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T-dual Superstring Lagrangian with double zweibeins

We present superstring Lagrangians with manifest T-duality. The Lagrangian version of the section conditions are necessary to make Lagrangians to be general coordinate invariant. We show the general solution of section conditions. The D-dimensional left and right moving currents are the 2D-dimensional chiral current which causes the chiral boson problem. We solve the problem by adding the unphysical 2D-dimensional anti-selfdual current with the selfduality constraints. The Lagrange multipliers of the selfduality constraints play the role of the worldsheet zweibein allowing the Weyl invariant and Lorentz symmetric worldsheet. Doubling the zweibein makes the type II kappa-symmetry splitting into two sets of the type I kappa-symmetries.