#  TIT/HEP-699 March 2024 Giant graviton expansions for line operator index

Yosuke Imamura111E-mail<EMAIL_ADDRESS>

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

We discuss giant graviton expansions for the Schur index of ${\cal N}=4$
$U(N)$ SYM with the insertion of Wilson lines of the fundamental and the anti-
fundamental representations. We first propose a double-sum giant graviton
expansion and numerically confirm that it correctly reproduces the line-
operator index. We also find that it reduces to a simple-sum expansion when we
treat the index as a Taylor series with respect to a specific fugacity.

###### Contents

  1. 1 Introduction
  2. 2 Leading corrections
  3. 3 Worldsheet fluctuations
  4. 4 Multiple-wrapping contributions
  5. 5 Simple-sum expansion
  6. 6 Discussion

##  1 Introduction

The superconformal index [1, 2] of ${\cal N}=4$ $U(N)$ SYM is defined by

$\displaystyle\mathop{\rm
tr}\nolimits\left[(-1)^{F}q^{J_{1}}p^{J_{2}}x^{R_{x}}y^{R_{y}}z^{R_{z}}\right]$
(1)

where $J_{1}$ and $J_{2}$ are spins and $R_{x}$, $R_{y}$, and $R_{z}$ are
$R$-charges. To preserve supersymmetry we impose the constraint $qp=xyz$ on
the fugacities. This can be regarded as the partition function in the
spacetime $S^{3}\times S^{1}$.

In this paper we are interested in the index with the insertion of Wilson line
operators [3, 4, 5, 6, 7, 8, 9]. We insert $1/2$-BPS Wilson lines of
fundamental and anti-fundamental representations at the north and south poles
of $S^{3}$, respectively. This breaks $so(4)$ rotational symmetry of $S^{3}$
to $so(3)$. We define the poles so that they are fixed under $J_{1}$ action.
Then, $J_{1}$ is the Cartan generator of the preserved rotational symmetry,
which we denote by $so(3)_{J_{1}}$. The insertion also breaks $so(6)_{R}$
symmetry to $so(5)_{R}$, and without loosing generality we assume that $R_{x}$
and $R_{y}$ are preserved. The symmetries generated by $J_{2}$ and $R_{z}$ are
broken, and we have to turn off the corresponding fugacities $p$ and $z$. Then
the superconformal index reduces to the Schur index [10]. In the following we
use $I_{N}$ for the Schur index without line operator insertion

$\displaystyle I_{N}=\mathop{\rm
tr}\nolimits\left[(-1)^{F}q^{J_{1}}x^{R_{x}}y^{R_{y}}\right],\quad q=xy.$ (2)

See [11, 12, 13] for analytic expressions of $I_{N}$ with finite $N$.

We denote the Schur index with the Wilson line operator insertion by $I_{{\rm
line},N}$. $I_{{\rm line},N}$ can be calculated by the localization formula

$\displaystyle I_{{\rm line},N}=\int_{U(N)}d\mu\left[\mathop{\rm
Pexp}\nolimits\left\\{\left(1-\frac{(1-x)(1-y)}{1-q}\right)\chi_{\rm
adj}\right\\}\chi_{\rm fund}\overline{\chi}_{\rm fund}\right].$ (3)

$\chi_{\rm adj}$, $\chi_{\rm fund}$, and $\overline{\chi}_{\rm fund}$ are the
$U(N)$ characters of the adjoint, the fundamental, and the anti-fundamental
representations, respectively. $\mathop{\rm Pexp}\nolimits$ is the plethystic
exponential and $\int_{U(N)}d\mu$ is the integral over gauge group $U(N)$ with
the Haar measure. The factor $\chi_{\rm fund}\overline{\chi}_{\rm fund}$ in
the integrand corresponds to the line operator insertion, and $I_{N}$ can be
calculated by (3) with the factor removed. We can also consider Wilson line
operators associated with more general representations and the corresponding
index can be calculated by replacing $\chi_{\rm fund}\overline{\chi}_{\rm
fund}$ in (3) by the product of the corresponding characters. In this paper,
however, we will not discuss such a generalization.

It is well known that in the large $N$ limit the superconformal index
$I_{\infty}$ (even before taking the Schur limit) coincides with the index of
the supergravity multiplet in $AdS_{5}\times S^{5}$ [2] via AdS/CFT
correspondence [14, 15, 16]. It is in the Schur limit given by

$\displaystyle I_{\infty}=I_{\rm sugra}=\mathop{\rm Pexp}\nolimits i_{\rm
sugra}$ (4)

with the supergravity letter index

$\displaystyle i_{\rm sugra}=\frac{x}{1-x}+\frac{y}{1-y}-\frac{q}{1-q}.$ (5)

The large $N$ limit of the line-operator index was calculated in [4]. See also
[8, 9]. On the AdS side, the line operators correspond to a fundamental string
extended along $AdS_{2}\subset AdS_{5}$ [17, 18]. Two line operators are
identified with the two boundaries of the worldsheet. $I_{{\rm line},\infty}$
is reproduced by taking account of the contribution from fluctuation modes on
the string worldsheet [4]:

$\displaystyle I_{{\rm line},\infty}=I_{\rm sugra}I_{\rm F1}=\mathop{\rm
Pexp}\nolimits(i_{\rm sugra}+i_{\rm F1}),$ (6)

where $i_{\rm F1}$ is the letter index of worldsheet fluctuations

$\displaystyle i_{\rm F1}=x+y-q.$ (7)

The explicit form of the index is

$\displaystyle I_{{\rm line},\infty}$
$\displaystyle=1+4t+10t^{2}+22t^{3}+44t^{4}+82t^{5}+146t^{6}+\cdots,$ (8)

where we used the unrefinement $x=y=t$ to simplify the equation. The variable
$t$ is also used for order counting. ${\cal O}(t^{n})$ indicates terms of the
form $x^{n-k}y^{k}$ or higher power terms.

