T-duality

Definition

Consider closed bosonic string theory, with Regge slope $\alpha' > 0$, on the spacetime $\R^{1,24} \times S^1_R$, where $S^1_R$ is a circle of radius $R > 0$. A closed string state has a momentum number $n \in \Z$, with momentum $n/R$ along the circle, a winding number $w \in \Z$, and oscillator levels $N, \tilde N \in \N$. Its mass $M$ in the $25$ non-compact dimensions satisfies $$M^2 = \frac{n^2}{R^2} + \frac{w^2 R^2}{\alpha'^2} + \frac{2}{\alpha'}\big(N + \tilde N - 2\big), \qquad N - \tilde N = n w.$$ T-duality is the equivalence of the theory at radius $R$ with the theory at radius $\alpha'/R$. It maps states by exchanging $n$ and $w$ and keeping $N$ and $\tilde N$. On the worldsheet fields it acts by $X_L \mapsto X_L$ and $X_R \mapsto -X_R$ on the left- and right-moving parts of the circle coordinate.