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Moduli Spaces, Non-Commutative Geometry and Deformed Differential Categories

Received: 25 October 2014     Accepted: 2 November 2014     Published: 5 November 2014
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Abstract

The following research plays a central role in deformation theory. If x, is a moduli space over a field k, of characteristic zero, then a formal neighborhood of any point xϵx, is controlled by a differential graded Lie algebra. Then using the derived categories language we give an analogous of the before sentence in the setting of non-commutative geometry, considering some aspects E∞— rings and derived moduli problems related with these rings. After is obtained a scheme to spectrum; by functor Spec and their ∞— category functor inside of the space Fun_(hoat_∞ )to these E∞— rings and their derived moduli in field theory.

Published in Pure and Applied Mathematics Journal (Volume 3, Issue 6-2)

This article belongs to the Special Issue Integral Geometry Methods on Derived Categories in the Geometrical Langlands Program

DOI 10.11648/j.pamj.s.2014030602.13
Page(s) 12-19
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2014. Published by Science Publishing Group

Keywords

Deformed Category, E∞— Rings, Formal Moduli Problem, Koszul Duality, Non-Commutative Geometry

References
[1] A. Grothendieck, On the De Rham Cohomology of algebraic varieties, Publ. Math.I.H.E.S. 29 (1966) 95-103.
[2] M. Kontsevich, Y. Soibelman, Deformations of algebras over operads and the Deligne conjecture, Conference Moshe Flato 1999, Vol 1 (Dijon), 255-307, Math. Phys. Stud., 21, Kluwer Acad. Publ., Dordrecht, 2000.
[3] J. Milnor, “On spaces having the homotopy type of a CW-complex” Trans. Amer. Math. Soc. 90 (1959), 272–280.
[4] C. Teleman, The quantization conjecture revised, Ann. Of Math. (2) 152 (2000), 1-43.
[5] P. G. Goerss, “Topological modular forms (aftern Hopkins, Miller, and Lurie),” Available at arXiv:0910.5130v1
[6] A. Efimov, V. Lunts, D. Orlov, Deformations theoryof objects in homotopy and derived categories I: General Theory, Adv. Math. 222 (2009), no. 2, 359-401.
[7] A. Efimov, V. Lunts, D. Orlov, Deformations theoryof objects in homotopy and derived categories II: Pro-representability of the deformation functor, Available at arXiv:math/0702839v3.
[8] A. Efimov, V. Lunts, D. Orlov, Deformations theoryof objects in homotopy and derived categories III: Abelian categories, Available at arXiv:math/0702840v3.
[9] V. Hinich, DG coalgebras as formal stacks, J. Pure Appl. Algebra, 162 (2001), 209-250.
[10] V. Hinich, DG Deformations of homotopy algebras, Communications in Algebra, 32 (2004), 473-494.
[11] B.Toën,The homotopy theory of dg-categories and derived Morita theory, Invent. Math. 167 (2007), no. 3, 615-667.
[12] B.Toën, M. Vaquié, Moduli of objects in dg-categories, Ann. Sci. cole Norm. Sup. (4) 40 (2007), no. 3, 387-444.
[13] B.Toën, G. Vezzosi, From HAG to derived moduli stacks. Axiomatic, enriched and motivic homotopy theory, 173-216. NATO Sci. Ser II, Math. Phys. Chem., 131, Kluwer Acad. Publ. Dordrecht, 2004.
[14] D. Ben-zvi and D. Nadler, The character theory of complex group, 5 (2011) arXiv:0904.1247v2[math.RT],
[15] B. Fresse, Koszul duality of En-operads, Available as arXiv:0904.3123v6
[16] F. Bulnes, Penrose Transform on Induced DG/H-Modules and Their Moduli Stacks in the Field Theory, Advances in Pure Mathematics 3 (2) (2013) 246-253. doi: 10.4236/apm.2013.32035.
[17] F. Bulnes, Cohomology of Moduli Spaces in Differential Operators Classification to the Field Theory (II), in: Proceedings of Function Spaces, Differential Operators and Non-linear Analysis., 2011, Tabarz Thur, Germany, Vol. 1 (12) pp001-022.
[18] E. Frenkel, C. Teleman, Geometric Langlands Correspondence Near Opers, Available at arXiv:1306.0876v1.
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  • APA Style

    Ivan Verkelov. (2014). Moduli Spaces, Non-Commutative Geometry and Deformed Differential Categories. Pure and Applied Mathematics Journal, 3(6-2), 12-19. https://doi.org/10.11648/j.pamj.s.2014030602.13

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    ACS Style

    Ivan Verkelov. Moduli Spaces, Non-Commutative Geometry and Deformed Differential Categories. Pure Appl. Math. J. 2014, 3(6-2), 12-19. doi: 10.11648/j.pamj.s.2014030602.13

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    AMA Style

    Ivan Verkelov. Moduli Spaces, Non-Commutative Geometry and Deformed Differential Categories. Pure Appl Math J. 2014;3(6-2):12-19. doi: 10.11648/j.pamj.s.2014030602.13

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  • @article{10.11648/j.pamj.s.2014030602.13,
      author = {Ivan Verkelov},
      title = {Moduli Spaces, Non-Commutative Geometry and Deformed Differential Categories},
      journal = {Pure and Applied Mathematics Journal},
      volume = {3},
      number = {6-2},
      pages = {12-19},
      doi = {10.11648/j.pamj.s.2014030602.13},
      url = {https://doi.org/10.11648/j.pamj.s.2014030602.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.s.2014030602.13},
      abstract = {The following research plays a central role in deformation theory. If x, is a moduli space over a field k, of characteristic zero, then a formal neighborhood of any point xϵx, is controlled by a differential graded Lie algebra. Then using the derived categories language we give an analogous of the before sentence in the setting of non-commutative geometry, considering some aspects E∞— rings and derived moduli problems related with these rings. After is obtained a scheme to spectrum; by functor Spec and their ∞— category functor inside of the space Fun_(hoat_∞ )to these E∞— rings and their derived moduli in field theory.},
     year = {2014}
    }
    

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    AB  - The following research plays a central role in deformation theory. If x, is a moduli space over a field k, of characteristic zero, then a formal neighborhood of any point xϵx, is controlled by a differential graded Lie algebra. Then using the derived categories language we give an analogous of the before sentence in the setting of non-commutative geometry, considering some aspects E∞— rings and derived moduli problems related with these rings. After is obtained a scheme to spectrum; by functor Spec and their ∞— category functor inside of the space Fun_(hoat_∞ )to these E∞— rings and their derived moduli in field theory.
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Author Information
  • Research Group-Tescha, Dept. of Mathematics, Baikov Institute, Baikov, Russia

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