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The presence of the antisymmetric background field $B_{\mu\nu}$ leads to thenoncommutativity of the Dp-brane manifold, while the linear dilaton field inthe form $\Phi(x)=\Phi_0+a_\mu x^\mu$, causes the appearance of the commutativeDp-brane coordinate, $x_c=a_\mu x^\mu$. In the present article we consider thecase where the conformal invariance is realized by inclusion of the Liouvilleterm. Then, the theory is conformally invariant even in the presence of theworld sheet conformal factor $F$, and it depends on the new parameter, thecentral charge $c$. As well as in the absence of the Liouville action, forparticular relations between background fields, the local gauge symmetriesappear in the theory. They turn some Neumann boundary conditions into theDirichlet ones, and decrease the number of the Dp-brane dimensions.Comment: 21 pages, 1 table, one sentence in the abstract is added, some  explanations in the Introduction and Concluding remarks are added in order to  improve the manuscript and to emphasize the new result

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

Gauge symmetries decrease the number of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>Dp</mml:mi></mml:math>-brane dimensions. II. Inclusion of the Liouville term

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