Gauge symmetries decrease the number of Dp-brane dimensions. II. Inclusion of the Liouville term

The presence of the antisymmetric background field $B_{\mu\nu}$ leads to the noncommutativity of the Dp-brane manifold, while the linear dilaton field in the form $\Phi(x)=\Phi_0+a_\mu x^\mu$, causes the appearance of the commutative Dp-brane coordinate, $x_c=a_\mu x^\mu$. In the present article we consider the case where the conformal invariance is realized by inclusion of the Liouville term. Then, the theory is conformally invariant even in the presence of the world sheet conformal factor $F$, and it depends on the new parameter, the central charge $c$. As well as in the absence of the Liouville action, for particular relations between background fields, the local gauge symmetries appear in the theory. They turn some Neumann boundary conditions into the Dirichlet ones, and decrease the number of the Dp-brane dimensions.