††institutetext: Department of Physics, The University of Tokyo
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan††institutetext: RIKEN
Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)
Wako, Saitama 351-0198, Japan

# Genus-$g$, $n$-point, $b$-boundary, $c$-crosscap correlation functions of
two-dimensional conformal field theory: Definition and general properties.

Xun Liu<EMAIL_ADDRESS>
###### Abstract

We propose a systematic definition of genus-$g$, $n$-point, $b$-boundary,
$c$-crosscap correlation functions $\mathcal{F}_{g,n,b,c}^{x}$ with $x$
boundary operators, for general two-dimensional conformal field theory. The
$\mathcal{F}_{g,n,b,c}^{x}$ are defined as the inner products of the surface
states with the tensor product of $n$ asymptotic states, $b$ boundary states
and $c$ crosscap states, where the $b$ defining boundary states carry a total
of $x$ boundary operators. The definition implies that
$\mathcal{F}_{g,n,b,c}^{x}$ are infinite linear combinations of genus-$g$,
$(n+b+c)$-point functions $\mathcal{F}_{g,(n+b+c)}$. A single pole structure
is identified in the expansion coefficients $\mathcal{F}_{g,n,b,c}^{0}$, which
is analogous to the single pole structure in the conformal block
decomposition. This linear property unifies boundary and bulk CFTs. The
consistency conditions of $\mathcal{F}_{g,n,b,c}^{x}$ are discussed:
conventional upper-half plane (UHP) and crosscap constraints can be
reformulated as extra constraints on a collection of bulk correlation
functions.

##  1 Introduction and Summary

The core of two-dimensional conformal field theory research is the
investigation of the conformal symmetries and CFT correlation functions (CFT
always refers to two-dimensional CFT in this work). Several techniques have
been developed to study the form and properties of CFT correlation functions
defined on various types of surfaces Fuchs:2009iz ; DiFrancesco:1997nk .
Despite the intensive studies of genus-$g$, $n$-point bulk correlation
functions $\mathcal{F}_{g,n}$, correlation functions defined on surfaces with
boundaries or crosscaps have received lesser attention. In this work, we
propose a systematic definition of general genus-$g$, $n$-point, $b$-boundary,
$c$-crosscap correlation functions $\mathcal{F}_{g,n,b,c}^{x}$ with $x$
boundary operators using the surface states Alvarez-Gaume:1987eux ;
Sen:2014pia , and investigate some of the basic properties of
$\mathcal{F}_{g,n,b,c}^{x}$. The primary motivation of this work is to define
$\mathcal{F}_{g,n,b,c}^{x}$, with some extended motivations are listed here:

  * •

The unification of bulk and boundary CFTs. Many essential elements in boundary
CFT (BCFT) are secondary elements induced from the concepts in bulk CFT. For
example, the boundary operators are defined by bulk-boundary OPEs Cardy:1991tv
or boundary condition changing Cardy:1989ir , and the set of symmetry
preserving conformal generators on the boundary determines the open-sector
chiral algebra $\mathcal{A}^{op}\subset\mathcal{A}$ Cardy:1984bb ;
Recknagel:1997sb . Based on these facts, we ask the following question: Can we
unify BCFT with bulk CFT? If any BCFT correlation function, with arbitrary
numbers of boundaries and boundary operators, can be reformulated as a linear
combination of bulk correlation functions, then we state that BCFTs can be
unified with bulk CFTs. In this work, we will answer this question using the
definition of $\mathcal{F}_{g,n,b,c}^{x}$.

  * •

The application of $\mathcal{F}_{g,n,b,c}^{x}$ to bosonic string theory. With
the definition of $\mathcal{F}_{g,n,b,c}^{x}$, we can consider the computation
of corresponding perturbative bosonic string amplitudes
$\mathcal{A}_{g,n,b,c}^{x}$, both on- and off-shell. To compute the off-shell
amplitudes, we need to define an unoriented open-closed bosonic string field
theory (SFT), where the Batalin-Vilkovisky (BV) structure is manifestly
satisfied as conventional oriented closed and open-closed SFT Zwiebach:1992ie
; Zwiebach:1997fe . The application to string amplitude computations is left
for future studies.

We summarize the main results of this paper and label their positions here:

  * •

The definition of $\mathcal{F}_{g,n,b,c}^{x}{}$ and their linear property
(Subsection 2.1): The path integral over a genus-$g$, $(n+b+c)$-holes Riemann
surface with general boundary conditions give a state
$|\Sigma_{g,(n+b+c)}\rangle\in\mathcal{H}^{\otimes(n+b+c)}$ Alvarez-
Gaume:1987eux . The genus-$g$, $n$-point, $b$-boundary, $c$-crosscap functions
with $x$ boundary operators are defined by

$\mathcal{F}_{g,n,b,c}^{x}:=\langle\Sigma_{g,(n+b+c)}|\otimes_{i=1}^{n}|\phi_{\alpha
i}\rangle\otimes_{j=1}^{b}|(b_{\\{j\\}})_{\cdots}\rangle_{\\{\omega_{b_{j}}\\}}^{r_{b_{j}}}\otimes_{k=1}^{c}|c_{k}\rangle_{\omega_{c_{k}}}^{r_{c_{k}}}\in\mathbb{C},$
(1)

where $\langle\Sigma_{g,(n+b+c)}|$ is the dual element of
$|\Sigma_{g,(n+b+c)}\rangle$, $|\phi_{\alpha i}\rangle$ are asymptotic states
corresponding to bulk vertex operators,
$|(b_{\\{j\\}})_{\cdots}\rangle_{\\{\omega_{b_{j}}\\}}^{r_{b_{j}}}$ are
boundary states with boundary operators ($\cdots$ denotes boundary operators,
$\\{j\\}$ include all boundary conditions on the $j^{\text{th}}$ boundary),
and $|c_{k}\rangle_{\omega_{c_{k}}}^{r_{c_{k}}}$ are crosscap states. By
expanding the defining boundary and crosscap states in (1) as linear
combinations of vectors in the CFT standard basis, we link
$\mathcal{F}_{g,n,b,c}^{x}$ to infinite linear combinations of the genus-$g$,
$(n+b+c)$-point functions $\mathcal{F}_{g,(n+b+c)}$.

  * •

Unifying the consistency conditions (Subsection 2.2): We discuss the
consistency conditions of $\mathcal{F}_{g,n,b,c}^{x}$. Conventional studies
have proved that the consistency conditions for CFT correlation functions
include

    * –

Bulk OPE associativity and the modular invariance of $T^{2}$ partition
functions Belavin:1984vu ; Cardy:1986ie ; Sonoda:1988fq .

    * –

The UHP sewing constraints Lewellen:1991tb ; Runkel:1998he .

    * –

The crosscap sewing constraints Fioravanti:1993hf ; Stanev:2001na .

We show how to impose the UHP and crosscap sewing constraints to corresponding
$\mathcal{F}_{g,n,b,c}^{x}$. Using the linear property of
$\mathcal{F}_{g,n,b,c}^{x}$, the UHP and crosscap sewing constraints become an
extra set of constraint conditions imposed on a collection of bulk correlation
functions.

  * •

Reformulating BCFT correlation functions as bulk correlation functions
(Subsection 2.3): Consider a correlation function $\mathcal{F}_{g,n,b,c}^{x}$
where $k\leq b$ of the defining boundary states carrying boundary operators.
Using the boundary-boundary OPE and the inversion of bulk-boundary OPE, we can
write $\mathcal{F}_{g,n,b,c}^{x}$ as a series of $\mathcal{F}_{g,n+k,b,c}^{0}$
with all defining boundary states having no boundary operator. Expanding
boundary states with no boundary operator and crosscap states can be achieved
in practice. Hence, we can write any $\mathcal{F}_{g,n,b,c}^{x}$ as linear
combinations of bulk correlation functions $\mathcal{F}_{g,(n+k+b+c)}$ in
practice. We show this procedure agrees with the expectation that when the CFT
data is known, all CFT correlation functions are known.

  * •

The single pole structure in Ishibashi state expansions (Subsection 3.2 \-
3.3): The core of boundary (with no boundary operator) and crosscap state
expansions is the expansion of Ishibashi states Ishibashi:1988kg ;
Onogi:1988qk . We prove the existence of a single pole structure in the
expansion coefficients of Ishibashi states corresponding to degenerate Verma
modules. This is a direct analog to the single pole structure in conformal
block decomposition Zamolodchikov:1984eqp ; Hadasz:2006qb ; Hadasz:2008dt ;
Hadasz:2009db ; Hadasz:2012im ; Cho:2017oxl . We demonstrate some
$\mathcal{A}$-Ishibashi state expansions at the lowest levels to check the
single pole structure.

Besides the main results, we discuss the closed- and open-sector plumbing
fixtures of $\mathcal{F}_{g,n,b,c}^{x}$ in Subsection 2.4 and define the
conformal blocks of $\mathcal{F}_{g,n,b,c}^{x}$ in Subsection 2.5. In
Subsection 2.6, we compare the non-vanishing conditions of
$\mathcal{F}_{0,n,b,0}^{0}$ with corresponding UHP functions
$\langle\phi_{1}\cdots\phi_{n}\psi_{1}\cdots\psi_{b}\rangle_{UHP}$ as a
consistency check for the definition of $\mathcal{F}_{g,n,b,c}^{x}$. The two
sets of non-vanishing conditions should be equivalent, that is they are
necessary and sufficient conditions to each other. We will prove the
equivalence in RCFTs for $b=1$ cases and partially for $b=2$ bases. In
Subsection 3.3, we find out that when the chiral algebra has simple
commutation relations, the Ishibashi state expansion is known up to
level-$\infty$. Examples include the $c_{Vir}\to\infty$ semi-classical limit
and free CFT models such as free boson and fermion CFTs Callan:1987px . Hence,
in Subsection 3.4, we are motivated to express the Virasoro Ishibashi states
in minimal models, whose expansions are complicated, as linear combinations of
free boson or free fermion Ishibashi states.

##  2 The definition of genus-$g$, $n$-point, $b$-boundary, $c$-crosscap
correlation functions $\mathcal{F}_{g,n,b,c}^{x}$ with $x$ boundary operators

In section 2, we define the genus-$g$, $n$-point, $b$-boundary, $c$-crosscap
correlation functions $\mathcal{F}_{g,n,b,c}^{x}$ with $x$ boundary operators
and discuss their basic properties. The discussions are not restricted to
rational CFTs (RCFTs), except for Subsection 2.6. Throughout this work, we
will use $\xi$ to denote descendant vectors in Verma modules and use $v$ to
denote primary vectors.

###  2.1 The definition of $\mathcal{F}_{g,n,b,c}^{x}$

Consider a CFT defined on a compact genus-$g$, $(n+b+c)$-boundary Riemann
surface $\Sigma_{g,0,(n+b+c),0}$ (a genus-$g$, $n$-punctured, $b$-boundary,
$c$-crosscap Riemann surface is denoted by $\Sigma_{g,n,b,c}$,
$g,n,b,c\in\mathbb{N}$), whose fields are denoted collectively by $\phi$. The
complex local coordinates at each hole are denoted by $t_{i}$, where the
centers of the holes are set to be the origin $t_{i}=0$, and the boundaries
are set to be the local unit circle $|t_{i}|=1$, $i=1,\cdots,(n+b+c)$. The
moduli space $\mathcal{M}_{g,0,(n+b+c),0}$ of $\Sigma_{g,0,(n+b+c),0}$ has
real dimension $[6g-6+3(n+b+c)]$, with $(n+b+c)$ real moduli are fixed to be
$1$ by the radii of the holes. The path integral over the Riemann surface
$\Sigma_{g,0,(n+b+c),0}$ will give a functional $\Phi$ depending on the
boundary conditions $\phi_{i}$ at all the $(n+b+c)$ boundaries. Each boundary
condition $\phi_{i}$ is an element in the CFT Hilbert space $\mathcal{H}$.
Hence, the functional $\Phi$ is an element in the $(n+b+c)$-fold tensor
product of the closed-sector CFT Hilbert space $\mathcal{H}$ Alvarez-
Gaume:1987eux ; Sen:2014pia , which is denoted by

$|\Sigma_{g,(n+b+c)}\rangle:=\Phi\Big{[}\phi_{1},\cdots,\phi_{n+b+c}\Big{]}\in\mathcal{H}^{\otimes(n+b+c)}.$
(2)

The dual element of $|\Sigma_{g,(n+b+c)}\rangle$ is called the surface state
of $\Sigma_{g,(n+b+c)}$, which is an element in the $(n+b+c)$-fold tensor
product of the dual CFT Hilbert space $\mathcal{H}^{\ast}$

$\langle\Sigma_{g,(n+b+c)}|\in(\mathcal{H}^{\ast})^{\otimes(n+b+c)}.$ (3)

The surface states consist of the geometric information of the
$[6g-6+2(n+b+c)]$ unfixed real moduli of the corresponding
$\Sigma_{g,0,(n+b+c),0}$. Every surface state $\langle\Sigma_{g,(n+b+c)}|$
corresponds to a point in the moduli space of genus-$g$, $(n+b+c)$-punctured
surface $\mathcal{M}_{g,(n+b+c)}$ and those points fully cover the moduli
space $\mathcal{M}_{g,(n+b+c)}$.

We define a genus-$g$, $n$-point, $b$-boundary, $c$-crosscap correlation
function $\mathcal{F}_{g,n,b,c}^{x}\in\mathbb{C}$ with $x$ boundary operators
as the inner product of the surface state $\langle\Sigma_{g,(n+b+c)}|$, with
the tensor product of $n$ asymptotic states $|\phi_{\alpha}\rangle$, $b$
propagated boundary states
$|(b_{j})_{\cdots}\rangle_{\\{\omega_{b_{j}}\\}}^{r_{b_{j}}}$ (carrying total
$x$ boundary operators and $\\{\omega_{b_{j}}\\}$ is the set of gluing
automorphisms of the boundary conditions on the $j^{\text{th}}$ boundary), and
$c$ propagated crosscap states $|c\rangle^{r}_{\omega}$

$\mathcal{F}_{g,n,b,c}^{{q}}[\otimes_{i}|\phi_{\alpha_{i}}\rangle,\otimes_{j}|(b_{j})_{\cdots}\rangle_{\\{\omega_{b_{j}}\\}}^{r_{b_{j}}},\otimes_{k}|c_{k}\rangle_{\omega_{c_{k}}}^{r_{c_{k}}}]:=\langle\Sigma_{g,(n+b+c)}|\otimes_{i=1}^{n}|\phi_{\alpha_{i}}\rangle\otimes_{j=1}^{b}|(b_{\\{j\\}})_{\cdots}\rangle_{\\{\omega_{b_{j}}\\}}^{r_{b_{j}}}\otimes_{k=1}^{c}|c_{k}\rangle^{r_{c_{k}}}_{\omega_{c_{k}}}\in\mathbb{C}.$
(4)

The asymptotic states $|\phi_{\alpha_{i}}\rangle$ correspond to bulk vertex
operators by the state-operator correspondence. They span the closed-sector
CFT Hilbert space $\mathcal{H}$ and will be called the standard basis of
$\mathcal{H}$. The detailed vectors in the standard basis depend on the chiral
algebra $\mathcal{A}$. The boundary and crosscap states in the definition (4)
are the propagated boundary and crosscap states $|b,c\rangle^{r}$

$|b,c\rangle^{r}=e^{(L_{0}+\bar{L}_{0}-\frac{c}{12})\ln{(r)}}|b,c\rangle\in\mathcal{H},\quad
r\in(0,1).$ (5)

Geometrically, taking the inner product (4) partially fill the $(n+b+c)$ holes
on the original surface $\Sigma_{g,0,(n+b+c),0}$ by $n$ local disks with a
single bulk puncture at the center $t_{i}=0$, $b$ local annulus with their
inner boundaries at $|t_{j}|=r_{b_{j}}$ (with total $x$ marked points on all
the $b$ inner boundaries), and $c$ local crosscaps in region $|t_{k}|\leq
r_{c_{k}}$. The resulting surface $\Sigma_{g,n,b,c}^{x}$ is a genus-$g$,
$n$-punctured, $b$-boundary, $c$-crosscaps Riemann surface with $x$ boundary
marked points. The correlation functions $\mathcal{F}_{g,n,b,c}^{x}$ depend on
$(6g-6+2n+3b+3c+q)$ real moduli, where $(6g-6+2n+2b+2c)$ are from the surface
state, $(b+c)$ are from the radii of boundaries and crosscaps, and $x$ are
from the positions of the boundary marked points.

The surfaces obtained from the above filling procedure live in the following
region of the moduli space $\mathcal{M}_{g,n,b,c}^{x}$

$\mathcal{M}_{g,(n+b+c)}\times(0,1)^{(b+c)}\times[0,2\pi)^{x}\subset\mathcal{M}_{g,n,b,c}^{x},$
(6)

where the boundary operators on the same boundary cannot have the same phase.
We are uncertain about whether this region can fully cover the moduli space
$\mathcal{M}_{g,n,b,c}^{x}$. Nevertheless, we assume the correlation functions
$\mathcal{F}_{g,n,b,c}^{x}$ defined are valid to the whole moduli space
$\mathcal{M}_{g,n,b,c}^{x}$.

##### The correlation function validness assumption:

The expression of CFT correlation functions $\mathcal{F}_{g,n,b,c}^{x}$,
defined on the surface at any point in the moduli space
$\mathcal{M}_{g,n,b,c}^{x}$, is valid over the whole moduli space
$\mathcal{M}_{g,n,b,c}^{x}$. The reason for making this assumption is the
following: the correlation functions $\mathcal{F}_{g,n,b,c}^{x}$ are solutions
to Ward identities of the conformal algebra $\mathcal{A}\otimes\mathcal{A}$.
The same correlation function $\mathcal{F}_{g,n,b,c}^{x}$ defined on surfaces
at different points of the moduli space $\mathcal{M}_{g,n,b,c}^{x}$ should
satisfy the same set of conformal Ward identities, which indicates that the
same $\mathcal{F}_{g,n,b,c}^{x}$ on surfaces at different points of the moduli
space should have the same expression.

##### The linear property:

The fact that boundary and crosscap states are both elements in the CFT
Hilbert space $\mathcal{H}$ (5) implies that correlation functions
$\mathcal{F}_{g,n,b,c}^{x}$ are infinite linear combinations of genus-$g$,
$(n+b+c)$-point functions $\mathcal{F}_{g,(n+b+c)}$. To show this explicitly,
we introduce the dual basis of the standard basis
$\\{\langle(\phi_{\beta})^{c}|\\}$ and define the Gram matrix as

$(G)_{\alpha\beta}:=\langle(\phi_{\beta})^{c}|\phi_{\alpha}\rangle.$ (7)

We can use the complete basis of $\mathcal{H}$ to express the identity
operator $I\otimes\bar{I}$ as

$I\otimes\bar{I}=\int_{\alpha,\beta}|\phi_{\alpha}\rangle\times(G)^{\alpha\beta}\times\langle(\phi_{\beta})^{c}|,$
(8)

where the integral indicates that $\mathcal{H}$ may be continuum. Formally,
the boundary or crosscap state expansions are realized by left acting the
$I\otimes\bar{I}$ in (8) on the corresponding states $|b,c\rangle^{r}$

$|b,c\rangle^{r}=\int_{\alpha,\beta}\>|\phi_{\alpha}\rangle\times(G)^{\alpha\beta}\times\langle(\phi_{\beta})^{c}|b,c\rangle^{r}.$
(9)

Applying the expansion (9) to the Ward identities of conformal algebra
$\mathcal{A}\otimes\mathcal{A}$, we can effectively write the contour integral
around a boundary or a crosscap as a collection of contour integrals around a
puncture. As an example, we write down a Ward identity of a
$\mathcal{F}_{g,n,b,c}^{x}$, where the contour integral is around boundary
$B_{i}$, having $x^{\prime}$ boundary operators

$\oint_{B_{i}}\frac{dt_{i}}{2\pi
i}f(t_{i})\Big{\langle}W^{a}(t_{i})\mathcal{F}_{g,n,b,c}^{x}(|(b)_{\cdots}\rangle_{\\{\omega\\}}^{r},\cdots)\Big{\rangle}=$
$\oint_{p_{i}}\frac{dt_{i}}{2\pi
i}\int_{\alpha,\beta}f(t_{i})\Big{\langle}W^{a}(t_{i})\mathcal{F}_{g,n+1,b-1,c}^{x-x^{\prime}}(|\phi_{\alpha}\rangle,\cdots)\Big{\rangle}\times\langle\phi_{\beta}^{c}|(b)_{\cdots}\rangle_{\\{\omega\\}}^{r}\times(G)^{\alpha\beta},$
(10)

where $p_{i}$ is the puncture positioned at the center of boundary $B_{i}$,
that is $t_{i}=0$. $W^{a}(t_{i})$ is a conformal generator of the chiral
algebra $\mathcal{A}$. $f(t_{i})$ is some meromorphic function defined on the
whole local disk $|t_{i}|\leq 1$, whose poles are only allowed to exist only
at the puncture $p_{i}$. The graphic representation of equation (10) is shown
in Figure 1.

$|(b)_{\cdots}\rangle_{\\{\omega\\}}^{r}$$=$$\int_{\alpha,\beta}$$\phi_{\alpha}$$\times$$(\phi_{\beta})^{c}$$|(b)_{\cdots}\rangle_{\\{\omega\\}}^{r}$$\times$$(G)^{\alpha\beta}$
Figure 1: The graphic representation of equation (10): the effective
expression of the contour integral around a boundary $B_{i}$, whose boundary
conditions and boundary operators are described by
$|(b)_{\cdots}\rangle_{\\{\omega\\}}^{r}$, as a collection of contour
integrals around puncture $p_{i}$ with all vertex operators
$\phi_{\alpha}\in\mathcal{H}$ inserted.

To compute a correlation function $\mathcal{F}_{g,n,b,c}^{x}$ explicitly, we
need to first remove all the boundary operators and then we will be able to
consider the expansions in practice by studying Ishibashi state expansions.
The Ishibashi state expansion coefficients will be studied intensively in
Section 3. The form of the propagated boundary and crosscap states (5)
indicates that when one of the radii is small $r\to 0$, the corresponding
boundary or crosscap state expansion is a perturbative expansion.

