Generalised $G_2$-structures and type IIB superstrings

TL;DR

The paper connects type IIB string vacua to generalized geometry by showing that compactifications on seven-manifolds with two covariantly constant spinors yield a generalized $G_2$-structure associated with the reduction $SO(7,7)\to G_2\times G_2$, while compactifications on six-manifolds give a generalized $SU(3)$-structure from $SU(3)\times SU(3)$. It develops both the spinorial and differential-form (through $\rho^{ev/od}$ and the Box operator) descriptions, and formulates integrability in terms of $d_H=e^H\wedge d$-cohomology via the conditions $d_H e^{-\phi}\rho=0$ and $d_H e^{-\phi}\Box_{\rho} e^{-\phi}\rho=0$, highlighting Hitchin-type variational principles. The work also clarifies the connection to the classical $SU(3)$-case in the static limit ($a=\pi/2$) and elaborates how the six- and seven-dimensional pictures are linked through a dimensional reduction, providing a unified framework for understanding SUSY-preserving backgrounds and duality-inspired generalized geometries.

Abstract

The recent mathematical literature introduces generalised geometries which are defined by a reduction from the structure group $SO(d,d)$ of the vector bundle $T^d\oplus T^{d*}$ to a special subgroup. In this article we show that compactification of IIB superstring vacua on 7-manifolds with two covariantly constant spinors leads to a generalised $G_2$-structure associated with a reduction from SO(7,7) to $G_2\times G_2$. We also consider compactifications on 6-manifolds where analogously we obtain a generalised SU(3)-structure associated with $SU(3)\times SU(3)$, and show how these relate to generalised $G_2$-structures.