The Weak Mixing Angle in String Theory and the Green-Schwarz Mechanism

Four-dimensional strings with the standard model gauge group $SU(3)\times SU(2)\times U(1)$ give model-dependent predictions for the tree level weak mixing-angle. In the presence of an extra pseudo-anomalous gauged- ${U(1)}_X$, the value of the weak angle may be computed purely in terms of the charges of the massless fermions of the theory, independently of the details of the massive string sector. I present the simplest such ${U(1)}_X$ which leads to the canonical result $sin^2\theta _W=3/8$ in the supersymmetric standard model. This is a sort of gauged Peccei-Quinn symmetry which requires the presence of just the minimal set of Higgs doublets and forbids dimension-four $B$ and $L$-violating terms. In this approach the cancellation of the $U(1)_X$ anomalies through a Green-Schwarz mechanism plays a crucial role. In a different context (that of non-string low-energy supersymmetric models) I briefly discuss wether this type of anomaly cancellation mechanism could be of phenomenological relevance close to the electroweak scale. (Talk at the 23-th Workshop of the INFN Eloisatron Project on Properties of Susy particles, Erice, October 1992)