Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2006-04-25
Adv.Theor.Math.Phys.10:759-851,2006
Physics
High Energy Physics
High Energy Physics - Theory
93 pages, no figures. Published version. See also companion paper arXiv:0705.0790
Scientific paper
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 $\it{any}$ 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-K\"ahler, 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.
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