Abstract
We study one-instantons over \({\overline{\mathbb{C}{\rm P}}^2_q}\), that is, anti-selfdual connections with instanton number 1 on the quantum projective plane \({\mathbb{C}{\rm P}^2_q}\) with orientation which is reversed with respect to the usual one. The orientation is fixed by a suitable choice of a basis element for the rank 1 free bimodule of top forms. The noncommutative family of solutions is foliated, each non-singular leaf being isomorphic to \({\overline{\mathbb{C}{\rm P}}^2_q}\) itself.
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References
Buchdahl N.P.: Instantons on CP2. J. Diff. Geom. 24, 19–52 (1986)
D’Andrea F., Dąbrowski L., Landi G.: The noncommutative geometry of the quantum projective plane. Rev. Math. Phys. 20, 979–1006 (2008)
D’Andrea F., Landi G.: Anti-selfdual connections on the quantum projective plane: monopoles. Commun. Math. Phys. 297, 841–893 (2010)
D’Andrea, F., Landi, G.: Geometry of the quantum projective plane. Noncommutative structures in mathematics and physics. In: 5th ECM Satellite Conference Proceedings. Royal Flemish Academy, Brussels, pp. 85–102 (2008)
Donaldson, S.K.: Vector bundles on the flag manifolds and the Ward correspondence. Geom. Today Progr. Math. 60, 109–119 (1985) (Birkhäuser)
Donaldson S.K., Kronheimer P.B.: The Geometry of Four-Manifolds. Oxford University Press, Oxford (1990)
Groisser D.: The geometry of the moduli space of CP2 instantons. Invent. Math. 99, 393–409 (1990)
Habermann L.: A family of metrics on the moduli space of CP2 instantons. Commun. Math. Phys. 149, 209–216 (1992)
Klimyk A., Schmüdgen K.: Quantum Groups and Their Representations. Springer, New York (1997)
Vaksman L., Soibelman Ya.: The algebra of functions on the quantum group SU(n + 1) and odd-dimensional quantum spheres. Leningrad Math. J. 2, 1023–1042 (1991)
Wells R.O.: Differential analysis on complex manifolds. GTM, vol 65. Springer, New York (1980)
Woronowicz S.L.: From multiplicative unitaries to quantum groups. Int. J. Math. 7, 127–149 (1996)
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Communicated by A. Connes
Dedicated to the memory of Tetsuya Masuda
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D’Andrea, F., Landi, G. Anti-Selfdual Connections on the Quantum Projective Plane: Instantons. Commun. Math. Phys. 333, 505–540 (2015). https://doi.org/10.1007/s00220-014-2192-9
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DOI: https://doi.org/10.1007/s00220-014-2192-9