domain $D$ with Dirichlet boundary condition from the Gaussian free field on
its Schottky double $S=D^{\mathrm{double}}.$ Recall that the Gaussian free
field can be defined as a Gaussian field indexed by the energy space. See
Subsection 3.5 for this definition on $\widehat{\mathbb{C}}.$ The energy space
$\mathcal{E}(D)$ can be embedded isometrically into $\mathcal{E}(S)$ in a
natural way. For example, for $\mu$ in the energy space
$\mathcal{E}(\mathbb{H}),$

$\displaystyle\|\mu\|_{\mathcal{E}(\mathbb{H})}^{2}$
$\displaystyle=\int_{\mathbb{H}\times\mathbb{H}}\log\frac{1}{|z-w|^{2}}\,\mu(z)\overline{\mu(w)}-\int_{\mathbb{H}\times\mathbb{H}}\log\frac{1}{|z-\bar{w}|^{2}}\,\mu(z)\overline{\mu(w)}$
$\displaystyle=\frac{1}{2}\int_{\mathbb{C}\times\mathbb{C}}\log\frac{1}{|z-w|^{2}}\,\nu(z)\overline{\nu(w)}=\frac{1}{2}\|\nu\|_{\mathcal{E}(\widehat{\mathbb{C}})}^{2},$

where

$\nu:=\begin{cases}\phantom{-}\mu\phantom{{}^{*}}\textrm{ in }\mathbb{H},\\\
-\mu^{*}\textrm{ in }\mathbb{H}^{*},\end{cases}$

and $E^{*}:=\\{\bar{z}\,|\,z\in E\\},\mu^{*}(E):=\mu(E^{*}).$ As a Fock space
field, the Gaussian free field $\Phi\equiv\Phi_{D}$ in $D$ with Dirichlet
boundary condition can be constructed from the Gaussian free field
$\Psi\equiv\Psi_{S}$ on $S:$

$\Phi_{D}(z)=\frac{1}{\sqrt{2}}\Psi_{S}(z,z^{*}),$

where $\iota:S\to S,z\mapsto z^{*}$ is the canonical involution in the
Schottky double $S$ of $D.$ For example,

$\Phi_{\mathbb{H}}(z)=\frac{1}{\sqrt{2}}\Psi_{\widehat{\mathbb{C}}}(z,\bar{z}),$

and

$\Phi_{\mathbb{D}}(z)=\frac{1}{\sqrt{2}}\Psi_{\widehat{\mathbb{C}}}(z,z^{*}),\qquad
z^{*}=1/\bar{z}.$

In the chordal case, the current fields and the stress energy tensors are
related as

$J_{\mathbb{H}}(z)=\frac{1}{\sqrt{2}}\big{(}J_{\mathbb{C}}(z)-{\bar{J}_{\mathbb{C}}(\bar{z})}\big{)}$

and

$A_{\mathbb{H}}(z)=\frac{1}{2}\big{(}A_{\mathbb{C}}(z)+{\bar{A}_{\mathbb{C}}(\bar{z})}+J_{\mathbb{C}}(z)\odot{\bar{J}_{\mathbb{C}}(\bar{z})}\big{)}.$

Similar statements hold in the radial case. Furthermore, the formal bosonic
fields are related as

$\Phi^{\pm}(z)=\frac{1}{\sqrt{2}}\big{(}\Psi^{\pm}(z)-\Psi^{\mp}(z^{*})\big{)}.$

More generally, for a double divisor
$(\boldsymbol{\sigma}^{+},\boldsymbol{\sigma}^{-})$ satisfying the neutrality
condition $(\mathrm{NC}_{0}),$ we have

(5.11)
$\Phi[\boldsymbol{\sigma}^{+},\boldsymbol{\sigma}^{-}]=\frac{1}{\sqrt{2}}\Psi[\boldsymbol{\sigma}^{+}+\boldsymbol{\sigma}_{*}^{-},\boldsymbol{\sigma}_{*}^{+}+\boldsymbol{\sigma}^{-}].$

In Subsection 9.1 we construct the Gaussian free field in $D$ with Neumann
boundary condition from $\Psi_{S}.$

###  5.4. Modification of Gaussian free field in a simply-connected domain

In Subsection 5.1 we introduce background charge modifications
$\Psi_{\boldsymbol{\beta}^{+},\boldsymbol{\beta}^{-}}$ of the Gaussian free
field on the Riemann sphere. Here the background charges
$\boldsymbol{\beta}^{\pm}$ satisfy the neutrality condition
$(\mathrm{NC}_{b^{\pm}})(b^{\pm}\in\mathbb{C}).$ In this subsection, from
$\Psi_{\boldsymbol{\beta}^{+},\boldsymbol{\beta}^{-}},$ we construct
background charge modifications of the Gaussian free field with Dirichlet
boundary condition in a simply-connected domain.

Let us fix a background charge parameter $b\in\mathbb{R}.$ This parameter $b$
is related to the central charge $c$ as $c=1-12b^{2}.$ For a double background
charge $(\boldsymbol{\beta}^{+},\boldsymbol{\beta}^{-})$ on $\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu\times D$ satisfying the
neutrality condition $(\mathrm{NC}_{b}):$

$\int(\boldsymbol{\beta}^{+}+\boldsymbol{\beta}^{-})=2b,$

we set $\boldsymbol{\beta}=\boldsymbol{\beta}^{+}+\boldsymbol{\beta}^{-}_{*}$
and define

$\Phi_{\boldsymbol{\beta}}(z)\equiv\Phi_{\boldsymbol{\beta}^{+},\boldsymbol{\beta}^{-}}(z)=\frac{1}{\sqrt{2}}\Psi_{\sqrt{2}\,\boldsymbol{\beta}^{+},\sqrt{2}\,\boldsymbol{\beta}_{*}^{-}}(z,z^{*}).$

Then $\Phi_{\boldsymbol{\beta}}$ is a $\mathrm{PPS}(ib,-ib)$ form and

$\varphi_{\boldsymbol{\beta}}(z):=\mathbf{E}\,\Phi_{\boldsymbol{\beta}}(z)=\sum(\beta_{k}^{+}+\beta_{k}^{-})\big{(}\arg(z-q_{k})+\arg(z-\bar{q}_{k})\big{)}+i\sum(\beta_{k}^{+}-\beta_{k}^{-})\log\Big{|}\frac{z-\bar{q}_{k}}{z-q_{k}}\Big{|}$

in the $\mathbb{H}$-uniformization.

We also define background charge modifications of formal bosonic fields as

$\Phi^{\pm}_{\boldsymbol{\beta}}(z)=\frac{1}{\sqrt{2}}\big{(}\Psi^{\pm}_{\sqrt{2}\boldsymbol{\beta}^{+},\sqrt{2}\boldsymbol{\beta}_{*}^{-}}(z)-\Psi^{\mp}_{\sqrt{2}\boldsymbol{\beta}^{+},\sqrt{2}\boldsymbol{\beta}_{*}^{-}}(z^{*})\big{)}.$

For a double divisor $(\boldsymbol{\tau}^{+},\boldsymbol{\tau}^{-})$
satisfying the neutrality condition $(\mathrm{NC}_{0}),$ we set
$\boldsymbol{\tau}=\boldsymbol{\tau}^{+}+\boldsymbol{\tau}_{*}^{-}.$ Then we
have

(5.12)
$\Phi_{\boldsymbol{\beta}}[\boldsymbol{\tau}]=\frac{1}{\sqrt{2}}\Psi_{\sqrt{2}\,\boldsymbol{\beta}^{+},\sqrt{2}\,\boldsymbol{\beta}_{*}^{-}}[\boldsymbol{\tau},\boldsymbol{\tau}_{*}].$

•  _Chordal case without spins._ For a simply-connected domain $D$ with marked
boundary points $q,q_{k}\in\partial D,$ a background charge
$\boldsymbol{\beta}=(2b-\sum\beta_{k})\cdot q+\sum\beta_{k}\cdot
q_{k},(\beta_{k}\in\mathbb{R}),$ and a conformal map $w\equiv
w_{D,q}:~{}(D,q)\to(\mathbb{H},\infty)$ we have

$\Phi_{\boldsymbol{\beta}}=\Phi+\varphi_{\boldsymbol{\beta}},\qquad\varphi_{\boldsymbol{\beta}}=-2b\arg
w^{\prime}+2\sum\beta_{k}\arg(w-w(q_{k})).$

We also define $J_{\boldsymbol{\beta}}:=\partial\Phi_{\boldsymbol{\beta}}.$
Then it is a $\mathrm{PS}(ib)$ form and

$J_{\boldsymbol{\beta}}=J+j_{\boldsymbol{\beta}},\qquad
j_{\boldsymbol{\beta}}=ib\frac{w^{\prime\prime}}{w^{\prime}}-i\sum\beta_{k}\frac{w^{\prime}}{w-w(q_{k})}.$

In particular, if $\boldsymbol{\beta}=2b\cdot q,$ then we have

(5.13) $\Phi_{\boldsymbol{\beta}}=\Phi-2b\arg w^{\prime},\qquad
J_{\boldsymbol{\beta}}=\partial\Phi_{\boldsymbol{\beta}}=J+ib\frac{w^{\prime\prime}}{w^{\prime}},$

and the 1-point function
$\varphi_{\boldsymbol{\beta}}=\mathbf{E}\,\Phi_{\boldsymbol{\beta}}=-2b\arg
w^{\prime}$ does not depend on the choice of the conformal map. Due to this
property, these modifications (5.13) are well-defined. As a $\mathrm{PS}(ib)$
form, the current field $J_{\boldsymbol{\beta}}$ is conformally invariant with
respect to $\mathrm{Aut}(D,q).$

•  _Radial case with a spin only at a marked interior point._ For a simply-
connected domain $D$ with a marked interior point $q\in D,$ we consider a
conformal map

$w\equiv w_{D,q}:~{}(D,q)\to(\mathbb{D},0),$

from $D$ onto the unit disc $\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}.$ Let
$q_{k}\in\partial D$ be marked boundary points. Given a background charge
$\boldsymbol{\beta}=\beta_{q}\cdot q+\overline{\beta_{q}}\cdot
q^{*}+\sum\beta_{k}\cdot q_{k}(\beta_{k}\in\mathbb{R})$ with the neutrality
condition $(\mathrm{NC}_{b}):$

$2\,\mathrm{Re}\,\beta_{q}+\sum\beta_{k}=2b,$

the 1-point function
$\varphi_{\boldsymbol{\beta}}=\mathbf{E}\,\Phi_{\boldsymbol{\beta}}$ is given
by

$\varphi_{\boldsymbol{\beta}}(z)=2\,\mathrm{Re}\,\beta_{q}\arg
z+2\,\mathrm{Im}\,\beta_{q}\log|z|+2\sum\beta_{k}\arg(z-q_{k}).$

in the $\mathbb{D}$-uniformization. Also we define
$J_{\boldsymbol{\beta}}:=\partial\Phi_{\boldsymbol{\beta}}.$ Then it is a
$\mathrm{PS}(ib)$ form and

$J_{\boldsymbol{\beta}}=J+ib\frac{w^{\prime\prime}}{w^{\prime}}-i{\beta_{q}}\frac{w^{\prime}}{w}-i\sum\beta_{k}\frac{w^{\prime}}{w-w(q_{k})}.$

In particular, if $\boldsymbol{\beta}=b\cdot q+b\cdot q^{*},$ then

(5.14)
$\Phi_{\boldsymbol{\beta}}=\Phi+\varphi_{\boldsymbol{\beta}},\qquad\varphi_{\boldsymbol{\beta}}=-2b\arg\frac{w^{\prime}}{w}.$

Since the multivalued function $\varphi_{\boldsymbol{\beta}}$ does not depend
on the choice of the conformal map, the “bosonic” field,
$\Phi_{\boldsymbol{\beta}}$ in $D^{*}:=D\setminus\\{q\\}$ is invariant with
respect to $\mathrm{Aut}(D,q).$ Also we have

(5.15) $J_{\boldsymbol{\beta}}=J+j_{\boldsymbol{\beta}},\qquad
j_{\boldsymbol{\beta}}=\partial\varphi_{\boldsymbol{\beta}}=ib\Big{(}\frac{w^{\prime\prime}}{w^{\prime}}-\frac{w^{\prime}}{w}\Big{)}.$

######  Remarks.

(a) For $b\neq 0,$ the 1-point function $\varphi_{\boldsymbol{\beta}}$ has
monodromy $4\pi b$ around $q.$

(b) For $b\neq 0,$ the 1-point function $j_{\boldsymbol{\beta}}$ has a simple
pole at $q,$ and $J$ is holomorphic in $D^{*}.$

(c) As a $\mathrm{PS}(ib)$ form, the current $J_{\boldsymbol{\beta}}$ is
conformally invariant with respect to $\mathrm{Aut}(D,q).$

For a symmetric background charge $\boldsymbol{\beta}=\sum\beta_{k}\cdot
q_{k}+\mkern 1.5mu\overline{\mkern-1.5mu\beta_{k}\mkern-1.5mu}\mkern
1.5mu\cdot q_{k}^{*}$ on $S,$ we have

$\varphi_{\boldsymbol{\beta}}(z):=\mathbf{E}\,\Phi_{\boldsymbol{\beta}}(z)=2\sum\mathrm{Re}\,\beta_{k}\big{(}\arg(z-q_{k})+\arg(z-\bar{q}_{k})\big{)}-2\sum\mathrm{Im}\,\beta_{k}\log\Big{|}\frac{z-\bar{q}_{k}}{z-q_{k}}\Big{|}$

and

(5.16)
$j_{\boldsymbol{\beta}}(z):=\mathbf{E}\,J_{\boldsymbol{\beta}}(z)=-i\sum\Big{(}\frac{\beta_{k}}{z-q_{k}}+\frac{\bar{\beta}_{k}}{z-\bar{q}_{k}}\Big{)}$

in the $\mathbb{H}$-uniformization if all $q_{k}$’s are in $D.$ Later, we
present the connection between boundary conformal field theory with symmetric
background charges and the chordal/radial SLE theory with forces and spins.
The following theorem is parallel with Theorem 5.2.

######  Theorem 5.4.

For a symmetric background charge $\boldsymbol{\beta}$ on $S,$ the bosonic
field $\Phi_{\boldsymbol{\beta}}$ in $D$ has a stress tensor
$(A_{\boldsymbol{\beta}},\bar{A}_{\boldsymbol{\beta}}):$

$A_{\boldsymbol{\beta}}=A+ib\partial J-j_{\boldsymbol{\beta}}J$

and its Virasoro field is

$T_{\boldsymbol{\beta}}=-\frac{1}{2}J_{\boldsymbol{\beta}}*J_{\boldsymbol{\beta}}+ib\partial
J_{\boldsymbol{\beta}}.$

######  Example.

For a background charge $\boldsymbol{\beta}=\beta_{q}\cdot q+\mkern
1.5mu\overline{\mkern-1.5mu\beta_{q}\mkern-1.5mu}\mkern 1.5mu\cdot
q^{*}+\sum\beta_{k}\cdot q_{k}(q\in D,q_{k}\in\partial
D,\beta_{k}\in\mathbb{R})$ satisfying the neutrality condition
$(\mathrm{NC}_{b})$, we have

$\mathbf{E}\,T_{\boldsymbol{\beta}}(z)=\sum_{k}\frac{\lambda_{k}}{(z-q_{k})^{2}}+\sum_{j<k}\frac{\beta_{j}\beta_{k}}{(z-q_{j})(z-q_{k})}+\frac{\lambda_{q}}{z^{2}}+\sum_{k}\frac{\beta_{q}\beta_{k}}{z(z-q_{k})}$

in the $(\mathbb{D},0)$-uniformization, where
$\lambda_{k}=\lambda_{b}(\beta_{k})$ and $\lambda_{q}=\lambda_{b}(\beta_{q}).$

###  5.5. OPE exponentials in a punctured domain

For simplicity, we consider the standard radial case
$\boldsymbol{\beta}=b\cdot q+b\cdot q^{*}.$ Our goal in this subsection is to
explain the statement that under the neutrality condition $(\mathrm{NC}_{0})$
on $\boldsymbol{\tau}=\boldsymbol{\tau}^{+}+\boldsymbol{\tau}^{-}_{*}$ the
modified multi-vertex field
$\mathcal{O}_{\boldsymbol{\beta}}[\boldsymbol{\tau}]\equiv\mathcal{O}_{\boldsymbol{\beta}}[\boldsymbol{\tau}^{+},\boldsymbol{\tau}^{-}]$
(with $\tau_{q}^{+}=\tau_{q}^{-}=0$) in $D^{*}$ can be viewed as the OPE
exponential of
$i\Phi_{\boldsymbol{\beta}}[\boldsymbol{\tau}^{+},\boldsymbol{\tau}^{-}].$
Non-chiral vertex fields $\mathcal{V}_{\boldsymbol{\beta}}^{(\sigma)}$ are
defined by

$\mathcal{V}_{\boldsymbol{\beta}}^{(\sigma)}:=\mathrm{e}^{*i\sigma\Phi_{\boldsymbol{\beta}}}=\sum_{n=0}^{\infty}\frac{(i\sigma)^{n}}{n!}\,\Phi_{\boldsymbol{\beta}}^{*n}.$

We have

$\mathcal{V}_{\boldsymbol{\beta}}^{(\sigma)}=\mathrm{e}^{i\sigma\varphi_{\boldsymbol{\beta}}}\mathcal{V}^{(\sigma)}=\mathrm{e}^{i\sigma\varphi_{\boldsymbol{\beta}}}C^{-\sigma^{2}}\mathrm{e}^{\odot
i\sigma\Phi},\qquad(\varphi_{\boldsymbol{\beta}}:=\mathbf{E}\,\Phi_{\boldsymbol{\beta}}).$

The non-random field $\varphi_{\boldsymbol{\beta}}$ is a
$\mathrm{PPS}(ib,-ib)$ form and $C$ is the conformal radius. In terms of a
conformal map $w:(D,q)\to(\mathbb{D},0),$ we have

$(C\,\|\,\mathrm{id}_{D})(z)=\frac{1-|w(z)|^{2}}{|w^{\prime}(z)|},$

where $\mathrm{id}_{D}$ is the identity chart of $D.$ Thus the conformal
radius is a $[-1/2,-1/2]$-differential and $\mathcal{V}^{(\sigma)}$ is a
$[\lambda,\lambda_{*}]$-differential with
$\lambda=\lambda_{b}(\sigma),\lambda_{*}=\lambda_{b}(-\sigma).$ Its 1-point
function in the $(\mathbb{D},0)$-uniformization is

$\mathbf{E}\,\mathcal{V}_{\boldsymbol{\beta}}^{(\sigma)}(z)=(1-|z|^{2})^{-\sigma^{2}}z^{\sigma
b}(\bar{z})^{-\sigma b}.$

Since $\mathcal{V}_{\boldsymbol{\beta}}^{(\sigma)}(z)$ and
$\mathcal{O}_{\boldsymbol{\beta}}[\sigma\cdot z,-\sigma\cdot z]$ contain the
same Wick’s exponential and they are differentials with the same conformal
dimensions and the same correlation functions in the
$(\mathbb{D},0)$-uniformization, we conclude that

$\mathcal{O}_{\boldsymbol{\beta}}[\sigma\cdot z,-\sigma\cdot
z]=\mathrm{e}^{*i\sigma\Phi_{\boldsymbol{\beta}}(z)}.$

Next, we explain the OPE product of OPE exponentials and why it corresponds to
the addition of divisors. Here is the typical example:

(5.17)
$\mathcal{V}_{\boldsymbol{\beta}}^{(\sigma_{1})}*\mathcal{V}_{\boldsymbol{\beta}}^{(\sigma_{2})}=\mathcal{V}_{\boldsymbol{\beta}}^{(\sigma_{1}+\sigma_{2})}.$

Here $*$ means the coefficients of the leading term in OPE expansion. An
alternative notation for such OPE multiplication is
$\mathcal{V}_{\boldsymbol{\beta}}^{(\sigma_{1})}\star\mathcal{V}_{\boldsymbol{\beta}}^{(\sigma_{2})},$
see [18, Section 15.2]. To see (5.17), we need to compute asymptotic behavior
of $\exp(-\sigma_{1}\sigma_{2}\mathbf{E}[\Phi(\zeta)\Phi(z)]):$

$\bigg{|}\frac{w(\zeta)-w(z)}{1-w(\zeta)\overline{w(z)}}\bigg{|}^{2\sigma_{1}\sigma_{2}}\sim\bigg{|}\frac{w^{\prime}(z)(\zeta-z)}{1-|w(z)|^{2}}\bigg{|}^{2\sigma_{1}\sigma_{2}},\qquad\zeta\to
z.$

This implies $\mathrm{e}^{\odot i\sigma_{1}\Phi}*\mathrm{e}^{\odot
i\sigma_{2}\Phi}=C^{-2\sigma_{1}\sigma_{2}}\mathrm{e}^{\odot
i(\sigma_{1}+\sigma_{2})\Phi}.$ The same argument shows that if $f_{1}$ and
$f_{2}$ are non-random fields, then

$(f_{1}\mathrm{e}^{\odot i\sigma_{1}\Phi})*(f_{2}\mathrm{e}^{\odot
i\sigma_{2}\Phi})=f_{1}f_{2}C^{-2\sigma_{1}\sigma_{2}}\mathrm{e}^{\odot
i(\sigma_{1}+\sigma_{2})\Phi}.$

Taking
$f_{j}=\mathbf{E}\,\mathcal{V}_{\boldsymbol{\beta}}^{(\sigma_{j})}=\mathrm{e}^{i\sigma_{j}\varphi_{\boldsymbol{\beta}}}C^{-\sigma_{j}^{2}},$
the above identity shows (5.17).

The computation is more transparent if we use formal OPE exponentials. Let us
formally define

$\mathcal{O}^{(\sigma)}(``\equiv"\mathrm{e}^{*i\sigma\Phi_{\boldsymbol{\beta}}^{+}})=(w^{\prime})^{\lambda}w^{\sigma
b}\mathrm{e}^{\odot i\sigma\Phi^{+}}$

(which depends on the choice of conformal map) so that

$\mathcal{O}[\sigma\cdot z-\sigma\cdot
z_{0}]=\mathcal{O}^{(\sigma)}(z)\mathcal{O}^{(-\sigma)}(z_{0})$

holds (formally) within formal correlations. In addition to the product of
1-point functions (and Wick’s exponential), we have the interaction term

$\mathrm{e}^{\sigma^{2}\mathbf{E}\,[\Phi^{+}(z)\Phi^{+}(z_{0})]}=(w(z)-w(z_{0}))^{-\sigma^{2}}$

in $\mathcal{O}^{(\sigma)}(z)\mathcal{O}^{(-\sigma)}(z_{0})$. As in the OPE
product of non-chiral vertex fields, we have (again formally)

(5.18)
$\mathcal{O}^{(\sigma_{1})}*\mathcal{O}^{(\sigma_{2})}=\mathcal{O}^{(\sigma_{1}+\sigma_{2})},$

where the OPE product $*$ could be further specified as the $\star$ product
(leading coefficient) or as $*_{\sigma_{1}\sigma_{2}}$ (coefficient of
$(\zeta-z)^{\sigma_{1}\sigma_{2}}$ in the operator product expansion of
$\mathcal{O}^{(\sigma_{1})}(\zeta)\mathcal{O}^{(\sigma_{2})}(z)$). Let us
compute (5.18):

$\displaystyle\mathbf{E}\,\mathcal{O}^{(\sigma_{1})}(\zeta)\mathcal{O}^{(\sigma_{2})}(z)$
$\displaystyle=(w(\zeta)-w(z))^{\sigma_{1}\sigma_{2}}w^{\prime}(\zeta)^{\lambda_{b}(\sigma_{1})}w(\zeta)^{\sigma_{1}b}w^{\prime}(z)^{\lambda_{b}(\sigma_{2})}w(z)^{\sigma_{2}b}$
$\displaystyle\sim(\zeta-z)^{\sigma_{1}\sigma_{2}}w^{\prime}(z)^{\sigma_{1}\sigma_{2}+\lambda_{b}(\sigma_{1})+\lambda_{b}(\sigma_{2})}w(z)^{(\sigma_{1}+\sigma_{2})b}.$

It is trivial to compute Wick’s exponential part in the operator product
expansion. The identity
$\lambda_{b}(\sigma_{1}+\sigma_{2})=\lambda_{b}(\sigma_{1})+\lambda_{b}(\sigma_{2})+\sigma_{1}\sigma_{2}$
now shows (5.18).

