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1. In this note we compute the cohomological obstruction to the existence of certain sheaves of vertex algebras on smooth varieties. These sheaves have been introduced and studied in the previous work by A.Vaintrob and two of the authors, cf.[MSV] and [MS1]. Hopefully our result clarifies to some extent the constructions of op. cit.Recall that in [MSV] we discussed two kinds of sheaves on smooth complex algebraic (or analytic) varieties X. First, we defined the sheaf of conformal vertex superalgebras Ω ch X , called chiral de Rham algebra. These sheaves are canonically defined for an arbitrary X. Second, for some varieties X one can define a purely even counterpart of Ω ch X , a sheaf of graded vertex algebras O ch X , called a chiral structure sheaf, cf. op. cit., §5. For example, one can define O ch X for curves, and for flag spaces G/B. For an arbitrary X, there arises certain cohomological obstruction to the existence of O ch X . The infinitesimal incarnation of this obstruction is calculated in op. cit., §5, A.

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We consider the interplay of infinite-dimensional Lie algebras of Virasoro type and moduli spaces of curves, suggested by string theory. We will see that the infinitesimal geometry of determinant bundles is governed by Virasoro symmetries. The Mumford forms are just invariants of these symmetries. The representations of Virasoro algebra define (twisted) ^-modules on moduli spaces; these ^-modules are equations on correlators in conformal field theory. Q>IS> ois the ^0-Atiyah algebra.Clearly, both the Atiyah algebras and the do-algebras form categories, £# ~ -+@}tf, £^~ -»e£/^ are functors between them, and we have Lemma. These functors are inverse to each other.So the Atiyah algebras are the same as the do-algebras. We have & E = @^E. 1.1.4. Let si be an JR-Atiyah algebra.The connections of j/ form a Hom0 x (^,#) = Ω^®jR-torsor To give an integrable connection is the same as to give a morphism of Atiyah algebras £/ &x -*&/ [V corresponds to a morphism τ + /ι-» 7(τ) + f,τe^X 9 fεO x ~\ Or it is the same as to give a ^-action on R together with the isomorphism j/ ~ £Γ X ex R ( = the semi-direct product with respect to this action).A connection V defines the ^-derivative (that we will also denote V] of the graded algebra Ω' ® R, F(ω®r) = d(ω)®r + ω F(r), F(r)(τ) = [F(τ),r], where ωeΩ', ΓG.R, τe^, P^eΩ 1 ®^. We have P 2 (*)-Cp *.A connection on jtf E is the same as a usual connection on E.1.1.5. Standard Operations on Atiyah Algebras. These are the following ones.(i) Push forward φ^. Let j/ be an .R-Atiyah algebra, and R' an d^-algebra. Consider a pair φ = (φ^,φ R ) of ίP^-linear Lie algebra maps φ R :R Lίe -^R' Lie . Assume that ad^φ R = φ^\ R and φ^(d) (/) = ε(α) (/) for fE& x -^R'. Define the .R'-Atiyah algebra φ^(^) to be the semi-direct product R' x s$ modulo the relations (φ R (a\ 0) = (0, α), a e R. One has canonical d/ x -linear Lie algebras map sέ-^φ^ .(ii) Tfte product. If j/ f are Λ Γ Atiyah algebras we get an R γ x ^-Atiyah algebra j/ t x s$ ' 2 . Fx (iii) T/ie opposite algebra for an .R-Atiyah algebra j/ is the R°-Atiyah algebra j3/° such that ^o = (^t fi/ )°; here K 0 , (^^)° is JR,^ with reversed multiplication. Explicitly, j/° = s$ as a sheaf, [ , ]^0 = -[ ]^, ε^o = -ε^, and the left ίP structure for j/° is the right one for j/.