#  Bootstrapping Boundaries and Branes

###### Abstract

The study of conformal boundary conditions for two-dimensional conformal field
theories (CFTs) has a long history, ranging from the description of impurities
in one-dimensional quantum chains to the formulation of D-branes in string
theory. Nevertheless, the landscape of conformal boundaries is largely
unknown, including in rational CFTs, where the local operator data is
completely determined. We initiate a systematic bootstrap study of conformal
boundaries in 2d CFTs by investigating the bootstrap equation that arises from
the open-closed consistency condition of the annulus partition function with
identical boundaries. We find that this deceivingly simple bootstrap equation,
when combined with unitarity, leads to surprisingly strong constraints on
admissible boundary states. In particular, we derive universal bounds on the
tension (boundary entropy) of stable boundary conditions, which provide a
rigorous diagnostic for potential D-brane decays. We also find unique
solutions to the bootstrap problem of stable branes in a number of rational
CFTs. Along the way, we observe a curious connection between the annulus
bootstrap and the sphere packing problem, which is a natural extension of
previous work on the modular bootstrap. We also derive bounds on the boundary
entropy at large central charge. These potentially have implications for end-
of-the-world branes in pure gravity on AdS3.

###### Contents

  1. 1 Introduction and Summary
    1. 1.1 Summary of the Main Results
  2. 2 Review of Boundary States and D-branes
    1. 2.1 Conformal Boundaries
    2. 2.2 Boundaries in Rational CFTs
    3. 2.3 Brane Tension and Boundary Entropy
    4. 2.4 Folding Trick and Conformal Interfaces
    5. 2.5 Boundaries from Generalized Symmetries
    6. 2.6 D-branes in Sigma Models
  3. 3 Bootstrap Methods for Boundaries
    1. 3.1 Linear Programming
    2. 3.2 Analytic Functionals
    3. 3.3 Annulus Partition Function as a Four-point Function of Twist Fields
  4. 4 Bounds on Stable Branes in Specific CFTs
    1. 4.1 $c=1$ and the $SU(2)_{1}$ CFT
    2. 4.2 $c=2$ and the $SU(3)_{1}$ CFT
    3. 4.3 $(G_{2})_{1}$ CFT
    4. 4.4 $Spin(8)_{1}$ CFT
    5. 4.5 $c=8$ and the $(E_{8})_{1}$ CFT
    6. 4.6 $(F_{4})_{1}$, $(E_{6})_{1}$ and $(E_{7})_{1}$ CFTs
    7. 4.7 $|\text{Monster}|^{2}$ CFT
  5. 5 Analytic Bounds and Optimal Solutions
    1. 5.1 Saturation at $c=8$
    2. 5.2 Saturation at $c=24$
  6. 6 Universal Bounds on Brane Tension
    1. 6.1 Universal Upper Bounds on Stable Branes
    2. 6.2 Universal Lower Bounds on Branes with a Bulk Gap
    3. 6.3 Entropy Bound for General $c$ from Analytic Functionals
    4. 6.4 Large $c$ and the End-of-the-world Branes in Pure Gravity
  7. 7 Outlook

##  1 Introduction and Summary

Boundaries are an integral part of the modern understanding of quantum field
theory (QFT). They are pervasive throughout all branches of physics and are
responsible for important boundary effects in many-body systems that arise in
particle physics, condensed matter and statistical mechanics. Systems with
boundaries exhibit universal behavior in the continuum limit and are often
described in QFT by boundary conditions constrained by the basic principles of
locality, unitarity and symmetries. The incorporation of boundaries introduces
an abundance of new observables in QFT with novel features that pose
outstanding challenges for theoretical understanding. At the same time,
studying the boundaries also provides new windows to the structure of bulk
dynamics in QFT.

The boundary effects are the most pronounced when the bulk system is in the
critical phase where the correlation length diverges, described by a conformal
field theory (CFT). The local boundary conditions of a CFT are organized into
universality classes under boundary renormalization group (RG) flows (e.g.
triggered by relevant deformations on the boundary) which are characterized by
RG fixed points that correspond to conformal boundaries. The coupled bulk-
boundary critical system is also known as boundary conformal field theory
(BCFT). By virtue of universality, the conformal boundaries are the natural
platform to start delineating the landscape of interesting boundary conditions
in QFT.111See [1] for a recent review on boundaries and more general defects
in CFTs. Thanks to the enhanced conformal symmetry, there is a systematic and
efficient way to explore the constraints from locality and unitarity, which
falls under the conformal bootstrap program (see [2] for a review).

Here we focus on conformal boundaries of 2d CFTs. They are important for
describing critical quantum impurities in one-dimensional quantum spin chains,
whose intricate dynamics is exemplified by the Kondo effect [3, 4, 5, 6, 7].
They also represent D-branes in string theory as boundary conditions for the
worldsheet CFT. Via the AdS/CFT correspondence, these conformal boundaries are
mapped to end-of-the-world branes for quantum gravity on AdS3 [8, 9, 10, 11],
which feature prominently in recent studies of black holes and the unitarity
of their evaporation (see e.g. [12, 13, 14, 15, 16, 17]).

Given the aforementioned broad applications, the 2d conformal boundaries are
arguably the most well-studied among all BCFTs. This is further facilitated by
the fact that they inherit a diagonal Virasoro subalgebra from the 2d CFT in
the bulk and consequently appear to have a more rigid structure compared to
their higher dimensional cousins. In fact it is believed that 2d BCFT is
completely specified by the (bulk and boundary) operator spectrum and
operator-product expansion (OPE) data, which are solutions to a set of
algebraic constraints, known as bootstrap equations, that ensure the
consistency of cutting and gluing of the Riemann surface with boundaries on
which the BCFT lives. However, explicit solutions to these bootstrap equations
are only known in minimal models [18, 19, 20], for special cases in free CFTs
(e.g. Dirichlet and Neumann boundary conditions for free compact bosons [21,
22]), in rational CFT (RCFT) for boundaries that preserve a larger chiral
algebra that extends the Virasoro algebra [23], and in Liouville theory [24,
25, 26]. Already for free bosons at $c=2$, there is no known classification of
the general conformal boundaries (see [27] for some interesting
examples222Here the bulk CFT is taken to be square torus with radius
$R=\sqrt{2{\alpha}^{\prime}}$ and no $B$-field which describes the quantum
brownian motion [27].).

Meanwhile, instead of directly solving the bootstrap equations for (B)CFTs, a
different strategy, known as the functional bootstrap [28], has led to steady
and promising progress in constraining the operator spectrum and OPE data of
general (B)CFT. The bootstrap equations are in general linear in the conformal
blocks whose coefficients encode the OPE data of the (B)CFT. The strategy is
to rule out certain operator spectrum and OPE data, by studying a small subset
of the bootstrap equations (usually those with positive expansions in the
conformal blocks in a given OPE channel) and identifying functionals acting on
the conformal blocks with desired properties to eliminate this possibility. In
2d CFT, this strategy has been explored extensively both numerically and
analytically to deduce universal bounds on the bulk CFT data by studying the
torus partition function [29, 30, 31, 32, 33, 34], the sphere four-point
functions [35, 36, 37, 38] and higher-genus partition functions [39]. The
numerical approach involves a scan over a vector space of functionals (that
are linear in derivatives with respect to the moduli of the CFT observable)
using semi-definite programming, while the analytical approach relies on a
clever choice of certain analytic functionals [40, 41, 42] which turn out to
have curious connections to the sphere packing problem [33, 34].

For 2d BCFT, there is a distinguished universal quantity among all the bulk-
boundary OPE data, given by the (vaccum) disk partition function with the
conformal boundary [43]. It is commonly denoted by $g$ and also known as the
$g$-function or brane tension in string theory applications. Similar to the
conformal $c$ anomaly in the bulk, the $g$-function is known to be a measure
of the boundary degrees of freedom and the $g$-theorem says $g$ must decrease
monotonically under boundary RG flows [43, 44]. A natural target for the
bootstrap program is to look for nontrivial bounds on $g$, which are of
interest in the context of both general 2d BCFTs and D-branes in string
theory. In this paper, we pursue this goal by performing a multi-angle
analysis of the bootstrap equation associated to the cylinder (annulus)
partition of the BCFT, extending the previous works [45, 46]. We find that
this deceivingly simple bootstrap equation, together with unitarity
(positivity), produces a wealth of constraints on the BCFT data including $g$
and boundary operator spectrum as well as the bulk $c$ anomaly and operator
spectrum.

###  1.1 Summary of the Main Results

In the rest of the paper, we start by reviewing general boundary states and
the $g$-function in Section 2 along with a description of boundaries in RCFTs
and sigma models which provide important reference points for the bootstrap
analysis. We also discuss methods to construct new boundaries in general CFTs
using (generalized) symmetries. In Section 3, we explain the functional method
to the annulus bootstrap equation and discuss both the linear programming
approach and the analytic functional approach. The latter is made possible by
an interesting relation between the annulus partition function with two
(possibly distinct) boundary conditions and a four-point function of the
${\mathbb{Z}}_{2}$ twist fields at the ends of two corresponding conformal
defect lines. In either case, we will see that there exist nontrivial upper
and lower bounds in the presence of a sufficiently large gap in the spectrum
of boundary and bulk scalar operators, respectively. The bounds we have
obtained from both the numerical and analytical approaches are presented in
Section 4,5 and 6, where we make comparisons to existing boundary states as
well as new boundary states we construct using the methods explained in
Section 2. Below we provide a brief summary.

#### Upper bound on the tension $g$ of stable elementary branes

We define stable elementary branes as conformal boundaries that are not a
direct sum of other conformal boundaries and furthermore for which the
boundary operator spectrum has a gap $h_{\rm gap}\geq 1$ above the vacuum. In
string theory, they correspond to elementary D-branes that are free from open
string tachyons. In Section 6.1, we derive from the linear programming method
a rigorous numerical upper bound on the tension $g$ of a stable elementary
brane, as a function of the central charge $c$ that for sufficiently large $c$
grows monotonically. This bound exists for $1\leq c<25$. We emphasize that our
bound is universal and does not depend on details of the BCFT nor of the
spectrum of the bulk CFT. It implies in particular that elementary bound
states of D-branes in bosonic string theory of sufficiently large tension must
decay due to open string tachyons.

#### Lower bound on the tension $g$ in the presence of a bulk scalar gap

A nontrivial lower bound on the tension $g$ of general conformal boundaries
exists provided that the gap in the spectrum of bulk scalar primaries is
sufficiently large [45]. This bound is less universal than the previously
described upper bound in the sense that it depends on some details of the
spectrum of the bulk CFT. In Section 6.2, as a proof of principle we compute
lower bounds on the $g$-function for general conformal boundary conditions as
a function of the central charge with a particular choice of the bulk gap that
is close to the maximal value allowed by modular invariance. This carves out a
narrow window of allowed values of the tension for stable boundaries in CFTs
with this large value of the gap.

#### Two-sided bounds for boundaries of specific CFTs and the uniqueness of
stable elementary branes

Stronger bounds on the tension $g$ are obtained when additional information
about the bulk CFT beyond the central charge $c$ is fed into the bootstrap
program. In Section 4, we carry out this analysis for the boundaries of a
number of RCFTs (and toroidal CFTs) for which little is known beyond the
rational branes, namely the special boundary conditions that preserve the
maximal chiral algebra. We discover that for $G_{1}$ WZW CFTs with

$\displaystyle
G=SU(2)\,,~{}SU(3)\,,~{}G_{2}\,,~{}F_{4}\,,~{}Spin(8)\,,~{}E_{6}\,,~{}E_{7}\,,~{}E_{8}\,,$
(1.1)

the stable elementary branes have the same $g$ and boundary spectrum as the
identity rational brane, providing strong evidence that all stable elementary
branes in these $G_{1}$ WZW CFTs are related by the ${G_{L}\times G_{R}\over
Z(G)}\rtimes{\rm Out}(G)$ global symmetry.333Here $G_{L}$ and $G_{R}$ are the
left and right moving $G$-symmetries and their diagonal center $Z(G)$ in the
quotient acts trivially. ${\rm Out}(G)$ is the outer-automorphism group of $G$
which acts as ${\mathbb{Z}}_{2}$ charge conjugation for $G=SU(N)$ with $N>2$
and as $S_{3}$ triality for $G=Spin(8)$. The exceptional groups have trivial
${\rm Out}(G)$ except for $G=E_{6}$ which has a ${\rm
Out}(E_{6})={\mathbb{Z}}_{2}$ symmetry.

Furthermore, for CFTs with a conformal manifold, we also explore the
dependence of the bounds on $g$ on the bulk marginal couplings, by tuning the
bulk scalar gap $\Delta_{\rm gap}$. Explicit results are presented for $c=1$,
$c=2$, and $c=8$ in Section 4, which are in agreement with known boundary
conditions. The latter two cases also provide guidance on finding new
conformal boundary conditions in the $c=2$ and $c=8$ compact boson theories
(and their discrete orbifolds).

#### Analytic bounds and optimal solutions

In Section 6.3, we use the analytic functional introduced in Sections 3.2 and
3.3 to derive an exact upper bound on $g$ when the boundary gap satisfies
$h_{\rm gap}\geq{c+8\over 16}$, and an exact lower bound on $g$ when the bulk
scalar gap satisfies $\Delta_{\rm gap}\geq{c+8\over 8}$. As we show in Section
5, both bounds are saturated at $c=8$ by the rational branes of the
$(E_{8})_{1}$ CFT, and at $c=24$ by the identity brane of the non-holomorphic
$|{\rm Monster}|^{2}$ CFT. In particular, the corresponding optimal solutions
to the bootstrap equation are uniquely determined by the analytic functional
and coincide with the annulus partition functions in the $(E_{8})_{1}$ and the
$|{\rm Monster}|^{2}$ CFT. We thus establish the following two theorems.

###### Theorem 1

All branes in the $(E_{8})_{1}$ CFT satisfy $g\geq 1$ and $h_{\rm gap}\leq 1$.
Moreover the following are equivalent

  1. 1.

The brane tension saturates the lower bound, $g=1$

  2. 2.

The boundary gap saturates the upper bound $h_{\rm gap}=1$,

  3. 3.

The cylinder partition function with the same boundary conditions is
$j(\tau)^{1/3}$.

###### Theorem 2

All branes in the $|{\rm Monster}|^{2}$ CFT satisfy $g\geq 1$ and $h_{\rm
gap}\leq 2$. Moreover the following are equivalent

  1. 1.

The brane tension saturates the lower bound $g=1$,

  2. 2.

The boundary gap saturates the upper bound $h_{\rm gap}=2$,

  3. 3.

The cylinder partition function with the same boundary conditions is
$j(\tau)-744$.

Here $j(\tau)$ is the modular $j$-invariant. This proves analytically that all
branes in the $(E_{8})_{1}$ CFT have tension $g\geq 1$ while the stable
elementary branes must have $g=1$, confirming the numerical bounds in Section
4.5. It also gives an immediate rigorous proof that all branes in the $|{\rm
Monster}|^{2}$ CFT must have tension $g\geq 1$, extending the numerical
results of [46]. Along the way, we also identify new boundary states for the
$|{\rm Monster}|^{2}$ CFT beyond those in [47].

#### A certainty relation between bulk and boundary gaps

So far we have been talking about upper and lower bounds on the tension $g$,
by imposing conditions on the gaps $\\{\Delta_{\rm gap},h_{\rm gap}\\}$ in the
bulk scalar and boundary operator spectrum. Conversely, nontrivial constraints
on $\\{\Delta_{\rm gap},h_{\rm gap}\\}$ can be deduced by obvious consistency
conditions on the $g$ bounds, namely the upper bound needs to be above the
lower bound. For example, we find from both numeric and analytic bootstrap
that for $c>24$, either $\Delta_{\rm gap}<{c+8\over 8}$ or $h_{\rm
gap}<{c+8\over 16}$. More generally, as explained in Footnote 35, whenever the
assumptions on the value of the boundary gap used to compute the upper bound
on $g$ and that of the bulk gap used to compute the lower bound are identical
$h_{\rm gap}={\Delta_{\rm gap}\over 2}$, then the resulting upper ($g_{+}$)
and lower ($g_{-}$) bounds on $g$ are inversely related, $g_{+}g_{-}=1$.
Mutual compatibility of these assumptions then requires $g_{+}\geq 1$.

#### Boundary entropy bounds at large $c$ and the end-of-the-world (ETW)
branes in pure AdS3 gravity

The bootstrap bounds on the boundary entropy for CFTs with large central
charge $c$ have implications for the end-of-the-world (ETW) branes in the
holographic dual quantum gravity on AdS3 [8, 9, 10, 11]. Motivated by a
natural extension of the pure gravity on AdS3 that incorporates ETW branes, we
illustrate this point by studying the large $c$ limit of the upper and lower
bounds on $g$ with a gap assumption $h_{\rm gap}={\Delta_{\rm gap}\over
2}={c\over 24}$ that is saturated by BTZ black holes (in the presence of an
ETW brane). We find that the boundary entropy $\log g$ is bounded from above
and below linearly in $c$, which limits the validity of the semi-classical ETW
solutions where the entropy can be tuned arbitarily.

##  2 Review of Boundary States and D-branes

###  2.1 Conformal Boundaries

To formulate 2d CFTs on manifolds with boundaries requires specifying the
boundary condition on each boundary component. In the context of string
theory, this amounts to understanding how the closed string worldsheet ends on
D-branes. The simplest setup involves the CFT on the upper half-space
${\mathbb{H}}$ with complex coordinate $z$ and a boundary at ${\mathop{\rm
Im}\nolimits}\,z=0$. By locality properties of 2d CFT with boundaries, it
suffices to understand the boundary conditions in this setup in order to
determine general CFT observables on Riemann surfaces with boundaries. Here we
are particularly interested in conformal boundary conditions for the CFT,
which requires the bulk stress tensor to remain traceless on ${\mathbb{H}}$
and satisfy the following “gluing condition” on the boundary,

$\displaystyle\left.T(z)\right|_{\mathop{\rm Im}\nolimits
z=0}=\left.\bar{T}(\bar{z})\right|_{\mathop{\rm Im}\nolimits z=0}\,.$ (2.1)

This ensures that the diagonal Virasoro algebra of the 2d CFT is preserved by
the boundary. In particular, this implies that the gravitational anomaly
$c_{L}-c_{R}$ of the CFT must vanish [48, 49, 50, 51].444Gravitational
anomalies produce a general obstruction to boundary conditions for QFTs [50,
52, 51]. In the following, we always have $c_{L}=c_{R}=c$ unless otherwise
noted.

