Monochromatic exponents of two-dimensional critical loop models

In two-dimensional critical loop models, the monochromatic exponents are a series of universal quantities that govern the behaviour of probabilities of connections by a number of disjoint paths. The first non-trivial case is the backbone exponent. The expression for this exponent is more complicated than those for other known critical exponents. In particular, in the special case of critical percolation, the backbone exponent is a transcendent number, while other known exponents are rational numbers.

The determination of the backbone exponent raises two questions:
 * Determining the other monochromatic exponents.
 * Finding a simple derivation. In particular, the known derivation couples the loop model to quantum Liouville gravity. Is it possible to avoid this step?

Original formulation
For $$4<\kappa<8$$ the parameter of a conformal loop ensemble, the backbone exponent $$\xi$$ obeys

\sin\left(\tfrac{8\pi}{\kappa}\right) \sqrt{\tfrac12 \kappa \xi +\left(1-\tfrac{\kappa}{4}\right)^2} = \sin\left(\tfrac{8\pi}{\kappa}\sqrt{\tfrac12 \kappa \xi +\left(1-\tfrac{\kappa}{4}\right)^2}\right) \quad ,\quad 0<\xi <1-\tfrac{2}{\kappa} $$ Depending on $$\kappa$$, this equation has one or two solutions. If there are two solutions, one of them is $$1-\tfrac{\kappa}{8}$$, and the exponent is the other one.

CFT variables
The parameter $$\kappa$$ is related to the central charge $$c$$ and the contractible loop weight $$n$$ by

c= 13-6\beta^2 -6\beta^{-2} \quad, \quad \beta =\frac{2}{\sqrt{\kappa}} \quad , \quad \frac12<\beta^2 <1 \quad , \quad n = -2\cos(\pi\beta^2) $$ The exponent is a conformal dimension, i.e. twice a chiral conformal dimension. The corresponding momentum $$P$$ and loop weight $$w$$ are defined by

\xi = \frac{c-1}{12} + 2P^2 \quad \iff \quad P = \sqrt{\tfrac{\xi}{2} + \kappa^{-1}\left(1-\tfrac{\kappa}{4}\right)^2} \quad, \quad w = 2\cos(2\pi \beta P) $$ The momentum is a natural variable in CFT, as it allows many formulas to simplify, including the fusion rules of degenerate fields, and the DOZZ formula for the three-point structure constants of Liouville theory. Special values of the momentums are

P_{(r,s)} = \frac{r}{2}\beta - \frac{s}{2}\beta^{-1} $$ where $$r,s$$ are Kac table indices, which are integer for degenerate representations of the Virasoro algebra. Many known critical exponents correspond to momentums $$P_{(r,s)}$$ with rational indices.

In terms of the CFT variables $$\beta, P$$, the equation that determines the backbone exponent reads

\varphi\left(4\pi\beta P\right) = \varphi\left(4\pi \beta P_{(1,0)}\right) \quad \text{with} \quad \varphi(x)=\frac{\sin(x)}{x} $$

Derivation
The derivation involves coupling SLE$$_\kappa$$ to what is called Liouville theory with parameter $$\beta=\frac{\gamma}{2}$$ and central charge $$c^f=1+6(\beta+\beta^{-1})^2$$ such that $$c^f+c=26$$ where $$c$$ is the loop model's central charge. However, the bulk cosmological constant vanishes, so this is a free bosonic CFT with conformal boundaries i.e. boundary conditions that break the affine symmetry. (Such boundary conditions are well-known in the case of compactified free bosons with $$c=1$$. )

The coupling between the loop model and the free boson involves fields with total dimension $$\Delta+\Delta^f=1$$. We write $$\Delta=\frac14(\beta-\beta^{-1})^2 +P^2$$ and $$\Delta^f = \frac14(\beta+\beta^{-1})^2 -P^2$$, then the momentum $$P$$ is related to the "Liouville" bulk or boundary momentum $$\alpha$$ by $$\alpha = Q+2P$$ with $$Q=\beta+\beta^{-1}$$. In terms of Kac table indices $$r,s$$ we have $$P^f_{(r,s)} = \frac{r}{2}\beta + \frac{s}{2}\beta^{-1}= P_{(r,-s)}$$ so that $$\Delta_{(r,s)}+\Delta^f_{(r,-s)}=1$$.

