Emergent String Conjecture in Quantum Gravity

• Emergent String Conjecture is a framework that classifies infinite-distance limits in quantum gravity by distinguishing between decompactification with KK towers and the emergence of a unique, tensionless string.
• The approach integrates geometric, thermodynamic, and amplitude-based analyses—using moduli space degenerations, fibration structures, and modular invariance—to rigorously define the dichotomy.
• This conjecture has practical implications for understanding UV completions of effective field theories and constrains possible degenerations of Calabi–Yau manifolds.

The Emergent String Conjecture posits a dichotomy governing the asymptotic structure of quantum gravity moduli spaces: every infinite-distance limit is characterized either by decompactification, manifesting as a Kaluza–Klein (KK) tower of states associated with emerging extra dimensions, or by the appearance of a unique, critical, asymptotically tensionless string accompanied by a tower of string oscillator excitations. This framework sharply refines the Swampland Distance Conjecture by asserting that in any infinite-distance regime admitting a finite-volume limit, the breakdown of the effective field theory is always signaled by one of these two mechanisms, tied closely to deep duality and geometric structures in string and M-theory, as well as to quantum and thermodynamic considerations.

1. Geometric and Moduli Space Classification of Infinite Distance Limits

The foundational analysis of infinite-distance limits is rooted in the detailed paper of the Kähler or complex structure moduli spaces associated with Calabi–Yau compactifications. In the classical setting—exemplified by M-theory or Type IIA compactified on Calabi–Yau threefolds—the Kähler form can be parameterized as

$J = \sum_{i \in I} T^{i} J_{i},$

with at least one $T^i \to \infty$ while maintaining finite overall volume $\mathcal{V}_Y = \frac{1}{6} \int_Y J^3$. The main geometric classification theorem demonstrates that every such finite-volume, infinite-distance limit forces $Y$ to acquire a universal fibration structure: $\pi : F \to Y \to B_2,$ where $F$ is the fiber, uniquely determined by the fastest-shrinking Kähler direction. Three fiber types arise:

• $F \simeq T^2$ (elliptic/genus one curve),
• $F$ is a K3 surface,
• $F$ is an abelian surface (topologically $T^4$).

Type IIB on Calabi–Yau threefolds also exhibits a sophisticated degeneration structure, most notably Tyurin degenerations: $V_0 = X_1 \cup_Z X_2,$ where $Z$ is a K3 surface, and the emergent light sectors arise from special Lagrangian cycles in $Z$.

The scaling behavior of couplings and tensions along these limits is encoded in Hodge-theoretic invariants, leading to a rigorous semiorthogonal decomposition of the cohomology and the identification of the infinite tower through asymptotic growth theorems for periods and modular invariants (Lee et al., 2019, Friedrich et al., 1 Apr 2025).

2. Physical Mechanisms: Decompactification vs. Emergent Strings

KK Tower (Decompactification Limits)

In limits where a genus–one fiber ($T^2$) shrinks, the system reduces to a higher-dimensional theory; Kaluza–Klein modes wrapping this vanishing cycle furnish an infinite tower of states whose mass gap scales as $M_{KK} \sim \lambda^{-1}$. This is the “decompactifying” phase yielding an effective increase in the spacetime dimension (e.g., the F-theory limit from 5d to 6d). The gauge couplings scale exponentially with the proper geodesic distance as $g^2_{\text{min}}\sim \exp(-2/\sqrt{3}\, d_M)$, where $d_M$ is the moduli space distance (Kaufmann et al., 16 Dec 2024).

Emergent String Tower (Tensor Limits)

Alternatively, when a higher-genus surface (K3 or $T^4$) shrinks, wrapped M5-branes (or dual D4/NS5-branes in the IIA frame) yield a unique critical string, generically heterotic or Type II. Its tower of oscillator excitations becomes the dominant infinite tower:

• String tension scales as $T \sim \lambda^{-1}$.
• The KK scale is parametrically equal to the string scale in the “equi-dimensional” tensor limit.

The universal result is that only one critical string, with central charge matching heterotic $(c_L,c_R)=(24,12)$ or Type II pattern, emerges in a suitable duality frame (Lee et al., 2019, Kaufmann et al., 16 Dec 2024, Friedrich et al., 1 Apr 2025). Wrapped D3-branes on special Lagrangian cycles supply a "species" scale matching the critical field content.

Quantum corrections—e.g., worldsheet instantons, string loop effects—can obstruct certain infinite distance limits (notably in Type IIA, F-theory, or $G_2$ models), enforcing or censoring nonphysical scenarios where the string tension would fall too far below the KK scale (Lee et al., 2019, Klaewer et al., 2020, Xu, 2020).

3. Universal Fibration Structure and Uniqueness

A central geometric consequence is the requirement of a unique fibration structure in the compactification manifold. Any infinite distance limit is dominated asymptotically by a single fastest-shrinking fiber:

• In J-class A limits, Oguiso’s theorem ensures the fiber is genus–one.
• In J-class B, the fiber is a surface, with uniqueness enforced by precise scaling hierarchies.

