Abstract
We investigate instantons on manifolds with Killing spinors and their cones. Examples of manifolds with Killing spinors include nearly Kähler 6-manifolds, nearly parallel G 2-manifolds in dimension 7, Sasaki-Einstein manifolds, and 3-Sasakian manifolds. We construct a connection on the tangent bundle over these manifolds which solves the instanton equation, and also show that the instanton equation implies the Yang-Mills equation, despite the presence of torsion. We then construct instantons on the cones over these manifolds, and lift them to solutions of heterotic supergravity. Amongst our solutions are new instantons on even-dimensional Euclidean spaces, as well as the well-known BPST, quaternionic and octonionic instantons.
References
E. Corrigan, C. Devchand, D. Fairlie and J. Nuyts, First order equations for gauge fields in spaces of dimension greater than four, Nucl. Phys. B 214 (1983) 452 [INSPIRE].
R.S. Ward, Completely solvable gauge field equations in dimension greater than four, Nucl. Phys. B 236 (1984) 381 [INSPIRE].
S.K. Donaldson and R.P. Thomas, Gauge theory in higher dimensions, in The geometric universe, S. Hugett et al. eds., Oxford University Press, Oxford U.K. (1998).
S. Donaldson and E. Segal, Gauge theory in higher dimensions, II, arXiv:0902.3239 [INSPIRE].
G. Tian, Gauge theory and calibrated geometry, Ann. Math. 151 (2000) 193 [math/0010015].
D. Fairlie and J. Nuyts, Spherically symmetric solutions of gauge theories in eight-dimensions, J. Phys. A 17 (1984) 2867 [INSPIRE].
S. Fubini and H. Nicolai, The octonionic instanton, Phys. Lett. B 155 (1985) 369 [INSPIRE].
E. Corrigan, P. Goddard and A. Kent, Some comments on the ADHM construction in 4k dimensions, Commun. Math. Phys. 100 (1985) 1.
T. Ivanova and A. Popov, Selfdual Yang-Mills fields in D = 7, 8, octonions and Ward equations, Lett. Math. Phys. 24 (1992) 85 [INSPIRE].
T. Ivanova and A. Popov, (Anti)selfdual gauge fields in dimension d ≥ 4, Theor. Math. Phys. 94 (1993) 225 [INSPIRE].
J. Broedel, T.A. Ivanova and O. Lechtenfeld, Construction of noncommutative instantons in 4k dimensions, Mod. Phys. Lett. A 23 (2008) 179 [hep-th/0703009] [INSPIRE].
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] [INSPIRE].
D. Harland and A.D. Popov, Yang-Mills fields in flux compactifications on homogeneous manifolds with SU(4)-structure, JHEP 02 (2012) 107 [arXiv:1005.2837] [INSPIRE].
A.S. Haupt, T.A. Ivanova, O. Lechtenfeld and A.D. Popov, Chern-Simons flows on Aloff-Wallach spaces and Spin(7)-instantons, Phys. Rev. D 83 (2011) 105028 [arXiv:1104.5231] [INSPIRE].
K.-P. Gemmer, O. Lechtenfeld, C. Nölle and A.D. Popov, Yang-Mills instantons on cones and sine-cones over nearly Kähler manifolds, JHEP 09 (2011) 103 [arXiv:1108.3951] [INSPIRE].
F. Xu, On instantons on nearly Kähler 6-manifolds, Asian J. Math. 13 (2009) 535.
F.P. Correia, Hermitian Yang-Mills instantons on Calabi-Yau cones, JHEP 12 (2009) 004 [arXiv:0910.1096] [INSPIRE].
F.P. Correia, Hermitian Yang-Mills instantons on resolutions of Calabi-Yau cones, JHEP 02 (2011) 054 [arXiv:1009.0526] [INSPIRE].
B.S. Acharya, J. Figueroa-O’Farrill, C. Hull and B.J. Spence, Branes at conical singularities and holography, Adv. Theor. Math. Phys. 2 (1999) 1249 [hep-th/9808014] [INSPIRE].
P. Koerber, D. Lüst and D. Tsimpis, Type IIA AdS 4 compactifications on cosets, interpolations and domain walls, JHEP 07 (2008) 017 [arXiv:0804.0614] [INSPIRE].
C. Bär, Real Killing spinors and holonomy, Commun. Math. Phys. 154 (1993) 509.
J.P. Gauntlett and O. Varela, Consistent Kaluza-Klein reductions for general supersymmetric AdS solutions, Phys. Rev. D 76 (2007) 126007 [arXiv:0707.2315] [INSPIRE].
A. Strominger, Heterotic solitons, Nucl. Phys. B 343 (1990) 167 [Erratum ibid. B 353 (1991) 565] [INSPIRE].
J.A. Harvey and A. Strominger, Octonionic superstring solitons, Phys. Rev. Lett. 66 (1991) 549 [INSPIRE].
M. Günaydin and H. Nicolai, Seven-dimensional octonionic Yang-Mills instanton and its extension to an heterotic string soliton, Phys. Lett. B 351 (1995) 169 [Addendum ibid. B 376 (1996) 329] [hep-th/9502009] [INSPIRE].
A. Belavin, A.M. Polyakov, A. Schwartz and Y. Tyupkin, Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B 59 (1975) 85 [INSPIRE].
C. Nölle, Homogeneous heterotic supergravity solutions with linear dilaton, J. Phys. A A 45 (2012) 045402 [arXiv:1011.2873] [INSPIRE].
