Physics and Geometry of Complex Structure Limits in Type IIB Calabi-Yau Compactifications (2509.07056v1)

Abstract: We provide a detailed geometric and physical interpretation of infinite distance limits in the complex structure moduli space of Type IIB compactifications on Calabi-Yau threefolds, motivated by the Emergent String Conjecture. In the framework of semi-stable degenerations, such limits are characterised by a simple fibration structure of the fastest vanishing three-cycles. The previously studied Hodge theoretic classification of infinite distance limits of type II, III, and IV is reflected in the number of one-cycles of the shrinking fibres. Complementing our recent work on limits of type II, we focus here on type III and type IV degenerations. Based on effective field theory considerations, these are expected to be decompactification limits to 6d and 5d, respectively. However, establishing the existence of the associated Kaluza-Klein tower(s) of states with both the appropriate mass scaling and the correct degeneracy requires explicit geometric input. We show that the aforementioned vanishing three-cycles are special Lagrangian three-tori, thus giving rise to towers of asymptotically massless BPS particles from multi-wrapped D3-branes with the degeneracy of a Kaluza-Klein tower. We furthermore relate the BPS index of these three-cycles to the Euler characteristic of the threefold. Finally, we systematically analyse infinite distance trajectories in multi-parameter limits described by so-called enhancement chains. We find that the primary singularity type encodes the gravitational duality frame of the limit whereas the secondary singularity type is related to the rank of the gauge group coupled to gravity. The specifics of the asymptotic physics depend crucially on whether or not the trajectory is induced by the backreaction of an EFT string.

• The paper presents a precise construction linking infinite distance limits to emergent towers of BPS states, establishing a universal BPS index based on the Euler characteristic.
• It classifies degeneration types (II, III, IV) in the complex structure moduli space using dual graphs and limiting mixed Hodge structures to interpret decompactification limits.
• The analysis connects geometric degenerations and enhancement chains with effective field theory spectra, supporting the Emergent String Conjecture and Swampland criteria.

Physics and Geometry of Complex Structure Limits in Type IIB Calabi-Yau Compactifications

Introduction and Motivation

This work provides a comprehensive analysis of infinite distance limits in the complex structure moduli space of Type IIB Calabi-Yau threefold compactifications, with a focus on the interplay between geometric degenerations and the emergent physical spectra. The paper is motivated by the Emergent String Conjecture, which posits that infinite distance limits in moduli space are accompanied by towers of states—either Kaluza-Klein (KK) or string excitations—becoming exponentially light. The paper systematically investigates the geometric realization of these towers, the associated BPS indices, and the structure of enhancement chains in multi-parameter degenerations, thereby elucidating the physical interpretation of type II, III, and IV limits in the context of the Swampland program.

Semi-Stable Degenerations and Mixed Hodge Structures

The analysis is grounded in the framework of semi-stable degenerations, where the Calabi-Yau threefold degenerates into a union of smooth components intersecting normally. The central fiber's intersection pattern is encoded in a dual graph, whose dimension directly reflects the type of limiting mixed Hodge structure (II, III, IV). The geometric mixed Hodge structure, constructed via Deligne's formalism, is related to the limiting mixed Hodge structure governing the asymptotic behavior of the effective field theory couplings.

Figure 1: A schematic depiction of a semi-stable degeneration, illustrating the splitting of the Calabi-Yau fiber into intersecting components.

The graded pieces of the mixed Hodge structure, crucial for identifying the vanishing cycles and their physical interpretation, are organized as in Figure 2.

Figure 2: An overview of the various graded spaces $\mathrm{Gr}_\ell H^3(V_0)$, which encode the cohomological data relevant for the physical towers of states.

