Three Point Amplitudes in Matrix Theory

Aidan Herderschee and Juan Maldacena

Institute for Advanced Study, Princeton, NJ 08540, USA

We compute the three graviton amplitude in the Banks-Fischler-Shenker-Susskind
matrix model for M-theory. Even though the three point amplitude is determined
by super Poincare invariance in eleven dimensional M-theory, it requires a
non-trivial computation in the matrix model. We consider a configuration where
all three gravitons carry non-zero longitudinal momentum. To simplify the
problem, we compactify one additional dimension and relate the amplitude to a
supersymmetric index computation. We find agreement with the expected answer
even at finite values of $N$.

###### Contents

  1. 1 Introduction and motivation
  2. 2 Two and three point amplitudes in M-theory
    1. 2.1 Amplitudes in gravity with compact directions
    2. 2.2 Rewriting the amplitudes
  3. 3 The BFSS Matrix model
    1. 3.1 The BFSS model after we compactify one spatial dimension
    2. 3.2 Introducing brane sources
    3. 3.3 The two point function and the boundary states
    4. 3.4 The three point functions and the amplitude
  4. 4 Conclusions
  5. A Computation of the three point amplitude when one particle has no light cone momentum

##  1 Introduction and motivation

The BFSS matrix model can be used to compute flat space scattering amplitudes
in M-theory [1, 2, 3, 4, 5, 6]. In this paper, we compute the simplest such
amplitude, the three point amplitude, obtaining the expected Lorentz invariant
answer. In fact, Lorentz invariance of the eleven dimensional bulk theory
uniquely fixes the three point amplitude. However, this is insufficient from
the perspective of the matrix model, because eleven dimensional Lorentz
symmetry is not present in the matrix model at finite $N$.

In general, it is difficult to compute amplitudes from the matrix model,
because the prescription involves a low energy computation that is strongly
coupled. Furthermore, the ground state wavefunctions to be scattered are not
known, except in particular limits [7, 8, 9, 10, 11, 12, 13]. Even though the
three point amplitude is fixed by symmetry in M-theory, in the matrix model it
corresponds to a highly non-trivial process where two strongly coupled clouds
of D0 branes merge into a single cloud of D0 branes. This process seems to be
inaccessible by the background field and worldline approximations previously
applied at four point where the clouds of D0 branes are always well separated
and their internal structure is largely irrelevant [14, 15, 16, 17, 18].

The crucial step in our calculation involves relating the three point
amplitude in the matrix model to an index. The index is simply a number and
therefore invariant under changes of the coupling. For this relation, it is
crucial that the kinematics for which the on-shell three-point amplitude is
non-vanishing preserve 1/4 of the supersymmetries. Notably, this configuration
involves making one of the momenta imaginary.111One could also say that we are
going to $(2,9)$ signature, but we prefer to view it as an analytic
continuation from the usual signature in this paper. Relating the amplitude to
the index involves two steps:

  1. 1.

Compactifying one of the transverse dimensions so that the Matrix model
becomes a 1+1 gauge theory or matrix field theory [19, 20].

  2. 2.

Introducing transverse M2 branes that are localized along the light-cone
direction and are separated along some of the transverse dimensions, including
the (Euclidean) time direction. The three separated M2 branes can be viewed as
sourcing the three gravitons being scattered. Introducing these M2 branes in
the bulk theory amounts to imposing particular boundary conditions in the 1+1D
matrix model.

Under T-duality, in the 1+1 matrix string description, these separated M2
branes become boundary conditions similar to those imposed by D1 branes ending
on D3 branes. The bulk scattering configuration can therefore be identified
with a three string network [21, 22, 23, 24, 25, 26, 27, 28] ending on three
separated D3 branes, see figure 1. The number of states for this string
network configuration was computed by Sen in [29, 30, 31]. We can view the
problem in an “open string channel”, where it has an index interpretation, and
also in a “closed string channel”, where it has an amplitude interpretation.
The connection between the two is worked out in detail to reproduce that the
amplitude depends on continuous parameters, as opposed to the index, which is
just a number.

Figure 1: This is a sketch of the main idea for this paper. The blue lines
represent the scattering amplitude configuration in Euclidean time. $N_{i}$
and $n_{i}$ indicate the quantized momenta in the longitudinal null direction
and the transverse compact dimension respectively. The vertical direction is a
non-compact transverse dimension. $N_{i}$ and $n_{i}$ can also be viewed as
the number of D1 branes and fundamental strings that are forming bound state.
As we explain below, the three point amplitude is then related to a string
network stretching between three separated D3 branes.

In the limit that $R_{9}/l_{p}\to 0$, the matrix model reduces to weakly
coupled IIA string theory. It was argued that the matrix string gives rise to
string perturbation theory in [32, 20], which reproduces the gravity three
point function, see also [33, 34, 35]. In this paper, we argue that this is
indeed the case for all values of the coupling, or all values of
$R_{9}/l_{p}$, enabling us to access the eleven dimensional limit
$R_{9}\to\infty$.

This paper is organized as follows. In section 2, we write the two and three
point amplitudes as determined by supersymmetry and Lorentz invariance. We
rewrite the amplitudes in a few different ways so as to take them to a form
that is most natural for the matrix model computation. In section 3, we recall
a few facts about the matrix model, introduce the setup for our computation,
and compute the three point amplitude. Finally, we present some conclusions
and discussion.

##  2 Two and three point amplitudes in M-theory

M-theory contains massless excitations that are in a short multiplet of the
supersymmetry algebra. In fact, a massless particle preserves half of the
supersymmetries, the ones that obey ${\not\\!p}\epsilon=0$.

In order to describe the kinematic configuration of the three point amplitude,
it is convenient to select four of the eleven dimensions and introduce the
following coordinates

$\begin{split}&ds^{2}=-2dx^{+}dx^{-}+2dzd\bar{z}~{},\\\
&~{}~{}\textrm{where}~{}~{}~{}\sqrt{2}z=x^{9}+ix^{8},~{}~{}\sqrt{2}\bar{z}=x^{9}-ix^{8},~{}~{}\sqrt{2}x^{\pm}=x^{0}\pm
x^{10}\end{split}$ (1)

We can now choose a configuration with momenta that are purely in the $p_{-}$
and $p_{z}$ directions, with $p_{+}=p_{\bar{z}}=0$ for all particles. This is
possible only if the momenta in the direction $p_{8}$ are purely imaginary.
For example, one possible configuration is

$p^{i}_{\mu}=(p_{+},p_{-},p_{z},p_{\bar{z}})~{},~{}~{}~{}~{}p^{1}_{\mu}=(0,p^{1}_{-},q_{z},0)~{},~{}~{}~{}~{}p^{2}_{\mu}=(0,p^{2}_{-},-q_{z},0)~{},~{}~{}~{}~{}p^{3}_{\mu}=-(0,p^{1}_{-}+p^{2}_{-},0,0)$
(2)

which clearly obeys momentum conservation. In addition, imposing
${{\not\\!p}}^{i}\epsilon=0$ for the three particles $i=1,2,3$ implies that
$\Gamma^{-}\epsilon=\Gamma^{z}\epsilon=0$, which are compatible conditions
that select 1/4 of the 32 original supersymmetries.222This means that out of
the 32 supersymmetric Ward identities [36] that we have for generic on-shell
amplitudes only 24 linearly independent ones remain for the three point
amplitude.

In order to write the scattering amplitudes, it is convenient to use an
efficient description for the whole supermultiplet. Since any on-shell three
point amplitude configuration can be related by Lorentz symmetry to one where
the momenta are in four of the dimensions, as in (2), we will use four
dimensional notation for the supersymmetries. This is the same notation that
is used in describing ${\cal N}=8$ supergravity in four dimensions. In that
case, it is convenient to use spinor helicity variables to write the momenta
as $p_{\alpha\dot{\beta}}=\lambda_{\alpha}\bar{\lambda}_{\dot{\beta}}$. The
supercharges obey an algebra of the form

$\\{Q_{\alpha},\bar{Q}_{\dot{\beta}}\\}=\lambda_{\alpha}\bar{\lambda}_{\dot{\beta}}$
(3)

and we can generate the whole multiplet by starting from the positive helicity
graviton and writing [37, 38, 39, 40]

$|\bar{\eta}\rangle=e^{b^{\dagger
I}\bar{\eta}_{I}}|+\rangle~{},~{}~{}~{}~{}~{}{\rm
with}~{}~{}~{}~{}~{}b^{\dagger
I}\equiv\bar{w}^{\dot{\beta}}\bar{Q}_{\dot{\beta}}^{\,I}$ (4)

where $\bar{w}$ and $w$ are spinors obeying333 Note that
$[\bar{w},\bar{\lambda}]$ denotes a “square bracket” defined as in (4) and not
a commutator.

