# Scalars on asymptotically locally AdS wormholes with $\mathcal{R}^{2}$ terms

Octavio Fierro1, Daniela Narbona2, Julio Oliva2, Constanza Quijada2, Guillermo
Rubilar2 1Departamento de Matemática y Física Aplicadas, Universidad Católica
de la Santísima Concepción, Alonso de Ribera 2850, Concepción, Chile,
2Departamento de Física, Universidad de Concepción, Casilla 160-C, Concepción,
Chile.

###### Abstract

In this paper we study the propagation of a probe scalar on an asymptotically
locally AdS wormhole solution of Einstein-Gauss-Bonnet theory in five
dimensions. The radial coordinate $\rho$ connects both asymptotic regions
located at $\rho\rightarrow\pm\infty$. The metric is characterized by a single
integration constant $\rho_{0}$ and the wormhole throat is located at
$\rho=0$. In the region $0<\rho<\rho_{0}$, both the gravitational pull as well
as the centrifugal contributions to the geodesic motion point in the same
direction and therefore they cannot balance. We explore the consequences of
the existence of this region on the propagation of a scalar probe. The cases
with $\rho_{0}=0$ as well as the limit $\rho_{0}\rightarrow+\infty$ lead to
exactly solvable differential eigenvalue problems, with shape-invariant
potentials of the Rosen-Morse and Scarf family, respectively. Here, we
numerically obtain the normal modes of a scalar field when $\rho_{0}\neq 0$,
with reflecting boundary conditions at both asymptotic regions. We also
explore the effect of a non-minimal coupling between the scalar curvature and
the scalar field. Remarkably, there is a particular value of the non-minimal
coupling parameter that leads to fully resonant spectra in the limit of
vanishing $\rho_{0}$ as well as when $\rho_{0}\rightarrow+\infty$, for purely
radial modes.

##  I Introduction

In the realm of four dimensional General Relativity, it is very difficult to
construct asymptotically flat, spherically symmetric wormhole geometries since
they generally require violations of different energy conditions Visser:1995cc
. In particular the averaged null energy condition must be circumvented for a
wormhole to exist if the throat is to provide a shorter path connecting points
of each asymptotic region through the bulk. The obstructions can be avoided,
for instance, by modifying of the gravitational theory (see e.g. Lobo:2012ai ,
Bueno:2018uoy  and references therein) by the inclusion of a NUT charge Ayon-
Beato:2015eca , or by constructing wormholes with long throats supported by
Casimir energies of quantum fields Maldacena:2018gjk . Recently, the inclusion
of a negative cosmological constant allowed to construct asymptotically AdS
wormholes which in the radial direction are foliated by warped AdS spacetimes
and are devoid of closed timelike curves Anabalon:2018rzq .

The general features that a static metric must fulfil in order to describe a
traversable, asymptotically flat wormhole were studied in the original papers
Ellis:1973yv , Morris:1988cz  and Morris:1988tu , and the metrics considered
in such references have been used as toy models to study the propagation of
probe fields on spacetimes with wormhole topology. For example, the
propagation of scalar and electromagnetic waves may correctly led to the
interpretation of the wormhole geometry as an extended particle Clement:1982ej
. Transmission and reflection coefficients for asymptotically flat,
ultrastatic wormholes in 2+1 and 3+1 dimensions have been studied in
references Kar:1994ty , Kar:1995jz  and Taylor:2014vsa  featuring resonances
for particular values of the wormhole parameters, and leading to almost
reflectionless effective potentials111There is also an extensive literature on
Euclidean wormholes leading to instantons and on the use of scalar probes to
test the stability of the configuration (see e.g. Betzios:2017krj  and
references therein). In this paper we will be interested in traversable,
Lorentzian wormholes.. In reference Taylor:2014vsa  it was also shown that a
non-minimally coupled scalar field may lead to an instability since the
effective Schrödinger potential for the perturbation turns out to be negative
definite. Even more, when the wormhole is asymptotically flat in both
asymptotic regions, outgoing boundary conditions at both infinities lead to a
quasinormal spectrum which can mimic that of a black hole of mass $M$ for a
given wormhole mass $M^{-1}$ Taylor:2014vsa .

More recently, by imposing boundary conditions at the asymptotic AdS boundary
as well as at the throat, a particular family of smooth wormholes in
Einstein’s theory coupled to a non-linear electrodynamics has also been shown
to be stable provided an effective Breitenlohner-Freedman bound BF1 ; BF2 ;
MT1  is fulfilled for the probe scalar field Kim:2018ang . Wormhole geometries
in asymptotically AdS spacetimes have also received large attention during the
last decades due to the AdS/CFT correspondence. The bulk dual of a maximally
entangled state of two independent CFTs is an eternal Schwarzschild-AdS black
hole with the two boundaries causally disconnected by an Einstein-Rosen bridge
Maldacena:2001kr . The inclusion of an interaction between the CFTs opens a
throat in the bulk allowing for a causal connection of the boundaries
Gao:2016bin , whose size increases with the angular momentum in the bulk
Caceres:2018ehr .

Asymptotically AdS wormholes with quadratic, stringy-like corrections do exist
in five dimensions DOT1 . The action for the Einstein-Gauss-Bonnet theory in
five dimensions reads

$I=\frac{1}{16\pi
G}\int\sqrt{-g}\left(R-2\Lambda+\alpha\left(R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}\right)\right)d^{5}x,$
(1)

where $\alpha$ is a coupling constant with mass dimension equal to $-2$. When
formulated in first order, the local Lorentz invariance of the theory is
enlarged to a local (A)dS group at the point $\Lambda\alpha=-3/4$ (see e.g.
MokhtarZanelliBook  and references therein). In this case the Lagrangian can
be written as a Chern-Simons form where the spin connection as well as the
vielbein transform as components of an (A)dS gauge connection. This structure
can be extended to higher odd-dimensions as well as to dimension three, but in
contrast to the latter, the higher dimensional case does describe a theory
with local degrees of freedom in the bulk. As shown in Banados:1993ur , all
these theories contain generalizations of the static $2+1$-dimensional BTZ
black hole BTZ1 -BTZ2 , characterized by an integration constant which can be
identified with the mass computed with the canonical formalism Banados:1993ur
. In DOT1  it was shown that this theory actually admits a larger family of
static solutions, including wormhole geometries in vacuum with two
asymptotically locally AdS2n+1 regions for $n\geq 2$. In five dimensions, the
line element of these solutions is given by

$ds^{2}=l^{2}\left[-\cosh^{2}\left(\rho-\rho_{0}\right)dt^{2}+d\rho^{2}+\cosh^{2}\rho\left(d\phi^{2}+d\Sigma_{2}^{2}\right)\right],$
(2)

where $-\infty<t<\infty$, $-\infty<\rho<\infty$, $0<\phi<2\pi$ and
$d\Sigma_{2}$ stands for the line-element of an Euclidean $2$-dimensional
manifold, whose local geometry is that of the hyperbolic space with radius
$3^{-1/2}$, globally equivalent to the quotient of the hyperbolic space by a
Fuchsian group, i.e. $\Sigma_{2}$ is homeomorphic to the quotient of $H_{2}$
by a freely acting, discrete subgroup of $SO(2,1)$. For any local purpose one
can consider complex projective coordinates such that

$d\Sigma_{2}^{2}=\frac{1}{3}\left(1-\frac{z\bar{z}}{4}\right)^{-2}dzd\bar{z}.$
(3)

The spacetime (2) represents a wormhole geometry with a throat located at
$\rho=0$, connecting two asymptotically locally AdS spacetimes with curvature
radius $l^{2}=4\alpha=-3/\Lambda$. Hereafter we set $l=1$. The constant
$\rho_{0}$ is an arbitrary integration constant and determines the apparent
mass of the wormhole as seen by an asymptotic observer at a given asymptotic
region DOT1 -DOT2 . The presence of $\rho_{0}\neq 0$, creates a region which
has interesting properties for particles with angular momentum, probing these
geometries. As shown in DOT2 , the effective potential for the geodesics has
two contributions, one of which is due to the angular momentum of the particle
around the $S^{1}$ factor parametrized by the coordinate $\phi$ in (2). This
centrifugal contribution always points outwards the throat located at
$\rho=0$, while the remaining contribution flips its direction at the surface
$\rho=\rho_{0}$ and always points towards it. Therefore, the gravitational
pull acting on a particle can be balanced by the centrifugal contribution only
if the particle is in one of the regions $-\infty<\rho<0$ or
$\rho_{0}<\rho<+\infty$. Geodesic probes turn out to be expelled from the
region $0<\rho<\rho_{0}$. And for probe strings propagating in this background
the surface $\rho=\rho_{0}/2$ defines the turning point of strings with both
ends attached to the same asymptotic region Ali:2009ky .

The main goal of the present work is to explore the effect of $\rho_{0}\neq 0$
on the propagation of massive and massless probe scalars, with both minimal
and non-minimal coupling with the curvature. We will be interested in the
normal frequencies of the scalars obtained by imposing reflective boundary
conditions at the AdS boundaries.

First we will study a minimally coupled massive scalar field on the wormhole
geometry (2), revisiting the case with $\rho_{0}=0$ which can be solved
analytically (see Correa:2008nq ). Here, by formulating the problem in a
Schrödinger form we show that the effective potential for the scalar probe
corresponds to a Rosen-Morse potential, which explains the integrability of
the spectrum Cooper:1994eh . Remarkably, we find that the limit
$\rho_{0}\rightarrow\infty$ also leads to an exact spectrum for the scalar,
which would be particularly useful for obtaining the normal frequencies of the
scalar on wormholes with large $\rho_{0}$ (a problem that could therefore be
treated in a perturbative manner). Then, we will numerically solve the
spectrum for normal frequencies of the scalar field for finite, non-vanishing
values of $\rho_{0}$, connecting the two exactly solvable models.

