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#  TIT/HEP-700 April 2024 Brane expansions for
anti-symmetric line operator index

Yosuke Imamura111E-mail<EMAIL_ADDRESS>and Masato Inoue222E-mail:
<EMAIL_ADDRESS>

Department of Physics, Tokyo Institute of Technology,
Tokyo 152-8551, Japan

Based on the D5-brane realization of Wilson line operators in anti-symmetric
representations, we propose brane expansion formulas for $I_{N,k}$, the Schur
index of ${\cal N}=4$ $U(N)$ SYM decorated by line operators in the anti-
symmetric representation of rank $k$. For the large $N$ index $I_{\infty,k}$
we propose a double-sum expansion, and for finite $N$ index $I_{N,k}$ we
propose a quadruple-sum expansion. Objects causing finite $k$ and finite $N$
corrections are disk D3-branes ending on the D5-brane.

###### Contents

  1. 1 Introduction
  2. 2 Finite $k$ corrections
    1. 2.1 Large $N$ and large $k$ limit
    2. 2.2 Full expansion
    3. 2.3 Reduced expansion
    4. 2.4 Half BPS limit
  3. 3 Finite $N$ corrections
    1. 3.1 Full expansion
    2. 3.2 Reduced expansion
    3. 3.3 Half BPS limit
  4. 4 Conclusions and discussion
  5. A Character expansion method

###### Contents

  1. 1 Introduction
  2. 2 Finite $k$ corrections
    1. 2.1 Large $N$ and large $k$ limit
    2. 2.2 Full expansion
    3. 2.3 Reduced expansion
    4. 2.4 Half BPS limit
  3. 3 Finite $N$ corrections
    1. 3.1 Full expansion
    2. 3.2 Reduced expansion
    3. 3.3 Half BPS limit
  4. 4 Conclusions and discussion
  5. A Character expansion method

##  1 Introduction

Wilson lines are the most basic and important observables in gauge theories.
They serve as order parameters for detecting phase transitions. They are also
useful in gaining deeper insights into phenomena such as dualities. In this
work, we study Wilson line operators of ${\cal N}=4$ $U(N)$ SYM within the
framework of the AdS/CFT correspondence [1, 2, 3].

Wilson line operators are labeled by representations of the gauge group. We
consider $U(N)$ gauge theories, and irreducible representations are specified
by partitions $\mu$ ($\ell(\mu)\leq N$), which are depicted as Young diagrams.
The number of boxes $k=|\mu|$ in $\mu$ is referred to as the rank of the
representation.

Different holographic realizations exist for Wilson lines. Wilson line
operators in the fundamental representation ($\mu=\\{1\\}$) are realized by
fundamental strings in the AdS space [4, 5]. If the rank of the representation
is of order $N$, the coincident strings are more naturally described by
D-branes. A Wilson line in the symmetric representation with $\mu=\\{k\\}$ is
related to a tubular D3-brane with a cross section $S^{2}\subset AdS_{5}$ [6]
(See also a discussion in Section 4). For the anti-symmetric representation
with $\mu=\\{1^{k}\\}$, the corresponding object is a tubular D5-brane with a
cross section $S^{4}\subset S^{5}$ [7]. In both cases, the charge of $k$
fundamental strings is carried by the electric flux on the D-brane. If the
rank is of order $N^{2}$, the strings or D-branes deform the background
spacetime significantly, and the line operators are naturally described by
bubbling geometries [8].

Among these realizations of line operators, we specifically focus on the anti-
symmetric line operators realized by D5-branes. We investigate them by
computing the Schur index [9], which is a specialization of the superconformal
index [10, 11]. See [12, 13, 14] for analytic formulas of the Schur index
without line operator insertion. These indices are regarded as the partition
function in $S^{3}\times S^{1}$, and line operator indices [15, 16, 17, 18,
19, 20, 21] are defined as the partition functions with insertion of BPS line
operators. We insert $1/2$ BPS Wilson line operators at the poles of $S^{3}$,
thereby breaking the rotational symmetry $so(4)$ of $S^{3}$ to $so(3)$. The
R-symmetry $so(6)$ is also broken to $so(5)$. Let $J_{1}$ be the Cartan
generators of the $so(3)$, and $R_{x}$ and $R_{y}$ be the Cartan generators of
the $so(5)$. The Schur index of ${\cal N}=4$ SYM is defined using these
generators by

$\displaystyle\mathop{\rm
tr}\nolimits[(-1)^{F}q^{J_{1}}x^{R_{x}}y^{R_{y}}],\quad q=xy.$ (1)

We can regard the index with line operator insertion as a correlator of the
line operators in the spacetime $S^{3}\times S^{1}$, and we use the notation
$\langle\cdots\rangle^{U(N)}$. We can calculate the index with line operators
in a representation $\mu$ and the cojugate of a representation $\nu$ inserted
at the north and the south poles, respectively, by the localizatin formula

$\displaystyle\langle
s_{\mu}(U)s_{\nu}(U^{-1})\rangle^{U(N)}=\int_{U(N)}dU\mathop{\rm
Pexp}\nolimits(i_{\rm vec}\chi_{\rm adj}^{U(N)}(U))s_{\mu}(U)s_{\nu}(U^{-1}),$
(2)

where $i_{\rm vec}$ is the letter index of the ${\cal N}=4$ vector multiplet

$\displaystyle i_{\rm vec}=1-\frac{(1-x)(1-y)}{1-q},$ (3)

and $\chi_{\rm adj}^{U(N)}(U)$ is the $U(N)$ adjoint character

$\displaystyle\chi_{\rm adj}^{U(N)}(U)=\mathop{\rm tr}\nolimits(U)\mathop{\rm
tr}\nolimits(U^{-1}).$ (4)

$\mathop{\rm Pexp}\nolimits$ is the plethystic exponential, and
$\int_{U(N)}dU$ is the integral over the $U(N)$ gauge group manifold with the
Haar measure. $s_{\mu}(U)$ is the $U(N)$ character of an irreducible
representation $\mu$. For a diagonalized matrix $U=\mathop{\rm
diag}\nolimits(z_{1},\ldots,z_{N})$ $s_{\mu}(U)$ are symmetric polynomials of
the $N$ gauge fugacities $z_{a}$ ($a=1,\ldots,N$), which are called Schur
polynomials. We will abuse this term to refer to $s_{\mu}(U)$, too. On the
left hand side in (2) we express the line operators by the corresponding Schur
polynomials.

The line operator index we are interested in is

$\displaystyle I_{N,k}=\langle
s_{\\{1^{k}\\}}(U)s_{\\{1^{k}\\}}(U^{-1})\rangle^{U(N)}\quad(0\leq k\leq N).$
(5)

It is well known that the large $N$ index without insertion, $I_{\infty,0}$,
is reproduced as the supergravity index on the AdS side [11];

$\displaystyle I_{\infty,0}=I_{\rm sugra}=\mathop{\rm
Pexp}\nolimits\left(\frac{x}{1-x}+\frac{y}{1-y}-\frac{q}{1-q}\right).$ (6)

For the insertion of fundamental line operators, it was confirmed in [16] that
the index $I_{\infty,1}$ is reproduced by including the contribution from the
fluctuation modes on a fundamental string extended along $AdS_{2}$ analyzed in
[22].

$\displaystyle\frac{I_{\infty,1}}{I_{\rm sugra}}=I_{\rm F1}=\mathop{\rm
Pexp}\nolimits(x+y-q)=1+\frac{x}{1-x}+\frac{y}{1-y}.$ (7)

For the finite $N$ index $I_{N,0}$ without line operator insertion, the finite
$N$ corrections are reproduced by D3-branes wrapped around topologically
trivial three-cycles in $S^{5}$, called giant gravitons [23, 24]. The finite
$N$ index can be given in the form of expansions with respect to the number of
giant gravitons, and such expansions are called giant graviton expansions [25,
26, 27, 28]. See also [29, 30, 31, 32].

Such expansions were generalized to the line operator index of the fundamental
representation $I_{N,1}$ in [33]. (See also [34].)

The purpose of this paper is to propose similar holographic relations for
$I_{N,k}$ with finite $N$ and finite $k$ based on the D5-brane realization of
anti-symmetric line operators.

We will heavily rely on numerical analysis in which we treat indices as Taylor
series. We use two different Taylor series with respect to two different
variables. One is the $y$-series in the form

$\displaystyle\sum_{i=0}^{\infty}f_{i}(x)y^{i},$ (8)

and the other is the $t$-series in the form

$\displaystyle\sum_{i=0}^{\infty}f_{i}(u)t^{i},$ (9)

where $t$ and $u$ are defined by the relations

$\displaystyle x=tu,\quad y=tu^{-1}.$ (10)

In the following we mainly use the variables $x$ and $y$. When we consider
$t$-expansion of a function, we implicitly regard it as a function of $t$ and
$u$, even if it is written in terms of $x$ and $y$.

##  2 Finite $k$ corrections

###  2.1 Large $N$ and large $k$ limit

Let us first consider the large $N$ index $I_{\infty,k}$ given by the formula
[18]

$\displaystyle\frac{I_{\infty,k}}{I_{\rm sugra}}$
$\displaystyle=\sum_{\lambda\vdash k}\frac{1}{z_{\lambda}}[I_{\rm
F1}]_{\lambda}.$ (11)

(See Appendix for a derivation.) $\sum_{\lambda\vdash k}$ is the summation
over partitions of an integer $k$, and $[\cdots]_{\lambda}$ is defined by

$\displaystyle[f(x,y)]_{\lambda}=\prod_{i=1}^{\ell(\lambda)}f(x^{\lambda_{i}},y^{\lambda_{i}})$
(12)

for a function $f(x,y)$ of fugacities $x$ and $y$. $z_{\lambda}$ is defined
for a partition
$\lambda=\\{\lambda_{1},\lambda_{2},\ldots,\lambda_{\ell(\lambda)}\\}=\\{\ldots,3^{r_{3}},2^{r_{2}},1^{r_{1}}\\}$
by $z_{\lambda}=\prod_{m=1}^{\infty}m^{r_{m}}r_{m}!$. The $t$-series
expansions of $I_{\infty,k}/I_{\rm sugra}$ for small $k$ are

$\displaystyle I_{\infty,0}/I_{\rm sugra}$ $\displaystyle=1,$ $\displaystyle
I_{\infty,1}/I_{\rm sugra}$
$\displaystyle=1+2t+2t^{2}+2t^{3}+2t^{4}+2t^{5}+2t^{6}+2t^{7}+{\cal
O}(t^{8}),$ $\displaystyle I_{\infty,2}/I_{\rm sugra}$
$\displaystyle=1+2t+5t^{2}+6t^{3}+9t^{4}+10t^{5}+13t^{6}+14t^{7}+{\cal
O}(t^{8}),$ $\displaystyle I_{\infty,3}/I_{\rm sugra}$
$\displaystyle=1+2t+5t^{2}+10t^{3}+15t^{4}+22t^{5}+31t^{6}+40t^{7}+{\cal
O}(t^{8}),$ $\displaystyle I_{\infty,4}/I_{\rm sugra}$
$\displaystyle=1+2t+5t^{2}+10t^{3}+20t^{4}+30t^{5}+48t^{6}+68t^{7}+{\cal
O}(t^{8}),$ $\displaystyle I_{\infty,5}/I_{\rm sugra}$
$\displaystyle=1+2t+5t^{2}+10t^{3}+20t^{4}+36t^{5}+58t^{6}+90t^{7}+{\cal
O}(t^{8}),$ $\displaystyle I_{\infty,6}/I_{\rm sugra}$
$\displaystyle=1+2t+5t^{2}+10t^{3}+20t^{4}+36t^{5}+65t^{6}+102t^{7}+{\cal
O}(t^{8}).$ (13)

We showed unrefined index with $x,y\rightarrow t$ to save the space. We can
clearly see the convergence in the limit $k\rightarrow\infty$.

