Uniform Spanning Tree in Topological Polygons,
Partition Functions for SLE(8), and
Correlations in $c=-2$ Logarithmic CFT

Let us summarize more precisely the main findings of the present work. We will consider uniformly chosen spanning trees on subsets of the square lattice, focusing on the behavior of the random chordal Peano curves between the tree and its dual. Specifically, we consider scaling limits of the UST model on polygons with an even number $2N$ of marked boundary points. Thus, we fix $N\geq 1$ and a polygon $(\Omega;x_{1},\ldots,x_{2N})$, that is, a bounded simply connected domain $\Omega\subset\mathbb{C}$ such that $\partial\Omega$ is locally connected, together with distinct marked boundary points $x_{1},\ldots,x_{2N}$ in counterclockwise order. We encode boundary conditions in link patterns (i.e., planar/non-crossing pair partitions)

where $\{a_{1},b_{1},\ldots,a_{N},b_{N}\}=\{1,2,\ldots,2N\}$. For convenience, we have chosen a particular ordering of the endpoints of the links $\{a_{r},b_{r}\}$. We shall denote by $\mathrm{LP}_{N}\ni\beta$ the set of link patterns $\beta$ of $N$ links.

• •

  (Theorem 1.3): We identify the scaling limits of the UST Peano curves with any boundary condition $\beta$. These are variants of the $\mathrm{SLE}_{8}$ process, whose partition functions (denoted $\mathcal{F}_{\beta}$) are determined by $\beta$, arising naturally from the discrete holomorphic observable that we employ to derive this result. Let us remark that, compared to the earlier results [LSW04, Dub06, HLW24], the choice and analysis of the general observable is significantly more intricate, and the boundary conditions for it are non-trivial.

• •

  (Theorem 1.4): We find the scaling limits of all crossing probabilities of the UST Peano curves as ratios of $\mathcal{F}_{\beta}$ and so-called $\mathrm{SLE}_{8}$ pure partition functions $\mathcal{Z}_{\alpha}$ (see [BBK05, Dub07, KL07, Pel19] and references therein for cases with $\kappa<8$). In contrast to earlier lattice-level results [KW11], we obtain a more complete description of the scaling limits, provide explicit formulas in terms of integrals of Coulomb gas type, and interpret these probabilities in terms of operators in a $c=-2$ log-CFT. Interestingly, the Coulomb gas integrals can also be written in terms of $a$-periods of a suitable hyperelliptic Riemann surface (see Proposition 2.4) — in particular, they have a determinantal structure.

• •

  (Theorems 1.1 & 1.2): We prove salient properties of the partition functions, some of them predicted from conformal field theory [BPZ84, Gur93, PR07], and others crucial for their probabilistic meaning.

Even though the log-CFT interpretation is rather heuristic, from the formal, algebraic viewpoint (in terms of the representation theory of the Virasoro algebra), we thus provide evidence that any CFT describing the planar UST model in the scaling limit must be a non-unitary, logarithmic CFT containing specific operator content (cf. Section 1.4). Our results furthermore provide a probabilistic construction for such a CFT in terms of correlation functions.

In order to state our scaling limit results, we first describe the objects that determine the scaling limit. We recommend readers only interested in the scaling limit results per se to first glance at Section 1.3 and then return to the present Section 1.2. However, let us point out that the results of the present section greatly differ from the previously considered cases of the critical Ising model ($\kappa=3$) [Izy15, PW23], multiple LERW ($\kappa=2$) [KW11, Kar20, KKP20], critical percolation ($\kappa=6$) [Dub06], and critical random-cluster models ($\kappa=16/3$) [Izy22, FPW24]. Indeed, when $\kappa<8$, for instance the $\mathrm{SLE}_{\kappa}$ pure partition functions can be uniquely classified as solutions to a certain PDE boundary value problem [FK15, PW19], whereas in the present case of $\kappa=8$, no classification is known and the techniques of [FK15, PW19] fail. Furthermore, while the Kac conformal weights $h_{1,1}(\kappa)$ and $h_{1,3}(\kappa)$ for $\kappa<8$ in Equation (1.16) determine unambiguously the asymptotic boundary conditions for such a PDE boundary value problem, giving rise to two distinct Frobenius exponents, these exponents coincide when $\kappa=8$. This seemingly innocent property lies at the heart of the logarithmic phenomena in the CFT describing the scaling limit of the UST model.