$[\bar{w},\bar{\lambda}]\equiv{\bar{w}}^{\dot{\beta}}\bar{\lambda}_{\dot{\beta}}=1~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}\langle
w,\lambda\rangle=w^{\alpha}\lambda_{\alpha}=1\ .$ (5)

In (4) $I$ runs over eight values. This implies that $b^{\dagger I}$ in (4)
and $b_{I}\equiv w^{\alpha}Q_{\alpha I}$ are canonically normalized fermion
creation and annihilation operators obeying $\\{b_{I},b^{\dagger
I^{\prime}}\\}=\delta_{I}^{I^{\prime}}$ thanks to (3) and (5).

Using this notation, the two point amplitude is

${\cal
A}_{2}\sim\sqrt{p_{-}}\delta(p_{-}+p^{\prime}_{-})\delta^{9}(\vec{p}+\vec{p}^{\prime})\delta^{8}(\bar{\eta}_{I}+\bar{\eta^{\prime}}_{I})$
(6)

(note that there is no $\delta(p_{+}+p_{+}^{\prime})$ term). This sets the
normalization of the states. All amplitudes in this paper will be computed up
to an overall numerical coefficient. In this paper, the symbol “$\sim$” means
that there is a numerical coefficient between the left and right hand sides,
such as factors of 2 and $\pi$, but all other factors are included.

For the three point amplitude, we will consider a configuration where all
$\bar{\lambda}^{i}$ are proportional to each other444The other amplitude is
similar up to exchanging the bar and unbarred quantities [41], and we will not
discuss it any further in this paper.

${\cal A}_{3}\sim\sqrt{G_{N,11}}\delta(\sum p_{-}^{i})\delta(\sum
p_{+}^{i})\delta^{9}(\sum\vec{p}^{\,i})\delta^{16}(\lambda_{\alpha}^{i}\bar{\eta}_{I}^{i}){1\over\left(\langle
1,2\rangle\langle 2,3\rangle\langle 1,3\rangle\right)^{2}}$ (7)

This form of the amplitude is completely determined by supersymmetry and
Lorentz invariance. In particular, supersymmetry fixes the sixteen delta
functions in (7). This is, of course, the amplitude in the gravity theory, but
it is also the amplitude in the full M-theory because it is fully fixed by
symmetries up to an overall constant.

###  2.1 Amplitudes in gravity with compact directions

For our problem, we will be interested in compactifying the $x^{-}$ direction.
In addition, we also find it convenient to compactify in the $x^{9}$
direction. The momenta are then 555Note that $p_{\pm}$ with the indices down
are negative semidefinite for physical ingoing particles.

$-p_{-}^{i}={N_{i}\over R_{-}}~{},~{}~{}~{}~{}~{}~{}p_{9}=ip_{8}={n_{i}\over
R_{9}}~{},~{}~{}~{}~{}~{}{\rm with}~{}~{}x^{-}\sim x^{-}+2\pi
R_{-}~{},~{}~{}~{}~{}~{}~{}~{}x^{9}\sim x^{9}+2\pi R_{9}$ (8)

$p_{9}$ is quantized and $p_{8}$ is fixed to this value by the kinematics of
the three point amplitude even though $x^{8}$ is non compact. These momenta
correspond to the spinor helicity variables

$\displaystyle\lambda_{i}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\sqrt{N_{i}\over R_{-}}\cr{n_{i}\over
R_{9}}\sqrt{R_{-}\over
N_{i}}\end{array}\right)~{},~{}~{}~{}~{}~{}\bar{\lambda}_{i}=\left(\begin{array}[]{c}\sqrt{N_{i}\over
R_{-}}\cr 0\end{array}\right)~{}~{}~{}~{}i=1,2$ (13)
$\displaystyle\lambda_{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\sqrt{N_{3}\over R_{-}}\cr{n_{3}\over
R_{9}}\sqrt{R_{-}\over
N_{3}}\end{array}\right)~{},~{}~{}~{}~{}\bar{\lambda}_{3}=-\left(\begin{array}[]{c}\sqrt{N_{3}\over
R_{-}}\cr
0\end{array}\right)~{},~{}~{}~{}N_{3}=N_{1}+N_{2}~{},~{}~{}~{}n_{3}=n_{1}+n_{2}$
(18)

We have also chosen the integers so that momentum conservation is obeyed.
These give momenta similar to (2), where $p_{+}^{i}=0=p_{\bar{z}}^{i}$. We are
writing the momenta in the all ingoing notation and that is the reason for the
extra minus sign in (18). We are viewing particles 1 and 2 as ingoing and 3
outgoing.

After these choices, the amplitude becomes, with
$G_{N,11}=16\pi^{7}l_{p}^{9}$,

$\displaystyle{\cal A}_{2}$ $\displaystyle\sim$
$\displaystyle\sqrt{N}R_{9}\delta^{8}(\vec{p}+\vec{p}^{\prime})\delta^{8}(\bar{\eta}_{I}+{\bar{\eta}}^{\prime}_{I})~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}N=N^{\prime}~{},~{}~{}~{}n=n^{\prime}$
(19) $\displaystyle{\cal A}_{3}$ $\displaystyle\sim$ $\displaystyle
l_{p}^{9/2}{R_{-}\over
R_{9}}{(N_{1}N_{2}N_{3})^{2}\over(n_{1}N_{2}-n_{2}N_{1})^{6}}\delta(\sum
p_{+}^{i})\delta^{8}(\sum\vec{p}^{\,i})\delta^{8}(\sum\sqrt{N_{i}}\bar{\eta}^{i}_{I})\delta^{8}(\sum{n_{i}\over\sqrt{N_{i}}}\bar{\eta}^{i}_{I})$
(20)

where we have used $\delta(n/R)\sim R\delta_{n}$ and conservation conditions
(18) for the integers. These remove the $p_{-}$ and $p_{9}$ delta functions.
The $p_{8}$ delta function is still present in the above expression.

###  2.2 Rewriting the amplitudes

In order to compare to the matrix model, it is natural to normalize the states
using the cannonical normalization than the relativistic one (6):

${\cal A}_{2}=\langle p|p^{\prime}\rangle~{},~{}~{}~{}~{}~{}~{}~{}{\cal
A}_{2}^{c}=~{}^{c}\langle
p|p^{\prime}\rangle^{c}~{},~{}~{}~{}~{}~{}|p\rangle^{c}={|p\rangle\over\sqrt{NR_{9}}}\
.$ (21)

We will also switch the variables ${\bar{\eta}}_{1},~{}{\bar{\eta}}_{2}$ to
“center of mass” and “relative” variables

${\bar{\eta}}^{1}_{I}={\bar{\eta}}^{cm}_{I}\sqrt{N_{1}\over
N_{3}}+{\bar{\eta}}^{r}_{I}\sqrt{N_{2}\over
N_{3}}~{},~{}~{}~{}~{}~{}~{}~{}{\bar{\eta}}^{2}_{I}={\bar{\eta}}^{cm}_{I}\sqrt{N_{2}\over
N_{3}}-{\bar{\eta}}^{r}_{I}\sqrt{N_{1}\over N_{3}}\ .$ (22)

At this point, this change of variables looks arbitrary. However, these new
variables are convenient for the matrix model computation. Applying this
redefinition, (22), with the canonical normalization (21), the amplitudes
become

$\displaystyle{\cal A}_{2}^{c}$ $\displaystyle\sim$ $\displaystyle\
\delta^{8}(\vec{p}+\vec{p}^{\prime})\delta^{8}(\bar{\eta}_{I}-{\bar{\eta}}^{\prime}_{I})~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}N=N^{\prime}~{},~{}~{}~{}n=n^{\prime}\
,$ (23) $\displaystyle{\cal A}_{3}^{c}$ $\displaystyle\sim$ $\displaystyle
l_{p}^{9/2}{R_{-}\over
R_{9}^{5/2}}{(n_{1}N_{2}-n_{2}N_{1})^{2}N_{3}^{2}\over(N_{1}N_{2})^{2}\sqrt{N_{1}N_{2}N_{3}}}\delta(\sum
p_{+}^{i})\delta^{8}(\sum\vec{p}^{\,i})\delta^{8}({\bar{\eta}}^{cm}_{I}+{\bar{\eta}}^{3}_{I})\delta^{8}({\bar{\eta}}^{r}_{I})\
.$ (24)

Equation (24) is simply a rewriting of the original amplitude in non-
relativistic normalization.

