Dual string description of Wilson loop in non-commutative gauge theory

The Wilson loop in some non-commutative gauge theories is studied by using the dual string description in which the corresponding string is on the curved background with B field. For the theory in which a constant B field is turned on along the brane worldvolume the Wilson loop always shows a Coulomb phase, as studied in the previous literature. We extend the examination to the theory with a non-constant B field, which duals to the gauge theory with non-constant non-commutativity, and re-examine the theory in the presence of a nonzero B field with one leg along the brane worldvolume and other transverse to it, which duals to a non-commutative dipole theory. The expectation value of the Wilson loop is calculated to the lowest order by evaluating the area of the string worldsheet. The results show that, while the non-commutativity could modify the Coulomb type potential in IR it may produce a strong repulsive force between the quark and anti-quark if they are close enough. In particular, we find that there presents a minimum distance between the quarks and that the distance is proportional to the value of the non-commutativity, exhibiting the nature of the non-commutative theory.