Conformal field theory of dipolar SLE with the Dirichlet boundary condition

Abstract

We develop a version of dipolar conformal field theory based on the central charge modification of the Gaussian free field with the Dirichlet boundary condition and prove that correlators of certain family of fields in this theory are martingale-observables for dipolar SLE. We prove the restriction property of dipolar SLE(8/3) and Friedrich-Werner’s formula in the dipolar case.

Appendix: Basic properties of conformal Fock space fields

Here, we include some definitions and concepts of conformal Fock space fields developed in [8] so that this article can be read as a self-contained one.

1.1 A.1 Fock space correlation functionals

This subsection is borrowed from [8, Section 1.3]. By definition, basic correlation functionals are formal expressions of the type (Wick’s product of $X_j(z_j)$)

where points $z_j\in D$ are not necessarily distinct and $X_j$’s are derivatives of the Gaussian free field, (i.e., $X_j=\partial ^j{\bar{\partial }}^k\varPhi$), or Wick’s exponentials

$$\begin{aligned} e^{\odot \alpha \varPhi }=\sum _{n=0}^\infty \frac{\alpha ^n}{n!}\varPhi ^{\odot n}. \end{aligned}$$

The constant $1$ is also included to the list of basic functionals. We write for the set of all points $z_j$ (the nodes of ) in the expression of

For derivatives $X_{jk}$ of the Gaussian free field and basic functionals of the form

we define the tensor product by

(7.1)

where the sum is taken over Feynman diagrams with vertices $v$ labeled by functionals $X_{jk}$ such that there are no contractions of vertices with the same $j,$ and the Wick’s product is taken over unpaired vertices $v^{\prime \prime }.$ By definition, ${\mathbf{E}}[X_{v}(z_{v})X_{v^{\prime }}(z_{v^{\prime }})]$ in (7.1) are given by the 2-point functions of derivatives of the Gaussian free field, e.g.,

$$\begin{aligned} {\mathbf{E}}[\partial ^j\varPhi (\zeta )\partial ^k\varPhi (z)] = \partial _\zeta ^j \partial _z^k{\mathbf{E}}[\varPhi (\zeta )\varPhi (z)] = 2 \partial _\zeta ^j \partial _z^k G(\zeta ,z). \end{aligned}$$

For example, the Feynman diagram with two edges $\{1,4\},\{3,5\}$ and two unpaired vertices $2,6$ corresponds to

The definition of tensor product can be extended to general correlation functionals by linearity. The tensor product of correlation functionals is commutative and associative, see [8, Proposition 1.1].

We define the correlation of of by linearity, ${\mathbf{E}}[1] = 1,$ and

$$\begin{aligned} {\mathbf{E}}[X_1(z_1)\odot \cdots \odot X_n(z_n)] = 0, \end{aligned}$$

where $X_j$ are derivatives of $\varPhi .$ For example, ${\mathbf{E}}[e^{\odot \alpha \varPhi (z)}] = 1$ and

$$\begin{aligned} {\mathbf{E}}[\varPhi (z_1)\cdots \varPhi (z_n)] = \sum \prod _k 2G(z_{i_k},z_{j_k}), \end{aligned}$$

where the sum is over all partitions of the set $\{1,\ldots ,n\}$ into disjoint pairs $\{i_k,j_k\}.$

If holds for all ${\fancyscript{Y}}$ with nodes outside we identify with and write We consider Fock space functionals modulo an ideal of Wick’s algebra. The concept of a correlation functional can be extended to the case when some of the nodes of lie on the boundary. For example, $e^{\odot \alpha \varPhi (z)} = 1$ for $z\in \partial D.$ The complex conjugation of is defined (modulo ${\fancyscript{N}}$) by the equation for all ${\fancyscript{Y}}$’s of the form $\varPhi (z_1)\odot \cdots \odot \varPhi (z_n).$ For example, if $J=\partial \varPhi$ in the half-plane $\mathbb{H }$ and if $z\in \partial \mathbb{H },$ then $J(z)$ is purely imaginary, i.e., $\overline{J(z)}=-J(z),$ and $J(z)\odot J(z)$ is real.