For each $\beta\in\mathrm{LP}_{N}$ as in (1.1), we define $\mathcal{F}_{\beta}\colon\mathfrak{X}_{2N}\to\mathbb{C}$, where

to be a “Coulomb gas integral function” (for $\kappa=8$)

where the branch of the multivalued integrand is chosen to be real and positive when

and the integration in (1.2) is understood so that the integration variables $u_{r}$ avoid the ramification points $x_{1},x_{2},\ldots,x_{2N}$ by encircling them from the upper half-plane (see Section 2). With such a branch choice, $\mathcal{F}_{\beta}$ actually takes values in $(0,\infty)$, see Theorem 1.1. This fact is crucial for the probabilistic interpretation of these functions — however, it is not obvious.

Quite a specific feature to the present case of $\kappa=8$ is that the function (1.2) also equals, up to a multiplicative factor, an integral of the Vandermonde determinant:

is a matrix which can also be written in terms of $a$-periods of a suitable hyperelliptic Riemann surface — see Proposition 2.4 and Equation (2.15, 2.16). Such a determinantal structure can be viewed as a feature of the fermionic nature of the UST model (CFT with $c=-2$).

Historically, these Coulomb gas integrals stem from conformal field theory [DF84, Dub06, KP20], where they have been used as a general ansatz to find formulas for correlation functions. Specifically to our case, we seek correlation functions of so-called degenerate fields, which should satisfy a system of “BPZ PDEs” attributed to Belavin, Polyakov & Zamolodchikov [BPZ84],

and the specific covariance property

for all Möbius maps $\varphi$ of the upper half-plane $\mathbb{H}:=\{z\in\mathbb{C}\colon\operatorname{Im}(z)>0\}$ such that $\varphi(x_{1})<\cdots<\varphi(x_{2N})$.

In short, the proof of Theorem 1.1 comprises Proposition 2.6, Proposition 2.8 (or Corollary 3.7), Proposition 2.9, Proposition 2.14, and the conclusion in Section 4.3.

• •

  The BPZ PDE system (PDE) can be verified by the following argument for the Coulomb gas integrals. The integration contours in (1.2) can be written as a closed surface in a suitable homology, and the integrand after being hit by the differential operator $\mathcal{D}^{(j)}$ in (PDE) gives an exact form. Therefore, Stokes’ theorem can be used (with care) to argue that (1.2) is a solution to (PDE). We perform this argument in Proposition 2.8. The strategy is explained in detail in [KP20, Section 1]²²2The results in [KP20] concern the case of irrational values of $\kappa$, but the same idea for this part also works for rational $\kappa$..

• •

  The Möbius covariance (COV) is immediate for translations and scalings, but the verification of it for special conformal transformations is surprisingly troublesome. In our special case where $\kappa=8$, the determinantal structure (1.3) of $\mathcal{F}_{\beta}$ is helpful (cf. Proposition 2.6).

• •

  In general, it is very difficult to establish positivity (POS) for Coulomb gas type integrals³³3For $\kappa\leq 6$, positivity results have been established via a construction of $\mathrm{SLE}_{\kappa}$ partition functions explicitly in terms of probabilistic quantities, such as the Brownian loop measure and multiple $\mathrm{SLE}_{\kappa}$ [KL07, Law09, PW19, Wu20].. Once again, in our special case the positivity is guaranteed by the determinantal structure, see Proposition 2.14. This property is absolutely essential for the probabilistic interpretation and usage of $\mathcal{F}_{\beta}$ for the scaling limit results stated below in Section 1.3; indeed, a priori $\mathcal{F}_{\beta}$ are complicated complex valued functions expressed as iterated integrals (1.2).