##  II Normal modes of the scalar probe

The dynamics of a minimally coupled scalar probe on the wormhole geometry (2)
if determined by the equation

$\left(\square-m^{2}\right)\Phi\left(x^{\mu}\right)=0\ .$ (4)

For simplicity, we will only consider angular momentum for the field $\Phi$
along the $S^{1}$ direction, parametrized by the coordinate $\phi$. Therefore,
we use the following ansatz for the scalar

$\Phi\left(x\right)=\mathcal{R}\left(\sum_{n=-\infty}^{\infty}\int d\omega\
R_{\omega,n}\left(\rho\right)\ e^{-i\omega t+in\phi}\right)\ ,$ (5)

and hereafter we will drop the dependence on $\omega$ and $n$ of the radial
function. Since the equation is linear, one finds decoupled, second order,
linear ODEs for each mode, which leads to

$\displaystyle 0$
$\displaystyle=\cosh^{2}\left(\rho\right)\frac{d^{2}R}{d\rho^{2}}+\left(3\cosh\left(\rho\right)\sinh\left(\rho\right)+\frac{\sinh\left(\rho-\rho_{0}\right)\cosh^{2}\left(\rho\right)}{\cosh\left(\rho-\rho_{0}\right)}\right)\frac{dR}{d\rho}$
$\displaystyle-\left(\cosh^{2}\left(\rho\right)m^{2}+n^{2}-\frac{\omega^{2}\cosh^{2}\left(\rho\right)}{\cosh^{2}\left(\rho-\rho_{0}\right)}\right)R\left(\rho\right).$
(6)

Introducing the inversion $z=(1-\tanh\rho)/{2}$, we map $-\infty<\rho<+\infty$
to $1>z>0$, and obtain the asymptotic behavior for the radial dependence of
the scalar field:

$R\left(z\right)\rightarrow
c_{1}z^{\Delta_{+}}\left[1+\mathcal{O}(z)\right]+c_{2}z^{\Delta_{-}}\left[1+\mathcal{O}(z)\right]\text{
as }z\rightarrow 0,$ (7)

and

$R\left(z\right)\rightarrow
d_{1}\left(1-z\right)^{\Delta_{+}}\left[1+\mathcal{O}(1-z)\right]+d_{2}\left(1-z\right)^{\Delta_{-}}\left[1+\mathcal{O}(1-z)\right]\text{
as }z\rightarrow 1,$ (8)

with

$\Delta_{\pm}=1\pm\frac{1}{2}\sqrt{m^{2}+4}\ ,$ (9)

$c_{1,2}$ and $d_{1,2}$ being integration constants.

Even though scalar fields on asymptotically AdS spacetimes can have negative
$m^{2}$, we will focus in the case $m^{2}\geq 0$ (for a thorough analysis of
the properties of a scalar field on the particular case of the wormhole with
$\rho_{0}=0$ and the shifted Breitenlohner-Freedman bound, see Correa:2008nq
). Reflective boundary conditions require the field to vanish at infinity and
therefore only the $\Delta_{+}$ branch in (7) and (8) is allowed. Under these
boundary condition the equation (6) defines a Sturm-Liouville problem which
can be manifestly seen by transforming the radial problem (6) into a
Schrödinger-like equation,

$-\frac{d^{2}u}{d\bar{\rho}^{2}}+U(\bar{\rho})u=\omega^{2}u\ ,$ (10)

where

$u=R\cosh^{3/2}(\rho)\quad\text{and}\quad\rho=\rho_{0}+\ln\left(\tan\left(\frac{\bar{\rho}}{2}\right)\right)\
.$ (11)

The radial coordinate $\bar{\rho}$ works effectively as a “tortoise”-like
coordinate in the sense that in terms of $(t,\bar{\rho})$ the two-dimensional
part of the metric (2) is manifestly conformally flat. The coordinate
$\bar{\rho}$ connects both asymptotically AdS regions $0<\bar{\rho}<\pi$. The
effective Schrödinger potential takes the form

$\displaystyle U(\bar{\rho})$
$\displaystyle=\frac{\left(4n^{2}-3\right)}{4}\left(\frac{\cos(\bar{\rho})\cosh(\rho_{0})-\sinh(\rho_{0})}{\cos(\bar{\rho})-\coth(\rho_{0})}\right)^{2}-\left(2n^{2}-3\right)\left(\frac{\cos(\bar{\rho})\cosh(\rho_{0})-\sinh(\rho_{0})}{\cos(\bar{\rho})\operatorname{sech}(\rho_{0})-\operatorname{csch}(\rho_{0})}\right)$
$\displaystyle\ \
+\frac{\cosh^{2}(\rho_{0})\left(4n^{2}-9\right)}{4}+\frac{\left(4m^{2}+15\right)}{4\sinh^{2}(\bar{\rho})}\,.$
(12)

Figure 1: Effective potential for the radial dependence of the scalar probe,
for different values of the parameters. As expected in the presence of a
negative cosmological constant, the potential diverges at the boundaries
$\bar{\rho}=0,\pi$.

This potential $U(\bar{\rho})$ (depicted in Fig. 1), parametrically depends on
the wormhole integration constant $\rho_{0}$, and for arbitrary values of the
latter, the equation cannot be solved analytically. Even though we will solve
the equation numerically to find the normal modes of the scalar on the
wormhole, it is interesting to note that there are two particular values of
$\rho_{0}$ for which the potential is shape invariant Cooper:1994eh  and (10)
can be solved analytically. Those values are $\rho_{0}=0$ and
$\rho_{0}\rightarrow+\infty$, and the corresponding potentials $U$ are given
by

$\displaystyle U^{\left(0\right)}(\bar{\rho})$
$\displaystyle:=\frac{15}{4}\frac{1}{\sin^{2}\bar{\rho}}+\frac{m^{2}}{\sin^{2}\bar{\rho}}-\frac{9}{4}+n^{2}+\mathcal{O}\left(\rho_{0}\right)\
,$ (13) $\displaystyle U^{\left(\infty\right)}(\bar{\rho})$
$\displaystyle:=-\frac{1}{4}\frac{6\cos\bar{\rho}-4m^{2}-9}{\sin^{2}\bar{\rho}}+\mathcal{O}\left(e^{-2\rho_{0}}\right)\
.$ (14)

The leading term of $U^{\left(0\right)}$ leads to an effective quantum
mechanical problem with a Rosen-Morse potential, which can be solved
analytically and with energies and bound states given by

$\displaystyle\omega_{\left(0\right),p}^{2}$
$\displaystyle=\left(\frac{1}{2}+\sqrt{4+m^{2}}+p\right)^{2}+n^{2}-\frac{9}{4}\
,$ (15) $\displaystyle u_{p}^{\left(0\right)}\left(\bar{\rho}\right)$
$\displaystyle=A_{p}^{\left(0\right)}\left(\sin\bar{\rho}\right)^{s+p}P_{p}^{\left(-s-p,-s-p\right)}\left(i\cot\left(\bar{\rho}\right)\right)\
,$ (16)

with $s=\sqrt{4+m^{2}}+{1}/{2}$, $A_{p}^{\left(0\right)}$ an arbitrary
integration constant that can be fixed by normalization, and $p=0,1,2,3,...$
as the mode number. For the particular value $\rho_{0}=0$ the wormhole
acquires a reflection symmetry with respect to the throat. $P$ are the
corresponding Jacobi polynomials.

Remarkably, for $\rho_{0}\rightarrow\infty$, the Schrödinger problem in (14)
can also be integrated analytically since it corresponds to a Scarf potential,
and leads to the following frequencies and eigenfunctions

$\displaystyle\omega_{\left(\infty\right),p}^{2}$
$\displaystyle=\left(A+p\right)^{2}\text{\ ,}$ (17) $\displaystyle
u_{p}^{\left(\infty\right)}\left(\bar{\rho}\right)$
$\displaystyle=A_{p}^{\left(\infty\right)}\left(1-\cos\left(\bar{\rho}\right)\right)^{\frac{A-B}{2}}\left(\cos\left(\bar{\rho}\right)+1\right)^{\frac{A+B}{2}}P_{p}^{\left(A-B-\frac{1}{2},A+B-\frac{1}{2}\right)}\left(\cos\left(\bar{\rho}\right)\right)\
,$ (18)

where the constant $A>B$ are given by

$\displaystyle 2A$ $\displaystyle=$ $\displaystyle
1+\sqrt{5+2m^{2}+2\sqrt{\left(m^{2}+1\right)\left(m^{2}+4\right)}}\ ,$ (19)
$\displaystyle 6B$ $\displaystyle=$
$\displaystyle\left(5+2m^{2}-2\sqrt{\left(m^{2}+1\right)\left(m^{2}+4\right)}\right)\sqrt{5+2m^{2}+2\sqrt{\left(m^{2}+1\right)\left(m^{2}+4\right)}}$
(20)

$A_{p}^{\left(\infty\right)}$ is an integration constant and $p=0,1,2,...$ ,
is again the mode number. It is worth to mention that in this case the
frequencies $\omega_{p}^{\left(\infty\right)}$ lead to an equispaced, fully
resonant spectrum. This result may be particularly relevant for obtaining the
spectrum of the scalar field on the wormhole, for large $\rho_{0}$, by
perturbative methods. The strict limit $\rho_{0}$ to infinity can be taken
after suitable regularization of the geometry DOT2 , leading to a wormhole
that connects two different asymptotic regions. Since such wormhole is not
asymptotically locally AdS we left its analysis for the Appendix.

