Infinite Distances in Field Space and Massless Towers of States

TL;DR

The paper tests the Swampland Distance Conjecture in the complex-structure moduli space of Calabi–Yau manifolds by linking infinite-distance points to infinite-order monodromies and, via the Nilpotent Orbit Theorem and limiting mixed Hodge structures, to an infinite tower of BPS states becoming massless. It shows that these towers can be identified with monodromy orbits acting on BPS charges, and that their masses decrease exponentially with proper field distance, consistent with the SDC; it also argues that infinite distance can emerge from integrating out the tower, with a corresponding emergence of vanishing gauge couplings and a diminishing species scale. The work ties these phenomena to global symmetries becoming emergent and provides a framework to study both the spectrum and gauge dynamics through mixed Hodge-theoretic data, offering a general, model-independent account of the SDC in CY moduli spaces. Overall, it provides both mathematical machinery and physical interpretation supporting the conjecture and its connections to WGC and emergent gravity ideas.

Abstract

It has been conjectured that in theories consistent with quantum gravity infinite distances in field space coincide with an infinite tower of states becoming massless exponentially fast in the proper field distance. The complex-structure moduli space of Calabi-Yau manifolds is a good testing ground for this conjecture since it is known to encode quantum gravity physics. We study infinite distances in this setting and present new evidence for the above conjecture. Points in moduli space which are at infinite proper distance along any path are characterised by an infinite order monodromy matrix. We utilise the nilpotent orbit theorem to show that for a large class of such points the monodromy matrix generates an infinite orbit within the spectrum of BPS states. We identify an infinite tower of states with this orbit. Further, the theorem gives the local metric on the moduli space which can be used to show that the mass of the states decreases exponentially fast upon approaching the point. We also propose a reason for why infinite distances are related to infinite towers of states. Specifically, we present evidence that the infinite distance itself is an emergent quantum phenomenon induced by integrating out at one-loop the states that become massless. Concretely, we show that the behaviour of the field space metric upon approaching infinite distance can be recovered from integrating out the BPS states. Similarly, at infinite distance the gauge couplings of closed-string Abelian gauge symmetries vanish in a way which can be matched onto integrating out the infinite tower of charged BPS states. This presents evidence towards the idea that also the gauge theory weak-coupling limit can be thought of as emergent.

Figures (3)

• Figure 1: Smooth path connecting a regular point $Q$ to a singular point $P$ which might be at infinite distance in moduli space.
• Figure 2: Figure showing the states in the theory at $z=z_0$ and $z=z_0 e^{2 \pi i}$. The labels $x$ and $y$ track each physical state. The state $y$ of charge ${\bf q}$ is BPS at $z=z_0$ and the state $x$ of charge $T{\bf q}$ is BPS at $z=z_0 e^{2 \pi i}$. In the last step the charges of all the states are simultaneously relabelled so that the spectrum matches the one before the monodromy.
• Figure 3: Effective theory of one light scalar field $\phi$ and S heavy ones $h_i$ up to $\Lambda_{UV}$.