The analytic expression for the large $k$ limit is obtained as follows. Notice
that we can regard the right hand side of (11) as the partition function of
$k$ identical particles. If we take the large $k$ limit with fixed energy,
almost all particles are in the ground state (Bose-Einstein condensate), and
the number of excited particles is not constrained. The ground state is given
by the term “$1$” in (7), and particles in the ground state do not contribute
to the index. Therefore, the large $k$ limit of the index becomes [16]

$\displaystyle\lim_{k\rightarrow\infty}\frac{I_{\infty,k}}{I_{\rm sugra}}$
$\displaystyle=\mathop{\rm Pexp}\nolimits(I_{\rm F1}-1)$
$\displaystyle=1+2t+5t^{2}+10t^{3}+20t^{4}+36t^{5}+65t^{6}+110t^{7}+{\cal
O}(t^{8}).$ (14)

In the second line we again showed the unrefined index.

This result agrees with the index of the fluctuation modes on the D5-brane.
The insertion of large $k$ anti-symmetric line operators is realized on the
AdS side by introducing a D5-brane extended along $AdS_{2}\times S^{4}$ [7].
The radus of $S^{4}$ is given by $L_{\rm AdS}\sin\theta_{\rm D5}$ with the
angle $\theta_{\rm D5}$ determined by the relation

$\displaystyle\frac{k}{N}=\frac{1}{\pi}(\theta_{\rm D5}-\sin\theta_{\rm
D5}\cos\theta_{\rm D5}).$ (15)

If $k$ is of order $N$, the radius of $S^{4}$ is comparable to the AdS radius
$L_{\rm AdS}$, and we can treat the brane as a semi-classical object. Mode
analysis of the fields on the D5-brane was performed in [22, 35], and the
corresponding index

$\displaystyle I_{\rm D5}$ $\displaystyle=\mathop{\rm
Pexp}\nolimits\left(\frac{x}{1-x}+\frac{y}{1-y}\right)$ (16)

agrees with (14).

###  2.2 Full expansion

In the large $k$ limit, the index is given by (14). We are interested in the
finite $k$ correction of ${\cal O}(t^{k+1})$ in the index $I_{\infty,k}$.

$\displaystyle\frac{I_{\infty,k}}{I_{\rm sugra}I_{\rm D5}}=1+{\cal
O}(t^{k+1}).$ (17)

What are objects causing the finite $k$ correction on the AdS side?

Let ${\cal C}_{x}$ and ${\cal C}_{y}$ be the $R_{x}$-fixed locus and
$R_{y}$-fixed locus in $S^{5}$, respectively. They are homologically trivial
$S^{3}$ in $S^{5}$. In the giant graviton expansion of the Schur index without
line operator insertion [26], D3-branes wrapped around ${\cal C}_{x}$ and
${\cal C}_{y}$ give finite $N$ corrections to the index. Let us define the
latitude $\theta$ ($0\leq\theta\leq\pi$) in $S^{5}$ so that two poles
$\theta=0$ and $\theta=\pi$ are located on the intersection ${\cal
C}_{x}\cap{\cal C}_{y}$. The metric of $S^{5}$ is

$\displaystyle ds_{S^{5}}^{2}=L_{\rm AdS}^{2}(d\theta^{2}+\sin^{2}\theta
ds_{S^{4}}^{2}).$ (18)

A cross section of the tubular D5-brane is $\theta=\theta_{\rm D5}$, and the
three-cycle ${\cal C}_{x}$ is divided by the D5-brane into two parts
$0\leq\theta\leq\theta_{\rm D5}$ and $\theta_{\rm D5}\leq\theta\leq\pi$, which
we denote by ${\cal D}_{x}$ and ${\cal D}^{\prime}_{x}$. Similarly, ${\cal
C}_{y}$ is divided into two parts ${\cal D}_{y}$ and ${\cal D}^{\prime}_{y}$.
(See Figure 1.)

Figure 1: The small circle is a cross-section ($S^{4}$) of the tubular
D5-brane. The $R_{x}$-fixed locus ${\cal C}_{x}$ shown as the horizontal
circle is divided into two parts ${\cal D}_{x}$ and ${\cal D}^{\prime}_{x}$ by
the D5-brane. Similarly, the $R_{y}$-fixed locus ${\cal C}_{y}$ shown as the
vertical circle is devided into ${\cal D}_{y}$ and ${\cal D}^{\prime}_{y}$.
Some important features are not shown correctly. For example, ${\cal D}_{x}$
and ${\cal D}_{y}$ intersect not at a point but along a segment, and ${\cal
D}_{x}$ and ${\cal D}^{\prime}_{y}$ are not separated as shown but in contact
at two points.

A suggestive fact is that the right hand side of (15) is the ratio of the
volumes

$\displaystyle\frac{k}{N}=\frac{\mathop{\rm vol}\nolimits({\cal
D}_{x})}{\mathop{\rm vol}\nolimits({\cal C}_{x})}=\frac{\mathop{\rm
vol}\nolimits({\cal D}_{y})}{\mathop{\rm vol}\nolimits({\cal C}_{y})}.$ (19)

It is essentially important in the giant graviton expansion that the mass of a
D3-brane wrapped around ${\cal C}_{x}$ and ${\cal C}_{y}$ is $N/L_{\rm AdS}$,
and the corresponding operator dimension is $N$. (19) means that the mass of a
D3-brane on ${\cal D}_{x}$ and ${\cal D}_{y}$ is $k/L_{\rm AdS}$, and this
fact suggests that such D3-branes give the finite $k$ corrections to the
index.

Based on analogy with giant graviton expansions for finite $N$ corrections to
$I_{N,0}$, we propose the expansion

$\displaystyle\frac{I_{\infty,k}}{I_{\rm sugra}I_{\rm
D5}}=\sum_{m_{x},m_{y}=0}^{\infty}x^{km_{x}}y^{km_{y}}{\cal F}_{m_{x},m_{y}},$
(20)

where $m_{x}$ and $m_{y}$ are the numbers of disk D3-branes on ${\cal D}_{x}$
and ${\cal D}_{y}$, respectively. In this section we consider the large $N$
limit with finite $k$, and then $k^{\prime}=N-k$ is infinite and D3-branes on
${\cal D}^{\prime}_{x}$ and ${\cal D}^{\prime}_{y}$ are decoupled.
$k$-independent functions ${\cal F}_{m_{x},m_{y}}$ are the indices of the
theories realized on the system consisting of $m_{x}$ D3-branes on ${\cal
D}_{x}$ and $m_{y}$ D3-branes on ${\cal D}_{y}$ (and the tubular D5-brane).

Because the branes causing the finite $k$ correction are not giant gravitons
but disk D3-branes, we call (20) (and similar expansions we will see below) “a
brane expansion” rather than a giant graviton expansion.

Let us first consider the functions ${\cal F}_{m,0}$, which are the index of
the theories realized on ${\cal D}_{x}$. Such an index of theories living on a
three-dimensional disk is known as the half-index [15], and for ${\cal N}=4$
$U(m)$ SYM it is given by

$\displaystyle\Pi_{m}=\int_{U(m)}dU\mathop{\rm Pexp}\nolimits(i_{\rm
half}\chi_{\rm adj}^{U(m)}(U)),\quad i_{\rm half}=\frac{x-q}{1-q}.$ (21)

Analytic form of the integral is333We did not analytically derive this
analytic form, but only checked the agreement of Taylor expansions up to a
very high order.

$\displaystyle\Pi_{m}=\mathop{\rm
Pexp}\nolimits\left(\frac{x^{m}-q}{1-q}+\frac{x-x^{m}}{1-x}\right).$ (22)

We need to apply the variable change

$\displaystyle\sigma_{x}:(x,y)\rightarrow(x^{-1},xy)$ (23)

to translate the index of the boundary theory to the index on the theory on
the $R_{x}$-fixed locus [25, 26, 28]. Namely, ${\cal F}_{m,0}$ should be given
by ${\cal F}_{m,0}=\sigma_{x}\Pi_{m}$. With the analytic expression (22) for
$\Pi_{m}$ it is easy to calculate $\sigma_{x}\Pi_{m}$ explicitly. For small
$m$ the explicit forms of $t$-expansion of ${\cal F}_{m,0}$ are

$\displaystyle{\cal F}_{1,0}$
$\displaystyle=\frac{-x}{1-\frac{y}{x}}+\frac{x^{2}(-1+\frac{y}{x}-\frac{y^{2}}{x^{2}})}{1-\frac{y}{x}}+\frac{x^{3}(-1+\frac{y}{x}-\frac{y^{2}}{x^{2}})}{1-\frac{y}{x}}+\frac{x^{4}(-1+\frac{y}{x}-\frac{y^{2}}{x^{2}})}{1-\frac{y}{x}}+{\cal
O}(t^{5}),$ $\displaystyle{\cal F}_{2,0}$
$\displaystyle=\frac{-\frac{x^{5}}{y}}{1-\frac{y^{2}}{x^{2}}}+\frac{\frac{x^{7}}{y^{2}}(-1+\frac{y^{2}}{x^{2}}-\frac{y^{3}}{x^{3}})}{1-\frac{y}{x}}+\frac{\frac{x^{9}}{y^{3}}(-1-\frac{y}{x}-\frac{y^{2}}{x^{2}}+\frac{y^{3}}{x^{3}}-\frac{y^{5}}{x^{5}}-\frac{y^{8}}{x^{8}})}{1-\frac{y^{2}}{x^{2}}}+{\cal
O}(t^{7}),$ $\displaystyle{\cal F}_{3,0}$
$\displaystyle=\frac{-\frac{x^{12}}{y^{3}}}{1-\frac{y^{3}}{x^{3}}}+\frac{\frac{x^{15}}{y^{5}}(-1-\frac{y^{2}}{x^{2}}+\frac{y^{3}}{x^{3}}-\frac{y^{6}}{x^{6}})}{1-\frac{y^{3}}{x^{3}}}+\frac{\frac{x^{18}}{y^{7}}(-1-\frac{y^{2}}{x^{2}}-2\frac{y^{4}}{x^{4}}+\frac{y^{5}}{x^{5}}-\frac{y^{8}}{x^{8}}-\frac{y^{12}}{x^{12}})}{1-\frac{y^{3}}{x^{3}}}+{\cal
O}(t^{12})$ (24)

${\cal F}_{0,m}$ are obtained from ${\cal F}_{m,0}$ by exchanging $x$ and $y$;
${\cal F}_{0,m}={\cal F}_{m,0}|_{x\leftrightarrow y}$.