###  2.2 The consistency conditions of $\mathcal{F}_{g,n,b,c}^{x}$

We discuss the consistency conditions for $\mathcal{F}_{g,n,b,c}^{x}$ in this
subsection. Conventional studies have proved the complete set of consistency
conditions of CFT correlation functions include:

  * •

The bulk OPE associativity and the modular invariance of $T^{2}$ partition
function Belavin:1984vu ; Cardy:1986ie ; Sonoda:1988fq .

  * •

The UHP sewing constraints Lewellen:1991tb ; Runkel:1998he  and the
constraints on the linear coefficients of boundary states (the Cardy
condition) Cardy:1989ir .

  * •

The crosscap sewing constraints Fioravanti:1993hf  and the constraints on the
linear coefficients of crosscap states Pradisi:1995qy ; Pradisi:1995pp .

Imposing the bulk OPE associativity and $T^{2}$ partition function modular
invariance to $\mathcal{F}_{g,n,b,c}^{x}$ is straightforward, which will not
be described. If we use the linear property of $\mathcal{F}_{g,n,b,c}^{x}$,
the bulk OPE associativity extends to all the bulk operators from the boundary
or crosscap state expansions. As a simple example, we show the bulk OPE
associativity of a $\mathcal{F}_{0,0,4,0}^{0}$ in Figure 2. This indicates
that $\mathcal{F}_{g,n,b,c}^{x}$ will satisfy similar duality identities to
that of bulk correlation functions $\mathcal{F}_{g,(n+b+c)}$ Moore:1988qv .
For example, we draw a pentagon identity for sphere five-boundary functions
$\mathcal{F}_{0,0,5,0}^{0}$ in Figure 3.

$\int$$|b_{1}\rangle^{r_{1}}_{\omega_{1}}$$|b_{2}\rangle^{r_{2}}_{\omega_{2}}$$|b_{3}\rangle^{r_{3}}_{\omega_{3}}$$|b_{4}\rangle^{r_{4}}_{\omega_{4}}$$=$$\int$$\int
c_{\alpha\beta 1}|\phi_{\alpha_{1}}\rangle$$\int c_{\alpha\beta
2}|\phi_{\alpha 2}\rangle$$\int c_{\alpha\beta 3}|\phi_{\alpha 3}\rangle$$\int
c_{\alpha\beta 4}|\phi_{\alpha 4}\rangle$$=$$\int$$\int c_{\alpha\beta
1}|\phi_{\alpha 1}\rangle$$\int c_{\alpha\beta 2}|\phi_{\alpha 2}\rangle$$\int
c_{\alpha\beta 3}|\phi_{\alpha 3}\rangle$$\int c_{\alpha\beta 4}|\phi_{\alpha
4}\rangle$$=$$\int$$|b_{1}\rangle_{\omega_{1}}^{r_{1}}$$|b_{2}\rangle_{\omega_{2}}^{r_{2}}$$|b_{3}\rangle_{\omega_{3}}^{r_{3}}$$|b_{4}\rangle_{\omega_{4}}^{r_{4}}$
Figure 2: The bulk OPE associativity of a $\mathcal{F}_{0,0,4,0}^{0}$, which
is the collection of the bulk OPE associativity of all $\mathcal{F}_{0,4}$ in
the expansion. $c_{\alpha\beta
i}=\langle(\phi_{\beta})^{c}|b_{i}\rangle^{r_{i}}_{\omega_{i}}\times(G)^{\alpha\beta}$.
$|b_{1}\rangle$$|b_{2}\rangle$$|b_{3}\rangle$$F$$|b_{1}\rangle$$|b_{2}\rangle$$|b_{3}\rangle$$F$$|b_{1}\rangle$$|b_{2}\rangle$$|b_{3}\rangle$$|b_{1}\rangle$$|b_{2}\rangle$$|b_{3}\rangle$$F$$F$$|b_{1}\rangle$$|b_{2}\rangle$$|b_{3}\rangle$$F$$|b_{1}\rangle$$|b_{2}\rangle$$|b_{3}\rangle$
Figure 3: The pentagon identity for sphere five-boundary functions, where the
two endpoints without labeling also correspond to boundary states.

We show how to impose the UHP and crosscap sewing constraints to
$\mathcal{F}_{g,n,b,c}^{x}$.

##### Imposing the UHP sewing constraints:

The UHP sewing constraints include the OPE associativity of the bulk and
boundary operators. Hence, we quickly review the two distinct methods of
introducing boundary operators in conventional studies:

  * •

When a bulk operator $\phi_{(i,\bar{i})}$ approaches a boundary with boundary
condition $b_{1}$, we can introduce the bulk-boundary OPE Cardy:1991tv

$\phi_{(i,\bar{i})}(z,\bar{z})=\int_{k}C_{(i\bar{i})k}^{b_{1}}|z-\bar{z}|^{h_{k}-h_{i}-\bar{h}_{\bar{i}}}\psi_{k}^{b_{1}b_{1}}(x),\quad
x=\frac{z+\bar{z}}{2},\quad\bar{i}=\omega^{-1}_{1}(i^{\ast}),$ (11)

where the integral indicates that the open-sector Hilbert space
$\mathcal{H}_{b_{1}b_{1}}$ can also be continuum.

  * •

Consider a finite cylinder with boundary conditions $b_{1}$ and $b_{2}$ on the
two ends Cardy:1989ir . Assuming the two boundary conditions have the same
open-sector chiral algebra $\mathcal{A}^{op}$, the open-sector Hilbert space
$\mathcal{H}_{b_{1}b_{2}}$ can be decomposed into a direct integral over the
chiral irreducible representations of $\mathcal{A}^{op}$

$\mathcal{H}_{b_{1}b_{2}}=\int_{k}^{\oplus}\>(\mathcal{H}^{k}_{b_{1}b_{2}})^{\oplus(n_{b_{1}b_{2}}^{k}\times
N_{1}\times N_{2})},\quad n_{b_{1}b_{2}}^{k}\in\mathbb{N},\quad
N_{1},\>N_{2}\in\mathbb{Z}_{+},$ (12)

where $N_{1}$ and $N_{2}$ are the Chan-Paton factors associated with boundary
conditions $b_{1}$ and $b_{2}$. Cutting the finite cylinder open to a strip
and extending its length to semi-infinite, by an exponential map from the
semi-infinite strip to the UHP, each irreducible representation in
$\mathcal{H}_{b_{1}b_{2}}$ is mapped to a boundary operator at the origin
$\psi_{k}^{b_{1}b_{2}}(0)\cong(\mathcal{H}^{k}_{b_{1}b_{2}})^{\oplus(n_{b_{1}b_{2}}^{k}\times
N_{1}\times N_{2})}$. The boundary operators $\psi_{k}^{b_{1}b_{2}}(0)$
interchange the two boundary conditions $b_{1}$ and $b_{2}$ on the two sides.

We apply these two methods to introduce boundary states with boundary operator
configurations. Starting from a correlation function with no boundary operator
$\mathcal{F}_{g,n,b,c}^{0}$, we can define a series of boundary states with a
single boundary operator by the bulk-boundary OPEs

$\int_{k}\>C_{(i,\bar{i})k}^{b_{1}}|z-\bar{z}|^{h_{k}-h_{i}-\bar{h}_{\bar{i}}}\mathcal{F}_{g,n-1,b,c}^{1}\big{[}|(b_{1})_{\psi_{k}^{b_{1}b_{1}}}\rangle_{\omega_{1}}^{r_{1}}\otimes\cdots\big{]}=\mathcal{F}_{g,n,b,c}^{0}\big{[}|b_{1}\rangle^{r_{1}}_{\omega_{1}}\otimes|\phi_{(i,\bar{i})}\rangle\otimes\cdots\big{]}.$
(13)

We count the number of real moduli contributed from the manifesting part on
both sides (13). On the RHS, the states contribute a total of five real
moduli. On the LHS, $|z-\bar{z}|$ and the boundary state also contribute a
total of five real moduli.

Another method is to combine $k\leq b$ boundaries together into one, which
will define a series of boundary states with $k$ boundary operators

$J\int_{i_{1}}\cdots\int_{i_{k}}\mathcal{F}_{g,n,b-k+1,c}^{k}\big{[}|(b_{1\cdots
k})_{\psi_{i_{1}}^{b_{1}b_{2}}\cdots\psi_{i_{k}}^{b_{k}b_{1}}}\rangle^{r}_{\omega_{1}\cdots\omega_{k}}\otimes\cdots\big{]}=\mathcal{F}_{g,n,b,c}^{0}\big{[}|b_{1}\rangle_{\omega_{1}}^{r_{1}}\otimes\cdots\otimes|b_{k}\rangle_{\omega_{k}}^{r_{k}}\otimes\cdots\big{]},$
(14)

where $J$ is the collection of all Jacobian from the coordinate
transformations needed for the combinations. Each boundary state induced
$|(b_{1\cdots
k})_{\psi_{i_{1}}^{b_{1}b_{2}}\cdots\psi_{i_{k}}^{b_{k}b_{1}}}\rangle^{r}_{\omega_{1}\cdots\omega_{k}}$
contains information about:

  * •

All the $k$ boundary conditions $b_{i}$ described by the $k$ boundary states
on the RHS of (14), including all the $k$ gluing automorphsims $\omega_{i}$.

  * •

$k$ boundary operators $\psi_{i_{a}}^{b_{a}b_{a+1}}$ that interchanging the
boundary conditions in a fixed order.

The $k$ boundary states on the RHS of (14) contribute a total of $(3k)$ real
moduli. $(k+3)$ of them are manifest, where $k$ are the positions of boundary
operators, two are from the center position, and one is the radius of the
boundary obtained. The rest $(2k-3)$ moduli are not manifest. We expect that
they are hidden in the Jacobian $J$ and also in the expansion coefficients
when we expand $\mathcal{F}_{g,n,b-k+1,c}^{k}$ as linear combinations of bulk
correlation functions.

Conventional boundary-boundary OPEs are given by

$\psi_{i}^{b_{1}b_{2}}(x_{1})\psi_{j}^{b_{2}b_{3}}(x_{2})=\int_{k}C_{ijk}^{b_{1}b_{2}b_{3}}|x_{12}|^{h_{k}-h_{i}-h_{j}}\psi_{k}^{b_{1}b_{3}}(x_{2}).$
(15)

Applying the boundary-boundary OPEs to $|(b_{1\cdots
k})_{\psi_{i_{1}}^{b_{1}b_{2}}\cdots\psi_{i_{k}}^{b_{k}b_{1}}}\rangle^{r}_{\omega_{1}\cdots\omega_{k}}$
repeatedly, we can reduce the number of boundary operators on the boundary
state

$\mathcal{F}_{g,n,b-k+1,c}^{k}\big{[}|(b_{1\cdots
k})_{\psi_{i_{1}}^{b_{1}b_{2}}\cdots\psi_{i_{k}}^{b_{k}b_{1}}}\rangle^{r}_{\omega_{1}\cdots\omega_{k}}\otimes\cdots\big{]}=\int_{s}C_{i_{1}i_{2}i_{s}}^{b_{1}b_{2}b_{3}}$
$|x_{12}|^{h_{i_{s}}-h_{i_{1}}-h_{i_{2}}}\times\mathcal{F}_{g,n,b-k+1,c}^{k-1}\big{[}|(b)_{\psi_{i_{s}}^{b_{1}b_{3}}\cdots\psi_{i_{k}}^{b_{k}b_{1}}}\rangle^{r}_{\omega_{1}\cdots\omega_{k}}\otimes\cdots\big{]}=\cdots.$
(16)

Now, we are ready to impose the UHP sewing constraints to
$\mathcal{F}_{g,n,b,c}^{x}$. We only write down the crossing symmetry equation
of UHP bulk two-point functions
$\langle\phi_{(i,\bar{i})}\phi_{(j,\bar{j})}\rangle_{UHP}$ and the crossing
symmetry equation of UHP boundary four-point functions
$\langle\psi_{i_{1}}^{b_{1}b_{2}}\psi_{i_{2}}^{b_{2}b_{3}}\psi_{i_{3}}^{b_{3}b_{4}}\psi_{i_{4}}^{b_{4}b_{1}}\rangle_{UHP}$
as examples. Other UHP sewing constraints need to be imposed similarly
Lewellen:1991tb ; Runkel:1998he .

$=$$\phi_{(i,\bar{i})}$$\phi_{(j,\bar{j})}$$\psi_{a}^{b_{1}b_{1}}$$\psi_{a^{\ast}}^{b_{1}b_{1}}$$\psi_{0}^{b_{1}b_{1}}$$\psi_{0}^{b_{1}b_{1}}$$\phi_{(i,\bar{i})}$$\phi_{(k,\bar{k})}$$\phi_{(j,\bar{j})}$$\int_{a}$$\int_{(k,\bar{k})}$
Figure 4: The crossing symmetry equation of the bulk two-point functions. Only
the boundary identity operator gives a non-vanishing UHP boundary one-point
function because of the translation and scaling invariance along the real
axis.

The crossing symmetry equation of UHP bulk two-point functions is the
equivalence of the OPE at two different limits, which is shown by Figure 4.
The equation expressing the RHS of Figure 4 is

$\mathcal{F}_{0,2,1,0}^{0}\big{[}|\phi_{(i,\bar{i})}\rangle,|\phi_{(j,\bar{j})}\rangle,|b_{1}\rangle_{\omega_{1}}^{r_{1}}\big{]}=\int_{(k,\bar{k})}C_{(i,\bar{i})(j,\bar{j})}^{(k,\bar{k})}(z_{12})^{h_{k}-h_{i}-h_{j}}(\bar{z}_{12})^{\bar{h}_{\bar{k}}-\bar{h}_{\bar{i}}-\bar{h}_{\bar{j}}}\mathcal{F}_{0,1,1,0}^{0}\big{[}|\phi_{(k,\bar{k})}\rangle,|b_{1}\rangle_{\omega_{1}}^{r_{1}}\big{]}$
$=\int_{(k,\bar{k})}C_{(i,\bar{i})(j,\bar{j})}^{(k,\bar{k})}(z_{12})^{h_{k}-h_{i}-h_{j}}(\bar{z}_{12})^{\bar{h}_{\bar{k}}-\bar{h}_{\bar{i}}-\bar{h}_{\bar{j}}}C_{(k,\bar{k})0}^{b_{1}}|z_{2}-\bar{z}_{2}|^{-h_{k}-\bar{h}_{\bar{k}}}\mathcal{F}_{0,0,1,0}^{0}\big{[}|b_{1}\rangle_{\omega_{1}}^{r_{1}}\big{]}.$
(17)

The equation expressing the LHS of Figure 4 is

$\mathcal{F}_{0,2,1,0}^{0}\big{[}|\phi_{i,\bar{i}}\rangle,|\phi_{j,\bar{j}}\rangle,|b_{1}\rangle_{\omega_{1}}^{r_{1}}\big{]}=\int_{a}C_{(i,\bar{i})a}^{b_{1}}|z_{1}-\bar{z}_{1}|^{h_{a}-h_{i}-\bar{h}_{\bar{i}}}C_{(j,\bar{j})a^{\ast}}^{b_{1}}|z_{2}-\bar{z}_{2}|^{h_{a}-h_{j}-\bar{h}_{\bar{j}}}\mathcal{F}_{0,0,1,0}^{2}\big{[}|(b_{1})_{\psi_{a}^{b_{1}b_{1}}\psi_{a^{\ast}}^{b_{1}b_{1}}}\rangle_{\omega_{1}}^{r_{1}}\big{]}$
$=\int_{a}C_{(i,\bar{i})a}^{b_{1}}|z_{1}-\bar{z}_{1}|^{h_{a}-h_{i}-\bar{h}_{\bar{i}}}C_{(j,\bar{j})a^{\ast}}^{b_{1}}|z_{2}-\bar{z}_{2}|^{h_{a}-h_{j}-\bar{h}_{\bar{j}}}C_{aa^{\ast}0}^{b_{1}b_{1}b_{1}}|x_{12}|^{-2h_{a}}\mathcal{F}_{0,0,1,0}^{0}(|b_{1}\rangle_{\omega_{1}}^{r_{1}}),$
(18)

where $x_{1}=\text{Re}(z_{1})$ and $x_{2}=\text{Re}(z_{2})$. Equating the
second line of (17) with the second line of (18), we have successfully imposed
the UHP bulk two-point function sewing constraint to
$\mathcal{F}_{0,2,1,0}^{0}$.

Consider a $\mathcal{F}_{0,0,4,0}^{0}$ and combine the four boundaries

$J\>\int_{i_{1},i_{2},i_{3},i_{4}}\mathcal{F}_{0,0,1,0}^{4}\big{[}|(b_{1\cdots
4})_{\psi_{i_{1}}^{b_{1}b_{2}}\psi_{i_{2}}^{b_{2}b_{3}}\psi_{i_{3}}^{b_{3}b_{4}}\psi_{i_{4}}^{b_{4}b_{1}}}\rangle_{\omega_{1}\cdots\omega_{4}}^{r}\big{]}=\mathcal{F}_{0,0,4,0}^{0}\big{[}|b_{1}\rangle_{\omega_{1}}^{r_{1}},|b_{2}\rangle_{\omega_{2}}^{r_{2}},|b_{3}\rangle_{\omega_{3}}^{r_{3}},|b_{4}\rangle_{\omega_{4}}^{r_{4}}\big{]}.$
(19)

We pick one of the $\mathcal{F}_{0,0,1,0}^{4}$ on the LHS of (19) and impose
the crossing symmetry equation of the UHP boundary four-point functions on it.

$\int_{i_{u}}$$\int_{i_{s}}$$=$$\psi_{i_{1}}^{b_{1}b_{2}}$$\psi_{i_{2}}^{b_{2}b_{3}}$$\psi_{i_{4}}^{b_{4}b_{1}}$$\psi_{i_{3}}^{b_{3}b_{4}}$$\psi_{i_{s}}^{b_{1}b_{3}}$$\psi_{i_{s^{\ast}}}^{b_{3}b_{1}}$$\psi_{0}^{b_{1}b_{1}}$$\psi_{i_{1}}^{b_{1}b_{2}}$$\psi_{i_{2}}^{b_{2}b_{3}}$$\psi_{i_{3}}^{b_{3}b_{4}}$$\psi_{i_{u}}^{b_{2}b_{4}}$$\psi_{i_{4^{\ast}}^{b_{1}b_{4}}}$$\psi_{i_{4}}^{b_{4}b_{1}}$$\psi_{0}^{b_{1}b_{1}}$
Figure 5: The crossing symmetry equation of a boundary UHP four-point
function.

The crossing symmetry equations of UHP boundary operator four-point functions
Figure 5. The equation expression of the LHS of Figure 5 is

$\mathcal{F}_{0,0,1,0}^{4}\big{[}|(b_{1\cdots
4})_{\psi_{i_{1}}^{b_{1}b_{2}}\psi_{i_{2}}^{b_{2}b_{3}}\psi_{i_{3}}^{b_{3}b_{4}}\psi_{i_{4}}^{b_{4}b_{1}}}\rangle_{\omega_{1}\cdots\omega_{4}}^{r}\big{]}=$
$\int_{i_{s}}C_{i_{1}i_{2}i_{s}}^{b_{1}b_{2}b_{3}}|x_{12}|^{h_{i_{s}}-h_{i_{1}}-h_{i_{2}}}C_{i_{3}i_{4}i_{s^{\ast}}}^{b_{3}b_{4}b_{1}}|x_{34}|^{h_{i_{s}}-h_{i_{3}}-h_{i_{4}}}C_{i_{s}i_{s^{\ast}}0}^{b_{1}b_{3}b_{1}}|x_{24}|^{-2h_{i_{s}}}\mathcal{F}_{0,0,1,0}^{0}\big{[}|b_{1}\rangle_{\omega_{1}}^{r}\big{]}.$
(20)

The RHS of Figure 5 is the

$\mathcal{F}_{0,0,1,0}^{4}\big{[}|(b_{1234})_{\psi_{i_{1}}^{b_{1}b_{2}}\psi_{i_{2}}^{b_{2}b_{3}}\psi_{i_{3}}^{b_{3}b_{4}}\psi_{i_{4}}^{b_{4}b_{1}}}\rangle_{\omega_{1}\cdots\omega_{4}}^{r}\big{]}=$
$\int_{i_{u}}C_{i_{2}i_{3}i_{u}}^{b_{2}b_{3}b_{4}}|x_{23}|^{h_{i_{u}}-h_{i_{2}}-h_{i_{3}}}C_{i_{1}i_{u}i_{4}^{\ast}}^{b_{1}b_{2}b_{4}}|x_{13}|^{h_{i_{4}}-h_{i_{1}}-h_{i_{u}}}C_{i_{4^{\ast}}i_{4}0}^{b_{1}b_{4}b_{1}}|x_{34}|^{-2h_{i_{4}}}\mathcal{F}_{0,0,1,0}^{0}\big{[}|b_{1}\rangle_{\omega_{1}}^{r}\big{]}.$
(21)

By equating the second line of (20) and the second line of (21), we have
successfully imposed the crossing symmetry equation of UHP boundary four-point
functions on $\mathcal{F}_{0,0,4,0}^{0}$.

##### Imposing the crosscap sewing constraints:

A crosscap can be obtained by identifying the opposite points on the boundary
of a disk, or by the antiholomorphic involution $\Omega$ of the Riemann sphere
$S^{2}$

$\Omega:\quad z\cong-\frac{1}{\bar{z}}.$ (22)

The crosscap involution acting on a bulk operator is

$\Omega\phi_{(i,\bar{i})}(z,\bar{z})=\epsilon_{i}J\>\phi_{(\bar{i},i)}\Big{(}-\frac{1}{\bar{z}},-\frac{1}{z}\Big{)},$
(23)

where $J$ is the Jacobian of the operator from the coordinate transformation
and $J=(z)^{-2h_{i}}(\bar{z})^{-2\bar{h}_{\bar{i}}}$ when $\phi_{(i,\bar{i})}$
is a Virasoro primary operator. $\epsilon_{i}=\pm 1$ is the eigenvalue under
crosscap involution. The crosscap involution $\Omega$ is fixed-point free,
hence only the identity operator is allowed in the expansion when a bulk
operator is approaching a crosscap Stanev:2001na . This indicates that the
only crosscap sewing constraint is the crossing symmetry equation of the
crosscap bulk two-point functions Fioravanti:1993hf .

