We define and study the structure of SUSY Lie conformal and vertex algebras. This leads to effective rules for computations with superfields.

We present a superfield formulation of the chiral de Rham complex (CDR), as introduced by Malikov, Schechtman and Vaintrob in 1999, in the setting of a general smooth manifold, and use it to endow CDR with superconformal structures of geometric origin. Given a Riemannian metric, we construct an N = 1 structure on CDR (action of the N = 1 superVirasoro, or Neveu-Schwarz, algebra). If the metric is Kähler, and the manifold Ricci-flat, this is augmented to an N = 2 structure. Finally, if the manifold is hyperkähler, we obtain an N = 4 structure. The superconformal structures are constructed directly from the Levi-Civita connection. These structures provide an analog for CDR of the extended supersymmetries of nonlinear σ-models.

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We translate the construction of the chiral operad by Beilinson and Drinfeld to the purely algebraic language of vertex algebras. Consequently, the general construction of a cohomology complex associated to a linear operad produces a vertex algebra cohomology complex. Likewise, the associated graded of the chiral operad leads to a classical operad, which produces a Poisson vertex algebra cohomology complex. The latter is closely related to the variational Poisson cohomology studied by two of the authors.such that for any Z-graded Lie superagebra g = j≥−1 g j , with g −1 = V , there is a unique grading preserving homomorphism g → W (V ) identical on V . It is easy to see that W j (V ) = Hom(S j+1 (V ), V ) ,

Abstract. We study the full sigma model with target the three-dimensional Heisenberg nilmanifold by means of a Hamiltonian formulation of double field theory. We show that the expected T -duality with the sigma model on a torus endowed with H-flux is a manifest symmetry of the theory. We compute correlation functions of scalar fields and show that they exhibit dilogarithmic singularities. We show how the reflection and pentagonal identities of the dilogarithm can be interpreted in terms of correlators with 4 and 5 insertions.