The number of allowed supersymmetries in superspace formulation is determined by the maximum dimension of the underlying space. In the case of four-dimensional space, the maximum number of supersymmetries is four. This is because supersymmetry requires a balance between bosonic and fermionic degrees of freedom, and in four dimensions, there are only four bosonic dimensions to match with four fermionic dimensions.
In superspace formulation, the four supersymmetries are represented by four fermionic coordinates, also known as Grassmann coordinates. These coordinates are used to transform the four bosonic coordinates, giving rise to the supersymmetric transformations. The resulting superspace has both bosonic and fermionic dimensions, allowing for the inclusion of both types of particles in a unified framework.
The maximum number of supersymmetries allowed in superspace formulation is determined by the dimension of the underlying space. In four-dimensional space, the maximum number is four. However, in higher dimensions, such as ten or eleven dimensions, more than four supersymmetries can be accommodated. This is why superspace formulation is often used in theories such as supergravity and string theory, which require higher-dimensional spaces.
The main limitation of superspace formulation with only 4 supersymmetries is that it cannot fully describe all possible supersymmetric theories. Some theories, such as certain types of supergravity, require more than four supersymmetries to be fully described. In these cases, alternative formulations, such as extended superspace, may be used.
Superspace formulation is one of several approaches to incorporating supersymmetry into theoretical physics. It is closely related to other formulations, such as the component formalism, which uses explicit calculations and equations to describe supersymmetric theories. Superspace formulation, on the other hand, provides a geometric and elegant way to describe supersymmetry, making it a popular choice among physicists.