A useful way to characterize the conformal boundary condition is to map the
half-space ${\mathbb{H}}$ to a unit disk by the conformal transformation

$\displaystyle z\to{z-i\over z+i}\,,$ (2.2)

and then the corresponding boundary condition on the unit circle defines a
state in the CFT Hilbert space ${\mathcal{H}}_{S^{1}}$ accordingly to the
usual radial quantization. We refer to such a state as a boundary state
$|B\rangle$. The condition (2.1) after the transformation (2.2) becomes

$\displaystyle(L_{n}-\bar{L}_{-n})|B\rangle=0\,,\quad{\rm
for~{}all~{}}n\in{\mathbb{Z}}\,.$ (2.3)

when expanded into Fourier modes on the circle. A basis of solutions to (2.3)
are coherent states in ${\mathcal{H}}_{S^{1}}$ known as Ishibashi states
$|\phi_{i}\rangle\\!\rangle$ which are in one-to-one correspondence with
scalar Virasoro primaries $|\phi_{i}\rangle\in{\mathcal{H}}_{S^{1}}$ of weight
$h=\bar{h}=h_{i}$ that satisfy
$\langle\phi_{i}|\phi_{j}\rangle={\delta}_{ij}$. We will not need the explicit
form of $|\phi_{i}\rangle\\!\rangle$ but the following property is useful.

Let us consider adding a concentric hole of radius $e^{-\pi t}$ to the unit
disk, and assigning two Ishibashi states $|\phi_{i}\rangle\\!\rangle$ and
$|\phi_{j}\rangle\\!\rangle$ to the two boundary circles of the annulus. The
corresponding CFT partition function on the annulus is equivalent by a
conformal transformation $z\to i\log z$ to the partition function on a
cylinder with length $\pi t$ and circumference $2\pi$, with the following
simple form

$\displaystyle\langle\\!\langle\phi_{i}|e^{-\pi tH_{\rm
cl}}|\phi_{j}\rangle\\!\rangle={\delta}_{ij}\chi_{\Delta_{i}\over 2}(t)\,,$
(2.4)

where $\Delta_{i}$ is the dimension of the scalar Virasoro primary associated
with the Ishibashi state $|\phi_{i}\rangle\\!\rangle$, $\chi_{h}$ denotes the
Virasoro character and takes the following form for a non-degenerate Virasoro
module of weight $h$,

$\chi_{h}(t)={q^{h-{c-1\over 24}}\over\eta(it)},~{}q\equiv e^{-2\pi t}$ (2.5)

where $\eta$ is the Dedekind eta function, and the “closed string” Hamiltonian
reads

$\displaystyle H_{\rm cl}=L_{0}+\bar{L}_{0}-{c\over 12}\,.$ (2.6)

The putative boundary state is in general a linear combination of the
Ishibashi states,

$\displaystyle|B\rangle=\sum_{i}B_{i}|\phi_{i}\rangle\\!\rangle\,,$ (2.7)

subject to two important constraints, the Cardy condition [18, 53] and the
bulk-boundary bootstrap equation (or Cardy-Lewellen equation) [19, 54].

The Cardy condition comes from the two different ways to decompose the
cylinder partition function with any two boundary states $|B_{\alpha}\rangle$
and $|B_{\beta}\rangle$ on the two ends. In the “closed string” channel, the
partition function computes the overlap of the two boundary states,

$\displaystyle Z_{{\alpha}{\beta}}(t)\equiv\langle B^{\alpha}|e^{-\pi tH_{\rm
cl}}|B_{\beta}\rangle=\sum_{i}(B^{\alpha}_{i})^{*}B^{\beta}_{i}\chi_{\Delta_{i}\over
2}(t)\,,$ (2.8)

while in the “open string” channel, it can be seen as the thermal partition
function for the Hilbert space ${\mathcal{H}}_{{\alpha}{\beta}}$ on an
interval of length $\pi$ with boundary conditions $B^{\alpha}$ and $B^{\beta}$
at the two ends, and a thermal circle of size $2\pi\over t$,

$\displaystyle Z_{{\alpha}{\beta}}(t)=\mathop{\rm
tr}\nolimits_{{\mathcal{H}}_{{\alpha}{\beta}}}e^{-{2\pi\over t}H_{\rm
op}}=\sum_{j}n^{j}_{{\alpha}{\beta}}\chi_{h_{j}}(t^{-1})\,.$ (2.9)

where $n^{j}_{{\alpha}{\beta}}\in{\mathbb{Z}}_{+}$ counts the primaries in
${\mathcal{H}}_{{\alpha}{\beta}}$ with conformal weight $h_{j}$. Here the
“open string” Hamiltonian is

$\displaystyle H_{\rm op}=L_{0}-{c\over 24}\,.$ (2.10)

The Cardy condition comes from the equality between (2.8) and (2.9)

$\displaystyle\sum_{i}(B^{\alpha}_{i})^{*}B^{\beta}_{i}\chi_{\Delta_{i}\over
2}(t)=\sum_{j}n^{j}_{{\alpha}{\beta}}\chi_{h_{j}}(t^{-1})\,,$ (2.11)

which is a natural extension of the usual modular crossing equation associated
with modular invariance of the torus partition function, and puts strong
constraints on the BCFT data.

Clearly a positive integer linear combination of admissible boundary states
remains a valid boundary state. The elementary (fundamental) boundaries
$|B_{\alpha}\rangle$ are defined by requiring
$n^{0}_{{\alpha}{\beta}}={\delta}_{{\alpha}{\beta}}$ so that there is a unique
dimension zero operator (i.e. the identity) on each elementary boundary
$|B_{\alpha}\rangle$ and no dimension zero boundary changing operators for
${\alpha}\neq{\beta}$.

Figure 1: The bootstrap constraint (2.12) arising from equating the boundary
and bulk OPE of the bulk two-point function
$\langle\phi_{i}(z_{1})\phi_{j}(z_{2})\rangle_{\alpha}$.

The constraint from the bulk-boundary bootstrap equation is most easily stated
for an elementary boundary $|B_{\alpha}\rangle$. The correlation function
$\langle\phi_{i}(z_{1})\phi_{j}(z_{2})\rangle_{\alpha}$ of two bulk primaries
in the presence of the boundary has two OPE channels: the the boundary and
bulk OPE channels as in Figure 1. The equivalence between the two leads to the
following bootstrap equation [19]

$\sum\limits_{l}\widetilde{C}^{\alpha}_{il}\widetilde{C}^{\alpha}_{jl}\,{\mathcal{G}}^{ii,jj}_{l}(\eta)=\sum\limits_{k}C_{ijk}B^{\alpha}_{k}\,{\mathcal{G}}^{ij,ij}_{k}(1-\eta)\,.$
(2.12)

The sum over $l$ on the LHS runs over boundary primary operators $\psi_{l}$
appearing in the bulk-boundary OPEs of $\phi_{i}$ and $\phi_{j}$, and
$\widetilde{C}^{\alpha}_{il}$ are the corresponding OPE coefficients. The sum
over $k$ on the RHS runs over bulk primaries appearing in the
$\phi_{i}\times\phi_{j}$ OPE and $C_{ijk}$ are the bulk OPE coefficients.
${\mathcal{G}}$ are the conformal blocks.

It is a nontrivial fact that, given OPE data that solve the bulk bootstrap
equations (namely sphere four-point crossing and torus one-point modular
covariance), the Cardy condition (2.11) and the bulk-boundary bootstrap
equation (2.12) are the necessary and sufficient conditions to define
correlation functions of local operators on general Riemann surfaces with
boundaries that are consistent with cutting and gluing [54, 55].555For more
general correlation functions that include boundary changing operators, the
necessary and sufficient constraints are also known and consist of four
“sewing constraints” [54]. In particular, the bulk-boundary bootstrap equation
can now involve in addition a boundary changing operator.  It is still to be
understood whether the complete set of solutions to the boundary states is
unique in general, but so far no counter-example is known.666See for example
[23] for a proof of uniqueness of the complete set of conformal boundary
states for Virasoro minimal models and rational boundary states for
$SU(2)_{k}$ and $SU(N)_{1}$ WZW models. Note that the equations (2.11) and
(2.12) are invariant under the overall rescaling $B^{\alpha}_{i}\to\lambda
B^{\alpha}_{i}$ with $|\lambda|=1$ but we do not regard this common phase
ambiguity as defining physically distinct sets of boundaries. In the
following, we fix this ambiguity by requiring that $B^{\alpha}_{0}>0$ which is
possible thanks to the reflection-positivity of the disk partition function
with boundary state $|B_{\alpha}\rangle$.

In the later sections of the paper, we will focus on the constraints coming
from the Cardy condition on boundary states in general 2d CFTs, and leave the
investigation of more general bulk-boundary bootstrap constraints to future.

###  2.2 Boundaries in Rational CFTs

It is a very difficult problem to solve the Cardy and bulk-boundary bootstrap
constraints for a general 2d CFT. One exception is when the CFT is rational:
in this case there is a finite number of conformal blocks with respect to a
chiral algebra ${\mathcal{V}}$ (and its anti-chiral counterpart
$\bar{\mathcal{V}}\cong{\mathcal{V}}$) that extends the Virasoro algebra. In
particular, there is a finite set ${\cal I}$ of
${\mathcal{V}}\times\bar{\mathcal{V}}$ primaries $\varphi_{a}$ with $a\in{\cal
I}$ and torus characters $\chi^{\mathcal{V}}_{a}(t)$. In this case, a
distinguished set of boundary states are those that preserve a diagonal
${\mathcal{V}}$ symmetry,

$\displaystyle(W_{n}-(-1)^{h_{W}}\rho(\bar{W}_{-n}))|B\rangle=0\,,\quad{\rm
for~{}all~{}}n\in{\mathbb{Z}}\,,$ (2.13)

for each strong generator $W(z)$ of weight $h_{W}$ (and $\bar{W}(\bar{z})$) of
the chiral algebra ${\mathcal{V}}$ (resp. $\bar{\mathcal{V}}$), where $\rho$
is an automorphism of ${\mathcal{V}}$ that leaves the Virasoro subalgebra
invariant. Correspondingly, the general solution is now a combination of the
Ishibashi states defined for the larger algebra
${\mathcal{V}}\otimes\bar{\mathcal{V}}$ which are in one-to-one correspondence
with the ${\mathcal{V}}\otimes\bar{\mathcal{V}}$ primaries,

$\displaystyle|B\rangle=\sum_{a\in{\cal
I}}B_{a}|\varphi_{a}\rangle\\!\rangle\,.$ (2.14)

In this case the bootstrap constraints for boundaries simplify dramatically.
In particular, the Cardy-Lewellen condition (2.12) can now be projected onto
the boundary-channel identity, leading to

$\displaystyle
B_{a}^{\alpha}B_{b}^{\alpha}=\sum_{k}C_{abc}(F^{aab}_{b})_{c1}B^{\alpha}_{0}B^{\alpha}_{k}$
(2.15)

where $C_{abc}$ are the bulk three-point functions coefficients and
$(F^{abc}_{d})_{ef}$ is the F-symbol.

When the CFT is defined by a diagonal (charge-conjugation) modular invariant
partition function, such as for the $c<1$ Virasoro minimal models, the
elementary boundaries $|B_{\alpha}\rangle\\!\rangle$ that solve (2.11) and
(2.15) are simply determined by the modular $S$-matrix,

$\displaystyle B^{{\alpha}}_{a}={S_{a{\alpha}}\over\sqrt{S_{0a}}}\,,$ (2.16)

where ${\alpha}$ and $a$ run over the same set ${\cal I}$, and the boundary
states are in one-to-one correspondence with the primaries. In particular the
bootstrap equation (2.15) reduces to the Verlinde algebra. The open string
spectrum that solves the Cardy condition (2.11) is determined by the RCFT
fusion coefficients $N_{abc}$,

$\displaystyle n^{a}_{{\alpha}{\beta}}=\sum_{b\in{\cal
I}}(B^{\alpha}_{b})^{*}B^{\beta}_{b}S_{ba}=N_{{\alpha}{\beta}a}\,,$ (2.17)

where the second equality follows from (2.16) and the Verlinde formula

$\displaystyle N_{abc}=\sum_{d\in{\cal I}}{S_{ad}S_{bd}S_{cd}^{*}\over
S_{0d}}\,.$ (2.18)

General RCFTs have a two-layer structure consisting of (anti)holomorphic data
(e.g. conformal blocks) from the chiral algebra ${\mathcal{V}}$ and
topological data (e.g. $S,T$ matrices and $F$-symbols) from an underlying
modular tensor category (MTC) ${\mathcal{C}}={\rm Rep}({\mathcal{V}})$ endowed
with a symmetric Frobenius algebra object $A\in{\mathcal{C}}$ that specifies
the modular invariant (the diagonal one corresponds to the identity
object).777Physically, objects in the category ${\mathcal{C}}$ correspond to
topological defect lines (TDLs) or Verlinde lines generating a non-invertible
global symmetry of the diagonal RCFT. The algebra object $A$ specifies a
subset of the TDLs that are non-anomalous and can be gauged (in a generalized
sense for the non-invertible TDLs). The RCFT obtained after gauging $A$ is the
theory with the new (non-diagonal) modular invariant specified by $A$.  It
turns out that the bootstrap constraints for boundaries lead to interesting
algebra structures that only depends on the topological MTC data of the RCFT
and can be solved entirely in terms of such data.

The Cardy condition for the RCFT (first equality of (2.17)) demands
$n^{a}_{{\alpha}{\beta}}$ is a nonnegative integer matrix (NIM) of size
$|{\cal I}|$, Furthermore, as a consequence of the Verlinde formula (2.18),
they satisfy the fusion rule (Verlinde algebra) of the RCFT,

$\displaystyle n^{a}\cdot n^{b}=\sum_{c\in{\cal I}}N_{abc}n^{c}\,.$ (2.19)

The matrices $n^{a}$ are thus called a “NIM-rep” of the RCFT fusion ring (for
the diagonal modular invariant the NIM-rep coincides with the fusion
coefficients). On the other hand, the bulk-boundary bootstrap equation becomes
[23],

$\displaystyle B^{\alpha}_{a}B^{\alpha}_{b}=\sum_{c\in{\cal
I}}M_{abc}B_{c}^{\alpha}B_{0}^{\alpha}\,,$ (2.20)

where $M_{abc}$ is determined by the MTC. The structure constants $M_{abc}$
defines a semi-simple commutative associative algebra that generalizes the
Verlinde algebra ($M_{abc}=N_{abc}$ for the diagonal modular invariant) and is
known as the classifying algebra for boundary conditions [56, 57].888In
contrast to the Verlinde algebra, the structure constants for a general
classifying algebra are neither positive nor integral. In particular, the
elementary rational boundaries correspond to one-dimensional representation
($B^{\alpha}_{a}/B^{\alpha}_{0}$) of the classifying algebra.

The solutions to these constraints are in principle known in terms of the
topological data from the MTC ${\mathcal{C}}$,999More precisely, the rational
boundary states (preserving the chiral algebra ${\mathcal{V}}$) correspond to
modules over ${\mathcal{C}}={\rm Rep}({\mathcal{V}})$ that are $A$-modules in
${\mathcal{C}}$ at the same time. Together with boundary changing operators,
they furnish a module category ${\rm Mod}_{\mathcal{C}}(A)$ over
${\mathcal{C}}$ [58]. The ${\mathcal{C}}$-module structure is natural because
in the RCFT ${\mathcal{C}}$ can be identified with a set of topological defect
lines which can fuse with a boundary to produce other boundary conditions
preserving the same chiral algebra. The $A$-module structure on the other hand
is required since we are gauging $A$ (see Footnote 7) and thus identifying
boundary states related by fusion with $A$.  but the latter is not known for
general chiral algebras. Thus, in line with the bootstrap philosophy, it maybe
useful to first attack the simpler constraints coming from the NIM-rep.
However only sporadic results of classification are known beyond the
$SU(2)_{k}$ WZW models.101010For $SU(2)_{k}$ WZW models, the classification of
NIM-reps reproduces all rational boundary states for the CFTs with modular
invariants of type A-D-E. However, it also produces a family of spurious
solutions for $k=2n-1$ corresponding to the tadpole graphs
$T_{n}=A_{2n}/{\mathbb{Z}}_{2}$ [59, 23].

###  2.3 Brane Tension and Boundary Entropy

Much like the conformal central charge $c$ of 2d CFT, there is a distinguished
universal quantity $g_{\alpha}$ for any conformal boundary
$|B_{\alpha}\rangle$, given by the coefficient of the identity Ishibashi state
in (2.7),

$\displaystyle|B_{\alpha}\rangle=g_{\alpha}|0\rangle\\!\rangle+\sum_{i:\,\Delta_{i}>0}B^{\alpha}_{i}|\phi_{i}\rangle\\!\rangle\,.$
(2.21)

The quantity $g_{\alpha}$ is often referred to as the $g$-function or brane
tension111111More precisely, when the CFT is a part of the worldsheet string
theory and the conformal boundary describes a D-brane. In this case the
D-brane effective tension and the $g$-function are related by  $\displaystyle
T\sim{1\over g_{s}}g_{\alpha},$ (2.22) where $g_{s}$ is the string coupling.
See Section 2.43 for more details.  for the boundary state
$|B_{\alpha}\rangle$. As explained before, $g_{\alpha}>0$ by unitarity and
$s_{\alpha}\equiv 2\log g_{\alpha}$ defines the boundary entropy.121212In what
follows we will often refer to $g$ itself as the boundary entropy as a sort of
shorthand. As alluded to, the $g$-function has several nice properties which
we elaborate below.

