TASI Lectures on de Sitter Vacua

TL;DR

This work surveys how type IIB string theory on Calabi-Yau orientifolds can yield four-dimensional cosmologies via flux compactifications. It builds from geometric foundations through the classical four-dimensional EFT (GVW superpotential, no-scale structure) to quantum corrections that lift moduli and enable stabilized vacua. A central thread is the ISD/GKP framework, warping, and the necessary tadpole constraints, which together facilitate controlled constructions of de Sitter vacua while highlighting remaining open problems. The notes emphasize both the technical tools for moduli stabilization and the current limits of calculability in quantum gravity contexts relevant for cosmology.

Abstract

These lectures provide a self-contained introduction to flux compactifications of type IIB string theory on Calabi-Yau orientifolds. The first lecture begins with geometric foundations, then presents vacuum solutions in Calabi-Yau compactifications, as well as the geometry and physics of the moduli problem. The second lecture develops the classical theory of type IIB flux compactifications, both in ten dimensions and in the four-dimensional effective theory. The third lecture turns to the quantum theory of flux compactifications, including perturbative and non-perturbative corrections. With this foundation, in the fourth lecture we give a detailed treatment of the candidate de Sitter vacua recently constructed in arXiv:2406.13751. These notes are intended to be accessible to graduate students working in adjacent fields, and so extensive background material is included throughout.

Figures (18)

• Figure 1: The CMB temperature map measured by Planck Planck:2013pxb. Figure courtesy of ESA and the Planck Collaboration.
• Figure 2: A schematic of the possible results of uplift from a supersymmetric AdS vacuum. The horizontal axis is the Calabi-Yau volume $\mathcal{V}=(T+\overline{T})^{3/2}$, and the vertical axis is the scalar potential $V$.
• Figure 3: The scalar potential on the positive $\mathrm{Im}(\tau)$ axis.
• Figure 4: Sketch of the KS throat. The warping is generated by $F_3$ and $H_3$ fluxes threading the $A$- and $B$-cycles of the deformed conifold, producing a smooth geometry that caps off at a finite three-sphere. The warping grows stronger near the tip, yielding a highly redshifted region suitable for localized supersymmetry breaking. Figure courtesy of S. Schreyer.
• Figure 5: Illustration of brane–flux annihilation in the KPV mechanism. The anti-D3-branes polarize into an NS5-brane that wraps a contractible $S^2$ inside the $S^3$ at the tip of the KS throat. As the NS5-brane moves across the $S^3$, its worldvolume flux decreases, annihilating anti-D3 charge against background three-form flux and ultimately producing $(M-p)$ D3-branes. Figure courtesy of S. Schreyer.

Theorems & Definitions (3)

• proof
• proof
• proof