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Emergent Strings from Infinite Distance Limits

Seung-Joo Lee, Wolfgang Lerche, Timo Weigand

TL;DR

The paper proposes and substantiates an Emergent String conjecture: in any equi-dimensional infinite-distance limit of a quantum gravity theory, the low-energy description either decompactifies or becomes a weakly coupled string theory with an infinite tower of string excitations. By classifying finite-volume infinite-distance limits of Calabi-Yau three-folds in the classical Kähler moduli space, it identifies three universal fibration types (T^2, K3, T^4) that govern the emergent physics in M-Theory and Type IIA compactifications. In M-Theory, Type T^2 limits realize standard F-Theory decompactification, while Type K3 and Type T^4 limits yield emergent, tensionless heterotic and Type II strings, respectively, with additional M2/M5-brane towers encoding further light states and dualities. In Type IIA, quantum corrections obstruct finite-volume infinite-distance limits for T^2 and K3 fibrations, though co-scaling the Planck or dilaton can restore equi-dimensional limits in the K3 case; Abelian (T^4) fibrations show a distinct, non-obstructed equi-dimensional limit via dual D-manifolds. The work thus weaves together geometric classifications, brane spectra, modular invariants, and mirror symmetry to support a coherent picture where emergent strings underpin the asymptotic structure of quantum gravity in these settings.

Abstract

As a refinement of the Swampland Distance Conjecture, we propose that a quantum gravitational theory in an infinite distance limit of its moduli space either decompactifies, or reduces to an asymptotically tensionless, weakly coupled string theory. We support our claim by classifying, as special cases, the behaviour of M-Theory and Type IIA string theory compactifications on Calabi-Yau three-folds at infinite distances in Kahler moduli space. The analysis comprises three parts: We first classify the possible infinite distance limits in the classical Kahler moduli space of a Calabi-Yau three-fold. Each such limit at finite volume is characterized by a universal fibration structure, for which the generic fiber shrinking in the limit is either an elliptic curve, a K3 surface, or an Abelian surface. In the second part we focus on M-Theory and investigate the nature of the towers of asymptotically massless states that arise from branes wrapped on the shrinking fibers. Depending on which of the three classes of fibrations are considered, we obtain decompactification to F-Theory, or a theory with a unique asymptotically tensionless, weakly coupled heterotic or Type II string, respectively. The latter probes a dual D-manifold which is in general non-geometric. In addition to the intrinsic string excitations, towers of states from M2-branes along non-contractible curves become light and correspond to further wrapping and winding modes of the tensionless heterotic or Type II string. In the third part of the analysis, we consider Type IIA string theory on Calabi-Yau three-folds and show that quantum effects obstruct taking finite volume infinite distance limits in the Kahler moduli space. The only possible infinite distance limit which is not a decompactification limit involves K3-fibrations with string scale fiber volume and gives rise to an emergent tensionless heterotic string.

Emergent Strings from Infinite Distance Limits

TL;DR

The paper proposes and substantiates an Emergent String conjecture: in any equi-dimensional infinite-distance limit of a quantum gravity theory, the low-energy description either decompactifies or becomes a weakly coupled string theory with an infinite tower of string excitations. By classifying finite-volume infinite-distance limits of Calabi-Yau three-folds in the classical Kähler moduli space, it identifies three universal fibration types (T^2, K3, T^4) that govern the emergent physics in M-Theory and Type IIA compactifications. In M-Theory, Type T^2 limits realize standard F-Theory decompactification, while Type K3 and Type T^4 limits yield emergent, tensionless heterotic and Type II strings, respectively, with additional M2/M5-brane towers encoding further light states and dualities. In Type IIA, quantum corrections obstruct finite-volume infinite-distance limits for T^2 and K3 fibrations, though co-scaling the Planck or dilaton can restore equi-dimensional limits in the K3 case; Abelian (T^4) fibrations show a distinct, non-obstructed equi-dimensional limit via dual D-manifolds. The work thus weaves together geometric classifications, brane spectra, modular invariants, and mirror symmetry to support a coherent picture where emergent strings underpin the asymptotic structure of quantum gravity in these settings.