• •

  Of the asymptotics properties in (ASY), the generic one, ($\mathcal{F}_{\beta}$-ASY_1,1), is very easy to verify (Lemma 2.10), whereas the logarithmic one, ($\mathcal{F}_{\beta}$-ASY_1,3), needs more detailed analysis (Lemma 2.11). These asymptotics properties are motivated by fusion rules in log-CFT, that in particular involve the logarithmic correction in one of the fusion channels. See Section 1.4 for discussion and literature.

• •

  Lastly, the linear independence of the functions $\mathcal{F}_{\beta}$ is a consequence of their different asymptotic properties, as explained in the end of Section 4.3.

Theorem 1.1 shows that $\{\mathcal{F}_{\beta}\colon\beta\in\mathrm{LP}_{N}\}$ forms a basis for a solution space⁴⁴4It is not known to us what the dimension of the full solution space to the PDE system (PDE) is. However, it is plausible that imposing the additional constraint (COV) and possibly a bound for the growth of the solutions analogous to [FK15, Part 1, Eq. (20)], the dimension of the restricted solution space would equal the Catalan number $\frac{1}{N+1}\binom{2N}{N}=|\mathrm{LP}_{N}|$. of the PDE system (PDE). There is another useful basis consisting of pure partition functions $\{\mathcal{Z}_{\alpha}\colon\alpha\in\mathrm{LP}_{N}\}$. To explain how these two bases are related, let us recall that a meander formed from two link patterns $\alpha,\beta\in\mathrm{LP}_{N}$ is the planar diagram obtained by placing $\alpha$ and the horizontal reflection of $\beta$ on top of each other⁵⁵5Some authors call these “meandric systems.”:

$\displaystyle\alpha\quad=\quad\vbox{\hbox{\includegraphics[scale={0.275}]{figures/alpha.pdf}}}\quad,\quad\beta\quad=\quad\vbox{\hbox{\includegraphics[scale={0.275}]{figures/beta.pdf}}}\quad\quad\Longrightarrow\quad\quad\vbox{\hbox{\includegraphics[scale={0.275}]{figures/meander.pdf}}}$

(1.5)

We define the (renormalized, symmetric) meander matrix entries $\{\mathcal{M}_{\alpha,\beta}\colon\alpha,\beta\in\mathrm{LP}_{N}\}$ as

We also set $\mathcal{M}_{\emptyset,\emptyset}\equiv 1$ by convention. By [DFGG97, Eq. (5.18)], the meander matrix (1.6) is invertible. We then define the pure partition functions $\mathcal{Z}_{\alpha}\colon\mathfrak{X}_{2N}\to\mathbb{C}$ as

The proof of Theorem 1.2 comprises Lemmas 4.4 and 4.5 in Section 4, and the conclusion in Section 4.3. The proof also uses Theorem 1.1. The main difficulties are to prove the asymptotic properties (ASY) in Section 4.2, and the positivity (POS) in Section 4.3 — importantly, both use the fact that $\mathcal{Z}_{\alpha}$ are related to UST crossing probabilities.

Generalizing the covariance property (COV), we extend the definitions of $\mathcal{F}_{\beta}$ and $\mathcal{Z}_{\alpha}$ to general polygons, whenever the derivatives of the associated conformal map in (COV) are defined. Thus, we set

and where $\varphi$ is any conformal map from $\Omega$ onto $\mathbb{H}$ with $\varphi(x_{1})<\cdots<\varphi(x_{2N})$, assuming that the marked boundary points $x_{1},\ldots,x_{2N}$ lie on sufficiently regular boundary segments (e.g. $C^{1+\epsilon}$ for some $\epsilon>0$).

In general, a partition function (with $\kappa=8$) will refer to a positive smooth function $\mathcal{Z}\colon\mathfrak{X}_{2N}\to\mathbb{R}_{>0}$ satisfying the BPZ PDE system (PDE) and Möbius covariance (COV). We can use any partition function to define a Loewner chain associated to $\mathcal{Z}$: in the upper half-plane $\mathbb{H}$, started from $x_{i}\in\mathbb{R}$, and with marked points $(x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{2N})$, this is the Loewner chain driven by the solution $W$ to the stochastic differential equations (SDEs)

This process is well-defined up to the first time when either $x_{i-1}$ or $x_{i+1}$ is swallowed (i.e., when the denominator in the SDE (1.8) blows up). The functions $\mathcal{F}_{\beta}$ and $\mathcal{Z}_{\alpha}$ are examples of partition functions.

