Group Theoretical Structure of $N=1$ and $N=2$ Two-Form Supegravity

Abstract

We clarifies the group theoretical structure of $N=1$ and $N=2$ two-form supergravity, which is classically equivalent to the Einstein supergravity. $N=1$ and $N=2$ two-form supergravity theories can be formulated as gauge theories. By introducing two Grassmann variables $\theta^A$ ($A=1,2$), we construct the explicit representations of the generators $Q^i$ of the gauge group, which makes to express any product of the generators as a linear combination of the generators $Q^iQ^j=\sum_k f^{ij}_k Q^k$. By using the expression and the tensor product representation, we explain how to construct finite-dimensional representations of the gauge groups. Based on these representations, we construct the Lagrangeans of $N=1$ and $N=2$ two-form supergravity theories.