A Penrose limit for type IIB $AdS_{6}$ solutions

Michael Gutperle and Nicholas Klein

Mani L. Bhaumik Institute for Theoretical Physics
Department of Physics and Astronomy
University of California, Los Angeles, CA 90095, USA

###### Abstract

In this paper a Penrose limit is constructed for type IIB $AdS_{6}\times
S^{2}\times\Sigma$ supergravity solutions. These solutions are dual to five
dimensional SCFTs related to (p,q) five brane webs, which can often be
described in terms of long quiver gauge theories. The null geodesic from which
the Penrose limit is constructed is localized at a unique point on the two
dimensional Riemann surface $\Sigma$, where the $AdS_{6}$ and $S^{2}$ metric
factors are extremal. The resulting pp-wave spacetime takes a universal form.
The world sheet action of the Green-Schwarz string is quadratic in the light
cone gauge and the spectrum of string excitations is obtained.

##  1 Introduction

The quantization of superstrings in the presence of Ramond-Ramond fields is a
technically challenging problem and not solved in general. This constitutes a
challenge to go beyond the semi-classical supergravity approximation in many
examples of AdS/CFT. Instead of solving the general problem one approach is to
consider limits or deformations of the supergravity solution which may lead to
a simpler quantization problem. One such limit is given by taking the Penrose
limit of an $AdS_{p}\times S^{q}$ background. The Penrose limit [1] (see also
[2, 3, 4]) corresponds to zooming in on the close vicinity of a null geodesic
in the spacetime and produces a plane wave geometry. In the case of the
Penrose limit of the type IIB $AdS_{5}\times S^{5}$ solution, where the null
geodesic sits at the center of $AdS_{5}$ and on a great circle of $S^{5}$, one
obtains a maximally supersymmetric plane wave [5, 6, 7]. It was proposed in
[8] that on the field theory side this limit corresponds to singling out a
special subset of CFT operators where both the conformal dimension $\Delta$
and a $U(1)$ R-charge $J$ are taken to be of order $\sqrt{N}$, with the
difference $\Delta-J$ finite, as $N\to\infty$. One important feature of this
type IIB plane wave background is that the Green-Schwarz string can be
quantized exactly [9, 10, 11]. The world sheet theory in the light cone gauge
corresponds to the action of eight massive bosons and fermions. Furthermore
the machinery of light-cone string field theory can be used to calculate
string interactions and compare the results to field theory [12, 13, 14, 15].
For an incomplete list discussing Penrose limits of other supergravity
backgrounds see [16, 17, 18, 19, 20].

The aim of this paper is to study the Penrose limit for a different AdS
solution, namely the type IIB solutions found in [21, 22], which are realized
as warped products of $AdS_{6}\times S^{2}$ over a Riemann surface $\Sigma$
with boundary. These solutions are supported by R-R and NS-NS three form
fluxes. Indeed the data characterizing the solution can be identified with the
$(p,q)$ five brane charges of semi-infinite five branes forming a five brane
web.

Unlike the type IIB $AdS_{5}\times S^{5}$ solution or the $AdS_{4|7}\times
S^{7|4}$ solution of M-theory these backgrounds are warped product geometries
and preserve only sixteen of the thirty-two supersymmetries of ten dimensional
type IIB supergravity. This fact makes finding a suitable Penrose limit more
challenging, since the radii of the $AdS_{6}$ and $S^{2}$ vary with the
location on the Riemann surface $\Sigma$ and therefore a general null geodesic
will also have a nontrivial dependence on the Riemann surface. There is a
special point on the Riemann surface, namely the critical point of a function
${\cal G}$. At this point the metric factors of the $AdS_{6}$ and $S^{2}$ are
extremized with respect to the coordinates on $\Sigma$. We choose the null-
geodesic on which the Penrose limit is based to be localized at this critical
point on $\Sigma$.

The structure of the paper is as follows: In section 2 we review the type IIB
supergravity solutions first found in [21, 22] which we use in the rest of the
paper. In section 3 we define the null geodesic on which the Penrose limit is
based and present the resulting type IIB plane wave background, including all
the other bosonic supergravity fields. In section 4 we discuss the
quantization of the Green-Schwarz superstring action in the light cone gauge
for this background. We discuss our results and possible directions for
further research in section 5. Some detailed calculations and supplementary
materials are relegated to appendices.

##  2 Type IIB $AdS_{6}$ solutions

Solutions of superstring theories with $AdS_{6}$ factors are candidates for
holographic duals of five dimensional superconformal field theories (SCFTs).
The AdS/CFT correspondence is an important tool in studying these theories.
Since the unique superconformal algebra $F(4)$ with an $SO(2,5)$ factor has
sixteen fermionic generators [23, 24, 25], the supergravity background
preserves only 16 of the 32 supersymmetries of maximally symmetric type II or
M-theory vacua. Furthermore the solutions are all realized as warped products
of the $AdS_{6}$ over a base space. The first such solution was obtained in
massive type IIA [26]. In this paper we will focus on type IIB solutions first
constructed in [21, 22], which is realized as a warped product of
$AdS_{6}\times S^{2}$ over a two dimensional Riemann surface $\Sigma$ with
boundary.

$\displaystyle ds^{2}$
$\displaystyle=f_{6}^{2}\,ds^{2}_{\mathrm{AdS}_{6}}+f_{2}^{2}\,ds^{2}_{\mathrm{S}^{2}}+4\hat{\rho}^{2}\,|dw|^{2}~{},$
(2.1)

where $w$ is a complex coordinate on $\Sigma$ and $ds^{2}_{AdS_{6}}$ and
$ds^{2}_{S^{2}}$ are the line elements for unit-radius $AdS_{6}$ and $S^{2}$,
respectively111We denote the metric factor of $\Sigma$ by $\hat{\rho}$ to
avoid confusion with $\rho$ which radial direction of $Ads_{6}$ used later..
The $AdS_{6}$ solutions are defined in terms of locally holomorphic functions
${\cal A}_{\pm}$ on a Riemann surface $\Sigma$. The metric functions read

$\displaystyle f_{6}^{2}$ $\displaystyle=\sqrt{6{\cal G}T}~{},$ $\displaystyle
f_{2}^{2}$ $\displaystyle=\frac{1}{9}\sqrt{6{\cal G}}\,T^{-\tfrac{3}{2}}~{},$
$\displaystyle\hat{\rho}^{2}$ $\displaystyle=\frac{\kappa^{2}}{\sqrt{6{\cal
G}}}T^{\tfrac{1}{2}}~{}.$ (2.2)

and they are expressed in terms of the holomorphic functions ${\cal A}_{\pm}$
as follows

$\displaystyle\kappa^{2}$ $\displaystyle=-|\partial_{w}{\cal
A}_{+}|^{2}+|\partial_{w}{\cal A}_{-}|^{2}~{},$
$\displaystyle\partial_{w}{\cal B}$ $\displaystyle={\cal
A}_{+}\partial_{w}{\cal A}_{-}-{\cal A}_{-}\partial_{w}{\cal A}_{+}~{},$
$\displaystyle{\cal G}$ $\displaystyle=|{\cal A}_{+}|^{2}-|{\cal
A}_{-}|^{2}+{\cal B}+\bar{{\cal B}}~{},$ $\displaystyle T^{2}$
$\displaystyle=\left(\frac{1+R}{1-R}\right)^{2}=1+\frac{2|\partial_{w}{\cal
G}|^{2}}{3\kappa^{2}\,{\cal G}}~{}.$ (2.3)

In order to obtain regular and geodesically complete solutions one has to
impose that ${\cal G}$ vanishes along the boundary of the Riemann surface,
which implies that the $S^{2}$ shrinks to zero size and the spacetime closes
off. In [22] a large class of regular solutions were constructed by choosing
$\Sigma$ to be the upper half plane and the holomorphic functions such that
$\partial_{w}{\cal A}_{\pm}$ have $L$ simple poles localized on the boundary
which is the real line.

$\displaystyle{\cal A}_{\pm}$ $\displaystyle={\cal
A}^{0}_{\pm}+\sum_{l=1}^{L}Z_{\pm}^{l}\ln(w-p_{l}),\quad\sum_{l=1}^{L}Z_{\pm}^{l}=0,\quad
Z_{\pm}^{l}=-{\bar{Z}}_{\mp}^{l}$ (2.4)

The regularity condition translates into $L$ conditions [22]

$\displaystyle{\cal A}^{0}_{+}Z_{-}^{k}-{\cal
A}^{0}_{-}Z^{k}_{+}+\sum_{l=1,l\neq
k}^{L}(Z_{+}^{l}Z_{-}^{k}-Z_{+}^{k}Z_{-}^{l})\ln|p_{l}-p_{k}|=0,\quad\quad
k=1,2,\cdots,L$ (2.5)

The other non vanishing type IIB supergravity fields of the solution are the
complex scalar $B$ which is related to the axion dilaton
$\tau=\chi+ie^{-\phi}$ via $B=(1+i\tau)/(1-i\tau)$ and the complex two form
antisymmetric tensor potential $C_{(2)}$

$\displaystyle B$ $\displaystyle=\frac{\partial_{w}{\cal
A}_{+}\,\partial_{\bar{w}}{\cal G}-R\,\partial_{\bar{w}}\bar{\cal
A}_{-}\partial_{w}{\cal G}}{R\,\partial_{\bar{w}}\bar{\cal
A}_{+}\partial_{w}{\cal G}-\partial_{w}{\cal A}_{-}\partial_{\bar{w}}{\cal
G}}~{},\quad C_{(2)}$
$\displaystyle=\frac{2i}{3}\left(\frac{\partial_{\bar{w}}{\cal
G}\partial_{w}{\cal A}_{+}+\partial_{w}{\cal G}\partial_{\bar{w}}\bar{\cal
A}_{-}}{3\kappa^{2}T^{2}}-\bar{\mathcal{A}}_{-}-\mathcal{A}_{+}\right){\rm
vol}_{S^{2}}~{}.$ (2.6)

As discussed in [22] the residues of ${\cal A}_{\pm}$ can be identified with
the charges of the $(p,q)$ five brane web which realizes the dual SCFT via

$\displaystyle Z_{+}^{l}={3\over 4}\alpha^{\prime}(p_{l}+iq_{l}),\quad
l=1,2,\cdots,L$ (2.7)

###  2.1 Explicit examples

For definiteness we consider two explicit examples of the regular supergravity
solution for which the dual SCFT are well studied, namely the $T_{N}$ theory
and the $+_{M,N}$ theory where the holomorphic functions ${\cal A}_{\pm}$ are
given by

$\displaystyle{\cal A}_{\pm}^{T_{N}}$
$\displaystyle=\frac{3N}{8\pi}\left[\pm\ln(w-1)+i\ln(2w)+(\mp
1-i)\ln(w+1)\right]~{},$ $\displaystyle{\cal A}_{\pm}^{+_{N,M}}$
$\displaystyle=\frac{3}{8\pi}\left[iN\left(\ln(2w-1)-\ln(w-1)\right)\pm
M\left(\ln(3w-2)-\ln w\right)\right]~{},$ (2.8)

The $T_{N}$ solution has three poles located at $w=\pm 1,0$ whereas the
$+_{M,N}$ has four poles located at $w=0,{1\over 2},{2\over 3},1$. The
relevant brane webs for the two theories are given by the junction of $N$ D5,
$N$ NS5 and $N$ (1,1) 5-branes for the $T_{N}$ theory and the intersection of
$N$ D5-branes and $M$ NS5-branes for the $+_{M,N}$ theory.

Figure 1: Left: brane web for the $T_{N}$ theory. Right: brane web for the
$+_{M,N}$ theory.

The quiver theories which at their conformal fixed points realize the SCFTs
are given by long linear quivers with $SU(n)$ gauge nodes and bi-fundamental
matter connecting them and a fundamental matter node at the end. For the
$T_{N}$ theories we have a linear quiver with $SU(k)$ nodes with increasing
$k$ along the quiver and fundamental representations attached to the end of
the quiver

$\displaystyle T_{N}:$ $\displaystyle\quad[2]-(2)-(3)-\cdots-(N-1)-[N]$ (2.9)

For the $+_{MN}$ theory we have $M-1$ $SU(N)$ nodes with matter in the
fundamental representation of the end nodes attached at the end of the quiver.

$\displaystyle+_{MN}:$ $\displaystyle\quad[N]-(N)-(N)-\cdots-(N)-[N]$ (2.10)

For the two example cases the function ${\cal G}$ defined in (2) takes the
following compact form

$\displaystyle{\cal G}_{T_{N}}$
$\displaystyle=\frac{9}{8\pi^{2}}N^{2}D\left(\frac{2w}{w+1}\right)~{},$
$\displaystyle{\cal G}_{+_{N,M}}$
$\displaystyle=\frac{9}{8\pi^{2}}NM\left[D\left(\frac{3w-2}{w}\right)+D\left(\frac{w}{2-3w}\right)\right]$
(2.11)

where $D$ is the Bloch-Wigner function given by

$\displaystyle D(u)$
$\displaystyle=\operatorname{Im}\left[Li_{2}(u)+\ln(1-u)\ln|u|\right]~{}.$
(2.12)

where $Li_{2}$ is the dilogarithm function.

##  3 Penrose limit and the plane wave background

The idea behind taking a Penrose limit of $AdS_{5}\times S^{5}$ solution of
type IIB is to consider the trajectory of a particle sitting at the center of
$AdS_{5}$ and moving very fast along a great circle on $S^{5}$, such that in
the limit the trajectory becomes a null geodesic. Zooming in on the vicinity
of the null geodesic produces a pp-wave solution which preserves 32
supercharges.

For the $AdS_{6}\times S^{2}$ solutions presented in section 2, the geometry
is more complicated since the $AdS_{6}$ and $S^{2}$ are warped over the two
dimensional surface $\Sigma$ and in general a nice Penrose limit does not
exist for geodesics through a generic point on $\Sigma$. Furthermore as
discussed in appendix B considering a geodesic with a nontrivial dependence on
$\Sigma$ looks daunting due to the complicated form of the geodesic equation.

Fortunately, we can find a special point on the Riemann surface $\Sigma$ by
considering critical points of the function ${\cal G}$ defined in (2), i.e
$w_{c}\in\Sigma$ for which

$\displaystyle\partial_{w}{\cal G}\mid_{w=w_{c}}=0$ (3.1)

It was observed in [27] that in all known examples, ${\cal G}$ has a unique
critical point on the Riemann surface $\Sigma$. For our two examples the
critical point on $w_{c}$ on $\Sigma$ is located at

$\displaystyle w_{c,T_{N}}={i\over\sqrt{3}},\quad\quad w_{c,+_{M,N}}={3+i\over
5}$ (3.2)

In [27] it was shown that the critical point $w_{c}$ correspond to the
location of BPS probe D3-branes that realize surface defects in the dual SCFT.
As we shall see in the following, the existence of the critical point is
essential for our construction of Penrose limit in the warped spacetime.

We can examine our solution near the critical point by expanding
$w=w_{c}+\varepsilon\zeta$. The most important expression, ${\cal G}$, can be
expanded in a power series in $\varepsilon$

$\displaystyle{\cal G}$
$\displaystyle=G_{0}+G_{2}\varepsilon^{2}|\zeta|^{2}+o(\varepsilon^{3})$ (3.3)

Using (2) we find the following expansions for the other functions which
appear in the metric factors

$\displaystyle\kappa^{2}$ $\displaystyle=-G_{2}+o(\varepsilon)$ $\displaystyle
T^{2}$ $\displaystyle=1-{2\over 3}{G_{2}\over
G_{0}}\varepsilon^{2}|\zeta|^{2}+o(\varepsilon^{3})$ $\displaystyle R$
$\displaystyle=-{G_{2}\over 6G_{0}}|\zeta^{2}|+o(\varepsilon)$ (3.4)

Using the expressions for the metric factors (2.2) it is straightforward to
show that the metric factors of the $AdS_{6}$ and $S_{2}$ are extremized at
$w=w_{c}$.

For the $T_{N}$ and $+_{N,M}$ examples we find the following expressions for
the expansion coefficients $G_{0}$ and $G_{2}$

$\displaystyle T_{N}:$ $\displaystyle\;\;G_{0}=Im(Li_{2}(e^{i\pi/3})),\quad
G_{2}=-{81\sqrt{3}\over 128\pi^{2}}N^{2}$ $\displaystyle+_{N,M}:$
$\displaystyle\;\;G_{0}={9CMN\over 4\pi^{2}},\quad\quad G_{2}=-{225MN\over
16\pi^{2}}$ (3.5)

where $C$ is Catalan’s constant.

###  3.1 Penrose limit

Using the expansions (3.3) and (3) the Einstein frame metric (2.1) can be
expanded to second order which corresponds to the metric close to the critical
point $w=w_{c}$.

$\displaystyle ds^{2}$
$\displaystyle=\sqrt{6G_{0}}\Big{\\{}(1+\varepsilon^{2}{G_{2}\over
3G_{0}}|\zeta|^{2})(-\cosh^{2}\rho dt^{2}+d\rho^{2}+\sinh^{2}\rho
ds_{S^{4}}^{2})$ $\displaystyle+{1\over 9}(1+\varepsilon^{2}{G_{2}\over
G_{0}}|\zeta|^{2})(d\theta^{2}+\cos^{2}\theta d\phi^{2})+{2\over
3}\varepsilon^{2}{|G_{2}|\over G_{0}}|d\zeta|^{2}\Big{\\}}+o(\varepsilon^{3})$
(3.6)

We can drop the expansion parameter $\varepsilon$ in the following since it
can be absorbed into a redefinition of the coordinate $\zeta$. The null
geodesic which gives the proper Penrose limit is defined by

$\displaystyle\zeta=0,\quad\rho=0,\quad\theta=0,\quad t-{\phi\over 3}=const$
(3.7)

Note that $\zeta=0$ implies that the null geodesic is localized at the
critical point $w=w_{c}$ on the Riemann surface. We express the metric in
terms of rescaled and relabelled coordinates

$\displaystyle t=x^{+}+{x^{-}\over(6G_{0})^{1\over
2}},\quad\quad\phi=3x^{+}-{3x^{-}\over(6G_{0})^{1\over 2}}$
$\displaystyle\rho={1\over(6G_{0})^{1\over
4}}r,\quad\quad\theta={3\over(6G_{0})^{1\over
4}}x^{6},\quad\quad\zeta=\varepsilon\left({3G_{0}\over
2|G_{2}|}\right)^{1\over 2}{1\over(6G_{0})^{1\over 4}}(x^{7}+ix^{8})$ (3.8)

The Penrose limit is obtained by taking $G_{0}\to\infty$ , which corresponds
to taking $N\to\infty$ (for the $T_{N}$ case) and $MN\to\infty$ (for the
$+_{N,M}$ example) and keeping the finite terms in the metric (3.1). All
expressions involving the ratio $G_{2}/G_{0}$ are finite in this limit. Note
that the curvature radius of the $AdS_{6}$ and $S^{2}$ factors are controlled
by $G_{0}$ and hence the limit of large $G_{0}$ is precisely the one taken for
a Penrose limit in $AdS_{p}\times S^{q}$ spacetimes, i.e. zooming in on a
region close to the null geodesic, which is much smaller than the curvature
radius of the spacetime.

The resulting metric is given by

$\displaystyle
ds^{2}=-4dx^{+}dx^{-}+(dx^{+})^{2}\Big{(}-r^{2}-9x_{6}^{2}-x_{7}^{2}-x_{8}^{2}\Big{)}+dr^{2}+r^{2}ds_{S^{4}}^{2}+dx_{6}^{2}+dx_{7}^{2}+dx_{8}^{2}$
(3.9)

where $r^{2}=x_{1}^{2}+x_{2}^{3}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}$. The only
difference between this metric and the one of the Penrose limit of
$AdS_{5}\times S^{5}$ that the coordinate $x_{6}$ is singled out and has a
different normalization in the $(dx^{+})^{2}$ term in the metric, which in the
world sheet action leads to a different mass term for the field associated
with $x_{6}$.

The energy in global AdS is associated with $i\partial_{t}$ whereas
$-i\partial_{\phi}$ is related to a $U(1)$ generator of the $SU(2)_{R}$
symmetry, which is realized as the group of isometries of the $S^{2}$. Hence
using the argument of [8] for the Penrose limit (3.1) we can identify the
light cone momentum with the conformal dimension and the R-charge of a dual
operator via

$\displaystyle 2p^{-}$
$\displaystyle=i\partial_{+}=i(\partial_{t}+3\partial_{\phi})=\Delta-3J$
$\displaystyle 2p^{+}$ $\displaystyle=i\partial_{-}={i\over
R^{2}}(\partial_{t}-3\partial_{\phi})={1\over R^{2}}(\Delta+3J)$ (3.10)

where we have identified the AdS radius with $R^{2}=\sqrt{6G_{0}}$. Since we
take $R$ to be very large in the Penrose limit excitations with finite
$p^{\pm}$ correspond to states with large energy and angular momentum which on
the CFT side correspond to operators with large conformal dimension $\Delta$
and R-charge $J$, which are close to the BPS bound $\Delta=3J$ [28, 29].

###  3.2 Other supergravity fields

In the Penrose limit, the complex scalar field $B$ of type IIB supergravity
becomes a constant

$\displaystyle B$ $\displaystyle=-{\partial_{w}{\cal
A}_{+}\over\partial_{w}{\cal A}_{-}}\mid_{w=w_{c}}+o(\varepsilon)$ (3.11)

Note that the standard RR axion $\chi$ and the dilaton $\phi$ is related to
the complex scalar $B$ via

$\displaystyle B={1+i\tau\over 1-i\tau},\quad\quad\tau=\chi+ie^{-\Phi}$ (3.12)

Consequently, the axion dilaton has the following expression in terms of the
holomorphic functions at the critical point $w_{c}$

$\displaystyle\tau=i{\partial_{w}{\cal A}_{-}+\partial_{w}{\cal
A}_{+}\over\partial_{w}{\cal A}_{-}-\partial_{w}{\cal A}_{+}}\mid_{w=w_{c}}$
(3.13)

For the two examples we consider in this paper one finds222In [21, 22]
different normalization for the dilaton is used, we have translated the
expression to the commonly used one.

$\displaystyle T_{N}:$ $\displaystyle\;\;\chi={1\over
2},\;\;e^{-\phi}={\sqrt{3}\over 2},$
$\displaystyle+_{N,M}:\;\;\chi=0,\;\;e^{-\phi}={N\over M}$ (3.14)

The complex rank two antisymmetric tensor potential $C^{(2)}$ is related to
the NS-NS and R-R two form potentials by

$\displaystyle C^{(2)}$ $\displaystyle=B^{NSNS}+iC^{RR}$
$\displaystyle=\Big{(}c_{0}-i{8\over 9}\Big{(}\partial_{w}{\cal
A}_{+}\mid_{w=w_{c}}\varepsilon\zeta+\partial_{\bar{w}}\bar{\cal
A}_{-}\mid_{w=w_{c}}\varepsilon\bar{\zeta}\Big{)}+o(\varepsilon^{2})\Big{)}\cos\theta
d\theta\wedge d\phi$ (3.15)

Note that the constant term $c_{0}$ is not important since in the bosonic part
of the worldsheet action, a constant NS-NS anti-symmetric tensor potential
will be a total derivative. In the fermionic worldsheet action only the field
strength, which does not depend on $c_{0}$, appears. In the Penrose limit the
finite part which survives (after using the rescaled variables)

$\displaystyle C^{(2)}=-4i{1\over|G_{2}|^{1\over 2}}\Big{(}\partial_{w}{\cal
A}_{+}\mid_{w=w_{c}}(x^{7}+ix^{8})+\partial_{\bar{w}}\bar{\cal
A}_{-}\mid_{w=w_{c}}(x^{7}-ix^{8})\Big{)}dx^{6}\wedge dx^{+}$ (3.16)

Note that in the scaling limit which produces the Penrose limit discussed
below (3.1) the $\partial A_{\pm}$ and $|G_{2}|$ scale in the same way and
hence the term above is the expression which survives the Penrose limit.

The NS-NS 3 form field strength $H^{3}_{NS}$ and the RR field strength
$F^{3}_{RR}$ are given by

$\displaystyle H^{3}_{NS}=Re\Big{(}dC_{2}\Big{)},\quad\quad
F^{3}_{RR}=Im\Big{(}dC_{2}\Big{)},$ (3.17)

For the examples we get

$\displaystyle T_{N}:$ $\displaystyle\quad H^{3}_{NS}=-{4\sqrt{2}\over
3^{1\over 4}}dx^{8}\wedge dx^{6}\wedge dx^{+},$ $\displaystyle\quad
F^{3}_{RR}=-{2\sqrt{2}\over 3^{3\over 4}}\Big{(}3dx^{7}+\sqrt{3}dx^{8})\wedge
dx^{6}\wedge dx^{+}$ (3.18)

and

$\displaystyle+_{N,M}:$ $\displaystyle\quad H^{3}_{NS}=-{4\over 5}\sqrt{M\over
N}(4dx^{7}+3dx^{8})\wedge dx^{6}\wedge dx^{+},$ $\displaystyle\quad
F^{3}_{RR}={4\over 5}\sqrt{N\over M}(-3dx^{7}+4dx^{8})\wedge dx^{6}\wedge
dx^{+}$ (3.19)

It is important to mention that the supergravity solutions of section 2 are
given in the Einstein frame. In order to quantize the Green-Schwarz string in
this background we have to transform to the string frame

$\displaystyle G_{\mu\nu}^{string}=e^{\phi/2}G_{\mu\nu}^{Einstein}$ (3.20)

Note that the particular combinations of the anti-symmetric tensor field
strength which appear in the string frame supersymmetry transformations as
well as the Green-Schwarz action are $H^{(3)}_{NS}$ and
$e^{\phi}(F^{(3)}_{R}-\chi H^{(3)}_{NS})$.

In the Penrose limit, the transformation to the Einstein frame is just a
constant rescaling of the metric. However, it is convenient for the string
frame metric in to be the canonical plane wave form (3.9). This can be
accomplished by a rescaling of all coordinates (except $x^{+}$) by a factor of
$e^{-\phi/4}$. This has the effect of introducing a factor of $e^{-\phi/2}$ in
each of the antisymmetric tensor fields. The end result is the following
combination of fields which appears in the string worldsheet:

$\displaystyle T_{N}:$ $\displaystyle\quad H^{3}_{NS,string}=-4dx^{8}\wedge
dx^{6}\wedge dx^{+},$ $\displaystyle e^{\phi}(F^{3}_{RR,string}-\chi
H^{3}_{NS,string})=-4dx^{7}\wedge dx^{6}\wedge dx^{+}$ (3.21)
$\displaystyle+_{N,M}:$ $\displaystyle\quad H^{3}_{NS,string}=-{4\over
5}(4dx^{7}+3dx^{8})\wedge dx^{6}\wedge dx^{+},$ $\displaystyle
e^{\phi}(F^{3}_{RR,string}-\chi H^{3}_{NS,string})={4\over
5}(-3dx^{7}+4dx^{8})\wedge dx^{6}\wedge dx^{+}$ (3.22)

We will drop ”string” from the subscripts moving forward since we will always
be working in the string frame and there will be no confusion.