Note that the last OPE has the form

(5.19) $X(\zeta)Y(z)=(\zeta-z)^{\alpha}\sum{C_{n}(z)}{(\zeta-z)^{n}},\qquad
C_{n}:=X*_{\alpha+n}Y.$

###  5.6. Insertions and changes of background charges

Denote by $\mathcal{F}_{\boldsymbol{\beta}}(D^{\prime})$ the _OPE family_ of
$\Phi_{\boldsymbol{\beta}}$ on $D^{\prime}\subseteq D,$ the algebra (over
$\mathbb{C}$) spanned by $1$ and the derivatives of
$\Phi_{\boldsymbol{\beta}},$ $\mathcal{O}_{\boldsymbol{\beta}}$ with nodes in
$D^{\prime}$ under the OPE multiplication $*.$ Let
$D^{*}:=D\setminus(\mathrm{supp}\,\boldsymbol{\beta}_{1}^{\pm}\cup\,\mathrm{supp}\,\boldsymbol{\beta}_{2}^{\pm}).$

######  Theorem 5.5.

Given background charges $\boldsymbol{\beta}_{1},\boldsymbol{\beta}_{2}$ on
$S$ with the neutrality conditions $(\mathrm{NC}_{b}),$ the image of
$\mathcal{F}_{\boldsymbol{\beta}_{1}}(D^{*})$ under the insertion of $e^{\odot
i\Phi[\boldsymbol{\beta}_{2}-\boldsymbol{\beta}_{1}]}$ is
$\mathcal{F}_{\boldsymbol{\beta}_{2}}(D^{*})$ within correlations.

Theorem 5.5 follows immediately from Theorem 5.3 and the Schottky double
construction (5.12) of $\Phi_{\boldsymbol{\beta}}$ from
$\Psi_{\sqrt{2}\,\boldsymbol{\beta}^{+},\sqrt{2}\,\boldsymbol{\beta}_{*}^{-}}.$

We now present a typical example of the insertion operators which create the
chordal/radial SLE curves. In the radial case with
$\check{\boldsymbol{\beta}\,}=b\cdot q+b\cdot q^{*},(q\in D),$ the insertion
of one-leg operator

$\Lambda_{p}\equiv\Lambda(p):=\mathcal{O}_{\check{\boldsymbol{\beta}\,}}[\boldsymbol{\tau}],\quad\boldsymbol{\tau}=a\cdot
p-\frac{a}{2}\cdot q-\frac{a}{2}\cdot q^{*},\quad(p\in\partial D)$

produces an operator

$\mathcal{X}\mapsto\widehat{\mathcal{X}}$

acting on Fock space functionals/fields by the rules

(5.20)
$\partial\mathcal{X}\mapsto\partial\widehat{\mathcal{X}},\quad{\bar{\partial}}\mathcal{X}\mapsto{\bar{\partial}}\widehat{\mathcal{X}},\quad\alpha\mathcal{X}+\beta\mathcal{Y}\mapsto\alpha\widehat{\mathcal{X}}+\beta\widehat{\mathcal{Y}},\quad\mathcal{X}\odot\mathcal{Y}\mapsto\widehat{\mathcal{X}}\odot\widehat{\mathcal{Y}}$

for Fock space functionals $\mathcal{X}$ and $\mathcal{Y}$ in $D^{*}$ and the
formula

(5.21)
$\widehat{\Phi}_{\check{\boldsymbol{\beta}\,}}(z)=\Phi_{\boldsymbol{\beta}}(z)=\Phi_{\check{\boldsymbol{\beta}\,}}(z)+a\arg\frac{(1-w(z))^{2}}{w(z)},$

where $w:(D,p,q)\to(\mathbb{D},1,0)$ is a conformal map and

(5.22)
$\boldsymbol{\beta}=\check{\boldsymbol{\beta}\,}+\boldsymbol{\tau}=a\cdot
p+\Big{(}b-\frac{a}{2}\Big{)}\cdot q+\Big{(}b-\frac{a}{2}\Big{)}\cdot q^{*}.$

In the chordal case with $\check{\boldsymbol{\beta}\,}=2b\cdot q,(q\in\partial
D),$ recall that the insertion of one-leg operator

$\Lambda_{p}\equiv\Lambda(p):=\mathcal{O}_{\check{\boldsymbol{\beta}\,}}[\boldsymbol{\tau}],\quad\boldsymbol{\tau}=a\cdot
p-a\cdot q,\quad(p\in\partial D)$

produces an operator $\mathcal{X}\mapsto\widehat{\mathcal{X}}$ by the rules
(5.20) and the formula

$\widehat{\Phi}_{\check{\boldsymbol{\beta}\,}}(z)=\Phi_{\boldsymbol{\beta}}(z)=\Phi_{\check{\boldsymbol{\beta}\,}}(z)+2a\arg
w(z),$

where $w:(D,p,q)\to(\mathbb{H},0,\infty)$ is a conformal map and

$\boldsymbol{\beta}=\check{\boldsymbol{\beta}\,}+\boldsymbol{\tau}=a\cdot
p+(2b-a)\cdot q.$

See [18, Section 14.2].

Let

$\widehat{\mathbf{E}}[\mathcal{X}]:=\frac{\mathbf{E}[\Lambda_{p}\mathcal{X}]}{\mathbf{E}[\Lambda_{p}]}=\mathbf{E}[\mathrm{e}^{\odot
i\Phi[\boldsymbol{\tau}]}\mathcal{X}],\quad\boldsymbol{\tau}=a\cdot
p-\frac{a}{2}\cdot q-\frac{a}{2}\cdot q^{*}.$

Let $\widehat{\mathcal{X}}\in\mathcal{F}_{\boldsymbol{\beta}}(D^{*})$
correspond to the string
$\mathcal{X}\in\mathcal{F}_{\check{\boldsymbol{\beta}}\,}(D^{*})$ under the
map given by (5.20) – (5.21). Then by Theorem 5.5 we have

(5.23) $\widehat{\mathbf{E}}[\mathcal{X}]=\mathbf{E}[\widehat{\mathcal{X}}].$

######  Examples.

Let $\varphi_{\boldsymbol{\beta}}:=\mathbf{E}\,\Phi_{\boldsymbol{\beta}}$ in
the radial case with $\boldsymbol{\beta}$ in (5.22).

(a) The current $J_{\boldsymbol{\beta}}$ is a pre-Schwarzian form of order
$ib,$

(5.24)
$J_{\boldsymbol{\beta}}=J+ia\frac{w^{\prime}}{1-w}+\Big{(}\frac{ia}{2}-ib\Big{)}\frac{w^{\prime}}{w}+ib\frac{w^{\prime\prime}}{w^{\prime}}.$

In the $(\mathbb{D},1,0)$-uniformization,

$j_{\boldsymbol{\beta}}(z)=\mathbf{E}\,J_{\boldsymbol{\beta}}(z)=ia\dfrac{1}{1-z}+\Big{(}\dfrac{ia}{2}-ib\Big{)}\dfrac{1}{z};$

(b) The Virasoro field $T_{\boldsymbol{\beta}}$ is a Schwarzian form of order
$\frac{1}{12}c,$

(5.25) $\displaystyle T_{\boldsymbol{\beta}}$
$\displaystyle=-\dfrac{1}{2}J_{\boldsymbol{\beta}}*J_{\boldsymbol{\beta}}+ib\partial
J_{\boldsymbol{\beta}}$ $\displaystyle=A-j_{\boldsymbol{\beta}}J+ib\partial
J+\frac{c}{12}S_{w}+h_{1,2}\frac{w^{\prime
2}}{w(1-w)^{2}}+h_{0,1/2}\frac{w^{\prime 2}}{w^{2}}\qquad(A=-\frac{1}{2}J\odot
J),$

where $h_{1,2}=\frac{1}{2}a^{2}-ab,$
$h_{0,1/2}=\frac{1}{8}{a^{2}}-\frac{1}{2}b^{2},$ (see (4.1) and (4.2)) and
$S_{w}=N_{w}^{\prime}-\frac{1}{2}{N_{w}^{2}},\,(N_{w}=(\log
w^{\prime})^{\prime})$ is the Schwarzian derivative of $w.$ In the
$(\mathbb{D},1,0)$-uniformization,

$\mathbf{E}\,T_{\boldsymbol{\beta}}(z)=\dfrac{h_{1,2}}{z(1-z)^{2}}+\dfrac{h_{0,1/2}}{z^{2}};$

(c) The operator produced by the insertion of one-leg operator can be extended
to the formal fields. For example, we have

$\Phi_{\boldsymbol{\beta}}^{+}=\Phi_{\check{\boldsymbol{\beta}\,}}^{+}+\dfrac{ia}{2}\log
w-ia\log(1-w);$

and

$\mathcal{O}_{\boldsymbol{\beta}}^{(\sigma)}=\Big{(}\frac{1-w}{\sqrt{w}}\Big{)}^{a\sigma}\mathcal{O}_{\check{\boldsymbol{\beta}\,}}^{(\sigma)}.$

##  6\. Extended OPE family

In this section we extend the OPE family
$\mathcal{F}_{\boldsymbol{\beta}}(D^{*})$
($D^{*}:=D\setminus(\mathrm{supp}\,\boldsymbol{\beta}^{+}\cup\,\mathrm{supp}\,\boldsymbol{\beta}^{-})$,
$\boldsymbol{\beta}=\boldsymbol{\beta}^{+}+\boldsymbol{\beta}^{-}_{*}$) of
$\Phi_{\boldsymbol{\beta}}$ by adding the generators obtained from the
operator product expansion of fields in
$\mathcal{F}_{\boldsymbol{\beta}}(D^{*})$ at the punctures
$q_{k}\in\mathrm{supp}\,\boldsymbol{\beta}^{+}\cup\,\mathrm{supp}\,\boldsymbol{\beta}^{-}$
or the rooting procedure. Examples include the OPE exponentials of
$\Phi_{\boldsymbol{\beta}}$ with nodes at
$q_{k}\in\mathrm{supp}\,\boldsymbol{\beta}^{+}\cup\,\mathrm{supp}\,\boldsymbol{\beta}^{-}.$

###  6.1. OPE at the puncture

Let us consider the radial case first. For simplicity we only consider the
“formal” holomorphic puncture differential in the case that
$\check{\boldsymbol{\beta}\,}=b\cdot q+b\cdot q^{*}:$

$\mathcal{O}_{q}^{(\tau)}=(w_{q}^{\prime})^{\tau^{2}/2}\mathrm{e}^{\odot
i\tau\Phi^{+}(q)}.$

This is a (formal) Fock space correlation functional with conformal dimension
$h_{q}=\tau_{q}^{2}/2$ at $q.$

######  Proposition 6.1.

We have

$\mathcal{O}^{(\sigma)}(z)\mathcal{O}_{q}^{(\tau)}\sim(z-q)^{\sigma(\tau+b)}\mathcal{O}_{q}^{(\sigma+\tau)}.$

###### Proof.

As $z\to q,$ we have

$\displaystyle\mathcal{O}^{(\sigma)}(z)\mathcal{O}_{q}^{(\tau)}$
$\displaystyle=w^{\sigma b}(w^{\prime})^{\lambda_{b}(\sigma)}\mathrm{e}^{\odot
i\sigma\Phi^{+}(z)}(w_{q}^{\prime})^{\tau^{2}/2}\mathrm{e}^{\odot
i\tau\Phi^{+}(q)}$
$\displaystyle=w^{\sigma(\tau+b)}(w^{\prime})^{\lambda_{b}(\sigma)}(w_{q}^{\prime})^{\tau^{2}/2}\mathrm{e}^{\odot
i\sigma\Phi^{+}(z)+i\tau\Phi^{+}(q)}$
$\displaystyle\sim(z-q)^{\sigma(\tau+b)}(w_{q}^{\prime})^{\sigma(\tau+b)+\lambda_{b}(\sigma)+\tau^{2}/2}\mathrm{e}^{\odot
i(\sigma+\tau)\Phi^{+}(q)}$
$\displaystyle=(z-q)^{\sigma(\tau+b)}\mathcal{O}_{q}^{(\sigma+\tau)}.$

Here we use $\mathrm{e}^{\odot i\sigma\Phi^{+}(z)}\mathrm{e}^{\odot
i\tau\Phi^{+}(q)}=w^{\sigma\tau}\mathrm{e}^{\odot
i(\sigma\Phi^{+}(z)+\tau\Phi^{+}(q))},$ $w(z)\sim(z-q)w_{q}^{\prime}$ as $z\to
q,$ and the identity
$\lambda_{b}(\sigma)+\tau^{2}/2+\sigma(\tau+b)=(\sigma+\tau)^{2}/2.$ The
notation $\sim$ means the first leading term of the operator product
expansion. ∎

We can express the above proposition in any of the following formulas:

$\mathcal{O}^{(\sigma)}*\mathcal{O}_{q}^{(\tau)}\equiv\mathcal{O}^{(\sigma)}*_{\sigma(\tau+b)}\mathcal{O}_{q}^{(\tau)}=\mathcal{O}_{q}^{(\sigma+\tau)},\qquad\mathcal{O}^{(\sigma)}\star\mathcal{O}_{q}^{(\tau)}=\mathcal{O}_{q}^{(\sigma+\tau)}.$

The point is that the arithmetic of divisors has the OPE nature, both in
$D^{*}:=D\setminus\\{q\\}$ and at the puncture $q.$

Let us mention a special case ($\tau=0$) of the proposition.

######  Corollary 6.2.

We have

$\mathcal{O}_{q}^{(\sigma)}=\mathcal{O}^{(\sigma)}*1_{q}.$

This OPE $\mathcal{O}^{(\sigma)}(z)\sim(z-q)^{\sigma
b}\mathcal{O}_{q}^{(\sigma)}\,(z\to q)$ is related to the “rooting” procedure
(see [18, Section 12.3] for this procedure in the chordal case):

$\mathcal{O}_{q}^{(\sigma)}=\lim_{\varepsilon\to 0}z_{\varepsilon}^{-\sigma
b}\,\mathcal{O}^{(\sigma)}(z_{\varepsilon}),$

where the point $z_{\varepsilon}$ is at distance $\varepsilon$ from $q=0$ in
the chart. Equivalently, this definition can be reached by applying the
following rooting rule to the formal field
$\mathcal{O}^{(\sigma)}(z)=w(z)^{\sigma
b}w^{\prime}(z)^{\lambda}\,\mathrm{e}^{\odot i\sigma\Phi(z)}:$

  1. (a)

the point $z$ in the terms $w^{\prime}(z)^{\lambda}$ and $\mathrm{e}^{\odot
i\sigma\Phi(z)}$ is replaced by the puncture $q;$

  2. (b)

the term $w(z)^{\sigma b}$ is replaced by $(w_{q}^{\prime})^{\sigma b}.$

In the chordal case with $\check{\boldsymbol{\beta}\,}=2b\cdot q,$ we consider
the “formal” boundary puncture differential

$\mathcal{O}_{q}^{(\tau)}=(w_{q}^{\prime})^{\lambda_{b}(-\tau)}\mathrm{e}^{\odot
i\tau\Phi^{+}(q).}$

In terms of a uniformizing map $w:(D,q)\to(\mathbb{H},0)$ we have
$\mathcal{O}^{(\sigma)}=(w^{\prime})^{\lambda_{b}(\sigma)}w^{2\sigma b}$ and

$\mathcal{O}^{(\sigma)}(z)\mathcal{O}_{q}^{(\tau)}\sim(z-q)^{\sigma(\tau+2b)}\mathcal{O}_{q}^{(\sigma+\tau)}$

as $z\to q.$ We use the identity
$\lambda_{b}(\sigma)+\lambda_{b}(-\tau)+\sigma(\tau+2b)=\lambda_{b}(-\sigma-\tau).$
This OPE can be expressed as

$\mathcal{O}^{(\sigma)}*\mathcal{O}_{q}^{(\tau)}\equiv\mathcal{O}^{(\sigma)}*_{\sigma(\tau+2b)}\mathcal{O}_{q}^{(\tau)}=\mathcal{O}_{q}^{(\sigma+\tau)},\qquad\mathcal{O}^{(\sigma)}\star\mathcal{O}_{q}^{(\tau)}=\mathcal{O}_{q}^{(\sigma+\tau)}.$

We refer to (5.19) to remind the readers what this type of OPE is. In the
special case $\tau=0,$ we have

$\mathcal{O}_{q}^{(\sigma)}=\mathcal{O}^{(\sigma)}*_{2\sigma b}1_{q}.$

This is related to the “rooting” procedure:

$\mathcal{O}_{q}^{(\sigma)}=\lim_{\varepsilon\to 0}z_{\varepsilon}^{-2\sigma
b}\mathcal{O}^{(\sigma)}(z_{\varepsilon}),$

where the point $z_{\varepsilon}$ is at distance $\varepsilon$ from $q=0$ in
the chart. In terms of a uniformizing map $w:(D,q)\to(\mathbb{H},\infty)$
satisfying $w(\zeta)\sim-1/(\zeta-q)$ as $\zeta\to q$ in a fixed boundary
chart at $q,$ then the rooting procedure becomes

$\mathcal{O}_{q}^{(\sigma)}=\lim_{\varepsilon\to
0}w^{\prime}(z_{\varepsilon})^{\sigma
b}\mathcal{O}^{(\sigma)}(z_{\varepsilon}),$

where the point $z_{\varepsilon}$ is at (spherical) distance $\varepsilon$
from $q$ in the chart $\phi,$ see [18, Section 12.3].

###  6.2. Definition of the extended OPE family

For simplicity, we consider the standard radial case with
$\boldsymbol{\beta}=b\cdot q+b\cdot q^{*}.$ If
$X\in\mathcal{F}_{\boldsymbol{\beta}}(D^{*}),$ $(D^{*}:=D\setminus\\{q\\}),$
then for all points $z\in D$ and all holomorphic (local) vector fields $v$ we
have

(6.1) $\frac{1}{2\pi
i}\oint_{(z)}vT_{\boldsymbol{\beta}}\,X(z)=\mathcal{L}_{v}^{+}X(z),\qquad\frac{1}{2\pi
i}\oint_{(z)}vT_{\boldsymbol{\beta}}\,\bar{X}(z)=\mathcal{L}_{v}^{+}\bar{X}(z).$

This is not a characterization of fields in
$\mathcal{F}_{\boldsymbol{\beta}}(D^{*}),$ e.g., $X\Phi(q)$ would satisfy
(6.1). Note that the fields in $\mathcal{F}_{\boldsymbol{\beta}}(D^{*})$ are
typically not defined at $q,$ e.g.,
$\Phi_{\boldsymbol{\beta}},J_{\boldsymbol{\beta}},$ and
$T_{\boldsymbol{\beta}}.$

A Fock space functional at $q$ is the value of a Fock space field at $q,$

$X_{q}=X(q).$

Here Fock space means the Fock space of the Gaussian free field $\Phi$ and the
functionals that depend on the chart at the puncture $q.$ A Fock space
functional at $q$ correlates with Fock space strings $\mathcal{X}$ in $D^{*}.$
In fact we can consider the (tensor) products $\mathcal{X}X_{q}.$ Such
products are well-defined Fock space functionals. Examples of Fock space
functional at $q$ include $1_{q},\Phi(q),$ and even formal functionals
$\mathcal{O}^{(\sigma)}(q)$ (with our usual proviso of neutrality).

######  Definition.

We say $X_{q}\in\mathcal{F}_{\boldsymbol{\beta}}(q)$ if
$X_{q}=X_{1}*(\cdots*(X_{n}*1_{q})\cdots)$ for some fields $X_{1},\cdots
X_{n}\in\mathcal{F}_{\boldsymbol{\beta}}(D^{*}).$ The (extended) OPE family
$\mathcal{F}_{\boldsymbol{\beta}}=\mathcal{F}_{\boldsymbol{\beta}}(D,q)$ is
the collection of Fock space strings

$\\{\mathcal{Y}=\mathcal{X}X_{q}\,|\,\mathcal{X}\in\mathcal{F}_{\boldsymbol{\beta}}(D^{*}),X_{q}\in\mathcal{F}_{\boldsymbol{\beta}}(q)\\}.$

For example, if $X=Y=J_{\check{\boldsymbol{\beta}\,}}$
($\check{\boldsymbol{\beta}\,}=b\cdot q+b\cdot q^{*}$), then both
$Y*(X*1_{q})$ and $(Y*X)*1_{q}$ are in $\mathcal{F}_{\boldsymbol{\beta}}(q),$
but they are different, see the last example in the next subsection.

###  6.3. Examples of the OPE family

It is clear that $1_{q}\in\mathcal{F}(q),$ so every $X\in\mathcal{F}(D^{*})$
belongs to the extended family. We now present some less obvious examples in
the standard radial case, $\check{\boldsymbol{\beta}\,}=b\cdot q+b\cdot
q^{*}.$

######  Example.

As mentioned before,

$\mathcal{O}_{q}^{(\sigma)}=\mathcal{O}^{(\sigma)}*1_{q}\in\mathcal{F}_{\check{\boldsymbol{\beta}\,}}(q),$

and more generally
$\mathcal{O}^{(\sigma)}*\mathcal{O}_{q}^{(\tau)}=\mathcal{O}^{(\sigma+\tau)}_{q}.$
We define the formal functional $\mathcal{O}_{q}^{(\tau,\tau_{*})}$ by

$\mathcal{O}_{q}^{(\tau,\tau_{*})}:=\mathcal{O}_{q}^{(\tau)}\overline{\mathcal{O}_{q}^{(\overline{\tau_{*}})}}=(w_{q}^{\prime})^{\tau^{2}/2}(\overline{w_{q}^{\prime}})^{\tau_{*}^{2}/2}\mathrm{e}^{\odot
i\tau\Phi^{+}(q)-i\tau_{*}\Phi^{-}(q)}.$

Let us discuss the vertex algebra at the puncture. We define the
multiplication of OPE exponentials at the puncture by

$\mathcal{O}_{q}^{(\sigma,\sigma_{*})}\mathcal{O}_{q}^{(\tau,\tau_{*})}=\mathcal{O}^{(\sigma,\sigma_{*})}*\mathcal{O}_{q}^{(\tau,\tau_{*})}$

so that
$\mathcal{O}_{q}^{(\sigma,\sigma_{*})}\mathcal{O}_{q}^{(\tau,\tau_{*})}=\mathcal{O}_{q}^{(\sigma+\tau,\sigma_{*}+\tau_{*})}.$

######  Examples.

(a) We have $(\Phi_{\check{\boldsymbol{\beta}\,}})_{q}=\Phi(q).$ It is because
$(z-q)w^{\prime}(z)/w(z)=1+o(z-q).$

(b) We have

$(J_{\check{\boldsymbol{\beta}\,}})_{q}=J(q)+\frac{ib}{2}N_{w}(q).$

This follows from
$J_{\check{\boldsymbol{\beta}\,}}=J+j_{\check{\boldsymbol{\beta}\,}}$ with
$j_{\check{\boldsymbol{\beta}\,}}=ib({w^{\prime\prime}}/{w^{\prime}}-{w^{\prime}}/{w}),$
and

$\Big{(}\frac{w^{\prime}}{w}\Big{)}*1_{q}=\lim_{z\to
q}\Big{(}\frac{w^{\prime}(z)}{w(z)}-\frac{1}{z-q}\Big{)}=\lim_{z\to
q}\frac{w^{\prime\prime}(z)}{w^{\prime}(z)+w(z)/(z-q)}=\frac{1}{2}\frac{w^{\prime\prime}(q)}{w^{\prime}(q)}.$

(c) We have

$\displaystyle(T_{\check{\boldsymbol{\beta}\,}})_{q}$
$\displaystyle=A(q)+2ib\partial
J(q)-\frac{ib}{2}N_{w}(q)J(q)+\Big{(}\frac{c}{12}-\frac{1}{3}\,b^{2}\Big{)}S_{w}(q)-\frac{3}{8}\,b^{2}N_{w}(q)^{2}.$

######  Example.

For $X_{q}=X*1_{q},$ $Y*X_{q}\neq(Y*X)*1_{q}$ in general. For example, with
the choice of $X=Y=J_{\check{\boldsymbol{\beta}\,}},$ we have
$Y*(X*1_{q})\neq(Y*X)*1_{q}.$ It is because

$\Big{(}\frac{w^{\prime\prime}}{w^{\prime}}-\frac{w^{\prime}}{w}\Big{)}^{2}*1_{q}=-\frac{4}{3}\,S_{w}(q)-\frac{1}{4}\,N_{w}(q)^{2}\neq\bigg{(}\Big{(}\frac{w^{\prime\prime}}{w^{\prime}}-\frac{w^{\prime}}{w}\Big{)}*1_{q}\bigg{)}^{2}=\frac{1}{4}\,N_{w}(q)^{2}.$

###  6.4. Insertions

In this subsection we extend the insertion of the one-leg operator to the
strings containing puncture functionals in particular to the fields in the
extended OPE family. For simplicity, we consider the standard radial case with
$\check{\boldsymbol{\beta}\,}=b\cdot q+b\cdot q^{*}.$ Let $\Lambda_{\zeta}$
denote the one-leg operator,