The $g$-function provides a measure of the boundary degrees of freedom.
Following [60], we can define the regularized dimension of the open string
Hilbert space ${\mathcal{H}}_{{\alpha}{\alpha}}$ from the limit $t\to\infty$
(i.e. $\tilde{q}=e^{-{2\pi\over t}}\to 1$) of the cylinder partition function,

$\displaystyle\dim_{\rm
reg}{\mathcal{H}}_{{\alpha}{\alpha}}\equiv\lim_{t\to\infty}e^{-{c\over 12}\pi
t}\sum_{i}n^{i}_{{\alpha}{\alpha}}\chi_{i}(t^{-1})=g_{\alpha}^{2}$ (2.23)

where the second equality follows from the Cardy condition (2.11).

The $g$-function also determines the subleading piece in the high-temperature
limit of the thermal entropy of a 2d CFT on an interval of length $L$ with
inverse temperature ${\beta}$ [43]. In general, with the same but not
necessarily conformal boundary conditions at the two ends of the interval, the
entropy is determined by the cylinder partition function $Z({\beta})$ and
takes following form

$\displaystyle S({\beta})\equiv(1-{\beta}\partial_{\beta})\log
Z({\beta})={\pi\over 3}c{L\over{\beta}}+s({\beta})+\dots{}$ (2.24)

where we have omitted terms that vanish in the high temperature limit
$L/{\beta}\to\infty$. Here $s({\beta})$ is the Affleck-Ludwig boundary entropy
which may depend on dimensional parameters of the non-conformal boundary. For
conformal boundaries, $s({\beta})$ is a constant. Indeed the cylinder
partition function with conformal boundaries $|B_{\alpha}\rangle$ is given by
(2.9) with $t={2L\over{\beta}}$,

$\displaystyle\log Z_{{\alpha}{\alpha}}({\beta})={\pi\over
6}c{L\over{\beta}}+2\log g_{\alpha}+\dots\,,$ (2.25)

so that $s({\beta})=2\log g_{\alpha}$ as promised. Furthermore, the boundary
entropy $s({\beta})$, well-defined away from the boundary conformal fixed
points, was proven to be monotonic under boundary RG flows [6, 44],131313The
$g$-function (more precisely $\log g$) also contributes to a universal
constant term in the ground state entanglement entropy of an interval
$x\in[0,r)$ in the large $r$ limit with the conformal boundary at $x=0$ [61].
Relatedly, an entropic proof of the $g$-theorem was given in [62]. providing a
boundary analog of Zamolochikov’s $c$-function in the case of bulk RG flows.
As a corollary, the $g$-function is independent of boundary marginal
couplings.141414The $g$-function can and does depend on bulk marginal
couplings in general. This is a common feature of the entropy associated with
odd dimensional conformal defects.

Finally, the $g$-function for conformal boundaries in RCFTs admit simple
expressions. For example, if the RCFT is defined by a diagonal modular
invariant with respect to a chiral algebra ${\mathcal{V}}$, then the
elementary Cardy boundaries $|B_{\alpha}\rangle$ preserving ${\mathcal{V}}$
have tension

$\displaystyle\sqrt{S_{00}}\leq
g_{\alpha}={S_{0{\alpha}}\over\sqrt{S_{00}}}\leq{1\over\sqrt{S_{00}}}\,,$
(2.26)

where the second inequality comes from the unitarity of the S-matrix and is
saturated only when the underlying MTC is trivial (e.g. the $(E_{8})_{1}$ CFT
or the non-holomorphic $|{\rm Monster}|^{2}$ CFT) and the first inequality is
a consequence of $S_{0{\alpha}}\geq S_{00}$ in MTCs and saturation happens
when ${\alpha}$ labels a simple current (abelian anyon).151515There are many
ways to see $S_{0{\alpha}}\geq S_{00}$ in RCFT. For example
${S_{0{\alpha}}\over S_{00}}=\langle{\mathcal{L}}_{\alpha}\rangle$ is the
quantum dimension of the Verlinde line ${\mathcal{L}}_{\alpha}$ which
satisfies $\langle{\mathcal{L}}_{\alpha}\rangle\geq 1$ using fusion (see
around (2.32)). Alternatively, the fusion matrix $N_{{\alpha}}$ (from
$a={\alpha}$ in (2.18)) is a non-negative integral irreducible square matrix
and the inequality follows from the Perron-Frobenius theorem (see for example
[63]).

The same inequality (2.26) also holds for general RCFTs. This is because, for
a complete set of elementary Cardy branes, the following matrix

$\displaystyle R_{a{\alpha}}=B^{\alpha}_{a}\sqrt{S_{0a}}$ (2.27)

is unitary as a consequence of the Cardy condition (2.17) with
$n^{0}_{{\alpha}{\beta}}={\delta}_{{\alpha}{\beta}}$.

###  2.4 Folding Trick and Conformal Interfaces

Conformal boundaries are closely related to conformal interfaces in CFTs by
the folding trick. Namely a conformal interface ${\cal
I}_{{\mathcal{T}}_{1}|{\mathcal{T}}_{2}}$ between two (possibly identical)
CFTs ${\mathcal{T}}_{1}$ and ${\mathcal{T}}_{2}$ corresponds to a conformal
boundary in the product CFT
${\mathcal{T}}_{1}\times\overline{{\mathcal{T}}_{2}}$ where the second factor
involves an orientation reversal. Such an interface is admissible only if the
neighboring CFTs have identical gravitational anomalies. There is a natural
notion of interface $g$-function or interface entropy as defined by those of
the boundary state obtained after folding, equivalently given by the
expectation value in the ground states of the neighboring CFTs,

$\displaystyle g_{{\cal
I}_{{\mathcal{T}}_{1}|{\mathcal{T}}_{2}}}=\langle\Omega_{1}|{\cal
I}_{{\mathcal{T}}_{1}|{\mathcal{T}}_{2}}|\Omega_{2}\rangle\,.$ (2.28)

Correspondingly, the interface $g$-function obeys the $g$-theorem under
interface RG flows.161616The $g$-theorem in 2d CFTs have been recently
generalized to line defects in higher dimensional CFTs in [64].  Moreover, the
bootstrap results we obtain in the latter sections for the $g$-function of
conformal boundaries will automatically apply to conformal interfaces when
intepreted accordingly.

Interfaces are categorized by the fraction of energy that is reflected and
transmitted at their insertions [65, 66, 67]. At the extremes are interfaces
that are totally reflective or transmissive. The former are also known as
factorized interfaces, and are in one-to-one correspondence with a pair of
conformal boundaries $B_{\alpha}^{1}$ and $B_{\beta}^{2}$ in the
${\mathcal{T}}_{1}$ and ${\mathcal{T}}_{2}$ CFTs respectively,

$\displaystyle{\cal
I}_{{\mathcal{T}}_{1}|{\mathcal{T}}_{2}}=|{\mathcal{B}}^{1}_{\alpha}\rangle\langle{\mathcal{B}}^{2}_{\beta}|\,.$
(2.29)

Consequently the factorized interfaces lead to decoupled observables and have
a $g$-function given by the product of the corresponding values for the two
boundary states.

On the other hand, the purely transmissive interfaces require the stress
energy tensor to be continous across their insertions, and thus correspond to
topological interfaces. Such interfaces are admissible only if the neighoring
CFTs ${\mathcal{T}}_{1}$ and ${\mathcal{T}}_{2}$ have identical central
charges $c_{L}$ and $c_{R}$. For example, for a conformal manifold
${\mathcal{M}}$ of CFTs, they are topological interfaces connecting CFTs at
different points on ${\mathcal{M}}$ [65, 68, 69, 70, 71]. Alternatively, if
the CFT ${\mathcal{T}}$ admits a non-anomalous symmetry $G$, there is natural
topological interface between ${\mathcal{T}}$ and its orbifold
${\mathcal{T}}/G$ (see e.g [68, 72]).171717Conventionally $G$ denotes an
invertible non-anomalous symmetry associated with a group. In terms of the
fusion category symmetry that generalizes invertible symmetries [73, 74, 75,
72], $G$ is replaced by a symmetric Frobenius algebra object $A$ in the fusion
category. The $g$-function of this interface is determined by the quantum
dimension of the algebra $\langle A\rangle$ as $g_{\cal I}=\sqrt{\langle
A\rangle}$. For invertible symmetry $G$, $\langle A\rangle=|G|$ is simply the
order of the group. A key feature of the topological interfaces is that they
have well-defined fusion products,

$\displaystyle{\cal I}_{{\mathcal{T}}_{1}|{\mathcal{T}}_{2}}\,{\cal
I}_{{\mathcal{T}}_{2}|{\mathcal{T}}_{3}}=\sum_{a}n_{a}{\cal I}^{a}_{{\cal
I}_{1}|{\cal I}_{3}}$ (2.30)

with $n_{a}\in{\mathbb{Z}}_{+}$. Evaluating the expectation value of both
sides of fusion equation in the CFT ground states and using cluster
decomposition, we conclude the interface $g$-functions (2.28) must obey the
abelianized fusion rule.

Topological interfaces between identical (isomorphic) CFTs define topological
defect lines (TDL) [76, 77, 78, 79]. They give rise to a vast generalization
of symmetries associated with groups in CFTs that is recently axiomatized in
[73, 74]. In particular they obey fusion rules as in (2.30) and are
mathematically described by fusion categories. The interface (defect)
$g$-function of a TDL ${\mathcal{L}}$ coincides with its quantum dimension,
namely the expectation value of ${\mathcal{L}}$ in the CFT ground state,

$\displaystyle g_{\mathcal{L}}=\langle{\mathcal{L}}\rangle\,.$ (2.31)

The group-like symmetries correspond to the case $g_{\mathcal{L}}=1$ and more
general (non-invertible) TDLs have $g_{\mathcal{L}}>1$. In fact this lower
bound on $g$ holds for general topological interfaces (and the corresponding
boundary states after folding),

$\displaystyle g_{{\cal I}^{\rm
top}_{{\mathcal{T}}_{1}|{\mathcal{T}}_{2}}}\geq 1\,.$ (2.32)

This follows from inspecting the fusion of a topological interface and its
orientation reversal which always contains the identity TDL which has $g=1$.

###  2.5 Boundaries from Generalized Symmetries

It should come as no surprise that symmetries are extremely powerful in
constraining not only the local operator content in a CFT but also its
defects.

#### Boundaries from Fusion with Topological Defects

In particular, boundaries of 2d CFTs form modules of the generalized fusion
category symmetries in the bulk [80, 81, 82]. This is because when a TDL
${\mathcal{L}}_{a}$ acts by fusion on a given boundary $|B_{\alpha}\rangle$,
it produces another boundary $|{\mathcal{L}}_{a}B_{\alpha}\rangle$ (possibly
reducible) that obey all the bootstrap axioms thanks to the topological nature
and locality properties of the TDLs. In particular, the resulting boundary
state $|{\mathcal{L}}_{a}B_{\alpha}\rangle$ generally has the following
decomposition into elementary branes $|B_{\beta}\rangle$,

$\displaystyle|{\mathcal{L}}_{a}B_{\alpha}\rangle\equiv{\mathcal{L}}_{a}|B_{\alpha}\rangle=\sum_{\beta}n^{a}_{{\alpha}{\beta}}|B_{\beta}\rangle\,,$
(2.33)

with $n^{a}_{{\alpha}{\beta}}\in{\mathbb{Z}}_{+}$.

To see this, let us recall the defect Hilbert space
${\mathcal{H}}^{{\mathcal{L}}_{a}}$ associated to a TDL ${\mathcal{L}}_{a}$
intersecting the spatial $S^{1}$ transversely at one point. Via the operator-
state correspondence, states in ${\mathcal{H}}^{{\mathcal{L}}_{a}}$ are in
one-to-one correspondence with operators that live at the end of
${\mathcal{L}}_{a}$. The defect Hilbert space also generalizes to the open-
string channel. Here we have an interval bounded by two conformal boundaries
$|B_{\alpha}\rangle$ and $|B_{\beta}\rangle$ and the same TDL
${\mathcal{L}}_{a}$ that passes through a point on the interval. This Hilbert
space ${\mathcal{H}}^{{\mathcal{L}}_{a}}_{{\alpha}{\beta}}$ now contain states
that represent operators at the trivalent junction between the boundaries and
the TDL. In particular, the weight zero operators correspond to topological
junctions whose multiplicity is precisely $n^{a}_{{\alpha}{\beta}}$ in (2.33).
Such topological junctions endow the set of boundary states
$\\{|B_{\alpha}\rangle\\}$ the structure of a module category over the fusion
category generated by the TDLs ${\mathcal{L}}_{a}$.

By considering consecutive fusions of TDLs with a boundary state, we see
$n^{a}$ must furnish a NIM-rep of the fusion algebra of the TDLs,

$\displaystyle{\mathcal{L}}_{a}{\mathcal{L}}_{b}=\sum_{c}{\mathcal{N}}_{ab}{}^{c}{\mathcal{L}}_{c}\,.$
(2.34)

The well-studied NIM-rep condition for RCFTs reviewed in Section 2.2
corresponds to the special case when the TDLs are Verlinde lines (i.e. objects
in the MTC ${\mathcal{C}}$) and the boundary states preserve the chiral
algebra ${\mathcal{V}}$.

In practice (and as we will see in examples), the TDLs provide a useful way to
generate consistent boundary states in a CFT given a subset of them. In
particular, the $g$-function of the resulting boundary state from fusion is
given by

$\displaystyle g_{{\mathcal{L}}_{a}B_{\alpha}}=\langle{\mathcal{L}}_{a}\rangle
g_{\alpha}\,.$ (2.35)

The cylinder partition function is also determined in terms of the interval
defect Hilbert spaces for the TDLs as

$\displaystyle\langle{\mathcal{L}}_{a}B_{\alpha}|e^{-\pi tH_{\rm
cl}}|{\mathcal{L}}_{a}B_{\alpha}\rangle=\sum_{{\mathcal{L}}_{b}\in{\mathcal{L}}_{a}\overline{{\mathcal{L}}}_{a}}\langle
B_{\alpha}|e^{-\pi tH_{\rm
cl}}|{\mathcal{L}}_{b}B_{\alpha}\rangle=\sum_{{\mathcal{L}}_{b}\in{\mathcal{L}}_{a}\overline{{\mathcal{L}}}_{a}}\mathop{\rm
tr}\nolimits_{{\mathcal{H}}^{{\mathcal{L}}_{b}}_{{\alpha}{\alpha}}}e^{-{2\pi\over
t}H_{\rm op}}\,.$ (2.36)

Note that the interval Hilbert spaces
${\mathcal{H}}^{{\mathcal{L}}_{b}}_{{\alpha}{\alpha}}$ and
${\mathcal{H}}_{{\alpha}{\alpha}^{\prime}}$ with
$|B_{{\alpha}^{\prime}}\rangle\equiv|{\mathcal{L}}_{b}B_{\alpha}\rangle$ are
equivalent.

A simple consequence of the above is that boundary states related by fusion
with invertible TDLs all have the same $g$-function and cylinder partition
function.

One type of TDLs that appear in many CFTs (and we will see examples later) is
the duality defect ${\mathcal{N}}$ which explains the self-duality of the
theory under gauging an abelian discrete group $G$. The fusion rules are

$\displaystyle{\mathcal{N}}^{2}=\sum_{g\in G}g\,,\quad
g{\mathcal{N}}={\mathcal{N}}g={\mathcal{N}}\,,$ (2.37)

and the quantum dimension $\langle{\mathcal{N}}\rangle=\sqrt{|G|}$. The
corresponding generalized symmetry is described by a Tambara-Yamagami fusion
category [83]. In Section 4, we will use these duality defects to illustrate
the power of TDLs in generating new boundary states. The simple relations for
the $g$-functions (2.35) and the cylinder partition functions (2.36) will come
in handy.

#### Boundaries from Discrete Gauging

An anomaly-free symmetry $G$ of the CFT ${\mathcal{T}}$ can be gauged to
produce another CFT ${\mathcal{T}}/G$. Correspondingly, the boundary states of
the gauged theory can be understood in terms of those before gauging $G$. This
relation is particularly simple when $G$ is a discrete group and the resulting
theory is a usual $G$-orbifold.181818Gauging a non-anomalous continous
symmetry $G$ corresponds to the coset construction in CFT. In this case, the
boundary states in the coset CFT can also be identified from those in the
parent theory by standard coset methods [84, 85, 86, 87, 88, 89, 90, 91, 92].
For the cases where ${\mathcal{T}}$ is a free theory or an RCFT, the boundary
states in the orbifold have been worked out [56, 93, 94, 95, 96, 97, 98, 99,
100, 101, 102]. Below we present the general construction.

There is an obvious class of boundary states in ${\mathcal{T}}/G$ that are
given by $G$-invariant combinations of the boundaries $|B_{\alpha}\rangle$ in
${\mathcal{T}}$,

$\displaystyle|(B_{\alpha})_{O}\rangle={1\over\sqrt{|G|}}\sum_{g\in
G}g|B_{\alpha}\rangle\,,$ (2.38)

which are known as regular branes. The overall normalization is chosen such
that the Cardy condition is obeyed in ${\mathcal{T}}/G$ and that
$|(B_{\alpha})_{O}\rangle$ is elementary for generic boundary moduli.

