Derived Categories in Langlands Geometrical Ramifications: Approaching by Penrose Transforms

Affiliation(s)

Research Department in Mathematics and Engineering, Tecnológico de Estudios Superiores de Chalco, Chalco, Mexico.

Research Department in Mathematics and Engineering, Tecnológico de Estudios Superiores de Chalco, Chalco, Mexico.

ABSTRACT

Some derived categories
and their deformed versions are used to develop a theory of the ramifications of
field studied in the geometrical Langlands program to obtain the correspondences
between moduli stacks and solution classes represented cohomologically under the
study of the kernels of the differential operators studied in their classification
of the corresponding field equations. The corresponding *D*-modules in this case may be viewed as sheaves of conformal blocks
(or co-invariants) (images under a version of the Penrose transform) naturally arising
in the framework of conformal field theory. Inside the geometrical Langlands correspondence
and in their cohomological context of strings can be established a framework of
the space-time through the different versions of the Penrose transforms and their
relation between them by intertwining operators (integral transforms that are isomorphisms
between cohomological spaces of orbital spaces of the space-time), obtaining the
functors that give equivalences of their corresponding categories.(For more
information,please refer to the PDF version.)

KEYWORDS

Geometrical Langlands Correspondence, Hecke Categories, Moduli Stacks, Penrose Transforms, Quasi-Coherent Sheaves

Geometrical Langlands Correspondence, Hecke Categories, Moduli Stacks, Penrose Transforms, Quasi-Coherent Sheaves

Cite this paper

Bulnes, F. (2014) Derived Categories in Langlands Geometrical Ramifications: Approaching by Penrose Transforms.*Advances in Pure Mathematics*, **4**, 253-260. doi: 10.4236/apm.2014.46034.

Bulnes, F. (2014) Derived Categories in Langlands Geometrical Ramifications: Approaching by Penrose Transforms.

References

[1] Bulnes, F. (2011) Cohomology of Moduli Spaces in Differential Operators Classification to the Field Theory (II). Proceedings of Function Spaces, Differential Operators and Non-linear Analysis, 1, 001-022.

[2] Bulnes, F. (2010) Cohomology of Moduli Spaces on Sheaves Coherent to Conformal Class of the Space-Time. Technical Report for XLIII-National Congress of Mathematics of SMM, (RESEARCH) Tuxtla GutiErrez.

[3] Bulnes, F. (2013) Penrose Transform on Induced DG/H-Modules and Their Moduli Stacks in the Field Theory. Advances in Pure Mathematics, 3, 246-253.

http://dx.doi.org/10.4236/apm.2013.32035

[4] Kashiwara, M. and Schmid, W. (1994) Quasi-Equivariant D-Modules, Equivariant Derived Category, and Representations of Reductive Lie Groups, in Lie Theory and Geometry. Progress in Mathematics, 123, 457-488.

[5] Bulnes, F. (2009) Integral Geometry and Complex Integral Operators Cohomology in Field Theory on Space-Time. Proceedings of 1st International Congress of Applied Mathematics-UPVT (Mexico), 1, 42-51.

[6] Kapustin, A., Kreuser, M. and Schlesinger, K.G. (2009) Homological Mirror Symmetry: New Developments and Perspectives. Springer, Berlin, Heidelberg.

[7] Bulnes, F. (2013) Geometrical Langlands Ramifications and Differential Operators Classification by Coherent D-Modules in Field Theory. Journal of Mathematics and System Sciences, 3, 491-507.

[8] Ben-Zvi, D. and Nadler, D. (2011) The Character Theory of Complex Group. arXiv:0904.1247v2[math.RT].

