Non-split almost complex and non-split Riemannian supermanifolds

Matthias Kalus

Archivum Mathematicum (2019)

  • Volume: 055, Issue: 4, page 229-238
  • ISSN: 0044-8753

Abstract

top
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.

How to cite

top

Kalus, Matthias. "Non-split almost complex and non-split Riemannian supermanifolds." Archivum Mathematicum 055.4 (2019): 229-238. <http://eudml.org/doc/294151>.

@article{Kalus2019,
abstract = {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.},
author = {Kalus, Matthias},
journal = {Archivum Mathematicum},
keywords = {supermanifold; almost complex structure; Riemannian metric; non-split},
language = {eng},
number = {4},
pages = {229-238},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Non-split almost complex and non-split Riemannian supermanifolds},
url = {http://eudml.org/doc/294151},
volume = {055},
year = {2019},
}

TY - JOUR
AU - Kalus, Matthias
TI - Non-split almost complex and non-split Riemannian supermanifolds
JO - Archivum Mathematicum
PY - 2019
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 055
IS - 4
SP - 229
EP - 238
AB - 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.
LA - eng
KW - supermanifold; almost complex structure; Riemannian metric; non-split
UR - http://eudml.org/doc/294151
ER -

References

top
  1. Batchelor, M., 10.1090/S0002-9947-1979-0536951-0, Trans. Amer. Math. Soc. 253 (1979), 329–338. (1979) MR0536951DOI10.1090/S0002-9947-1979-0536951-0
  2. Batchelor, M., 10.1090/S0002-9947-1980-0554332-9, Trans. Amer. Math. Soc. 258 (1980), 257–270. (1980) MR0554332DOI10.1090/S0002-9947-1980-0554332-9
  3. Berezin, F.A., Leites, D.A., Supermanifolds, Dokl. Akad. Nauk SSSR 224 (3) (1975), 505–508. (1975) MR0402795
  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. (1999) Zbl1170.58302MR1701597
  5. Donagi, R., Witten, E., Supermoduli space in not projected, Proc. Sympos. Pure Math., 2015, String-Math 2012, pp. 19–71. (2015) MR3409787
  6. Green, P., 10.1090/S0002-9939-1982-0660609-6, Proc. Amer. Math. Soc. 85 (4) (1982), 587–590. (1982) MR0660609DOI10.1090/S0002-9939-1982-0660609-6
  7. Kostant, B., Graded manifolds, graded Lie theory and prequantization, Lecture Notes in Math., vol. 570, Springer Verlag, 1987, pp. 177–306. (1987) MR0580292
  8. Koszul, J.-L., 10.1016/0926-2245(94)00011-5, Differential Geom. Appl. 4 (2) (1994), 151–161. (1994) MR1279014DOI10.1016/0926-2245(94)00011-5
  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. (1988) MR0954833
  10. Massey, W.S., 10.1090/S0002-9904-1961-10690-3, Bull. Amer. Math. Soc. 67 (1961), 559–564. (1961) MR0133137DOI10.1090/S0002-9904-1961-10690-3
  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. (1995) MR1373431
  12. McHugh, A., 10.1063/1.528373, J. Math. Phys. 30 (5) (1989), 1039–1042. (1989) MR0992576DOI10.1063/1.528373
  13. Rothstein, M., 10.1090/S0002-9939-1985-0801334-0, Proc. Amer. Math. Soc. 95 (2) (1985), 255–260. (1985) MR0801334DOI10.1090/S0002-9939-1985-0801334-0
  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. (1991) MR1134168
  15. Thomas, E., 10.2307/2373409, Amer. J. Math. 89 (1967), 887–908. (1967) MR0220310DOI10.2307/2373409
  16. Vaintrob, A.Yu., 10.1007/BF01077827, Functional Anal. Appl. 18 (2) (1984), 135–136. (1984) MR0745703DOI10.1007/BF01077827
  17. Varadarajan, V.S., Supersymmetry for mathematicians: An introduction, Courant Lect. Notes Math. 11, New York University, Courant Institute of Mathematical Sciences, 2004. (2004) MR2069561
  18. Vishnyakova, E., The splitting problem for complex homogeneous supermanifolds, J. Lie Theory 25 (2) (2015), 459–476. (2015) MR3346067

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.