# Cosmologies inside hyperbolic black holes

Simon F<EMAIL_ADDRESS>Centre for Particle Theory, Department
of Mathematical Sciences Durham University, South Road, Durham DH1 3LE, U.K.

###### Abstract

Models with closed FRW cosmologies on the worldvolume of a constant-tension
brane inside a black hole provide an interesting setup for studying cosmology
holographically. However, in more than two worldvolume dimensions, there are
limitations on such models with flat spatial slices. I show that these
limitations can be avoided by considering instead hyperbolic slices. This also
naturally makes contact with previous work on Euclidean wormholes.

##  1 Introduction

An approach to understanding closed universes with big-bang/big-crunch
cosmologies holographically was proposed in [1] (and further developed in [2,
3, 4, 5, 6, 7]). The idea is to consider an asymptotically AdSd+1 black hole
spacetime with a $d$-dimensional dynamical end of the world (ETW) brane behind
the horizon providing an inner boundary of the spacetime, as depicted in
figure 1. Starting from the $t=0$ surface, the ETW brane falls into the black
hole and terminates at the singularity, so its worldvolume geometry is a big-
bang/big-crunch cosmology. The state in the bulk on the $t=0$ surface is dual
to some state in the dual $d$-dimensional CFT on the asymptotic boundary on
the right in figure 1, which therefore includes a description of the cosmology
on the ETW brane worldvolume.

An appealing feature of this model is that the state in the bulk on the $t=0$
surface can be constructed from a Euclidean path integral. In the Euclidean
section, the ETW brane moves outward away from $t=0$, and eventually meets the
asymptotic boundary. Such solutions were proposed in [8, 9] as duals of
boundary conformal field theories (BCFTs). The state in the $d$-dimensional
CFT dual to the $t=0$ slice in the bulk is then constructed by starting with a
$(d-1)$-dimensional boundary state specified by the BCFT and evolving through
some period of Euclidean time. Such states have been extensively discussed in
recent investigations of black holes, see e.g. [10, 11, 12, 13].

Figure 1: Penrose diagram of an AdS black hole with the left asymptotic region
terminating in an ETW brane (shown in red). The worldvolume geometry of the
ETW brane is a big-bang/big-crunch cosmology.

To have a controlled description of the cosmology on the ETW brane, we want to
have a separation of scales between the scale that controls the curvature of
the ETW brane and the bulk curvature, such that there is a good effective
description of the dynamics in terms of ordinary Einstein gravity localised on
the ETW brane. This can be achieved by taking the radial position $r_{0}$ of
the brane at $t=0$ to be much larger than the horizon scale $r_{H}$, which can
be achieved by increasing the tension $T$ of the ETW brane [14, 15, 16].

However, in simple examples of this construction this separation of scales is
incompatible with the path integral construction of the state. If we are in
$d>2$, and we take the bulk solution to be an uncharged black hole with flat
spatial slices, when we increase $T$ for fixed bulk black hole geometry, there
is a critical value $T=T_{*}$ at which the ETW brane intersects the asymptotic
boundary at $t=0$. Increasing $T$ beyond $T_{*}$, the ETW brane will self-
intersect before it reaches the asymptotic boundary. Thus, for $T>T_{*}$, we
lose the Euclidean path integral construction of the state dual to the $t=0$
slice.222The $d=2$ case, where the bulk black hole is a BTZ solution [17], is
special; there the ETW brane always intersects the boundary at a quarter the
period of the Euclidean solution. One resolution of this problem, proposed in
[2], is to take the bulk solution to be a charged black hole. In [5], it was
suggested that the problem could be avoided by introducing an additional
interface brane, but in [18] (see also [19]) we found that the self-
intersection problem persists in the presence of the interface brane.

In this paper, we will see that another way to avoid this problem is to
consider hyperbolic spatial slices, with constant negative curvature. Bulk
black hole solutions with hyperbolic horizons were obtained in [20, 21, 22];
these include a locally-AdS solution, which is a natural higher-dimensional
analogue of the BTZ black hole. It is perhaps not surprising that considering
ETW branes behind the horizon of these hyperbolic black hole then gives us a
higher-dimensional scenario with behaviour similar to the $d=2$ case.

Another way to see why considering hyperbolic spatial slices is helpful is to
consider the worldvolume of the ETW brane. In the Euclidean black hole this is
a wormhole; for hyperbolic spatial sections this is
$\Sigma_{g}\times\mathbb{R}$, and we will see that when we consider the
locally-AdS “topological” black hole, the worldvolume geometry is precisely
the Maldacena-Maoz wormhole of [23]. This is a solution of Einstein’s
equations with a negative cosmological constant. By contrast, there is no
classical solution describing a wormhole with flat spatial sections; the bulk
spacetime thus plays a more essential role in the construction of the
solutions where the ETW brane is a wormhole with flat spatial sections, and it
seems natural that it’s harder to construct solutions in the limit where the
effective dynamics is approximated by Einstein gravity localised on the ETW
brane.

As in the charged black hole case, these hyperbolic black holes have an
extremal limit, where the temperature goes to zero for fixed horizon area. But
the more important effect from the ETW brane perspective is that the curvature
term dominates over the mass term in the metric at large distances. As a
result, in the limit of interest, the mass term is negligible and the dynamics
of the ETW brane is as in the locally-AdS case, which gives a finite time for
the ETW brane to reach the Euclidean boundary, as in the $d=2$ BTZ black hole
example.

In the next section, we review some of the previous work on these models. In
section 3, we introduce the generalization to consider hyperbolic spatial
sections. Section 4 presents some concluding remarks. In appendix A, I include
some speculations on the existence of other bulk saddle-points with the same
boundary conditions as the hyperbolic black hole; this is a side comment
independent of the main discussion.

##  2 End of the world brane cosmology

The holographic model of cosmology we consider was first proposed in [1]. The
model consists of an AdS black hole bulk with one asymptotic region, with a
dynamical constant-tension ETW brane behind the horizon, as pictured in figure
1. The induced geometry on the ETW brane worldvolume is that of a closed FRW
universe, with the radial position playing the role of the scale factor. The
bulk action is

$I=\frac{1}{16\pi
G}\left[\int_{\mathcal{M}}d^{d+1}x\sqrt{-g}\,(R-2\Lambda)+2\int_{\partial\mathcal{M}}d^{d}y\sqrt{-h}\,K-2(d-1)\int_{\mathcal{Q}}d^{d}y\sqrt{-h}\,T\right],$
(1)

where $\Lambda=-\frac{d(d-1)}{2L^{2}}$ is a cosmological constant, $K$ is the
trace of the extrinsic curvature and $T$ is the tension of the ETW brane with
worldvolume $\mathcal{Q}$, which we take to be one component of the boundary
$\partial\mathcal{M}$ of the spacetime, the other component corresponding to
the asymptotically AdS conformal boundary. We consider a $(d+1)$-dimensional
bulk spacetime, dual to a $d$-dimensional CFT on the boundary. Previous work
considered an AdS-Schwarzchild black hole bulk solution

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+\frac{r^{2}}{L^{2}}dx^{a}dx_{a},\quad\quad
f(r)\equiv\frac{r^{2}}{L^{2}}-\frac{\mu}{r^{d-2}},$ (2)

where $a=1,\ldots,d-1$. This has a horizon at $r=r_{H}$, where $r_{H}^{d}=\mu
L^{2}$. The brane has stress-energy tensor $8\pi GT_{ab}=(1-d)Th_{ab}$, and
the action implies that the boundary condition for the bulk metric at
$\mathcal{Q}$ is

$K_{ab}-Kh_{ab}=(1-d)Th_{ab}$ (3)

The $tt$ component of this equation leads to the brane equation of motion

$\left(\frac{dr}{dt}\right)^{2}=\frac{f^{2}(r)}{T^{2}r^{2}}\left(T^{2}r^{2}-f(r)\right).$
(4)

In the Lorentzian black hole geometry, the brane will reach a maximum radius
$r_{0}$, with $(r_{0})^{d}=\frac{r_{H}^{d}}{1-T^{2}L^{2}}$, which we take to
occur at $t=0$. Note $r_{0}>r_{H}$ for $T>0$, and $r_{0}\to\infty$ as $T\to
L^{-1}$. To the future and past of this, $r(t)$ decreases, as pictured in
figure 1. The brane worldvolume geometry is thus a closed FRW big-bang/big-
crunch cosmology, where the brane radius $r(t)$ plays the role of the scale
factor, and the brane equation of motion (4) corresponds to the Friedmann
equation in this worldvolume cosmology.

The state on the $t=0$ slice can be obtained by a Euclidean path integral. In
the Euclidean black hole, the motion of the ETW brane is

$\left(\frac{dr}{d\tau}\right)^{2}=\frac{f^{2}(r)}{T^{2}r^{2}}\left(f(r)-T^{2}r^{2}\right).$
(5)

This now has a minimum at $r=r_{0}$. Since the ETW brane is inside the black
hole in the Lorentzian geometry, it is at its minimum radius in the Euclidean
solution at $\tau=\beta/2$, where $\beta=2\pi L^{2}/r_{H}$ is the periodicity
in Euclidean time $\tau$. It reaches the AdS boundary at a time

$\tau^{ETW}=\frac{\beta}{2}-\int_{r_{0}}^{\infty}\frac{dr}{f(r)}\frac{Tr}{\sqrt{f(r)-T^{2}r^{2}}}.$
(6)

To avoid self-intersections in the Euclidean solution, we need $\tau^{ETW}>0$.
However, setting $r=r_{0}x$, we have

$\sigma^{ETW}=\frac{2\tau^{ETW}}{\beta}=1-\frac{d}{2\pi}TL(y_{0})^{\frac{d-2}{2}}\int_{1}^{\infty}\frac{dx}{x^{2}(1-(y_{0})^{-d}x^{-d})\sqrt{1-x^{-d}}},$
(7)

where $y_{0}=r_{0}/r_{H}$, so $(y_{0})^{-d}=1-T^{2}L^{2}$. It is clear that
for $d>2$ we can’t take $r_{0}\to\infty$ while keeping $\tau^{ETW}>0$. There
must then be some critical value $T=T_{*}<L^{-1}$ such that $\tau^{ETW}=0$,
and if we consider $T>T_{*}$ we will have self-intersecting branes in the
Euclidean solution.