1.2 A.2 Fock space fields

This subsection is borrowed from [8, Section 1.4]. Basic Fock space fields $X_\alpha$ are formal expressions written as Wick’s products of derivatives of the Gaussian free field $\varPhi$ and Wick’s exponential $e^{\odot \alpha \varPhi },$ e.g., $1,\, \varPhi \odot \varPhi \odot \varPhi ,\, \partial ^2 \varPhi \odot {\bar{\partial }}\varPhi ,\, \partial \varPhi \odot e^{\odot \alpha \varPhi },$ etc. A general Fock space field is a linear combination of basic fields $X_\alpha ,$

$$\begin{aligned} X=\sum _\alpha f_\alpha X_\alpha , \end{aligned}$$

where $f_\alpha$’s are arbitrary (smooth) functions in $D.$ If $X_1,\ldots ,X_n$ are Fock space fields and $z_1,\ldots ,z_n$ are distinct points in $D,$ then is a correlation functional.

We define the differential operators $\partial$ and ${\bar{\partial }}$ on Fock space fields by specifying their action on basic fields so that the action on $\varPhi$ is consistent with the definition of $\partial \varPhi , {\bar{\partial }}\varPhi$ and so that

$$\begin{aligned} \partial (X\odot Y)=(\partial X)\odot Y+ X\odot (\partial Y),\quad {\bar{\partial }}(X\odot Y)=({\bar{\partial }}X)\odot Y+ X\odot ({\bar{\partial }}Y). \end{aligned}$$

We extend this action to general Fock space fields by linearity and by Leibniz’s rule with respect to multiplication by smooth functions. Then (modulo ${\fancyscript{N}}$)

$$\begin{aligned} {\mathbf{E}}[(\partial X)(z){\fancyscript{Y}}] = \partial _z {\mathbf{E}}[X(z){\fancyscript{Y}}],\quad (z\not \in S_{\fancyscript{Y}}), \end{aligned}$$

for all correlation functionals ${\fancyscript{Y}}.$

By definition, $X$ is holomorphic in $D$ if ${\bar{\partial }}X\approx 0,$ i.e., all correlation functions ${\mathbf{E}}[X(\zeta ) {\fancyscript{Y}}]$ are holomorphic in $\zeta \in D{\setminus } S_{\fancyscript{Y}}.$ For example, $J=\partial \varPhi ,X = J\odot J$ are holomorphic fields.

1.3 A.3 Operator product expansion

This subsection is borrowed from [8, Sections 3.1–3.2]. Operator product expansion (OPE) is the expansion of the tensor product of two fields near diagonal. For example,

$$\begin{aligned} \varPhi (\zeta ) \varPhi (z) = \log \frac{1}{|\zeta -z|^2} + 2c(z) + \varPhi ^{\odot 2}(z) + o(1) \quad \hbox {as }\zeta \rightarrow z,\;\zeta \ne z,\qquad \end{aligned}$$

(7.2)

where $c=\log C$ is the logarithm of conformal radius $C,$ i.e., $c(z) = u(z,z),u(\zeta ,z) = G(\zeta ,z) + \log |\zeta -z|.$ The meaning of the convergence is the following: the equation

holds for all Fock space correlation functionals in $D$ satisfying To derive (7.2) we use Wick’s formula (7.1),

$$\begin{aligned} \varPhi (\zeta ) \varPhi (z) = {\mathbf{E}}[\varPhi (\zeta ) \varPhi (z)] + \varPhi (\zeta )\odot \varPhi (z) \end{aligned}$$

and the relation

$$\begin{aligned} {\mathbf{E}}[\varPhi (\zeta ) \varPhi (z)] = 2G(\zeta ,z) = \log \frac{1}{|\zeta -z|^2} + 2c(z) + o(1). \end{aligned}$$

The convergence of $\varPhi (\zeta )\odot \varPhi (z)$ to $\varPhi ^{\odot 2}(z)$ means (by definition) that

for every such that

If the field $X$ is holomorphic (i.e., all correlation functions ${\mathbf{E}}[X(\zeta ) {\fancyscript{Y}}]$ are holomorphic in $\zeta \in D{\setminus } S_{\fancyscript{Y}}$), then the operator product expansion is then defined as a (formal) Laurent series expansion

$$\begin{aligned} X(\zeta )Y(z)= \sum {C_n(z)}{(\zeta -z)^n},\quad \zeta \rightarrow z. \end{aligned}$$

(7.3)

Since the function $\zeta \mapsto {\mathbf{E}}\,X(\zeta )Y(z){\fancyscript{Z}}$ is holomorphic in a punctured neighborhood of $z,$ it has a Laurent series expansion.

