Article
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
supermanifold; almost complex structure; Riemannian metric; non-split
supermanifold; almost complex structure; Riemannian metric; non-split
Summary:
Non-split almost complex supermanifolds and non-split Riemannian supermanifolds are studied. The first obstacle for a splitting is parametrized by group orbits on an infinite dimensional vector space. For almost complex structures, the existence of a splitting is equivalent to the existence of local coordinates in which the almost complex structure can be represented by a purely numerical matrix, i.e. containing no Grassmann variables. For Riemannian metrics, terms up to degree 2 are allowed in such a local matrix representation, in order to preserve non-degeneracy. It is further shown that non-split structures appear in the almost complex case as deformations of a split reduction and in the Riemannian case as the deformation of an underlying metric. In contrast to non-split deformations of complex supermanifolds, these deformations can be restricted by cut-off functions to local deformations. A class of examples of nowhere split structures constructed from almost complex manifolds of dimension $6$ and higher, is provided for both cases.
References:
[1] Batchelor, M.: The structure of supermanifolds. Trans. Amer. Math. Soc. 253 (1979), 329–338. DOI 10.1090/S0002-9947-1979-0536951-0 | MR 0536951
[2] Batchelor, M.: Two approaches to supermanifolds. Trans. Amer. Math. Soc. 258 (1980), 257–270. DOI 10.1090/S0002-9947-1980-0554332-9 | MR 0554332
[3] Berezin, F.A., Leites, D.A.: Supermanifolds. Dokl. Akad. Nauk SSSR 224 (3) (1975), 505–508. MR 0402795
[4] Deligne, P., Morgan, J.W.: Notes on supersymmetry (following Joseph Bernstein). Quantum fields and strings: a course for mathematicians, vol. 1, AMS, 1999, pp. 41–97. MR 1701597 | Zbl 1170.58302
[5] Donagi, R., Witten, E.: Supermoduli space in not projected. Proc. Sympos. Pure Math., 2015, String-Math 2012, pp. 19–71. MR 3409787
[6] Green, P.: On holomorphic graded manifolds. Proc. Amer. Math. Soc. 85 (4) (1982), 587–590. DOI 10.1090/S0002-9939-1982-0660609-6 | MR 0660609
[7] Kostant, B.: Graded manifolds, graded Lie theory and prequantization. Lecture Notes in Math., vol. 570, Springer Verlag, 1987, pp. 177–306. MR 0580292
[8] Koszul, J.-L.: Connections and splittings of supermanifolds. Differential Geom. Appl. 4 (2) (1994), 151–161. DOI 10.1016/0926-2245(94)00011-5 | MR 1279014
[9] Manin, Y.I.: Gauge field theory and complex geometry. Grundlehren der Mathematischen Wissenschaften, vol. 289, Springer-Verlag, Berlin, 1988, Translated from the Russian by N. Koblitz and J.R. King. MR 0954833
[10] Massey, W.S.: Obstructions to the existence of almost complex structures. Bull. Amer. Math. Soc. 67 (1961), 559–564. DOI 10.1090/S0002-9904-1961-10690-3 | MR 0133137
[11] McDuff, D., Salamon, D.: Introduction to symplectic topology. oxford mathematical monographs. oxford science publications ed., The Clarendon Press, Oxford University Press, New York, 1995. MR 1373431
[12] McHugh, A.: A Newlander-Nirenberg theorem for supermanifold. J. Math. Phys. 30 (5) (1989), 1039–1042. DOI 10.1063/1.528373 | MR 0992576
[13] Rothstein, M.: Deformations of complex supermanifolds. Proc. Amer. Math. Soc. 95 (2) (1985), 255–260. DOI 10.1090/S0002-9939-1985-0801334-0 | MR 0801334
[14] Rothstein, M.: The structure of supersymplectic supermanifolds. Lecture Notes in Phys., Springer, Berlin, 1991, Differential geometric methods in theoretical physics (Rapallo, 1990), pp. 331–343. MR 1134168
[15] Thomas, E.: Complex structures on real vector bundles. Amer. J. Math. 89 (1967), 887–908. DOI 10.2307/2373409 | MR 0220310
[16] Vaintrob, A.Yu.: Deformations of complex structures on supermanifolds. Functional Anal. Appl. 18 (2) (1984), 135–136. DOI 10.1007/BF01077827 | MR 0745703
[17] Varadarajan, V.S.: Supersymmetry for mathematicians: An introduction. Courant Lect. Notes Math. 11, New York University, Courant Institute of Mathematical Sciences, 2004. MR 2069561
[18] Vishnyakova, E.: The splitting problem for complex homogeneous supermanifolds. J. Lie Theory 25 (2) (2015), 459–476. MR 3346067
Search
Partner of