Here we are interested in the effects of a finite value of $\rho_{0}$ on the
propagation of a minimally coupled scalar field. Therefore, we are obligated
to integrate equation (10) numerically, with reflective boundary conditions at
both infinities which can be achieved only for a countably infinity number of
frequencies $\omega_{p}$. This is done by using a variation of the so-called
Shooting Method (see, for instance, Newman2013 ), which in our case consists
in varying the value of $\omega$ in (10) and seeking those values which fulfil
the boundary condition at $\bar{\rho}=\pi$. More specifically, the Schrödinger
equation is recast as a first order ODE system, which is then integrated “from
left to right”, i.e. from $\bar{\rho}=0$ to $\bar{\rho}=\pi$ for each value of
$\omega$. The integration is performed using the standard 4th order Runge-
Kutta method Newman2013 , which thus determines the value of the solution at
the right boundary, i.e. $u(\bar{\rho}=\pi,\omega)$. The later can be
considered as a function of $\omega$, for which the sought after values
$\omega_{p}$ are zeros of, in virtue of the right boundary condition
$u(\bar{\rho}=\pi,\omega_{p})$. After a solution for $\omega_{p}$ is found,
the value of $p$ is determined by counting the number of nodes of the function
$u(\bar{\rho},\omega_{p})$.

##  III Spectra for the minimally coupled scalar probe

Since the problem for the normal modes with $\rho_{0}\neq 0$ is a Sturm-
Liouville problem, with potential fixed by the angular momentum of the field
$n$, as well as $\rho_{0}$ and $m^{2}$, the eigenfunctions and
eigenfrequencies will both be labelled by an integer $p$. In all the plots
shown below, we have included small black bars near $\rho_{0}=0$ and
$\rho_{0}$ large, marking the analytic values for the frequencies obtained in
those cases. In what follows we plot some of the numerically obtained spectra,
which present interesting features as $\rho_{0}$ varies.
The frequencies of the fundamental mode ($p=0$) are shown on the left panel of
FIG. 2, while the right panel shows a set of frequencies for the tenth
overtone ($p=10$). While the fundamental mode is a monotonic function of
$\rho_{0}$, we observe that the excited modes have a maximum frequency for a
critical value of $\rho_{0}$ and then decay to a given value. As expected, the
frequencies increase with the angular momentum of the field, and consistently
with the asymptotic expression for $U^{(\infty)}$ in equation (14), the
frequencies merge regardless the value of $n$.

FIG.3 shows the fundamental and ten excited modes for the “s-wave” of the
scalar ($n=0$) on the left panel, while the frequencies for a spinning scalar
probe with $n=10$ have been plotted on the right-panel, both as a function of
$\rho_{0}$. It is particularly interesting to notice that the “s-wave”
fundamental and excited frequencies remain almost constant regardless the
value of the integration constant $\rho_{0}$.

Finally, FIG.4 shows the fundamental and first overtones for different values
of the angular momentum and the mass of the scalar. Even though the effective
Schrödinger problem is one-dimensional, there are degeneracies due to the fact
the the effective potential depends on $n$, therefore different values of $n$
lead to different one-dimensional Sturm-Liouville problems.

Figure 2: Spectra for the fundamental mode $p=0$ (left-panel) and tenth
overtone $p=10$ (right-panel) as a function of $\rho_{0}$ for different even
values of the angular momentum of the scalar probe $n=0,2,4,...,10,$ for
$m^{2}=10$. The fundamental mode has a monotonic behaviour with $\rho_{0}$
while excited frequencies present a maximum for a given critical value of
$\rho_{0}$ which depends on the angular momentum of the field.

Figure 3: Spectra for vanishing angular momentum $n=0$ (left-panel) and a
spinning scalar probe with $n=10$ (right-panel) as a function of $\rho_{0}$
for the fundamental mode and the first 10 excited states ($m^{2}=10$).

Figure 4: Fundamental and first two overtones for the scalar probe with
$n=0,1,2,3,4$, and $m^{2}=6$ (left-panel) and $m^{2}=30$ (right-panel). For a
given value of $\rho_{0}$ the same frequency can be obtained for different
modes since the effective potential depends explicitly on $n$.

##  IV Scalars probes with non-minimal coupling

As shown in Correa:2008nq , for the case $\rho_{0}=0$ the equation for a
scalar non-minimally coupled with the scalar curvature can also be solved in
an analytic manner. This is remarkable since the Ricci scalar of the wormhole
background is a non-trivial function of the radial coordinate. Here we explore
the effect of a non-minimal coupling on the spectrum of the scalar for
$\rho_{0}\neq 0$. The equation for the scalar in this case is given by

$\left(\square-m^{2}-\xi R\right)\Phi\left(x^{\mu}\right)=0\,.$ (21)

This scalar probe is invariant under local Weyl rescalings for $\xi=3/16$.
When $\rho_{0}$ vanishes, the equation for the radial dependence of the non-
minimally coupled scalar can be obtained from that of the minimally coupled
one by shifting the mass as well as the eigenvalues of the Laplace operator on
the manifold $\Sigma_{3}=S^{1}\times H_{2}/\Gamma$. For non-vanishing
$\rho_{0}$, one cannot rely on this shift, and one is forced to numerically
find the spectrum of normal modes. Using the separation (5) and the change of
variables (11), one leads to a Schrödinger-like equation of the form (10) with
a potential

$\displaystyle U_{\xi}(\bar{\rho})$
$\displaystyle=\frac{\left(4n^{2}-3\right)}{4}\left(\frac{\cos(\bar{\rho})\cosh(\rho_{0})-\sinh(\rho_{0})}{\cos(\bar{\rho})-\coth(\rho_{0})}\right)^{2}-\left(2n^{2}-3+6\xi\right)\left(\frac{\cos(\bar{\rho})\cosh(\rho_{0})-\sinh(\rho_{0})}{\cos(\bar{\rho})\operatorname{sech}(\rho_{0})-\operatorname{csch}(\rho_{0})}\right)$
$\displaystyle\ \
+\frac{\cosh^{2}(\rho_{0})\left(4n^{2}-9+24\xi\right)}{4}+\frac{\left(4m^{2}+15\right)}{4\sinh^{2}(\bar{\rho})}.$
(22)

Even with the non-minimal coupling with the scalar curvature, in the extremal
values $\rho_{0}=0$ and $\rho_{0}\rightarrow\infty$ one also recovers shape
invariant potentials, since

$\displaystyle U_{\xi}^{\left(0\right)}\left(\bar{\rho}\right)$
$\displaystyle=\left(\frac{15}{4}+m^{2}-20\xi\right)\frac{1}{\sin^{2}\bar{\rho}}+n^{2}-\frac{9}{4}+6\xi+O\left(\rho_{0}\right)$
(23) $\displaystyle U_{\xi}^{\left(\infty\right)}\left(\bar{\rho}\right)$
$\displaystyle=\frac{1}{4}\frac{6(4\xi-1)\cos\bar{\rho}+4m^{2}+9-56\xi}{\sin^{2}\bar{\rho}}+O\left(e^{-2\rho_{0}}\right)\,,$
(24)

corresponding to a Rosen-Morse potential and a Scarf potential, respectively.

It is interesting to note that for modes without angular momentum ($n=0$) and
$\xi=3/8$ both potentials lead to a fully resonant, equispaced spectra, which
might enhance the energy transfer between modes when non-linearities are
included turbulentwormholes , as it happens in AdS. Note that in a truly
quantum-mechanical problem nor the Rosen-Morse, neither the Scarf potential
lead to equispaced energies since in that case the eigenvalue is quadratic in
the principal quantum number, and actually the harmonic oscillator is the
unique potential with such property. Nevertheless, in our relativistic theory
the eigenvalue is quadratic in the frequencies, therefore all the potentials
which are quadratic in the principal quantum number, may lead to a equispaced
set of $\omega_{n}$ (see kleingordonization ).

The asymptotic behaviours for the function $R(z)$ with $z=(1-\tanh(\rho))/2$,
that defined the radial dependence of the field (5) is the same as that given
in equations (7) and (8), replacing the conformal weights by

$\Delta^{\xi}_{\pm}=1\pm\frac{1}{2}\sqrt{m^{2}-20\xi+4}.$ (25)

Regarding the asymptotic behaviour one can define an effective mass
$m_{\mathrm{eff}}^{2}:=m^{2}-20\xi$ and we will be focused in the region
$m_{\mathrm{eff}}^{2}\geq 0$, in which the reflective boundary conditions lead
to a unique set of normal frequencies. Below we present the spectra for
different values of the parameters characterizing the potential, i.e.
$(\xi,\rho_{0},m^{2},n)$.

Figure 5: Fundamental and first overtones for the non-minimally coupled scalar
probe without angular momentum ($n=0$, left), and $n=10$ (right), for
$m^{2}=10$ and $\xi=3/8$. The figure in the left smoothly connects two fully
resonant spectra for $\rho_{0}=0$ and $\rho_{0}\rightarrow\infty$.

Figure 6: First three modes spectra for $\xi=1/4$ (left) and $\xi=3/8$ (right)
for $m^{2}=10$ for $n=0,1,2,3,4$.

