Y-algebroids and $E_{7(7)} \times \mathbb{R}^+$-generalised geometry

TL;DR

This paper introduces Y-algebroids, a new mathematical framework generalizing existing algebroids, to better describe the symmetry structures in string theory and M-theory compactifications, especially in four dimensions.

Contribution

The paper defines Y-algebroids, classifies M-exact E7-algebroids, and connects them to E7(7) generalised geometry, extending the understanding of fluxes and dualities in string/M-theory.

Findings

Classified M-exact E7-algebroids matching generalized tangent spaces.

Linked Y-algebroids to E7(7) generalised geometry with flux twists.

Reformulated generalized Leibniz parallelisable spaces and dualities in this framework.

Abstract

We define the notion of Y-algebroids, generalising the Lie, Courant, and exceptional algebroids that have been used to capture the local symmetry structure of type II string theory and M-theory compactifications to $D \geq 5$ dimensions. Instead of an invariant inner product, or its generalisation arising in exceptional algebroids, Y-algebroids are built around a specific type of tensor, denoted $Y$, that provides exactly the necessary properties to also describe compactifications to $D=4$ dimensions. We classify "M-exact" $E_7$-algebroids and show that this precisely matches the form of the generalised tangent space of $E_{7(7)} \times \mathbb{R}^+$-generalised geometry, with possible twists due to 1-, 4- and 7-form fluxes, corresponding physically to the derivative of the warp factor and the M-theory fluxes. We translate the notion of generalised Leibniz parallelisable spaces, relevant to consistent truncations, into this language, where they are mapped to so-called exceptional Manin pairs. We also show how to understand Poisson-Lie U-duality and exceptional complex structures using Y-algebroids.