Algebraic structures in exceptional geometry

TL;DR

This paper explores the complex algebraic structures such as Borcherds superalgebras and $L_$ algebras that emerge in exceptional field theory, providing insights into the mathematical foundations of U-duality symmetries in M-theory.

Contribution

It presents a comprehensive overview of the algebraic structures appearing in exceptional field theory, highlighting their roles in understanding U-duality symmetries.

Findings

Identification of Borcherds superalgebras in EFT

Connection of tensor hierarchy algebras to EFT structures

Discussion of $L_$ algebras in the context of EFT

Abstract

Exceptional field theory (EFT) gives a geometric underpinning of the U-duality symmetries of M-theory. In this talk I give an overview of the surprisingly rich algebraic structures which naturally appear in the context of EFT. This includes Borcherds superalgebras, Cartan type superalgebras (tensor hierarchy algebras) and $L_\infty$ algebras. This is the written version of a talk based mainly on refs. [1-6], presented at ISQS25, Prague, June 2017, at QTS-10/LT-12, Varna, June 2017, at SQS 2017, Dubna, Aug. 2017, and at the 9th Mathematical Physics Meeting, Belgrade, Sept. 2017.