C.G. Callan Jr., J.A. Harvey and A. Strominger, World sheet approach to heterotic instantons and solitons, Nucl. Phys. B 359 (1991) 611 [INSPIRE].
C.G. Callan Jr., J.A. Harvey and A. Strominger, Supersymmetric string solitons, hep-th/9112030 [INSPIRE].
O. Hijazi, A conformal lower bound on the smallest eigenvalue of the Dirac operator and Killing spinors, Commun. Math. Phys. 104 (1986) 151.
H. Baum, T. Friedrich, R. Grunewald and I. Kath, Twistor and Killing spinors on Riemannian manifolds, Teubner-Verlag, Germany (1991).
S.K. Donaldson, Anti-self-dual Yang-Mills connections over complex algebraic surfaces 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 in stable vector bundles, Commun. Pure Appl. Math. 39 (1986) 257.
K.K. Uhlenbeck and S.-T. Yau, A note on our previous paper, Commun. Pure Appl. Math. 42 (1989) 703.
T. Friedrich and S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math. 6 (2002) 303 [math/0102142].
I. Agricola, The Srni lectures on non-integrable geometries with torsion, Arch. Math. 42 (2006) 5 [math/0606705].
T. Friedrich, I. Kath, A. Moroianu and U. Semmelmann, On nearly parallel G2-structures, J. Geom. Phys. 23 (1997) 259.
W. Ziller, Examples of riemannian manifolds with non-negative sectional curvature, math/0701389.
L. Verdiani and W. Ziller, Positively curved homogeneous metrics on spheres, Math. Z. 261 (2009) 473 [arXiv:0707.3056] [INSPIRE].
O. Dearricott, Positive sectional curvature on 3-Sasakian manifolds, Ann. Global Anal. Geom. 25 (2004) 59.
C.P. Boyer and K. Galicki, Sasakian geometry, Oxford University Press, Oxford U.K. (2008).
J.B. Butruille, Homogeneous nearly Kähler manifolds, in Handbook of pseudo-riemannian geometry and supersymmetry, V. Cortés ed., European Mathematical Society (2010), math/0612655.
M. Fernández, S. Ivanov, V. Muñoz and L. Ugarte, Nearly hypo structures and compact nearly Kähler 6-manifolds with conical singularities, math/0602160.
J. Sparks, Sasaki-Einstein Manifolds, Surveys Diff. Geom. 16 (2011) 265 [arXiv:1004.2461] [INSPIRE].
D. Fabbri et al., 3D superconformal theories from Sasakian seven manifolds: new nontrivial evidences for AdS 4 /CF T 3, Nucl. Phys. B 577 (2000) 547 [hep-th/9907219] [INSPIRE].
L. Castellani, R. D’Auria and P. Fré, SU(3) × SU(2) × U(1) from D = 11 supergravity, Nucl. Phys. B 239 (1984) 610 [INSPIRE].
J.P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Sasaki-Einstein metrics on S 2 × S 3, Adv. Theor. Math. Phys. 8 (2004) 711 [hep-th/0403002].
C.P. Boyer and K. Galicki, Sasakian geometry and Einstein metrics on spheres, CRM Proc. Lecture Notes 40 (2006) 47 [math/0505221].
C.P. Boyer and K. Galicki, Einstein metrics on rational homology spheres, J. Diff. Geom. 74 (2006) 353 [math/0311355].
J.P. Gauntlett, D. Martelli, J.F. Sparks and D. Waldram, A new infinite class of Sasaki-Einstein manifolds, Adv. Theor. Math. Phys. 8 (2006) 987 [hep-th/0403038] [INSPIRE].
M. Cvetič, H. Lü, D.N. Page and C. Pope, New Einstein-Sasaki spaces in five and higher dimensions, Phys. Rev. Lett. 95 (2005) 071101 [hep-th/0504225] [INSPIRE].
H. Lü, C. Pope and J.F. Vazquez-Poritz, A new construction of Einstein-Sasaki metrics in D ≥ 7,Phys. Rev. D 75 (2007) 026005 [hep-th/0512306] [INSPIRE].
I. Agricola and T. Friedrich, 3-Sasakian manifolds in dimension seven, their spinors and G 2 structures, J. Geom. Phys. 60 (2010) 326 [arXiv:0812.1651].
C.P. Boyer and K. Galicki, 3-Sasakian manifolds, Surveys Diff. Geom. 7 (1999) 123 [hep-th/9810250] [INSPIRE].
J.H. Eschenburg, New examples of manifolds with strictly positive curvature, Invent. Math. 66 (1982) 469.
C. LeBrun and S. Salamon, Strong rigidity of positive quaternion-Kähler manifolds, Invent. Math. 118 (1994) 109.
S. Kobayashi and K. Nomizu, Foundations of differential geometry, volume 1, Interscience Publishers, U.S.A. (1963).
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] [INSPIRE].
E. Bergshoeff and M. de Roo, The quartic effective action of the heterotic string and supersymmetry, Nucl. Phys. B 328 (1989) 439 [INSPIRE].
S. Ivanov, Heterotic supersymmetry, anomaly cancellation and equations of motion, Phys. Lett. B 685 (2010) 190 [arXiv:0908.2927] [INSPIRE].
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Harland, D., Nölle, C. Instantons and Killing spinors. J. High Energ. Phys. 2012, 82 (2012). https://doi.org/10.1007/JHEP03(2012)082
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DOI: https://doi.org/10.1007/JHEP03(2012)082