The regular branes by themselves do not form a complete basis for boundaries
in the orbifold theory ${\mathcal{T}}/G$ if $G$ does not freely on the
boundary states in ${\mathcal{T}}$. In particular the regular brane
$|(B_{\alpha})_{O}\rangle$ is no longer elementary when $|B_{\alpha}\rangle$
is preserved by a subgroup of $G$, and so should decompose into other
elementary branes, known as fractional branes. This is because there are
twisted sector operators which correspond to extra Ishibashi states in the
orbifold theory. They give rise to extra boundary states (fractional branes)
of the form

$\displaystyle|(B_{\alpha},r_{N})_{O}\rangle=\sum_{h\in
N}{\chi_{r_{N}}^{N}(h)\over|N|}\sum_{g\in
C_{G}(h)}{\sqrt{|G|}\over|C_{G}(h)|}g|B_{\alpha}\rangle_{h}\,,$ (2.39)

for each boundary state $|B_{\alpha}\rangle$ that is invariant under the
subgroup $N\subset G$. Here $r_{N}$ denotes irreducible representations of $N$
and $\chi^{N}_{r_{N}}$ is the corresponding group character. The second sum in
(2.39) is over the centralizer of $h$ in $G$ denoted by $C_{G}(h)$ which
implements the projection to $G$-invariant combinations of the (twisted)
boundary states in the CFT ${\mathcal{T}}$. The twisted boundary states
$|B_{\alpha}\rangle_{h}$ is a combination of the $h$-twisted Ishibashi states
and obeys the generalized Cardy condition,

$\displaystyle{}_{h}\langle B_{\alpha}|e^{-\pi tH_{\rm
cl}}|B_{\beta}\rangle_{h^{\prime}}={\delta}_{h,h^{\prime}}\mathop{\rm
tr}\nolimits_{{\mathcal{H}}_{{\alpha}{\beta}}}\left(he^{-{2\pi\over t}H_{\rm
op}}\right)\,,$ (2.40)

where the RHS involves the symmetry defect $h$ acting on the interval Hilbert
space ${\mathcal{H}}_{{\alpha}{\beta}}$.191919Note that the symmetry defect
$h$ ends topologically on the $h$-invariant boundaries. In (2.39), we have
identified the $h=1$ case with the untwisted boundary state
$|B_{\alpha}\rangle$. One can verify that the regular branes (2.38) and the
fractional branes (2.39) solve the Cardy conditions in the orbifold
${\mathcal{T}}/G$. Moreover, the regular branes decompose, when stabilized by
$N\subset G$, as

$\displaystyle|(B_{\alpha})_{O}\rangle=\sum_{r_{N}}\chi_{r_{N}}^{N}(1)|(B_{\alpha},r_{N})_{O}\rangle\,,$
(2.41)

where the sum is over irreducible representations of $N$ and the equality
follows from the identity

$\displaystyle\sum_{r_{N}}\chi_{r_{N}}^{N}(h_{1})\chi_{r_{N}}^{N}(h_{2}^{-1})=|N|{\delta}_{h_{1},h_{2}}\,,$
(2.42)

where ${\delta}_{h_{1},h_{2}}=1$ if $h_{1}$ and $h_{2}$ are conjugate and
otherwise ${\delta}_{h_{1},h_{2}}=0$.

Another perspective on the relation between (twisted) boundary states in the
CFT ${\mathcal{T}}$ and its orbifold ${\mathcal{T}}/G$ is provided by the
topological interface ${\cal I}_{{\mathcal{T}}|{\mathcal{T}}/G}$ introduced in
Section 2.4, which also makes manifest the reversibility of the orbifold
procedure.202020The orbifold ${\mathcal{T}}/G$ has a dual symmetry which can
be gauged to recover the original CFT ${\mathcal{T}}$. For abelian $G$, this
is known as the quantum $G$ symmetry of the orbifold, while for non-abelian
$G$, the dual symmetry is a fusion category ${\rm Rep}(G)$ [73]. Fusion of the
topological interface ${\cal I}_{{\mathcal{T}}|{\mathcal{T}}/G}$ with
$G$-twisted boundaries in ${\mathcal{T}}$ produce boundary states in the
orbifold ${\mathcal{T}}/G$. Alternatively, fusing ${\cal
I}_{{\mathcal{T}}|{\mathcal{T}}/G}$ with boundaries in ${\mathcal{T}}/G$
twisted by the dual symmetry recovers boundary states in ${\mathcal{T}}$. The
relations (2.38) and (2.39) encode the mapping between the (twisted) boundary
states in one theory and its orbifold.

Symmetries in 2d CFTs are generally discribed by a fusion category
${\mathcal{C}}$ (or a suitable extension thereof) [73, 74]. The general
discrete gauging is defined with respect to a symmetric Frobenius algebra
object $A\in{\mathcal{C}}$ [103, 104, 105], which can be thought of as a non-
anomalous topological defect in the fusion category. The generalized gauging
is implemented by decorating the spacetime Riemann surface with a mesh of $A$.
For group-like symmetries, ${\mathcal{C}}$ is the group algebra
${\mathbb{C}}[G]$ and the usual $G$-orbifold corresponds to $A=\oplus_{g\in
G}\,g$. It is natural to ponder the mapping between boundary states under the
generalized gauging. For rational boundaries in RCFTs, this question is
addressed in [106, 107]. An exposition for the most general case would be
interesting but out of the scope of this paper.

###  2.6 D-branes in Sigma Models

Let us now consider the case when the CFT has a sigma model description on a
$d$-dimensional target space manifold ${\mathcal{M}}$, defined by the
following action,

$\displaystyle S_{\sigma}={1\over
4\pi\alpha^{\prime}}\int_{\Sigma}d^{2}\sigma\sqrt{g}(g^{ab}G_{\mu\nu}+i\epsilon^{ab}B_{\mu\nu})\partial_{a}X^{\mu}\partial_{b}X^{\nu}$
(2.43)

where $X^{\mu}$ denotes target space coordinates with $\mu=1,2,\dots,d$,
$G_{\mu\nu}$ and $B_{\mu\nu}$ are the metric and the $B$-field on
${\mathcal{M}}$. The worldsheet $\Sigma$ has coordinates $\sigma^{a}$ with
$a=1,2$ and $g_{ab}$ is the worldsheet metric and $\epsilon_{ab}$ is the
worldsheet two-form. For example, $G$-WZW models can be described by such a
sigma model where ${\mathcal{M}}$ is the group manifold $G$ with a quantized
volume and $B$-field (the torodial CFTs correspond to the case when $G$ is
abelian).

Given such a geometric perspective on the CFT, a subset of the boundaries also
acquires natural geometric intepretations. Indeed, for $d<26$, one can think
of the sigma model (2.43) as defining the internal part of the bosonic string
theory. The D$p$-branes in the string theory that extend on a $p$-dimensional
submanifold ${\mathcal{S}}_{p}\subset{\mathcal{M}}$ and also along the time
direction are described by conformal boundaries of the internal sigma models
(they specify where the worldsheets can end in the target space
${\mathcal{M}}$). In this case, the boundary $g$-function controls the
effective tension of the brane, which is in turn determined by the DBI
action,212121We expect the formula (2.44) to hold regardless of the string
theory embedding as long as the backgrounds $(G,B)$ in (2.43) give an exact
solution to the vanishing beta function.

$\displaystyle
g_{{\mathcal{S}}_{p}}=(\pi\sqrt{2{\alpha}^{\prime}})^{{d-2p\over
2}}{\int_{{\mathcal{S}}_{p}}\det^{1\over 2}(G_{ij}+B_{ij})\over{\rm
vol}({\mathcal{M}})^{1\over 2}}\,.$ (2.44)

Here $G_{ij}$ and $B_{ij}$ are the induced metric and $B$-field on
${\mathcal{S}}_{p}\subset{\mathcal{M}}$. The normalization is fixed by
comparing to explicit answers for the toroidal CFTs [60] where the target
space coordinates have periodicity $X^{\mu}\sim X^{\mu}+2\pi$ and the D-branes
wrap sub-tori,

$\displaystyle g_{{\mathcal{S}}_{p}}=\left({\alpha}^{\prime}\over
2\right)^{{d-2p\over 4}}{|\det(G_{ij}+B_{ij})|^{1\over 2}\over|\det
G_{\mu\nu}|^{1\over 4}}\,.$ (2.45)

The D0-branes correspond to case when the numerators in (2.44) and (2.45) are
1.

After reducing to the transverse noncompact $(26-d)$-dimensional spacetime,
these D-branes give rise to point-like particles and their masses (in the
Einstein frame) are determined by the $g$-function of the branes [60],

$\displaystyle m_{{\mathcal{S}}_{p}}\sim\sqrt{{\alpha}^{\prime 11-{d\over
2}}\over G_{26-d}}g_{{\mathcal{S}}_{p}}\,,$ (2.46)

where $G_{26-d}$ is the effective Newton constant in the noncompact
directions.

The D-branes for a given sigma model are related to one another by target
space dualities such as T-dualities (see [108] for a review). Given the
discussion in Section 2.5, one expects these dualites to originate from
topological defect lines that act on the branes by fusion. In fact there is a
natural geometric construction of such defects as bi-branes for the sigma
model [109], namely D-branes in the doubled target space
${\mathcal{M}}\times{\mathcal{M}}$, which is reminiscent of the folding trick
in the CFT. The most general bi-branes define conformal defects in the CFT,
while a subset of them are topological and explicitly constructed in [109] for
WZW models (further examples of duality defects can be found in [110, 111,
112, 113, 72]).

The above discussions generalize to supersymmetric sigma models that define
supersymmetric CFTs (SCFT) straightforwardly. The preserved supersymmetry
opens the door to an immense realm of CFTs defined by highly nontrivial target
space ${\mathcal{M}}$ such as Calabi-Yau (CY) manifolds. The D-branes provide
fine probes of the geometry due to their non-perturbative nature [114]. In
particular, $g$-functions of stable D-branes encode the volumes of minimal-
volume cycles in the CY target space.222222Here stability means the absence of
supersymmetric relevant deformations of the boundary state. As before, we can
embed the CY sigma model into the type II superstring theory. Then an obvious
class of stable D-branes is given by the BPS branes wrapping calibrated cycles
in the CY. Their stability, as supersymmetric boundary states for the SCFT, is
ensured by the spacetime conserved charges due to the nontrivial homology
classes of the cycles they wrap. However, not all homology classes of a CY
manifold admit calibrated cycles, and consequently the minimal-volume
representative would have to be non-BPS (see recent discussions in [115,
116]). Unlike the calibrated cycles whose volumes, thanks to the BPS
condition, are determined by protected data of the CY, the non-BPS minimal-
volumes carry much more nontrivial information involving the CY metric about
which little is known. It was argued in [115] that the Weak Gravity Conjecture
[117] leads to interesting upper bounds on the $g$-function of such non-BPS
stable D-branes and it will be interesting to compare with bootstrap bounds on
boundary states in the SCFT. We hope to report on this in the future.

##  3 Bootstrap Methods for Boundaries

Combined with unitarity, the cylinder crossing equation (2.11) places powerful
constraints on the allowed CFT data in the presence of boundaries. In what
follows we will consider the Cardy condition associated with the crossing
equation for the cylinder with identical boundary conditions $B^{\alpha}$ on
either end for a unitary, compact CFT with central charge $c>1$ and a non-
degenerate ground state on the strip Hilbert space
$\mathcal{H}_{\alpha\alpha}$. In this setting the Cardy condition can be
written as

$g_{\alpha}^{2}\widehat{\chi}_{\rm vac}(t)-\widehat{\chi}_{\rm
vac}(t^{-1})+\sum_{\begin{subarray}{c}i\in\mathcal{I}^{\ell=0}_{\rm
bulk}\end{subarray}}|B^{\alpha}_{i}|^{2}\widehat{\chi}_{\Delta_{i}\over
2}(t)-\sum_{\begin{subarray}{c}j\in\mathcal{I}^{\alpha\alpha}_{\rm
bdy}\end{subarray}}n^{j}_{\alpha\alpha}\widehat{\chi}_{h_{j}}(t^{-1})=0,$
(3.1)

where $\mathcal{I}_{\rm bulk}^{\ell=0}$ is the spectrum of scalar Virasoro
primaries in the Hilbert space of the bulk CFT on the circle, and
$\mathcal{I}^{\alpha\alpha}_{\rm bdy}$ is the spectrum of Virasoro primary
operators in the strip Hilbert space $\mathcal{H}_{\alpha\alpha}$. Here, the
crossing equation has been written in terms of the reduced Virasoro characters

$\displaystyle\widehat{\chi}_{h}(t)$ $\displaystyle=t^{1\over 4}q^{h-{c-1\over
24}}$ (3.2) $\displaystyle\widehat{\chi}_{\rm vac}(t)$
$\displaystyle=t^{1\over 4}q^{-{c-1\over 24}}(1-q).$

Notice that the vacuum character includes the subtraction of the null
descendant of the identity operator. For the case of boundary conditions in
unitary CFTs with central charge $c>1$ that are the subject of this paper,
this is the only degenerate character needed.

We have focused on the case of identical boundaries in order to harness the
constraints of unitarity. In this setting the coefficients
$|B_{i}^{\alpha}|^{2}$ and $g_{\alpha}^{2}$ are non-negative real numbers, and
the $n^{j}_{\alpha\alpha}$ are non-negative integers (the latter constraint is
of course not special to the case of identical boundaries). We will see that
the Cardy condition (3.1) together with unitarity and certain simple
assumptions about the bulk and/or boundary spectrum lead to highly nontrivial
constraints on the allowed boundary CFT data. Indeed we will see that in
certain cases these simple inputs are sufficient to uniquely specify the
admissable conformal boundary states.

###  3.1 Linear Programming

In the pioneering work of [28] it was realized that one can derive powerful
constraints on CFT data in the presence of unitarity by acting on crossing
equations like the Cardy condition (3.1) with simple linear functionals. To
illustrate the strategy, consider the following setup. Suppose that the
boundary entropy is given by some positive number $B^{\alpha}_{0}=g_{\alpha}$,
and that there are nonzero gaps above the identity operator in the bulk scalar
primary spectrum and in the boundary primary spectrum, respectively given by
$\Delta_{\rm gap}$ and $h_{\rm gap}^{\alpha\alpha}$. If one can construct a
linear functional $\omega$ whose action on the reduced characters has the
following positivity properties

$\displaystyle\omega\left[g_{\alpha}^{2}\widehat{\chi}_{\rm
vac}(t)-\widehat{\chi}_{\rm vac}(t^{-1})\right]$ $\displaystyle>0$ (3.3)
$\displaystyle\omega\left[\widehat{\chi}_{\Delta_{i}\over 2}(t)\right]$
$\displaystyle\geq 0,\quad\Delta_{i}\geq\Delta_{\rm gap}$
$\displaystyle\omega\left[\widehat{\chi}_{h_{j}}(t^{-1})\right]$
$\displaystyle\leq 0,\quad h_{j}\geq h^{\alpha\alpha}_{\rm gap},$

then by non-negativity of the coefficients $|B^{\alpha}_{i}|^{2}$ and
$n^{j}_{\alpha\alpha}$, the proposed spectrum is incompatible with unitarity.
That is, given such a functional one can rigorously conclude that the boundary
entropy $g_{\alpha}$ cannot be realized by any unitary cylinder partition
function with the proposed gaps in its bulk and boundary spectra.

A conventional basis for such a functional in numerical approaches to
bootstrap consists of derivatives evaluated at the crossing-symmetric point.
For example, if we write $t=e^{z}$, then such a functional could be written as

$\omega=\sum_{m=0}^{N}\omega_{m}\left.\partial_{z}^{m}\right|_{z=0},$ (3.4)

for some real coefficients $\omega_{m}$, where $N$ is referred to as the
“derivative order” of the functional. Such functionals can be constructed
using semidefinite programming (in this paper we will make use of the SDPB
solver [118]), and the corresponding bounds on the CFT data will improve as
the derivative order $N$ is increased and the space of functionals is more
fully explored.

In sections 4 and 6 we will numerically construct such functionals to derive
novel upper and lower bounds on the boundary entropy $g_{\alpha}$ of possible
boundary conditions given some physically motivated assumptions. Let us
briefly illustrate how one computes such bounds. The strategy is similar to
how one bounds particular squared OPE coefficients in the bootstrap of four-
point functions of local operators [119]. Suppose one can find linear
functionals $\omega^{\pm}$ such that

$\displaystyle\omega^{\pm}\left[\widehat{\chi}_{\Delta_{i}\over 2}(t)\right]$
$\displaystyle\geq 0,\quad\Delta_{i}\geq\Delta_{\rm gap}$ (3.5)
$\displaystyle\omega^{\pm}\left[\widehat{\chi}_{h_{j}}(t^{-1})\right]$
$\displaystyle\leq 0,\quad h_{j}\geq h^{\alpha\alpha}_{\rm gap},$

normalized such that

$\omega^{\pm}\left[\widehat{\chi}_{\rm vac}(t)\right]=\pm 1.$ (3.6)

Then for a fixed derivative order $N$, the best upper and lower bounds on the
possible values of the boundary entropy $g_{\alpha}$ are given by optimizing
the action of $\omega^{+}$ and $\omega^{-}$, respectively, on the vacuum
character in the open string channel:

$\displaystyle g_{\alpha}^{2}$
$\displaystyle\leq\min_{\\{\omega_{m}\\}}\left(\omega^{+}\left[\widehat{\chi}_{\rm
vac}(t^{-1})\right]\right)$ (3.7) $\displaystyle g_{\alpha}^{2}$
$\displaystyle\geq\max_{\\{\omega_{m}\\}}\left(-\omega^{-}\left[\widehat{\chi}_{\rm
vac}(t^{-1})\right]\right).$

This optimization procedure is also implemented in SDPB.

It is not at all obvious that functionals satisfying the positivity conditions
(3.5) should exist in the first place. For one thing, we note that if $h_{\rm
gap}^{\alpha\alpha}<{c-1\over 24}$ and $\Delta_{\rm gap}<{c-1\over 12}$, then
it is impossible to find such a functional. The reason is that the Virasoro
characters in the cross-channel can be expanded in a complete basis of
characters in the original channel

$\widehat{\chi}_{h}(t^{-1})=\int_{{c-1\over
12}}^{\infty}d\Delta{\cos\left(4\pi\sqrt{(h-{c-1\over 24})({\Delta\over
2}-{c-1\over 24})}\right)\over\sqrt{\Delta-{c-1\over
12}}}\widehat{\chi}_{\Delta\over 2}(t).$ (3.8)

In particular if $h<{c-1\over 24}$ then the density of states in the integrand
on the right hand side is positive everywhere, and so for functionals formed
out of a finite number of derivatives evaluated at the crossing-symmetric
point, we can commute the action of the functional with that of the integral
and arrive at a contradiction. Similar remarks were made in [45], where it was
noticed that one needed to assume $\Delta_{\rm gap}>{c-1\over 12}$ in order to
compute a lower bound on $g_{\alpha}$ in the absence of any assumptions about
the boundary spectrum.