There are only finitely many terms in the principle (or singular) part of the Laurent series (7.3). We use the notation $\sim$ for the singular part of the operator product expansion,

$$\begin{aligned} X(\zeta )Y(z)\sim \sum _{n<0} {C_n(z)}{(\zeta -z)^n}. \end{aligned}$$

We also write $\hbox {Sing}_{\zeta \rightarrow z}\,X(\zeta )Y(z)$ for $\sum _{n<0} {C_n(z)}{(\zeta -z)^n}.$ It is clear that we can differentiate operator product expansions (7.3) both in $\zeta$ and $z$; and the differentiation preserves singular parts. For example,

$$\begin{aligned} J(\zeta )\varPhi (z)\sim -\frac{1}{\zeta -z},\quad J(\zeta )J(z)\sim -\dfrac{1}{(\zeta -z)^2}. \end{aligned}$$

The coefficients in the operator product expansions [e.g., $2c(z) + \varPhi ^{\odot 2}(z)$ in (7.2), $C_n(z)$ in (7.3)] are called OPE coefficients. OPE coefficients of Fock space fields are Fock space fields (as functions of $z$). In particular, if $X$ is holomorphic, then we define the $*_n$ product by $X *_{n} Y=C_n.$ We write $*$ for $*_{0}$ and call $X*Y$ the OPE multiplication, or the OPE product of $X$ and $Y.$

Vertex fields are defined as OPE-exponentials of $\varPhi \!\!:$

$$\begin{aligned} {\fancyscript{V}}^\alpha =e^{*\alpha \varPhi }=\sum _{n=0}^\infty \frac{\alpha ^n}{n!}\varPhi ^{*n}. \end{aligned}$$

They can be expressed in terms of Wick’s calculus: ${\fancyscript{V}}^\alpha =C^{\alpha ^2}e^{\odot \alpha \varPhi },$ see [8, Proposition 3.3]. Here, $C=e^c$ is the conformal radius, see (7.2). The Virasoro field is defined as OPE square of $J = \partial \varPhi \!\!:$

$$\begin{aligned} T=-\frac{1}{2} J* J. \end{aligned}$$

Then by Wick’s calculus,

$$\begin{aligned} T=-\frac{1}{2}~J\odot J+\frac{1}{12}S, \end{aligned}$$

where $S(z) = S(z,z), S(\zeta ,z) := -12\partial _\zeta \partial _z u(\zeta ,z),$ and $u(\zeta ,z) = G(\zeta ,z) + \log |\zeta -z|.$ Thus $T$ is a Schwarzian form of order $\frac{1}{12}.$ In terms of a conformal map $w:D\rightarrow \mathbb{H },S = S_w,$ the Schwarzian derivative of $w.$

1.4 A.4 Conformal Fock space fields

This subsection is borrowed from [8, Sections 4.2–4.4]. We use Lie derivative of a conformal field to define the stress tensor and to state Ward’s identities.

A general conformal Fock space field is a linear combination of basic fields $X_\alpha ,$

$$\begin{aligned} X=\sum _\alpha f_\alpha X_\alpha , \end{aligned}$$

where $f_\alpha$’s are non-random conformal fields, see Sect. 2.1. A non-random conformal field $f$ is said to be invariant with respect to some conformal automorphism $\tau$ of $M$ if

$$\begin{aligned} (f\,\Vert \,\phi )=(f\,\Vert \,\phi \circ \tau ^{-1}) \end{aligned}$$

for all charts $\phi .$ For example, suppose $D$ is a planar domain and let us write $f$ for $(f\,\Vert \,\hbox {id}_D).$ Then $f$ is a $\tau$-invariant $[\lambda ,\lambda _*]$-differential if

$$\begin{aligned} f(z) =f(\tau z)~\tau ^{\prime }(z)^\lambda ~\overline{\tau ^{\prime }(z)}^{\lambda _*}. \end{aligned}$$

It is because $\tau$ is the transition map between the charts $\phi \circ \tau ^{-1}$ and $\phi =\hbox {id}_D.$ By definition, a random conformal field (or a family of conformal fields) is $\tau$-invariant if all correlations are invariant as non-random conformal fields.

