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**Morita equivalence and \(T\)-duality (or \(B\) versus \(\Theta\)).**
*(English)*
Zbl 1060.81612

Summary: \(T\)-duality in M(atrix) theory has been argued to be realized as Morita equivalence in Yang-Mills theory on a noncommutative torus (NCSYM). Even though the two have the same structure group, they differ in their action since Morita equivalence makes crucial use of an additional modulus on the NCSYM side, the constant Abelian magnetic background. In this paper, we reanalyze and clarify the correspondence between M(atrix) theory and NCSYM, and provide two resolutions of this puzzle. In the first of them, the standard map is kept and the extra modulus is ignored, but the anomalous transformation is offset by the M(atrix) theory “rest term”. In the second, the standard map is modified so that the duality transformations agree, and a SO\((d)\) symmetry is found to eliminate the spurious modulus. We argue that this is a true symmetry of supersymmetric Born-Infeld theory on a noncommutative torus, which allows to freely trade a constant magnetic background for noncommutativity of the base-space. We also obtain a BPS mass formula for this theory, invariant under \(T\)-duality, \(U\)-duality, and continuous SO\((d)\) symmetry.

### MSC:

81T75 | Noncommutative geometry methods in quantum field theory |

46L60 | Applications of selfadjoint operator algebras to physics |

58B34 | Noncommutative geometry (à la Connes) |

81R60 | Noncommutative geometry in quantum theory |

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |