# Monodromy of Complexified Planar Kepler Problem

Shanzhong Sun Partially supported by NSFC (No. 11771303), Email:
<EMAIL_ADDRESS>Department of Mathematics, Capital Normal University, Beijing
100048 P. R. China  Peng You Partially supported by NSFC (No. 11771303),
Email<EMAIL_ADDRESS>Department of Mathematics, Capital Normal University,
Beijing 100048 P. R. China

###### Abstract

The planar Kepler problem is complexified and we show that this holomorphic
completely integrable Hamiltonian system has nontrivial monodromy.

##  1 Introduction

The Kepler problem has been studied from a variety of different perspectives
and many beautiful structures continue to be unraveled even recently. However
there are still mysteries to be clarified. We try to approach this subject by
complexifying the problem.

If a material point moves in plane $\mathbb{R}^{2}$ with coordinates $(x,y)$,
it is convenient to identify $\mathbb{R}^{2}$ with the Gauss complex plane
$\mathbb{C}$ and write it in terms of complex variable $z:=x+iy\in\mathbb{C}$.
It is an old and nice idea to use complex variables in the planar $N$-body
problem especially in the planar Kepler problem. In fact, Cauchy developed his
theory of functions of one complex variable in order to study the Kepler
equation. Here by complexified mechanical problem, we mean the coordinates of
the mass point are $(x,y)\in\mathbb{C}^{2}$ instead of $\mathbb{R}^{2}$.

Recently, Behtash et al ([5]) argued that in the semi-classical analysis of
path integrals of supersymmetric quantum field theory and quantum mechanics,
both the complexification of the action and the measure of the theory are
necessary to get right nonperturbative structure of the model. It is obvious
and promising from this viewpoint that we need to complexify the Newton
equations in classical mechanics to get holomorphic Newton equations in
complexified configuration spaces. They claim that complexified solutions
including both multi-valued and even singular ones to holomorphic Newton
equations are responsible for consistency of the semi-classical theory,
interesting physical properties like hidden topological angles and remarkable
features to the resurgent cancellation of ambiguities to get exact solutions.
There is also a huge literature on the complexification of path integration
for various physical motivations (see [5] and references therein).

In fact this is an idea dating back at least to Balian-Bloch ([2]) as a
computational tool to get the finer structures of the spectral problems of the
Schrödinger operators in quantum mechanics, and please refer to Pham ([11])
for the further developments and mathematical justifications.

Back to $N$-body problem in celestial mechanics, Albouy and Kaloshin ([1])
proved the generic finiteness of central configurations for planar $N$-body
problem by complexifying each real coordinate and using complex algebraic
geometry.

Inspired by these progresses in quantum physics and mathematics, it seems
natural to develop complexified classical mechanics and to investigate the
holomophic Newton equations. The relation to real Newton equations and their
physical implications would be interesting problems. Other than the harmonic
oscillator, the first example coming to our minds is the complexified Kepler
problem and this is one of the main motivations of the current work.

Once we got this idea, we check it in the mathematical literature.
Unfortunately we only find the following references: Beukers-Cushman([7]),
Bates-Cushman([3]) and Bender-Hook-Kooner([6]) with related references
therein. In the latter the authors proposed through extensive numerical
studies that complex classical systems can exhibit tunneling-like behavior as
in quantum mechanics.

As shown by concrete examples appearing in quantum molecular spectroscopy,
monodromies of singularities in completely integrable Hamiltonian systems have
quantum effects (please refer to Cushman et al([10]) as an interesting
example, and see also [8] to get a flavor of the subject).

We prove that the complexified Kepler problem is a completely integrable
holomorphic Hamiltonian system and the period lattice of the regular fibers of
the energy-momentum map and the monodromy group are determined which is a new
phenomenon compared with the traditional real Kepler problem.

The paper is organized as follows. We recall the classical planar Kepler
problem with some facts used later in §2. The problem is complexified in §3,
and it becomes a holomorphic Hamiltonian system. The Hamiltonian action of
$SO(2,\mathbb{C})$ and the geometrical properties of the energy-momentum
mapping are established in §3. In §4, we derive the period lattice on each
smooth fiber of the energy-momentum mapping by rewriting the holomorphic
symplectic form to get holomorphic $1$-forms $\omega_{1},\,\omega_{2}$ and
considering the generalized Abel-Jacobi map. The main result about the
monodromy of the complexified planar Kepler problem is obtained in §5, we show
that the problem has nontrivial monodromy which is different from the
classical Kepler problem. Finally, in §6 some possible directions to study
further are discussed.

######  Acknowledgements.

We would like to thank Yong Li, Volodya Roubtsov and David Sauzin for their
valuable suggestions.

##  2 Classical planar Kepler problem

We recall the classical planar Kepler Hamiltonian system first. On the phase
space
$T_{0}\mathbb{R}^{2}=(\mathbb{R}^{2}\setminus\\{0\\})\times\mathbb{R}^{2}$
with coordinates $(x_{1},x_{2},y_{1},y_{2})$ and symplectic form
$\omega=dx_{1}\wedge dy_{1}+dx_{2}\wedge dy_{2}$, consider the Kepler
Hamiltonian

$H:T_{0}\mathbb{R}^{2}\rightarrow\mathbb{R};(x_{1},x_{2},y_{1},y_{2})\mapsto\frac{1}{2}(y_{1}^{2}+y_{2}^{2})-\frac{1}{\sqrt{x_{1}^{2}+x_{2}^{2}}}.$

The integral curves of the Hamiltonian vector field $X_{H}$ on the
$T_{0}\mathbb{R}^{2}$ satisfy the equations

$\displaystyle\dot{x}$ $\displaystyle=y$ $\displaystyle\dot{y}$
$\displaystyle=-x\|x\|^{-3},$

where $x=(x_{1},x_{2}),y=(y_{1},y_{2}),\|x\|=\sqrt{x_{1}^{2}+x_{2}^{2}}$.

The Hamiltonian vector field $X_{H}$ has well known integrals: the
Hamiltonian, i.e., the total energy

$H(x_{1},x_{2},y_{1},y_{2})=\frac{1}{2}(y_{1}^{2}+y_{2}^{2})-\frac{1}{\sqrt{x_{1}^{2}+x_{2}^{2}}}$

and the angular momentum

$J(x_{1},x_{2},y_{1},y_{2})=x_{1}y_{2}-x_{2}y_{1}.$

A Hamiltonian system $(M,\omega,f_{1})$ on a smooth symplectic manifold
$(M,\omega)$ of dimension $2n$ is a Liouville integrable system with
$n$-degrees of freedom if there are Poisson commuting functions
$(f_{1},...,f_{n})$, that is, $\\{f_{i},f_{j}\\}=0$, whose differentials are
linearly independent on an open dense subset $W$ of $M$ and whose associated
Hamiltonian vector fields $X_{f_{i}}$ are complete. By definition,
$(T_{0}\mathbb{R}^{2},\omega,H,J)$ is a Liouville integrable system.

##  3 Complexified planar Kepler problem

In this section, we describe the complex Hamiltonian system given by the
complexified planar Kepler problem and its energy-momentum map.

Let’s start from the complexified planar Kepler problem. As a first step, we
are required to calculate the square roots of $x_{1}^{2}+x_{2}^{2}$ in the
Kepler Hamiltonian $H$. As in complex analysis we construct the Riemann
surface of $\sqrt{z}$ as $\sqrt{z}$’s maximal domain to remove the
multivaluedness, we ”double the $(x_{1},x_{2})$ configuration space” to make
the function $\sqrt{x_{1}^{2}+x_{2}^{2}}$ single valued and holomorphic.

