Deformations of \(\text{G}_2\)-instantons on nearly \(\text{G}_2\) manifolds. (English) Zbl 1539.53035
A \(\text{G}_2\)-structure on a 7-manifold is nearly parallel if the defining 3-form \(\varphi\) satisfies the equation \(d\varphi=\tau_0\ast\varphi\), for some (constant) function \(\tau_0\). A 7-manifold admits a \(\text{G}_2\)-structure if and only if it is orientable and spin. Via this link with spin geometry, nearly parallel \(\text{G}_2\)-structures can be equivalently characterized as the \(\text{G}_2\)-structures with a real spinor \(\eta\) satisfying \(\nabla_X\eta=-\frac{\tau_0}{8}X\cdot\eta\); hence, \(\eta\) is a Killing spinor, and the manifold is Einstein with positive scalar curvature.
The author considers instantons on a nearly parallel \(\text{G}_2\)-manifold \(M\). These are connections on principal \(K\)-bundles over \(M\) whose curvature 2-form \(F\) satisfies the conditions \[ F\cdot\eta=0\Leftrightarrow \ F\wedge\varphi=\ast F \Leftrightarrow F\wedge\ast\varphi=0\Leftrightarrow F\lrcorner\varphi=0\,. \] Instantons on nearly parallel \(\text{G}_2\)-manifolds solve the Yang-Mills equation \(d^\ast_\nabla F=0\), albeit they need not be minimizers of the Yang-Mills functional. Nearly parallel \(\text{G}_2\)-instantons on certain Aloff-Wallach spaces were constructed by G. Ball and G. Oliveira [Geom. Topol. 23, No. 2, 685–743 (2019). (2019; Zbl 1418.53027)], who also classified invariant \(\text{G}_2\)-instantons with gauge group \(\mathrm{U}(1)\) and \(\mathrm{SO}(3)\) on these spaces.
The author is interested in deformations of such instantons. He shows that the instantons are rigid if, for example, \(K\) is abelian. The author then proceeds to a systematic analysis of normal homogeneous nearly parallel \(\text{G}_2\)-manifolds, where he describes the deformation space of the characteristic connection (which is an instanton).
The author considers instantons on a nearly parallel \(\text{G}_2\)-manifold \(M\). These are connections on principal \(K\)-bundles over \(M\) whose curvature 2-form \(F\) satisfies the conditions \[ F\cdot\eta=0\Leftrightarrow \ F\wedge\varphi=\ast F \Leftrightarrow F\wedge\ast\varphi=0\Leftrightarrow F\lrcorner\varphi=0\,. \] Instantons on nearly parallel \(\text{G}_2\)-manifolds solve the Yang-Mills equation \(d^\ast_\nabla F=0\), albeit they need not be minimizers of the Yang-Mills functional. Nearly parallel \(\text{G}_2\)-instantons on certain Aloff-Wallach spaces were constructed by G. Ball and G. Oliveira [Geom. Topol. 23, No. 2, 685–743 (2019). (2019; Zbl 1418.53027)], who also classified invariant \(\text{G}_2\)-instantons with gauge group \(\mathrm{U}(1)\) and \(\mathrm{SO}(3)\) on these spaces.
The author is interested in deformations of such instantons. He shows that the instantons are rigid if, for example, \(K\) is abelian. The author then proceeds to a systematic analysis of normal homogeneous nearly parallel \(\text{G}_2\)-manifolds, where he describes the deformation space of the characteristic connection (which is an instanton).
