Lie conformal algebra complex and integrable hierarchies

Lie conformal algebras encode the singular operator product expansion of chiral
fields in conformal field theory. As for Lie algebras, there exists a cohomology
theory which parametrizes first order deformations and abelian extensions.

A recent application of Lie conformal algebras is to the classification and
construction of integrable hierarchies. An important component in this theory is
the variational complex, defined as a reduction of the de Rham complex on phase
space.

We explain how to use Lie conformal algebra cohomology to explicitly construct
the variational complex. We also describe how to endow these complexes with a
calculus structure. This is joint work with Victor Kac and Alberto De Sole.