Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 002, 47 pages      arXiv:1906.09834      https://doi.org/10.3842/SIGMA.2020.002

The Schwarz-Voronov Embedding of ${\mathbb Z}_{2}^{n}$-Manifolds

Andrew James Bruce, Eduardo Ibarguengoytia and Norbert Poncin
Mathematics Research Unit, University of Luxembourg, Maison du Nombre 6, avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg

Received July 10, 2019, in final form December 30, 2019; Published online January 08, 2020

Abstract
Informally, ${\mathbb Z}_2^n$-manifolds are 'manifolds' with ${\mathbb Z}_2^n$-graded coordinates and a sign rule determined by the standard scalar product of their ${\mathbb Z}_2^n$-degrees. Such manifolds can be understood in a sheaf-theoretic framework, as supermanifolds can, but with significant differences, in particular in integration theory. In this paper, we reformulate the notion of a ${\mathbb Z}_2^n$-manifold within a categorical framework via the functor of points. We show that it is sufficient to consider ${\mathbb Z}_2^n$-points, i.e., trivial ${\mathbb Z}_2^n$-manifolds for which the reduced manifold is just a single point, as 'probes' when employing the functor of points. This allows us to construct a fully faithful restricted Yoneda embedding of the category of ${\mathbb Z}_2^n$-manifolds into a subcategory of contravariant functors from the category of ${\mathbb Z}_2^n$-points to a category of Fréchet manifolds over algebras. We refer to this embedding as the Schwarz-Voronov embedding. We further prove that the category of ${\mathbb Z}_2^n$-manifolds is equivalent to the full subcategory of locally trivial functors in the preceding subcategory.

Key words: supergeometry; superalgebra; ringed spaces; higher grading; functor of points.

