Generalising G2 geometry: involutivity, moment maps and moduli

Ashmore, Anthony; Strickland-Constable, Charles; Tennyson, David; Waldram, Daniel

doi:10.1007/jhep01(2021)158

Title: Generalising G₂ geometry: involutivity, moment maps and moduli

Abstract

We analyse the geometry of generic Minkowski $N$ = 1, $D$ = 4 flux compactifications in string theory, the default backgrounds for string model building. In M-theory they are the natural string theoretic extensions of G2 holonomy manifolds. In type II theories, they extend the notion of Calabi-Yau geometry and include the class of flux backgrounds based on generalised complex structures first considered by Graña et al. (GMPT). Using E₇₍₇₎× $$\mathbb{R}$$⁺generalised geometry we show that these compactifications are characterised by an SU(7) ⊂ E₇₍₇₎structure defining an involutive subbundle of the generalised tangent space, and with a vanishing moment map, corresponding to the action of the diffeomorphism and gauge symmetries of the theory. The Kähler potential on the space of structures defines a natural extension of Hitchin’s G₂functional. Using this framework we are able to count, for the first time, the massless scalar moduli of GMPT solutions in terms of generalised geometry cohomology groups. It also provides an intriguing new perspective on the existence of G₂manifolds, suggesting possible connections to Geometrical Invariant Theory and stability.

Authors:

Ashmore, Anthony ^[1]; Strickland-Constable, Charles ^[2]; Tennyson, David ^[3]; Waldram, Daniel ^[3]

Univ. of Pennsylvania, Philadelphia, PA (United States). Dept. of Physics & Astronomy
Univ. of Hertfordshire, Hatfield (United Kingdom). School of Physics, Astronomy and Mathematics
Imperial College, London (United Kingdom). Dept. of Physics

Publication Date:: Tue Jan 26 00:00:00 EST 2021

Research Org.:: Univ. of Pennsylvania, Philadelphia, PA (United States)

Sponsoring Org.:: USDOE Office of Science (SC); Science and Technology Facilities Council (STFC)

OSTI Identifier:: 1851030

Grant/Contract Number:: SC0007901; ST/P000762/1

Resource Type:: Accepted Manuscript

Journal Name:: Journal of High Energy Physics (Online)

Additional Journal Information:: Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2021; Journal Issue: 1; Journal ID: ISSN 1029-8479

Publisher:: Springer Nature

Country of Publication:: United States

Language:: English

Subject:: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Physics; Differential and Algebraic Geometry; Flux compactifications

Citation Formats


                    Ashmore, Anthony, Strickland-Constable, Charles, Tennyson, David, and Waldram, Daniel. Generalising G2 geometry: involutivity, moment maps and moduli.  United States: N. p., 2021. 
Web.  doi:10.1007/jhep01(2021)158.

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                    Ashmore, Anthony, Strickland-Constable, Charles, Tennyson, David, & Waldram, Daniel. Generalising G2 geometry: involutivity, moment maps and moduli.  United States.  https://doi.org/10.1007/jhep01(2021)158

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                    Ashmore, Anthony, Strickland-Constable, Charles, Tennyson, David, and Waldram, Daniel. Tue .  
"Generalising G2 geometry: involutivity, moment maps and moduli".  United States.  https://doi.org/10.1007/jhep01(2021)158.  https://www.osti.gov/servlets/purl/1851030.

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@article{osti_1851030,

  title        = {Generalising G2 geometry: involutivity, moment maps and moduli},

  author       = {Ashmore, Anthony and Strickland-Constable, Charles and Tennyson, David and Waldram, Daniel},

  abstractNote = {We analyse the geometry of generic Minkowski N = 1, D = 4 flux compactifications in string theory, the default backgrounds for string model building. In M-theory they are the natural string theoretic extensions of G2 holonomy manifolds. In type II theories, they extend the notion of Calabi-Yau geometry and include the class of flux backgrounds based on generalised complex structures first considered by Graña et al. (GMPT). Using E7(7)× $\mathbb{R}$+generalised geometry we show that these compactifications are characterised by an SU(7) ⊂ E7(7)structure defining an involutive subbundle of the generalised tangent space, and with a vanishing moment map, corresponding to the action of the diffeomorphism and gauge symmetries of the theory. The Kähler potential on the space of structures defines a natural extension of Hitchin’s G2functional. Using this framework we are able to count, for the first time, the massless scalar moduli of GMPT solutions in terms of generalised geometry cohomology groups. It also provides an intriguing new perspective on the existence of G2manifolds, suggesting possible connections to Geometrical Invariant Theory and stability.},

