AdS Distance Conjecture Overview

• AdS Distance Conjecture is a principle that relates the geometry of AdS vacua to the emergence of an infinite tower of light states in quantum gravity.
• It predicts that the mass scale of the tower scales as m ∼ |Λ|^α, with α typically 1/2 for supersymmetric cases and 1/d in non-supersymmetric scenarios.
• The conjecture underpins key Swampland criteria by connecting effective field theory breakdown, black hole instabilities, and holographic dualities in AdS settings.

The Anti-de Sitter Distance Conjecture (AdS Distance Conjecture, ADC) posits a fundamental relationship between the geometry of the moduli space of AdS vacua in quantum gravity and the emergence of an infinite tower of light states. Specifically, as one approaches the limit where the AdS curvature vanishes ($|\Lambda|\to 0$) or equivalently the AdS radius becomes infinite, there necessarily appears an infinite tower of states whose mass scale approaches zero in a manner controlled by the AdS curvature. The ADC is a central facet within the broader Swampland program, linking quantum gravity consistency conditions, string theory vacua, and holography.

1. Formal Statement and Framework

The AdS Distance Conjecture asserts that for any infinite family of $d$-dimensional AdS vacua, labeled by a cosmological constant $\Lambda_{\rm AdS} < 0$, there exists an infinite tower of states with masses scaling as

$m_{\rm tower} \sim |\Lambda_{\rm AdS}|^\alpha,$

with $\alpha = \mathcal{O}(1)$ as $|\Lambda_{\rm AdS}| \to 0$ (Lust et al., 2019, Herráez et al., 25 Mar 2025, Montero et al., 2022, Nam, 2022). In supersymmetric examples, $\alpha = 1/2$ is typically realized; non-supersymmetric or circle-reduced vacua yield $\alpha = 1/d$ (Gonzalo et al., 2021, Herráez et al., 25 Mar 2025).

A refined form considers the geodesic distance $\Delta\phi$ in field space. The tower mass behaves exponentially: $$m_{\rm tower} \sim m_0\, e^{-\alpha\Delta\phi}, \qquad \Delta\phi\to\infty.$$ Proper field-space distances are computed using kinetic terms of scalar fields and, in full generality, also include contributions from metric, flux, and higher-form deformations (Lust et al., 2019, Shiu et al., 2022, Li et al., 2023).

Version	Scaling Law	Context
Strong ADC	$m_{\rm tower} \sim |\Lambda|^{1/2}$	Supersymmetric vacua, e.g. AdS$_5\times S^5$
Mild ADC	$m_{\rm tower} \sim |\Lambda|^{\alpha},\ \alpha >0$	Non-supersymmetric, e.g. circle compactifications

2. Mathematical Formulation: Scaling and Distance

The ADC is often packaged in terms of either scaling with cosmological constant or in terms of field-space/metric distance:

• Curvature scaling:

$$m_{\rm tower} \sim |\Lambda|^{\alpha} \ \rightarrow\ m_{\rm tower} \sim L_{\rm AdS}^{-\alpha}$$

where $L_{\rm AdS}^{-2} \sim |\Lambda|$.

• Field-space distance:

$m_{\rm tower} \sim e^{-\alpha\,\Delta}$

where $\Delta$ is the proper geodesic distance along the trajectory in moduli space enlarging the AdS radius.

For pure AdS in string theory (e.g., AdS$_5 \times S^5$), the KK tower masses scale as $m_{\rm KK}\sim 1/L_{\rm AdS}$, in agreement with $\alpha = 1/2$ (Lust et al., 2019, Lavdas et al., 2020). In generic flux models, the exponent can differ depending on the mechanism and the presence of discrete symmetries (Buratti et al., 2020, Shiu et al., 2022).

3. Derivation from Black Hole and Instability Scales

A robust, model-independent derivation is based on the physics of black hole instabilities rather than abstract field-space distances (Herráez et al., 25 Mar 2025). The central observation is that the breakdown of effective field theory (EFT) for semi-classical neutral black holes in AdS occurs at scales tied directly to $\Lambda_{\rm AdS}$. There exist two critical temperature/energy scales for Schwarzschild–AdS$_d$ black holes:

• The Hawking–Page transition: $T_{\rm HP} \sim |\Lambda_{\rm AdS}|^{1/2}$
• The Jeans instability: $T_{\rm J} \sim |\Lambda_{\rm AdS}|^{1/d}$

Hence, the instability scale (physically associated with the light tower) is bounded: $$|\Lambda_{\rm AdS}|^{1/2} \lesssim \Lambda_{\rm BH} \lesssim |\Lambda_{\rm AdS}|^{1/d}$$ implying

$\frac{1}{d} \leq \alpha \leq \frac12$

This identifies the ADC scaling with the emergence of a new tower of states triggered by gravitational instabilities, circumventing the need for explicit distance computations in field space (Herráez et al., 25 Mar 2025).

