Flow Equations for Uplifting Half-Flat to Spin(7) Manifolds

TL;DR

This paper develops flow equations to uplift half-flat six-folds to Spin(7) eight-folds via fibrations, explicitly demonstrates the uplift of the Iwasawa manifold, and explores potential inclusion of elliptic functions in the deformation of SU(3) structure manifolds.

Contribution

It introduces generalized flow equations for uplifting half-flat manifolds to Spin(7) structures and explicitly constructs new Spin(7) metrics from known examples.

Findings

Explicit uplift of the Iwasawa manifold to Spin(7) using two methods.

Proposal of new Spin(7) metrics based on these uplift techniques.

Discussion on including elliptic functions in deformation functions of SU(3) structure manifolds.

Abstract

In this short supplement to [1], we discuss the uplift of half-flat six-folds to Spin(7) eight-folds by fibration of the former over a product of two intervals. We show that the same can be done in two ways - one, such that the required Spin(7) eight-fold is a double G_2 seven-fold fibration over an interval, the G_2 seven-fold itself being the half-flat six-fold fibered over the other interval, and second, by simply considering the fibration of the half-flat six-fold over a product of two intervals. The flow equations one gets are an obvious generalization of the Hitchin's flow equations (to obtain seven-folds of G_2 holonomy from half-flat six-folds [2]). We explicitly show the uplift of the Iwasawa using both methods, thereby proposing the form of the new Spin(7) metrics. We give a plausibility argument ruling out the uplift of the Iwasawa manifold to a Spin(7) eight fold at the "edge", using the second method. For $Spin(7)$ eight-folds of the type $X_7\times S^1$, $X_7$ being a seven-fold of SU(3) structure, we motivate the possibility of including elliptic functions into the "shape deformation" functions of seven-folds of SU(3) structure of [1] via some connections between elliptic functions, the Heisenberg group, theta functions, the already known $D7$-brane metric [3] and hyper-K\"{a}hler metrics obtained in twistor spaces by deformations of Atiyah-Hitchin manifolds by a Legendre transform in [4].