$\Lambda_{\zeta}=\mathcal{O}_{\check{\boldsymbol{\beta}\,}}[a\cdot\zeta-\frac{a}{2}\cdot
q-\frac{a}{2}\cdot
q^{*}]=\mathcal{O}^{(a)}(\zeta)\mathcal{O}^{(-a/2)}(q)\overline{\mathcal{O}^{(-a/2)}(q)}.$

If $q$ is a node of $\mathcal{X},$ e.g.,

$\mathcal{X}=\mathcal{Y}X_{q},\qquad
X_{q}=X*1_{q}\in\mathcal{F}_{\boldsymbol{\beta}}(q),$

then the accurate definition of the product $\Lambda_{\zeta}\mathcal{X}$ is

$\Lambda_{\zeta}\mathcal{X}:=\mathcal{O}^{(a)}(\zeta)\mathcal{Y}(X*\mathcal{O}_{q}),\qquad\mathcal{O}_{q}:=\mathcal{O}^{(-a/2)}(q)\overline{\mathcal{O}^{(-a/2)}(q)}.$

######  Proposition 6.3.

Let $\mathcal{X}=\mathcal{Y}X_{q},$ $X_{q}=X*1_{q}$ and
$\widehat{X}_{q}=\widehat{X}*1_{q}.$ Then we have

$\widehat{\mathbf{E}}\,\mathcal{Y}X_{q}=\mathbf{E}\,\widehat{\mathcal{Y}}\widehat{X}_{q}.$

###### Proof.

By definition,

$\widehat{\mathbf{E}}\,\mathcal{Y}X_{q}=\frac{\mathbf{E}\,\Lambda_{p}\mathcal{Y}X_{q}}{\mathbf{E}\,\Lambda_{p}}=\frac{\mathbf{E}\,\mathcal{O}^{(a)}(p)\mathcal{Y}(X*\mathcal{O}_{q})}{\mathbf{E}\,\Lambda_{p}}.$

In follows from (5.23) that

$\frac{\mathbf{E}\,\mathcal{Y}X(z)\Lambda_{p}}{\mathbf{E}\,\Lambda_{p}}=\mathbf{E}\,\widehat{\mathcal{Y}}\widehat{X}(z)1_{q}.$

Comparing the OPE coefficients at the puncture, we have

$\frac{\mathbf{E}\,\mathcal{O}^{(a)}(p)\mathcal{Y}(X*\mathcal{O}_{q})}{\mathbf{E}\,\Lambda_{p}}=\mathbf{E}\,\widehat{\mathcal{Y}}\widehat{X}_{q}.$

∎

######  Examples.

Let $\check{\boldsymbol{\beta}\,}=b\cdot q+b\cdot q^{*}$ and
$\boldsymbol{\beta}=a\cdot p+(b-a/2)\cdot q+(b-a/2)\cdot q^{*}.$

(a) Recall that (see (5.21) and Subsection 6.3, Example (a))

$\widehat{\Phi}_{\check{\boldsymbol{\beta}\,}}=\Phi_{\check{\boldsymbol{\beta}\,}}+a\arg\frac{(1-w)^{2}}{w}=\Phi_{\boldsymbol{\beta}},$

and $(\Phi_{\check{\boldsymbol{\beta}\,}})_{q}=\Phi(q).$ Since $(\arg
w)*1_{q}=\arg w_{q}^{\prime},$ we have

$(\widehat{\Phi}_{\check{\boldsymbol{\beta}\,}})_{q}=\Phi(q)-a\arg
w_{q}^{\prime}.$

(b) It follows from (5.24) and Example (b) in Subsection 6.3 that

$(\widehat{J}_{\check{\boldsymbol{\beta}\,}})_{q}=J(q)+i\Big{(}\frac{a}{4}+\frac{b}{2}\Big{)}N_{w}(q)+iaw^{\prime}(q).$

(c) Note
$\widehat{\mathcal{O}}_{q}^{(\sigma)}:=\widehat{\mathcal{O}}^{(\sigma)}*1_{q}=(w_{q}^{\prime})^{-a\sigma/2}\mathcal{O}_{q}^{(\sigma)},$
cf.

$\widehat{\mathcal{O}}^{(\sigma)}=\Big{(}\frac{1-w}{\sqrt{w}}\Big{)}^{a\sigma}\mathcal{O}^{(\sigma)}.$

We have
$\widehat{\mathbf{E}}\,\mathcal{O}_{q}^{(\sigma)}=\mathbf{E}\,\widehat{\mathcal{O}}_{q}^{(\sigma)}$
because the differentials on both sides have the same dimension and are equal
to 1 in $(\mathbb{D},0,1).$

It is not difficult to see that Theorem 5.5 extends to Theorem 1.1 using the
argument in the proof of Proposition 6.3. We remark that Proposition 6.3 is a
special case ($\check{\boldsymbol{\beta}\,}=b\cdot q+b\cdot q^{*}$ and
$\boldsymbol{\beta}=a\cdot p+(b-a/2)\cdot q+(b-a/2)\cdot q^{*}$) of Theorem
1.1.

For two background charges $\boldsymbol{\beta},\check{\boldsymbol{\beta}\,}$
satisfying the neutrality condition $(\mathrm{NC}_{b}),$ let

$\widehat{\mathbf{E}}\,\mathcal{X}_{\check{\boldsymbol{\beta}\,}}:=\mathbf{E}\,\mathcal{X}_{\check{\boldsymbol{\beta}\,}}V^{\odot}[\boldsymbol{\beta}-\check{\boldsymbol{\beta}\,}]=\frac{\mathbf{E}\,\mathcal{X}_{\check{\boldsymbol{\beta}\,}}\mathcal{O}_{\check{\boldsymbol{\beta}\,}}[\boldsymbol{\beta}-\check{\boldsymbol{\beta}\,}]}{\mathbf{E}\,\mathcal{O}_{\check{\boldsymbol{\beta}\,}}[\boldsymbol{\beta}-\check{\boldsymbol{\beta}\,}]}.$

If the nodes of $\mathcal{X}_{\check{\boldsymbol{\beta}\,}}$ intersect with
the those of $\boldsymbol{\beta}-\check{\boldsymbol{\beta}\,}$, then the
product of $\mathcal{X}_{\check{\boldsymbol{\beta}\,}}$ and
$\mathcal{O}_{\check{\boldsymbol{\beta}\,}}[\boldsymbol{\beta}-\check{\boldsymbol{\beta}\,}]$
should be understood as the sense of OPE’s.

The insertion of $V^{\odot}[\boldsymbol{\beta}-\check{\boldsymbol{\beta}\,}]$
is an operator $\mathcal{X}\mapsto\widehat{\mathcal{X}}$ on Fock space
functionals/fields given by the rules (5.20) and the formula

$\widehat{\Phi}_{\check{\boldsymbol{\beta}\,}}=\Phi_{\boldsymbol{\beta}}.$

######  Example.

Suppose that a divisor $\boldsymbol{\tau}$ on $S$ satisfies the neutrality
condition $(\mathrm{NC}_{0}).$ Then we have

(6.2)
$\widehat{\mathcal{O}}_{\check{\boldsymbol{\beta}\,}}[\boldsymbol{\tau}]=\mathcal{O}_{\boldsymbol{\beta}}[\boldsymbol{\tau}].$

Since both sides contain the same Wick’s exponential
$V^{\odot}[\boldsymbol{\tau}]=\mathrm{e}^{\odot i\Phi[\boldsymbol{\tau}]},$
all we need is to check that both sides have the same correlation functions.
It follows from Theorem 1.1 and the algebra of OPE exponentials that

$\mathbf{E}\,\widehat{\mathcal{O}}_{\check{\boldsymbol{\beta}\,}}[\boldsymbol{\tau}]=\widehat{\mathbf{E}}\,\mathcal{O}_{\check{\boldsymbol{\beta}\,}}[\boldsymbol{\tau}]=\frac{\mathbf{E}\,\mathcal{O}_{\check{\boldsymbol{\beta}\,}}[\boldsymbol{\tau}]\mathcal{O}_{\check{\boldsymbol{\beta}\,}}[\boldsymbol{\beta}-\check{\boldsymbol{\beta}\,}]}{\mathbf{E}\,\mathcal{O}_{\check{\boldsymbol{\beta}\,}}[\boldsymbol{\beta}-\check{\boldsymbol{\beta}\,}]}=\frac{\mathbf{E}\,\mathcal{O}_{\check{\boldsymbol{\beta}\,}}[\boldsymbol{\tau}+\boldsymbol{\beta}-\check{\boldsymbol{\beta}\,}]}{\mathbf{E}\,\mathcal{O}_{\check{\boldsymbol{\beta}\,}}[\boldsymbol{\beta}-\check{\boldsymbol{\beta}\,}]}=\frac{C[\boldsymbol{\beta}+\boldsymbol{\tau}]}{C[\boldsymbol{\beta}]}=\mathbf{E}\,\mathcal{O}_{\boldsymbol{\beta}}[\boldsymbol{\tau}].$

##  7\. Ward identities and BPZ equations

We represent the Ward functionals as the Lie derivative operators within
correlations of fields in the extended OPE family. Combining these
representations (Ward identities) with the expression of the Virasoro fields
in terms of the Ward functionals, we derive the Belavin-Polyakov-Zamolodchikov
equations (BPZ equations) for correlations involving the Virasoro fields or
the Virasoro generators.

###  7.1. Ward’s identity

Let
$D^{*}:=D\setminus(\mathrm{supp}\,\boldsymbol{\beta}^{+}\cup\,\mathrm{supp}\,\boldsymbol{\beta}^{-})$
and $\boldsymbol{\beta}=\boldsymbol{\beta}^{+}+\boldsymbol{\beta}^{-}_{*}$ as
before.

######  Lemma 7.1.

For a non-random local vector field $v,$ all fields in
$\mathcal{F}_{\boldsymbol{\beta}}(D^{*})$ satisfy local Ward’s identity (5.4)
in $D_{\mathrm{hol}}(v)\cap D^{*}.$

###### Proof.

For simplicity, let us consider the case of _holomorphic_ fields $X$ and $Y$
only. We need to show that if $X$ and $Y$ satisfy the residue form (5.4) of
Ward’s identity at the nodes, then so does
$X*_{\alpha+n}Y\in\mathcal{F}_{\boldsymbol{\beta}}(D^{*}).$ (The case
$\alpha=0$ was covered in [18, Proposition 5.8].) In this case, we have
$\mathcal{L}_{v}^{+}X=(vT)*_{-1}X$ and $\mathcal{L}_{v}^{+}Y=(vT)*_{-1}Y.$ We
need to check that

$\mathcal{L}_{v}^{+}(X*_{\alpha+n}Y)=(vT)*_{-1}(X*_{\alpha+n}Y).$

By Leibniz’s rule, the left-hand side is
$(\mathcal{L}_{v}^{+}X)*_{\alpha+n}Y+X*_{\alpha+n}(\mathcal{L}_{v}^{+}Y).$
Since $X$ and $Y$ satisfy the residue form of Ward’s identity, it is equal to

$[(vT)*_{-1}X]*_{\alpha+n}Y+X*_{\alpha+n}[(vT)*_{-1}Y].$

Let us show
$[(vT)*_{-1}X]*_{\alpha+n}Y+X*_{\alpha+n}[(vT)*_{-1}Y]=(vT)*_{-1}(X*_{\alpha+n}Y).$

Let $C_{-},$ $C,$ and $C_{+}$ be three concentric circles centered at $z$ with
increasing radii, $\varepsilon_{-}<\varepsilon<\varepsilon_{+}(\ll 1).$ In the
correlations with the string $\mathcal{X}$ whose nodes are outside of the
discs, we have

$\displaystyle(vT)*_{-1}(X*_{\alpha+n}Y)(z)=\frac{1}{2\pi i}\,\oint_{\zeta\in
C_{+}}v(\zeta)T(\zeta)\,(X*_{\alpha+n}Y)(z)\,\mathrm{d}$ $\displaystyle\zeta$
$\displaystyle=\frac{1}{(2\pi i)^{2}}\,\oint_{\zeta\in C_{+}}\oint_{\eta\in
C}v(\zeta)T(\zeta)\,(\eta-z)^{-\alpha-n-1}X(\eta)\,Y(z)\,\mathrm{d}\eta\,\mathrm{d}$
$\displaystyle\zeta.$

For $\alpha\notin\mathbb{Z},$ the multivalued factor $(\eta-z)^{-\alpha}$ is
canceled out by the $\mathrm{OPE}_{\alpha}$ of $X(\eta)$ and $Y(z).$ In a
similar way, we compute $X*_{\alpha+n}[(vT)*_{-1}Y](z)$ as

$\qquad\frac{1}{(2\pi i)^{2}}\,\oint_{\zeta\in C_{-}}\oint_{\eta\in
C}v(\zeta)T(\zeta)\,(\eta-z)^{-\alpha-n-1}X(\eta)\,Y(z)\,\mathrm{d}\eta\,\mathrm{d}\zeta.$

Subtracting, we express
$(vT)*_{-1}(X*_{\alpha+n}Y)(z)-X*_{\alpha+n}[(vT)*_{-1}Y](z)$ as

$\displaystyle\frac{1}{(2\pi i)^{2}}\,\oint_{\eta\in
C}\oint_{\zeta\in[C_{+}-C_{-}]}v(\zeta)T(\zeta)\,(\eta-z)^{-\alpha-n-1}X(\eta)\,Y(z)\,\mathrm{d}\zeta\,\mathrm{d}\eta$
$\displaystyle=\frac{1}{(2\pi i)^{2}}\,\oint_{\eta\in
C}(\eta-z)^{-\alpha-n-1}\oint_{(\eta)}v(\zeta)T(\zeta)X(\eta)\,\mathrm{d}\zeta\,Y(z)\,\mathrm{d}\eta.$

The last integral simplifies to $[(vT)*_{-1}X]*_{\alpha+n}Y(z).$ ∎

The next lemma states that within correlations of fields in
$\mathcal{F}_{\boldsymbol{\beta}}(D^{*})$ the residue form of Ward’s identity
holds at the marked points $q_{k}$’s where the background charges are placed.

######  Lemma 7.2.

We have

$\frac{1}{2\pi
i}\oint_{(q_{k})}vA_{\boldsymbol{\beta}}=\mathcal{L}_{v}^{+}(q_{k})=v(q_{k})\partial_{q_{k}}$

within correlations of fields in $\mathcal{F}_{\boldsymbol{\beta}}(D^{*}).$

###### Proof.

From the relation

$A_{\boldsymbol{\beta}}=A+ib\partial J-j_{\boldsymbol{\beta}}J,\qquad
j_{\boldsymbol{\beta}}(\zeta)=\mathbf{E}\,J_{\boldsymbol{\beta}}(\zeta),$

and the fact that the meromorphic function $j_{\boldsymbol{\beta}}$ has a
simple pole at $q_{k}$ with
$\operatorname{Res}_{q_{k}}j_{\boldsymbol{\beta}}=-i\beta_{k},$ it follows
that

$\frac{1}{2\pi i}\oint_{(q_{k})}vA_{\boldsymbol{\beta}}=-\frac{1}{2\pi
i}\oint_{(q_{k})}vj_{\boldsymbol{\beta}}J=i\beta_{k}v(q_{k})J(q_{k}).$

We now claim that

(7.1)
$i\beta_{k}\mathbf{E}\,J(q_{k})\mathcal{X}_{\boldsymbol{\beta}}=\partial_{q_{k}}\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}}$

for any string $\mathcal{X}_{\boldsymbol{\beta}}$ of fields in
$\mathcal{F}_{\boldsymbol{\beta}}(D^{*}).$ Then lemma follows immediately.
Note that the second identity

$\mathcal{L}_{v}^{+}(q_{k})\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}}=v(q_{k})\partial_{q_{k}}\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}}$

is obvious since $\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}}$ is a scalar
with respect to $q_{k}$.

To prove the claim, let us consider a reference background charge
$\boldsymbol{\beta}_{0}$ such that

$\mathrm{supp}\,\boldsymbol{\beta}\cap\mathrm{supp}\,\boldsymbol{\beta}_{0}=\emptyset.$

Differentiating the relation (Theorem 5.5)
$\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}}=\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}_{0}}\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]},$ we have

(7.2)
$\partial_{q_{k}}\mathbf{E}\mathcal{X}_{\boldsymbol{\beta}}=\partial_{q_{k}}\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}_{0}}\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}=i\beta_{k}\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}_{0}}J(q_{k})\odot\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}.$

It follows from Wick’s calculus that

(7.3) $J(q_{k}^{\prime})\odot\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}=J(q_{k}^{\prime})\,\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}-i\mathbf{E}[J(q_{k}^{\prime})\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]]\,\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}.$

(Indeed, if one applies Wick’s calculus to
$J(q_{k}^{\prime})\,\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]},$ the term
$J(q_{k}^{\prime})\odot\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}$ comes from $0$ contraction
and the term
$i\mathbf{E}[J(q_{k}^{\prime})\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]]\,\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}$ comes from $1$
contraction.) Here we take $q_{k}^{\prime}\neq q_{k}$ so that the term
$J_{\boldsymbol{\beta}_{0}}(q_{k}^{\prime})\,\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}$ in the above makes sense
but send $q_{k}^{\prime}$ to $q_{k}$ at the end. By Wick’s calculus, we find
the non-random factor in the last term as

(7.4)
$i\mathbf{E}[J(q_{k}^{\prime})\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]]=\mathbf{E}[J(q_{k}^{\prime})\,\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}]=j_{\boldsymbol{\beta}}(q_{k}^{\prime})-j_{\boldsymbol{\beta}_{0}}(q_{k}^{\prime}).$

Alternatively, the last identity in the above follows from Theorem 5.5:

$\displaystyle
j_{\boldsymbol{\beta}}(q_{k}^{\prime})=\mathbf{E}\,J_{\boldsymbol{\beta}}(q_{k}^{\prime})$
$\displaystyle=\mathbf{E}[J_{\boldsymbol{\beta}_{0}}(q_{k}^{\prime})\,\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}]$
$\displaystyle=\mathbf{E}[(J(q_{k}^{\prime})+j_{\boldsymbol{\beta}_{0}}(q_{k}^{\prime}))\,\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}]=\mathbf{E}[J(q_{k}^{\prime})\,\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}]+j_{\boldsymbol{\beta}_{0}}(q_{k}^{\prime}).$

Combining (7.3) and (7.4), we have

$J(q_{k}^{\prime})\odot\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}=J(q_{k}^{\prime})\,\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}+\big{(}j_{\boldsymbol{\beta}_{0}}(q_{k}^{\prime})-j_{\boldsymbol{\beta}}(q_{k}^{\prime})\big{)}\,\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}.$

In application to $\mathcal{X}_{\boldsymbol{\beta}_{0}}$ within correlations,

$\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}_{0}}J(q_{k}^{\prime})\odot\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}=\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}_{0}}J_{\boldsymbol{\beta}_{0}}(q_{k}^{\prime})\,\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}-j_{\boldsymbol{\beta}}(q_{k}^{\prime})\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}_{0}}\,\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}.$

Applying Theorem 5.5 again to each term on the right-hand side, we obtain

$\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}_{0}}J(q_{k}^{\prime})\odot\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}=\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}}J_{\boldsymbol{\beta}}(q_{k}^{\prime})-j_{\boldsymbol{\beta}}(q_{k}^{\prime})\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}}=\mathbf{E}\,J(q_{k}^{\prime})\mathcal{X}_{\boldsymbol{\beta}}.$

Sending $q_{k}^{\prime}$ to $q_{k},$ we have

$\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}_{0}}J(q_{k})\odot\mathrm{e}^{\odot
i\Phi[\boldsymbol{\beta}-\boldsymbol{\beta}_{0}]}=\mathbf{E}\,J(q_{k})\mathcal{X}_{\boldsymbol{\beta}}.$

The claim (7.1) now follows from the above equation and (7.2). We now finish
the proof of the lemma. ∎

Combining the above two lemmas, we claim that all fields in
$\mathcal{F}_{\boldsymbol{\beta}}(D^{*})$ satisfy Ward’s identity (5.4) in
$D_{\mathrm{hol}}(v)\cap D$ within correlations. Applying the same arguments
as in the proof of Lemma 7.1 to this claim, we extend this claim to
$\mathcal{F}_{\boldsymbol{\beta}}(D).$

######  Theorem 7.3.

For a non-random local vector field $v,$ all fields in
$\mathcal{F}_{\boldsymbol{\beta}}(D)$ satisfy local Ward’s identity (5.4) in
$D_{\mathrm{hol}}(v)\cap D$ within correlations.

###  7.2. Ward equations on the Riemann sphere

In this subsection we express the Virasoro fields $T_{\boldsymbol{\beta}}$ in
terms of Lie derivative within correlations of fields in
$\mathcal{F}_{\boldsymbol{\beta}}(S).$

Given a meromorphic vector field with poles $\xi_{l},$ we define the Ward
functional $W_{\boldsymbol{\beta}}^{+}(v)$ by

$\mathbf{E}\,W_{\boldsymbol{\beta}}^{+}(v)\mathcal{X}:=-\frac{1}{2\pi
i}\sum_{l}\oint_{(\xi_{l})}v\,\mathbf{E}\,A_{\boldsymbol{\beta}}\mathcal{X}$

for any tensor product $\mathcal{X}$ of Fock space fields such that the set
$S_{\mathcal{X}}$ of all nodes of $\mathcal{X}$ does not intersect the set of
poles of $v.$

######  Proposition 7.4.

In correlations with any string of fields in the extended OPE family
$\mathcal{F}_{\boldsymbol{\beta}}(S),$ we have

$W_{\boldsymbol{\beta}}^{+}(v)=\mathcal{L}_{v}^{+}.$

###### Proof.

Since the sum of all residues of the meromorphic $1$-differential
$\zeta\mapsto
v(\zeta)\mathbf{E}\,A_{\boldsymbol{\beta}}(\zeta)\mathcal{X}_{\boldsymbol{\beta}}$
is zero, we have

$\mathbf{E}\,W_{\boldsymbol{\beta}}^{+}(v)\mathcal{X}_{\boldsymbol{\beta}}=\frac{1}{2\pi
i}\sum_{j}\oint_{(z_{j})}v\,\mathbf{E}\,A_{\boldsymbol{\beta}}\mathcal{X}_{\boldsymbol{\beta}}+\frac{1}{2\pi
i}\sum_{k}\oint_{(q_{k})}v\,\mathbf{E}\,A_{\boldsymbol{\beta}}\mathcal{X}_{\boldsymbol{\beta}}.$

It follows from Ward’s identity (Theorem 7.3) that

$\frac{1}{2\pi
i}\sum_{j}\oint_{(z_{j})}v\,\mathbf{E}\,A_{\boldsymbol{\beta}}\mathcal{X}_{\boldsymbol{\beta}}+\frac{1}{2\pi
i}\sum_{k}\oint_{(q_{k})}v\,\mathbf{E}\,A_{\boldsymbol{\beta}}\mathcal{X}_{\boldsymbol{\beta}}=\sum_{j}\mathbf{E}\,\mathcal{L}_{v}^{+}(z_{j})\mathcal{X}_{\boldsymbol{\beta}}+\sum_{k}\mathbf{E}\,\mathcal{L}_{v}^{+}(q_{k})\mathcal{X}_{\boldsymbol{\beta}}=\mathbf{E}\,\mathcal{L}_{v}^{+}\mathcal{X}_{\boldsymbol{\beta}},$

which completes the proof. ∎

Given $\xi\in\mathbb{C},$ let us consider the vector fields $k_{\xi},v_{\xi}$
given by

$k_{\xi}(z)=\frac{1}{\xi-z},\qquad v_{\xi}(z)=z\frac{\xi+z}{\xi-z}$

in the identity chart of $\mathbb{C}.$ Ward’s equations on the Riemann sphere
now follow from the previous theorem.

######  Corollary 7.5.

In the $\widehat{\mathbb{C}}$-uniformization, for any tensor product
$\mathcal{X}_{\boldsymbol{\beta}}$ of fields in the extended OPE family
$\mathcal{F}_{\boldsymbol{\beta}},$ we have

(7.5)
$\mathbf{E}\,T_{\boldsymbol{\beta}}(\xi)\mathcal{X}_{\boldsymbol{\beta}}=\mathbf{E}\,T_{\boldsymbol{\beta}}(\xi)\,\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}}+\mathbf{E}\,\mathcal{L}_{k_{\xi}}^{+}\mathcal{X}_{\boldsymbol{\beta}}$

and

(7.6)
$2\xi^{2}\mathbf{E}\,T_{\boldsymbol{\beta}}(\xi)\mathcal{X}_{\boldsymbol{\beta}}=2\xi^{2}\mathbf{E}\,T_{\boldsymbol{\beta}}(\xi)\,\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}}+\mathbf{E}\,\mathcal{L}_{v_{\xi}}^{+}\mathcal{X}_{\boldsymbol{\beta}}.$

###  7.3. Ward’s equations in a simply-connected domain

To derive Ward’s equations in $\mathbb{H}$ and $\mathbb{D}$, we modify Ward’s
functional to be defined in a simply-connected domain. Ward’s equations in
$\mathbb{H}$ (resp. in $\mathbb{D}$) represent the insertion of the Virasoro
fields within correlations of fields in $\mathcal{F}_{\boldsymbol{\beta}}$ in
terms of Lie derivative operators with respect to the chordal (resp. radial)
Loewner vector fields.

