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Star Product Formula of Theta Functions

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Abstract

As a noncommutative generalization of the addition formula of theta functions, we construct a class of theta functions which are closed with respect to the Moyal star product of a fixed noncommutative parameter. These theta functions can be regarded as bases of the space of holomorphic homomorphisms between holomorphic line bundles over noncommutative complex tori.

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References

  1. Bayen F., Flato M., Frønsdal C., Lichnerowicz A., Sternheimer D. (1978). Deformation theory and quantization I, II. Ann. phys. 111, 61–110, 111–151,

    Google Scholar 

  2. Dabrowski L., Krajewski T., Landi G. (2000). Some properties of non-linear sigma models in noncommutative geometry. Int. J. Mod. Phys. B 14: 2367 hep-th/0003099

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Dabrowski L., Krajewski T., Landi G. (2003). Non-linear sigma-models in noncommutative geometry: fields with values in finite spaces. Mod. Phys. Lett. A 18:2371 math.QA/0309143

    Article  ADS  MathSciNet  Google Scholar 

  4. Dieng, M., Schwarz, A.: Differential and complex geometry of two-dimensional noncommutative tori. math.QA/0203160

  5. Fukaya, K.: Mirror symmetry of abelian varieties and multi theta functions. J. Algebraic Geom. 11, 393–512, (2002) preprint, Kyoto University (1998)

    Google Scholar 

  6. Ganor O.J., Ramgoolam S., Taylor W.I. (1997). Branes, fluxes and duality in M(atrix)-theory. Nucl. Phys. B 492: 191 hep-th/9611202

    Article  ADS  MathSciNet  Google Scholar 

  7. Gutt S., Rawnsley J. (1999). Equivalence of star products on a symplectic manifold. an introduction to Deligne’s Cech cohomology classes. J. Geom. Phys. 29:347–392

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. Ho P. (1998). Twisted bundle on quantum torus and BPS states in matrix theory. Phys. Lett. B 434: 41 hep-th/9803166

    Article  ADS  MathSciNet  Google Scholar 

  9. Igusa, J.: Theta functions, Die Grundlehren der mathematischen Wissenschaften, Band 194. Springer,Berlin, Heidelberg Newyork x+232 pp (1972)

  10. Kajiura H. (2002). Kronecker foliation, D1-branes and Morita equivalence of noncommutative two-tori. JHEP 0208: 050 hep-th/0207097

    Article  ADS  MathSciNet  Google Scholar 

  11. Kajiura, H.: Homological mirror symmetry on noncommutative two-tori. hep-th/0406233.

  12. Kajiura, H.: Categories of holomorphic line bundles on higher dimensional noncommutative complex tori. hep-th/0510119

  13. Kim, E., Kim, H.: Moduli spaces of standard holomorphic bundles on a noncommutative complex torus. math.QA/0312228

  14. Konechny, A., Schwarz, A.: Introduction to M(atrix) theory and noncommutative geometry. hep-th/0012145 and references therein

  15. Kontsevich, M.: Homological algebra of mirror symmetry. In Proceedings of the international congress of mathematicians. vol. 1, 2 (Zürich, 1994), pp. 120–139, Birkhäuser, Basel, (1995) math.AG/9411018

  16. Morariu, B., Zumino, B.: Super Yang-mills on the noncommutative torus. hep-th/9807198

  17. Moyal J.E. (1949). Quantum mechanics as a statistical theory. Proc. Cambridge Phil. Soc. 45:99–124

    Article  MATH  MathSciNet  Google Scholar 

  18. Mumford, D.: Tata lectures on theta. I. In: Musili, C., Nori,M., Previato, E. Stillman, M.(eds.): Progress in mathematics, vol. 28. Birkhäuser, Boston, xiii+235 pp. (1983)

  19. Omori H., Maeda Y., Yoshioka A. (1991). Weyl manifolds and deformation quantization. Adv. Math. 85:225–255

    Article  MathSciNet  Google Scholar 

  20. Polishchuk A., Schwarz A. (2003). Categories of holomorphic vector bundles on noncommutative two-tori. Commun. Math. Phys. 236:135 math.QA/0211262

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. Polishchuk A., Zaslow E. (1998). Categorical mirror symmetry: the elliptic curve”. Adv. Theor. Math. Phys. 2:443–470 math.AG/9801119

    MATH  MathSciNet  Google Scholar 

  22. Schwarz A. (2001). Theta functions on noncommutative tori. Lett. Math. Phys. 58: 81–90 math.QA/0107186

    Article  MATH  ADS  MathSciNet  Google Scholar 

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  1. Yukawa Institute for Theoretical Physcics, Kyoto University, Kyoto, 606-8502, Japan

    Hiroshige Kajiura

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  1. Hiroshige Kajiura
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Correspondence to Hiroshige Kajiura.

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Kajiura, H. Star Product Formula of Theta Functions. Lett Math Phys 75, 279–292 (2006). https://doi.org/10.1007/s11005-006-0051-0

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  • DOI: https://doi.org/10.1007/s11005-006-0051-0

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