If both $m_{x}$ and $m_{y}$ are positive, we naively expect that ${\cal
F}_{m_{x},m_{y}}$ is the index of a $U(m_{x})\times U(m_{y})$ gauge theory
with bi-fundamenal fields living on the intersection ${\cal D}_{x}\cap{\cal
D}_{y}$. The intersection is a segment, and we need to carefully determine the
boudnary conditions at the ends of the segment to calculate the letter index
of the bi-fundamental fields. We will not examine the boundary conditions
here. In fact, numerical analysis shows that only ${\cal F}_{m_{x},0}$ and
${\cal F}_{0,m_{y}}$ are sufficient to reprodcue the finite $k$ index in (11),
and we do not have to include the contributions from intersecting branes. This
means ${\cal F}_{m_{x},m_{y}}=0$ for $m_{x},m_{y}\geq 1$, and strongly
suggests that the supersymmetry is spontaneously broken in the theory realized
on ${\cal D}_{x}\cup{\cal D}_{y}$.

In summary, the brane expansion of $I_{\infty,k}$ is

$\displaystyle\frac{I_{\infty,k}}{I_{\rm sugra}I_{\rm
D5}}=1+\sum_{m=1}^{\infty}x^{mk}{\cal F}_{m,0}+\sum_{m=1}^{\infty}y^{mk}{\cal
F}_{0,m}.$ (25)

We numerically confirmed that this relation holds for $k\leq 20$ up to ${\cal
O}(t^{20})$.

###  2.3 Reduced expansion

For the Schur index without line insertion the double sum expansion in [26] is
reduced to simple-sum expansion if we treat the index as a $y$-series [28].
See also [36, 37] for the mechanism of the reduction. This reduction mechanism
works for the brane expansion of the line-operator index (25). We can easily
confirm that $y$-series expansion of ${\cal
F}_{0,m}=(\sigma_{x}\Pi_{m})|_{x\leftrightarrow y}$ ($m\geq 1$) is zero, and
(25) reduces to the simple-sum expansion

$\displaystyle\frac{I_{\infty,k}}{I_{\rm sugra}I_{\rm
D5}}=\sum_{m=0}^{\infty}x^{km}{\cal F}_{m,0}.$ (26)

${\cal F}_{m,0}$ expressed as $y$-series are shown for small $m$ below.

$\displaystyle{\cal F}_{1,0}$
$\displaystyle=\frac{-x}{1-x}-y+(-\tfrac{1}{x}-1)y^{2}+(-\tfrac{1}{x^{2}}-\tfrac{1}{x})y^{3}+{\cal
O}(y^{4}),$ $\displaystyle{\cal F}_{2,0}$
$\displaystyle=\frac{x^{3}}{(1-x)(1-x^{2})}+\frac{x}{1-x}y+\frac{\frac{1}{x}+x}{1-x}y^{2}+\frac{\frac{1}{x^{3}}+\frac{1}{x}}{1-x}y^{3}+{\cal
O}(y^{4}),$ $\displaystyle{\cal F}_{3,0}$
$\displaystyle=\frac{-x^{6}}{(1-x)(1-x^{2})(1-x^{3})}+\frac{-x^{3}}{(1-x)(1-x^{2})}y+\frac{\\-1+x-x^{2}}{(1-x)^{2}}y^{2}+{\cal
O}(y^{3}).$ (27)

We numerically confirmed that (26) holds for $k\leq 20$ up to ${\cal
O}(y^{10})$ and ${\cal O}(x^{20})$.

###  2.4 Half BPS limit

We can simplify the problem by taking the half-BPS limit $y\rightarrow 0$.
Then, functions appearing in (26) take simple forms

$\displaystyle\frac{I_{\infty,n}}{I_{\rm sugra}}=\Pi_{n}=\mathop{\rm
Pexp}\nolimits(x+\cdots+x^{n}),\quad I_{\rm D5}=I_{\rm sugra}=\mathop{\rm
Pexp}\nolimits\frac{x}{1-x},$ (28)

and we can analytically confirm (26). In fact, the simple-sum brane expansion
(26) is identical with the simple-sum giant graviton expansion of the Schur
index without line insertion [28] in the half BPS limit.

##  3 Finite $N$ corrections

###  3.1 Full expansion

If $N$ is finite, in addition to disk D3-branes on ${\cal D}_{x}$ and ${\cal
D}_{y}$, ones on ${\cal D}^{\prime}_{x}$ and ${\cal D}^{\prime}_{y}$ also
contribute to the index. We expect the quadruple-sum brane expansion

$\displaystyle\frac{I_{N,k}}{I_{\rm sugra}I_{\rm
D5}}=\sum_{m_{x},m_{y},m^{\prime}_{x},m^{\prime}_{y}=0}^{\infty}x^{km_{x}+k^{\prime}m^{\prime}_{x}}y^{km_{y}+k^{\prime}m^{\prime}_{y}}{\cal
F}_{m_{x},m_{y},m^{\prime}_{x},m^{\prime}_{y}},$ (29)

where $k^{\prime}=N-k$. Functions ${\cal
F}_{m_{x},m_{y},m^{\prime}_{x},m^{\prime}_{y}}$ are not all independent but
satisfy

$\displaystyle{\cal F}_{m_{x},m_{y},m^{\prime}_{x},m^{\prime}_{y}}={\cal
F}_{m^{\prime}_{x},m^{\prime}_{y},m_{x},m_{y}}={\cal
F}_{m_{y},m_{x},m^{\prime}_{y},m^{\prime}_{x}}|_{x\leftrightarrow y}$ (30)

and by definition

$\displaystyle{\cal F}_{m_{x},m_{y},0,0}={\cal F}_{m_{x},m_{y}}.$ (31)

We have not yet fully understood how to calculate the functions ${\cal
F}_{m_{x},m_{y}.m^{\prime}_{x},m^{\prime}_{y}}$ for general wrapping numbers.
In this subsection, we will calculate the $t$-series expansion of $I_{N,k}$ on
the gauge theory side for different values of $N$ and $k$, and we extract
functions ${\cal F}_{m_{x},m_{y},m^{\prime}_{x},m^{\prime}_{y}}$ for small
wrapping numbers by assuming the expansion (29).

Let $m_{\rm tot}$ be the total wrapping number; $m_{\rm
tot}=m_{x}+m_{y}+m^{\prime}_{x}+m^{\prime}_{y}$. There are four functions for
$m_{\rm tot}=1$, and they are all related to ${\cal F}_{1,0,0,0}={\cal
F}_{1,0}=\sigma_{x}\Pi_{1}$ by (30).

For $m_{\rm tot}=2$, there are four essentially different functions

$\displaystyle{\cal F}_{2,0,0,0},\quad{\cal F}_{1,1,0,0},\quad{\cal
F}_{1,0,1,0},\quad{\cal F}_{1,0,0,1}.$ (32)

The first two are the same as the functions given in the previous section

$\displaystyle{\cal F}_{2,0,0,0}$ $\displaystyle={\cal
F}_{2,0}=\sigma_{x}\Pi_{2},$ $\displaystyle{\cal F}_{1,1,0,0}$
$\displaystyle={\cal F}_{1,1}=0,$ (33)

and the other two extracted from the comparison to $I_{N,k}$ are

$\displaystyle{\cal F}_{1,0,1,0}$
$\displaystyle=\frac{x^{2}(1+\frac{y}{x})}{(1-\frac{y}{x})^{2}}+\frac{x^{3}(2-2\frac{y}{x}+2\frac{y^{2}}{x^{2}}+2\frac{y^{3}}{x^{3}})}{(1-\frac{y}{x})^{2}}$
$\displaystyle\quad+\frac{x^{4}(3-4\frac{y}{x}+4\frac{y^{2}}{x^{2}}-6\frac{y^{3}}{x^{3}}+6\frac{y^{4}}{x^{4}}+3\frac{y^{5}}{x^{5}})}{(1-\frac{y}{x})^{2}}+{\cal
O}(t^{5}),$ $\displaystyle{\cal F}_{1,0,0,1}$
$\displaystyle=\frac{2xy}{(1-\frac{y}{x})(1-\frac{x}{y})}+\frac{2(x^{3}+y^{3})}{(1-\frac{y}{x})(1-\frac{x}{y})}$
$\displaystyle\quad+\frac{2(\frac{x^{5}}{y}+x^{4}-x^{3}y+x^{2}y^{2}-xy^{3}+y^{4}+\frac{y^{5}}{x})}{(1-\frac{y}{x})(1-\frac{x}{y})}+{\cal
O}(t^{5}).$ (34)

These are expected to be the indices of $U(1)\times U(1)$ gauge theories with
bi-fundamental fields.

The brane system corresponding to ${\cal F}_{1,0,1,0}$ consists of two disks
${\cal D}_{x}$ and ${\cal D}^{\prime}_{x}$, and they are adjacent on ${\cal
D}_{x}\cap{\cal D}^{\prime}_{x}=S^{2}$. There exists a bi-fundamenal
hypermultiplet on $S^{2}$. Indeed, we can confirm that ${\cal F}_{1,0,1,0}$ in
(34) is given by

$\displaystyle{\cal F}_{1,0,1,0}={\cal F}_{1,0}^{2}\sigma_{x}I_{\rm 3d},$ (35)

where $I_{\rm 3d}$ is the 3d index [38, 39] of a charged hypermultiplet

$\displaystyle I_{\rm 3d}=\int\frac{dz}{2\pi iz}\mathop{\rm
Pexp}\nolimits\left[i_{\rm 3d}\left(z+\frac{1}{z}\right)\right],\quad i_{\rm
3d}=\frac{y^{\frac{1}{2}}(1-x)}{1-q}.$ (36)

${\cal F}_{1,0,0,1}$ is the index of the theory realized on ${\cal
D}_{x}\cup{\cal D}^{\prime}_{y}$. Two disks ${\cal D}_{x}$ and ${\cal
D}^{\prime}_{y}$ are touching each other at two points, and bi-fundamental
fields live on the points. We find the relation

$\displaystyle{\cal F}_{1,0,0,1}=2{\cal F}_{1,0}{\cal F}_{0,1}$ (37)

holds, and identify the factor $2$ as the contribution from the bi-fundamental
fields. Indeed, a pair of bi-fundamental fermion with the letter index
$-(z+\frac{1}{z})$ gives this factor

$\displaystyle 2=\int\frac{dz}{2\pi iz}\mathop{\rm
Pexp}\nolimits\left[-\left(z+\frac{1}{z}\right)\right].$ (38)

Using these functions ${\cal F}_{m_{x},m_{y},m^{\prime}_{x},m^{\prime}_{y}}$
for $m_{\rm tot}\leq 2$, the quadruple-sum expansion (29) correctly reproduces
$I_{N,k}$ up to order ${\cal O}(t^{n})$ with
$n=\min(2k+k^{\prime},k+2k^{\prime})$.