##  3 Hyperbolic spatial sections

The previous discussion was for branes with flat spatial sections. Consider
now the generalization to hyperbolic spatial sections. This changes relatively
little in the previous analysis; the relevant black hole solution is now [20,
21, 22]

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+\frac{r^{2}}{L^{2}}ds^{2}_{\Sigma_{g}},\quad\quad
f(r)=\frac{r^{2}}{L^{2}}-1-\frac{\mu}{r^{d-2}},$ (8)

where $\Sigma_{g}$ is a $(d-1)$-dimensional compact manifold of constant
negative curvature. For $\mu\in(\mu_{min},\infty)$, where

$\mu_{min}=-\frac{2}{d}\left(\frac{d-2}{d}\right)^{\frac{d-2}{2}}L^{d-2},$ (9)

the black hole has a horizon at $r=r_{H}>0$ where $f(r_{H})=0$; we can write

$\mu=r_{H}^{d-2}(\frac{r_{H}^{2}}{L^{2}}-1).$ (10)

The inverse temperature of the black hole is

$\beta=\frac{4\pi r_{H}L^{2}}{dr_{H}^{2}-(d-2)L^{2}}.$ (11)

As $\mu\to\mu_{min}$, $r_{H}^{2}\to\frac{d-2}{d}L^{2}$, and $f(r)$ develops a
double root. The temperature goes to zero.333The possibility of having a
vanishing temperature seems like it would be useful in constructing ETW brane
solutions, but as we will see, the more important effect is the $-1$ in
$f(r)$, which dominates over the mass term at large radial distances.

For $\mu=0$, the solution is locally AdSd+1. The metric with $\Sigma_{g}$
replaced by a non-compact hyperbolic space $H^{d-1}$ is simply AdS written in
a Rindler-like coordinate system (the coordinate transformation between these
coordinates and Poincaré coordinates basically reduces to the relation between
Rindler coordinates and hyperbolic coordinates on the conformal boundary). The
geometry of interest here is then obtained by quotienting by a discrete
isometry group $\Gamma$ such that $H^{d-1}/\Gamma=\Sigma_{g}$. This $\mu=0$
solution is therefore referred to as the topological black hole. The $\mu=0$
solution has a finite temperature, which is related to the Rindler
acceleration temperature in the boundary theory if we don’t do the quotient.

We consider an ETW brane inside this black hole, with a turnaround at
$r=r_{0}$ at $t=0$. Considering the Euclidean continuation, the brane equation
of motion is (5), and the brane reaches the AdS boundary at

$\tau^{ETW}=\frac{\beta}{2}-\int_{r_{0}}^{\infty}\frac{dr}{f(r)}\frac{Tr}{\sqrt{f(r)-T^{2}r^{2}}}.$
(12)

The key difference from the previous flat case is the behaviour of the square
root factor in the denominator: if we set $r=r_{0}x$, in both cases, we pull
out a factor of $1/\sqrt{r_{0}^{2}(1-T^{2}L^{2})}$, but whereas in the flat
case we had $1-T^{2}L^{2}\sim 1/r_{0}^{d}$, so this factor goes like
$r_{0}^{\frac{d-2}{2}}$, which blows up as $r_{0}\to\infty$, for the
hyperbolic black hole,
$r_{0}^{2}(1-T^{2}L^{2})=L^{2}-\mu\frac{L^{2}}{r_{0}^{d-2}}$. The last term
becomes negligible if $\mu$ remains finite as $r_{0}\to\infty$, so
$1/\sqrt{r_{0}^{2}(1-T^{2}L^{2})}$ has a finite limit as $r_{0}\to\infty$.

This can be simply illustrated by considering the case $\mu=0$. As noted
above, we still have a finite temperature at $\mu=0$: (11) gives $\beta=2\pi
L$. Setting $r=r_{0}x$, we have

$\tau^{ETW}=\pi
L-TL^{2}\int_{1}^{\infty}\frac{xdx}{\left(x^{2}-\frac{L^{2}}{r_{0}^{2}}\right)\sqrt{x^{2}-1}}.$
(13)

As $TL\to 1$, $r_{0}\to\infty$, and we can do the integral exactly; it’s equal
to $\frac{\pi}{2}$, so $\tau^{ETW}\to\frac{\pi L}{2}$. Thus, there’s no self-
intersection problem here; we can freely take the ETW brane to large radius in
arbitrary dimensions.444The integrand is larger, and hence $\tau^{ETW}$
smaller, for finite $r_{0}$; but we are mostly interested in the region of
large $r_{0}$, so we will not analyse in detail the positivity of $\tau^{ETW}$
for finite $r_{0}$. It is interesting to note that for $\mu=0$, the bulk
geometry is locally AdSd+1, and the brane is locally AdSd, so the solution we
are considering here is just a quotient of the sub-critical braneworld
introduced in [16].

Generalizing to $\mu\neq 0$, so long as we hold $\mu$ fixed as we take the
limit as $r_{0}\to\infty$ the $\frac{\mu}{r^{d-2}}$ term in $f(r)$ becomes
negligible in the limit, so the integral will converge to the above $\mu=0$
expression. The only difference for $\mu\neq 0$ is then that the black hole
has a different temperature, so

$\tau^{ETW}=\frac{\beta}{2}-\frac{\pi L}{2}.$ (14)

There is thus a good solution in the $r_{0}\to\infty$ limit for all $\mu$ such
that $\beta>\pi L$. This corresponds to $\mu<\mu_{max}$ where

$\mu_{max}=(r_{H}^{max})^{d-2}\left(\frac{(r_{H}^{max})^{2}}{L^{2}}-1\right),\quad\frac{r_{H}^{max}}{L}=\frac{2+\sqrt{4+d(d-2)}}{d}.$
(15)

For fixed $T,L$, the boundary interval $\tau^{ETW}$ is determined by $\mu$;
from the boundary perspective we would like to invert this relationship and
fix $\tau^{ETW}$ and determine $\mu$. In the limit as $r_{0}\to\infty$, it is
easy to calculate that $\tau^{ETW}$ runs from $+\infty$ to $0$ monotonically
as $\mu\in(\mu_{min},\mu_{max})$. Thus, there is a one to one map from
possible values of $\tau^{ETW}$ to $\mu$ in this range for large $r_{0}$.

Consider the induced geometry on the ETW brane: in general this is

$ds_{b}^{2}=\frac{dr^{2}}{f(r)-T^{2}r^{2}}+r^{2}d\Sigma_{g}^{2}.$ (16)

If $\mu=0$, setting $r=r_{0}\cosh\rho$, we have

$ds_{b}^{2}=r_{0}^{2}(d\rho^{2}+\cosh^{2}\rho d\Sigma_{g}^{2}).$ (17)

Thus, for the topological black hole, the induced geometry on the brane is
precisely the Maldacena-Maoz wormhole [23]. In general, if we take
$r_{0}\to\infty$ at fixed $\mu$, the mass term in $f(r)$ is negligible at the
brane position, and the induced geometry on the brane will be approximately
the same. This is natural, as the effective theory on the brane in this limit
reduces to Einstein gravity, and the Maldacena-Maoz wormhole is a classical
solution of this theory. This offers another perspective on the advantages of
considering the brane with hyperbolic cross-sections.

At finite $r_{0}$, the induced geometry on the brane is modified for $\mu\neq
0$; these modifications can be interpreted from the brane effective theory
prespective as due to the effective stress tensor on the brane dual to the
bulk geometry. Note that for $\mu<0$, this effective stress tensor will have a
negative energy density. This can be interpreted as a Casimir energy due to
putting the theory on a compact hyperbolic space. For $\mu=0$, the energy
density on the brane vanishes due to a cancellation between the negative
Casimir energy and the positive thermal energy associated with the finite
temperature.

##  4 Discussion

We have seen that considering spatial manifolds with negative curvature
provides a good environment for implementing the holographic cosmology
proposal of [1] in higher dimensions. The bulk solution is a hyperbolic black
hole, whose structure in higher dimensions is more similar to the structure of
the BTZ black hole in three dimensions. There are many questions about the
holographic cosmology which it will be interesting to investigate in this
context in the future: how the degrees of freedom on the brane are encoded in
the boundary theory, what predictions these models make for the cosmological
evolution on the brane, the role of the negative Casimir energy of the CFT,
and others.

In top-down models, the CFT usually includes scalar fields; conformally
invariant scalar fields have a coupling to the curvature of the background
which leads to an instability when the background is negatively curved. In the
bulk this is related to an instability for probe branes at large radial
position to run away towards the boundary [24]. This would seem to be a
significant issue with the models we are considering, and indeed if we
consider the zero temperature black hole the bulk solution is unstable; branes
at any radial position run away to the boundary. However, if we consider the
finite temperature solutions (for example in AdS5) there is a non-trivial
radial potential for probe (D3)-branes, and branes close to the horizon will
collapse into the black hole [25]. Thus, branes must tunnel through this
potential barrier to escape to infinity, and the dual CFT at finite
temperature is meta-stable. The model should provide a useful way of
describing the cosmology holographically on timescales short compared to the
decay time.

