Redefining B-twisted topological sigma models

We take a fresh look at the relation between generalised K\"ahler geometry and $N=(2,2)$ supersymmetric sigma models in two dimensions formulated in terms of $(2,2)$ superfields. Dual formulations in terms of different kinds of superfield are combined to give a formulation with a doubled target space and both the original superfield and the dual superfield. For K\"ahler geometry, we show that this doubled geometry is Donaldson's deformation of the holomorphic cotangent bundle...

We find the conditions under which a Riemannian manifold equipped with a closed three-form and a vector field define an on--shell N=(2,2) supersymmetric gauged sigma model. The conditions are that the manifold admits a twisted generalized Kaehler structure, that the vector field preserves this structure, and that a so--called generalized moment map exists for it. By a theorem in generalized complex geometry, these conditions imply that the quotient is again a twisted generali...

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...

We construct a (1,2) heterotic sigma model whose target space geometry consists of a transitive Lie algebroid with complex structure on a Kaehler manifold. We show that, under certain geometrical and topological conditions, there are two distinguished topological half--twists of the heterotic sigma model leading to A and B type half--topological models. Each of these models is characterized by the usual topological BRST operator, stemming from the heterotic (0,2) supersymmetr...

We study nonanticommutative deformations of N=2 two-dimensional Euclidean sigma models. We find that these theories are described by simple deformations of Zumino's Lagrangian and the holomorphic superpotential. Geometrically, this deformation can be interpreted as a fuzziness in target space controlled by the vacuum expectation value of the auxiliary field. In the case of nonanticommutative deformations preserving Euclidean invariance, we find that a continuation of the defo...

We propose a 4-dimensional version of topological sigma B-model, governing maps from a smooth compact 4-manifold M to a Calabi-Yau target manifold X. The theory depends on on complex structure of X, while is independent of Kaehler metric of X. The theory is also a 4-dimensiona topological field theory in the sense that the theory is independent of variation of Riemannian metric of the source 4-manifold M, potentially leading to new smooth invariant of 4-manifolds. We argue th...

(Minor corrections and reference added)

Non-linear sigma models with extended supersymmetry have constrained target space geometries, and can serve as effective tools for investigating and constructing new geometries. Analyzing the geometrical and topological properties of sigma models is necessary to understand the underlying structures of string theory. The most general two-dimensional sigma model with manifest N=(2,2) supersymmetry can be parametrized by chiral, twisted chiral and semichiral superfields. In the ...

Generalized complex geometry is a new mathematical framework that is useful for describing the target space of N=(2,2) nonlinear sigma-models. The most direct relation is obtained at the N=(1,1) level when the sigma model is formulated with an additional auxiliary spinorial field. We revive a formulation in terms of N=(2,2) semi-(anti)chiral multiplets where such auxiliary fields are naturally present. The underlying generalized complex structures are shown to commute (unlike...

I stress how the form of sigma models with (2, 2) supersymmetry differs depending on the number of manifest supersymmetries. The differences correspond to different aspects/formulations of Generalized K\"ahler Geometry.