We take two copies of
$\mathbb{C}^{2}\setminus\\{(x_{1},x_{2})\in\mathbb{C}^{2}:x_{1}^{2}+x_{2}^{2}=0\\}$,
one is denoted by $Q_{I}$ and the other by $Q_{II}$. For convenience purpose
we introduce the new coordinates $\xi_{1}=x_{1}+ix_{2}$,
$\xi_{2}=x_{1}-ix_{2}\in\mathbb{C}$. Cut $Q_{I}$ and $Q_{II}$ along
$x_{1}^{2}+x_{2}^{2}=\xi_{1}\xi_{2}=t\in\mathbb{R}_{<0}$ as follows: for each
fixed $\xi_{2}\neq 0$, we get two copies of $\mathbb{C}\setminus\\{0\\}$ in
$Q_{I}$ and $Q_{II}$ respectively denoted simply by $I$ and $II$, cut both $I$
and $II$ along the half line $\xi_{1}=t\xi_{2}^{-1}$, $t<0$ to get $I_{\pm}$
and $II_{\pm}$. Glue $I_{+}$ with $II_{-}$ and $I_{-}$ with $II_{+}$
respectively as shown in Figure 1.

Figure 1: Glue $I$ and $II$ together for each fixed $\xi_{2}\neq 0$.

Then let $\xi_{2}$ vary, we obtain the glued space $Q=Q_{I}\coprod Q_{II}$. In
the following we will use subscript $k$ in $(x_{1},x_{2})_{k}$ to indicate
which $Q_{k}$, $k=I,II$ the point $(x_{1},x_{2})$ is in.

The configuration space $Q$ is a complex manifold of dimension $2$. The phase
space $M=T^{*}Q$ is a trivial bundle. Furthermore $M$ is a holomorphic
symplectic manifold with the canonical symplectic form $\omega$ which in
natural coordinate $(x_{1},x_{2},y_{1},y_{2})$ can be written as

$\omega=dx_{1}\wedge dy_{1}+dx_{2}\wedge dy_{2}.$

Now the Hamiltonian
$H=\frac{1}{2}(y_{1}^{2}+y_{2}^{2})-\frac{1}{\sqrt{x_{1}^{2}+x_{2}^{2}}}$
makes sense on $M$ and is holomorphic. The complexified planar Kepler problem
becomes a holomorphic Hamiltonian system $(M,\omega,H)$.

The group

$SO(2,\mathbb{C})=\\{A=\begin{pmatrix}a&-b\\\ b&a\end{pmatrix}\in
GL(2,\mathbb{C}):a^{2}+b^{2}=1\\}$

which preserves $H$, acts on $M$ by

$\tilde{\Phi}:SO(2,\mathbb{C})\times M\rightarrow
M;(A,((x_{1},x_{2})_{k},(y_{1},y_{2})))\mapsto((A\begin{pmatrix}x_{1}\\\
x_{2}\end{pmatrix})_{k},A\begin{pmatrix}y_{1}\\\ y_{2}\end{pmatrix}),k=I,II.$

As in the usual planar Kepler problem, the action $\tilde{\Phi}$ is a
Hamiltonian action of $SO(2,\mathbb{C})$ on the holomorphic symplectic
manifold $(M,\omega)$ with moment map the (complexified) angular momentum

$J(x_{1},x_{2},y_{1},y_{2})=x_{1}y_{2}-x_{2}y_{1}.$

Similar to the real case, we can prove that $(M,\omega,H,J)$ is a Liouville
completely integrable system in the holomorphic sense.

It is convenient to use the coordinates $(\xi_{1},\xi_{2},\eta_{1},\eta_{2})$
on $M$ where

$\xi_{1}=x_{1}+ix_{2},\ \xi_{2}=x_{1}-ix_{2},\ \eta_{1}=y_{1}+iy_{2},\
\eta_{2}=y_{1}-iy_{2}.$

Note that $\xi_{1}\neq 0$ and $\xi_{2}\neq 0$ because
$\xi_{1}\xi_{2}=x_{1}^{2}+x_{2}^{2}\neq 0$. In terms of these new coordinates
the $SO(2,\mathbb{C})$-action $\tilde{\Phi}$ becomes the
$\mathbb{C}^{*}$-action

$\Phi:\mathbb{C}^{*}\times M\rightarrow
M:(\lambda,((\xi_{1},\xi_{2})_{k},(\eta_{1},\eta_{2}))\mapsto((\lambda\xi_{1},\lambda^{-1}\xi_{2})_{k},(\lambda\eta_{1},\lambda^{-1}\eta_{2})),\
k=I,II,$

Where $\lambda=a+ib$, $A=\begin{pmatrix}a&-b\\\ b&a\end{pmatrix}\in
SO(2,\mathbb{C})$. The action of a group element $\lambda$ on a point $P$ is
usually written as $\Phi_{\lambda}(P)$.

We have

######  Proposition 3.1.

1\. The $\mathbb{C}^{*}$-action $\Phi$ on $M$ is free and proper.

2\. The map

$\pi:M\rightarrow\mathbb{C}^{4};((\xi_{1},\xi_{2})_{k},(\eta_{1},\eta_{2}))\mapsto(c_{1},c_{2},c_{3},c_{4}),k=I,II.$

is the quotient map $M\rightarrow M/\mathbb{C}^{*}$ of the
$\mathbb{C}^{*}$-action $\Phi$, where

$c_{1}=\begin{cases}\sqrt{\xi_{1}\xi_{2}}&(\xi_{1},\xi_{2})_{I}\in Q_{I}\\\
-\sqrt{\xi_{1}\xi_{2}}&(\xi_{1},\xi_{2})_{II}\in Q_{II}\end{cases},\
c_{2}=\xi_{1}\eta_{2},\ c_{3}=\frac{i}{2}(\xi_{1}\eta_{2}-\xi_{2}\eta_{1}),\
c_{4}=\frac{1}{2}\eta_{1}\eta_{2}-\frac{1}{c_{1}}.$

3\. The image of the map $\pi$ is the subset $R$ of $\mathbb{C}^{4}$ defined
by

$\\{(c_{1},c_{2},c_{3},c_{4})\in\mathbb{C}^{4}:2c_{4}c_{1}^{2}+2c_{1}=c_{2}^{2}+2ic_{3}c_{2},c_{1}\neq
0\\}.$

Note that $c_{4}$ and $c_{3}$ are just the Hamiltonian and the angular
momentum in terms of these new coordinates.

4\. The map $\pi:M\rightarrow R$ is the projection map of a
$\mathbb{C}^{*}$-bundle over $R$.

###### Proof.

1\. If
$((\lambda\xi_{1},\lambda^{-1}\xi_{2})_{k},(\lambda\eta_{1},\lambda^{-1}\eta_{2}))=((\xi_{1},\xi_{2})_{k},(\eta_{1},\eta_{2}))$,
then $\lambda=1$ for $\xi_{1}\neq 0$. Hence $\mathbb{C}^{*}$ acts freely on
$M$. The action $\Phi$ is proper if and only if
$L=\\{\lambda\in\mathbb{C}^{*}:K\cap\Phi_{\lambda}(K)\neq\emptyset\\}$ is
compact for every compact set $K$ in $M$. Since $K$ is compact, there exist
two positive numbers $\rho_{1}<\rho_{2}$ such that $\rho_{1}<|\xi_{1}|,\
|\xi_{2}|<\rho_{2}$ on $K$. For every $\lambda\in L$, there is a point
$((\xi_{1},\xi_{2})_{k},(\eta_{1},\eta_{2}))$ such that
$((\lambda\xi_{1},\lambda^{-1}\xi_{2})_{k},(\lambda\eta_{1},\lambda^{-1}\eta_{2}))$
is also in $K$. Because $|\xi_{1}|,\ |\xi_{2}|<\rho_{2}$ and
$\rho_{1}<|\lambda\xi_{1}|,\ |\lambda^{-1}\xi_{2}|$, we get
$\frac{\rho_{1}}{\rho_{2}}<|\lambda|<\frac{\rho_{2}}{\rho_{1}}$. If
$\lambda_{n}(\in L)\rightarrow\lambda$, then there exist $P_{n}\in K$ such
that $\Phi_{\lambda_{n}}(P_{n})\in K$. Since $K$ is compact, we have a
subsequence of $\\{P_{n}\\}_{n\geq 1}$ converging to a point $P\in K$, we
still use $\\{P_{n}\\}_{n\geq 1}$ to denote this subsequence. Because of the
continuity of $\Phi$ and compactness of $K$, $\Phi_{\lambda}(P)\in K$. Then
$L$ is compact.