We now consider UST on a scaled square lattice (see the precise formulation in Section 3). Suppose that $(\Omega^{\delta,\diamond};x_{1}^{\delta,\diamond},\ldots,x_{2N}^{\delta,\diamond})$ is a sequence of medial polygons on $\delta(\mathbb{Z}^{2})^{\diamond}$. A common notion of convergence for such a sequence is termed after Carathéodory (see Section 3.2). However, for our purposes, the following stronger notion of convergence phrased in terms of a curve metric is relevant. The set $X$ of planar oriented curves, that is, continuous mappings from $[0,1]$ to $\mathbb{C}$ modulo reparameterization, is a complete and separable metric space

where the infimum is taken over all increasing homeomorphisms $\psi_{1},\psi_{2}\colon[0,1]\to[0,1]$. We will assume that $(\Omega^{\delta,\diamond};x_{1}^{\delta,\diamond},\ldots,x_{2N}^{\delta,\diamond})$ converges to a polygon $(\Omega;x_{1},\ldots,x_{2N})$ in the following sense:

where $(x\,y)$ denotes the counterclockwise boundary arc between $x$ and $y$. Note that such a convergence also implies convergence in the Carathéodory sense.

Next, let $\Omega^{\delta}\subset\delta\mathbb{Z}^{2}$ be the graph on the primal lattice corresponding to $\Omega^{\delta,\diamond}$. Consider the primal polygon $(\Omega^{\delta};x_{1}^{\delta},\ldots,x_{2N}^{\delta})$ and spanning trees on it, with the following boundary conditions (see Figure 1.1): first, every other boundary arc is wired,

and second, these $N$ wired arcs are further wired together according to a non-crossing partition outside of $\Omega^{\delta}$. Note that there is a bijection between non-crossing partitions of the $N$ wired boundary arcs and planar link patterns with $N$ links, illustrated in Figure 3.2. Hence, we may encode the boundary condition by a label $\beta\in\mathrm{LP}_{N}$, and we thus speak of the UST with boundary condition (b.c.) $\beta$. Let $\Upsilon_{\delta}$ be a uniformly chosen spanning tree on $\Omega^{\delta}$ with b.c. $\beta$. Then, there exist $N$ curves on the medial lattice $\Omega^{\delta,\diamond}$ running along the tree and connecting $\{x_{1}^{\delta,\diamond},\ldots,x_{2N}^{\delta,\diamond}\}$ pairwise. We call them UST Peano curves. The goal of this section is to describe the scaling limit of these Peano curves.

Our first result identifies the scaling limits of the UST Peano curves with $\mathrm{SLE}_{8}$ type processes having specific partition functions, given exactly by the functions $\mathcal{F}_{\beta}$ of Theorem 1.1, defined in (1.2). For $N=1$, we have $\mathcal{F}_{\vbox{\hbox{\includegraphics[scale={0.2}]{figures/link-0.pdf}}}}(x_{1},x_{2})=\pi\,(x_{2}-x_{1})^{1/4}$ (cf. Lemma B.1) and the limit process in Theorem 1.3 is the chordal $\mathrm{SLE}_{8}$ in $\Omega$ between $x_{1}$ and $x_{2}$ [LSW04].

The main issue to prove Theorem 1.3 is to identify the limit and show its uniqueness. To this end, one can use a discrete holomorphic observable, which is natural when only one or two curves are present (in that case, there is at most one free parameter after fixing three parameters via conformal invariance). Dubédat proposed a formula for the simplest case of

in his article [Dub06] but without proof. His formula is different from ours at first sight, but it follows from Proposition 2.4 that they are actually the same. The recent work [HLW24] concerns the case of $N=2$, which is solvable by an ordinary differential equation⁶⁶6H.W. learned the observable in [HLW24] from a master course delivered by S. Smirnov in 2015, but we are not able to identify a published reference.. However, the general $\beta$ involving non-trivial conformal moduli is significantly more difficult.

