transition
BPS excitation with $n_y<0$	Twisted sector primary	Reduction of left-moving $S^3$ angular momentum
Fractional winding $w_e$	$w + w_y/\kappa$	Joining $\kappa$-wound strands
Orbifold fixed point localization	Twisted-sector ground	Recombination at orbifold locus

Orbifold singularities in the bulk geometry permit twisted-sector ground states. These states function as “ergo-strings”, mediating transitions between CFT sectors. The worldsheet formalism thus provides a powerful dictionary between geometry and dual CFT states.

5. Microstate Geometry and Implications for Stability

Supersymmetric ergoregions occur generically in large classes of microstate geometries constructed from bound states of NS5-branes, F1 strings, and momentum. Whereas conventional ergoregions in non-supersymmetric settings are associated with superradiant instabilities (amplification of waves due to negative-energy modes), the presence of global supersymmetry and a globally null Killing vector in these geometries eliminates linear instability (Martinec et al., 7 Aug 2025). The construction ensures that all “runaway” modes are restricted to BPS excitations, and the backgrounds remain dynamically stable at the linear level.

The ergoregion’s topology, localization of geodesics, and detailed worldsheet spectrum control the possible dynamics and long-lived excitations. As shown through the explicit mapping to CFT transitions, ergoregions allow for intricate scrambling of charge and momentum without energy divergence or instability.

6. Applications in AdS$_3$/CFT$_2$ and the Role of Orbifold Singularities

The stringy structure of supersymmetric ergoregions in AdS$_3$/CFT$_2$ backgrounds leads to novel classes of heavy BPS states and provides precise tools to analyze nonperturbative features beyond supergravity. The gauged WZW model gives access to string excitations not captured in the classical supergravity analysis, particularly those localized at orbifold fixed points.

A schematic diagram (adapted from Fig. 1 of (Martinec et al., 7 Aug 2025)) shows localization of BPS and non-BPS wavefunctions in terms of allowed latitudinal oscillations on $SU(2)$ harmonics, illustrating the semiclassical orbits of strings and the localization of microstate transitions.

7. Broader Implications and Future Directions

Supersymmetric ergoregions challenge traditional expectations about energy bounds and excitation spectra in rotating gravitational backgrounds. The existence of negative-energy BPS excitations, the role of twisted sector operators, and the holographic mapping to CFT transitions collectively advance the paper of black hole microstate structure and energy extraction in string theory. Continued research may explore the behavior of nearly BPS excitations (long-lived modes), extensions to other AdS/CFT setups, and possible implications for information scrambling and gravitational wave phenomenology.

These findings demonstrate that supersymmetric ergoregions are central not only to the classification of microstate geometries but also to the interplay between geometry, worldsheet string theory, and holography in the AdS$_3$/CFT$_2$ context (Martinec et al., 7 Aug 2025).