For flat spatial slices, the Euclidean ETW brane geometry can be analytically
continued in one of the spatial directions to obtain an eternal traversable
wormhole geometry. The relation between the problem of constructing ETW brane
cosmologies with flat slices and the challenges in constructing eternal
traversable wormholes was explored in [5, 18, 19], following [26].555The
construction of wormholes using a coupling between theories was earlier
explored in [27]. See also [28, 7, 29] for more recent work on these
wormholes. It is therefore interesting to ask if there is a connection between
the geometries with hyperbolic spatial slices studied here and Lorentzian
wormholes. The metric on $\Sigma_{g}$ will not have a translation symmetry,
but if it has a surface of time reflection symmetry, we can define a
Lorentzian solution by analytic continuation. This Lorentzian geometry will be
a quotient of AdS similar to those considered in [30], which gives again a
time-dependent cosmology with big bang big crunch “singularities” where the
quotient degenerates. The spatial slices of this brane geometry will be a
wormhole; whether it is traversable or not depends on the ratio of the time
needed to cross the radial spatial direction between the two boundaries and
the finite lifetime of the cosmology. In the simple $\mu=0$ case, both times
should be of order one in units of the AdS scale, so understanding whether
these wormholes are traversable seems to require a detailed case by case
analysis.

Finally, as we have emphasized connections to the $d=2$ case in this work, it
is worth noting that for $d=2$ the Euclidean boundary conditions with an
interval of Euclidean time between two brane boundaries can be filled in in
two different ways: we can have a piece of the bulk BTZ black hole bounded by
a connected ETW brane, or a piece of vacuum AdS bounded by disconnected ETW
branes [9]. We have discussed the analogue of the connected solution; is there
an analogue of the disconnected one? This requires an analogue of thermal AdS
for the hyperbolic black hole. No such solution is known, but in the appendix
I suggest such solutions may exist. They are more difficult to construct
analytically; it might be interesting to study them numerically.

## Acknowledgements

I thank Mark van Raamsdonk for helpful discussions. This work is supported in
part by STFC through grant ST/T000708/1, and by a grant from the Simons
Foundation. This work was performed in part at the Aspen Center for Physics,
which is supported by National Science Foundation grant PHY-1607611.

##  Appendix A Phase transitions for hyperbolic black holes

In the main body of the paper, we have considered adding ETW branes to the
hyperbolic black holes introduced in [20, 21, 22]. It is interesting to
consider whether there might be other bulk solutions for the same boundary
conditions, which could lead to a non-trivial phase structure analogous to the
Hawking-Page phase transition [31].

The thermal partition function for a field theory on $S^{d}$ is given by a
path integral on $S^{1}_{\beta}\times S^{d}$, where the size of the circle is
fixed by the inverse temperature. There are two possible bulk saddle-points in
the holographic description of this path integral: Euclidean Schwarzschild-
AdS, and thermal AdS. In the first the $S^{1}$ is contractible in the interior
of the spacetime, and in the second the $S^{d}$ is. There is an exchange of
dominance between these two saddle-points which occurs at an order one value
of the ratio of the size of the $S^{1}$ to the size of the $S^{d}$ [31]. This
corresponds to a phase transition in the thermal partition function, the
Hawking-Page phase transition, which is interpreted in the dual as a
deconfinement transition [32].

If we consider the field theory on $\mathbb{R}^{d}$, there is no phase
transition; the dominant bulk saddle is the planar AdS black hole for all
temperatures. However, if we consider compactifying the space to $T^{d}$, the
story is more subtle. In supersymmetric theories, with supersymmetry-
preserving boundary conditions on the torus, there is no phase transition, but
if we consider antiperiodic boundary conditions on one or more cycles on the
torus, there is a non-trivial phase structure. With antiperiodic boundary
conditions on a spatial cycle, at zero temperature there is an AdS soliton
solution [33], which reproduces the expected negative Casimir energy of the
field theory. At finite temperature we consider the field theory path integral
on $S^{1}_{\beta}\times T^{d}$, and there are multiple bulk saddle-points: the
planar black hole and thermal AdS solitons for any circles with antiperiodic
boundary conditions. There is a phase transition when the ratio of the size of
the thermal circle $S^{1}_{\beta}$ to the size of the smallest circle which is
contractible in an AdS soliton is equal to one.666There are also transitions
between different AdS solitons if we vary the relative size of different
circles in the torus.

For the field theory on a compact hyperbolic space $\Sigma_{g}$, the finite
temperature partition function is given by a path integral on
$S^{1}_{\beta}\times\Sigma_{g}$. The only known bulk saddle is the hyperbolic
black hole, and in the previous literature this has been taken as indicating
the absence of a phase transition in this case. However, this may simply
reflect our ignorance. By analogy to the torus case discussed above, we might
expect that (assuming we consider antiperiodic boundary conditions for the
fermions) there would be bulk solutions where a non-contractible cycle in
$\Sigma_{g}$ becomes contractible in the bulk. Indeed, if we consider a two-
dimensional field theory on a Riemann surface $\Sigma_{g}$, such handlebody
solutions have been explicitly constructed [34]. Constructing such solutions
in the higher-dimensional case with $S^{1}_{\beta}\times\Sigma_{g}$ boundary
where $\Sigma_{g}$ has two or more dimensions will be challenging, as the
solutions will only have a $U(1)$ symmetry, corresponding to translations
along the $S^{1}$. The explicit constructions of [34] were only possible
because the bulk solution was locally AdS3, and in higher dimensions bulk
vacuum solutions are not required to be locally AdS. But there is no obvious
reason to expect such solutions not to exist. Indeed, given the experience
with the torus case and Riemann surface boundaries in AdS3, it would seem more
surprising if they did not than if they did.

If such solutions where a cycle in $\Sigma_{g}$ is contractible exist, there
would then be multiple bulk saddle-points for the path integral on
$S^{1}_{\beta}\times\Sigma_{g}$, and there could be phase transitions due to
the competition between them and the hyperbolic black hole, as a function of
the temperature and the moduli of $\Sigma_{g}$. There is however an
interesting difference between this case and the torus: for the torus, the
lowest-energy black hole has zero energy, while we expect the field theory to
have a negative energy density (for antiperiodic fermions) at sufficiently low
temperature due to the Casimir energy, so there is a clear argument that a
different phase (the AdS soliton) must take over at low temperatures. For the
theory on $\Sigma_{g}$, the low temperature black hole has negative energy, so
this could already reproduce the Casimir energy in the field theory. Thus,
even assuming that such solutions where a cycle in $\Sigma_{g}$ is
contractible exist, the question of whether they dominate in the thermal
partition function at low temperatures would remain open. This seems a very
interesting question for future numerical investigation.

Returning to the solutions with ETW branes considered in the body of the
paper, if such solutions where a cycle in $\Sigma_{g}$ is contractible exist,
for a field theory on an interval cross $\Sigma_{g}$ ended by brane boundaries
there might then be solutions with disconnected ETW branes bounding a piece of
such a solution.

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Improving Modular Bootstrap Bounds with Integrality

A. Liam Fitzpatrick, Wei Li

Department of Physics, Boston University, Boston, MA 02215, USA

We implement methods that efficiently impose integrality – i.e., the condition
that the coefficients of characters in the partition function must be integers
– into numerical modular bootstrap. We demonstrate the method with a number of
examples where it can be used to strengthen modular bootstrap results. First,
we show that, with a mild extra assumption, imposing integrality improves the
bound on the maximal allowed gap in dimensions of operators in theories with a
$U(1)^{c}$ symmetry at $c=3$, and reduces it to the value saturated by the
$SU(4)_{1}$ WZW model point of $c=3$ Narain lattices moduli space. Second, we
show that our method can be used to eliminate all but a discrete set of points
saturating the bound from previous Virasoro modular bootstrap results.
Finally, when central charge is close to $1$, we can slightly improve the
upper bound on the scaling dimension gap.

###### Contents

  1. 1 Introduction
  2. 2 Methods
    1. 2.1 Setup
    2. 2.2 Degeneracy bounds on individual operator
    3. 2.3 Degeneracy bounds in a interval
  3. 3 Review of Previous Bootstrap Results
  4. 4 Bootstrapping the $c=3$ Narain CFT
  5. 5 Eliminating Extremal Solutions
  6. 6 Lowering the Spectrual Gap Near $c\sim 1$
  7. A Partition function for extremal theories

##  1 Introduction

Unitary Conformal Field Theories (CFTs) must satisfy a number of highly
restrictive conditions, which can be used to derive bounds on their spectrum
and OPE coefficients. Two particularly powerful conditions are crossing
symmetry of correlators, and, in the case of 2d CFTs, modular invariance of
the torus partition function. Combined with the constraints of conformal
symmetry and unitarity, they lead to the conformal bootstrap and the modular
bootstrap, respectively Rattazzi:2008pe ; Hellerman:2009bu . These two
conditions are structurally equivalent, so that usually analytic or numeric
methods developed to study one of them can be applied to the other. One major
way in which they differ is that the partition function is at heart a sum over
states, and consequently the degeneracies of operators that appear in the
partition function must be integers. This additional ‘integrality’ constraint
is harder to implement in practice, but despite this a few encouraging results
have appeared in the literature using integrality to constrain the space of 2d
CFTs Benjamin:2020zbs ; Kaidi:2020ecu .

