Flow equations for uplifting halfflat to Spin(7) manifolds
Abstract
In this supplement to the paper by Franzen et al. [Fortschr. Phys. (to be published)], we discuss the uplift of halfflat sixfolds to Spin(7) eightfolds 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) eightfold is a double G_{2} sevenfold fibration over an interval, the G_{2} sevenfold itself being the halfflat sixfold fibered over the other interval, and second, by simply considering the fibration of the halfflat sixfold over a product of two intervals. The flow equations one gets are an obvious generalization of the Hitchin's flow equations [to obtain sevenfolds of G_{2} holonomy from halfflat sixfolds [Hitchin (2001)]]. We explicitly show the uplift of the Iwasawa using both methods, thereby proposing the form of new Spin(7) metrics. We give a plausibility argument ruling out the uplift of the Iwasawa manifold to a Spin(7) eightfold at the "edge," using the second method. For Spin(7) eightfolds of the type X_{7}×S^{1}, X_{7} being a sevenfold of SU(3) structure, we motivate the possibility of including elliptic functions into the "shape deformation" functions of sevenfolds of SU(3) structure of Franzen et al. via some connections between elliptic functions, the Heisenberg group, theta functions, the already known D7brane metric [Greene et al., Nucl. Phys. B 337, 1 (1990)], and hyperKähler metrics obtained in twistor spaces by deformations of AtiyahHitchin manifolds by a Legendre transform [Chalmers, Phys. Rev. D 58, 125011 (1998)].
 Publication:

Journal of Mathematical Physics
 Pub Date:
 March 2006
 DOI:
 10.1063/1.2178156
 arXiv:
 arXiv:hepth/0507147
 Bibcode:
 2006JMP....47c3504M
 Keywords:

 11.30.Hv;
 11.27.+d;
 02.30.Ks;
 02.10.De;
 02.20.Qs;
 Flavor symmetries;
 Extended classical solutions;
 cosmic strings domain walls texture;
 Delay and functional equations;
 Algebraic structures and number theory;
 General properties structure and representation of Lie groups;
 High Energy Physics  Theory
 EPrint:
 12 pages, LaTeX