The purpose of this paper is to generalize the holographic relation (6) to
finite $N$ by including contributions of giant gravitons [19, 20, 21, 22].

##  2 Leading corrections

Let us first look at some numerical results for small $N$:

$\displaystyle I_{{\rm line},0}$ $\displaystyle=0,$ $\displaystyle I_{{\rm
line},1}$ $\displaystyle=1+2t+t^{2}+2t^{3}+2t^{4}+0t^{5}+3t^{6}+\cdots,$
$\displaystyle I_{{\rm line},2}$
$\displaystyle=1+4t+7t^{2}+8t^{3}+11t^{4}+16t^{5}+15t^{6}+\cdots,$
$\displaystyle I_{{\rm line},3}$
$\displaystyle=1+4t+10t^{2}+18t^{3}+25t^{4}+36t^{5}+52t^{6}+\cdots,$
$\displaystyle I_{{\rm line},4}$
$\displaystyle=1+4t+10t^{2}+22t^{3}+39t^{4}+58t^{5}+87t^{6}+\cdots,$
$\displaystyle I_{{\rm line},5}$
$\displaystyle=1+4t+10t^{2}+22t^{3}+44t^{4}+76t^{5}+117t^{6}+\cdots,$
$\displaystyle I_{{\rm line},6}$
$\displaystyle=1+4t+10t^{2}+22t^{3}+44t^{4}+82t^{5}+139t^{6}+\cdots.$ (9)

Again we showed unrefined indices. We defined $I_{{\rm line},0}=0$ because we
cannot insert non-trivial line operators in $N=0$ theory. (In other words, we
cannot introduce strings ending on D-branes if there are no D-branes.) These
are converging to (8) in the large $N$ limit, and the leading term of the
finite $N$ correction is

$\displaystyle I_{{\rm line},N}-I_{{\rm line},\infty}$
$\displaystyle=-(N+1)t^{N}+{\cal O}(t^{N+1}).$ (10)

This is similar to the finite $N$ correction of $I_{N}$ [11]:

$\displaystyle I_{N}-I_{\infty}=-(N+2)t^{N+1}+{\cal O}(t^{N+2}).$ (11)

Let us first review how (11) is reproduced on the AdS side [23, 24]. In [24]
the following expansion for the Schur index was proposed.

$\displaystyle\frac{I_{N}}{I_{\infty}}=\sum_{m_{x},m_{y}=0}^{\infty}x^{m_{x}N}F^{(x)}_{m_{x}}\cdot(xy)^{m_{x}m_{y}}\cdot
y^{m_{y}N}F^{(y)}_{m_{y}}.$ (12)

The non-negative integers $m_{x}$ and $m_{y}$ are wrapping numbers of giant
gravitons, and this kind of expansions are named in [25] giant graviton
expansions. See also [26, 27, 28, 29, 30].

The summand in (12) consists of three factors. The first factor
$x^{m_{x}N}F^{(x)}_{m_{x}}$ is the contribution from a stack of $m_{x}$ D3
giants wrapped around the $R_{x}$-fixed locus in $S^{5}$. $x^{m_{x}N}$ comes
from the $R$-charge of the giants, and $F^{(x)}_{m_{x}}$ is the Schur index of
the field theory realized on the worldvolume of the giants. The last factor
$y^{m_{y}N}F^{(y)}_{m_{y}}$ is a similar contribution from $m_{y}$ giants on
the $R_{y}$-fixed locus in $S^{5}$. Because the worldvolumes of the giants are
$S^{3}\times S^{1}$, which are isomorphic to the AdS boundary, $F^{(x)}_{m}$
and $F^{(y)}_{m}$ are essentially the same as $I_{m}$. Due to the different
actions of the symmetry generators they are related by [24]

$\displaystyle F^{(x)}_{m}=\sigma_{x}I_{m},\quad F^{(y)}_{m}=\sigma_{y}I_{m},$
(13)

where $\sigma_{x}$ and $\sigma_{y}$ are the following simple variable changes.

$\displaystyle\sigma_{x}:(x,y)\rightarrow(x^{-1},xy),\quad\sigma_{y}:(x,y)\rightarrow(xy,y^{-1}).$
(14)

The explicit forms of $F^{(x)}_{N}$ for small $N$ are [24]

$\displaystyle F^{(x)}_{0}$ $\displaystyle=1,$ $\displaystyle F^{(x)}_{1}$
$\displaystyle=\frac{-x}{1-\frac{y}{x}}+x^{2}(-1+\tfrac{y}{x})+x^{3}(-1+\tfrac{y^{3}}{x^{3}})+x^{4}(-1+\tfrac{y^{2}}{x^{2}}-\tfrac{y^{3}}{x^{3}}+\tfrac{y^{4}}{x^{4}})+\cdots$
$\displaystyle F^{(x)}_{2}$
$\displaystyle=\frac{\frac{x^{5}}{y}(-1+2\frac{y}{x})}{(1-\frac{y}{x})(1-\frac{y^{2}}{x^{2}})}+\tfrac{x^{7}}{y^{2}}(-1+2\tfrac{y^{2}}{x^{2}})+\tfrac{x^{9}}{y^{3}}(-1+2\tfrac{y^{3}}{x^{3}}+2\tfrac{y^{6}}{x^{6}})+\cdots$
$\displaystyle F^{(x)}_{3}$
$\displaystyle=\frac{\frac{x^{12}}{y^{3}}(-2+3\frac{y}{x}+3\frac{y^{2}}{x^{2}}-5\frac{y^{3}}{x^{3}})}{(1-\frac{y}{x})(1-\frac{y^{2}}{x^{2}})(1-\frac{y^{3}}{x^{3}})}+\frac{\frac{x^{15}}{y^{5}}(-2+3\frac{y^{2}}{x^{2}}+3\frac{y^{3}}{x^{3}}-5\frac{y^{5}}{x^{5}})}{1-\frac{y^{2}}{x^{2}}}+\cdots.$
(15)

The same expansions were also obtained in [30] using analytic results in [6].

