Abstract
We consider LieG-valued Yang-Mills fields on the space \( \mathbb{R} \times {{G} \left/ {H} \right.} \), where G/H is a compact nearly Kähler six-dimensional homogeneous space, and the manifold \( \mathbb{R} \times {{G} \left/ {H} \right.} \) carries a G 2-structure. After imposing a general G-invariance condition, Yang-Mills theory with torsion on \( \mathbb{R} \times {{G} \left/ {H} \right.} \) is reduced to Newtonian mechanics of a particle moving in \( {\mathbb{R}^6} \), \( {\mathbb{R}^4} \) or \( {\mathbb{R}^2} \) under the influence of an inverted double-well-type potential for the cases G/H = SU(3)/ U(1)×U(1), Sp(2)/ Sp(1)×U(1) or G2/SU(3), respectively. We analyze all critical points and present analytical and numerical kink-and bounce-type solutions, which yield G-invariant instanton configurations on those cosets. Periodic solutions on S 1×G/H and dyons on \( i\mathbb{R} \times {{G} \left/ {H} \right.} \) are also given.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M.B. Green, J.H. Schwarz and E. Witten, Superstring theory, Cambridge University Press, Cambridge U.K. (1987).
E. Corrigan, C. Devchand, D.B. Fairlie and J. Nuyts, First Order Equations for Gauge Fields in Spaces of Dimension Greater Than Four, Nucl. Phys. B 214 (1983) 452 [SPIRES].
R.S. Ward, Completely Solvable Gauge Field Equations in Dimension Greater Than Four, Nucl. Phys. B 236 (1984) 381 [SPIRES].
S.K. Donaldson, Anti-self-dual Yang-Mills connections on a complex algebraic surface and stable vector bundles, Proc. Lond. Math. Soc. 50 (1985) 1.
S.K. Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. J. 54 (1987) 231.
K.K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian-Yang-Mills connections on stable bundles over compact Kähler manifolds, Comm. Pure Appl. Math. 39 (1986) 257.
K.K. Uhlenbeck and S.-T. Yau, A note on our previous paper: On the existence of Hermitian YangMills connections in stable vector bundles, Comm. Pure Appl. Math. 42 (1989) 703.
M. Mamone Capria and S.M. Salamon, Yang-Millsfifields on quaternionic spaces, Nonlinearity 1 (1988) 517.
R. Reyes Carrión, A generalization of the notion of instanton, Differ. Geom. Appl. 8 (1998) 1 [SPIRES].
L. Baulieu, H. Kanno and I.M. Singer, Special quantum field theories in eight and other dimensions, Commun. Math. Phys. 194 (1998) 149 [hep-th/9704167] [SPIRES].
G. Tian, Gauge theory and calibrated geometry. I, Annals Math. 151 (2000) 193 [math/0010015].
T. Tao and G. Tian, A singularity removal theorem for Yang-Mills fields in higher dimensions, J. Amer. Math. Soc. 17 (2004) 557.
S.K. Donaldson and R.P. Thomas, Gauge theory in higher dimensions, in The Geometric Universe, Oxford University Press, Oxford U.K. (1998).
S. Donaldson and E. Segal, Gauge Theory in higher dimensions, II, arXiv:0902.3239 [SPIRES].
A.D. Popov, Non-Abelian Vortices, super-Yang-Mills Theory and Spin(7)-Instantons, Lett. Math. Phys. 92 (2010) 253 [arXiv:0908.3055] [SPIRES].
D. Harland and A.D. Popov, Yang-Mills fields in flux compactifications on homogeneous manifolds with SU(4)-structure, arXiv:1005.2837 [SPIRES].
D.B. Fairlie and J. Nuyts, Spherically symmetric solutions of gauge theories in eight dimensions, J. Phys. A 17 (1984) 2867 [SPIRES].