However, if $h_{\rm gap}^{\alpha\alpha}>{c-1\over 24}$ or $\Delta_{\rm
gap}>{c-1\over 12}$, then one may be able to compute bounds on the boundary
entropy $g_{\alpha}$ using the functional method. In particular, we will see
that the existence of any upper bound on $g_{\alpha}$ requries $h_{\rm
gap}^{\alpha\alpha}>{c-1\over 24}$, while the existence of a nontrivial lower
bound requires $\Delta_{\rm gap}>{c-1\over 12}$. Let us start by explaining
why the existence of an upper bound requries a gap in the spectrum of boundary
primaries. To derive an upper bound, we require a functional $\omega^{+}$ that
satisfies the positivity conditions (3.5) and that moreover acts positively on
the vacuum character in the closed-string channel as in (3.6). The closed
string vacuum character can be expanded in a complete basis of boundary
characters with a positive density of states232323Here we have made use of the
Liouville parameterization of the central charge via $c=1+6(b+b^{-1})^{2}$.
For $c>1$ the density of states on the right-hand side of (3.9) is positive.
To see this, note that for $c>25$ we have $0<b<1$, while for $1<c<25$ $b$ is a
pure phase.

$\widehat{\chi}_{\rm vac}(t)=\int_{c-1\over
24}^{\infty}{dh\over\sqrt{h-{c-1\over 24}}}\,2\sqrt{2}\sinh\left(2\pi
b\sqrt{h-{c-1\over 24}}\right)\sinh\left(2\pi b^{-1}\sqrt{h-{c-1\over
24}}\right)\widehat{\chi}_{h}(t^{-1}).$ (3.9)

So if the gap in the boundary spectrum is sufficiently small, in particular if
$h_{\rm gap}^{\alpha\alpha}<{c-1\over 24}$, then it is impossible to satisfy
(3.5) and (3.6) simultaneously since $\omega^{+}$ acts non-positively on each
boundary character on the right-hand side of (3.9).

A nearly identical argument shows that if the gap in the spectrum of bulk
Virasoro scalar primaries is sufficiently small, in particular if $\Delta_{\rm
gap}<{c-1\over 12}$, then although it may be possible to find a functional
$\omega^{-}$ satisfying (3.5) and (3.6), the lower bound on the boundary
entropy one infers from (3.7) will be trivially satisfied in a unitary theory.
The reason is that if the bulk gap is $\Delta_{\rm gap}<{c-1\over 12}$, then a
similar decomposition to (3.9) implies that the action of the functional
$\omega^{-}$ satisfying (3.5) on the vacuum character in the open string
channel is necessarily non-negative

$\omega^{-}\left[\widehat{\chi}_{\rm vac}(t^{-1})\right]\geq 0\,\text{ if
}\Delta_{\rm gap}<{c-1\over 12}.$ (3.10)

Thus (3.7) implies a trivial lower bound on the boundary entropy
$g_{\alpha}^{2}$, as the right-hand side is non-positive.

The arguments requiring the existence of sufficiently large gaps in order for
bounds on $g$ to exist can be rephrased in terms of the existence of solutions
to the cylinder crossing equation that are noncompact (i.e. those with a
continuous spectrum and that do not possess a vacuum state) in one
channel.242424We can be slightly more explicit as follows. The upper bound on
$g$ requires $h_{\rm gap}>{c-1\over 24}$ because otherwise
$g^{2}\widehat{\chi}_{\rm vac}(t)=g^{2}\int_{c-1\over
24}^{\infty}dh\rho(h)\widehat{\chi}_{h}(t^{-1})$ is a valid solution to
crossing for arbitrarily large $g$, where $\rho(h)$ is a positive density of
states. Similarly, for the lower bound to exist we need $\Delta_{\rm
gap}>{c-1\over 12}$ in order to rule out the solution $\widehat{\chi}_{\rm
vac}(t^{-1})=\int_{c-1\over
12}^{\infty}d\Delta\rho(\Delta)\widehat{\chi}_{\Delta\over 2}(t)$, which has
vanishing $g$. It is possible that bounds on $g$ could exist in the absence of
these assumptions provided one was able to insist on the presence of the
vacuum state in both channels and compactness of the bulk and boundary
spectra. We do not presently know how to accomplish this, and thus avoid
spurious noncompact solutions to crossing, in numerical
bootstrap.252525Although we note that there is significant motivation to do so
from other perspectives. For example, one could address the question of
whether ${c-1\over 12}$ is really the optimal upper bound on the twist gap in
compact CFT [31]. More ambitiously, one might hope to bound the total Hodge
numbers of Calabi-Yau threefolds by studying modular bootstrap constraints on
CY sigma models [120].

###  3.2 Analytic Functionals

In the following section, we will see that numerical bounds on the boundary
entropy are often nearly saturated by interesting physical theories. In such
cases, it is natural to try to prove analytically that in fact the bound is
exactly saturated. Having an analytic proof of exact saturation means
explicitly constructing a functional $\omega$ satisfying equations (3.3) with
the first line replaced by the equality

$\omega\left[g_{\alpha}^{2}\widehat{\chi}_{\rm vac}(t)-\widehat{\chi}_{\rm
vac}(t^{-1})\right]=0\,,$ (3.11)

where $g_{\alpha}^{2}$ is the exact value in the saturating theory. It further
follows that such $\omega$ must vanish on the entire spectrum exchanged in
both channels. We can thus read off the bulk and boundary spectrum of the
saturating theory from the zeros of
$\omega[\widehat{\chi}_{\frac{\Delta}{2}}(t)]$ and
$\omega[\widehat{\chi}_{h}(t^{-1})]$ respectively.

Various examples of analytic functionals proving optimal bootstrap bounds have
appeared in the literature, starting with the case of the correlator bootstrap
in 1D CFTs [40, 41, 42]. Other examples include the correlator bootstrap in
general dimension [121, 122, 123, 124, 125, 126], the correlator bootstrap in
the presence of conformal boundaries [127, 128] and the modular bootstrap of
the torus partition function [33, 129]. To construct analytic functionals for
the annulus partition function, we will closely follow the strategy of [33].
The main idea is to think of the partition function as a suitable four-point
function of twist fields. This will allow us to uplift the 1D correlator
functionals to the present setting. We will now briefly review the 1D
correlator functionals of [40, 41, 33] and explain their uplift to the annulus
partition function. Applications to concrete bounds on boundary entropy appear
in Section 5.

Consider a four-point function of pairwise identical scalar primaries inserted
at locations $x_{1}<x_{2}<x_{3}<x_{4}$ along the real line

$\langle\phi_{a}(x_{1})\phi_{a}(x_{2})\phi_{b}(x_{3})\phi_{b}(x_{4})\rangle=\langle\phi_{a}(x_{1})\phi_{a}(x_{2})\rangle\langle\phi_{b}(x_{3})\phi_{b}(x_{4})\rangle\,\mathcal{G}(z)\,,$
(3.12)

where $z=(x_{12}x_{34})/(x_{13}x_{24})$ is the 1D cross-ratio. Let us assume
all external operators have the same dimension $\Delta_{\phi}$. The equality
between the s- and t-channel OPE takes the form

$z^{-2\Delta_{\phi}}{\mathcal{G}}(z)=\sum\limits_{{\mathcal{O}}\in\phi_{a}\times\phi_{a}}\\!\\!\\!\\!f_{aa{\mathcal{O}}}f_{bb{\mathcal{O}}}z^{-\Delta_{\phi}}G_{\Delta_{{\mathcal{O}}}}(z)=\sum\limits_{{\mathcal{P}}\in\phi_{a}\times\phi_{b}}\\!\\!\\!\\!(f_{ab{\mathcal{P}}})^{2}(1-z)^{-\Delta_{\phi}}G_{\Delta_{{\mathcal{P}}}}(1-z)\,.$
(3.13)

Here

$G_{\Delta}(z)=z^{\Delta}{}_{2}F_{1}(\Delta,\Delta;2\Delta;z)$ (3.14)

are the 1D conformal blocks and $f_{abc}$ are the structure constants. It is
convenient to symmetrize and antisymmetrize the equation under crossing
$z\leftrightarrow 1-z$

$\displaystyle\sum\limits_{{\mathcal{O}}\in\phi_{a}\times\phi_{a}}\\!\\!\\!\\!f_{aa{\mathcal{O}}}f_{bb{\mathcal{O}}}F^{+}_{\Delta_{{\mathcal{O}}}}(z)-\sum\limits_{{\mathcal{P}}\in\phi_{a}\times\phi_{b}}\\!\\!\\!\\!(f_{ab{\mathcal{P}}})^{2}F^{+}_{\Delta_{{\mathcal{P}}}}(z)=0$
(3.15)
$\displaystyle\sum\limits_{{\mathcal{O}}\in\phi_{a}\times\phi_{a}}\\!\\!\\!\\!f_{aa{\mathcal{O}}}f_{bb{\mathcal{O}}}F^{-}_{\Delta_{{\mathcal{O}}}}(z)+\sum\limits_{{\mathcal{P}}\in\phi_{a}\times\phi_{b}}\\!\\!\\!\\!(f_{ab{\mathcal{P}}})^{2}F^{-}_{\Delta_{{\mathcal{P}}}}(z)=0\,,$

where

$F^{\pm}_{\Delta}=z^{-\Delta_{\phi}}G_{\Delta}(z)\pm(1-z)^{-\Delta_{\phi}}G_{\Delta}(1-z)\,.$
(3.16)

A general functional $\omega$ for the full crossing equation (3.13) is a sum
of functionals $\omega^{+}$, $\omega^{-}$ acting on the first and second line
of (3.15). For each $\Delta_{\phi}>0$, there exist distinguished functionals
$\beta^{+}_{\Delta_{\phi}}$ and $\beta^{-}_{\Delta_{\phi}}$, acting
respectively on crossing-symmetric and antisymmetric functions and satisfying

  1. 1.

$\beta^{+}_{\Delta_{\phi}}[F^{+}_{\Delta}(z)]$ and
$\beta^{-}_{\Delta_{\phi}}[F^{-}_{\Delta}(z)]$ each have a simple zero at
$\Delta=2\Delta_{\phi}+1$ with unit slope.

  2. 2.

$\beta^{+}_{\Delta_{\phi}}[F^{+}_{\Delta}(z)]$ and
$\beta^{-}_{\Delta_{\phi}}[F^{-}_{\Delta}(z)]$ each have double zeros at
$\Delta=2\Delta_{\phi}+2n+1$ for all $n=1,2,\ldots$.

It turns out that these properties fix $\beta^{+}_{\Delta_{\phi}}$ and
$\beta^{-}_{\Delta_{\phi}}$ uniquely. The functionals take the form of contour
integrals in the complex $z$ plane. The construction was reviewed in Section 5
of [33], to which we refer the reader for details.262626Note that
$\beta^{-}_{\Delta_{\phi}}$ is denoted $\beta_{\Delta_{\phi}}$ there.

We can combine $\beta^{+}_{\Delta_{\phi}}$ and $\beta^{-}_{\Delta_{\phi}}$ in
a natural way as follows. Consider the functional

$\omega_{\Delta_{\phi}}[{\mathcal{F}}(z)]=\beta^{+}_{\Delta_{\phi}}\left[\frac{{\mathcal{F}}(z)+{\mathcal{F}}(1-z)}{2}\right]+\beta^{-}_{\Delta_{\phi}}\left[\frac{{\mathcal{F}}(z)-{\mathcal{F}}(1-z)}{2}\right]\,.$
(3.17)

It follows from the structure of zeros articulated in points 1 and 2 above
that

  1. 1.

$\omega_{\Delta_{\phi}}[z^{-\Delta_{\phi}}G_{\Delta}(z)]$ has a simple zero
with unit slope at $\Delta=2\Delta_{\phi}+1$ and double zeros at
$\Delta=2\Delta_{\phi}+2n+1$ for all $n=1,2,\ldots$.

  2. 2.

$\omega_{\Delta_{\phi}}[(1-z)^{-\Delta_{\phi}}G_{\Delta}(1-z)]$ has double
zeros at $\Delta=2\Delta_{\phi}+2n+1$ for all $n=0,1,\ldots$.

The key point is that in the first case, the slopes at
$\Delta=2\Delta_{\phi}+1$ add up, but they cancel out in the second case,
yielding a double zero. Besides the structure of double zeros,
$\omega_{\Delta_{\phi}}$ enjoys the following remarkable positivity properties

  1. 1.

$\omega_{\Delta_{\phi}}[z^{-\Delta_{\phi}}G_{\Delta}(z)]\geq 0$ for all
$\Delta\geq 2\Delta_{\phi}+1$.

  2. 2.

$\omega_{\Delta_{\phi}}[(1-z)^{-\Delta_{\phi}}G_{\Delta}(1-z)]\leq 0$ for all
$\Delta\geq 0$.

These inequalities ensure that after we uplift $\omega_{\Delta_{\phi}}$ to a
functional for the annulus modular bootstrap, it will satisfy the inequalities
listed in (3.5).

###  3.3 Annulus Partition Function as a Four-point Function of Twist Fields

Let us explain how we can represent the annulus partition function
$Z_{\alpha\beta}(t)$ as a four-point function of local primary operators
inserted on a line. This will allow us to apply $\omega_{\Delta_{\phi}}$ from
the previous subsection directly to the problem at hand. The construction is
closely related to the following familiar interpretation of the torus
partition function of a 2D CFT as the four-point function of twist fields in
the symmetric product orbifold
$({\mathcal{T}}\times{\mathcal{T}})/\mathbb{Z}_{2}$, where ${\mathcal{T}}$
stands for a single copy of the theory. A complex torus of modulus $\tau$ can
be represented as the set of solutions of

$y^{2}=x(x-\lambda(\tau))(x-1)$ (3.18)

in $(x,y)\in\mathbb{C}^{2}$, where

$\lambda(\tau)=\frac{\theta_{2}(\tau)^{4}}{\theta_{3}(\tau)^{4}}$ (3.19)

is the modular lambda function. This means we have a two-to-one covering of
the sphere by the torus $(x,y)\mapsto x$. $\lambda(\tau)$ computes the cross-
ratio of the locations of the branch points. Physically, this can be realized
by considering the product orbifold
$({\mathcal{T}}\times{\mathcal{T}})/\mathbb{Z}_{2}$ in a plane, with
$\mathbb{Z}_{2}$ twist operators inserted at $x_{1}=0$, $x_{2}=\lambda(\tau)$,
$x_{3}=1$ and $x_{4}=\infty$. Going around any of the twist operators switches
the two copies of ${\mathcal{T}}$, or equivalently switches the two sheets of
the above covering.

Let us now mimic this construction for the annulus. First, note that by sewing
two copies of an annulus of modulus $t$ along the boundaries, we get a torus
with modulus $\tau=it$. This implies that the annulus corresponds to a single
sheet of the above covering. Indeed, the covering restricts to a biholomorphic
map between the open annulus of modulus $t$ and the complex plane with the
intervals $[0,\lambda(it)]$ and $[1,\infty)$ removed. The two boundary circles
of the annulus get mapped to the two intervals. If $t\in\mathbb{R}$, then
$\lambda(it)\in(0,1)$, and thus the four points are collinear. The map is
illustrated in Figure 2.

Figure 2: The complex plane with intervals $[0,z]$ and $[1,\infty)$ removed is
biholomorphic to the open annulus of modulus $t$, where $z=\lambda(it)$. This
allows us to think of the cylinder partition function as a four-point function
of twist fields in the plane.

To proceed, note that for each conformal boundary condition $\alpha$ of theory
${\mathcal{T}}$, we can define a conformal line defect
${\mathcal{D}}_{\alpha}$ by cutting spacetime along a line and imposing the
boundary condition $\alpha$ along both sides of the cut. In other words,
${\mathcal{D}}_{\alpha}$ is a factorized conformal interface between two
copies of ${\mathcal{T}}$ in neighboring half-spaces. This defect can end at a
point, say $z=0$. In that case, it can be mapped using $w=\sqrt{z}$ to the
situation where we have ${\mathcal{T}}$ in a half-plane with boundary
condition $\alpha$. Therefore, the local operators that can reside at the
endpoint of ${\mathcal{D}}_{\alpha}$ are the same as local operators on the
boundary. The operator living at the end of ${\mathcal{D}}_{\alpha}$ which
corresponds to the identity operator on the boundary will be denoted
$\phi_{\alpha}$. It is a primary operator with conformal dimension
$\Delta_{\phi}=c/16$, which can be shown by considering the correlator of the
stress tensor in the presence of ${\mathcal{D}}_{\alpha}$ stretching along a
segment, see [61].

It follows that the annulus partition function $Z_{\alpha\beta}(t)$ can be
computed as the four-point function
$\langle\phi_{\alpha}(x_{1})\phi_{\alpha}(x_{2})\phi_{\beta}(x_{3})\phi_{\beta}(x_{4})\rangle$
in the plane, where we insert ${\mathcal{D}}_{\alpha}$ along the segment
$[x_{1},x_{2}]$ and ${\mathcal{D}}_{\beta}$ along the segment $[x_{3},x_{4}]$.
Without loss of generality, we can take $x_{i}\in\mathbb{R}$ with
$x_{1}<x_{2}<x_{3}<x_{4}$. Their cross-ratio satisfies $z=\lambda(it)$. The
precise relationship is as follows (c.f. [130] for the torus
example)272727Similar identities have appeared in [131] in the study of finite
dimensional representations of the mapping class group for a punctured Riemann
surface. However the physical picture is different. While our relation (3.20)
here is between quantities in the same CFT, the identities in [131] relate the
(normalized) four-point conformal blocks and torus characters of two distinct
chiral algebras. In particular, there is a curious relation between characters
of the $G_{1}$ WZW CFT in the Deligne-Cvitanović exceptional series and
certain Virasoro four-point blocks in a sequence of minimal models related to
the ${G_{1}\times G_{1}\over G_{2}}$ coset [131].