Classification of Infinite Distance Limits: Types II, III, IV

The paper classifies infinite distance limits according to the type of limiting mixed Hodge structure:

• Type II: The degeneration is characterized by a one-dimensional dual graph (tree), with vanishing cycles corresponding to fibrations of $T^1$ over a two-dimensional base. These limits are associated with emergent string scenarios, as previously established.
• Type III: The dual graph is two-dimensional, and the relevant vanishing cycles are $T^2$-fibrations over a one-dimensional base. The physical interpretation is a decompactification to six dimensions, with two independent KK towers.
• Type IV: The dual graph is three-dimensional (homeomorphic to $S^3$), and the vanishing cycles are $T^3$-fibrations over a point. This limit corresponds to a decompactification to five dimensions, with a single KK tower.

  Figure 3: The limiting mixed Hodge structures for a $\mathrm{II} \to \mathrm{III} \to \mathrm{IV}$ enhancement chain, indicating the graded spaces associated with the lightest towers.

Physical Interpretation: Towers of BPS States and BPS Indices

A central result is the explicit geometric construction of the vanishing special Lagrangian 3-cycles responsible for the towers of asymptotically massless BPS states. For type III and IV limits, these cycles are shown to be special Lagrangian three-tori, supporting multi-wrapped D3-brane states with the degeneracy structure of KK towers. The mass scaling of these states is determined by the growth theorem for the Hodge norm, yielding exponential suppression in the moduli space distance, with exponents $\gamma=1$ (type III) and $\gamma=\sqrt{3/2}$ (type IV).

The BPS index for these towers is computed and found to be universally given by minus the Euler characteristic of the Calabi-Yau threefold, $\Omega_{\rm BPS}(n\Gamma) = -\chi(V)$ for all $n>0$. This result is robust and independent of the specific degeneration, provided the vanishing cycle is a $T^3$.

Enhancement Chains and Multi-Parameter Limits

The paper extends to multi-parameter limits, where the moduli approach infinity at different rates, leading to enhancement chains of singularities. The primary singularity type in the chain determines the emergent gravitational duality frame, while the secondary type encodes the rank of the gauge group decoupled from gravity. The analysis distinguishes between EFT string-induced trajectories (linear in saxions) and more general paths, showing that the physical interpretation of the limit is sensitive to the scaling hierarchy among the moduli.

Figure 4: Illustrated are a type $\mathrm{II}$ divisor and a type $\mathrm{III}$ divisor intersecting in a type $\mathrm{IV}$ singularity, with growth sectors and physical interpretations indicated.

Figure 5: An enhancement $\mathrm{II}_{b_1}\to\mathrm{II}_{b_2}$, showing the change in secondary singularity type and the associated movement of states in the graded spaces.

Geometric-Physical Correspondence and Species Scale

The work provides a detailed correspondence between the geometric data of the degeneration (dual graph, graded cohomology) and the physical spectrum (towers of BPS states, species scale, gauge group structure). The species scale is shown to be determined by the scaling of the saxions responsible for the first infinite distance limit in the enhancement chain, and the analysis recovers the expected relations between KK scales and higher-dimensional Planck scales for decompactification limits.

Gromov-Hausdorff Perspective

The appendix discusses the Gromov-Hausdorff limit of the Ricci-flat metric on the degenerating Calabi-Yau, showing that the asymptotic geometry is captured by the dual graph of the central fiber. This provides a differential-geometric underpinning for the role of the dual graph in the mixed Hodge structure and the physical interpretation of the degeneration.

Figure 6: Semi-stable degeneration limit, illustrating the geometric transition and the emergence of the dual graph structure.

Conclusion

This paper establishes a precise geometric and physical dictionary for infinite distance limits in the complex structure moduli space of Type IIB Calabi-Yau compactifications. The identification of vanishing special Lagrangian tori as the geometric origin of KK towers, the universal BPS index formula, and the detailed analysis of enhancement chains provide strong evidence for the Emergent String Conjecture and clarify the structure of the Swampland at infinite distance. The results have significant implications for the classification of effective field theories in string theory, the understanding of moduli space boundaries, and the interplay between geometry and quantum gravity constraints. Future directions include extending the analysis to more general degenerations, exploring the implications for the landscape/swampland dichotomy, and leveraging the geometric insights for explicit model building and moduli stabilization.