$\eta$-symbols in exceptional field theory

Abstract

We present the universal form of $\eta$-symbols that can be applied to an arbitrary $E_{d(d)}$ exceptional field theory (EFT) up to $d=7$. We then express the $Y$-tensor, which governs the gauge algebra of EFT, as a quadratic form of the $\eta$-symbols. The usual definition of the $Y$-tensor strongly depends on the dimension of the compactification torus while it is not the case for our $Y$-tensor. Furthermore, using the $\eta$-symbols, we propose a universal form of the linear section equation. In particular, in the SL(5) EFT, we explicitly show the equivalence to the known linear section equation.