Nontrivial RR two-form field strength and SU(3)-structure

TL;DR

This work analyzes ${\cal N}=1$ Type II flux compactifications with a nontrivial RR two-form and dilaton by framing the six-dimensional internal geometry as an $SU(3)$-structure $X$ arising from a KK reduction of a seven-manifold $Y$ with $G_2$-structure. It derives a set of supersymmetry constraints in terms of a generalized monopole equation $\beta\,d\varphi = \frac{1}{2}\check{F}\lrcorner (*\psi_3)$, a primitivity condition $J\lrcorner F=0$, and a Killing spinor equation, and shows that integrability of the almost complex structure requires $F^{(1,1)}=0$ with $\beta=2\alpha$. The authors compute the intrinsic torsion of the induced $SU(3)$-structure in terms of the underlying $G_2$-torsion, obtaining explicit components such as $W_2^+=-\check{F}_0^{(1,1)}$, $W_4=-(\beta-2\alpha)d\varphi$, and $W_5=-(\beta-3\alpha)d\varphi$ for generic ${\cal N}=1$ backgrounds with nontrivial RR flux and dilaton. In the torsion-free $G_2$ limit, the results specialize to a primitivity constraint $F^{(0)}=0$ and the stated $SU(3)$-torsion relations, clarifying when the six-manifold is Calabi–Yau or merely a non-Calabi–Yau $SU(3)$-structure. These findings provide a concrete geometric framework for flux compactifications and connect the $G_2$-torsion data to explicit $SU(3)$-structure torsion in string theory contexts.

Abstract

We discuss how in the presence of a nontrivial RR two-form field strength and nontrivial dilaton the conditions of preserving supersymmetry on six-dimensional manifolds lead to generalized monopole and Killing spinor equations. We show that the manifold is Kähler in the ten-dimensional string frame if F_0^{(1,1)}=0. We then determine explicitly the intrinsic torsion of the SU(3)-structure on six-manifolds that result via Kaluza-Klein reduction from seven-manifolds with G_2-structure of generic intrinsic torsion. Lastly we give explicitly the intrinsic torsion of the SU(3)-structure for an N=1 supersymmetric background in the presence of nontrivial RR two-form field strength and nontrivial dilaton.