Abstract
Using the Weil–Brezin–Zak transform of solid state physics, we describe line bundles over elliptic curves in terms of Weyl operators. We then discuss the connection with finitely generated projective modules over the algebra A θ of the noncommutative torus. We show that such A θ -modules have a natural interpretation as Moyal deformations of vector bundles over an elliptic curve E τ , under the condition that the deformation parameter θ and the modular parameter τ satisfy a non-trivial relation.
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References
Atiyah M.F.: Vector bundles over an elliptic curve. Proc. Lond. Math. Soc. 7, 414–452 (1957)
Birkenhake, C., Lange, H.: Complex Abelian Varieties. 2nd ed. Springer, Berlin (2004)
Connes A.: C *-algèbres et géométrie différentielle. C. R. Acad. Sci. Paris 290A, 599–604 (1980)
Connes A.: Noncommutative Geometry. Academic Press, New York (1994)
Connes A., Rieffel M.A.: Yang-Mills for noncommutative two-tori. Contemp. Math. 62, 237–266 (1987)
Drinfeld V.G.: Quasi-Hopf algebras. Leningrad Math. J. 1, 1419–1457 (1989)
Fiore, G.: On twisted symmetries and quantum mechanics with a magnetic field on noncommutative tori, PoS(CNCFG2010)018. http://pos.sissa.it/archive/conferences/127/018/CNCFG2010_018
Fiore G.: On quantum mechanics with a magnetic field on \({\mathbb{R}^n}\) and on a torus \({\mathbb{T}^n}\) , and their relation. Int. J. Theor. Phys. 52, 877–896 (2013)
Folland, G.B.: Harmonic analysis in phase space. Ann. Math. Stud. 122; Princeton University Press, Princeton (1989)
Gayral V., Gracia-Bondía J.M., Iochum B., Schücker T., Várilly J.C.: Moyal planes are spectral triples. Commun. Math. Phys. 246, 569–623 (2004)
Lechner, G., Waldmann, S.: Strict deformation quantization of locally convex algebras and modules. preprint arXiv:1109.5950 [math.QA]
Mahanta S., Suijlekom W.D.: Noncommutative tori and the Riemann–Hilbert correspondence. J. Noncommut. Geom. 3, 261–287 (2009)
Manin, Y.: Real multiplication and noncommutative geometry. In: The Legacy of Niels Henrik Abel, pp. 685–727. Springer, Berlin (2004)
Mumford D.: Tata Lectures on Theta I. Birkäuser, Basel (1983)
Plazas, J.: Arithmetic structures on noncommutative tori with real multiplication. Int. Math. Res. Not. (2008), rnm147
Polishchuk A., Schwarz A.: Categories of holomorphic vector bundles on noncommutative two-tori. Commun. Math. Phys. 236, 135–159 (2003)
Polishchuk A.: Noncommutative two-tori with real multiplication as noncommutative projective varieties. J. Geom. Phys. 50, 162–187 (2004)
Rieffel M.A.: The cancellation theorem for projective modules over irrational rotation C *-algebras. Proc. Lond. Math. Soc. 47, 285–302 (1983)
Rieffel, M.A.: Deformation quantization and operator algebras. Proc. Symp. Pure Math. 51, 411–423 (1990)
Rieffel, M.A.: Deformation quantization for actions of \({\mathbb{R}^d}\) . Mem. Am. Math. Soc. 106 (1993)
Swan R.G.: Vector bundles and projective modules. Trans. Am. Math. Soc. 105, 264 (1962)
Várilly, J.C.: An introduction to noncommutative geometry. EMS Lect. Ser. in Math. (2006)
Vlasenko, M.: The graded ring of quantum theta functions for noncommutative torus with real multiplication. Int. Math. Res. Not. 15825 (2006)
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D’Andrea, F., Fiore, G. & Franco, D. Modules Over the Noncommutative Torus and Elliptic Curves. Lett Math Phys 104, 1425–1443 (2014). https://doi.org/10.1007/s11005-014-0718-x
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DOI: https://doi.org/10.1007/s11005-014-0718-x