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Penrose Transform on Induced *D _{G/H}*-Modules and Their Moduli Stacks in the Field Theory

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We consider generalizations of the Radon-Schmid transform on coherent *D*_{G/H}-Modules, with the intention of obtaining the equivalence between geometric objects (vector bundles) and algebraic objects (*D*-Modules) characterizing conformal classes in the space-time that determine a space moduli [1] on coherent sheaves for the securing solutions in field theory [2]. In a major context, elements of derived categories like *D*-branes and heterotic strings are considered, and using the geometric Langlands program, a moduli space is obtained of equivalence between certain geometrical pictures (non-conformal world sheets [3]) and physical stacks (derived sheaves), that establishes equivalence between certain theories of super symmetries of field of a Penrose transform that generalizes the implications given by the Langlands program. With it we obtain extensions of a cohomology of integrals for a major class of field equations to corresponding Hecke category.

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*D*-Modules and Their Moduli Stacks in the Field Theory,"

_{G/H}*Advances in Pure Mathematics*, Vol. 3 No. 2, 2013, pp. 246-253. doi: 10.4236/apm.2013.32035.

[1] | F. Bulnes, “Cohomology of Moduli Spaces in Differential Operators Classification to the Field Theory (II),” Proceedings of FSDONA-11 (Function Spaces, Differential Operators and Nonlinear Analysis, 2011), Tabarz Thur, Vol. 1, No. 12, 2011, pp. 1-22. |

[2] | F. Bulnes, “Cohomology of Moduli Spaces on Sheaves Coherent to Conformal Class of the Space-Time,” XLIII-National Congress of Mathematics of SMM, (RESEAR-CH) Tuxtla GutiErrez, Chiapas, 2010. |

[3] | F. Bulnes, “Electromagnetic Gauges and Maxwell Lagrangians Applied to the Determination of Curvature in the Space-Time and Their Applications,” Journal of Electromagnetic Analysis and Applications, Vol. 4, No. 6, 2012, pp. 252-266. doi:10.4236/jemaa.2012.46035 |

[4] | 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 |

[5] | T. N. Bailey and M. G. Eastwood, “Complex Para-Conformal Manifolds Their Differential Geometry and Twistor Theory,” Forum Mathematicum, Vol. 3, No. 1, 1991, pp. 61-103. doi:10.1515/form.1991.3.61 |

[6] | 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 |

[7] | R. Dijkgraaf and E. Witten, “Topological Gauge Theories and Group Cohomology,” 1989. |

[8] | R. Hartshorne, “Deformation Theory (Graduate Texts in Mathematics),” Springer, New York, 2010. |

[9] | M. Kontsevich, “Deformation Quantization of Poisson manifolds I, First European Congress of Mathematics,” Progress in Mathematics, Vol. II, 1992, pp. 97-121. doi:10.1007/978-1-4419-1596-2 |

[10] | M. Kontsevich, “Formality Conjecture,” Math & Physics Study, Vol. 20, Kluwer Academic Publishers, Dordrecht, 1997, pp. 139-156. |

[11] | L. Mason and D. Skinner, “Heterotic Twistor-String Theory,” Nuclear Physics B, Vol. 795, No. 1, 2007, pp. 105- 137. |

[12] | S. Gindikin, “Penrose Transform at Flag Domains,” Oxford University Press, Oxford, 1998, pp. 383-393. |

[13] | M. Kashiwara and W. Schmid, “Quasi-equivariant D-Modules, Equivariant Derived Category, and Representations of Reductive Lie Groups,” Lie Theory and Geometry, Vol. 123, Birkhauser, Boston, 1994, pp. 457-488. |

[14] | K. Fukaya, “Floer Homology and Mirror Symmetry I,” Kyoto University, Kyoto, pp. 606-8224. |

[15] | M. K. Murray and J. W. Rice, “Algebraic Local Cohomology and the Cousin-Dolbeault Complex,” Preprint. |

[16] | J. W. Rice, “Cousin Complexes and Resolutions of Representations,” In: M. Eastwood, J. Wolf and R. Zierau Eds., The Penrose Transform and Analytic Cohomology in Representation Theory, Vol. 154, 1993. doi:10.1090/conm/154/01364 |

[17] | A. Grothendieck, “On the De Rham Cohomology of Algebraic Varieties,” Publications MathEmatiques de l’IHES, Vol. 29, No. 1, 1966, pp. 95-103. doi:10.1007/BF02684807 |

[18] | M. Kashiwara, “On the Maximally Overdetermined Systems of Linear Differential Equation I,” Publications of The Research Institute for Mathematical Sciences, Vol. 10, No. 2, 1975, pp. 563-579. doi:10.2977/prims/1195192011 |

[19] | M. Kashiwara, “On the Rationality of the Roots of b-Functions,” Lettre a Malgrange Janvier, Seminaire Grenoble, 1975-1976. |

[20] | M. Kashiwara and W. Schmid, “Quasi-Equivariant D-Modules, Equivariant Derived Category, and Representations of Reductive Lie Groups, in Lie Theory and Geometry,” Progress in Mathematics, Birkhauser, Boston, Vol. 123, 1994, pp. 457-488. |

[21] | J.-P. Ramis and G. Ruget, “Complex Dualisant en Geometrie Analytique,” Publications MathEmatiques de l’IHES, Vol. 38, 1971, pp. 77-91. |

[22] | J.-P. Ramis and G. Ruget, “Dualite et Residu,” Inventiones Mathematicae, Vol. 26, No. 2, 1974, pp. 89-131. |

[23] | A. Kapustin, M. Kreuser and K. G. Schlesinger, “Homological Mirror Symmetry: New Developments and Perspectives,” Springer. Berlin, Heidelberg, 2009. |

[24] | Z. Mebkhout, “Local Cohomology of Analytic Spaces,” Research Institute for Mathematical Sciences, Vol. 12, No. 2, 1977, pp. 247-256. |

[25] | J. Lurie, Higher Algebra. http://www.math.harvard.edu/?lurie/ |

[26] | C. Samann and M. Wolf, “Constraint and Super Yang-Mills Equations on the Deformed Superspace R^(4|16)_\ hbar,” Journal of High Energy Physics, Vol. 2004, No. 3, 2004, p. 48. |

[27] | D. Ben-zvi and D. Nadler, “The Character Theory of Complex Group,” Representation Theory, 5 Jun 2011. |

[28] | C. Marastoni and T. Tanisaki. “Radon Transforms for Quasi-Equivariant D-Modules on Generalized Flag Ma- nifolds,” Differential Geometry and Its Applications, Vol. 18, No. 2, 2003, pp. 147-176. doi:10.1016/S0926-2245(02)00145-6 |

[29] | C. Samann and A. D. Popov, “On Supertwistors, the Penrose-Ward Transform and N=4 Super Yang-Mills Theory,” Advances in Theoretical and Mathematical Physics, Vol. 9, 2005, pp. 931-988. |

[30] | R. J. Baston and M. G. Eastwood. “The Penrose Trans- form,” Oxford Mathematical Monographs, The Claren- don Press Oxford University Press, New York, 1989. |

[31] | E. Frenkel, “Ramifications of the Geometric Langlands Program,” Representation Theory and Complex Analysis, Vol. 1931, 2004, pp. 51-135. |

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