2\. From the definition of the map $\pi$, the following equalities hold for
points of $M$

$\displaystyle\xi_{1}\xi_{2}$ $\displaystyle=c_{1}^{2};$
$\displaystyle\xi_{1}\eta_{2}$ $\displaystyle=c_{2};$
$\displaystyle\xi_{2}\eta_{1}$ $\displaystyle=c_{2}+2ic_{3};$
$\displaystyle\eta_{1}\eta_{2}$ $\displaystyle=2c_{4}+\frac{2}{c_{1}}.$

Let $(c_{1},c_{2},c_{3},c_{4})$ be in the image of the map $\pi$. If two
points $((\xi_{1},\xi_{2})_{k},(\eta_{1},\eta_{2}))$ and
$((\tilde{\xi}_{1},\tilde{\xi}_{2})_{\tilde{k}},(\tilde{\eta}_{1},\tilde{\eta}_{2}))$
are mapped to the same $(c_{1},c_{2},c_{3},c_{4})$, we have $k=\tilde{k}$,
$\xi_{1}\xi_{2}=\tilde{\xi}_{1}\tilde{\xi}_{2}$,
$\xi_{1}\eta_{2}=\tilde{\xi}_{1}\tilde{\eta}_{2}$,
$\xi_{2}\eta_{1}=\tilde{\xi}_{2}\tilde{\eta}_{1}$. Then there exists a unique
$\lambda\in\mathbb{C}^{*}$ such that $\tilde{\xi}_{1}=\lambda\xi_{1}$,
$\tilde{\xi}_{2}=\lambda^{-1}\xi_{2}$, $\tilde{\eta}_{1}=\lambda\eta_{1}$,
$\tilde{\eta}_{2}=\lambda^{-1}\eta_{2}$. Noting that $c_{i}(i=1,2,3,4)$ is
invariant under the $\mathbb{C}^{*}$-action $\Phi$, we see that the inverse
image of $(c_{1},c_{2},c_{3},c_{4})$ under the map $\pi$ is a single
$\mathbb{C}^{*}$-orbit.

3\. Observe that the identity
$(\xi_{1}\xi_{2})(\eta_{1}\eta_{2})=(\xi_{1}\eta_{2})(\xi_{2}\eta_{1})$ gives
rise to

$2c_{4}c_{1}^{2}+2c_{1}=c_{2}^{2}+2ic_{3}c_{2},\ c_{1}\neq 0.$ (1)

For $(c_{1},c_{2},c_{3},c_{4})$ satisfying the condition (1), the point

$((1,c_{1}^{2})_{k},(\frac{c_{2}+2ic_{3}}{c_{1}^{2}},c_{2})),\
k=\begin{cases}I&-\frac{\pi}{2}<\arg c_{1}\leq\frac{\pi}{2}\\\
II&\frac{\pi}{2}<\arg c_{1}\leq\frac{3\pi}{2}\end{cases}$

is a preimage of $(c_{1},c_{2},c_{3},c_{4})$ under the quotient map $\pi$.

4\. It is a general fact that if $\mathbb{C}^{*}$ acts freely and properly on
a complex manifold $N$, then the quotient map $N\rightarrow N/\mathbb{C}^{*}$
is the projection map of a $\mathbb{C}^{*}$-bundle over $N/\mathbb{C}^{*}$. In
particular, this holds when $N=M$. ∎

For any fixed value $c=(c_{3},c_{4})$, we consider the algebraic curve

$2c_{4}c_{1}^{2}+2c_{1}=c_{2}^{2}+2ic_{3}c_{2}.$

It is easy to check that the curve is smooth if and only if $c$ does not lie
in the discriminant $\Delta$:

$\Delta:=\\{(c_{3},c_{4})\in\mathbb{C}^{2}:1+2c_{4}c_{3}^{2}=0\\}.$

We slightly change the form of the algebraic curve as

$2c_{4}c_{1}^{2}+2c_{1}-c_{3}^{2}=(c_{2}+ic_{3})^{2},$ (2)

which falls into one of the three cases:

1\. If $c_{4}=0$, the curve is in the form

$2c_{1}-c_{3}^{2}=(c_{2}+ic_{3})^{2},$

which is homeomorphic to $\mathbb{C}$.

2\. If $c_{4}\neq 0$ and the discriminant of the left hand side of (2)
$4+8c_{4}c_{3}^{2}\neq 0$, the curve is in the form

$2c_{4}(c_{1}-r)(c_{1}-s)=(c_{2}+ic_{3})^{2},r\neq s,$

which is homeomorphic to $\mathbb{C}^{*}$.

3\. If $c_{4}\neq 0$ and the discriminant of the left hand side of (2)
$4+8c_{4}c_{3}^{2}=0$, the curve is in the form

$2c_{4}(c_{1}+\frac{1}{2c_{4}})^{2}=(c_{2}+ic_{3})^{2},$

which is homeomorphic to a conical surface.

In the first two cases, the curve is smooth. In the third case, the curve is
singular.

For any fixed value $c=(c_{3},c_{4})$, we define $R_{c}$ as the algebraic
curve with points such that $c_{1}=0$ removed

$\\{(c_{1},c_{2})\in\mathbb{C}^{2}:2c_{4}c_{1}^{2}+2c_{1}=c_{2}^{2}+2ic_{3}c_{2}\\}\setminus\\{(0,0),(0,-2ic_{3})\\}$

and $M_{c}$ as the $c$-level set of the energy-momentum map

$EM:M\rightarrow\mathbb{C}^{2};((\xi_{1},\xi_{2})_{k},(\eta_{1},\eta_{2}))\mapsto(c_{3},c_{4}),\
k=I,II,$

where $c_{3}$, $c_{4}$ are defined in Proposition 3.1.

##  4 Period lattice

In this section we obtain a period lattice for the smooth fibers of the
energy-momentum map of the complexified Kepler problem.

First, we choose good coordinates on $M$ and express the symplectic form
$\omega$ in these coordinates. Let $V$ be the open subset of $M$ defined by

$\\{((\xi_{1},\xi_{2})_{k},(\eta_{1},\eta_{2}))\in M,k=I,II:c_{2}+ic_{3}\neq
0\\}.$

The variables $(\xi_{1},c_{1},c_{3},c_{4})$ can be used as local coordinates
at every point $p$ of $V$, because the condition $c_{2}+ic_{3}\neq 0$ implies
the Jacobian of the energy-momentum map at $p$ is nonsingular. We now write
the symplectic form $\omega$ in these coordinates.

