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

TL;DR

The paper analyzes infinite-distance limits in the vector multiplet moduli space of Type IIB Calabi–Yau compactifications, linking geometric degenerations to emergent gravitational descriptions via the Emergent String Conjecture. By combining asymptotic Hodge theory with semi-stable degenerations, it identifies special Lagrangian 3-cycles that shrink to generate towers of BPS D3-brane states, and shows Type III/IV limits realize decompactification to 6d/5d with two or one KK-type towers, respectively; in these limits the BPS degeneracies are fixed by the Euler characteristic $-\chi(V)$. The work further develops the notion of enhancement chains, distinguishing how primary and secondary singularity types control the gravitational frame and gauge-sector decoupling, and provides a geometric mechanism to predict spectra and state degeneracies in the asymptotic regime. Overall, the study provides a coherent geometric and EFT-based framework that corroborates the Emergent String Conjecture for Type IIB CY compactifications and offers detailed rules for interpreting multi-parameter limits and their associated towers.

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.

Figures (6)

• Figure 1: A schematic depiction of a semi-stable degeneration \ref{['eq:4foldV']}-\ref{['eq:def-V0']} with $N=2$. The fibre $V_z$ corresponding to the Calabi--Yau threefold over a generic point in the moduli space is depicted in pink, while the central fibre $V_0$ is depicted in blue. The latter splits into two components $V_1$ and $V_2$ which intersect over a complex surface $V_{12}$.
• Figure 2: An overview of the various graded spaces \ref{['eq:Gr-general']} that make up the geometric mixed Hodge structure on the cohomology $H^3(V_0)$ of the central fibre $V_0$. The relation $\mathrm{Gr}_0\cong H^3(\Pi(V_0))$ is elaborated upon around equation \ref{['eq:Gr0-dualgraph']}.
• Figure 3: The limiting mixed Hodge structures for a $\rm{II}\to\rm{III}\to\rm{IV}$ enhancement chain. Along each step in the chain, the states indicated in pink correspond to the lightest towers and subsequently determine the would-be species scales as summarised in equation \ref{['eq:species-hierarchy']}. In addition, the states in blue comprise the subleading KK tower.
• Figure 4: Illustrated are a type $\rm{II}$ divisor $(s_1\to\infty)$ and type $\rm{III}$ divisor ($s_2\to \infty$) intersecting in a type $\rm{IV}$ singularity. The two growth sectors corresponding to the $\rm{II}\to\rm{IV}$ ($s_1>s_2$) and $\rm{III}\to\rm{IV}$$(s_2> s_1)$ enhancements are depicted in pink and blue, respectively. Within each growth sector we have indicated the qualitatively different trajectories $\mathrm{(a)}-\mathrm{(f)}$, whose physical interpretation is listed on the right.
• Figure 5: An enhancement $\mathrm{II}_{b_1}\to\mathrm{II}_{b_2}$, with $b_2>b_1$, that only involves a change in the secondary singularity type. The change is induced by $\delta i^{2,2}=b_2-b_1$ states in $\mathrm{Gr}_3(\Delta_{k_1})$ on the left-hand side moving to $\mathrm{Gr}_2(\Delta_{k_1 k_2})$ and $\mathrm{Gr}_4(\Delta_{k_1 k_2})$ on the right-hand side.