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SL(2;R) Duality of the Noncommutative DBI Lagrangian

We study the action of the $SL(2;R)$ group on the noncommutative DBI Lagrangian. The symmetry conditions of this theory under the above group will be obtained. These conditions determine the extra U(1) gauge field. By introducing some consistent relations we observe that the noncommutative (or ordinary) DBI Lagrangian and its $SL(2;R)$ dual theory are dual of each other. Therefore, we find some $SL(2;R)$ invariant equations. In this case the noncommutativity parameter, its $T$-dual and its $SL(2;R)$ dual versions are expressed in terms of each other. Furthermore, we show that on the effective variables, $T$-duality and $SL(2;R)$ duality do not commute. We also study the effects of the $SL(2;R)$ group on the noncommutative Chern-Simons action.