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Spin(7)-Instantons, Cayley Submanifolds and Fueter Sections

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Abstract

We prove an existence theorem for Spin(7)-instantons, which are highly concentrated near a Cayley submanifold; thus giving a partial converse to Tian’s foundational compactness theorem (Ann Math (2) 151(1):193–268, 2000). As an application, we show how to construct Spin(7)-instantons on Spin(7)-manifolds with suitable local K3 Cayley fibrations. This recovers an example constructed by Lewis (Spin(7) instantons, Ph.D. Thesis, 1998).

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References

  1. Brendle, S.: Complex anti-self-dual instantons and Cayley submanifolds. arXiv:math/0302094v2 (2003)

  2. Brendle, S.: On the construction of solutions to the Yang–Mills equations in higher dimensions. arXiv:math/0302093v3 (2003)

  3. Corrigan E., Devchand C., Fairlie D.B., Nuyts J.: First-order equations for gauge fields in spaces of dimension greater than four. Nucl. Phys. B 214(3), 452–464 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  4. 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. Zbl0904.57001 (1990)

  5. Donaldson, S.K.: An application of gauge theory to four-dimensional topology. J. Differ. Geom. 18(2), 279–315. Zbl0507.57010 (1983)

  6. Donaldson, S.K., Segal, E.P.: Gauge theory in higher dimensions, II, Surveys in differential geometry. Volume XVI. Geometry of special holonomy and related topics, pp. 1–41. Zbl1256.53038 (2011)

  7. Donaldson, S.K., Thomas, R.P.: Gauge Theory in Higher Dimensions. The Geometric Universe (Oxford, 1996), pp. 31–47. Zbl0926.58003 (1998)

  8. Haydys, A.: Gauge theory, calibrated geometry and harmonic spinors. J. Lond. Math. Soc. 86(2), 482–498. Zbl1256.81080 (2012)

  9. Harvey, R., Lawson, H.B., Jr.: Calibrated geometries. Acta Math. 148, 47–157. Zbl0584.53021 (1982)

  10. Joyce, D.D.: Compact manifolds with special holonomy, Oxford Mathematical Monographs. Oxford University Press, Oxford. Zbl1200.53003 (2000)

  11. Joyce, D.D.: Compact 8-manifolds with holonomy Spin(7). Invent. Math. 123(3), 507–552. Zbl0858.53037 (1996)

  12. Joyce, D.D.: A new construction of compact 8-manifolds with holonomy Spin(7). J. Differ. Geom. 53(1), 89–130. Zbl1040.53062 (1999)

  13. Lewis, C.: Spin(7) instantons, Ph.D. Thesis (1998)

  14. McLean, R.C.: Deformations of calibrated submanifolds. Comm. Anal. Geom. 6(4), 705–747. Zbl0929.53027 (1998)

  15. Nakajima, H.: Compactness of the moduli space of Yang–Mills connections in higher dimensions. J. Math. Soc. Jpn. 40(3), 383–392. Zbl0647.53030 (1988)

  16. Nakajima, H.: Moduli spaces of anti-self-dual connections on ALE gravitational instantons. Invent. Math. 102(2), 267–303. Zbl0714.53026 (1990)

  17. Pacard, F., Ritoré, M.: From constant mean curvature hypersurfaces to the gradient theory of phase transitions. J. Differ. Geom. 64(3), 359–423. Zbl1070.58014 (2003)

  18. Price, P.: A monotonicity formula for Yang–Mills fields. Manuscr. Math. 43(2–3), 131–166. Zbl0521.58024 (1983)

  19. Salamon, D.A, Walpuski, T.: Notes on the octonians. arXiv:1005.2820v1 (2010)

  20. Tanaka, Y.: A construction of Spin(7)-instantons. Ann. Glob. Anal. Geom. 42(4), 495–521. Zbl1258.53022 (2012)

  21. Taubes, C.H.: Self-dual Yang–Mills connections on non-self-dual 4-manifolds. J. Differ. Geom. 17(1), 139–170. Zbl0484.53026 (1982)

  22. Taubes, C.H.: Stability in Yang–Mills theories. Comm. Math. Phys. 91(2), 235–263. Zbl0524.58020 (1983)

  23. Tian, G.: Gauge theory and calibrated geometry. I. Ann. Math. (2) 151(1), 193–268. Zbl0957.58013 (2000)

  24. Uhlenbeck, K.K.: Connections with L p bounds on curvature. Comm. Math. Phys. 83(1), 31–42. Zbl0499.58019 (1982)

  25. Uhlenbeck, K.K.: Removable singularities in Yang–Mills fields. Comm. Math. Phys. 83(1), 11–29. Zbl0491.58032 (1982)

  26. Walpuski, T.: Gauge theory on G2-manifolds. Ph.D. Thesis. https://spiral.imperial.ac.uk/bitstream/10044/1/14365/1/Walpuski-T-2013-PhD-Thesis.pdf (2013)

  27. Walpuski, T.: G2-instantons on generalised Kummer constructions. Geom. Topol. 17(4), 2345–2388. Zbl1278.53051 (2013)

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  1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Thomas Walpuski

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  1. Thomas Walpuski
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Correspondence to Thomas Walpuski.

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Communicated by N. A. Nekrasov

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Walpuski, T. Spin(7)-Instantons, Cayley Submanifolds and Fueter Sections. Commun. Math. Phys. 352, 1–36 (2017). https://doi.org/10.1007/s00220-016-2724-6

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  • DOI: https://doi.org/10.1007/s00220-016-2724-6

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