Research PaperNo Access

Transversality for the moduli space of Spin(7)-instantons

Vicente Muñoz

Departamento de Algebra, Geometría y Topología, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, Spain

E-mail Address: vicente.munoz@uma.es

Search for more papers by this author

and

C. S. Shahbazi

Institut für Theoretische Physik, Leibniz Universität Hannover, Germany

E-mail Address: carlos.shahbazi@itp.uni-hannover.de

Search for more papers by this author

https://doi.org/10.1142/S0129055X20500130Cited by:3 (Source: Crossref)

Abstract

We construct the moduli space of Spin(7)-instantons on a hermitian complex vector bundle over a closed 8-dimensional manifold endowed with a (possibly non-integrable) Spin(7)-structure. We find suitable perturbations that achieve regularity of the moduli space, so that it is smooth and of the expected dimension over the irreducible locus.

Keywords:

AMSC: 53C38, 53C07, 53C25, 58D27

References

1. M. F. Atiyah and R. Bott, The Yang–Mills equations over Riemann surfaces, Phil. Trans. R. Soc. Lond. A 308 (1983) 523–615. Crossref, Web of Science, ADS, Google Scholar
2. M. F. Atiyah, N. Hitchin and I. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978) 425–461. Crossref, Web of Science, ADS, Google Scholar
3. R. Bryant and S. Salamon, On the construction of some complete metrics with exceptional holonomy, Duke Math. J. 58 (1989) 829–850. Crossref, Web of Science, Google Scholar
4. M. Čadek and J. Vanžura, Almost quaternionic structures on eight-manifolds, Osaka J. Math. 35 (1998) 165–190. Web of Science, Google Scholar
5. Y. Cao, J. Gross and D. Joyce, Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi–Yau 4-folds, arXiv:1811.09658 [math-dg]. Google Scholar
6. B. Charbonneau and D. Harland, Deformations of nearly Kähler instantons, Comm. Math. Phys. 348(3) (2016) 959–990. Crossref, Web of Science, ADS, Google Scholar
7. A. Clarke, M. García-Fernández and C. Tipler, Moduli of $G_{2}$ -structures and the Strominger system in dimension 7, arXiv:1607.01219 [math-dg]. Google Scholar
8. E. Corrigan, C. Devchand, D. Fairlie and J. Nuyts, First order equations for gauge fields in spaces of dimension greater than four, Nuclear Phys. B 214 (1983) 452–464. Crossref, Web of Science, ADS, Google Scholar
9. S. Donaldson and P. Kronheimer, The Geometry of Four-Manifolds, Oxford Mathematical Monographs (The Clarendon Press, Oxford University Press, 1997). Google Scholar
10. S. Donaldson and R. Thomas, Gauge theory in higher dimensions, in The Geometric Universe: Science, Geometry, and the Work of Roger Penrose (Oxford University Press, 1998), pp. 31–47. Crossref, Google Scholar
11. S. Donaldson and E. Segal, Gauge theory in higher dimensions II, Surveys Diff. Geom. 16 (2011) 1–41. Crossref, Google Scholar
12. M. Fernández, A classification of Riemannian manifolds with structure group Spin(7), Annali Mat. Pura Appl. 143 (1986) 101–122. Crossref, Web of Science, Google Scholar
13. M. Fernández, A new example of compact Riemannian manifold with structure group Spin(7), Portugaliae Math. 44 (1987) 161–165. Google Scholar
14. M. Fernández and A. Gray, Riemannian manifolds with structure group $G_{2}$ , Annali Mat. Pura Appl. 132 (1982) 19–45. Crossref, Web of Science, Google Scholar
15. M. Fernández, S. Ivanov, L. Ugarte and R. Villacampa, Compact supersymmetric solutions of the heterotic equations of motion in dimensions 7 and 8, Adv. Theor. Math. Phys. 15 (2011) 245–284. Crossref, Web of Science, Google Scholar
16. J. M. Figueroa-O’Farrill, Gauge theory and the division algebras, J. Geom. Phys. 32 (1999) 227–240. Crossref, Web of Science, ADS, Google Scholar
17. R. Friedman and J. Morgan, Gauge Theory and the Topology of Four-Manifolds (Amer. Math. Soc., 1997). Crossref, Google Scholar
18. T. Friedrich and S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math. 6 (2002) 303–336. Crossref, Google Scholar
19. H. Glöckner and K. Neeb, Banach–Lie quotients, enlargibility, and universal complexifications, J. Reine Angew. Math. 560 (2003) 1–28. Crossref, Web of Science, Google Scholar
20. A. S. Haupt, T. A. Ivanova, O. Lechtenfeld and A. D. Popov, Chern–Simons flows on Aloff–Wallach spaces and $S p i n (7)$ -instantons, Phys. Rev. D 83 (2011) 105028. Crossref, Web of Science, ADS, Google Scholar
