= One-loop stiffness of a spherical nonlinear sigma model
{title2=$g_{\rm eff}^{-1}=g^{-1}-(n-2)I_D$}
For the unit-sphere <nonlinear sigma model> with action $\int d^Dx\,|\nabla\sigma|^2/(2g)$, parameterize a slowly rotating background by $n_0=(\cos\theta,\sin\theta,0,\ldots)$ and tangent vectors $e_1=(-\sin\theta,\cos\theta,0,\ldots)$, $e_a$ for $a\geq2$. Write $\sigma=n_0\sqrt{1-\pi^2}+\sum_{a=1}^{n-1}e_a\pi_a$. For constant slow $\nabla\theta$, the coefficient of $|\nabla\theta|^2$ in $|\nabla\sigma|^2$ is $1-\sum_{a=2}^{n-1}\pi_a^2$. The fast <Gaussian field theory> gives $\langle\pi_a^2\rangle=gI_D$, where $I_D=\int_{\Lambda/b<|p|<\Lambda}d^Dp/[(2\pi)^Dp^2]$. The omitted linear term integrates to zero, and terms cubic in the fast variables have zero one-loop average. Thus precisely $n-2$ tangent components reduce the <phase stiffness> $1/g$. Restoring the cutoff in $D=2+\epsilon$ gives $dg/d\log b=-\epsilon g+(n-2)S_D\Lambda^\epsilon g^2/(2\pi)^D+O(g^3)$. This weak-coupling expansion requires $n\geq2$; the discrete target $S^0$ has no tangent expansion.
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