We first consider the special case that no background charge is placed on the
boundary $\partial D.$ The general case can be treated through a suitable
limit procedure. In this special case, $A_{\boldsymbol{\beta}}$ is continuous
on the boundary. (The continuity on the boundary should be understood in terms
of standard boundary charts.) As we mentioned before, we mostly concern
ourselves with a symmetric background charge $\boldsymbol{\beta}.$ In this
case, $A_{\boldsymbol{\beta}}$ is real on the boundary. (Again, we understand
the real-valuedness on the boundary in terms of standard boundary charts.)

Given a meromorphic vector field $v$ in $D$ (continuous up to the boundary)
with poles $\xi_{l}$’s ($\xi_{l}\in D$), we define the Ward functional
$W_{\boldsymbol{\beta}}^{+}(v)$ by

$W_{\boldsymbol{\beta}}^{+}(v):=\lim_{\varepsilon\to 0}\Big{(}\frac{1}{2\pi
i}\int_{\partial
D_{\varepsilon}}vA_{\boldsymbol{\beta}}-\frac{1}{\pi}\int_{D_{\varepsilon}}({\bar{\partial}}v)A_{\boldsymbol{\beta}}\Big{)},$

where $D_{\varepsilon}=D\setminus\bigcup B(\xi_{l},\varepsilon).$ Both
integrals are coordinate independent since $vA_{\boldsymbol{\beta}}$ is a
$[1,0]$-differential and $({\bar{\partial}}v)A_{\boldsymbol{\beta}}$ is a
$[1,1]$-differential. Their correlations with Fock space functionals
$\mathcal{X}$ are well-defined provided that any node of $\mathcal{X}$ is not
a pole of $v.$

Somewhat symbolically, the Ward functional $W_{\boldsymbol{\beta}}^{+}(v)$ can
be represented as

$W_{\boldsymbol{\beta}}^{+}(v)=\frac{1}{2\pi i}\int_{\partial
D}vA_{\boldsymbol{\beta}}-\sum_{l}\frac{1}{2\pi
i}\oint_{(\xi_{l})}vA_{\boldsymbol{\beta}},$

or

$W_{\boldsymbol{\beta}}^{+}(v)=\frac{1}{2\pi i}\int_{\partial
D}vA_{\boldsymbol{\beta}}-\frac{1}{\pi}\int_{D}({\bar{\partial}}v)A_{\boldsymbol{\beta}},$

where ${\bar{\partial}}v$ in the last integral should be understood in the
sense of distributions. If a background charge is placed on the boundary, then
the first integral should be taken in the sense of the Cauchy principal value.

From now on, we only consider the case that $\boldsymbol{\beta}$ is symmetric.
Then
$W_{\boldsymbol{\beta}}=(A_{\boldsymbol{\beta}},\overline{A_{\boldsymbol{\beta}}}).$
We now prove Theorem 1.2.

###### Proof of Theorem 1.2.

We may consider the case that
$\mathrm{supp}\,\boldsymbol{\beta}^{\pm}\subseteq D.$ It is enough to show
that

$W_{\boldsymbol{\beta}}^{+}(v)=\mathcal{L}_{v}^{+}.$

in correlations with any string of fields in the extended OPE family
$\mathcal{F}_{\boldsymbol{\beta}}(D).$

Let $D_{j}\,(D^{\prime}_{k},D^{\prime\prime}_{l})$ be a sufficiently small
open disk centered at $z_{j}\,(q_{k},\xi_{l}),$ respectively. Applying Stokes’
theorem to

$\mathrm{d}\big{(}v(z)\mathbf{E}\,A_{\boldsymbol{\beta}}(z)\mathcal{X}_{\boldsymbol{\beta}}\,\mathrm{d}z\big{)}=-{\bar{\partial}}\big{(}v(z)\mathbf{E}\,A_{\boldsymbol{\beta}}(z)\mathcal{X}_{\boldsymbol{\beta}}\big{)}\,\mathrm{d}z\wedge\mathrm{d}\bar{z}=2i{\bar{\partial}}\big{(}v(z)\mathbf{E}\,A_{\boldsymbol{\beta}}(z)\mathcal{X}_{\boldsymbol{\beta}}\big{)}\,\mathrm{d}x\wedge\mathrm{d}y$

over
$D\setminus(\bigcup_{j}D_{j}\cup\bigcup_{k}D^{\prime}_{k}\cup\bigcup_{l}D^{\prime\prime}_{l}),$
we have

$\mathbf{E}\,W_{\boldsymbol{\beta}}^{+}(v)\mathcal{X}_{\boldsymbol{\beta}}=\frac{1}{2\pi
i}\sum_{j}\oint_{(z_{j})}v\,\mathbf{E}\,A_{\boldsymbol{\beta}}\mathcal{X}_{\boldsymbol{\beta}}+\frac{1}{2\pi
i}\sum_{j}\oint_{(q_{k})}v\,\mathbf{E}\,A_{\boldsymbol{\beta}}\mathcal{X}_{\boldsymbol{\beta}}.$

Theorem now follows from Ward’s identity (Theorem 7.3) as in the proof of
Proposition 7.4. ∎

We recall the representation of a stress tensor $A$ in terms of Ward’s
functionals $W_{\boldsymbol{\beta}}^{+}(v)$ with the meromorphic vector field
$v=k_{\zeta}(\zeta\in\mathbb{C}).$

######  Proposition 7.6 (Proposition 5.11 in [18]).

Let $A$ be a holomorphic quadratic differential in $\mathbb{H},$ and
$W=(A,\bar{A}).$ Suppose $A$ is continuous and real on the boundary (including
$\infty$). Then

(7.7)
$(A\,\|\,\mathrm{id})(\xi)=W^{+}(k_{\xi})+\overline{W^{+}(k_{\bar{\xi}})}.$

Applying the above proposition to $A_{\boldsymbol{\beta}}$ in the case that
$\mathrm{supp}\,\boldsymbol{\beta}^{\pm}\subseteq D,$ we obtain (7.7) for
$A=A_{\boldsymbol{\beta}}$ and $W^{+}=W^{+}_{\boldsymbol{\beta}}.$ Combining
the above proposition with Theorem 1.2 and using a limit procedure in the
general case that $\mathrm{supp}\,\boldsymbol{\beta}^{\pm}\subseteq\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu,$ we derive Ward’s
equation in $\mathbb{H}.$

######  Corollary 7.7 (Ward’s equations in $\mathbb{H}$).

In the $\mathbb{H}$-uniformization, for any tensor product
$\mathcal{X}_{\boldsymbol{\beta}}$ of fields in the extended OPE family
$\mathcal{F}_{\boldsymbol{\beta}},$ we have

(7.8)
$\mathbf{E}\,T_{\boldsymbol{\beta}}(\xi)\mathcal{X}_{\boldsymbol{\beta}}=\mathbf{E}\,T_{\boldsymbol{\beta}}(\xi)\,\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}}+\mathbf{E}\,(\mathcal{L}_{k_{\xi}}^{+}+\mathcal{L}_{k_{\bar{\xi}}}^{-})\mathcal{X}_{\boldsymbol{\beta}}.$

######  Example.

Let $\mathcal{X}_{\boldsymbol{\beta}}=X_{1}(z_{1})\cdots X_{n}(z_{n})$ be the
tensor product of $[\lambda_{j}^{+},\lambda_{j}^{-}]$-differentials $X_{j}$ in
$\mathcal{F}_{\boldsymbol{\beta}}.$ If $\xi\in\mathbb{R},$ then we have

$\displaystyle\mathbf{E}\,T_{\boldsymbol{\beta}}(\xi)\mathcal{X}_{\boldsymbol{\beta}}$
$\displaystyle=\mathbf{E}\,T_{\boldsymbol{\beta}}(\xi)\,\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}}+\sum_{k}\Big{(}\frac{\partial_{q_{k}}}{\xi-
q_{k}}+\frac{{\bar{\partial}}_{q_{k}}}{\xi-\bar{q}_{k}}\Big{)}\,\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}}$
$\displaystyle+\sum_{j}\Big{(}\frac{\partial_{j}}{\xi-
z_{j}}+\frac{\lambda_{j}^{+}}{(\xi-
z_{j})^{2}}+\frac{{\bar{\partial}}_{j}}{\xi-\bar{z}_{j}}+\frac{\lambda_{j}^{-}}{(\xi-\bar{z}_{j})^{2}}\Big{)}\,\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}}.$

Given a meromorphic vector field $v$ with a local flow $z_{t}$ on
$\widehat{\mathbb{C}},$ we define its reflected vector field $v^{*}$ with
respect to the unit circle $\partial\mathbb{D}$ by the vector field of the
reflected flow $z_{t}^{*}$ of $z_{t}.$ Indeed, if $z_{t}$ is the flow of $v$,
then $z_{t}^{*}=1/\bar{z}_{t}$ is the reflected flow and its vector field
$v^{*}$ is given by the equation
$\dot{z}^{*}=-\overline{\dot{z}/z^{2}}=-\overline{v(z)}(z^{*})^{2}.$ Thus we
have $v^{*}(z)=-z^{2}\overline{v({1}/{\bar{z}})}.$ Using a similar method as
in the previous corollary, we obtain the following form of Ward’s equations in
the radial case.

######  Corollary 7.8 (Ward’s equations in $\mathbb{D}$).

In the $\mathbb{D}$-uniformization, for any tensor product
$\mathcal{X}_{\boldsymbol{\beta}}$ of fields in the extended OPE family
$\mathcal{F}_{\boldsymbol{\beta}},$

(7.9)
$2\zeta^{2}\mathbf{E}\,T_{\boldsymbol{\beta}}(\zeta)\mathcal{X}_{\boldsymbol{\beta}}=2\zeta^{2}\mathbf{E}\,T_{\boldsymbol{\beta}}(\zeta)\,\mathbf{E}\,\mathcal{X}_{\boldsymbol{\beta}}+\mathbf{E}\,(\mathcal{L}_{v_{\zeta}}^{+}+\mathcal{L}_{v_{\zeta^{*}}}^{-})\mathcal{X}_{\boldsymbol{\beta}}.$

Let us emphasize that the above formulas apply to the strings without nodes at
$\zeta.$ The case where $\zeta$ is a node is discussed in the following
subsection. We use OPE calculus and local operators. Theorem 1.3 now follows
from Corollaries 7.7 – 7.8 and the next lemma.

######  Lemma 7.9.

We have

(7.10)
$\mathbf{E}\,T_{\boldsymbol{\beta}}(z)=\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{+}_{k_{z}}\mathcal{P}_{\boldsymbol{\beta}}+\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{-}_{k_{\bar{z}}}\mathcal{P}_{\boldsymbol{\beta}}$

in the identity chart of $\mathbb{H}$ and

(7.11)
$2z^{2}\mathbf{E}\,T_{\boldsymbol{\beta}}(z)=\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{+}_{v_{z}}\mathcal{P}_{\boldsymbol{\beta}}+\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{-}_{v_{z^{*}}}\mathcal{P}_{\boldsymbol{\beta}}$

in the identity chart of $\mathbb{D}.$

###### Proof.

Let
$\lambda_{k}^{\pm}=\lambda(\beta_{k}^{\pm}),(\lambda(\sigma)=\sigma^{2}/2-\sigma
b).$ Since $\mathcal{P}_{\boldsymbol{\beta}}=C_{(b)}[\boldsymbol{\beta}]$ is a
$[\lambda_{j}^{+},\lambda_{j}^{-}]$-differential at $q_{j},$

$\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{+}_{k_{z}}\mathcal{P}_{\boldsymbol{\beta}}=\sum_{j}\frac{\lambda_{j}^{+}}{(z-q_{j})^{2}}+\frac{\partial_{j}\log\mathcal{P}_{\boldsymbol{\beta}}}{z-q_{j}},\qquad\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{-}_{k_{\bar{z}}}\mathcal{P}_{\boldsymbol{\beta}}=\sum_{j}\frac{\lambda_{j}^{-}}{(z-\bar{q}_{j})^{2}}+\frac{{\bar{\partial}}_{j}\log\mathcal{P}_{\boldsymbol{\beta}}}{z-\bar{q}_{j}}$

in the identity chart of $\mathbb{H}.$ Differentiating
$\mathcal{P}_{\boldsymbol{\beta}}=C_{(b)}[\boldsymbol{\beta}],$ we have

$\partial_{j}\log\mathcal{P}_{\boldsymbol{\beta}}=\sum_{k\neq
j}\frac{\beta_{j}^{+}\beta_{k}^{+}}{q_{j}-q_{k}}+\sum_{k}\frac{\beta_{j}^{+}\beta_{k}^{-}}{q_{j}-\bar{q}_{k}},\qquad{\bar{\partial}}_{j}\log\mathcal{P}_{\boldsymbol{\beta}}=\sum_{k\neq
j}\frac{\beta_{j}^{-}\beta_{k}^{-}}{\bar{q}_{j}-\bar{q}_{k}}+\sum_{k}\frac{\beta_{j}^{-}\beta_{k}^{+}}{\bar{q}_{j}-q_{k}}.$

On the other hand, it follows from (5.16) and
$\mathbf{E}\,T_{\boldsymbol{\beta}}=-\frac{1}{2}j_{\boldsymbol{\beta}}^{2}+ibj^{\prime}_{\boldsymbol{\beta}}$
that

$\displaystyle\mathbf{E}\,T_{\boldsymbol{\beta}}(z)$
$\displaystyle=\sum_{j}\bigg{(}\frac{\lambda_{j}^{+}}{(z-q_{j})^{2}}+\frac{\lambda_{j}^{-}}{(z-\bar{q}_{j})^{2}}\bigg{)}$
$\displaystyle+\sum_{j<k}\bigg{(}\frac{\beta_{j}^{+}\beta_{k}^{+}}{(z-q_{j})(z-q_{k})}+\frac{\beta_{j}^{-}\beta_{j}^{-}}{(z-\bar{q}_{j})(z-\bar{q}_{k})}\bigg{)}+\sum_{j,k}\frac{\beta_{j}^{+}\beta_{k}^{-}}{(z-q_{j})(z-\bar{q}_{k})}.$

As a rational function in $z,$

$\displaystyle\sum_{j}\frac{1}{z-q_{j}}$ $\displaystyle\bigg{(}\sum_{k\neq
j}\frac{\beta_{j}^{+}\beta_{k}^{+}}{q_{j}-q_{k}}+\sum_{k}\frac{\beta_{j}^{+}\beta_{k}^{-}}{q_{j}-\bar{q}_{k}}\bigg{)}+\sum_{j}\frac{1}{z-\bar{q}_{j}}\bigg{(}\sum_{k\neq
j}\frac{\beta_{j}^{-}\beta_{k}^{-}}{\bar{q}_{j}-\bar{q}_{k}}+\sum_{k}\frac{\beta_{j}^{-}\beta_{k}^{+}}{\bar{q}_{j}-q_{k}}\bigg{)}$
$\displaystyle=\sum_{j<k}\bigg{(}\frac{\beta_{j}^{+}\beta_{k}^{+}}{(z-q_{j})(z-q_{k})}+\frac{\beta_{j}^{-}\beta_{j}^{-}}{(z-\bar{q}_{j})(z-\bar{q}_{k})}\bigg{)}+\sum_{j,k}\frac{\beta_{j}^{+}\beta_{k}^{-}}{(z-q_{j})(z-\bar{q}_{k})},$

which shows (7.10). The formula (7.11) can be shown similarly. ∎

###  7.4. BPZ equations

Recall the definition of Virasoro generators $L_{n}\equiv
L_{n}^{\boldsymbol{\beta}}$: they are operators $X\mapsto L_{n}X$ acting on
fields, $L_{n}^{\boldsymbol{\beta}}X=T_{\boldsymbol{\beta}}*_{(-n-2)}X,$ i.e.,

(7.12)
$T_{\boldsymbol{\beta}}(\zeta)X(z)=\cdots+\frac{(L_{0}X)(z)}{(\zeta-z)^{2}}+\frac{(L_{-1}X)(z)}{\zeta-z}+(L_{-2}X)(z)+\cdots\qquad(\zeta\to
z).$

We write $L(z)=L_{-2}(z)$ in $(\mathbb{H},\infty)$ and

(7.13) $L(z)=2z^{2}L_{-2}(z)+3zL_{-1}(z)+L_{0}(z)$

in $(\mathbb{D},0).$

Recall the Belavin-Polyakov-Zamolodchikov equations (BPZ equations [7]) in the
chordal case with $\boldsymbol{\beta}=2b\cdot q,$ see [18, Proposition 5.13].

######  Proposition 7.10.

Let $Y,X_{1},\cdots,X_{n}\in\mathcal{F}_{\boldsymbol{\beta}}$ and let $X$ be
the tensor product of $X_{j}$’s. Then

$\mathbf{E}\,(T_{\boldsymbol{\beta}}*Y)(z)\,X=\mathbf{E}\,Y(z)\mathcal{L}^{+}_{k_{z}}X+\mathbf{E}\,\mathcal{L}^{-}_{k_{\bar{z}}}[Y(z)X],\qquad\boldsymbol{\beta}=2b\cdot
q,$

where all fields are evaluated in the identity chart of $(\mathbb{H},\infty).$

Somewhat symbolically, we have

$L(z)=\check{\mathcal{L}}^{+}_{k_{z}}+\mathcal{L}^{-}_{k_{\bar{z}}},$

where $\check{\mathcal{L}}^{+}_{k_{z}}$ means
$\mathcal{L}^{+}_{k_{z}}(\mathbb{H}\setminus\\{z\\}),$ e.g.,
$\check{\mathcal{L}}^{+}_{k_{z}}Y(z)X:=Y(z)\mathcal{L}^{+}_{k_{z}}X.$ We can
neglect the check mark if we come to terms that we never differentiate at the
poles of $v.$

The previous proposition can be extended to the general background charge, see
the next two theorems. We only present the proof of BPZ equations in the
radial case. The chordal case can be proved in a similar way.

######  Theorem 7.11 (BPZ equations in $\mathbb{H}$).

Let $Y,X_{1},\cdots,X_{n}\in\mathcal{F}_{\boldsymbol{\beta}}$ and let $X$ be
the tensor product of $X_{j}$’s. Then

$\mathbf{E}\,(T_{\boldsymbol{\beta}}*Y)(z)\,X=\mathbf{E}\,Y(z)\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{+}_{k_{z}}\mathcal{P}_{\boldsymbol{\beta}}X+\mathbf{E}\,\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{-}_{k_{\bar{z}}}[\mathcal{P}_{\boldsymbol{\beta}}Y(z)X],$

where all fields are evaluated in the identity chart of $\mathbb{H}.$

######  Theorem 7.12 (BPZ equations in $\mathbb{D}$).

Let $Y,X_{1},\cdots,X_{n}\in\mathcal{F}_{\boldsymbol{\beta}}$ and let $X$ be
the tensor product of $X_{j}$’s. Then

$\mathbf{E}\,L(z)Y(z)\,X=\mathbf{E}\,Y(z)\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{+}_{v_{z}}\mathcal{P}_{\boldsymbol{\beta}}X+\mathbf{E}\,\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{-}_{v_{z^{*}}}[\mathcal{P}_{\boldsymbol{\beta}}Y(z)X],$

where all fields are evaluated in the identity chart of $\mathbb{D}$ and
$L(z)$ is given by (7.13).

###### Proof.

Recall Ward’s OPE, $\operatorname{Sing}_{\zeta\to
z}T_{\boldsymbol{\beta}}(\zeta)Y(z)=\mathcal{L}^{+}_{k_{\zeta}}Y(z)$ (see [18,
Proposition 5.3]). It follows immediately that

(7.14) $2\zeta^{2}\operatorname{Sing}_{\zeta\to
z}T_{\boldsymbol{\beta}}(\zeta)Y(z)=\mathcal{L}^{+}_{2\zeta^{2}k_{\zeta}}Y(z).$

Denote
$\delta_{\zeta}(\eta)=2\zeta^{2}k_{\zeta}(\eta)-v_{\zeta}(\eta)=2\zeta+\eta.$
By (7.12), we observe that

$\mathcal{L}_{\delta_{\zeta}}^{+}Y(z)=\frac{1}{2\pi
i}\oint_{(z)}\delta_{\zeta}(\eta)T_{\boldsymbol{\beta}}(\eta)Y(z)\,\mathrm{d}\eta=L_{0}Y(z)+(2\zeta+z)L_{-1}Y(z).$

It follows from (7.14) that

(7.15) $2\zeta^{2}\operatorname{Sing}_{\zeta\to
z}T_{\boldsymbol{\beta}}(\zeta)Y(z)-\mathcal{L}^{+}_{v_{\zeta}}Y(z)=L_{0}Y(z)+(2\zeta+z)L_{-1}Y(z).$

On the other hand, by Theorem 1.3,

$2\zeta^{2}\,\mathbf{E}\,T_{\boldsymbol{\beta}}(\zeta)Y(z)X=\mathbf{E}\,\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{+}_{v_{\zeta}}\mathcal{P}_{\boldsymbol{\beta}}Y(z)X+\mathbf{E}\,\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{-}_{v_{\zeta^{*}}}\mathcal{P}_{\boldsymbol{\beta}}Y(z)X,$

where the first term on the right-hand side is

$\mathbf{E}\,\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{+}_{v_{\zeta}}\mathcal{P}_{\boldsymbol{\beta}}Y(z)X=\mathbf{E}\,\mathcal{L}^{+}_{v_{\zeta}}Y(z)X+\mathbf{E}\,Y(z)\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{+}_{v_{\zeta}}\mathcal{P}_{\boldsymbol{\beta}}X.$

Combining the last three equations and subtracting all singular terms, we get

$\displaystyle\mathbf{E}\,\big{(}2\zeta^{2}\operatorname{Reg}_{\zeta\to
z}(T_{\boldsymbol{\beta}}(\zeta)Y(z))X$
$\displaystyle+L_{0}Y(z)X+(2\zeta+z)L_{-1}Y(z)X\big{)}$
$\displaystyle=\mathbf{E}\,Y(z)\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{+}_{v_{\zeta}}\mathcal{P}_{\boldsymbol{\beta}}X+\mathbf{E}\,\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{-}_{v_{\zeta^{*}}}\mathcal{P}_{\boldsymbol{\beta}}Y(z)X,$

where $\operatorname{Reg}_{\zeta\to z}T_{\boldsymbol{\beta}}(\zeta)Y(z)$ is
the regular part of the operator product expansion. Theorem now follows by
taking the limit $\zeta\to z$ in both sides. For instance, the left-hand side
converges to $\mathbf{E}\,L(z)Y(z)\,X.$ ∎

###  7.5. Null vectors

After we briefly review the level two degeneracy equations for current primary
fields, we apply these equations and the BPZ equations to the (tensor/OPE)
products of the one-leg operators and the fields in the extended OPE family.

We denote by $\\{J_{n}\\}$ and $\\{L_{n}\\}$ the current generators and the
Virasoro generators, the modes of $J_{\boldsymbol{\beta}}$ and
$T_{\boldsymbol{\beta}}$ in $\mathcal{F}_{\boldsymbol{\beta}}$ theory,
respectively:

$J_{n}(z):=\frac{1}{2\pi
i}\oint_{(z)}(\zeta-z)^{n}J_{\boldsymbol{\beta}}(\zeta)\,\mathrm{d}\zeta,\qquad
L_{n}(z):=\frac{1}{2\pi
i}\oint_{(z)}(\zeta-z)^{n+1}T_{\boldsymbol{\beta}}(\zeta)\,\mathrm{d}\zeta.$

As operators acting on fields in $\mathcal{F}_{\boldsymbol{\beta}},$ we have
the following equations:

$[J_{m},J_{n}]=n\delta_{m+n,0},\quad[L_{m},J_{n}]=-nJ_{m+n}+ibm(m+1)\delta_{m+n,0},$

and

$[L_{m},L_{n}]=(m-n)L_{m+n}+\frac{c}{12}m(m^{2}-1)\delta_{m+n,0},$

where $c$ is the central charge, $c=1-12b^{2}.$

By definition, $X\in\mathcal{F}_{\boldsymbol{\beta}}$ is (Virasoro) primary if
$X$ is a $[\lambda,\lambda_{*}]$-differential; equivalently

(7.16) $L_{\geq 1}X=0,\qquad L_{0}X=\lambda X,\qquad L_{-1}X=\partial X,$

and similar equations hold for $\bar{X}.$ Here $L_{\geq k}X=0$ means that
$L_{n}X=0$ for all $n\geq k.$ A (Virasoro) primary field $X$ is called
_current primary_ with charges $\sigma$ and $\sigma_{*}$ if $J_{\geq
1}X=J_{\geq 1}\bar{X}=0,$ and

$J_{0}X=-i\sigma X,\quad J_{0}\bar{X}=i\bar{\sigma}_{*}\bar{X}.$

Let us recall the characterization of level two degenerate current primary
fields.