The middle factor in (12), $(xy)^{m_{x}m_{y}}$, is the contribution from the
bi-fundamental fields arising from open strings stretched between the two
loci. Although the letter index of the bi-fundamental fields depends on the
gauge fugacities, its plethystic exponential becomes the factor independent of
the gauge fugacities. Due to this accidental simplification, the gauge
fugacity integrals of $U(m_{x})$ and $U(m_{y})$ gauge groups are factorized.

Let us focus on the leading correction from $(m_{x},m_{y})=(1,0)$ and $(0,1)$.

$\displaystyle\frac{I_{N}}{I_{\infty}}=1+x^{N}F^{(x)}_{1}+y^{N}F^{(y)}_{1}+\cdots,$
(16)

where $F^{(x)}_{1}$ is the index of the $U(1)$ vector multiplet on a giant
graviton wrapped around the $R_{x}$-fixed locus:

$\displaystyle F^{(x)}_{1}$ $\displaystyle=\sigma_{x}I_{1}=\mathop{\rm
Pexp}\nolimits\frac{\frac{1}{x}-2y+xy}{1-y}$ $\displaystyle=\mathop{\rm
Pexp}\nolimits(\tfrac{1}{x}+\tfrac{y}{x}+{\cal O}(t^{1})).$ (17)

The first term $\frac{1}{x}$ and the second term $\frac{y}{x}$ in the letter
index are ${\cal O}(t^{-1})$ and ${\cal O}(t^{0})$, respectively, and these
two terms are enough to obtain the leading term in the $t$-expansion of
$F^{(x)}_{1}$

$\displaystyle
F^{(x)}_{1}=\frac{1}{1-\frac{1}{x}}\frac{1}{1-\frac{y}{x}}(1+{\cal
O}(t^{1}))=\frac{-x}{1-\frac{y}{x}}+{\cal O}(t^{2}).$ (18)

The negative order term $\frac{1}{x}$ in the letter index corresponds to the
unwrapping mode (tachyonic mode) of the giant graviton. Its plethystic
exponential $\frac{1}{1-\frac{1}{x}}=\frac{-x}{1-x}$ includes the factor $-x$,
which has two effects: it changes the overall sign and it raises the order of
the leading correction by one. This is the mechanism generating the negative
leading correction of ${\cal O}(t^{N+1})$ in (11).

Let us turn to the leading correction of $I_{{\rm line},N}$ shown in (10). A
common feature with (10) is that it is negative. We expect this is again due
to the unwrapping modes because even when we introduce the string worldsheet
corresponding to the line operators giant gravitons still have unwrapping
modes. However, the order of (10) seems contrary to the fact that the
unwrapping modes raises the order by one. To match the order of the
correction, we need extra factor of ${\cal O}(t^{-1})$. One possible source of
this factor is a backreaction of the introduction of the string worldsheet.
Unfortunately, we cannot give a clear explanation for this factor at present,
and in the following, we simply assume the existence of such a factor.

Now, let us start the analysis of giant graviton contributions to $I_{{\rm
line},N}$. We consider the situation in which the string worldsheet and a
single giant graviton coexist. There are two cases. One is the case in which
the string and the giant graviton do not correlate. Namely, the string does
not end on the worldvolume of the giant. The other is the case that the string
worldsheet is divided into two parts by the giant graviton and the two semi-
infinite strings are ending on the worldvolume of the giant. In the former
case, we obtain the contribution

$\displaystyle I_{{\rm line},N}=\cdots+I_{\rm
sugra}(x^{N}F_{1}^{(x)}+y^{N}F_{1}^{(y)})I_{\rm F1}+\cdots.$ (19)

We also have similar contributions from other wrapping numbers
$(m_{x},m_{y})$, and they are summed up to $I_{N}I_{\rm F1}$. Let us subtract
this contribution from $I_{{\rm line},N}$. Furthermore, we remove the
supergravity contribution by dividing the difference by $I_{\rm
sugra}=I_{\infty}$. Numerical analysis on the gauge theory side shows

$\displaystyle\frac{I_{{\rm line},N}-I_{N}I_{\rm
F1}}{I_{\infty}}=x^{N}F^{\prime(x)}+y^{N}F^{\prime(y)}+{\cal O}(t^{2N+2}),$
(20)

where $F^{\prime(x)}$ and $F^{\prime(y)}$ are $N$-independent functions.
$F^{\prime(x)}$ is

$\displaystyle F^{\prime(x)}$
$\displaystyle=\frac{-1}{1-\frac{y}{x}}+x\frac{-1-\frac{y^{2}}{x^{2}}}{1-\frac{y}{x}}+x^{2}\frac{-1+\frac{y}{x}+\frac{y^{2}}{x^{2}}-\frac{y^{3}}{x^{3}}-\frac{y^{4}}{x^{4}}}{1-\frac{y}{x}}$
$\displaystyle+x^{3}(-1+2\tfrac{y^{3}}{x^{3}}+2\tfrac{y^{4}}{x^{4}}+\tfrac{y^{5}}{x^{5}})+x^{4}(-1-\tfrac{y^{3}}{x^{3}}+3\tfrac{y^{5}}{x^{5}}+2\tfrac{x^{6}}{y^{6}}+\tfrac{x^{7}}{y^{7}})$
$\displaystyle+{\cal O}(t^{5})$ (21)

and $F^{\prime(y)}$ is obtained from $F^{\prime(x)}$ by exchanging $x$ and
$y$. We can rewrite (21) in the simple form

$\displaystyle F^{\prime(x)}$ $\displaystyle=\frac{1}{x}\mathop{\rm
Pexp}\nolimits\left(\frac{\frac{1}{x}-2y+xy}{1-y}+2(y-xy)\right).$ (22)

This is close to (17). The first term in the parentheses is the same as the
letter index in (17), and interpreted as the contribution from a giant
graviton wrapped on the $R_{x}$-fixed locus. The prefactor $1/x$ is the
mismatch factor mentioned above, and we have no clear explanation for it. We
accept the existence of such factors, and later we will insert such factors by
hand.