We will do one more change of variables. It will be more convenient for us to
express the supermultiplet in terms of supercharges acting on a different
“vacuum” than the one in (4). We want to pick a vacuum that is annihilated by
a supercharge obeying

$\Gamma^{-+456789}\epsilon=\epsilon\ .$ (25)

This selects the supersymmetries that are preserved by an M2 brane along the
directions 123. It is useful to split the eight supercharges we had in (4)
into two groups with opposite eigenvalues under the projector
$\Gamma^{-+456789}\epsilon=\pm\epsilon$. . We will denote them in terms of the
indices $J$ and $K$, each of them going from one to four (while the original
index $I$ went from one to eight). In other words,
${\bar{\eta}}_{I}\to({\bar{\eta}}_{J},~{}{\bar{\eta}}_{K})$. After making this
replacement, we then “Fourier transform” the ${\bar{\eta}}_{K}$ to $\eta^{K}$.
In other words, we define a “new” amplitude

$\hat{\cal
A}_{n}({\bar{\eta}}^{i}_{J},\eta^{iK})=\int\prod_{i}d^{4}{\bar{\eta}}^{i}_{K}e^{\eta^{iK}{\bar{\eta}}_{K}^{i}}{\cal
A}_{n}({\bar{\eta}}^{i}_{J},{\bar{\eta}}^{i}_{K})\ .$ (26)

which is an amplitude of the supersymmetric coherent states

$|{\bar{\eta}}_{J},\eta^{K}\rangle=e^{[w,\bar{Q}^{J}]\bar{\eta}_{J}}e^{\langle
w,Q_{K}\rangle\eta^{K}}|0\rangle$ (27)

where $|0\rangle$ is a particular state with spin zero in four of the
dimensions, namely $|0\rangle=\prod_{K=1}^{4}[\bar{w},\bar{Q}^{K}]|+\rangle$,
with $|+\rangle$ as in (4).666This is the supergravity analog of the non-
chiral superspace studied in [42] for super Yang-Mills.

Defining the $\eta^{K}$ version of the center of mass and relative variables
as in (22), the amplitudes can be rewritten as

$\displaystyle\hat{\cal A}_{2}^{c}$ $\displaystyle\sim$
$\displaystyle\delta^{8}(\vec{p}+\vec{p}^{\prime})\delta^{4}(\bar{\eta}_{J}+{\bar{\eta}}^{\prime}_{J})\delta^{4}(\eta^{K}-{\eta^{\prime}}^{K})~{},~{}~{}~{}~{}~{}~{}N=N^{\prime}~{},~{}~{}~{}~{}~{}n=n^{\prime}$
(28) $\displaystyle\hat{\cal A}_{3}^{c}$ $\displaystyle\sim$
$\displaystyle\delta^{8}(\sum\vec{p}^{\,i})\delta^{4}(\bar{\eta}^{cm}_{J}+{\bar{\eta}}^{3}_{J})\delta^{4}(\eta^{cmK}-\eta^{3K})\times$
(29) $\displaystyle\times l_{p}^{9/2}{R_{-}\over
R_{9}^{5/2}}{(n_{1}N_{2}-n_{2}N_{1})^{2}N_{3}^{2}\over(N_{1}N_{2})^{2}\sqrt{N_{1}N_{2}N_{3}}}\delta(\sum
p_{+}^{i})\delta^{4}({\bar{\eta}}^{r}_{J})\ .$ (30)

Note that the first line in (29) has a form similar to the two point function
(28), a fact which will be exploited later.

Let us emphasize that we have done nothing nontrivial yet. The amplitudes (28)
and (29) are trivial rewritings of the amplitudes (6) and (7) in terms of
convenient variables. The goal of Section 3 is to derive (28) and (29) from a
matrix model computation.

##  3 The BFSS Matrix model

We first review the BFSS model and emphasize some aspects that will be
important in later computations. The BFSS matrix model is defined by

$S={1\over R_{-}}\int dt\textrm{Tr}\left[{1\over
2}\sum_{a=1}^{9}(D_{t}X^{a})^{2}+{1\over
4}{R_{-}^{2}\over(2\pi)^{2}l_{p}^{6}}\sum_{a,b=1}^{9}[X^{a},X^{b}]^{2}+{\rm
fermions}\right]$ (31)

where $t=x^{+}$, and $X^{a}$ are normalized so that they are matrix versions
of the position coordinates, with dimensions of length. The matrices are
$N\times N$ in the adjoint of the $U(N)$ gauge group [2, 43]. The fermions are
in the 16 of SO(9). We also used that $G_{N,11}=16\pi^{7}l_{p}^{9}$.

The $U(1)$ part of the theory decouples and describes the center of mass
degrees of freedom, which consist of nine positions and 16 Majorana fermions.
These fermions are the Goldstone fermions of the 16 supercharges that are not
preserved by a massless superparticle with null momentum. The physics of these
nine variables and 16 fermions reproduces what we expect for a superparticle
in light cone gauge. In particular, they give rise to the two point amplitude
(28) and the “center of mass” part of the three point amplitude. We will
discuss this in more detail below. The $SU(N)$ degrees of freedom are believed
to give rise to a zero energy bound state at threshold [44] as well as a
continuum. The continuum is related to configurations where the matrices
develop large diagonal vacuum expectation values that break
$U(N)\to\prod_{i=1}^{\ell}U(N_{i})$ which are then viewed as a scattering
state of $\ell$ gravitons with null momentum $-p_{-}^{i}=N_{i}/R_{-}$. Each of
these $U(N_{i})$ factors has a $U(1)_{i}$ part and an $SU(N_{i})$ part. When
we define the asymptotic scattering states we focus on the zero energy bound
state in the $SU(N_{i})$ parts. Then the $U(1)_{i}$ parts describe massless
superparticles.

The 32 original supersymmetries are still realized in this theory: 16 of them
linearly and 16 non-linearly. The latter are realized as shifts of the 16
Majorana fermions of the $U(1)$ part of the theory. The quantization of these
sixteen fermions in each $U(1)$ fills out the $2^{8}$ states of a massless
superparticle in eleven dimensions. We can use the same method as in (4) to
label the asymptotic states in terms of Grassmann variables. This means that
we can define a matrix model amplitude, in canonical normalization, in terms
of scattering states ${\cal A}_{MM}^{c}$. The BFSS conjecture [2] states that
the large $N$ limit of these amplitudes, keeping $-p^{i}_{-}=N_{i}/R_{-}$
fixed, precisely matches the M-theory amplitudes with $x^{-}$ non-compact.

The matrix model does not realize the full super-Poincare symmetry, but it
does realize a subgroup of these symmetries. This subgroup is constraining
enough to fix the full polarization dependence of the three point amplitude.
To see this, first note that, for the three point amplitude, we can use
rotation symmetry, momentum conservation and Galilean symmetry (which is a
part of the original Lorentz group that commutes with $x^{-}$ translations) to
transform to a momentum configuration that lies in just four of the eleven
dimensions. This justifies our use of four dimensional notation. This is
completely parallel to our discussion of the M-theory amplitude. The only new
point is that we do not need to use the broken boosts that involve the $x^{-}$
coordinate, the boosts that would change $N_{i}$.

The next observation is that the algebra of the supercharges is not modified
relative to the non-compact situation. In other words, the supercharge algebra
acting on the external states is the same as in (3). All the various
polarization states can be obtained by the action of the same operators
discussed in (4) and (5). In the matrix model, the $b_{I}$ and $b^{\dagger I}$
are the fermionic creation and annihilation operators for each $U(1)_{i}$
factor [45]. Once we combine the various polarization states into a field as
in (4), we can then view the supercharges as acting as

$Q_{\alpha
I}=\sum_{i}\lambda^{i}_{\alpha}\bar{\eta}_{I}^{i}~{},~{}~{}~{}~{}~{}~{}\bar{Q}_{\dot{\beta}}^{I}=\sum_{i}\bar{\lambda}^{i}_{\dot{\beta}}\partial_{\bar{\eta}^{i}_{I}}\
.$ (32)

For three point kinematics, equation (32) gives us the action of only half of
the $\bar{Q}$ supersymmetries. However, this is sufficient because the three
point function is actually invariant under the other half. We now try to write
the most general Grassman expression for the the three point function that
will be invariant under (32):

$\delta^{16}(\sum_{i}\lambda^{i}_{\alpha}\bar{\eta}_{I}^{i})\ .$ (33)

In principle, equation (33) could be multiplied by an extra function of the
$\bar{\eta}^{i}_{I}$. However, for the three point function, there is no
function we can add that is non-zero on the support of the delta function (33)
which is invariant under the $\bar{Q}$ supersymmetries in (33). Of course,
(33) is invariant due to momentum conservation. This means that (33) is the
only combination that is invariant under supersymmetry. This determines
completely the polarization dependence of the three point amplitude. Let us
emphasize that despite the fact that we are using four dimensional notation,
equation (33) encodes all the polarization data in the full eleven dimensional
theory. In other words, the $\bar{\eta}_{I}$ variables have spin in the other
7 dimensions.

