SL ( 2 , Z ) Duality and 4 - Dimensional Noncommutative Theories

We investigate how the four-dimensional noncommutative open string/Yang-Mills theory behaves under a general non-perturbative quantum SL(2, Z) symmetry transformation. We discuss this by considering D3 branes in a constant background of axion, dilaton, and electric and magnetic fields (including both E ⊥ B and E||B cases) in the respective decoupling limit. We find that the value of axion, whether rational or irrational, determines the nature of the resulting theory under SL(2, Z) as well as its properties such as the coupling constant and the number of noncommutative directions. In particular, a strongly coupled theory with an irrational value of axion can never be physically equivalent to a weakly coupled theory while this is usually true for a theory with a rational value of axion. A noncommutative Yang-Mills (NCYM) (resulting from D3 branes with pure magnetic flux) is physically equivalent to a noncommuta-tive open string (NCOS) but if the value of axion is irrational, we also have noncommutative space-space directions in addition to the usual noncommutative space-time directions for NCOS. We also find in general that a NCOS cannot be physically equivalent to a NCYM but to another NCOS if the value of axion is irrational. We find another new decoupling limit for possible light-like NCYM whose SL(2, Z) duality is a light-like ordinary Yang-Mills if the value of the axion is rational. Various related questions are also discussed.