The remaining part, $2(y-xy)$ in the parentheses is expected to be the
contribution from the two semi-infinite string worldsheets ending on the giant
graviton. In the next section we will show that this is indeed obtained from
the mode analysis on the semi-infinite strings.

##  3 Worldsheet fluctuations

Fluctuation modes on a superstring worldsheet extended along $AdS_{2}\subset
AdS_{5}$ are studied in [31, 32]. They form a short multiplet of
$osp(4^{*}|4)$, and the spectrum is

$\displaystyle\varphi(1,\bm{1},\bm{5})+\psi(\tfrac{3}{2},\bm{2},\bm{4})+\phi(2,\bm{3},\bm{1}).$
(23)

$\varphi$ and $\phi$ are scalar fields corresponding to five directions in
$S^{5}$ and three transverse directions in $AdS_{5}$, respectively. $\psi$ is
the fermion field with $8$ degrees of freedom. We showed in (23) the
representations under $so(2,1)_{\rm conf}\times so(3)_{J_{1}}\times
so(5)_{R}$. The value $\nu$ in the first slot in each representation is the
energy eigenvalue of the conformal primary state normalized by the AdS radius.
In addition, there are infinitely many conformal descendants with energies
$\nu+1,\nu+2,\ldots$ in each conformal representation.

The presence of a giant graviton wrapped around the $R_{x}$-fixed locus breaks
$so(5)_{R}$ to $so(2)_{R_{x}}\times so(3)_{R_{y}}$. Under this symmetry
breaking $\varphi$ and $\psi$ split into
$(\varphi_{3},\varphi_{X},\varphi_{\overline{X}})$ and $(\psi_{+},\psi_{-})$,
respectively. See Table 1 for quantum numbers of each component.

The $so(2,1)_{\rm conf}\times so(3)_{J_{1}}\times so(5)_{R}$ quantum numbers
of the supercharges preserved by the string worldsheet are
$(\pm\tfrac{1}{2},\bm{2},\bm{4})$. Among the real $16$ supercharges only eight
with $H=R_{x}$ are preserved after the introduction of a giant wrapped around
the $R_{x}$-fixed locus, and the superconformal representation (23) is
decomposed into small representations labeled by $n=H-R_{x}=0,1,2,\ldots$. We
denote them by ${\cal R}_{n}$, and their components are shown in Table 1.

Table 1: Components of the representation ${\cal R}_{n}$ and their quantum numbers are shown. The last column shows the mode functions of the scalar fields. | $E$ | $so(3)_{J_{1}}$ | $R_{x}$ | $so(3)_{R_{y}}$ | $n$ | mode
---|---|---|---|---|---|---
$\varphi_{\overline{X}}$ | $n-1$ | $\bm{1}$ | $-1$ | $\bm{1}$ | $n\geq 2$ | $f_{n-2}^{(1)}$
$\psi_{-}$ | $n-\frac{1}{2}$ | $\bm{2}$ | $-\frac{1}{2}$ | $\bm{2}$ | $n\geq 2$ |
$\phi$ | $n$ | $\bm{3}$ | $0$ | $\bm{1}$ | $n\geq 2$ | $f^{(2)}_{n-2}$
$\varphi_{3}$ | $n$ | $\bm{1}$ | $0$ | $\bm{3}$ | $n\geq 1$ | $f^{(1)}_{n-1}$
$\psi_{+}$ | $n+\frac{1}{2}$ | $\bm{2}$ | $+\frac{1}{2}$ | $\bm{2}$ | $n\geq 1$ |
$\varphi_{\overline{X}}$ | $n+1$ | $\bm{1}$ | $+1$ | $\bm{1}$ | $n\geq 0$ | $f^{(1)}_{n}$

Before the introduction of the giant graviton, there are three BPS modes
saturating the bound $H-J_{1}-R_{x}-R_{y}\geq 0$ and contributing to the
index. One of them with $(H,J_{1},R_{x},R_{y})=(1,0,+1,0)$ belongs to ${\cal
R}_{0}$, and the other two with $(H,J_{1},R_{x},R_{y})=(1,0,0,+1)$ and
$(\tfrac{3}{2},+\tfrac{1}{2},+\tfrac{1}{2},+\tfrac{1}{2})$ belong to ${\cal
R}_{1}$. These three modes give the three terms $x$, $y$, and $-xy$ in $i_{\rm
F1}$, respectively.

When the string worldsheet is attached on the worldvolume of the giant
graviton, some representations ${\cal R}_{n}$ are removed by the boundary
conditions. To determine which of them are allowed, let us explicitly write
down the mode functions for the scalar fields.