Reviewer: Giovanni Bazzoni (Como)
MSC:
| 53C10 | \(G\)-structures |
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
| 53C27 | Spin and Spin\({}^c\) geometry |
| 53C30 | Differential geometry of homogeneous manifolds |
| 83C60 | Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism |
Citations:
Zbl 1418.53027References:
| [1] | Agricola, I.; Becker-Bender, J.; Kim, H., Twistorial eigenvalue estimates for generalized Dirac operators with torsion, Adv. Math., 243, 296-329 (2013) · Zbl 1287.58015 · doi:10.1016/j.aim.2013.05.001 |
| [2] | Agricola, I.; Chiossi, SG; Friedrich, T.; Höll, J., Spinorial description of \({\rm SU}(3)\)- and \({\rm G}_2\)-manifolds, J. Geom. Phys., 98, 535-555 (2015) · Zbl 1333.53037 · doi:10.1016/j.geomphys.2015.08.023 |
| [3] | Agricola, I.; Friedrich, T., On the holonomy of connections with skew-symmetric torsion, Mathematische Annalen, 328, 4, 711-748 (2004) · Zbl 1055.53031 · doi:10.1007/s00208-003-0507-9 |
| [4] | Agricola, I.; Ferreira, AC, Einstein manifolds with skew torsion, Q. J. Math., 65, 3, 717-741 (2014) · Zbl 1368.53034 · doi:10.1093/qmath/hat050 |
| [5] | Agricola, I.; Friedrich, T.; Höll, J., \({\rm Sp}(3)\) structures on 14-dimensional manifolds, J. Geom. Phys., 69, 12-30 (2013) · Zbl 1291.53036 · doi:10.1016/j.geomphys.2013.02.010 |
| [6] | Agricola, I., Connections on naturally reductive spaces, their Dirac operator and homogeneous models in string theory, Comm. Math. Phys., 232, 3, 535-563 (2003) · Zbl 1032.53041 · doi:10.1007/s00220-002-0743-y |
| [7] | Agricola, I.; Höll, J., Cones of \(G\) manifolds and Killing spinors with skew torsion, Ann. Mat. Pura Appl., 194, 3, 673-718 (2015) · Zbl 1319.53043 · doi:10.1007/s10231-013-0393-z |
| [8] | Atiyah, MF; Hitchin, NJ; Singer, IM, Self-duality in four-dimensional Riemannian geometry, Proc. R. Soc. London Ser. A, 362, 1711, 425-461 (1978) · Zbl 0389.53011 · doi:10.1098/rspa.1978.0143 |
| [9] | Alexandrov, B.; Semmelmann, U., Deformations of nearly parallel \({\rm G}_2\)-structures, Asian J. Math., 16, 4, 713-744 (2012) · Zbl 1266.53048 · doi:10.4310/AJM.2012.v16.n4.a6 |
| [10] | Bär, C., Real Killing spinors and holonomy, Comm. Math. Phys., 154, 3, 509-521 (1993) · Zbl 0778.53037 · doi:10.1007/BF02102106 |
| [11] | Berger, M.: Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes. Bull. Soc. Math. France 83, 279-330 (1955) · Zbl 0068.36002 |
| [12] | Baum, H., Friedrich, T., Grunewald, R., Kath, I.: Twistor and Killing spinors on Riemannian manifolds (1991) · Zbl 0705.53004 |
| [13] | Bismut, J-M, A local index theorem for non-Kähler manifolds, Math. Ann., 284, 4, 681-699 (1989) · Zbl 0666.58042 · doi:10.1007/BF01443359 |
| [14] | Ball, G.; Oliveira, G., Gauge theory on Aloff-Wallach spaces, Geom. Topol., 23, 2, 685-743 (2019) · Zbl 1418.53027 · doi:10.2140/gt.2019.23.685 |
| [15] | Bryant, R.L.: Some remarks on \(G_2\)-structures. In: Proceedings of Gökova Geometry-Topology Conference 2005, Gökova Geometry/Topology Conference (GGT), Gökova, 75- 109 (2006) · Zbl 1115.53018 |
| [16] | Bryant, RL, Metrics with exceptional holonomy, Ann. Math., 126, 3, 525-576 (1987) · Zbl 0637.53042 · doi:10.2307/1971360 |
| [17] | Charbonneau, B.; Harland, D., Deformations of nearly Kähler instantons, Commun. Math. Phys., 348, 3, 959-990 (2016) · Zbl 1429.53037 · doi:10.1007/s00220-016-2675-y |
| [18] | Chrysikos, I., Killing and twistor spinors with torsion, Ann. Global Anal. Geom., 49, 2, 105-141 (2016) · Zbl 1365.53049 · doi:10.1007/s10455-015-9483-z |
| [19] | Clarke, A., Instantons on the exceptional holonomy manifolds of Bryant and Salamon, J. Geom. Phys., 82, 84-97 (2014) · Zbl 1293.53032 · doi:10.1016/j.geomphys.2014.04.006 |
| [20] | Cleyton, R.; Swann, A., Einstein metrics via intrinsic or parallel torsion, Mathematische Zeitschrift, 247, 3, 513-528 (2004) · Zbl 1069.53041 · doi:10.1007/s00209-003-0616-x |
| [21] | Donaldson, S.K., Kronheimer, P.B.: The Geometry of Four-Manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, Oxford Science Publications(1990) · Zbl 0820.57002 |
| [22] | Driscoll, J.: Deformations of asymptotically conical \(\rm G_2\)-instantons, to appear in Communications in Mathematical Physics (2020) |
