# T-Branes and $G_2$ Backgrounds

## TL;DR

This paper explores the geometry of G_2 compactifications in M- and string theory, focusing on T-brane phenomena, gauge theory solutions, and localized matter, providing explicit examples and algebraic characterizations.

## Contribution

It introduces a gauge theory framework capturing T-brane effects in G_2 backgrounds, generalizes twisted connected sums, and reveals how codimension six singularities can support chiral matter.

## Key findings

- Gauge theory solutions include non-trivial flux and non-commuting matrix coordinates.
- Explicit examples of G_2 backgrounds with T-branes and localized matter profiles.
- Algebraic method for characterizing localized matter in T-brane configurations.

## Abstract

Compactification of M- / string theory on manifolds with $G_2$ structure yields a wide variety of 4D and 3D physical theories. We analyze the local geometry of such compactifications as captured by a gauge theory obtained from a three-manifold of ADE singularities. Generic gauge theory solutions include a non-trivial gauge field flux as well as normal deformations to the three-manifold captured by non-commuting matrix coordinates, a signal of T-brane phenomena. Solutions of the 3D gauge theory on a three-manifold are given by a deformation of the Hitchin system on a marked Riemann surface which is fibered over an interval. We present explicit examples of such backgrounds as well as the profile of the corresponding zero modes for localized chiral matter. We also provide a purely algebraic prescription for characterizing localized matter for such T-brane configurations. The geometric interpretation of this gauge theory description provides a generalization of twisted connected sums with codimension seven singularities at localized regions of the geometry. It also indicates that geometric codimension six singularities can sometimes support 4D chiral matter due to physical structure "hidden" in T-branes.