The scalar fields satisfy the wave equations

$\displaystyle(\Delta_{AdS_{2}}-2)\phi=0,\quad\Delta_{AdS_{2}}\varphi=0.$ (24)

These equations can be easily solved as follows. We define the $AdS_{2}$ as a
subspace in $\mathbb{R}^{2,1}$ with the metric

$\displaystyle ds^{2}=-2dzdz^{*}+dx^{2}$ (25)

by the embedding

$\displaystyle z=\tfrac{1}{\sqrt{2}}\cosh\rho e^{it},\quad
z^{*}=\tfrac{1}{\sqrt{2}}\cosh\rho e^{-it},\quad x=\sinh\rho.$ (26)

We introduce conformal generators

$\displaystyle K$ $\displaystyle=i(x\partial_{z^{*}}+z\partial_{x}),$
$\displaystyle P$ $\displaystyle=i(x\partial_{z}+z^{*}\partial_{x}),$
$\displaystyle H$ $\displaystyle=-z\partial_{z}+z^{*}\partial_{z^{*}}.$ (27)

satisfying

$\displaystyle[H,P]=P,\quad[H,K]=-K,\quad[K,P]=H.$ (28)

The mode function $f_{0}^{(\nu)}$ of the conformal primary state satisfying
$Kf_{0}^{(\nu)}=0$ and $Hf_{0}^{(\nu)}=\nu f_{0}^{(\nu)}$ is given by

$\displaystyle f_{0}^{(\nu)}(t,\rho)=\frac{1}{z^{\nu}}\propto\frac{e^{-i\nu
t}}{\cosh^{\nu}\rho}.$ (29)

The mode functions $f_{k}^{(\nu)}$ for conformal descendant states are
obtained by applying an arbitrary number of the raising operator $P$ to
$f_{0}^{(\nu)}$;

$\displaystyle f_{k}^{(\nu)}(t,\rho)=P^{k}f_{0}^{(\nu)}(t,\rho)\quad
k=0,1,2,\ldots.$ (30)

Using the relation between the scalar field Laplacian $\Delta_{AdS_{2}}$ and
the Casimir operator of the conformal algebra

$\displaystyle\Delta_{AdS_{2}}=H^{2}-(PK+KP),$ (31)

we can easily find that the mode functions $f^{(\nu)}_{k}$ satisfy

$\displaystyle\Delta_{AdS_{2}}f_{k}^{(\nu)}=\nu(\nu-1)f_{k}^{(\nu)},$ (32)

and $\phi$ and $\varphi$ satisfying (24) can be expanded by $f_{k}^{(2)}$ and
$f_{k}^{(1)}$, respectively.

An important fact is that the mode functions $f^{(\nu)}_{k}$ are even for even
$k$ and odd for odd $k$ under the reflection of the worldsheet
$\rho\rightarrow-\rho$. This means that for a field with Dirichlet (Neumann)
boundary condition only modes with odd (even) $k$ are allowed. We can easily
find that scalar modes in ${\cal R}_{n}$ with odd $n$ satisfy the boundary
conditions imposed on the giant graviton worldvolume, and the fluctuations in
${\cal R}_{n}$ with even $n$ are not allowed.

Concerning the three BPS saturating modes, only the two in ${\cal R}_{1}$ are
consistent with the boundary conditions, and they give the letter index
$y-xy$. Furthermore, we have two semi-infinite strings ending on the giant.
Therefore, total contribution to the index is

$\displaystyle I^{(x)}_{\rm F1}=\mathop{\rm Pexp}\nolimits i^{(x)}_{\rm
F1},\quad i^{(x)}_{\rm F1}=2(y-xy).$ (33)

With the presence of the factor $1/x$, this correctly reproduces (22).

$\displaystyle F^{\prime(x)}=\frac{1}{x}F^{(x)}_{1}I^{(x)}_{\rm F1}.$ (34)

$F^{\prime(y)}$ is also obtained in a similar way.

##  4 Multiple-wrapping contributions

Now, based on the analysis of the single-wrapping contribution in the previous
sections, we propose the following giant graviton expansion for $I_{{\rm
line},N}$.

$\displaystyle\frac{I_{{\rm line},N}}{I_{\infty}}$ $\displaystyle=I_{\rm
F1}\sum_{m_{x},m_{y}=0}^{\infty}x^{m_{x}N}F^{(x)}_{m_{x}}\cdot(xy)^{m_{x}m_{y}}\cdot
y^{m_{y}N}F^{(y)}_{m_{y}}$
$\displaystyle+\sum_{m_{x},m_{y}=0}^{\infty}x^{m_{x}N}F^{(x)}_{{\rm
line},m_{x}}\frac{I_{\rm F1}^{(x)}}{x}\cdot(xy)^{m_{x}m_{y}}\cdot
y^{m_{y}N}F^{(y)}_{m_{y}}$
$\displaystyle+\sum_{m_{x},m_{y}=0}^{\infty}x^{m_{x}N}F^{(x)}_{m_{x}}\cdot(xy)^{m_{x}m_{y}}\cdot
y^{m_{y}N}F^{(y)}_{{\rm line},m_{y}}\frac{I_{\rm F1}^{(y)}}{y}.$ (35)

For each pair of wrapping numbers $(m_{x},m_{y})$ we have to consider three
contributions. The first line comes from giants on which string worldsheet
does not end. The second line is the contribution from giants with semi-
infinite strings ending on the $R_{x}$-fixed locus, and the third line comes
from giants with semi-infinite strings ending on the $R_{y}$-fixed locus. We
do not have to consider the case with two semi-infinite strings ending on
different loci because the contribution from such a configuration vanishes
when we perform the factorized gauge fugacity integrals.

$\frac{I_{\rm F1}^{(x)}}{x}$ in the second line is the contribution from semi-
infinite strings (33) together with the mismatch factor $1/x$. $F^{(x)}_{{\rm
line},m_{x}}$ is the contribution from the field theory realized on the
worldvolume of $m_{x}$ coincident giants on the $R_{x}$-fixed locus. We take
account of the attached strings as the line operator insertion. This is not
necessary when $m_{x}=1$ because the effect of the insertion is trivial and
$F^{(x)}_{{\rm line},1}=F^{(x)}_{1}$. We define $F^{(x)}_{{\rm
line},0}=F^{(y)}_{{\rm line},0}=0$. For $(m_{x},m_{y})=(1,0)$ the second line
in (35) reduces to the single-wrapping contribution $x^{N}F^{\prime(x)}$ in
(20).