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References

  1. Aizawa N., Isaac P.S., Segar J., ${\mathbb Z}_2\times{\mathbb Z}_2$ generalizations of ${\mathcal N}=1$ superconformal Galilei algebras and their representations, J. Math. Phys. 60 (2019), 023507, 11 pages, arXiv:1808.09112.
  2. Aizawa N., Kuznetsova Z., Tanaka H., Toppan F., $\mathbb{Z}_2\times\mathbb{Z}_2$-graded Lie symmetries of the Lévy-Leblond equations, Prog. Theor. Exp. Phys. (2016), 123A01, 26 pages, arXiv:1609.08224.
  3. Aizawa N., Segar J., ${\mathbb Z}_2\times{\mathbb Z}_2$ generalizations of ${\mathcal N}=2$ super Schrödinger algebras and their representations, J. Math. Phys. 58 (2017), 113501, 14 pages, arXiv:1705.10414.
  4. Albuquerque H., Majid S., Quasialgebra structure of the octonions, J. Algebra 220 (1999), 188-224, arXiv:math.QA/9802116.
  5. Albuquerque H., Majid S., Clifford algebras obtained by twisting of group algebras, J. Pure Appl. Algebra 171 (2002), 133-148, arXiv:math.QA/0011040.
  6. Azarmi S., Foliations associated with the structure of a manifold over a Grassmann algebra of even degree exterior forms, Russian Math. 56 (2012), 76-78.
  7. Balduzzi L., Carmeli C., Fioresi R., The local functors of points of supermanifolds, Expo. Math. 28 (2010), 201-217, arXiv:0908.1872.
  8. Balduzzi L., Carmeli C., Fioresi R., A comparison of the functors of points of supermanifolds, J. Algebra Appl. 12 (2013), 1250152, 41 pages, arXiv:0902.1824.
  9. Berezin F.A., Introduction to superanalysis, Mathematical Physics and Applied Mathematics, Vol. 9, D. Reidel Publishing Co., Dordrecht, 1987.
  10. Bruce A.J., On a ${\mathbb Z}_2^n$-graded version of supersymmetry, Symmetry 11 (2019), 116, 20 pages, arXiv:1812.02943.
  11. Bruce A.J., Ibarguengoytia E., The graded differential geometry of mixed symmetry tensors, Arch. Math. (Brno) 55 (2019), 123-137, arXiv:1806.04048.
  12. Bruce A.J., Poncin N., Functional analytic issues in ${\mathbb Z}_2^n$-geometry, Rev. Un. Mat. Argentina, to appear, arXiv:1807.11739.
  13. Bruce A.J., Poncin N., Products in the category of $\mathbb Z^n_2$-manifolds, J. Nonlinear Math. Phys. 26 (2019), 420-453, arXiv:1807.11740.
  14. Conway J.B., A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, Vol. 96, Springer-Verlag, New York, 2007.
  15. Covolo T., Grabowski J., Poncin N., The category of $\mathbb{Z}_2^n$-supermanifolds, J. Math. Phys. 57 (2016), 073503, 16 pages, arXiv:1602.03312.
  16. Covolo T., Grabowski J., Poncin N., Splitting theorem for $\mathbb{Z}_2^n$-supermanifolds, J. Geom. Phys. 110 (2016), 393-401, arXiv:1602.03671.
  17. Covolo T., Kwok S., Poncin N., Differential calculus on ${\mathbb Z}_{2}^{n}$-supermanifolds, arXiv:1608.00949.
  18. Covolo T., Kwok S., Poncin N., The Frobenius theorem for ${\mathbb Z}_{2}^{n}$-supermanifolds, arXiv:1608.00949.
  19. Covolo T., Ovsienko V., Poncin N., Higher trace and Berezinian of matrices over a Clifford algebra, J. Geom. Phys. 62 (2012), 2294-2319, arXiv:1109.5877.
  20. Di Brino G., Pištalo D., Poncin N., Koszul-Tate resolutions as cofibrant replacements of algebras over differential operators, J. Homotopy Relat. Struct. 13 (2018), 793-846, arXiv:1801.03770.
  21. Di Brino G., Pištalo D., Poncin N., Homotopical algebraic context over differential operators, J. Homotopy Relat. Struct. 14 (2019), 293-347, arXiv:1706.05922.
  22. Drühl K., Haag R., Roberts J.E., On parastatistics, Comm. Math. Phys. 18 (1970), 204-226.
  23. Goodman R., Wallach N.R., Symmetry, representations, and invariants, Graduate Texts in Mathematics, Vol. 255, Springer, Dordrecht, 2009.
  24. Green H.S., A generalized method of field quantization, Phys. Rev. 90 (1953), 270-273.
  25. Greenberg O.W., Messiah A.M.L., Selection rules for parafields and the absence of para particles in nature, Phys. Rev. 138 (1965), B1155-B1167.
  26. Grothendieck A., Introduction to functorial algebraic geometry, Part 1: Affine algebraic geometry, Summer School in Buffalo, 1973, Lecture notes by Federico Gaeta.
  27. Hamilton R.S., The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 65-222.
  28. Kol'ař I., Michor P.W., Slov'ak J., Natural operations in differential geometry, Springer-Verlag, Berlin, 1993.
