Conformal field theory for annulus SLE: partition functions and martingale-observables

Abstract

We implement a version of conformal field theory in a doubly connected domain with numerous conformal types to connect it to the theory of annulus SLE of various types, including the standard annulus SLE, the reversible annulus SLE, and the annulus SLE with several force points. This implementation considers the statistical fields generated under the OPE multiplication by the Gaussian free field and its central/background charge modifications with a weighted combination of Dirichlet and excursion-reflected boundary conditions. We derive the Eguchi–Ooguri version of Ward’s equations and Belavin–Polyakov–Zamolodchikov equations for those statistical fields and use them to show that the correlations of fields in the OPE family under the insertion of the one-leg operators are martingale-observables for various annulus SLEs. We find Coulomb gas (Dotsenko–Fateev integral) solutions to the parabolic partial differential equations for partition functions of conformal field theory for the reversible annulus SLE.

Appendix A: Representation of Green’s function

In this “Appendix” we present an analytic representation of Green’s function in a doubly connected domain with various boundary conditions.

For a parameter $\beta \in (-1/2,1/2)$, we say a function F defined in a domain ${\bar{D}}$ satisfies RH$_\beta$ boundary condition (b.c.) on $l \subset \partial D$ if

$$\begin{aligned} \Bigg [ \cos (\beta \pi )\partial _n-\sin (\beta \pi ) \partial _\tau \Bigg ] F(\cdot )=0 \text { on } l \subset \partial D. \end{aligned}$$

Here $\partial _n$ is the (inwards) normal derivative and $\partial _\tau$ is the tangential one. In particular, the case $\beta =0$ corresponds to the Neumann b.c.

Let us define

$$\begin{aligned} f_\beta (r,z):=\frac{\Theta '(r,0)}{\Theta (r,\pi -2\beta \pi )} \frac{\Theta (r,z+\pi -2\beta \pi )}{\Theta (r,z)}. \end{aligned}$$

By the quasi-periodicities (3.4) of the theta function, we have

$$\begin{aligned} f_\beta (r,z+2\pi )=f_\beta (r,z), \qquad f_\beta (r,z+2ir)=e^{i(2\beta -1)\pi }f_\beta (r,z). \end{aligned}$$

(A.1)

Since $\Theta (r,\cdot )$ is an odd function,

$$\begin{aligned} f_\beta (r,z)=-f_{-\beta }(r,-z). \end{aligned}$$

(A.2)

On the other hand, $f_\beta$ satisfies

$$\begin{aligned} f_\beta (r,z)=\frac{1}{z}+\frac{H(r,\pi -2\beta \pi )}{2}+O(z),\qquad \text {as }z\rightarrow 0. \end{aligned}$$

(A.3)

Thus one can observe that $f_0$ has an alternative expression

$$\begin{aligned} f_0(r,z)= \frac{1}{2} (H(2r,z)-H_I(2r,z)). \end{aligned}$$

(A.4)

It is also easy to show that $f_\beta (r,\cdot )$ is a conformal map from the cylinder ${\mathcal {C}}_r$ to the upper half-plane minus a slit with argument $(1/2-\beta )\pi$.

For given $w \in {\mathcal {C}}_r$, a meromorphic function

$$\begin{aligned} g_\beta (z):=f_\beta (r,z-{\bar{w}})-f_\beta (r,z-w) \end{aligned}$$

satisfies following properties:

• $g_\beta (z+2\pi )=g_\beta (z)$, $g_\beta (r,z+2ir)=e^{i(2\beta -1)\pi }g_\beta (r,z)$;

• $g_\beta (r,\cdot )$ has simple poles at w (resp., ${\bar{w}}$) with residue $-1$ (resp., 1);

• $g_\beta (r,\cdot )$ maps ${\mathbb {R}}$ to $i {\mathbb {R}}$;

• $g_\beta (r,\cdot )$ maps ${\mathbb {R}}_r$ to a line passing through 0 with angle $\beta$ with ${\mathbb {R}}$.

All of these properties are immediate consequences of (A.1), (A.2) and (A.3). For instance, the last property follows from that for $x\in {\mathbb {R}}$,

$$\begin{aligned} g_\beta (x+ir)&=\overline{f_\beta (r,x-ir-w)}-f_\beta (r,x+ir-w) \\&=e^{i(2\beta -1)\pi }\overline{f_\beta (r,x+ir-w)}-f_\beta (r,x+ir-w) \\&=-e^{i\beta \pi } \Bigg (e^{i\beta \pi } \overline{f_\beta (r,x+ir-w)}+ e^{-i\beta \pi } f_\beta (r,x+ir-w) \Bigg ) \in e^{i\beta \pi } {\mathbb {R}}. \end{aligned}$$

We denote by $F_\beta (r,\cdot )$ a primitive of $f_\beta (r,\cdot )$, i.e., $\partial _z F_\beta (r,z)=f_\beta (r,z)$. To our knowledge, for general $\beta$, there is no known expression for $F_\beta (r,z)$ in terms of well-known special functions. On the other hand, by (A.4), we have

$$\begin{aligned} F_0(r,z)=\log \Bigg (\frac{\Theta (2r,z)}{\Theta _I(2r,z)}\Bigg ) \end{aligned}$$

(A.5)

up to an additive constant.

We now present Green’s function $G_\beta$ in ${\mathcal {C}}_r$ with zero Dirichlet b.c. on ${\mathbb {R}}$ which satisfies RH$_\beta$ condition on ${\mathbb {R}}_r$. The associated (non-symmetric) stochastic process is called obliquely reflected Brownian motion (ORBM), see [7] and references therein. We remark that ORBM gives a geometric interpolation between reflected Brownian motion ($\beta =0$) and ERBM ($\beta \rightarrow \pm 1/2$).

We claim that Green’s function $G_\beta$ in ${\mathcal {C}}_r$ with zero Dirichlet b.c. on ${\mathbb {R}}$ which satisfies RH$_\beta$ condition on ${\mathbb {R}}_r$ is expressed as

$$\begin{aligned} G_\beta (z_1,z_2)= \textrm{Re}\,[ F_\beta (r,z_1-{\bar{z}}_2)-F_\beta (r,z_1-z_2) ]. \end{aligned}$$

(A.6)

In particular, by (A.5), Green’s function $G_0$ with Dirichlet–Neumann mixed boundary condition is given by

$$\begin{aligned} G_0(z_1,z_2)=\log \Bigg | \frac{\Theta /\Theta _I(2r,z_1-{\bar{z}}_2)}{\Theta /\Theta _I(2r,z_1-z_2)} \Bigg |. \end{aligned}$$

To show (A.6), it suffices to check the followings:

• $G_\beta (z_1,z_2)+\log |z_1-z_2|$ is harmonic function in both variables;

• $G_\beta (z_1,z_2)=0$ if $z_1$ or $z_2$ is on ${\mathbb {R}}$;

• $G_\beta (\cdot ,z_2)$ satisfies RH$_\beta$ condition on ${\mathbb {R}}_r$.

All of these requirements easily follow from the properties presented above. We remark that $G_\beta$ satisfies the asymmetric relation $G_\beta (z_1,z_2)=G_{-\beta }(z_2,z_1)$.