The conclusion is that the only unknown part of the three point amplitude is a
function of the $N_{i}$. These cannot be changed by any of the symmetries that
are realized in the matrix model. In other words, the three point amplitude in
the matrix model is (7), up to an overall multiplicative function of the
$N_{i}$. We will show that this function is a constant independent of the
$N_{i}$, which we expect to be one, but we did not check the overall numerical
value.

###  3.1 The BFSS model after we compactify one spatial dimension

We will not attack the problem of computing the three point amplitude
directly. We first consider a slightly different problem which leads to a
couple of simplifications. In particular, we compactify one of the spatial
dimensions, $x^{9}\to x^{9}+2\pi R_{9}$. The matrix model is replaced by a 1
+1 dimensional matrix gauge theory [19, 2, 20] with the Lagrangian

$\displaystyle S$ $\displaystyle=$ $\displaystyle{1\over
R_{-}(2\pi\tilde{R}_{9})}\int
dt\int_{0}^{2\pi\tilde{R}_{9}}d\tilde{x}_{9}\textrm{Tr}\left[-{1\over
2}(D_{\alpha}X^{a})^{2}+{1\over
4}{R_{-}^{2}\over(2\pi)^{2}l_{p}^{6}}[X^{a},X^{b}]^{2}-{1\over
4}{(2\pi)^{2}l_{p}^{6}\over R_{-}^{2}}F^{2}+{\rm
fermions}\right]~{}~{}~{}~{}~{}~{}$ (34) $\displaystyle{\rm
with}~{}~{}~{}~{}~{}~{}~{}\tilde{R}_{9}={l_{p}^{3}\over R_{-}R_{9}}$ (35)

where $\tilde{x}^{9}$ is T-dual to the original coordinate $x^{9}$ [19, 4, 5,
33]. The two dimensional gauge theory has a quantized total electric flux

$n={\partial L\over\partial F_{01}}={2\pi l_{p}^{6}\over
R_{-}^{3}\tilde{R}_{9}}\textrm{Tr}[F_{01}]\ .$ (36)

This integer should be viewed as the momentum in the compact $x^{9}$
direction. Indeed, we can check that the energy of such a state would be given
by $E=-p_{+}={1\over 2}{n^{2}\over R_{9}^{2}}{R_{-}\over N}$, in agreement
with $2p_{+}p_{-}={n^{2}\over R_{9}^{2}}$.

The compactification is desirable for two reasons. First, in the case is that
the $(N,n)$ are coprime777”coprime” means that they do not share any non-
trivial common factor. the ground state is gapped [44]. The presence of the
gap is useful because it allows the bound state to be more cleanly isolated
from the continuum. Second, after recalling that a real value of $n$
corresponds to real momentum along $x^{9}$ and imaginary momentum along
$x^{8}$, we can now go to to Euclidean time. Namely, we take $t=x^{+}$ in (34)
imaginary, $t=i\tau$, so that $X^{8}\propto n\tau$ with no $i$. Euclidean time
naturally leads to a simple picture for the scattering process, which is given
in figure 2. Notice that euclidean time in the matrix model corresponds to
imaginary $x^{+}$ in the original eleven dimensional description.

Figure 2:  The scattering amplitude configuration in Euclidean time. As
$\tau\to-\infty$ the matrix splits into two groups $N_{1}$ and $N_{2}$. For
$\tau\to+\infty$ they join into a single bound state. For suitable choices of
the electric fields, $n_{i}$, the bound states are gapped. We can view this
configuration as a (limit of) a simple BPS string network [27]. The arrows
indicate that the D1 branes extend to infinity.

This configuration can be viewed as a limit of an $(N,n)$ string network
consisting of branes with $N$ units of D1 brane charge and $n$ units of
fundamental string charge. We say it is a limit because we are only
considering the field theory in (34), which is a decoupling limit of the
fields on the D1 brane theory. This description as a string network is not
strictly necessary but it helps us connect the problem to the work by Sen in
[31].

###  3.2 Introducing brane sources

In order, to relate the computation to an index, we modify the D1 boundary
conditions so that the D1 branes terminate at finite Euclidean time. In other
words, in the field theory defined by (34), we want to introduce boundary
conditions in the Euclidean time direction. These boundary conditions amount
to demanding that the D1 branes terminate on D3 branes extended along the
directions $123\tilde{9}$. We have Dirichlet boundary conditions for the
directions tranverse to the D3 branes. For the longitudinal directions, we
have a more complicated boundary condition involving a Nahm pole [46, 47]

$X^{A}(\tau)={L^{A}\over(\tau-\tau_{\rm bdy})}+{\rm
subleading}~{},~{}~{}~{}~{}~{}~{}~{}~{}{\rm a}~{}~{}\tau\to\tau_{\rm
bdy}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}A=1,2,3$ (37)

where $L^{A}$ are $N_{3}\times N_{3}$ matrices obeying the $SU(2)$ commutation
relations. For the right boundary we impose that we have a single irreducible
representation of $SU(2)$ with dimension $N_{3}$ or spin $j$ with $2j+1=N_{3}$
[47]. On the left side, the $L^{A}$ are given by two irreducible
representations, one with dimension $N_{1}$ and one with dimension $N_{2}$,
each forming a sub-block of the matrix $L^{A}$. In addition, for the
$U(1)_{i}$ part of all the $U(N_{i})$, we have Neumann boundary conditions
along the longitudinal directions (the directions 1,2,3). For $X^{8}$, we have
the Dirichlet boundary conditions

$X^{8}(\tau_{l})=\left(\begin{array}[]{cc}x^{8}_{1}{\bf 1}_{N_{1}}&0\\\
0&x^{8}_{2}{\bf
1}_{N_{2}}\end{array}\right)~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{8}(\tau_{r})=x^{8}_{3}{\bf
1}_{N_{3}}$ (38)

where ${\bf 1}_{N}$ is the identity matrix of dimension $N$. The other
boundary conditions are simply $X^{a}=0$ for $a=3,4,5,6$. In addition, we have
a boundary condition for the electric flux. On the right side, at
$\tau=\tau_{r}$ we impose that the total electric flux is $n_{3}$. On the left
side, there are two $U(1)$ electric fields, one involving the identity in the
first $N_{1}$ entries and the second involving the identity in the second
$N_{2}$ entries of the matrix. We demand that these are $n_{1}$ and $n_{2}$
respectively.

Figure 3:  The problem with D3 brane sources. We now have the 1+1 field theory
with a set of D3 boundary conditions. In (a), we see the problem for the two
point function. In (b), for the three point function. We have
$N_{3}=N_{1}+N_{2}$ and $n_{3}=n_{1}+n_{2}$. We have also defined
$x_{r}^{8}=x^{8}_{1}-x^{8}_{2}$ to be the relative separation in the $x^{8}$
direction, which is a boundary condition in the $SU(N)$ part of the 1+1
dimensional theory on the interval. The time $\tau_{*}$ is well defined in the
classical theory, but it is an integration variable in the quantum theory.

In the original M-theory problem, these boundary conditions amount to adding
M2 branes extended along the directions 123. The supersymmetries preserved by
these M2 branes obey the condition (25). If we consider the gravitons sourced
by such M2 branes, we see that they will be annihilated by such
supersymmetries. In other words, the graviton has some momentum $p$ and, in
addition, the particular polarization state is annihilated by supersymmetries
determined by the M2 brane. This is precisely the vacuum state (27). In other
words, the vacuum, $|0\rangle$ in (27), is annihilated by 8 of the
supercharges broken by the momentum $p$, the 8 that are preserved by the M2
branes. These are the supercharges $\langle w,Q_{J}\rangle$ and
$[\bar{w},\bar{Q}^{K}]$ where the indices $K$ and $J$ are defined as in the
remarks after (25). Here we used the M2 brane language, but this can be easily
translated into the field theory language of (34). The reason we chose to
write the amplitude as in (29) is because the $|0\rangle$ ground state is the
one produced by these brane sources.

Let us emphasize that, even though we talk about branes, the computation we
are doing just involves the field theory (34) with particular boundary
conditions.

Figure 4: Two equal views. In the “open string channel” we view the theory as
defined on a spatial interval and propagating along a circle of Euclidean
time. In the “closed string channel” we view the theory as defined on a closed
circle and propagating between two boundary states. The red line indicates the
spatial interval and the green direction the Euclidean time direction.