In this paper, we will develop and implement new methods to incorporate the
constraint of intergrality into the numerically modular bootstrap. In some
cases, we will focus on the integer constraint on the number of specific
operators in the partition function, whereas in other cases we will focus on
bound the total degeneracy number of operators with dimensions in a interval
$\Delta_{a}\leq\Delta\leq\Delta_{b}$. A key part of the strategy will be to
obtain two-sided bounds (i.e., an upper- and a lower-bound) on these
degeneracies, in which case we can impose the constraint that _there must be
an integer inside the upper and lower bound_. We will describe these methods
in detail in Sec.2. Starting from Sec.4, we will apply these methods to
various examples. In some cases, we do not assume any symmetry beyond Virasoro
symmetry, whereas in other we assume an additional $U(1)^{c}$ current algebra.
Our major results are as follows:

_Bootstrapping the $c=3$ Narain CFT_

A frequent target for modular bootstrap bounds is to determine the largest
possible allowed gap in dimensions of operators between the identity (i.e.,
vacuum state) and the next lowest-dimension operator. It is an open question
whether or not the optimal bound in the case of two-dimensional CFTs with
$U(1)^{c}\times U(1)^{c}$ algebra (for integer $c$) can be saturated by Narain
lattices CFTs. Previous modular bootstrap analyses Collier:2016cls ;
Hartman:2019pcd ; Afkhami-Jeddi:2020ezh  found that this question could be
answered positively at some small values of $c$, specifically at $c=1,2,4$. A
natural question is whether the optimal bound can be improved so that it is
also saturated by Narain CFTs at $c=3$, or whether there exist unitary $c=3$
CFTs with a larger gap. In Sec.4 we will show that the condition of
integrality, together with a mild additional assumption on the spectrum,
implies that the optimal bound on the gap is indeed saturated by a $c=3$
Narain CFT.

_Eliminating Extremal Solutions & Lowering the Spectrual Gap In Virasoro
Bootstrap_

The authors of Collier:2016cls  found that, using only Virasoro symmetry, the
optimal spectral gap converged to $\Delta_{\text{mod}}={c\over 6}+{1\over 3}$.
Using the extremal function method El-Showk:2012vjm ; El-Showk:2016mxr , this
means that at the limit of the bound, there is a family of modular invariant
partition functions with positive coefficients for each Virasoro character in
the partition function. However, only four values of $c$ produced partition
functions which corresponded to known CFTs Collier:2016cls . In Sec.5, we use
integrality to show that no other points can correspond to true CFTs. More
precisely, we take the spectrum given by the extremal functional method, and
derive two-sided bounds on the degeneracies of states for several operators in
the spectrum. The four values of $c$ identified in Collier:2016cls  are the
only four values where all two-sided bounds contain integers.

Finally in Sec.6, we analyze the regime of $c$ close to 1 from above, and show
that one can obtain a slight improvement on the bound on the gap itself by
imposing integrality.

##  2 Methods

###  2.1 Setup

We begin by considering the partition function $Z(\tau,\overline{\tau})$ of a
$2d$ CFT on a torus. All modular transformations can be generated from the $S$
and $T$-transformations:

$\displaystyle\text{S-transform:~{}}Z(\tau,\overline{\tau})=Z\Big{(}-{1\over\tau},-{1\over\overline{\tau}}\Big{)},$
$\displaystyle\text{T-transform:~{}}Z(\tau,\overline{\tau})=Z(\tau+1,\overline{\tau}+1).$
(2.1)

The partition function can moreover be decomposed into a sum over Virasoro
characters with non-negative integer coefficients $n_{h,\overline{h}}$ that
count the degeneracy of states with weight $h,\overline{h}$:

$Z(\tau,\overline{\tau})=\chi_{vac}(\tau)\overline{\chi}_{vac}(\overline{\tau})+\sum_{h,\overline{h}}n_{h,\overline{h}}\chi_{h}(\tau)\overline{\chi}_{h}(\overline{\tau})$
(2.2)

The vacuum $\chi_{vac}$ and non-degenerate Virasoro character $\chi_{h}$ are
given by

$\chi_{vac}(\tau)={q^{-{c-1\over
24}}\over\eta(\tau)}(1-q),\quad\chi_{h}(\tau)={q^{h-{c-1\over
24}}\over\eta(\tau)},$ (2.3)

where $q=e^{2\pi i\tau}$ and $\eta(\tau)$ is the Dedekind eta function. For
the rest of paper, we will further assume the partition function is parity-
invariant $Z(\tau,\overline{\tau})=Z(\overline{\tau},\tau)$. In this case we
can label the Virasoro primaries by their scaling dimension
$\Delta=h+\overline{h}$ and their spin $\ell=|h-\overline{h}|$. Then the
character decomposition becomes

$Z(\tau,\overline{\tau})=\chi_{vac}(\tau)\overline{\chi}_{vac}(\overline{\tau})+\sum_{\Delta,\ell}n_{\Delta,\ell}\Big{[}\chi_{{\Delta+\ell\over
2}}(\tau)\overline{\chi}_{{\Delta-\ell\over
2}}(\overline{\tau})+\chi_{{\Delta-\ell\over
2}}(\tau)\overline{\chi}_{{\Delta+\ell\over 2}}(\overline{\tau})\Big{]}.$
(2.4)

As first introduced in Friedan:2013cba , we can define reduced partition
function $\widehat{Z}(\tau,\overline{\tau})$ to get rid of the Dedekind eta
function,

$\widehat{Z}(\tau,\overline{\tau})=|\tau|^{{1\over
2}}|\eta(\tau)|^{2}Z(\tau,\overline{\tau})$ (2.5)

The extra factor of $|\tau|^{{1\over 2}}$ is present because the combination
$|\tau|^{{1\over 2}}|\eta(\tau)|^{2}$ is invariant under $\tau\to-1/\tau$ and
consequently so is $\widehat{Z}(\tau,\overline{\tau})$.

The $T$-transformation just tells us that spin $\ell$ must be an integer
(2.4), so we only have to consider the $S$-transformation. Combining (2.4) and
(2.1) we have,

$\widehat{F}_{\text{vac}}(\tau,\overline{\tau})+\sum_{\Delta,\ell}n_{\Delta,\ell}\widehat{F}_{\Delta,\ell}(\tau,\overline{\tau})=0,$
(2.6)

where

$\displaystyle\widehat{F}_{\text{vac}}(\tau,\overline{\tau})=\widehat{\chi}_{\text{vac}}(\tau)\widehat{\overline{\chi}}_{\text{vac}}(\overline{\tau})-(\tau\to-{1\over\tau},\overline{\tau}\to-{1\over\overline{\tau}}),$
$\displaystyle\widehat{F}_{\Delta,\ell}(\tau,\overline{\tau})=\widehat{\chi}_{{\Delta+\ell\over
2}}(\tau)\widehat{\overline{\chi}}_{{\Delta-\ell\over
2}}(\overline{\tau})+\widehat{\chi}_{{\Delta-\ell\over
2}}(\tau)\widehat{\overline{\chi}}_{{\Delta+\ell\over
2}}(\overline{\tau})-(\tau\to-{1\over\tau},\overline{\tau}\to-{1\over\overline{\tau}}).$
(2.7)

The idea of the modular bootstrap is to find a linear functional
$\omega[\widehat{F}_{\Delta,\ell}(\tau,\overline{\tau})]=\omega(\Delta,\ell)$
such that

$\displaystyle\omega[\widehat{F}_{\text{vac}}]\geq 0,$
$\displaystyle\omega(\Delta,0)\geq 0,\quad\text{For~{}}\Delta\geq\Delta^{*},$
$\displaystyle\omega(\Delta,\ell)\geq
0,\quad\text{For~{}}\Delta\geq\ell\text{~{}and~{}}\ell\geq 1.$ (2.8)

This linear functional $\omega$ acting on (2.6) has a positive contribution
from $\omega[\widehat{F}_{\text{vac}}]$ that must be canceled by negative
contributions, which can only come from scalar states with dimension lower
than $\Delta_{*}$. In other words, the scalar gap must be smaller than
$\Delta^{*}$. To implement semi-definite programmingSimmons-Duffin:2015qma ;
Landry:2019qug  on the modular crossing equation, we will apply the following
linear functionals $\omega^{(m,n)}$ as introduced in Afkhami-Jeddi:2021iuw ;
Afkhami-Jeddi:2019zci ,

$\displaystyle\omega^{(k,\overline{k})}(\Delta,\ell)=\mathcal{D}_{\beta}^{k}\mathcal{D}_{\overline{\beta}}^{\overline{k}}\widehat{F}_{\Delta,\ell}(i\beta,-i\overline{\beta})\Big{|}_{\beta=\overline{\beta}=1}$
$\displaystyle=e^{-4\pi(\Delta-{c-1\over
12})}L_{2k-1}^{(-1/2)}\Big{[}4\pi\Big{(}{\Delta+\ell\over 2}-{c-1\over
24}\Big{)}\Big{]}L_{2\overline{k}-1}^{(-1/2)}\Big{[}4\pi\Big{(}{\Delta-\ell\over
2}-{c-1\over 24}\Big{)}\Big{]}+(k\leftrightarrow\overline{k}),$
$\displaystyle\omega^{(k,\overline{k})}[\widehat{F}_{\text{vac}}]=\omega^{(k,\overline{k})}(0,0)-2\omega^{(k,\overline{k})}(1,1)+\omega^{(k,\overline{k})}(2,0).$
(2.9)

where $k+\overline{k}$ are odd integers and $L_{p}^{(\alpha)}(x)$ are the
generalized Laguerre polynomials. Collecting different components of
$(k,\overline{k})$ we get a truncated crossing equation:

$\vec{F}_{\text{vac}}+\sum_{\Delta,\ell}n_{\Delta,\ell}\vec{F}_{\Delta,\ell}=\vec{0},\quad\vec{F}_{\text{vac}}=\begin{pmatrix}\omega^{(1,0)}[\widehat{F}_{\text{vac}}]\\\
\omega^{(2,1)}[\widehat{F}_{\text{vac}}]\\\ \vdots\\\
\omega^{(k,\overline{k})}[\widehat{F}_{\text{vac}}]\end{pmatrix},\quad\quad\vec{F}_{\Delta,\ell}=\begin{pmatrix}\omega^{(1,0)}(\Delta,\ell)\\\
\omega^{(2,1)}(\Delta,\ell)\\\ \vdots\\\
\omega^{(k,\overline{k})}(\Delta,\ell)\end{pmatrix}$ (2.10)

For the rest of the paper, we will use $\Lambda_{\text{max}}$ to denote the
maximal derivative order $k+\overline{k}\leq\Lambda_{\text{max}}$ in (2.10).

