ON THE COHOMOLOGY OF THE LIE SUPERALGEBRA OF CONTACT VECTOR FIELDS ON S-1 vertical bar m

摘要

The main result of this article is the explicit calculation of the first cohomology space H-1 (K(3), (G psi DO(S-1 vertical bar 3)) of the Lie superalgebra K(3) of contact vector fields on the supercircle S-1 vertical bar 3 with coefficients in the module of superpseudodifferential operators G psi DO(S-1 vertical bar 3). For the supercicles of dimensional 1 vertical bar 0, 1 vertical bar 1, and 1 vertical bar 2, the first cohomology space is computed, respectively, in the following articles: [2, 3, 14]. The case m >= 4 is still out of reach, but we give a lower bound for the dimension of the cohomology space and exhibit three nontrivial, 1-cocycles.