##  V Conclusions

In this work we have obtained the spectra of normal modes for a scalar probe
on an asymptotically locally AdS wormhole in five dimensions. The spacetime is
a solution of General Relativity supplemented with quadratic terms in the
curvature, and is characterized by an integration constant $\rho_{0}$ that
defines the contribution to the energy content of the spacetime as seen by an
asymptotic observer. In Correa:2008nq  it was shown that the scalar probe can
be solved analytically when $\rho_{0}=0$. Here we have shown that this is
indeed the case by mapping the problem with the radial dependence to a
Schrödinger-like problem in the well-known Rosen-Morse potential. Remarkably,
we have shown that when $\rho_{0}\rightarrow\infty$ the spectrum can be also
obtained in an analytic manner and it does not depend on the angular momentum
of the scalar. This exact result can be useful to obtain the spectrum on a
wormhole with a large value of $\rho_{0}$ by perturbation theory. As discussed
in the Appendix the exact limit $\rho_{0}\rightarrow\infty$ can be taken after
suitable regularizations leading to a geometry that also describes a wormhole.
We left some of the details of this case to the Appendix since only one of the
asymptotic regions in this case is locally AdS. It is important to mention
that since the $g_{tt}$ component of the metric is not a constant, the
wormhole is not ultrastatic. Even more, for $\rho_{0}\neq 0$ the minimum of
the $g_{tt}$ component of the metric does not coincide with the throat.

Recently in reference Kim:2018ang , the authors considered a scalar probe
propagating on the so-called natural wormholes that can be constructed by
smoothly matching spherically symmetric solutions of G.R., coupled to Born-
Infeld. In the effective radial eigenvalue problem, they imposed reflective
boundary condition at one of the asymptotic regions and ingoing boundary
conditions at the throat, leading to complex, quasinormal frequencies. In our
case, we have defined our differential eigenvalue problem by ensuring that the
field is non-divergent at both asymptotic regions, requiring reflective
boundary conditions when $\rho\rightarrow\pm\infty$.

The frequencies of the scalar field are given with respect to the coordinate
time $t$, which in our case coincides with the proper time of a static
observer located at $\rho=\rho_{0}$. Note that such observer is a geodesic
observer. In an asymptotically flat spacetime, as for example in the
Schwarzschild black hole, the frequencies are referred to an observer located
at the asymptotic region, which is independent of the value of the black hole
mass. Nevertheless, when a negative cosmological constant is included, as for
example in the four-dimensional Schwarzschild-AdS black hole, in
Schwarzschild-like coordinates, a time dependence of the form $e^{-i\omega t}$
in a scalar probe implies that the frequencies correspond to those measured by
an observer located at $r=\left(2Ml^{2}\right)^{1/3}$. Note that such an
observer is non-geodesic, in contraposition to our geodesic observed measuring
frequencies in the wormhole. From the holographic view point this coordinate
time is actually the time in the dual CFT.

We have also included a non-minimal coupling between the scalar probe and the
scalar curvature, which allows, for some particular values, to connect two
fully resonant, equispaced spectra. It is worth to mention here also that
fully resonant, equispaced spectra play a central role in the turbulent energy
transfer that leads to non-perturbative AdS instability, see e.g. Turbulent1
-Turbulent6 . This interesting phenomenology has also been observed in other
non-linear models as gravitating scalars on a spherical cavity in 3+1
Maliborski:2012gx , on systems governed by the Gross-Pitaevskii equation
Biasi:2018iiu , on conformal dynamics on the Einstein Universe Bizon:2016uyg
and on vortex precession in Bose-Einstein condensates Biasi:2017pdp .

It would be interesting to explore the sector with negative squared masses.
Since there is an asymptotically locally AdS asymptotic behavior, it is
natural to expect the existence of an effective Breitenlohner-Freedman bound
that may depend on $\rho_{0}$ and that would have to be obtained numerically.
We expect to report on this problem in the future.

## Aknowledgements

The authors want to thank Eloy-Ayon Beato, Francisco Correa, Danilo Diaz, Oleg
Evnin and Nicolas Grandi for enlightening comments. This work was partially
supported by CONICYT Fellowships 21161099 and 22171243, FONDECYT grant number
1181047. O.F. is supported by the Project DINREG 19/2018 of the Dirección de
Investigación of the Universidad Católica de la Santísima Concepción.

##  VI Appendix

Here we discuss some features of the scalar probe on the spacetime obtained
after a suitable regularization in the limit $\rho_{0}\rightarrow+\infty$. It
is useful to return to Schwarzschild-like coordinates, that cover part of the
wormhole spacetime (2). In such coordinates the metric reads DOT2

$ds^{2}=-\left(\frac{r}{l^{2}}+a\sqrt{\frac{r^{2}}{l^{2}}-1}\right)^{2}d\bar{t}^{2}+\left(\frac{r^{2}}{l^{2}}-1\right)^{-1}dr^{2}+r^{2}(d\phi^{2}+d\Sigma_{2}^{2}),$
(26)

where $r=l\cosh(\rho)$, $t=\bar{t}/(l\cosh(\rho_{0}))$ and the integration
constant $\rho_{0}$ relates to $a$ by $\rho_{0}:=-\tanh^{-1}(a)$. Now we
consider that $\rho_{0}\rightarrow\infty$. In this case $a=-1$ and returning
to the proper radial coordinate $\rho$ one obtains

$ds^{2}=l^{2}\left[-e^{-2\rho}dt^{2}+d\rho^{2}+\cosh^{2}(\rho)(d\phi^{2}+d\Sigma_{2}^{2})\right],$
(27)

where $t=\bar{t}/l$. This solution also describes a wormhole geometry with a
traversable throat located at $\rho=0$. Note that this spacetime is
asymptotically locally AdS only when $\rho\rightarrow-\infty$. Hereafter we
set $l=1$.

The radial equation for the non-minimally coupled scalar probe on this
wormhole, for the ansatz $\Phi=e^{-i\omega t}e^{in\phi}R(\rho)$ reduces to

$\frac{d^{2}}{d\rho^{2}}R(\rho)-(1-3\tanh(\rho))\frac{d}{d\rho}R(\rho)+\left(e^{2\rho}\omega^{2}-m^{2}+(14-6\tanh(\rho))\xi-\frac{n^{2}}{\cosh^{2}(\rho)}\right)R(\rho)=0.$
(28)

The asymptotic behavior of the solution is given by

$\displaystyle R(\rho)$ $\displaystyle\overset{\rho\rightarrow-\infty}{\sim}$
$\displaystyle A_{1}\ e^{(2-\sqrt{4+m^{2}-20\xi})\rho}+A_{2}\
e^{(2+\sqrt{4+m^{2}-20\xi})\rho}+\mathcal{O}(e^{2\rho}),$ (29) $\displaystyle
R(\rho)$ $\displaystyle\overset{\rho\rightarrow\infty}{\sim}$ $\displaystyle
B_{1}\ e^{-3\rho/2-i\omega e^{\rho}}+B_{2}\ e^{-3\rho/2+i\omega
e^{\rho}}+\mathcal{O}(e^{-\rho}).$ (30)

As expected the behavior of the scalar when $\rho\rightarrow-\infty$ is a
power law behavior on the areal coordinate $r\sim e^{\rho}$, typical of
scalars on asymptotically AdS regions. While the behavior at the other non-AdS
asymptotic region corresponds to that of an ingoing and outgoing wave. The
equation can be solved analytically in terms on confluent Heun functions Heun
. The equation can be recast in a Schrödinger form by scaling the radial
function and considering the ansatz $\Phi=e^{-i\omega
t}e^{in\phi}u(\rho)/\cosh^{3/2}(\rho)$ and using the new radial coordinate
$\bar{\rho}=e^{\rho}$, with $\bar{\rho}\in(0,\infty)$, leading to

$-\frac{d^{2}}{d\bar{\rho}^{2}}u(\bar{\rho})+Uu(\bar{\rho})=\omega^{2}u(\bar{\rho}),$
(31)

with the effective potential explicitly given in terms of the coordinate
$\bar{\rho}$ by

$U:=\frac{m^{2}}{\bar{\rho}^{2}}-\frac{4(2\bar{\rho}^{2}+5)}{\bar{\rho}^{2}(\bar{\rho}^{2}+1)}\xi+\frac{4n^{2}}{(\bar{\rho}^{2}+1)^{2}}+\frac{3}{4}\frac{\bar{\rho}^{4}+2\bar{\rho}^{2}+5}{(\bar{\rho}^{2}+1)^{2}\bar{\rho}^{2}}.$
(32)

Note that this potential depends explicitly on the angular momentum of the
scalar $n$ while $U^{(\infty)}_{\xi}$ defined in (24), does not. This confirms
what we previously discuss in the paper regarding the fact that the exactly
solvable potential obtained for the leading term of (24), has only to be
thought as a tool to perturbatively obtain the frequencies for large
$\rho_{0}$.

Note that here, at least on one side of the wormhole it is clear how to
recognize ingoing and outgoing modes. It would be interesting in this case to
compute the transmission and reflection coefficients of the wormhole,
mimicking what is done in the computation of greybody factors in
asymptotically AdS black holes GBF .

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# Online and Global Network Optimization
Towards the Next-Generation of Routing Platforms

Jérémie Leguay, Moez Draief, Symeon Chouvardas, Stefano Paris,
Georgios S. Paschos, Lorenzo Maggi, Meiyu Qi
Email<EMAIL_ADDRESS>Mathematical and Algorithmic Science Lab,
France Research Center, Huawei Technologies.

###### Abstract

The computation power of SDN controllers fosters the development of a new
generation of control plane that uses compute-intensive operations to automate
and optimize the network configuration across layers. From now on, cutting-
edge optimization and machine learning algorithms can be used to control
networks in real-time. This formidable opportunity transforms the way routing
systems should be conceived and designed. This paper presents a candidate
architecture for the next generation of routing platforms built on three main
pillars for admission control, re-routing and monitoring that would have not
been possible in legacy control planes.

###### Index Terms:

Software Defined Networking, Routing Systems, Admission Control, Network
Monitoring, Network Optimization, Online Algorithms, Machine Learning.