######  Proposition 4.1.

The symplectic form $\omega$ on $M$ restricted to $V$ is

$\left((\frac{i}{c_{1}}+\frac{-c_{3}}{c_{1}(c_{2}+ic_{3})})dc_{1}+\frac{-i}{\xi_{1}}d\xi_{1}\right)\wedge
dc_{3}+\frac{c_{1}}{c_{2}+ic_{3}}dc_{1}\wedge dc_{4}.$

###### Proof.

The canonical symplectic form $\omega$ on $M$ can be written as

$\frac{1}{2}(d\xi_{2}\wedge d\eta_{1}+d\xi_{1}\wedge d\eta_{2}).$

Since

$\xi_{2}=\frac{c_{1}^{2}}{\xi_{1}},\
\eta_{1}=\frac{c_{2}+2ic_{3}}{\xi_{2}}=\frac{(c_{2}+2ic_{3})\xi_{1}}{c_{1}^{2}},\
\eta_{2}=\frac{c_{2}}{\xi_{1}}$

on $V$, we obtain

$d\xi_{2}=\frac{2c_{1}}{\xi_{1}}dc_{1}+\frac{-c_{1}^{2}}{\xi_{1}^{2}}d\xi_{1},$
$d\eta_{1}=\frac{\xi_{1}}{c_{1}^{2}}dc_{2}+\frac{2i\xi_{1}}{c_{1}^{2}}dc_{3}+\frac{c_{2}+2ic_{3}}{c_{1}^{2}}d\xi_{1}+\frac{-2(c_{2}+2ic_{3})\xi_{1}}{c_{1}^{3}}dc_{1}$

and

$d\eta_{2}=\frac{1}{\xi_{1}}dc_{2}+\frac{-c_{2}}{\xi_{1}^{2}}d\xi_{1}.$

Hence

$\omega|_{V}=\frac{1}{c_{1}}dc_{1}\wedge dc_{2}+\frac{2i}{c_{1}}dc_{1}\wedge
dc_{3}+\frac{-i}{\xi_{1}}d\xi_{1}\wedge dc_{3}.$

Differentiating $c_{2}^{2}+2ic_{3}c_{2}=2c_{4}c_{1}^{2}+2c_{1}$ gives

$dc_{2}=\frac{2c_{4}c_{1}+1}{c_{2}+ic_{3}}dc_{1}+\frac{-ic_{2}}{c_{2}+ic_{3}}dc_{3}+\frac{c_{1}^{2}}{c_{2}+ic_{3}}dc_{4},$

which substituted into $\omega|_{V}$ yields

$\omega|_{V}=\left((\frac{i}{c_{1}}+\frac{-c_{3}}{c_{1}(c_{2}+ic_{3})})dc_{1}+\frac{-i}{\xi_{1}}d\xi_{1}\right)\wedge
dc_{3}+\frac{c_{1}}{c_{2}+ic_{3}}dc_{1}\wedge dc_{4}.$

∎

Let

$\omega_{1}=(\frac{i}{c_{1}}+\frac{-c_{3}}{c_{1}(c_{2}+ic_{3})})dc_{1}+\frac{-i}{\xi_{1}}d\xi_{1},\
\omega_{2}=\frac{c_{1}}{c_{2}+ic_{3}}dc_{1}.$

The differential forms $\omega_{1}$, $\omega_{2}$ restricted to the subset of
$M_{c}$ where $c_{2}+ic_{3}\neq 0$ can be extended holomorphically to $M_{c}$.
In fact, $3$-forms $\omega_{1}\wedge dc_{3}\wedge dc_{4}=\omega\wedge dc_{4}$
and $\omega_{2}\wedge dc_{4}\wedge dc_{3}=\omega\wedge dc_{3}$ are holomorphic
on $M$. We also use $\omega_{1}$ and $\omega_{2}$ to denote the extended
$1$-forms.

We construct the period lattice on $M_{c}$ as follows. Assuming
$c\notin\Delta$. Fix a point $p_{0}\in M_{c}$ and consider the $Abel$-$Jacobi$
map

$AJ:M_{c}\rightarrow\mathbb{C}^{2}:p\mapsto(\int_{\gamma}\omega_{1},\int_{\gamma}\omega_{2}),$

where $\gamma$ is a path from $p_{0}$ to $p$ on $M_{c}$. Since $\omega_{1}$
and $\omega_{2}$ are closed forms, the map $AJ$ depends only on the homology
class of the path on $M_{c}$ from $p_{0}$ to $p$. The map $AJ$ is multivalued
since $p_{0}$ and $p$ can be joined by nonhomologous paths, say $\gamma$ and
$\tilde{\gamma}$. The difference in the corresponding values of $AJ$ is given
by a vector $(\int_{\Gamma}\omega_{1},\int_{\Gamma}\omega_{2})$, where
$\Gamma$ is the closed path tracing out $\gamma$ first and then tracing out
$\tilde{\gamma}$ backwards. Thus the multivaluedness of $AJ$ is given by
vectors $(\int_{\Gamma}\omega_{1},\int_{\Gamma}\omega_{2})$ where $\Gamma$
ranges over $H_{1}(M_{c},\mathbb{Z})$. Let

$L_{c}=\\{(\int_{\Gamma}\omega_{1},\int_{\Gamma}\omega_{2})\in\mathbb{C}^{2}:\Gamma\in
H_{1}(M_{c},\mathbb{Z})\\}.$

When $c_{3}\neq 0$, the first homology groups of $M_{c}$ and $R_{c}$ are
$H_{1}(M_{c},\mathbb{Z})=\mathbb{Z}^{4}$ and
$H_{1}(R_{c},\mathbb{Z})=\mathbb{Z}^{3}$ respectively.

######  Theorem 4.2.

Let $\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$ be a $\mathbb{Z}$-basis of
$H_{1}(R_{c},\mathbb{Z})$ where $\gamma_{1}$ is the generator of the first
homology group of the smooth algebraic curve
$2c_{4}c_{1}^{2}+2c_{1}=c_{2}^{2}+2ic_{3}c_{2}$. Define

$\mu=\int_{\gamma_{1}}\left(\frac{i}{c_{1}}+\frac{-c_{3}}{c_{1}(c_{2}+ic_{3})}\right)dc_{1},\
\nu=\int_{\gamma_{1}}\left(\frac{c_{1}}{c_{2}+ic_{3}}\right)dc_{1}.$

Then $L_{c}$ is a lattice in $\mathbb{C}^{2}$ called the period lattice.
$L_{c}$ has $\mathbb{Z}$-rank two and is generated by the vectors

$(\mu,\nu),\ (2\pi,0).$

###### Proof.

The point $(c_{1},c_{2})=(0,-2ic_{3})$ is the removable singularity of the
differential
$\left(\frac{i}{c_{1}}+\frac{-c_{3}}{c_{1}(c_{2}+ic_{3})}\right)dc_{1}$ on
$R_{c}$, then

$\int_{\gamma_{3}}\left(\frac{i}{c_{1}}+\frac{-c_{3}}{c_{1}(c_{2}+ic_{3})}\right)dc_{1}=0.$

The point $(c_{1},c_{2})=(0,0)$ is the pole of the differential
$\left(\frac{i}{c_{1}}+\frac{-c_{3}}{c_{1}(c_{2}+ic_{3})}\right)dc_{1}$ on
$R_{c}$ which is first order. If $c_{3}\neq 0$, the residue at the pole
$(0,0)$ is $2i$, then

$\int_{\gamma_{2}}\left(\frac{i}{c_{1}}+\frac{-c_{3}}{c_{1}(c_{2}+ic_{3})}\right)dc_{1}=-4\pi.$

If $c_{3}=0$, the residue at the pole $(0,0)$ is $i$, then

$\int_{\gamma_{2}}\left(\frac{i}{c_{1}}+\frac{-c_{3}}{c_{1}(c_{2}+ic_{3})}\right)dc_{1}=-2\pi.$