Note one interesting property of specific solutions (3.2) and (3.22): The
orientation of the forms in the $7,8$ directions can be parameterized as
$n_{7}dx^{7}+n_{8}dx^{8}$, for both $T_{N}$ and $+_{MN}$ examples the $n_{a}$
associated with the $H^{(3)}_{NS}$ and $e^{\phi}(F^{(3)}_{R}-\chi
H^{(3)}_{NS})$ tensors are orthogonal to each other and square to four. This
means that we could rotate $x^{7,8}$ so that the fields which appear in the
worldsheet action take exactly the same form in both $T_{N}$ and $+_{MN}$
theories and therefore give the same string spectrum. In fact, this statement
also holds for all other solutions [30, 31] that we have checked. We
conjecture that all global solutions (2.1) - (2.6) share this property and in
the quantization the follows, take our fields to have the form of the $T_{N}$
example (3.2).

##  4 Light cone Green-Schwarz string spectrum

In this section we quantize the Green-Schwarz string the pp-wave spacetime
obtained in the previous section. As shown in [10, 9, 11, 32] the fermionic
part of the Green-Schwarz string becomes quadratic in fermionic fields for the
pp-wave spacetimes in the light cone gauge, which makes the free string
spectrum exactly solvable.

###  4.1 Bosonic Spectrum

We will start by examining the bosonic spectrum for the Green-Schwarz string
in the plane wave background with NS-NS flux. The general action is given by

$\displaystyle S_{b}=-\frac{1}{4\pi\alpha^{\prime}}\int
d^{2}\sigma\sqrt{h}\left(h^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}G_{\mu\nu}+\varepsilon^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}B_{\mu\nu}\right)$
(4.1)

As in the $AdS_{5}\times S^{5}$ case, this action simplifies considerably
after light cone gauge-fixing, leaving us with a free theory which can be
easily quantized following [33]. We set the target space coordinate
$x^{+}=\tau$ and worldsheet metric to $h^{\sigma\tau}=0$,
$h^{\sigma\sigma}=-h^{\tau\tau}=1$ and after plugging in the background (3.9),
(3.2), we are left with the following action for the eight remaining
transverse scalar fields

$\displaystyle
S_{b}=\frac{1}{4\pi\alpha^{\prime}}\int_{0}^{2\pi\alpha^{\prime}p^{+}}d^{2}\sigma\left(\sum_{I=1}^{8}\left(\left(\partial_{\tau}X^{I}\right)^{2}-\left(\partial_{\sigma}X^{I}\right)^{2}-m_{I}^{2}\left(X^{I}\right)^{2}\right)+4X^{6}\partial_{\sigma}X^{7}-4X^{7}\partial_{\sigma}X^{6}\right)$
(4.2)

where $m_{I}^{2}=1$ for all $I\neq 6$ in which case $m_{6}^{2}=9$. The
equations of motion deduced from this action are given by

$\displaystyle\partial_{\tau}^{2}X^{I}-\partial_{\sigma}^{2}X^{I}+X^{I}$
$\displaystyle=0\quad I=1,...,5,8$ (4.3)
$\displaystyle\partial_{\tau}^{2}X^{6}-\partial_{\sigma}^{2}X^{6}+9X^{6}-4\partial_{\sigma}X^{7}$
$\displaystyle=0$ (4.4)
$\displaystyle\partial_{\tau}^{2}X^{7}-\partial_{\sigma}^{2}X^{7}+X^{7}+4\partial_{\sigma}X^{6}$
$\displaystyle=0$ (4.5)

Together with the periodicity condition
$X^{I}(\sigma+2\pi\alpha^{\prime}p^{+},\tau)=X^{I}(\sigma,\tau)$, the first
set of equations (4.3) lead to the following familiar solutions for
$I=1,...,5,8$

$\displaystyle
x^{I}(\sigma,\tau)=x_{0}^{I}\cos\tau+\frac{p_{0}^{I}}{p^{+}}\sin\tau+\sqrt{\frac{\alpha^{\prime}}{2}}\sum_{n=1}^{\infty}\frac{1}{\sqrt{\omega^{I}_{n}}}[$
$\displaystyle\alpha_{n}^{I}e^{-\frac{i}{\alpha^{\prime}p^{+}}\left(\omega^{I}_{n}\tau+n\sigma\right)}+\tilde{\alpha}_{n}^{I}e^{-\frac{i}{\alpha^{\prime}p^{+}}\left(\omega^{I}_{n}\tau-n\sigma\right)}+$
$\displaystyle\alpha_{n}^{I\dagger}e^{\frac{i}{\alpha^{\prime}p^{+}}\left(\omega^{I}_{n}\tau+n\sigma\right)}+\tilde{\alpha}_{n}^{I\dagger}e^{\frac{i}{\alpha^{\prime}p^{+}}\left(\omega^{I}_{n}\tau-n\sigma\right)}]$
(4.6)

where the frequencies will determine the energy of the string excitations

$\displaystyle\omega_{n}^{I}=\sqrt{n^{2}+(\alpha^{\prime}p^{+})^{2}}$ (4.7)

To solve the remaining two equations of motion (4.4) and (4.5) for $I=6,7$ we
use an ansatz of the following form

$\displaystyle x^{I}(\sigma,\tau)=\sqrt{{\alpha^{\prime}\over
2}}A_{0}(\tau)+\sqrt{{\alpha^{\prime}\over
2}}\sum_{n>0}[A_{n}^{I}(\tau,\sigma)+\tilde{A}_{n}^{I}(\tau,\sigma)]$ (4.8)

where $A^{I}_{n}$ and $\tilde{A}^{I}_{n}$ for $I=6,7$ can be expressed in
terms of eigenmodes

$\displaystyle A_{n}^{I}(\tau,\sigma)$
$\displaystyle=\sum_{J=6,7}{1\over\sqrt{\omega_{n}^{J}}}\Big{\\{}(V_{n})^{I}_{J}\alpha_{n}^{J}e^{-{i\over\alpha^{\prime}p^{+}}(\omega_{n}^{J}\tau+n\sigma)}+(\bar{V}_{n})^{I}_{J}(\alpha^{\dagger})_{n}^{J}e^{{i\over\alpha^{\prime}p^{+}}(\omega_{n}^{J}\tau+n\sigma)}\Big{\\}}$
$\displaystyle\tilde{A}_{n}^{I}(\tau,\sigma)$
$\displaystyle=\sum_{J=6,7}{1\over\sqrt{\omega_{n}^{J}}}\Big{\\{}(V_{-n})^{I}_{J}\alpha_{n}^{J}e^{-{i\over\alpha^{\prime}p^{+}}(\omega_{n}^{J}\tau-n\sigma)}+(\bar{V}_{-n})^{I}_{J}(\alpha^{\dagger})_{n}^{J}e^{{i\over\alpha^{\prime}p^{+}}(\omega_{n}^{J}\tau-n\sigma)}\Big{\\}}$
(4.9)

For the zero mode $A_{0}$ there only one eigenfunction which can be obtained
from $A_{n}^{I}$ in (4.1) by setting $n=0$. The eigenfrequencies for $J=6,7$

$\displaystyle\omega_{n}^{6}$
$\displaystyle=\sqrt{n^{2}+(\alpha^{\prime}p^{+})^{2}}+2\alpha^{\prime}p^{+}$
$\displaystyle\omega_{n}^{7}$
$\displaystyle=\left|\sqrt{n^{2}+(\alpha^{\prime}p^{+})^{2}}-2\alpha^{\prime}p^{+}\right|$
(4.10)

and $(V_{n})^{I}_{\;J},J=6,7$ are the orthonormal set of vectors which satisfy

$\displaystyle\left(\begin{array}[]{cc}{-(\omega^{J}_{n})^{2}+n^{2}\over(\alpha^{\prime}p^{+})^{2}}+9&-{4in\over\alpha^{\prime}p^{+}}\\\
+{4in\over\alpha^{\prime}p^{+}}&{-(\omega^{J}_{n})^{2}+n^{2}\over(\alpha^{\prime}p^{+})^{2}}+1\end{array}\right)\left(\begin{array}[]{c}(V_{n})^{6}_{\;J}\\\
(V_{n})^{7}_{\;J}\end{array}\right)=0,\quad\quad J=6,7$ (4.15)

With this mode equation the the canonical quantization conditions

$\displaystyle[x^{I}(\sigma,\tau),p^{J}(\sigma^{\prime},\tau)]=i\delta^{IJ}\delta(\sigma-\sigma^{\prime})$
(4.16)

where $p^{I}={1\over 2\pi\alpha^{\prime}}\partial_{\tau}x^{I}$, yield the
commutation relation for the creation and annihilation operators

$\displaystyle[\alpha_{n}^{I},\alpha_{m}^{J\dagger}]=[\tilde{\alpha}_{n}^{I},\tilde{\alpha}_{m}^{J\dagger}]=\delta^{IJ}\delta_{mn}$
(4.17)

where now we allow the indices $I,J$ to run through all eight transverse
oscillators $1,2,\cdots,8$.

The light-cone Hamiltonian can be expressed in terms of the creation and
annihilation operators (4.1) and (4.8)

$\displaystyle H_{l.c.}^{\text{b }}=\frac{1}{\alpha^{\prime}p^{+}}$
$\displaystyle\sum_{I=0}^{8}\left[\omega^{I}_{0}\alpha_{0}^{I\dagger}\alpha_{0}^{I}+\sum_{n=0}^{\infty}\omega_{n}^{I}\left(\alpha_{n}^{I\dagger}\alpha_{n}^{I}+\tilde{\alpha}_{n}^{I\dagger}\tilde{\alpha}_{n}^{I}\right)\right]+{\nu_{\rm
bos}}$ (4.18)

Here $\nu_{\rm bos}$ denotes a normal ordering constant for the bosonic
creation and annihilation operators. The zero mode modes
$x^{I}_{0},p^{I}_{0},I=1,2,\cdots,5,8$ have been expressed in terms of
creation and annihilation operators

$\displaystyle\tilde{\alpha}_{0}^{I}=\frac{1}{\sqrt{2p^{+}}}p_{0}^{I}-i\sqrt{\frac{p^{+}}{2}}x_{0}^{I}$
(4.19)

to maintain consistent normalization. It will be convenient to write $\nu_{\rm
bos}$ in terms of the oscillator frequencies:

$\displaystyle\nu_{\rm bos}={1\over
2\alpha^{\prime}p^{+}}\sum_{I=1}^{8}\big{(}\omega^{I}_{0}+2\sum_{n}\omega_{n}^{I}\big{)}$
(4.20)

###  4.2 Fermionic Spectrum

We can now turn our attention to the fermionic part of the Green-Schwarz
action for type IIB strings, the term quadratic in the world sheet fermions is
given by (see appendix A for details on the notation).

$\displaystyle S_{f}^{(2)}=\frac{i}{4\pi\alpha^{\prime}}\int
d^{2}\sigma\sqrt{h}\left(h^{ij}\delta_{ab}-\varepsilon^{ij}\left(\sigma_{3}\right)_{ab}\right)\partial_{i}x^{M}\bar{\theta}^{a}\Gamma_{M}\left(D_{j}\right)_{c}^{b}\theta^{c}$
(4.21)

This Green-Schwarz action possesses a $\kappa$ symmetry which is required to
obtain spacetime supersymmetry for the on-shell string modes [9]. To obtain
these physical fermionic modes, we can gauge fix (as in flat space) by
choosing

$\displaystyle\Gamma^{+}\theta^{a}=0$ (4.22)

where $\Gamma^{\pm}={1\over\sqrt{2}}(\Gamma^{0}\pm\Gamma^{9})$. After
enforcing the light cone gauge condition $x^{+}=\tau$ and worldsheet metric to
$h^{\sigma\tau}=0$, $h^{\sigma\sigma}=-h^{\tau\tau}=1$, the fermionic part of
the Green-Schwarz action reduces to the quadratic one in the plane wave
background [9, 10, 11] and takes the following form

$\displaystyle S_{f}=\frac{i}{4\pi\alpha^{\prime}}\int
d^{2}\sigma\left(-\left(\theta^{a}\right)^{T}\Gamma^{+}\Gamma_{+}\left(D_{\tau}\right)_{c}^{a}\theta^{c}-\left(\theta^{a}\right)^{T}\Gamma^{+}\Gamma_{+}\sigma_{ab}^{(3)}\left(D_{\sigma}\right)_{c}^{b}\theta^{c}\right)$
(4.23)

after inserting the antisymmetric tensor fields (3.2) for the $T_{N}$ theory,
the action becomes

$\displaystyle S_{f}$ $\displaystyle=-{i\over 4\pi\alpha^{\prime}}\int
d^{2}\sigma(\theta^{a})^{T}\Gamma^{+}\Gamma_{+}\Big{(}\delta_{ab}\partial_{\tau}+(\sigma_{3})_{ab}\partial_{\sigma}-{1\over
4}H_{+67}\Gamma^{67}(\sigma_{3})_{ab}+{1\over
4}\tilde{F}_{+68}\Gamma^{68}(\sigma_{1})_{ab}\Big{)}\theta^{b}$
$\displaystyle=-{i\over 4\pi\alpha^{\prime}}\int
d^{2}\sigma(\theta^{a})^{T}\Gamma^{+}\Gamma_{+}\Big{(}\delta_{ab}\partial_{\tau}+(\sigma_{3})_{ab}\partial_{\sigma}+\Gamma^{67}(\sigma_{3})_{ab}-\Gamma^{68}\
(\sigma_{1})_{ab}\Big{)}\theta^{b}$ (4.24)

where we have used the fact that $\Gamma^{+}\Gamma_{+}=1+\Gamma^{09}$ and
hence $(\Gamma^{+}\Gamma_{+})^{2}=2\Gamma^{+}\Gamma_{+}$. The chirality
condition $\Gamma^{11}\theta^{a}=\theta^{a}$ together with the light cone
gauge condition (4.22) projects a general thirty-two component spinor onto an
eight dimensional subspace. It is possible to find a representation of the
Gamma matrices in this subspace and combine them with the $\sigma$ matrices
encoding the two chiral spinors such that the action can be written as

$\displaystyle S_{f}$ $\displaystyle=-{i\over 2\pi\alpha^{\prime}}\int
d^{2}\sigma\theta^{T}1_{4}\otimes\Big{(}1_{2}\otimes
1_{2}\;\partial_{\tau}+1_{2}\otimes\sigma_{3}\;\partial_{\sigma}+\sigma_{3}\otimes\sigma_{3}-\sigma_{1}\otimes\sigma_{1}\Big{)}\theta$
(4.25)

Due to the common $1_{4}$ factor in all the terms it follows that there is a
fourfold degeneracy and equation of motion for each degenerate spinor
component is given by

$\displaystyle\Big{(}1_{2}\otimes
1_{2}\;\partial_{\tau}+1_{2}\otimes\sigma_{3}\;\partial_{\sigma}+i\sigma_{3}\otimes\sigma_{3}-i\sigma_{1}\otimes\sigma_{1}\Big{)}\theta^{\alpha}=0$
(4.26)

where $\alpha=1,...,4$ labels four degenerate spinor components acting on the
first $1_{4}$ factor in the tensor product. We can proceed just as in the
bosonic case with an ansatz of the form

$\displaystyle\theta^{\alpha}(\sigma,\tau)={1\over\sqrt{2p^{+}}}\sum_{n=0}^{\infty}\Big{(}\theta_{n}^{\alpha,+}(\tau,\sigma)+\theta_{n}^{\alpha,-}(\tau,\sigma)\Big{)}$
(4.27)

Where

$\displaystyle\theta_{n}^{\alpha,\pm}$
$\displaystyle=V^{\pm}_{n}e^{-{i\over\alpha^{\prime}p^{+}}(\omega_{n}^{\pm}\tau+n\sigma)}\beta_{n}^{\alpha,\pm}+V^{\pm}_{-n}e^{{i\over\alpha^{\prime}p^{+}}(\omega_{n}^{\pm}\tau+n\sigma)}(\beta_{n}^{\alpha,\pm})^{\dagger}$
$\displaystyle\quad+V^{\pm}_{-n}e^{-{i\over\alpha^{\prime}p^{+}}(\omega_{n}^{\pm}\tau-n\sigma)}\tilde{\beta}_{n}^{\alpha,\pm}+V^{\pm}_{n}e^{{i\over\alpha^{\prime}p^{+}}(\omega_{n}^{\pm}\tau-n\sigma)}\tilde{(}\beta_{n}^{\alpha,\pm})^{\dagger}$
(4.28)

the positive eigenfrequencies are given by

$\displaystyle\omega_{n}^{+}=\sqrt{(\alpha^{\prime}p^{+})^{2}+n^{2}}+\alpha^{\prime}p^{+}$
$\displaystyle\omega_{n}^{-}=\sqrt{(\alpha^{\prime}p^{+})^{2}+n^{2}}-\alpha^{\prime}p^{+}$
(4.29)

The vectors $V^{\pm}_{n}$ satisfy the following equation

$\displaystyle\Big{(}-\omega_{n}^{\pm}\;1_{2}\otimes
1_{2}\;-n\;1_{2}\otimes\sigma_{3}+\sigma_{3}\otimes\sigma_{3}-\sigma_{1}\otimes\sigma_{1}\Big{)}V_{n}^{\pm}=0$
(4.30)

ensuring that the mode expansion of $\theta$ (4.2) satisfies the equation of
motion (4.26). For each $n>0$ the four vectors $V^{\pm}_{n},V^{\pm}_{-n}$ can
be chosen to form a orthonormal basis. The canonical anti-commutation
relations following from the action (4.25)

$\displaystyle\\{\theta^{T}(\tau,\sigma),\theta(\tau,\sigma^{\prime})\\}=i2\pi\alpha^{\prime}\delta(\sigma-\sigma^{\prime})1_{16\times
16}$ (4.31)

imply the following algebra for the fermionic creation and annihilation
operators

$\displaystyle\\{\beta_{n}^{\alpha,\pm},(\beta_{m}^{\gamma,\pm})^{\dagger}\\}=\\{\tilde{\beta}_{n}^{\alpha,\pm},(\tilde{\beta}_{m}^{\gamma,\pm})^{\dagger}\\}=\delta^{\alpha\gamma}\delta_{m,n}$
(4.32)

with all other anti commutators vanishing.

Interestingly, we can see from (4.2) that our spectrum contains eight
fermionic zero modes with $\omega^{-}_{n}=0$ when $n=0$. The zero modes are
associated with $\theta$ in the original fermionic action (4.2) for which

$\displaystyle\left(\Gamma^{67}\sigma^{(3)}-\Gamma^{68}\sigma^{(1)}\right)\theta=\Gamma^{67}\sigma^{(3)}\left(1-i\Gamma^{78}\sigma^{2}\right)\theta=0$
(4.33)

where we have dropped the index $a$ for notational simplicity. Notice that
$\left(1-i\Gamma^{78}\sigma^{2}\right)$ is a projector and hence has an eight
dimensional eigenspace with eigenvalue zero, matching precisely with the eight
zero modes found above.

Lastly, we can write the fermionic light cone Hamiltonian

$\displaystyle H_{l.c.}^{f}=\frac{1}{\alpha^{\prime}p^{+}}$
$\displaystyle\sum_{\alpha}\left[\omega_{0}^{+}(\beta_{0}^{\alpha,+})^{\dagger}\beta_{0}^{\alpha,+}+\sum_{n=0}^{\infty}\sum_{\gamma=\pm}\omega^{\alpha,\gamma}_{n}\left((\beta_{n}^{\alpha,\gamma})^{\dagger}\beta_{n}^{\alpha+}+(\tilde{\beta}_{n}^{\alpha,\gamma})^{\dagger}\tilde{\beta}_{n}^{\alpha,\gamma}\right)\right]+\nu_{f}$
(4.34)

Where $\nu_{f}$ is the constant obtained by normal ordering the fermionic
creation and anilhilation operators.

$\displaystyle\nu_{f}$ $\displaystyle=-{1\over
2\alpha^{\prime}p^{+}}\Big{(}4\omega_{0}^{+}+8\sum_{n>0}(\omega_{n}^{+}+\omega_{n}^{-})\Big{)}$

###  4.3 Light cone spectrum

The Hamiltonian of the light cone string is the sum of the bosonic (4.18) and
fermionic (4.34) parts

$\displaystyle H_{l.c.}$
$\displaystyle=\frac{1}{\alpha^{\prime}p^{+}}\left(\sum_{I=0}^{8}\left[\omega^{I}_{0}\alpha_{0}^{I\dagger}\alpha_{0}^{I}+\sum_{n=1}^{\infty}\omega_{n}^{I}\left(\alpha_{n}^{I\dagger}\alpha_{n}^{I}+\tilde{\alpha}_{n}^{I\dagger}\tilde{\alpha}_{n}^{I}\right)\right]\right.$
$\displaystyle\left.\quad\quad+\sum_{\alpha}\left[\omega_{0}^{+}(\beta_{0}^{\alpha,+})^{\dagger}\beta_{0}^{\alpha,+}+\sum_{n=1}^{\infty}\sum_{\gamma=\pm}\omega^{\alpha,\gamma}_{n}\left((\beta_{n}^{\alpha,\gamma})^{\dagger}\beta_{n}^{\alpha+}+(\tilde{\beta}_{n}^{\alpha,\gamma})^{\dagger}\tilde{\beta}_{n}^{\alpha,\gamma}\right)\right]\right)+\nu$
(4.35)

The normal ordering constant is the sum of the bosonic (4.20) and fermionic
(4.2) contribution and the bosonic and fermionic normal ordering contributions
cancel up to a finite number of terms

$\displaystyle\nu$ $\displaystyle=\nu_{b}+\nu_{f}$
$\displaystyle={1\over\alpha^{\prime}p^{+}}\Big{(}2\alpha^{\prime}p^{+}+4\sum_{0<n<\sqrt{3}\alpha^{\prime}p^{+}}\big{(}2\alpha^{\prime}p^{+}-\sqrt{n^{2}+(\alpha^{\prime}p^{+})^{2}}\big{)}\Big{)}$
(4.36)

It is interesting to note that in the limit of vanishing
$\alpha^{\prime}p^{+}$ the finite normal ordering constant goes to zero, in
agreement with the zero normal ordering constant for the light cone string in
flat space. On the other hand for large $\alpha^{\prime}p^{+}$, a large but
finite number of modes contribute to the normal ordering constant333The fact
that the normal ordering constant is not divergent can be understood from the
one loop ultraviolet finiteness of the Green-Schwarz string..

In the light cone gauge the level matching constraint is obtained by
considering the variation of the action with respect to $h^{\tau\sigma}$ in
terms of the bosonic and fermionic modes one finds the constraint

$\displaystyle\sum_{n=1}^{\infty}\omega_{n}^{I}\left(\alpha_{n}^{I\dagger}\alpha_{n}^{I}-\tilde{\alpha}_{n}^{I\dagger}\tilde{\alpha}_{n}^{I}\right)+\sum_{n=1}^{\infty}\sum_{\gamma=\pm}\omega^{\alpha,\gamma}_{n}\left((\beta_{n}^{\alpha,\gamma})^{\dagger}\beta_{n}^{\alpha+}-(\tilde{\beta}_{n}^{\alpha,\gamma})^{\dagger}\tilde{\beta}_{n}^{\alpha,\gamma}\right)=0$
(4.37)

The spectrum of the light cone string has several features which distinguish
it from the Penrose limit of $AdS_{5}\times S^{5}$ discussed in [8]. First due
to the absence of world sheet supersymmetry the bosonic and fermionic
worldsheet energies $\omega^{I}_{n}$ and $\omega^{\alpha,\pm}_{n}$ are not the
same. As discussed in appendix A.1 this is a consequence of the fact that
there are no ”supernumerary” [34] supersymmetries in the pp-wave background we
consider. A second difference is the presence of fermionic zero modes
associated with the $\beta_{0}^{\alpha,-},(\beta_{0}^{\alpha,-})^{\dagger}$
modes. These modes imply that even though the bosonic and fermionic
oscillators have different energies that there are an equal number of bosonic
and fermionic states at each energy level due to the 8 bosonic and 8 fermionic
states created by $(\beta_{0}^{\alpha,-})^{\dagger},\;\alpha=1,2,3,4$. The
third feature is the presence of a normal ordering constant and the dependence
on $\alpha^{\prime}p^{+}$. We discuss some possible reasons for the different
properties of the lightcone string action in the next section. Presently, we
have not been able to find an identification of the string spectrum with BMN
like operators on the field theory side.

##  5 Discussion

In this paper we have constructed a Penrose limit for type IIB warped
$AdS_{6}\times S^{2}$ solutions. We have constructed the limit by zooming in
on a null geodesic at the center of the $AdS_{6}$ and along a great circle of
the $S^{2}$. This construction only works if the geodesic is localized at the
critical point $w=w_{c}$ of the function ${\cal G}$ on the two dimensional
Riemann surface $\Sigma$. We constructed the pp-wave supergravity solution one
obtains from taking the Penrose limit. After appropriate coordinate
transformation the resulting pp-wave background is universal for all the cases
we have considered. It would be interesting to consider a more general class
of examples, such as the solutions including seven branes [35] or O7-branes
[36] and confirm that the behavior of the pp-wave at the critical point is of
the same form. Another observation is that all known regular solutions have a
unique critical point $w_{c}$ on the Riemann surface. This statement has not
been proven but checked in all cases constructed in [37]. It is interesting
that this critical point $w_{c}$ is also relevant for supersymmetric embedding
of D3-branes in the $AdS_{6}\times S^{2}$ solutions which realizes BPS co-
dimension two defects in the dual SCFT [27].

We presented the quadratic bosonic and fermionic world sheet actions of the
Green-Schwarz string in this background and calculated the spectrum of bosonic
and fermionic excitations of the string in the light cone gauge. The spectrum
has some interesting features which are different from other cases such as the
Penrose limit of $AdS_{5}\times S^{5}$. Namely that the frequencies associated
with the bosonic and fermionic creation and annihilation operators do not
coincide, which is a consequence of the absence of any ”extra” supersymmetries
which would be associated with linearly realized supersymmetries. An
additional new feature is the presence of fermionic zero modes, which makes
sure that there are equal number of fermions and bosons at each level of the
light cone hamiltonian.