$\displaystyle\langle\phi_{\alpha}(x_{1})\phi_{\alpha}(x_{2})\phi_{\beta}(x_{3})\phi_{\beta}(x_{4})\rangle$
$\displaystyle=\langle\phi_{\alpha}(x_{1})\phi_{\alpha}(x_{2})\rangle\langle\phi_{\beta}(x_{3})\phi_{\beta}(x_{4})\rangle\,{\mathcal{G}}(z)$
(3.20)
$\displaystyle=\langle\phi_{\alpha}(x_{1})\phi_{\alpha}(x_{2})\rangle\langle\phi_{\beta}(x_{3})\phi_{\beta}(x_{4})\rangle\left[\tfrac{z^{2}}{2^{8}(1-z)}\right]^{\frac{c}{24}}Z_{\alpha\beta}(t)\,.$

The expansion of $Z_{\alpha\beta}(t)$ into bulk characters becomes the
s-channel OPE of the four-point function ${\mathcal{G}}(z)$. Indeed, in that
case we are quantizing the theory on a circle surrounding the interval
$[x_{1},x_{2}]$, and computing the overlap of states created by
${\mathcal{D}}_{\alpha}$ and ${\mathcal{D}}_{\beta}$. The conformal blocks for
this situation are precisely the bulk characters. The bulk character of a
scalar Virasoro primary of dimension $\Delta$ can be expanded in the s-channel
1D conformal blocks (3.14) of dimensions $\Delta+2n$, $n=0,1,\ldots$ with
positive coefficients. More concretely,

$\left[\tfrac{z^{2}}{2^{8}(1-z)}\right]^{\frac{c}{24}}\chi_{\frac{\Delta}{2}}(t(z))=\sum\limits_{n=0}^{\infty}a_{n}\,G_{\Delta+2n}(z)$
(3.21)

with $a_{n}>0$. Here $t(z)$ is the inverse of $z=\lambda(it)$, namely

$t(z)=\frac{K(1-z)}{K(z)}\,,$ (3.22)

where

$K(z)=\frac{\pi}{2}{}_{2}F_{1}(1/2,1/2;1;z)$ (3.23)

is the complete elliptic integral of the first kind.

Similarly, the expansion of $Z_{\alpha\beta}(t)$ in the boundary channel is
the same as the t-channel OPE of ${\mathcal{G}}(z)$. In that case, the Hilbert
space in radial quantization is that of ${\mathcal{T}}$ on a cylinder with
${\mathcal{D}}_{\alpha}$ and ${\mathcal{D}}_{\beta}$ inserted along the time
direction at opposite points of the $S^{1}$. This Hilbert space is
${\mathcal{H}}_{\alpha\beta}\otimes{\mathcal{H}}_{\alpha\beta}$. The primaries
in the full Hilbert space are pairs of primaries in
${\mathcal{H}}_{\alpha\beta}$, i.e. $({\mathcal{P}}_{i},{\mathcal{P}}_{j})$.
However, only the primaries with ${\mathcal{P}}_{i}={\mathcal{P}}_{j}$ can
contribute in the t-channel OPE of (3.20). This is because the overlap of the
in-state with $({\mathcal{P}}_{i},{\mathcal{P}}_{j})$ is equal to the boundary
two-point function
$\langle\beta\,{\mathcal{P}}_{i}\,\alpha\,{\mathcal{P}}_{j}\,\beta\rangle\sim\delta_{ij}$,
where $\alpha,\beta$ denote the boundary conditions, see Figure 3.

Figure 3: The overlap of the t-channel initial state with the state
$|{\mathcal{P}}_{i}\rangle\otimes|{\mathcal{P}}_{j}\rangle\in{\mathcal{H}}_{\alpha\beta}\otimes{\mathcal{H}}_{\alpha\beta}$
is equal to the boundary two-point function
$\langle\beta\,{\mathcal{P}}_{i}\,\alpha\,{\mathcal{P}}_{j}\,\beta\rangle\sim\delta_{ij}$.

The boundary character of a primary of dimension $h$ can be expanded in
t-channel 1D conformal blocks of dimensions $2h+2n$ with positive coefficients

$\left[\tfrac{z^{2}}{2^{8}(1-z)}\right]^{\frac{c}{24}}\chi_{h}(t(z)^{-1})=\left[\tfrac{z^{2}}{2^{8}(1-z)}\right]^{\frac{c}{24}}\chi_{h}(t(1-z))=\left(\tfrac{z}{1-z}\right)^{\frac{c}{8}}\sum\limits_{n=0}^{\infty}a_{n}\,G_{2h+2n}(1-z)\,.$
(3.24)

The upshot of the present discussion for the functional bootstrap is the
following. Consider any functional $\omega$ for the 1D correlator bootstrap,
which acts on the crossing equation in the normalization of (3.13), where
$\Delta_{\phi}=c/16$ is the external dimension. There is a canonical way to
apply $\omega$ to the modular crossing equation for $Z_{\alpha\beta}(\tau)$ at
central charge $c$, namely

$\omega\left[\left[2^{8}z(1-z)\right]^{-\frac{c}{24}}Z_{\alpha\beta}(t(z))\right]$
(3.25)

with $Z_{\alpha\beta}$ expanded in bulk and boundary characters. If $\omega$
satisfies positivity properties when acting on 1D conformal blocks, then
positivity of the coefficients in (3.21) and (3.24) guarantees that some
positivity is preserved when we apply $\omega$ to the modular crossing
equation using (3.25). We will discuss a concrete incarnation of this idea in
Section 5.

##  4 Bounds on Stable Branes in Specific CFTs

In this section we will study upper and lower bounds on the boundary entropy
of stable boundary conditions in certain specific CFTs from semidefinite
programming, following the logic outlined in section 3.1. By stable, we mean
that the spectrum of boundary primaries includes no relevant operators

$h^{\alpha\alpha}_{\rm gap}\geq 1.$ (4.1)

In this sense the corresponding boundary conditions are stable against
boundary renormalization group flow. In the case that the bulk CFT is part of
a worldsheet string theory, the corresponding D-branes are free from open-
string tachyons and thus perturbatively stable against decay. In general, we
believe any compact CFT should have at least one stable brane. This is because
we expect a lower bound on the boundary entropy $\log g$ and the existence of
a stable brane follows by the $g$-theorem. Although we don’t have a complete
proof, we will provide plenty of evidence for this lower bound here and also
in Section 6. In particular, the identity Cardy brane of RCFTs is always
stable, and as we will see later in this section, they turn out to saturate
various annulus bootstrap bounds.

In what follows we will pay particular attention to CFTs where the gap in
spectrum of bulk scalar Virasoro primaries saturates the modular bootstrap
bound of [31]. In particular, we will see that in these cases the upper and
lower bounds on the boundary entropy for stable branes coincide to a high
degree of numerical precision, providing evidence that the boundary entropy
for stable branes in these CFTs is uniquely specified. Indeed, in section 5 we
will see that in certain cases we are able to explicitly construct the
analytic functional that proves optimality of the upper and lower bounds on
the boundary entropy and from which we can reconstruct the spectrum of
boundary primary operators.

For sufficiently small values of the central charge, the lowest-lying local
primary operator in the Hilbert space of the bulk CFT on the circle must
necessarily be a scalar. Indeed, for $1\leq c\leq 4$, the upper bound on the
dimension of the lightest Virasoro primary from modular invariance in unitary
CFTs [31] converged to the following to remarkable numerical precision

$\Delta_{\rm gap}\leq{c\over 6}+{1\over 3},\quad 1\leq c\leq 4.$ (4.2)

To the best of our knowledge, the physical origin of this uniform bound has
not been understood. For $c>4$, the bound on the dimension of the lightest
scalar primary is weaker than the bound on overall gap in the spectrum of
Virasoro primaries. In [31], several rational CFTs were identified that
saturated the upper bound on the scalar gap at the following values of the
central charge

$c=1,\,2,\,{14\over 5},\,4,\,8$ (4.3)

corresponding to the $SU(2)_{1}$, $SU(3)_{1}$, $(G_{2})_{1}$, $Spin(8)_{1}$,
and $(E_{8})_{1}$ WZW models, respectively. These diagonal theories have the
property that there are only two characters with respect to the maximal chiral
algebra except for the $(E_{8})_{1}$ case which has only one
character.282828One might notice that these groups belong to the Deligne-
Cvitanović exceptional series, and might further wonder whether the WZW models
at level one corresponding to the remainder of the Deligne-Cvitanović
exceptional series, namely the $(F_{4})_{1}$, $(E_{6})_{1}$ and the
$(E_{7})_{1}$ WZW models at central charges $c=26/5,\,6,\,7$ respectively,
also saturate bootstrap bounds. Indeed, in [132] it was shown that if one
allows for the presence of conserved currents, the level-one WZW models
corresponding to the entirety of the Deligne-Cvitanović exceptional series
saturate the bound on the gap in the spectrum of twists $t$ of local primary
operators, where $t=\Delta-|\ell|$.

In this section we will study the upper and lower bounds on the boundary
entropy for stable boundary conditions in these CFTs, and will find convincing
numerical evidence that in these settings the boundary entropy, and indeed the
entire cylinder partition function, is uniquely specified. In some special
cases we will find that it is enough to specify the value of the bulk scalar
gap in order to pin down $g$. In some other cases we will find that one needs
to further input more information about the spectrum of the bulk CFT, for
example the dimensions of the first $N$ low-lying scalar operators, in order
for the upper and lower bounds to convincingly coincide. We implement this in
semidefinite programming as follows. We will seek functionals $\omega^{\pm}$
that, rather than acting non-negatively on all closed string scalar characters
above the gap as in the first line of (3.5), instead only act non-negatively
on the specific characters corresponding to the first $N$ low-lying scalars,
and on all closed-string characters with dimension above that of the
$(N+1)^{\rm th}$ operator:

$\displaystyle\omega^{\pm}\left[\widehat{\chi}_{\Delta_{i}\over 2}(t)\right]$
$\displaystyle\geq 0,\quad i=1,\ldots,N,~{}\Delta_{i}\in\mathcal{I}_{\rm
bulk}^{\ell=0}$ (4.4)
$\displaystyle\omega^{\pm}\left[\widehat{\chi}_{\Delta\over 2}(t)\right]$
$\displaystyle\geq 0,\quad\Delta\geq\Delta_{N+1}.$

For the particular diagonal rational CFTs that saturate the bound on the
scalar gap, we will find evidence that the converging upper and lower bounds
on the boundary entropy are saturated by the cylinder partition function
associated to the identity Cardy brane

$|B^{\mathds{1}}\rangle=\sum_{j\in{\cal
I}}\sqrt{S_{0j}}|\varphi_{j}\rangle\\!\rangle\,,$ (4.5)

as well as those obtained from a global symmetry rotation by ${G\times G\over
Z(G)}\rtimes{\rm Out}(G)$ (see Section 2.5). Here, it is understood that $S$
is the modular $S$ matrix associated with the maximal chiral algebra of the
bulk rational CFT and that ${\cal I}$ is the spectrum of local operators that
are primary with respect to the maximal chiral algebra (necessarily scalar by
the diagonal condition). The corresponding cylinder partition function is then
given by the identity character of the maximal chiral algebra in the open-
string channel

$Z_{\mathds{1}\mathds{1}}(t)=\sum_{j\in{\cal
I}}S_{0j}\widetilde{\chi}_{\Delta_{j}\over 2}(t)=\widetilde{\chi}_{\rm
vac}(t^{-1}),$ (4.6)

where $\widetilde{\chi}$ are the characters of representations of the maximal
chiral algebra. The saturating value of the $g$ function is correspondingly
given in terms of the identity-identity element of the modular $S$ matrix as

$g_{\mathds{1}}=\sqrt{S_{00}}.$ (4.7)

###  4.1 $c=1$ and the $SU(2)_{1}$ CFT

We start by computing bounds on the boundary entropy for unitary, compact CFTs
with $c=1$. To the best of our knowledge, the only known unitary, compact CFTs
with $c=1$ are the free compact boson and its orbifolds [133]. Boundary states
for this family of CFTs have been classified [134, 135, 136, 21, 22, 137, 98,
100, 138] which we will review below.

We start with the circle branch of the $c=1$ CFTs, which is characterized by a
global symmetry $(U(1)_{m}\times U(1)_{w})\rtimes{\mathbb{Z}}_{2}^{C}$ where
$U(1)_{m}\times U(1)_{w}$ are the momentum and winding symmetries respectively
while the ${\mathbb{Z}}_{2}^{C}$ acts by charge conjugation (flips the sign on
the boson). The circle branch is a one-dimensional conformal manifold
parametrized by the radius $R$ of the compact boson.292929Here we are working
in units where $\alpha^{\prime}=2$. At generic $R$, the bulk spectrum of
Virasoro primaries consists of $\mathfrak{u}(1)$ primaries

$\displaystyle|(m,w)\rangle~{}{\rm with}~{}h={1\over 2}\left({m\over
R}+{wR\over 2}\right)^{2},~{}\bar{h}={1\over 2}\left({m\over R}-{wR\over
2}\right)^{2}$ (4.8)

that carry momentum and winding charges $m,w\in{\mathbb{Z}}$, as well as
$\mathfrak{u}(1)$ Kac-Moody descendants of the vacuum which are normal ordered
symmetric Schur polynomials in the holomorphic and anti-holomorphic
$\mathfrak{u}(1)$ currents,

$\displaystyle|n;\bar{n}\rangle~{}~{}{\rm
with}~{}h=n^{2},~{}\bar{h}=\bar{n}^{2}$ (4.9)

labelled by $n,\bar{n}\in{\mathbb{Z}}_{+}$.

The conformal boundaries fall into three disconnected families: the Dirichlet
boundaries $|D(\theta)\rangle$ with $\theta\in[0,2\pi)$,

$\displaystyle|D(\theta)\rangle=$
$\displaystyle{1\over\sqrt{R}}\left(\sum_{n\in{\mathbb{Z}}_{+}}|n;n\rangle\\!\rangle+\sum_{m\in{\mathbb{Z}}}e^{im\theta}|(m,0)\rangle\\!\rangle\right)\,,$
(4.10)

the Neumann boundaries $|N(\phi)\rangle$ with $\phi\in[0,2\pi)$,

$\displaystyle|N(\phi)\rangle=$ $\displaystyle\sqrt{R\over
2}\left((-1)^{n}\sum_{n\in{\mathbb{Z}}_{+}}|n;n\rangle\\!\rangle+\sum_{w\in{\mathbb{Z}}}e^{iw\phi}|(0,w)\rangle\\!\rangle\right)\,,$
(4.11)

and the exotic boundaries $|E(x)\rangle$ with $x\in[-1,1]$ [137],

$\displaystyle|E(x)\rangle=$
$\displaystyle{\mathcal{N}}\sum_{n\in{\mathbb{Z}}_{\geq
0}}P_{n}(x)|n;n\rangle\\!\rangle\,,$ (4.12)

where $P_{n}(x)$ denotes the Legendre polynomials and ${\mathcal{N}}$ is an
arbitrary constant. Here $|\cdot\rangle\\!\rangle$ denotes the Virasoro
Ishibashi states. While the Dirichlet and Neumann boundary states preserve the
$\mathfrak{u}(1)$ KM algebra and can be written in terms of $\mathfrak{u}(1)$
KM Ishibashi states, the exotic boundaries only preserve the Virasoro algebra.
Furthermore the exotic boundaries have a continous open-string spectrum
starting at $h=0$ and do not have a normalizable vacuum (similar to the
properties of a non-compact boson) and thus do not obey the Cardy condition
strictly with integral degeneracies.303030Similar occurence of non-compactness
for defects in compact CFTs was recently noticed for topological defect lines
on the $c=1$ orbifold branch [139].

To gain a better intuition of these exotic branes, it is helpful to consider
the dense rational points on the $c=1$ circle branch with $R={M\over
N}\sqrt{2}$ for coprime integers $M,N>0$. This is related to the $SU(2)_{1}$
WZW CFT at $R=\sqrt{2}$ by first gauging a ${\mathbb{Z}}_{N}$ momentum
symmetry and then gauging a ${\mathbb{Z}}_{M}$ winding symmetry.
Correspondingly, this gauging picture allows one to classify the boundary
states at the rational points from those at $R=\sqrt{2}$. The conformal
boundaries at the $SU(2)_{1}$ point are classified in [21] and they are in
one-to-one correspondence with group elements of $SU(2)$. Consequently, the
boundary states at the $R={M\over N}\sqrt{2}$, include in addition to the
Dirichlet and Neumann branes, a family $|h\rangle$ parametrized by
$h\in{SU(2)/({\mathbb{Z}}_{M}\times{\mathbb{Z}}_{N})}$ where
${\mathbb{Z}}_{M}$ acts by $h\to e^{{\pi i\over M}\sigma_{3}}he^{-{\pi i\over
M}\sigma_{3}}$ and ${\mathbb{Z}}_{N}$ acts by $h\to e^{{\pi i\over
N}\sigma_{3}}he^{{\pi i\over N}\sigma_{3}}$. The $|h\rangle$ branes, in the
limit $M,N\to\infty$, away from the fixed loci of
${\mathbb{Z}}_{M}\times{\mathbb{Z}}_{N}$, approach the one parameter family of
exotic branes $|E(x)\rangle$ at irrational points on the circle branch.
Nevertheless, for finite $M,N$, they define ordinary boundary states that obey
all of the BCFT axioms [22, 138]. In particular, they have boundary tension
$g=2^{-{1\over 4}}\sqrt{MN}\geq 1$ and always contain relevant operators in
the open string spectrum.