$F^{(x)}_{{\rm line},m}$ and $F^{(y)}_{{\rm line},m}$ are related to $I_{{\rm
line},m}$ by the variable changes (14).

$\displaystyle F^{(x)}_{{\rm line},m}=\sigma_{x}I_{{\rm line},m},\quad
F^{(y)}_{{\rm line},m}=\sigma_{y}I_{{\rm line},m}.$ (36)

It is not so easy to perform these variable changes because we need analytic
continuation [25, 33]. Here we use the method in [24]. Namely, we go back to
the localization formula (3) with the variable changes (14) applied to the
integrand, and perform the integral with contours carefully chosen in the
consistent way with the analyticity. The results for small $N$ are

$\displaystyle F^{(x)}_{{\rm line},0}$ $\displaystyle=0,$ $\displaystyle
F^{(x)}_{{\rm line},1}$ $\displaystyle=F^{(x)}_{1},$ $\displaystyle
F^{(x)}_{{\rm line},2}$
$\displaystyle=\frac{-\frac{x^{4}}{y}}{1-\frac{y^{2}}{x^{2}}}+\frac{\frac{x^{6}}{y^{2}}(-1-\frac{y}{x}+5\frac{y^{2}}{x^{2}}-\frac{y^{3}}{x^{3}}-\frac{y^{4}}{x^{4}}+\frac{y^{5}}{x^{5}})}{(1-\frac{y}{x})(1-\frac{y^{2}}{x^{2}})}$
$\displaystyle+\frac{\frac{x^{8}}{y^{3}}(-1-2\frac{y}{x}+6\frac{y^{3}}{x^{3}}+\frac{y^{4}}{x^{4}}-4\frac{y^{5}}{x^{5}}-\frac{y^{8}}{x^{8}})}{1-\frac{y^{2}}{x^{2}}}$
$\displaystyle+\tfrac{x^{10}}{y^{4}}(-1-2\tfrac{y}{x}-\tfrac{y^{2}}{x^{2}}+5\tfrac{y^{4}}{x^{5}}-4\tfrac{y^{7}}{x^{7}}+\tfrac{y^{8}}{x^{8}}+\tfrac{y^{10}}{x^{10}})+{\cal
O}(t^{7}),$ $\displaystyle F^{(x)}_{{\rm line},3}$
$\displaystyle=\frac{-\frac{x^{10}}{y^{3}}}{1-\frac{y^{3}}{x^{3}}}+\frac{\frac{x^{13}}{y^{5}}(-1-2\frac{y^{2}}{x^{2}}+\frac{y^{3}}{x^{3}}+5\frac{y^{4}}{x^{4}}-\frac{y^{6}}{x^{6}}+\frac{y^{8}}{x^{8}})}{(1-\frac{y^{2}}{x^{2}})(1-\frac{y^{3}}{x^{3}})}+{\cal
O}(t^{9}).$ (37)

$F^{(y)}_{{\rm line},N}$ are obtained from these by exchanging $x$ and $y$. It
is demanding to calculate higher order terms in $F_{{\rm line},3}^{(x)}$, and
we have obtained only the first two terms shown in (37). (The method in [30]
may be useful to compute these functions more efficiently.) With this result
we checked (35) correctly reproduces the index up to ${\cal O}(t^{3N+7})$ for
$N\leq 3$.

##  5 Simple-sum expansion

As was first found in [25], the double-sum expansion (12) for the Schur index
and the triple-sum expansion in [26] for the superconformal index reduce to
simple-sum expansions associated with only one cycle if we adopt an
appropriate expansion scheme. See also [33, 34] for the mechanism of the
reduction to simple-sum expansions. It is the case for the line-operator
index, too. If we treat the indices as $y$-series rather than $t$-series,

$\displaystyle I_{N}(x,y)=\sum_{k=0}^{\infty}f_{N,k}(x)y^{k},\quad I_{{\rm
line},N}(x,y)=\sum_{k=0}^{\infty}f_{{\rm line},N,k}(x)y^{k},$ (38)

then the contributions from $m_{y}\geq 1$ decouple, and the double-sum
expansion (35) reduces to the simple-sum expansion

$\displaystyle\frac{I_{{\rm line},N}}{I_{\infty}}$ $\displaystyle=I_{\rm
F1}\sum_{m_{x}=0}^{\infty}x^{m_{x}N}F^{(x)}_{m_{x}}+\frac{I_{\rm
F1}^{(x)}}{x}\sum_{m_{x}=0}^{\infty}x^{m_{x}N}F^{(x)}_{{\rm line},m_{x}}$ (39)

associated with only the $R_{x}$-fixed locus. The variable change $\sigma_{x}$
in (14) acts independently on each term in the $y$-series (38), and replace
the coefficient functions $f_{N,k}(x)$ and $f_{{\rm line},N,k}(x)$ by
$x^{k}f_{N,k}(x^{-1})$ and $x^{k}f_{{\rm line},N,k}(x^{-1})$, respectively.
Because the coefficient functions are rational functions, it is easy to obtain
$F_{N}^{(x)}$ and $F_{{\rm line},N}^{(x)}$ from $I_{N}$ and $I_{{\rm line},N}$
by the relations in (13) and (36). In particular, the leading terms in (38),
which gives the Coulomb branch limit $y\rightarrow 0$ of the indices, are
given by

$\displaystyle f_{N,0}=\prod_{i=1}^{N}\frac{1}{1-x^{i}},\quad f_{{\rm
line},N,0}=\frac{1}{1-x}\prod_{i=1}^{N-1}\frac{1}{1-x^{i}},$ (40)

and it is easy to analytically prove the relation (39) in the Coulomb branch
limit. One can also numerically check that (39) holds even without taking the
Coulomb branch limit.

