Emergent Strings in Type IIB Calabi--Yau Compactifications

TL;DR

This work provides a geometric realization of emergent-string limits in the vector multiplet sector of Type IIB Calabi–Yau compactifications by focusing on Tyurin degenerations, where the CY splits along a K3. It shows that the tensionless EFT string in these limits corresponds to a unique critical heterotic string on T^2×K3, accompanied by an infinite tower of BPS states arising from D3-branes on special Lagrangian 3-cycles; the BPS spectrum is conjectured to be governed by modular-type forms. Mirror symmetry is used to connect these Type IIB results to emergent strings in Type IIA on K3-fibred CYs, with explicit SYZ-type relations mapping worldsheets across dualities. Beyond Tyurin degenerations, the authors propose geometric constraints on Type II degenerations, including bounds on the parameter b and the possible intersection lattices, which align with the Emergent String Conjecture and echo Kulikov-type degenerations in K3 geometry. Overall, the paper strengthens evidence for the universality of emergent-string limits in quantum gravity and offers a concrete bridge between complex-structure degenerations, worldsheet theories, and dual heterotic descriptions.

Abstract

We study infinite distance limits in the complex structure moduli space of Type IIB compactifications on Calabi--Yau threefolds, in light of the Emergent String Conjecture. We focus on the so-called type II limits, which, based on the asymptotic behaviour of the physical couplings in the low-energy effective theory, are candidates for emergent string limits. However, due to the absence of Type IIB branes of suitable dimensionality, the emergence of a unique critical string accompanied by a tower of Kaluza--Klein states has so far remained elusive. By considering a broad class of type II$_b$ limits, corresponding to so-called Tyurin degenerations, and studying the asymptotic behaviour of four-dimensional EFT strings in this geometry, we argue that the worldsheet theory of the latter describes a unique critical heterotic string on $T^2\times\mathrm{K3}$ with a gauge bundle whose rank depends on $b$. In addition, we establish the presence of an infinite tower of BPS particles arising from wrapped D3-branes by identifying a suitable set of special Lagrangian 3-cycles in the geometry. The associated BPS invariants are conjectured to be counted by generalisations of modular forms. As a consistency check, we further show that in special cases mirror symmetry identifies the EFT strings with the well-understood emergent string limits in the Kähler moduli space of Type IIA compactifications on K3-fibred Calabi--Yau threefolds. Finally, we discuss the implications of the Emergent String Conjecture for type II limits which do not correspond to Tyurin degenerations, and predict new constraints on the possible geometries of type II degenerations which resemble those arising in the Kulikov classification of degenerations of K3 surfaces.

Figures (8)

• Figure 1: A local patch in a complex two-dimensional vector multiplet moduli space $\mathcal{M}_{\mathrm{V}}$ containing two divisors $\Delta_1$ and $\Delta_2$ which intersect at a point $\Delta_{12}$.
• Figure 2: The Hodge--Deligne diamond corresponding to a type $\mathrm{II}_b$ singularity. The horizontal rows indicated in black, blue, and red correspond to the graded spaces $\mathrm{Gr}_4$, $\mathrm{Gr}_3$, and $\mathrm{Gr}_2$, respectively. Here we put $b'=h^{2,1}-b-1$.
• Figure 3: A schematic depiction of a Tyurin degeneration \ref{['eq:4foldV']}. The fibre $V_z$ corresponding to the Calabi-Yau threefold over a generic point in the moduli space is depicted in red, while the central fibre $V_0$ is depicted in orange. For a Tyurin degeneration, the latter splits into two components $X_1$ and $X_2$ which intersect over a K3 surface $Z$, here indicated in blue.
• Figure 4: A geometric depiction of the set-up described around equation \ref{['eq:hatVfibration']}. The Type IIA NS5-brane (indicated in green) is located at a generic point in the base $\widehat{\mathbb{P}}^1_b$ and wraps the generic K3-fibre $\widehat{Z}$. The emergent string is obtained in the limit where the volume of the base becomes large, while the volume of the generic fibre stays constant.
• Figure 5: (a) The Tyurin degeneration \ref{['TyurinhetV']} in which the base of the K3-fibration splits into two components $\widehat{\mathbb{P}}^1_1$ and $\widehat{\mathbb{P}}^1_2$ intersecting over a single point. (b) The same situation as in Figure (a), but with the intersection point removed, giving rise to two asymptotically cylindrical geometries the 1-cycles $\widehat{\gamma}_1$ and $\widehat{\gamma}_2$ at the boundary.

Theorems & Definitions (2)

• Conjecture 1
• Conjecture 2