This ensures only one weakly-coupled string or KK tower emerges, guaranteeing a unique dominant charge-to-mass vector in moduli space, formally related to the eigenvectors of the anomaly or Chern–Simons matrix (Kaufmann et al., 16 Dec 2024, Etheredge et al., 30 May 2024).

4. Spectral, Thermodynamic, and Amplitude-Based Evidence

Analysis based on black hole thermodynamics, scattering amplitudes, and density of states provides a bottom-up rationale for the conjecture:

• The one-particle density of states, $\rho(E)$, for $E \ll M_\mathrm{min}$ (the threshold for black holes) is forced to be either polynomial (KK tower) or exponential (string/Hagedorn tower) (Bedroya et al., 30 Apr 2024).
• Tension bounds for $p$-branes ($T \gtrsim \Lambda^{p+1}$) exclude non-string-like infinite towers below the quantum gravity cutoff.
• Scattering amplitude analyses confirm that only linear Regge trajectories (string excitations) or KK-like towers are compatible with unitarity and causality (Basile et al., 2023).
• Modular invariance and spectral properties of the CFT underlying worldsheet approaches encode the same dichotomy via the scaling of the species scale and the vanishing of the gravitational cutoff (Aoufia et al., 6 May 2024).

This convergence of geometric, thermodynamic, and worldsheet arguments demonstrates strong universality, independent of supersymmetry or string-theory-specific assumptions.

5. Duality Structure and Non-Geometric Extensions

String dualities enforce the universal dichotomy in all frames: T-duality and mirror symmetry relate decompactification and emergent string limits across Type II, heterotic, M, and F-theory compactifications (Baume et al., 2019, Friedrich et al., 1 Apr 2025). In non-geometric settings—where the internal sector is described by a general modular-invariant CFT or a dual D-manifold—regulated modular integrals and the role of the species scale confirm that emergent strings or KK towers saturate the effective field theory cutoff (Aoufia et al., 6 May 2024, Lee et al., 2019).

Mirror symmetry is particularly powerful: emergent string limits in Type IIB complex structure moduli space match, under suitable dualities, to well-understood heterotic or NS5-brane emergent string limits in IIA Kähler moduli space (Friedrich et al., 1 Apr 2025).

6. Constraints, Taxonomy, and Generalizations

The Emergent String Conjecture formulated as a taxonomy restricts the possible infinite-distance limits to a finite set, each associated with distinct "charge-to-mass" polytopes in moduli space. The global gluing of these regions in moduli space must satisfy nontrivial compatibility rules, with explicit restrictions on the proliferation of light towers and the spectrum of emergent critical strings (Etheredge et al., 30 May 2024).

In addition, the conjecture leads to nontrivial constraints on the possible degenerations of Calabi–Yau spaces, bounding, for example, the integer $b$ labeling Type II degenerations by $0 \leq b \leq 19$—mirroring classic results like the Kulikov classification for K3 surfaces and pointing to new swampland constraints on allowed geometric transitions (Friedrich et al., 1 Apr 2025).

7. Extensions Beyond Supersymmetry and String Theory

Recent investigations demonstrate the validity of the Emergent String Conjecture in non-supersymmetric and bottom-up scenarios:

• Holographic RG flows in broken supersymmetry models yield either a KK tower or the full tower of higher-spin excitations of an emergent string, with anomalous dimension suppression matching the infinite-distance limit (Basile, 2022).
• Thermodynamic and density-of-state arguments, divorced from explicit string constructions, reinforce that only KK or string towers survive as parametrically light in consistent quantum gravitational EFTs (Bedroya et al., 30 Apr 2024, Basile et al., 2023).

Summary Table: Infinite-Distance Limits and Physical Realizations

Limit Class	Geometric Fiber	Emerging Tower	Physical Interpretation	Example Frame
Decompactification	$T^2$	KK tower	Extra spacetime dimension opens	M/F-theory, IIA
String Limit (K3)	K3 surface	Critical heterotic string	Tensionless string emerges	M-theory, IIB/IIB-K3
String Limit ($T^4$)	Abelian surface ($T^4$)	Type II string	Tensionless Type II string	M-theory, IIA/IIB
Non-geometric	D-manifold/CFT	String/KK-like tower	Mirror/T-dual realization	Generalizations

All known explicit string and M-theory models, including non-geometric and flux scenarios, fit into this dichotomy, with universal control by a unique fibration structure, spectral scaling, and modular-invariant properties of the underlying compactification.

Outlook

The Emergent String Conjecture is emerging as a robust organizing principle across quantum gravity, yielding new geometric constraints, guiding the construction of consistent low-energy theories, and unifying thermodynamic, geometric, worldsheet, and amplitude-based approaches. It restricts the possible phenomenology of infinite-distance physics, with prospective applications in cosmology, moduli stabilization, and the UV-completion of effective field theories. Open questions persist: in particular, the precise role of certain obstructions, the taxonomy of non-geometric towers, and the extension of these principles to more general corners of the string landscape.