Topological mass in seven dimensions and dualities in four dimensions

The massive topologically and self dual theories en seven dimensions are considered. The local duality between these theories is established and the dimensional reduction lead to the different dualities for massive antisymmetric fields in four dimensions. The topological Chern Simons terms have played an important role in several physical models. For instance, they appear in the eleven dimensional supergravity in a natural way [1] and arise in the anomalies cancellation for gauge and string theories. Also, they allow the formulation of genuine gauge theory for gravity.[2]. If they are included in the conventional theories for odd dimension D = 4k− 1, it is possible formulate massive theories which are compatible with gauge invariance. Initially, this goal was formulated in three dimensions provide gauge invariant theories for massive spin 1 and spin 2 fields [3]. These are called massive topologically theories. Alternatively, other formulations describe the same physical dynamics but in a non-gauge invariant way: the self dual theories [4][5] Eventually, it was established that they are essentially two ways for describing the same physics[6], i.e., they are related by duality[7]. Furthermore, the dual equivalence can be shown from the hamiltonian framework[8]. The self dual theories are gauged fixed versions of the topologically massive theories. Also, there exist analogues dualities for antisymmetric fields in four dimensions, [9],[10], which constitute alternative manners to describe massive scalar and vectorial fields through of gauge invariant topological BF terms. In this work, we will consider the dual equivalence between the topologically massive and self dual theories in seven dimensions. The basic field is a third order antisymmetric tensor. Recently, it was recognized the importance of these theories, when the eleven dimensional supergravity is dimensionally reduced in a consistent way on AdS7⊗S4[11]. We perform the Hamiltonian analysis in order to achieve the local canonical duality between them. Besides, the duality ∗ E-mail:[email protected] and [email protected]