††institutetext: School of Natural Sciences, Institute for Advanced Study,
1 Einstein Drive, Princeton, NJ 08540 USA

# Holography for cylindrical gravitational waves

Robert F. Penna<EMAIL_ADDRESS>
###### Abstract

We study a two dimensional nonlinear sigma model whose classical solutions
describe cylindrical gravitational wave scattering with zero cosmological
constant. We quantize this sigma model and compute its two-particle tree-level
$S$-matrix. We discuss an $\mathrm{SL}(2,\mathbb{R})$ symmetry of the sigma
model called the Ehlers group and directly verify the associated $S$-matrix
conservation law at tree level. Finally, we use the holographic dictionary to
define a dual boundary theory and compute its two-point correlation function.
This is an example of holography with zero cosmological constant.

##  1 Introduction

Cylindrical gravitational waves are vacuum solutions of general relativity
with two commuting Killing vectors. The Killing vectors generate the
isometries of the cylinder. So cylindrical gravitational waves in four
spacetime dimensions are effectively two dimensional. In generalized Weyl
coordinates, the metric only depends on $t$ and $r$ (time and space). The
classical scattering of cylindrical gravitational waves with zero cosmological
constant can be completely described using a two dimensional nonlinear sigma
model Breitenlohner:1986um ; Nicolai:1991tt . The sigma model action comes
from a dimensional reduction of the Einstein-Hilbert action to two spacetime
dimensions.

In this paper, we quantize this sigma model and compute its two-particle tree-
level $S$-matrix. We think of this as a toy model for a quantum gravity
$S$-matrix. The asymptotic in- and out-states are cylindrical gravitational
waves. This calculation is itself a new result.

The sigma model we are discussing is known to be classically integrable.
Whether or not it is quantum integrable is an interesting open question that
we will not address in this work. The reason it is classically integrable is
that it has a hidden Kac-Moody symmetry called the Geroch group
Breitenlohner:1986um ; Geroch:1970nt ; Geroch:1972yt ; Lu:2007jc ; Lu:2007zv ;
Nicolai:1991tt ; Schwarz:1995af ; Schwarz:1995td . We will not address the
fate of the Geroch group in the quantum theory. However, we will discuss an
important $\mathrm{SL}(2,\mathbb{R})$ subgroup of the Geroch group called the
Ehlers group. We will show that two of the three generators of the Ehlers
group are spontaneously broken in the quantum theory, but the third generator
is unbroken and gives a bona fide $S$-matrix conservation law. We will derive
this $S$-matrix conservation law and directly verify it at tree level.

Finally, we will introduce a holographically dual version of the sigma model
and compute its two-point correlation function. Known examples of holographic
duality usually have a negative cosmological constant111Recently, there has
been a lot of interest in using the “Celestial Holography” program
Pasterski:2021raf  to understand holography with zero cosmological constant.
One of the main insights of Celestial Holography is that boost eigenstates are
better than momentum eigenstates for certain calculations in asymptotically
flat spacetime. However, cylindrical gravitational waves are in a rotation
eigenstate (by virtue of their rotational symmetry), and this rotation does
not commute with boosts orthogonal to the cylinder. So boost eigenstates do
not seem very useful for our problem. so it is notable that our example has
zero cosmological constant. We will essentially define the boundary theory
using the holographic dictionary. In the future, it would be very interesting
to find an independent definition of the boundary theory.

There have been a number of works Kuchar:1971xm ; Ashtekar:1996bb ;
Angulo:2000ad ; Niedermaier:2000ud ; BarberoG:2003ffm ; BarberoG:2004kdm ;
BarberoG:2008fcp , starting with an early work of Kuchař, on the problem of
quantizing _polarized_ cylindrical gravitational waves. Polarized cylindrical
gravitational waves are described by a free theory. They do not have
nontrivial $S$-matrices. The interesting observables in the free theory are
the $n$-point correlation functions. These have been studied previously by
Niedermaier Niedermaier:2000ud  and by Barbero G., Garay, and Villaseñor
BarberoG:2008fcp .