For $m_{\rm tot}=3$, there are five essentially different functions:

$\displaystyle{\cal F}_{3,0,0,0},\quad{\cal F}_{2,1,0,0},\quad{\cal
F}_{2,0,1,0},\quad{\cal F}_{2,0,0,1},\quad{\cal F}_{1,1,1,0}.$ (39)

The first two are given by ${\cal F}_{3,0,0,0}={\cal
F}_{3,0}=\sigma_{x}\Pi_{3}$ and ${\cal F}_{2,1,0,0}={\cal F}_{2,1}=0$, and we
can extract the other three by assuming the quadruple-sum expansion (29). The
leading terms in the $t$-expansions are

$\displaystyle{\cal F}_{2,0,1,0}$
$\displaystyle=\frac{x^{4}}{(1-\frac{y}{x})^{2}(1-\frac{y^{2}}{x^{2}})}+{\cal
O}(t^{5}),$ $\displaystyle{\cal F}_{2,0,0,1}$
$\displaystyle=\frac{-x^{3}y}{(1-\frac{y}{x})^{2}(1-\frac{y^{2}}{x^{2}})}+{\cal
O}(t^{5}),$ $\displaystyle{\cal F}_{1,1,1,0}$
$\displaystyle=\frac{-xy^{3}}{(1-\frac{y}{x})^{3}}+{\cal O}(t^{5}).$ (40)

Although it would also be possible to obtain higher order terms in (40) and
functions for $m_{\rm tot}\geq 4$ by continuing a similar analysis, we stop
here the analysis of the quadruple-sum expansion (29).

###  3.2 Reduced expansion

We can reduce the quadruple-sum expansion (29) to the double-sum expansion

$\displaystyle\frac{I_{N,k}}{I_{\rm sugra}I_{\rm
D5}}=\sum_{m,m^{\prime}=0}^{\infty}x^{km+k^{\prime}m^{\prime}}{\cal
F}_{m,0,m^{\prime},0},$ (41)

by treating the index as a $y$-series. In this expansion only the
$R_{x}$-fixed locus ${\cal C}_{x}$ contributes to the index, and the summation
is taken over the two wrapping numbers $m$ and $m^{\prime}$ asscoiated with
${\cal D}_{x}$ and ${\cal D}^{\prime}_{x}$, respectively.

${\cal F}_{m,0,m^{\prime},0}$ is the index of the theory on ${\cal C}_{x}$
divided by the D5-brane to ${\cal D}_{x}$ and ${\cal D}^{\prime}_{x}$. We take
the S-duality frame in which the tubuler D5-brane becomes an NS5-brane. Then,
the gauge theory on ${\cal C}_{x}$ is the quiver gauge theory with gauge group
$U(m)\times U(m^{\prime})$. The vector multiplets live in two disks ${\cal
D}_{x}$ and ${\cal D}^{\prime}_{x}$, and the bi-fundamental hypermultiplet
lives on the wall dividing ${\cal C}_{x}$ into the two parts. The functions
${\cal F}_{m,0,m^{\prime},0}$ are given by the localization formula

$\displaystyle{\cal F}_{m,0,m^{\prime},0}$
$\displaystyle=\sigma_{x}\Bigg{[}\int_{U(m)}dU\int_{U(m^{\prime})}dU^{\prime}\mathop{\rm
Pexp}\nolimits\bigg{(}i_{\rm half}\chi_{\rm adj}^{U(m)}(U)$
$\displaystyle\quad\quad+i_{\rm half}\chi_{\rm
adj}^{U(m^{\prime})}(U^{\prime})+i_{\rm 3d}\chi^{U(m)\times
U(m^{\prime})}_{\rm bifund}(U,U^{\prime})\bigg{)}\Bigg{]},$ (42)

where the bi-fundamental character is defined by

$\displaystyle\chi^{U(m)\times U(m^{\prime})}_{\rm
bifund}(U,U^{\prime})=\mathop{\rm tr}\nolimits(U)\mathop{\rm
tr}\nolimits(U^{\prime-1})+\mathop{\rm tr}\nolimits(U^{-1})\mathop{\rm
tr}\nolimits(U^{\prime}).$ (43)

The first few terms in the $y$-series expansions of the functions for small
$m_{\rm tot}$ are

$\displaystyle{\cal F}_{1,0,1,0}$
$\displaystyle=\frac{x^{2}}{{(1-x)}^{2}}+\frac{x(3-x)}{1-x}y+\frac{5-x+x^{2}-x^{3}}{1-x}y^{2}+\mathcal{O}(y^{3}),$
$\displaystyle{\cal F}_{2,0,1,0}$
$\displaystyle=\frac{-x^{4}}{(1-x)^{2}(1-x^{2})}+\frac{x^{2}(-2-2x+x^{2})}{(1-x)(1-x^{2})}y$
$\displaystyle\quad+\frac{-2-2x+2x^{2}-2x^{3}+x^{4}}{(1-x)^{2}}y^{2}+\mathcal{O}(y^{3}),$
$\displaystyle{\cal F}_{3,0,1,0}$
$\displaystyle=\frac{x^{7}}{(1-x)^{2}(1-x^{2})(1-x^{3})}+\frac{x^{4}(2+x+2x^{2}-x^{3})}{(1-x)(1-x^{2})(1-x^{3})}y$
$\displaystyle\quad+\frac{x(2+2x+3x^{2}-x^{3}+2x^{4}+x^{6}-x^{7})}{(1-x)(1-x^{2})(1-x^{3})}y^{2}+{\cal
O}(y^{3}),$ $\displaystyle{\cal F}_{2,0,2,0}$
$\displaystyle=\frac{x^{6}}{(1-x)^{2}(1-x^{2})^{2}}+\frac{x^{3}(1+2x-x^{2})}{(1-x)^{2}(1-x^{2})}y$
$\displaystyle\quad+\frac{2x(1+3x-3x^{2}+2x^{4}-x^{5})}{(1-x)^{2}(1-x^{2})}y^{2}+{\cal
O}(y^{3}).$ (44)

See (27) for ${\cal F}_{m,0,0,0}={\cal F}_{m,0}=\sigma_{x}\Pi_{m}$.

We numerically confirmed the expansion (41) works well. For example, for $N=2$
and $k=1$, if we sum up the contributions from $m_{\rm tot}\leq 4$, the
discrepancy is

$\displaystyle\frac{I_{2,1}}{I_{\rm sugra}I_{\rm D5}}-\sum_{m+m^{\prime}\leq
4}(\cdots)={\cal O}({x}^{14})y^{0}+{\cal O}({x}^{10})y^{1}+{\cal
O}(x^{7})y^{2}+{\cal O}(x^{4})y^{3}+{\cal O}(y^{4}),$ (45)

and a lot of terms are correctly reproduced.

###  3.3 Half BPS limit

The half-BPS limit $y\rightarrow 0$ of $I_{N,k}$ for general values of $N$ and
$k$ is given by [20]

$\displaystyle
I_{N,k}=\prod_{m=1}^{k}\frac{1}{1-x^{m}}\prod_{m=1}^{N-k}\frac{1}{1-x^{m}}.$
(46)

The expansion (41) reproduces this factorized form. In the half-BPS limit the
letter index $\sigma_{x}i_{\rm 3d}$ of the hypermultiplet on the wall
vanishes, and the wall contribution becomes trivial. As the result, the
functions ${\cal F}_{m,0,m^{\prime},0}$ are factorized to the product of
${\cal F}_{m,0}$ and ${\cal F}_{m^{\prime},0}$;

$\displaystyle{\cal F}_{m,0,m^{\prime},0}={\cal F}_{m,0}{\cal
F}_{m^{\prime},0}=\sigma_{x}(\Pi_{m}\Pi_{m^{\prime}}).$ (47)

Then, the double-sum expansion (41) factorizes into two copies of the simple-
sum expansion (26) for the large $N$ index

$\displaystyle\frac{I_{N,k}}{I_{\rm sugra}I_{\rm D5}}$
$\displaystyle=\left(\sum_{m=0}^{\infty}x^{km}\sigma_{x}\Pi_{m}\right)\left(\sum_{m^{\prime}=0}^{\infty}x^{k^{\prime}m^{\prime}}\sigma_{x}\Pi_{m^{\prime}}\right)=\frac{I_{\infty,k}}{I_{\rm
sugra}I_{\rm D5}}\frac{I_{\infty,k^{\prime}}}{I_{\rm sugra}I_{\rm D5}}.$ (48)

This correctly reproduces the factorized form (46).

##  4 Conclusions and discussion

In this paper we investigated the brane expansions for the index $I_{N,k}$ of
the ${\cal N}=4$ $U(N)$ SYM with the line operators of the rank $k$ anti-
symmetric representations. Our analysis was based on the D5-brane realization
of the line-operators.

The large $N$ index $I_{\infty,k}$ with finite $k$ is given by the double-sum
brane expansion (20), and the finite $N$ index $I_{N,k}$ is given by the
quadruple-sum brane expansion (29). By taking an appropriate expansion scheme
they reduce to the simple-sum expansion (26) and the double-sum expansion
(41), respectively. The objects causing the finite $k$ and finite $N$
corrections are disk D3-branes ending on the D5-brane.

For the quadruple-sum expansion (29) for finite $N$, the brane indices ${\cal
F}_{m_{x},m_{y},m^{\prime}_{x},m^{\prime}_{y}}$ are labeled by the four
wrapping numbers, and the corresponding brane systems consist of branes
extended on the four disks ${\cal D}_{x}$, ${\cal D}_{y}$, ${\cal
D}^{\prime}_{x}$, and ${\cal D}^{\prime}_{y}$. Due to the complication of the
systems, we did not give a general formula to calculate such contributions.
One interesting fact we found is the contributions ${\cal
F}_{m_{x},m_{y}}={\cal F}_{m_{x},m_{y},0,0}$ for $m_{x},m_{y}\geq 1$ are
vanishing. This indicates the supersymmetry is spontaneously broken in the
theory realized on ${\cal D}_{x}\cup{\cal D}_{y}$. In connection with this,
another interesting fact we found is ${\cal F}_{1,1,1,0}\neq 0$. This means
that the corresponding brane system ${\cal D}_{x}\cup{\cal D}_{y}\cup{\cal
D}^{\prime}_{x}$ preserves supersymmetry even though it includes the
supersymmetry breaking configuration ${\cal D}_{x}\cup{\cal D}_{y}$ as a
subsystem. This indicates that whether the supersymmetry is broken depends on
the global structure of the brane system. This situation reminds us of the
$s$-rules in Hanany-Witten type brane systems [40]. It would be interesting to
investigate the supersymmetry breaking/preserving mechanism on the brane
systems with general wrapping numbers.

There are different directions of generalization of the results in this work.

One is generalization to multiple tubular D5-branes. In this work we focused
on the anti-symmetric Wilson lines with $\mu=\\{1^{k}\\}$. We can consider a
more general line labeled by
$\mu=\\{n^{k_{1}-k_{2}},(n-1)^{k_{2}-k_{3}},\ldots,1^{k_{n}}\\}$. The Young
diagram has $n$ columns, and the line corresponds to concentric $n$ D5-branes.
The $n$ D5-branes divide each of ${\cal C}_{x}$ and ${\cal C}_{y}$ into $n+1$
zones. If we treat the index as $y$-series, only branes on ${\cal C}_{x}$ will
contribute to the index, and we will have a gauge theory depicted as a linear
quiver with $n+1$ nodes. The full expansion will be more complicated and the
theory on a brane system is in general a quiver gauge theory with $2(n+1)$
nodes.