The map

$M_{c}\rightarrow
R_{c}\times\mathbb{C}^{*}:((\xi_{1},\xi_{2})_{k},(\eta_{1},\eta_{2}))\mapsto(c_{1},c_{2},\xi_{1})$

is a diffeomorphism. Let $\Gamma$ be any closed path in $M_{c}$. Without loss
of generality we can assume that $\Gamma$ lies in the $M_{c}\cap V$. The image
in $R_{c}$ of the curve $\Gamma$ under the projection $\pi$ is homologous to
$n_{1}\gamma_{1}+n_{2}\gamma_{2}+n_{3}\gamma_{3}$. Hence

$\int_{\Gamma}\omega_{2}=\int_{\pi(\Gamma)}\frac{c_{1}}{c_{2}+ic_{3}}dc_{1}=n_{1}\int_{\gamma_{1}}\frac{c_{1}}{c_{2}+ic_{3}}dc_{1}+n_{2}\int_{\gamma_{2}}\frac{c_{1}}{c_{2}+ic_{3}}dc_{1}+n_{3}\int_{\gamma_{3}}\frac{c_{1}}{c_{2}+ic_{3}}dc_{1}=n_{1}\nu,$

and

$\displaystyle\int_{\Gamma}\omega_{1}=$
$\displaystyle\int_{\Gamma}\frac{-i}{\xi_{1}}d\xi_{1}+\int_{\pi(\Gamma)}\left(\frac{i}{c_{1}}+\frac{-c_{3}}{c_{1}(c_{2}+ic_{3})}\right)dc_{1}$
$\displaystyle=$
$\displaystyle\int_{\Gamma}\frac{-i}{\xi_{1}}d\xi_{1}+n_{1}\int_{\gamma_{1}}\left(\frac{i}{c_{1}}+\frac{-c_{3}}{c_{1}(c_{2}+ic_{3})}\right)dc_{1}+n_{2}\int_{\gamma_{2}}\left(\frac{i}{c_{1}}+\frac{-c_{3}}{c_{1}(c_{2}+ic_{3})}\right)dc_{1}$
$\displaystyle+n_{3}\int_{\gamma_{3}}\left(\frac{i}{c_{1}}+\frac{-c_{3}}{c_{1}(c_{2}+ic_{3})}\right)dc_{1}$
$\displaystyle=$ $\displaystyle n_{1}\mu+2m\pi,$

where $m$ is some integer. The last equality holds because we can let $\Gamma$
encircle $\xi_{1}=0$ as many times as we want. This ends the proof. ∎

##  5 Monodromy

In this section we show that the complexified planar Kepler problem has
monodromy. We will prove that the monodromies are due to chasing the
nontrivial loops around the discriminant on the one hand and the $c_{3}$ axis
on the other where the fiber $M_{c}$ changes its topology.

###  5.1 Monodromy around the discriminant $\Delta$

More precisely, we will show that there is a noncontractible loop $\Gamma$ in
$\mathbb{C}^{2}\setminus\Delta$ such that the bundle of period lattices

$\phi:\coprod_{c\in\Gamma}L_{c}\rightarrow\Gamma$

over $\Gamma$ has classifying map given by $\begin{pmatrix}-1&0\\\
-2&1\end{pmatrix}$.

######  Theorem 5.1.

Consider the periods of the differential forms $\omega_{1}$ and $\omega_{2}$
on $R_{c}$ as analytic functions of $c=(c_{3},c_{4})$. Then there is a closed
path $\Gamma$ in the $c$-space $\mathbb{C}^{2}\setminus\Delta$ such that after
analytic continuation along one circuit of $\Gamma$, the period of
$\alpha=\left(\frac{i}{c_{1}}+\frac{-c_{3}}{c_{1}(c_{2}+ic_{3})}\right)dc_{1}$
turns into $-4\pi-\int_{\gamma}\alpha$ and the period of $\omega_{2}$ changes
the sign.

###### Proof.

The idea of the proof is that we normalize the curve $R_{c}$ into
$2u(u-1)=v^{2}$, then we find the change of the periods of $\alpha$ and
$\omega_{2}$ along a well chosen closed path $\Gamma$ in
$\mathbb{C}^{2}\setminus\Delta$.

If $(c_{3},c_{4})\notin\mathbb{C}^{2}\setminus\Delta$ and $c_{4}\neq 0$, the
curve has the form

$2c_{4}(c_{1}-r)(c_{1}-s)=(c_{2}+ic_{3})^{2},$

where $r=\frac{-1-\sqrt{1+2c_{3}^{2}c_{4}}}{2c_{4}}$,
$s=\frac{-1+\sqrt{1+2c_{3}^{2}c_{4}}}{2c_{4}}$. Taking the substitution

$c_{1}=(s-r)u+r,\ c_{2}=(s-r)\sqrt{c_{4}}v-ic_{3}$

transforms the equation $2c_{4}(c_{1}-r)(c_{1}-s)=(c_{2}+ic_{3})^{2}$ into the
equation $2u(u-1)=v^{2}$. Under the same transformation, the $1$-forms
$\alpha$ and $\omega_{2}$ are changed into

$\alpha=\left(\frac{i}{u-\frac{-r}{s-r}}+\frac{-c_{3}}{\sqrt{c_{4}}(s-r)}\frac{1}{v(u-\frac{-r}{s-r})}\right)du,\
\omega_{2}=\left(\frac{s-r}{\sqrt{c_{4}}}\frac{u-\frac{-r}{s-r}}{v}\right)du$

and the pole $(0,0)$ of $\alpha$ will be transformed to the point
$(\frac{-r}{s-r},\frac{ic_{3}}{(s-r)\sqrt{c_{4}}})$.

Take a point $(1,\frac{-1}{2})\in\Delta$ and a circle $\Gamma$ with the center
$(1,\frac{-1}{2})$ and the radius $r_{1}<\frac{1}{2}$ in the complex plane
$\\{(c_{3},c_{4})\in\mathbb{C}^{2}\,|\,c_{3}=1\\}$. Then on closed path
$\Gamma$ the period of $\alpha$ and $\omega_{2}$ will be

$\int_{\gamma}\alpha=\int_{\gamma}\left(\frac{i}{u-(\frac{1}{2}+\frac{1}{2\sqrt{1+2c_{4}}})}+\frac{-\sqrt{c_{4}}}{\sqrt{1+2c_{4}}}\frac{1}{\sqrt{2u(u-1)}}\frac{1}{u-(\frac{1}{2}+\frac{1}{2\sqrt{1+2c_{4}}})}\right)du$

and

$\int_{\gamma}\omega_{2}=\int_{\gamma}\left(\frac{\sqrt{1+2c_{4}}}{c_{4}\sqrt{c_{4}}}\frac{u-(\frac{1}{2}+\frac{1}{2\sqrt{1+2c_{4}}})}{\sqrt{2u(u-1)}}\right)du.$

As shown in Figure 2, the pole $\frac{1}{2}+\frac{1}{2\sqrt{1+2c_{4}}}$ of
$\alpha$ will go along a semicircle on the circle with the radius
$\frac{1}{\sqrt{r_{1}}}$ and the center $\frac{1}{2}$ in the $u$-plane, when
we let $c_{4}$ go around $\Gamma$ once. By choosing $\gamma$ as a circle with
the center $\frac{1}{2}$ and the radius larger than $\frac{1}{\sqrt{r_{1}}}$,
we get

$\int_{\gamma}\alpha=-2\pi+\frac{-\sqrt{c_{4}}}{\sqrt{1+2c_{4}}}\int_{\gamma}\left(\frac{1}{\sqrt{2u(u-1)}}\frac{1}{u-(\frac{1}{2}+\frac{1}{2\sqrt{1+2c_{4}}})}\right)du.$

When $c_{4}$ goes through $\Gamma$ once, the integral
$\int_{\gamma}\left(\frac{1}{\sqrt{2u(u-1)}}\frac{1}{u-(\frac{1}{2}+\frac{1}{2\sqrt{1+2c_{4}}})}\right)du$
does not change but $\frac{-\sqrt{c_{4}}}{\sqrt{1+2c_{4}}}$ changes sign, then
$\int_{\gamma}\alpha$ turns into $-4\pi-\int_{\gamma}\alpha$. For the same
reason, $\int_{\gamma}\omega_{2}$ changes the sign after analytic
continuation.