The proof of Theorem 1.3 is given in Section 3. Roughly, it follows the standard strategy: first, we need precompactness (tightness) of the family $(\eta_{i}^{\delta})_{\delta>0}$ (from well-known arguments, cf. Lemma 3.1); second, we construct a discrete martingale observable (Sections 3.1–3.2); and third, we identify the subsequential limits $\phi_{\beta}$ through the observable (Section 3.4). The identification step involves deriving the expansion of the observable $\phi_{\beta}(z)$ as $z$ approaches one of the marked points to a certain precision, and relating the expansion coefficients explicitly to the partition function $\mathcal{F}_{\beta}$ (Lemmas 3.5 & 3.6 in Section 3.3). The observable $\phi_{\beta}$ from Proposition 3.4 also gives the scaling limit distribution of the loop-erased random walk branch in the UST, see [LW23].

Interestingly enough, the scaling limit of the observable $\phi_{\beta}$ (see Proposition 3.4) can be written, on the one hand, as an abelian integral (see Equation (3.10)),

involving the $a$-periods discussed in Section 2,

and on the other hand, as a degenerate Schwarz-Christoffel conformal map (see Appendix C),

where the accessory parameters $\mu_{1},\ldots,\mu_{N-2}\in\mathbb{R}$ are mapped to the tips of the slits in the image of $\phi_{\beta}$ — see Figure 1.2 for an example.

In our second scaling limit result, we identify the scaling limits of the crossing probabilities of the Peano curves as ratios of pure partition functions $\mathcal{Z}_{\alpha}$ with the partition functions $\mathcal{F}_{\beta}$, the latter arising from the choice of the boundary condition $\beta$. Of course, the internal crossing pattern encoded in $\alpha$ must be compatible with the b.c. $\beta$, which is exactly what the meander matrix (1.6) ensures. We denote by $\mathbb{P}_{\beta}^{\delta}$ the law of the UST with b.c. $\beta\in\mathrm{LP}_{N}$.

Let us briefly summarize the strategy for the proof of Theorem 1.4, given in detail in Sections 4.1–4.2. Roughly, it requires two inputs. First, the scaling limit of the law of $\eta_{i}^{\delta}$ is given by the Loewner chain associated to $\mathcal{F}_{\beta}$, which is provided by Theorem 1.3. Second, the scaling limit of the probability $\mathbb{P}_{\beta}^{\delta}[\vartheta_{\mathrm{UST}}^{\delta}=\alpha]$, denoted by $p^{\alpha}_{\beta}$, exists and is conformally invariant (Proposition 4.1). Our proof of this second fact relies on the earlier work by Kenyon & Wilson [KW11]. Lastly, with these two inputs at hand, we consider the conditional law of $\eta_{i}^{\delta}$ given $\{\vartheta_{\mathrm{UST}}^{\delta}=\alpha\}$ as in Proposition D.2. It turns out that the family of such conditional laws is precompact and independent of the boundary condition $\beta$ (Lemma D.3). Now, for any subsequential limit $\tilde{\eta}_{i}$, from the preceding two inputs we conclude that the law of $\tilde{\eta}_{i}$ is given by the Loewner chain associated to $p^{\alpha}_{\beta}\,\mathcal{F}_{\beta}$ (Proposition D.2). As the law of $\tilde{\eta}_{i}$ is independent of the boundary condition, we then conclude that $p^{\alpha}_{\beta}\,\mathcal{F}_{\beta}=\mathcal{Z}_{\alpha}$.

Let us also remark that, while [KW11] provides a systematic method to calculate the probability $\mathbb{P}_{\beta}^{\delta}[\vartheta_{\mathrm{UST}}^{\delta}=\alpha]$ in the discrete model (see the summary in Section 4.1), this does not yet give the result in the scaling limit. Indeed, by taking the limit $\delta\to 0$, we merely obtain a formula for the left-hand side of (1.15). However, it is far from clear why the answer from [KW11] is the same as the right-hand side of (1.15). Importantly, the latter is an explicit formula for the scaling limit of the crossing probability (1.15), which can furthermore be directly related to log-CFT, as we motivate next.

