Penrose Transform on D-Modules, Moduli Spaces and Field Theory


We consider a generalization of the Radon-Schmid transform on coherent D-modules of sheaves of holomorphic complex bundles inside a moduli space, with the purpose of establishing the equivalences among geometric objects (vector bundles) and algebraic objects as they are the coherent D-modules, these last with the goal of obtaining conformal classes of connections of the holomorphic complex bundles. The class of these equivalences conforms a moduli space on coherent sheaves that define solutions in field theory. Also by this way, and using one generalization of the Penrose transform in the context of coherent D-modules we find conformal classes of the space-time that include the heterotic strings and branes geometry.

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F. Bulnes, "Penrose Transform on D-Modules, Moduli Spaces and Field Theory," Advances in Pure Mathematics, Vol. 2 No. 6, 2012, pp. 379-390. doi: 10.4236/apm.2012.26057.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] C. Marastoni and T. Tanisaki. “Radon Transforms for Quasi-Equivariant D-Modules on Generalized Flag Manifolds,” Differential Geometry and Its Applications, Vol. 18, No. 2, 2003, pp. 147-176. doi:10.1016/S0926-2245(02)00145-6
[2] A. Borel, et al, “Algebraic D-Modules,” 2nd Edition, Academic Press, Boston, 1987.
[3] S. Gindikin and G. Henkin, “Integral Geometry for -Cohomology in q-Linear Concave Domains in CPn,” Functional Analysis and Its Applications, Vol. 12, 1978, pp. 247-261.
[4] R. J. Baston and M. G. Eastwood, “Invariant Operators,” Twistor in Physics, Cambridge, 1981.
[5] F. Bulnes, “Integral Geometry and Complex Integral Operators Cohomology in Field Theory on Space-Time,” Proceedings of 1st International Congress of Applied Mathematics-UPVT, Government of State of Mexico, Mexico City, 2009, pp. 42-51.
[6] M. Kashiwara, “Representation Theory and D-Modules on Flag Varieties,” Astérisque, Vol. 173-174, No. 9, 1989, pp. 55-109.
[7] V. Knapp, “Harish-Chandra Modules and Penrose Transforms,” American Mathematical Society, Mount Holyoke College, 27 June - 3 July 1992, pp. 1-17.
[8] S. Alexakis, “On Conformally Invariant Differential Operators,” In: A. Juhl, Ed., Families of Conformal Covariant Differential Operators, Q-Curvature and Holography, Cornell University, Ithaca, 2007, pp. 1-50
[9] C. R. Graham, “Non Existence of Curved Conformally Invariant Operators,” 1980, in press.
[10] S. Gindikin, “Penrose Transform at Flag Domains,” The Erwin Schr?dinger International Institute for Mathematical Physics, Boltzmanngasse 9, A-1090, Wien, 2011.
[11] W. Schmid, “Homogeneous Complex Manifolds and Representations of Semisimple Lie Groups,” In: Representation Theory and Harmonic Analysis on Semisimple Lie Groups, American Mathematical Society, Providence, 1989, pp. 223-286.
[12] A. D’Agnolo and P. Schapira, “Radon-Penrose Transform for D-Modules,” Journal of Functional Analysis, Vol. 139, No. 2, 1996, pp. 349-382. doi:10.1006/jfan.1996.0089
[13] M. Kashiwara and W. Schmid, “Quasi-Equivariant D-Modules, Equivariant Derived Category, and Representations of Reductive Lie Groups, in Lie Theory and Geometry,” Birkh?user, Boston, 1994, pp. 457-488.
[14] R. Penrose, “Twistor Quantization and Curved Space-Time,” International Journal of Theoretical Physics, Vol. 1, No. 1, 1968, pp. 61-99. doi:10.1007/BF00668831
[15] R. J. Baston and M. G. Eastwood, “The Penrose Transform,” The Clarendon Press Oxford University Press, New York, 1989.
[16] M. G. Eastwood, R. Penrose and R. O. Wells Jr, “Cohomology and Massless Fields,” Communications in Mathematical Physics, Vol. 78, No. 3, 1981, pp. 305-351. doi:10.1007/BF01942327