The crosscap sewing constraint is the equivalence of the bulk-bulk OPEs in the
crosscap two-point functions at two different limits. One limit is when
$\phi_{(i,\bar{i})}(z_{1},\bar{z}_{1})$ is approaching
$\phi_{(j,\bar{j})}(z_{2},\bar{z}_{2})$, the OPE at this limit in
$\mathcal{F}_{0,2,0,1}^{0}$ is

$\mathcal{F}_{0,2,0,1}^{0}\big{[}|\phi_{(i,\bar{i})}\rangle,|\phi_{(j,\bar{j})}\rangle,|c_{1}\rangle_{\omega_{1}}^{r_{1}}\big{]}=\int_{(k,\bar{k})}C_{(i,\bar{i})(j,\bar{j})}^{(k,\bar{k})}(z_{12})^{h_{k}-h_{i}-h_{j}}(\bar{z}_{12})^{\bar{h}_{\bar{k}}-\bar{h}_{\bar{i}}-\bar{h}_{\bar{j}}}\mathcal{F}_{0,1,0,1}^{0}\big{[}|\phi_{(k,\bar{k})}\rangle,|c_{1}\rangle_{\omega_{1}}^{r_{1}}\big{]}.$
(24)

The other limit is when the image
$\Omega\phi_{(j,\bar{j})}(z_{2},\bar{z}_{2})=\epsilon_{j}J\phi_{(\bar{j},j)}(-\frac{1}{\bar{z}_{2}},-\frac{1}{z_{2}})$
is approaching $\phi_{(i,\bar{i})}(z_{1},\bar{z}_{1})$. The OPE at this limit
is

$\mathcal{F}_{0,2,0,1}^{0}\big{[}|\phi_{(i,\bar{i})}\rangle,|\phi_{(j,\bar{j})}\rangle,|c_{1}\rangle_{\omega_{1}}^{r_{1}}\big{]}=(-1)^{h_{i}-\bar{h}_{\bar{i}}+h_{j}-\bar{h}_{\bar{j}}}\>J\int_{(l,\bar{l})}C_{(i,\bar{i})(\bar{j},j)}^{(l,\bar{l})}\epsilon_{j}$
$(z_{1}+\frac{1}{\bar{z}_{2}})^{h_{l}-h_{i}-\bar{h}_{\bar{j}}}(\bar{z}_{1}+\frac{1}{z_{2}})^{\bar{h}_{\bar{l}}-\bar{h}_{\bar{i}}-h_{j}}\mathcal{F}_{0,1,0,1}^{0}\big{[}|\phi_{(l,\bar{l})}\rangle,|c_{1}\rangle_{\omega_{1}}^{r_{1}}\big{]},$
(25)

where $(-1)^{h_{i}-\bar{h}_{\bar{i}}+h_{j}-\bar{h}_{\bar{j}}}$ is the from the
braiding matrices of chiral four-point blocks. Equating the RHS of (24) and
(25), we have successfully imposed the crosscap sewing constraints on
$\mathcal{F}_{0,2,0,1}^{0}$.

##### UHP and crosscap sewing constraints as constraint conditions on bulk
correlation functions:

Using the linear property of $\mathcal{F}_{g,n,b,c}^{x}$, the UHP and crosscap
sewing constraint becomes constraint conditions on a collection of bulk
correlation functions.

##### Constraint conditions on boundary and crosscap states:

Boundary (with no boundary operators) and crosscap states are spanned by the
Ishibashi states. The linear coefficients of them are constrained by the
partition functions on finite cylinders, Mobius strips, and Klein bottle
Cardy:1989ir ; Pradisi:1995qy ; Stanev:2001na ; Angelantonj:2002ct ;
Brunner:2002em . The direct channel finite cylinder partition function
$A_{12}$, with the boundary conditions $b_{1}$ and $b_{2}$ at the two ends,
must be the following linear combination of chiral characters

$A_{12}=\int_{j}n_{12}^{j}N_{1}N_{2}\>\chi_{j},\quad
n_{12}^{j}\in\mathbb{N},\quad N_{1},N_{2}\in\mathbb{Z}^{+}.$ (26)

The direct channel Klein bottle partition function $K$ must be the following
linear combinations of chiral characters Stanev:2001na ; Angelantonj:2002ct

$K=\int_{j}K^{j}\>\chi_{j},\quad K^{j}=0,\>\pm 1.$ (27)

The direct channel Mobius strip partition function $M_{1}$, with boundary
condition $b_{1}$ on the boundary, must be the following linear combinations
of hatted chiral characters $\hat{\chi}$ Stanev:2001na ; Angelantonj:2002ct

$M_{1}=\sum_{j}M_{1}^{j}N_{1}\>\hat{\chi}_{j},\quad M_{1}^{j}=0,\>\pm 1,\quad
N_{1}\in\mathbb{Z}^{+}.$ (28)

Detailed reviews can be found in Stanev:2001na ; Angelantonj:2002ct ;
Brunner:2002em .

We require that only boundary and crosscap satisfying the above constraint
conditions can be involved in the definition (4) of physical
$\mathcal{F}_{g,n,b,c}^{x}$. For boundary states with boundary operators, we
require every boundary condition must be a Cardy boundary condition.

###  2.3 Arbitrary BCFT correlation functions as linear combinations of bulk
correlators

Boundary states with boundary operator configurations are elements in the
closed-sector CFT Hilbert space $\mathcal{H}$. By the definition of
$\mathcal{F}_{g,n,b,c}^{x}$, correlation functions with arbitrary numbers of
boundaries and boundary operators can be written as infinite linear
combinations of bulk operators. However, the above statement is abstract, we
want to show this procedure explicitly.

Consider a correlation function $\mathcal{F}_{g,n,b,c}^{x}$ with all $x\geq 2$
boundary operators on the same boundary

$\mathcal{F}_{g,n,b,c}^{x}\big{[}|(b_{1\cdots
x})_{\psi_{i_{1}}^{b_{1}b_{2}}\cdots\psi_{i_{x}}^{b_{x}b_{1}}}\rangle_{\omega_{1}\cdots\omega_{x}}^{r},\cdots\big{]}.$
(29)

Applying the boundary-boundary OPE repeatedly to the boundary state, we obtain
a series of $\mathcal{F}_{g,n,b,c}^{1}$

$\mathcal{F}_{g,n,b,c}^{x}\big{[}|(b_{1\cdots
x})_{\psi_{i_{1}}^{b_{1}b_{2}}\cdots\psi_{i_{x}}^{b_{x}b_{1}}}\rangle_{\omega_{1}\cdots\omega_{x}}^{r},\cdots\big{]}=\int_{i_{a_{1}}}\cdots\int_{i_{a_{x-1}}}C_{i_{1}i_{2}i_{a_{1}}}^{b_{1}b_{2}b_{3}}|x_{12}|^{h_{i_{a_{1}}}-h_{i_{1}}-h_{i_{2}}}\cdots$
$C_{i_{a_{q-2}}i_{q}i_{a_{q-1}}}^{b_{1}b_{x}b_{1}}|x_{(x-1)x}|^{h_{i_{a_{x-1}}}-h_{i_{x}}-h_{i_{a_{x-2}}}}\mathcal{F}_{g,n,b,c}^{1}\big{[}|(b_{1})_{\psi_{i_{a_{x-1}}}}\rangle_{\omega_{1}}^{r},\cdots\big{]}$
(30)

Next, we define the inversion of bulk-boundary OPE as

$\psi_{i}^{b_{1}b_{1}}\big{(}\frac{z+\bar{z}}{2}\big{)}=\int_{(a,\bar{a})}\>^{\text{in}}C_{(a,\bar{a})b_{1}}^{i}\>\phi_{(a,\bar{a})}(z,\bar{z}),$
(31)

where we have assumed that for all admissible boundary conditions $b_{i}{}$
and all chiral representations in $\mathcal{H}_{b_{i}b_{i}}$, there always
exist a set of bulk operators such that inversion (31) can be realized. The
coefficients ${}^{\text{in}}C_{(a,\bar{a})b_{1}}^{i}$ have to cancel the
dependence of $|z-\bar{z}|$ on the RHS of (31). Applying the inversion to the
boundary state with single boundary operators, we will be able to write any
component on the RHS of (30) as

$\mathcal{F}_{g,n,b,c}^{1}\big{[}|(b_{1})_{\psi_{i_{a_{q-1}}}^{b_{1}b_{1}}}\rangle_{\omega_{1}}^{r},\cdots\big{]}=\int_{(a,\bar{a})}\>^{in}C_{(a,\bar{a})b_{1}}^{i_{a_{q-1}}}\>\mathcal{F}_{g,n+1,b,c}^{0}\big{[}|b_{1}\rangle_{\omega_{1}}^{r},|\phi_{(a,\bar{a})}\rangle\cdots\big{]}.$
(32)

Substituting (32) into (30), we finish removing all boundary operators on the
corresponding boundary. If there are total $k\leq b$ defining boundary states
that have boundary operators, by repeating the boundary operator removing
procedure, we can write the original correlation function
$\mathcal{F}_{g,n,b,c}^{x}$ as linear combinations of
$\mathcal{F}_{g,n+k,b,c}^{0}$ with all defining boundary states having no
boundary operator. The expansion of boundary states with no boundary operator
can be realized in practice. Hence, by expanding all boundary and crosscap
states in all the $\mathcal{F}_{g,n+k,b,c}^{0}$, we can write correlation
functions with arbitrary boundary operators as linear combinations of bulk
correlation functions explicitly. From the discussion in this subsection, we
state that BCFT can be unified with bulk CFT.

##### CFT data and correlation functions:

We expect that all correlation functions are known in principle if the set of
CFT data is known. The CFT data includes

  * •

The CFT Hilbert space $\mathcal{H}$ giving the modular invariant $T^{2}$
partition function.

  * •

Bulk-bulk, bulk-boundary, boundary-boundary OPE coefficients, and the crosscap
one-point functions.

We shall see that the fact that any $\mathcal{F}_{g,n,b,c}^{x}$ can be written
as linear combinations of bulk correlation functions agree with this
expectation explicitly. To know the form of an arbitrary
$\mathcal{F}_{g,n,b,c}^{x}$, we first need to know bulk-boundary and boundary-
boundary OPE coefficients to express $\mathcal{F}_{g,n,b,c}^{x}$ as a
combination of $\mathcal{F}_{g,n+k,b,c}^{0}$ (knowing the bulk-boundary OPE
coefficients is equivalent to knowing its inversion coefficients). Next, we
need to expand $\mathcal{F}_{g,n+k,b,c}^{0}$ obtained in terms of bulk
correlation functions $\mathcal{F}_{g,n+k+b+c}$, where we need to know the
explicit forms of boundary and crosscap states. We know which Ishibashi states
exist if we know the CFT Hilbert space $\mathcal{H}$, and the boundary-bulk
OPE coefficients and crosscap one-point functions will determine the explicit
form of the boundary and crosscap states. Finally, after expanding
$\mathcal{F}_{g,n+k,b,c}^{0}$ as infinite linear combinations of
$\mathcal{F}_{g,n+k+b+c}$, each $\mathcal{F}_{g,n+k+b+c}$ is known in
principle if we know the CFT Hilbert space $\mathcal{H}$ and all the bulk-bulk
OPE coefficients. Hence, any $\mathcal{F}_{g,n,b,c}^{x}$ is known in principle
if the set of the CFT data is known.

###  2.4 The plumbing fixture of $\mathcal{F}_{g,n,b,c}^{x}$

We describe both the closed and open-sector plumbing fixture of
$\mathcal{F}_{g,n,b,c}^{x}$ Lewellen:1991tb . The closed-sector plumbing
fixtures are the same as the plumbing fixtures for genus-$g$, $n$-point
functions $\mathcal{F}_{g,n}$. To perform the open-sector plumbing fixtures,
we have to create the identity operator $I$ of the corresponding open-sector
Hilbert space. The plumbing fixture of surfaces with lower dimensional moduli
spaces will not fully cover the whole moduli space of the surface obtained.
Nevertheless, correlation functions obtained by the plumbing fixtures will be
valid over the whole moduli space by the validness assumption.

####  2.4.1 Closed-sector plumbing fixtures

Geometrically, the closed-sector plumbing fixture is realized by cutting two
bulk punctures and identifying the boundaries of the holes created. This
procedure creates a tube. Algebraically, we create the identity operator
$I\otimes\bar{I}$ of closed-sector Hilbert space $\mathcal{H}$ (8) using the
bulk operators inserted at two punctures.

##### The separating plumbing fixtures:

Consider two Riemann surfaces $\Sigma_{g_{1},n_{1},b_{1},c_{1}}^{x_{1}}$ and
$\Sigma_{g_{2},n_{2},b_{2},c_{2}}^{x_{2}}$. We pick the $n_{1}^{\text{th}}$
and $n_{2}^{\text{th}}$ punctures from two surfaces respectively, cut two
punctures out by removing regions $t_{n_{1}}^{(1)}\leq\sqrt{|q|}$ and
$t_{n_{2}}^{(2)}\leq\sqrt{|q|}$, and sew two surfaces together by identifying
two boundaries by

$t_{n_{1}}^{(1)}t_{n_{2}}^{(2)}=q\in\mathbb{C},\quad 0<|q|<1.$ (33)

This procedure is denoted by

$\Sigma_{(g_{1}+g_{2}),(n_{1}+n_{2}-2),(b_{1}+b_{2}),(c_{1}+c_{2})}^{(x_{1}+x_{2})}=\Sigma_{g_{1},n_{1},b_{1},c_{1}}^{x_{1}}\\#\Sigma_{g_{2},n_{2},b_{2},c_{2}}^{x_{2}}.$
(34)

The correlation functions
$\mathcal{F}_{(g_{1}+g_{2}),(n_{1}+n_{2}-2),(b_{1}+b_{2}),(c_{1}+c_{2})}^{(x_{1}+x_{2})}$
obtained by plumbing fixture can be expressed schematically by

$\mathcal{F}_{(g_{1}+g_{2}),(n_{1}+n_{2}-2),(b_{1}+b_{2}),(c_{1}+c_{2})}^{(x_{1}+x_{2})}\big{[}\cdots\big{]}=\int_{\alpha,\beta}\mathcal{F}_{g_{1},n_{1},b_{1},c_{1}}^{x_{1}}\big{[}\cdots,|\phi_{\alpha}\rangle\big{]}\times(G)^{\alpha\beta}\times\mathcal{F}_{g_{2},n_{2},b_{2},c_{2}}^{x_{2}}\big{[}\cdots,|\phi_{\beta}\rangle\big{]}.$
(35)

We introduce an auxiliary vector $\langle S|$ such that

$I\otimes\bar{I}=\int_{\alpha,\beta}(G)^{\alpha\beta}\langle
S|\phi_{\alpha}\rangle\otimes|\phi_{\beta}\rangle,\quad\langle
S|\in(\mathcal{H}^{\ast})^{\otimes 2},\quad\langle
S|\phi_{\alpha}\rangle=\langle(\phi_{\alpha})^{c}|,\>\forall|\phi_{\alpha}\rangle\in\mathcal{H}.$
(36)

A formal identification of the surface states can be obtained

$\langle\Sigma_{g_{1},n_{1},b_{1},c_{1}}|\otimes\langle\Sigma_{g_{2},n_{2},b_{2},c_{2}}|=\langle\Sigma_{(g_{1}+g_{2}),(n_{1}+n_{2}-2),(b_{1}+b_{2}),(c_{1}+c_{2})}|\otimes\langle
S|.$ (37)

##### The non-separating plumbing fixtures:

Consider a Riemann surface with no less than two punctures
$\Sigma_{g,n,b,c}^{x}$, $n\geq 2$. We can cut two of the bulk punctures out,
perform the boundary identification $t_{n-1}t_{n}=q\in\mathbb{C}$, and obtain
a new surface $\Sigma_{g+1,n-2,b,c}^{x}$. Such a non-separating plumbing
fixture procedure will be denoted as

$\Sigma_{g+1,n-2,b,c}^{x}=\\#\Sigma_{g,n,b,c}^{x}.$ (38)

Similar to the separating cases, there is a schematic expression for the
correlation functions on the surface $\Sigma_{g+1,n-2,b,c}^{x}$ obtained

$\mathcal{F}_{g+1,n-2,b,c}^{q}\big{[}\cdots\big{]}=\int_{\alpha,\beta}\mathcal{F}_{g,n,b,c}^{q}\big{[}\cdots,|\phi_{\alpha}\rangle,|\phi_{\beta}\rangle\big{]}\times(G)^{\alpha\beta},$
(39)

The formal identifications of the surface states for non-separating plumbing
fixtures are

$\langle\Sigma_{g+1,n-2,b,c}|\otimes\langle S|=\langle\Sigma_{g,n,b,c}|.$ (40)

##### Discussion: The necessity of cutting punctures in the plumbing fixture
procedure.

We end the discussion on closed-sector plumbing fixtures by discussing one
question: Is it necessary to cut two bulk punctures to create the boundaries
for identification? Geometrically, the plumbing fixture can be done by
identifying two boundaries that already exist on the surfaces or cutting two
crosscaps to create the boundaries for sewing.

The answer to this question is no. Because such operations are not admissible
algebraically. CFT boundary states cannot reproduce the closed-sector identity
operator $I\otimes\bar{I}$ as (8), even when the boundary conditions are
complete Pradisi:1996yd ; Behrend:1998fd ; Behrend:1999bn . To show this, we
write down the form of the twisted boundary Ishibashi states Onogi:1988qk

$|b,i\rangle\rangle_{\omega}=\sum_{N}|i,N\rangle\otimes
V_{\omega}U|i,N\rangle\in\mathcal{H}_{i}\otimes\mathcal{H}_{\omega^{-1}(i^{\ast})}.$
(41)

In Ishibashi states, the same $L$ and $R$ orthonormal basis will always
couple, which is different from the vectors in the standard basis where
arbitrary $L\otimes R$ couplings are allowed. Hence, the identity operator
$I\otimes\bar{I}$ in (8) cannot be reproduced from the boundary states.
Similarly, crosscap states also cannot reproduce $I\otimes\bar{I}$. We must
cut the bulk punctures to perform the closed-sector plumbing fixture of
$\mathcal{F}_{g,n,b,c}^{x}$, as shown in Figure 6.

$\int$ Figure 6: The closed-sector plumbing fixture: creating a tube by
cutting punctures and sewing.

####  2.4.2 Open-sector plumbing fixtures

We discuss the open-sector plumbing fixtures in $\mathcal{F}_{g,n,b,c}^{x}$,
which are shown in Figure 7 Lewellen:1991tb . Algebraically, this procedure
will create the identity operator from the boundary operators in the open-
sector Hilbert space $\mathcal{H}_{b_{1}b_{2}}$. Recall that
$\mathcal{H}_{b_{1}b_{2}}$ can be decomposed as a direct integral of the
irreducible representations of the open-sector chiral algebra
$\mathcal{A}^{op}$

$\mathcal{H}_{b_{1}b_{2}}=\int_{i}^{\oplus}\>(\mathcal{H}_{b_{1}b_{2}}^{i})^{\oplus\>(n_{b_{1}b_{2}}^{i}\times
N_{1}\times N_{2})},\quad n_{b_{1}b_{2}}^{i}\in\mathbb{N},\quad
N_{1},\>N_{2}\in\mathbb{Z}_{+}.$ (42)

We denote all vectors in
$(\mathcal{H}_{b_{1}b_{2}}^{i})^{\oplus\>(n_{b_{1}b_{2}}^{i}\times N_{1}\times
N_{2})}$ collectively as $|\psi_{i}^{b_{1}b_{2}}\rangle$. The identity
operator $I$ of $\mathcal{H}_{b_{1}b_{2}}$ can be expressed as

$I=\int_{i}|\psi_{i}^{b_{1}b_{2}}\rangle(G)^{ii}\langle\psi_{i^{\ast}}^{b_{2}b_{1}}|,\quad(G)_{ii}:=\langle\psi_{i^{\ast}}^{b_{2}b_{1}}|\psi_{i}^{b_{1}b_{2}}\rangle,\quad(G)^{ii}:=(G_{ii})^{-1},$
(43)

where $\langle\psi_{i^{\ast}}^{b_{2}b_{1}}|$ is the BPZ dual of
$|\psi_{i}^{b_{1}b_{2}}\rangle$. The open-sector plumbing fixtures include
three different scenarios Lewellen:1991tb ; Zwiebach:1997fe . We write down
the expression for correlation functions obtained from the three different
open-sector plumbing fixtures respectively:

  * •

The separating open-sector plumbing fixtures. Such a procedure decreases the
number of total boundaries by one. The correlation functions on the surfaces
obtained are

$\mathcal{F}_{g_{1}+g_{2},n_{1}+n_{2},b_{1}+b_{2}-1,c_{1}+c_{2}}^{x_{1}+x_{2}-2}\big{[}|(b_{1\cdots
2\cdots})_{\cdots}\rangle_{\omega_{1}\cdots\omega_{2}\cdots}^{r},\cdots\big{]}=$
$\int_{i}\mathcal{F}_{g_{1},n_{1},b_{1},c_{1}}^{x_{1}}\big{[}|(b_{12\cdots})_{\psi_{i}^{b_{1}b_{2}}\cdots}\rangle_{\omega_{1}\omega_{2}\cdots}^{r_{1}},\cdots\big{]}\times(G)^{ii}\times\mathcal{F}_{g_{2},n_{2},b_{2},c_{2}}^{x_{2}}\big{[}|(b_{12\cdots})_{\psi_{i}^{b_{1}b_{2}}\cdots}\rangle_{\omega_{1}\omega_{2}\cdots}^{r_{2}},\cdots\big{]}.$
(44)

  * •

The non-separating open-sector plumbing fixtures with two boundary operators
on the same boundary. Such a procedure increases the number of boundaries by
one. The correlation functions on the surfaces obtained are

$\mathcal{F}_{g,n,b+1,c}^{x-2}\big{[}|(b_{1\cdots})_{\cdots}\rangle_{\omega_{1}\cdots}^{r_{1}},|(b_{2\cdots})_{\cdots}\rangle_{\omega_{2}\cdots}^{r_{2}},\cdots\big{]}=$
$\int_{i}\>(G)^{ii}\times\mathcal{F}_{g,n,b,c}^{x}\big{[}|(b_{12\cdots
21\cdots})_{\psi_{i}^{b_{1}b_{2}}\cdots\psi_{i^{\ast}}^{b_{2}b_{1}}\cdots}\rangle_{\omega_{1}\omega_{2}\cdots\omega_{2}\omega_{1}\cdots}^{r},\cdots\big{]}.$
(45)

  * •

The non-separating open-sector plumbing fixtures with two boundary operators
on different boundaries. Such a procedure increases the number of handles by
one and decreases the number of boundaries by one. The correlation functions
on the surfaces obtained are

$\mathcal{F}_{g+1,n,b-1,c}^{x-2}\big{[}|b_{1\cdots
2\cdots}\rangle_{\omega_{1}\cdots\omega_{2}\cdots}^{r},\cdots\big{]}=$
$\int_{i}\>(G)^{ii}\times\mathcal{F}_{g,n,b,c}^{x}\big{[}|(b_{12\cdots})_{\psi_{i}^{b_{1}b_{2}}\cdots}\rangle_{\omega_{1}\omega_{2}\cdots}^{r_{1}},|(b_{12\cdots})_{\psi_{i}^{b_{1}b_{2}}\cdots}\rangle_{\omega_{1}\omega_{2}\cdots}^{r_{2}},\cdots\big{]}.$
(46)

$b_{1}$$b_{2}$$\int$$b_{1}$$b_{2}$$b_{1}$$b_{2}$ Figure 7: The open-sector
plumbing fixture: creating a strip by cutting boundary operators and sewing.

