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.

Suppose that a finite group G acts on a smooth complex variety X. Then this action lifts to the Chiral de Rham complex Ω ch X of X and to its cohomology by automorphisms of the vertex algebra structure. We define twisted sectors for Ω ch X (and their cohomologies) as sheaves of twisted vertex algebra modules supported on the components of the fixed-point sets X g , g ∈ G. Each twisted sector sheaf carries a BRST differential and is quasi-isomorphic to the de Rham complex of X g . Putting the twisted sectors together with the vacuum sector and taking G-invariants, we recover the additive and graded structures of Chen-Ruan orbifold cohomology. Finally, we show that the orbifold elliptic genus is the partition function of the direct sum of the cohomologies of the twisted sectors.

Abstract. We construct symmetric monoidal categories LRF , LF G of rooted forests and Feynman graphs. These categories closely resemble finitary abelian categories, and in particular, the notion of Ringel-Hall algebra applies. The Ringel-Hall Hopf algebras of LRF , LF G, H LRF , H LF G are dual to the corresponding Connes-Kreimer Hopf algebras on rooted trees and Feynman diagrams. We thus obtain an interpretation of the Connes-Kreimer Lie algebras on rooted trees and Feynman graphs as Ringel-Hall Lie algebras.

We extend the geometric approach to vertex algebras developed by the first author to twisted modules, allowing us to treat orbifold models in conformal field theory. Let V be a vertex algebra, H a finite group of automorphisms of V , and C an algebraic curve such that H ⊂ Aut(C). We show that a suitable collection of twisted V -modules gives rise to a section of a certain sheaf on the quotient X = C/H. We introduce the notion of conformal blocks for twisted modules, and analyze them in the case of the Heisenberg and affine Kac-Moody vertex algebras. We also give a chiral algebra interpretation of twisted modules.

Abstract. We construct Wakimoto modules for twisted affine Lie algebras, and interpret this construction in terms of vertex algebras and their twisted modules. Using the Wakimoto construction, we prove the Kac-Kazhdan conjecture on the characters of irreducible modules with generic critical highest weights in the twisted case. We provide explicit formulas for the twisted fields in the case of A (2) 2 .