TASI Lectures on Supergravity and String Vacua in Various Dimensions

TL;DR

The notes map the landscape of consistent quantum gravity theories across 11D/10D, 8D, and 6D, using anomaly cancellation and Green-Schwarz-type mechanisms as primary consistency probes. They reveal string universality in 10D/11D, where every consistent high-dimensional SUSY gravity theory has a string realization, and show how lower dimensions expand the allowed gauge content while remaining tightly constrained by dualities, lattices, and F-theory geometry. A central thrust is constructing a bottom-up link between low-energy data (gauge groups, matter, and anomaly lattices) and top-down string constructions (heterotic, Type II, M-theory, and F-theory), culminating in a mapping to elliptically fibered Calabi–Yau geometries. The work highlights the swampland, connectivity of moduli spaces, and finite structure in higher dimensions, while outlining key open questions for four-dimensional vacua and broader consistency conditions beyond anomalies. Overall, it provides a unified framework for understanding how gravity, supersymmetry, and geometry organize the space of consistent quantum vacua in higher dimensions and guides the search for viable 4D theories.

Abstract

These lectures aim to provide a global picture of the spaces of consistent quantum supergravity theories and string vacua in higher dimensions. The lectures focus on theories in the even dimensions 10, 8, and 6. Supersymmetry, along with with anomaly cancellation and other quantum constraints, places strong limitations on the set of physical theories which can be consistently coupled to gravity in higher-dimensional space-times. As the dimensionality of space-time decreases, the range of possible supergravity theories and the set of known string vacuum constructions expand. These lectures develop the basic technology for describing a variety of string vacua, including heterotic, intersecting brane, and F-theory compactifications. In particular, a systematic presentation is given of the basic elements of F-theory. In each dimension, we summarize the current state of knowledge regarding the extent to which supergravity theories not realized in string theory can be shown to be inconsistent.

Figures (9)

• Figure 1: Venn diagram of supergravity theories. The set ${\cal G}$ of apparently consistent quantum gravity theories (largest set, red + green regions) contains the set ${\cal V}$ of known string vacua in any particular dimension (smallest set, green region). Intermediate sets ${\cal G}_*$, ${\cal V}_*$ denote the mathematically complete sets of consistent gravity theories and string vacua respectively. Set inclusions satisfy ${\cal G} \supseteq {\cal G}_* \supseteq {\cal V}_* \supseteq {\cal V}$.
• Figure 2: Hexagon diagrams give rise to gravitational, gauge, and mixed gauge-gravitational anomalies in ten dimensions.
• Figure 3: The Green-Schwarz mechanism: A tree diagram describing exchange of a $B$ field can cancel the anomalous part of the one-loop hexagon diagram in ten dimensions in special cases.
• Figure 4: K3 as an elliptic fibration over the sphere $S^2$. The fibration is singular at 24 points. The product of the monodromies around the individual points must be the identity, as the associated curve is contractable.
• Figure 5: An orbifold limit of the K3 surface can be viewed as the space $T^4/\mathbb{Z}_2$, which is locally flat except at 16 orbifold points (marked with "x"'s). The product of the regions on the two 2-tori that have not been marked in gray gives a fundamental domain for the orbifold space.