It is also important to find precise line operators corresponding to tubular
D3-branes, and investigate the finite $k$ and finite $N$ corrections to the
corresponding line operator index. Although it is often claimed in the
literature that the line operator in a symmetric representation is
holographically dual to a tubular D3-brane configuration found in [6],
fluctuation modes on the D3-brane analyzed in [22, 35] do not reproduce the
index of the symmetric line operators, which is identical in the large $N$
limit to the index (11) of the anti-symmetric line operator with the same rank
[18]. This means some modification is necessary for line operators dual to the
tubular D3-branes. Regardless of whatever correspond to tubular D3-branes, it
would be interesting to analyze the line-operator index for the tubular
D3-brane insertion by using brane expansions. We point out that D-strings are
strong candidates for causing finite $k$ corrections. Let us introduce a
coordinate $u$ in $AdS_{5}$ so that the metric of $AdS_{5}$ is given by

$\displaystyle ds_{AdS_{5}}^{2}=L_{\rm
AdS}^{2}(\cosh^{2}uds_{AdS_{2}}^{2}+du^{2}+\sinh^{2}uds_{S^{2}}^{2}).$ (49)

For a tubular D3-brane with string charge $k$, its worldvolume is given by
$u=u_{\rm D3}$ [6] where $u_{\rm D3}$ is given by

$\displaystyle\sinh u_{\rm D3}=\frac{k\sqrt{4\pi g_{s}N}}{4N}.$ (50)

The mass of a D-string stretched between the two poles of $S^{2}$ is

$\displaystyle T_{\rm D1}\int_{-u_{\rm D3}}^{u_{\rm D3}}L_{\rm AdS}\cosh
udu=2T_{\rm D1}L_{\rm AdS}\sinh u_{\rm D3}=\frac{k}{L_{\rm AdS}},$ (51)

and the dimension of the corresponding operator is $k$. This fact strongly
suggests that D-strings cause finite $k$ corrections to the line operator
index. In addition, D3 giants will give finite $N$ corrections. It would be
interesting to study brane expansions including the contribution from
D-strings and D3 giants.

It is expected that when the numbers of columns and rows of Young diagrams
become of order $N$ the holographic correspondent should be bubbling geometry
[8]. Recently, relation between the Schur indices of line operators with such
large representations and bubbling geometries were discussed in [21]. It is
interesting to investigate the finite $N$ corrections in such background
geometries by using brane expansions.

We hope to revisit these issues in future works.

## Acknowledgments

The authors thank A. Sei and D. Yokoyama for valuable discussions and
comments. The work of Y. I. was supported by JSPS KAKENHI Grant Number
JP21K03569.

##  Appendix A Character expansion method

In this appendix we derive some useful formulas using character expansion
method [41, 42]. Let us introduce the power sum symmetric polynomials

$\displaystyle p_{\lambda}(U)=\prod_{i=1}^{\ell(\lambda)}\mathop{\rm
tr}\nolimits U^{\lambda_{i}}=[\mathop{\rm tr}\nolimits(U)]_{\lambda}$ (52)

labeled by partitions $\lambda$, and define the index with insertion of line
operators associated with these polynomials. We start from the localization
formula (2) with $s_{\mu}$ and $s_{\nu}$ replaced by $p_{\lambda}$ and
$p_{\lambda^{\prime}}$:

$\displaystyle I_{N,\lambda,\lambda^{\prime}}$ $\displaystyle=\langle
p_{\lambda}(U)p_{\lambda^{\prime}}(U^{-1})\rangle^{U(N)}$
$\displaystyle=\int_{U(N)}dU\mathop{\rm Pexp}\nolimits(i_{\rm vec}\mathop{\rm
tr}\nolimits(U)\mathop{\rm
tr}\nolimits(U^{-1}))p_{\lambda}(U)p_{\lambda^{\prime}}(U^{-1}).$ (53)

By using the definition of the plethystic exponential we can rewrite this as

$\displaystyle
I_{N,\lambda,\lambda^{\prime}}=\int_{U(N)}dU\sum_{\lambda^{\prime\prime}}\frac{1}{z_{\lambda^{\prime\prime}}}[i_{\rm
vec}]_{\lambda^{\prime\prime}}p_{\lambda+\lambda^{\prime\prime}}(U)p_{\lambda^{\prime}+\lambda^{\prime\prime}}(U^{-1}),$
(54)

where $\lambda+\lambda^{\prime}$ is the union of two partitions $\lambda$ and
$\lambda^{\prime}$, which is given by arranging
$\\{\lambda_{1},\lambda_{2},\ldots,\lambda_{\ell(\lambda)},\lambda^{\prime}_{1},\lambda^{\prime}_{2},\ldots,\lambda^{\prime}_{\ell(\lambda^{\prime})}\\}$
in descending order. The polynomials $p_{\lambda}(U)$ are expressed as linear
combinations of the Schur polynomials $s_{\mu}(U)$ by the formula

$\displaystyle p_{\lambda}(U)=\sum_{\mu,\ell(\mu)\leq
N}\chi_{\mu}(\lambda)s_{\mu}(U)$ (55)

given by Frobenius. $\chi_{\mu}(\lambda)$ is the character of an irreducible
representation of the symmetric group $S_{n}$ ($n=|\lambda|$) corresponding to
the partition $\mu$ evaluated at a permutation of cycle type $\lambda$. With
this formula and the orthonormal relation of the Schur polynomials

$\displaystyle\int_{U(N)}dUs_{\mu}(U)s_{\nu}(U)=\delta_{\mu,\nu},$ (56)

we obtain the expression

$\displaystyle
I_{N,\lambda,\lambda^{\prime}}=\sum_{\lambda^{\prime\prime}}\frac{1}{z_{\lambda^{\prime\prime}}}[i_{\rm
vec}]_{\lambda^{\prime\prime}}\sum_{\mu,\ell(\mu)\leq
N}\chi_{\mu}(\lambda+\lambda^{\prime\prime})\chi_{\mu}(\lambda^{\prime}+\lambda^{\prime\prime}).$
(57)

In the large $N$ limit, the bound $\ell(\mu)\leq N$ in the $\mu$-sum in (57)
dissapears, and we can use the orthogonal relation of the characters
$\chi_{\mu}(\lambda)$:

$\displaystyle\sum_{\mu}\chi_{\mu}(\lambda)\chi_{\mu}(\lambda^{\prime})=z_{\lambda}\delta_{\lambda,\lambda^{\prime}}.$
(58)

Then, we obtain

$\displaystyle I_{\infty,\lambda,\lambda^{\prime}}$
$\displaystyle=\delta_{\lambda,\lambda^{\prime}}\sum_{\lambda^{\prime\prime}}\frac{z_{\lambda}}{z_{\lambda^{\prime\prime}}}[i_{\rm
vec}]_{\lambda^{\prime\prime}}=z_{\lambda}\delta_{\lambda,\lambda^{\prime}}[I_{\rm
F1}]_{\lambda}I_{\rm sugra}.$ (59)

For $\lambda=\lambda^{\prime}=\\{1\\}$ this gives (7).

By using the inverse relation to (55)

$\displaystyle
s_{\mu}(U)=\sum_{\lambda\vdash|\mu|}\frac{1}{z_{\lambda}}\chi_{\mu}(\lambda)p_{\lambda}(U),$
(60)

We can express $\langle s_{\mu}(U)s_{\nu}(U^{-1})\rangle^{U(N)}$ as a linear
combination of $I_{N,\lambda,\lambda^{\prime}}$. For $N=\infty$, combining
(59) and (60) we obtain

$\displaystyle\langle s_{\mu}(U)s_{\nu}(U^{-1})\rangle^{U(\infty)}$
$\displaystyle=I_{\rm
sugra}\sum_{\lambda\vdash|\mu|}\frac{1}{z_{\lambda}}[I_{\rm
F1}]_{\lambda}\chi_{\mu}(\lambda)\chi_{\nu}(\lambda).$ (61)

This is non-vanishing only when $|\mu|=|\nu|$.

The line operator indices for the symmetric and anti-symmetric representations
can be obtained by substituting $\sigma_{\\{1^{k}\\}}(\lambda)=\pm 1$ and
$\sigma_{\\{k\\}}(\lambda)=1$ for $\lambda\vdash k$ into (61). The common
result is shown in (11).

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# Propagation Stability Concepts for Network Synchronization Processes

Sandip Roy, Subir Sarker, and Mengran Xue The first two authors are with
School of Electrical Engineering and Computer Science, Washington State
University, Pullman, WA 99164, USA. The third author is with Raytheon BBN
Technologies. Correspondence<EMAIL_ADDRESS>
###### Abstract

A notion of disturbance propagation stability is defined for dynamical network
processes, in terms of decrescence of an input-output energy metric along
cutsets away from the disturbance source. A characterization of the
disturbance propagation notion is developed for a canonical model for
synchronization of linearly-coupled homogeneous subsystems. Specifically,
propagation stability is equivalenced with the frequency response of a certain
local closed-loop model, which is defined from the subsystem model and local
network connections, being sub-unity gain. For the case where the subsystem is
single-input single-output (SISO), a further simplification in terms of the
subsystem’s open loop Nyquist plot is obtained. An extension of the
disturbance propagation stability concept toward imperviousness of subnetworks
to disturbances is briefly developed, and an example focused on networks with
planar subsystems is considered.

##  I Introduction

The synchronization of coupled systems is common in nature and in the
engineered world. For this reason, there has been a substantial cross-
disciplinary research effort to define and characterize synchronization
phenomena in networks of coupled systems [1, 2, 3]. These studies define
synchronization in terms of the internal asymptotic stability of a manifold,
on which each coupled system has an identical state or output. However,
synchronization in a practical sense requires not only that the coupled
systems come to a common state/output, but also that this equilibrium is
impervious to external disturbances. Based on this recognition, a body of
recent work has considered the disturbance responses of network
synchronization processes [4, 5, 6]. Broadly, these efforts model disturbances
as impinging on all or a subset of subsystems within a network, and evaluate
their potential impacts on network-wide synchrony according to a performance
metric (typically, a $H_{2}$ or $H_{\infty}$ gain). The studies in this
direction have largely focused on evaluating the metrics from a graph-
theoretic perspective, and designing controllers to bound the performance
metrics within a threshold.

Engineers working with large-scale built networks often do not think about
coordination in terms of either internal stability characteristics or global
(network-wide) disturbance response metrics. Rather, they are concerned with
the extent of propagation of local disturbances, whether arising from
exogenous inputs or state deviations. For instance, bulk power grid operators
often distinguish well-damped networks where oscillatory disturbance responses
remain localized from poorly-damped ones where such disturbances have network-
wide impact [7]. Similar assessments of network performance in terms of
disturbance propagation are of interest in disciplines ranging from air
traffic management to infectious-disease epidemiology and cyber-security [8].

The spatial propagation of disturbances among interconnected systems in
cascade or line-topology configurations has been extensively researched under
the heading of string stability [9, 10]. The string stability concept has also
been extended to general directional networks in the mesh stability literature
[11]. The recent study [12] has recognized the need for disturbance-
propagation stability notions for general (bi-directionally connected)
dynamical networks, and therefore has proposed a definition for network
stability in terms of boundedness of input-to-state or input-to-output gains
which parallels the basic string stability definition. It has also introduced
an alternate network stability definition for tree-like graphs which captures
monotonic decresence of the propagative response away from the disturbance
source; this definition is a generalization of the strict string stability
concept [10]. Other recent studies have also examined stability notions for
linear and nonlinear network processes, which are concerned with disturbance
propagation (e.g., [13]). Additionally, a number of recent studies have
characterized local-input-to-output properties of dynamical networks (e.g.,
gains, zeros), without however explicitly considering propagation [14, 15].
However, definitions for propagation stability are not yet mature, and the
formal analysis of propagation is incomplete even for networks of coupled
linear systems.