##  6 Discussion

In this paper we discussed the Schur index with the insertion of fundamental
and anti-fundamental line operators for ${\cal N}=4$ $U(N)$ SYM with finite
$N$. As expected, we found that the finite $N$ corrections are reproduced by
summing up the contributions from giant gravitons. Unfortunately, we have not
yet have clear explanation for the factors $1/x$ and $1/y$ appearing when the
semi-infinite strings end on giant gravitons. A potential origin of the
factors is a back-reaction of the fundamental strings.

An important and interesting problem is the generalization to other
representations of the line operators. It is known that symmetric and anti-
symmetric representations with rank $k$ of order $N$ are holographically
realized by a D3-brane wrapped on $AdS_{2}\times S^{2}$ [35] and a D5-brane
wrapped on $AdS_{2}\times S^{4}$ [36], respectively. More general
representations with the rank $k$ of order $N^{2}$ are realized by bubbling
geometries [37]. In the large $N$ limit the matching of the indices for these
cases were studied in [4, 9]. It is interesting to study how the finite $N$
corrections arise on the AdS side as giant graviton contributions.

Another important direction is to consider line operator indices in other
theories [4, 7]: ${\cal N}=4$ SYM with $SO$ and $Sp$ gauge groups and ${\cal
N}=2$ SCFTs which have holographic duals. Giant graviton expansions for such
theories without operator insertions have been studied in [23, 38, 34], and we
expect they can be generalized to line operator indices.

In this paper we proposed the giant graviton expansions based on the analysis
on the AdS side. It is also important to understand how the expansion arises
on the gauge theory side in a similar way to the derivation of the giant
graviton expansion for the superconformal index in [25].

We look forward to revisiting these aspects in the near future.

## Acknowledgments

The author thanks Tadashi Okazaki and Daisuke Yokoyama for valuable
discussions. This work was supported by JSPS KAKENHI Grant Number JP21K03569.

