Vertex Algebroids Ii

In this note we determine the obstruction to triviality of the stack of exact vertex algebroids thereby recovering the result of [GMS]. The stack EVAOX of exact vertex OX-algebroids is a torsor under the stack in Picard groupoids ECAOX of exact Courant OX -algebroids. The latter is equivalent to the stack of torsors under ΩX −→Ω . Therefore, ECAOX -torsors are classified by H (X; ΩX −→Ω ). The goal of the present note is to determine the class of EVAOX . The first step toward this goal is to replace EVAOX by the equivalent ECAOX -torsor CEXT OX (AΩ1X )〈 , 〉, whose (locally defined) objects are certain Courant algebroids which are extensions by ΩX of the LieOX -algebroidAΩ1X , the Atiyah algebra of the sheaf Ω 1 X . Any such extension induces an AΩ1X -invariant symmetric pairing 〈 , 〉 on the Lie algebra EndOX (Ω 1 X). The objects of CEXT OX (AΩ1X)〈 , 〉 are those for which 〈 , 〉 is given by the negative of the trace of the product of endomorphisms. We show that EVAOX and CEXT OX (AΩ1X )〈 , 〉 are anti-equivalent as ECAOX -torsors by adapting the strategy of [BD] to the present setting and making use of (the degree zero part of) the unique vertex ΩX -algebroid constructed in [B]. It follows that the classes of EVAOX and CEXT OX (AΩ1X)〈 , 〉 in H (X; ΩX −→Ω ) are negatives of each other. The advantage of passing to CEXT OX (AΩ1X)Tr has to do with the fact that Courant algebroids are objects of “classical” nature. In particular they are OX -modules (as opposed to vertex algebroids). The second step is the determination of the class of CEXT OX (AΩ1X)〈 , 〉 which is achieved a the more general framework. Namely, we consider a transitive Lie OX algebroid A, locally free of finite rank over OX , and denote by g the kernel of the anchor map. Thus, g is a sheaf of Lie algebras in OX -modules. If  is a Courant algebroid, such that the associated Lie algebroid is identified with A, then  is an extension of A by ΩX . The symmetric bilinear form on  induces an A-invariant symmetric bilinear form on on g. Suppose A, g as above, and 〈 , 〉 an A-invariant symmetric bilinear form on on g. Let CEXT OX (A)〈 , 〉 denote the stack of whose (locally defined) objects are pairs, consisting of a Courant algebroid  together with an identification of the associated Lie algebroid with A, such that the symmetric bilinear form induced on g coincides with 〈 , 〉. We show that, if A admits a flat connection locally on X, then CEXT OX (A)〈 , 〉 has a natural structure of an ECAOX -torsor and calculate its characteristic class in H (X; ΩX −→Ω ). It turns out that