One of the exciting features of the BMN correspondence was the identification
of the light cone vacuum with a special class of BPS operators in the N=4 SYM
theory, which can be viewed as a long spin chain. Furthermore, the action of
creation operators on the string vacuum can be related to the insertion of
impurities into the spin chain and diagonalizing the Hamiltonian. It is an
interesting question whether a similar identification can be found for the
light cone vacuum and excited states of the Green-Schwarz string in this plane
wave background. This identification is more challenging for two reasons.
While the Penrose limit identifies the light cone string vacuum with BPS
states which satisfy $\Delta=3J$, where $J$ is a $U(1)$ inside the $SU(2)_{R}$
symmetry of the SCFT and therefore related to protected BPS multiplet, these
theories are realized as conformal fixed points of long quiver theories and
hence strongly coupled. For $d=4,{\cal N}=4$ SYM the identification of these
operators was possible in terms of the scalar fields of the SYM theory. For
long quiver theories it is not clear how to construct these operators from the
fundamental fields of the nodes of the quiver and the bi-fundamental matter. A
class of such operators corresponding to ”stringy” long meson and baryon
operators was given in [30] which are constructed by taking products of the
bi-fundamental hyper multiplets, from one end of the quiver to the other444See
[38] for a related construction in theories dual to six dimensional SCFTs..
Since our null geodesic is at a fixed location on $\Sigma$ is is natural to
speculate that the BMN like operator would be associated with a single node in
the quiver unlike the string like operators in [30] which stretch across the
Riemann surface $\Sigma$. It would be interesting to investigate what
mechanism on the field theory side singles out a specific node in the quiver
theory. One could also investigate more general null geodesics which are not
at a fixed point on $\Sigma$ and consider the their Penrose limits. A very
preliminary discussion of this issue can be found in appendix B.

In [39, 40, 41] sphere partition functions and expectation values of BPS
Wilson line operators for five dimensional SCFTs were calculated using
localization techniques. For long quivers in the limit of large number of
nodes and large gauge groups, the eigenvalues for the nodes where replaced by
a continuous distribution and the saddle point of the path integral was
reduced to an analogue electrostatic problem. It would be interesting to see
whether these methods could be adapted to identify the BMN operators or
whether the two dimensional electrostatic problem can be related to the light
cone world sheet in some way and explain some of the curious properties of the
light cone spectrum described in section 4.3.

We leave these interesting questions for future work.

## Acknowledgements

The work of M.G. was supported, in part, by the National Science Foundation
under grant PHY-19-14412. We are grateful to the Mani L. Bhaumik Institute for
Theoretical Physics for support. We are grateful to C. Uhlemann for useful
discussions and comments on a draft of this paper.

##  Appendix A Supersymmetry of the plane wave background

The fermionic supersymmetry transformations in the string frame can be
expressed in terms of the following two operators [42], that are also used to
construct the part of the Green-Schwarz string action which is quadratic in
space time fermions.

$\displaystyle D_{\;b}^{a}$ $\displaystyle=-{1\over 4\cdot
3!}H_{mnp}\Gamma^{mnp}(\sigma^{3})_{\;a}^{b}-{1\over 4\cdot
3!}e^{\phi}(F_{mnp}-\chi H_{mnp})\Gamma^{mnp}(\sigma^{1})_{\;a}^{b}$
$\displaystyle(D_{M})_{\;b}^{a}$ $\displaystyle=\partial_{M}+{1\over
4}\omega_{mn,M}\Gamma^{mn}\delta_{a}^{b}-{1\over
8}e^{m}_{M}H_{mnp}\Gamma^{np}(\sigma^{3})_{\;a}^{b}+{1\over 8\cdot
3!}e^{\phi}(F_{npq}-\chi H_{npq})\Gamma^{npq}\Gamma_{M}(\sigma^{1})_{\;a}^{b}$
(A.1)

Here $\sigma^{a},a=1,3$ are the Pauli matrices which act on two component
Majorna-Weyl spinors555The dictionary to go from the two component (with the
matrices $\sigma$ to formulation of using complex Weyl spinors spinor is given
by $\varepsilon=\varepsilon^{1}+i\varepsilon^{2}$ and
$\sigma^{1}\varepsilon\to i\varepsilon^{*},\quad
i\sigma^{2}\to-i\varepsilon,\quad\sigma^{3}\varepsilon\to\varepsilon^{*}$..
The supersymmetry transformations in the two component formalism are given by

$\displaystyle\delta\lambda^{a}=D_{\;b}^{a}\varepsilon^{b},\quad\quad\delta\psi_{M}^{a}=(D_{M})_{\;b}^{a}\varepsilon^{b}$
(A.2)

The susy transformations for the gravitino in the string frame are is
different from the one in the Einstein frame given in [43], this is because
the transformation $g_{MN}\to e^{\phi/2}g_{MN}$ induces a mixing of the
dilatino supersymmetry transformation with the gravitino supersymmetry
transformation [44]. The supersymmetry transformations (A.2) are given in the
string frame.

The string world sheet action which is quadratic in fermions is given by

$\displaystyle S^{(2)}_{RR}={1\over 4\pi\alpha^{\prime}}\int
d^{2}\sigma\sqrt{h}\Big{(}h^{ij}\delta_{ab}-\varepsilon^{ij}(\sigma^{3})_{ab}\Big{)}\partial_{i}x^{M}\bar{\theta}^{a}\Gamma_{M}(D_{j})^{b}_{\;c}\theta^{c}$
(A.3)

where $\theta^{a}$, $a=1,2$ are the two ten dimensional Weyl-Majorana spinor
world sheet fields with the same chirality as the supersymmetry transformation
parameters $\varepsilon^{a}$ in (A.2) . The operator $(D_{j})^{b}_{c}$ is the
pullback of the covariant derivative to the world sheet is related to (A) by

$\displaystyle(D_{j})^{b}_{\;c}$
$\displaystyle=\partial_{j}\delta^{a}_{\;b}+\partial_{j}x^{M}\Big{(}{1\over
4}\omega_{mn,M}\Gamma^{mn}\delta_{a}^{b}-{1\over
8}e^{m}_{M}H_{mnp}\Gamma^{np}(\sigma^{3})_{\;a}^{b}$
$\displaystyle\quad+{1\over 8\cdot 3!}e^{\phi}(F_{npq}-\chi
H_{npq})\Gamma^{npq}\Gamma_{M}(\sigma^{1})_{\;a}^{b}\Big{)}$ (A.4)

###  A.1 Integrability of supersymmetry transformations

To examine the supersymmetries of of the background (3.9), (3.2), we will
first need to work out the spin connection.

$\displaystyle X^{\pm}={1\over 2}(T\pm X),\quad g_{+-}=g_{-+}=-2,\quad
g^{+-}=g^{-+}=-{1\over 2}$ (A.5)

The frame forms fields are given by

$\displaystyle e^{i}_{\mu}dx^{\mu}=dx^{i},\quad
e^{+}_{\mu}dx^{\mu}=dx^{+},\quad e^{-}_{\mu}dx^{\mu}=dx^{-}+{1\over
4}\sum_{k}m_{k}x_{k}^{2}dx^{+}$ (A.6)

where $m_{k}=1$ for all $x^{k}$ except for $k=6$ for which $m_{6}=3$. We can
calculate the spin connection using the Cartan structure equations

$\displaystyle de^{a}+\omega^{a}_{\;\;b}\wedge e^{b}=0$ (A.7)

The the frame forms given above the only nontrivial one is

$\displaystyle de^{-}={1\over 2}\sum_{k}m_{k}x_{k}dx^{k}\wedge dx^{+}$ (A.8)

which implies that the connection 1-form is given by

$\displaystyle\omega^{-}_{\;\;k}={1\over 2}m_{k}x^{k}dx^{+}$ (A.9)

and we find that the only non vanishing components of the spin connection are
given by

$\displaystyle\omega_{+k,+}=-m_{k}x^{k}$ (A.10)

Note that because (A.10) has two legs along the $x^{+}$ directions, the spin
connection will not contribute to the fermion action (A.3), since the light-
cone gauge $\Gamma^{+}\theta^{a}=0$ condition will make it vanish.

In the following we will present the explicit form of the supersymmetry
transformations in the string frame two component formalism following from
(A.2). For the antisymmetric tensor fields of the $T_{N}$ example (3.2) the
dilatino variation takes the form666As discussed in section 3 any Penrose
limit can be brought into the $T_{N}$ form by a simple rotation in the 7-8
plane..

$\displaystyle\delta\lambda^{a}$
$\displaystyle=\Big{(}\Gamma^{67}(\sigma^{3})^{a}_{\;b}+\Gamma^{68}(\sigma^{1})^{a}_{\;b}\Big{)}\Gamma^{+}\varepsilon^{b}$
(A.11)

and the gravitino variations are given by

$\displaystyle\delta\psi_{+}^{a}$
$\displaystyle=\partial_{+}\varepsilon^{a}+{1\over
2}\sum_{k}m_{k}x^{k}\Gamma^{k+}\varepsilon^{a}+\Gamma^{67}(\sigma^{3})^{a}_{\;b}\varepsilon^{b}-{1\over
2}\Gamma^{68}\Gamma^{+}\Gamma_{+}(\sigma^{1})^{a}_{\;b}\varepsilon^{b}$
$\displaystyle\delta\psi_{i}^{a}$
$\displaystyle=\partial_{i}\varepsilon^{a}-{1\over
2}\Gamma^{68+}\Gamma_{i}(\sigma^{1})^{a}_{\;b}\varepsilon^{b},\quad
i=1,2,3,4,5$ $\displaystyle\delta\psi_{6}^{a}$
$\displaystyle=\partial_{6}\varepsilon^{a}+\Gamma^{7+}(\sigma^{3})^{a}_{\;b}\varepsilon^{b}-{1\over
2}\Gamma^{8+}(\sigma^{1})^{a}_{\;b}\varepsilon^{b}$
$\displaystyle\delta\psi_{7}^{a}$
$\displaystyle=\partial_{7}\varepsilon^{a}-\Gamma^{6+}(\sigma^{3})^{a}_{\;b}\varepsilon^{b}-{1\over
2}\Gamma^{678+}(\sigma^{1})^{a}_{\;b}\varepsilon^{b}$
$\displaystyle\delta\psi_{8}^{a}$
$\displaystyle=\partial_{8}\varepsilon^{a}+{1\over
2}\Gamma^{6+}(\sigma^{1})^{a}_{\;b}\varepsilon^{b}$ (A.12)

It was pointed out in [34] that pp waves have $16+N_{sup}$ Killing spinors; 16
of which must occur in any background while the remaining $0\leq N_{sup}\leq
16$ so-called ”supernumerary” Killing spinors occur only in special
backgrounds. After light-cone gauge fixing, only these extra spinors give rise
to linearly-realized supersymmetries. Such linearly realized supersymmetries
also act as two dimensional world sheet supersymmetries implying a degeneracy
of the world sheet energies of bosonic and fermionic excitations. In our case,
the sixteen $\varepsilon$ which satisfy $\Gamma^{+}\varepsilon=0$ are the
”automatic” supersymmetries. It’s easy to check that for those the conditions
(A.1) are integrable as only $\delta\psi_{+}$ is not automatically vanishing
and can easily be integrated.

For the ”supernumerary” Killing spinors the vanishing of the dilatino
variation (A.11) imposes a projection condition on the supersymmetry
transformation parameters $\varepsilon$. The integrability condition in the
$i,+$ directions of $\delta\psi_{M}=0$ becomes

$\displaystyle(\partial_{+}\partial_{i}-\partial_{i}\partial_{+})\varepsilon^{a}$
$\displaystyle={1\over 2}\Gamma^{i+}\big{(}1-{1\over
2}\Gamma_{+}\Gamma^{+}\big{)}\varepsilon^{a}$ $\displaystyle={1\over
4}\Gamma^{i+}\Gamma^{+}\Gamma_{+}\varepsilon^{a}$ $\displaystyle=0$ (A.13)

Where we used the identities

$\displaystyle\Gamma^{+}\Gamma_{+}=1+\Gamma^{09},\quad\Gamma_{+}\Gamma^{+}=1-\Gamma^{09}$
(A.14)

Using the dilatino projection condition the $+,7$ and $+,8$ integrability
conditions are also satisfied. However the $+,6$ condition takes the following
form

$\displaystyle(\partial_{+}\partial_{6}-\partial_{6}\partial_{+})\varepsilon^{a}$
$\displaystyle={1\over 2}\Gamma^{6+}\big{(}\big{(}3+{1\over
2}\Gamma_{+}\Gamma^{+}\big{)}\delta^{a}_{\;b}-2i\Gamma^{78}\big{(}1+{1\over
2}\Gamma_{+}\Gamma^{+}\big{)}(\sigma^{2})^{a}_{\;b}\big{)}\varepsilon^{b}$
(A.15)

Where the factor in parenthesis is not a projector and consequently there are
no ”supernumerary” Killing spinors in our background. This is to be contrasted
with the Penrose limit of $AdS_{5}\times S^{5}$ where there are 16 additional
supersymmetries [6].

##  Appendix B Null geodesics

In this appendix we consider null geodesics in the warped $AdS_{6}\times
S^{2}$ metric

$\displaystyle ds^{2}$
$\displaystyle=f_{6}^{2}(-\cosh\rho^{2}dt+d\rho^{2}+\sinh\rho^{2}ds_{S^{4}}^{2})+f_{2}^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})+4\hat{\rho}^{2}\,|dw|^{2}~{},$ (B.1)

We consider geodesics which are located at the center of $AdS_{6}$ at $\rho=0$
and at the equator of the two sphere at $\theta={\pi\over 2}$. These choices
preserve the $U(1)$ of the two-sphere which we identify with a $U(1)$ inside
the $SU(2)_{R}$ symmetry of the dual SCFT as well as the symmetries of the
four-sphere of $AdS_{6}$, which means that the dual state has no angular
momentum.