21. D. Harland, T. A. Ivanova, O. Lechtenfeld and A. D. Popov, Yang–Mills flows on nearly Kähler manifolds and $G_{2}$ -instantons, Comm. Math. Phys. 300 (2010) 185–204. Crossref, Web of Science, ADS, Google Scholar
22. R. Harvey and H. Lawson, Calibrated geometries, Acta Math. 148 (1982) 47–157. Crossref, Web of Science, Google Scholar
23. A. Haydys, Gauge theory, calibrated geometry and harmonic spinors, J. Lond. Math. Soc. 86 (2012) 482–498. Crossref, Web of Science, Google Scholar
24. S. Ivanov, Connections with torsion, parallel spinors and geometry of Spin(7)-manifolds, Math. Res. Lett. 11(2–3) (2004) 171–186. Crossref, Web of Science, Google Scholar
25. S. Ivanov, Heterotic supersymmetry, anomaly cancellation and equations of motion, Phys. Lett. B 685 (2010) 190–196. Crossref, Web of Science, ADS, Google Scholar
26. D. Joyce, Compact 8-manifolds with holonomy Spin(7), Invent. Math. 123 (1996) 507–552. Crossref, Web of Science, ADS, Google Scholar
27. D. Joyce, A new construction of compact 8-manifolds with holonomy Spin(7), J. Differential Geom. 53 (1999) 89–130. Crossref, Web of Science, Google Scholar
28. D. Joyce, Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs (Oxford University Press, 2000). Crossref, Google Scholar
29. D. Joyce, Conjectures on counting associative 3-folds in $G_{2}$ -manifolds, in Modern Geometry: A Celebration of the Work of Simon Donaldson, Proc. Sympos. Pure Math., Vol. 99 (Amer. Math. Soc., 2018), pp. 97–160. Crossref, Google Scholar
30. P. Kronheimer, Four-manifold invariants from higher-rank bundles, J. Differential Geom. 70 (2005) 59–112. Crossref, Web of Science, Google Scholar
31. H. Lawson, The Theory of Gauge Fields in Four Dimensions, CBMS Regional Conference Series in Mathematics, Vol. 58 (Amer. Math. Soc., 1985). Crossref, Google Scholar
32. H. Lawson and M. Michelsohn, Spin Geometry, Princeton Mathematical Series, Vol. 38 (Princeton University Press, 1989). Google Scholar
33. C. Lewis, Spin(7) instantons, Oxford University D.Phil. thesis (1998). Google Scholar
34. P. Mitter and C. Viallet, On the bundle of connections and the gauge orbit manifold in Yang–Mills theory, Comm. Math. Phys. 79 (1981) 457–472. Crossref, Web of Science, ADS, Google Scholar
35. V. Muñoz, Spin(7)-instantons, stable bundles and the Bogomolov inequality for complex $4$ -tori, J. Math. Pures Appl. 102 (2014) 124–152. Crossref, Web of Science, Google Scholar
36. V. Muñoz, On rotation of complex structures, J. Geom. Phys. 87 (2015) 344–353. Crossref, Web of Science, ADS, Google Scholar
37. V. Muñoz and C. S. Shahbazi, Orientability of the moduli space of Spin(7)-instantons, Pure Appl. Math. Q. 13(3) (2017) 453–476. Crossref, Web of Science, Google Scholar
38. X. Ossa, M. Larfors and E. Svanes, Infinitesimal moduli of $G_{2}$ holonomy manifolds with instanton bundles, J. High Energy Phys. 2016(11) (2016), Art. 016. Google Scholar
39. T. Ramadas, Spin(7)-instantons and the Hodge Conjecture for certain abelian four-folds: A modest proposal, in Vector Bundles and Complex Geometry, Contemporary Math, Vol. 522 (Amer. Math. Soc., 2010), pp. 155–170. Crossref, Google Scholar
40. R. Reyes-Carrión, A generalization of the notion of instanton, Differential Geom. Appl. 8 (1998) 1–20. Crossref, Web of Science, Google Scholar
41. S. Smale, An infinite dimensional version of Sard’s theorem, Amer. J. Math. 87 (1965) 861–866. Crossref, Web of Science, Google Scholar
42. A. Strominger, Superstrings with torsion, Nuclear Phys. B 274 (1986) 253–284. Crossref, Web of Science, ADS, Google Scholar
43. Y. Tanaka, A construction of Spin(7)-instantons, Ann. Global Anal. Geom. 42 (2012) 495–521. Crossref, Web of Science, Google Scholar
44. G. Tian, Smoothness of the universal deformation space of compact Calabi–Yau manifolds and its Peterson–Weil metric, in Mathematical Aspects of String Theory (World Scientific, 1987), pp. 629–646. Link, Google Scholar
45. G. Tian, Gauge theory and calibrated geometry, I, Ann. of Math. 151 (2000) 193–268. Crossref, Web of Science, Google Scholar
46. T. Walpuski, Spin(7)-instantons, Cayley submanifolds and Fueter sections, Comm. Math. Phys. 352 (2014) 1–36. Crossref, Web of Science, ADS, Google Scholar
47. R. Ward, Completely solvable gauge-field equations in dimension greater than four, Nuclear Phys. B. 236 (1984) 381–396. Crossref, Web of Science, ADS, Google Scholar