(0,2) Gauged Linear Sigma Model on Supermanifold

We construct (0, 2), D = 2 gauged linear sigma model on supermanifold with both an Abelian and non-Abelian gauge symmetry. For the purpose of checking the exact supersymmetric (SUSY) invariance of the Lagrangian density, it is convenient to introduce a new operator Û for the Abelian gauge group. The Û operator provides consistency conditions for satisfying the SUSY invariance. On the other hand, it is not essential to introduce a similar operator in order to check the exact SUSY invariance of the Lagrangian density of non-Abelian model, contrary to the Abelian one. However, we still need a new operator in order to define the (0,2) chirality conditions for the (0,2) chiral superfields. The operator Ûa can be defined from the conditions assuring the (0,2) supersymmetric invariance of the Lagrangian density in superfield formalism for the (0,2) U(N) gauged linear sigma model. We found consistency conditions for the Abelian gauge group which assure (0,2) supersymmetric invariance of Lagrangian density and agree with (0,2) chirality conditions for the superpotential. The supermanifold M becomes the super weighted complex projective space WCP in the U(1) case, which is considered as an example of a Calabi-Yau supermanifold. The superpotential W (φ, ξ) for the non-Abelian gauge group satisfies more complex condition for the SU(N) part, except the U(1) part of U(N), but does not satisfy a quasi-homogeneous condition. This fact implies the need for taking care of constructing the Calabi-Yau supermanifold in the SU(N) part. Because more stringent restrictions are imposed on the form of the superpotential than in the U(1) case, the superpotential seems to define a certain kind of new supermanifolds which we cannot identify exactly with one of the mathematically well defined objects.