##  I Introduction

Software-Defined Networking (SDN) technologies [1] are radically transforming
the network architecture of data centers, network overlays, and carrier
networks. They provide programmable data planes that can be configured from a
remote controller platform. SDN controllers are implemented on top of
commodity servers which provide a tremendous computational power compared to
legacy network devices. They also use modern distributed computing
technologies (e.g., loosely consistent databases, parallel computing,
consensus algorithms, transactional updates) to gather and keep a global view
of the network status in real-time and to push consistent configuration
updates to the network equipment.

While initial SDN deployments focused mainly on automated provisioning of
tunnels and virtual networks, essentially virtualizing known concepts, it
becomes clear today that SDN presents an opportunity for a paradigm shift in
network optimization. Departing from the classical autonomic routing systems
we migrate to a new centralized routing paradigm, where a powerful controller
is aware of the static parameters and time-varying conditions, and centrally
orchestrates the entire network reaching unprecedented levels of efficiency.
Indeed, the necessary ingredients are in place: (i) the power of modern SDN
controllers, and (ii) the recent advances in optimization and machine learning
theories can be married to produce the desired outcome. This proof of concept
was recently demonstrated in large scale datacenter interconnects with the B4
controller [2].

This paper presents a candidate architecture for compute-intensive control
planes where new types of algorithms can be deployed for online and global
network optimization. It fully exploits the processing power and the central
position of the controller to globally optimize routing and admission control
using efficient mathematical tools ranging from online algorithms,
combinatorial optimization, control theory and machine learning. It improves
the accuracy of network monitoring at low overhead costs by using data
completion techniques and online learning algorithms to better coordinate the
data collection effort. Finally, it tightens the link between monitoring and
routing which can now co-exist on the same platform.

Our candidate architecture for the next-generation of routing platforms relies
on three fundamental building blocks:

Admission control with experts. The centralized nature of SDN architectures
unleashes the potential of using online algorithms which have been initially
defined for online covering and packing problems with provable guarantees
compared to an offline optimal. Due to the unpredictable nature of network
traffic, it is difficult to choose a priori which online algorithm should be
used. We thus perform admission control with expert advice. This setting is
classical in machine learning, and it is commonly called boosting algorithms.
Past decisions and past rewards obtained by the different algorithms are used
to track and follow the advice of the best algorithm in hindsight.

An iterative routing solver. To maintain the best flow configuration in the
network, we propose to use an iterative routing solver which outputs a
sequence of feasible solutions. As the routing solver may work on multi-
commodity problems with millions of variables and constraints in large
networks, the convergence to the optimal can take a few iterations. To avoid
pushing the solution of each iteration into network equipments, we propose a
control framework that operates on top of the routing solver and decides which
feasible configuration to apply. Given a reconfiguration budget, it aims at
minimizing the flow allocation cost.

Parsimonious monitoring. A severe complication that arises in network
monitoring is the incredible amount of measurements collected. This big data
includes important information for network control, but due to its immense
size is difficult to handle. Although the computational power available at SDN
controllers is an helpful ally, intelligent data handling is also necessary.
In fact, recently developed algorithmic tools from machine learning, graph
theory, and signal processing can be efficiently combined to enable the SDN
controller to have an accurate view of the time-varying network conditions and
their evolution. We show how to combine 1) perform parsimonious and adaptive
network measurements, 2) recover missing information and infer global traffic
properties from the measurements, and 3) predict future variations in network
traffic using spatial and temporal correlations.

The rest of this article is organized as follows. After describing the overall
architecture of the routing platform., we present the core elements for
admission control, re-routing and real-time monitoring. For each of them we
discuss their operational benefits and future directions.

##  II Architecture

We have in mind a classical SDN controller architecture depicted in Fig. 1.
The controller is connected through north bound interface to the application
layer enabling a wealth of applications like bandwidth on demand, load
balancing, etc, while the decided routing commands are dispatched to the
network devices using SDN protocols such as OpenFlow via the south bound
interface. The latter is also used to collect measurements. This work is
primarily concerned with highlighting the architecture of the SDN controller.
In Fig. 1 we depict the three fundamental building blocks of our architecture,
namely 1) the monitoring system, 2) the analysis system, and 3) the routing
system. Below we explain in detail each system operation and functionality.

SDN controllers classically receive service requests from applications and
send routing commands to programmable switches. At the same time, they monitor
the network status to quickly react to system changes. Although it might be
tempting to reuse control plane algorithms from legacy routing software, their
formidable computation and strategic location spur the development of a new
generation of control plane that uses compute-intensive operations to automate
and optimize the network in real-time. This routing platform can embed a set
of complementary optimization and machines learning algorithms which seat
together and tightly cooperate. Fig. 1 presents a candidate architecture with
three main sub-systems for routing, monitoring and analysis.

Figure 1: Global Routing Optimization Platform with three main pillars for
admission control, re-routing and monitoring.

Routing system. When connection requests arrive at ingress nodes or when a
group of demands have to be rerouted after a failure, the controller has to
find a set of feasible paths satisfying multiple constraints (e.g., network
capacity, QoS, protection). In addition, smart decisions must be taken to
accept or reject service request to optimize long term objectives such as
throughput maximization or load balancing. Although this Fast Connection Setup
problem refers to a single commodity flow problems, it is still quite
complicated to solve. Hence at this stage, the goal is not to optimize the
network, but rather to quickly find a good path allocation.

The network configuration obtained as a result of fast connection setups and
of flow departures poses significant concerns on the evolution over time of
the global objective function (e.g., routing cost). Therefore, periodic or
event-based reconfiguration of the overall flow allocation can help reduce the
optimality gap. We call this mechanism Garbage collection of network resources
since it mirrors the way Java reorganizes the memory. For this, an iterative
routing solver works on an evolving instance of the routing problem to provide
a sequence of better configurations. A deployment policy, which works on top
of the iterative solver, decides which configuration should be applied, taking
into account a fixed reconfiguration budget.

Monitoring system. The main objective it to feed the routing system with the
most accurate view of the network in terms of traffic and topology. The
central position of the controller allows to implement completely new
monitoring systems compared to legacy architecture where equipments were
statically configured to report periodic traffic samples and statistics. The
data collection effort can from now on be driven by the controller to avoid
redundant and low value measurements. Traffic sampling can be adapted in real
time to maximize the accuracy of traffic estimation algorithms while keeping
the overhead low or below a given budget. It can be used [3] to better
classify flows in real time (e.g., heavy hitters identification, DDoS
detection) and to guide the routing system for the provisioning of custom flow
configurations.

Analysis system. On top of the monitoring system and at a larger time scale,
SDN controllers can embed traffic and fault prediction algorithms that can
spare the routing system from taking myopic decisions. They can exploit this
estimation and its uncertainty to optimize network resources in a robust
fashion [4].

##  III Admission Control with Experts

A key task of the SDN controller is the Admission Control (AC) of incoming
connection requests, in an online fashion. Its goal is to gracefully manage
service requests when the network becomes highly utilized, as new incoming
requests arrive. Non-myopic accept or reject decisions have to be made to
maximize a given profit, such as the total accepted throughput, the financial
revenue generated or the quality of service experienced by users.

The existing admission control implementations are in their majority
threshold-based. They use max-, min-, exclusive- and non-exclusive-limits on
resource portions that the network operator can define for different classes
of flows. At any time, routers inspect the sum of exclusive and non-exclusive
portions and perform a threshold decision. If the sum is higher than a certain
(engineered) percentage of the network capacity, then new connections are
rejected. The main problem here lies in defining the threshold in a dynamic
fashion, as the optimal configuration depends on the network traffic
conditions, which fluctuate over time. Several proposals on how to optimize
the configuration of thresholds can be found in the literature. Worst-case
strategies have been put forward for the sake of network planning. They are
based on sophisticated stochastic models using queuing theory and require some
a priori information on the stochastic model for the traffic evolution. Such
information is typically extracted offline with the aid of long-term traffic
estimators available at the controller.

Admission Control can be formulated as an online packing problem, as the goal
for instance is to maximize the number of accepted requests subject to some
constraints. The challenge in this context comes from the online nature of the
optimization problem. New variables and additional constraints are revealed
sequentially, as soon as an arrival or departure of a flow occurs in the
system. The theory on online algorithms has evolved significantly in the last
decade for this type of problems. Algorithms with theoretic guarantees over
the offline optimal, which has full information on the future state of the
system, have been proposed for online packing problems.These guarantees hold
only when consistent decisions are taken by all devices within the network.
The centralized nature of SDN lies the ground to apply online algorithms for
admission control [5].

Figure 2: Admission control module with expert advice from Greedy, Agrawal and
AAP-PD (Primal Dual). Performance evaluation results on GEANT network (see [5]
for more details).

###  III-A Competitive online algorithms

Motivated by the considerations above, we first review and adapt, well-known
and recent algorithms from the online literature to the admission control
problem in SDN. We then test their performance under different traffic
conditions to understand and highlight the strengths and weaknesses of each of
them. Traditionally, online algorithms for admission control fall into two
main categories: i) worst-case and ii) average-case.

Worst-case algorithms are characterized by max-min performance guarantees
under specific worst-case scenarios where a malicious adversary chooses the
worst possible sequence of connection requests. Due to their conservative
nature, they generally underperform under more standard traffic conditions.
One of the first online algorithms for admission control has been AAP. The
algorithm is log competitive, meaning that it cannot reject O(log n) times
more requests than the offline optimal (n being the number of nodes.
Buchbinder et al. proposed in [6] a primal dual framework to derive algorithms
for online packing and covering problems with performance guarantees in the
worst-case scenario. Such framework developed the theory behind the initial
intuition of AAP.