Abstract

As a refinement of the Swampland Distance Conjecture, we propose that a quantum gravitational theory in an infinite distance limit of its moduli space either decompactifies, or reduces to an asymptotically tensionless, weakly coupled string theory. We support our claim by classifying, as special cases, the behaviour of M-Theory and Type IIA string theory compactifications on Calabi-Yau three-folds at infinite distances in Kahler moduli space. The analysis comprises three parts: We first classify the possible infinite distance limits in the classical Kahler moduli space of a Calabi-Yau three-fold. Each such limit at finite volume is characterized by a universal fibration structure, for which the generic fiber shrinking in the limit is either an elliptic curve, a K3 surface, or an Abelian surface. In the second part we focus on M-Theory and investigate the nature of the towers of asymptotically massless states that arise from branes wrapped on the shrinking fibers. Depending on which of the three classes of fibrations are considered, we obtain decompactification to F-Theory, or a theory with a unique asymptotically tensionless, weakly coupled heterotic or Type II string, respectively. The latter probes a dual D-manifold which is in general non-geometric. In addition to the intrinsic string excitations, towers of states from M2-branes along non-contractible curves become light and correspond to further wrapping and winding modes of the tensionless heterotic or Type II string. In the third part of the analysis, we consider Type IIA string theory on Calabi-Yau three-folds and show that quantum effects obstruct taking finite volume infinite distance limits in the Kahler moduli space. The only possible infinite distance limit which is not a decompactification limit involves K3-fibrations with string scale fiber volume and gives rise to an emergent tensionless heterotic string.

Paper Structure

This paper contains 53 sections, 9 theorems, 427 equations, 3 figures.

Table of Contents

  1. Introduction and Summary
  2. Emergent strings
  3. Summary of results
  4. Large Distance Limits in the Classical Kähler Moduli Space of Calabi-Yau three-folds
  5. Kähler form structure of infinite distance limits
  6. $J$-class A: $J_0^3 = 0$, but $J_0^2 \neq 0$.
  7. $J$-class B: $J_0^2 = 0$.
  8. Finite volume limits at infinite distance as vanishing fiber limits
  9. Infinite distance limits as weak coupling limits
  10. M-Theory on Calabi-Yau 3-folds in finite volume infinite distance limits
  11. The Type $T^2$ Limit as an F-Theory limit
  12. Potential towers from M2-branes
  13. Potential tensionless strings from M5-branes
  14. Nested limits in 6d
  15. The Type K3 limit and an Emergent Heterotic String
  16. ...and 38 more sections

Key Result

Theorem 1

Consider an infinite distance limit in the classical Kähler moduli space of a Calabi-Yau three-fold $Y$, subject to the constraint that the total volume remains finite. The Kähler form must be of $J$-class A or $J$-class B, as summarized in Section subsec_Kahlercone. Furthermore, the geometry of $~Y$ necessarily falls into one of the following three classes, which are mutually exclusive and hence

Figures (3)

  • Figure 1: Classification of large distance finite volume limits in classical Kähler moduli space, in terms of fibration types of Calabi-Yau three-folds, $Y$. Shown are also the corresponding dual stringy geometries that emerge from M-Theory on $Y$ in the respective limits. Thus only limits of Type K3 and $T^4$ are equi-dimensional. As will be discussed later, for Type IIA strings on $Y$ the situation is quite different.
  • Figure 2: Basic infinite distance limits in the quantum Kähler moduli space for Type IIA string theory on a Calabi-Yau three-fold. All further possible limits are decompactifications limits.
  • Figure 3: Schematic representation of (part of) the complex structure moduli space of the three-fold $X$, where the coordinates $z_1$ and $z_2$ govern the quantum volumes of fiber and base, respectively, of the mirror fibration, $Y$. The locus $z_2=0$ corresponds to a partial long distance limit where the base volume is large, and the $z_1$-axis coincides with the moduli space of the fiber. The large complex structure point, $L_1$, corresponds to large total volume ${\cal V}_Y$, and $L_2$ to the limit where, in classical geometry, the fiber would have zero volume. For more details see the text.

Theorems & Definitions (9)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Proposition 1
  • Proposition 2
  • Proposition 3