In the research of critical phenomena in planar models, Polyakov’s conformal invariance conjecture [Pol70], later formalized by Belavin, Polyakov & Zamolodchikov [BPZ84], has proven in the last couple of decades to be a remarkable idea that has lead to many breakthroughs in contemporary mathematics (see, e.g., [LSW01, Sch06, Smi06]). Even before its mathematical fruition, from the assumption of conformal invariance, many properties of critical planar models such as critical exponents and the KPZ formula were correctly derived (see, e.g., [Nie87, Dup04, Car84]). It has now become customary to speak of a conformal field theory, or briefly, CFT, associated to each critical lattice model even in the mathematics literature.

The best understood such a CFT description is arguably that for the minimal model of the critical planar Ising model (having central charge $c=1/2$), whose bulk correlation functions can now be claimed to have been fully constructed [CHI21]. For other models, even though numerous results towards their conformal invariance have now been established, their full description in terms of conformal field theories is at its infancy. In particular, unfortunately but interestingly, CFTs pertaining to a full description of (non-local) observables in even the simplest lattice models such as percolation and self-avoiding polymers [Car99, MR07, RS07] ($c=0$), or spanning trees, critical dense polymers, and the Abelian sandpile model [PR07, Rue13] ($c=-2$), or the Ising model at the presence of boundary conditions, are still poorly understood even in the physics literature. Namely, whenever one wishes to find a CFT description for the boundary critical phenomena in these models, such as for the interfaces and boundary conditions, one immediately runs outside of the realm of minimal models. So-called logarithmic conformal field theory (log-CFT) has been proposed as a framework for the complete description of the scaling limits of such models. These conformal field theories are non-unitary, and in particular, they lack reflection positivity. However, as we shall see in the present article, such theories can still have a probabilistic origin.

Conformal invariance features in CFT via the effect of infinitesimal local conformal transformations, represented in terms of the Virasoro algebra, on the fields in the theory. Thus, the representation theory of the Virasoro algebra, or extensions thereof, plays a fundamental role in understanding CFT mathematically [Sch08]. (For instance, the minimal models such as in [CHI21] comprise only finitely many irreducible Virasoro representations, which makes them amenable to a complete solution.) The Virasoro algebra is the infinite-dimensional Lie algebra spanned by $\{\mathrm{L}_{n}\colon n\in\mathbb{Z}\}\cup\{\mathrm{C}\}$ such that

The generator $\mathrm{L}_{0}$ of scalings in the Virasoro algebra becomes non-diagonalizable in log-CFTs, that is, it has non-trivial Jordan blocks (generalized eigenspaces of dimension greater than one) in the representations relevant to the theory. The logarithmic behavior in the correlation functions stems from this property⁷⁷7This results from the appearance of fields transforming in non-semisimple representations of the Virasoro algebra, making the algebraic content of log-CFTs notoriously difficult.. For a survey on log-CFT, see [CR13].

Let us now very briefly describe the representations of the Virasoro algebra relevant to CFT interpretations of the present work. Each universal highest weight module $\mathrm{V}_{c,h}$ (Verma module) is generated by a highest weight vector $v_{c,h}$ that is an eigenvector of the Virasoro generator $\mathrm{L}_{0}$ with some eigenvalue $h\in\mathbb{C}$, called conformal weight, and where the central element $\mathrm{C}$ acts as a constant $c\in\mathbb{C}$, called the central charge. Of interest to us are those Verma modules that contain so-called singular vectors, which result in correlation functions satisfying BPZ PDEs, such as (PDE) for $\kappa=8$. These Verma modules have been classified by Feĭgin and Fuchs [FF84]: they belong to a special series indexed by two integers $r,s\geq 1$, and a parameter $t(\kappa)=\kappa/4\in\mathbb{C}\setminus\{0\}$, such that $h=h_{r,s}(\kappa)$ and $c=c(\kappa)$ are given by

In this case, the smallest such $\ell=rs$ is the lowest level at which a singular vector occurs in $\mathrm{V}_{c,h}$. The $\mathrm{L}_{0}$-eigenvalues $h_{r,s}(\kappa)$ are often termed Kac conformal weights.