S. Fubini and H. Nicolai, The octonionic instanton, Phys. Lett. B 155 (1985) 369 [SPIRES].
T.A. Ivanova and A.D. Popov, Selfdual Yang-Mills fields in D =7, 8, octonions and Ward equations, Lett. Math. Phys. 24 (1992) 85 [SPIRES].
T.A. Ivanova and A.D. Popov, (Anti)selfdual gauge fields in dimension d≥4, Theor. Math. Phys. 94 (1993) 225 [SPIRES].
T.A. Ivanova and O. Lechtenfeld, Yang-Mills Instantons and Dyons on Group Manifolds, Phys. Lett. B 670 (2008) 91 [arXiv:0806.0394] [SPIRES].
T.A. Ivanova, O. Lechtenfeld, A.D. Popov and T. Rahn, Instantons and Yang-Mills Flows on Coset Spaces, Lett. Math. Phys. 89 (2009) 231 [arXiv:0904.0654] [SPIRES].
T. Rahn, Yang-Mills Equations of Motion for the Higgs Sector of SU(3)-Equivariant Quiver Gauge Theories, J. Math. Phys. 51 (2010) 072302 [arXiv:0908.4275] [SPIRES].
D. Harland, T.A. Ivanova, O. Lechtenfeld and A.D. Popov, Yang-Mills flows on nearly Kähler manifolds and G 2 -instantons, Commun. Math. Phys. 300 (2010) 185 [arXiv:0909.2730] [SPIRES].
M. Graña, Flux compactifications in string theory: A comprehensive review, Phys. Rept. 423 (2006) 91 [hep-th/0509003] [SPIRES].
M.R. Douglas and S. Kachru, Flux compactification, Rev. Mod. Phys. 79 (2007) 733 [hep-th/0610102] [SPIRES].
R. Blumenhagen, B. Körs, D. Lüst and S. Stieberger, Four-dimensional String Compactifications with D-branes, Orientifolds and Fluxes, Phys. Rept. 445 (2007) 1 [hep-th/0610327] [SPIRES].
A. Strominger, Superstrings with Torsion, Nucl. Phys. B 274 (1986) 253 [SPIRES].
C.M. Hull, Anomalies, ambiguities and superstrings, Phys. Lett. B 167 (1986) 51 [SPIRES].
C.M. Hull, Compactifications of the heterotic superstring, Phys. Lett. B 178 (1986) 357 [SPIRES].
D. Lüst, Compactification of ten-dimensional superstring theories over Ricci flat coset spaces, Nucl. Phys. B 276 (1986) 220 [SPIRES].
B. de Wit, D.J. Smit and N.D. Hari Dass, Residual Supersymmetry of Compactified D = 10 Supergravity, Nucl. Phys. B 283 (1987) 165 [SPIRES].
J.-B. Butruille, Homogeneous nearly Kähler manifolds, math/0612655.
F. Xu, SU(3)-structures and special lagrangian geometries, math/0610532.
A. Tomasiello, New string vacua from twistor spaces, Phys. Rev. D 78 (2008) 046007 [arXiv:0712.1396] [SPIRES].
C. Caviezel et al., The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets, Class. Quant. Grav. 26 (2009) 025014 [arXiv:0806.3458] [SPIRES].
A.D. Popov, Hermitian-Yang-Mills equations and pseudo-holomorphic bundles on nearly Kähler and nearly Calabi-Yau twistor 6-manifolds, Nucl. Phys. B 828 (2010) 594 [arXiv:0907.0106] [SPIRES].
A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Y.S. Tyupkin, Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B 59 (1975) 85 [SPIRES].
R. Rajaraman, Solitons and instantons, North-Holland, Amsterdam Netherlands (1984).
N. Manton and P. Sutcliffe, Topological solitons, Cambridge University Press, Cambridge U.K. (2004).
J.-X. Fu, L.-S. Tseng and S.-T. Yau, Local Heterotic Torsional Models, Commun. Math. Phys. 289 (2009) 1151 [arXiv:0806.2392] [SPIRES].