###  2.5 Conformal blocks of $\mathcal{F}_{g,n,b,c}^{0}$

We define the conformal blocks of correlation functions
$\mathcal{F}_{g,n,b,c}^{0}$. Consider $\mathcal{F}_{g,n,b,c}^{0}$ where all
the defining boundary and crosscap states are twisted Ishibashi states
$|V_{h}\rangle\rangle_{\omega}^{r}\in V_{h}\otimes V_{\omega}UV_{h}$. $V_{h}$
is an $\mathcal{A}$-algebra Verma module whose primary conformal weight is
$h$. We will first define the (anti-)holomorphic conformal blocks
$F_{g,(n+b+c)}$ ($\bar{F}_{g,(n+b+c)}$) for bulk correlation functions
$\mathcal{F}_{g,(n+b+c)}$, and then define the blocks $F_{g,n,b,c}$ for
$\mathcal{F}_{g,n,b,c}^{0}$ in the same manner. Using the linear property of
$\mathcal{F}_{g,n,b,c}^{0}$, conformal blocks $F_{g,n,b,c}$ are infinite
linear combinations of products of holomorphic and anti-holomorphic conformal
blocks $\sum F_{g,(n+b+c)}\times\bar{F}_{g,(n+b+c)}$.

To define the conformal blocks of $\mathcal{F}_{g,(n+b+c)}^{0}$, we consider
punctured Riemann surfaces $\Sigma_{g,(n+b+c)}$ in regions obtained by
plumbing fixtures of the sphere three-point vertices
$\cdots\\#(\mathcal{V}_{0,3}\\#\cdots\\#\mathcal{V}_{0,3})\in\mathcal{M}_{g,(n+b+c)}$.
Surfaces in those regions are obtained by $(2g-3+n+b+c)$ separating and $g$
non-separating closed-sector plumbing fixtures of total $(2g-2+n+b+c)$ three
punctured spheres $\Sigma_{0,3}$, indicating that the surfaces in those
regions can be decomposed into $(3g-3+n+b+c)$ internal propagators, $n$
external propagators, and $(2g-2+n+b+c)$ sphere three-point interacting
vertices. Figure 8 is a simple graphic example showing such a decomposition.
We write down the conformal block decomposition of $\mathcal{F}_{g,(n+b+c)}$
according to the geometric decomposition

$\mathcal{F}_{g,(n+b+c)}=\prod_{A=1}^{3g-3+n+b+c}\int_{(\mathcal{H}_{iA}\otimes\bar{\mathcal{H}}_{\bar{i}A})\in\mathcal{H}}\prod_{B=1}^{2g-2+n+b+c}C(B)\>\times$
$\Big{[}F_{g,(n+b+c)}(\prod_{A=1}^{3g-3+n+b+c}\mathcal{H}_{iA})\times\bar{F}_{g,(n+b+c)}(\prod_{A=1}^{3g-3+n+b+c}\mathcal{H}_{\bar{i}A})\Big{]},$
(47)

where $A$ labels the internal propagators and $B$ labels the three-point
vertices. $C(B)$ are the sphere three-point structure constants of each sphere
three-point vertices and
$F_{g,(n+b+c)}(\prod_{A=1}^{3g-3+n+b+c}\mathcal{H}_{iA})$
$\Big{(}\bar{F}_{g,(n+b+c)}(\prod_{A=1}^{3g-3+n+b+c}\mathcal{H}_{\bar{i}A})\Big{)}$
are the holomorphic (anti-holomorphic) genus-$g$, $(n+b+c)$-point conformal
blocks respectively. The integrals in (47) will run over all irreducible
representations in the closed-sector CFT Hilbert space $\mathcal{H}$ at every
internal propagator. Detailed field configurations on all the sphere three-
point vertices will determine the channel of the conformal blocks.

Figure 8: A Feynman graph showing a punctured Riemann surface’s decomposition
into external propagators, internal propagators, and sphere three-point
vertices. Each propagator (line) will carry indices of irreducible
representations in the CFT Hilbert space and each interacting vertex will be
labeled by the three irreducible representations from the three lines joining.

Now, we define the conformal block decomposition of
$\mathcal{F}_{g,n,b,c}^{0}$ similar to (47)

$\mathcal{F}_{g,n,b,c}^{0}=\prod_{A=1}^{3g-3+n+b+c}\int_{(\mathcal{H}_{iA}\otimes\bar{\mathcal{H}}_{\bar{i}A})}\prod_{B=1}^{2g-2+n+b+c}C(B)F_{g,n,b,c}\Big{[}\prod_{A=1}^{3g-3+n+b+c}(\mathcal{H}_{iA}\otimes\mathcal{H}_{\bar{i}A})\Big{]},$
(48)

where $F_{g,n,b,c}$ contain both holomorphic and anti-holomrophic components.
Expanding $\mathcal{F}_{g,n,b,c}^{0}$ as bulk correlation functions
$\mathcal{F}_{g,(n+b+c)}$, we can write
$F_{g,n,b,c}\Big{[}\prod_{A=1}^{3g-3+n+b+c}(\mathcal{H}_{iA}\otimes\mathcal{H}_{\bar{i}A})\Big{]}$
as infinite linear combinations of conformal blocks of different
$\mathcal{F}_{g,(n+b+c)}$

$F_{g,n,b,c}\Big{[}\prod_{A=1}^{3g-3+n+b+c}(\mathcal{H}_{iA}\otimes\mathcal{H}_{\bar{i}A})\Big{]}=C_{g,n,b,c}\Big{[}F_{g,(n+b+c)}(\prod_{A=1}^{3g-3+n+b+c}\mathcal{H}_{iA})\times\text{c.c.}+\cdots\Big{]},$
(49)

where $C_{g,n,b,c}$ is some overall constant factor. The c.c. in (49) denotes
the complex conjugate. The
$F_{g,(n+b+c)}(\prod_{A=1}^{3g-3+n+b+c}\mathcal{H}_{iA})\times\text{c.c.}$
term in (49) is the contribution from the $\mathcal{F}_{g,(n+b+c)}$ where all
the $(b+c)$ bulk vertex operators from Ishibashi state expansions are primary
operators. The $\cdots$ in (49) denotes the conformal blocks from all other
$\mathcal{F}_{g,(n+b+c)}$. The conformal blocks in different channels can be
transformed by linear transformations called the duality transformations such
as the fusing $F$ or the braiding $B$ transformations.

In general, we can only give such a schematic definition (48) of the conformal
blocks $F_{g,n,b,c}$. More explicit definitions require us to specify the
conformal algebra $\mathcal{A}\otimes\mathcal{A}$ and the geometric
information of the surface. We shall show the Virasoro conformal blocks of
sphere four-boundary functions $\mathcal{F}_{0,0,4,0}^{0}$ and torus-boundary
functions $\mathcal{F}_{1,0,1,0}^{0}$ as simple examples. Especially, we show
how to write them as combinations of chiral sphere three-point functions.

Two types of single poles can exist in the conformal blocks $F_{g,n,b,c}$. One
type is from the inversion Gram matrices in genus-$g$, $(n+b+c)$-point blocks
$F_{g,(n+b+c)}$ when the Verma modules of the intermediate propagators are
degenerate. Another type is from the inversion Gram matrices in Ishibashi
state expansion coefficients of $\mathcal{F}_{g,n,b,c}^{0}$ when Ishibashi
states correspond to degenerate Verma modules.

##### Virasoro conformal blocks for sphere four-boundary functions:

Consider a CFT having Virasoro conformal algebra
$\mathfrak{Vir}\otimes\mathfrak{Vir}$. The sphere four-boundary function
$\mathcal{F}_{0,0,4,0}^{0}$ with four defining Virasoro Ishibashi states
$|V_{i}\rangle\rangle\in(V_{i}\otimes UV_{i})$ is

$\mathcal{F}_{0,0,4,0}^{0}\Big{(}\otimes_{i=1}^{4}|V_{i}\rangle\rangle^{r_{i}}\Big{)}:=\langle
V_{0,4}|V_{1}\rangle\rangle^{r_{1}}\otimes|V_{2}\rangle\rangle^{r_{2}}\otimes|V_{3}\rangle\rangle^{r_{3}}\otimes|V_{4}\rangle\rangle^{r_{4}},\quad
r_{i}\in(0,1).$ (50)

The $s$-channel conformal block decomposition of the above sphere four-
boundary function is

$\mathcal{F}_{0,0,4,0}^{0}\Big{(}\otimes_{i=1}^{4}|V_{i}\rangle\rangle^{r_{i}}\Big{)}=\int_{(h_{s},\bar{h}_{s})}C_{12(h_{s},\bar{h}_{s})}C_{3(h_{s},\bar{h}_{s})4}\>F_{0,0,4,0}^{s}\big{[}(h_{s},\bar{h}_{s})\big{]},$
(51)

where we have used $(h_{s},\bar{h}_{s})$ to label the $s$-channel intermediate
irreducible representations.

Now, we want to write the $s$-channel sphere four-boundary blocks
$F_{0,0,4,0}^{s}$ as combinations of chiral sphere three-point functions. The
centers of the four boundaries are fixed at $z_{1}=0$, $z_{2}=z$, $z_{3}=1$
and $z_{4}=\infty$. First, we expand the function $\mathcal{F}_{0,0,4,0}^{0}$
as linear combinations of sphere four-point functions $\mathcal{F}_{0,4}$

$\mathcal{F}_{0,0,4,0}^{0}\Big{(}\otimes_{i=1}^{4}|V_{i}\rangle\rangle^{r_{i}}\Big{)}=\prod_{i=1}^{4}r_{i}^{2h_{i}-\frac{c_{Vir}}{12}}\Big{\\{}\mathcal{F}_{0,4}\big{[}\prod_{i=1}^{4}(v_{i}\otimes\bar{v}_{i})\big{]}+\cdots\Big{\\}}.$
(52)

The $s$-channel conformal block decomposition of $\mathcal{F}_{0,4}$ in the
expansion are

$\langle\xi_{4}\otimes\bar{\xi}_{4}|\phi_{\xi_{3}\otimes\bar{\xi}_{3}}(1,1)\phi_{\xi_{2}\otimes\bar{\xi}_{2}}(z,\bar{z})|\xi_{1}\otimes\bar{\xi}_{1}\rangle=\int_{(h_{s},\bar{h}_{s})}C_{12(h_{s},\bar{h}_{s})}\times$
$C_{3(h_{s},\bar{h}_{s})4}\times
z^{h_{s}-h_{\xi_{1}}-h_{\xi_{2}}}F_{0,4}^{s}\big{(}h_{s};\prod_{i=1}^{4}\xi_{i}\big{)}\times\bar{z}^{\bar{h}_{s}-\bar{h}_{\bar{\xi}_{1}}-\bar{h}_{\bar{\xi}_{2}}}\bar{F}_{0,4}^{s}\big{(}\bar{h}_{s};\prod_{i=1}^{4}\bar{\xi}_{i}\big{)},$
(53)

where $C_{12(h_{s},\bar{h}_{s})}$ and $C_{3(h_{s},\bar{h}_{s})4}$ are sphere
three-point structure constants. The $s$-channel holomorphic Virasoro sphere
four-point blocks are

$F_{0,4}^{s}\big{(}h_{s};\prod_{i=1}^{4}\xi_{i}\big{)}=\sum_{|M|=|N|=l\geq
0}z^{l}\rho(\xi_{4},\xi_{3},L_{-M}v_{h_{s}}|1)(G_{c_{Vir},h_{s}}^{l})^{MN}\rho(L_{-N}v_{h_{s}},\xi_{2},\xi_{1}|z),$
(54)

where $(G_{c_{Vir},h_{s}}^{l})^{MN}$ is the level-$l$ inversion Gram matrix of
Virasoro Verma module $V_{h_{s}}$. $\rho(\xi_{3},\xi_{2},\xi_{1}|z)$ are
defined by Teschner:2001rv

$\langle\xi_{3}\otimes\bar{\xi}_{3}|\phi_{{\xi_{2}\otimes\bar{\xi}_{2}}}(z,\bar{z})|\xi_{1}\otimes\bar{\xi}_{1}\rangle=C_{123}\rho(\xi_{3},\xi_{2},\xi_{1}|z)\rho(\bar{\xi}_{3},\bar{\xi}_{2},\bar{\xi}_{1}|\bar{z}).$
(55)

From the linear property of $\mathcal{F}_{0,0,4,0}^{0}$, we obtain

$F_{0,0,4,0}^{s}(h_{s},\bar{h}_{s})=\prod_{i=1}^{4}r_{i}^{2h_{i}-\frac{c_{Vir}}{12}}z^{h_{s}-h_{1}-h_{2}}\bar{z}^{\bar{h}_{s}-h_{1}-h_{2}}\Big{[}F_{0,4}^{s}(h_{s},\prod_{i=1}^{4}v_{i})\times\bar{F}_{0,4}^{s}(\bar{h}_{s},\prod_{\bar{i}=1}^{4}\bar{v}_{\bar{i}})+\cdots\Big{]},$
(56)

where $\cdots$ denotes terms
$F_{0,4}^{s}(h_{s},\prod_{i=1}^{4}\xi_{i})\times\bar{F}_{0,4}^{s}(\bar{h}_{s},\prod_{\bar{i}=1}^{4}\bar{\xi}_{\bar{i}})$
with proper expansion coefficients. Substituting the form of Virasoro four-
point blocks $F_{0,4}^{s}$ (54) into (56), we obtain the four-boundary blocks
$F_{0,0,4,0}^{s}$ in terms of chiral sphere three-point functions $\rho$.

The sphere four-boundary conformal blocks in other channels can be defined in
similar manners. The duality transformations between sphere four-boundary
blocks are infinite combinations of products of duality transformations of
chiral sphere four-point blocks $F_{0,4}$ ($\bar{F}_{0,4}$).

##### Virasoro conformal blocks for torus one-boundary functions:

Another simple example is the Virasoro conformal blocks of torus one-boundary
functions. Consider the following torus one-boundary function

$\langle\Sigma_{1,1}|V_{1}\rangle\rangle^{r}=r^{2h_{1}-\frac{c_{Vir}}{12}}\Big{[}\langle\phi_{v_{1}\otimes\bar{v}_{1}}(1,1)\rangle_{T^{2}}+\cdots\Big{]}.$
(57)

The block decomposition of the above function is

$\langle\Sigma_{1,1}|V_{1}\rangle\rangle^{r_{1}}=\int_{(h,\bar{h})}C_{1(h,\bar{h})(h,\bar{h})}\times
F_{1,0,1,0}(h,\bar{h}).$ (58)

We want to write $F_{1,0,1,0}(h,\bar{h})$ as combinations of chiral sphere
three-point functions. First, we write down the conformal block decomposition
of a torus one-point functions

$\langle\phi_{\xi_{1}\otimes\bar{\xi}_{1}}(1,1)\rangle_{T^{2}}=\int_{(h,\bar{h})}C_{1(h,\bar{h})(h,\bar{h})}F_{1,1}(h,\xi_{1})\times\bar{F}_{1,1}(\bar{h},\bar{\xi}_{1}),$
(59)

with the holomorphic block being

$F_{1,1}(h,\xi_{1})=q^{h-\frac{c_{Vir}}{24}}\sum_{|M|=|N|=l\geq
0}q^{l}(G_{c_{Vir},h}^{l})^{MN}\rho(L_{-M}v_{h},\xi_{1},L_{-N}v_{h}|1).$ (60)

Using the linear property of $\mathcal{F}_{1,0,1,0}^{0}$, we obtain

$F_{1,0,1,0}^{0}(h,\bar{h})=r^{2h_{1}-\frac{c_{Vir}}{12}}\Big{[}F_{1,1}(h,v_{1})\times\bar{F}_{1,1}(\bar{h},\bar{v}_{1})+\cdots\Big{]},$
(61)

where $\cdots$ denotes contributions from
$F_{1,1}(h,\xi_{1})\times\bar{F}_{1,1}(\bar{h},\bar{\xi}_{1})$. Substituting
the torus one-point blocks (60) into (61), we can express the torus one-
boundary blocks in terms of chiral sphere three-point functions.

###  2.6  $\mathcal{F}_{0,n,b,0}^{0}$ non-vanishing conditions vs upper half-
plane correlation function non-vanishing conditions in RCFTs

As a consistency check of our definition of $\mathcal{F}_{g,n,b,c}^{x}$, we
compare the non-vanishing conditions of $\mathcal{F}_{0,n,b,0}^{0}$ with
corresponding UHP bulk $n$-point, boundary $b$-point functions
$\langle\phi_{1}\cdots\phi_{n}\psi_{1}\cdots\psi_{b}\rangle_{UHP}$ in RCFTs.
The non-vanishing conditions on both sides should be equivalent, that is they
are sufficient and necessary conditions to each other since we expect the two
types of correlation functions to be equivalent up to a non-singular Jacobian
$J$

$J\>\prod_{i=1}^{b}\int_{i}\>\langle\phi_{1}\cdots\phi_{n}\psi_{1}\cdots\psi_{b}\rangle=\mathcal{F}_{0,n,b,0}^{0}\>[\cdots].$
(62)

The advantage of comparing the non-vanishing conditions is that they are
completely determined by the representation theory. We can realize the
comparison without any explicit computation of the correlation functions. Yet,
comparing the non-vanishing conditions gives a non-trivial consistency check
to the definition of $\mathcal{F}_{g,n,b,c}^{x}$. Fulfilling the general
comparisons for arbitrary numbers of boundaries $b$ is difficult. In this
work, we will start the comparison from the simplest $b=1$ case and increase
the number of boundaries by one for each step. In Subsection 2.6, we will
restrict the discussion to rational CFT (RCFT), where only a finite number of
irreducible representations of the chiral algebra $\mathcal{A}$ exists in the
closed-sector Hilbert space $\mathcal{H}$. Note that the non-vanishing
condition in this subsection means the existence of non-zero components, the
complete non-vanishing condition will further require all non-zero components
not to cancel each other, which will be discussed in future studies.

##### A quick review on relevant results in RCFTs:

We have a quick review of some relevant results in RCFTs. Bulk correlation
functions in CFT are non-zero if and only if they are admissible by the fusion
rules Moore:1988ss . The fusion coefficients describing the chiral sphere
three-point interaction $N_{ij}^{k}$ in RCFTs are given by the Verlinde
formula Verlinde:1988sn

$N_{ij}^{k}=\sum_{m}\frac{S_{im}}{S_{0m}}S_{jm}(S_{mk})^{\ast},$ (63)

where $S_{ij}$ is the modular $S$ matrix describing the behaviour of
characters under modular $S$ transformations

$\chi_{i}\big{(}-\frac{1}{\tau}\big{)}=\sum_{j}S_{ij}\>\chi_{j}(\tau).$ (64)

The multiplicity of the open-sector chiral representations
$n_{b_{1}b_{2}}^{i}$ in (42) are given by a formula similar to the Verlinde
formula Cardy:1989ir ; Behrend:1999bn

$n_{b_{1}b_{2}}^{i}=\sum_{m}\frac{S_{im}}{S_{0m}}K_{b_{1}}^{m}(K_{b_{2}}^{m})^{\ast},$
(65)

where $K_{b_{1}}^{m}$ are the expansion coefficients of boundary states in
terms of twisted Ishibashi states

$|b_{1}\rangle_{\omega}=\sum_{j}\frac{K_{b_{1}}^{j}}{\sqrt{S_{0j}}}|j\rangle\rangle_{\omega}.$
(66)

If the boundary conditions have the same labeling as the chiral representation
and set $K_{b_{1}}^{j}$ as corresponding $S_{ij}$, we arrive at the celebrated
$n=N$ relation Cardy:1989ir .

##### The non-vanishing conditions on both sides in general:

We first give the general non-vanishing conditions on both sides. We first
expand the $\mathcal{F}_{0,n,b,0}^{0}$ of interest in terms of the bulk
functions $\mathcal{F}_{0,(n+b)}$, then $\mathcal{F}_{0,(n+b)}$ will have a
non-zero element if and only if one of the $\mathcal{F}_{0,(n+b)}$ in the
expansion have admissible fusion rules Moore:1988ss . A UHP correlation
function is non-zero if and only if there exists the boundary identity
operator term after finishing all bulk-bulk, bulk-boundary, and boundary-
boundary OPEs Lewellen:1991tb ; Runkel:1998he . We will start the comparison
from the $b=1$ cases and increase the number of boundaries by one each step.