We have used the parameterization by $\kappa$ to make connection with $\mathrm{SLE}_{\kappa}$ theory (see [Pel19] for references). For example, we have $h_{1,1}(\kappa)=0$, $h_{1,2}(\kappa)=\frac{6-\kappa}{2\kappa}$, and $h_{1,3}(\kappa)=\frac{8-\kappa}{\kappa}$. It is believed that the $\mathrm{SLE}_{\kappa}$ curve is in some sense generated by a CFT (primary) field $\Phi_{1,2}$ of weight $h_{1,2}(\kappa)$, that generates a representation which is a quotient of the Verma module $\mathrm{V}_{c(\kappa),\,h_{1,2}(\kappa)}$ (by the universality property). The field $\Phi_{1,2}$ is also known as a boundary condition changing operator [Car84, BB03, Car03]. Note that when $\kappa=8$, we have $h_{1,1}(8)=0=h_{1,3}(8)$. This results in a logarithmic correction in the asymptotics of the correlation functions for the field $\Phi_{1,2}$, that we observe rigorously in Theorems 1.1 and 1.2. One way to see this heuristically is via so-called fusion of the field with itself, that we next describe⁸⁸8For the sake of keeping the exposition to the point, we only discuss fusion of the simplest primary fields $\Phi_{1,2}$, and refer to the vast CFT literature for more general features (keeping in mind that most of it is written in the physics level of rigor)..

Generically, one expects a property of type $\Phi_{1,2}\boxtimes\Phi_{1,2}=\Phi_{1,1}\boxplus\Phi_{1,3}$ to hold, where “$\boxtimes$” denotes some kind of an operator product (OPE) and “$\boxplus$” indicates the possible outcomes (but does not represent a direct sum). That is, the following heuristic operator product asymptotic expansion should hold:

where $c_{1},c_{2}\in\mathbb{C}$ are structure constants, and the exponents are $\Delta_{1,1}=2h_{1,2}(\kappa)-h_{1,1}(\kappa)=\frac{6-\kappa}{\kappa}$ for the so-called identity channel, and $\Delta_{1,3}=2h_{1,2}(\kappa)-h_{1,3}(\kappa)=-\frac{2}{\kappa}$ for the other channel. Caution is in order here: (1.17) is to be understood in terms of correlation functions of the fields, as the latter are not defined pointwise. In other words, specifying (1.17) means specifying the Frobenius series of the correlation functions. When $\kappa=8$, the exponents coincide: $\Delta_{1,1}=\Delta_{1,3}=-1/4$. This results in a phenomenon similar to the solution of the hypergeometric equation when the roots of its indicial exponents coincide (or differ by an integer) — one of the linearly independent solutions has a logarithm. We invite the reader to compare this with the statements (ASY) in Theorems 1.1 and 1.2.

To accommodate this phenomenon, one could write the right-hand side of (1.17) in the less restrictive form $c_{1}(z_{1},z_{2})\,\Phi_{1,1}(z_{2})+c_{2}(z_{1},z_{2})\,\Phi_{1,3}(z_{2})$, for some functions $c_{1}(z_{1},z_{2})$ and $c_{2}(z_{1},z_{2})$ allowing logarithmic terms in the expansion. Our results show that, for any CFT (boundary) fields describing the scaling limit of the UST Peano curves, that is, $\mathrm{SLE}_{8}$ curves, the formal OPE product (1.17) has the explicit form