[9] Ben-Zvi, D., Francis, J. and Nadler, D. (2010) Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry. Journal of the American Mathematical Society, 23, 909-966.

http://dx.doi.org/10.1090/S0894-0347-10-00669-7

[10] D’Agnolo, A. and Schapira, P. (1996) Radon-Penrose Transform for D-Modules. Journal of Functional Analysis, 139, 349-382.

http://dx.doi.org/10.1006/jfan.1996.0089

[11] Marastoni, C. and Tanisaki, T. (2003) Radon Transforms for Quasi-Equivariant D-Modules on Generalized Flag Manifolds. Differential Geometry and Its Applications, 18, 147-176.

http://dx.doi.org/10.1016/S0926-2245(02)00145-6

[12] Bulnes, F. (2012) Penrose Transform on D-Modules, Moduli Spaces and Field Theory. Advances in Pure Mathematics, 2, 379-390.

http://dx.doi.org/10.4236/apm.2012.26057

[13] Bulnes, F. (2009) On the Last Progress of Cohomological Induction in the Problem of Classification of Lie Groups Representations. Proceeding of Masterful Conferences, International Conference of Infinite Dimensional Analysis and Topology (Ivano-Frankivsk, Ukraine), 1, 21-22.

[14] Knapp, A.W. and Wallach, N. (1976) Szego Kernels Associated with Discrete Series. Inventiones Mathematicae, 34, 163-200.

http://dx.doi.org/10.1007/BF01403066

[15] Bulnes, F. (2013) Orbital Integrals on Reductive Lie Groups and their Algebras. Intech, Rijeka.

http://www.intechopen.com/books/orbital-integrals-

on-reductive-lie-groups-and-their-algebras/orbital-

integrals-on-reductive-lie-groups-and-their-algebrasB

[16] Gindikin, S. (1978) Penrose Transform at Flag Domains. The Erwin Schrodinger International Institute for Mathematical Physics, Wien.

[17] Frenkel, E. (2004) Ramifications of the Geometric Langlands Program. CIME Summer School “Representation Theory and Complex Analysis”, Venice.

[18] Mebkhout, Z. (1977) Local Cohomology of Analytic Spaces. Proceedings of the Oji Seminar on Algebraic Analysis and the RIMS Symposium on Algebraic Analysis, 12, 247-256.

[19] Verdier, J.L. (1967) A Duality Theorem in the Etale Cohomoloy of Schemes. Proceedings of a Conference on Local Fields: NUFFIC Summer School, Berlin, New York, 184-198.

[20] Verdier, J.L. (1995) DualitE Dans la Cohomologie des Espaces Localement Compacts. SEminaire Bourbaki, Vol. 9, SociEtE MathEmatique de France, Paris.

[21] Samman, S. and Lechtenfeld, O. (2006) Matrix Models and D-Branes in Twistor String Theory. JHEP, hep-th/0511130.

[22] Grothendieck, A. (1966) On the De Rham Cohomology of Algebraic Varieties. Publications MathEmatiques de l’IHES, 29, 95-103.

http://dx.doi.org/10.1007/BF02684807

[23] Bulnes, F. and Shapiro, M. (2007) On a General Theory of Integral Operators to Geometry and Analysis. Applied Mathematics, 3, Special Section SEPI-IPN, IM-UNAM, Mexico City.

[1] Bulnes, F. (2011) Cohomology of Moduli Spaces in Differential Operators Classification to the Field Theory (II). Proceedings of Function Spaces, Differential Operators and Non-linear Analysis, 1, 001-022.

[2] Bulnes, F. (2010) Cohomology of Moduli Spaces on Sheaves Coherent to Conformal Class of the Space-Time. Technical Report for XLIII-National Congress of Mathematics of SMM, (RESEARCH) Tuxtla GutiErrez.

[3] Bulnes, F. (2013) Penrose Transform on Induced DG/H-Modules and Their Moduli Stacks in the Field Theory. Advances in Pure Mathematics, 3, 246-253.

http://dx.doi.org/10.4236/apm.2013.32035

[4] Kashiwara, M. and Schmid, W. (1994) Quasi-Equivariant D-Modules, Equivariant Derived Category, and Representations of Reductive Lie Groups, in Lie Theory and Geometry. Progress in Mathematics, 123, 457-488.