######  Proposition 7.13 (Proposition 11.2 in [18]).

Let $\mathcal{O}$ be a current primary field in
$\mathcal{F}_{\boldsymbol{\beta}},$ and let $\sigma,\sigma_{*}$ be charges of
$\mathcal{O}.$ If $2\sigma(b+\sigma)=1,\,\eta=-1/(2\sigma^{2}),$ then

$(L_{-2}+\eta L_{-1}^{2})\mathcal{O}=0.$

If the parameters $a$ and $b$ are related to the SLE parameter $\kappa$ as

$a=\pm\sqrt{2/\kappa},\qquad b=a(\kappa/4-1),$

then $2a(a+b)=1$ and the formal fields
$\mathcal{O}^{(a)},\mathcal{O}^{(-a-b)}$ satisfy

(7.17)
$L_{-2}\,\mathcal{O}^{(a)}=\frac{\kappa}{4}\,\partial_{z}^{2}\mathcal{O}^{(a)},\qquad
L_{-2}\,\mathcal{O}^{(-a-b)}=\frac{4}{\kappa}\,\partial_{z}^{2}\mathcal{O}^{(-a-b)}$

(in all charts). Since $\mathcal{O}^{(a)},\mathcal{O}^{(-a-b)}$ are formal
fields, we need to assume the neutrality condition $(\mathrm{NC}_{0})$ on
$\mathcal{O}^{(a)}X,\mathcal{O}^{(-a-b)}X$ for some product $X$ of formal
fields. For example, the one-leg operator $\Lambda_{p}$ (both in the chordal
case and in the radial one) satisfy the level two degeneracy equation. As
mentioned before, we will use this level two degeneracy equation for
$\Lambda_{p}$ to establish the connection between the chordal/radial SLE
theory and conformal field theory.

•  _Standard chordal case with $\boldsymbol{\beta}=2b\cdot q$._ Recall the BPZ
equations in $(\mathbb{H},\infty):$

$L(z)=\check{\mathcal{L}}^{+}_{k_{z}}+\mathcal{L}^{-}_{k_{\bar{z}}},\qquad
k_{\zeta}(\eta)=\frac{1}{\zeta-\eta},$

which we apply to the field $Y(z)X$ in $\mathcal{F}_{\boldsymbol{\beta}},$
where $X$ has no node at $z.$ In all applications $Y$ is holomorphic, so we
have

(7.18)
$\big{(}L(z)Y(z)\big{)}X=Y(z)(\check{\mathcal{L}}^{+}_{k_{z}}+\mathcal{L}^{-}_{k_{\bar{z}}})X=Y(z)(\mathcal{L}^{+}_{k_{z}}+\mathcal{L}^{-}_{k_{\bar{z}}})X.$

If $X$ is also holomorphic (in particular, in the “boundary” situation), then
(7.18) becomes

$\big{(}L(z)Y(z)\big{)}X=Y(z)\mathcal{L}^{+}_{k_{z}}X.$

Let us discuss both sides in the equation (7.18).

We study the cases that $Y$ is the null vector

$Y(z)=\mathcal{O}^{(a)}(z),\qquad Y(z)=\mathcal{O}^{(-a-b)}(z).$

Of course, $Y$ is a formal field, so we need to assume the neutrality
condition $(\mathrm{NC}_{0})$ in $Y(z)X.$ Recall that the conformal dimensions
$h,h^{\prime}$ of $\mathcal{O}^{(a)},\mathcal{O}^{(-a-b)},$ respectively at
$z$ are

(7.19) $h:=\frac{3a^{2}}{2}-\frac{1}{2}=\frac{6-\kappa}{2\kappa},\qquad
h^{\prime}:=\frac{3}{8a^{2}}-\frac{1}{2}=\frac{3\kappa-8}{16}.$

Recall the level two degeneracy equation (7.17) for the formal field
$\mathcal{O}^{(a)}$. In the boundary case $z=x\in\mathbb{R}$ we have

(7.20)
$L_{-2}\,\mathcal{O}^{(a)}(x)=\frac{\kappa}{4}\,\partial_{x}^{2}\mathcal{O}^{(a)}(x)$

in the $(\mathbb{H},\infty)$-uniformization because
$\partial_{x}=\partial+{\bar{\partial}}$ and $\mathcal{O}^{(a)}$ is
holomorphic.

In the case that $X=\mathcal{O}_{\boldsymbol{\beta}}[\boldsymbol{\tau}]$ with
$\boldsymbol{\tau}=\sum\tau_{j}\cdot z_{j},$ we have

$Y(z)\mathcal{L}^{+}_{k_{z}}X=\big{(}\sum
k_{z}(z_{j})\partial_{z_{j}}+\lambda_{j}\partial_{z_{j}}k_{z}(z_{j})\big{)}Y(z)X,$

where $\lambda_{j}=\lambda_{b}(\tau_{j})=\frac{1}{2}\tau_{j}^{2}-b\tau_{j}.$

•  _Standard radial case with $\boldsymbol{\beta}=b\cdot q+b\cdot q^{*}$._ We
apply the BPZ equation to the tensor product $Y(z)X$ of fields in
$\mathcal{F}_{\boldsymbol{\beta}}(D,q)$ with holomorphic $Y:$

(7.21)
$\big{(}L(z)Y(z)\big{)}X=Y(z)\mathcal{P}_{q}^{-1}(\mathcal{L}^{+}_{v_{z}}+\mathcal{L}^{-}_{v_{z^{*}}})\mathcal{P}_{q}X,\qquad
v_{\zeta}(\eta)=\eta\frac{\zeta+\eta}{\zeta-\eta}.$

Let us discuss both sides in the equation (7.21).

We consider the $(\mathbb{D},0)$-uniformization. For $Y=\mathcal{O}^{(a)},$ we
have

(7.22)
$L(z)\mathcal{O}^{(a)}=\Big{(}\frac{\kappa}{2}\,z^{2}\partial_{z}^{2}+3z\partial_{z}+h\Big{)}\mathcal{O}^{(a)},$

where $h$ is given by (7.19). In particular, for
$z=\mathrm{e}^{i\theta}\in\partial\mathbb{D},$

$L(\mathrm{e}^{i\theta})\mathcal{O}^{(a)}=\Big{(}\\!-\kappa\big{(}\frac{1}{2}\partial_{\theta}^{2}+ih\partial_{\theta}\big{)}+h\Big{)}\mathcal{O}^{(a)}.$

In the case of $O^{(-a-b)}$ the corresponding differential operator is

$L(\mathrm{e}^{i\theta})=-\kappa^{\prime}\big{(}\frac{1}{2}\partial_{\theta}^{2}+ih^{\prime}\partial_{\theta}\big{)}+h^{\prime},$

where $\kappa^{\prime}=16/\kappa$ and $h^{\prime}$ is given by (7.19).

By Lemma 7.9, the right hand-side becomes

$Y(z)(\mathcal{L}^{+}_{v_{z}}+\mathcal{L}^{-}_{v_{z^{*}}})X+2z^{2}Y(z)X\,\mathbf{E}\,T_{\boldsymbol{\beta}}(z).$

##  8\. Connection to the theory of forward chordal/radial SLE with forces and
spins

In Subsections 8.1 – 8.2 we define the one-leg operators for chordal/radial
SLEs with forces and spins as the OPE exponentials with specific background
charge $\boldsymbol{\beta}$ at the marked points. The insertion of a one-leg
operator changes the boundary values of Fock space fields. As the PDEs for
correlation functions of the fields in the extended OPE family
$\mathcal{F}_{\boldsymbol{\beta}}(D),$ the BPZ-Cardy equations for such
insertion operations are used to prove the connection (Theorem 1.5) between
the theory $\mathcal{F}_{\boldsymbol{\beta}}$ and the chordal/radial
$\mathrm{SLE}[\boldsymbol{\beta}]$ theory. A dipolar theory studied in [19] is
the special of this theory since it is well known ([33, 37]) that dipolar
$\mathrm{SLE}(\kappa)$ with two force points $q_{\pm}$ is equivalent to
chordal $\mathrm{SLE}(\kappa,\boldsymbol{\rho})$ with
$\boldsymbol{\rho}=\frac{1}{2}(\kappa-6)\cdot q_{+}+\frac{1}{2}(\kappa-6)\cdot
q_{-}.$ In Subsection 8.6 we present field theoretic approach to chordal
$\mathrm{SLE}(\kappa,\boldsymbol{\rho})$ restriction martingale-observable
introduced in [13, Lemma 4].

###  8.1. One-leg operators for chordal SLEs with forces and spins

Let $(D,p,q)$ be a simply-connected domain with two distinct marked boundary
points $p,q\in\partial D,$ and $\boldsymbol{\beta}$ be a symmetric background
charge on $S=D^{\mathrm{double}}$ with

$\boldsymbol{\beta}=a\cdot p+\boldsymbol{\eta},\quad
p\notin\mathrm{supp}\,\boldsymbol{\eta}$

satisfying the neutrality condition $(\mathrm{NC}_{b}):$

$\int\boldsymbol{\beta}=2b.$

Here the parameters $a$ and $b$ are related to the SLE parameter $\kappa$ as

$a=\pm\sqrt{2/\kappa},\qquad b=a(\kappa/4-1).$

Let $\boldsymbol{\tau}=a\cdot p-a\cdot q.$ For any divisor
$\boldsymbol{\sigma},$ we denote
$\check{\boldsymbol{\sigma}\,}=\boldsymbol{\sigma}-\boldsymbol{\tau}$ and
$\hat{\boldsymbol{\sigma}\,}=\boldsymbol{\sigma}+\boldsymbol{\tau}.$

•  _Insertions._ The insertion of the one-leg operator

$\Lambda_{p}\equiv\Lambda(p):=\mathcal{O}_{\check{\boldsymbol{\beta}\,}}[\boldsymbol{\tau}]$

produces an operator $\mathcal{X}\mapsto\widehat{\mathcal{X}}$ acting on Fock
space functionals/fields by the rules (5.20) and the formula

$\widehat{\Phi}_{\check{\boldsymbol{\beta}\,}}(z)=\Phi_{\check{\boldsymbol{\beta}\,}}(z)+2a\arg
w(z)=\Phi_{\boldsymbol{\beta}}(z),$

where $w:(D,p,q)\to(\mathbb{H},0,\infty)$ is a conformal map.

We denote

$\widehat{\mathbf{E}}\,\mathcal{X}:=\frac{\mathbf{E}\,\Lambda_{p}\mathcal{X}}{\mathbf{E}\,\Lambda_{p}}=\frac{\mathbf{E}\,\Lambda_{p}^{\mathrm{eff}}\mathcal{X}}{\mathbf{E}\,\Lambda_{p}^{\mathrm{eff}}}=\mathbf{E}\,\mathcal{X}\mathrm{e}^{\odot
ia\Phi^{+}(p,q)},$

where $\Lambda_{p}^{\mathrm{eff}}$ is the effective one-leg operator

$\Lambda_{p}^{\mathrm{eff}}=\mathcal{P}_{\check{\boldsymbol{\beta}\,}}\Lambda_{p}=C_{(b)}[\check{\boldsymbol{\beta}\,}]\mathcal{O}_{\check{\boldsymbol{\beta}\,}}[\boldsymbol{\tau}]=C_{(b)}[\boldsymbol{\beta}]V^{\odot}[\boldsymbol{\tau}]=V_{\check{\boldsymbol{\beta}\,}}[\boldsymbol{\beta}].$

Theorem 1.1 says that

$\widehat{\mathbf{E}}\,\mathcal{X}=\mathbf{E}\,\widehat{\mathcal{X}}$

for any string $\mathcal{X}$ of fields in
$\mathcal{F}_{\check{\boldsymbol{\beta}\,}}.$ We define the partition function
associated with a symmetric background charge $\boldsymbol{\beta}$ by

$Z_{\boldsymbol{\beta}}:=\big{|}C_{(b)}[\boldsymbol{\beta}]\big{|}.$

In the $\mathbb{H}$-uniformization, $C_{(b)}[\boldsymbol{\beta}]$ is non-
negative (up to a phase), see Example (b) in Subsection 2.5 and thus we have
$Z_{\boldsymbol{\beta}}=C_{(b)}[\boldsymbol{\beta}].$

•  _BPZ equations._ Let $\xi\in\partial D$ and
$\boldsymbol{\beta}_{\xi}=\boldsymbol{\beta}-a\cdot p+a\cdot\xi.$ Let $X$ be
the tensor product $X=X_{1}(z_{1})\cdots X_{n}(z_{n})$ of fields $X_{j}$ in
$\mathcal{F}_{\check{\boldsymbol{\beta}\,}}$ $(z_{j}\in D).$ It follows from
Theorem 7.11 that

(8.1)
$\frac{1}{2a^{2}}\partial_{\xi}^{2}\mathbf{E}\Lambda_{\xi}^{\mathrm{eff}}X=\mathbf{E}\,\check{\mathcal{L}}_{k_{\xi}}\Lambda_{\xi}^{\mathrm{eff}}X,\qquad
k_{\xi}(z):=\frac{1}{\xi-z},$

in the $\mathbb{H}$-uniformization. Here
$\partial_{\xi}=\partial+\bar{\partial}$ is the operator of differentiation
with respect to the real variable $\xi$ and $\check{\mathcal{L}}_{k_{\xi}}$ is
taken over the finite notes of $X$ and
$\mathrm{supp}\,\boldsymbol{\beta}_{\xi}\setminus\\{\xi\\}.$ In particular,
the partition function $Z_{\xi}\equiv Z_{\boldsymbol{\beta}_{\xi}}$ satisfies
the null vector equation

(8.2)
$\frac{1}{2a^{2}}\partial_{\xi}^{2}Z_{\xi}=\check{\mathcal{L}}_{k_{\xi}}Z_{\xi},$

in the $\mathbb{H}$-uniformization since
$Z_{\xi}=\mathbf{E}\Lambda_{\xi}^{\mathrm{eff}}$ up to a phase in the
$\mathbb{H}$-uniformization. Here $\check{\mathcal{L}}_{k_{\xi}}$ is taken
over $\mathrm{supp}\,\boldsymbol{\beta}_{\xi}\setminus\\{\xi\\}.$

•  _BPZ-Cardy equations._ For $\xi\in\mathbb{R}$ and a tensor product
$X=X_{1}(z_{1})\cdots X_{n}(z_{n})$ of fields $X_{j}$ in
$\mathcal{F}_{\check{\boldsymbol{\beta}\,}}$ $(z_{j}\in\mathbb{H})$, we denote

$R_{\xi}(\boldsymbol{z},\boldsymbol{q})\equiv\widehat{\mathbf{E}}_{\xi}X:=\frac{\mathbf{E}\,\Lambda_{\xi}X}{\mathbf{E}\,\Lambda_{\xi}}=\frac{\mathbf{E}\,\Lambda_{\xi}^{\mathrm{eff}}X}{\mathbf{E}\,\Lambda_{\xi}^{\mathrm{eff}}}=\mathbf{E}\,X\mathrm{e}^{\odot
ia\Phi^{+}(\xi,q)}.$

The following is the first half of Theorem 1.4.

######  Proposition 8.1 (BPZ-Cardy equations in $\mathbb{H}$).

In the identity chart of $\mathbb{H},$ we have

(8.3)
$\frac{1}{2a^{2}}\big{(}\partial_{\xi}^{2}\widehat{\mathbf{E}}_{\xi}\,X+2(\partial_{\xi}\log
Z_{\xi})\partial_{\xi}\widehat{\mathbf{E}}_{\xi}\,X\big{)}=\check{\mathcal{L}}_{k_{\xi}}\widehat{\mathbf{E}}_{\xi}\,X,\qquad
k_{\xi}(z):=\frac{1}{\xi-z},$

where $\partial_{\xi}=\partial+{\bar{\partial}}$ and
$\check{\mathcal{L}}_{k_{\xi}}$ is taken over the finite notes of $X$ and
$\mathrm{supp}\,\boldsymbol{\beta}_{\xi}\setminus\\{\xi\\}.$

###### Proof.

By differentiation,

$\frac{\partial_{\xi}^{2}\mathbf{E}\,\Lambda_{\xi}^{\mathrm{eff}}X}{\mathbf{E}\,\Lambda_{\xi}^{\mathrm{eff}}}=\partial_{\xi}^{2}\widehat{\mathbf{E}}_{\xi}\,X+2\frac{\partial_{\xi}Z_{\xi}}{Z_{\xi}}\partial_{\xi}\widehat{\mathbf{E}}_{\xi}\,X+\frac{\partial_{\xi}^{2}Z_{\xi}}{Z_{\xi}}\widehat{\mathbf{E}}_{\xi}\,X.$

On the other hand, it follows from the BPZ equations (8.1) that the left-hand
side of the above becomes
$2a^{2}{\mathbf{E}\,\check{\mathcal{L}}_{k_{\xi}}\Lambda_{\xi}^{\mathrm{eff}}X}/{\mathbf{E}\,\Lambda_{\xi}^{\mathrm{eff}}}.$
By Leibniz’s rule for Lie derivatives, we have

$\frac{\mathbf{E}\,\check{\mathcal{L}}_{k_{\xi}}\Lambda_{\xi}^{\mathrm{eff}}X}{\mathbf{E}\,\Lambda_{\xi}^{\mathrm{eff}}}=\check{\mathcal{L}}_{k_{\xi}}\widehat{\mathbf{E}}_{\xi}\,X+\frac{\check{\mathcal{L}}_{k_{\xi}}Z_{\xi}}{Z_{\xi}}\,\widehat{\mathbf{E}}_{\xi}\,X.$

Combining all of the above, we find

$\frac{1}{2a^{2}}\big{(}\partial_{\xi}^{2}\widehat{\mathbf{E}}_{\xi}\,X+2\frac{\partial_{\xi}Z_{\xi}}{Z_{\xi}}\partial_{\xi}\widehat{\mathbf{E}}_{\xi}\,X+\frac{\partial_{\xi}^{2}Z_{\xi}}{Z_{\xi}}\widehat{\mathbf{E}}_{\xi}\,X\big{)}=\check{\mathcal{L}}_{k_{\xi}}\widehat{\mathbf{E}}_{\xi}\,X+\frac{\check{\mathcal{L}}_{k_{\xi}}Z_{\xi}}{Z_{\xi}}\,\widehat{\mathbf{E}}_{\xi}\,X.$

We remark that the coefficient of $\widehat{\mathbf{E}}_{\xi}\,X$ vanishes
since the partition function $Z_{\xi}$ satisfies the null vector equation
(8.2). ∎

###  8.2. One-leg operators for radial SLEs with forces and spins

Let $(D,p,q)$ be a simply-connected domain $D$ with a marked boundary point
$p\in\partial D,$ and a marked interior point $q\in D.$ As in the previous
subsection, we consider a symmetric background charge $\boldsymbol{\beta}$ on
$S=D^{\mathrm{double}}$ with

$\boldsymbol{\beta}=a\cdot p+\boldsymbol{\eta},\quad
p\notin\mathrm{supp}\,\boldsymbol{\eta}$

satisfying the neutrality condition $(\mathrm{NC}_{b}).$

Let $\eta\in\mathbb{R}.$ We introduce the one-leg operator $\Lambda_{p}^{s}$
with spin $s=i\eta{a^{2}}/2$ as

$\Lambda_{p}^{s}\equiv\mathcal{O}_{\check{\boldsymbol{\beta}\,}}[\boldsymbol{\tau}],\qquad\boldsymbol{\tau}=a\cdot
p-\frac{a+i\delta}{2}\cdot q-\frac{a-i\delta}{2}\cdot q^{*},\qquad\delta=\eta
a,$

where $\check{\boldsymbol{\beta}\,}=\boldsymbol{\beta}-\boldsymbol{\tau}$ is a
symmetric background charge.

•  _Insertions._ As in the chordal theory, the insertion of the one-leg
operator $\Lambda_{p}^{s}$ produces an operator
$\mathcal{X}\mapsto\widehat{\mathcal{X}}$ acting on Fock space
functionals/fields by the rules (5.20) and the formula

$\widehat{\Phi}_{\check{\boldsymbol{\beta}\,}}(z)=\Phi_{\boldsymbol{\beta}}(z).$

We denote

$\widehat{\mathbf{E}}[\mathcal{X}]:=\frac{\mathbf{E}[\Lambda_{p}^{s}\mathcal{X}]}{\mathbf{E}[\Lambda_{p}^{s}]}=\frac{\mathbf{E}[(\Lambda_{p}^{s})^{\mathrm{eff}}\mathcal{X}]}{\mathbf{E}[(\Lambda_{p}^{s})^{\mathrm{eff}}]}=\mathbf{E}\,\mathcal{X}\mathrm{e}^{\odot
i\Phi[\boldsymbol{\tau}]},$

where $(\Lambda_{p}^{s})^{\mathrm{eff}}$ is the effective one-leg operator
with spin $s$:
$(\Lambda_{p}^{s})^{\mathrm{eff}}=\mathcal{P}_{\check{\boldsymbol{\beta}\,}}\Lambda_{p}^{s}.$
Then we have
$\widehat{\mathbf{E}}[\mathcal{X}]=\mathbf{E}[\widehat{\mathcal{X}}].$

######  Examples.

We consider the case

$\boldsymbol{\beta}^{s}=a\cdot p+\Big{(}b-\frac{a+i\delta}{2}\Big{)}\cdot
q+\Big{(}b-\frac{a-i\delta}{2}\Big{)}\cdot q^{*},\qquad\delta=\eta a.$

Let $\boldsymbol{\beta}=\boldsymbol{\beta}^{0}.$

(a) The bosonic $\Phi_{\boldsymbol{\beta}^{s}}$ is a pre-pre-Schwarzian form
of order $(ib,-ib),$

$\Phi_{\boldsymbol{\beta}^{s}}=\Phi_{\boldsymbol{\beta}}-\delta\log|w|,$

where $w:(D,p,q)\to(\mathbb{D},1,0)$ is a conformal map;

(b) The current $J_{\boldsymbol{\beta}^{s}}$ is a pre-Schwarzian form of order
$ib,$

$J_{\boldsymbol{\beta}^{s}}=J_{\boldsymbol{\beta}}-\frac{\delta}{2}\frac{w^{\prime}}{w};$

(c) The Virasoro field $T_{\boldsymbol{\beta}^{s}}$ is a Schwarzian form of
order $\frac{1}{12}c,$

$\displaystyle T_{\boldsymbol{\beta}^{s}}$
$\displaystyle=-\dfrac{1}{2}J_{\boldsymbol{\beta}^{s}}*J_{\boldsymbol{\beta}^{s}}+ib\partial
J_{\boldsymbol{\beta}^{s}}$
$\displaystyle=A-j_{\boldsymbol{\beta}^{s}}J+ib\partial
J+\frac{c}{12}S_{w}+h_{1,2}\frac{w^{\prime
2}}{w(1-w)^{2}}+h_{0,1/2}^{s}\frac{w^{\prime 2}}{w^{2}}+s\frac{w^{\prime
2}}{w(1-w)},$

where $j_{\boldsymbol{\beta}^{s}}=\mathbf{E}\,J_{\boldsymbol{\beta}^{s}},$
$h_{1,2}=\frac{1}{2}a^{2}-ab$ and
$h_{0,1/2}^{s}=\frac{1}{8}(a+i\delta)^{2}-\frac{1}{2}b^{2}.$

Let $X$ be a tensor product of fields in the extended family
$\mathcal{F}_{\check{\boldsymbol{\beta}\,}}(D).$ We assume that $\zeta$ is not
a node of $X.$ Denote

$R_{\zeta}\equiv\widehat{\mathbf{E}}_{\zeta}X=\frac{\mathbf{E}\,(\Lambda^{s}_{\zeta})^{\mathrm{eff}}X}{\mathbf{E}\,(\Lambda^{s}_{\zeta})^{\mathrm{eff}}},\quad
C_{\zeta}=\mathbf{E}\,(\Lambda^{s}_{\zeta})^{\mathrm{eff}}=C_{(b)}[\boldsymbol{\beta}_{\zeta}],\quad
Z_{\zeta}=|C_{\zeta}|,$

where $\boldsymbol{\beta}_{\zeta}=\boldsymbol{\beta}-a\cdot p+a\cdot\zeta.$
The following is the second half of Theorem 1.4.