Figure 2: Variation of periods in $u$-plane: encompassing the discriminant.
The red curve is the loci of the poles when traveling along $\Gamma$, and the
outer circle is the common cycle $\gamma$ to calculate the family of periods.

∎

###  5.2 Monodromy around $c_{3}$

In this subsection we will show that there is a noncontractible loop $\Gamma$
in $\mathbb{C}^{2}\setminus\\{(c_{3},c_{4}=0):c_{3}\in\mathbb{C}\\}$ such that
the bundle of period lattices

$\phi:\coprod_{c\in\Gamma}L_{c}\rightarrow\Gamma$

over $\Gamma$ has classifying map given by $\begin{pmatrix}-1&0\\\
-2&1\end{pmatrix}$.

######  Theorem 5.2.

Consider the periods of the differential forms $\omega_{1}$ and $\omega_{2}$
on $R_{c}$ as analytic functions of $c=(c_{3},c_{4})$. Then there is a loop
$\Gamma$ in the $c$-space
$\mathbb{C}^{2}\setminus\\{(c_{3},0):c_{3}\in\mathbb{C}\\}$ such that after
analytic continuation along one circuit of $\Gamma$, the period of
$\alpha=\left(\frac{i}{c_{1}}+\frac{-c_{3}}{c_{1}(c_{2}+ic_{3})}\right)dc_{1}$
turns into $-4\pi-\int_{\gamma}\alpha$ and the period of $\omega_{2}$ changes
the sign.

###### Proof.

The proof is similar to that of the Theorem 5.1, and we only sketch the key
points and the main differences.

The same transformation as constructed in the proof of the Theorem 5.1 works
also here. Take the point $(1,0)$ in $\\{(c_{3},0):c_{3}\in\mathbb{C}\\}$ and
a circle $\Gamma$ with the center $(1,0)$ and the radius smaller than
$\frac{1}{2}$ in the complex plane $c_{3}=1$. On $\Gamma$, the period of
$\alpha$ and $\omega_{2}$ will be

$\int_{\gamma}\alpha=\int_{\gamma}\left(\frac{i}{u-(\frac{1}{2}+\frac{1}{2\sqrt{1+2c_{4}}})}+\frac{-\sqrt{c_{4}}}{\sqrt{1+2c_{4}}}\frac{1}{\sqrt{2u(u-1)}}\frac{1}{u-(\frac{1}{2}+\frac{1}{2\sqrt{1+2c_{4}}})}\right)du$

and

$\int_{\gamma}\omega_{2}=\int_{\gamma}\left(\frac{\sqrt{1+2c_{4}}}{c_{4}\sqrt{c_{4}}}\frac{u-(\frac{1}{2}+\frac{1}{2\sqrt{1+2c_{4}}})}{\sqrt{2u(u-1)}}\right)du.$

As shown in Figure 3, the pole $\frac{1}{2}+\frac{1}{2\sqrt{1+2c_{4}}}$ of
$\alpha$ goes along a closed loop which encircles $1$, when we let $c_{4}$ go
around $\Gamma$ once. By choosing $\gamma$ as a circle with sufficiently large
radius, we get

$\int_{\gamma}\alpha=-2\pi+\frac{-\sqrt{c_{4}}}{\sqrt{1+2c_{4}}}\int_{\gamma}\left(\frac{1}{\sqrt{2u(u-1)}}\frac{1}{u-(\frac{1}{2}+\frac{1}{2\sqrt{1+2c_{4}}})}\right)du.$

When $c_{4}$ goes through $\Gamma$ once, the integral
$\int_{\gamma}\left(\frac{1}{\sqrt{2u(u-1)}}\frac{1}{u-(\frac{1}{2}+\frac{1}{2\sqrt{1+2c_{4}}})}\right)du$
does not change but $\frac{-\sqrt{c_{4}}}{\sqrt{1+2c_{4}}}$ changes sign, then
$\int_{\gamma}\alpha$ turns into $-4\pi-\int_{\gamma}\alpha$. For the same
reason, $\int_{\gamma}\omega_{2}$ changes the sign after analytic
continuation.

Figure 3: Variation of periods in $u$-plane: encompassing the $c_{3}$-axis.

∎

##  6 Conclusions and outlooks

As we have seen, the monodromies are due to the existence of the discriminant
loci where the level curves $M_{c}$ have a singularity and the degeneracy of
the level curve along the $c_{3}$ coordinate axis in $c=(c_{3},c_{4})$ space.
In each case, we construct a loop turning around the singular loci and show
that there is nontrivial monodromy. We observed that the monodromy are the
same for the both cases, and in fact we can construct a loop encompassing both
singular loci with trivial monodromy (i.e., the monodromy effects cancelled
due to algebraic additivity of the periods). We tend to believe that the
fundamental group of the singular loci

$\pi_{1}(\mathbb{C}^{2}\backslash(\\{(c_{3},c_{4})\in\mathbb{C}^{2}\,|\,1+2c_{4}c_{3}^{2}=0\\}\cup\\{(c_{3},c_{4}=0)\in\mathbb{C}^{2}\,|\,c_{3}\in\mathbb{C}\\}))\cong\mathbb{Z}^{2},$

although we could not establish it at present. If this is the case, then the
monodromy group would be

$\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}.$

The next step would be to understand the dynamical meaning of the nontrivial
monodromy. To name a few, is there any quantum mechanics implications as
suggested by quantum molecular spetroscopy? Is there any physical explanation
about the unipotent property of the monodromy matrix?

The monodromy of the complexified Kepler problem shows the potential to the
relation with the quantum behavior. Maybe we can benefit more from the
complexification point of view, for example the unification of the positive
and negative orbits and the regularization problems which we left for the
future research.

An even more interesting question would be about complexified $3$-dimensional
Kepler problem. As is well known the classical spatial Kepler problem has one
more constant of motions the eccentricity or Laplace-Runge-Lenz vector due to
the hidden $SO(4)$ symmetry which has far reaching implications about the
spectrum of the hydrogen atoms.

Notice that the phase space of the complexified planar Kepler problem as well
as any complexified classical mechanical system has beautiful geometry and is
a holomorphic symplectic manifold a.k.a. hyperKähler manifold ([4]). The
interaction between the dynamics of the holomorphic Newton equations and the
geometry of the underlying phase space would be a fascinating direction to
pursue.

## References

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  * [3] L. M. Bates and R. H. Cushman, Scattering monodromy and the $A_{1}$ singularity, Central European Journal of Mathematics, 5(3), (2007), 429-451.
  * [4] A. Beauville, Holomorphic symplectic geometry: a problem list. Complex and differential geometry, Springer Proc. Math., 8, Springer, Heidelberg, 2011(49-63).
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  * [8] M. S. Child, Quamtum monodromy and molecule spectroscopy, Contemporary Physics, 55:3 (2014)212-221.
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# An Intelligent Security centered Resource-Efficient Resource Management
Model for Cloud Computing Environments

Deepika Saxena
Department of Computer Applications
National Institute of Technology
Kurukshetra, India
<EMAIL_ADDRESS>
& Ashutosh Kumar Singh
Department of Computer Applications
National Institute of Technology
Kurukshetra, India
<EMAIL_ADDRESS>The authors would like to thank National Institute of
Technology Kurukshetra, India for financially supporting this research work.

