International Congress of Mathematicians

Proposition 2.5 (1) dc n = 0;(2) c n induces an isomorphismDeformations of Chiral Algebras 113P ch (K[l] r ,..., K[1Y, SK[\] r ) s » -+ F"- 1 (def sx )[l - n].2.6.1.For a chiral algebra M denote by H M the graded Lie algebra of cohomologyof defM-2.6.2.Assume that K is finitely generated. Letbe the dual module. ThenK* = hom(K,V x )®(ux)~ 1H SK *^ ® n (P*(K r ,... ,K r ;SK r ®V x )e n ) s »[l-n] = ® n (A n K^' ® 0x SK) r [l-n\.2.6.3.We will postpone the calculation of the *-Lie bracket on H$K>• until we showin the next section that H M has in fact a richer structure.3. Algebraic structure on the cohomology of thedeformation pro-*-Lie algebraWe will keep the agreements and the notations from 2.2.1..3.1. Cup productWe will define a chiral operation U £ F^ def r_ 1 -i(2) and then we will studythe induced map on cohomology.3.1.1.whereRecall thatdef M [-l] = ® n (a r ,a n = P ch (N[l],...,N[l];M®V x ).Let i n : X —t X n be the diagonal embedding and let p % : X n —t X U n C X n bethe complement to the union of all pairwise diagonals p % x = p>>x and j n : U n —¥ X nbe the open embedding. Let U n>m C X n+m be the complement to the diagonalsp l x = pix, where 1 < i < n, n+l