Infinite Distance Networks in Field Space and Charge Orbits

TL;DR

The paper develops a global, geometrical understanding of the Swampland Distance Conjecture in the complex structure moduli space of Calabi–Yau threefolds by exploiting the orbit theorems of mixed Hodge structures. It shows that infinite-distance loci are classified by discrete monodromy data and can intersect to form networks governed by precise enhancement rules, with the Sl(2)-orbit framework providing commuting algebras that organize towers of states into charge orbits. A key advance is the construction of infinite, massless charge orbits at locus intersections, including multi-divisor configurations, offering a path toward a global view of the distance conjecture in field space. While the link to populated BPS towers remains to be fully proven, the work provides a robust, general mechanism to identify the towers and their mass behavior across complex, multi-parameter moduli spaces, with implications for quantum gravity constraints and mirror-symmetric large-volume regions.

Abstract

The Swampland Distance Conjecture proposes that approaching infinite distances in field space an infinite tower of states becomes exponentially light. We study this conjecture for the complex structure moduli space of Calabi-Yau manifolds. In this context, we uncover significant structure within the proposal by showing that there is a rich spectrum of different infinite distance loci that can be classified by certain topological data derived from an associated discrete symmetry. We show how this data also determines the rules for how the different infinite distance loci can intersect and form an infinite distance network. We study the properties of the intersections in detail and, in particular, propose an identification of the infinite tower of states near such intersections in terms of what we term charge orbits. These orbits have the property that they are not completely local, but depend on data within a finite patch around the intersection, thereby forming an initial step towards understanding global aspects of the distance conjecture in field spaces. Our results follow from a deep mathematical structure captured by the so-called orbit theorems, which gives a handle on singularities in the moduli space through mixed Hodge structures, and is related to a local notion of mirror symmetry thereby allowing us to apply it also to the large volume setting. These theorems are general and apply far beyond Calabi-Yau moduli spaces, leading us to propose that similarly the infinite distance structures we uncover are also more general.

Figures (23)

• Figure 1: Figure illustrating the relation between the distance conjecture and monodromy. The point $P$ is at infinite distance and the monodromy about it is denoted by T. The monodromy determines the local singular geometry of the moduli space, which leads to the exponential behaviour of the mass of the tower of states. The monodromy also acts on the spectrum of states picking out a specific infinite set of states.
• Figure 2: Figure showing an example field space with multiple infinite distance loci. The example is the (resolved) complex structure moduli space of the (mirror of the) two parameter Calabi-Yau $\mathbb{P}^{1,1,2,2,2}[8]$ as studied in Candelas:1993dm. Each infinite distance locus is denoted by a solid line and assigned a type labelled by $\mathrm{II}$, $\mathrm{III}$, or $\mathrm{IV}$. We also show special finite distance loci with dashed lines, and these are associated to type I. Some well-known loci are labelled explicitly, the finite distance conifold and orbifold loci, and the infinite distance large complex-structure point.
• Figure 3: Figure showing an example intersecting network for the (mirror of the) Calabi-Yau $\mathbb{P}^{1,1,1,6,9}[18]$ as studied in Candelas:1994hw. In this case we focus in one a particular region of the network, within the box, and show the more refined data for each locus including the sub-index. At the points of intersections the type of a locus can be modified. We show the types associated to each intersection point in the focused region.
• Figure 4: Graphs of allowed type enhancements for field spaces with $h^{2,1}$ complex fields. In terms of Calabi-Yau geometries, $h^{2,1}$ is the associated Hodge number. An arrow denotes that a starting type of locus may enhance over a sub-locus, corresponding to an intersection, to the end type. Note that the enhancement relations are not transitive. For example, in the $h^{2, 1} = 2$ case, there is a chain of $\mathrm{II}_0 \to \mathrm{II}_1 \to \mathrm{IV}_2$ enhancements, but there is no direct enhancement from $\mathrm{II}_0$ to $\mathrm{IV}_2$.
• Figure 5: Two normally intersecting divisors of the discriminant locus $\Delta$. The singularity of the Calabi-Yau threefold, here depicted as genus-two Riemann surface, worsens at the intersection.