This paper is a review about why the precise shape of Calabi‑Yau threefolds matters for building models of particle physics and cosmology from string theory. Calabi‑Yau (CY) threefolds are the six extra dimensions that string theorists use to make a four‑dimensional universe. The authors explain how the details of those shapes — in particular the topology of surfaces (divisors) and curves inside them — control the low‑energy scalar potentials that determine inflation and other cosmological dynamics.

The review explains how moduli stabilization works in two steps used in many constructions. First, fluxes fix the complex‑structure moduli and the axio‑dilaton, leaving a number W0. Second, the remaining Kähler moduli (which measure sizes of cycles) are fixed by smaller corrections. Two popular schemes are discussed: KKLT (a widely used non‑perturbative approach) and the Large Volume Scenario (LVS), where a balance of corrections can give an exponentially large overall volume. The paper stresses that generating the needed corrections depends on having the right kinds of divisors. For example, rigid del Pezzo surfaces are often required for non‑perturbative terms in the superpotential, and a special “Wilson” divisor is needed for so‑called poly‑instanton effects.

The authors also survey the computational progress and datasets that make explicit model building possible. There are two main CY datasets: the Complete Intersection Calabi‑Yau (CICY) list and toric hypersurface CYs from the Kreuzer–Skarke database. One projective CICY database contains 7,890 examples with their Hodge numbers. A useful, more compact collection called the AGHJN dataset covers toric CYs with h1,1 (the number of Kähler moduli) from 1 to 6 and includes many model‑building data like intersection numbers and Kähler cones. Tools such as PALP, cohomCalg, SAGE and the newer CYTools have helped compute divisor topologies and other quantities needed for global constructions.