This field theory is defined on a cylinder given by the product of a circle
along the $\tilde{9}$ direction and an interval $[\tau_{l},\tau_{r}]$ along
the $\tau$ direction. In this case, we can view the theory in what is called
the “open string channel” and the “closed string channel,” see figure 4. In
the first case we view $\tilde{x}^{9}$ as the Euclidean time direction and in
the second we view $\tau$ as a Euclidean time direction. Note that we are not
talking about the conformal field theory of a fundamental string here, but the
non-conformal theory described by (34).

In the open string channel, we are essentially computing an index. The reason
is that the fermions obey periodic boundary conditions along the
$\tilde{x}^{9}$ circle. In the closed string channel, we can view the problem
as the overlap between a left and right boundary state up to evolution by the
Hamiltonian on the $\tilde{R}_{9}$ circle over a period of Euclidean time:

$Tr\left[(-1)^{F}\exp\left(-2\pi\tilde{R}_{9}H_{\rm
open}\right)\right]=\langle B_{l}|\exp\left(-(\tau_{r}-\tau_{l})H_{\rm
closed}\right)|B_{r}\rangle\ .$ (39)

For the case under study, both the left and right hand sides are zero due to
fermion zero modes. However, the equality continues to be true when we insert
operators. We will therefore insert an operator that soaks up the fermion zero
modes.

###  3.3 The two point function and the boundary states

We will first discuss the problem involving the two point function. In this
case, we have two D3 branes, see figure 3(a). It is convenient to split the
dynamics into that of the $U(1)$ degree of freedom and the $SU(N)$ part. Once
we include a non-zero electric flux $n$, coprime with $N$, we have a gapped
system for the $SU(N)$ part [44] and we can forget about it for this problem.
In other words, all the contributions for the two point function come from the
$U(1)$ part. The configuration with the boundary conditions preserves 8 of the
kinematic supercharges (the non-linearly realized ones) and 8 of the linearly
realized ones. This means that we have 8 fermion zero modes coming from the
kinematic supercharges. This makes the naively defined index vanish. We can
get something non-vanishing by computing an index of the form [31]

${\cal I}_{4}\sim\textrm{Tr}[(-1)^{F}J^{4}]\sim 1$ (40)

which contains insertions of the angular momentum operator $J$. Each operator
$J$ soaks up two fermion zero modes. Again, “$\sim$” denotes an equality up to
numerical factors (of $2$, $\pi$, etc).

In the closed string channel, we have a boundary state of the form

$|B\rangle={\cal N}|0_{\parallel}\rangle\int d^{5}p|p\rangle$ (41)

where ${\cal N}$ is a normalization factor that we will need to fix. The state
$|0_{\parallel}\rangle$ denotes the zero momentum state in the 123 directions,
where we have Neumann boundary conditions. The integral over $p$ enforces the
Dirichlet boundary conditions along the other five directions. Notice that we
integrate over five and not six directions because one of the directions
orthogonal to the D3 brane is the $\tau$ direction.

The open string channel is a computation of the index with an insertion of
$J^{4}$, which takes the schematic form

${\cal Z}_{o}(J^{4})\sim
Tr\left[(-1)^{F}J^{4}e^{-(2\pi\tilde{R}_{9})H_{open}}\right]\sim
Tr\left[\chi_{o}^{8}e^{-(2\pi\tilde{R}_{9})H_{open}}\right]=\langle
B_{l}|\chi_{o}^{8}e^{-\Delta\tau H_{cl}}|B_{r}\rangle$ (42)

The operators $\chi_{o}$ are eight different fermion zero modes canonically
normalized in the open string channel. Each angular momentum generator
contains a pair of these fermion zero modes. When we say canonically
normalized we mean the following. The fermions on the worldvolume have an
action of the form

$S=C\int dtd\tilde{x}^{9}[\psi\partial_{t}\psi+\cdots]\sim C\Delta\tau\int
d\tilde{x}_{9}\psi_{0}\partial_{t}\psi_{0}=\int
d\tilde{x}_{9}\chi_{o}\partial_{t}\chi_{o}~{},~{}~{}~{}~{}~{}~{}\chi_{o}=\sqrt{C\Delta\tau}\psi_{0}$
(43)

where $\psi_{0}$ is the constant part of $\psi$ on the interval and $C$ is a
constant that will not matter for us. Inserting $\chi_{o}$ into (42) is
equivalent to inserting eight different fermions $\psi$ at any points on the
two dimensional cylinder since only the zero mode contributes. On the other
hand, when we go to the closed string channel, it is natural to define the
canonically normalized fermion, $\chi_{cl}$, as

$S=C\int dtd\tilde{x}^{9}[\psi\partial_{\tilde{x}^{9}}\psi+\cdots]\sim
C\tilde{R}_{9}\int d\tau\psi_{0}\partial_{\tilde{x}^{9}}\psi_{0}=\int
d\tau\chi_{cl}\partial_{\tilde{x}^{9}}\chi_{cl}~{},~{}~{}~{}~{}~{}~{}\chi_{cl}=\sqrt{C\tilde{R}_{9}}\psi_{0}$
(44)

where $\psi_{0}$ is now the constant mode on the circle. In the closed string
channel, the action of the broken supercharges that generate the multiplet
translate into insertions of $\chi_{cl}$. These are the operators $b_{I}$ and
$b^{\dagger I}$ discussed around (4). These operators can be replaced by
$\psi$ insertions anywhere on the cylinder. However, these differ by
normalization factors from similar insertions in the open string channel

$\chi_{o}\sim\sqrt{\Delta\tau\over R_{9}}\chi_{cl}\ .$ (45)

Therefore, we have an identity of the form

${\cal Z}_{o}(J^{4})\sim
Tr\left[(-1)^{F}J^{4}e^{-(2\pi\tilde{R}_{9})H_{open}}\right]\sim{(\Delta\tau)^{4}\over\tilde{R}_{9}^{4}}\langle
B_{l}|\chi_{cl}^{8}e^{-H_{cl}\Delta\tau}|B_{r}\rangle\ .$ (46)

In this discussion, we have suppressed the fermion indices, but there are 8
fermions in the open string channel. In contrast, there are 16 fermions in the
closed string channel. However, we have only 8 insertions. This is the
necessary number of insertions to amount to 8 supercharge insertions, which is
what we have when we look at the 8 fermionic delta functions in (28).

Now, let us evaluate the overlap in the right hand side of (46). The action
for the $U(1)$ center of mass degree of freedom in the closed string channel,
from (34), involves a term

$S={N\over R_{-}}\int dt{1\over 2}\left({dx\over dt}\right)^{2}$ (47)

where $x$ is the center of mass position. This leads to an energy
$H_{cl}={1\over 2}{p^{2}N\over R_{-}}$. We then get

$\langle B_{l}|\chi_{cl}^{8}e^{-H_{cl}\Delta\tau}|B_{r}\rangle={\cal
N}^{2}V_{\parallel}\int d^{5}pe^{-{1\over 2}p^{2}{R_{-}\over
N}\Delta\tau}\sim{\cal
N}^{2}V_{\parallel}{N^{5/2}\over(R_{-}\Delta\tau)^{5/2}}\ .$ (48)

Notice that all oscillator modes are cancelling (among bosons and fermions) in
this computation. Here, $V_{\parallel}$ is the volume of the 123 directions
which arises due to the overlap of two $|0_{\parallel}\rangle$ states. This
infinite factor is also present in the left hand side of (42) from the
integral over the bosonic center of mass degrees of freedom. In fact, the left
hand side (42) is

${\cal Z}_{o}(J^{4})\sim{\cal I}_{4}\times V_{\parallel}\int
d^{3}k\exp\left(-{1\over 2}{k^{2}R_{-}\tilde{R}_{9}^{2}2\pi\over
N\Delta\tau}\right)\sim
V_{\parallel}{(N\Delta\tau)^{3/2}\over(R_{-}\tilde{R}_{9})^{3/2}}$ (49)

where we used (40) and that in the open string channel the center of mass
motion modes have the action

$S={N\Delta\tau\over R_{-}\tilde{R}_{9}}\int d\tilde{x}_{9}{1\over
2}(\partial_{\tilde{9}}\vec{x})^{2}\ .$ (50)

Equating the left and right hand sides of (46), given in (49) and (48), we get

$V_{\parallel}{(N\Delta\tau)^{3/2}\over(R_{-}\tilde{R}_{9}^{2})^{3/2}}\sim{\cal
N}^{2}{(\Delta\tau)^{4}\over\tilde{R}_{9}^{4}}V_{\parallel}{N^{5/2}\over(R_{-}\Delta\tau)^{5/2}}~{},~{}~{}~{}~{}\longrightarrow~{}~{}~{}~{}\boxed{{\cal
N}=\sqrt{R_{-}\tilde{R}_{9}\over N}}\ .$ (51)

A consistency check is that there is no $\Delta\tau$ dependence in ${\cal N}$.