On the other hand, ii) average-case algorithms show high expected performance
over random traffic conditions, but cannot guarantee good performance in
specific adversarial scenarios. The Primal-Beats-Dual (PBD) algorithm has been
introduced by Kesselheim et al. in [7] . It computes the optimal (fractional)
solution of the relaxed Linear Program (LP), by considering all the past
requests and scaling the capacity of the graph. It then attempts to route the
request over a path randomly selected, with a probability proportional to the
value of the computed fractional solution. PBD suffers from computation
complexity. Agrawal et al. [8] have proposed a fast algorithm with
multiplicative updates to solve this issue. It applies to i.i.d. and random
order inputs. It implements an efficient stochastic gradient descent that we
adapted for our admission control problem.

###  III-B Using expert advices

As observed in practice and highlighted on Fig. 5 which plots a performance
evaluation on GEANT, worst-case and average-case online algorithms for
admission control are better than Greedy, the naive strategy which accepts
every demand until bottlenecks appears. However, there is no algorithm
outperforming all the remaining ones under all traffic conditions. Luckily,
modern SDN control platforms, which are running on top of commodity servers
and are built upon cutting-edge distributed computing technologies, enable the
parallel execution of different algorithms to solve a single decision problem.
Such algorithmic architecture, called boosting or prediction with expert
advice setting, is commonly used in machine learning. It executes all the
algorithms in parallel and attempts to track the best one in an online
fashion.

The bulk of the literature on experts focuses on proving theoretical
performance bounds in the basic non-reactive scenario where the action taken
by the decision maker does not affect the state of the system, which is
definitely not the case for admission control. We identified that the
Strategic Expert meta-Algorithm (SEA) [9] applies to out reactive setting.
Under some stationary conditions on the system, SEA is proven to perform at
least as well as the oracle which steadily selects the algorithm with best
average performance. As shown on Fig. 5, this approach achieves very good
performance in practice. In the worst case, SEA performs as Follow-the-Leader
meta-Algorithm (FLA) which alway picks the most instantaneous rewarding
decision (i.e., Greedy). However, we believe there is room to improve these
encouraging results by better combining the advices and integrating traffic
predictions.

The advent of Software Defined Networks (SDN) advocates a paradigm shift in
the implementation of admission control (AC) protocols for at least two main
reasons. Firstly, SDN control is centralized, which finally allows to utilize
online AC algorithms developed for online packing and covering problems.
Secondly, we can now afford to run several online admission control algorithms
in parallel and integrate real-time traffic predictions in our decision
making.

##  IV Iterative Routing Solver

Traditionally, routers collaboratively share topological information and
establish end-to-end paths using traffic engineering protocols such as OSPF
and MPLS with TE extensions [10]. However, TE solutions that rely on hot
standby mechanisms to quickly handle topological changes (in case of link
failure) suffer from possibly long recovering times, while other mechanisms at
IP level, more reactive, do not offer a fine-tuned control of paths and may
lead to sub-optimal flow allocations. Hence these approaches fall short of
optimizing network utilization while meeting QoS requirements and guaranteeing
resilience at the same time. Conversely, SDN architectures unleash the
potential to reconfigure in real-time custom paths for each flow, so that
network resources are optimized and performance requirements are met (in QoS,
resilience).

Figure 3: Routing optimization module based on an iterative routing solver and
a policy to control its output (top). Evolution of the optimality gap obtained
using the iterative routing solver (red points) and the control policy (dashed
green curve). The black curve is the interpolation of the points computed by
the solver at each iteration. Solid dots show where the solution computed by
the solver is applied.

To maintain the best network flow configuration, the SDN controller has to
solve variants of the classical Multi-Commodity Flow (MCF) problem in real-
time, which may involve millions of variables and constraints in large
networks. Several and different constraints can be imposed on each commodity
such as node or edge inclusion, protection mechanisms with a maximally
disjoint backups paths, or a maximum end-to-end delay. In addition, some flows
may receive a special treatment such as elephant flows which must be split
over multiple paths while only a few split configurations are possible due
equipment limitations (e.g., hash functions, group tables). As the overall
problem instance itself evolves over time due to time-varying demands or
topological failures, the SDN controller solves a sequence of routing problems
and needs to constantly reconsider the flow configuration. Finally, to satisfy
application requirements, the controller needs to solve these problems under
tight timing constraints. Due to the large problem size, the routing solver
has to iterate to deliver a sequence of feasible configurations. The sequence
of sub-optimal network configurations obtained as a result of the iterative
execution of the solver typically exhibits diminishing returns.

At each time slot the SDN controller has the option to reconfigure the network
flows as per the current computed solution. However, flow reconfigurations
degrade QoS and introduce inertia into the system, hence we may avoid them and
neglect to apply the most recent computed configuration. Ideally, the
objective function of the problem includes a cost that models the distance
between two resource allocations (e.g., the number of edge or flow changes) so
that the iterative solver finds a good trade-off between the allocation and
the reconfiguration costs. On the other hand, using the existing configuration
may quickly become costly since the new flows that are coming, or the ones
that need to be repaired after a network failure, are configured with quick
suboptimal routing decisions.

We propose a control framework that operates on top of the iterative SDN
routing solver to limit the number of network reconfigurations and to reduce
at the same time the flow allocation cost. The control policy applies to
switches the solution of the SDN routing solver only when the benefit is
larger than a threshold, which is dynamically and optimally updated over time.
At any time instant such a dynamic threshold combines the gap between the
current and optimal configurations and the number of reconfigurations applied
so far. Therefore the control policy strives to balance the surcharge and the
reconfiguration rate.

Figure 4: Performance comparison on GEANT in term of routing cost for Shortest
Path (SP), the offline optimal (LP), and the policies which limit network
reconfigurations (top). Number of routing entry changes at each
reconfiguration with the two policies (bottom).

The goal of minimizing the number of flow reconfigurations clearly conflicts
with the goal of minimizing the average optimality gap. It is possible to
apply the best found solution at a frequency less than one, which will incur
less reconfigurations but it will increase the optimality gap. We recently
proposes [11] a drift-plus-penalty policy using a virtual queue as a
fundamental tool, which is essentially a counter on which a dynamic threshold
is applied.

In order to showcase the validity of our approach, we implemented around an
iterative routing solver the control framework with the drift-plus penalty
(DP) control policy. As the goal is to stay below a target reconfiguration
rate while minimizing the routing cost, we compared the performance of our
policy against a Periodic Policy (PP) that selects reconfiguration points
periodically.

Numerical results obtained in real-life network conditions show that SDN
controllers can effectively track the evolution of traffic, thus opening a new
era for the dynamic control of online routing solvers. Fig. 4 presents a
performance comparison between a greedy Shortest Path (SP) where the
allocation is never reconfigured, the offline solution of the Linear Program
(LP), and the online policies with a limited reconfiguration budget. We can
clearly see that DP drastically limits reconfigurations while minimizing the
allocation cost.

This framework can be extended towards a number of interesting directions.
First, the development of scalable multi-commodity flow solvers that could
work on large evolving problems, without the need to converge to the optimal
configuration before the system changes. Second, the integration of traffic
predictions to avoid myopic decisions with regards to the evolution of traffic
so that reconfigurations could be even further limited. Third, other metrics
that model the distance between two configurations in terms of equipment and
flow changes can be designed to be included in the objective function.
Finally, traffic and network uncertainty can be considered [12] using
stochastic and robust optimization techniques.

##  V Parsimonious Monitoring

The inherent centralized nature of SDN controllers requires a non-negligible
amount of signaling between SDN switches, monitoring the traffic over links,
and the SDN controller, exploiting such information to take routing decisions.
When both the network size and the frequency at which new requests arrive are
large, the need to reduce the signaling traffic arises. Hence, parsimonious
collection of statistics, e.g., flow statistics, link loads, just to name a
few, becomes essential. For this reason, we propose to undersample the
measurement collection and to exploit techniques which spring from the
compressed sensing algorithmic family, such as matrix completion techniques,
to estimate the missing information. The rationale behind this approach is
that the observed traffic exhibits inherent time and space dependencies and
that one can take advantage of such dependencies to collect partial
information on the traffic characteristics and yet be able to complete the
missing information.

Figure 5: Parsimonious monitoring process driven by the controller.

We propose a framework to parsimoniously and intelligently collect information
on link loads. We assume that one has only access to the data from a few links
at any given time instant (see bottom of Fig. 5) and that we wish to recover
all the link loads at every instant to inform the routing system. In this
context, there are clear spatial and temporal dependencies in the traffic
patterns that can be exploited. More specifically, these dependencies are
manifested in the low dimensionality of the observed data. The space-time
traffic matrix capturing the link loads evolution over time is inherently low
rank. To complete the missing data, we use the matrix with the lowest rank
consistent with the observed data. In Fig. 6.a, we report the approximation
error of the aforementioned matrix completion algorithm and compare it to
naive approach that completes the missing load for a given link using the most
recent available measurement for that link when we increase the number of
observed links per time-epoch.

(a) Rejected demands

(b) Monitoring Error

Figure 6: Left: percentage of rejected demand vs. measurement portion $\xi$
under Random traffic conditions, for Agrawal’s AC algorithm with matrix
completion (Agrawal MC), with full information (Agrawal Full info), and
without information about the presence of best-effort traffic (Agrawal No
info). Right: Total estimation error over all matrix elements for SVT matrix
completion and the naive strategy, vs. the portion $\xi$ of measured links.

