Radial Hilbert-space data and Liouville Hamiltonian

Let $0<\gamma <2$, $\mu >0$ and $Q=2\mathrm{/}\gamma +\gamma \mathrm{/}2$. Let $D=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$ and $T=\mathrm{\partial }D$, with angular coordinate $\theta \in [0,2\pi )$. Let ${\mathrm{\Omega }}_T=({\mathbb{R}}^2{)}^{{\mathbb{N}}_{>0}}$ with the product of standard two-dimensional Gaussian laws. Write its coordinate pairs as $(x_n,y_n)$ and set ${\phi }_n=(x_n+i{y}_n)\mathrm{/}(2\sqrt{n})$, ${\phi }_{-n}=\overline{{\phi }_n}$; the real circle Gaussian free field is the random distribution $\phi (\theta )={\sum }_{n\mathrm{\ne }0}{\phi }_n{e}^{in\theta }$, with covariance $\mathbb{E}[\phi (\theta )\phi ({\theta }^{\mathrm{\prime }})]=\log (1\mathrm{/}\mathrm{\mid }{e}^{i\theta }-e^{i{\theta }^{\mathrm{\prime }}}\mathrm{\mid })$ interpreted distributionally. Let $P_T$ be the product Gaussian law and ${\mathcal{H}}_T=L^2({\mathrm{\Omega }}_T,P_T)$.

For a finite sequence of nonnegative integers $k=(k_n)$ and $l=(l_n)$, let $H_{k,l}$ be the product of the normalized Hermite polynomials in the coordinates $(x_n,y_n)$, with the one-variable normalization fixed by $\mathbb{E}[H_m(Z)H_j(Z)]={\delta }_{mj}$ for a standard normal $Z$. On their linear span define the nonnegative number operator $\mathcal{P}{H}_{k,l}=({\sum }_{n\ge 1}n(k_n+l_n))H_{k,l}$.

A radial Liouville data package in the function-potential regime has $0<\gamma <\sqrt{2}$ and includes the circle-GMC limit, when it exists, of

$m_{\gamma ,N}(d\theta )=\exp \text{\hspace{0.17em}⁣}(\gamma {\phi }^{(N)}(\theta )-\frac{{\gamma }^2}{2}\mathbb{E}[{\phi }^{(N)}(\theta {)}^2])d\theta ,{\phi }^{(N)}(\theta )={\sum }_{0<\mathrm{\mid }n\mathrm{\mid }\le N}{\phi }_n{e}^{in\theta },$

and its total mass $V=m_{\gamma }(T)$. It also includes the closed nonnegative quadratic-form realization, when it exists, of

$H=-\frac{1}{2}{\mathrm{\partial }}_c^2+\frac{1}{2}{Q}^2+\mathcal{P}+\mu e^{\gamma c}V$

on $\mathcal{H}=L^2(\mathbb{R}\times {\mathrm{\Omega }}_T,dc\otimes P_T)$. “Friedrichs realization” means the self-adjoint operator associated with the closure of the lower-bounded form initially evaluated on $C_c^{\mathrm{\infty }}(\mathbb{R})$ tensored with finite Hermite polynomials. This definition does not assert that the form is closable for every parameter or that $H$ has a particular spectrum.

The cutoff measures above remain defined for $0<\gamma <2$, as do the conditional disk data below. The displayed multiplication-potential formula defines the radial Liouville Hamiltonian only in the function-potential regime $0<\gamma <\sqrt{2}$. A zero raw circle limit does not define the interacting Liouville Hamiltonian for $\sqrt{2}\le \gamma <2$; the source's full-range operator is instead the radial semigroup generator of §2.3. No construction of that additional operator is asserted here.

For $x,y\in D$, define the Dirichlet Green kernel $G_D(x,y)=\log \mid \frac{1-\bar{y}x}{x-y}\mid$. Let $X_D$ be an independent centered Dirichlet Gaussian free field with covariance $G_D$; its normalized exponential mollifications are $M_{\gamma ,D,\epsilon }(dx)=e^{\gamma X_{D,\epsilon }(x)-{\gamma }^2\mathbb{E}[X_{D,\epsilon }(x{)}^2]\mathrm{/}2}dx$, and $M_{\gamma ,D}$ is their Radon-measure limit in the vague topology (convergence against every continuous compactly supported function on $D$) when it exists. The harmonic extension of the boundary distribution is $P\phi (r{e}^{i\theta })={\sum }_{n\mathrm{\ne }0}{r}^{\mathrm{\mid }n\mathrm{\mid }}{\phi }_n{e}^{in\theta }$ for $0\le r<1$, with convergence in distributions. For real $\alpha ,\beta <Q$ and $0<\mathrm{\mid }z\mathrm{\mid }<1$, the radial two-insertion vector, when the conditional expectation exists, is

$U_{\alpha ,\beta }(0,z)(c,\phi )=e^{(\alpha +\beta -Q)c+\beta (P\phi )(z)+\alpha \beta G_D(0,z)}(1-\mathrm{\mid }z{\mathrm{\mid }}^2{)}^{{\beta }^2\mathrm{/}2}\mathbb{E}\text{\hspace{0.17em}⁣}\left[\exp \text{\hspace{0.17em}⁣}(-\mu e^{\gamma c}{\int }_D{e}^{\gamma (P\phi )(x)}(1-\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{{\gamma }^2\mathrm{/}2}{e}^{\gamma \alpha G_D(x,0)+\gamma \beta G_D(x,z)}{M}_{\gamma ,D}(dx))\mid \phi \right],$

where $P\phi$ is the harmonic extension of the boundary distribution to $D$, $M_{\gamma ,D}$ is the GMC of the independent Dirichlet GFF on $D$, and the conditional expectation is over that interior field. The existence of this vector, its relation to sphere correlation functions, and the self-adjoint spectral resolution of $H$ are separate theorem obligations.

Probabilistic Liouville CFT on the sphere

Spherical Gaussian multiplicative chaos interaction data

This radial package records the objects in the cited scattering proof. It is not a proof that the probabilistic sphere measure satisfies reflection positivity or that the displayed quadratic form has the source's self-adjoint extension and continuous spectrum.