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X = (X L + X R )/2 as the string coordinate on a circle with radius RT-dualityX ~ = (X L - X R )/2 as the coordinate on the dual circle with radius 1/R. The theory which includes X and X ~ is under the chiral constraint suchthat X L is left-moving and X R is right-moving:dX L = *dX L ,dX R = -*dX Rso that the physical degrees of freedom are not doubled. It’s natural to consider vertex operators e kX L and e kX R in the string theory,therefore to include X and its dual X ~ . The first one to consider such theoryis Tseytlin (Nucl.Phys.B350 (1991) 395-440 & Phys. Lett. B242, 163, ’92).Here comes your footerPage 6