In this thesis, we first examine quantum field theory at zero dimension. By studying supersymmetric transformations, we describe the concept of localization in di?erent approaches. Then we discuss the supersymmetry breaking and study the index of Witten. Subsequently, the Sigma model examines the quantum field theory of a one-dimensional line on the real line and the circle in both the operator and path integral approaches, and we obtain the corresponding partition function. We study the Morse index and express its relation to the Witten index. We consider the supersymmetry breaking on compact and orientable curves, such as a circle, a torus, and a sphere using the Witten index, and we explicitly calculate the ground states of manifolds which their Witten-index is zero. In the end, we briefly study the T duality.