We also use these link-load estimates to run the admission control algorithm
in a scenario where guaranteed services coexist with best effort traffic which
fluctuates over time, see Fig. 6.b. More precisely, online routing algorithms
ideally need a pervasive and seamless knowledge of the residual capacity on
each link. For this reason, we propose a method to parsimoniously monitor the
best-effort traffic on subsets of links and then estimate the missing entries
via matrix completion techniques, e.g., Singular Value Thresholding (SVT). We
show that the introduction of traffic estimation techniques is beneficial both
in terms of reduction of rejected demands. We evaluated the effectiveness of
all the proposed solutions on the GEANT topology.

The above paragraph assumes that we only have access to link loads and
leverage these direct measurements to complete the missing information. In
many realistic SDN settings, the controller can retrieve several counters
about flows. In OpenFlow for instance, a counter is maintained for every
single rules in flow tables. Each entry can correspond to a specific flow by
matching on the traditional 5-tuple or to a given flow aggregate. Based on
this information, the goal is for example to recover the complete origin
destination traffic matrices over time. A simple way of modeling such a
problem is to see it as a linear system where we the unknown vector is the per
flow traffic and what we observe the vector of aggregate flows as the (matrix)
product of the aggregation matrix and the individual flow vector. It is worth
noting that the number of observed aggregate flows is generally much smaller
than the actual number of flows in the system. Indeed, rules are stored in
TCAM of hardware switches. This type of memory can perform high-speed packet
lookups but are extremely expensive and thus limited in space.

In a recent paper, a system called iSTAMP [13], such an under-determined
system is solved by adding a sparsity constraint to the flow vector (analogous
to low-rank constraints for link-load matrices). The rationale behind this
assumption is that a small number of flows explain the observed traffic. The
iStamp proposal goes a step further and extends the setting to adaptive
measurements aiming at monitoring the most informative flows to complete the
information gleaned from aggregate measurements. In particular, given the
opportunity to directly observe a few individual flows, which flows should one
measure to optimally improve the error in solving the above under-determined
linear sysem. To this end, a multi-armed bandit approach is proposed to select
the best (called most rewarding in [13]) such individual flows in terms of
their estimated size. This approach is shown to have a very good performance
in the context of heavy-hitters identification.

Another popular approach to monitor networks is to sample packets [14, 15].
Due to the vast amount of information sampling every packet and tracking each
flow of the network is not feasible. To circumvent this, modern systems
implement efficient mechanisms for picking the relevant packets to decide
which specific flow aggregates to measure. They rely on hash functions
(mapping the data into an array of buckets) to build a compact summary of the
stream of packets. These summaries and associated estimated counters are then
exploited to uncover traffic statistical patterns such heavy hitters or
potential sources of denial of service attacks among other traffic patterns.

One of the aims of this paper is to provide some preliminary ideas, and
hopefully to spur new theoretical research, on the new and interesting field
on parsimonious monitoring for admission control. In particular, we envision
to explore the field of adaptive sampling through which we will identify and
then sample the most informative pieces of information so that to fulfill our
monitoring task. Furthermore, we envision to explore the very interesting
field of monitoring of data streams via sketching and to reveal how such
techniques could be combined with the compressed sensing rationale.

##  VI Perspectives

As seen in this paper, the next-generation of routing systems for online and
global network optimization is likely to embed computer-intensive machine
learning and optimization algorithms. Beyond admission control, re-routing and
monitoring it will also embed algorithms for fault prediction and analysis to
drive real-time decisions in a robust fashion.

The tremendous computation power of SDN controllers does not mean that a
single entity will solve everything in real-time. Routing platforms will still
follow multi-domain and hierarchical organizations which can help to break
down large network optimization problems. In this context, all the algorithms
will have to be extended to interact with their peers or superiors to reach a
global optimization objective.

## References

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$\lambda\operatorname{Tr}_{E/F}\gamma+N_{E/F}\gamma\equiv
0\pmod{\mathfrak{p}_{F}^{2e_{F}+1-m_{1}}},$

and it follows that $v_{F}(N_{E/F}\gamma)\geq a_{n+1}$. Since $E/F$ is totally
ramified, we have

$v_{F}(N_{E/F}\gamma)=v_{E}(\gamma),$

so indeed $v_{E}(\gamma)\geq a_{n+1}$, and by induction this is true for all
$n$.

It is easy to see that if $a_{n}<m_{1}-1$, then $a_{n}<a_{n+1}$, so there is
some $n$ with $a_{n}\geq m_{1}-1$, and therefore $v_{E}(\gamma)\geq m_{1}-1$,
as required. ∎

Lemma A.2 tells us that $\omega_{1}$ is well-defined.

######  Lemma A.3.

We have

    1. (a)

$N_{E/F}\omega_{1}=N_{E/F}\omega$.

    2. (b)

$\omega_{1}\equiv\lambda\pmod{\mathfrak{p}_{E}^{m_{1}}}$.

###### Proof.

If $v_{E}(\omega-\lambda)\geq m_{1}$, then there is nothing to prove, so let
us assume that $v_{E}(\omega-\lambda)=m_{1}-1$. The first claim follows from
that fact that $N_{E/F}\rho=-b$. Recall that we write $\gamma=\omega-\lambda$.
By Lemma A.2 we have $v_{E}(\gamma)=m_{1}-1$, so
$\gamma/\pi_{F}^{m_{1}/2-1}=u+v\rho$ for $u,v\in\mathcal{O}_{F}$ with
$v_{F}(u)\geq 1$ and $v_{F}(v)=0$. We have

$\displaystyle N_{E/F}\omega-\lambda^{2}$
$\displaystyle=\lambda\operatorname{Tr}_{E/F}\gamma+N_{E/F}\gamma$
$\displaystyle=\lambda\pi_{F}^{\frac{m_{1}}{2}-1}(2u+av)+\pi_{F}^{m_{1}-2}(u^{2}+uva-v^{2}b)$
$\displaystyle\equiv\lambda
av\pi_{F}^{\frac{m_{1}}{2}-1}-v^{2}b\pi_{F}^{m_{1}-2}\pmod{\mathfrak{p}_{F}^{m_{1}}}.$

We know that
$N_{E/F}\omega\equiv\lambda^{2}\pmod{\mathfrak{p}_{F}^{2e_{F}+1-m_{1}}}$, and
$m_{1}\leq e_{F}$, so in fact

$N_{E/F}\omega\equiv\lambda^{2}\pmod{\mathfrak{p}_{F}^{m_{1}}},$

and it follows that

$\lambda av\pi_{F}^{-\frac{m_{1}}{2}}-v^{2}b\pi_{F}^{-1}\equiv
0\pmod{\mathfrak{p}_{F}}.$

Since $v_{F}(v)=0$, it follows that

$v\equiv\frac{\lambda
a}{b\pi_{F}^{\frac{m_{1}}{2}-1}}\pmod{\mathfrak{p}_{F}},$

and therefore

$\gamma\equiv\frac{\lambda a\rho}{b}\pmod{\mathfrak{p}_{E}^{m_{1}}},$

so

$\omega\equiv\lambda(1+\frac{a\rho}{b})\pmod{\mathfrak{p}_{E}^{m_{1}}},$

and it follows that

$\displaystyle\omega_{1}$ $\displaystyle=\frac{\omega b}{\rho^{2}}$
$\displaystyle\equiv\lambda(\frac{b}{\rho^{2}}+\frac{a}{\rho})\pmod{\mathfrak{p}_{E}^{m_{1}}}$
$\displaystyle=\lambda\frac{b+a\rho}{\rho^{2}}$ $\displaystyle=\lambda.$

∎

  7. (7)

Write $\omega_{1}=r_{1}+s_{1}\rho$, for $r_{1},s_{1}\in\mathcal{O}_{F}$.

######  Lemma A.4.

We have

    1. (a)

$v_{F}(s_{1})\geq\frac{m_{1}}{2}$.

    2. (b)

$v_{E}(\omega_{1}-\bar{\omega}_{1})=2v_{F}(s_{1})+m_{1}$.

###### Proof.

Since $\omega_{1}\equiv\lambda\pmod{\mathfrak{p}_{E}^{m_{1}}}$, we have

$(r_{1}-\lambda)+s_{1}\rho\equiv 0\pmod{\mathfrak{p}_{E}^{m_{1}}},$

so $v_{F}(s_{1})\geq\frac{m_{1}}{2}$. The second claim is obvious. ∎

It follows that $\omega_{2}$ and $\lambda_{2}$ are well-defined, and their
definitions are equivalent to

$\omega_{2}=\begin{cases}\omega_{1}\quad\text{if
$v_{F}(s_{1})=\frac{m_{1}}{2}$},\\\ \omega_{1}(1+\rho)^{2}\quad\text{if
$v_{F}(s_{1})>\frac{m_{1}}{2}$},\end{cases}$

and

$\lambda_{2}=\begin{cases}\lambda\quad\text{if
$v_{F}(s_{1})=\frac{m_{1}}{2}$},\\\ \lambda(1+a-b)\quad\text{if
$v_{F}(s_{1})>\frac{m_{1}}{2}$}.\end{cases}$

Write $\omega_{2}=r_{2}+s_{2}\rho$.

######  Lemma A.5.

We have

    1. (a)

$N_{E/F}\omega_{2}\equiv\lambda_{2}^{2}\pmod{\mathfrak{p}_{F}^{2e_{F}+1-m_{1}}}$.

    2. (b)

$\omega_{2}\equiv\lambda_{2}\pmod{\mathfrak{p}_{E}^{m_{1}}}$.

    3. (c)

$v_{F}(s_{2})=\frac{m_{1}}{2}$.