  doi          = {10.1007/jhep01(2021)158},

  journal      = {Journal of High Energy Physics (Online)},

  number       = 1,

  volume       = 2021,

  place        = {United States},

  year         = {Tue Jan 26 00:00:00 EST 2021},

  month        = {Tue Jan 26 00:00:00 EST 2021}

}

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Uhlenbeck, K.; Yau, S. T.
Communications on Pure and Applied Mathematics, Vol. 39, Issue S1
DOI: 10.1002/cpa.3160390714

A note on our previous paper: On the existence of Hermitian Yang–Mills connections in stable vector bundles
journal, July 1989

Uhlenbeck, K.; Yau, S. T.
Communications on Pure and Applied Mathematics, Vol. 42, Issue 5
DOI: 10.1002/cpa.3160420505

Supersymmetric AdS5 solutions of massive IIA supergravity
journal, June 2015

Apruzzi, Fabio; Fazzi, Marco; Passias, Achilleas
Journal of High Energy Physics, Vol. 2015, Issue 6
DOI: 10.1007/jhep06(2015)195

Supergravity as generalised geometry I: type II theories
journal, November 2011

Coimbra, André; Strickland-Constable, Charles; Waldram, Daniel
Journal of High Energy Physics, Vol. 2011, Issue 11
DOI: 10.1007/jhep11(2011)091

AdS 5 Solutions of Type IIB Supergravity and Generalized Complex Geometry
journal, June 2010

Gabella, Maxime; Gauntlett, Jerome P.; Palti, Eran
Communications in Mathematical Physics, Vol. 299, Issue 2
DOI: 10.1007/s00220-010-1083-y

Scherk-Schwarz reduction of M-theory on G ₂ -manifolds with fluxes
journal, October 2005

Dall'Agata, Gianguido; Prezas, Nikolaos
Journal of High Energy Physics, Vol. 2005, Issue 10
DOI: 10.1088/1126-6708/2005/10/103

On the geometry and the moduli space of β-deformed quiver gauge theories
journal, July 2008

Butti, Agostino; Forcella, Davide; Martucci, Luca
Journal of High Energy Physics, Vol. 2008, Issue 07
DOI: 10.1088/1126-6708/2008/07/053

Topological strings in generalized complex space
journal, January 2007

Pestun, Vasily
Advances in Theoretical and Mathematical Physics, Vol. 11, Issue 3
DOI: 10.4310/atmp.2007.v11.n3.a3

New supersymmetric flux vacua with intermediate SU(2) structure
text, January 2008

Andriot, David
arXiv
DOI: 10.48550/arxiv.0804.1769

Supersymmetric Backgrounds and Generalised Special Holonomy
text, January 2014

Coimbra, André; Strickland-Constable, Charles; Waldram, Daniel
arXiv
DOI: 10.48550/arxiv.1411.5721

Generalized geometric vacua with eight supercharges
text, January 2016

Graña, Mariana; Ntokos, Praxitelis
arXiv
DOI: 10.48550/arxiv.1605.06383

Fluxes in M-theory on 7-manifolds: G-structures and Superpotential
text, January 2003

Behrndt, Klaus; Jeschek, Claus
arXiv
DOI: 10.48550/arxiv.hep-th/0311119

N=1 geometries for M-theory and type IIA strings with fluxes
text, January 2003

Dall'Agata, Gianguido; Prezas, Nikolaos
arXiv
DOI: 10.48550/arxiv.hep-th/0311146

M-theory compactification, fluxes and AdS_4
text, January 2004

Lukas, Andre; Saffin, P. M.
arXiv
DOI: 10.48550/arxiv.hep-th/0403235

Generalized N=1 Orientifold Compactifications and the Hitchin functionals
text, January 2006

Benmachiche, Iman; Grimm, Thomas W.
arXiv
DOI: 10.48550/arxiv.hep-th/0602241

Gravity duals to deformed SYM theories and Generalized Complex Geometry
text, January 2006

Minasian, Ruben; Petrini, Michela; Zaffaroni, Alberto
arXiv
DOI: 10.48550/arxiv.hep-th/0606257