[17] R. Penrose, “Solutions of the Zero-Rest-Mass Equations,” Journal of Mathematical Physics, Vol. 10, 1969, pp. 38-39. doi:10.1063/1.1664756
[18] F. Bulnes, “Cohomology of Moduli Spaces in Differential Operators Classification to the Field Theory (II),” Proceedings of FSDONA-11 (Function Spaces, Differential Operators and Non-linear Analysis), Tabarz Thur, Vol. 1 No. 12, 2011, pp. 1-22.
[19] T. N. Bailey and M. G. Eastwood, “Complex Para-Conformal Manifolds Their Differential Geometry and Twistor Theory,” Forum Mathematicum, Vol. 3, 1991, pp. 61- 103. doi:10.1515/form.1991.3.61
[20] M. F. Atiyah, “Magnetic Monopoles in Hyperbolic Space,” Oxford University Press, Oxford, 1987, pp. 1-34.
[21] S. Gindikin and G. Henkin, “The Penrose Transforms and Complex Integral Geometry Problems,” VINITI, Moscow, pp. 57-112.
[22] A. Kapustin, M. Kreuser and K. G. Schlesinger, “Homological Mirror Symmetry: New Developments and Perspectives,” Springer, Berlin, 2009.
[23] M. Kashiwara and T. Oshima, “Systems of Differential Equations with Regular Singularities and Their Boundary Value Problems,” Annals of Mathematics, Vol. 106, 1977, pp. 145-200. doi:10.2307/1971163
[24] F. Warner, “Differential Manifolds,” 2nd Edition, Springer-Verlag, Berlin, 1966.
[25] A. Grothendieck, “Techniques de Construction en Géométrie Analytique. I. Description Axiomatique de l’Espace de Teichmüller et de Ses Variantes,” Séminaire Henri Cartan, Vol. 13, No. 1, 1960, pp. 1-33.
[26] D. Mili?i?, “Algebraic D-Modules and Representation Theory of Semi-Simple Lie Groups,” American Mathematical Society, Providence, pp. 133-168.
[27] L. Kefeng, “Recent Results of Moduli Spaces on Riemann Surfaces,” JDG Conferences, Harvard, 27 June-3 July 2005.
[28] C. R. LeBrun, “Twistors, Ambitwistors and Conformal Gravity,” Twistor in Physics, Cambridge, 1981.
[29] S. Gindikin, “The Penrose Transforms of Flags Domains in F(CP2),” Contemporary Mathematical Physics, Vol. 175, No. 2, 1996, pp. 49-56.
[30] F. Bulnes, and M. Shapiro, “General Theory of Integrals to Analysis and Geometry,” SEPI-IPN, IM-UNAM, Mexico, 2007.
[31] S. Schroer, “Some Calabi-Yau Threefolds with Obstructed Deformations over the Witt Vectors,” Journal Compositio Mathematica, Vol. 140, No. 6, 2004, pp. 1579- 1592.
[32] F. Bulnes, “Research on Curvature of Homogeneous Spaces,” Department of Research in Mathematics and Engineering TESCHA, Government of State of Mexico, Mexico,2010.
[33] L. Mason and D. Skinner, “Heterotic Twistor-String Theory,” Oxford University, Oxford, 2007. arXiv:0708.2276v1 [hep-th]
[34] J. A. Wolf, “The Stein Condition for Cycle Spaces of Open Orbits on Complex Flag Manifolds,” Annals of Ma thematics, Vol. 136, No. 3, 1992, pp. 541-555. doi:10.2307/2946599
[35] F. Bulnes, “Design of Measurement and Detection De- vices of Curvature through of the Synergic Integral Operators of the Mechanics on Light Waves,” Proceedings of IMECE/ASME on Electronics and Photonics, Vol. 5, 2009, pp. 91-103. doi:10.1115/IMECE2009-10038
[36] F. Bulnes, “On the Last Progress of Cohomological In- duction in the Problem of Classification of Lie Groups Representations,” Proceeding of Masterful Conferences, International Conference of Infinite Dimensional Analysis and Topology, Ivano-Frankivsk, 27 May 1 June 2009, pp. 21-22.
[37] A. W. Knapp and N. Wallach, “Szego Kernels Associated with Discrete Series,” Inventiones Mathematicae, Vol. 34, No. 3, 1976, pp. 163-200. doi:10.1007/BF01403066
[38] S. Helgason, “The Radon Transform,” In: H. Bass, J. Oesterlé, A. Weinstein and Birkh?user, Eds., Progress in Mathematics, Birkh?user, Boston Mass, 1980.

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