In conclusion, we have used the two point function to find the normalization
constant ${\cal N}$ in the boundary state.

###  3.4 The three point functions and the amplitude

We now turn to the three point amplitude. We will consider the configuration
in figure 3(b), where $N_{1}+N_{2}\to N_{3}$. On the left side we have two
sub-blocks of sizes $N_{1}$ and $N_{2}$ which are widely separated and on the
right we have a single block of size $N_{3}$.

There are again two pictures of the underlying process. The first picture is
the open string picture, which involves computing an index ${\cal I}_{6}$ (up
to the center of mass). The six angular momentum insertions soak up twelve
fermionic zero modes, eight from the center of mass motion, or the $U(1)$
part, and four from the breaking of half of the eight supercharges of the
$SU(N)$ theory by our configuration.

The “closed string channel” picture is one that can be viewed as involving the
amplitude, at least in the large $\Delta\tau$ limit. From now on we consider
this limit of large $\Delta\tau=\tau_{r}-\tau_{l}$ and large $x_{r}^{8}$ (see
figure 3), with $x_{r}^{8}/\Delta\tau$ fixed. The left boundary state can be
decomposed into a product of two boundary states, $\langle B_{l}|=\langle
B_{1}|\langle B_{2}|$, for each of the two sub-blocks of size $N_{1}$ and
$N_{2}$. We split the euclidean time evolution into three periods:

  1. 1.

The initial time interval when the $N_{1}$ and $N_{2}$ blocks are well
separated.

  2. 2.

The times when $N_{1}$ and $N_{2}$ merge into the $N_{3}$ block.

  3. 3.

The late time interval when the $N_{3}$ block propagates by itself.

The middle period is centered around some time $\tau_{*}$ and has some time
width that depends on the fundamental constants but it is much smaller than
the overall time interval $\Delta\tau$. We then have an expression of the form

$\begin{split}{\cal Z}_{o}(J^{6})&=\langle B_{1}|\langle
B_{2}|J^{6}\int_{0}^{\Delta\tau}d\tau^{\star}e^{-\Delta\tau_{1}(H_{1}+H_{2})}{\hat{\bf
A}^{c}}_{M}e^{-(\Delta\tau_{3})H_{3}}|B_{3}\rangle~{},\\\
&~{}~{}\Delta\tau_{1}\equiv\tau_{*}-\tau_{l}~{},~{}~{}\Delta\tau_{3}\equiv\tau_{r}-\tau_{*}\
,\end{split}$ (52)

where we suppressed the polarization degrees of freedom for notational
simplicity. The $J^{6}$ insertion will become clearer below.

In (52), ${\hat{\bf A}^{c}}_{M}$ is the amplitude without the energy
conserving delta function

$\hat{\cal A}^{c}_{M}=\langle p_{1}|\langle p_{2}|{\hat{\bf
A}^{c}}_{M}|p_{3}\rangle\delta(\sum p_{+}^{i})~{}.$ (53)

This results from doing the path integral in the full theory in the middle
region with the boundary conditions set by the momenta in the other two
regions. The location of this middle region, which we call $\tau_{*}$ becomes
an integration variable. This energy conserving delta function in the full
amplitude comes from the integration over $\tau_{*}$, schematically

$\delta(\sum p_{+}^{i})~{}~{}\longrightarrow~{}~{}~{}\int d\tau_{*}\ .$ (54)

In other words, in the index-amplitude relation (52), it is the amplitude
stripped of the momentum conserving delta function that appears on the right
hand side, not the actual scattering amplitude, because the amplitude event is
occurring at a definite time, $\tau_{\star}$.

The basic idea is that we will be able to compute the left hand side of (52),
and then determine the amplitude that appears in the right hand side of (52).

In order to analyze the problem, it is useful to view this configuration as
resulting from a configuration for the center of mass $U(1)$ degree of freedom
and a configuration for the $SU(N)$ part. On the right side, we have the
$SU(N)$ degrees of freedom forming a single bound state, of size $N_{3}$. The
$SU(N_{3})$ theory on the right hand side is completely gapped [44]. On the
left side, we have the degrees of freedom forming two separate bound states
with sizes $N_{1}$ and $N_{2}$. In this case, there is a massless degree of
freedom describing the separation between these two blocks. We have the
breaking pattern $SU(N_{3})\to SU(N_{1})\times SU(N_{2})\times U(1)$. Each of
the $SU(N_{i})$ is individually gapped, but the $U(1)$ part gives rise to
gapless modes.

When we think about the left two sub-blocks, we have the action

$\displaystyle S$ $\displaystyle=$ $\displaystyle{1\over
R_{-}\tilde{R}_{9}}\int d\tau d\tilde{x}_{9}{1\over 2}\left[N_{1}(\partial
x_{1})^{2}+N_{2}(\partial x_{2})^{2}\right]$ (55) $\displaystyle=$
$\displaystyle{1\over R_{-}\tilde{R}_{9}}\int d\tau d\tilde{x}_{9}{1\over
2}\left[N_{3}(\partial x_{cm})^{2}+N_{r}(\partial x_{r})^{2}\right]$ (56)

where we defined

$x_{1}=x_{cm}+x_{r}{N_{2}\over
N_{3}}~{},~{}~{}~{}~{}~{}~{}x_{2}=x_{cm}-x_{r}{N_{1}\over
N_{3}}~{},~{}~{}~{}~{}N_{r}={N_{1}N_{2}\over N_{3}}\ .$ (57)

Note that $N_{r}$ is not an integer. Similarly, we can define U(1) fluxes
which are

$n_{cm}=n_{1}+n_{2}=n_{3}~{},~{}~{}~{}~{}~{}~{}~{}~{}n_{r}={(n_{1}N_{2}-n_{2}N_{1})\over
N_{3}}\ .$ (58)

For the boundary states of blocks 1 and 2, we write

$\displaystyle\langle B_{1}|\langle B_{2}|$ $\displaystyle\sim$
$\displaystyle{\cal N}_{1}{\cal N}_{2}\langle 0_{1\parallel}|\int
d^{5}p_{1}\langle p_{1}|\times\langle 0_{2\parallel}|\int d^{5}p_{2}\langle
p_{2}|$ (59) $\displaystyle\sim$ $\displaystyle{\cal N}_{1}{\cal N}_{2}\langle
0_{cm\parallel}|\int d^{5}p_{cm}\langle p_{cm}|\times\langle
0_{r\parallel}|\int d^{5}p_{r}\langle p_{r}|$ (60) $\displaystyle\sim$
$\displaystyle\langle B_{cm}|\times\langle B_{r}|~{},~{}~{}~{}~{}~{}{\rm
with}~{}~{}~{}~{}\langle B_{r}|\equiv{{\cal N}_{1}{\cal N}_{2}\over{\cal
N}_{3}}\langle 0_{r\parallel}|\int d^{5}p_{r}\langle p_{r}|$ (61)

where in the second line the two-body state is re-written in terms of its
center of mass and relative momenta, $p_{cm}=p_{1}+p_{2}$ and
$p_{r}=p_{1}-p_{2}$ respectively. Note that the factor of ${\cal N}_{3}$ comes
in when we define $\langle B_{cm}|$.

The equality (52) can be separated into two equalities, one for the center of
mass, the $U(1)$ part, and one for the relative part, the $SU(N_{3})$ part. In
preparation for this splitting, it is useful to write the matrix model
amplitude as

$\displaystyle{\hat{A}}^{c}_{M}$ $\displaystyle=$ $\displaystyle\hat{\cal
A}_{Mcm}^{c}\hat{\cal A}^{c}_{Mr}$ (62) $\displaystyle\hat{\cal A}_{Mcm}^{c}$
$\displaystyle=$
$\displaystyle\delta^{8}(\sum\vec{p}^{\,i})\delta^{4}(\bar{\eta}^{cm}_{J}+{\bar{\eta}}^{3}_{J})\delta^{4}(\eta^{cmK}-\eta^{3K})$
(63) $\displaystyle\hat{\cal A}_{Mr}^{c}$ $\displaystyle=$
$\displaystyle\delta(\sum p^{i}_{+})\delta^{4}(\bar{\eta}^{r}_{J})A(p_{r})$
(64)

where $A(p_{r})$ is an off-shell amplitude that is a function of $p_{r}$,
$N_{i}$ and $n_{i}$. The delta function for energy conservation fixes the
value of $p_{r}$ in terms of $N_{i}$ and $n_{i}$, but we leave it general now
to make the derivation below a bit more clear. The goal is to determine the
on-shell value of $A$. Finally, note that the dependence on the Grassmann
variables is fixed by supersymmetry from the discussion around (33); our
choice of “vacuum” is the state produced by the M2 branes. This state is a
combination of the graviton and the three form potential that is invariant
under the $SO(3)\times SO(6)$ subgroup of $SO(9)$ that leaves the M2 brane
invariant (that preserves the 123 subspace). The $SO(9)$ is the symmetry
around the null momentum of the particle.