##### The $b=1$ comparisons:

We start from the the non-vanishing condition comparison of
$\mathcal{F}_{0,1,1,0}$ and
$\langle\phi_{i,\bar{i}}(z,\bar{z})\rangle_{b_{1}}$. The UHP bulk one-point
functions have the following form

$\langle\phi_{i,\bar{i}}(z,\bar{z})\rangle_{b_{1}}=\frac{A_{i\bar{i}}^{b_{1}}}{|z-\bar{z}|^{h_{i}+h_{\bar{i}}}}\delta_{h_{i}h_{\bar{i}}}\delta_{\bar{i}\omega^{-1}(i^{\ast})},$
(67)

where $\omega$ is the gluing automorphism of the boundary condition $b_{1}$.
$\langle\phi_{i,\bar{i}}(z,\bar{z})\rangle_{b_{1}}$ is non-zero if and only if
the bulk operator $\phi_{(i,\bar{i})}$ satisfies the $\omega$-mirror
reflection condition $\bar{i}=\omega^{-1}(i^{\ast})$, the level-matching
condition $h_{i}=h_{\bar{i}}$, and the overall constant is non-zero
$A_{i\bar{i}}^{b_{1}}\neq 0$.

On the
$\mathcal{F}_{0,1,1,0}=\langle\Sigma_{0,2}|\phi_{i,\bar{i}}\rangle\otimes|b_{1}\rangle_{\omega}^{r}$
side, we first need to expand the boundary state $|b_{1}\rangle_{\omega}^{r}$
in terms of $\omega$-twisted Ishibashi states using (66).
$\langle\Sigma_{0,2}|\phi_{i,\bar{i}}\rangle\otimes|b_{1}\rangle_{\omega}^{r}$
has a non-zero component if there exist a representation $j$, such that
$K_{b_{1}}^{j}\neq 0$ and

$N_{ij}^{0}\times N_{\bar{i}\omega^{-1}(j^{\ast})}^{0}\neq
0,\quad\Longrightarrow\quad i=j^{\ast},\>\bar{i}=\omega^{-1}(j),$ (68)

which is identical to the $\delta_{\bar{i}\omega^{-1}(i^{\ast})}$ in (67). The
$\delta_{h_{i}h_{\bar{i}}}$ constraint in (67) is ensured by the existence of
$j$ satisfying (68) and the gluing condition
$(L_{0}-\bar{L}_{0})|b_{1}\rangle_{\omega}^{r}=0$ . By mapping the UHP to a
semi-infinite cylinder, we can show that Recknagel:2013uja

$A_{i\bar{i}}^{b_{1}}\quad\propto\quad K_{b_{1}}^{i^{\ast}}.$ (69)

Combining all the discussions, we conclude that the non-vanishing conditions
of $\langle\phi_{i,\bar{i}}\rangle_{UHP}$ and $\mathcal{F}_{0,1,1,0}$ are
equivalent. The equivalence of
$\mathcal{F}_{0,1,1,0}$-$\langle\phi_{1}\rangle_{UHP}$ non-vanishing
conditions indicates the equivalence of the
$\mathcal{F}_{0,n,1,0}$-$\langle\phi_{1}\cdots\phi_{n}\rangle_{UHP}$ non-
vanishing conditions. Because the extra conditions on both sides are from the
fusion rules of the same set of bulk operators.

##### The $b=2$ comparisons:

The sphere two-boundary function
$\mathcal{F}_{0,0,2,0}^{0}=\langle\Sigma_{0,2}|b_{1}\rangle_{\omega_{1}}^{r_{1}}\otimes|b_{2}\rangle_{\omega_{2}}^{r_{2}}$
has a non-zero component if there exist a pair of representations $(i,j)$ in
the boundary state expansions such that

$K_{b_{1}}^{i}\neq 0,\quad K_{b_{2}}^{j}\neq 0;\quad N_{ij}^{0}\times
N_{\omega^{-1}_{1}(i^{\ast})\omega^{-1}_{2}(j^{\ast})}^{0}\neq 0.$ (70)

(70) indicates that $i=j^{\ast}$ and
$\omega^{-1}_{1}(i^{\ast})=\omega^{-1}_{2}(j)$ (this does not imply
$\omega_{1}=\omega_{2}$).

The corresponding UHP functions have a non-zero component when there exists a
pair of representations $(i^{\prime},j^{\prime})$ such that

$n_{b_{1}b_{2}}^{i^{\prime}}\neq 0,\quad n_{b_{2}b_{1}}^{j^{\prime}}\neq
0,\quad N_{i^{\prime}j^{\prime}}^{0}\neq 0.$ (71)

Our objective is to show that we can derive (71) from (70) and vice versa.

We will consider the $b=2$ comparison in the RCFT models with maximally
extended symmetry. Maximally extended RCFTs have Hilbert spaces bearing the
following form Dijkgraaf:1988tf ; Moore:1988ss

$\mathcal{H}=\bigoplus_{i}\mathcal{H}_{i}\otimes\mathcal{H}_{\lambda(i)},$
(72)

where $\lambda$ is a fusion rule automorphism such that
$N_{ij}^{k}=N_{\lambda(i)\lambda(j)}^{\lambda(k)}$, $\lambda(0)=0$. All the
Ishibashi states and boundary states in maximally extended RCFT models have
the same gluing automorphism $\omega=[\lambda^{-1}]^{\ast}$.

##### From (71) to (70):

$n_{b_{1}b_{2}}^{i^{\prime}}\neq 0$ in (71) indicates the existence of a
representation $m$ such that $K_{b_{1}}^{m}\neq 0$,
$(K_{b_{2}}^{m})^{\ast}\neq 0$ and $S_{i^{\prime}m}/S_{0m}\neq 0$. By setting
$i=m$ and $j=m^{\ast}$, we have shown the existence of $(i,j)$ satisfying

$K_{b_{1}}^{i}\neq 0,\quad K_{b_{2}}^{j}\neq 0,\quad N_{ij}^{0}\neq 0,$ (73)

which is not the $\mathcal{F}_{0,0,2,0}^{0}$ non-vanishing condition (70) yet.
In maximally extended RCFTs, $\omega_{1}=\omega_{2}$, and since $\omega$ are
fusion rule automorphisms $N_{ij}^{k}=N_{\omega(i)\omega(j)}^{\omega(k)}$,
with $\omega(0)=0$, indicating that
$N_{\omega^{-1}_{1}(i)\omega_{1}^{-1}(k)}^{0}=N_{ij}^{0}\neq 0$. We arrive at
the $\mathcal{F}_{0,0,2,0}$ non-vanishing condition (70).

##### From (70) to (71):

To show the $n_{b_{1}b_{2}}^{i^{\prime}}\neq 0$ condition in (71), we have to
first show the existence of a representation $m$ such that

$\frac{S_{i^{\prime}m}}{S_{0m}}\neq 0,\quad K_{b_{1}}^{m}\neq
0,\quad(K_{b_{2}}^{m})^{\ast}\neq 0.$ (74)

We can set $i=m$, then (70) ensures $K_{b_{1}}^{m}\neq 0$ and
$(K_{b_{2}}^{m})^{\ast}\neq 0$. Finally, we need to show the existence of
$i^{\prime}$ such that $S_{i^{\prime}m}\neq 0$. This is always true, which can
be seen from the definition of modular $S$ matrices

$\chi_{i^{\prime}}\big{(}-\frac{1}{\tau}\big{)}=\sum_{m}S_{i^{\prime}m}\>\chi_{m}(\tau).$
(75)

If all $S_{i^{\prime}m}=0$, then the RHS of (75) is zero. However, the fact
that the LHS is non-zero leads to a contradiction. This show that
$n_{b_{1}b_{2}}^{i^{\prime}}$ has at least one non-zero component. However, to
obtain $n_{b_{1}b_{2}}^{i^{\prime}}\neq 0$, we still need to show that all the
non-vanishing components do not cancel each other. This is difficult to show
without specifying a model. We expect it is necessary to exhaust all the Cardy
solutions in RCFT models Behrend:1999bn  to show it (left for future studies).
Once we have achieved this step, we can complete the derivation from (70) to
(71) by simply setting $j^{\prime}=(i^{\prime})^{\ast}$.

We have only partially proved the
$\mathcal{F}_{0,0,2,0}^{0}$-$\langle\psi_{i^{\prime}}^{b_{1}b_{2}}\psi_{j^{\prime}}^{b_{2}b_{1}}\rangle_{UHP}$
non-vanishing condition equivalence. If we can finish the proof for
$\mathcal{F}_{0,0,2,0}^{0}$-$\langle\psi_{i^{\prime}}^{b_{1}b_{2}}\psi_{j^{\prime}}^{b_{2}b_{1}}\rangle_{UHP}$
non-vanishing condition equivalence, we will be able to generalize it to
$\mathcal{F}_{0,n,2,0}-\langle\phi_{n}\cdots\phi_{1}\psi_{i^{\prime}}^{b_{1}b_{2}}\psi_{j^{\prime}}^{b_{2}b_{1}}\rangle_{UHP}$
non-vanishing condition equivalence straightforwardly. The reason is the same
as the $b=1$ case.

##### The $b\geq 3$ comparisons:

The $b\geq 3$ comparisons are complicated in general. We use the $b=3$ cases
as an example. Consider the following correlation function
$\mathcal{F}_{0,0,3,0}^{0}$

$\mathcal{F}_{0,0,3,0}^{0}=\langle\Sigma_{0,3}|b_{1}\rangle^{r_{1}}_{\omega_{1}}\otimes|b_{2}\rangle^{r_{2}}_{\omega_{2}}\otimes|b_{3}\rangle^{r_{3}}_{\omega_{3}},$
(76)

which has a non-zero component if there exists a representation triple
$(i,j,k)$ in the boundary state expansions such that

$K_{b_{1}}^{i}\neq 0,\quad K_{b_{2}}^{j}\neq 0,\quad K_{b_{3}}^{k}\neq 0;\quad
N_{ij}^{k^{\ast}}\times
N_{\omega_{1}^{-1}(i^{\ast})\omega_{2}^{-1}(j^{\ast})}^{\omega_{3}^{-1}(k)}\neq
0.$ (77)

The UHP side is a collection of UHP boundary operator three-point functions

$\int_{i^{\prime},\>j^{\prime},\>k^{\prime}}\langle\psi_{i^{\prime}}^{b_{1}b_{2}}\psi_{j^{\prime}}^{b_{2}b_{3}}\psi_{k^{\prime}}^{b_{3}b_{1}}\rangle_{UHP}.$
(78)

The non-vanishing condition of the UHP three boundary operator collection is:
there exists a representation triple $(i^{\prime},j^{\prime},k^{\prime})$ such
that

$n_{b_{1}b_{2}}^{i^{\prime}}\neq 0,\quad n_{b_{2}b_{3}}^{j^{\prime}}\neq
0,\quad n_{b_{3}b_{1}}^{k^{\prime}}\neq 0,\quad
N_{i^{\prime}j^{\prime}}^{k^{\prime\ast}}\neq 0.$ (79)

It is unapparent how (77) and (79) are related. Similar for the cases with
$b\geq 4$. It seems to be inevitable to use detailed models to study the non-
vanishing condition comparisons for $b\geq 3$ cases.

##### The summary of the comparison:

When $b=1$, $\mathcal{F}_{0,n,1,0}^{0}$ and corresponding UHP bulk $n$-point
functions have equivalent non-vanishing conditions. When $b=2$, we have
partially shown the equivalence between $\mathcal{F}_{0,n,2,0}^{0}$ and the
corresponding UHP functions in maximally extended RCFTs. When $b\geq 3$, the
comparison becomes complicated, with no conclusions made at present.

##  3 The expansion of $\mathcal{F}_{g,n,b,c}^{0}$ as linear combinations of
$\mathcal{F}_{g,(n+b+c)}$

In this section, we will focus on expanding genus-$g$, $n$-point,
$b$-boundary, $c$-crosscap correlation functions $\mathcal{F}_{g,n,b,c}^{0}$
in terms of genus-$g$, $(n+b+c)$-point functions $\mathcal{F}_{g,(n+b+c)}$.
Both the boundary states (with no boundary operator) and crosscap states are
spanned by Ishibashi states. Hence, the core of the
$\mathcal{F}_{g,n,b,c}^{0}$ expansions is the expansion of Ishibashi states.
We shall mainly focus on the expansions of boundary Ishibashi states in this
work, where the expansions of crosscap Ishibashi states are completely
parallel. Formally, Ishibashi states are expanded by left acting the identity
operator $I\otimes\bar{I}$ (8) on Ishibashi states. When the Ishibashi state
corresponds to the degenerate Verma modules, the terms from null sub-modules
will have single poles in their expansion coefficients. The single poles
structure in expansion coefficients is an analog to the single pole structure
in conformal block decomposition Zamolodchikov:1984eqp ; Hadasz:2006qb ;
Hadasz:2008dt ; Hadasz:2009db ; Hadasz:2012im ; Cho:2017oxl . We will show the
single pole structures explicitly when the degenerate Verma modules have a
unique null vector, and when they are the Virasoro and $N=1$ minimal
degenerate Verma modules.

In practice, we must first specify the choice of the chiral algebra
$\mathcal{A}$. The chiral algebras of interest are mainly Virasoro, $N=1$,
$N=2$, and Kac-Moody algebra since they are closely related to the CFTs on
string worldsheets Blumenhagen:2013fgp . Next, we write the level-$l$
orthonormal basis as linear combinations of level-$l$ standard basis and
impose the orthonormal condition to obtain the level-$l$ expansion. In
general, the expansion must be performed level-by-level. Nevertheless, when
the chiral algebra has simple commutation relations, we can obtain the
explicit Ishibashi state expansion up to level-$\infty$. This includes free
CFT models such as the free boson and the free fermion CFT Callan:1987px , and
specific limits of parameters such as the semi-classical $c_{Vir}\to\infty$
limits.

Since the Ishibashi state expansion is simple for free CFT models. We are
motivated to express Ishibashi states with complicated expansions as linear
combinations of free CFT Ishibashi states to simplify the expansion. In this
work, we will write the Ishibashi states in the Virasoro minimal models as
linear combinations of free CFT Ishibashi states.

###  3.1 Formal Ishibashi state expansions

Consider a twisted Ishibashi state related to $\mathcal{A}$ Verma module
$|V_{i}\rangle\rangle_{\omega}\in(V_{i}\otimes
V_{\omega}UV_{\omega^{-1}(i^{\ast})})$. The expansion is performed by left
acting $I\otimes\bar{I}$ in (8) on $|V_{i}\rangle\rangle_{\omega}$. We use the
simplest Virasoro algebra case $\mathfrak{Vir}\otimes\mathfrak{Vir}$ to
visualize the Ishibashi state expansion

$|V_{i}\rangle\rangle=\int_{(h,\bar{h})}\sum_{|M|=|N|=l}^{l\geq
0}\sum_{|\bar{M}|=|\bar{N}|=\bar{l}}^{\bar{l}\geq
0}|L_{-M}v_{h}\otimes\bar{L}_{-\bar{M}}\bar{v}_{\bar{h}}\rangle(G_{c_{Vir},h}^{l})^{MN}(G_{c_{Vir},\bar{h}}^{\bar{l}})^{\bar{M}\bar{N}}\langle
L_{-N}v_{h}\otimes\bar{L}_{-\bar{N}}\bar{v}_{\bar{h}}|V_{i}\rangle\rangle.$
(80)

Only the vectors
$|L_{-M}v_{h}\otimes\bar{L}_{-\bar{M}}\bar{v}_{\bar{h}}\rangle\in(V_{i}\otimes
UV_{i^{\ast}})$ will give non-zero contributions to the expansion (80).

###  3.2 The single pole structure in expansion coefficients

The inversion Gram matrices $(G_{c,h}^{l})^{MN}$ will result in single poles
in the coefficients when the Verma module $V_{h}$ is degenerate. In this work,
we shall prove the single pole structure in Ishibashi state expansion
coefficients explicitly. The null vector structures of degenerate Verma
modules heavily depend on the chiral algebra $\mathcal{A}$ and the value of
parameters. In this work, we focus on the single pole structures in two cases:

  * •

The degenerate Verma modules $V_{h}$ that have a unique null vector.

  * •

The degenerate Verma modules $V_{h}$ in the Virasoro minimal
$\mathcal{M}(p,p^{\prime})$ or $N=1$ minimal $\mathcal{SM}(p,p^{\prime})$
models. Those degenerate Verma modules have two infinite classes of null
vectors Belavin:1984vu ; Friedan:1984rv ; Kiritsis:1986rv .

####  3.2.1 Degenerate Verma modules with a unique null vector

The simplest degenerate Verma modules are the modules having a unique null
vector $\chi$. Consider the degenerate Virasoro Verma module $V_{r,s}$ with
primary conformal weight $h_{r,s}$ given by

$h_{r,s}=\frac{c_{Vir}-1}{24}+\frac{1}{4}(r\alpha_{+}^{Vir}+s\alpha_{-}^{Vir})^{2},\quad\alpha_{\pm}^{Vir}=\frac{\sqrt{1-c_{Vir}}\pm\sqrt{25-c_{Vir}}}{\sqrt{24}}.$
(81)

When $c_{Vir}\neq c_{Vir}^{p,p^{\prime}}$, $V_{r,s}$ has a single level-$rs$
null vector $\chi_{rs}$, generating a non-degenerate null sub-module
$V_{h_{r,s}+rs}$. The values $c_{Vir}^{p,p^{\prime}}$ are the solutions to the
rational conditions (95)

$c_{Vir}^{p,p^{\prime}}=1-\frac{6(p-p^{\prime})^{2}}{pp^{\prime}},$ (82)

where $p$ and $p^{\prime}$ are two positive integers that are coprime.

When $l\geq rs$, the determinants of the level-$l$ Gram matrices are zero
DiFrancesco:1997nk

$\det\>(G_{c_{Vir},h}^{l})_{MN}\quad\propto\quad(h-h_{r,s})^{p(l-rs)}=0,$ (83)

where $p(N)$ is the partition number of integer $N$ and $p(N):=0$ when $N<0$.
The zero determinants indicate the existence of singular terms in the
$|V_{r,s}\rangle\rangle$ expansion. However, the form of the determinants (83)
alone is not enough to determine which terms are singular and the order of the
singular terms. To determine the singular structure in detail, it is necessary
to perform specific linear transformations on the level-$l\geq rs$ standard
basis. Before detailed discussions, we need to introduce some relevant facts
and notations. The level-$rs$ null vector $\chi_{rs}$ is normalized as

$|\chi_{rs}\rangle=\sum^{M}_{|M|=rs}\chi_{rs}^{M}|L_{-M}v_{r,s}\rangle,\quad\chi_{rs}^{\\{1,1,\cdots,1\\}}=1,$
(84)

where $|v_{r,s}\rangle$ is the primary vector of the Verma module $V_{r,s}$.
We define vectors $|\chi_{rs}^{h}\rangle$ as

$|\chi_{rs}^{h}\rangle=\sum^{M}_{|M|=rs}\chi_{rs}^{M}|L_{-M}v_{h}\rangle,$
(85)

where $|v_{h}\rangle$ is a general primary vector with conformal weight $h$.
The norm of $|\chi_{rs}\rangle$ is zero, which is from a first-order zero at
the $h\to h_{r,s}$ limit

$\lim_{h\to
h_{r,s}}\>\langle\chi_{rs}^{h}|\chi_{rs}^{h}\rangle=[A_{rs}^{c_{Vir}}]^{-1}(h-h_{r,s}),$
(86)

where $A_{rs}^{c_{Vir}}$ are non-zero finite numbers, whose forms can be found
in Zamolodchikov:2003yb ; Cho:2017oxl . An immediate conclusion following (86)
is that the norms of null descendants $|L_{-M}\chi_{rs}\rangle$ process the
same first-order zero

$\lim_{h\to h_{r,s}}\>\langle
L_{-M}\chi_{rs}^{h}|L_{-M}\chi_{rs}^{h}\rangle=[A_{rs}^{c_{Vir}}]^{-1}(h-h_{r,s})\times\langle
L_{-M}v_{h_{r,s}+rs}|L_{-M}v_{h_{r,s}+rs}\rangle,$ (87)

where $\langle L_{-M}v_{h_{r,s}+rs}|L_{-M}v_{h_{r,s}+rs}\rangle$ are non-zero
finite numbers since the null sub-module $V_{h_{r,s}+rs}$ is non-degenerate.

We introduce the null basis $\\{|h_{r,s},l,\text{nl}\rangle\\}$ for
level-$l\geq rs$ by fixing the last $p(l-rs)$ basis vectors to be
$|L_{-M}\chi_{rs}^{h}\rangle$, where $|M|=l-rs$. For the level-$l\geq rs$, the
order of all entities in the Gram matrix of the null basis
$(H_{c_{Vir},h}^{l})_{ab}$ are

$\lim_{h\to
h_{r,s}}(H_{c_{Vir},h}^{l})_{ab}=\begin{pmatrix}O(h-h_{r,s})^{0}&\cdots&O(h-h_{r,s})^{0}&O(h-h_{r,s})^{\frac{1}{2}}&\cdots&O(h-h_{r,s})^{\frac{1}{2}}\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
O(h-h_{r,s})^{0}&\cdots&O(h-h_{r,s})^{0}&O(h-h_{r,s})^{\frac{1}{2}}&\cdots&O(h-h_{r,s})^{\frac{1}{2}}\\\
O(h-h_{r,s})^{\frac{1}{2}}&\cdots&O(h-h_{r,s})^{\frac{1}{2}}&O(h-h_{r,s})^{1}&\cdots&O(h-h_{r,s})^{1}\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
O(h-h_{r,s})^{\frac{1}{2}}&\cdots&O(h-h_{r,s})^{\frac{1}{2}}&O(h-h_{r,s})^{1}&\cdots&O(h-h_{r,s})^{1}\end{pmatrix}.$
(88)

$(H_{c_{Vir},h_{r,s}}^{l})_{ab}$ is a block diagonal matrix with a
$[p(l)-p(l-rs)]\times[p(l)-p(l-rs)]$ sized non-zero block and a $p(l-rs)\times
p(l-rs)$ sized zero block. From (88), we conclude that the determinant of
$(H_{c_{Vir},h_{r,s}}^{l})_{ab}$ is at order $(h-h_{r,s})^{p(l-rs)}$. (88)
also indicates that the determinant of the $p(l-rs)\times p(l-rs)$ sized zero
block has determinant at order $(h-h_{r,s})^{p(l-rs)}$ and the
$[p(l)-p(l-rs)]\times[p(l)-p(l-rs)]$ sized non-zero block has a non-zero
determinant.

