Modave Lecture notes: Introduction to Exceptional Field Theory

TL;DR

This paper provides an introductory overview of Exceptional Field Theory, covering its mathematical foundations, relation to supergravity, hidden symmetries, and applications to consistent truncations, aimed at researchers in mathematical physics.

Contribution

It offers a comprehensive introduction to ExFT, including construction, properties, and applications, with detailed discussion of $ ext{E}_{d(d)}$ groups and their role in string dualities.

Findings

Explains the role of $ ext{E}_{d(d)}$ groups in supergravity and string dualities.

Details the construction of $ ext{E}_{7(7)}$-ExFT and other $ ext{E}_{d(d)}$-ExFTs.

Demonstrates applications of ExFT to consistent truncations.

Abstract

These notes are based on lectures given at the XIX Modave School on Mathematical Physics and present an introduction to Exceptional Field Theory. We cover the standard Kaluza-Klein reductions on tori, with applications to supergravity. We review the supergravity action of type IIA/IIB and eleven-dimensional supergravities. We motivate the appearance of the hidden symmetry groups $\mathrm{E}_{d(d)}$ through the lens of string dualities. We explore the construction and properties of $\mathrm{E}_{7(7)}$-ExFT and review other $\mathrm{E}_{d(d)}$-ExFT for $3\leq d\leq 8$, as well as double field theories. These notes conclude with some applications of ExFT to consistent truncations.