On generalized $G_2$-structures and $T$-duality

TL;DR

This paper explores how $T$-duality transforms classical $G_2$-structures into integrable generalized $G_2$-structures on seven-dimensional manifolds, revealing new structures that can be closed without being traditional $G_2$-structures.

Contribution

It demonstrates the construction of integrable generalized $G_2$-structures via $T$-duality, including examples that admit closed generalized structures without classical counterparts.

Findings

Existence of closed generalized $G_2$-structures not compatible with classical $G_2$-structures

Construction of integrable generalized $G_2$-structures through $T$-duality

New examples of manifolds with generalized $G_2$-structures

Abstract

This is a short note on generalized $G_2$-structures obtained as a consequence of a $T$-dual construction given in a previous work of the authors together with Leonardo Soriani. Given classical $G_2$-structure on certain seven dimensional manifolds, either closed or co-closed, we obtain integrable generalized $G_2$-structures which are no longer an usual one, and with non-zero three form in general. In particular we obtain manifolds admitting closed generalized $G_2$-structures not admitting closed (usual) $G_2$-structures.