We consider however the possibility that the geodesic moves along the Riemann
surface $\Sigma$, i.e. the coordinates $t,\phi,w$ and $\bar{w}$ depend on an
affine parameter $\lambda$ and the geodesic equation and null condition. With
these choices the geodesic equation

$\displaystyle{d^{2}x^{\mu}\over
d\lambda^{2}}+\Gamma^{\mu}_{\;\nu\rho}{dx^{\nu}\over d\lambda}{dx^{\rho}\over
d\lambda}=0$ (B.2)

takes the following form

$\displaystyle{d\over d\lambda}\Big{(}{dt\over
d\lambda}f^{2}_{6}(w,\bar{w})\Big{)}$ $\displaystyle=0$ (B.3)
$\displaystyle{d\over d\lambda}\Big{(}{d\phi\over
d\lambda}f^{2}_{2}(w,\bar{w})\Big{)}$ $\displaystyle=0$ (B.4)
$\displaystyle{d^{2}w\over d\lambda^{2}}-\left({dt\over
d\lambda}\right)^{2}{\partial_{\bar{w}}f^{2}_{6}\over
4\hat{\rho}^{2}}+\left({d\phi\over
d\lambda}\right)^{2}{\partial_{\bar{w}}f^{2}_{2}\over
4\hat{\rho}^{2}}+\left({dw\over
d\lambda}\right)^{2}{\partial_{w}\hat{\rho}^{2}\over\hat{\rho}^{2}}$
$\displaystyle=0$ (B.5) $\displaystyle{d^{2}\bar{w}\over
d\lambda^{2}}-\left({dt\over d\lambda}\right)^{2}{\partial_{w}f^{2}_{6}\over
4\hat{\rho}^{2}}+\left({d\phi\over
d\lambda}\right)^{2}{\partial_{w}f^{2}_{2}\over
4\hat{\rho}^{2}}+\left({d\bar{w}\over
d\lambda}\right)^{2}{\partial_{\bar{w}}\hat{\rho}^{2}\over\hat{\rho}^{2}}$
$\displaystyle=0\ $ (B.6)

Equations (B.3) and (B.4) can be integrated once

$\displaystyle{dt\over d\lambda}$ $\displaystyle={c_{6}\over
f_{6}^{2}},\quad{d\phi\over d\lambda}={c_{2}\over f_{2}^{2}}\ $ (B.7)

The condition that the geodesic is null $g_{\mu\nu}{dx^{\mu}\over
d\lambda}{dx^{\nu}\over d\lambda}=0$, becomes

$\displaystyle 4\hat{\rho}^{2}{dw\over d\lambda}{d\bar{w}\over
d\lambda}+{c_{2}^{2}\over f_{2}^{2}}-{c_{6}^{2}\over f_{6}^{2}}=0$ (B.8)

As discussed in section 3.1 our goal is to describe dual operators close to
the BPS bound $\Delta=3J$ which leads to a condition on the coordinates $t$
and $\phi$

$\displaystyle{\partial\phi\over\partial t}=3$ (B.9)

by identifying the generators of $t$ and $\phi$ translations with the scaling
and R symmetry generators. Using (B.7) this condition implies

$\displaystyle 3$ $\displaystyle={c_{2}\over c_{6}}{f_{6}^{2}\over
f_{2}^{2}}=9{c_{2}\over c_{6}}T^{2}$ (B.10)

Where (2.2) was used. It is easily confirmed that the choice used in the body
of the paper, where $w$ is independent of $\lambda$ and fixed at the critical
point $w=w_{c}$, is indeed a null geodesic satisfying (B.9) for
$c_{6}=3c_{2}$, due to the fact that $T|_{w=w_{c}}=1$ and
$\partial_{w}f_{2}|_{w=w_{c}}=\partial_{w}f_{6}|_{w=w_{c}}=0$.

We briefly discuss the possibility of more general geodesics where $w,\bar{w}$
depend on the affine parameter $\lambda$. The condition (B.10) implies that
$T$ should be constant along such a geodesic. We have numerically searched for
such geodesics for some examples and found that $T$ varies along the null
geodesics. This provides evidence that one has to drop either the condition
(B.9), which may make the field theory identification of the Penrose limit
challenging or consider more general trajectories on $S^{2}$. It would be very
interesting to investigate the geodesics further and determine whether they
are integrable. This is a quite challenging problem due to the complicated
dependence of all the metric factors on the coordinates of $\Sigma$

We leave the investigation of these interesting questions for future work.

##  Appendix C Plane wave limit for $AdS_{6}$ solution of massive type IIA

In [26] a solution of massive type IIA was found which is a warped product of
$AdS_{6}$ over part of a four sphere $S^{4}$. This background was constructed
by taking a near horizon limit of a D4-D8 semi-localized brane solution. In
this appendix we briefly compare the Penrose limit for this supergravity
background [45] to the one for the type IIB $AdS_{6}$ solutions described in
the body of this paper. The metric and four form antisymmetric tensor field
strength of the solution are given by

$\displaystyle ds^{2}$ $\displaystyle={9\over 2}W(\xi)\Big{(}-\cosh^{2}\rho
dt^{2}+d\rho^{2}+\sinh^{2}\rho ds_{S^{4}}^{2}+{4\over
9}\big{(}d\xi^{2}+\sin^{2}\xi ds_{S_{3}}^{2}\big{)}\Big{)}$ $\displaystyle
F_{4}$ $\displaystyle={20\sqrt{2}\over 3}(\cos\xi)^{1\over
3}\sin\xi^{3}d\xi\wedge\omega_{S_{3}},\quad\quad e^{\phi}=(\cos\xi)^{-{5\over
6}}$ (C.1)

Here the warp factor is given by $W(\xi)=\big{(}\cos\xi\big{)}^{1\over 6}$ in
the string frame. The range of the warping coordinate $\xi\in[0,\pi/2]$ means
that the three sphere warped over $\xi$ produces only half of a four-sphere.
The critical point of the warp factor of the three sphere is at $\xi=\pi/2$
and this corresponds to a Penrose limit considered in [45], which is analogous
to the one discussed in our paper. There is however one important difference
since $\xi=\pi/2$ corresponds to the $S^{3}$ along the equator of the $S^{4}$
and therefore to the strong coupling region where the supergravity
approximation breaks down. In contrast, the critical point in the type IIB
$AdS_{6}$ is a regular point where the supergravity approximation is valid for
large $N$. The same conclusion can be drawn from considering the T-dual of the
Brandhuber-Oz solution [46] for which the holomorphic functions ${\cal
A}_{\pm}$ and ${\cal G}$ are [21]

$\displaystyle{\cal A}_{\pm}={a\over 2}w^{2}\mp bw,\quad\quad{\cal G}={ab\over
3}\big{(}1+(w+\bar{w})^{3}\big{)}$ (C.2)

Here the condition $\partial_{w}{\cal G}=0$ is solved by $Im(w)=0$ which under
T-duality is mapped to $\xi=\pi/2$ on the type IIA side.

## References

  * [1] R. Penrose, _Any Space-Time has a Plane Wave as a Limit, in: Differential Geometry and Relativity. Mathematical Physics and Applied Mathematic vol 3_ , .
  * [2] R. Gueven, _Plane Waves in Effective Field Theories of Superstrings_ , _Phys. Lett. B_ 191 (1987) 275–281.
  * [3] K. Sfetsos, _Gauging a nonsemisimple WZW model_ , _Phys. Lett. B_ 324 (1994) 335–344, [hep-th/9311010].
  * [4] K. Sfetsos and A. A. Tseytlin, _Four-dimensional plane wave string solutions with coset CFT description_ , _Nucl. Phys. B_ 427 (1994) 245–272, [hep-th/9404063].
  * [5] J. M. Figueroa-O’Farrill and G. Papadopoulos, _Homogeneous fluxes, branes and a maximally supersymmetric solution of M theory_ , _JHEP_ 08 (2001) 036, [hep-th/0105308].
  * [6] M. Blau, J. M. Figueroa-O’Farrill, C. Hull and G. Papadopoulos, _Penrose limits and maximal supersymmetry_ , _Class. Quant. Grav._ 19 (2002) L87–L95, [hep-th/0201081].
  * [7] M. Blau, J. M. Figueroa-O’Farrill, C. Hull and G. Papadopoulos, _A New maximally supersymmetric background of IIB superstring theory_ , _JHEP_ 01 (2002) 047, [hep-th/0110242].
  * [8] D. E. Berenstein, J. M. Maldacena and H. S. Nastase, _Strings in flat space and pp waves from N=4 superYang-Mills_ , _JHEP_ 04 (2002) 013, [hep-th/0202021].
  * [9] R. R. Metsaev, _Type IIB Green-Schwarz superstring in plane wave Ramond-Ramond background_ , _Nucl. Phys. B_ 625 (2002) 70–96, [hep-th/0112044].
  * [10] R. R. Metsaev and A. A. Tseytlin, _Exactly solvable model of superstring in Ramond-Ramond plane wave background_ , _Phys. Rev. D_ 65 (2002) 126004, [hep-th/0202109].
  * [11] S. Mizoguchi, T. Mogami and Y. Satoh, _Penrose limits and Green-Schwarz strings_ , _Class. Quant. Grav._ 20 (2003) 1489–1502, [hep-th/0209043].
  * [12] N. R. Constable, D. Z. Freedman, M. Headrick, S. Minwalla, L. Motl, A. Postnikov et al., _PP wave string interactions from perturbative Yang-Mills theory_ , _JHEP_ 07 (2002) 017, [hep-th/0205089].
  * [13] N. Beisert, C. Kristjansen, J. Plefka, G. W. Semenoff and M. Staudacher, _BMN correlators and operator mixing in N=4 superYang-Mills theory_ , _Nucl. Phys. B_ 650 (2003) 125–161, [hep-th/0208178].
  * [14] M. Spradlin and A. Volovich, _Superstring interactions in a p p wave background_ , _Phys. Rev. D_ 66 (2002) 086004, [hep-th/0204146].
  * [15] N. R. Constable, D. Z. Freedman, M. Headrick and S. Minwalla, _Operator mixing and the BMN correspondence_ , _JHEP_ 10 (2002) 068, [hep-th/0209002].
  * [16] T. Nishioka and T. Takayanagi, _On Type IIA Penrose Limit and N=6 Chern-Simons Theories_ , _JHEP_ 08 (2008) 001, [0806.3391].
  * [17] J. Gomis and H. Ooguri, _Penrose limit of N = 1 gauge theories_ , _Nucl. Phys. B_ 635 (2002) 106–126, [hep-th/0202157].
  * [18] M. Cvetic, H. Lu and C. N. Pope, _M theory p p waves, Penrose limits and supernumerary supersymmetries_ , _Nucl. Phys. B_ 644 (2002) 65–84, [hep-th/0203229].
  * [19] M. Cvetic, H. Lu and C. N. Pope, _Penrose limits, PP waves and deformed M2 branes_ , _Phys. Rev. D_ 69 (2004) 046003, [hep-th/0203082].
  * [20] U. Gursoy, C. Nunez and M. Schvellinger, _RG flows from spin(7), CY 4 fold and HK manifolds to AdS, Penrose limits and pp waves_ , _JHEP_ 06 (2002) 015, [hep-th/0203124].
  * [21] E. D’Hoker, M. Gutperle, A. Karch and C. F. Uhlemann, _Warped $AdS_{6}\times S^{2}$ in Type IIB supergravity I: Local solutions_, _JHEP_ 08 (2016) 046, [1606.01254].
  * [22] E. D’Hoker, M. Gutperle and C. F. Uhlemann, _Warped $AdS_{6}\times S^{2}$ in Type IIB supergravity II: Global solutions and five-brane webs_, _JHEP_ 05 (2017) 131, [1703.08186].
  * [23] V. G. Kac, _Lie Superalgebras_ , _Adv. Math._ 26 (1977) 8–96.
  * [24] W. Nahm, _Supersymmetries and their Representations_ , _Nucl. Phys. B_ 135 (1978) 149.
  * [25] S. Shnider, _THE SUPERCONFORMAL ALGEBRA IN HIGHER DIMENSIONS_ , _Lett. Math. Phys._ 16 (1988) 377–383.
  * [26] A. Brandhuber and Y. Oz, _The D-4 - D-8 brane system and five-dimensional fixed points_ , _Phys. Lett. B_ 460 (1999) 307–312, [hep-th/9905148].
  * [27] M. Gutperle and C. F. Uhlemann, _Surface defects in holographic 5d SCFTs_ , 2012.14547.
  * [28] S. Minwalla, _Restrictions imposed by superconformal invariance on quantum field theories_ , _Adv. Theor. Math. Phys._ 2 (1998) 783–851, [hep-th/9712074].
  * [29] C. Cordova, T. T. Dumitrescu and K. Intriligator, _Multiplets of Superconformal Symmetry in Diverse Dimensions_ , _JHEP_ 03 (2019) 163, [1612.00809].
  * [30] O. Bergman, D. Rodríguez-Gómez and C. F. Uhlemann, _Testing AdS 6/CFT5 in Type IIB with stringy operators_, _JHEP_ 08 (2018) 127, [1806.07898].
  * [31] M. Gutperle, C. Marasinou, A. Trivella and C. F. Uhlemann, _Entanglement entropy vs. free energy in IIB supergravity duals for 5d SCFTs_ , _JHEP_ 09 (2017) 125, [1705.01561].
  * [32] J. G. Russo and A. A. Tseytlin, _On solvable models of type 2B superstring in NS NS and RR plane wave backgrounds_ , _JHEP_ 04 (2002) 021, [hep-th/0202179].
  * [33] J. Polchinski, _String theory. Vol. 1: An introduction to the bosonic string_.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 12, 2007, 10.1017/CBO9780511816079.
  * [34] M. Cvetic, H. Lu, C. N. Pope and K. S. Stelle, _Linearly realised world sheet supersymmetry in pp wave background_ , _Nucl. Phys. B_ 662 (2003) 89–119, [hep-th/0209193].
  * [35] E. D’Hoker, M. Gutperle and C. F. Uhlemann, _Warped $AdS_{6}\times S^{2}$ in Type IIB supergravity III: Global solutions with seven-branes_, _JHEP_ 11 (2017) 200, [1706.00433].
  * [36] C. F. Uhlemann, _AdS 6/CFT5 with O7-planes_, _JHEP_ 04 (2020) 113, [1912.09716].
  * [37] M. Fluder and C. F. Uhlemann, _Evidence for a 5d F-theorem_ , _JHEP_ 02 (2021) 192, [2011.00006].
  * [38] F. Baume, J. J. Heckman and C. Lawrie, _6D SCFTs, 4D SCFTs, Conformal Matter, and Spin Chains_ , _Nucl. Phys. B_ 967 (2021) 115401, [2007.07262].
  * [39] C. F. Uhlemann, _Wilson loops in 5d long quiver gauge theories_ , _JHEP_ 09 (2020) 145, [2006.01142].
  * [40] C. F. Uhlemann, _Exact results for 5d SCFTs of long quiver type_ , _JHEP_ 11 (2019) 072, [1909.01369].
  * [41] M. Fluder and C. F. Uhlemann, _Precision Test of AdS 6/CFT5 in Type IIB String Theory_, _Phys. Rev. Lett._ 121 (2018) 171603, [1806.08374].
  * [42] M. Grana, _D3-brane action in a supergravity background: The Fermionic story_ , _Phys. Rev. D_ 66 (2002) 045014, [hep-th/0202118].
  * [43] J. H. Schwarz, _Covariant Field Equations of Chiral N=2 D=10 Supergravity_ , _Nucl. Phys. B_ 226 (1983) 269.
  * [44] S. F. Hassan, _T duality, space-time spinors and RR fields in curved backgrounds_ , _Nucl. Phys. B_ 568 (2000) 145–161, [hep-th/9907152].
  * [45] Z. W. Chong, H. Lu and C. N. Pope, _Rotating strings in massive type IIA supergravity_ , hep-th/0402202.
  * [46] Y. Lozano, E. Ó Colgáin and D. Rodríguez-Gómez, _Hints of 5d Fixed Point Theories from Non-Abelian T-duality_ , _JHEP_ 05 (2014) 009, [1311.4842].