The purpose of this article is to define and characterize a notion of
disturbance propagation stability in the context of a canonical network model
for synchronization of coupled homogeneous linear subsystems. The main
contributions of the study are two-fold:

1) A general definition is introduced for (strict) propagation stability,
based on decrescence of response norms across cutsets in the network’s
digraph, or equivalently along paths away from the disturbance source.

2) Propagation stability is characterized in terms of the closed-loop
frequency responses of the subsystem model with a proportional feedback
controller applied. For the case of single-input single-output subsystems,
these conditions are further simplified to conditions on the subsystem’s open-
loop frequency response and/or transfer function.

The analysis of propagation stability depends on a new, local characterization
of synchronization models. This characterization is markedly different from
the spectral decomposition that has been exhaustively used to understand both
internal and global external stability [1, 2], and provides a further
assessment/categorization of synchronization processes.

The remainder of the article is organized as follows. The network
synchronization model is described in Section II, and disturbance propagation
stability notions are defined in Section III. The main characterizations of
propagation stability are presented in Section IV. Finally, in Section V, an
example is given which focuses on the impact of damping on propagation
stability when the subsytems are planar devices.

##  II Model

A network with $N$ identical, interconnected devices or nodes or subsystems,
labeled $1,\ldots,N$, is considered. Each subsystem $i\in 1,\ldots,N$ has a
state ${\bf x}_{i}\in R^{n}$ and output ${\bf y}_{i}\in R^{m}$ which are
governed by the following linear or linearized state-space equations:

$\displaystyle\dot{\bf x}_{i}=A{\bf x}_{i}+B\left(\alpha\sum_{j\neq
i}g_{ij}({\bf y}_{j}-{\bf y}_{i})+\gamma_{i}{\bf w}_{i}\right)$ (1)
$\displaystyle{\bf y}_{i}=C{\bf x}_{i}.$

Here, $A$, $B$, and $C$ are a subsystem’s state, input, and output matrices,
respectively; the scalars $g_{ij}\geq 0$ are coupling weights; the vector
${\bf w}_{i}\in R^{m}$ represents an external disturbance input at subsystem
$i$; and $\alpha$ is a global coupling-strength parameter which allows tuning
of the network connectivity (see [1]). Our focus here is on an external
disturbance impinging on a single source node $s\in 1,\ldots,N$, which is
modeled by setting $\gamma_{s}=1$ and $\gamma_{i}=0$ for $i\neq s$.

The model (1) is a standard representation for the small-signal dynamics of
synchronizing coupled oscillators [1, 2], with two distinctions. First, a
disturbance is applied at a single subsystem, to allow for analysis of
propagative impacts. Second, the subsystems are modeled as being
interconnected through commensurately-dimensioned inputs and outputs, rather
than through an explicit inner-coupling term or alternately through a
designable protocol. This format is used to stress that the network is made up
of input-output devices with fixed connections, but the formulation
encompasses the scenarios with an inner coupling or a designable protocol.

Analyses of (1) often are phrased in terms of a graph that represents the
network interconnections. For our development, a weighted digraph $\Gamma$ is
defined with $N$ vertices corresponding to the $N$ subsystems. A directed edge
is drawn from vertex $j$ to vertex $i$ if $g_{ij}>0$, reflecting a direct
influence of the output of subsystem $j$ on the state evolution of subsystem
$i$. The edge is assigned a weight of $g_{ij}$. We use the notation ${\cal V}$
for the set of vertices, and ${\cal E}$ for the set of edges. Additionally, it
is convenient to define an (asymmetric) Laplacian matrix $L=[l_{ij}]\in
R^{N\times N}$. Each off-diagonal entry $l_{ij}$ is given by $-g_{ij}$, while
the diagonal entries are selected so that each row sums to $0$ (i.e.
$l_{ii}=\sum_{j\neq i}g_{ij}$).

For the model (1), the synchronization manifold where the states ${\bf
x}_{1},\ldots,{\bf x}_{n}$ are identical is known to be asympotically stable
under broad conditions on the network graph, the subsystem model, and the
coupling-strength parameter. More precisely, stability can be related to
Hurwitz stability of the $N$ complex matrices $A+\lambda_{i}BC$, where
$0=\lambda_{1},\lambda_{2},\ldots,\lambda_{N}$ are the eigenvalues of the
Laplacian matrix $L$. From this analysis, stability can be distilled to a
simple test on the Laplacian matrix’s spectrum via the master stability
function construct, see e.g. [1] for details. A number of other
characteristics of (1), including global disturbance stability and
controllability via external stimulation, can also be related to the matrices
$A+\lambda_{i}BC$ [4, 16].

From here on we refer to the model (1) as the network synchronization model.
The model is approximative of a number of synchronization phenomena, including
the swing dynamics of the bulk power grid, multi-vehicle formation flight, and
the nonlinear dynamics of electrical oscillator networks.

##  III Propagation Stability Definition

A notion of propagation stability is defined based on the spatial patterns of
output energies (squared two norms) at network subsystems over a time interval
$[0,T]$, when an exogenous disturbance input $w_{s}(t)$ is applied at a single
node. In our development, the disturbance is assumed to satisfy the Dirichlet
conditions (absolute integrability over any period, finite number of
discontinuities and minima/maxima, bounded over any interval), but otherwise
may be arbitrary. In defining stability, the squared two-norm metric
$E_{i}(T)=\int_{t=0}^{T}{\bf y}_{i}^{T}(t){\bf y}_{i}(t)\,dt$ is considered
for each network subsystem $i$. Conceptually, the network can be viewed as
propagation stable, if these energies are attenuated away from the disturbance
source with respect to the network graph. However, since the network’s graph
in general has a spatially inhomogeneous structure, defining attenuation
requires some care.

Figure 1: Illustration of the propagation stability concept.

One natural way to assess disturbance propagation is to consider the energy
metric $E_{i}(T)$ for vertex-cutsets in the network’s graph (i.e., sets of
vertices whose removal partition the graph). If the metric value for at least
one cutset vertex is larger than the metric values for vertices that are
separated from the source by the cutset, then the response can be viewed as
being attenuated away from the source (see Figure 1). To formalize this
notion, let us consider a set of vertices ${\cal V}_{C}\in{\cal V}$. We refer
to ${\cal V}_{C}$ as a separating cutset for the source $s$, if the remaining
vertices ${\cal V}\setminus{\cal V}_{C}$ can be partitioned into two subsets
${\cal V}_{1}$ and ${\cal V}_{B}$ such that: 1) there are no edges from
vertices in ${\cal V}_{1}$ to vertices in ${\cal V}_{B}$ and 2) ${\cal
V}_{A}={\cal V}_{1}\cup{\cal V}_{C}$ contains the source vertex $s$. Using
this notation, the following definition for propagation stability is proposed:

###### Definition 1

The network synchronization model is propagation stable if the following two
conditions hold.

  1. 1.

The synchronization manifold is asymptotically stable in the sense of
Lyapunov.

  2. 2.

For every source location $s$, disturbance signal ${\bf w}_{s}(t)$, separating
cutset ${\cal V}_{C}$ for $s$, time horizon $T>0$, and vertex $b\in{\cal
V}_{B}$, the following majorization holds: $E_{b}(T)\leq\max_{c\in{\cal
V}_{C}}E_{c}(T)$.

The definition captures that the output signal energy for a subsystem
associated with a graph cutset upper bounds the signal energy for all
subsystems beyond the cutset, and hence the output signal energy is attenuated
at cutsets away from the source. An immediate consequence of the definition is
that the maximum response energy among subsystems at a distance $r$ from the
source location on the network graph, say $E(r)$, is a non-increasing function
of $r$.

With some further thought, one also sees that propagation stability can be
equivalently expressed in terms of paths in the network graph. In particular,
the network model is propagation stable if and only if there is at least one
path in the network graph from the disturbance source to each other vertex
such that the response energy is non-increasing along the path. From this
equivalent form, it is evident that propagation stability is a generalization
of the classical strict string-stability definition and a recent definition
for (strict) network stability in tree-like networks [10, 12], in that
decrescence or attenuation along paths away from the disturbance source is
enforced. We stress that our definition only requires attenuation along one
path for each pair of vertices, so as to encompass the varying attenuation
patterns that may arise in inhomogeneously-structured networks.

In some circumstances, the propagation stability definition may be too rigid
to capture attenuative dynamics in a network. In particular, it is possible
that small regions in a network may be susceptible to disturbance
amplification, but a substantial portion of the network nevertheless
attenuates disturbances. This partial notion of propagation attenuation is
captured in the following definition:

###### Definition 2

Consider a set ${\cal V}_{D}$ of vertices in the network graph (respectively,
subsystems in the network model). Assume that the induced subgraph of $\Gamma$
defined by ${\cal V}_{D}$ is strongly connected. The subnetwork defined by
${\cal V}_{D}$ is said to be propagation impervious if the following two
conditions hold.

  1. 1.

The synchronization manifold for the network synchronization model is
asymptotically stable in the sense of Lyapunov.

  2. 2.

For every source location $s$, disturbance signal ${\bf w}_{s}(t)$, separating
cutset ${\cal V}_{C}$ for $s$ contained within ${\cal V}_{D}$, time horizon
$T>0$, and vertex $b\in{\cal V}_{B}\cap{\cal V}_{D}$, the following
majorization holds: $E_{b}\leq\max_{c\in{\cal V}_{C}}E_{c}$.

The definition asserts that disturbances that enter the propagation impervious
subnetwork (whether from an outside or an inside source) then exhibit a
spatial attenuation within that subnetwork. The definition for propagation
imperviousness also aligns with concepts in the strict string stability
literature [12], which allow for amplification in a radius around the
disturbance source provided that attenuation is guaranteed elsewhere.

##  IV Propagation Stability Analysis

Propagation stability is concerned with the ability of network subsystems to
attenuate impinging disturbances, such that a disturbance response falls off
spatially in the network regardless of the source. Thus, one might expect
propagation stability to be related to local characteristics of the network
synchronization model, specifically the structure of each subsystem and its
interconnections with its neighbors. In the following development, we present
several conditions for propagation stability and imperviousness, which are
phrased in terms of local (subsystem-level) frequency response
characteristics. Formally, these conditions are expressed in terms of the
following Laplace-domain local feedback transfer matrices $H_{i}(s)$, defined
for each subsystem $i\in 1,\ldots,N$:

$H_{i}(s)=(\alpha\sum_{j\in{\cal N}(i)}g_{ij})C(sI-A+\alpha\sum_{j\in{\cal
N}(i)}g_{ij}BC)^{-1}B.$ (2)

Here, the notation ${\cal N}_{i}$ refers to the set of incoming neighbors of
vertex $i$ in the graph, i.e. the set containing vertices $j$ such that
$g_{ij}>0$.