The part involving $\hat{\cal A}_{Mcm}^{c}$ in (63) is the same as what
appeared in the two point function. This part involves only the $U(1)$ degrees
of freedom and the discussion is exactly the same as in the two point
function. We subsequently focus on the $SU(N)$ problem, which schematically
involves

$\widetilde{{\cal Z}}_{o}(J^{2})\sim\int d\tau^{\star}\langle
B_{r}|J^{2}e^{-\Delta\tau_{1}H}\hat{\bf A}_{Mr}^{c}|0\rangle$ (65)

where the $|0\rangle$ on the right hand side is the $SU(N_{3})$ ground state.
The tilde means we are considering the $SU(N)$ problem.

We now analyze this problem in more detail. This is a completely self
contained problem in the 1+1 dimensional $SU(N)$ gauge theory on an interval.
On the right side, there is an $SU(N_{3})$ gapped ground state, as long as an
appropriate non-zero value of $n_{3}$ is chosen such that it is coprime with
$N_{3}$. The details of the D3 brane on the right side do not matter much.
They just give a constant that was already included in the normalization
factor ${\cal N}_{3}$. On the left side, the $SU(N_{3})$ is broken to
$SU(N_{1})\times SU(N_{2})\times U(1)_{r}$ by the boundary conditions. These
boundary conditions involve the statement that the relative flux in the
$U(1)_{r}$ part to be $n_{r}$ and the $8^{th}$ position to be
$x_{r}=x^{8}_{r}$, see figure 3. We see that the vacuum expectation value of
$X^{8}_{r}$ is non-zero near the boundary and then it becomes zero at some
intermediate value of Euclidean time $\tau_{*}$. This is just a classical
picture. In the full quantum picture, there can be fluctuations around this
configuration. The presence of a non-zero gradient for $X^{8}_{r}$ as well as
the flux $n_{r}$ breaks 4 the 8 supercharges that are preserved by the
boundary condition. This leads to four fermion zero modes that are soaked up
by the insertion of $J^{2}$ in (65).

Notice that, in contrast with the two point function case, the relative fields
all vanish on the right side. This implies that there is no volume factor and
that

$\tilde{\cal Z}_{o}(J^{2})\sim(n_{1}N_{2}-n_{2}N_{1})$ (66)

This follows from a logic very similar to that in Sen’s computation [31]. It
is identical to that computation if we are willing to consider the index in
the full string theory. On the other hand, if we interpret the computation in
[31] as a computation of dyons in ${\cal N}=4$ SYM, then our problem is
slightly different because we are considering the problem from the point of
view of the $1+1$ field theory. These two problems are connected by the Nahm
transform [46, 47], so we expect that we can phrase Sen’s computation of the
index as a computation in the 1+1 dimensional field theory.

Let us now turn to the right hand side of (65). The insertion of $J^{2}$
translates into the insertion of four fermions that are canonically normalized
in the open string channel, $\chi_{o}$. It is convenient to translate these to
fermion zero modes canonically normalized in the closed string channel

${\chi}_{o}=\sqrt{\Delta\tau_{*}\over\tilde{R}_{9}}\chi_{cl}~{},~{}~{}~{}~{}~{}~{}J^{2}\sim\chi_{o}^{4}\sim{(\Delta\tau_{*})^{2}\over\tilde{R}_{9}^{2}}\chi_{cl}^{4}~{},~{}~{}~{}~{}~{}\Delta\tau_{*}=\tau_{*}-\tau_{l}$
(67)

where $\Delta\tau_{*}$ is the Euclidean time between the left boundary and the
point where the vev becomes zero. The location of this point is integrated
over as we discuss below in more detail. The insertion of $\chi_{cl}$ then
corresponds to the action of the supercharges and is related to the four
Grassmann delta functions in (64).

Now, let us discuss the contributions from the bosonic momentum eigenstates in
the boundary state. On the left side, we have $\langle B_{r}|$ as in (61). On
the right side, we have a state with $X_{r}=0$, so we expect that we are
simply expanding a delta function that contains all momenta. Along the
longitudinal directions (directions 1,2,3), the state $|0\rangle_{\parallel}$
selects the zero momentum part. For the other five dimensions (directions
4,5,6,7,8), we still need to do an integral, which is of the form

$\int d^{5}p_{r}e^{ip_{r}.x_{r}}\exp(-{1\over 2}{p^{2}R_{-}\over
N_{r}}\Delta\tau_{*})A(p_{r})\sim\left({N_{r}\over
R_{-}\Delta\tau_{*}}\right)^{5/2}\exp\left(-{1\over 2}{x_{r}^{2}N_{r}\over
R_{-}\Delta\tau_{*}}\right)A(p_{r}^{s})$ (68)

where $p_{r}^{s}$ is the saddle point for $p_{r}$, namely
$p_{8r}^{s}=i{N_{r}\over R_{-}}{x_{r}^{8}\over\Delta\tau_{*}}$. The term
involving $x_{r}$ in (68) arises because we are fixing the boundary value of
$x_{r}=x_{r}^{8}$ to a non-zero value at the left boundary, which involves a
displacement in the $X^{8}$ coordinate, see figure 3. We can interpret the
exponential in (68) as the energy due to the gradient in $X^{8}$. There is an
extra contribution to the energy from the electric flux, which leads to an
extra factor of the form

$\exp\left(-{1\over 2}{n_{r}^{2}R^{3}_{-}\Delta\tau_{*}\tilde{R}_{9}^{2}\over
N_{r}l_{p}^{6}}\right)\ .$ (69)

Putting together all the $\tau_{*}$ dependent terms from (67) (68) (69) and
integrating over $\tau_{*}$, we find a integral amiable to a saddle point
approximation:

$\int_{0}^{\infty}d(\Delta\tau_{*}){1\over(\Delta\tau_{*})^{1/2}}\exp\left(-{a\over\Delta\tau_{*}}-b\Delta\tau_{*}\right)f(\Delta\tau_{*})\sim
b^{-1/2}\exp(-\sqrt{ab})f(\Delta\tau_{*}^{s})$ (70)

where $\Delta\tau_{*}^{s}$ is the saddle point of the $\Delta\tau_{*}$
integral.888Note that $A(p_{r}^{s})$ depends on $\tau_{\star}$ because
$p_{r}^{s}$ depends on $\tau_{\star}$. Putting all of this together, the right
hand side of (65) is then

$\langle B_{r}|J^{2}e^{-\Delta\tau_{1}H}\hat{\bf
A}_{Mr}^{c}|0\rangle\sim{{\cal N}_{1}{\cal N}_{2}\over{\cal
N}_{3}}{1\over\tilde{R}_{9}^{2}}\left({N_{r}\over
R_{-}}\right)^{5/2}{N_{r}^{{1\over
2}}l_{p}^{3}\over\tilde{R}_{9}n_{r}R_{-}^{3/2}}\exp\left(-x_{r}{n_{r}\over
R_{9}}\right)A(p_{r}^{s})$ (71)

where $A(p_{r}^{s})$ was defined in (64). The saddle point approximations in
(68) and (70) together localize $p_{r}^{s}$ to its on-shell value

$p_{8r}^{s}=i{n_{r}\over R_{9}}={N_{2}p^{1}_{8}-N_{1}p^{2}_{8}\over N_{3}}\ .$
(72)

The remaining exponential in (71) could be viewed as the energy contribution
at the saddle point of the $\tau_{*}$ integral (70). This is just the
classical energy of the configuration in figure 3. Since the configuration is
BPS, we expect that its energy should be equal to the appropriate “central
charge” of the supersymmetry algebra. This central charge can depend on the
boundary conditions. In fact, we see that the exponential in (71) depends in a
simple way on the boundary value $x_{r}$ and the electric field flux $n_{r}$.
We therefore argue that the exponential term in (71) should be removed once we
subtract the central charge and will drop it.

