Duality is an exact symmetry of string perturbation theory.

Abstract

Both $\ensuremath{\beta}$ duality and $R$ duality in the toroidal compactification of heterotic strings are shown to be symmetries of the contribution of arbitrary genus to the free energy. This symmetry relates physics at radius $R$ and coupling $\ensuremath{\kappa}$ with physics at radius $\frac{{\ensuremath{\alpha}}^{\ensuremath{'}}}{R}$ and coupling $\ensuremath{\kappa}{(\frac{{\ensuremath{\alpha}}^{\ensuremath{'}}}{{R}^{2}})}^{\frac{d}{2}}$ where $d$ is the number of dimensions compactified. Some comments on the possible breaking of duality by nonperturbative effects are also included.