######  Proposition 8.2 (BPZ-Cardy equations in $\mathbb{D}$).

In the $(\mathbb{D},0)$-uniformization we have

$-\frac{2}{a^{2}}\Big{(}\frac{1}{2}\partial_{\theta}^{2}+\big{(}\partial_{\theta}\log
Z_{\zeta}\big{)}\partial_{\theta}\Big{)}\widehat{\mathbf{E}}_{\zeta}X=\check{\mathcal{L}}_{v_{\zeta}}\widehat{\mathbf{E}}_{\zeta}X,\qquad(\zeta=\mathrm{e}^{i\theta},\theta\in\mathbb{R}),$

where the Lie derivative $\check{\mathcal{L}}_{v_{\zeta}}$ is taken over the
finite notes of $X$ and
$\mathrm{supp}\,\boldsymbol{\beta}_{\zeta}\setminus\\{\zeta\\}.$

###### Proof.

Let $\zeta=\mathrm{e}^{i\theta}\in\partial D.$ Applying the BPZ equation in
$(\mathbb{D},0)$ to $(\Lambda^{s}_{\zeta})^{\mathrm{eff}}X,$ we have

$\mathbf{E}\,L(\zeta)(\Lambda^{s}_{\zeta})^{\mathrm{eff}}X=\check{\mathcal{L}}_{v_{\zeta}}\mathbf{E}\,(\Lambda^{s}_{\zeta})^{\mathrm{eff}}X,$

where $L$ is the differential operator in Subsection 7.13:

$L(z)=\frac{\kappa}{2}z^{2}\partial_{z}^{2}+3z\partial_{z}+h,\textrm{ or
}L(\mathrm{e}^{i\theta})=-\kappa\big{(}\frac{1}{2}\partial_{\theta}^{2}+ih\partial_{\theta}\big{)}+h.$

In particular, $C_{\zeta}=\mathbf{E}\,(\Lambda^{s}_{\zeta})^{\mathrm{eff}}$
satisfies the null vector equation

(8.4) $L(\zeta)C_{\zeta}=\check{\mathcal{L}}_{v_{\zeta}}C_{\zeta}.$

By differentiation,

$\frac{\mathbf{E}\,L(\zeta)(\Lambda^{s}_{\zeta})^{\mathrm{eff}}X}{\mathbf{E}\,(\Lambda^{s}_{\zeta})^{\mathrm{eff}}}=(L(\zeta)-h)\widehat{\mathbf{E}}_{\zeta}X+\kappa\,\zeta^{2}\frac{\partial_{\zeta}C_{\zeta}}{C_{\zeta}}\partial_{\zeta}\widehat{\mathbf{E}}_{\zeta}X+\frac{L(\zeta)C_{\zeta}}{C_{\zeta}}\widehat{\mathbf{E}}_{\zeta}X.$

It follows from Leibniz rule for Lie derivative that

$\frac{\check{\mathcal{L}}_{v_{\zeta}}\mathbf{E}\,(\Lambda^{s}_{\zeta})^{\mathrm{eff}}X}{\mathbf{E}\,(\Lambda^{s}_{\zeta})^{\mathrm{eff}}}=\check{\mathcal{L}}_{v_{\zeta}}\widehat{\mathbf{E}}_{\zeta}X+\frac{\check{\mathcal{L}}_{v_{\zeta}}C_{\zeta}}{C_{\zeta}}\,\widehat{\mathbf{E}}_{\zeta}X.$

Combining all of the above, we have

(8.5)
$-\kappa\Big{(}\frac{1}{2}\partial_{\theta}^{2}+i\big{(}h+\zeta\frac{\partial_{\zeta}C_{\zeta}}{C_{\zeta}}\big{)}\partial_{\theta}\Big{)}\widehat{\mathbf{E}}_{\zeta}X=\check{\mathcal{L}}_{v_{\zeta}}\widehat{\mathbf{E}}_{\zeta}X$

in the $(\mathbb{D},0)$-uniformization. Note that the coefficient of
$\widehat{\mathbf{E}}_{\zeta}X$ vanishes since $C_{\zeta}$ satisfies the null
vector equation (8.4).

Let

$\boldsymbol{\beta}_{\zeta}^{0}=a\cdot\zeta+\Big{(}b-\frac{a}{2}\Big{)}\cdot
q+\Big{(}b-\frac{a}{2}\Big{)}\cdot q^{*},\qquad
C_{\zeta}^{0}=C_{(b)}[\boldsymbol{\beta}_{\zeta}^{0}].$

Then we have

$\zeta\frac{\partial_{\zeta}C_{\zeta}^{0}}{C_{\zeta}^{0}}=-h,\quad
h=h_{1,2}:=\frac{1}{2}a^{2}-ab.$

and rewrite (8.5) as

$-\kappa\Big{(}\frac{1}{2}\partial_{\theta}^{2}+\big{(}\partial_{\theta}\log\frac{C_{\zeta}}{C^{0}_{\zeta}}\big{)}\partial_{\theta}\Big{)}\widehat{\mathbf{E}}_{\zeta}X=\check{\mathcal{L}}_{v_{\zeta}}\widehat{\mathbf{E}}_{\zeta}X.$

It is easy to see that

$\partial_{\theta}\log
Z_{\zeta}=\partial_{\theta}\log\frac{C_{\zeta}}{C^{0}_{\zeta}}$

in the $\mathbb{D}$-uniformization. ∎

###  8.3. Martingale-observables for chordal/radial SLE with forces and spins

We now prove Theorem 1.5.

###### Proof of Theorem 1.5.

We first consider the chordal case. Let $g_{t}$ be the chordal
$\mathrm{SLE}[\boldsymbol{\beta}]$ map driven by the real process $\xi_{t}:$

$\mathrm{d}\xi_{t}=\sqrt{\kappa}\,\mathrm{d}B_{t}+\lambda(t)\,\mathrm{d}t,\quad\lambda(t)=(\lambda\,\|\,g_{t}^{-1}),\quad\lambda=\kappa\,\partial_{\xi}\log
Z_{\xi},$

Then we have

$M_{t}=m(\xi_{t},t),\quad m(\xi,t)=(R_{\xi}\,\|\,g_{t}^{-1}).$

It follows from Itô’s formula that

$\displaystyle\mathrm{d}M_{t}$
$\displaystyle=\sqrt{\kappa}\,\partial_{\xi}\big{|}_{\xi=\xi_{t}}m(\xi,t)\,\mathrm{d}B_{t}+\kappa\,\partial_{\xi}\big{|}_{\xi=\xi_{t}}m(\xi,t)\,\partial_{\xi}\big{|}_{\xi=\xi_{t}}(\log
Z_{\xi}\,\|\,g_{t}^{-1})\,\mathrm{d}t$
$\displaystyle+\frac{\kappa}{2}\,\partial_{\xi}^{2}\big{|}_{\xi=\xi_{t}}m(\xi,t)\,\mathrm{d}t+L_{t}\,\mathrm{d}t,$

where

$L_{t}:=\frac{\mathrm{d}}{\mathrm{d}s}\Big{|}_{s=0}\big{(}R_{\xi_{t}}\,\|\,g_{t+s}^{-1}\big{)}.$

At each time $t,$ $f_{s,t}:=g_{t+s}\circ g_{t}^{-1}$ satisfy the differential
equations

$\frac{\mathrm{d}}{\mathrm{d}s}f_{s,t}(\zeta)=\frac{2}{f_{s,t}(\zeta)-\xi_{t+s}}.$

As $s\to 0,$

$f_{s,t}=\mathrm{id}-2sk_{\xi_{t}}+o(s),\qquad k_{\xi}(z)=\frac{1}{\xi-z}.$

In terms of the time-dependent flows $f_{s,t},$ $L_{t}$ is rewritten as

$L_{t}=\frac{\mathrm{d}}{\mathrm{d}s}\Big{|}_{s=0}\big{(}R_{\xi_{t}}\,\|\,g_{t}^{-1}\circ
f_{s,t}^{-1}\big{)}=-2\big{(}\check{\mathcal{L}}_{k_{\xi_{t}}}R_{\xi_{t}}\,\|\,g_{t}^{-1}\big{)}.$

The last equality follows from the fact any fields in
$\mathcal{F}_{\boldsymbol{\beta}}$ depend smoothly on local charts. Thus the
drift term of $\mathrm{d}M_{t}$ simplifies to

$\Big{(}\frac{\kappa}{2}\,\partial_{\xi}^{2}\big{|}_{\xi=\xi_{t}}m(\xi,t)+\kappa\,\partial_{\xi}\big{|}_{\xi=\xi_{t}}m(\xi,t)\frac{\partial_{\xi}|_{\xi=\xi_{t}}(Z_{\xi}\,\|\,g_{t}^{-1})}{(Z_{\xi}\,\|\,g_{t}^{-1})}-2\big{(}\check{\mathcal{L}}_{k_{\xi_{t}}}R_{\xi_{t}}\,\|\,g_{t}^{-1}\big{)}\Big{)}\,\mathrm{d}t.$

It vanishes by the BPZ-Cardy equations (Proposition 8.1).

Next, we consider the radial case. For $\mathcal{X}=X_{1}(z_{1})\cdots
X_{n}(z_{n}),X_{j}\in\mathcal{F}_{\check{\boldsymbol{\beta}\,}}(D),$ denote

$R_{\zeta}(z_{1},\cdots,z_{n})\equiv\widehat{\mathbf{E}}_{\zeta}[X_{1}(z_{1})\cdots
X_{n}(z_{n})].$

Then the process $M_{t}(z_{1},\cdots,z_{n})$ is represented by

$M_{t}=m(\zeta_{t},t),\qquad
m(\zeta,t)=\big{(}R_{\zeta}\,\|\,g_{t}^{-1}\big{)},$

where $g_{t}:(D_{t},\gamma_{t},q)\to(\mathbb{D},\zeta_{t},0)$ is the radial
$\mathrm{SLE}[\boldsymbol{\beta}]$ map driven by the real process
$\theta_{t}:$

$\mathrm{d}\theta_{t}=\sqrt{\kappa}\,\mathrm{d}B_{t}+\lambda(t)\,\mathrm{d}t,\quad\lambda(t)=(\lambda\,\|\,g_{t}^{-1}),\quad\lambda=\kappa\,\partial_{\theta}\log
Z_{\boldsymbol{\beta}_{\zeta}},\quad\zeta_{t}=\mathrm{e}^{i\theta_{t}}.$

Using a similar argument in the chordal case, we find the drift term of
$\mathrm{d}M_{t}$ as

$\Big{(}\frac{\kappa}{2}\partial_{\theta}^{2}+\kappa\frac{\partial_{\theta}(Z_{\zeta}\,\|\,g_{t}^{-1})}{(Z_{\zeta}\,\|\,g_{t}^{-1})}\big{)}\partial_{\theta}\Big{)}\Big{|}_{\zeta=\zeta_{t}}~{}m(\zeta,t)\,\mathrm{d}t\\\
+\
\big{(}\check{\mathcal{L}}_{v_{\zeta_{t}}}R_{\zeta_{t}}\,\|\,g_{t}^{-1}\big{)}\mathrm{d}t.$

By the BPZ-Cardy equations (Proposition 8.2) in the radial case, $M_{t}$ is
driftless. ∎

######  Example.

Non-chiral vertex fields

$\mathcal{V}_{\boldsymbol{\beta}}^{(\sigma)}:=\mathrm{e}^{*i\sigma\Phi_{\boldsymbol{\beta}}}=\mathcal{O}_{\boldsymbol{\beta}}[\sigma\cdot
z,-\sigma\cdot z]$

have real conformal dimensions $[\lambda^{+},\lambda^{-}]$ at $z:$

$\lambda^{+}=\frac{\sigma^{2}}{2}-\sigma
b,\qquad\lambda^{-}=\frac{\sigma^{2}}{2}+\sigma b.$

If the conformal spin $s:=\lambda^{+}-\lambda^{-}$ is 1, then
$\mathcal{V}_{\boldsymbol{\beta}}^{(\sigma)}$ transforms as a (formal) vector
field and its flow lines can be identified with
$\mathrm{SLE}[\boldsymbol{\beta}],$ see [26] for a chordal case and [27] for a
radial case with spin at $q.$

###  8.4. Examples of radial SLE martingale-observables

In this subsection we present examples of radial SLE martingale-observables
including Schramm-Sheffield’s observables, Friedrich-Werner’s formula, and the
restriction formula in the standard radial case with the following background
charge $\boldsymbol{\beta}$:

(8.6) $\boldsymbol{\beta}=a\cdot p+\Big{(}b-\frac{a}{2}\Big{)}\cdot
q+\Big{(}b-\frac{a}{2}\Big{)}\cdot q^{*},$

######  Example (Schramm-Sheffield’s observables).

In the standard chordal case with

$\boldsymbol{\beta}=a\cdot p+(2b-a)\cdot q,$

the 1-point functions of the bosonic fields

$\varphi_{\boldsymbol{\beta}}(z)=\mathbf{E}[\Phi_{\boldsymbol{\beta}}(z)]=2a\arg
w(z)-2b\arg w^{\prime}(z),\quad w:(D,p,q)\to(\mathbb{H},0,\infty)$

were introduced as SLE martingale-observables by Schramm and Sheffield, see
[32]. Similarly, the 1-point functions
$\varphi_{\boldsymbol{\beta}}=\mathbf{E}\,\Phi_{\boldsymbol{\beta}}$ (with
standard radial background charge $\boldsymbol{\beta}$ in (8.6)) of the
bosonic fields are martingale-observables of radial SLEs:

$\varphi_{\boldsymbol{\beta}}(z)=\mathbf{E}[\Phi_{\boldsymbol{\beta}}(z)]=2a\arg(1-w(z))-a\arg
w(z)-2b\arg\frac{w^{\prime}(z)}{w(z)},$

where $w:(D,p,q)\to(\mathbb{D},1,0)$ is a conformal map. By Itô’s calculus, we
have

$\displaystyle\varphi_{t}(z)$
$\displaystyle=\mathbf{E}[\Phi_{D_{t},\gamma_{t},q}(z)]=2a\arg(1-w_{t}(z))-a\arg
w_{t}(z)-2b\arg\frac{w_{t}^{\prime}(z)}{w_{t}(z)}$
$\displaystyle=2a\arg(1-w(z))-a\arg
w(z)-2b\arg\frac{w^{\prime}(z)}{w(z)}+\sqrt{2}\int_{0}^{t}\mathrm{Re}\,\frac{1+w_{s}(z)}{1-w_{s}(z)}\,\mathrm{d}B_{s}.$

One can use the $2$-point martingale-observables

$\mathbf{E}[\Phi_{\boldsymbol{\beta}}(z_{1})\Phi_{\boldsymbol{\beta}}(z_{2})]=2G(z_{1},z_{2})+\varphi_{\boldsymbol{\beta}}(z_{1})\varphi_{\boldsymbol{\beta}}(z_{2})$

or Hadamard’s variation formula

(8.7)
$\mathrm{d}G_{D_{t}}(z_{1},z_{2})=-\,\mathrm{Re}\,\frac{1+w_{t}(z_{1})}{1-w_{t}(z_{1})}\,\mathrm{Re}\,\frac{1+w_{t}(z_{2})}{1-w_{t}(z_{2})}\,\mathrm{d}t=-\frac{1}{2}\,\mathrm{d}\langle\varphi_{\boldsymbol{\beta}}(z_{1}),\varphi_{\boldsymbol{\beta}}(z_{2})\rangle_{t}$

to construct a coupling of radial SLE and the Gaussian free field such that

$\mathbf{E}[\,\Phi_{D,p,q}\,|\,\gamma[0,t]\,]=\Phi_{D_{t},\gamma_{t},q},$

see [14].

Let us recall the _restriction property_ of radial $\mathrm{SLE}(8/3)$ (see
[21, Section 6.5]):

  * •

the law of $\mathrm{SLE}(8/3)$ in $\mathbb{D}$ conditioned to avoid a fixed
hull $K$ is identical to the law of $\mathrm{SLE}(8/3)$ in
$\mathbb{D}\setminus K;$

  * •

equivalently, there exist $\lambda$ and $\mu$ such that for all $K,$

$\mathbf{P}(\mathrm{SLE}(8/3)\textrm{ avoids
}K)=|\Psi_{K}^{\prime}(1)|^{\lambda}(\Psi_{K}^{\prime}(0))^{\mu},$

where $\Psi_{K}$ is the conformal map $(\mathbb{D}\setminus
K,0)\to(\mathbb{D},0)$ satisfying $\Psi_{K}^{\prime}(0)>0.$ (The restriction
exponents $\lambda$ and $\mu$ of radial $\mathrm{SLE}(8/3)$ are equal to $5/8$
and $5/48,$ respectively.)

Let $\kappa\leq 4.$ On the event $\gamma[0,\infty)\cap K=\emptyset,$ a
conformal map $h_{t}:\Omega_{t}=g_{t}(D_{t}\setminus K)\to\mathbb{D}$ is
defined by

$h_{t}=\widetilde{g}_{t}\circ\Psi_{K}\circ g_{t}^{-1},$

where $\widetilde{g}_{t}$ is a Loewner map from
$\widetilde{D}_{t}=D\setminus\widetilde{\gamma}[0,t]$ onto $\mathbb{D},$ and
$\widetilde{\gamma}(t)=\Psi_{K}\circ\gamma(t).$ Let $\Lambda^{\mathrm{eff}}$
be the effective one-leg operator, i.e.,
$\Lambda^{\mathrm{eff}}=\mathcal{P}_{q}\Lambda$ and

$M_{t}:=\frac{(Z_{\boldsymbol{\beta}_{t}}\|\,\mathrm{id}_{\Omega_{t}})}{(Z_{\boldsymbol{\beta}_{t}}\|\,\mathrm{id}_{\mathbb{H}})},\quad\boldsymbol{\beta}_{t}=a\cdot\zeta_{t}+(b-\frac{a+\beta}{2})\cdot
q+(b-\frac{a+\beta}{2})\cdot
q^{*},\qquad(\zeta_{t}=\mathrm{e}^{i\sqrt{\kappa}B_{t}}).$

Then

$M_{t}=|h_{t}^{\prime}(\zeta_{t})|^{\lambda}h_{t}^{\prime}(0)^{\mu}=\Big{(}\zeta_{t}\frac{h_{t}^{\prime}(\zeta_{t})}{h_{t}(\zeta_{t})}\Big{)}^{\lambda}h_{t}^{\prime}(0)^{\mu},$

where exponents are given by

$\lambda=\lambda_{b}(\Lambda^{\mathrm{eff}})\equiv\frac{a^{2}}{2}-ab=\frac{6-\kappa}{2\kappa},\qquad\mu=H_{q}(\Lambda^{\mathrm{eff}})\equiv\frac{a^{2}}{4}-b^{2}=\frac{(\kappa-2)(6-\kappa)}{8\kappa}.$

Restriction property of radial $\mathrm{SLE}(8/3)$ follows from the local
martingale property of $M_{t}$ (by optional stopping theorem). This is a
special case of the following formula:

(8.8) $\textrm{the drift term of
}\mathrm{d}M_{t}=-\frac{c}{6}\zeta_{t}^{2}S_{h_{t}}(\zeta_{t})M_{t}\,\mathrm{d}t.$

In Subsection 8.6 we use the CFT argument to prove (8.8) for radial
$\mathrm{SLE}[\boldsymbol{\beta}].$

We now prove Friedrich-Werner’s formula in the radial case.

######  Theorem 8.3.

Let $\theta_{j}\in(0,2\pi)$ all distinct. Then we have

$\lim_{t\to 0}\frac{1}{t^{n}}\mathbf{P}(\mathrm{SLE}(8/3)\textrm{ hits all
}[r_{t}\mathrm{e}^{i\theta_{j}},\mathrm{e}^{i\theta_{j}}])=(-2)^{n}\mathrm{e}^{2i\sum_{j=1}^{n}\theta_{j}}\mathbf{E}\,[\,T_{\boldsymbol{\beta}}(\mathrm{e}^{i\theta_{1}})\cdots
T_{\boldsymbol{\beta}}(\mathrm{e}^{i\theta_{n}})\,\|\,\mathrm{id}_{\mathbb{D}}\,],$

where $\boldsymbol{\beta}=a\cdot p+(b-a/2)\cdot q+(b-a/2)\cdot q^{*},$
$a=\frac{1}{2}\sqrt{3},$ $b=-\frac{1}{6}\sqrt{3},$ $p=1,q=0,$ and
$r_{t}=1-2\sqrt{t}.$

###### Proof.

Let $z_{j}=\mathrm{e}^{i\theta_{j}}.$ We apply Ward’s equations to the
function

$\mathbf{E}\,[\,T_{\boldsymbol{\beta}}(z)T_{\boldsymbol{\beta}}(z_{1})\cdots
T_{\boldsymbol{\beta}}(z_{n})\,\|\,\mathrm{id}_{\mathbb{D}}\,]=\mathbf{E}\,[\,T_{\check{\boldsymbol{\beta}\,}}(z)\Lambda^{\mathrm{eff}}(1)T_{\check{\boldsymbol{\beta}\,}}(z_{1})\cdots
T_{\check{\boldsymbol{\beta}\,}}(z_{n})\,\|\,\mathrm{id}_{\mathbb{D}}\,],$

by replacing $T_{\check{\boldsymbol{\beta}\,}}(z)$ on the right-hand side with
the corresponding Ward’s functional. Denote
$\boldsymbol{z}=(z_{1},\cdots,z_{n}),$
$\boldsymbol{z}_{j}=(z_{1},\cdots,\widehat{z}_{j}\cdots,z_{n}),$ and

$R(\zeta;\boldsymbol{z})=\mathbf{E}\,[\,\Lambda^{\mathrm{eff}}(\zeta)\,T_{\check{\boldsymbol{\beta}\,}}(z_{1})\cdots
T_{\check{\boldsymbol{\beta}\,}}(z_{n})\,].$

The non-random field $R(\zeta;\boldsymbol{z})$ is a boundary differential of
conformal dimension $\lambda=\frac{1}{2}{a^{2}}-ab=5/8$ with respect to
$\zeta$ and of conformal dimension $2$ with respect to $z_{j}.$ It is also a
differential of conformal dimension $\mu=\frac{1}{4}a^{2}-b^{2}=5/48$ with
respect to $q=0.$ It follows from Ward’s equation for $\Lambda^{\mathrm{eff}}$
that

(8.9)
$R(\zeta;z,\boldsymbol{z})=\frac{1}{2z^{2}}\,\mathcal{L}({v_{z}},\mathbb{D})\,R(\zeta;\boldsymbol{z}),\quad(\textrm{in
}\mathrm{id}_{\mathbb{D}}),$

where $v_{z}(\eta)=\eta(z+\eta)/(z-\eta).$

Let

$U(\theta_{1},\cdots,\theta_{n})=\lim_{t\to
0}\frac{(-1)^{n}}{(2t)^{n}}\mathrm{e}^{-2i\sum_{j=1}^{n}\theta_{j}}\mathbf{P}(\mathrm{SLE}(8/3)\textrm{
hits all }[r_{t}\mathrm{e}^{i\theta_{j}},\mathrm{e}^{i\theta_{j}}])$

(if the limit exists). Define the non-random field
$T(\zeta;z_{1},\cdots,z_{n})$ as follows:

  * •

$T$ is a boundary differential of conformal dimension $\lambda=5/8$ with
respect to $\zeta,$ and of conformal dimension $2$ with respect to $z_{j};$

  * •

$T$ is a differential of conformal dimension $\mu=\frac{1}{4}a^{2}-b^{2}=5/48$
with respect to $q=0;$

  * •

$(T(\mathrm{e}^{i\varphi};\mathrm{e}^{i\theta_{1}},\cdots,\mathrm{e}^{i\theta_{n}})\,\|\,\mathrm{id}_{\mathbb{D}})=U(\theta_{1}-\varphi,\cdots,\theta_{n}-\varphi).$

We now claim that if the limit $U(\theta_{1},\cdots,\theta_{n})$ exists then
the limit $U(\theta,\theta_{1},\cdots,\theta_{n})$ exists and

(8.10)
$T(1;z,\boldsymbol{z})=\frac{1}{2z^{2}}\,\mathcal{L}({v_{z}},\mathbb{D})\,T(1;\boldsymbol{z})\qquad
z,z_{j}\in\partial\mathbb{D}.$

By (8.9) and (8.10), $T(1;\cdot)$ and $R(1;\cdot)$ satisfy the same recursive
equation (see the remark at the end of this subsection) and are therefore
equal since $T(1;\cdot)=R(1;\cdot)=1$ for $n=0.$ Thus