4. Field Space Geometry, Hodge Theory, and Action Metrics

The proper definition of "distance" between AdS vacua, especially in string compactifications with fluxes and internal moduli, involves metrics on configuration space that incorporate geometric and flux variations (Lust et al., 2019, Li et al., 2023). The "action metric" approach defines a positive-definite line element on the space of deformations:

• External Weyl (AdS) rescalings alone yield a negative-definite kinetic term (the conformal factor problem).
• Inclusion of internal volume moduli and Freund–Rubin flux variations can compensate, rendering the total action metric positive for physical deformations.
• The positivity of the action metric aligns with the absence of strong scale separation in such vacua (Li et al., 2023). In models engineered for strong scale separation, the negative contribution dominates and the metric becomes ill-defined, in tension with quantum gravity consistency.

For Type IIA Calabi–Yau orientifold vacua, mixed Hodge theory yields universal lower bounds for the scaling exponent, with $\delta \geq 1/3$ for CY$_3$ and $\delta\geq 1/4$ for CY$_4$ (Castellano et al., 2021).

5. Explicit Realizations and Refinements

String duals and AdS/CFT: For AdS$_5\times S^5$ (dual to 4d $\mathcal{N}=4$ SYM), the Zamolodchikov distance (CFT side) matches the metric distance in AdS moduli space. At infinite distance, higher-spin states become exactly massless (Baume et al., 2020, Perlmutter et al., 2020).

Refined versions:

• In the presence of discrete $\mathbb{Z}_k$ gauge/domain-wall symmetries, the scaling can be refined:

$$\Lambda \sim \frac{M_{\rm cutoff}^2}{k} \implies M_{\rm cutoff} \sim \sqrt{k|\Lambda|},\qquad \frac{M_{\rm KK}^2}{|\Lambda|}\sim k$$

so that parametric scale separation is governed by $k$, the order of the discrete symmetry (Buratti et al., 2020).

Scale-separated vacua: In massive Type IIA DGKT vacua, open-string scalars interpolate between vacua with different four-form flux quanta. The field-space trajectory realizes parametrically large AdS radii, with the KK tower of mass $$m_{\rm tower} \sim N^{-1/4} \sim e^{-\alpha \Delta\phi}$$ (here $N$ is the flux), directly verifying the exponential behavior predicted by the conjecture even in scale-separated AdS (Shiu et al., 2022).

Top-down and bottom-up constructions consistently confirm $\alpha=1/2$ in explicit AdS vacua, and always demonstrate that the EFT cutoff coincides (up to order-one factors) with the light tower's scale (Lust et al., 2019, Lavdas et al., 2020, Nam, 2022).

6. Swampland, Holography, and Connections to Other Conjectures

The ADC is deeply linked with other Swampland principles:

• Weak Gravity Conjecture: The descent of a mass scale in the tower enforces bounds on gauge couplings as required by the Weak Gravity Conjecture (Buratti et al., 2020).
• No Global Symmetries: Taking the AdS radius to infinity in pure (gauged R-symmetry) supergravity leads to exact global symmetries in the flat-space limit, violating Swampland criteria unless a light tower emerges with $\alpha > 2/3$ (Montero et al., 2022).
• Refined de Sitter Conjecture: The structure of the potential at an AdS minimum constrains towers through the ADC (Lust et al., 2019).
• CFT Distance Conjecture: Via AdS/CFT, approaching infinite distance in the conformal manifold (e.g., decoupling a gauge group) produces an emergent higher-spin symmetry and a vanishing mass gap (Perlmutter et al., 2020).

7. Open Problems, Dimensionality, and Limitations

There is evidence that the ADC, in its black hole instability form, is only operative for $3\le d \le 10$. Above $d=10$, no Jeans instability arises for self-gravitating radiation, suggesting an upper dimensional bound for the ADC's applicability (Herráez et al., 25 Mar 2025). The role of discrete symmetries in refining scaling relations, and potential violations or limitations in non-supersymmetric compactifications and scale-separated models, remain active areas of research.

Limitations of the conjecture include reliance on trajectories that maintain EFT control, realization of positivity in action metrics, and the need for explicit UV completions to resolve moduli-space singularities. For de Sitter space, parallel conjectures are less precisely established; a lower bound on the tower mass $m_{\rm tow}\gtrsim \Lambda_{\rm dS}^{1/2}$ is suggested but not universally demonstrated (Herráez et al., 25 Mar 2025).

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References:

• "AdS and the Swampland" (Lust et al., 2019)
• "Black Hole Transitions, AdS and the Distance Conjecture" (Herráez et al., 25 Mar 2025)
• "A Gravitino Distance Conjecture" (Castellano et al., 2021)
• "On scale separation in type II AdS flux vacua" (Font et al., 2019)
• "Pure Supersymmetric AdS and the Swampland" (Montero et al., 2022)
• "Towards AdS Distances in String Theory" (Li et al., 2023)
• "AdS Distance Conjecture and dS vacua" (Nam, 2022)
• "Discrete Symmetries, Weak Coupling Conjecture and Scale Separation in AdS Vacua" (Buratti et al., 2020)
• "AdS scale separation and the distance conjecture" (Shiu et al., 2022)
• "Massive gravitons on the Landscape and the AdS Distance Conjecture" (Lavdas et al., 2020)
• "Tackling the SDC in AdS with CFTs" (Baume et al., 2020)
• "A CFT Distance Conjecture" (Perlmutter et al., 2020)
• "AdS Swampland Conjectures and Light Fermions" (Gonzalo et al., 2021)