By rewriting the identity operator $I\otimes\bar{I}$ in terms of null basis,
it is clear that all the singular terms in the $|V_{r,s}\rangle\rangle$
expansion are from the null sub-modules $V_{h_{r,s}+rs}$. To determine the
order, we write down the singular terms in the level-$l$ expansion

$\text{level $l$ singular
terms}=\sum_{a,\bar{a}\geq[p(j)-p(j-rs)+1]}^{b,\bar{b}\geq[p(j)-p(j-rs)+1]}|(h_{r,s},l,a,\text{nl})\otimes(h_{r,s},\bar{l},\bar{a},\text{nl})\rangle$
$[H_{c_{Vir},h_{r,s}}^{l}]^{ab}[H_{c_{Vir},h_{r,s}}^{\bar{l}}]^{\bar{a}\bar{b}}\langle(h_{r,s},l,b,\text{nl})\otimes(h_{r,s},\bar{l},\bar{b},\text{nl})|V_{r,s}\rangle\rangle.$
(89)

All entities in the zero blocks process are at the order $O(h-h_{r,s})$
implying that all entities in the inversion zero blocks are at the order
$O(h-h_{r,s})^{-1}$. Hence the holomorphic and the anti-holomorphic inversion
Gram matrices contribute the order
$O(h-h_{r,s})^{-1}O(\bar{h}-h_{r,s})^{-1}=O(h-h_{r,s})^{-2}$ ($h=\bar{h}$ is
from the $(L_{0}-\bar{L}_{0})|V_{r,s}\rangle\rangle=0$ gluing condition). The
inner products
$\langle(h_{r,s},l,b,\text{nl})\otimes(h_{r,s},\bar{l},\bar{b},\text{nl})|V_{r,s}\rangle\rangle$
contribute the order $O(h-h_{r,s})^{1/2}O(\bar{h}-h_{r,s})^{1/2}=O(h-h_{r,s})$
since the Ishibashi state $|V_{r,s}\rangle\rangle$ is the summation over
orthonormal vectors. Combining all contributions, we conclude that all the
singular terms in (89) are at the order $O(h-h_{r,s})^{-1}$.

The block diagonalization of the Gram matrix under a linear transformation
indicates that the expansion of $|V_{r,s}\rangle\rangle$ factorizes to a
regular term corresponding to the vectors in the irreducible representation
$|\mathcal{H}_{r,s}\rangle\rangle\in\mathcal{H}_{r,s}\otimes
U\mathcal{H}_{r,s}$, and a singular term corresponding to the vectors in the
null sub-module $|V_{h_{r,s}+rs}\rangle\rangle\in V_{h_{r,s}+rs}\otimes
UV_{h_{r,s}+rs}$

$|V_{r,s}\rangle\rangle=|\mathcal{H}_{r,s}\rangle\rangle+|V_{h_{r,s}+rs}\rangle\rangle=|\mathcal{H}_{r,s}\rangle\rangle+\frac{A_{rs}^{c_{Vir}}}{(h-h_{r,s})}\Big{[}|\chi_{rs}\otimes
U\bar{\chi}_{rs}\rangle+\cdots\Big{]}.$ (90)

which is a reflection of the decoupling of null sub-modules

$\mathcal{H}_{r,s}=V_{r,s}/V_{h_{r,s}+rs}.$ (91)

The equation (90) shows that the single pole structure originates from
$(\langle\chi_{rs}^{h}|\chi_{rs}^{h}\rangle)^{-1}$.

We check whether equation (90) can correctly reproduce the irreducible
character by taking the overlap of irreducible Ishibashi states ($q:=e^{2\pi
i\tau}$, $\tau$ is the modular parameter of a finite cylinder)

$\langle\langle\mathcal{H}_{r,s}|q^{\frac{1}{2}(L_{0}+\bar{L}_{0}-\frac{c_{Vir}}{12})}|\mathcal{H}_{r,s}\rangle\rangle=\langle\langle
V_{r,s}|q^{\frac{1}{2}(L_{0}+\bar{L}_{0}-\frac{c_{Vir}}{12})}|V_{r,s}\rangle\rangle$
$-2\langle\langle
V_{r,s}|q^{\frac{1}{2}(L_{0}+\bar{L}_{0}-\frac{c_{Vir}}{12})}|V_{h_{r,s}+rs}\rangle\rangle+\langle\langle
V_{h_{r,s}+rs}|q^{\frac{1}{2}(L_{0}+\bar{L}_{0}-\frac{c_{Vir}}{12})}|V_{h_{r,s}+rs}\rangle\rangle.$
(92)

The crossed term $\langle\langle
V_{r,s}|q^{\frac{1}{2}(L_{0}+\bar{L}_{0}-\frac{c_{Vir}}{12})}|V_{h_{r,s}+rs}\rangle\rangle=q^{(1-c_{Vir})/24}q^{h_{r,s}+rs}/\eta(q)$
is non-zero since $V_{h_{r,s}+rs}\subset V_{r,s}$, leading to the correct
reproduction of the irreducible character from the overlap

$\chi_{r,s}(q)=\frac{q^{(1-c_{Vir})/24}(q^{h_{r,s}}-2q^{h_{r,s}+rs}+q^{h_{r,s}+rs})}{\eta(q)}=\frac{q^{(1-c_{Vir})/24}(q^{h_{r,s}}-q^{h_{r,s}+rs})}{\eta(q)},$
(93)

where $\eta(q):=q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^{n})$ is the Dedekind
$\eta$-function.

Properties similar to (90) should be satisfied by Ishibashi state
$|V_{\text{de}}\rangle\rangle$ related to general degenerate
$\mathcal{A}$-Verma modules with a single null vector $|\chi\rangle$

$|V_{\text{de}}\rangle\rangle_{\omega}=|\mathcal{H}\rangle\rangle_{\omega}+|V_{\text{nul}}\rangle\rangle_{\omega}=|\mathcal{H}\rangle\rangle_{\omega}+\frac{1}{(\langle\chi|\chi\rangle)}\Big{[}|\chi\rangle\otimes
V_{\omega}U|\bar{\chi}\rangle+\cdots\Big{]}.$ (94)

The single poles can be expressed by different variables such as the conformal
weight $h$, the Virasoro central charge $c_{Vir}$ and the Kac-Moody level $k$,
whenever the norm of the corresponding null vector $|\chi\rangle$ depends on
those variables.

####  3.2.2 Minimal model degenerate Verma modules

Models describing physical systems often have more complicated null vector
structures than a unique null vector. For example, the degenerate Verma
modules $V_{r,s}^{p,p^{\prime}}$ in Virasoro minimal models
$\mathcal{M}(p,p^{\prime})$ and $N=1$ minimal models
$\mathcal{SM}(p,p^{\prime})$ have two infinite classes of null vectors
Belavin:1984vu ; Friedan:1984rv ; Kiritsis:1986rv ; DiFrancesco:1997nk .

In the sequels, we consider the Virasoro minimal models
$\mathcal{M}(p,p^{\prime})$ as an example, while $N=1$ minimal models have
similar structures. Virasoro minimal models $\mathcal{M}(p,p^{\prime})$ are
RCFTs with finite numbers of irreducible representations
$\mathcal{H}_{r,s}^{p,p^{\prime}}$. The finiteness of irreducible
representations leads to the following rational condition

$p^{\prime}\alpha_{+}^{Vir}+p\alpha_{-}^{Vir}=0.$ (95)

The rational conditions indicate that the minimal model degenerate module
$V_{r,s}^{p,p^{\prime}}$ has two distinct null vectors at level-$rs$ and
level-$(p^{\prime}-r)(p-s)$ respectively

$h_{r,s}^{p,p^{\prime}}=h_{(p^{\prime}-r)(p-s)}^{p,p^{\prime}},$ (96)
$h_{r,s}+rs=h_{p^{\prime}+r,p-s}^{p,p^{\prime}}=h_{p^{\prime}-r,p+s}^{p,p^{\prime}},\quad
h_{r,s}+(p^{\prime}-r)(p-s)=h_{r,2p-s}^{p,p^{\prime}}=h_{2p^{\prime}-r,s}.$
(97)

(97) indicates that

$\big{(}V_{p^{\prime}+r,p-s}^{p,p^{\prime}}\cup
V_{r,2p-s}^{p,p^{\prime}}\big{)}\subset V_{r,s}^{p,p^{\prime}}.$ (98)

An irreducible representation is obtained by factoring out the direct sum of
these two null sub-modules

$\mathcal{H}_{r,s}^{p,p^{\prime}}=V_{r,s}^{p,p^{\prime}}/\big{[}V_{p^{\prime}+r,p-s}^{p,p^{\prime}}\oplus
V_{r,2p-s}^{p,p^{\prime}}\big{]}.$ (99)

where the element $\big{[}V_{p^{\prime}+r,p-s}^{p,p^{\prime}}\oplus
V_{r,2p-s}^{p,p^{\prime}}\big{]}$ is complicated since
$V_{p^{\prime}+r,p-s}^{p,p^{\prime}}$ and $V_{r,2p-s}^{p,p^{\prime}}$ share
the same two null sub-modules. The two null vectors of
$V_{p^{\prime}+r,p-s}^{p,p^{\prime}}\equiv
V_{p^{\prime}-r,p+s}^{p,p^{\prime}}$ are at levels $(p^{\prime}+r)(p-s)$ and
$(p^{\prime}-r)(p+s)$, indicating

$\big{(}V_{2p^{\prime}+r,s}^{p,p^{\prime}}\cup
V_{r,2p+s}^{p,p^{\prime}}\big{)}\subset V_{p^{\prime}+r,p-s}^{p,p^{\prime}}.$
(100)

Similarly, the two null vectors of $V_{r,2p-s}^{p,p^{\prime}}\equiv
V_{2p^{\prime}-r,s}^{p,p^{\prime}}$ at the levels $r(2p-s)$ and
$s(2p^{\prime}-r)$, indicating

$\big{(}V_{p^{\prime}-r,3p-s}^{p,p^{\prime}}\cup
V_{3p^{\prime}-r,p-s}^{p,p^{\prime}}\big{)}\subset V_{r,2p-s}^{p,p^{\prime}}.$
(101)

The two null sub-modules of $V_{p^{\prime}+r,p-s}^{p,p^{\prime}}$ and
$V_{r,2p-s}^{p,p^{\prime}}$ can be identified:
$V_{2p^{\prime}+r}^{p,p^{\prime}}\equiv V_{p^{\prime}-r,3p-s}^{p,p^{\prime}}$
and $V_{r,2p+s}^{p,p^{\prime}}\equiv V_{3p^{\prime}-r,p-s}^{p,p^{\prime}}$.
The direct sum is obtained by taking the following coset

$V_{p^{\prime}+r,p-s}^{p,p^{\prime}}\oplus
V_{r,2p-s}^{p,p^{\prime}}=\big{[}V_{p^{\prime}+r,p-s}^{p,p^{\prime}}\cup
V_{r,2p-s}^{p,p^{\prime}}\big{]}/\big{[}V_{2p^{\prime}+r,s}^{p,p^{\prime}}\oplus
V_{r,2p+s}^{p,p^{\prime}}\big{]}.$ (102)

The direct sum $\big{[}V_{2p^{\prime}+r,s}^{p,p^{\prime}}\oplus
V_{r,2p+s}^{p,p^{\prime}}\big{]}$ is again complicated, since the two modules
$V_{2p^{\prime}+r,s}^{p,p^{\prime}}$ and $V_{r,2p+s}^{p,p^{\prime}}$ also
share two null sub-modules.

This procedure can be iterated infinite times, with two new null sub-modules
appearing at each step. Hence, the Verma modules in minimal models
$V_{r,s}^{p,p^{\prime}}$ contain two infinite classes of null vectors, as
shown in Figure 9. From this null vector structure, the irreducible
representations $\mathcal{H}_{r,s}^{p,p^{\prime}}$ can be written as

$\mathcal{H}_{r,s}^{p,p^{\prime}}=V_{r,s}^{p,p^{\prime}}+\sum_{k=1}^{\infty}(-1)^{k}\big{[}V_{r+kp^{\prime},(-1)^{s}+\frac{p}{2}[1-(-1)^{k}]}^{p,p^{\prime}}\cup
V_{r,kp+(-1)^{k}s+\frac{p}{2}[1-(-1)^{k}]}^{p,p^{\prime}}\big{]},$ (103)

with the corresponding irreducible characters $\chi_{r,s}^{p,p^{\prime}}(q)$
are given by

$\chi_{r,s}^{p,p^{\prime}}(q)=\frac{q^{\frac{1-c_{Vir}}{24}}}{\eta(q)}\Big{[}q^{h_{r,s}}+\sum_{k=1}^{\infty}(-1)^{k}(q^{h_{r+k^{\prime}p,(-1)^{k}s+[1-(-1)^{k}]\frac{p}{2}}}+q^{h_{r,kp+(-1)^{k}s+[1-(-1)^{k}]\frac{p}{2}}})\Big{]}.$
(104)

$(r,s)$$(r,2p-s)$$(p^{\prime}+r,p-s)$$(r,2p+s)$$(2p^{\prime}+r,s)$$(r,kp+(-1)^{k}s+p[1-(-1)^{k}]/2)$$(r+kp^{\prime},(-1)^{k}s+p[1-(-1)^{k}]/2)$
Figure 9: The two infinite classes of null vectors of the degenerate Verma
module $V_{r,s}^{p,p^{\prime}}$ in the Virasoro minimal model
$\mathcal{M}(p,p^{\prime})$.

From the form of irreducible characters $\mathcal{H}_{r,s}^{p,p^{\prime}}$
(103), we will write down the irreducible representations

$|\mathcal{H}_{r,s}^{p,p^{\prime}}\rangle\rangle=|V_{r,s}^{p,p^{\prime}}\rangle\rangle+\sum_{k=1}^{\infty}(-1)^{k}\big{[}|V_{r+kp^{\prime},(-1)^{s}+\frac{p}{2}[1-(-1)^{k}]}\rangle\rangle+|V_{r,kp+(-1)^{k}s+\frac{p}{2}[1-(-1)^{k}]}\rangle\rangle\big{]}.$
(105)

We want to show all the single poles in the expansion of
$|V_{r,s}^{p,p^{\prime}}\rangle\rangle$. At the first floor of the infinite
double null vector tower in Figure 9, Ishibashi state
$|V_{r,s}^{p,p^{\prime}}\rangle\rangle$ factorizes as

$|V_{r,s}^{p,p^{\prime}}\rangle\rangle=|\mathcal{H}_{r,s}^{p,p^{\prime}}\rangle\rangle+\Big{[}|V_{(p^{\prime}+r),(p-s)}^{p,p^{\prime}}\rangle\rangle+|V_{r,(2p-s)}^{p,p^{\prime}}\rangle\rangle\Big{]}-\cdots=|\mathcal{H}_{r,s}^{p,p^{\prime}}\rangle\rangle+$
$\frac{A_{rs}^{p,p^{\prime}}}{(h-h_{r,s}^{p,p^{\prime}})}\Big{[}|\chi_{rs}^{p,p^{\prime}}\otimes
U\bar{\chi}_{rs}^{p,p^{\prime}}\rangle+\cdots\Big{]}+\frac{A_{(p^{\prime}-r)(p-s)}^{p,p^{\prime}}}{(h-h_{(p^{\prime}-r)(p-s)}^{p,p^{\prime}})}\Big{[}|\chi_{(p^{\prime}-r)(p-s)}^{p,p^{\prime}}\otimes
U\bar{\chi}_{(p^{\prime}-r)(p-s)}^{p,p^{\prime}}\rangle+\cdots\Big{]}-\cdots,$
(106)

where the $-\cdots$ denotes the elimination of vectors from
$V_{(p^{\prime}+r),(p-s)}^{p,p^{\prime}}\cap V_{r,(2p-s)}^{p,p^{\prime}}$.
$|\chi_{l}^{p,p^{\prime}}\rangle$ denotes the level-$l$ null vector in the
Virasoro minimal model $\mathcal{M}(p,p^{\prime})$. In the $k^{\text{th}}$
floor in the infinity double null vector tower, the Ishibashi state factorizes
as

$(-1)^{k}\big{(}|V_{r+kp^{\prime},(-1)^{k}s+[1-(-1)^{k}]\frac{p}{2}}^{p,p^{\prime}}\rangle\rangle+|V_{r,kp+(-1)^{k}s+[1-(-1)^{k}]\frac{p}{2}}^{p,p^{\prime}}\rangle\rangle-\cdots\big{)}=$
$(-1)^{k}\Big{\\{}|\mathcal{H}_{r+kp^{\prime},(-1)^{k}s+[1-(-1)^{k}]\frac{p}{2}}^{p,p^{\prime}}\rangle\rangle+|\mathcal{H}_{r,kp+(-1)^{k}s+[1-(-1)^{k}]\frac{p}{2}}^{p,p^{\prime}}\rangle\rangle+$
$\frac{A_{r+kp^{\prime},(-1)^{k}s+[1-(-1)^{k}]\frac{p}{2}}^{p,p^{\prime}}}{(h-h_{r+kp^{\prime},(-1)^{k}s+[1-(-1)^{k}]\frac{p}{2}}^{p,p^{\prime}})}\Big{[}\cdots\Big{]}+\frac{A_{r,kp+(-1)^{k}s+[1-(-1)^{k}]\frac{p}{2}}^{p,p^{\prime}}}{(h-h_{r,kp+(-1)^{k}s+[1-(-1)^{k}]\frac{p}{2}}^{p,p^{\prime}})}\Big{[}\cdots\Big{]}-\cdots\Big{\\}}.$
(107)

In summary, there exist single poles at
$h\to(h_{r+kp^{\prime},(-1)^{k}s+[1-(-1)^{k}]\frac{p}{2}}^{p,p^{\prime}}=h_{r,kp+(-1)^{k}s+[1-(-1)^{k}]\frac{p}{2}}^{p,p^{\prime}})$,
$\forall k\in\mathbb{N}$ limits, in the expansion of
$|V_{r,s}^{p,p^{\prime}}\rangle\rangle$.

We expect the minimal characters $\chi_{r,s}^{p,p^{\prime}}(q)$ can be
reproduced from the self-overlap of
$|\mathcal{H}_{r,s}^{p,p^{\prime}}\rangle\rangle$. The computation is not
shown here since we focus on finding the single poles in the expansion
coefficients.

###  3.3 Ishibashi state expansions in practice

We demonstrate some general Ishibashi state expansion at the lowest levels and
check the single pole structures in the expansions. We will compute some of
the Virasoro, $N=1$, $N=2$, and Kac-Moody Ishibashi states expansions. We will
also describe the cases where the full expansions up to level-$\infty$ are
known, including the celebrated free CFT Ishibashi states Callan:1987px  and
the $c_{Vir}\to\infty$ semi-classical limit.

####  3.3.1 Virasoro Ishibashi state expansions

Consider a general Virasoro Ishibashi state expansion $|V_{h}\rangle\rangle\in
V_{h}\otimes UV_{h}$ up level-$2$. At level-$0$ and level-$1$, the expansion
is simple

$|v_{h}\otimes U\bar{v}_{h}\rangle+\frac{1}{2h}|L_{-1}v_{h}\otimes
U\bar{L}_{-1}\bar{v}_{h}\rangle.$ (108)

The coefficient for the $|L_{-1}v_{h}\otimes U\bar{L}_{-1}v_{h}\rangle$ term
has a single pole at the $h\to h_{1,1}=0$ limit, which is from
$(\langle\chi_{1}^{h}|\chi_{1}^{h}\rangle)^{-1}$. The level-$2$ expansion is
less simple. Nevertheless, the result can be neatly summarized as

$\frac{9}{64(h+\frac{1}{2})(h-h_{2,1})(h-h_{1,2})}|\chi_{2}^{h}\otimes
U\bar{\chi}_{2}^{h}\rangle+\frac{1}{8(h-h_{1,1})(h+\frac{1}{2})}|L_{-1}\chi_{1}^{h}\otimes
U\bar{L}_{-1}\bar{\chi}_{1}^{h}\rangle,$ (109)

where the vector $|\chi_{2}^{h}\rangle$ is

$|\chi_{2}^{h}\rangle=\Big{[}-\frac{2(2h+1)}{3}L_{-2}+L_{-1}^{2}\Big{]}|v_{h}\rangle.$
(110)

The single poles at the $h\to h_{2,1}$ and $h_{1,2}$ limits are from
$(\langle\chi_{2}^{h}|\chi_{2}^{h}\rangle)^{-1}$. The single pole at the $h\to
h_{1,1}$ limit is from $(\langle
L_{-1}\chi_{1}^{h}|L_{-1}\chi_{1}^{h}\rangle)^{-1}$. There exists an extra off
Kac table single pole at the $h\to-\frac{1}{2}$ limit from both
$(\langle\chi_{2}^{h}|\chi_{2}^{h}\rangle)^{-1}$ and $(\langle
L_{-1}\chi_{1}^{h}|L_{-1}\chi_{1}^{h}\rangle)^{-1}$.

##### Virasoro Ishibashi state expansion at the $c_{Vir}\to\infty$ limit:

We end the practice of Virasoro Ishibashi state expansions by considering the
semi-classical limit $c_{Vir}\to\infty$, with $h$ fixed to be finite. At this
limit, the non-zero elements in the inversion Gram matrices are
$(G_{c_{Vir},h}^{l})^{L_{-1}^{l},L_{-1}^{l}}$, $l\in\mathbb{N}$, which
indicates that we only need to compute the norms of $|L_{-1}^{l}v_{h}\rangle$
to determine the full Virasoro Ishibashi state expansion up to level-$\infty$
at the $c_{Vir}\to\infty$ limit

$\langle L_{-1}^{l}v_{h}|L_{-1}^{l}v_{h}\rangle=l!(2h)_{l},$
$\lim_{c_{Vir}\to\infty}|V_{h}\rangle\rangle=\sum_{l=0}^{\infty}\frac{1}{l!(2h)_{l}}|L_{-1}^{l}v_{h}\otimes
U\bar{L}_{-1}^{l}\bar{v}_{h}\rangle,$ (111)

where $a_{l}:=a(a+1)\cdots(a+l-1)$ are the Pochhammer symbols. The single
poles in coefficients only exist at the $h\to h_{1,1}$ limit, which are from
$(\langle L_{-1}^{l}v_{1,1}|L_{-1}^{l}v_{1,1}\rangle)^{-1}$,
$l\in\mathbb{Z}^{+}$.