Suppose a non-random smooth vector field $v$ is holomorphic in some open set $U\subset M.$ For a conformal Fock space field $X,$ we define the Lie derivative ${\fancyscript{L}}_vX$ in $U$ as

$$\begin{aligned} ({\fancyscript{L}}_v X\,\Vert \, \phi ) = \frac{\hbox {d}}{\hbox {d}t}\Big |_{t=0} (X\,\Vert \, \phi \circ \psi _{-t}), \end{aligned}$$

where $\psi _t$ is a local flow of $v,$ and $\phi$ is an arbitrary chart.

Lie derivative of a differential is a differential but Lie derivative of a Schwarzian form is a quadratic differential:

• ${\fancyscript{L}}_vX=(v\partial +{\bar{v}}\bar{\partial }+\lambda v^{\prime }+\lambda _*\overline{v^{\prime }})X$ for a $[\lambda ,\lambda _*]$-differential $X;$

• ${\fancyscript{L}}_vX=(v\partial +v^{\prime })X +\mu v^{\prime }$ for a pre-Schwarzian form $X$ of order $\mu ;$

• ${\fancyscript{L}}_vX=(v\partial +2v^{\prime })X +\mu v^{\prime \prime \prime }$ for a Schwarzian form $X$ of order $\mu .$

We recall basic properties of Lie derivatives:

• ${\fancyscript{L}}_v$ is an $\mathbb{R }$-linear operator on Fock space fields;

• ${\mathbf{E}}[{\fancyscript{L}}_vX]={\fancyscript{L}}_v({\mathbf{E}}[X]);$

• ${\fancyscript{L}}_v ({\bar{X}})=\overline{({\fancyscript{L}}_vX)}$;

• ${\fancyscript{L}}_v(\partial X)=\partial ({\fancyscript{L}}_v X)$ and ${\fancyscript{L}}_v({\bar{\partial }} X)={\bar{\partial }}({\fancyscript{L}}_v X);$

• Leibniz’s rule applies to Wick’s products, OPE products, and tensor products.

We define the $\mathbb{C }$-linear part ${\fancyscript{L}}_v^+$ and anti-linear part ${\fancyscript{L}}_v^-$ of the Lie derivative ${\fancyscript{L}}_v$ by

$$\begin{aligned} 2{\fancyscript{L}}_v^+ = {\fancyscript{L}}_v-i{\fancyscript{L}}_{iv},\quad 2{\fancyscript{L}}_v^- = {\fancyscript{L}}_v+i{\fancyscript{L}}_{iv}. \end{aligned}$$

1.5 A.5 Stress tensor

This subsection is borrowed from [8, Sections 5.2–5.3]. A Fock space field $X$ in $D$ is said to have a (symmetric) stress tensor $(A,{\bar{A}})$ ($X\in {\fancyscript{F}}(A,{\bar{A}})$) if $A$ is a holomorphic quadratic differential and if Ward’s OPE holds for $X,$ i.e., on a given chart $\phi :U\rightarrow \phi U,$

$$\begin{aligned} \hbox {Sing}_{\zeta \rightarrow z}[A(\zeta )X(z)]= ({\fancyscript{L}}_{k_\zeta }^+X)(z), \quad \hbox {Sing}_{\zeta \rightarrow z}[A(\zeta ){\bar{X}}(z)]= ({\fancyscript{L}}_{k_\zeta }^+{\bar{X}})(z), \end{aligned}$$

where the (local) vector field $k_\zeta$ is defined by $(k_\zeta \,\Vert \,\phi )(\eta ) = 1/(\zeta -\eta ).$ Ward’s family ${\fancyscript{F}}(A,{\bar{A}})$ is closed under differentiation and OPE multiplication, see [8, Proposition 5.8]. In the case of differentials or forms, it is enough to verify Ward’s OPEs in just one chart. For example, a $[\lambda , \lambda _*]$-differential $X$ is in ${\fancyscript{F}}(A,{\bar{A}})$ if and only if the following operator product expansions hold in every/some chart:

$$\begin{aligned} A(\zeta )X(z)\sim \frac{\lambda X(z)}{(\zeta -z)^2}+\frac{\partial X(z)}{\zeta -z},\quad A(\zeta ){\bar{X}}(z)\sim \frac{\bar{\lambda }_*{\bar{X}}(z)}{(\zeta -z)^2}+\frac{\partial {\bar{X}}(z)}{\zeta -z}. \end{aligned}$$

