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

Abstract

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 $\mathbf{E}\perp \mathbf{B}$ and {\bf E}$||${\bf 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 noncommutative 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.

22 pages, latex

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• Strong Coupling Effects in non-commutative spaces from OM Theory and supergravity
• Open String Attractors
• Branched SL(2,Z) Duality
• Supergravity and "New" Six-Dimensional Gauge Theories
• Duality Rotations and BPS Monopoles with Space and Time Noncommutativity
• On decoupled theories in (5+1) dimensions from (F,D1,NS5,D5) supergravity configuration
• A Note on Duality Symmetry in Nonlinear Gauge Theories
• Dual Non-Lorentzian Backgrounds for Matrix Theories
• A note on half-supersymmetric bound states in M-theory and type IIA
• The Galilean Nature of V-duality for Noncommutative Open String and Yang-Mills Theories
• (NS5,D5,D3) bound state, OD3, OD5 limits and SL(2,Z) duality
• SL(2;R) Duality of the Noncommutative DBI Lagrangian