###### Proof.

If $v_{F}(s_{1})=\frac{m_{1}}{2}$, then there is nothing to prove, so assume
that $v_{F}(s_{1})>\frac{m_{1}}{2}$. The first claim follows from the fact
that $N_{E/F}(1+\rho)=1+a-b$. the second claim follows from the fact that

$(1+\rho)^{2}\equiv 1+a-b\pmod{\mathfrak{p}_{E}^{m_{1}}}.$

Since $(1+\rho)^{2}=(1+b)+(2+a)\rho$, we have

$\omega_{2}=(1+b)\omega_{1}+(2+a)\rho\omega_{1},$

and therefore

$\bar{\omega}_{2}=(1+b)\bar{\omega}_{1}+(2+a)\bar{\rho}\bar{\omega}_{1},$

so

$\omega_{2}-\bar{\omega}_{2}=(1+b)(\omega_{1}-\bar{\omega}_{1})+(2+a)(\rho\omega_{1}-\bar{\rho}\bar{\omega_{1}}).$

We know that $v_{E}(\omega_{1}-\bar{\omega}_{1})>2m_{1}$ and
$v_{E}(2+a)=m_{1}$, so it suffices to show that

$v_{E}(\rho\omega_{1}-\bar{\rho}\bar{\omega}_{1})=m_{1}.$

Since $\omega_{1}=r_{1}+s_{1}\rho$, we have

$\rho\omega_{1}-\bar{\rho}\bar{\omega}_{1}=(\rho-\bar{\rho})(r_{1}+s_{1}(\rho+\bar{\rho})).$

It is easy to see that $v_{E}(\rho-\bar{\rho})=m_{1}$. Since
$v_{E}(\omega_{1})=0$, we have $v_{E}(r_{1})=0$, so
$v_{E}(r_{1}+s_{1}(\rho+\bar{\rho}))=0$, and therefore

$v_{E}(\rho\omega_{1}-\bar{\rho}\bar{\omega}_{1})=m_{1},$

as required. ∎

  8. (8)

We know that $s_{2}\neq 0$ since $N_{E/F}\omega_{2}\in dF^{\times 2}$, so
$\omega_{2}\not\in F^{\times}$. Similarly, $v_{E}(\omega_{2})=0$ so $r_{2}\neq
0$. It follows that $q$ and $n$ are well-defined.

  9. (9)

Since $\omega_{2}\not\in F^{\times}$, we have $\omega_{2}\neq\lambda_{2}$,
which implies that $q+\rho\neq 0$, and therefore the output is well-defined.
Since $n\in F^{\times}$, we have

$N_{E/F}\Big{(}\frac{\omega_{2}n}{(q+\rho)^{2}}\Big{)}\in dF^{\times 2}.$

######  Lemma A.6.

We have $v_{F}(r_{2}-\lambda_{2})=\frac{m_{1}}{2}$.

###### Proof.

We have

$N_{E/F}\omega_{2}=r_{2}^{2}+r_{2}s_{2}a-s_{2}^{2}b,$

so

$(r_{2}^{2}-\lambda_{2}^{2})+r_{2}s_{2}a-s_{2}^{2}b\equiv
0\pmod{\mathfrak{p}_{F}^{2e_{F}+1-m_{1}}}.$

We know that $v_{F}(s_{2}^{2}b)=m_{1}+1$ and $v_{F}(r_{2}s_{2}a)=m_{1}$, so we
need

$v_{F}(r_{2}^{2}-\lambda_{2}^{2})=m_{1}.$

Since $m_{1}\leq e_{F}$, we have

$v_{F}(r_{2}-\lambda_{2})=v_{F}(r_{2}+\lambda_{2}),$

and the result follows. ∎

We also have

$\displaystyle\omega_{2}n-(q+\rho)^{2}$ $\displaystyle=(b-\rho^{2})+\frac{\rho
bs_{2}}{r_{2}}-q\rho\frac{q+r_{2}}{r_{2}}$
$\displaystyle=(b-\rho^{2})+\frac{\rho
bs_{2}}{r_{2}}+\rho\frac{\lambda_{2}^{2}-r^{2}}{rs}$
$\displaystyle=\frac{\rho}{r_{2}s_{2}}(\lambda_{2}^{2}-r_{2}^{2}+\frac{r_{2}s_{2}}{\rho}(b-\rho^{2})+bs_{2}^{2})$
$\displaystyle=\frac{\rho}{r_{2}s_{2}}(\lambda_{2}^{2}-r_{2}^{2}+bs_{2}^{2}-ars_{2})$
$\displaystyle=\frac{\rho}{r_{2}s_{2}}(\lambda_{2}^{2}-N_{E/F}\omega_{2}).$

Since $v_{E}(r_{2})=0$ and $v_{E}(s_{2})=m_{1}$, we have
$v_{E}(\frac{\rho}{r_{2}s_{2}})=1-m_{1}$. Since

$v_{E}(\lambda_{2}^{2}-N_{E/F}\omega_{2})\geq 4e_{F}+2-2m_{1},$

it follows that

$\omega_{2}n\equiv(q+\rho)^{2}\pmod{\mathfrak{p}_{E}^{4e_{F}+3-3m_{1}}}.$

Lemma A.6 tells us that $v_{F}(q)=0$, and therefore $v_{E}(q+\rho)=0$, so it
follows that

$\frac{\omega_{2}n}{(q+\rho)^{2}}\equiv
1\pmod{\mathfrak{p}_{E}^{4e_{F}+3-3m_{1}}}.$

Theorem 4.33 then follows by [Mon24, Lemma 4.6].

##  Appendix B Explicit helper functions

  * •

Given a $p$-adic field $F$ with residue characteristic $p$ and residue degree
$q$, we define functions $A,B:\mathbb{R}\to\mathbb{R}$ by

$A(t)=\begin{cases}q^{1-\lfloor\frac{t}{2}\rfloor}\cdot\frac{q^{\lfloor\frac{t}{2}\rfloor-1}-1}{q-1}\quad\text{if
$p=2$},\\\
q^{-p(p-2)}\cdot\frac{q^{(p-1)(p-2)}-1}{q^{p-2}-1}\cdot\frac{q^{-(p-1)^{2}(\lfloor\frac{t}{p}\rfloor)}-1}{q^{-(p-1)^{2}}-1}\quad\text{if
$p\neq 2$},\end{cases}$

and

$B(t)=\begin{cases}0\quad\text{if $p=2$},\\\
q^{-(\lfloor\frac{t}{p}\rfloor-1)}\cdot\frac{q^{-(p-2)(t+1)}-q^{-(p-2)((\lfloor\frac{t}{p}\rfloor-1)p+2)}}{q^{-(p-2)}-1}\quad\text{if
$p\neq 2$}.\end{cases}$

  * •

Define the explicit function $N_{(1^{2}1^{2})}^{\neq}$ by

$N_{(1^{2}1^{2})}^{\neq}(m)=\begin{cases}2(q-1)^{2}q^{\frac{m}{2}-2}(e_{F}-1)-\mathbbm{1}_{4\mid
m}(q-1)q^{\frac{m}{4}-1}\quad\text{if $2\leq m\leq 4e_{F}$ and $m$ is
even},\\\ 4(q-1)q^{\frac{m-1}{2}-1}\quad\text{if $2e_{F}+3\leq m\leq 4e_{F}+1$
and $m$ is odd},\\\ q^{e_{F}}(2q^{e_{F}}-1)\quad\text{if $m=4e_{F}+2$},\\\
0\quad\text{otherwise}.\end{cases}$

  * •

$N^{C_{2}}(m_{1})=\begin{cases}2q^{\frac{m_{1}}{2}-1}(q-1)\quad\text{if $0\leq
m_{1}\leq 2e_{F}$ and $m_{1}$ is even},\\\ 2q^{e_{F}}\quad\text{if
$m_{1}=2e_{F}+1$},\\\ 0\quad\text{otherwise}.\end{cases}$

  * •

Let $m_{1}$ be an even integer with $2\leq m_{1}\leq e_{F}$. For each integer
$m_{2}$, define

$N^{C_{4}}(m_{1},m_{2})=\begin{cases}q^{m_{1}-1}\quad\text{if
$m_{2}=3m_{1}-2$},\\\
q^{\lfloor\frac{m_{1}+m_{2}}{4}\rfloor}-q^{\lfloor\frac{m_{1}+m_{2}-2}{4}\rfloor}\quad\text{if
$3m_{1}\leq m_{2}\leq 4e_{F}-m_{1}$ and $m_{2}$ is even},\\\
q^{e_{F}}\quad\text{if $m_{2}=4e_{F}-m_{1}+2$},\\\
0\quad\text{otherwise}.\end{cases}$

Suppose that $m_{1}=2e_{F}+1$ or $m_{1}$ is even with $e_{F}<m_{1}\leq
2e_{F}$. Then define

$N^{C_{4}}(m_{1},m_{2})=\begin{cases}2q^{e_{F}}\quad\text{if
$m_{2}=m_{1}+2e_{F}$},\\\ 0\quad\text{otherwise}.\end{cases}$

  * •

Let $m_{1}$ be either $2e_{F}+1$ or an even integer with $2\leq m_{1}\leq
2e_{F}$. Define

$N^{V_{4}}(m_{1},m_{2})=\begin{cases}2q^{\frac{m_{2}}{2}-1}(q-1)\quad\text{if
$m_{2}<m_{1}$ and $m_{2}$ is even},\\\ q^{\frac{m_{2}}{2}-1}(q-2)\quad\text{if
$m_{2}=m_{1}$ and $m_{1}$ is even},\\\
q^{\frac{m_{1}+m_{2}}{4}-1}(q-1)\quad\text{if $m_{2}>m_{1}$ and $m_{1}\equiv
m_{2}\pmod{4}$},\\\ q^{e_{F}}\quad\text{if $m_{2}>m_{1}$ and
$m_{1}+m_{2}=4e_{F}+2$},\\\ 0\quad\text{otherwise}.\end{cases}$

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