In Sec.4, we will also consider the bootstrap 2D CFTs with $U(1)^{c}\times
U(1)^{c}$ algebra. In that case, the only modification is that we have to
change to Virasoro character to $U(1)^{c}$ character:

$\chi_{\text{vac}}^{U(1)^{c}}={1\over\eta(\tau)^{c}},\quad\chi_{h}^{U(1)^{c}}={q^{h}\over\eta(\tau)^{c}}.$
(2.11)

For the linear functionals, we will use the eigenfunctions found in Afkhami-
Jeddi:2020ezh  as our basis,

$\displaystyle\omega^{(k,\overline{k})}_{U(1)^{c}}(\Delta,\ell)=e^{-2\pi\Delta}\bigg{(}L_{k}^{(c/2-1)}\Big{[}4\pi\Big{(}{\Delta+\ell\over
2}\Big{)}\Big{]}L_{\overline{k}}^{(c/2-1)}\Big{[}4\pi\Big{(}{\Delta-\ell\over
2}\Big{)}\Big{]}+(k\leftrightarrow\overline{k})\bigg{)},$ (2.12)
$\displaystyle\omega^{(k,\overline{k})}_{U(1)^{c}}[\widehat{F}_{\text{vac}}^{U(1)^{c}}]={1\over
2}\omega^{(k,\overline{k})}_{U(1)^{c}}(0,0).$

###  2.2 Degeneracy bounds on individual operator

Let us first review the standard method for bounding the degeneracy numbers
$n_{\Delta_{\mathcal{O}},\ell_{\mathcal{O}}}$. From Sec.2.1, modular
invariance implies the following crossing equation:

$\vec{F}_{\text{vac}}+n_{\Delta_{\mathcal{O}},\ell_{\mathcal{O}}}\vec{F}_{\Delta_{\mathcal{O}},\ell_{\mathcal{O}}}+\sum_{\\{\Delta,\ell\\}\in\mathcal{S}}n_{\Delta,\ell}\vec{F}_{\Delta,\ell}=\vec{0}$
(2.13)

where $\mathcal{S}$ contains all the operators (excluding the vacuum and
$\mathcal{O}$) in the CFT . If there exist a vector $\vec{\alpha}$ satisfying
the condition

$\text{For
all}~{}\\{\Delta,\ell\\}\in\mathcal{S},\quad\vec{\alpha}\cdot\vec{F}_{\Delta,\ell}\geq
0,$ (2.14)

then we can act with $\vec{\alpha}$ on both sides of (2.13), which produces a
upper bound or lower bound of $n_{\Delta_{\mathcal{O}},\ell_{\mathcal{O}}}$
depending on the sign of
$\vec{\alpha}\cdot\vec{F}_{\Delta_{\mathcal{O}},\ell_{\mathcal{O}}}$. For
convenience, we choose
$\vec{\alpha}\cdot\vec{F}_{\Delta_{\mathcal{O}},\ell_{\mathcal{O}}}=\pm 1$, in
which case

$\vec{\alpha}\cdot\vec{F}_{\Delta_{\mathcal{O}},\ell_{\mathcal{O}}}=1\Rightarrow
n_{\Delta_{\mathcal{O}},\ell_{\mathcal{O}}}\leq-\vec{\alpha}\cdot\vec{F}_{\text{vac}},\quad\vec{\alpha}\cdot\vec{F}_{\Delta_{\mathcal{O}},\ell_{\mathcal{O}}}=-1\Rightarrow
n_{\Delta_{\mathcal{O}},\ell_{\mathcal{O}}}\geq\vec{\alpha}\cdot\vec{F}_{\text{vac}}.$
(2.15)

In these two cases, the strongest upper and lower bound are both coming from
maximizing $\vec{\alpha}\cdot\vec{F}_{\text{vac}}$.

The exact same technique has been applied to the conformal bootstrap for
correlators, where the result is a bound on OPE coefficients Caracciolo:2009bx
. But in the modular bootstrap case, integrality imposes an additional
constraint:

The upper and lower bound of $n_{\Delta_{\mathcal{O},\ell_{\mathcal{O}}}}$
must contain a integer (2.16)

One of the first things one might be tempted to try is simply to implement the
above strategy for two-sided bounds on
$n_{\Delta_{\mathcal{O},\ell_{\mathcal{O}}}}$, and then impose (2.16) in order
to improve the bound on the gap. Unfortunately, this strategy fails. The
reason is that, for the optimal gap, $\Delta_{\mathcal{O}}$ is at the infimum
of the range of dimensions $\Delta$ of operators in $\mathcal{S}$, but
$\vec{\alpha}\cdot\vec{F}_{\Delta,\ell}\geq 0$ for $\Delta\in\mathcal{S}$ by
construction. Then, by continuity,
$\vec{\alpha}\cdot\vec{F}_{\Delta_{\mathcal{O}},\ell_{\mathcal{O}}}$ cannot be
negative, so we cannot use (2.15) to get lower bounds on
$n_{\Delta_{\mathcal{O}},\ell_{\mathcal{O}}}$.

So if we want to have a two-sided bound on
$n_{\Delta_{\text{mod}},\ell_{\mathcal{O}}}$, we will need to make an
additional assumption on the spectrum. One option is to impose a second gap
$\Delta_{*}$ in the spin sector $\ell=\ell_{\mathcal{O}}$. Now the operator
set $\mathcal{S}$ will become,

$\mathcal{S}=\\{(\Delta,\ell)~{}|~{}\text{If
}\ell=\ell_{\mathcal{O}},\Delta\geq\Delta_{*},~{}\text{else
}\Delta\geq\text{max}(\Delta_{\text{mod}},\ell),~{}\ell\geq 0\\}$ (2.17)

In Fig.1, we show two plots for functionals that produces upper and lower
bound given by (2.15).

Figure 1: (a) Functionals that produces upper bound
$n_{\Delta_{\text{mod}},\ell_{\mathcal{O}}=0}$ (b) Functionals that produces
lower bound $n_{\Delta_{\text{mod}},\ell_{\mathcal{O}}=0}$, with additional
assumption that the next operator has scaling dimension larger than
$\Delta_{*}$

Although this method by itself does not help to shrink the previous modular
bootstrap bound, it is still a powerful method when we know some information
about the next operator. In Sec.4 and Sec.5, we will show two applications of
this method. The first application will be to lower the bootstrap down to a
$c=3$ Narain theory under the assumption on the second gap in the $\ell=1$
sector. The second application is to rigorously eliminate all but a small set
of extremal solutions from the Virasoro bootstrap in $1\leq c\leq 4$.

###  2.3 Degeneracy bounds in a interval

In this section we provide a method to impose the integrality on the interval
of operators, inspired by the method from Collier:2017shs  for bounds on the
CFT spectral density. Let $\mathcal{F}_{\ell}(\Delta_{a},\Delta_{b})$ be the
integrated density of operators in an interval
$\Delta\in[\Delta_{a},\Delta_{b}]$:

$\mathcal{F}_{\ell}(\Delta_{a},\Delta_{b})=\sum_{\Delta_{a}\leq\Delta\leq\Delta_{b}}n_{\Delta,\ell}.$
(2.18)

If we can obtain a two sided bound on
$A\leq\mathcal{F}_{\ell}(\Delta_{a},\Delta_{b})\leq B$, then we can impose the
additional constraint that $[A,B]$ must contain an integer.

