Tame Complexity of Effective Field Theories in the Quantum Gravity Landscape

TL;DR

The paper introduces tame geometry, particularly sharp o-minimality, as a quantitative framework to bound the apparently infinite data of EFTs in quantum gravity. It formulates the Finite Complexity Conjecture, asserting that EFTs up to a fixed cutoff $\Lambda$ admit uniform, finite tame complexity and that the space of such EFTs has a finite EFT covering, with EFT domains capturing local descriptions across moduli spaces. By connecting to string compactifications and dualities, the authors illustrate how differential constraints and dual representations reorganize infinite Wilsonian data into finite information, and they define notions of EFT domains, coverings, and volumes that yield well-defined, finite counting. The approach provides a unified language for finiteness phenomena across the swampland program and offers a route to quantitative bounds on the landscape, with concrete examples in higher-dimensional supergravity and Calabi–Yau moduli spaces. If developed further, this framework could yield explicit complexity bounds and volume-counting formulas, deepening our understanding of quantum-gravity constraints on low-energy physics.

Abstract

Effective field theories consistent with quantum gravity obey surprising finiteness constraints, appearing in several distinct but interconnected forms. In this work we develop a framework that unifies these observations by proposing that the defining data of such theories, as well as the landscape of effective field theories that are valid at least up to a fixed cutoff, admit descriptions with a uniform bound on complexity. To make this precise, we use tame geometry and work in sharply o-minimal structures, in which tame sets and functions come with two integer parameters that quantify their information content; we call this pair their tame complexity. Our Finite Complexity Conjectures are supported by controlled examples in which an infinite Wilsonian expansion nevertheless admits an equivalent finite-complexity description, typically through hidden rigidity conditions such as differential or recursion relations. We further assemble evidence from string compactifications, highlighting the constraining role of moduli space geometry and the importance of dualities. This perspective also yields mathematically well-defined notions of counting and volume measures on the space of effective theories, formulated in terms of effective field theory domains and coverings, whose finiteness is naturally enforced by the conjectures.

Figures (7)

• Figure 1: An example of a tame set $X\subseteq \mathbb{R}^2$ (left) and a cylindrical cell decomposition of $\mathbb{R}^2$ adapted to $X$ (right). The vertical boundaries of the cells ensure that the cell decomposition behaves appropriately under the linear projection to the horizontal axis.
• Figure 2: EFT covering of $\mathcal{N}=2$$SU(2)$ Seiberg-Witten moduli space with finitely many EFT domains (discs and punctured discs). Around each of the three infinite distant limits there exists a different EFT description of finite complexity in terms of different combinations of variables $a$ and $a_D$, capturing the electric-magnetic duality discussed in Seiberg:1994rs. In addition, three other discs centered around regular points are included to cover the full moduli space.
• Figure 3: Depiction of the moduli space $\mathcal{M}_{\Lambda}$ of Type IIA supergravity in 10 dimensions. The boundaries of the region are located at the points in which $\Lambda_{\rm QG}$ (given by in this case by the lightest state of the tower) falls below $\Lambda$. The shaded areas correspond to inaccessible regions of the moduli space of the compactification that do not admit an EFT with the chosen cutoff.
• Figure 4: Depiction of the moduli space $\mathcal{M}_{\Lambda}$ of Type IIB supergravity in 10 dimensions as a subset of the hyperbolic plane. The solid blue lines depict the boundaries of the fundamental domain of $\mathbb{H}/\text{SL}(2,\mathbb{Z})$. The region of the fundamental domain where the EFT supergravity description is valid is highlighted with shaded blue while the dark shaded red corresponds to the region of the fundamental domain where the masses of the tower of D1 oscillations fall below the cutoff $\Lambda$. Finally, the region outside the fundamental domain where the masses of the S-dual string tower fall below the cutoff has also been highlighted.
• Figure 5: Shape of the moduli space $\mathcal{M}_{\Lambda}$ of the 9-dimensional theory obtained from compactifying M-theory on a torus for different values of $\Lambda$ (in 9-dimensional Planck units). The moduli space has been parametrized using the volume of the torus $U=R_{10}R_{11}$ and its complex structure $\tau=R_{11}/R_{10}$. The towers whose light states set the boundary of validity of the effective theory along each direction are highlighted.