The ten-dimensional supergravity theory is a geometric low-energy effective theory and the equations of motion for its fields can be obtained from string theory by computing β functions. With d compact dimensions, an O ( d , d ; Z ) geometric structure can be added to it giving the supergravity theory with T-duality manifest. In this paper, this is constructed through the use of a suitable star product whose role is the one to implement the weak constraint on the fields and the gauge parameters in order to have a closed gauge symmetry algebra. The consistency of the action here proposed is based on the orthogonality of the momenta associated with fields in their triple star products in the cubic terms defined for d ≥ 1 . This orthogonality holds also for an arbitrary number of star products of fields for d = 1 . Finally, we extend our analysis to the double sigma model, non-commutative geometry and open string theory.
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