International Congress of Mathematicians

and use this to define operatorsHilbert Schemes of Points on Surface 487p n :H -> End(H); p n (a)(y) := pi^(pr* 2 (a) Upr{(y) D [Z,,„]).Let p- n (a) := ( —l) n p_ n (a)t, where t denotes the adjoint with respect to / 5 [ n] , andPo(a) := 0. By [45],[32] the p n (a) fulfill the commutation relations of a Heisenbergalgebra:[Pn(a),Pm(ß)] = (^iT^nôn-n-J a-ßJidH, n, m £ Z, a, ß £ H. (3.1)J SWe can interpret this as follows. Let H = H + ® FA be the decomposition into evenand odd cohomology. Put S*(H) := @ i>0 S % (H + ) ® @ i>0 A*(FA). The Fock spaceassociated to H is F(H) := S*(H ® tQ[t]). Using the above theorems one readilyshows that there is an isomorphism of graded vector spaces F(H) —¥ H. With thisH becomes an irreducible module under the Heisenberg-Clifford algebra.The ring structure of the H*(S^) is connected to the Heisenberg algebraaction. Given an action of a Heisenberg algebra, a standard construction givesan action of the corresponding Virasoro algebra. The important fact however,proven in [37] is that the Virasoro algebra generators have a geometrical interpretationtying them to the ring structure of the cohomology of the S^. LetÖ : S —¥ S x S be the diagonal embedding, and let Ö* : H*(S) —¥ H*(S x S) bethe corresponding pushforward. Let p v p n - v 8(a) : H*(S) —¥ End(H) be definedas p v p n - v (ß x 7) := p„(/?)p n _„(7) applied to 5* (a) £ H x H. For B / 0 defineL n (a) := YaPvP-v8*(a). These operatorssatisfy the relations of the Virasoro algebra:Yi^ — fi / f \[L n (a),L m (ßj] = (n -m)L n+m (ab) + 6 n _ TO ——— f / c 2 (S , )a6jid H . (3.2)Let d : H —^ H be the operator which on each H*(S^) is the multiplication withci(€>M), where O^ = nJZ n (Sj) is the tautological vector bundle associated tothe trivial line bundle on S. The tie given in [37] to the ring structure is:[d,p n (a)] = nL n (a) + (f\p n (K s a), n>0,a£ H*(S). (3.3)In [42], for each a £ H*(S), classes a^ £ H*(S^) are defined as generatizationsof the Chern characters ch(F^) of tautological bundles, which are studied in [37].The homogeneous components of the a^ generate the ring H*(S^). [37],[42]relate the multiplication by the a M to the higher order commutators with d: Let«[*] : H -> H be the operator which on every H*(S^) is the multiplication with«["], then[a^,pi(ßj] =eMad(d))pi(aß), (3.4)where for an operator A : H —t H, ad(d)A = [d, A].(3.2),(3.3),(3.4) determine the cohomology rings of the S^. In case Kg = 0this is used in [38],[39] to give an elementary description of the cohomology ringsH*(S^) in terms of the symmetric group, which we will relate below to orbifoldcohomology rings.