Geodesic
On Integrations and Cross Ratios on Supermanifolds
Dimitry Leites1, 2
1(1) Division of Science and Mathematics, New York University, Abu Dhabi, United Arab Emirates
2(2) Department of Mathematics, University of Stockholm, Sweden
Journal of Lie Theory, Tome 33 (2023) no. 2, pp. 527-546 
Dimitry Leites. On Integrations and Cross Ratios on Supermanifolds. Journal of Lie Theory, Tome 33 (2023) no. 2, pp. 527-546. http://geodesic.mathdoc.fr/item/JOLT_2023_33_2_a3/
@article{JOLT_2023_33_2_a3,
     author = {Dimitry Leites},
     title = {On {Integrations} and {Cross} {Ratios} on {Supermanifolds}},
     journal = {Journal of Lie Theory},
     pages = {527--546},
     year = {2023},
     volume = {33},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JOLT_2023_33_2_a3/}
}
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Voir la notice de l'article provenant de la source Heldermann Verlag

(A) The conventional integration theory on supermanifolds had been constructed in order to have (an analog of) the Stokes formula in which a sub-supermanifold is of codimension 1 = (1|0). I review other integrations and formulate related open problems:
(1) On the 1|1-dimensional superstring associated with the trivial bundle, in presence of a contact structure there is a special integration useful in describing super versions of elliptic functions. It is needed to construct a~particular spinor representation of the Neveu-Schwarz superalgebra.
(2) Versions of the Stokes formula with "over-supermanifold" of codimension (0|-1) due to Shander and Palamodov should be developed further.
(3) Apply Shander's integration with odd parameters over chains to inverse problems.
(4) Establish existence of conjectural integrations (apparently, not leading to any analog of the Stokes formula) related to various (super)traces on various Lie superalgebras and the corresponding (super)determinants.