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We study gauge transformations of Dirac structures and the relationship between gauge and
Morita equivalences of Poisson manifolds. We describe how the symplectic structure of a
symplectic groupoid is affected by a gauge transformation of the Poisson structure on its
identity section, and prove that gauge-equivalent integrable Poisson structures are
Morita equivalent. As an example, we study certain generic sets of Poisson structures on
Riemann surfaces: we find complete gauge-equivalence invariants...
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