Using (57), (58), and (51), and equating (71) to (66) we get

$\displaystyle\tilde{\cal Z}_{o}(J^{2})$ $\displaystyle\sim$
$\displaystyle\langle B_{r}|J^{2}e^{-\Delta\tau_{1}H}\hat{\bf
A}_{Mr}^{c}|0\rangle$ (73) $\displaystyle(n_{1}N_{2}-n_{2}N_{1})$
$\displaystyle\sim$ $\displaystyle\sqrt{N^{3}\over
N_{1}N_{2}}\sqrt{R_{-}\tilde{R}_{9}}{1\over\tilde{R}_{9}^{2}}\left({N_{r}\over
R_{-}}\right)^{5/2}{N_{r}^{{1\over
2}}l_{p}^{3}\over\tilde{R}_{9}n_{r}R_{-}^{3/2}}A(p_{r}^{s})$ (74)

which then leads to

$\boxed{A(p_{r}^{s})\sim{l_{p}^{9/2}R_{-}\over
R_{9}^{5/2}}{1\over\sqrt{N_{1}N_{2}N_{3}}}{(n_{1}N_{2}-n_{2}N_{1})^{2}N_{3}^{2}\over
N_{1}^{2}N_{2}^{2}}}$ (75)

Equation (75) agrees precisely with (29). One might worry that (75) is only
approximate because we are using perturbative techniques in the initial time
interval even though we are claiming the result is valid for generic values of
$N$ and $n$. In principle, the integrals over $p_{r}$ and $\tau_{\star}$ could
be corrected by fluctuations around the saddle point, where the precise of
corrections are fixed by $N$ and $n$. However, the actual widths of the
$p_{r}$ and $\tau_{\star}$ saddle points can be taken to be arbitrarily small
by taking $x_{r},~{}\Delta\tau$ to be arbitrarily large. Therefore, the
perturbative calculation for the initial time region is exact in the large
$x_{r},~{}\Delta\tau$ limit.

In conclusion, the amplitude computed using the matrix model coincides with
the amplitude expected in gravity. Having obtained the amplitude when $R_{9}$
is finite, we can also take the limit that $R_{9}\to\infty$ and
$n^{i}\to\infty$ keeping $p^{i}_{9}=n^{i}/R_{9}$ fixed (also $p_{8}$ remains
fixed). In this limit, we recover the three point amplitude in the 0+1
dimensional matrix model. We can further take the limit $N^{i}\to\infty$,
keeping $p^{i}_{-}={N^{i}/R_{-}}$ fixed. This gives the amplitude in the non-
compact eleven dimensions. Note that the amplitude agrees precisely even at
finite $N$ and $n$. For the BFSS conjecture, this agreement was not necessary,
only the large $N$ limit had to agree with gravity. Perhaps there is some
argument, beyond this explicit computation, that explains why the amplitudes
had to agree even at finite $N$ and $n$.

Notice that in the weakly coupled IIA limit, $R_{9}\to 0$, these amplitudes do
not go over to the fundamental string amplitudes when $n_{i}$ are non-zero.
They are amplitudes that involve scattering of bound states of D0 branes in
the IIA theory. It is amusing that in the large $N$ limit, these amplitudes
become very similar to the problem of scattering D0 branes in the original
BFSS matrix model.

##  4 Conclusions

In this paper, we have computed the three graviton amplitude in the BFSS
model. Crucially, all three particles have non-zero longitudinal momentum, or
$N_{i}>0$, so this computation probes strongly coupled regime of the matrix
model. We have shown that the amplitude agrees with the expected form in
M-theory. To do the computation, we used that the three point boundary
conditions preserve some supersymmetries and related the amplitude to an
index, a protected quantity, which we in turn used to fix the amplitude. The
fact that certain supersymmetric amplitudes can be computed by relating them
to other problems has been used before many times, e.g. [48, 49]. To make the
identification, we found it useful to view the amplitude as arising from the
exchange of gravitons sourced by M2 brane “operators.” For the case of the two
point function, a somewhat similar discussion was contained in [17]. In this
paper, we considered the case where the gravitons interact as they go between
different brane sources.

This three point amplitude can be used to derive the soft theorems [50, 51],
as we show in a separate publication [52].

There are a number of future directions.

It is natural to wonder whether other supersymmetric amplitudes could be
computed by using a similar method. For example, can other string network
indices be used to compute higher point amplitudes? Particularly interesting
amplitudes are those related to higher curvature corrections of the
gravitational action. In addition, string networks like the ones we considered
sometimes have moduli. For example, in the network we considered, we could add
a small triangle at the interaction vertex so that the three individual three
point amplitudes are sewn together. For the particular case of the three point
amplitudes, such configurations cannot become very large because of the
polarizations. However, for higher point amplitudes we would need to worry
about such configurations, see [27, 31].

We could also consider three point amplitudes in different theories. For
example, the graviton three point function in heterotic string theory has an
$\alpha^{\prime}$ correction. It would be interesting to see whether this is
reproduced by a similar computation in the corresponding matrix model [53,
54]. Here the three point function has a correction that depends on a
continuous parameter $(\alpha^{\prime})$, which seems to be a problem for the
strategy pursued in this paper.

Since the three point amplitude is an essential element in the BCFW [55]
construction of gravity amplitudes, one could wonder whether a similar
argument could show that all gravity amplitudes are reproduced by the matrix
model. Of course, we expect this to work only for the low energy gravity
amplitudes rather than the full M-theory amplitudes which would have a more
complicated analytic structure as a function of momenta. One significant
hurdle is arguing that the contour at infinity in the BCFW derivation should
vanish in the supergravity regime.

Finally, it might also be possible to use the supersymmetric localization
methods developed in [56, 57] for the plane wave matrix model [58] to compute
amplitudes. It seems that the amplitude computation in that model might be
related to the a tunneling computation between two different vacua of the
model, but we did not manage to do the computation in that way.

In this paper, we did not compute the overall numerical normalization of the
amplitude, namely the factors of $2$ and $\pi$. It should also be determined
by our method if one keeps track of all such factors in all of our formulas.
Note that for the special amplitude in appendix A, this factor is indeed
correct.

### Acknowledgments

We thank O. Aharony, T. Banks, M. Dine, N. Miller, S. Mizera, N. Seiberg, A.
Strominger, A. Tropper, N. Valdes, H. Verlinde, and T. Wang for discussions.

J.M. is supported in part by U.S. Department of Energy grant DE-SC0009988.
A.H. is supported by the Simons Foundation.

##  Appendix A Computation of the three point amplitude when one particle has
no light cone momentum

In this appendix, we discuss the computation of the three point function in
the special case when $N_{2}=0$, $N_{1}=N_{3}=N$, see figure 5. In this case,
the gravity mode with $N_{2}=0$ can be viewed simply as a graviton with some
momentum $p_{z}\not=0,~{}p_{\bar{z}}=0$ in two of the transverse dimensions.
We can also choose the polarization vector $\epsilon_{++}\not=0$ and the rest
of the components equal to zero. Then, we simply have the original D0 brane
action interacting with a background gravitational plane wave metric of the
form

$ds^{2}=2dx^{+}dx^{-}+2dzd\bar{z}+d\vec{y}^{2}+(dx^{+})^{2}\epsilon_{++}e^{ip_{z}z}$
(76)

This induces and extra term in the Matrix model Lagrangian (31) that is simply
a potential term of the form

$\int dt\epsilon_{++}\textrm{Tr}[e^{ip_{z}Z}]$ (77)

Figure 5: The scattering configuration for the three point function where one
of the particles has zero momentum. It leads to the insertion of the operator
(77) in the matrix quantum mechanics.

The computation of the amplitude involves expanding the path integral to
leading order in $\epsilon_{++}$ and this results to an insertion of (77) as
an operator. In other words, we need to compute

$\langle N,\vec{p}_{1}|\int
dt\epsilon_{++}\textrm{Tr}[e^{ip_{z}Z}]|N,\vec{p}_{3}\rangle$ (78)

The computation of this expression involves the $U(1)$ degrees of freedom,
which simply give rise to the 9 spatial momentum delta functions. The integral
over $dt$ produces the energy conservation delta function. This $U(1)$ part
reproduces the amplitude because it is simply the superparticle action in
lightcone gauge interacting with a plane wave, which is merely one way in
which we could compute the amplitude in gravity. Notice that this also implies
that the overall coefficient also reproduces gravity.

What remains is to compute

${1\over N}\langle 0|\textrm{Tr}[\exp(ip_{z}Z)]|0\rangle$ (79)

in the $SU(N)$ theory. Here $|0\rangle$ is the bound state. We do not know it
explicitly. However, we know it is $SO(9)$ invariant. On the other hand, the
$Z$ that appears in the exponent carries $SO(2)$ charge. So, if we expand the
exponential, all the nonzero powers of $Z$ have a zero expectation value by
the symmetries of the problem. Therefore, the full expectation value (79) is
one. One might be worried that this might be somewhat ill defined for very
high powers of $Z$ because the bound state wavefunction decays only as a power
law in the flat directions. In fact, we can give a more solid argument by
introducing a small mass term $\mu$, so that we consider a computation in the
plane wave matrix model [58]. This computation can be performed using the
localization method in [56, 57], after identifying the bound state with the
maximal $SU(2)$ representation in the discussion in [56, 57].

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