# Geometry of $\mathbb{R}^{+}\times E_{3(3)}$ Exceptional Field Theory and F-theory

## TL;DR

This paper explores a specific solution in exceptional field theory that models F-theory's monodromies of 7-branes, deriving flux constraints and connecting equations of motion to a generalized Ricci tensor.

## Contribution

It introduces a novel solution to the section condition in $ ext{E}_{3(3)}$ EFT, linking it to F-theory and deriving explicit flux and geometric expressions.

## Key findings

- Describes how fields depend on stringy coordinates to model 7-brane monodromies.
- Derives flux constraints from embedding tensor formalism.
- Shows F-theory equations of motion emerge from generalized Ricci tensor.

## Abstract

We consider a non trivial solution to the section condition in the context of $\mathbb{R}^{+}\times E_{3(3)}$ exceptional field theory and show that allowing fields to depend on the additional stringy coordinates of the extended internal space permits to describe the monodromies of (p,q) 7-branes in the context of F-theory. General expressions of non trivial fluxes with associated linear and quadratic constraints are obtained via a comparison to the embedding tensor of eight dimensional gauged maximal supergravity with gauged trombone symmetry. We write an explicit generalised Christoffel symbol for $E_{3(3)}$ EFT and show that the equations of motion of F-theory, namely the vanishing of a 4 dimensional Ricci tensor with two of its dimensions fibered, can be obtained from a generalised Ricci tensor and an appropriate type IIB ansatz for the metric.