Now we introduce our method to get an upper and lower bound on
$\mathcal{F}_{\ell}(\Delta_{a},\Delta_{b})$ using semi-definite
programmingSimmons-Duffin:2015qma ; Landry:2019qug . Combining (2.10) and
(2.18) we obtain the following vector equation,

$\begin{pmatrix}0\\\
\vec{F}_{\text{vac}}\end{pmatrix}+\sum_{\\{\Delta,\ell\\}\in\mathcal{S}}n_{\Delta,\ell}\begin{pmatrix}\theta(\Delta-\Delta_{b})\theta(\Delta_{a}-\Delta)\delta_{s,\ell}\\\
\vec{F}_{\Delta,\ell}\end{pmatrix}=\begin{pmatrix}\mathcal{F}_{s}(\Delta_{a},\Delta_{b})\\\
\vec{0}\end{pmatrix}$ (2.19)

where $\theta(x)$ and $\delta_{s,l}$ are Heaviside step function and Kronecker
delta function. We will seek a vector $\vec{\beta}$,

$\vec{\beta}\equiv\begin{pmatrix}\pm 1\\\ \vec{\alpha}\end{pmatrix},$ (2.20)

such that,

$\vec{\beta}\cdot\begin{pmatrix}\theta(\Delta-\Delta_{b})\theta(\Delta_{a}-\Delta)\delta_{s,\ell}\\\
\vec{F}_{\Delta,\ell}\end{pmatrix}\geq 0~{}~{}\text{for
}\\{\Delta,\ell\\}\in\mathcal{S}.$ (2.21)

Then, acting $\vec{\beta}$ on (2.19) we have,

$\pm\mathcal{F}_{s}(\Delta_{a},\Delta_{b})=\vec{\alpha}\cdot\vec{F}_{\text{vac}}+\sum_{\\{\Delta,\ell\\}\in\mathcal{S}}(n_{\Delta,\ell})\vec{\beta}\cdot\begin{pmatrix}\theta(\Delta-\Delta_{b})\theta(\Delta_{a}-\Delta)\delta_{s,\ell}\\\
\vec{F}_{\Delta,\ell}\end{pmatrix}\geq\vec{\alpha}\cdot\vec{F}_{\text{vac}}$
(2.22)

So depending on which sign we choose for the top component of $\vec{\beta}$,
we can produce either an upper or lower bound on
$\mathcal{F}_{s}(\Delta_{a},\Delta_{b})$. The strongest bound in both cases
will come from maximizing $\vec{\alpha}\cdot\vec{F}_{\text{vac}}$. Below we
show two typical functionals that give rise to upper and lower bound of
$\mathcal{F}_{s}(\Delta_{a},\Delta_{b})$.

In general, we find that by choosing an appropriate interval
$[\Delta_{a},\Delta_{b}]$ and spin sector $s$, one can always find an upper
and lower bound of $\mathcal{F}_{s}(\Delta_{a},\Delta_{b})$. So unlike in
Sec.2.2, we don’t need to put additional assumption of the next operator to
get a lower bound. Then the natural question is whether this method can lower
the previous bootstrap bound? For the problem of bounding the lowest scaling
dimension, we find that the answer is yes, though the actual amount by which
we will end up improving the bound in our applications will be modest. We will
discuss this in more detail in Sec.6.

##  3 Review of Previous Bootstrap Results

Without assuming any symmetry beyond Virasoro, in Collier:2016cls  the authors
found that for $1\leq c\leq 4$, the modular bootstrap bound on the overall
spectral gap converges to

$\Delta_{\text{mod}}={c\over 6}+{1\over 3},\quad c\in[1,4].$ (3.1)

Moreover, they identified the following extremal solutions in terms of known
theories:

  * •

At $c=1$, the bound is $\Delta_{\text{mod}}={1\over 2}$. The theory saturating
this bound is free compact boson at self-dual radius, or equivalently
$SU(2)_{1}$ WZW model.

  * •

At $c=2$, the bound is $\Delta_{\text{mod}}={2\over 3}$. The theory saturating
this bound is the $SU(3)_{1}$ WZW model

  * •

At $c={14\over 5}$, the bound is $\Delta_{\text{mod}}={4\over 5}$. The theory
saturating this bound is $(G_{2})_{1}$ WZW model

  * •

At $c=4$, the bound is $\Delta_{\text{mod}}=1$. The theory saturating this
bound is $SO(8)_{1}$ WZW model

More generally, by looking at the spectrum of the truncated crossing equation
at sufficient high derivative order, we observed the spectrum of the extremal
solution for the range $1\leq c\leq 4$ will have the following pattern:

$\displaystyle\ell=0,\quad\Delta=\\{\Delta_{\phi},2,2+\Delta_{\phi},4,4+\Delta_{\phi},\dots,2n,2n+\Delta_{\phi}\\}$
$\displaystyle\ell=1,\quad\Delta=\\{1,1+\Delta_{\phi},3,3+\Delta_{\phi},5,5+\Delta_{\phi},\dots,2n+1,2n+1+\Delta_{\phi}\\}$
$\displaystyle\ell=2,\quad\Delta=\\{2,2+\Delta_{\phi},4,4+\Delta_{\phi},\dots,2n,2n+\Delta_{\phi}\\}$
$\displaystyle\ell=3,\quad\Delta=\\{3,3+\Delta_{\phi},5,5+\Delta_{\phi},\dots,2n+1,2n++1+\Delta_{\phi}\\}$
$\displaystyle\vdots$ (3.2)

where $\Delta_{\phi}={c\over 6}+{1\over 3}$.111The spectrum suggests that it’s
likely to be RCFT with two primaries: one is the identity $\mathbf{1}$ and the
other one is a scalar $\phi$ with conformal weight $h=\overline{h}={c\over
12}+{1\over 6}$. And for these four WZW models shown in the table above, they
all have exactly two chiral algebra primaries. In Fig.2, we show the first few
operators from $\Lambda_{max}=25$ extremal solutions to verify this pattern.
The convergence of the spectrum means that this is indeed the spectrum of a
modular invariant partition function with positive coefficients when
decomposed into the characters. But in general we find, most of them don’t
have integer degeneracy numbers. In Sec.5, we will obtain two-sided bounds on
the degeneracies and show that, assuming the spectrum obeys this pattern,
$c=1,2,{14\over 5},4$ are the only legitimate solutions.

Figure 2: This plot shows the scaling dimension $\Delta$ of the first three
operators in $\ell=0,1,2$. It’s extracting from extremal solutions of Virasoro
Bootstrap at $\Lambda_{max}=25,L_{max}=40$ (a) $\ell=0$ (b) $\ell=1$ (c)
$\ell=2$

We can also consider not just Virasoro symmetry, but assume the extra
conserved currents. We focus on the case of 2d CFTs with a $U(1)^{c}$ chiral
algebra. In Afkhami-Jeddi:2020ezh , the authors showed that in this case, the
modular bootstrap bounds at $c=1,2,4$, the bound are saturated by Narain
lattices CFTs.

At $c=3$ the authors found the spectral gap
$\Delta_{\text{mod}}^{(25)}=0.8227$ while the Narain CFTs at $c=3$ have at
most a spectral gap $\Delta^{\text{Narain}}_{\text{mod}}={3\over 4}$. One
might wonder whether the bound rom Afkhami-Jeddi:2020ezh  could be improved
from 0.8227 to 0.75 simply by increasing the truncation on the number of
derivatives ($\Lambda_{max}=25$ in their case) in the modular bootstrap
analysis. However, it turns out that this is not the case. In Fig.3 we show
the modular bootstrap bound up to $\Lambda_{max}=63$. Extrapolating to
$\Lambda_{\rm max}=\infty$, we estimate that the infinite truncation modular
bootstrap bound is $\Delta_{\text{mod}}^{\Lambda_{max}=\infty}\sim 0.819$,
which is still not close to $0.75$. In Sec.4, we will argue that the bound can
nevertheless be decreased to $0.75$ using integrality together with the
assumption that the first two states in the spin-1 sector follow the same
pattern as seen in (2) from the extremal spectra.

Figure 3: This plot shows the spectral gap from $U(1)^{c=3}$ bootstrap from
$\Lambda_{max}=11$ to $\Lambda_{max}=63$. Black and red dashed curves
correspond to two different fitting functions:
$\Delta_{\text{mod}}=a+{b\over\Lambda_{max}}$ and
$\Delta_{\text{mod}}=a+b\Big{(}{1\over\Lambda_{max}}\Big{)}^{c}$

##  4 Bootstrapping the $c=3$ Narain CFT

In this section we will apply integrality to the bootstrap bound on the
spectral gap in theories with a $U(1)^{c}$ chiral algebra, at $c=3$. In the
previous section, we saw that without any additional assumptions beyond
unitarity, the modular bootstrap bound at infinite derivative order appears to
asymptote to $\Delta_{\rm mod}\approx 0.819$, whereas the largest possible gap
of a $c=3$ Narain CFT is $3/4$. In this section, we will have to make one
additional assumption, which is that the structure seen in the first two
states in the spin-1 sector in the extremal spectrum (see (3) from the modular
bootstrap bound holds for the true theory that maximizes the spectral gap. To
be precise, we assume that the spin-1 sector has a spin-1 current, followed by
a gap to a spin-1 state with dimension $1+\Delta_{\phi}$, where
$\Delta_{\phi}$ is the spectral gap. Aside from the fact that this is the
extremal spectrum seen from the modular bootstrap, it is also what one would
expect if extremal theories have an enhanced symmetry (the presence of at
least one extra spin-1 current in the spectrum, which exist in addition to the
$U(1)^{c}$ currents). We do not, however, rule out the possibility that there
might be a $c=3$ theory with a gap between $3/4$ and $0.819$ that has a spin-1
state with dimension between $1$ and $1+\Delta_{\phi}$.