###### Abstract

This paper proposes a conceptual model for a secure and performance efficient
workload management model in cloud environments. In this model, a resource
management unit is employed for energy and performance proficient allocation
of virtual machines while ensuring the secure processing of users’
applications by defending against the data breaches due to unauthorized access
of virtual machines in real-time. The resource management unit is guided by a
secure virtual machine management unit which is designed to generate
information regarding unauthorized access or inter-communication links among
active virtual machines. Also, a workload analyser unit operates concurrently
to estimate resource utilization information to assist the resource management
unit in performance efficient allocation of virtual machines. Contrary to
prior works which engages access control mechanisms, and encryption and
decryption of data before transfer, and use of tunnelling for prevention of an
unauthorised access to virtual machines which raises excess computational cost
overhead, the proposed model operates diversely for efficiently serving the
same purpose.

_Keywords_ Workload prediction $\cdot$ Resource allocation $\cdot$ Security
$\cdot$ VM consolidation $\cdot$ VM migration

##  1 Introduction

Cloud Computing enabled with seamless pool of hardware and software resources
with advanced computation and numerous benefits viz., service availability,
reliability, elasticity, and maximum scalability at minimum upfront investment
has driven business enterprise, scientific and research organisations towards
the cloud datacentres [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. The users
requests or applications are executed on different virtual machines hosted on
different severs within the datacentre [13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30]. The cloud virtualization technology
facilitates elasticity for satisfying dynamically varying resource demands of
users by allocating and deallocating required number of virtual machines on
the shared physical servers [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41].
However, the sharing of physical resources and processing of confidential
applications on the third-party cloud servers prompts a biggest concern of
‘security’ among all the cloud users [42, 43, 44, 45, 46, 47, 48, 49, 50, 51,
52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 9].

The virtual machines dedicated for the processing of applications of a user
needs to communicate for data exchange among them [65, 66, 43, 67, 68, 69, 70,
71, 72, 73, 74, 75, 76]. Most of the data hacks and security attacks occur by
reason of data transfer leveraging the security loopholes or vulnerabilities
of compute, storage, and network devices within datacentre [77, 78, 79, 80,
81, 82, 83, 84]. Also, the co-location of virtual machines of multiple users
on a common physical server often leads to malicious activities like, data
hampering and leakage of cloud user’s sensitive information [85, 86, 87, 88,
89, 90, 91, 92, 93, 94, 7, 95]. A malicious user or attacker may initiate one
or group of virtual machines and exploit multiple network routes to attain co-
residency with the target virtual machine [96, 97, 98, 99, 100, 22, 101, 102,
103, 104, 105]. The vulnerability and susceptibilities of hypervisor or
virtual machine management layer allow malicious user to easily compromise all
the virtual machines hosted on it. Therefore, the key challenge for cloud
datacentre is securing user data distributed on the shared physical servers
while maintaining scalability of resources along with high performance and
cost optimization [106, 107, 108, 109, 110, 111, 112, 113, 114].

The existing approaches furnished for securing real-time data communications
among virtual machines and other different entities includes encryption and
decryption of data before transfer, use of virtual private networks,
tunnelling, and use of passwords, captchas for prevention of an unauthorised
access to virtual machines of benign user etc. Also, minimization of the
physical server sharing and periodic migration of virtual machines are some
practices employed for reduction of security attacks and safeguarding online
data execution and communication. Despite the adoption of aforementioned
security upgrading measures, the misconfiguration and mismanagement at cloud
platform is the topmost cause for leakage of terabytes of sensitive data of
millions of cloud users across the world. Hence, a robust security mechanism
during user data processing and communication at virtual machine placement and
physical resource management level is needed to combat the aforementioned
bottleneck problem of security threats in cloud environments.

##  2 Proposed Model

This section discusses an overview of the resource distribution and allocation
of virtual machines for securing data execution and unauthorized access within
the cloud datacenter in a simplified format that are further explained in the
detailed description of the paper. The proposed conceptual model addresses the
challenges of security breaches, virtual machine hijack and data leakage,
network security attacks, unauthorized data access along with performance
degradation due to server under/over-utilization by proposing a novel virtual
machine security ingrained cloud resource allocation model. The architectural
view of proposed model is illustrated in Fig. 1.

Figure 1: Proposed model architectural overview

A resource management unit is employed to receive application execution
requests from numerous users across the globe. These applications maybe
received in any possible format such as high-performance computing
applications, scientific workflows, military, medicinal or research-based
applications, online business and marketing etc. The resource management unit
divides these applications into one or more components or tasks to be executed
concurrently on different virtual machines. All the virtual machines engaged
in processing of the component of a common application needs to communicate
and exchange information with each other builds a virtual network of virtual
machines with authorized access links. The resource management unit store and
maintains this information of authorized virtual machine links in an
Authorized Virtual machine Access Database (AVAD).

The resource management unit assigns different tasks to distinct virtual
machines deployed on various servers within the datacenter. During execution
of tasks by different virtual machine, the resources viz., CPU, memory, disk,
bandwidth utilization are recorded and Workload/Resource Usage Data Store is
created and updated periodically. The information saved in this data store is
provided as input for training and re-training of a machine learning based
workload analyser. This workload analyser is employed to estimate the future
resource utilization on different virtual machines and servers. The resource
management unit utilizes estimated resource usage information in making
various proactive decisions confined to load management such as creation,
termination, or migration of different virtual machines, handling of
over/under-load conditions of servers etc., for an efficient management of
resources. The proposed model employed a Virtual Machine (VM) Security
Management Unit to ensure secure execution of user application on group of
virtual machines hosted on same or different servers. This unit allows
monitoring of inter-communication links among active virtual machines in real-
time and enable detection of attacker virtual machines for mitigation of
security breaches within the datacenter. During task execution, the resource
usage is collected to update workload data store and inter-communication links
are monitored and stored in Current VM Access Links (CVAL) log. All the live
inter-communication links among virtual machines are maintained in a log which
helps to determine unauthorized access links and terminate attacker/malicious
virtual machines before the occurrence of any security breaches. The proposed
model is capable avoiding data security attacks because of new/unknown as well
as known attacker virtual machine.

The present approach illustrates a general cloud computing environment
accompanying role of management of resources within cloud environment wherein,
cloud infrastructure comprises of clusters of servers connected via router
networking device for communicating with one another as per the requirement.
Each cluster comprises of numerous physical server machines of same or varying
resource capacity. The resources may include Central Processing Unit (CPU),
RAM, Disk memory, and bandwidth etc. The virtual machines which maybe in the
form of computing, storage or networking instances are hosted on servers for
processing of cloud user application requests. The cloud users {$User$ 1,
$User$ 2, …, $User$ n} may send application requests to a Resource Management
Unit for processing at the cloud infrastructure. The Resource Management Unit
receives requests in the form of applications from various cloud users {$User$
1, $User$ 2, …, $User$ n}. Moreover, it facilitates the virtual machine
management including virtual machine scheduling, placement, migration, and
handling over/under-loading conditions by reason of inefficient resource
distribution among virtual machines. Further, it serves tasks management by
performing the function of user application division into sub-components or
tasks, assignment and scheduling of these tasks’ execution on different
virtual machines. The output processed tasks belonging to an application are
integrated to generate the response for the user retrieved via Resource
Management Unit.