From the point of view of Virasoro representation theory, if a Verma module $\mathrm{V}_{c,h}$ contains a singular vector, we may take one at the lowest level and form the quotient module of $\mathrm{V}_{c,h}$ by this submodule. This quotient module $\mathrm{S}_{c,h}$ is unique and simple (irreducible). Minimal models are CFTs whose fields are constrained to live in such representations, whence the quotienting results in strict truncation of the operator content of the theory, forcing it to be finite. More precisely, one can also parametrize the special series of central charges and Kac weights by two coprime integers $p,p^{\prime}\geq 1$ as in [DFMS97, Chapter 7, Eq. (7.65)]. When $c(\kappa)\leq 1$, which is of interest to us, we have $t(\kappa)=p/p^{\prime}$. A minimal model of type $M(p,p^{\prime})$ comprises fields $\{\Phi_{r,s}\colon r,s\in\mathbb{Z}_{>0},\;1\leq r\leq p^{\prime}-1,\;1\leq s\leq p-1,\;pr>p^{\prime}s\}$. In particular, with $\kappa=8$, we find that the minimal model $M(2,1)$ is empty. Such a model would have central charge $c=-2$, and it is perhaps the most studied example of a log-CFT [Gur93, Kau00]. In particular, extending it beyond the minimal model to include, for instance, fields $\Phi_{1,s}$ with $s\geq 1$, we can consider a theory in particular containing $\Phi_{1,1}$, $\Phi_{1,2}$, and $\Phi_{1,3}$. Note that the conformal weights of the fields $\Phi_{1,s}$ with $\kappa=8$ read $h_{1,s}(8)\in\{0,-\frac{1}{8},0,\frac{3}{8},1,\frac{15}{8},3,\frac{35}{8},6,\frac{63}{8},10,\ldots\}$. In such a model, one can consider fusion of representations of the Virasoro algebra and ask whether all relevant Virasoro modules are contained in the theory. From the literature of fusion products in CFT [Gur93, GK96], in the case of $c=-2$ the fusion of two simple modules $\mathrm{S}_{1,2}$ (corresponding to $\Phi_{1,2}$ with $\kappa=8$) is described by the exact sequence

The resulting object $\mathrm{M}:=\mathrm{S}_{1,2}\boxtimes\mathrm{S}_{1,2}$ is a so-called staggered module [Roh96, KR09] of the Virasoro algebra, which in this case is non-trivial but relatively innocent. Gurarie showed in [Gur93] that this staggered module exists, and Gaberdiel & Kausch constructed it explicitly [GK96] as the fusion (1.19). We will not get into details of this construction, but only note that such a module is unique [KR09, Example 2 and Corollary 3.5]⁹⁹9See also [Kyt09, Section 3.6] relating local martingales for certain $\mathrm{SLE}$ variants to log-CFT.. From its structure (1.19), we see that the staggered module $\mathrm{M}$ contains the simple module $\mathrm{S}_{1,1}$ as a submodule, and it projects onto the module $\mathrm{S}_{1,3}$ such that $\mathrm{M}/\mathrm{S}_{1,1}\cong\mathrm{S}_{1,3}$. Loosely speaking, the asymptotics in Theorems 1.1 and 1.2 should correspond to this fusion. Note that in our formulas, also the structure constants “$-1$” and “$\pi$”, that cannot be inferred from the representation theory, are explicit as in (1.18). Following [Gur93], the asymptotic property without a logarithm corresponds to the identity field $\Phi_{1,1}$ that generates the simple submodule $\mathrm{S}_{1,1}$, and the asymptotic property with the logarithm to its “logarithmic partner” $\Phi_{1,3}$ that generates a projective Virasoro module $\mathrm{S}_{1,3}$, which also happens to be simple in this case. However, the fusion module $\mathrm{M}$ is not a direct sum of these pieces, whereas it is indecomposable but not semisimple. Indeed, $\mathrm{L}_{0}$ is not diagonalizable, and we have an off-diagonal action from $\mathrm{S}_{1,3}$ to $\mathrm{S}_{1,1}$, so $\mathrm{S}_{1,3}$ cannot be a Virasoro submodule of $\mathrm{M}$: namely in $\mathrm{M}$, we have $\mathrm{L}_{0}\,\Phi_{1,3}=\Phi_{1,1}$ and $\mathrm{L}_{n}\,\Phi_{1,3}=0$ for all $n\geq 1$. This is drastically different from the case of unitary CFTs.