Abstract
The instanton equations on vector bundles over Calabi-Yau and hyper-Kähler cones can be reduced to matrix equations resembling Nahm’s equations. We complement the discussion of Hermitian Yang-Mills (HYM) equations on Calabi-Yau cones, based on regular semi-simple elements, by a new set of (singular) boundary conditions which have a known instanton solution in one direction. This approach extends the classic results of Kronheimer by probing a relation between generalised Nahm’s equations and nilpotent pairs/tuples. Moreover, we consider quaternionic instantons on hyper-Kähler cones over generic 3-Sasakian manifolds and study the HYM moduli spaces arising in this set-up, using the fact that their analysis can be traced back to the intersection of three Hermitian Yang-Mills conditions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Gross, D. Huybrechts and D. Joyce, Calabi-Yau manifolds and related geometries, Lectures from the Summer School in Nordfjordeid, Nordfjordeid Norway (2001), Springer-Verlag, Berlin Germany (2003).
N.J. Hitchin, A. Karlhede, U. Lindström and M. Roček, HyperKähler Metrics and Supersymmetry, Commun. Math. Phys. 108 (1987) 535 [INSPIRE].
G.W. Gibbons and P. Rychenkova, Cones, triSasakian structures and superconformal invariance, Phys. Lett. B 443 (1998) 138 [hep-th/9809158] [INSPIRE].
B. de Wit, B. Kleijn and S. Vandoren, Rigid N = 2 superconformal hypermultiplets, hep-th/9808160 [INSPIRE].
S.K. Donaldson, Self-dual connections and the topology of smooth 4-manifolds, Bull. Am. Math. Soc. 8 (1983) 81.
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 [INSPIRE].
S.K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. 3 (1985) 1.
K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Commun. Pure Appl. Math. 39 (1986) 1.
M. Mamone Capria and S.M. Salamon, Yang-Mills fields on quaternionic spaces, Nonlinearity 1 (1988) 517.
T. Nitta, Vector bundles over quaternionic Kähler manifolds, Tohoku Math. J. 40 (1988) 425.
C. Bartocci and M. Jardim, Hyperkähler Nahm transforms, CRM Proc. Lecture Notes 38 (2004) 103 [math/0312045].
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].
I. Bauer, T.A. Ivanova, O. Lechtenfeld and F. Lubbe, Yang-Mills instantons and dyons on homogeneous G 2 -manifolds, JHEP 10 (2010) 044 [arXiv:1006.2388] [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. Nolle and A.D. Popov, Yang-Mills instantons on cones and sine-cones over nearly Kähler manifolds, JHEP 09 (2011) 103 [arXiv:1108.3951] [INSPIRE].
D. Harland and C. Nölle, Instantons and Killing spinors, JHEP 03 (2012) 082 [arXiv:1109.3552] [INSPIRE].
T.A. Ivanova and A.D. Popov, Instantons on Special Holonomy Manifolds, Phys. Rev. D 85 (2012) 105012 [arXiv:1203.2657] [INSPIRE].
S. Bunk, T.A. Ivanova, O. Lechtenfeld, A.D. Popov and M. Sperling, Instantons on sine-cones over Sasakian manifolds, Phys. Rev. D 90 (2014) 065028 [arXiv:1407.2948] [INSPIRE].
S. Bunk, O. Lechtenfeld, A.D. Popov and M. Sperling, Instantons on conical half-flat 6-manifolds, JHEP 01 (2015) 030 [arXiv:1409.0030] [INSPIRE].
M. Sperling, Instantons on Calabi-Yau cones, Nucl. Phys. B 901 (2015) 354 [arXiv:1505.01755] [INSPIRE].
A.S. Haupt, Yang-Mills solutions and Spin(7)-instantons on cylinders over coset spaces with G 2 -structure, JHEP 03 (2016) 038 [arXiv:1512.07254] [INSPIRE].
D. Gaiotto and E. Witten, Supersymmetric Boundary Conditions in N = 4 Super Yang-Mills Theory, J. Statist. Phys. 135 (2009) 789 [arXiv:0804.2902] [INSPIRE].
D. Gaiotto and E. Witten, S-duality of Boundary Conditions In N = 4 Super Yang-Mills Theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].
D. Xie, M5 brane and four dimensional N = 1 theories I, JHEP 04 (2014) 154 [arXiv:1307.5877] [INSPIRE].
J.J. Heckman, P. Jefferson, T. Rudelius and C. Vafa, Punctures for theories of class \( {\mathcal{S}}_{\Gamma} \), JHEP 03 (2017) 171 [arXiv:1609.01281] [INSPIRE].
A. Hashimoto, P. Ouyang and M. Yamazaki, Boundaries and defects of \( \mathcal{N}=4 \) SYM with 4 supercharges. Part I: Boundary/junction conditions, JHEP 10 (2014) 107 [arXiv:1404.5527] [INSPIRE].
