$\alpha$-quantum wedge in Liouville Quantum Gravity

Question

We consider a proper simply connected subdomain $D\subset\mathbb{C}$ and let $\overline{\mathcal{D}}(D)$ be the space of smooth functions in $D$ with finite Dirichlet energy, which we consider modulo constants. The closure of $\overline{\mathcal{D}}(D)$ with respect to the Dirichlet inner product

$$\langle f,g\rangle_\nabla:=\frac{1}{2\pi}\int_D\nabla f\cdot\nabla g$$

will be denoted by $\overline{H}^1(D)$. Let $\{\overline{f}_j\}_{j\in\mathbb{N}}$ be an orthonormal basis for $\overline{H}^1(D)$ and let $\{X_j\}_{j\in\mathbb{N}}$ be independent $\mathcal{N}(0,1)$ random variables. The random series

$$\overline{h}_n:=\sum_{j\in\mathbb{N}}X_j\overline{f}_j$$

converges almost surely in the space of distributions modulo constants, and the law of the limit does not depend on the choice of orthonormal basis $\{\overline{f}_j\}_{j\in\mathbb{N}}$. The Neumann Gaussian free field (NGFF) is defined to be the limit $h$ of this random series.

Let $S=\mathbb{R}\times(0,\pi)$ be an infinite strip. We can decompose $\overline{H}^1(S)$ as follows: Let $\overline{\mathcal{H}}_{\text{rad}}$ be the subspace of $\overline{H}^1(S)$ obtained by closing the smooth functions which are constant on each vertical segment, modulo constants and let $\mathcal{H}_{\text{circ}}$ be the subspace obtained by closing off the smooth functions which have mean zero on all vertical segments. We have that

$$\overline{H}^1(S)=\overline{\mathcal{H}}_{\text{rad}}\oplus\mathcal{H}_{\text{circ}}.$$

This allows us to write the NGFF on $S$ as $h=h_{\text{rad}}^{\text{GFF}}+h_{\text{circ}}^{\text{GFF}}$, where $h_{\text{rad}}^{\text{GFF}},h_{\text{circ}}^{\text{GFF}}$ are independent, $h_{\text{rad}}^{\text{GFF}}(z)=B_{2s}$ if $\text{Re}(z)=s$, where $B$ is an independent standard Brownian motion modulo constants, and $h_{\text{circ}}^{\text{GFF}}(z)$ has mean zero on each vertical segment.

Now let $\tilde{\mathcal{M}}$ be the space of signed Radon measures $\rho$ on $S$ with finite and equal positive and negative mass, so that $\int_D\rho(dx)=0$, such that

$$\overline{f}\mapsto\int_D\overline{f}(x)\rho(dx)$$

defines a continuous linear functional on $\overline{H}^1(S)$. We define

$$\mathcal{M}:=\{\rho:\rho=\tilde{\rho}+f\text{ with }\tilde{\rho}\in\tilde{\mathcal{M}}\text{ and }f\in\mathcal{D}_0(S)\},$$

where $\mathcal{D}_0(S)$ is the space of smooth test functions that are compactly supported in $S$.

An $\alpha$-quantum wedge is defined as follows: Let

$$h_{\text{rad}}=\begin{cases}B_{2s}+(\alpha+Q),&\text{if Re}(z)=\text{ and }s\geq0;\\\widehat{B}_{-2s}+(\alpha-Q)s,&\text{if Re}(z)=s\text{ and }s<0,\end{cases}$$

where $B$ is a standard Brownian motion, and $\widehat{B}$ is an independent Brownian motion conditioned so that $\widehat{B}_{2t}+(Q-\alpha)t>0$ for all $t>0$ and $Q$ is just a constant $Q=\frac{2}{\gamma}+\frac{\gamma}{2}$ and $\gamma\in[0,2)$. Let $h_{\text{circ}}$ be a stochastic process indexed by $\mathcal{M}$ that is independent of $h_{\text{rad}}$ and has the same law as $h_{\text{circ}}^{\text{GFF}}$. We call $h=h_{\text{rad}}+h_{\text{circ}}$ an $\alpha$-quantum wedge in $S$.

Can someone help me understand why the $\alpha$-quantum wedge is important for LQG? To my eyes the definition includes a lot of formalities that don't necessarily form a good picture or vision in my head. Why is this construction necessary/useful?

Edit: I want to explain why this is a math question, and not a physics question, since I've been asked. Liouville quantum gravity (LQG) is a "poorly" named field of math. It was inspired by Polyakov's 1984 seminal work The Quantum Geometry of Bosonic Strings. From that paper blossomed the field of Liouville conformal field theory (LCFT), a 2D theory of quantum gravity. When physicists first tried to solve LCFT, they did so through a nonrigorous analytic continuation argument, from which they found the DOZZ formula -- A sort of good guess. When mathematicians came to tackle the problem (just like how the Feynman path integral is formalized by Brownian bridges), they discovered that a two dimension version of the Brownian bridge, called the Gaussian free field (GFF), was involved. Now, LQG has branched off significantly from LCFT: The rigorous proof of the DOZZ formula was recently completed, and the next goal is to work on the conformal bootstrap (this is work in LCFT; I'll be publishing notes soon on the rigorous proof of the DOZZ formula, since it takes hundreds of pages across multiple papers and books). LQG on the other hand, focuses more on the random geometry. We consider things like random models on graphs, and discover that their scaling limits are Liouville quantum gravity surfaces, explicitly construct a random metrics, and as you saw above, notions of GFF, etc. This is only the tip of the iceberg for LQG, which has deep connections to imaginary geometry, SLE and other random processes, etc., and even the motivating LCFT.