The fundamental quantity is the joint moment (4.8), which comes from conformal welding (3.15), and reads in our notations
 * $$ g(P, p) = \nu_\kappa\left[|\psi'(i)|^{-2\Delta_P} |\varphi'(1)|^{\Delta_{(3,1)}-\Delta_p}\right]

$$ where $$\nu_\kappa$$ is the SLE measure, $$\psi,\varphi$$ are conformal transformations, and $$P,p$$ are a bulk and a boundary momentum. It turns out that up to simple factors, this quantity is an integral transform of a certain boundary 3-point structure constant, with respect to a boundary cosmological constant. This is written in Eq. (4.9), which has the form
 * $$ g(P, p) \left\langle V_P B_p\right\rangle \propto \int_0^\infty d\mu \frac{\mu^{-2\beta^{-1}P}}{\mu(\mu+1)} \left\langle B_{P_{(1,0)}}B_{P_{(1,0)}}B_p\right\rangle_{\mu,1,1}

$$ where $$V_P$$ and $$B_p,B_{P_{(1,0)}}$$ are bulk and boundary fields in "Liouville" theory.

The monochromatic momentum is a solution of (1.6), which (up to a detail) reads

g(P, P_{(3,1)}) = 1 $$ And $$g(P, P_{(3,1)})$$ can be computed by performing the integeral. The appearance of boundary fields of momentum $$P_{(1,0)}$$ is somewhat natural, since the exponent is related to bulk fields of the same momentum. The appearance of the boundary momentum $$P_{(3,1)}=P^f_{(3,-1)}$$ is more mysterious. Apparently $$2\Delta_{(3,1)}$$ is the dimension of the set of touching points of the boundary touching loop.

Monochromatic exponents
Monochromatic exponents are labelled by integers $$k\in\mathbb{N}^*$$. They govern probabilities of existence of connections with $$k$$ disjoint paths. Let $$ P^\text{mono}_k$$ be the corresponding momentums. In particular, the backbone exponent corresponds to $$k=2$$.

Numerical estimates for the monochromatic exponents for percolation:

\xi_2 = 0.3569 \pm 0.0006\quad, \quad \xi_3 = 0.77 \pm 0.02 \quad , \quad \xi_4 = 1.33\pm 0.03 \quad , \quad \xi_5= 2.1\pm 0.2 \quad , \quad \xi_6 = 3\pm 0.3 $$

Polychromatic exponents
Polychromatic exponents $$ P^\text{poly}_k$$ are defined similarly to the monochromatic exponents, however the disjoint paths need not all be on clusters, they can also be on dual clusters. The exponents do not change if we assume that the clusters and dual clusters alternate. Clusters and dual clusters are separated by loops, so the polychromatic exponents have an interpretation in loop models.

The corresponding momentums are
 * $$ P^\text{poly}_k \underset{k\geq 2}{=} P_{(\frac{k}{2},0)} = \frac{k}{4}\beta

$$ In critical loop models, these momentums correspond to the conformal dimensions of the so-called watermelon operators, which create $$k$$ loops. The case $$k=1$$ is special, with

P^\text{poly}_1 = P^\text{mono}_1 = P_{(0,\frac12)} $$ In the case of percolation $$\kappa=6$$ i.e. $$\beta^2=\frac23$$, we have the Beffara-Nolin inequalities

P^\text{poly}_k < P^\text{mono}_k < P^\text{poly}_{k+1} $$

Boundary touching probabilities
The expectation value of a certain conformal radius $$R$$ to a power $$\lambda = -\xi$$ is

P^2<P_{(0,\frac12)}^2 \implies \mathbb{E}(R^{-\xi}) = -\frac{\cos(\pi \beta^2)}{\cos(2\pi \beta P)} = \frac{n}{w} $$ where the momentum $$P$$ corresponds to the conformal dimension $$\xi$$. This results is then refined: we have $$\mathbb{E}(R^{-\xi})= \mathbb{E}_1(R^{-\xi})+\mathbb{E}_2(R^{-\xi})$$ depending on whether a certain loop touches the boundary of the unit disk or not, with (Theorem 1.2)

