TASI Lectures on Compatification and Duality

TL;DR

These notes survey moduli spaces of theories with $16$ or $32$ supercharges and their dual descriptions across perturbative, brane, and curved-background frameworks. They develop the interlocking roles of S-, T-, and U-dualities, orientifold constructions, F-theory, and M-theory in organizing compactifications, with K3 and Calabi–Yau manifolds as central geometric examples, underpinned by holonomy and algebraic geometry. A unifying theme is that the spectrum and moduli of these theories assemble into symmetric spaces such as $G/K$ with $G=E_{d+1(d+1)}$ or $O(d,d+16)$, and that dualities map seemingly distinct theories into equivalent descriptions in different regimes. The work highlights how geometric and algebraic structures—elliptic fibrations, ADE algebras from wrapped branes, and Torelli-type moduli—drive a rich web of dualities that connects heterotic, type II, M-/F-theory compactifications across dimensions.

Abstract

We describe the moduli spaces of theories with 32 or 16 supercharges, from several points of view. Included is a review of backgrounds with D-branes (including type I' vacua and F-theory), a discussion of holonomy of Riemannian metrics, and an introduction to the relevant portions of algebraic geometry. The case of K3 surfaces is treated in some detail.

Figures (6)

• Figure 1: Compactifications of M-theory
• Figure 2: Compactifications of Type I and Heterotic Strings
• Figure 3: A generic function $H(y)$
• Figure 4: The function $H(y)$ for gauge algebra $\mathfrak{e}_8 \oplus \mathfrak{e}_8\oplus\mathfrak{u}(1)^{\oplus2}$
• Figure 5: Constraints on invariants of surfaces of general type