##  3 Secure Resource Management

Fig. 2 portrays a complete view of a secure and performance efficient load
management model in cloud environment in accordance with the embodiment of the
proposed model. The cloud users leverages services of cloud infrastructure by
sending requests in the form of different applications to be received by the
resource management unit. These applications are divided into tasks in
accordance with the tasks management, which are assigned for execution to
different virtual machines subject to the resource requirement of a task is
lesser than or equals to the resource capacity of virtual machine in
accordance with the virtual machine management. The virtual machines utilize
physical resource capacity of the servers upon which they are hosted.

Figure 2: Security focused resource management unit

The resource utilization or received workload information by different virtual
machines on different servers are recorded in a Workload/Resource Usage Data
Store. This information is used for periodic training and re-training of a
machine learning algorithm based Workload Analyser Unit which is further
described in detail in accordance with an embodiment of the proposed model as
depicted in Fig. 3. The Workload Analyser Unit estimates future resource
requirement on each server for successful execution of user requests by the
respective virtual machines avoiding any performance degradation due to sudden
upsurge or plunge in the forthcoming workload. This predicted resource usage
information is passed to the resource management unit to provide it with the
knowledge of the expected variance in the forthcoming workload and adjust the
entire load with proficiency by handing all the under-utilization and over-
utilization of resources proactively. The resource management unit creates
virtual networks of a group of active virtual machines by considering the
trivial fact that all the virtual machines involved in execution of a common
application are in communication, exchanging data with each other, are legally
authorized to access data of one another. The resource management unit 208
also builds and maintains Authorized virtual machine (VM) Access Database
(AVAD) by utilizing the link access information of aforesaid virtual networks
of virtual machines. A virtual machine (VM) Security Management Unit in
accordance with the embodiment of the proposed model, which is capable of
providing an effective security during execution of user application within
the cloud infrastructure. It comprises of a monitor which observes live
communication links among active virtual machine continuously in real-time.
This monitor analyses virtual machine link access patterns wherein, the group
of virtual machines involved in exchanging data among them generates live
virtual networks of active virtual machines. The knowledge produced by these
access links is utilized to create Online VM Access Log which is further
utilized by the secure VM allocation unit which makes decisions of shifting,
terminating, creating, re-allocating virtual machines in the favour of
security of user data under processing.

The virtual networks for live communication among virtual machines are
monitored and recorded in a log named Current VM Access Links Log (CVAL). The
information captured in this log about inter-communication among active
virtual machines is analysed and compared with Authorised VM Access Database
(AVAD) which is generated by resource management unit. All the mis-matched
link are identified for detection and termination of unauthorized access links
and attacker virtual machines prior to occurrence of successful security
breaches. The present approach utilizes the knowledge produced by comparative
analysis between AVAD and CVAL to furnish active virtual machines security
decisions in real-time.

##  4 Resource Management

A block diagram as illustrated in Fig. 3 is provided to elucidate an
implementation of high performance and energy-proficient resource management
in accordance with resource management unit presented in Fig. 2 of the
proposed model. The historical and live resource utilization information from
the cloud infrastructure is stored in a resource usage data store. This is
information is pre-processed by applying feature selection and normalization
techniques which maybe performed using any appropriate machine learning
method; such as Decision Tree, Boruta Forest, ChiSquare, or Random Forest etc.
for feature selection and Linear scaling, Clipping, Max-Min algorithm, Z-score
etc. for normalization of input data within a specific range.

Figure 3: Machine learning driven resource estimation unit

The normalized resource usage data is provided to Machine Learning based
Resource Usage Analyser for training and re-training periodically in real-
time. This analyser maybe developed using any of the prediction methods such
as Neural Network perceptron trained with help of an optimization algorithm,
Deep Neural Network, Long Short Term Memory based Recurrent Neural Network
(LSTM-RNN) etc. The periodic training of the prediction module rejuvenates the
prediction model for improved and precise prediction of the respective
resource utilization within cloud infrastructure. This prediction information
is generated in the form of patterns of resource usage within a virtual
machine and thus produces a pool of predicted resource usage of VMs. This
estimated resource usage information is passed to resource management unit
which corresponds to resource management unit for directing resource
provisioning decisions. In case, the predicted capacity of resource usage is
more than the physically available resource capacity of a server in the
cluster, an overloading is expected. The highest resource capacity virtual
machines are migrated from the respective server for alleviation of such an
overloading problem that may degrade the performance of entire cloud
infrastructure. The highest resource capacity virtual machines are selected
for migration in order to avoid the necessity of frequent migration. Also, in
case of much lesser predicted capacity of resource usage than the actual
capacity of server, an underloading problem is estimated wherein, such an
underloaded server maybe turned off after shifting the virtual machines hosted
on them to some other active server subject to resource capacity constraints.

The resource usage information from workload data store maybe use for periodic
training/re-training of machine learning based resource usage analyser for
estimation of future workload and resource utilization. The information of
inter-access link among active virtual machines in CVAL are compared with that
of AVAD. If the outcome of the comparison block is ‘NO’, any security breach
is detected proactively which requires enough time for surpassing the security
defence and actual occurrence of the successful attack and terminate the
unauthorised links among respective VMs before initiation of data
transmission, otherwise, no post-action is needed. Accordingly, the knowledge
of predicted resource usage and unauthorized inter-communication links are
passed to resource management unit that may allocate/deallocate or migrate
virtual machines based on server resource capacity and security status of
virtual machines.

##  5 Illustrations

The effectiveness of the present approach against a co-residency attack is
provided in accordance with an engagement of the proposed model. Let us assume
that User 1 application’s component tasks are executing on multiple virtual
machines, among which one of the virtual machines is co-located with an
attacker’s virtual machine who may launch a side-channel or co-residency
attack on all the virtual machines of User 1 by reason of networking links
among them. Such an attack requires sufficient time for breaching the security
boundaries of target virtual machine to launch a successful attack. The
proposed model continuously monitors, records and analyses the inter-
communication links among active virtual machines which successfully points
out the unauthorized link as soon as it is developed. As depicted in proposed
approach, the attacker virtual machine is detected and terminated providing
successful defence against the security breach prior to its occurrence.

The working strategy of the present approach against a multiple virtual
machines hijack attack in accordance with an embodiment of the proposed model
can be illustrated as follows. Suppose an attacker launched multiple virtual
machines for maximum data breaches simultaneously by exploiting co-residency
effect. Since, the proposed model continuously monitors, records and analyses
the inter-communication links among active virtual machines which successfully
points out the multiple unauthorized links as soon as they are developed. The
VM Security Analysis unit detects and terminates all attacker virtual machines
involved in data breaches via unauthorized access links before occurrence of
successful attack.

Furthermore, the virtual machine security mechanism of the present approach
against a grouped virtual machines attack targeting a single virtual machine
in accordance with the proposed model can be realised with the subsequent
example. Consider an attacker launched multiple virtual machines to reach the
target virtual machine via unauthorized link using network cascading effect.
Since, the proposed model continuously monitors, records and analyses the
inter-communication links among active virtual machines which successfully
points out the multiple unauthorized links as soon as they are developed. The
VM Security Analysis unit detects and terminates all attacker virtual machines
involved in such an attack via unauthorized access links prior to its success.

##  6 Conclusions

The proposed conceptual model for security and performance efficiency of cloud
computing environment is a novel approach accomplishing two conflicting
objectives concurrently. This paper presented an architectural and operational
design of the proposed model with essential illustrations to validate the
effectiveness of the proposed approach. In future, the authors envisions to
implement this model and compare it with existing state-of-the-art approaches.

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