# End-Host Distribution in Application-Layer Multicast: Main Issues and
Solutions

Genge Béla and Haller Piroska  Department of Electrical Engineering
“Petru Maior” University of Târgu Mureş
Târgu Mureş, Mureş, Romania, 540088
<EMAIL_ADDRESS>
###### Abstract

Application-layer multicast implements the multicast functionality at the
application layer. The main goal of application-layer multicast is to
construct and maintain efficient distribution structures between end-hosts. In
this paper we focus on the implementation of an application-layer multicast
distribution algorithm. We observe that the total time required to measure
network latency over TCP is influenced dramatically by the TCP connection
time. We argue that end-host distribution is not only influenced by the
quality of network links but also by the time required to make connections
between nodes. We provide several solutions to decrease the total end-host
distribution time.

_Keywords— Multicast; Overlay networks; PlanetLab_

##  I Introduction

For several years now group communications have been receiving significant
attention from both the industry and scientific communities [1, 2]. The main
goal of group communication is to enable the exchange of information between
group members that can be located across the entire globe.

One of the main application of group communications is in the field of
_multicast_. Historically speaking, the first multicast applications were
implemented over the IP layer, also known as _IP multicast_ [3]. However,
after nearly a decade of research in the field of IP multicast, it was never
fully adopted because of several technical and administrative issues [4].

Later, there have been several proposals for other multicast implementations
that would be easier to deploy over the already existing and well-established
Internet protocols and would require little or no modifications in existing
routers. Such a survey of existing solutions was provided by El-Sayed et al
[5].

One of the directions that has been clearly adopted over the last few years is
_application-layer multicast_ , which implements the multicast functionality
at the application layer. The main goal of application-layer multicast is to
construct and maintain efficient distribution structures between _end-hosts_.
These structures are constructed using an _overlay_ network providing the
necessary infrastructure for data transfer between end-hosts.

Today’s research focuses on the many aspects of application-layer multicast,
including construction of overlay networks [6, 7], optimization issues [8] or
security [9]. In our previous work [10] we have addressed the problem of
optimally distributing end-hosts (i.e. EH) to overlay network hosts (i.e. OH)
in order to minimize network latency and to distribute the load of OH. Based
on a heuristic algorithm we proved that the algorithm ensures a local optimal
distribution of EH in real time and thus can be used to provide a feasible
solution to the distribution problem.

In this paper we focus on the actual deployment of the algorithm proposed in
our previous work in a real and globally-scaled distributed system:
_PlanetLab_ [11]. _PlanetLab_ is a “geographically distributed overlay network
designed to support the deployment and evaluation of planetary-scale network
services” [11]. Using PlanetLab, researchers can test their algorithms and
systems in a real environment where nodes can become unreachable, network
bandwidth can fluctuate and node processing capabilities can drop
dramatically.

In order to test the real applicability of our previously proposed algorithm
we have developed an overlay network in PlanetLab where nodes are connected in
a complete graph model. There are several advantages for using such a graph
model. First, there is no need for implementing complex routing algorithms
[12], which greatly simplifies the implementation and functionality of the
overlay. Second, maintaining routing tables is not more complex than
maintaining connections with all the other nodes. As a downside of this
topology, there is a large number of connections that must be maintained,
which grows exponentially with the number of OH. However, the simplicity of
the routing algorithms between OH makes this topology a great candidate for
using it as a leaf component in hierarchical topologies [13, 14].

Existing research [6, 7, 15] focuses on measuring the delay between nodes
after the overlay has been constructed or measuring the overlay construction
time after TCP connections are done. In deploying our algorithm we have
observed that the total time required to measure network latency over TCP is
influenced dramatically by the TCP connection time. In this paper we also
argue that end-host distribution is not only influenced by the quality of
network links but also by the time required to make connections between nodes.

The paper is structured as follows. In Section II we provide an overall
presentation of the overlay network, we discuss our previous work and we
identify the main problems for deploying the previously proposed algorithm. In
Section III we present the measurement results that were done with nodes
spread across 23 countries and we provide 3 solutions for improving the
performance of the measurements. Finally, we conclude with an overview of the
proposed solutions and we mention some future solutions that could also be
implemented.

##  II Problem Statement

The measurements that follow in the next sections are based on a complete
graph overlay topology where EH are distributed using an heuristic algorithm.
An example of such a topology is given in figure Fig. 1, where we have
illustrated the presence of 3 host types:

  * •

End-hosts (i.e. EH);

  * •

Overlay-hosts (i.e. OH);

  * •

Monitor-hosts (i.e. MH ).

EH are the producers and consumers of data transferred by the overlay
containing the OH. MH are used to monitor the load of each OH and to
distribute the connection of EH. The heuristic algorithm we proposed in our
previous work is used to distribute EH to OH in order to minimize latency and
to distribute the load of OH.

Figure 1: Multicast topology

The distribution algorithm uses the measured latency between all OH pairs, the
load of each OH and the measured latency between each EH and OH pairs. The
algorithm is run by the MH each time a new EH must be connected. At this time,
the EH must provide the MH its measurement results on the network latency it
recorded to each OH. Based on this data and the reported load received from
each OH, the MH runs the distribution algorithm.

As mentioned in our previous work, after all data is available, the algorithm
executes very fast. For instance, from the simulations we run, for 100 OH the
algorithm execution time for distributing a single EH is about 3.7 ms. This
execution time provides a real-time applicability of the proposed algorithm.

We have chosen to deploy the proposed multicast in PlanetLab because it
provides globally-available network services that can be used to run any
application type that can run on a Linux OS. From the beginning of the
implementation process we had to deal with several problems. First of all,
network connections between PlanetLab nodes or even node CPUs can be heavily
loaded, sometimes even leading to SYN_ACK timeouts for TCP connections.
Second, nodes can be rebooted at anytime by PlanetLab Central coordinators in
order to ensure a software update, for software maintenance or simply because
of some hardware problems. These problems must be handled by the MH in order
to ensure that EH are not distributed to such nodes and that already
distributed EH nodes are redistributed if necessary (i.e. on OH failure).

We also encountered several problems on the EH side. The proposed algorithm
heavily relies on the measurement data provided by EH. This means that when
joining the network, all EH must first measure the latency with all OH and
then send this data to MH. The problem with this approach is that in some
cases the response time from OH is very long, in the order of seconds as shown
in the next sections. This leads to an overall distribution time in the order
of seconds or even minutes, which is unacceptable.

##  III Measurement Issues and Solutions

###  III-A Overlay Construction Time

Although the construction of the overlay is done only once, we consider that
measuring the construction time can provide useful perspective of the time
required to re-construct the overlay in possible future developments. The
constructing of the overlay network is not made instantly. In order to
evaluate the performance and the general usability of the proposed overlay, we
have measured the time needed to construct the complete graph between the
overlay nodes.

TABLE I: Country and OH node count Country | Node count | Country | Node count
---|---|---|---
Austria | 1 | Italy | 6
Canada | 2 | Korea | 2
France | 4 | Poland | 3
Germany | 9 | Romania | 2
Greece | 1 | Spain | 2
Hungary | 1 | Switzerland | 1
Israel | 1 | US | 5

Deploying and starting applications on PlanetLab nodes can be done
automatically using applications such as _multicopy_ or _multiquery_ that are
part of the CoDeploy project [16]. These allow a parallel deployment and
execution of commands on a set of nodes. We have considered 5 settings with a
different number of OH nodes. The OH applications were deployed on nodes from
14 countries (for the maximum number of 40 OH nodes), as shown in Table I.
After starting the OH applications, each OH connects to all other OH according
to Alg. 1, where OH corresponds to the set of OH, Cout is the set of outgoing
connections and Cin is the set of incoming connections.

Algorithm 1 Complete connections for one OH

Let $t_{1}$ = @Get_curr_time()

Let $\textsf{Cout}=\phi$

{Start connection sequences}

for all $oh\in\textsf{OH}$ do

$c$ = @Start_conn_sequence($oh$)

$\textsf{Cout}=\textsf{Cout}\cup\\{c\\}$

end for

{Wait for completion}

@Wait_for_completion( Cout )

{Now eliminate duplicate connections}

Let Cin = @Get_incoming_connections()

for all $c\in\textsf{Cout}$ do

if $\exists
c^{\prime}\in\textsf{Cin}:$@Src_address($c^{\prime}$)=@Dest_address($c$) then

$(Meas_{out},Meas_{in})$=@Run_measurements($c,c^{\prime}$)

if $Meas_{out}<Meas_{in}$ then

@End_connection($c$)

$\textsf{Cout}=\textsf{Cout}\setminus\\{c\\}$

end if

end if

end for

{Calculate complete connection time}

Let $t_{2}$ = @Get_curr_time()

Let $G_{Time}=t_{2}-t_{1}$

Figure 2: Complete graph construction time

At first, each OH starts the connection process to other OH nodes. Then, it
waits for the connection process to complete. This process leads to duplicate
connections between each OH node pair. In order to eliminate duplicate
connections we measure the connection latency in each direction by sending a
single package of 1500Bytes and we eliminate the connection with the maximum
latency.

According to Alg. 1, each OH calculates a complete connection time $G_{Time}$.
The complete graph construction time is the maximum of these values, as shown
in Fig. 2. As we can see in Fig. 2 the construction of the overlay is greatly
influenced by the number of nodes. However, the variation is not linear
because the overlay also depends on other factors such as the quality of
network connections and the load of nodes. The result shown in Fig. 2 has the
following explanation. In the first OH set (i.e. 3 nodes), all 3 nodes are
located in European countries, with a minimum load. In the next OH set (i.e.
10 nodes) we have added additional nodes from Europe, one node from the US and
one node from Asia. This almost doubled the graph construction time because
the node from Asia was heavily loaded, with the CPU running at over 80% almost
all the time. In the next set (i.e. 20 nodes) we have added additional nodes
from Asia, Canada and Europe which, because of network connection latencies
and heavily loaded nodes (i.e. from Israel and Germany) has led to a quadruple
time. In the next two sets (i.e. 30 and 40 nodes) we have added additional
nodes from Europe and US, leading to the results shown in Fig. 2.

