Les Houches Lectures on Constructing String Vacua

TL;DR

The notes deliver a comprehensive framework for constructing and analyzing string vacua with a focus on F-theory flux vacua and IIB realizations. They integrate a detailed physical-geometric pathway: from fundamental challenges (Dine–Seiberg, no-go theorems) to a practical toolkit (toric geometry, index theorems) and to concrete stabilization mechanisms (KKLT and large-volume Swiss cheese) within warped, fluxed backgrounds. A key contribution is the development of continuum-index techniques to estimate the count and distribution of flux vacua across complex-structure moduli spaces, including finite-tadpole constraints, warping effects, and nonperturbative contributions from instantons and gaugino condensation. The framework links topological data (Euler characteristics, Chern classes) to effective 4d physics, enabling approximate, scalable predictions about vacua abundance, coupling distributions, and hierarchies, while outlining explicit geometric constructions for realistic-like models. Overall, the work blends deep geometric tools with statistical methods to map the string landscape and identify viable routes to stabilized vacua with small cosmological constants and controlled effective theories.$

Abstract

These lectures give a detailed introduction to constructing and analyzing string vacua suitable for phenomenological model building, with particular emphasis on F-theory flux vacua. Topics include (1) general challenges and overview of some proposed scenarios, (2) an extensive introduction to F-theory and its relation to M-theory and perturbative IIB string theory, (3) F-theory flux vacua and moduli stabilization scenarios, (4) a practical geometrical toolkit for constructing string vacua from scratch, (5) statistics of flux vacua, and (6) explicit models.

Figures (17)

• Figure 1: On the left: two possible behaviors for the effective potential to first order. On the right: including higher order corrections.
• Figure 2: F-theory from M-theory. Starting point: $T^2$ fibration over $M_9$.
• Figure 3: F-theory realization of $(p,q)$ 7-branes.
• Figure 4: Domain wall as a source of flux; the domain wall can be thought of as wrapping the fluxlines (or more formally the Poincaré dual) of the flux jump it sources. On the left we have a domain wall producing three units of a brane-type flux; on the right two units of a bulk-type flux.
• Figure 5: F-theory on K3 ellipticaly fibered over a sphere. The dots indicate the 24 degeneration loci of the elliptic fiber.