####  3.3.2 $N=1$ Ishibashi states expansions

$N=1$ superconformal field theories (SCFTs) Hilbert spaces factorize into the
NS and the R sectors Friedan:1984rv . $N=1$ Ishibashi state expansions must be
considered separately in the two sectors. $N=1$ Ishibashi states
$|V_{h}^{N=1}\rangle\rangle_{\pm}$ allow two types of gluing automorphisms
$V_{\omega}=\pm I$, corresponding to the two types of gluing conditions

$\big{(}G_{r}\mp i\bar{G}_{-r}\big{)}|V_{h}^{N=1}\rangle\rangle_{\pm}=0.$
(112)

##### NS sector expansion:

We expand a general NS $N=1$ Ishibashi state
$|V_{h}^{\text{NS};N=1}\rangle\rangle_{\pm}$ up to level-$\frac{3}{2}$. The
expansion up to level-$1$ is simple

$|v_{h}\otimes
U\bar{v}_{h}\rangle+\frac{1}{2(h-h_{1,1}^{N=1})}|G_{-\frac{1}{2}}v_{h}\otimes(\pm
U)\bar{G}_{-\frac{1}{2}}\bar{v}_{h}\rangle+\frac{1}{2(h-h_{1,1}^{N=1})}|L_{-1}v_{h}\otimes
U\bar{L}_{-1}\bar{v}_{h}\rangle.$ (113)

The single poles at the $h\to h_{1,1}^{N=1}$ limit are from
$(\langle\chi_{\frac{1}{2}}^{h;N=1,\text{NS}}|\chi_{\frac{1}{2}}^{h;N=1,\text{NS}}\rangle)^{-1}$
and
$(\langle
G_{-\frac{1}{2}}\chi_{\frac{1}{2}}^{h;N=1,\text{NS}}|G_{-\frac{1}{2}}\chi_{\frac{1}{2}}^{h;N=1,\text{NS}}\rangle)^{-1}$,
where
$|\chi_{\frac{1}{2}}^{h;N=1,\text{NS}}\rangle=G_{-\frac{1}{2}}|v_{h}\rangle$.
The less simple level-$\frac{3}{2}$ expansion can be summarized as

$\frac{(h+\frac{\hat{c}}{2})}{4(h-h_{1,1}^{N=1})(h-h_{3,1}^{N=1})(h-h_{1,3}^{N=1})}|\chi_{\frac{3}{2}}^{h;N=1,\text{NS}}\otimes(\pm
U)\bar{\chi}_{\frac{3}{2}}^{h;N=1,\text{NS}}\rangle+\frac{1}{(2h+\hat{c})}|G_{-\frac{3}{2}}v_{h}\otimes(\pm
U)\bar{G}_{-\frac{3}{2}}\bar{v}_{h}\rangle,$ (114)

where $\hat{c}=\frac{2}{3}c_{Vir}$. The single poles at $h\to h_{1,1}^{N=1}$,
$h_{3,1}^{N=1}$ and $h_{1,3}^{N=1}$ limits are from
$(\langle\chi_{\frac{3}{2}}^{h;N=1,\text{NS}}|\chi_{\frac{3}{2}}^{h;N=1,\text{NS}}\rangle)^{-1}$,
where

$|\chi_{\frac{3}{2}}^{h;N=1,\text{NS}}\rangle=\Big{[}\frac{6h}{(c+3h)}G_{-\frac{3}{2}}-G_{-\frac{1}{2}}L_{-1}\Big{]}|v_{h}\rangle.$
(115)

The off Kac table single pole at the $h\to-\hat{c}/2$ limit is from $(\langle
G_{-\frac{3}{2}}v_{h}|G_{-\frac{3}{2}}v_{h}\rangle)^{-1}$.

##### The $\hat{c}\to\infty$ limit of NS $N=1$ Ishibashi state expansions:

At the $\hat{c}\to\infty$ limit, the non-zero contributions to NS $N=1$
Ishibashi state expansion are from $|L_{-1}^{l}v_{h}\rangle$ and
$|G_{-\frac{1}{2}}L_{-1}^{l}v_{h}\rangle$. We only need to compute the norms
of those vectors to fully determine the NS $N=1$ Ishibashi state expansion up
to level-$\infty$ at the $\hat{c}\to\infty$ limit

$\lim_{\hat{c}\to\infty}|V_{h}^{\text{NS};N=1}\rangle\rangle_{\pm}=\sum_{l=0}^{\infty}\Big{[}\frac{|L_{-1}^{l}v_{h}\otimes
U\bar{L}_{-1}^{l}\bar{v}_{h}\rangle}{l!(2h)_{l}}+\frac{|G_{-\frac{1}{2}}L_{-1}^{l}v_{h}\otimes(\pm
U)\bar{G}_{-\frac{1}{2}}\bar{L}_{-1}^{l}\bar{v}_{h}\rangle}{l!(2h)_{l+1}}\Big{]}.$
(116)

The poles at the $h\to h_{1,1}^{N=1}=0$ limit are from $(\langle
L_{-1}^{l+1}v_{h}|L_{-1}^{l+1}v_{h}\rangle)^{-1}$ and $(\langle
G_{-\frac{1}{2}}L_{-1}^{l}v_{h}|G_{-\frac{1}{2}}L_{-1}^{l}v_{h}\rangle)^{-1}$,
$l\in\mathbb{N}$.

##### R sector expansions:

In the R sector, the Hilbert space is further separated into the spin $(+)$
and spin $(-)$ sub-sectors. The spin eigenvalues $(-1)^{\\#G}$ are determined
by the total number of supercurrent modes $\\#G$ acting on the spin $(+)$
primary vector $|v_{h}^{+}\rangle$. Vectors from different spin sub-sectors
are orthogonal to each other.

The R sector $N=1$ Ishibashi state expansion at level-$0$ is (including both
spin $(\pm)$ sub-sectors)

$|v_{h}^{+}\otimes
U\bar{v}_{h}^{+}\rangle+\frac{1}{(h-\frac{\hat{c}}{16})}|v_{h}^{-}\otimes(\pm
U)\bar{v}_{h}^{-}\rangle,$ (117)

where $|v_{h}^{-}\rangle:=G_{0}|v_{h}^{+}\rangle$, which is a null vector when
$h=\frac{\hat{c}}{16}$. The spin $(+)$ sub-sector expansion at level-$1$ is

$\frac{(h+\frac{3\hat{c}}{16})}{2(h-h_{2,1}^{N=1})(h-h_{1,2}^{N=1})}|\chi_{1}^{h;N=1,\text{R},+}\otimes
U\bar{\chi}_{1}^{h;N=1,\text{R},+}\rangle+\frac{1}{2(h+\frac{3\hat{c}}{16})(h-\frac{\hat{c}}{16})}|G_{-1}v_{h}^{-}\otimes
U\bar{G}_{-1}\bar{v}_{h}^{-}\rangle,$ (118)

where $|\chi_{1}^{h;N=1,\text{R},+}\rangle$ is the spin $(+)$ level-$1$ null
vector

$|\chi_{1}^{h;N=1,\text{R},+}\rangle=-\frac{3}{4(h+\frac{3\hat{c}}{16})}G_{-1}|v^{-}_{h}\rangle+L_{-1}|v^{+}_{h}\rangle.$
(119)

The poles at the $h\to h_{2,1}^{N=1}$ and $h_{1,2}^{N=1}$ limits are from
$(\langle\chi_{1}^{h;N=1,\text{R},+}|\chi_{1}^{h;N=1,\text{R},+}\rangle)^{-1}$
and the off Kac table poles at $h\to\frac{\hat{c}}{16}$ and
$-\frac{3\hat{c}}{16}$ limits are from $(\langle
G_{-1}v_{-}|G_{-1}v_{-}\rangle)^{-1}$. The spin $(-)$ sub-sector expansion at
level-$1$ is

$\frac{(h+\frac{3\hat{c}}{16})}{2(h-h_{2,1}^{N=1})(h-h_{1,2}^{N=1})(h-\frac{\hat{c}}{16})}|\chi_{1}^{h;N=1,\text{R},-}\otimes(\pm
U)\bar{\chi}_{1}^{h;N=1,\text{R},-}\rangle+\frac{1}{2(h+\frac{3\hat{c}}{16})}|G_{-1}v^{+}_{h}\otimes(\pm
U)\bar{G}_{-1}\bar{v}^{+}_{h}\rangle,$ (120)

where $|\chi_{1}^{h;\text{R},-}\rangle$ is the spin $(-)$ level-$1$ null
vector

$|\chi_{1}^{h;N=1,\text{R},-}\rangle=|L_{-1}v^{-}_{h}\rangle-\frac{3(h-\frac{\hat{c}}{16})}{4(h+\frac{3\hat{c}}{16})}|G_{-1}v^{+}_{h}\rangle.$
(121)

The poles at the $h\to\frac{\hat{c}}{16}$, $h_{2,1}^{N=1}$ and $h_{1,2}^{N=1}$
limits are from
$(\langle\chi_{1}^{h;N=1,\text{R},-}|\chi_{1}^{h;N=1,\text{R},-}\rangle)^{-1}$,
and the pole at $h\to\frac{-3\hat{c}}{16}$ limit is from $(\langle
G_{-1}v_{h}^{+}|G_{-1}v_{h}^{+}\rangle)^{-1}$.

##### The $\hat{c}\to\infty$ limit of the $N=1$ R Ishibashi state expansions:

At the $\hat{c}\to\infty$ limit, the non-zero contributions in the R sector
Ishibashi state expansions are from $|L_{-1}^{l}v_{h}^{+}\rangle$. The $N=1$ R
sector Ishibashi state expansion at $\hat{c}\to\infty$ limit is

$\lim_{\hat{c}\to\infty}|V_{h}^{N=1,\text{R}}\rangle\rangle_{\pm}=\sum_{l=0}^{\infty}\>\frac{1}{l!(2h)_{l}}|L_{-1}^{l}v_{h}^{+}\otimes
U\bar{L}_{-1}^{l}\bar{v}_{h}^{+}\rangle.$ (122)

####  3.3.3 $N=2$ NS Ishibashi state expansions

We also demonstrate the expansion of $N=2$ NS sector twisted Ishibashi states
$|V_{h,q}\rangle\rangle_{\omega}$ up to level-$1$. Two types of gluing
conditions allowed are the so-called A-type and B-type gluing conditions
Ooguri:1996ck . For simplicity, we do not write down the explicit actions of
$V_{\omega}$ on conformal generators.

The expansion of $|V_{h,q}\rangle\rangle_{\omega}$ up to level-$\frac{1}{2}$
is

$|v_{h,q}\otimes
V_{\omega}Uv_{h,q}\rangle+\frac{1}{(2h-q)}|G_{-\frac{1}{2}}^{+}v_{h,q}\otimes
V_{\omega}U\bar{G}_{-\frac{1}{2}}^{+}\bar{v}_{h,q}\rangle+\frac{1}{(2h+q)}|G_{-\frac{1}{2}}^{-}v_{h,q}\otimes
V_{\omega}U\bar{G}_{-\frac{1}{2}}^{-}\bar{v}_{h,q}\rangle.$ (123)

The poles at $h\to\pm\frac{q}{2}$ are from $(\langle
G_{-\frac{1}{2}}^{\mp}v_{h,q}|G_{-\frac{1}{2}}^{\mp}v_{h,q}\rangle)^{-1}$. The
level-$1$ expansion is

$\Big{\\{}\frac{2(c_{Vir}-3)h+2c_{Vir}-(3+c_{Vir})q}{(2h-q)[2c_{Vir}h+c_{Vir}-3(2h+q^{2})]}\Big{\\}}|\chi_{1}^{N=2;\text{NS},1}\otimes
V_{\omega}U\bar{\chi}_{1}^{N=2;\text{NS},1}\rangle$
$+\Big{[}\frac{3(2+2h-q)}{[2(c_{Vir}-3)h+2c_{Vir}-(3+c_{Vir})q]}\Big{]}|\chi_{1}^{N=2;\text{NS},2}\otimes
V_{\omega}U\bar{\chi}_{1}^{N=2;\text{NS},2}\rangle$
$+\frac{1}{(2h+q)(2+2h-q)}|G_{-\frac{1}{2}}^{+}G_{-\frac{1}{2}}^{-}v_{h,q}\otimes
V_{\omega}U\bar{G}_{-\frac{1}{2}}^{+}\bar{G}_{-\frac{1}{2}}^{-}\bar{v}_{h,q}\rangle,$
(124)

where

$|\chi_{1}^{N=2;\text{NS},1}\rangle=|L_{-1}v_{h,q}\rangle-\frac{3(2h-q)(q-1)|J_{-1}v_{h,q}\rangle}{2(c_{Vir}-3)+2c_{Vir}-q(c+3)}-\frac{(c_{Vir}-3q)|G_{-\frac{1}{2}}^{+}G_{-\frac{1}{2}}^{-}v_{h,q}\rangle}{2(c_{Vir}-3)+2c_{Vir}-q(c_{Vir}+3)},$
(125)
$|\chi_{1}^{N=2;\text{NS},2}\rangle=\Big{[}J_{-1}-\frac{1}{(2+2h-q)}G_{-\frac{1}{2}}^{+}G_{-\frac{1}{2}}^{-}\Big{]}|v_{h,q}\rangle.$
(126)

The poles at the $h\to\frac{q}{2}$ and
$h\to\frac{3q^{2}-c_{Vir}}{2(c_{Vir}-3)}$ limits are from
$(\langle\chi_{1}^{N=2;\text{NS},1}|\chi_{1}^{N=2;\text{NS},1}\rangle)^{-1}$.
The pole at the $h\to\frac{c_{Vir}q+3q-2c_{Vir}}{2(c_{Vir}-3)}$ limit is from
$(\langle\chi_{1}^{N=2;\text{NS},2}|\chi_{1}^{N=2;\text{NS},2}\rangle)^{-1}$.
The poles at the $h\to-\frac{q}{2}$ and $h\to\frac{q-2}{2}$ limits are from
$(\langle
G_{-\frac{1}{2}}^{+}G_{-\frac{1}{2}}^{-}v_{h,q}|G_{-\frac{1}{2}}^{+}G_{-\frac{1}{2}}^{-}v_{h,q}\rangle)^{-1}$.

##### The $c_{Vir}\to\infty$ limit of the $N=2$ NS Ishibashi state expansion:

The non-zero elements that will contribute to $N=2$ NS Ishibashi state
expansion at the $c_{Vir}\to\infty$ limit are $L_{-1}^{l}|v_{h,q}\rangle$,
$G_{-\frac{1}{2}}^{+}L_{-1}^{l}|v_{h,q}\rangle$,
$G_{-\frac{1}{2}}^{-}L_{-1}^{l}|v_{h,q}\rangle$, and
$G_{-\frac{1}{2}}^{+}G_{-\frac{1}{2}}^{-}L_{-1}^{l}|v_{h,q}\rangle$. The basis
vectors at level-$(l+\frac{1}{2})$ are orthogonal, but the basis vectors at
level-$(l+1)$ are not. We need to compute the explicit Gram matrices to obtain
the expansion. The result of the expansion is

$\lim_{c_{Vir}\to\infty}|V_{h,q}\rangle\rangle=\sum_{l=0}^{\infty}\frac{|G_{-\frac{1}{2}}^{+}L_{-1}^{l}v_{h,q}\otimes
V_{\omega}U\bar{G}_{-\frac{1}{2}}^{+}L_{-1}^{l}\bar{v}_{h,q}\rangle}{l!(2h-q)(2h)_{l}}+\frac{|G_{-\frac{1}{2}}^{-}L_{-1}^{l}v_{h,q}\otimes
V_{\omega}U\bar{G}_{-\frac{1}{2}}^{-}L_{-1}^{l}\bar{v}_{h,q}\rangle}{l!(2h+q)(2h)_{l}}.$
$+\frac{|L_{-1}^{l}v_{h,q}\otimes
V_{\omega}U\bar{L}_{-1}^{l}\bar{v}_{h,q}\rangle}{l!(2h)_{l}}+\frac{(2h+q)|\chi_{l+1}^{N=2,\text{NS},c_{Vir}\to\infty}\otimes
V_{\omega}U\bar{\chi}_{l+1}^{N=2,\text{NS},c_{Vir}\to\infty}\rangle}{2h[4h^{2}+2h(2+l-q)+lq]l!(2h)_{l}-(l+1)!(2h)_{l+1}},$
(127)

where $|\chi_{l+1}^{N=2,\text{NS},c_{Vir}\to\infty}\rangle$ are

$|\chi_{l+1}^{N=2,\text{NS},c_{Vir}\to\infty}\rangle:=\Big{[}L_{-1}^{l+1}-\frac{2h}{2h+q}G_{-\frac{1}{2}}^{+}G_{-\frac{1}{2}}^{-}L_{-1}^{l}\Big{]}|v_{h,q}\rangle.$
(128)

The poles exist at $h\to 0$, $h\pm q/2$, and $h\to h_{l}$ limits, where
$h_{l}$ are the solutions of equations

$2h[4h^{2}+2h(2+l-q)+lq]=(l+1)(2h+l),\quad l\in\mathbb{Z}_{+}.$ (129)

If we also set $q\to\infty$, the expansion can be further be simplified

$\lim_{c_{Vir},q\to\infty}|V_{h,q}\rangle\rangle=\sum_{l=0}^{\infty}\frac{|L_{-1}^{l}v_{h,q}\otimes
V_{\omega}U\bar{L}_{-1}^{l}\bar{v}_{h,q}\rangle}{l!(2h)_{l}}+\frac{|\chi_{l+1}^{N=2,\text{NS},c_{Vir}\to\infty}\otimes
V_{\omega}U\bar{\chi}_{l+1}^{N=2,\text{NS},c_{Vir}\to\infty}\rangle}{2h(l-2h)l!(2h)_{l}}.$
(130)

####  3.3.4 Kac-Moody Ishibashi state expansions

Now, we consider Kac-Moody Ishibashi state expansions. The descendant vectors
of Kac-Moody Verma modules are

$L_{-N}J_{-N_{a}}^{a}\cdots|v\rangle.$ (131)

The Virasoro generators in the Kac-Moody algebra are constructed by the
Sugawara construction

$L_{n}=\frac{1}{2(k+g)}\sum_{a}\Big{\\{}\sum_{m\leq-1}J_{m}^{a}J_{n-m}^{a}+\sum_{m\geq
0}J_{n-m}^{a}J_{m}^{a}\Big{\\}}:=\frac{1}{2(k+g)}\sum_{a}\sum_{m}:J_{m}^{a}J_{n-m}^{a}:\>,$
(132)

where the constant $k$ is the level of the Kac-Moody algebra and $g$ is the
dual Coxeter number of the corresponding Lie algebra. Hence, all Kac-Moody
Verma modules are spanned by the current descendants
$J^{a}_{-N_{a}}\cdots|v\rangle$. We shall use the current descendant basis as
the standard basis to perform the Kac-Moody Ishibashi state expansions.

Consider a Kac-Moody algebra $\hat{h}$ related to a finite Abelian Lie algebra
$h\cong u(1)^{\dim{h}}$, with generators $H^{a}$, $a=1,\cdots,\dim\>h$.
Abelian current algebras have simple commutation relations

$[H_{n}^{a},H_{m}^{b}]=kn\delta_{ab}\delta_{m+n},$ (133)

where the level $k$ can always be absorbed by a rescale of the generators
$H^{a}\to\tilde{H}^{a}=H^{a}/\sqrt{k}$. From the commutation relation, we can
show that the current descendant basis
$\prod_{a=1}^{\dim\>h}\tilde{H}_{-N_{a}}^{a}|v\rangle$ is orthogonal. We only
need to compute the norms of all current descendants to obtain the full
Ishibashi state expansion up to level-$\infty$

$\Big{\langle}\prod_{a=1}^{\dim
h}\tilde{H}_{-N^{a}}^{a}v\Big{|}\prod_{a=1}^{\dim
h}\tilde{H}_{-N^{a}}^{a}v\Big{\rangle}=\prod_{a=1}^{\dim
h}\prod_{i}m_{i}^{a}!(n_{i}^{a})^{m_{i}^{a}},\quad
N^{a}=\\{(n_{i}^{a})^{m_{i}^{a}},\cdots\\}.$ (134)

From (134), the Abelian Kac-Moody Ishibashi state expansions up to
level-$\infty$ can be summarized as

$|V\rangle\rangle_{\omega}=\exp{\Big{\\{}\sum_{i=1}^{\dim
h}\sum_{n=0}^{\infty}\frac{1}{n}\tilde{H}_{-n}^{a}V_{\omega}U\bar{\tilde{H}}_{-n}^{a}\Big{\\}}}|v\rangle.$
(135)

The isomorphism $h\cong u(1)^{\dim{h}}$ indicates that the gluing
automorphisms of Abelian Ishibashi states are $V_{\omega}=\\{\pm I,\cdots,\pm
I\\}\in\mathbb{Z}_{2}^{\dim{h}}$. When $\dim\>h=1$, we recover the celebrated
free boson Ishibashi states Callan:1987px . For uncompactified free boson CFT,
the Ishibashi state for the momentum $p$ $U(1)$ Fock space is

$|p\rangle\rangle_{D,N}=\exp\Big{\\{}\sum_{n=0}^{\infty}\pm\frac{1}{n}\alpha_{-n}\bar{\alpha}_{-n}\Big{\\}}|p\rangle,$
(136)

where the $+$ gluing automorphism corresponds to the Dirichlet boundary
condition and the $-$ gluing automorphism corresponds to the Neumann boundary
condition. The only Neumann Ishibashi state allowed by the gluing condition,
$(\alpha_{0}+\bar{\alpha}_{0})|p\rangle\rangle_{N}=p|p\rangle\rangle_{N}=0$,
is the zero-momentum Neumann Ishibashi state $|0\rangle\rangle_{N}$. For a
compactified free boson CFT, whose target space is sphere $S^{1}$ with radius
$R$, its $U(1)$ Fock spaces are labeled by momentum - winding number pairs
$(p,w)\in\mathbb{Z}^{2}$. The Dirichlet boundary condition requires zero
winding number:
$(\alpha_{0}-\bar{\alpha}_{0})|(p,w)\rangle\rangle_{D}=w|(p,w)\rangle\rangle_{D}=0$.
Similarly, the Neumann boundary condition requires zero momentum:
$(\alpha_{0}+\bar{\alpha}_{0})|(p,w)\rangle\rangle_{N}=p|(p,w)\rangle\rangle_{N}=0$.
The two types of Ishibashi states are

$|(p,0)\rangle\rangle_{D}=\exp{\Big{\\{}\sum_{n=0}^{\infty}\frac{1}{n}\alpha_{-n}\bar{\alpha}_{-n}\Big{\\}}}|(p,0)\rangle,$
(137)
$|(0,w)\rangle\rangle_{N}=\exp{\Big{\\{}-\sum_{n=0}^{\infty}\frac{1}{n}\alpha_{-n}\bar{\alpha}_{-n}\Big{\\}}}|(0,w)\rangle,$
(138)

respectively.

