Determine the extended Lie algebroid cohomology in degree three

Determine the degree-three extended Lie algebroid cohomology group H^3_DR(T) for the Yangian action Lie algebroid associated with the negative-loop group G_- and the polynomial current algebra g[u], in order to classify the vertex algebroid extensions arising from closed three-forms.

Background

The paper explains that vertex algebroid extensions of an action Lie algebroid can be modified by closed three-forms in the extended de Rham complex, with isomorphism classes forming a torsor under the corresponding third cohomology group. For the Yangian action Lie algebroid, ordinary algebraic de Rham cohomology of G_- vanishes in positive degrees, but the authors note that the relevant extended Lie algebroid cohomology need not be understood.

Because the degree-three extended cohomology is not computed, the paper restricts its main construction to the family of extensions determined by invariant bilinear forms, namely the level-twisted chiral cotangent Yangians. Determining this cohomology would clarify whether additional inequivalent vertex algebroid extensions exist.

References

Although one can still define extensions of $(G_-, )$ using closed three-forms in $Z3_{\mathrm{DR}()$, unfortunately we do not know much about the extended Lie algebroid cohomology $H3_{\mathrm{DR}(\mathcal{T})$.

— Chiral Extensions of Cotangent Yangian  (2610.10202 - Zeng, 7 Oct 2026) in Section 4, subsection “Action Lie algebroid extensions,” paragraph preceding Definition [Chiral cotangent Yangian]