Deformations of calibrated D-branes in flux generalized complex manifolds
text, January 2006

Koerber, Paul; Martucci, Luca
arXiv
DOI: 10.48550/arxiv.hep-th/0610044

The geometry of three-forms in six and seven dimensions
preprint, January 2000

Hitchin, Nigel
arXiv
DOI: 10.48550/arxiv.math/0010054

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Title: Generalising G2 geometry: involutivity, moment maps and moduli

Abstract

Citation Formats

Topological M-theory as unification of form theories of gravity journal, January 2005

On the existence of hermitian-yang-mills connections in stable vector bundles journal, January 1986

Deformations of calibrated D-branes in flux generalized complex manifolds journal, December 2006

SU(3) × SU(3) compactification and mirror duals of magnetic fluxes journal, April 2007

Generalised geometry for M-theory journal, July 2007

M-theory, exceptional generalised geometry and superpotentials journal, September 2008

E 11 and the embedding tensor journal, September 2007

Generalised geometry for string corrections journal, November 2014

The α ′ expansion on a compact manifold of exceptional holonomy journal, June 2014

Hierarchies from fluxes in string compactifications journal, November 2002

Reformulating supersymmetry with a generalized Dolbeault operator journal, February 2008

Matter from G2 manifolds journal, October 2002

Higgs bundles for M-theory on G2-manifolds journal, March 2019

Generalised structures for N = 1 $$ \mathcal{N}=1 $$ AdS backgrounds journal, November 2016

Hyperkähler metrics and supersymmetry journal, December 1987

Unified field theories with U(N) internal symmetries: G journal, October 1977

The Infinitesimal Moduli Space of Heterotic G 2 Systems journal, November 2017

Solitons, superpotentials and calibrations journal, May 2000

Hitchin functionals in N = 2 supergravity journal, January 2006

Kahler independence of the G 2-MSSM journal, September 2010

Kählerian Coset Spaces of Semisimple Lie Groups journal, December 1954

Five-brane effective action in M-theory journal, May 1997

Properties of conformal supergravity journal, June 1978

Stability and BPS branes journal, September 2005

Gravity duals to deformed SYM theories and generalized complex geometry journal, December 2006

M-theory compactifications on manifolds with G 2 structure journal, April 2005

The Topological G<sub>2</sub> String journal, January 2008

Warped generalized geometry compactifications, effective theories and non-perturbative effects journal, August 2008

A Note on Fluxes and Superpotentials in M-theory Compactifications on Manifolds of G 2 Holonomy journal, July 2002

A note on our previous paper: On the existence of Hermitian Yang–Mills connections in stable vector bundles journal, July 1989

Generalized structures of Script N = 1 vacua journal, November 2005

Marginal deformations of heterotic G2 sigma models journal, February 2018

M theory, Joyce orbifolds and super Yang–Mills journal, January 1999

A scan for new Script N = 1 vacua on twisted tori journal, May 2007

Higher curvature Bianchi identities, generalized geometry, and L ∞ algebras journal, November 2019

Generalized orientifold compactifications and the Hitchin functionals journal, July 2006

Supersymmetric backgrounds from generalized Calabi-Yau manifolds journal, August 2004

G -Structures and Wrapped NS5-Branes journal, May 2004

E 7(7) formulation of N = 2 backgrounds journal, July 2009

Generalized Calabi-Yau Manifolds journal, September 2003

Supersymmetry breaking by fluxes journal, October 2018

Compactification and supersymmetry of chiral N = 2, D = 10 supergravity journal, January 1985

$ \mathcal{N} $ = 2 vacua in generalized geometry journal, November 2012

The Hitchin Functionals and the Topological B-Model at One Loop journal, October 2005

On the ricci curvature of a compact kähler manifold and the complex monge-ampére equation, I journal, May 1978

Local Geometry of the G 2 Moduli Space journal, August 2008

Generalized geometric vacua with eight supercharges journal, August 2016

$M$-theory dynamics on a manifold of $G_2$ holonomy journal, January 2002

From ten to four and back again: how to generalize the geometry journal, August 2007

Stable D-branes, calibrations and generalized Calabi-Yau geometry journal, August 2005

On the geometry and the moduli space of β-deformed quiver gauge theories journal, July 2008

An obstruction to the existence of Einstein K�hler metrics journal, October 1983

Deformation of Dirac Structures via L∞ Algebras journal, June 2018

Supersymmetric backgrounds and generalised special holonomy journal, May 2016

Scherk-Schwarz reduction of M-theory on G 2 -manifolds with fluxes journal, October 2005