Therefore the potential stable branes on the circle branch are either
Dirichlet or Neumann branes.

| $|D(\theta)\rangle$ | $|N(\phi)\rangle$
---|---|---
$\Delta_{\rm gap}$ | ${1\over R^{2}}$
$h_{\rm gap}$ | ${R^{2}\over 2}$ | ${2\over R^{2}}$
$g$ | ${1\over\sqrt{R}}$ | $\sqrt{R\over 2}$
Table 1: Bulk and boundary gaps and brane tensions on the $c=1$ circular
branch

The orbifold branch of the $c=1$ CFTs are described by gauging the
${\mathbb{Z}}_{2}^{C}$ symmetry of the compact boson. The conformal boundaries
in the orbifold theory are built from ${\mathbb{Z}}_{2}^{C}$ invariant
combinations of the ordinary boundary states for the compact boson as well as
${\mathbb{Z}}_{2}^{C}$ twisted boundary states (see Section 2.5 for a general
discussion of branes in orbifold). Restricting to the former leads to so-
called regular branes (2.38) in the ${\mathbb{Z}}_{2}^{C}$ orbifold which only
have overlaps with untwisted sector bulk operators,

$\displaystyle|D_{O}(\theta)\rangle=$
$\displaystyle{1\over\sqrt{2}}(|D(\theta)\rangle+|D(-\theta)\rangle)\,,\quad|N_{O}(\phi)\rangle={1\over\sqrt{2}}(|N(\phi)\rangle+|N(-\phi)\rangle)\,,$
(4.13)

for $0<\phi,\theta<\pi$. Including the ${\mathbb{Z}}_{2}^{C}$ twisted sector
generates another eight branes localized at the ${\mathbb{Z}}_{2}^{C}$ fixed
points (known as fractional branes (2.39))

$\displaystyle|D_{O}(\theta)_{\pm}\rangle=$
$\displaystyle{1\over\sqrt{2}}|D(\theta)\rangle\pm{1\over
2^{1/4}}|D(\theta)\rangle\\!\rangle_{C}\,,$ (4.14)
$\displaystyle|N_{O}(\phi)_{\pm}\rangle=$
$\displaystyle{1\over\sqrt{2}}|N(\phi)\rangle\pm{1\over
2^{1/4}}|N(\phi)\rangle\\!\rangle_{C}\,,$

with $\phi,\theta=0,\pi$ where $|D(\theta)\rangle\\!\rangle_{C}$ and
$|N(\phi)\rangle\\!\rangle_{C}$ are specific combinations of the twised sector
Ishibashi states whose explicit forms can be found in [140]. At rational
points on the orbifold branch $R={M\over N}\sqrt{2}$, the $|h\rangle$ branes
on the circle branch with $h\in
SU(2)/({\mathbb{Z}}_{M}\times{\mathbb{Z}}_{N})$ leads to additional regular
branes,

$\displaystyle|h_{O}\rangle={1\over\sqrt{2}}(|h\rangle+C|h\rangle)={1\over\sqrt{2}}(|h\rangle+|\sigma_{1}h\sigma_{1}\rangle)\,.$
(4.15)

These branes have tension $g=2^{1\over 4}\sqrt{MN}$ and are always unstable.
In the limit $M,N\to\infty$, they give rise to analogs of the exotic branes
$|E(x)\rangle$ at irrational points on the orbifold branch.

| $|D_{O}(\theta)\rangle$ | $|N_{O}(\phi)\rangle$ | $|D_{O}(\theta)_{\pm}\rangle$ | $|N_{O}(\phi)_{\pm}\rangle$
---|---|---|---|---
$\Delta_{\rm gap}$ | $\min\left({1\over 8},{1\over R^{2}}\right)$
$h_{\rm gap}$ | ${R^{2}\over 2\pi^{2}}\min\left(\theta^{2},(\pi-\theta)^{2}\right)$ | ${2\over R^{2}\pi^{2}}\min\left(\phi^{2},(\pi-\phi)^{2}\right)$ | ${R^{2}\over 2}$ | ${2\over R^{2}}$
$g$ | $\sqrt{2\over R}$ | ${\sqrt{R}}$ | ${1\over\sqrt{2R}}$ | ${\sqrt{R}\over 2}$
Table 2: Bulk and boundary gaps and brane tensions on the $c=1$ orbifold
branch

Finally we come to the exceptional orbifolds that live at three isolated
points T,O and I on the $c=1$ moduli space [133]. They are described by the
orbifold of the $SU(2)_{1}$ CFT by the non-anomalous non-abelian global
symmetries $G=A_{4},S_{4},A_{5}\subset SO(3)$ respectively, which are related
to the exceptional simple Lie algebras $E_{6},E_{7},E_{8}$ by the Mckay
correspondence. These CFTs are rational with maximal chiral algebras given by
W-algebras of the type ${\mathcal{W}}(2,9,16)$, ${\mathcal{W}}(2,16)$ and
${\mathcal{W}}(2,36)$ respectively [141]. Their torus partition functions are
defined by the diagonal modular invariant with respect to these chiral
algebras. Consequently they contain elementary rational branes that are one-
to-one correspondence with the chiral primaries: 21 for $G=A_{4}$, 28 for
$G=S_{4}$ and 37 for $G=A_{5}$. Among them, the stable rational branes are
listed in Tables 3, 4 and 5, which are in one-to-one correspondence with
irreducible representations of $\tilde{G}$ which is the binary lift of $G$ to
$SU(2)$ (equivalently the stable rational branes are labelled by the
irreducible linear and projective representations of $G$). The values of their
brane tensions are listed there and also incorporated in the bootstrap bound
plot in Figure 4.

Besides the branes that preserve the diagonal W-algebras, the orbifold CFTs
all contain regular branes (2.38) defined by

$\displaystyle|[h]_{G}\rangle\equiv{1\over\sqrt{G}}\sum_{g\in
G}|ghg^{-1}\rangle$ (4.16)

which have tension $g=2^{-{1\over 4}}\sqrt{|G|}$ and live on a continous
moduli space parametrized by $G$-orbits $[h]_{G}$ in $SU(2)$. Such regular
branes are always unstable. They are elementary for generic $h\in SU(2)$ but
split into elementary fractional branes at fixed orbits where $[h]_{G}$ has a
nontrivial stablizer $G_{h}\subset G$. In particular, for $h\in\tilde{G}$, the
regular branes (now labelled by conjugacy classes of $\tilde{G}$) break down
to elementary rational branes of the exceptional orbifold CFT, which are in
general unstable. For the special cases $h=1$ and $h=-1$, the resulting
fractional rational branes are stable and labelled respectively by irreducible
linear and projective representations of $G$, as in Tables 3-5. The fractional
branes may admit marginal deformations and come with conformal families, in
which case the brane tension stays the same while the boundary spectrum can
develop a tachyon depending on the marginal parameter. In this case, we expect
the fractional branes to be stable for a closed subset of the boundary
conformal manifold. The above exhausts all stable branes in the exceptional
orbifold CFTs that are accessible from fractionalizations and marginal
deformations of the regular branes (4.16). Because brane tensions are constant
under boundary marginal deformations, the tension of all such stable branes
coincides with those listed in Tables 3-5.

| $|1_{0}\rangle,|1_{1}\rangle,|1_{2}\rangle$ | $|3\rangle$ | $|2_{0}\rangle,|2_{1}\rangle,|2_{2}\rangle$
---|---|---|---
$\Delta_{\rm gap}$ | $1\over 18$
$h_{\rm gap}$ | $9$ | $1,1$ | $1$
$g$ | ${1\over\sqrt{12\sqrt{2}}}$ | ${3\over\sqrt{12\sqrt{2}}}$ | ${2\over\sqrt{12\sqrt{2}}}$
Table 3: Stable rational branes in the $SU(2)_{1}/A_{4}$ orbifold CFT. They correspond to irreducible linear (left) and projective (right) representations of $A_{4}$. The label $|r_{i}\rangle$ keeps track of the multiple inequivalent $r$-dimensional representations of $G$. For each stable brane, we list the lowest scaling dimensions of the boundary operators, keeping track of their multiplicities. | $|1_{0}\rangle,|1_{1}\rangle$ | $|2\rangle$ | $|3_{0}\rangle,|3_{1}\rangle$ | $|2_{0}\rangle,|2_{1}\rangle$ | $|4\rangle$
---|---|---|---|---|---
$\Delta_{\rm gap}$ | $1\over 32$
$h_{\rm gap}$ | $16$ | $9$ | $1$ | $1$ | $1,1$
$g$ | ${1\over\sqrt{24\sqrt{2}}}$ | ${2\over\sqrt{24\sqrt{2}}}$ | ${3\over\sqrt{24\sqrt{2}}}$ | ${2\over\sqrt{24\sqrt{2}}}$ | ${4\over\sqrt{24\sqrt{2}}}$
Table 4: Stable rational branes in the $SU(2)_{1}/S_{4}$ orbifold CFT. The notations are parallel to that in Table 3. | $|1\rangle$ | $|3\rangle$ | $|3^{\prime}\rangle$ | $|4\rangle$ | $|5\rangle$ | $|2\rangle$ | $|2^{\prime}\rangle$ | $|4\rangle$ | $|6\rangle$
---|---|---|---|---|---|---|---|---|---
$\Delta_{\rm gap}$ | $1\over 50$
$h_{\rm gap}$ | $36$ | $1$ | $4$ | $1$ | $1$ | $1$ | $9$ | $1$ | $1,1$
$g$ | ${1\over\sqrt{60\sqrt{2}}}$ | ${3\over\sqrt{60\sqrt{2}}}$ | ${3\over\sqrt{60\sqrt{2}}}$ | ${4\over\sqrt{60\sqrt{2}}}$ | ${5\over\sqrt{60\sqrt{2}}}$ | ${2\over\sqrt{60\sqrt{2}}}$ | ${2\over\sqrt{60\sqrt{2}}}$ | ${4\over\sqrt{60\sqrt{2}}}$ | ${6\over\sqrt{60\sqrt{2}}}$
Table 5: Stable rational branes in the $SU(2)_{1}/A_{5}$ orbifold CFT. The
notations are parallel to that in Table 3.

From Table 1 and Table 2, we see the bulk scalar gap $\Delta_{\rm gap}$ is
sensitive to the target space modulus $R$ for the $c=1$ boson and its
${\mathbb{Z}}_{2}^{C}$ orbifold. We can thus study the bounds on the $g$
function as a function of the target space modulus by treating the bulk scalar
gap as a proxy for the latter. The gap $\Delta_{\rm gap}$ is maximized at the
self-dual radius, $R=\sqrt{2}$, where there is an enhancement of the chiral
algebra and the saturating theory is the $SU(2)_{1}$ WZW model. That this
bound is optimal in the space of all unitary, compact $c=1$ CFTs was
numerically observed in [31] and proven in [129] by construction of an
analytic functional whose action on the Virasoro characters is non-negative
for values of the dimension above the gap and that has zeros on the spectrum
of local operators in the saturating theory.

Figure 4: Upper (red) and lower (black) bounds on the $g$-function for stable
boundary conditions in $c=1$ CFTs as a function of the gap in the bulk scalar
sector. The dashed blue and magenta lines denote the curves $\Delta_{\rm
gap}^{1/4}$ and $2^{-1/2}\Delta_{\rm gap}^{1/4}$, corresponding to the
$g$-functions of the Dirichlet boundary conditions on the circle and orbifold
branches, respectively. The latter truncates at $\Delta_{\rm gap}={1\over 8}$
due to the moduli-independent gap in the twisted sector. The green points
denote the $g$-function of the stable rational branes in the exceptional
orbifold theories.

We are now in a position to compute bounds on the $g$-function for stable
boundary conditions in $c=1$ CFTs using the algorithm described in Section
3.1. An additional subtlety particular to $c=1$ is that in the regions where
the functional must act positively, we must additionally impose positivity of
the functional when acting on the degenerate characters at special values of
the weights where the representation of the Virasoro algebra becomes
degenerate. In particular, at weights given by $h={n^{2}\over 4}$ for
$n\in\mathbb{Z}_{\geq 0}$, the $c=1$ Virasoro algebra has null states at level
$n+1$. The corresponding degenerate characters are given by

$\widehat{\chi}_{\rm deg}^{(n)}(t)=t^{1\over 4}q^{n^{2}\over 4}(1-q^{n+1}).$
(4.17)

All bounds computed on the $g$ function for $c=1$ have this additional
positivity property incorporated.

In Figure 4 we present numerical upper and lower bounds on $g$ for stable
boundary conditions as a function of the bulk gap. The resulting bounds
exhibit some interesting features. We see that the upper bound is insensitive
to the gap in the spectrum of bulk scalar primaries (we will return to this
shortly). For sufficiently large bulk gap, the lower bound is well-
approximated by the $g$-function of the stable Dirichlet boundary condition on
the $S^{1}$ branch [134] (c.f. Table 1)

$g_{\rm Dirichlet}=(2R)^{-{1\over 2}}=\Delta_{\rm gap}^{1\over 4}.$ (4.18)

The bounds are also consistent with the stable boundary conditions on the
orbifold branch, including the exceptional orbifolds, as described in Tables
1-5. In particular, at the maximal value of the gap in the $\mathbb{Z}_{2}$
orbifold branch, $\Delta_{\rm gap}={1\over 8}$, the lower bound appears to be
saturated by the stable Dirichlet boundary condition:

$\Delta_{\rm gap}={1\over 8}:~{}g\geq 2^{-5/4}$ (4.19)

to within at least 40 digits of precision. On the other hand, in the case of
the exceptional orbifolds, the lower bounds are not quite saturated by the
corresponding minimal-tension rational branes. However, we find that if we
feed the detailed information of the low-lying bulk spectrum into the numerics
as described around equation (4.4), then the lower bounds are indeed realized
by these minimal tension rational branes. For instance, if we incorporate the
bulk scalars with dimension $\Delta\leq 2$, then the lower bounds obtained at
$N=127$ derivatives improve to the following

$\displaystyle A_{5}:~{}g$ $\displaystyle\geq
0.1085592604054384293742693646631371565$ (4.20) $\displaystyle S_{4}:~{}g$
$\displaystyle\geq 0.1716472619922598129858564379334175563$ $\displaystyle
A_{4}:~{}g$ $\displaystyle\geq 0.2427458858536617101418603137455420728,$

which should be compared to the numerical values of
${1\over\sqrt{60\sqrt{2}}}$, ${1\over\sqrt{24\sqrt{2}}}$ and
${1\over\sqrt{12\sqrt{2}}}$, respectively.

We also notice that at the maximal value of the bulk scalar gap $\Delta_{\rm
gap}={1\over 2}$, corresponding to the free boson theory at the self-dual
radius $R=\sqrt{2}$, the upper and lower bounds precisely coincide at

$g=2^{-{1\over 4}},$ (4.21)

so that the $g$-function is uniquely specified for stable boundary conditions.
The saturating $g$-function is realized by the cylinder partition function
associated with the identity Cardy brane, which is the identity character of
the $SU(2)_{1}$ current algebra in the open string channel

$Z^{SU(2)_{1}}_{\mathds{1}\mathds{1}}(t^{-1})={\Theta_{0,1}(it)\over\eta(it)}={\theta_{3}(2it)\over\eta(it)}\,,$
(4.22)

where

$\Theta_{m,k}(it)\equiv\sum_{n\in\mathbb{Z}}q^{k(n+{m\over 2k})^{2}}\,.$
(4.23)

We also notice in Figure 4 that the upper bound is completely flat, apparently
independent of the assumed gap in the spectrum of bulk scalars. The reason is
ultimately that the bounds must also apply to general linear combinations of
solutions to the cylinder crossing equation. For example, consider the
following fictitious cylinder partition function

$Z_{\rm fict}(t)=\sum_{i}a_{i}Z_{\alpha_{i}\alpha_{i}}(t)\,,$ (4.24)

where $\alpha_{i}$ label different stable boundary conditions (not necessarily
associated with the same bulk CFT, however the bulk central charge is assumed
to be the same) and the $a_{i}$ are positive coefficients such that
$\sum_{i}a_{i}=1$. Suppose the boundary condition $\alpha_{1}$ leads to the
largest value of $g=g_{1}$. Then if we take $a_{1}\gg a_{i>1}$, the
$g$-function associated to $Z_{\rm fict}$ will be well approximated by
$g_{1}$, while the gap in the bulk scalar sector will be given by the minimum
of the bulk gaps among the cylinder partition functions associated to the
boundary conditions $\alpha_{i}$. So in situations where the bulk gap can be
made arbitrarily small while admitting stable conformal boundaries, the upper
bound on $g$ must be insensitive to the value of the bulk gap. We emphasize
that this argument is quite general and does not only apply to $c=1$, indeed
we will observe this insensitivity of the upper bound to the assumed gap in
the bulk spectrum in the other examples that follow. Note that integrality of
the degeneracies in the boundary spectrum could in principle be used to rule
out these fictitious solutions to the crossing equations, since these linear
combinations do not in general preserve integrality of degeneracies of
operators in the boundary channel.

In practice, we also observe experimentally (in the present $c=1$ case and in
all examples that follow) that the lower bound on the $g$-function is
completely insensitive to the assumption of the gap in the boundary spectrum.
As far as we can tell, this phenomenon cannot similarly be simply explained by
the fact that the bounds should also apply to linear combinations of cylinder
partition functions. In cases where there is a boundary conformal manifold,
the boundary gap can be tuned (in general within a finite range) without
affecting the $g$-function.313131This happens for example for the Dirichlet
and Neumann boundaries for the ${\mathbb{Z}}_{2}^{C}$ orbifold at $c=1$ (see
Table 2).