| [23] | Dwivedi, S., Singhal, R.: Deformation theory of nearly \(G_2\) manifolds, to appear in Communications in Analysis and Geometry. arXiv:2007.02497 (2020-07) |
| [24] | Friedrich, T.; Ivanov, S., Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math., 6, 2, 303-335 (2002) · Zbl 1127.53304 · doi:10.4310/AJM.2002.v6.n2.a5 |
| [25] | Friedrich, T.; Kath, I., Einstein manifolds of dimension five with small first eigenvalue of the Dirac operator, J. Diff. Geom., 29, 2, 263-279 (1989) · Zbl 0633.53069 |
| [26] | Friedrich, T.; Kath, Ines, 7-dimensional compact riemannian manifolds with killing spinors, Commun. Math. Phys., 133, 3, 543-561 (1990) · Zbl 0722.53038 · doi:10.1007/BF02097009 |
| [27] | Friedrich, T.; Kath, I.; Moroianu, A.; Semmelmann, U., On nearly parallel \(G_2\)-structures, J. Geometry Phys., 23, 3, 259-286 (1997) · Zbl 0898.53038 · doi:10.1016/S0393-0440(97)80004-6 |
| [28] | Friedrich, T., The second Dirac eigenvalue of a nearly parallel \({\rm G}_2\)-manifold, Adv. Appl. Clifford Algebra, 22, 2, 301-311 (2012) · Zbl 1269.53048 · doi:10.1007/s00006-011-0301-9 |
| [29] | Goette, S., Equivariant \(\eta \)-invariants on homogeneous spaces, Math. Z., 232, 1, 1-42 (1999) · Zbl 0941.58016 · doi:10.1007/PL00004757 |
| [30] | Gray, Alfred, Weak holonomy groups, Mathematische Zeitschrift, 123, 4, 290-300 (1971) · Zbl 0222.53043 · doi:10.1007/BF01109983 |
| [31] | Grunewald, Ralf, Six-dimensional Riemannian manifolds with a real Killing spinor, Ann. Global Anal. Geom., 8, 1, 43-59 (1990) · Zbl 0704.53050 · doi:10.1007/BF00055017 |
| [32] | Hijazi, O.: Caractérisation de la sphère par les premières valeurs propres de l’opérateur de Dirac en dimensions [(3,4,7\]\) et \(8\), C. R. Acad. Sci. Paris Sér. I Math., 303(9), 417- 419 (1986) · Zbl 0606.53024 |
| [33] | Harland, D.; Nölle, C., Instantons and Killing spinors, J. High Energy Phys., 3, 82 (2012) · Zbl 1309.81162 · doi:10.1007/JHEP03(2012)082 |
| [34] | Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 9, Second printing, revised (1978) · Zbl 0447.17001 |
| [35] | Joyce, DD, Compact Manifolds with Special Holonomy. Oxford Mathematical Monographs (2000), Oxford: Oxford University Press, Oxford · Zbl 1027.53052 |
| [36] | Karigiannis, S., Flows of \(G_2\)-structures. I, Q. J. Math., 60, 4, 487-522 (2009) · Zbl 1190.53025 · doi:10.1093/qmath/han020 |
| [37] | Karigiannis, S.: Some notes on \(G_2\) and \({\rm Spin}(7)\) geometry, 11, 129-146 (2010) · Zbl 1248.53040 |
| [38] | Knapp, A.W.: Representation theory of semisimple groups: An overview based on examples (pms-36). Princeton University Press, REV - Revised (1986) · Zbl 0604.22001 |
| [39] | Kostant, Bertram, A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups, Duke Math. J., 100, 3, 447-501 (1999) · Zbl 0952.17005 · doi:10.1215/S0012-7094-99-10016-0 |
| [40] | Lawson, H.B., Michelson, Marie-Louise, Spin geometry (PMS-38), Princeton University Press, Princeton (1989) · Zbl 0688.57001 |
| [41] | Moroianu, A.; Semmelmann, U., The Hermitian Laplace operator on nearly Kähler manifolds, Comm. Math. Phys., 294, 1, 251-272 (2010) · Zbl 1210.58028 · doi:10.1007/s00220-009-0903-4 |
| [42] | Earp, S.; Henrique, N.; Walpuski, T., \({\rm G}_2\)-instantons over twisted connected sums, Geom. Topol., 19, 3, 1263-1285 (2015) · Zbl 1317.53034 · doi:10.2140/gt.2015.19.1263 |
| [43] | Walpuski, T., \({\rm G}_2\)-instantons over twisted connected sums: an example, Math. Res. Lett., 23, 2, 529-544 (2016) · Zbl 1350.53032 · doi:10.4310/MRL.2016.v23.n2.a11 |
| [44] | Waldron, A.: \({\rm G}_2\)-instantons on the 7-sphere, (2020-02), arXiv e-prints, arXiv:2002.02386 |
| [45] | Wang, MY, Parallel spinors and parallel forms, Ann. Global Anal. Geom., 7, 1, 59-68 (1989) · Zbl 0688.53007 · doi:10.1007/BF00137402 |
| [46] | Wilking, B., The normal homogeneous space \(({\rm SU}(3)\times{\rm SO}(3))/{\rm U}^\bullet (2)\) has positive sectional curvature, Proc. Am. Math. Soc., 127, 4, 1191-1194 (1999) · Zbl 0948.53025 · doi:10.1090/S0002-9939-99-04613-4 |
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