  29. Konechny A., Schwarz A., On $(k\oplus l|q)$-dimensional supermanifolds, in Supersymmetry and Quantum Field Theory (Kharkov, 1997), Lecture Notes in Phys., Vol. 509, Springer, Berlin, 1998, 201-206, arXiv:hep-th/9706003.
  30. Konechny A., Schwarz A., Theory of $(k\oplus l|q)$-dimensional supermanifolds, Selecta Math. (N.S.) 6 (2000), 471-486.
  31. Leites D. (Editor), Seminar on supersymmetry, Vol. 1, Algebra and calculus: main chapters, MCCME, Moscow, 2011.
  32. Mac Lane S., Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York, 1998.
  33. Mohammadi M., Salmasian H., The Gelfand-Naimark-Segal construction for unitary representations of $\mathbb{Z}_2^n$-graded Lie supergroups, in 50th Seminar ''Sophus Lie'', Banach Center Publ., Vol. 113, Polish Acad. Sci. Inst. Math., Warsaw, 2017, 263-274, arXiv:1709.06546.
  34. Molotkov V., Banach supermanifolds, in Differential Geometric Methods in Theoretical Physics (Shumen, 1984), World Sci. Publishing, Singapore, 1986, 117-125.
  35. Poncin N., Towards integration on colored supermanifolds, in Geometry of Jets and Fields, Banach Center Publ., Vol. 110, Polish Acad. Sci. Inst. Math., Warsaw, 2016, 201-217.
  36. Rittenberg V., Wyler D., Generalized superalgebras, Nuclear Phys. B 139 (1978), 189-202.
  37. Rittenberg V., Wyler D., Sequences of ${\mathbb Z}_{2}\oplus {\mathbb Z}_{2}$ graded Lie algebras and superalgebras, J. Math. Phys. 19 (1978), 2193-2200.
  38. Rogers A., Supermanifolds. Theory and applications, World Sci. Publ. Co. Pte. Ltd., Hackensack, NJ, 2007.
  39. Rudin W., Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.
  40. Sachse C., Global analytic approach to super Teichmüller spaces, Ph.D. Thesis, Universität Leipzig, 2007, arXiv:0902.3289.
  41. Saunders D.J., The geometry of jet bundles, London Mathematical Society Lecture Note Series, Vol. 142, Cambridge University Press, Cambridge, 1989.
  42. Schmitt T., Supergeometry and quantum field theory, or: what is a classical configuration?, Rev. Math. Phys. 9 (1997), 993-1052, arXiv:hep-th/9607132.
  43. Schwarz A.S., Supergravity, complex geometry and $G$-structures, Comm. Math. Phys. 87 (1982), 37-63.
  44. Schwarz A.S., On the definition of superspace, Theoret. and Math. Phys. 60 (1984), 657-660.
  45. Sharko V.V., Functions on manifolds, Translations of Mathematical Monographs, Vol. 131, Amer. Math. Soc., Providence, RI, 1993.
  46. Shurygin V.V., On the cohomology of manifolds over local algebras, Russian Math. 40 (1996), no. 9, 67-81.
  47. Shurygin V.V., The structure of smooth mappings over Weil algebras and the category of manifolds over algebras, Lobachevskii J. Math. 5 (1999), 29-55.
  48. Shurygin V.V., Smooth manifolds over local algebras and Weil bundles, J. Math. Sci. 108 (2002), 249-294.
  49. Toën B., Vezzosi G., Homotopical algebraic geometry. I. Topos theory, Adv. Math. 193 (2005), 257-372, arXiv:math.AG/0207028.
  50. Toën B., Vezzosi G., Homotopical algebraic geometry. II. Geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008), x+224 pages, arXiv:math.AG/0404373.
  51. Tolstoy V.N., Super-de Sitter and alternative super-Poincaré symmetries, in Lie Theory and its Applications in Physics, Springer Proc. Math. Stat., Vol. 111, Springer, Tokyo, 2014, 357-367, arXiv:1610.01566.
  52. Trèves F., Topological vector spaces, distributions and kernels, Academic Press, New York - London, 1967.
  53. Vishnevskiǐ V.V., Shirokov A.P., Shurygin V.V., Spaces over algebras, Kazanskiǐ Gosudarstvennyǐ Universitet, Kazan, 1985.
  54. Voronov A.A., Maps of supermanifolds, Theoret. and Math. Phys. 60 (1984), 660-664.
  55. Waelbroeck L., Topological vector spaces and algebras, Lecture Notes in Math., Vol. 230, Springer-Verlag, Berlin - New York, 1971.
  56. Weil A., Théorie des points proches sur les variétés différentiables, in Géométrie différentielle. Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1953, Centre National de la Recherche Scientifique, Paris, 1953, 111-117.
  57. Weyl H., The classical groups. Their invariants and representations, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997.
  58. Yang W., Jing S., A new kind of graded Lie algebra and parastatistical supersymmetry, Sci. China Ser. A 44 (2001), 1167-1173, arXiv:math-ph/0212004.

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