Answer 1

I think the question of the relevance of the quantum cone in LQG is two-fold. Firstly, how is the quantum cone related to the 'classical' LQG which uses a Gaussian Free Field? It turns out that the quantum cone is just an instance of a quantum surface, which generalises the notion of LQG to allow 'conformally invariant' random measures associated to variants of GFFs. Next, why is this quantum surface something we care about in particular? I hope to illustrate its relevance through a key property it enjoys about cutting and welding.

Quantum Surfaces and LQG

Liouville Quantum Gravity is a random measure defined on a domain $D \subseteq \mathbb{C}$. Formally, given a GFF $h$ and $\gamma \in [0,2)$, the LQG measure associated with $h$ is a random area meaure of the form $e^{\gamma h} d^2 z$, where $d^2z$ is Lebesgue measure. Since $h$ is a distribution, a regularisation procedure is required in order to make sense of this rigorously.

LQG measures enjoy a type of conformal invariance property. Suppose we have GFFs $h, h'$ in domains $D, D'$ repsectively, and let $$\tilde{h} = h \circ \psi + Q \log|\psi'|,$$ where $Q = \frac{2}{\gamma} + \frac{\gamma}{2}$ and $\psi: S \rightarrow D'$ is conformal. Then the LQG measure $\mu_{h'}$ on $D'$ associated to $h'$ is equal in law to the (GMC) measure $\mu_{\tilde{h}}$ associated to the field $\tilde{h}$. So we have to transform GFFs in this way to get their LQG measures to be consistent under conformal mappings.

Note that we can generalise this procedure. Given any random field $h$ on a domain $D$ (not just GFFs), we can consider the equivalence class of pairs $(h,D)$ under $\sim$, such that $(h,D) \sim (h',D')$ if and only if there is some conformal map $\psi : D \rightarrow D'$ such that $h' = h \circ \psi + Q \log |\psi'|$. We call an equivalence class of pairs $(h,D)$ a quantum surface, and the intention is to associate each quantum surface to the (class of) random measures $e^{\gamma h} d^2 z$.

Thus, the $\alpha$-quantum wedge is just a specific choice of the random field $h$ in the domain $S$, with $h$ defined as you have described. We make sense of quantum wedges in other domains under the appropriate transformation.

Now this begs the question of why this choice of field is a good one to think about. In short, we can couple quantum wedges with SLE to cut them into independent quantum wedges. I will motivate this by describing a similar result for GFFs.

Motivation: Level Lines of the GFF

Consider the following result about 'level lines' of the zero-boundary Gaussian Free Field as motivation:

There exists a coupling $(h^0,\eta)$ of a zero-boundary GFF $h^0$ on $\mathbb{H}$ with an $\text{SLE}_4$ $\eta$ in $\mathbb{H}$ from $0$ to $\infty$, and a constant $\lambda > 0$, such that the following is satisfied:

• Let $f : \mathbb{H} \rightarrow \mathbb{R}$ be the harmonic function on $\mathbb{H}$ taking the value $\lambda$ on $(0,\infty)$ and $-\lambda$ on $(-\infty,0)$. Then, for all $t \geq 0$, conditional on $\eta[0,t]$, the restriction of the field $h^0 + f$ to $\mathbb{H}\backslash \eta[0,t]$ can be decomposed as $h^t + f_t$, whre $h^t$ is a zero-boundary GFF on $\mathbb{H} \backslash \eta[0,t]$, and $f_t(z) = f(g_t(z)-U_t)$. Here, $g_t$ and $U_t$ are the Loewner flow and driving function associated to $\eta$, respectively.

The result also holds when we set $t = \infty$, in which case the result says that the SLE $\eta$ 'cuts' the GFF $h^0 + f$ into two independent GFFs (with different, but constant, boundary conditions); one on each connected component of $\mathbb{H}\backslash [0,\infty)$.

On the other hand, we can 'weld together' two independent GFFs to create a GFF on $\mathbb{H}$, by first tracing out an $\text{SLE}_4$ and then mapping the two GFFs to the two connected component that $\eta$ creates (and adding the correct boundary conditions).

An analogous result also holds for quantum cones; SLE can cut apart quantum cones to create independent quantum cones, and we can glue them back together again.

Cutting and Welding

We may also occasionally want to make reference to specific 'marked points' on the boundary (when we relate quantum wedges to SLE, we are going to need to specify where the SLE begins and ends). In this case, we add these marked points to our ordered pair, $(h,D,x,y)$, say, and also assert that $(h,D,x,y)\sim (h',D',x',y')$ iff $(h,D)\sim (h',D')$ via the map $\psi$, and furthermore that $x' = \psi(x), y' = \psi(y)$.

In the setting which you described, where quantum cone is in the strip $S$, the 'canonical' choice of marked points is $-\infty, \infty$. Another common choice of domain to define the quantum cone is $\mathbb{H}$, in which case the canoncial choice of marked points is $0$ and $\infty$.

We will also need the notion of the weight of a quantum wedge. Given $\alpha$ and $\gamma$, this is just $W:= \gamma(\gamma/2 + Q -\alpha)$. It turns out this is the most convenient choice of parameter to articulate the 'welding' and 'cutting' property of quantum wedges. We may now state the key result of quantum wedges, due to Duplantier, Miller and Sheffield:

Fix $\gamma \in [0,2)$ and set $\kappa = \gamma^2$.

Let $h$ be a quantum wedge in $\mathbb{H}$ with weight $W_1 + W_2$. Let $\eta$ be an $\text{SLE}_\kappa(W_1-2; W_2-2)$ in $\mathbb{H}$ from $0$ to $\infty$. Then the restriction of $h$ to each of the two connected components of $\mathbb{H}\backslash [0,\infty)$ are independent quantum cones of weights $W_1$ and $W_2$, respectively.

Overall, quantum wedges provide another avenue thorugh which SLE is related to LQG.