Non-Abelian Kac-Moody Ishibashi state expansions are complicated in general.
In this work, we only perform the level-$0$ expansion of $\widehat{su(2)}_{k}$
Ishibashi state to show the single poles in expansion coefficients when the
corresponding $\widehat{su(2)}_{k}$ Verma module is integral. Consider the
$\widehat{su(2)}_{k}$ current algebra in a rescaled Cartan-Weyl basis, whose
commutation relations are

$[\hat{j}_{m}^{3},\hat{j}_{n}^{3}]=\frac{mk}{2}\delta_{m+n,0},\quad[\hat{j}_{m}^{3},\hat{j}_{n}^{\pm}]=\pm\hat{j}_{m+n}^{\pm},\quad[\hat{j}_{m}^{+},\hat{j}_{n}^{-}]=km\delta_{m+n,0}+2\hat{j}_{m+n}^{3}.$
(139)

The $\widehat{su(2)}_{k}$ Kac-Moody primary states $|v_{h,q}\rangle$ are
labeled by a pair of numbers $(h,q)$ and satisfy

$\hat{j}_{n}^{3}|v_{h,q}\rangle=\hat{j}_{n}^{\pm}|v_{h,q}\rangle=0,\quad\forall
n>0,\quad\hat{j}_{0}^{3}|v_{h,q}\rangle=\frac{q}{2}|v_{h,q}\rangle,\quad\hat{j}_{0}^{+}|v_{h,q}\rangle=0.$
(140)

The level-$0$ vectors in the $\widehat{su(2)}_{k}$ Verma module $V_{h,q}$ are
$(j_{0}^{-})^{m}|v_{h,q}\rangle$, $m\in\mathbb{N}$. From the
$\widehat{su(2)}_{k}$ current algebra, we can show that all level-$0$ vectors
are orthogonal. Hence, we only need to compute the norm of
$(j_{0}^{-})^{m}|v_{h,q}\rangle$ to determine the level-$0$ expansion of
$|V_{h,q}\rangle\rangle_{\omega}$

$\langle v_{h,q}|(j_{0}^{+})^{m}(j_{0}^{-})^{m}|v_{h,q}\rangle=m!(q-m+1)_{m},$
$\sum_{m=0}^{\infty}\frac{1}{m!(q-m+1)_{m}}|(j_{0}^{-})^{m}v_{h,q}\otimes
V_{\omega}U(\bar{j}_{0}^{-})^{m}\bar{v}_{h,q}\rangle.$ (141)

When $q\in\mathbb{N}$, that is when the Verma module $V_{h,q}$ is integral,
$(j_{0}^{-})^{q+1}|v_{h,q}\rangle$ is a level-$0$ null vector. The expansion
coefficients of terms $(j_{0}^{-})^{m}|v_{h,q}\rangle$ with $m\geq(q+1)$,
process single poles from $(\langle
v_{h,q}|(j_{0}^{+})^{m}(j_{0}^{-})^{m}|v_{h,q}\rangle)^{-1}$.

###  3.4 Free boson and free fermion expression of Virasoro minimal Ishibashi
states

Many CFT models can be constructed effectively by the free boson or the free
fermion CFT. We require effective construction to be capable of the
reformulation of the defining information of the original CFT, that is:

  * •

We can express the characters of the irreducible representations in the
original CFT as linear combinations of free boson or free fermion characters.

  * •

We can compute correlation functions in the origin CFT using free boson or
free fermion vertex operators.

These effective constructions will be called the bosonization or
fermionization of the original CFT. A canonical example of the bosonization is
the Coulomb-gas formalism of the Virasoro minimal models, where the Virasoro
correlation functions can be computed by the charged free boson vertex
operators with insertions of the screening operators Dotsenko:1984nm ;
Dotsenko:1984ad ; Dotsenko:1985hi . The Virasoro irreducible representations
$\mathcal{H}_{r,s}^{p,p^{\prime}}$ are identified with the BRST cohomology
spaces of the screening operators and the minimal model characters
$\chi_{r,s}^{p,p^{\prime}}(q)$ can be expressed by an infinite combination of
$U(1)$ charged Fock space characters via the Rocha-Caridi’s $K$-function
expression Felder:1988zp ; Felder:1989ve ; Kapec:2020xaj .

The motivations of considering the bosonization or fermionization of the CFT
in the computations of $\mathcal{F}_{g,n,b,c}^{0}$ are two-fold:

  * •

The free boson and free fermion Ishibashi state expansion is known up to
level-$\infty$, which implies that all expansion coefficients of
$\mathcal{F}_{g,n,b,c}^{0}$ are known in the free models.

  * •

The computation of the bulk correlation functions in the expansion may be
simpler using the free boson or free fermion vertex operators.

In this work, we shall focus on the first motivation. Hence, we will ignore
whether the correlation functions in the original CFT can be computed using
the free CFT vertex operators and only focus on the relation between the
characters. We apply the bosonization and fermionization in the following
three cases:

  * •

Expressing the Ising model Ishibashi states as linear combinations of free
fermion Ishibashi states.

  * •

Reviewing the Coulomb-gas formalism of Virasoro Ishibashi states Kawai:2002vd
; Kawai:2002pz .

  * •

Expressing the Virasoro Ishibashi states as linear combinations of
compactified free boson Ishibashi states, where both the Dirichlet and the
Neumann boundary conditions are admissible.

####  3.4.1 Free fermion expression of Ising model Ishibashi states

Consider the fermionization of the Ising model $\mathcal{M}(4,3)$. The Ising
model characters can be expressed by the free fermion characters as

$\chi_{1,1}^{4,3}(q)=\frac{1}{2}\text{Tr}_{\text{NS}}\big{[}1+(-1)^{F}\big{]}q^{L_{0}-\frac{1}{48}},\quad\chi_{2,1}^{4,3}(q)=\frac{1}{2}\text{Tr}_{\text{NS}}\big{[}1-(-1)^{F}\big{]}q^{L_{0}-\frac{1}{48}},$
(142)
$\chi_{1,2}^{4,3}(q)=\frac{1}{2}\text{Tr}_{\text{R}}\big{[}1\pm(-1)^{F}\big{]}q^{L_{0}+\frac{1}{24}}.$
(143)

The Ising irreducible representation $\mathcal{H}_{1,1}^{4,3}$ consists of
fermion number even vectors in the NS sector and $\mathcal{H}_{2,1}^{4,3}$
consists of fermion number odd vectors in the NS sector, indicating

$|\mathcal{H}_{1,1}^{4,3}\rangle\rangle=\frac{1}{2}\big{(}|\psi,\text{NS}\rangle\rangle_{+}+|\psi,\text{NS}\rangle\rangle_{-}\big{)},\quad|\mathcal{H}_{2,1}^{4,3}\rangle\rangle=\frac{1}{2}\big{(}|\psi,\text{NS}\rangle\rangle_{+}-|\psi,\text{NS}\rangle\rangle_{-}\big{)}.$
(144)

The Ising irreducible representation $\mathcal{H}_{1,2}^{4,3}$ can be
expressed by either fermion number even or odd vectors in the R sector,
indicating

$|\mathcal{H}_{1,2}^{4,3}\rangle\rangle=\frac{1}{2}\big{(}|\psi,\text{R}\rangle\rangle_{+}\pm|\psi,\text{R}\rangle\rangle_{-}\big{)},$
(145)

where $+$ corresponds to the fermion number even expression and $-$
corresponds to the fermion number odd expression. The NS and R sector free
fermion Ishibashi states are

$|\psi,\text{NS}\rangle\rangle_{\pm}=\exp{\Big{[}\pm
i\sum_{r\in\mathbb{N}+\frac{1}{2}}\psi_{-r}\bar{\psi}_{-r}\Big{]}}|I\otimes\bar{I}\rangle,$
(146) $|\psi,\text{R}\rangle\rangle_{\pm}=\exp{\Big{[}\pm
i\sum_{r\in\mathbb{Z}_{+}}\psi_{-r}\bar{\psi}_{-r}\Big{]}}(1\pm\psi_{0}\bar{\psi}_{0})|R\otimes\bar{R}\rangle,$
(147)

where $|R\otimes\bar{R}\rangle$ is the fermion number even,
$(h,\bar{h})=(\frac{1}{16},\frac{1}{16})$ R sector ground state.

The overlaps of (144) and (145) reproduce the correct Ising characters. In the
NS sector, two overlaps are the traces over fermion even and odd vectors
respectively

$\frac{1}{4}(\>_{+}\langle\langle\psi,\text{NS}|+\>_{-}\langle\langle\psi,\text{NS}|)q^{L_{0}+\bar{L}_{0}-\frac{c_{Vir}}{12}}(|\psi,\text{NS}\rangle\rangle_{+}+|\psi,\text{NS}\rangle\rangle_{-})=\chi_{1,1}^{4,3}(q),$
(148)
$\frac{1}{4}(\>_{+}\langle\langle\psi,\text{NS}|-\>_{-}\langle\langle\psi,\text{NS}|)q^{L_{0}+\bar{L}_{0}-\frac{c_{Vir}}{12}}(|\psi,\text{NS}\rangle\rangle_{+}-|\psi,\text{NS}\rangle\rangle_{-})=\chi_{2,1}^{4,3}(q).$
(149)

In the R sector, the overlap is the trace over either fermion number even or
odd vectors

$\frac{1}{4}(\>_{+}\langle\langle\psi,\text{R}|\pm\>_{-}\langle\langle\psi,\text{R}|)q^{L_{0}+\bar{L}_{0}-\frac{c_{Vir}}{12}}(|\psi,\text{R}\rangle\rangle_{+}\pm|\psi,\text{R}\rangle\rangle_{-})=\chi_{1,2}^{4,3}(q).$
(150)

####  3.4.2 Free boson expression of Virasoro minimal Ishibashi states

##### Review on the Coulomb-gas formalism:

The Rocha-Caridi’s $K$-function expressions of minimal characters are given by
DiFrancesco:1997nk

$\chi_{r,s}^{p,p^{\prime}}(q)=K_{r,s}^{p,p^{\prime}}(q)-K_{r,-s}^{p,p^{\prime}}(q),\quad
K_{r,s}^{p,p^{\prime}}(q):=\frac{1}{\eta(q)}\sum_{n\in\mathbb{Z}}q^{\frac{(2pp^{\prime}n+\lambda_{r,s})^{2}}{4pp^{\prime}}},\quad\lambda_{r,s}=pr-p^{\prime}s.$
(151)

The terms in the $K$-functions equal some charged $U(1)$ Fock space characters
with specific values of $U(1)$ charges. The charged $U(1)$ Fock spaces
$F_{\alpha,\alpha_{0}}$ are generated by acting negative $U(1)$ current modes
on the $U(1)$ primary vectors $|\alpha,\alpha_{0}\rangle$. Taking the trace
over charged Fock spaces $F_{\alpha,\alpha_{0}}$, we obtain charged $U(1)$
characters

$\chi_{\alpha,\alpha_{0}}(q)=\text{Tr}_{F_{\alpha,\alpha_{0}}}q^{L_{0}-\frac{c_{Vir}}{24}}=\frac{1}{\eta(q)}q^{(\alpha-\alpha_{0})^{2}}.$
(152)

By comparing the charged $U(1)$ character $\chi_{\alpha,\alpha_{0}}(q)$ with
terms in $K_{r,s}^{p,p^{\prime}}(q)$ and $K_{r,-s}^{p,p^{\prime}}(q)$, we find
out the values of $U(1)$ charge $\alpha_{r,s}^{n}$ that contribute to
$K_{r,s}^{p,p^{\prime}}(q)$

$\alpha_{r,s}^{n}=\frac{2pp^{\prime}n+\lambda_{r,s}}{2\sqrt{pp^{\prime}}}+\alpha_{0},\quad
n\in\mathbb{Z},$ (153)

and the values $\alpha_{r,-s}^{n}$ that contribute to
$K_{r,-s}^{p,p^{\prime}}(q)$

$\alpha_{r,-s}^{n}=\frac{2pp^{\prime}n+\lambda_{r,-s}}{2\sqrt{pp^{\prime}}}+\alpha_{0},\quad
n\in\mathbb{Z}.$ (154)

Virasoro minimal Ishibashi states can now be written as infinite linear
combinations of Dirichlet free boson Ishibashi states Kawai:2002vd ;
Kawai:2002pz

$|\mathcal{H}_{r,s}^{p,p^{\prime}}\rangle\rangle=\sum_{n\in\mathbb{Z}}|\alpha_{r,s}^{n}\rangle\rangle_{D}+\sum_{n\in\mathbb{Z}}|\alpha_{r,-s}^{n}\rangle\rangle_{D}.$
(155)

The reproduction of minimal characters from the overlaps requires a
renormalization of Ishibashi states $|\alpha_{r,-s}^{n}\rangle\rangle_{D}$
Kawai:2002vd ; Kawai:2002pz

${}_{D}\langle\langle\alpha_{r,-s}^{n}|q^{\frac{1}{2}(L_{0}+\bar{L}_{0}-\frac{c_{Vir}}{12})}|\alpha_{r,-s}^{n}\rangle\rangle_{D}=-\frac{1}{\eta(q)}q^{(\alpha_{r,-s}^{n}-\alpha_{0})^{2}}.$
(156)

From the orthogonality of the Ishibashi state overlaps
${}_{D}\langle\langle\alpha,\alpha_{0}|q^{\frac{1}{2}(L_{0}+\bar{L}_{0}-\frac{c_{Vir}}{12})}|\alpha^{\prime},\alpha_{0}\rangle\rangle_{D}=\pm\delta_{\alpha\alpha^{\prime}}\chi_{\alpha,\alpha_{0}}(q)$,
the minimal characters can be reproduced

$\langle\langle\mathcal{H}_{r,s}^{p,p^{\prime}}|q^{\frac{1}{2}(L_{0}+\bar{L}_{0}-\frac{c_{Vir}}{12})}|\mathcal{H}_{r,s}^{p,p^{\prime}}\rangle\rangle=\chi_{r,s}^{p,p^{\prime}}(q).$
(157)

The unnatural renormalization (156) does not spoil Ishibashi states being the
solution to gluing conditions. Yet, it seems to be ad hoc. We want to ask
whether there exists an expression such that the irreducible characters can be
reproduced without the renormalization (156). The hint may be hidden in the
BRST complex of the Coulomb-gas formalism Felder:1988zp ; Kapec:2020xaj . From
the BRST complex, we can embed all the Fock spaces involved into the center
Fock space, by the action of the BRST charge $Q$. We suspect that we might
obtain an expression of $|\mathcal{H}_{r,s}^{p,p^{\prime}}\rangle\rangle$
without the unnatural renormalization by investigating the null vector
structure in the center Fock space. The details of this question are left for
future studies.

##### Minimal Ishibashi states as linear combinations of compactified free
boson Ishibashi states:

In the Coulomb-gas formalism, only the Dirichlet boundary condition is
admissible. Hence, we are motivated to express Virasoro Ishibashi states using
compactified free boson Ishibashi states, where the $T$-duality interchanges
the Dirichlet and Neumann boundary conditions. Consider the compactified free
boson CFT whose target space is a circle $S^{1}$ with radius
$R=\sqrt{2pp^{\prime}}$. The Dirichlet Ishibashi states carry momentum -
winding number pairs with zero winding number $(k,0)$. The primary conformal
weight and the character of irreducible representations
$\mathcal{H}_{(k,0)}^{R=\sqrt{2pp^{\prime}}}$ are

$h_{k,0}^{R=\sqrt{2pp^{\prime}}}=\frac{k^{2}}{2R^{2}}=\frac{k^{2}}{4pp^{\prime}},\quad\chi_{(k,0)}^{R=\sqrt{2pp^{\prime}}}=\frac{1}{\eta(q)}q^{\frac{k^{2}}{4pp^{\prime}}}.$
(158)

The momentum eigenvalues that contribute to $K_{r,s}^{p,p^{\prime}}(q)$ and
$K_{r,-s}^{p,p^{\prime}}(q)$ are $k_{1}=2pp^{\prime}n+\lambda_{r,s}$ and
$k_{2}=2pp^{\prime}n+\lambda_{r,-s}$, $n\in\mathbb{Z}$ respectively. This
enables us to write

$|\mathcal{H}_{r,s}^{p,p^{\prime}}\rangle\rangle=\sum_{k_{1}}|(k_{1},0)^{R=\sqrt{2pp^{\prime}}}\rangle\rangle_{D}+\sum_{k_{2}}|(k_{2},0)^{R=\sqrt{2pp^{\prime}}}\rangle\rangle_{D},$
(159)

where the reproduction of minimal characters $\chi_{r,s}^{p,p^{\prime}}(q)$
from the overlaps requires the renormalization of Ishibashi states
$|(k_{2},0)\rangle\rangle_{D}$

${}_{D}\langle\langle(k_{2},0)|q^{\frac{1}{2}(L_{0}+\bar{L}_{0}-\frac{c_{Vir}}{12})}|(k_{2},0)\rangle\rangle_{D}=-\frac{1}{\eta(q)}q^{\frac{k_{2}^{2}}{4pp^{\prime}}}.$
(160)

By the $T$-duality, we can express the Virasoro Ishibashi states in terms of
Neumann Ishibashi states of compactified free boson on $S^{1}$ with radius
$R=\sqrt{\frac{2}{pp^{\prime}}}$. The Neumann Ishibashi states carry momentum
- winding number pairs with zero momentum $(0,w)$. The primary conformal
weights and the character of irreducible representations
$\mathcal{H}_{(0,w)}^{R=\sqrt{\frac{2}{pp^{\prime}}}}$ are

$h_{(0,w)}^{R=\sqrt{\frac{2}{pp^{\prime}}}}=\frac{w^{2}R^{2}}{8}=\frac{w^{2}}{4pp^{\prime}},\quad\chi_{(0,w)}^{R=\sqrt{\frac{2}{pp^{\prime}}}}=\frac{1}{\eta(q)}q^{\frac{w^{2}}{4pp^{\prime}}}.$
(161)

The winding number eigenvalues that contribute to $K_{r,s}^{p,p^{\prime}}(q)$
and $K_{r,-s}^{p,p^{\prime}}(q)$ are $w_{1}=2pp^{\prime}n+\lambda_{r,s}$ and
$w_{2}=2pp^{\prime}n+\lambda_{r,-s}$, $n\in\mathbb{Z}$ respectively. This
enables us to write

$|\mathcal{H}_{r,s}^{p,p^{\prime}}\rangle\rangle=\sum_{w_{1}}|(0,w_{1})^{R=\sqrt{\frac{2}{pp^{\prime}}}}\rangle\rangle_{N}+\sum_{w_{2}}|(0,w_{2})^{R=\sqrt{\frac{2}{pp^{\prime}}}}\rangle\rangle_{N}.$
(162)

To correctly reproduce the minimal characters, we also need to renormalize
states $|(0,w_{2})^{R=\sqrt{\frac{2}{pp^{\prime}}}}\rangle\rangle_{N}$ as

${}_{N}\langle\langle(0,w_{2})^{R=\sqrt{\frac{2}{pp^{\prime}}}}|q^{\frac{1}{2}(L_{0}+\bar{L}_{0}-\frac{c_{Vir}}{12})}|(0,w_{2})^{R=\sqrt{\frac{2}{pp^{\prime}}}}\rangle\rangle_{N}=-\frac{1}{\eta(q)}q^{\frac{w_{2}^{2}}{4pp^{\prime}}}.$
(163)

Similar to the Coulomb-gas formalism, we want to ask whether there exists a
compactified free boson expression of
$|\mathcal{H}_{r,s}^{p,p^{\prime}}\rangle\rangle$, without the unnatural
renormalization.

###  3.5 Summary on Ishibashi state expansions

In this section, we have studied the expansion coefficients of the Ishibashi
state intensively. We have proved the existence of a single pole structure in
the expansion coefficients when the Ishibashi states correspond to degenerate
Verma modules. The single poles originate from the norm of null vectors. We
have demonstrated the expansion in some $\mathcal{A}$ Ishibashi states at the
lowest levels. At the semi-classical limit $c_{Vir}\to\infty$ or for some
simple chiral algebra, the full Ishibashi state expansions up to
level-$\infty$ are known. Because of this phenomenon, we are motivated to
express Ishibashi states with complicated expansion as linear combinations of
free boson and free fermion Ishibashi states. We have shown this in Virasoro
minimal models.

##  4 Open Questions

The main results of this work are summarized in Section 1. We list some open
questions for future studies:

  * •

Complete the non-vanishing conditions comparison between
$\mathcal{F}_{0,n,b,0}^{0}$ and UHP correlation functions.

  * •

From the lowest level Ishibashi state expansions, the vectors
$|\chi^{h}\rangle$ seem to be closely related to the orthonormal vectors in
the Verma modules. We would like to clarify whether this relationship exists.
If it exists, it will be helpful in general Ishibashi state expansions.

  * •

Find the free boson expressions for Virasoro Ishibashi states without the
unnatural renormalization.

  * •

The application of the bosonization and fermionization technique in more CFT
models. For example, we can consider the vertex representation Frenkel:1980rn
; Segal:1981ap ; DiFrancesco:1997nk  or the Wakimoto free field representation
Wakimoto:1986gf ; DiFrancesco:1997nk  of the WZW models.

  * •

Perform a detailed computation of $\mathcal{F}_{g,n,b,c}^{x}$. As a first
step, one may need to consider the computations at various limits where they
can be simplified. For example, the semi-classical $c_{Vir}\to\infty$ limit or