## References

  * [1] C. Romelsberger, “Counting chiral primaries in N = 1, d=4 superconformal field theories,” Nucl. Phys. B 747, 329-353 (2006) doi:10.1016/j.nuclphysb.2006.03.037 [arXiv:hep-th/0510060 [hep-th]].
  * [2] J. Kinney, J. M. Maldacena, S. Minwalla and S. Raju, “An Index for 4 dimensional super conformal theories,” Commun. Math. Phys. 275, 209-254 (2007) doi:10.1007/s00220-007-0258-7 [arXiv:hep-th/0510251 [hep-th]].
  * [3] T. Dimofte, D. Gaiotto and S. Gukov, “3-Manifolds and 3d Indices,” Adv. Theor. Math. Phys. 17, no.5, 975-1076 (2013) doi:10.4310/ATMP.2013.v17.n5.a3 [arXiv:1112.5179 [hep-th]].
  * [4] D. Gang, E. Koh and K. Lee, “Line Operator Index on $S^{1}\times S^{3}$,” JHEP 05, 007 (2012) doi:10.1007/JHEP05(2012)007 [arXiv:1201.5539 [hep-th]].
  * [5] N. Drukker, “The $\mathcal{N}=4$ Schur index with Polyakov loops,” JHEP 12, 012 (2015) doi:10.1007/JHEP12(2015)012 [arXiv:1510.02480 [hep-th]].
  * [6] Y. Hatsuda and T. Okazaki, “Exact $\mathcal{N}$ = 2∗ Schur line defect correlators,” JHEP 06, 169 (2023) doi:10.1007/JHEP06(2023)169 [arXiv:2303.14887 [hep-th]].
  * [7] Z. Guo, Y. Li, Y. Pan and Y. Wang, “N=2 Schur index and line operators,” Phys. Rev. D 108, no.10, 106002 (2023) doi:10.1103/PhysRevD.108.106002 [arXiv:2307.15650 [hep-th]].
  * [8] Y. Hatsuda and T. Okazaki, “Large N and large representations of Schur line defect correlators,” JHEP 01, 096 (2024) doi:10.1007/JHEP01(2024)096 [arXiv:2309.11712 [hep-th]].
  * [9] Y. Hatsuda and T. Okazaki, “Excitations of bubbling geometries for line defects,” [arXiv:2311.13740 [hep-th]].
  * [10] A. Gadde, L. Rastelli, S. S. Razamat and W. Yan, “Gauge Theories and Macdonald Polynomials,” Commun. Math. Phys. 319, 147-193 (2013) doi:10.1007/s00220-012-1607-8 [arXiv:1110.3740 [hep-th]].
  * [11] J. Bourdier, N. Drukker and J. Felix, “The exact Schur index of $\mathcal{N}=4$ SYM,” JHEP 11, 210 (2015) doi:10.1007/JHEP11(2015)210 [arXiv:1507.08659 [hep-th]].
  * [12] Y. Pan and W. Peelaers, “Exact Schur index in closed form,” Phys. Rev. D 106, no.4, 045017 (2022) doi:10.1103/PhysRevD.106.045017 [arXiv:2112.09705 [hep-th]].
  * [13] Y. Hatsuda and T. Okazaki, “$\mathcal{N}$ = 2∗ Schur indices,” JHEP 01, 029 (2023) doi:10.1007/JHEP01(2023)029 [arXiv:2208.01426 [hep-th]].
  * [14] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2, 231-252 (1998) doi:10.4310/ATMP.1998.v2.n2.a1 [arXiv:hep-th/9711200 [hep-th]].
  * [15] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators from noncritical string theory,” Phys. Lett. B 428, 105-114 (1998) doi:10.1016/S0370-2693(98)00377-3 [arXiv:hep-th/9802109 [hep-th]].
  * [16] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2, 253-291 (1998) doi:10.4310/ATMP.1998.v2.n2.a2 [arXiv:hep-th/9802150 [hep-th]].
  * [17] S. J. Rey and J. T. Yee, “Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity,” Eur. Phys. J. C 22, 379-394 (2001) doi:10.1007/s100520100799 [arXiv:hep-th/9803001 [hep-th]].
  * [18] J. M. Maldacena, “Wilson loops in large N field theories,” Phys. Rev. Lett. 80, 4859-4862 (1998) doi:10.1103/PhysRevLett.80.4859 [arXiv:hep-th/9803002 [hep-th]].
  * [19] J. McGreevy, L. Susskind and N. Toumbas, “Invasion of the giant gravitons from Anti-de Sitter space,” JHEP 0006, 008 (2000) doi:10.1088/1126-6708/2000/06/008 [hep-th/0003075].
  * [20] M. T. Grisaru, R. C. Myers and O. Tafjord, “SUSY and goliath,” JHEP 0008, 040 (2000) doi:10.1088/1126-6708/2000/08/040 [hep-th/0008015].
  * [21] A. Hashimoto, S. Hirano and N. Itzhaki, “Large branes in AdS and their field theory dual,” JHEP 0008, 051 (2000) doi:10.1088/1126-6708/2000/08/051 [hep-th/0008016].
  * [22] A. Mikhailov, “Giant gravitons from holomorphic surfaces,” JHEP 0011, 027 (2000) doi:10.1088/1126-6708/2000/11/027 [hep-th/0010206].
  * [23] R. Arai and Y. Imamura, “Finite $N$ Corrections to the Superconformal Index of S-fold Theories,” PTEP 2019, no.8, 083B04 (2019) doi:10.1093/ptep/ptz088 [arXiv:1904.09776 [hep-th]].
  * [24] R. Arai, S. Fujiwara, Y. Imamura and T. Mori, “Schur index of the ${\cal N}=4$ $U(N)$ supersymmetric Yang-Mills theory via the AdS/CFT correspondence,” Phys. Rev. D 101, no.8, 086017 (2020) doi:10.1103/PhysRevD.101.086017 [arXiv:2001.11667 [hep-th]].
  * [25] D. Gaiotto and J. H. Lee, “The Giant Graviton Expansion,” [arXiv:2109.02545 [hep-th]].
  * [26] Y. Imamura, “Finite-N superconformal index via the AdS/CFT correspondence,” PTEP 2021, no.12, 123B05 (2021) doi:10.1093/ptep/ptab141 [arXiv:2108.12090 [hep-th]].
  * [27] S. Murthy, “Unitary matrix models, free fermions, and the giant graviton expansion,” Pure Appl. Math. Quart. 19, no.1, 299-340 (2023) doi:10.4310/PAMQ.2023.v19.n1.a12 [arXiv:2202.06897 [hep-th]].
  * [28] J. H. Lee, “Exact stringy microstates from gauge theories,” JHEP 11, 137 (2022) doi:10.1007/JHEP11(2022)137 [arXiv:2204.09286 [hep-th]].
  * [29] M. Beccaria and A. Cabo-Bizet, “On the brane expansion of the Schur index,” JHEP 08, 073 (2023) doi:10.1007/JHEP08(2023)073 [arXiv:2305.17730 [hep-th]].
  * [30] M. Beccaria and A. Cabo-Bizet, “Giant graviton expansion of Schur index and quasimodular forms,” [arXiv:2403.06509 [hep-th]].
  * [31] N. Drukker, D. J. Gross and A. A. Tseytlin, “Green-Schwarz string in AdS(5) x S**5: Semiclassical partition function,” JHEP 04, 021 (2000) doi:10.1088/1126-6708/2000/04/021 [arXiv:hep-th/0001204 [hep-th]].
  * [32] A. Faraggi and L. A. Pando Zayas, “The Spectrum of Excitations of Holographic Wilson Loops,” JHEP 05, 018 (2011) doi:10.1007/JHEP05(2011)018 [arXiv:1101.5145 [hep-th]].
  * [33] Y. Imamura, “Analytic continuation for giant gravitons,” PTEP 2022, no.10, 103B02 (2022) doi:10.1093/ptep/ptac127 [arXiv:2205.14615 [hep-th]].
  * [34] S. Fujiwara, Y. Imamura, T. Mori, S. Murayama and D. Yokoyama, “Simple-Sum Giant Graviton Expansions for Orbifolds and Orientifolds,” PTEP 2024, no.2, 023B02 (2024) doi:10.1093/ptep/ptae006 [arXiv:2310.03332 [hep-th]].
  * [35] N. Drukker and B. Fiol, “All-genus calculation of Wilson loops using D-branes,” JHEP 02, 010 (2005) doi:10.1088/1126-6708/2005/02/010 [arXiv:hep-th/0501109 [hep-th]].
  * [36] S. Yamaguchi, “Wilson loops of anti-symmetric representation and D5-branes,” JHEP 05, 037 (2006) doi:10.1088/1126-6708/2006/05/037 [arXiv:hep-th/0603208 [hep-th]].
  * [37] E. D’Hoker, J. Estes and M. Gutperle, “Gravity duals of half-BPS Wilson loops,” JHEP 06, 063 (2007) doi:10.1088/1126-6708/2007/06/063 [arXiv:0705.1004 [hep-th]].
  * [38] R. Arai, S. Fujiwara, Y. Imamura and T. Mori, “Finite $N$ corrections to the superconformal index of orbifold quiver gauge theories,” JHEP 10, 243 (2019) doi:10.1007/JHEP10(2019)243 [arXiv:1907.05660 [hep-th]].