[5] Bulnes, F. (2009) Integral Geometry and Complex Integral Operators Cohomology in Field Theory on Space-Time. Proceedings of 1st International Congress of Applied Mathematics-UPVT (Mexico), 1, 42-51.

[6] Kapustin, A., Kreuser, M. and Schlesinger, K.G. (2009) Homological Mirror Symmetry: New Developments and Perspectives. Springer, Berlin, Heidelberg.

[7] Bulnes, F. (2013) Geometrical Langlands Ramifications and Differential Operators Classification by Coherent D-Modules in Field Theory. Journal of Mathematics and System Sciences, 3, 491-507.

[8] Ben-Zvi, D. and Nadler, D. (2011) The Character Theory of Complex Group. arXiv:0904.1247v2[math.RT].

[9] Ben-Zvi, D., Francis, J. and Nadler, D. (2010) Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry. Journal of the American Mathematical Society, 23, 909-966.

http://dx.doi.org/10.1090/S0894-0347-10-00669-7

[10] D’Agnolo, A. and Schapira, P. (1996) Radon-Penrose Transform for D-Modules. Journal of Functional Analysis, 139, 349-382.

http://dx.doi.org/10.1006/jfan.1996.0089

[11] Marastoni, C. and Tanisaki, T. (2003) Radon Transforms for Quasi-Equivariant D-Modules on Generalized Flag Manifolds. Differential Geometry and Its Applications, 18, 147-176.

http://dx.doi.org/10.1016/S0926-2245(02)00145-6

[12] Bulnes, F. (2012) Penrose Transform on D-Modules, Moduli Spaces and Field Theory. Advances in Pure Mathematics, 2, 379-390.

http://dx.doi.org/10.4236/apm.2012.26057

[13] Bulnes, F. (2009) On the Last Progress of Cohomological Induction in the Problem of Classification of Lie Groups Representations. Proceeding of Masterful Conferences, International Conference of Infinite Dimensional Analysis and Topology (Ivano-Frankivsk, Ukraine), 1, 21-22.

[14] Knapp, A.W. and Wallach, N. (1976) Szego Kernels Associated with Discrete Series. Inventiones Mathematicae, 34, 163-200.

http://dx.doi.org/10.1007/BF01403066

[15] Bulnes, F. (2013) Orbital Integrals on Reductive Lie Groups and their Algebras. Intech, Rijeka.

http://www.intechopen.com/books/orbital-integrals-

on-reductive-lie-groups-and-their-algebras/orbital-

integrals-on-reductive-lie-groups-and-their-algebrasB

[16] Gindikin, S. (1978) Penrose Transform at Flag Domains. The Erwin Schrodinger International Institute for Mathematical Physics, Wien.

[17] Frenkel, E. (2004) Ramifications of the Geometric Langlands Program. CIME Summer School “Representation Theory and Complex Analysis”, Venice.

[18] Mebkhout, Z. (1977) Local Cohomology of Analytic Spaces. Proceedings of the Oji Seminar on Algebraic Analysis and the RIMS Symposium on Algebraic Analysis, 12, 247-256.

[19] Verdier, J.L. (1967) A Duality Theorem in the Etale Cohomoloy of Schemes. Proceedings of a Conference on Local Fields: NUFFIC Summer School, Berlin, New York, 184-198.

[20] Verdier, J.L. (1995) DualitE Dans la Cohomologie des Espaces Localement Compacts. SEminaire Bourbaki, Vol. 9, SociEtE MathEmatique de France, Paris.

[21] Samman, S. and Lechtenfeld, O. (2006) Matrix Models and D-Branes in Twistor String Theory. JHEP, hep-th/0511130.

[22] Grothendieck, A. (1966) On the De Rham Cohomology of Algebraic Varieties. Publications MathEmatiques de l’IHES, 29, 95-103.

http://dx.doi.org/10.1007/BF02684807

[23] Bulnes, F. and Shapiro, M. (2007) On a General Theory of Integral Operators to Geometry and Analysis. Applied Mathematics, 3, Special Section SEPI-IPN, IM-UNAM, Mexico City.