$U(\theta_{1},\cdots,\theta_{n})=R(1;\mathrm{e}^{i\theta_{1}},\cdots,\mathrm{e}^{i\theta_{n}}).$

To verify the induction argument for existence of the limit
$U(\theta_{1},\cdots,\theta_{n})$ and show (8.10), denote
$\boldsymbol{\theta}=(\theta_{1},\cdots,\theta_{n}).$ We write
$\mathbf{P}(\boldsymbol{\theta})$ for the probability that radial
$\mathrm{SLE}(8/3)$ path hits all segments
$[r_{t}\mathrm{e}^{i\theta_{j}},\mathrm{e}^{i\theta_{j}}]\,(1\leq j\leq n)$
and $\mathbf{P}(\boldsymbol{\theta}\,|\,\neg\theta)$ for the same probability
conditioned on the event that the path avoids
$[r_{t}\mathrm{e}^{i\theta},\mathrm{e}^{i\theta}].$ By the induction
hypothesis,

(8.11) $\mathbf{P}(\boldsymbol{\theta})\approx(-2t)^{n}z_{1}^{2}\cdots
z_{n}^{2}\,T(1;\boldsymbol{z}),$

where $z_{j}=\mathrm{e}^{i\theta_{j}}.$ On the other hand, by the restriction
property of radial $\mathrm{SLE}(8/3),$ we have

(8.12)
$1-\mathbf{P}(\theta)=|\psi_{t}^{\prime}(1)|^{\lambda}\,\psi_{t}^{\prime}(0)^{\frac{1}{6}\lambda}$

and

(8.13)
$\mathbf{P}(\boldsymbol{\theta}\,|\,\neg\theta)\approx(-2t)^{n}\,T(\psi_{t}(1);\psi_{t}(z_{1}),\cdots,\psi_{t}(z_{n}))\prod_{j=1}^{n}z_{j}^{2}\psi_{t}^{\prime}(z_{j})^{2},$

where $\psi_{t}$ is a slit map from
$(\mathbb{D}\setminus[r_{t}\mathrm{e}^{i\theta},\mathrm{e}^{i\theta}],0)$ onto
$(\mathbb{D},0)$ with $\psi_{t}^{\prime}(0)>0$. It follows from (8.11) –
(8.13) and
$\mathbf{P}(\theta,\boldsymbol{\theta})=\mathbf{P}(\boldsymbol{\theta})-\mathbf{P}(\boldsymbol{\theta}\,|\,\neg\theta)(1-\mathbf{P}(\theta))$
$(z=\mathrm{e}^{i\theta})$ that

$\frac{\mathbf{P}(\theta,\boldsymbol{\theta})}{z^{2}z_{1}^{2}\cdots
z_{n}^{2}(-2t)^{n+1}}=\frac{\psi_{t}^{\prime}(0)^{\lambda/6}|\psi_{t}^{\prime}(1)|^{\lambda}\prod_{j=1}^{n}\psi_{t}^{\prime}(z_{j})^{2}T(\psi_{t}(1);\psi_{t}(z_{1}),\cdots,\psi_{t}(z_{n}))-T(1;\boldsymbol{z})}{2tz^{2}}$

up to $o(t)$ terms. Thus the limit $U(\theta,\theta_{1},\cdots,\theta_{n})$
exists. Since $\psi_{t}^{\prime}(0)=\mathrm{e}^{t},$ we have (8.10):

$T(1;z,\boldsymbol{z})=\frac{1}{2z^{2}}\Big{(}\mathcal{L}(v_{z},\mathbb{D}\setminus\\{0\\})+\frac{\lambda}{6}\Big{)}T(1;\boldsymbol{z})=\frac{1}{2z^{2}}\,\mathcal{L}({v_{z}},\mathbb{D})\,T(1;\boldsymbol{z}).$

∎

######  Remark.

The formula (8.9) holds for all $\kappa.$ Setting $R(z,\boldsymbol{z})\equiv
R(1;z,\boldsymbol{z}),$ the formula (8.9) at $\zeta=1$ gives the following
recursive formula for $R:$

$\displaystyle R(z,\boldsymbol{z})$
$\displaystyle=\frac{1}{2z^{2}}\Big{(}2n\frac{1+z}{1-z}+2\lambda\frac{z}{(1-z)^{2}}+(\frac{a^{2}}{4}-b^{2})+\frac{1+z}{1-z}\sum_{j=1}^{n}z_{j}\partial_{z_{j}}\Big{)}R(\boldsymbol{z})$
$\displaystyle+\frac{1}{2z^{2}}\sum_{j=1}^{n}\Big{(}z_{j}\frac{z+z_{j}}{z-z_{j}}\partial_{z_{j}}+2\frac{z^{2}+2zz_{j}-z_{j}^{2}}{(z-z_{j})^{2}}\Big{)}R(\boldsymbol{z})+\frac{c}{2}\sum_{j=1}^{n}\frac{1}{(z-z_{j})^{4}}R(\boldsymbol{z}_{j}).$

###  8.5. 1-point vertex observables

In this subsection we discuss some basic examples of 1-point vertex
observables, including Lawler-Schramm-Werner’s derivative exponents of radial
SLEs on the boundary. Let

$M^{(\tau^{+},\tau^{-};\tau_{q}^{+},\tau_{q}^{-})}(z)=\widehat{\mathbf{E}}\,\mathcal{O}_{\check{\boldsymbol{\beta}\,}}[\boldsymbol{\tau}]=\mathbf{E}\,\mathcal{O}_{\boldsymbol{\beta}}[\boldsymbol{\tau}],\qquad\boldsymbol{\tau}=\tau^{+}\cdot
z+\tau^{-}\cdot z^{*}+\tau_{q}^{+}\cdot q+\tau_{q}^{-}\cdot q^{*}.$

Then by (6.2) we have

$M^{(\tau^{+},\tau^{-};\tau_{q}^{+},\tau_{q}^{-})}=(w_{q}^{\prime})^{h_{q}^{+}}(\overline{w_{q}^{\prime}})^{h_{q}^{-}}(w^{\prime})^{h^{+}}(\overline{w^{\prime}})^{h^{-}}w^{\nu^{+}}(\bar{w})^{\nu^{-}}(1-w)^{a\tau^{+}}(1-\bar{w})^{a\tau^{-}}(1-|w|^{2})^{\tau^{+}\tau^{-}},$

where the exponents are $\nu^{\pm}=\tau^{\pm}(\tau_{q}^{\pm}+b-a/2)$ and the
dimensions are

$h^{\pm}=\lambda_{b}(\tau^{\pm}),\qquad
h_{q}^{\pm}=\frac{\tau_{q}^{\pm}(\tau_{q}^{\pm}-a)}{2}.$

The last formulas come from
$h_{q}^{\pm}=\lambda_{b}(\tau_{q}^{\pm}+b-a/2)-\lambda_{b}(\tau_{q}^{\pm}).$

•  _Constant fields._ The simplest examples of 1-point vertex fields are
constant fields, i.e., vertex fields with $\tau^{\pm}=0.$ By the neutrality
condition, $\tau_{q}^{-}=-\tau_{q}^{+}.$ Since
$w_{t}^{\prime}(0)=\mathrm{e}^{t-i\sqrt{\kappa}B_{t}},$

$M_{t}^{(0,0;\tau_{q},-\tau_{q})}=\mathrm{e}^{\tau_{q}^{2}t+ia\sqrt{\kappa}\tau_{q}B_{t}}=\mathrm{e}^{\tau_{q}^{2}t+i\sqrt{2}\tau_{q}B_{t}}.$

This is a martingale.

•  _Real fields._ The 1-point vertex fields are real if and only if
$\tau^{+}=\tau^{-}$ and $\tau_{q}^{+}=\tau_{q}^{-}.$ By the neutrality
condition, $\tau_{q}^{+}=-\tau^{+}.$ Thus the only real fields are

$M^{(\tau,\tau;-\tau,-\tau)}=|w_{q}^{\prime}|^{\tau^{2}+a\tau}|w^{\prime}|^{\tau^{2}-2b\tau}|w|^{\tau(2b-a-2\tau)}|1-w|^{2a\tau}(1-|w|^{2})^{\tau^{2}}.$

When $\tau=-a,$ there is no covariance at $q.$ In this special case,

$M^{(-a,-a;a,a)}=\bigg{|}\frac{w^{\prime}}{w}\bigg{|}^{1-2/\kappa}\bigg{(}\frac{1-|w|^{2}}{|1-w|^{2}}\bigg{)}^{2/\kappa}.$

######  Example.

If $\kappa=2,$ then $M^{(-a,-a;a,a)}$ coincides with the Lawler-Schramm-Werner
observable

$M=\frac{1-|w|^{2}}{|1-w|^{2}}=\frac{P_{\mathbb{D}}(1,w)}{P_{\mathbb{D}}(1,0)}=\frac{P_{D}(p,z)}{P_{D}(p,q)},$

where $P_{D}$ is the Poisson kernel of a domain $D.$ As mentioned in
Subsection 1.2, this 1-point field is an important observable in the theory of
LERW.

######  Example.

If $\kappa=4,$ then

$M=\bigg{|}\frac{w^{\prime}}{w}\bigg{|}^{1/2}\bigg{(}\frac{1-|w|^{2}}{|1-w|^{2}}\bigg{)}^{1/2}$

is Beffara’s type observable for radial $\mathrm{SLE}(4),$ see [3]. In the
chordal case, Beffara’s observables are real martingale-observables of
conformal dimensions

$h^{+}=h^{-}=\frac{1}{2}-\frac{\kappa}{16},\qquad h_{q}=0$

with the estimate

$\mathbf{P}(z,\varepsilon)\asymp\varepsilon^{1-\frac{1}{8}\kappa}~{}M(z),$

where $\mathbf{P}(z,\varepsilon)$ is the probability that the
$\mathrm{SLE}(\kappa)$ curve ($\kappa<8$) hits the disc at $z$ of size
$\varepsilon(\ll 1)$ measured in a local chart $\phi.$ See [6]. In [3],
Feynman-Kac formula is used to construct radial SLE martingale-observables
with the desired dimensions.

•  _1-point vertex fields without covariance at $q$._ A 1-point vertex field
$M$ has no covariance at $q$ if and only if

$(\tau_{q}^{+},\tau_{q}^{-})=(0,0),\quad(a,0),\quad(0,a),\quad\textrm{or}\quad(a,a).$

The first case is just the non-chiral vertex field $M=M^{(\tau,-\tau;0,0)}.$

When $\tau^{-}=0,$ the 1-point vertex observable

$M^{(\tau,0;\tau_{q}^{+},\tau_{q}^{-})}=(w^{\prime}_{q})^{h_{q}^{+}}(\overline{w^{\prime}_{q}})^{h_{q}^{-}}(w^{\prime})^{h}w^{\nu}(1-w)^{a\tau}$

is holomorphic. Recall the expression for the exponents and the dimensions

$h=\frac{1}{2}\tau^{2}-b\tau,\quad\nu=\tau(b-\frac{1}{2}a+\tau_{q}^{+}),\quad
h_{q}^{\pm}=\frac{\tau_{q}^{\pm}(\tau_{q}^{\pm}-a)}{2}.$

•  _Holomorphic 1-point fields without spin at $q$._ A holomorphic 1-point
field $M$ has no spin at $q$ if and only if $h_{q}^{+}=h_{q}^{-}.$
Equivalently,

$\tau_{q}^{+}=\tau_{q}^{-}\quad\textrm{or}\quad\tau_{q}^{+}+\tau_{q}^{-}=a.$

Case 1. $\tau_{q}^{+}=\tau_{q}^{-}.$ By the neutrality condition, we have

$\boldsymbol{\tau}=\tau\cdot z-\frac{\tau}{2}\cdot q-\frac{\tau}{2}\cdot
q^{*}.$

The holomorphic 1-point field
$\mathcal{O}_{\boldsymbol{\beta}}[\boldsymbol{\tau}]$ is a generalization of
the one-leg operator $\Lambda_{z}.$ In this case, its conformal dimensions are

$h=\frac{\tau^{2}}{2}-b\tau,\quad
h_{q}^{\pm}=h_{q}=\frac{\tau^{2}}{8}+\frac{a\tau}{4}.$

Thus

$M_{t}=\mathrm{e}^{2h_{q}t}\Big{(}\frac{w_{t}^{\prime}}{w_{t}}\Big{)}^{\frac{1}{2}\tau^{2}-b\tau}\Big{(}\frac{w_{t}}{(1-w_{t})^{2}}\Big{)}^{-\frac{1}{2}{a\tau}}.$

######  Example (Derivative exponents on the boundary [24]).

On the unit circle, (up to constant)

$M_{t}(\mathrm{e}^{i\theta})=\mathrm{e}^{2h_{q}t}|w_{t}^{\prime}(\mathrm{e}^{i\theta})|^{h}\Big{(}\sin^{2}\frac{\theta_{t}}{2}\Big{)}^{\frac{1}{2}a\tau},$

where $w_{t}(\mathrm{e}^{i\theta})=\mathrm{e}^{i\theta_{t}}.$ Given $h,$ the
equation $h=\frac{1}{2}\tau^{2}-b\tau$ is solved by

$\tau_{\pm}=\frac{a}{4}\big{(}\kappa-4\pm\sqrt{(\kappa-4)^{2}+16\kappa
h}\big{)}.$

With the choice of $\tau=\tau_{+},$ Lawler, Schramm, and Werner proved that
$M_{t}(\mathrm{e}^{i\theta})$ is a martingale. They applied the optional
stopping theorem to $M_{t}(\mathrm{e}^{i\theta})$ and used the estimate

$\mathbf{E}[M_{t}(\mathrm{e}^{i\theta})]\asymp\mathrm{e}^{2h_{q}t}\,\mathbf{E}[|w^{\prime}_{t}(\mathrm{e}^{i\theta})|^{h}\mathbf{1}_{\\{\tau_{\mathrm{e}^{i\theta}}>t\\}}]$

to derive the derivative exponents:

$\mathbf{E}[|w^{\prime}_{t}(\mathrm{e}^{i\theta})|^{h}\mathbf{1}_{\\{\tau_{\mathrm{e}^{i\theta}}>t\\}}]\asymp\mathrm{e}^{-2h_{q}t}\Big{(}\sin^{2}\frac{\theta}{2}\Big{)}^{\frac{1}{2}a\tau}.$

(Recall that $\tau_{z}$ is the first time when a point $z$ is swallowed by the
hull of SLE, see Subsection 1.2.) From the derivative exponent for $\kappa=6,$
they obtained the annulus crossing exponent for $\mathrm{SLE}_{6}$ and
combined it with other exponents to prove Mandelbrot’s conjecture that the
Hausdorff dimension of the planar Brownian frontier is $4/3$. See [23] and
references therein.

######  Example.

The field $M$ is a scalar if $\tau=2b.$ In this case,

$M_{t}=\mathrm{e}^{t(\kappa-4)/{8}}\Big{(}w_{t}+\frac{1}{w_{t}}-2\Big{)}^{(\kappa-4)/(2\kappa)}.$

Its derivative $\widetilde{M}=\partial M$ has the conformal dimensions
$[1,0;h_{q},h_{q}].$ It is not a vertex observable. If
$\widetilde{M}=M^{(\tilde{\tau}^{+},\tilde{\tau}^{-};\tilde{\tau}_{q}^{+},\tilde{\tau}_{q}^{-})}$
with $\tilde{h}^{+}=\lambda_{b}(\tilde{\tau}^{+})=1,$ then
$\tilde{\tau}^{+}=-2a$ or $\tilde{\tau}^{+}=2a+2b.$ As we will see below, the
holomorphic 1-differentials without spin at $q$ are not forms of $\partial
M^{(2b,0,-b,-b)}.$ Unlike the chordal case (see [18, Proposition 15.2]), the
holomorphic differential observables are not necessarily vertex observables.

######  Example.

If we take $\tau=2b-a$ so that $h=h_{1,2}:=\frac{1}{2}a^{2}-ab$ (there are
only two values of $\tau$ such that $h=h_{1,2},$ namely $\tau=a,2b-a$), then

$\boldsymbol{\tau}=(2b-a)\cdot z+(-b+\frac{a}{2})\cdot q+(-b+\frac{a}{2})\cdot
q^{*},$

and

$h=\frac{6-\kappa}{2\kappa}=-\frac{a\tau}{2},\quad
h_{q}=\frac{(2-\kappa)(6-\kappa)}{16\kappa}=-h_{0,1/2},\quad\nu=0.$

In this case, we have

$M_{t}=\mathrm{e}^{2h_{q}t}\Big{(}\frac{w_{t}^{\prime}}{(1-w_{t})^{2}}\Big{)}^{(6-\kappa)/(2\kappa)}.$

If $\kappa=2,$ then

$M=\frac{w^{\prime}}{(1-w)^{2}}$

and its anti-derivative is $1/(1-w).$ See the next example.

Case 2. $\tau_{q}^{+}+\tau_{q}^{-}=a.$ Let $\tau_{q}=\tau_{q}^{+}.$ It follows
from the neutrality condition that

$\boldsymbol{\tau}=-a\cdot z+\tau_{q}\cdot q+(a-\tau_{q})\cdot q^{*}.$

Thus we have

$M=(w^{\prime})^{(\kappa-2)/(2\kappa)}w^{a(-\tau_{q}+\frac{1}{2}a-b)}(1-w)^{-2/\kappa}|w_{q}^{\prime}|^{\tau_{q}^{2}-a\tau_{q}}.$

######  Example.

The fields $M$ have no covariance at $q$ if and only if $\tau_{q}=0$ or $a.$
In these cases, we have

$M^{(-a,0;a,0)}=\Big{(}\frac{w^{\prime}}{w}\Big{)}^{(\kappa-2)/(2\kappa)}(1-w)^{-2/\kappa}$

and

$M^{(-a,0;0,a)}=(w^{\prime})^{(\kappa-2)/(2\kappa)}w^{(6-\kappa)/(2\kappa)}(1-w)^{-2/\kappa}.$

For example, if $\kappa=6,$ then

$M^{(-a,0;0,a)}=\Big{(}\dfrac{w^{\prime}}{1-w}\Big{)}^{1/3},$

and if $\kappa=2,$ then both $M^{(-a,0;a,0)}$ and $M^{(-a,0;0,a)}$ produce the
same observable

$M^{(-a,0;a,0)}=1+M^{(-a,0;0,a)}=\frac{1}{1-w}.$

•  _Holomorphic differentials without spin at $q$._ Let $\tau=\tau^{+},$
$\tau^{-}=0.$ A holomorphic 1-point field $M$ is a 1-differential with respect
to $z$ if and only if $\tau=-2a$ or $\tau=2(a+b).$ Furthermore, if $M$ has no
spin at $q,$ then $\tau_{q}^{+}=\tau_{q}^{-}.$ (If $\tau=-2a$ or
$\tau=2(a+b),$ then the other possibility $\tau_{q}^{+}+\tau_{q}^{-}=a$ never
happens because of the neutrality condition.)

Case 1. $\tau=-2a.$ By the neutrality condition, we have

$\boldsymbol{\tau}=-2a\cdot z+a\cdot q+a\cdot q^{*}$

and

$M=w^{\prime}w^{2/\kappa-1}(1-w)^{-4/\kappa}.$

######  Example.

If $\kappa=2,$ then

$M^{(-2a,0;a,a)}=\frac{w^{\prime}}{(1-w)^{2}}$

and its anti-derivative is $1/(1-w).$ See the previous example.

######  Example.

If $\kappa=4,$ then $M^{(-2a,0;a,a)}=w^{\prime}(1-w)^{-1}w^{-1/2}$ and its
anti-derivative is

$\log(1+\sqrt{w})-\log(1-\sqrt{w}).$

Its imaginary part is a bosonic observable for a twisted conformal field
theory.

Case 2. $\tau=2(a+b).$ By the neutrality condition, we have

$\boldsymbol{\tau}=(2a+2b)\cdot z-(a+b)\cdot q-(a+b)\cdot q^{*}$

and

$M_{t}=\mathrm{e}^{t(\kappa+4)/8}w_{t}^{\prime}w_{t}^{-3/2}(1-w_{t}).$

###  8.6. Restriction formulas

Let $\kappa\leq 4.$ We first consider the chordal
$\mathrm{SLE}[\boldsymbol{\beta}]$ with

$\boldsymbol{\beta}=a\cdot p+\sum\beta_{k}\cdot
q_{k}+(2b-a-\sum\beta_{k})\cdot q,\qquad p,q,q_{k}\in\partial D.$

Fix a hull $K.$ Let $g_{t}$ be the chordal $\mathrm{SLE}[\boldsymbol{\beta}]$
map with the hull $K_{t}.$ On the event $K_{\infty}\cap K=\emptyset,$ a
conformal map $h_{t}:\Omega_{t}=g_{t}(D_{t}\setminus K)\to\mathbb{H}$ is
defined by

$h_{t}=\widetilde{g}_{t}\circ\Psi_{K}\circ g_{t}^{-1},$

where $\widetilde{g}_{t}$ is a Loewner map from
$\widetilde{D}_{t}=D\setminus\widetilde{K}_{t}$ onto $\mathbb{H},$
$\widetilde{K}_{t}=\Psi_{K}(K_{t}),$ and $\Psi_{K}$ is the conformal map
$(\mathbb{H}\setminus K,0,\infty)\to(\mathbb{H},0,\infty)$ satisfying
$\Psi_{K}^{\prime}(\infty)=1.$ Let

$M_{t}:=\frac{(Z_{\boldsymbol{\beta}_{t}}\|\,\mathrm{id}_{\Omega_{t}})}{(Z_{\boldsymbol{\beta}_{t}}\|\,\mathrm{id}_{\mathbb{H}})}=\frac{\mathbf{E}\,(\Lambda^{\mathrm{eff}}_{\Omega_{t}}\|\,\mathrm{id})(\xi_{t},\boldsymbol{q}(t))}{\mathbf{E}\,(\Lambda^{\mathrm{eff}}_{\mathbb{H}}\|\,\mathrm{id})(\xi_{t},\boldsymbol{q}(t))},$

where

$\boldsymbol{\beta}_{t}=a\cdot\xi_{t}+\sum\beta_{k}\cdot
q_{k}(t),\qquad\mathrm{d}q_{k}(t)=\frac{2}{q_{k}(t)-\xi_{t}}\,\mathrm{d}t,\qquad
q_{k}(0)=q_{k}$

and $\Lambda^{\mathrm{eff}}=\mathcal{P}_{\boldsymbol{\beta}}\,\Lambda$ is the
effective one-leg operator. Then the process $M_{t}$ is expressed in terms of
$h_{t},$ $\xi_{t},$ and $q_{k}(t)$ as

(8.14)
$M_{t}=h_{t}^{\prime}(\xi_{t})^{\lambda}\prod_{j}h_{t}^{\prime}(q_{j}(t))^{\lambda_{j}}\Big{(}\frac{h_{t}(\xi_{t})-h_{t}(q_{j}(t))}{\xi_{t}-q_{j}(t)}\Big{)}^{a\beta_{j}}\prod_{j<k}\Big{(}\frac{h_{t}(q_{j}(t))-h_{t}(q_{k}(t))}{q_{j}(t)-q_{k}(t)}\Big{)}^{\beta_{j}\beta_{k}}.$

We now present the conformal field theoretic proof for the restriction formula
([13]) of chordal $\mathrm{SLE}[\boldsymbol{\beta}]:$

(8.15)
$\mathrm{d}M_{t}=\frac{c}{6}S_{h_{t}}(\xi_{t})M_{t}\,\mathrm{d}t+\textrm{martingale
terms.}$

Let

$F(z,\boldsymbol{q},t):=\frac{G(z,\boldsymbol{q},t)}{H(z,\boldsymbol{q})},\quad\begin{cases}G(z,\boldsymbol{q},t):=&\mathbf{E}\,(\Lambda^{\mathrm{eff}}_{\Omega_{t}}\|\mathrm{id})(z,\boldsymbol{q})(=\mathbf{E}\,(\Lambda^{\mathrm{eff}}_{\mathbb{H}}\|h_{t}^{-1})(z,\boldsymbol{q})),\\\
H(z,\boldsymbol{q}):=&\mathbf{E}\,(\Lambda^{\mathrm{eff}}_{\mathbb{H}}\|\mathrm{id})(z,\boldsymbol{q}).\end{cases}$

Recall that the driving process $\xi_{t}$ satisfies

$\mathrm{d}\xi_{t}=\sqrt{\kappa}B_{t}+\kappa\frac{H_{\xi}(\xi_{t},\boldsymbol{q}(t))}{H(\xi_{t},\boldsymbol{q}(t))}\,\mathrm{d}t.$

It follows from Itô’s formula that the drift term of $\mathrm{d}M_{t}/M_{t}$
equals

