ar X iv : 0 70 5 . 40 21 v 3 [ he p - th ] 3 M ay 2 00 8 ( 0 , 2 ) Gauged Linear Sigma Model on a Supermanifold

We construct a (0, 2), D = 2 gauged linear sigma model on a supermanifold in both the Abelian gauge group and the non-Abelian gauge group. For the purpose of checking the exact supersymmetric (SUSY) invariance of the Lagrangian density, it is convenient to introduce a new operatorˆU for the Abelian gauge group. Thê U operator provides consistency conditions for satisfying the SUSY invariance. Contrary to the Abelian gauge group, it is not essential to introduce the new operator in order to check the exact SUSY invari-ance of the Lagrangian density. However, in order to introduce the (0,2) chiral superfields, we need thê U operator, because we can not define the (0,2) chirality conditions of the (0,2) chiral superfields without introducing the new operator by usingˆU and the enlarged operatorˆU a was obtained from the conditions that yield the (0,2) supersymmetric invari-ance of the Lagrangian density of the (0,2) U(N) gauged linear sigma model in superfield formalism. We found that the consistency conditions for the Abelian gauge group which assure the (0,2) supersymmetric invariance of Lagrangian density agree with (0,2) chiral-ity conditions for superpotential. The supermanifold M m|n becomes the super weighted complex projective space W CP m−1|n in U(1) case, which is considered as an example of Calabi-Yau supermanifold. The superpotential W (φ, ξ) for the non-Abelian gauge group does not satisfy quasi-homogeneous condition for the SU(N) except the U(1) part of the U(N). In the SU(N) part, we must take care of constructing the Calabi-Yau supermanifold, because the form of the superpotential is imposed more stringent restrictions than that of U(1) case.