The first main result of our development is a sufficient condition for
propagation stability:

###### Theorem 1

The network synchronization is propagation stable if: 1) the synchronization
manifold is asympotically stable in the sense of Lyapunov and 2)
$\sup_{\omega}\sigma_{max}(H_{i}(j\omega))\leq 1$ for all $i=1,\ldots,N$.

###### Proof:

The synchronization manifold is asymptotically stable by assumption, hence the
first criterion for propagation stability is met.

To verify that the second criterion for propagation stability is met, first
consider the zero-state output ${\bf y}_{i}(t)$ over the interval $[0,T]$ for
any subsystem $i$ other than the source $s$. We notice that, if ${\bf
y}_{j}(t)$ for $j\in{\cal N}(i)$ is known over the interval $[0,T]$, this is
sufficient to compute ${\bf y}_{i}(t)$ over the interval. Specifically, ${\bf
y}_{i}(t)$ can be computed over $[0,T]$ by first solving the following
equation for $\overline{\bf y}_{i}(t)$:

$\displaystyle\dot{\overline{\bf x}}_{i}=A\overline{\bf
x}_{i}+B\left(\alpha\sum_{j\neq i}g_{ij}(\overline{\bf y}_{j}-\overline{\bf
y}_{i})+\gamma_{i}{\bf w}_{i}\right)$ (3) $\displaystyle\overline{\bf
y}_{i}=C\overline{\bf x}_{i}$

where (without any loss of information) $\overline{\bf y}_{j}(t)$ for
$j\in{\cal N}(i)$ is set equal to the disturbance response ${\bf y}_{j}(t)$
(for Equation 1) over the interval $[0,T]$ and is set to zero for $t\geq T$,
and the initial conditions are assumed to be zero. The disturbance response
${\bf y}_{i}(t)$ is then equal to $\overline{\bf y}_{i}(t)$ over the interval
$[0,T]$,

Transforming (3) to the Laplace domain and solving for $\overline{Y}_{i}(s)$
yields:

$\overline{Y}_{i}(s)=H_{i}(s)\sum_{j\in{\cal
N}(i)}\frac{g_{ij}}{\sum_{j\in{\cal N}(i)}g_{ij}}\overline{Y}_{j}(s),$ (4)

where the notation $\overline{Y}_{j}(s)$ is used for the Laplace transform of
$\overline{\bf y}_{j}(t)$. Thus, $\overline{\bf y}_{i}(t)$ can be found by
filtering the signal $\overline{\bf z}_{i}(t)=\sum_{j\in{\cal
N}(i)}\frac{g_{ij}}{\sum_{j\in{\cal N}(i)}g_{ij}}\overline{\bf y}_{j}(t)$ with
the transfer function $H_{i}(s)$. From this relationship, the energy
$E_{i}(T)$ can be bounded in terms of $E_{j}(T)$, $j\in{\cal N}(i)$, as
follows. First:

$\displaystyle E_{i}$ $\displaystyle=$ $\displaystyle\int_{t=0}^{T}{\bf
y}_{i}^{T}(t){\bf y}_{i}(t)\,dt=\int_{t=0}^{T}\overline{\bf
y}_{i}^{T}(t)\overline{\bf y}_{i}(t)\,dt$ $\displaystyle\leq$
$\displaystyle\int_{t=0}^{\infty}\overline{\bf y}_{i}^{T}(t)\overline{\bf
y}_{i}(t)\,dt.$

Then, from Parseval’s theorem, it follows that:

$\int_{t=0}^{\infty}\overline{\bf y}_{i}^{T}(t)\overline{\bf
y}_{i}(t)\,dt=\frac{1}{2\pi}\int_{\omega=-\infty}^{\infty}\overline{Y}_{i}^{T}(j\omega)\overline{Y}_{i}(j\omega)\,d\omega$
(6)

From the relationship between $\overline{Y}_{i}(s)$ and $\overline{Z}_{i}(s)$,
this can be further characterized as:

$\int_{t=0}^{\infty}\overline{\bf y}_{i}^{T}(t)\overline{\bf
y}_{i}(t)\,dt=\frac{1}{2\pi}\int_{\omega=-\infty}^{\infty}\overline{Z}_{i}^{T}(j\omega)H_{i}^{T}(j\omega)H_{i}(j\omega)\overline{Z}_{i}(j\omega)\,d\omega$
(7)

However, since the maximum singular value of $H_{i}(j\omega)$ is assumed to be
less than or equal to $1$, it is immediate that

$\displaystyle\int_{t=0}^{\infty}\overline{\bf y}_{i}^{T}(t)\overline{\bf
y}_{i}(t)\,dt$ $\displaystyle\leq$
$\displaystyle\frac{1}{2\pi}\int_{\omega=-\infty}^{\infty}\overline{Z}_{i}^{T}(j\omega)\overline{Z}_{i}(j\omega)\,d\omega$
$\displaystyle=$ $\displaystyle\int_{t=0}^{\infty}\overline{\bf
z}_{i}^{T}\overline{\bf z}_{i}\,dt,$

where the final equality is based on another application of Parseval’s
theorem. Then, noting that the signal ${\bf z}_{i}(t)$ is non-zero only on
$[0,T]$ and exploiting convexity, it follows that:

$\displaystyle\int_{t=0}^{\infty}\overline{\bf z}_{i}^{T}\overline{\bf
z}_{i}\,dt$ $\displaystyle=\int_{t=0}^{T}\overline{\bf z}_{i}^{T}\overline{\bf
z}_{i}\,dt$ (9) $\displaystyle\leq\sum_{j\in{\cal
N}(i)}\frac{g_{ij}}{\sum_{j\in{\cal N}(i)}g_{ij}}\int_{t=0}^{T}\overline{\bf
y}_{j}^{T}\overline{\bf y}_{j}\,dt$ (10) $\displaystyle\leq\max_{j\in{\cal
N}(i)}\int_{t=0}^{T}\overline{\bf y}_{j}^{T}\overline{\bf
y}_{j}dt=\max_{j\in{\cal N}(i)}E_{j},$ (11)

where the last equivalence depends on recognizing that $\overline{\bf
y}_{j}={\bf y}_{j}$ over the interval $[0,T]$, Combining these inequalities
and equivalences, we get:

$E_{i}(T)\leq\max_{j\in{\cal N}(i)}E_{j}(T).$ (12)

Here, Equation (12) is valid for any subsystem $i$ other than the source $s$,
and for any interval $T$. Also, the inequality is strict unless all neighbors
in $j\in{\cal N}(i)$ have identical $E_{j}(T)$, since the second inequality in
(11) is strict except in this circumstance.

The second criterion for propagation stability can be proved through iterative
application of Equation (12). To develop the proof, consider any cutset ${\cal
V}_{C}$ and vertex $b\in{\cal V}_{B}$. Define a set ${\cal R}$ which initially
contains only the vertex $b$. Then, add to ${\cal R}$ all incoming neighbors
$i$ of $b$ which satisfy $E_{i}(T)\geq E_{b}(T)$; from Equation (12), there
exists at least one such vertex. If the new set ${\cal R}$ includes a vertex
in ${\cal V}_{C}$, then the majorization in the theorem statement is proven.
Otherwise, ${\cal R}$ can again be augmented to include additional vertices
$i$ which satisfy $E_{i}(T)\geq E_{b}(T)$; from (12) and the strictness
argument presented thereafter, at least one such vertex must exist. This
process can be iterated until at least one vertex in ${\cal V}_{C}$ is
included. Therefore, $E_{i}(T)\geq E_{b}(T)$ for some $i\in{\cal V}_{C}$, or
equivalently $E_{b}(t)\leq max_{i\in{\cal V}_{C}}E_{i}$. Since this inequality
holds for all source locations $s$, time horizons $T$, separating cutsets
${\cal V}_{C}$, and vertices $b\in{\cal V}_{B}$, the theorem is verified.

∎

Theorem 1 is tight in a certain sense, for the case that the subsystem model
is single-input single-output (SISO). Specifically, for this case, consider
that the condition of the theorem is not met, i.e.
$\max_{\omega}\sigma_{max}(H_{i}(j\omega))=\max_{\omega}|H_{i}(j\omega)|>1$
for some vertex $i$. Then for some network graphs, the network synchronization
model will not be propagation stable. To see why, consider a network graph for
which the vertex $i$ has only a single incoming edge, say from vertex $j$. It
is easy to check in this case that $Y_{i}(s)=H_{i}(s)Y_{j}(s)$, i.e. the
disturbance response at vertex $i$ is the filtration of the disturbance
response at vertex $j$ by $H_{i}(s)$. Then consider a single tone disturbance
$w_{s}(t)$ (where $s\neq i$) at a frequency $\omega$ such that
$|H_{i}(j\omega)|>1$. For this disturbance, the response at each subsystem is
also asymptotically a single-tone frequency at the same frequency $\omega$.
Since $|H_{i}(j\omega)|>1$, it thus follows that the amplitude of the
sinusoidal response at vertex $i$ is larger than that at vertex $j$. Thus, for
a sufficiently long time horizon $T$, $E_{i}(T)>E_{j}(T)$. Therefore, since
vertex $j$ is a cutset that separates the source vertex $s$ from vertex $i$,
the network model is not propagation stable. If the subsystem model is multi-
input multi-output (MIMO), then characterization of response amplitudes for a
single tone input is more complex, because the response vector at a subsystem
may not coincide with the maximum amplification direction. However, the
network certainly may be susceptible to disturbance amplification if
$|H_{i}(j\omega)|>1$.

Conditions for propagation imperviousness in a region of the network graph can
also be developed using a parallel argument to the proof of Theorem 1. Here is
the result:

###### Theorem 2

Consider a network synchronization model. For this model, consider a set
${\cal V}_{D}$ of the vertices in the network graph (respectively, subsystems
in the network model), such that the induced subgraph defined by ${\cal
V}_{D}$ is strongly connected. The subnetwork defined by ${\cal V}_{D}$ is
propagation impervious if the following two conditions hold: 1) the
synchronization manifold is asympotically stable in the sense of Lyapunov and
2) $\sup_{\omega}\sigma_{max}(H_{i}(j\omega))\leq 1$ for all $i\in{\cal
V}_{D}$.

Since the proof is analogous to that of Theorem 1, it is omitted.

Since propagation stability depends critically on the gains of the local
transfer matrices $H_{i}(s)$, it is useful to further interpret these matrices
from a system-theoretic standpoint. Of interest, the transfer matrices can be
interpreted as the closed-loop reference-signal-to-output transfer function
when a certain static feedback controller is applied to the subsystem model.
Specifically, the local transfer matrix $H_{i}(s)$ is the transfer function
from $r(t)$ to $y(t)$ of the following closed-loop system model:

$\displaystyle\dot{\bf x}=A{\bf x}+B{\bf u}$ (13) $\displaystyle{\bf y}=C{\bf
x}$ $\displaystyle{\bf u}=(\alpha\sum_{j\neq i}g_{ij})({\bf r}-{\bf y}).$ (14)

The local transfer matrices are thus seen to capture the closed-loop dynamics
of the subsystem model, when an identical proportional feedback controller
with gain $k_{i}=\alpha\sum_{j\neq i}g_{ij}$ applied at each channel. This
feedback control interpretation is useful for characterizing propagation
stability in terms of only the subsystem model, as we will do for the SISO
case in the following section.

