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 waysone, 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 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{sub 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{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; SU3 GROUPS
Citation Formats
Misra, Aalok, and Jefferson Physical Laboratory Harvard University, Cambridge, Massachusetts 02138. Flow equations for uplifting halfflat 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 halfflat 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 halfflat to Spin(7) manifolds". United States.
doi:10.1063/1.2178156.
@article{osti_20768744,
title = {Flow equations for uplifting halfflat 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 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 waysone, 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 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{sub 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{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 D7brane metric [Greene et al., Nucl. Phys. B 337, 1 (1990)], and hyperKaehler metrics obtained in twistor spaces by deformations of AtiyahHitchin 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}
}

We study the effective action of the heterotic string compactified on particular halfflat manifolds which arise in the context of mirror symmetry with NeveuSchwarzNeveuSchwarz flux. We explicitly derive the superpotential and Kaehler potential at lowest order in {alpha}{sup '} by a reduction of the bosonic action. The superpotential contains new terms depending on the Kaehler moduli which originate from the intrinsic geometrical flux of the halfflat manifolds. A generalized Gukov formula, valid for all manifolds with SU(3) structure, is derived from the gravitino mass term. For the halfflat manifolds it leads to a superpotential in agreement with our explicit bosonicmore »

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