$\eta$-symbols in exceptional field theory

TL;DR

This paper introduces a universal formulation of $\\eta$-symbols applicable to all $E_{d(d)}$ exceptional field theories up to $d=7$, providing a dimension-independent expression for the $Y$-tensor and the linear section equation.

Contribution

It presents a universal, dimension-independent form of $\\eta$-symbols, and expresses the $Y$-tensor as a quadratic form, unifying the gauge algebra description across different EFTs.

Findings

Universal $\\eta$-symbols applicable to $E_{d(d)}$ EFTs up to $d=7

Dimension-independent $Y$-tensor expressed as quadratic form of $\\eta$-symbols

Explicit equivalence of the linear section equation in SL(5) EFT

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.