$\Big{(}\frac{\dot{F}}{F}+\frac{\kappa}{2}\frac{F_{\xi\xi}}{F}+\sum_{j}\frac{F_{q_{j}}}{F}\dot{q}_{j}(t)+\kappa\frac{F_{\xi}}{F}\frac{H_{\xi}}{H}\Big{)}\,\mathrm{d}t$

evaluated at $(\xi_{t},\boldsymbol{q}(t),t).$ We rewrite the drift term of
$\mathrm{d}M_{t}/M_{t}$ in terms of $G$ and $H:$

$\textrm{the drift term of
}\frac{\mathrm{d}M_{t}}{M_{t}}=\Big{(}\frac{\dot{G}}{G}+\frac{\kappa}{2}\Big{(}\frac{G_{\xi\xi}}{G}-\frac{H_{\xi\xi}}{H}\Big{)}+\sum_{j}\Big{(}\frac{G_{q_{j}}}{G}-\frac{H_{q_{j}}}{H}\Big{)}\frac{2}{q_{j}(t)-\xi_{t}}\Big{)}\,\mathrm{d}t.$

Here we use

$\frac{\dot{F}}{F}=\frac{\dot{G}}{G},\qquad\frac{F_{q_{j}}}{F}=\frac{G_{q_{j}}}{G}-\frac{H_{q_{j}}}{H},\qquad\frac{F_{\xi\xi}}{F}=\frac{G_{\xi\xi}}{G}-\frac{H_{\xi\xi}}{H}-2\frac{F_{\xi}}{F}\frac{H_{\xi}}{H}.$

By Ward’s equations we have

$\displaystyle\dot{G}(z,\boldsymbol{q}(t),t)=$
$\displaystyle-2h_{t}^{\prime}(\xi_{t})^{2}h_{t}^{\prime}(z)^{\lambda}\prod_{j}h_{t}^{\prime}(q_{j}(t))^{\lambda_{j}}\mathbf{E}\big{(}A_{\mathbb{H}}(h_{t}(\xi_{t}))\Lambda^{\mathrm{eff}}_{\mathbb{H}}(h_{t}(z),h_{t}(q_{1}(t)),\cdots)\|\mathrm{id}\big{)}$
$\displaystyle+2\mathbf{E}\,(\mathcal{L}_{v_{\xi_{t}}}\Lambda^{\mathrm{eff}}_{\Omega_{t}}\|\mathrm{id})(z,\boldsymbol{q}(t)).$

It follows from conformal invariance that

$\dot{G}(z,\boldsymbol{q}(t),t)=-2(\mathbf{E}\,A_{\Omega_{t}}(\xi_{t})\Lambda^{\mathrm{eff}}_{\Omega_{t}}(z,\boldsymbol{q}(t))\|\mathrm{id})+2\mathbf{E}\,(\mathcal{L}_{k_{\xi_{t}}}\Lambda^{\mathrm{eff}}_{\Omega_{t}}\|\mathrm{id})(z,\boldsymbol{q}(t)).$

Sending $z$ to $\xi_{t},$

$\dot{G}(\xi_{t},\boldsymbol{q}(t),t)=-2(\mathbf{E}\,A_{\Omega_{t}}*_{\xi}\Lambda^{\mathrm{eff}}_{\Omega_{t}}(\xi_{t},\boldsymbol{q}(t))\|\mathrm{id})+2\mathbf{E}\,(\check{\mathcal{L}}_{k_{\xi_{t}}}\Lambda^{\mathrm{eff}}_{\Omega_{t}}\|\mathrm{id})(\xi_{t},\boldsymbol{q}(t)),$

where $\check{\mathcal{L}}_{k_{\xi_{t}}}$ does not apply to $\xi_{t}.$ By the
level two degeneracy equation for $\Lambda^{\mathrm{eff}}$ with respect to
$\xi$,

$\frac{\kappa}{2}\Big{(}\frac{G_{\xi\xi}}{G}-\frac{H_{\xi\xi}}{H}\Big{)}=2\frac{(\mathbf{E}\,T_{\Omega_{t}}*_{\xi}\Lambda^{\mathrm{eff}}_{\Omega_{t}}(\xi_{t},\boldsymbol{q}(t))\|\mathrm{id})}{(\mathbf{E}\,\Lambda^{\mathrm{eff}}_{\Omega_{t}}(\xi_{t},\boldsymbol{q}(t))\|\mathrm{id})}-2\frac{(\mathbf{E}\,T_{\mathbb{H}}*_{\xi}\Lambda^{\mathrm{eff}}_{\mathbb{H}}(\xi_{t},\boldsymbol{q}(t))\|\mathrm{id})}{(\mathbf{E}\,\Lambda^{\mathrm{eff}}_{\mathbb{H}}(\xi_{t},\boldsymbol{q}(t))\|\mathrm{id})}.$

It follows from Ward’s equation that

$(\mathbf{E}\,T_{\mathbb{H}}*_{\xi}\Lambda^{\mathrm{eff}}_{\mathbb{H}}(\xi_{t},\boldsymbol{q}(t))\|\mathrm{id})=\mathbf{E}\,(\check{\mathcal{L}}_{k_{\xi_{t}}}\Lambda^{\mathrm{eff}}_{\mathbb{H}}\|\mathrm{id})(\xi_{t},\boldsymbol{q}(t)).$

The formula for Lie derivatives of differentials gives

$\frac{\mathbf{E}\,(\check{\mathcal{L}}_{k_{\xi_{t}}}\Lambda^{\mathrm{eff}}_{\Omega_{t}}\|\mathrm{id})(\xi_{t},\boldsymbol{q}(t))}{(\mathbf{E}\,\Lambda^{\mathrm{eff}}_{\Omega_{t}}(\xi_{t},\boldsymbol{q}(t))\|\mathrm{id})}-\frac{\mathbf{E}\,(\check{\mathcal{L}}_{k_{\xi_{t}}}\Lambda^{\mathrm{eff}}_{\mathbb{H}}\|\mathrm{id})(\xi_{t},\boldsymbol{q}(t))}{(\mathbf{E}\,\Lambda^{\mathrm{eff}}_{\mathbb{H}}(\xi_{t},\boldsymbol{q}(t))\|\mathrm{id})}=-\sum_{j}\Big{(}\frac{G_{q_{j}}}{G}-\frac{H_{q_{j}}}{H}\Big{)}\frac{1}{q_{j}(t)-\xi_{t}}.$

Combining all of the above, we have

$\displaystyle\textrm{the drift term of }\frac{\mathrm{d}M_{t}}{M_{t}}$
$\displaystyle=-2\frac{(\mathbf{E}\,A_{\Omega_{t}}*_{\xi}\Lambda^{\mathrm{eff}}_{\Omega_{t}}(\xi_{t},\boldsymbol{q}(t))\|\mathrm{id})}{(\mathbf{E}\,\Lambda^{\mathrm{eff}}_{\Omega_{t}}(\xi_{t},\boldsymbol{q}(t))\|\mathrm{id})}\,\mathrm{d}t+2\frac{(\mathbf{E}\,T_{\Omega_{t}}*_{\xi}\Lambda^{\mathrm{eff}}_{\Omega_{t}}(\xi_{t},\boldsymbol{q}(t))\|\mathrm{id})}{(\mathbf{E}\,\Lambda^{\mathrm{eff}}_{\Omega_{t}}(\xi_{t},\boldsymbol{q}(t))\|\mathrm{id})}\,\mathrm{d}t$
$\displaystyle=\frac{c}{6}S_{h_{t}}(\xi_{t})\,\mathrm{d}t.$

We now present a conformal field theoretic proof for the restriction formula
of radial $\mathrm{SLE}[\boldsymbol{\beta}]$ $(\kappa=8/3).$ For this purpose,
we let $\zeta,q_{k}\in\partial D,q\in D,$

$\boldsymbol{\beta}=\boldsymbol{\tau}+\check{\boldsymbol{\beta}\,},\quad\boldsymbol{\tau}=a\cdot\zeta-\frac{a}{2}\cdot
q-\frac{a}{2}\cdot q^{*}\quad\check{\boldsymbol{\beta}\,}=\sum\beta_{k}\cdot
q_{k}+(b-\frac{\beta}{2})\cdot q+(b-\frac{\beta}{2})\cdot
q^{*},\quad\beta=\sum\beta_{k},$

and consider the effective one-leg operator $\Lambda^{\mathrm{eff}},$

$\Lambda^{\mathrm{eff}}(\zeta):=\mathcal{P}_{\check{\boldsymbol{\beta}\,}}\,\Lambda(\zeta)=C_{(b)}[\boldsymbol{\beta}]\,\mathrm{e}^{\odot
i\Phi[\boldsymbol{\tau}]}.$

Fix a hull $K$ such that $K$ does not intersect
$\zeta=1,q_{1},\cdots,q_{k},\cdots.$ Let $g_{t}$ be the radial
$\mathrm{SLE}[\boldsymbol{\beta}]$ map with the hull $K_{t}.$ On the event
$K_{\infty}\cap K=\emptyset,$ a conformal map
$h_{t}:\Omega_{t}=g_{t}(D_{t}\setminus K)\to\mathbb{D}$ is defined by

$h_{t}=\widetilde{g}_{t}\circ\Psi_{K}\circ g_{t}^{-1},$

where $\widetilde{g}_{t}$ is a radial Loewner map from
$\widetilde{D}_{t}=D\setminus\widetilde{K}_{t}$ onto $\mathbb{D},$
$\widetilde{K}_{t}=\Psi_{K}(K_{t}),$ and $\Psi_{K}$ is the conformal map
$(\mathbb{D}\setminus K,0)\to(\mathbb{D},0)$ satisfying
$\Psi_{K}^{\prime}(0)>0.$ Let

(8.16)
$M_{t}:=\frac{(Z_{\boldsymbol{\beta}_{t}}\|\,\mathrm{id}_{\Omega_{t}})}{(Z_{\boldsymbol{\beta}_{t}}\|\,\mathrm{id}_{\mathbb{H}})},\quad\boldsymbol{\beta}_{t}=a\cdot\zeta_{t}+(b-\frac{a+\beta}{2})\cdot
q+(b-\frac{a+\beta}{2})\cdot q^{*}+\sum\beta_{k}\cdot q_{k}(t),$

where $q_{k}(t)$ satisfies $q_{k}(0)=q_{k}$ and

$\mathrm{d}q_{k}(t)=q_{k}(t)\frac{\zeta_{t}+q_{k}(t)}{\zeta_{t}-q_{k}(t)}\,\mathrm{d}t.$

We now represent $M_{t}$ in terms of the effective one-leg operator:

$M_{t}=F(\zeta_{t},\boldsymbol{q}(t),t),\quad
F(z,\boldsymbol{q},t)=\frac{G(z,\boldsymbol{q},t)}{H(z,\boldsymbol{q})},$

where
$G(z,\boldsymbol{q},t)=\mathbf{E}(\Lambda_{\mathbb{D}}^{\mathrm{eff}}\|h_{t}^{-1})(z),$
and
$H(z,\boldsymbol{q})=\mathbf{E}(\Lambda_{\mathbb{D}}^{\mathrm{eff}}\|\mathrm{id})(z).$

######  Theorem 8.4.

We have

(8.17) $\textrm{the drift term of
}\frac{\mathrm{d}M_{t}}{M_{t}}=-\frac{c}{6}\,\zeta_{t}^{2}\,S_{h_{t}}(\zeta_{t})\,\mathrm{d}t.$

###### Proof.

Recall that

$\mathrm{d}\zeta_{t}=i\sqrt{\kappa}\,\zeta_{t}\,\mathrm{d}B_{t}-\frac{\kappa}{2}\zeta_{t}\,\mathrm{d}t-\kappa\zeta_{t}^{2}\Big{(}\frac{H_{\zeta}}{H}+\frac{h}{\zeta_{t}}\Big{)}\,\mathrm{d}t=i\sqrt{\kappa}\,\zeta_{t}\,\mathrm{d}B_{t}-\kappa\zeta_{t}^{2}\frac{H_{\zeta}}{H}\,\mathrm{d}t-3\zeta_{t}\,\mathrm{d}t.$

It follows from Itô’s formula that

$\mathrm{d}M_{t}=\dot{F}\,\mathrm{d}t-\frac{\kappa}{2}\zeta_{t}^{2}\,F_{\zeta\zeta}\,\mathrm{d}t+F_{\zeta}\,\mathrm{d}\zeta_{t}+\sum
F_{q_{k}}\,\mathrm{d}q_{k}(t)$

evaluated at $(\zeta_{t},\boldsymbol{q}(t),t).$ The drift term of
$\mathrm{d}M_{t}/M_{t}$ is

$\Big{(}\frac{\dot{F}}{F}-\frac{\kappa}{2}\zeta_{t}^{2}\,\frac{F_{\zeta\zeta}}{F}-\Big{(}\kappa\zeta_{t}^{2}\frac{H_{\zeta}}{H}+3\zeta_{t}\Big{)}\frac{F_{\zeta}}{F}+\sum_{k}\Big{(}q_{k}(t)\frac{\zeta_{t}+q_{k}(t)}{\zeta_{t}-q_{k}(t)}\Big{)}\frac{F_{q_{k}}}{F}\Big{)}\Big{|}_{(\zeta_{t},\boldsymbol{q}(t),t)}.$

We rewrite the drift term of $\mathrm{d}M_{t}/M_{t}$ in terms of $G$ and $H:$

(8.18)
$\Big{(}\frac{\dot{G}}{G}-\frac{\kappa}{2}\zeta_{t}^{2}\,\Big{(}\frac{G_{\zeta\zeta}}{G}-\frac{H_{\zeta\zeta}}{H}\Big{)}-3\zeta_{t}\Big{(}\frac{G_{\zeta}}{G}-\frac{H_{\zeta}}{H}\Big{)}+\sum_{k}\Big{(}q_{k}(t)\frac{\zeta_{t}+q_{k}(t)}{\zeta_{t}-q_{k}(t)}\Big{)}\Big{(}\frac{G_{q_{k}}}{G}-\frac{H_{q_{k}}}{H}\Big{)}\Big{)}$

evaluated at $(\xi_{t},\boldsymbol{q}(t),t).$

Using the similar method in [18, Section 14.5], we represent $\dot{G}$ in
terms of the Lie derivatives:

(8.19) $\displaystyle\dot{G}(z,\boldsymbol{q},t)$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}s}\Big{|}_{s=0}(\mathbf{E}\,\Lambda^{\mathrm{eff}}_{\mathbb{D}}\,\|\,h_{t+s}^{-1})(z)=\frac{\mathrm{d}}{\mathrm{d}s}\Big{|}_{s=0}(\mathbf{E}\,\Lambda^{\mathrm{eff}}_{\mathbb{D}}\,\|\,h_{t}^{-1}\circ
f_{s,t}^{-1})(z)$
$\displaystyle=(\mathbf{E}\,\mathcal{L}(v,\mathbb{D})\,\Lambda^{\mathrm{eff}}_{\mathbb{D}}\,\|\,h_{t}^{-1})(z),$

where $f_{s,t}=h_{t+s}\circ h_{t}^{-1}$ and

$(v\,\|\,\mathrm{id}_{\mathbb{D}})=\frac{\mathrm{d}}{\mathrm{d}s}\Big{|}_{s=0}f_{s,t}=\dot{h}_{t}\circ
h_{t}^{-1}.$

We only need to compute the vector field $v.$ We represent $v$ as the
difference of two Loewner vector fields associated with the flows in the
domains $\mathbb{D}$ and $\Omega_{t}.$ Applying the chain rule to
$h_{t}=\widetilde{g}_{t}\circ\Psi_{K}\circ g_{t}^{-1}$ and computing the
capacity changes, we have

$\dot{h}_{t}(z)=|h_{t}^{\prime}(\zeta_{t})|^{2}v_{\widetilde{\zeta}_{t}}(h_{t}(z))-h_{t}^{\prime}(z)v_{\zeta_{t}}(z),\quad\Big{(}v_{\zeta}(z)=z\frac{\zeta+z}{\zeta-z}\Big{)},$

where $\widetilde{\zeta}_{t}=h_{t}(\zeta_{t}).$ By the above equation and
$(v\,\|\,\mathrm{id}_{\mathbb{D}})=\dot{h}_{t}\circ h_{t}^{-1},$

(8.20)
$(v\,\|\,\mathrm{id}_{\mathbb{D}})(z)=|h_{t}^{\prime}(\zeta_{t})|^{2}v_{\widetilde{\zeta}_{t}}(z)-h_{t}^{\prime}(h_{t}^{-1}(z))v_{\zeta_{t}}(h_{t}^{-1}(z)).$

It follows from (8.19) and (8.20) that

$\displaystyle\dot{G}(z,\boldsymbol{q}(t),t)$
$\displaystyle=|h_{t}^{\prime}(\zeta_{t})|^{2}h_{t}^{\prime}(z)^{\lambda}\prod\big{(}h_{t}^{\prime}(q_{k}(t))\big{)}^{\lambda_{k}}\,\mathbf{E}\,\big{(}\mathcal{L}(v_{\widetilde{\zeta}_{t}},\mathbb{D})\,\Lambda^{\mathrm{eff}}_{\mathbb{D}}(h_{t}(z)\big{)}\,\|\,\mathrm{id}_{\mathbb{D}})$
$\displaystyle-\big{(}\mathbf{E}\,\mathcal{L}(v_{\zeta_{t}})\,\Lambda^{\mathrm{eff}}_{\Omega_{t}}\,\|\,\mathrm{id}_{\Omega_{t}}\big{)}(z).$

By Ward’s equation,

$\displaystyle\dot{G}(z,\boldsymbol{q}(t),t)$
$\displaystyle=2|h_{t}^{\prime}(\zeta_{t})|^{2}\prod\big{(}h_{t}^{\prime}(q_{k}(t))\big{)}^{\lambda_{k}}h_{t}^{\prime}(z)^{\lambda}\widetilde{\zeta}_{t}^{2}(\mathbf{E}\,T_{\mathbb{D}}(\widetilde{\zeta}_{t})\,\Lambda^{\mathrm{eff}}_{\mathbb{D}}(h_{t}(z))\,\|\,\mathrm{id}_{\mathbb{D}})$
$\displaystyle-(\mathbf{E}\,\mathcal{L}(v_{\zeta_{t}})\,\Lambda^{\mathrm{eff}}_{\Omega_{t}}\,\|\,\mathrm{id}_{\Omega_{t}}\big{)}(z).$

It follows from conformal invariance that

$\displaystyle\dot{G}(z,\boldsymbol{q}(t),t)$
$\displaystyle=2\zeta_{t}^{2}(\mathbf{E}\,T_{\Omega_{t}}(\zeta_{t})\,\Lambda^{\mathrm{eff}}_{\Omega_{t}}(z)\,\|\,\mathrm{id}_{\Omega_{t}})-\frac{c}{6}\zeta_{t}^{2}S_{h_{t}}(\zeta_{t})\,(\mathbf{E}\,\Lambda^{\mathrm{eff}}_{\Omega_{t}}\,\|\,\mathrm{id}_{\Omega_{t}}\big{)}(z)$
$\displaystyle-(\mathbf{E}\,\mathcal{L}(v_{\zeta_{t}})\,\Lambda^{\mathrm{eff}}_{\Omega_{t}}\,\|\,\mathrm{id}_{\Omega_{t}}\big{)}(z).$

Sending $z$ to $\zeta_{t}$ and applying (7.15) (and
$T*_{-1}\Lambda^{\mathrm{eff}}=\partial\Lambda^{\mathrm{eff}},T*_{-2}\Lambda^{\mathrm{eff}}=h\Lambda^{\mathrm{eff}}$,
see (7.16)),

$\frac{\dot{G}(\zeta_{t},\boldsymbol{q}(t),t)}{G(\zeta_{t},\boldsymbol{q}(t),t)}=-\frac{c}{6}\zeta_{t}^{2}S_{h_{t}}(\zeta_{t})+\frac{L(\zeta_{t})G(\zeta_{t},\boldsymbol{q}(t),t)}{G(\zeta_{t},\boldsymbol{q}(t),t)}-\frac{\check{\mathcal{L}}_{v_{\zeta_{t}}}G(\zeta_{t},\boldsymbol{q}(t),t)}{G(\zeta_{t},\boldsymbol{q}(t),t)},$

where $L(z)=2z^{2}L_{-2}(z)+3zL_{-1}(z)+L_{0}(z).$ By (8.18) and the level two
degeneracy equation, the drift of $\mathrm{d}M_{t}/M_{t}$ simplifies to

$\displaystyle-\frac{c}{6}\zeta_{t}^{2}S_{h_{t}}(\zeta_{t})$
$\displaystyle-\frac{\check{\mathcal{L}}_{v_{\zeta_{t}}}G(\zeta_{t},\boldsymbol{q}(t),t)}{G(\zeta_{t},\boldsymbol{q}(t),t)}+\Big{(}\frac{\kappa}{2}\zeta_{t}^{2}\,\frac{H_{\zeta\zeta}}{H}+3\zeta_{t}\frac{H^{\prime}}{H}+h\Big{)}\Big{|}_{(\zeta_{t},\boldsymbol{q}(t),t)}$
$\displaystyle+\sum_{k}\Big{(}q_{k}(t)\frac{\zeta_{t}+q_{k}(t)}{\zeta_{t}-q_{k}(t)}\Big{)}\Big{(}\frac{G_{q_{k}}}{G}-\frac{H_{q_{k}}}{H}\Big{)}\Big{|}_{(\zeta_{t},\boldsymbol{q}(t),t)}.$

It follows from the null vector equation that

$\frac{\kappa}{2}\zeta_{t}^{2}\,\frac{H_{\zeta\zeta}}{H}+3\zeta_{t}\frac{H^{\prime}}{H}+h=\frac{\check{\mathcal{L}}_{v_{\zeta_{t}}}H}{H}.$

Thus we find the drift term of $\mathrm{d}M_{t}/M_{t}$ as

$-\zeta_{t}^{2}\frac{c}{6}S_{h_{t}}(\zeta_{t})-\Big{(}\frac{\check{\mathcal{L}}_{v_{\zeta_{t}}}G}{G}-\frac{\check{\mathcal{L}}_{v_{\zeta_{t}}}H}{H}\Big{)}+\sum_{k}\Big{(}q_{k}(t)\frac{\zeta_{t}+q_{k}(t)}{\zeta_{t}-q_{k}(t)}\Big{)}\Big{(}\frac{G_{q_{k}}}{G}-\frac{H_{q_{k}}}{H}\Big{)}$

evaluated at $(\xi_{t},\boldsymbol{q}(t),t).$ Since

$\frac{\check{\mathcal{L}}_{v_{\zeta_{t}}}G}{G}-\frac{\check{\mathcal{L}}_{v_{\zeta_{t}}}H}{H}=\sum_{k}\Big{(}q_{k}(t)\frac{\zeta_{t}+q_{k}(t)}{\zeta_{t}-q_{k}(t)}\Big{)}\Big{(}\frac{G_{q_{k}}}{G}-\frac{H_{q_{k}}}{H}\Big{)},$

we have (8.17). ∎

##  9\. Conformal field theory with Neumann boundary condition

In this section we briefly implement a version of conformal field theory from
background charge modifications of the Gaussian field with Neumann boundary
condition. In the last subsection we present the connection of this theory to
the theory of the backward SLEs.

###  9.1. Gaussian free field with Neumann boundary condition

The Gaussian free field $N$ in a planar domain $D$ with Neumann boundary
condition is an isometry $N:\mathcal{E}_{N}(D)\to L^{2}(\Omega,\mathbf{P})$
from the Neumann energy space $\mathcal{E}_{N}(D)$ such that the image
consists of centered Gaussian random variables. Here $(\Omega,\mathbf{P})$ is
a probability space and $\mathcal{E}_{N}(D)$ is the completion of smooth
functions up to the boundary with mean zero (or the neutrality condition
$(\mathrm{NC}_{0})$), compact supports in $D\,\cup\,\partial D,$ and Neumann
boundary condition $\nabla f\cdot n=0$ (where $n$ is normal to $\partial D$)
with respect to the norm

$\|f\|^{2}_{\mathcal{E}_{N}}=\iint
2G_{N}(\zeta,z)\,f(\zeta)\,\overline{f(z)}~{}\mathrm{d}A(\zeta)\,\mathrm{d}A(z),$

where $A$ is the normalized area measure and $G_{N}$ is the Neumann Green’s
function for $D.$ In the upper half-plane, we have

$G_{N}(\zeta,z)=\log\frac{1}{|(\zeta-z)(\zeta-\bar{z})|}.$

As in the Dirichlet case, $N$ can be constructed from the Gaussian free field
$\Psi$ on its Schottky double $S=D^{\mathrm{double}}.$ The Neumann energy
space $\mathcal{E}_{N}(D)$ can be embedded isometrically into $\mathcal{E}(S)$
in a natural way. For example, for $\mu$ in the energy space