New supersymmetric flux vacua with intermediate SU(2) structure journal, August 2008

A massive class of $$ \mathcal{N}=2 $$ AdS4 IIA solutions journal, October 2018

Topological Strings on Noncommutative Manifolds journal, April 2004

Moduli spaces and brane solitons for M-theory compactifications on holonomy G2 manifolds journal, November 2001

Poincarésupergravity as broken superconformal gravity journal, May 1978

Topological sigma-models with H-flux and twisted generalized complex manifolds journal, January 2007

Exceptional Calabi-Yau spaces: the geometry of N=2 backgrounds with flux: Exceptional Calabi-Yau spaces journal, January 2017

Anti Self-Dual Yang-Mills Connections Over Complex Algebraic Surfaces and Stable Vector Bundles journal, January 1985

Flux and Freund-Rubin superpotentials in M theory journal, June 2005

Supersymmetric M-theory compactifications with fluxes on seven-manifolds and G -structures journal, July 2003

$ {E_d}_{(d)}\times {{\mathbb{R}}^{+}} $ generalised geometry, connections and M theory journal, February 2014

Fluxes in M-theory on 7-manifolds: G-structures and superpotential journal, August 2004

Compactification and supersymmetry in d = 11 supergravity journal, December 1984

Gauged supergravities, tensor hierarchies, and M-theory journal, February 2008

Supersymmetric D-branes and calibrations on general Script N = 1 backgrounds journal, November 2005

Finiteness of Ricci flatN=2 supersymmetric σ-models journal, September 1986

$\mathrm{G}_{2}$ -manifolds and associative submanifolds via semi-Fano $3$ -folds journal, July 2015

Supersymmetric backgrounds, the Killing superalgebra, and generalised special holonomy journal, November 2016

Fine structure of reductive pseudo-K�hlerian spaces journal, September 1991

Compactification of D = 11 supergravity on spaces of exceptional holonomy journal, September 1995

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Title: Generalising G₂ geometry: involutivity, moment maps and moduli

Topological M-theory as unification of form theories of gravity
journal, January 2005

On the existence of hermitian-yang-mills connections in stable vector bundles
journal, January 1986

Deformations of calibrated D-branes in flux generalized complex manifolds
journal, December 2006

SU(3) × SU(3) compactification and mirror duals of magnetic fluxes
journal, April 2007

Generalised geometry for M-theory
journal, July 2007

M-theory, exceptional generalised geometry and superpotentials
journal, September 2008

E ₁₁ and the embedding tensor
journal, September 2007

Generalised geometry for string corrections
journal, November 2014

The α ′ expansion on a compact manifold of exceptional holonomy
journal, June 2014

Hierarchies from fluxes in string compactifications
journal, November 2002

Reformulating supersymmetry with a generalized Dolbeault operator
journal, February 2008

Matter from G2 manifolds
journal, October 2002

Higgs bundles for M-theory on G2-manifolds
journal, March 2019

Generalised structures for N = 1 $$ \mathcal{N}=1 $$ AdS backgrounds
journal, November 2016

Hyperkähler metrics and supersymmetry
journal, December 1987

Unified field theories with U(N) internal symmetries: G
journal, October 1977

The Infinitesimal Moduli Space of Heterotic G 2 Systems
journal, November 2017

Solitons, superpotentials and calibrations
journal, May 2000

Hitchin functionals in N = 2 supergravity
journal, January 2006

Kahler independence of the G 2-MSSM
journal, September 2010

Kählerian Coset Spaces of Semisimple Lie Groups
journal, December 1954

Five-brane effective action in M-theory
journal, May 1997

Properties of conformal supergravity
journal, June 1978

Stability and BPS branes
journal, September 2005

Gravity duals to deformed SYM theories and generalized complex geometry
journal, December 2006

M-theory compactifications on manifolds with G ₂ structure
journal, April 2005

The Topological G<sub>2</sub> String
journal, January 2008

Warped generalized geometry compactifications, effective theories and non-perturbative effects
journal, August 2008

A Note on Fluxes and Superpotentials in M-theory Compactifications on Manifolds of G ₂ Holonomy
journal, July 2002

A note on our previous paper: On the existence of Hermitian Yang–Mills connections in stable vector bundles
journal, July 1989

