Two-Dimensional Twisted Sigma Models And The Theory of Chiral Differential Operators

Abstract

In this paper, we study the perturbative aspects of a twisted version of the two-dimensional $(0,2)$ heterotic sigma model on a holomorphic gauge bundle $\mathcal{E}$ over a complex, hermitian manifold $X$. We show that the model can be naturally described in terms of the mathematical theory of ``Chiral Differential Operators". In particular, the physical anomalies of the sigma model can be reinterpreted in terms of an obstruction to a global definition of the associated sheaf of vertex superalgebras derived from the free conformal field theory describing the model locally on $X$. One can also obtain a novel understanding of the sigma model one-loop beta function solely in terms of holomorphic data. At the $(2,2)$ locus, where the obstruction vanishes for $any$ smooth manifold $X$, we obtain a purely mathematical description of the half-twisted variant of the topological A-model and (if $c_1(X)=0$) its elliptic genus. By studying the half-twisted $(2,2)$ model on $X={\mathbb{CP}}^1$, one can show that a subset of the infinite-dimensional space of physical operators generates an underlying super-affine Lie algebra. Furthermore, on a non-Kähler, parallelised, group manifold with torsion, we uncover a direct relationship between the modulus of the corresponding sheaves of chiral de Rham complex, and the level of the underlying WZW theory.

arXiv comment93 pages, no figures. Published version. See also companion paper arXiv:0705.0790

References in corpus (8)

• Phases of $N=2$ Theories In Two Dimensions
• Mirror Manifolds And Topological Field Theory
• Criteria for Conformal Invariance of (0,2) Models
• Chiral de Rham complex and the half-twisted sigma-model
• Two-Dimensional Models With (0,2) Supersymmetry: Perturbative Aspects
• Mirror symmetry in two steps: A-I-B
• Chiral Poincaré duality
• Elliptic Genera of singular varieties, orbifold elliptic genus and chiral deRham complex

Cited by in corpus (5)

• Notes on certain other (0,2) correlation functions
• N=(0,2) Supersymmetry and a Nonrenormalization Theorem
• Partition Functions, Localization, and the Chiral de Rham complex
• Y-Formalism and Curved Beta-Gamma Systems
• Heterotic Flux Geometry from Chiral Gauge Dynamics