Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2606.08427

An elliptic approach to Reid's fantasy

Lara Anderson·James Gray·Richard Nally·Washington Taylor
View original source ↗Read source document ↗Export metadata ↓

Abstract

It is a long-standing problem to prove that the number of distinct topological types of Calabi-Yau threefolds is finite. A related proposition, Reid's fantasy, conjectures that all Calabi-Yau threefolds are connected in a single moduli space through extremal transitions. Finiteness of topological types has been proven for the class of elliptic and genus one fibered Calabi-Yau threefolds, which recently have been shown to constitute the vast majority of known Calabi-Yau threefolds; the moduli space of elliptic CY3's is connected. In this letter, we demonstrate that all non-fibered Calabi-Yau threefolds in two of the largest known classes (toric hypersurfaces and complete intersections in products of projective spaces) are connected to fibered Calabi-Yau threefolds through a simple class of geometric transitions involving the shrinking of a single divisor from a fibered geometry. This suggests that non-fibered Calabi-Yau threefolds are rare special cases that are reached by simplifying fibered Calabi-Yau threefolds, and points to a natural path towards proving finiteness and Reid's fantasy for Calabi-Yau threefolds.

Explore related subjects

hep-thmath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Lara Anderson, James Gray, Richard Nally, Washington Taylor. 2026-06-07. An elliptic approach to Reid's fantasy. https://arxiv.org/abs/2606.08427

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Gaillard-Zumino non-invertible symmetries

We uncover an infinite class of novel zero-form non-invertible symmetries in a broad family of four-dimensional models, studied years ago by Gaillard and Zumino (GZ), which includes several extended supergravities as particular subcases. The GZ models consist of abelian gauge fields coupled to a neutral sector, typically including a set of scalars, whose equations of motion are classically invariant under a continuous group $\mathscr{G}$ acting on the electric and magnetic field strengths via symplectic transformations. The standard lore holds that, at the quantum level, these symmetries are broken to an integral subgroup $\mathscr{G}_\mathbb{Z}$. We show that, in fact, a much larger subgroup $\mathscr{G}_\mathbb{Q}$ survives, albeit through non-invertible topological defects. We explicitly construct these defects and compute some of their fusion rules. As illustrative examples, we consider the axion-dilaton-Maxwell model and the bosonic sector of a class of $\mathcal{N}=2$ supergravities of the kind that appear in type II Calabi-Yau compactifications. Finally, we comment on how (part of) these non-invertible zero-form symmetries can be broken by gauging the $\mathscr{G}_\mathbb{Z}$ subgroup of invertible symmetries.

hep-th↗
arXiv2026

On the resolution of categorical symmetries in (Non-) Unitary Rational CFTs

We explore several aspects of categorical symmetry-resolved entanglement entropy (SREE) directly within two-dimensional rational conformal field theory (RCFT) (without invoking any SymTFT construction arXiv:2409.02806). We derive a general formula applicable whenever the action of the relevant topological defect lines on the annulus Hilbert space is known. This framework accommodates weakly and strongly symmetric boundaries, cloaking states, and fusion rings with multiplicities. We verify the formula in a range of diagonal unitary and non-unitary examples, including theories with generalized Haagerup-Izumi modular data. Furthermore, we extend the analysis to non-diagonal RCFTs. The $\frac{1}{2}E_6$ example demonstrates that closed-channel modular data and NIM-rep multiplicities alone do not suffice to determine the defect action on the complete open-channel Hilbert space.

hep-th↗
arXiv2026

Reflecting boundary conditions in critical loop models

In critical loop models, we call a boundary sticky if loops can attach to it, and reflecting otherwise. Using analytic bootstrap methods, we show that reflecting boundaries are characterised by one complex parameter, analogous to the boundary cosmological constant in Liouville theory. We determine disc 1-point functions, and write an explicit formula for disc 2-point functions as infinite combinations of conformal blocks. We also sketch the lattice interpretation of reflecting boundaries.

hep-th↗