###  4.2 $c=2$ and the $SU(3)_{1}$ CFT

For our next class of examples we study bounds on the $g$-function for $c=2$
CFTs. Already at $c=2$ there is no known classification of boundary states. A
family of $c=2$ CFTs is provided by sigma models with $T^{2}$ target space,
the theory of two free bosons based on an even self-dual lattice
$\Lambda\in\mathbb{R}^{2,2}$. This family of CFTs, whose operator content
organizes into representations of a $U(1)^{2}\times U(1)^{2}$ chiral algebra,
is parameterized by a four-dimensional moduli space, labelled by two elements
$\sigma,\rho$ of $\mathbb{H}/PSL(2,\mathbb{Z})$, which are respectively the
complex structure and the complexified Kahler structure of the target $T^{2}$.
These parameters can be written in terms of the metric $G$ and $B$-field flux
of the target $T^{2}$ as

$\displaystyle\rho$ $\displaystyle=B+i\sqrt{\det G}$ (4.25)
$\displaystyle\sigma$ $\displaystyle={G_{12}\over G_{11}}+i{\sqrt{\det G}\over
G_{11}}.$

If we work in a $T$-duality frame where $\sigma,\rho$ lie in the standard
fundamental domain of $\mathbb{H}/PSL(2,\mathbb{Z})$, then the gap in the
spectrum of primary operators is given in terms of the target space moduli as

$\Delta_{\rm gap}={1\over 2\operatorname{Im}\rho\operatorname{Im}\sigma}.$
(4.26)

Like the $c=1$ case, this is maximized at a point of enhanced symmetry, namely
the $\mathbb{Z}_{3}$ point where $\sigma=\rho=e^{\pi i\over 3}$. Here,
$\Delta_{\rm gap}={2\over 3}$, and the saturating theory is the $SU(3)_{1}$
WZW model. This turns out to be the maximal value of the gap in the space of
all unitary, compact $c=2$ CFTs [31].

Figure 5: Upper (red) and lower (black) bounds on the $g$-function for stable
boundary conditions in $c=2$ CFTs as a function of the gap in the bulk scalar
sector.

In Figure 5 we plot bounds on the $g$-function associated with stable boundary
conditions for $c=2$ CFTs as a function of the gap in the spectrum of bulk
primaries. Unlike the case of $c=1$, the upper and lower bounds are not
coincident at the maximal value of the gap. In order to explore whether the
$g$-function is uniquely specified for stable boundary conditions in the
theory with the maximal gap, we incorporate the information of a number of
low-lying bulk scalar primaries into the crossing equation as described in
Section 3.1. The bulk scalars come in two towers of states spaced by even
integers

$\Delta\in\\{2\mathbb{Z}_{\geq 0}\\}\cup\\{2/3+2\mathbb{Z}_{\geq 0}\\}.$
(4.27)

# of bulk scalars incorporated | lower bound | upper bound
---|---|---
0 | 0.7560986262 | 0.7677246687
2 | 0.7596267328 | 0.7598741664
10 | 0.7598352810 | 0.7598356865
50 | 0.7598356856 | 0.7598356857
Table 6: Upper and lower bounds on the $g$ function for stable boundary
conditions in the $SU(3)_{1}$ WZW model at $N=127$ derivative order with an
increasing number of low-lying bulk scalars incorporated. The bounds appear to
be convergent upon $3^{-{1\over 4}}=0.75983568565\ldots$

In Table 6 we present the resulting upper and lower bounds on $g$. As the
number of low-lying bulk scalars fed into the crossing equation is increased,
the bounds appear convergent upon

$\displaystyle g=3^{-{1\over 4}}$ (4.28)

to within a high degree of numerical precision. This is precisely the boundary
entropy associated with the identity rational brane $|B_{\mathds{1}}\rangle$
of the $SU(3)_{1}$ WZW model, with the saturating cylinder partition function
given by the identity character of the $SU(3)_{1}$ current algebra

$\displaystyle
Z^{SU(3)_{1}}_{\mathds{1}\mathds{1}}(t^{-1})={\Theta_{0,3}(it)\Theta_{0,1}(it)+\Theta_{3,3}(it)\Theta_{1,1}(it)\over\eta(it)^{2}}$
(4.29)

The same properties hold for the boundary conditions obtained from fusion with
the ${SU(3)\times SU(3)\over{\mathbb{Z}}_{3}}\rtimes{\mathbb{Z}}_{2}$
invertible global symmetries on $|B_{\mathds{1}}\rangle$. They include in
particular the other two rational branes $|B_{\varphi}\rangle$ and
$|B_{\bar{\varphi}}\rangle$ corresponding to the two $h={2\over 3}$ chiral
primaries $\varphi,\bar{\varphi}$ which are generated from the
${\mathbb{Z}}_{3}$ center symmetry acting on $|B_{\mathds{1}}\rangle$.

Indeed, we see that the spectrum of this cylinder partition function is
realized by the action of the corresponding numerical extremal functional
acting on the bulk and boundary characters, as plotted in Figure 6. In
particular we notice that the boundary spectrum is supported on positive
integer weights, consistent with the fact that the saturating cylinder
partition function is the $SU(3)_{1}$ vacuum character in the open string
channel.

A novel feature of this functional compared to extremal functionals that have
been studied previously in modular bootstrap is that the bulk spectrum is
supported on multiple towers of states with dimensions separated by even
integers. It would be very interesting to construct this functional
analytically using techniques of [33].

(a) Bulk spectrum

(b) Boundary spectrum

Figure 6: Action of the functional $\omega^{-}$ that computes the lower bounds
on $g$ for the $SU(3)_{1}$ WZW model on the bulk and boundary characters. We
plot the absolute value of the functional acting on bulk characters because
when we feed in the information of the low-lying scalars into the numerics,
the resulting functional has single rather than double zeros at these
dimensions.

###  4.3 $(G_{2})_{1}$ CFT

The next example found to saturate modular bootstrap bounds on the gap that we
will consider is the $(G_{2})_{1}$ WZW model, with central charge $c={14\over
5}$. The gap in the spectrum of primary operators in this theory is

$\Delta_{\rm gap}={4\over 5},$ (4.30)

which was found to be optimal in the space of unitary, compact $c={14\over 5}$
CFTs in [31].

As in the case of $c=2$, it appears from the numerical bounds on $g$ that the
information of the bulk gap alone is insufficient to uniquely specify the
boundary entropy for stable boundary conditions in the $(G_{2})_{1}$ WZW
model. However, as shown in Table 7, when we incorporate the low-lying scalars
with dimensions

$\Delta\in\\{2\mathbb{Z}_{\geq 0}\\}\cup\\{4/5+2\mathbb{Z}_{\geq 0}\\}$ (4.31)

into the cylinder crossing equation, the resulting bounds on $g$ are
increasingly convergent upon

$g=\left({1\over 2}-{\sqrt{5}\over 10}\right)^{1\over 4}\,.$ (4.32)

# of bulk scalars incorporated | lower bound | upper bound
---|---|---
0 | 0.7152115481 | 0.7409519929
2 | 0.7241441983 | 0.7252206408
10 | 0.7250670675 | 0.7250731872
50 | 0.7250731746 | 0.7250731771
Table 7: Upper and lower bounds on the $g$ function for stable boundary
conditions in the $(G_{2})_{1}$ WZW model at $N=127$ derivative order with an
increasing number of low-lying bulk scalars incorporated. The bounds appear to
be convergent upon $\left({1\over 2}-{\sqrt{5}\over 10}\right)^{1\over
4}=0.72507317708\ldots$

The $(G_{2})_{1}$ WZW model has two rational branes $|B_{\mathds{1}}\rangle$
and $|B_{\varphi}\rangle$ corresponding to the identity and the $h={2\over 5}$
chiral primary $\varphi$.323232The $(G_{2})_{1}$ CFT has a fusion category
symmetry described by the Fibonacci category with a single simple TDL $W$
satisfying the fusion rule $W^{2}=1+W$.The quantum dimension of $W$ is
$\langle W\rangle={\sqrt{5}+1\over 2}$. The two rational branes are related by
fusion as $|B_{\varphi}\rangle=W|B_{\mathds{1}}\rangle$.  Their $g$-functions
are given by

$\displaystyle g_{\mathds{1}}=\left({1\over 2}-{\sqrt{5}\over
10}\right)^{1\over 4}\,,\quad
g_{\varphi}=\left(1+{2\over\sqrt{5}}\right)^{1\over 4}\,.$ (4.33)

We see the former precisely coincide with our bootstrap bound (4.32). Indeed
the second boundary state $|B_{\varphi}\rangle$ is not stable, as it contains
a $h={2\over 5}$ relevant operator in the open-string spectrum.

We can further compare the entire spectrum obtained from the optimal
functional with the cylinder parition function for the stable brane
$|B_{\mathds{1}}\rangle$. The explicit form of the latter can be obtained
using the following isomorphism [142]

$\displaystyle(G_{2})_{1}\cong{SU(3)_{1}\times\text{three-state
Potts}\over{\mathbb{Z}}_{3}}\,.$ (4.34)

Then the saturating cylinder partition function is given by the identity
character of the $(G_{2})_{1}$ chiral algebra, which can be written as

$\displaystyle Z^{(G_{2})_{1}}_{\mathds{1}\mathds{1}}(t^{-1})=\chi_{0}^{\rm
Potts}(t)\chi_{\bf 0}^{SU(3)_{1}}(t)+\chi_{2/3}^{\rm Potts}(t)\chi_{\bf
3}^{SU(3)_{1}}(t)+\chi_{2/3}^{\rm
Potts}(t)\chi_{\bf\bar{3}}^{SU(3)_{1}}(t)\,,$ (4.35)

where $\chi_{h}^{\rm Potts}$ are W-algebra characters for primaries of weight
$h$ in the three-state Potts CFT,

$\displaystyle\chi_{0}^{\rm Potts}(t)=$ $\displaystyle{q^{1\over
20}\over\eta(\tau)^{2}}\sum_{r,s=0}^{\infty}(-1)^{r+s}q^{{r(r+1)\over
2}+{s(s+1)\over 2}+4rs}\left(1-q^{4(1+r+s)}\right)\,,$ (4.36)
$\displaystyle\chi_{2/3}^{\rm Potts}(t)=$ $\displaystyle{q^{43\over
60}\over\eta(\tau)^{2}}\sum_{r,s=0}^{\infty}(-1)^{r+s}q^{{r(r+1)\over
2}+{s(s+1)\over 2}+4rs}\left(q^{2r+s}-q^{1+2r+3s)}\right)\,.$

and $\chi_{\bf r}^{SU(3)_{1}}$ are $SU(3)_{1}$ Kac-Moody characters,

$\displaystyle\chi_{\bf 0}^{SU(3)_{1}}(t)=$
$\displaystyle{\Theta_{0,3}(it)\Theta_{0,1}(it)+\Theta_{3,3}(it)\Theta_{1,1}(it)\over\eta(it)^{2}}\,,$
(4.37) $\displaystyle\chi_{\bf
3}^{SU(3)_{1}}(t)=\chi_{\bf\bar{3}}^{SU(3)_{1}}(t)=$
$\displaystyle{\Theta_{2,3}(it)\Theta_{0,1}(it)+\Theta_{1,3}(it)\Theta_{1,1}(it)\over\eta(it)^{2}}\,.$

Indeed, the spectrum of this saturating cylinder partition function is
explicitly realized by the action of the numerical extremal functional on the
bulk and boundary characters, as plotted in Figure 7.

(a) Bulk spectrum

(b) Boundary spectrum

Figure 7: Action of the functional $\omega^{-}$ that computes the lower bounds
on $g$ for the $(G_{2})_{1}$ WZW model on the bulk and boundary characters.

###  4.4 $Spin(8)_{1}$ CFT

In [31], a kink was found in the curve describing the upper bound on the gap
in the spectrum of Virasoro primaries at $c=4$, where the optimal bound on the
gap was found to be $\Delta_{\rm gap}=1$. This bound is saturated by the
$Spin(8)_{1}$ WZW model.333333This is also sometimes written as the
$SO(8)_{1}$ WZW model in the literature. In general the $Spin(n)_{k}$ and
$SO(n)_{k}$ WZW models are related by a ${\mathbb{Z}}_{2}$ orbifold
$\displaystyle SO(n)_{k}={Spin(n)_{k}/{\mathbb{Z}}_{2}}\,,$ (4.38) where the
${\mathbb{Z}}_{2}\subset Z(Spin(n))$ is the kernel of the projection from the
double cover $Spin(n)$ to $SO(n)$. For $k=1$, the WZW model is self-dual under
this ${\mathbb{Z}}_{2}$ gauging,  $\displaystyle SO(n)_{1}\cong Spin(n)_{1}.$
(4.39) Once again, we find that the information of the bulk gap is
insufficient to pin down the $g$-function associated to stable boundary
conditions in this CFT. By incorporating the information of low-lying bulk
scalars with

$\Delta\in\mathbb{Z}_{\geq 0}$ (4.40)

into the cylinder crossing equation, the bounds on $g$ appear to be
increasingly convergent upon

$\displaystyle g={1\over\sqrt{2}}.$ (4.41)

See Table 8 for the numerical bounds.

# of bulk scalars incorporated | lower bound | upper bound
---|---|---
0 | 0.6801484630 | 0.7368753088
2 | 0.7021195531 | 0.7077424981
10 | 0.7069544116 | 0.7071069574
50 | 0.7071058400 | 0.7071067812
100 | 0.7071058572 | 0.7071067812
Table 8: Upper and lower bounds on the $g$ function for stable boundary
conditions in the $Spin(8)_{1}$ WZW model at $N=127$ derivative order with an
increasing number of low-lying bulk scalars incorporated. The bounds appear to
be convergent upon ${1\over\sqrt{2}}=0.7071067811865\ldots$.

Yet again, we find that the saturating value for the $g$-function of stable
boundary conditions is saturated by the identity rational brane
$|B_{\mathds{1}}\rangle$ of the $Spin(8)_{1}$ WZW model. The corresponding
cylinder partition function is given by the vacuum character of the
$Spin(8)_{1}$ chiral algebra

$\displaystyle
Z^{Spin(8)_{1}}_{\mathds{1}\mathds{1}}(t^{-1})={\theta_{3}(it)^{4}+\theta_{4}(it)^{4}\over
2\eta(it)^{4}}.$ (4.42)

The $Spin(8)_{1}$ CFT has three other rational branes
$|{\mathcal{B}}_{\varphi_{i}}\rangle$ with $i=1,2,3$ associated with the three
chiral primaries $\varphi_{i}$ of weight $h={1\over 2}$ that are permuted
under the $S_{3}$ triality symmetry. In addition, they are also related by
fusing the three ${\mathbb{Z}}_{2}$ generators of the
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ center symmetry with
$|B_{\mathds{1}}\rangle$. Consequently, all rational branes of the
$Spin(8)_{1}$ CFT have the same $g$-function and cylinder partition function.

The $Spin(8)_{1}$ is self-dual under gauging the
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ center symmetry, thus it also has
duality defects ${\mathcal{N}}_{i}$ for each ${\mathbb{Z}}_{2}$ subgroup
generated by $\eta_{i}$ satisfying

$\displaystyle{\mathcal{N}}^{2}_{i}=1+\eta_{i}\,,~{}~{}{\mathcal{N}}_{i}\eta_{i}=\eta_{i}{\mathcal{N}}_{i}={\mathcal{N}}_{i}\,,~{}~{}\eta_{i}^{2}=1\,,~{}~{}\eta_{1}\eta_{2}=\eta_{3}\,.$
(4.43)

They give rise to conformal boundaries
$|{\mathcal{N}}_{i}\rangle\equiv{\mathcal{N}}_{i}|B_{\mathds{1}}\rangle$ by
fusion. As reviewed in Section 2.5, these boundary states has
$g_{{\mathcal{N}}_{i}}=\langle{\mathcal{N}}_{i}\rangle g_{\mathds{1}}=1$ and
cylinder partition function (as in (2.36)),

$\displaystyle
Z^{Spin(8)_{1}}_{{\mathcal{N}}_{i}{\mathcal{N}}_{i}}(t^{-1})=Z^{Spin(8)_{1}}_{\mathds{1}\mathds{1}}(t^{-1})+Z^{Spin(8)_{1}}_{\mathds{1}\varphi_{i}}(t^{-1})\,.$
(4.44)

The second term on the RHS contains operators of dimension $h={1\over 2}$
which follows from the $Spin(8)_{1}$ fusion rule343434They are the lowest
weight boundary changing operators between boundaries $|B_{\mathds{1}}\rangle$
and $|B_{\varphi_{i}}\rangle$. and thus $|{\mathcal{N}}_{i}\rangle$ are
elementary but unstable, in agreement with our bootstrap bound.

###  4.5 $c=8$ and the $(E_{8})_{1}$ CFT

Another kink was identified on the modular bootstrap bounding curve of [31],
this time for the bound on the spectrum of scalar Virasoro primaries (which is
weaker than the overall gap bound for $c>4$), at $c=8$, with $\Delta_{\rm
gap}^{\ell=0}=2$. This bound is saturated by the $(E_{8})_{1}$ WZW model. In
Figure 8 we plot the bounds on the $g$-function for stable boundary conditions
as a function of the bulk scalar gap and again find that they are nearly
coincident at the maximal value of the bulk gap. Indeed, at $\Delta_{\rm
gap}=2$, we find the following upper and lower bounds on $g$

$N=127:~{}0.9999998668<g<1.0000001332.$ (4.45)

Thus it is tempting to conjecture that at this value of the bulk gap, $g$ is
uniquely specified for stable conformal boundaries to be

$g=1$ (4.46)

for stable boundary conditions in the $(E_{8})_{1}$ WZW model. Indeed, the
lower bound is insensitive to the value of the gap in the spectrum of boundary
primaries, so this suggests that _all_ boundary conditions (not necessarily
stable) for the $(E_{8})_{1}$ WZW model satisfy $g\geq 1$. In the next section
we will prove that this is the case by constructing analytically the extremal
functional that proves optimality of this bound.

Figure 8: Upper (red) and lower (black) bounds on the $g$-function for stable
boundary conditions in $c=8$ CFTs.

Once again, this value of $g$ is realized by the identity Cardy brane in the
$(E_{8})_{1}$ WZW model, with the cylinder partition function given by the
identity character of the $(E_{8})_{1}$ chiral algebra

$\displaystyle