# Flow equations for uplifting half-flat 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 half-flat 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{sub 2} sevenfold fibration over an interval, the G{sub 2} sevenfold itself being the half-flat sixfold fibered over the other interval, and second, by simply considering the fibration of the half-flat 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{sub 2} holonomy from half-flat 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{sub 7}xS{sup 1}, X{sub 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 Franzenmore »

- Authors:

- Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247 667 (India)
- (United States)

- Publication Date:

- OSTI Identifier:
- 20768744

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 47; Journal Issue: 3; Other Information: DOI: 10.1063/1.2178156; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DEFORMATION; EQUATIONS; LEGENDRE POLYNOMIALS; MATHEMATICAL MANIFOLDS; MEMBRANES; METRICS; SPIN; SU-3 GROUPS

### Citation Formats

```
Misra, Aalok, and Jefferson Physical Laboratory Harvard University, Cambridge, Massachusetts 02138.
```*Flow equations for uplifting half-flat to Spin(7) manifolds*. United States: N. p., 2006.
Web. doi:10.1063/1.2178156.

```
Misra, Aalok, & Jefferson Physical Laboratory Harvard University, Cambridge, Massachusetts 02138.
```*Flow equations for uplifting half-flat to Spin(7) manifolds*. United States. doi:10.1063/1.2178156.

```
Misra, Aalok, and Jefferson Physical Laboratory Harvard University, Cambridge, Massachusetts 02138. Wed .
"Flow equations for uplifting half-flat to Spin(7) manifolds". United States.
doi:10.1063/1.2178156.
```

```
@article{osti_20768744,
```

title = {Flow equations for uplifting half-flat to Spin(7) manifolds},

author = {Misra, Aalok and Jefferson Physical Laboratory Harvard University, Cambridge, Massachusetts 02138},

abstractNote = {In this supplement to the paper by Franzen et al. [Fortschr. Phys. (to be published)], we discuss the uplift of half-flat 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{sub 2} sevenfold fibration over an interval, the G{sub 2} sevenfold itself being the half-flat sixfold fibered over the other interval, and second, by simply considering the fibration of the half-flat 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{sub 2} holonomy from half-flat 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{sub 7}xS{sup 1}, X{sub 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 D7-brane metric [Greene et al., Nucl. Phys. B 337, 1 (1990)], and hyper-Kaehler metrics obtained in twistor spaces by deformations of Atiyah-Hitchin manifolds by a Legendre transform [Chalmers, Phys. Rev. D 58, 125011 (1998)].},

doi = {10.1063/1.2178156},

journal = {Journal of Mathematical Physics},

number = 3,

volume = 47,

place = {United States},

year = {Wed Mar 15 00:00:00 EST 2006},

month = {Wed Mar 15 00:00:00 EST 2006}

}