AdS/CFT Approach to Melvin Field Deformed Wilson Loop

We first apply the transformation of mixing azimuthal and internal coordinate to the 11D M-theory with a stack N M2-branes to find the spacetime of a stack of N D2-branes with Melvin one-form in 10D IIA string theory, after the Kaluza-Klein reduction. Next, we apply the Melvin twist to the spacetime and perform the T duality to obtain the background of a stack of N D3-branes. In the near-horizon limit the background becomes the Melvin field deformed $AdS_5 \times S^5$ with NS-NS B field. In the AdS/CFT correspondence it describes a non-commutative gauge theory with non-constant non-commutativity in the Melvin field deformed spacetime. We study the Wilson loop therein by investigating the classical Nambu-Goto action of the corresponding string configuration and find that, contrast to that in the undeformed spacetime, the string could be localized near the boundary. Our result shows that while the geometry could only modify the Coulomb type potential in IR there presents a minimum distance between the quarks. We argue that the mechanism behind producing a minimum distance is coming from geometry of the Melvin field deformed background and is not coming from the space non-commutativity.