M. Becker, L.-S. Tseng and S.-T. Yau, New Heterotic Non-Kähler Geometries, arXiv:0807.0827 [SPIRES].
K. Becker and S. Sethi, Torsional Heterotic Geometries, Nucl. Phys. B 820 (2009) 1 [arXiv:0903.3769] [SPIRES].
I. Benmachiche, J. Louis and D. Martinez-Pedrera, The effective action of the heterotic string compactified on manifolds with SU(3) structure, Class. Quant. Grav. 25 (2008) 135006 [arXiv:0802.0410] [SPIRES].
M. Fernandez, S. Ivanov, L. Ugarte and R. Villacampa, Non-Kähler Heterotic String Compactifications with non-zero fluxes and constant dilaton, Commun. Math. Phys. 288 (2009) 677 [arXiv:0804.1648] [SPIRES].
G. Papadopoulos, New half supersymmetric solutions of the heterotic string, Class. Quant. Grav. 26 (2009) 135001 [arXiv:0809.1156] [SPIRES].
H. Kunitomo and M. Ohta, Supersymmetric AdS 3 solutions in Heterotic Supergravity, Prog. Theor. Phys. 122 (2009) 631 [arXiv:0902.0655] [SPIRES].
G. Douzas, T. Grammatikopoulos and G. Zoupanos, Coset Space Dimensional Reduction and Wilson Flux Breaking of Ten-Dimensional N =1, E 8 Gauge Theory, Eur. Phys. J. C 59 (2009) 917 [arXiv:0808.3236] [SPIRES].
A. Chatzistavrakidis and G. Zoupanos, Dimensional Reduction of the Heterotic String over nearly-Kähler manifolds, JHEP 09 (2009) 077 [arXiv:0905.2398] [SPIRES].
A. Chatzistavrakidis, P. Manousselis and G. Zoupanos, Reducing the Heterotic Supergravity on nearly-Kähler coset spaces, Fortschr. Phys. 57 (2009) 527 [arXiv:0811.2182] [SPIRES].
S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. 1, Interscience Publishers, New York U.S.A. (1963).
Y.A. Kubyshin, I.P. Volobuev, J.M. Mourao and G. Rudolph, Dimensional reduction of gauge theories, spontaneous compactification and model building, Lect. Notes Phys. 349 (1990) 1 [SPIRES].
D. Kapetanakis and G. Zoupanos, Coset space dimensional reduction of gauge theories, Phys. Rept. 219 (1992) 1 [SPIRES].
O. Lechtenfeld, A.D. Popov and R.J. Szabo, Quiver gauge theory and noncommutative vortices, Prog. Theor. Phys. Suppl. 171 (2007) 258 [arXiv:0706.0979] [SPIRES].
O. Lechtenfeld, A.D. Popov and R.J. Szabo, SU(3)-Equivariant Quiver Gauge Theories and Nonabelian Vortices, JHEP 08 (2008) 093 [arXiv:0806.2791] [SPIRES].
S. Chiossi and S. Salamon, The intrinsic torsion of SU(3) and G 2 structures, math/0202282 [SPIRES].
S.J. Avis and C.J. Isham, Vacuum solutions for a twisted scalar field, Proc. Roy. Soc. Lond. A 363 (1978) 581 [SPIRES].
N.S. Manton and T.M. Samols, Sphalerons on a circle, Phys. Lett. B 207 (1988) 179 [SPIRES].
J.-Q. Liang, H.J.W. Muller-Kirsten and D.H. Tchrakian, Solitons, bounces and sphalerons on a circle, Phys. Lett. B 282 (1992) 105 [SPIRES].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1006.2388
Rights and permissions
About this article
Cite this article
Bauer, I., Ivanova, T.A., Lechtenfeld, O. et al. Yang-Mills instantons and dyons on homogeneous G 2-manifolds. J. High Energ. Phys. 2010, 44 (2010). https://doi.org/10.1007/JHEP10(2010)044
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2010)044