###  IV-A SISO Case: Subsystem-Based Characterization

The interpretation of the local transfer matrix $H_{i}(s)$ as a closed-loop
model allows easy development of conditions for propagation stability phrased
in terms of either the frequency response or the transfer function of the
subsystem model, when the subsystem is SISO. To develop this analysis, let us
denote the transfer function for the subsystem model as $T(s)=C(sI-A)^{-1}B$.
Then, for the SISO case, the local transfer matrix can be written as

$H_{i}(s)=\frac{k_{i}T(s)}{1+k_{i}T(s)},$ (15)

where $k_{i}=\alpha\sum_{j\neq i}g_{ij}$.

We now develop a check for propagation stability in terms of the subsystem
frequency response $T(j\omega)$. To do so, recall from Theorem $1$ that
propagation stability requires $|H_{i}(j\omega)|\leq 1$ for all $\omega$ and
$i=1,\ldots,N$, which we call the local requirement, in addition to stability
of the synchronization manifold. Through substitution, the local requirement
can equivalently be written as
$|\frac{k_{i}T(j\omega)}{1+k_{i}T(j\omega)}|\leq 1$ for all $i=1,\ldots,N$.
Noting $T(j\omega)$ is a complex number for each $\omega$, say
$a(\omega)+jb(\omega)$, the expression for the local requirement can be
written as
$|\frac{k_{i}(a(\omega)+jb(\omega))}{1+k_{i}(a(\omega)+jb(\omega))}|\leq 1$.
From this expression, it is apparent that the local requirement is met if and
only if $|k_{i}a(\omega)|\leq|1+k_{i}a(\omega)|$ for all $\omega$ and
$i=1,\ldots,N$. However, this inequality holds true if and only if
$a(\omega)\geq\frac{1}{-2k_{i}}$ for all $\omega$ and all $i=1,\ldots,N$.
Thus, the local requirement is seen to be met if and only if the following
condition holds:

$Re(T(j\omega))\geq-\frac{1}{2\alpha\max_{i}\sum_{j\neq i}g_{ij}}.$ (16)

Thus, we see that the local requirement for propagation stability is met if
and only if the frequency response $T(j\omega)$ of the local subsystem model
lies entirely to the right of $-\frac{1}{2\alpha\max_{i}\sum_{j\neq i}g_{ij}}$
in the complex plane.

For the SISO subsystem case, the characterization of the local requirement
provides a means to verify propagation stability entirely in terms of the
frequency response of the subsystem model. Specifically, both the local
requirement and the standard condition for asymptotic stability of the
synchronization manifold can be determined from the Nyquist plot of the
subsystem model. For the local requirement to be met, the Nyquist plot must
lie entirely to the right of the vertical line with intercept
$-\frac{1}{2\alpha\max_{i}\sum_{j\neq i}g_{ij}}$. Meanwhile, the
characterization of asymptotic stability of the synchronization manifold in
terms of the Nyquist plot is well known [17]. Briefly, asymptotic stability of
the manifold can be equivalenced with simultaneous Hurwitz stability of
matrices $A+\lambda_{i}BC$, $i=2,\ldots,N$, where the $\lambda_{i}$
($i=2,\ldots,N$) are the non-zero eigenvalues of the Laplacian matrix $L$. For
the SISO subsystem case, the matrices $A+\lambda_{i}BC$ are the state matrices
of the closed-loop system, when a proportional feedback controller with gain
$\lambda_{i}$ is applied to the subsystem. Thus, it is seen that Hurwitz
stability of each matrix $A+\lambda_{i}BC$ can be determined through the
standard Nyquist criterion, i.e. by comparing encirclements of the point
$\frac{-1}{\lambda_{i}}$ by the Nyquist plot with the number of open-loop
right-half-plane poles. Since this stability analysis for the synchronization
manifold has been developed extensively in previous work, details are omitted.

The propagation stability analysis for the SISO subsystem case gives insight
into the role of the coupling constant $\alpha$. As the coupling is weakened,
the permissible region for the Nyquist plot such that the local requirement
holds is widened, including more of the left half of the complex plane. In
fact, by decreasing the coupling, the local requirement can be met for any
subsystem model unless the Nyquist plot diverges in the left-half-plane (which
corresponds to subsystem models with repeated poles on the $j\omega$ axis).
This ability to meet the local requirement by reducing the coupling is
logical, since reduced couplings should attenuate propagation of a disturbance
through the network. However, scaling down the coupling may also influence the
stability of the synchronization manifold. Analyses using the master stability
function have shown the intermediate couplings are needed for stability of the
synchronization manifold, for many subsystem models (e.g., several chaotic
oscillators like the Rossler oscillator). These characterizations, which can
also be verified through the Nyquist analyses presented above, indicate that
the local requirement and manifold stability may conflict for some subsystem
models: a small gain may be needed for the local requirement to hold, while a
larger gain is needed for stability of the synchronization manifold.

The frequency-domain analysis developed above immediately yields conditions
for propagation stability in terms of the subsystem transfer function. First,
general characterizations of propagation stability can be obtained in the
cases where the subsystem model is either strictly unstable (has open right-
half-plane poles) or strictly stable:

###### Corollary 1

Consider the network synchronization model, and assume that the subsystem
model is SISO.

  * •

If the subsystem model has an open right-half-plane pole, then the network
synchronization model is not propagation stable for any network graph $\Gamma$
and coupling constant $\alpha$.

  * •

If the subsystem model is strictly stable (has all poles strictly in the open
left-half-plane), then there exists a positive constant $\overline{\alpha}$
such that the network synchronization model is propagation stable for
$\alpha\leq\overline{\alpha}$.

The result follows immediately from the Nyquist-plot-based characterization of
propagation stability (both internal stability and the local requirement),
hence details are omitted.

Keener characterizations of propagation stability can be developed by
considering what transfer functions meet the local requirement for propagation
stability. First, we note that the local requirement is met regardless of the
coupling strength and the network graph, if the Nyquist plot lies in the
closed right half of the complex plane. The restriction of the Nyquist plot in
the closed right half plane is a standard notion in systems theory – positive
realness – which holds true for a system if and only if that system is
passive. Thus, we can easily obtain a general characterization for propagation
stability when the subsystem model is passive:

###### Corollary 2

Consider the network synchronization model, and assume that the subsystem
model is SISO and passive. Also, assume that the synchronization manifold is
asymptotically stable. Then the network synchronization model is necessarily
propagation stable.

In the case where the subsystem model is not passive, propagation stability
depends on the network graph and the coupling strength, as well as the extent
to which the subsystem’s Nyquist plot encroaches on the left half of the
complex plane. Specifically, the subsystem’s frequency response is allowed to
have a real part which is lower bounded by
$-\frac{1}{2\alpha\max_{i}\sum_{j\neq i}g_{ij}}$ rather than by zero: thus,
the graph vertex with largest total incoming edge weight along with the
coupling strength decide the region where the Nyquist plot may lie. The
requirement on the subsystem model in this case may be viewed as an ”almost
passivity” requirement, i.e. a requirement that the system could be made
passive using a direct feed-through term subject to a bound. The requirement
is formalized in the following theorem, which summarizes the frequency-
response-based characterization of propagation stability:

###### Theorem 3

Consider the network synchronization model, and assume that the subsystem
model is SISO. Also, assume that the synchronization manifold is
asymptotically stable. If
$Re(T(j\omega))\geq-\frac{1}{2\alpha\max_{i}\sum_{j\neq i}g_{ij}}$ for all
$\omega$, then the network synchronization model is propagation stable.

##  V Example

The propagation stability analysis is illustrated for an example model with
planar subsystems. Specifically, let us consider a synchronization network
process whose subsystem model $(C,A,B)$ is defined as follows:
$A=\begin{bmatrix}0&1\\\ 0&d\end{bmatrix}$, $B=\begin{bmatrix}0\\\
1\end{bmatrix}$, and $C=\begin{bmatrix}1&0\end{bmatrix}$, where we refer to
the positive scalar $d$ as a damping constant. Several common network models
are well-approximated by this form, including the classical model for the bulk
power grid’s swing dynamics, mass-spring-damper network models, and models for
vehicle teams engaged in formation flight. These networks are known to exhibit
a dichotomy of responses to sinusoidal or periodic disturbances, depending on
the damping constant $d$. When the system is sufficiently damped,
periodic/sinusoidal disturbances cause only localized responses; on the other
hand, if the damping is low, disturbances at certain frequencies incur
network-wide responses.

The propagation stability concept provides a means for assessing how the
damping influences the disturbance response pattern for the defined class of
synhronization network processes. In particular, for this example, the
frequency-domain criterion for propagation stability can readily be phrased in
terms of the damping constant $d$. If the criterion is met, then disturbance
responses are necessarily localized. If not, the network model potentially may
be susceptible to network-wide responses for disturbances over certain
frequency ranges,

The criterion for propagation stability includes a standard requirement for
stability of the synchronization manifold, and an additional local
requirement. For the network model considered in this example, stability of
the synchronization manifold has been precisely characterized in prior work.
Provided that the eigenvalues of the Laplacian are real (which encompasses the
symmetric and diagonally symmetrizable cases), stability of the
synchronization manifold holds for any damping. If the Laplacian has complex
eigenvalues, then a sufficiently large damping is needed for stability of the
synchronization manifold.

In our development here, we assume that the criterion for stability of the
synchronization manifold is met, and focus on relating the local requirement
with the damping ratio. The local requirement is met if the subsystem model
$Re(T(j\omega))\geq-\frac{1}{2\alpha\max_{i}\sum_{j\neq i}g_{ij}}$ for all
$\omega$. The transfer function for the planar subsystem model is
$T(s)=\frac{1}{s^{2}+ds}$, and hence the frequency response is
$T(j\omega)=\frac{1}{-\omega^{2}+jd\omega}=\frac{-\omega^{2}-jd\omega}{\omega^{4}+d^{2}\omega^{2}}$.
It immediately follows that $Re(T(j\omega))=\frac{-1}{\omega^{2}+d^{2}}$.
Thus, we find that $Re(T(j\omega))$ is bounded by the interval
$[-\frac{1}{d^{2}},0]$, with the lower bound achieved asymptotically as
$\omega\rightarrow 0$. The local requirement is therefore met if
$-\frac{1}{d^{2}}\geq-\frac{1}{2\alpha\max_{i}\sum_{j\neq i}g_{ij}}$, or
equivalently $d\geq\sqrt{2\alpha\max_{i}\sum_{j\neq i}g_{ij}}$. Thus, a
sufficiently large damping relative to the largest (weighted) out-degree in
the network graph guarantees propagation stability. If the damping ratio is
not sufficiently large, the network is potentially susceptible to wide-area
responses, for low-frequency disturbances.

Thus, we have shown that an additional requirement of sufficient damping is
needed to ensure propagation stability in addition to stability of the
synchronization manifold. In the case where the criterion is not met,
disturbances with certain frequency components have the potential for
amplification across the network. On the other hand, when the damping
requirement is met, disturbances are restricted to have local spheres of
influence. We note that the condition for propagation stability is phrased
entirely in terms of the subsystem model and local graph properties, which
then allows the development of a condition on the damping for propagation
stability.

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