A. Hashimoto, P. Ouyang and M. Yamazaki, Boundaries and defects of \( \mathcal{N}=4 \) SYM with 4 supercharges. Part II: Brane constructions and 3d \( \mathcal{N}=2 \) field theories, JHEP 10 (2014) 108 [arXiv:1406.5501] [INSPIRE].
C.P. Boyer and K. Galicki, 3-Sasakian manifolds, Surveys Diff. Geom. 7 (1999) 123 [hep-th/9810250] [INSPIRE].
C.P. Boyer, K. Galicki and B.M. Mann, The geometry and topology of 3-Sasakian manifolds, J. Reine Angew. Math. 455 (1994) 183.
N. Hitchin, Hyperkähler manifolds, Séminaire Bourbaki 34 (1991-1992) 137.
H. Baum, T. Friedrich, R. Grunewald and I. Kath, Teubner-Texte zur Mathematik. Vol. 124: Twistors and Killing spinors on Riemannian manifolds, B.G. Teubner, Leipzig Germany (1991).
M.F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. Lond. A 308 (1982) 523 [INSPIRE].
A. Deser, O. Lechtenfeld and A.D. Popov, σ-model limit of Yang-Mills instantons in higher dimensions, Nucl. Phys. B 894 (2015) 361 [arXiv:1412.4258] [INSPIRE].
J.C. Geipel, O. Lechtenfeld, A.D. Popov and R.J. Szabo, Sasakian quiver gauge theories and instantons on cones over round and squashed seven-spheres, arXiv:1706.07383 [INSPIRE].
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] [INSPIRE].
B.P. Dolan and R.J. Szabo, Equivariant Dimensional Reduction and Quiver Gauge Theories, Gen. Rel. Grav. 43 (2010) 2453 [arXiv:1001.2429] [INSPIRE].
O. Lechtenfeld, A.D. Popov, M. Sperling and R.J. Szabo, Sasakian quiver gauge theories and instantons on cones over lens 5-spaces, Nucl. Phys. B 899 (2015) 848 [arXiv:1506.02786] [INSPIRE].
J.C. Geipel, O. Lechtenfeld, A.D. Popov and R.J. Szabo, Sasakian quiver gauge theories and instantons on the conifold, Nucl. Phys. B 907 (2016) 445 [arXiv:1601.05719] [INSPIRE].
J.C. Geipel, Sasakian quiver gauge theory on the Aloff-Wallach space X 1,1, Nucl. Phys. B 916 (2017) 279 [arXiv:1605.03521] [INSPIRE].
P.B. Kronheimer, A hyper-Kählerian structure on coadjoint orbits of a semisimple complex group, J. London Math. Soc. 2 (1990) 193.
P.B. Kronheimer, Instantons and the geometry of the nilpotent variety, J. Diff. Geom. 32 (1990) 473 [INSPIRE].
S.K. Donaldson, Nahm’S equations and the classification of monopoles, Commun. Math. Phys. 96 (1984) 387 [INSPIRE].
O. Lechtenfeld, Instantons and Chern-Simons flows in 6, 7 and 8 dimensions, Phys. Part. Nucl. 43 (2012) 569 [arXiv:1201.6390] [INSPIRE].
M. Sperling, Two aspects of gauge theories: higher-dimensional instantons on cones over Sasaki-Einstein spaces and Coulomb branches for 3-dimensional N = 4 gauge theories, Ph.D. Thesis, Hannover University, Hannover Germany (2016).
V. Ginzburg, Principal nilpotent pairs in a semisimple Lie algebra, Invent. Math. 140 (2000) 511 [math/9903059].
D.I. Panyushev, Nilpotent pairs in semisimple Lie algebras and their characteristics, Int. Math. Res. Notices 2000 (2000) 1 [math/9906049].
D.I. Panyushev, Nilpotent pairs, dual pairs, and sheets, J. Algebra 240 (2001) 635 [math/9904014].
A.G. Elashvili and D.I. Panyushev, A classification of the principal nilpotent pairs in simple Lie algebras and related problems, J. London Math. Soc. 63 (2001) 299 [math/9909082].
O. Biquard, Sur les équations de Nahm et la structure de Poisson des algèbres de Lie semi-simples complexes, Math. Ann. 304 (1996) 253.
A.G. Kovalev, Nahm’s equations and complex adjoint orbits, Quart. J. Math. 47 (1996) 41.
J. Fuchs and C. Schweigert, Symmetries, Lie algebras and representations: A graduate course for physicists, Cambridge University Press, Cambridge U.K. (2003).
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1709.01944
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Geipel, J.C., Sperling, M. Instantons on Calabi-Yau and hyper-Kähler cones. J. High Energ. Phys. 2017, 103 (2017). https://doi.org/10.1007/JHEP10(2017)103
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2017)103