Let $X$ be a form of order $\mu .$ Then $X\in {\fancyscript{F}}(A,{\bar{A}})$ if and only if the following operator product expansion holds in every/some chart:

$$\begin{aligned} A(\zeta )X(z)&\sim \frac{\mu }{(\zeta -z)^2}+\frac{\partial X(z)}{\zeta -z} \quad \hbox {for a pre-pre-Schwarzian form }X;\\ A(\zeta )X(z)&\sim \frac{2\mu }{(\zeta -z)^3}+\frac{X(z)}{(\zeta -z)^2}+\frac{\partial X(z)}{\zeta -z}\quad \hbox {for a pre-Schwarzian form }X;\\ A(\zeta )X(z)&\sim \frac{6\mu }{(\zeta -z)^4}+ \frac{2X(z)}{(\zeta -z)^2}+\frac{\partial X(z)}{\zeta -z}\quad \hbox {for a Schwarzian form }X. \end{aligned}$$

For example, Gaussian free field $\varPhi$ has a stress tensor

$$\begin{aligned} A=-\frac{1}{2} J\odot J,\quad J = \partial \varPhi . \end{aligned}$$

This holomorphic quadratic differential $A$ coincides with the Virasoro field $T$ in the upper half-plane uniformization. While $A$ itself does not belong to ${\fancyscript{F}}(A,{\bar{A}}),$ the Virasoro field $T$ is in ${\fancyscript{F}}(A,{\bar{A}}).$ We review the abstract theory of Virasoro field in the next subsection.

1.6 A.6 Virasoro field

This subsection is borrowed from [8, Lecture 7 and Appendix 11]. A Fock space field $T$ is said to be the Virasoro field for Ward’s family ${\fancyscript{F}}(A,{\bar{A}})$ if

• $T\in {\fancyscript{F}}(A,{\bar{A}}),$ and

• $T-A$ is a non-random holomorphic Schwarzian form.

We define Virasoro primary fields and current primary fields in terms of Virasoro generators $L_n$ (3.6) and current generator $J_n$ (4.9).

Proposition 7.1

(Proposition 7.5 in [8]) Let $X$ be a Fock space field. Any two of the following assertions imply the third one (but neither one implies the other two)

• $X\in {\fancyscript{F}}(A,{\bar{A}})$;

• $X$ is a $[\lambda ,\lambda _*]$-differential;

• $L_{\ge 1}X=0,\; L_0X=\lambda X, \; L_{-1}X=\partial X,$ and similar equations hold for ${\bar{X}}.$

Here, $L_{\ge k}X = 0$ means that $L_nX = 0$ for all $n\ge k.$ We call fields satisfying all three conditions (Virasoro) primary fields in ${\fancyscript{F}}(A,{\bar{A}}).$

A (Virasoro) primary field $X$ is called current primary if

$$\begin{aligned} J_{\ge 1}X = J_{\ge 1}{\bar{X}} = 0, \end{aligned}$$

and

$$\begin{aligned} J_0X = -iqX, \quad J_0{\bar{X}} = i{\bar{q}}_*{\bar{X}} \end{aligned}$$

for some numbers $q$ and $q_*.$ They are called “charges” of $X$. Here, $J_{\ge k}X = 0$ means that $J_nX = 0$ for all $n\ge k.$ We use the following proposition to prove Proposition 4.6 (level two degeneracy equations for $\varPsi$).

Proposition 7.2

(Proposition 11.2 in [8]) For a current primary field $V$ with charges $q,q_*$ in ${\fancyscript{F}}_{(b)},$

$$\begin{aligned} \left( L_{-2}-\frac{1}{2q^2} L_{-1}^2\right) V =0 \end{aligned}$$

provided $2q(b+q) = 1.$