P^2<P_{(1,0)}^2 \implies \mathbb{E}_1(R^{-\xi}) = \frac{2\cos(\pi \beta^2)\sin\left(2\pi (\beta-\beta^{-1})P\right)}{\sin(2\pi \beta^{-1}P)} $$

P^2<P_{(0,\frac12)}^2 \implies \mathbb{E}_2(R^{-\xi}) = \frac{\cos(\pi \beta^2)\sin\left(2\pi (\beta^{-1}-2\beta)P\right)}{\cos(2\pi \beta P)\sin(2\pi \beta^{-1}P)} $$ A further refinement, corresponding to removing boundary-touching loops, is (Theorem 1.3)

P^2<P_{(1,0)}^2 \implies \widetilde{\mathbb{E}}_2(R^{-\xi}) = \frac{\sin\left(2\pi (2\beta-\beta^{-1})P\right)}{\sin(2\pi \beta^{-1}P)} $$ i.e. $$\mathbb{E}_2(R^{-\xi}) =\mathbb{E}(R^{-\xi})\widetilde{\mathbb{E}}_2(R^{-\xi})$$. Then given a parameter $$a$$ (a sort of loop weight), the corresponding nested path exponent $$\xi_{NP}$$ is a solution of $$\widetilde{\mathbb{E}}_2(R^{-\xi_{NP}})=a$$. The nested loop exponent $$\xi_{NL}$$ is a solution of $$\mathbb{E}(R^{-\xi_{NL}})=a$$ i.e. it is the dimension associated to the loop weight $$w=\frac{n}{a}$$.

A guess for higher monochromatic exponents
We assume that the difference between the monochromatic and polychromatic exponents decreases as $$\frac{1}{k}$$. This leads to the formula

P^\text{mono}_k = \frac{k}{4}\beta + \frac{2P^\text{mono}_2-\beta}{k} = \frac{k^2-4}{4k}\beta +\frac{2}{k} P_2^\text{mono} $$ This leads to the numerical values

\xi_2 = 0.3567 \quad, \quad \xi_3 = 0.7692 \quad , \quad \xi_4 = 1.3511 \quad , \quad \xi_5= 2.100 \quad , \quad \xi_6 = 3.017 $$ where we take the exact value of $$\xi_2$$, and deduce the other exponents. These are very close to the numerical estimates. It would be interesting to have more precise numerical estimates, and estimates for other values of the central charge.

Relation with modular kernels
The modular S-matrix is

S_{P,P'} = \cos(4\pi PP') $$ In rational CFTs, this matrix is relevant to the computation of the annulus partition function. However, it is not quite what appears in the computation of the backbone exponent.

A quantity that is more closely related to that computation is the modular kernel

S_{P,P'}[\Delta=1] = \frac{\sin(4\pi PP')}{P'} $$ This kernel describes the modular properties of 1-point conformal blocks on a torus with a primary field of conformal dimension 1. (Why would such a field appear?) If $$P'$$ is the momentum related to the backbone exponent, then $$P=P_{(2,0)}$$ is the momentum of a watermelon operator that creates 4 loops.

Relation with monodromies of BPZ equations
The function $$\varphi(x)$$ that defines the backbone exponent can be rewritten in terms of Euler's Gamma function. The Gamma function appears in the monodromy of solutions of BPZ differential equations, which are obeyed by correlation functions that involve degenerate fields. This suggests that the backbone exponent might be determined from a simple condition on the monodromy of a degenerate field around the field that creates the two paths.

To be more specific, we have $$\varphi(\pi x)= \frac{1}{\Gamma(1-x)\Gamma(1+x)}$$, so the equation that defines the backbone exponent can be rewritten as

\prod_\pm \frac{\Gamma(1\pm 4\beta P)}{\Gamma(1\pm 2\beta^2)} = 1 $$ The BPZ equations for a four-point functions of the type $$\left\langle V_{\langle r,1\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle$$ with $$r=2,3,\dots$$ have monodromies that involve Gamma functions of combinations of $$\beta P_i$$.