Radius deformation as a Higgs mechanism ($m_W^2=(R/{\alpha }^{\prime }-1/R{)}^2$)

At the self-dual circle, left and right dimension-one currents generate $SU(2{)}_L\times SU(2{)}_R$. The compact-coordinate vertex $\partial Y_L\bar{\partial }{Y}_R$ is a Cartan–Cartan component of a scalar field in the $(3,3)$ representation. Its expectation value changes the circle radius and leaves $U(1{)}_L\times U(1{)}_R$ unbroken. The four extra charged vectors acquire the displayed mass, and four scalar field polarizations become their longitudinal modes. This realizes the Higgs mechanism geometrically. In a bosonic string the ever-present tachyon and additional scalar field instabilities prevent interpreting the whole family as a stable vacuum.