Uplifting the Iwasawa

TL;DR

This paper constructs explicit metrics for uplifting the Iwasawa manifold to seven-dimensional manifolds with G_2 holonomy or SU(3) structure, exploring deformation types and their geometric implications.

Contribution

It provides explicit new metrics for uplifting the Iwasawa manifold to G_2 and SU(3) structures using Hitchin's Flow equations, and analyzes deformation constraints.

Findings

Uplift to G_2 manifold via size deformation is possible.

Uplift to SU(3) structure involves specific size and shape deformations.

Certain deformations prevent uplift while retaining standard complex structure.

Abstract

The Iwasawa manifold is uplifted to seven-folds of either G_2 holonomy or SU(3) structure, explicit new metrics for the same having been constructed in this work. We uplift the Iwasawa manifold to a G_2 manifold through "size" deformation (of the Iwasawa metric), via Hitchin's Flow equations, showing also the impossibility of the uplift for "shape" and "size" deformations (of the Iwasawa metric). Using results of [1], we also uplift the Iwasawa manifold to a 7-fold with SU(3) structure through "size" and "shape" deformations via generalisation of Hitchin's Flow equations. For seven-folds with SU(3)-structure, the result could be interpreted as M5-branes wrapping two-cycles embedded in the seven-fold - a warped product of either a special hermitian six-fold or a balanced six-fold with the unit interval. There can be no uplift to seven-folds of SU(3) structure involving non-trivial "size" and "shape" deformations (of the Iwasawa metric) retaining the "standard complex structure" - the uplift generically makes one move in the space of almost complex structures such that one is neither at the standard complex structure point nor at the "edge". Using the results of [2] we show that given two "shape deformation" functions, and the dilaton, one can construct a Riemann surface obtained via Weierstrass representation for the conformal immersion of a surface in R^l, for a suitable l, with the condition of having conformal immersion being a quadric in CP^{l-1}.