Generalized structures of Script N = 1 vacua
journal, November 2005

Marginal deformations of heterotic G2 sigma models
journal, February 2018

M theory, Joyce orbifolds and super Yang–Mills
journal, January 1999

A scan for new Script N = 1 vacua on twisted tori
journal, May 2007

Higher curvature Bianchi identities, generalized geometry, and $L_{\infty}$ algebras
journal, November 2019

Generalized orientifold compactifications and the Hitchin functionals
journal, July 2006

Supersymmetric backgrounds from generalized Calabi-Yau manifolds
journal, August 2004

G -Structures and Wrapped NS5-Branes
journal, May 2004

E ₇₍₇₎ formulation of N = 2 backgrounds
journal, July 2009

Generalized Calabi-Yau Manifolds
journal, September 2003

Supersymmetry breaking by fluxes
journal, October 2018

Compactification and supersymmetry of chiral N = 2, D = 10 supergravity
journal, January 1985

$ \mathcal{N} $ = 2 vacua in generalized geometry
journal, November 2012

The Hitchin Functionals and the Topological B-Model at One Loop
journal, October 2005

On the ricci curvature of a compact kähler manifold and the complex monge-ampére equation, I
journal, May 1978

Local Geometry of the G 2 Moduli Space
journal, August 2008

Generalized geometric vacua with eight supercharges
journal, August 2016

$M$-theory dynamics on a manifold of $G_2$ holonomy
journal, January 2002

From ten to four and back again: how to generalize the geometry
journal, August 2007

Stable D-branes, calibrations and generalized Calabi-Yau geometry
journal, August 2005

On the geometry and the moduli space of β-deformed quiver gauge theories
journal, July 2008

An obstruction to the existence of Einstein K�hler metrics
journal, October 1983

Deformation of Dirac Structures via L∞ Algebras
journal, June 2018

Supersymmetric backgrounds and generalised special holonomy
journal, May 2016

Scherk-Schwarz reduction of M-theory on G ₂ -manifolds with fluxes
journal, October 2005

New supersymmetric flux vacua with intermediate SU(2) structure
journal, August 2008

A massive class of $$ \mathcal{N}=2 $$ AdS4 IIA solutions
journal, October 2018

Topological Strings on Noncommutative Manifolds
journal, April 2004

Moduli spaces and brane solitons for M-theory compactifications on holonomy G2 manifolds
journal, November 2001

Poincarésupergravity as broken superconformal gravity
journal, May 1978

Topological sigma-models with H-flux and twisted generalized complex manifolds
journal, January 2007

Exceptional Calabi-Yau spaces: the geometry of N=2 backgrounds with flux: Exceptional Calabi-Yau spaces
journal, January 2017

Anti Self-Dual Yang-Mills Connections Over Complex Algebraic Surfaces and Stable Vector Bundles
journal, January 1985

Flux and Freund-Rubin superpotentials in M theory
journal, June 2005

Supersymmetric M-theory compactifications with fluxes on seven-manifolds and G -structures
journal, July 2003

$ {E_d}_{(d)}\times {{\mathbb{R}}^{+}} $ generalised geometry, connections and M theory
journal, February 2014

Fluxes in M-theory on 7-manifolds: G-structures and superpotential
journal, August 2004

Compactification and supersymmetry in d = 11 supergravity
journal, December 1984

Gauged supergravities, tensor hierarchies, and M-theory
journal, February 2008

Supersymmetric D-branes and calibrations on general Script N = 1 backgrounds
journal, November 2005

Finiteness of Ricci flatN=2 supersymmetric σ-models
journal, September 1986

$\mathrm{G}_{2}$ -manifolds and associative submanifolds via semi-Fano $3$ -folds
journal, July 2015

Supersymmetric backgrounds, the Killing superalgebra, and generalised special holonomy
journal, November 2016

Fine structure of reductive pseudo-K�hlerian spaces
journal, September 1991

Compactification of D = 11 supergravity on spaces of exceptional holonomy
journal, September 1995

All AdS7 solutions of type II supergravity
journal, April 2014

Supergravity Description of Field Theories on Curved Manifolds and a no go Theorem
journal, February 2001

New supersymmetric vacua on solvmanifolds
journal, February 2016

AdS 5 Solutions of Type IIB Supergravity and Generalized Complex Geometry
journal, June 2010