Gaussian free field on the Riemann sphere in the Liouville convention

Let $S=\widehat{\mathbb{C}}=\mathbb{C}\cup \{\mathrm{\infty }\}$ with its usual two complex charts. In the finite chart write $z=x+iy$, put $\mathrm{\mid }z{\mathrm{\mid }}_{+}=\max \{1,\mathrm{\mid }z\mathrm{\mid }\}$, and use the finite positive background measure $d{A}_0(z)=\mathrm{\mid }z{\mathrm{\mid }}_{+}^{-4}dxdy$. It has total mass $\pi +2\pi {\int }_1^{\mathrm{\infty }}{r}^{-3}dr=2\pi$. Under $w=1\mathrm{/}z$, the Jacobian $d{x}_{z}d{y}_z=\mathrm{\mid }w{\mathrm{\mid }}^{-4}d{x}_{w}d{y}_w$ gives the same coordinate expression $d{A}_0(w)=\mathrm{\mid }w{\mathrm{\mid }}_{+}^{-4}d\mathrm{\Re }(w)d\mathrm{\Im }(w)$, which is continuous and positive but not smooth across $\mathrm{\mid }w\mathrm{\mid }=1$. Independently, use the smooth round metric $g_{\mathrm{r}\mathrm{d}}=4(d{x}^2+d{y}^2)\mathrm{/}(1+\mathrm{\mid }z{\mathrm{\mid }}^2{)}^2$ in the finite chart; in the $w$ chart it has the same expression. Write $d_{\mathrm{r}\mathrm{d}}$ for its geodesic distance and $d{A}_{\mathrm{r}\mathrm{d}}$ for its area measure. The covariance kernel below is interpreted globally in the two charts as a locally integrable function off its logarithmic diagonal singularity.

A centered spherical Gaussian free field in the Liouville convention is a probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ and a real linear map $X:C^{\mathrm{\infty }}(S;\mathbb{R})\to L^2(\mathrm{\Omega },\mathbb{P})$ such that every finite vector $(X(f_1),\dots ,X(f_n))$ is centered Gaussian and, for all real smooth test functions $f,h$,

$\mathbb{E}[X(f)X(h)]={\int }_S{\int }_{S}f(z)h(w)K(z,w)d{A}_{\mathrm{r}\mathrm{d}}(z)d{A}_{\mathrm{r}\mathrm{d}}(w),K(z,w)=\log \frac{1}{\mathrm{\mid }z-w\mathrm{\mid }}+\log \mathrm{\mid }z{\mathrm{\mid }}_{+}+\log \mathrm{\mid }w{\mathrm{\mid }}_{+}\mathrm{.}$

This finite-chart formula is global: when $\mathrm{\mid }z\mathrm{\mid },\mathrm{\mid }w\mathrm{\mid }>1$, it becomes $K=\log (1\mathrm{/}\mathrm{\mid }{z}^{-1}-w^{-1}\mathrm{\mid })$ in the $w=1\mathrm{/}z$ chart; when $\mathrm{\mid }z\mathrm{\mid }>1$ and $\mathrm{\mid }w\mathrm{\mid }\le 1$, it becomes $K=\log (1\mathrm{/}\mathrm{\mid }1-w\mathrm{/}z\mathrm{\mid })$, and the other mixed case follows by symmetry. These formulas agree on overlaps and have only the stated diagonal logarithmic singularity.

Require that, outside one null set, $f\mapsto X(f)$ is a continuous distribution on $C^{\mathrm{\infty }}(S)$ with its usual test-function topology. The point notation $X(z)$ is only shorthand for this random distribution and its covariance kernel; it does not assert a pointwise value. Fix $0<{\epsilon }_0<1$ and a finite constant $M$ large enough for the constructions below. A smoothing family is a family of nonnegative smooth local averaging kernels ${\rho }_{\epsilon }(z,w)$, for $0<\epsilon <{\epsilon }_0$, supported where $d_{\mathrm{r}\mathrm{d}}(z,w)\le \epsilon$, satisfying ${\rho }_{\epsilon }(z,w)\le M{\epsilon }^{-2}$, with unit integral in $w$ against $d{A}_{\mathrm{r}\mathrm{d}}(w)$ and satisfying ${\int }_S{\rho }_{\epsilon }(z,w)f(w)d{A}_{\mathrm{r}\mathrm{d}}(w)\to f(z)$ for every smooth $f$. For every two such families $\rho ,\sigma$, the mixed regularized covariance is $K_{\epsilon ,\delta }^{\rho ,\sigma }(z,w):=\mathbb{E}[X({\rho }_{\epsilon }(z,\cdot ))X({\sigma }_{\delta }(w,\cdot ))]$ and converges to $K(z,w)$ off the diagonal, with $K_{\epsilon ,\delta }^{\rho ,\sigma }(z,w)\le C+{\log }^{+}\text{\hspace{0.17em}⁣}\frac{1}{d_{\mathrm{r}\mathrm{d}}(z,w)+\epsilon +\delta }$ uniformly in $z,w,\epsilon ,\delta ,\rho ,\sigma$ for one finite $C$ depending only on $M$ and the fixed sphere geometry, as verified below. Set $X_{\epsilon }^{\rho }(z)=X({\rho }_{\epsilon }(z,\cdot ))$; this is a smooth Gaussian field for each $\epsilon >0$. Thus the local L2 lemma has its mixed-family covariance hypothesis. The test pairing uses the smooth measure $d{A}_{\mathrm{r}\mathrm{d}}$; the distinct measure $d{A}_0$ remains the interaction/background measure in GMC and is not inserted into the smoothing pairing. No division by the nonsmooth density of $d{A}_0$ is used.

This convention fixes the background measure and additive normalization used by the probabilistic Liouville sphere construction. The constant mode is not part of $X$ and is introduced separately in that construction.

Full local two-dimensional CFT data