Born sigma model for branes in exceptional geometry

In double field theory, the physical space has been understood as a subspace of the doubled space. Recently, the doubled space has been defined as the para-Hermitian manifold and the physical space is realized as a leaf of a foliation of the doubled space. This construction naturally introduces the fundamental 2-form, which plays an important role in a reformulation of string theory known as the Born sigma model. In this paper, we present the Born sigma model for $p$ -branes in M-theory and type IIB theory by extending the fundamental 2-form into $U$ -duality-covariant $(p+1)$ -forms.