###  III-B EH Connection Measurement Issues

When EH nodes are started, each node first connects to all OH nodes in order
to measure the network latency. The measured values are then sent to the MH
that applies the heuristic algorithm developed in our previous work [10] to
determine the OH node where each EH must connect. We have identified two
components that significantly influence the measured values: connection time
and network latency. Let EH be the set of EH. Then, the total measurement time
$M_{i}$ needed to be executed by an EH is:

$M_{i}=\max_{oh_{j}}\\{Conn(eh_{i},oh_{j})+CummLat(eh_{i},oh_{j})\\}$ (1)

where $eh_{i}\in\textsf{EH}$, $i=\overline{1,|\textsf{EH}|}$ and
$oh_{j}\in\textsf{OH}$, $j=\overline{1,|\textsf{OH}|}$. $Conn$ denotes the
time needed to establish a connection between $eh_{i}$ and $oh_{j}$. $CummLat$
denotes the cumulated round-trip latency calculated by measuring the time
difference between sent and received packages:

$\displaystyle CummLat(eh_{i},oh_{j})$ $\displaystyle=$ $\displaystyle
Lat_{1}(eh_{i},oh_{j})+$ (2) $\displaystyle Lat_{2}(eh_{i},oh_{j})+$
$\displaystyle Lat_{3}(eh_{i},oh_{j})$

where $Lat_{1}$, $Lat_{2}$ and $Lat_{3}$ denote the round-trip latency of 3
packages.

We have considered several scenarios, with EH count ranging from 10 to 1000.
EH nodes were deployed on nodes from 23 countries (for the maximum number of
1000 EH nodes), as shown in Table II.

TABLE II: Country and EH node count Country | Node count | Country | Node count
---|---|---|---
Argentina | 10 | Japan | 10
Australia | 10 | Korea | 20
Austria | 40 | Netherlands | 20
Belgium | 20 | Poland | 40
Canada | 100 | Portugal | 10
China | 20 | Romania | 20
Finland | 10 | Russia | 20
France | 110 | Spain | 40
Germany | 160 | Switzerland | 10
Greece | 10 | Taiwan | 10
Hungary | 20 | US | 240
Italy | 60 |  |

Figure 3: Average EH measurement time

Each EH calculates its own $M_{i}$ value that is sent to the MH that
calculates an average measurement time, illustrated in Fig. 3. We can see that
the number of OH nodes clearly influences the overall measurement time. There
are several values that break the linear trajectory. For instance, in the case
of 40 OH nodes, when running 50 EH nodes the average time is 39382ms and when
running 100 EH nodes the average time is reduced to 21571ms. The explanation
for this behavior lies in the way that the measurements were done. Because
PlanetLab offers a set of resources over the Internet that are shared among
researchers, time measurements can change dramatically from one execution to
another. Moreover, the measurements we made span across 10 days. We have
actually seen that in one day a given node can be extremely loaded because
other researchers may also be running experiments, and the next day the node
can show a minimum load. This is in fact the expected behavior of nodes
running in a real networking environment that greatly differs from the
controlled laboratory environments.

The values shown in Fig. 3 include both the connection time and the network
latency. However, as we can see from Fig. 4, the latency is only a small part
of the measurement time, with average values ranging from 68.59ms to 925.86ms.

Figure 4: Average EH-OH measured latency

The values shown in Fig. 3 clearly show that we should improve the performance
of the measuring algorithm. At this stage, the average time needed to measure
the network latency for 1000 EH nodes in the 40 OH node setting is 89000ms,
which corresponds to almost 1.5 minutes. However, this is the average time,
which is much smaller than the maximum time needed for an EH to make the
measurements. The maximum measurement time is shown in Fig. 5, where we can
see that the maximum time needed to make the measurements is in fact 561192ms,
which is almost 9.5 minutes. The values from Fig. 5 show that the time needed
for all nodes to make the measurements are influenced by the number of OH and
by the number of EH, leading to the value of 9.5 minutes, which is
unacceptable.

Figure 5: Maximum EH measurement time

The total accessing distribution time of EH is also influenced by the response
time from the MH. In all our measurements the MH resides on a single node from
Romania. In Fig. 6 we can see the average response time from the MH.
Interestingly, the response time is not influenced by the number of OH or by
the number of EH, but by the number of simultaneous requests that are
received. EH nodes connect to MH only after completing the measurements, this
is why when a large number of EH connect simultaneously to the MH we get the
peaks from the figure. From the measurements we have also seen that after
receiving the measurement data the distribution algorithm is running under 1ms
for each request, thus the values shown in Fig. 6 are given by message
processing and network delay.

Figure 6: Average MH response time

After an EH successfully connects to the OH, it can stay connected for an
unlimited time. However, if the connection is interrupted, it will reconnect
to the designated OH. If the designated OH is no longer available, it must
execute the measurement and distribution all over again. In case of new EH
nodes, these are distributed by the MH without redistributing the already
connected EH nodes.

As mentioned earlier, in case of OH failure, disconnected EH nodes initiate a
new measurement and distribution process. However, in case of network failures
between OH nodes, a reconnect mechanism is activated for each OH node that
tries to re-establish connection with all other OH nodes, effectively trying
to reconstruct the overlay.

###  III-C EH Connection Measurement Solutions

As illustrated in the previous section, making network measurements at the
application layer is mainly influenced by the connection time between nodes.
The network latency factor, as opposed to the connection time, has a minimum
impact on the total time.

When EH use the proposed overlay, their main goal is not to make measurements
but to actually use it to effectively distribute data. The time needed to make
the measurements should thus be reduced to a minimum possible.

In this section we propose 3 solutions to the measurement problem. After
implementing them, we have repeated the measurements for the 1000 EH setup,
where the modifications would have a greater impact.

The first solution involves reducing the reconnect process count to 0, meaning
that if a connect attempt fails, the EH removes the OH from its list. EH nodes
usually try to connect over and over again to OH nodes until successful. This
process dramatically increases the overall measurement time, as shown in the
previous section. By eliminating the reconnections, we are in fact eliminating
OH that are overloaded or to which we have a poor connection. The improvements
can be immediately seen, as shown in Fig. 7. In this case, for the maximum
setting, with 40 OH nodes, the average measurement time drops from 89000ms to
22027ms, improving the overall measurement 4 times.

Figure 7: Average improved EH measurement time for 1000 EH

The problem with the first solution is that a connection must be timed out by
the OS to eliminate the OH from the solution. As a second solution we propose
an application-controlled connection timeout, opposed to network OS timeout.
In this case we timed out connections that exceeded 10 seconds, decreasing the
average measurement time from 89000ms to 12284ms and improving the overall
measurement 7 times, as shown in Fig. 7. The 10 seconds were chosen based on
the observation that a lower timeout leads to an increased number of OH nodes
eliminated from the solution. This problem is discussed in detail later in
this section.

The third solution involves partitioning the OH and EH nodes into sub-groups,
thus reducing the total number of OH/EH and the total number of EH/OH. The
partitioning can be seen in Table III. As we can see from Fig. 7, the average
time required for measurements is reduced to 6459ms for 40 OH nodes, improving
the overall measurement time over 13 times.

TABLE III: Sub-group partitioning Sub-Group | 3 OH | 10 OH | 20 OH | 30 OH | 40 OH
---|---|---|---|---|---
| 1OH/EH | 2OH/EH | 4OH/EH | 6OH/EH | 8OH/EH
Grp1 | 333 EH | 200 EH | 200 EH | 200 EH | 200 EH
Grp2 | 333 EH | 200 EH | 200 EH | 200 EH | 200 EH
Grp3 | 333 EH | 200 EH | 200 EH | 200 EH | 200 EH
Grp4 | - | 200 EH | 200 EH | 200 EH | 200 EH
Grp5 | - | 200 EH | 200 EH | 200 EH | 200 EH

The direct effect of the first two solutions is that the number of OH nodes
for which EH nodes test the connection reduces significantly with the
reduction of the timeout. For instance, by using the OS timeout, which can
range from a few seconds to a few minutes we have less eliminated OH nodes
than using a fixed timeout of 10 seconds, as shown in Fig. 8. In case of only
one connection (i.e. OS timeout) the tested percentage is 100% for 3 OH nodes,
however, this drops to 95% for 10 and 20 nodes and then rises to 96.66% for 30
nodes and to 97.43% for 40 nodes. In case of application-layer timeout we have
a 98.1% for 3 OH nodes which drops to 71.79% for 40 OH nodes.

Although the partitioning-based solution provides the best timings, it can
limit sub-groups to a set of OH nodes that may not provide the optimal
solution for the entire group. While the application-layer timeout mechanism
seems to be the next best approach, care must be taken in choosing the timeout
value because a larger connection-time does not necessarily mean that the
specific node is heavily loaded, but several other factors can also influence
this value, such as a momentarily busy OS, or a momentarily busy application.

Other solutions could also be applied, such as using UDP for determining the
network latency between EH and OH. Such a solution would eliminate the
overhead given by TCP connection. However, because the overlay uses TCP for
forwarding data, making measurements by connecting to OH nodes via TCP
provides a more precise view on the future behavior of OH nodes.

Figure 8: Average percentage of measured connections

##  IV Conclusions and Future Work

We presented several issues and solutions for deploying application-layer
overlay networks. Based on our measurements conducted over PlanetLab, a real
network testing platform, we have concluded that distributing EH nodes can not
be based only on the measured network latency, but must also include other
elements such as connection time or EH geographical location to reduce the
time required to make the actual latency measurements.

The identified problems have several solutions. In this paper we have proposed
3 such solutions: a first one that eliminates reconnections, a second one that
uses application-layer timeouts and a third one that constructs sub-groups for
reducing the number of OH/EH and EH/OH. By using these solutions we have shown
that the measurement time can be reduced up to 13 times for 1000 EH and 40 OH.

As future work, we intend to use UDP for the initial measurements. However,
special care must be taken because a lower timing for UDP packages does not
necessarily imply lower timings for TCP packages. A study must be made to
determine the correspondence between UDP and TCP timings and how could UDP-
based measurements be used to forecast the overhead introduced by TCP
connections. This study must also take into consideration UDP packet losses
that may also influence the total measurement time.

## References

  * [1] F. Bacelli, A. Chaintreau, Z. Liu, A. Riabov, and S. Sahu, “Scalability of Reliable Group Communication Using Overlays”, Proceedings of INFOCOMM, 2004.
  * [2] S. M. Venilla, and V. Sankaranarayanan, “Threat Analysis for P-LeaSel, a Multicast Group Communication Model”, Asian Journa of Information Technology, Vol. 7, 2008, pp. 64–68.
  * [3] S. Deering, and D. Cheriton, “Multicast Routing in Datagram Internetworks and Extended LANS”, ACM Transactions on Computer Systems, Vol. 8, No. 2, 1990, pp. 85–111.
  * [4] C. Diot, B.N. Levine, B. Lyles, H. Kassem, and D. Balensiefen, “Deployment issues for the IP multicast service and architecture”, IEEE Network Magazine, Vol. 14, No. 1, 2000, pp. 78–88.
  * [5] A. El-Sayed, and V. Roca, “A Survey of Proposals for an Alternative Group Communication Service”, IEEE Network, Vol. 17, No. 1, 2003, pp. 46–51.
  * [6] V. Roca, and A. El-Sayed, “A Host-Based Multicast (HBM) Solution for Group Communications”, Proceedings of the First International Conference on Networking, LNCS, Vol. 2093, 2001, pp. 610–619.
  * [7] K. Ragab, and A. Yonezawa, “A Self-organized Clustering-based Overlay Network for Application Level Multicast”, Journal of Networks, Vol. 4, No. 2, 2009, pp. 85–91.
  * [8] S. Jaggi, P. Sanders, P. A. Chou, M. Effros, S. Egner, K. Jain, and L. Tolhuizen, “Polynomial Time Algorithms for Multicast Network Code Construction”, IEEE Transactions on Information Theory, Vol. 51, No. 6, 2005, pp. 1973–1982.
  * [9] N. Shanthi, and L. Ganesan, “Security In Multicast Mobile Ad-Hoc Networks”, International Journal of Computer Science and Network Security, Vol. 8, No. 7, 2008, pp. 326–330.
  * [10] H. Piroska, and R. Balint, “Optimal server distribution in multimedia communication”, IN the Proc. of the 4th International Conference on RoEduNet, 2005, pp. 142–147.
  * [11] A. Bavier, M. Bowman, B. Chun, D. Culler, S. Karlin, S. Muir, L. Peterson, T. Roscoe, T. Spalink, and M. Wawrzoniak, “Operating System Support for Planetary-Scale Network Services”, Networked Systems Design and Implementation, 2004.
  * [12] T. L. Huang, and D. T. Lee, “A distributed multicast routing algorithm for real-time applications in wide area networks”, Journal of Parallel and Distributed Computing, Vol. 67, Issue 5, 2007, pp. 516–530.
  * [13] W. Jia, W. Tu, and J. Wu, “Hierarchical Multicast Tree Algorithms for Application Layer Mesh Networks”, Networking and Mobile Computing, LNCS, Vol. 3619, 2005, pp. 549–559.
  * [14] W. Yong, W. Seng, and H. Xianying, “A new Hierarchical Application Layer Multicast algorithm for large-scale video broadcasting”, In the Proc. of the 2nd IEEE International Conference on Computer Science and Information Technology, 2009, pp.610–613.
  * [15] S. Ratnasamy, M. Handley, R. Karp, and S. Shenker, “Application-Level Multicast Using Content-Addressable Networks”, Networked Group Communication, LNCS, Vol. 2233, 2001, pp. 14–29.
  * [16] K. Park, and V. Pai, “Deploying Large File Transfer on an HTTP Content Distribution Network”, In Proceedings of the First Workshop on Real, Large Distributed Systems (WORLDS ’04), 2004.