Nevertheless, with this extra assumption, we will find that imposing
integrality on the $U(1)^{c}$ bootstrap at $c=3$ exactly implies that the
spectral gap is maximized by a Narain CFT. Because we assume the absence of
any operators in the $\ell=1$ sector between $\Delta=1$ and
$\Delta=1+\Delta_{\phi}$, we can obtain two-sided bounds on the degeneracy
$n_{\Delta=1,\ell=1}$ using the method introduced in Sec.2.2. In Fig.4, we
these bounds on $n_{\Delta=1,\ell=1}$ at derivative order $\Lambda_{max}=23$.
Without imposing integrality, $\Delta_{g}$ at this derivative order is only
bounded to be below $\Delta_{g}=0.7803$. However, at this point the two-sided
bounds on the degeneracy do not contain an integer. As the gap $\Delta_{g}$ is
decreased, the upper bound does not change, but the lower-bound slowly
decreases until it crosses an integer at approximately $\Delta_{g}\approx
0.7608$:

$\Delta_{g}\leq 0.7803\rightarrow\Delta_{g}\leq
0.7608\quad(\Lambda_{max}=23).$ (4.1)

Figure 4: Left: The upper and lower bounds on $n_{\Delta=1,\ell=1}$ for
$U(1)^{c=3}$ bootstrap, at $\Lambda_{max}=23$. Right: The lower bounds on
$n_{\Delta=1,\ell=1}$ for $U(1)^{c=3}$ bootstrap for various derivative
orders.

We also compute the lower bound of $n_{\Delta=1,\ell=1}$ for various
derivative orders in Fig.4. And in Fig.5, we show the scalar gap after
imposing integrality constraint for derivative orders up to
$\Lambda_{max}=51$, and use this to extrapolate to $\Lambda_{\rm max}=\infty$.
At $\Lambda_{max}=51$, the result is 222To get the result at
$\Lambda_{max}=51$, one need to be careful about precision. A workable SDPB
parameters setting is :precision=1024, maxComplementarity=1e+100,
primalErrorThreshold=1e-40, dualErrorThreshold=1e-40,
dualityGapThreshold=1e-30

$\Delta_{g}=0.750746,\quad(\Lambda_{max}=51),$ (4.2)

and we find that the bound extrapolates at $\Lambda_{\rm max}=\infty$ to

$\Delta_{g}=0.749\pm 0.001,\quad(\Lambda_{max}=\infty),$ (4.3)

where the error has been estimated from the variation in different ways to do
the extrapolation, see Fig. 5.

Figure 5: Scalar gap $\Delta_{g}$ after imposing integrality constraint for
derivative orders up to $\Lambda_{max}=51$. Black, gray and red dashed curves
are using three different fitting functions
$\Delta_{g}=a+{b\over\Lambda_{max}},\Delta_{g}=a+{b\over\Lambda_{max}^{2}},\Delta_{g}=a+{b\over\Lambda_{max}^{c}}$
respectively

We can also compare approximated spectrum coming the extremal functional with
the exact $U(1)^{c=3}$ Narain theory. The first few operators from the
extremal functional (at $\Lambda_{max}=51$) at $\ell\leq 2$ are 333We found
multiple spurious operators (operators that don’t appear in the $c=3$ Narain
theory) in the extremal spectrum. For example, at $\ell=0,\Delta\approx 1.366$
there is an operator with degeneracy $n\approx 0.256$ from the
$\Lambda_{max}=51$ extremal spectrum. We checked that those operator have a
small degeneracy, and their degeneracy converges to $0$ when
$\Lambda_{max}\to\infty$. We believe these spurious operators will disappear
at infinite derivative order. Therefore to avoid confusion, we do not list
them in (4) ,

$\displaystyle\ell=0,\quad\Delta=\\{0.7507,1.002,2.003,2.749,2.995,4.025,\dots\\}$
$\displaystyle\ell=1,\quad\Delta=\\{1,1.7508,2.002,2.992,3.735,3.987,\dots\\}$
$\displaystyle\ell=2,\quad\Delta=\\{2.007,2.760,3.006,4.006,4.728,\dots\\}$
(4.4)

For comparison, the first few operators from $U(1)^{c=3}$ Narain theory are

$\displaystyle\ell=0,\quad\Delta=\\{0.75,1,2,2.75,3,4,\dots\\}$
$\displaystyle\ell=1,\quad\Delta=\\{1,1.75,2,3,3.75,4,\dots\\}$
$\displaystyle\ell=2,\quad\Delta=\\{2,2.75,3,4,4.75,\dots\\}$ (4.5)

##  5 Eliminating Extremal Solutions

Our next application of integrality will be to the set of theories that
saturate the Virasoro modular bootstrap bound in the range $1\leq c\leq 4$
Collier:2016cls . The extremal functional method already gives an estimate for
the degeneracy numbers of these partition functions, but here we will go
farther and use two-sided bound to rigorously exclude all theories in this
range outside of very small numerical windows around the points
$c=1,2,{14\over 15}$, and $c=4$ discussed previously.

To begin, consider the upper and lower bounds of following the scalar
operators and two conserved currents operators,

$\mathcal{O}_{\Delta=\Delta_{\phi},\ell=0},\quad\mathcal{O}^{{}^{\prime}}_{\Delta=1,\ell=1},\quad\mathcal{O}^{{}^{\prime\prime}}_{\Delta=2,\ell=2}$
(5.1)

The extremal functional method applied to theories that saturate the bootstrap
bound on the spectral gap tells us the dimensions of the second lowest-
dimension operators in the spin $\ell=0,1,$ and $2$ sectors, as observed in
equation (3). They are, respectively:

$\ell=0,1,2:\quad\Delta_{\text{gap}}=2,1+\Delta_{\phi},2+\Delta_{\phi},\textrm{
resp.}$ (5.2)

In Fig.6 we show the two-sided bounds for three operators in
$\Lambda_{max}=15$. One can see that for three operators, the upper and lower
bound is extremely close. First, in Table.1, we compare the bounds with the
extremal theories at $c=1,2,{14\over 5},4$, where one can see that these two
sided bounds fix the degeneracy numbers of these three operators to extremely
high precision.

Figure 6: Panels (a)(b)(c) are the the upper and lower degeneracy bound for operators $\mathcal{O}_{\Delta=\Delta_{\phi},\ell=0},\mathcal{O}^{{}^{\prime}}_{\Delta=1,\ell=1},\mathcal{O}^{{}^{\prime\prime}}_{\Delta=2,\ell=2}$ respectively. The red solid line corresponds to the lower bound while the black dashed line is the upper bound. The inserted plots show the difference between the upper and lower bounds for each operator Theories |  | $n_{{c\over 6}+{1\over 3},0}$ | $n_{1,1}$ | $n_{2,2}$
---|---|---|---|---
$c=1$ | Exact | 4 | 6 | 0
| Bound | $(4-5.017*10^{-30},4+5.57*10^{-30})$ | $(6-8.96*10^{-28},6+1.16*10^{-29})$ | $(-2.666*10^{-28},2.201*10^{-28})$
$c=2$ | Exact | 18 | 16 | 16
| Bound | (17.99997,18.00002) | (15.9987,16.0005) | (15.9981,16.0029)
$c={14\over 5}$ | Exact | 49 | 28 | 54
| Bound | (48.9996,49.0007) | (27.9987,28.0035) | (53.9986,54.0209)
$c=4$ | Exact | 192 | 56 | 210
| Bound | (191.986,192.030) | (55.985,56.030) | (209.987,210.154)
Table 1: The comparison between the degeneracy with the two-sided bounds of
the extremal theories. The degeneracy number of the extremal theories are
computed in appendix.A

Next we use the result in Fig.6 to determine the allowed regions for the
central charge. To see how these regions are determine, consider the bound on
the degeneracy number of the first scalar operator in the region around
$c\sim{14\over 5}$. We show the two-sided bound in Fig.7, where it is clear
that the only integer within this range is $n_{{4\over 5},0}=49$, and the
allowed region $c\in[c_{\text{min}},c_{\text{max}}]$ is given by when the
upper and lower bounds pass through $n_{{4\over 5},0}=49$:

$(N_{i},c_{\text{min}},c_{\text{max}})=(49,2.79999,2.800007).$ (5.3)

Figure 7: Two sided bound of $n_{{4\over 5},0}$ near $c={14\over 5}$. The
dashed red region is the only allowed region of central charge within this
range

We can repeat this process for any degeneracy number in any range of the
central charge. Below we list the first few regions using three different
bounds of
$\mathcal{O}_{\Delta=\Delta_{\phi},\ell=0},\mathcal{O}^{{}^{\prime}}_{\Delta=1,\ell=1},\mathcal{O}^{{}^{\prime\prime}}_{\Delta=2,\ell=2}$
respectively, at derivative order $\Lambda_{max}=15$:

$\displaystyle\mathcal{O}_{\Delta=\Delta_{\phi},\ell=0},~{}~{}N\in[4,192],~{}~{}A_{1}=\\{(N_{i},c_{\text{min}},c_{\text{max}})\\}=\\{(4,1,1+3.78\times
10^{-9}),\dots$
$\displaystyle,(5,1.12834993,1.12834995),\dots(192,3.9999,4.00007)\\}$
$\displaystyle\mathcal{O}^{{}^{\prime}}_{\Delta=1,\ell=1},~{}~{}N\in[6,56],~{}~{}A_{2}=\\{(N_{i},c_{\text{min}},c_{\text{max}})\\}=\\{(6,1,1+1.52\times
10^{-9})\dots$ $\displaystyle(7,1.241873,1.241875),\dots(56,3.9999,4.0005)\\}$
$\displaystyle\mathcal{O}^{{}^{\prime\prime}}_{\Delta=2,\ell=2},~{}~{}N\in[0,210]~{}~{}A_{3}=\\{(N_{i},c_{\text{min}},c_{\text{max}})\\}=\\{(0,1,1+3.78\times
10^{-9})\dots$
$\displaystyle(1,1.135886,1.135889)\dots(210,3.9993,4.00006)\\}$ (5.4)

where $A_{1},A_{2},A_{3}$ are allowed regions selected by operators
$\mathcal{O}_{\Delta=\Delta_{\phi},\ell=0},\mathcal{O}^{{}^{\prime}}_{\Delta=1,\ell=1},\mathcal{O}^{{}^{\prime\prime}}_{\Delta=2,\ell=2}$.
In other words, for the degeneracy number $N$ of, for instance, the operator
with spin 0 and dimension $\Delta_{\phi}$, the set $A_{1}$ indicates for each
possible integer value of $N\in[4,193]$ the range of $c$s that are consistent
with that value of $N$. In Fig.8 we show some example plots of allowed and
disallowed regions. Any consistent value of the central charge $c$ must then
lie within one of the allowed regions for all three sets $A_{1},A_{2},A_{3}$,
since the degeneracy numbers of all three operators must be integers. The
intersection of these three regions $(A_{1}\cap A_{2})\cap A_{3}$ reduces to a
strikingly small set for the allowed ranges of the central charge for the
extremal theories:

$(c_{\text{min}},c_{\text{max}})=\\{(1,1+1.52\times
10^{-9}),(1.9999,2.000006),(2.7997,2.80002),(3.99986,4.00006)\\}.$ (5.5)

These four ranges exactly correspond to the four theories listed in Sec.3

Figure 8: Examples of allowed and disallowed region of $c$. LHS: Allowed
region of $c$ is between the black dashed line. RHS: No allowed region within
this range of $c$

##  6 Lowering the Spectrual Gap Near $c\sim 1$

In Sec.2.3, we noted that the degeneracy in an interval
$\mathcal{F}_{\ell}(\Delta_{a},\Delta_{b})$ is a stronger condition than just
considering the degeneracy for a specific operator, and in particular it will
have two-sided bounds in general. In this section, we will use this method to
slightly lower the spectral gap near $c\sim 1$.

We choose two examples, $c=1.01$ and $c=2$ to illustrate how this works.
Define $\Delta_{\text{mod}}$ as the gap of the lowest scaling dimension
including all spin sectors. Using only the Virasoro modular bootstrap, the
authors in Collier:2016cls  found that the bound on $\Delta_{\rm mod}$
converged to $\Delta_{\text{mod}}={c\over 6}+{1\over 3}$, which in our case is
${301\over 600}$ and ${2\over 3}$ respectively. We consider the bound on the
integrated degeneracy number
$\mathcal{F}_{0}\Big{(}\Delta_{\text{mod}},\Delta_{\text{mod}}+{1\over
20}\Big{)}$ in the interval
$\Delta\in[\Delta_{\text{mod}},\Delta_{\text{mod}}+{1\over 20}]$ in the spin
$\ell=0$ sector. At $\Lambda_{max}=15$, we find the following two-sided
bounds:

$\displaystyle c=1+{1\over 100},\quad
4.072268\leq\mathcal{F}_{0}\Big{(}{301\over 600},{301\over 600}+{1\over
20}\Big{)}\leq 4.072272\Rightarrow\text{Inconsistent!}$ $\displaystyle
c=2,\quad 17.944\leq\mathcal{F}_{0}\Big{(}{2\over 3},{2\over 3}+{1\over
20}\Big{)}\leq 18.014\Rightarrow\mathcal{F}_{0}\Big{(}{2\over 3},{2\over
3}+{1\over 20}\Big{)}=18.$ (6.1)

From this, we can immediately rule out the case when $c=1+{1\over 100}$ with
$\Delta_{\text{mod}}={301\over 600}$. By contrast, at $c=2$ the extremal
theory with $\Delta_{\text{mod}}={2\over 3}$ corresponds to the
$SU(3)_{1}~{}WZW$ model, and passes the consistency check as expected. At
$c=1.01$, we can now go farther and improve the bound on $\Delta_{mod}$. In
Fig.9, we plot the upper and lower bound of
$\mathcal{F}_{0}\Big{(}\Delta_{a},\Delta_{a}+{1\over 20}\Big{)}$ as a function
of $\Delta_{a}$ at $\Lambda_{max}=15$. We obtain an improved value for the
bound on $\Delta_{\text{mod}}$ from the point where
$\mathcal{F}_{0}\Big{(}\Delta_{a},\Delta_{a}+{1\over 20}\Big{)}$ first
includes $4$. This occurs at the value $\Delta_{a}\approx 0.5008$, whereas the
previous bound was $\Delta_{\rm mod}={301\over 600}\approx 0.5017$.

Figure 9: The upper and lower bounds on
$\mathcal{F}_{0}\Big{(}\Delta_{a},\Delta_{a}+{1\over 20}\Big{)}$ for Virasoro
bootstrap, at $c=1+{1\over 100},\Lambda_{max}=15$

In Fig.10 we show the result more generally for a range of $c$ near $c\sim 1$
at $\Lambda_{max}=15$.444For the case where $c$ is very large, this method
gives significantly less improvement in the bound; for example, at $c=3$ we
found that at $\Lambda_{max}=15$, the decrease in the spectral gap is less
than $10^{-5}$.

Figure 10: (a) The comparison between the integrality result at
$\Lambda_{max}=15$ with the convergent bound $\Delta_{\text{mod}}={c\over
6}+{1\over 3}$ from Collier:2016cls  (b) The exact difference between the
integrality result at $\Lambda_{max}=15$ and $\Delta_{\text{mod}}={c\over
6}+{1\over 3}$

### Acknowledgments

We thank David Simmons-Duffin and Yuan Xin for helpful conversations, and Yuan
Xin for comments on the earlier draft. ALF and WL are supported by the US
Department of Energy Office of Science under Award Number DE-SC0015845, and
the Simons Collaboration on the Non-Perturbative Bootstrap. ALF thanks the
Aspen Center of Physics for hospitality as this work was completed.

##  Appendix A Partition function for extremal theories

In the section we list the partition function extremal theories with
$c=1,2,{15\over 4},4$ in Sec.5. The partition function for extremal theory in
$c=1,2,4$ can be found in Collier:2016cls . The partition function
$(G_{2})_{1}$ $WZW$ model at $c={14\over 5}$ can be found in Rayhaun:2023pgc ,

$\displaystyle(q\overline{q})^{-{1\over
24}}|\eta(\tau)|^{2}\Big{[}Z_{\text{ext}}^{c=1}(\tau,\overline{\tau})-\chi_{vac}(\tau)\overline{\chi}_{vac}(\overline{\tau})\Big{]}=4(q\overline{q})^{{1\over
4}}+3(q+\overline{q})+3q\overline{q}+\dots$
$\displaystyle(q\overline{q})^{{1\over
24}}|\eta(\tau)|^{2}\Big{[}Z_{\text{ext}}^{c=2}(\tau,\overline{\tau})-\chi_{vac}(\tau)\overline{\chi}_{vac}(\overline{\tau})\Big{]}=18(q\overline{q})^{{1\over
3}}+8(q+\overline{q})+\overline{q}^{{4\over 3}}q^{{1\over
3}}+8(q^{2}+\overline{q}^{2})+48q\overline{q}+\dots$
$\displaystyle(q\overline{q})^{{3\over
40}}|\eta(\tau)|^{2}\Big{[}Z_{\text{ext}}^{c={14\over
5}}(\tau,\overline{\tau})-\chi_{vac}(\tau)\overline{\chi}_{vac}(\overline{\tau})\Big{]}=49(q\overline{q})^{{2\over
5}}+14(q+\overline{q})+27(q^{2}+\overline{q}^{2})+168q\overline{q}+\dots$
$\displaystyle(q\overline{q})^{{1\over
8}}|\eta(\tau)|^{2}\Big{[}Z_{\text{ext}}^{c=4}(\tau,\overline{\tau})-\chi_{vac}(\tau)\overline{\chi}_{vac}(\overline{\tau})\Big{]}=192(q\overline{q})^{{1\over
2}}+28(q+\overline{q})+105(q^{2}+\overline{q}^{2})+728q\overline{q}+\dots$
(A.1)

Expanding in the first few order in $q,\overline{q}$, we can extract the
degeneracy for operators $\mathcal{O}_{\Delta={c\over 6}+{1\over
3},\ell=0},\mathcal{O}^{{}^{\prime}}_{\Delta=1,\ell=1},\mathcal{O}^{{}^{\prime\prime}}_{\Delta=2,\ell=2}$.
Explicitly,

$(q\overline{q})^{{c-1\over
12}}|\eta(\tau)|^{2}\Big{[}Z_{\text{ext}}(\tau,\overline{\tau})-\chi_{vac}(\tau)\overline{\chi}_{vac}(\overline{\tau})\Big{]}=n_{{c\over
6}+{1\over 3},0}(q\overline{q})^{{c\over 12}+{1\over 6}}+{n_{1,1}\over
2}(q+\overline{q})+{n_{2,2}\over 2}(q^{2}+\overline{q}^{2})+\dots$ (A.2)

Thus for these four extremal theories, we have,

$\displaystyle c=1:\quad n_{{1\over 2},0}=4,\quad n_{1,1}=6,\quad n_{2,2}=0$
$\displaystyle c=2:\quad n_{{2\over 3},0}=18,\quad n_{1,1}=16,\quad
n_{2,2}=16$ $\displaystyle c={14\over 5}:\quad n_{{4\over 5},0}=49,\quad
n_{1,1}=28,\quad n_{2,2}=54$ $\displaystyle c=4:\quad n_{1,0}=192,\quad
n_{1,1}=56,\quad n_{2,2}=210$ (A.3)

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