Niedermaier Niedermaier:2000ud  investigated an $\mathrm{SL}(2,\mathbb{R})$
symmetry of the free theory which is different from the Ehlers
$\mathrm{SL}(2,\mathbb{R})$ that we will be studying. He used this symmetry as
the basis for a holographic model of polarized cylindrical gravitational
waves. His construction has some overlap with our discussion in section 4. A
novelty of our discussion is that it extends to the interacting theory. Also,
his theory is somewhat different from the free limit of our sigma model. In
particular, he studies a thermal version of the free theory on a double cover
of two-dimensional anti-de Sitter space. We do not study the thermal version
of the theory, and we study the theory on a half-plane version of two-
dimensional Minkowski space.

There have been several attempts to define a quantum version of the Geroch
group and study its representation theory Korotkin:1997ps ; Korotkin:1997fi ;
Niedermaier:1999bh ; Fuchs:2017jyk ; Peraza:2019agr ; Runov:2020ckc . The
basic idea is that the Kac-Moody algebra should be deformed into a kind of
Yangian. Representations of this Yangian should correspond, in some sense, to
Hilbert spaces for interacting cylindrical gravitational waves. The Hilbert
spaces so defined look quite different from the conventional kinds of Hilbert
spaces that we will be using in our work. Eventually, it would be interesting
to understand the connection between our work and this literature but for now
the relationship is not very clear.

Finally, we should mention that several papers have discussed the sigma model
we will be studying from the perspective of the “asymptotic safety” program
Niedermaier:2002eq ; Niedermaier:2003fz ; BarberoG:2003njp . Those papers are
concerned with how the Lagrangian changes under renormalization. All of our
calculations are at tree-level, so we will not make contact with their
results.

We can now give an overview of our paper. In section 2, we will review the
derivation of the classical sigma model from a dimensional reduction of
general relativity. In section 3, we will quantize the sigma model and compute
its two-particle tree-level $S$-matrix. Along the way, we will describe the
action of the Ehlers group and directly verify the associated $S$-matrix
conservation law. In section 4, we will introduce the holographically dual
boundary theory and compute its two-point correlation function. Finally, in
section 5, we will discuss our results and outline some possible next steps.

##  2 The classical sigma model

Our starting point is the four-dimensional Einstein-Hilbert action with zero
cosmological constant:

$\frac{1}{16\pi G}\int d^{4}x\sqrt{-g_{(4)}}R_{(4)}\,.$ (1)

The label $(4)$ is there to remind us that these are the metric and curvature
scalar in four spacetime dimensions. The metric signature is $(-+++)$.

The first step in the derivation of the sigma model is a standard Kaluza-Klein
reduction to three spacetime dimensions. We have in mind cylindrical
gravitational waves and, in this step, we are reducing along the length of the
cylinder. This brings in a new scale, $\ell$, which is the length of the
cylinder. We will assume that $\ell$ is a large but finite constant. The
reason we want $\ell$ to be large and finite is that $G/\ell$ is going to
become the coupling constant of the sigma model and we want it to be small but
nonzero. So what we are really studying are toroidal gravitational waves, but
we will stick to calling them cylindrical gravitational waves because that is
what is more commonly discussed in the literature.

After Kaluza-Klein reduction, we have a metric, a vector, and a scalar in
three spacetime dimensions. Let $z$ be the coordinate along the cylinder and
let $X^{1}$ be the scalar field in three dimensions. The relationship between
$X^{1}$ and the four dimensional metric is

$g^{(4)}_{zz}\equiv e^{-X^{1}/2}\,.$ (2)

The factor of $-1/2$ in the exponent is for convenience. The definitions of
the three dimensional metric and the three dimensional vector are standard so
we will not bother to review them here.

The three dimensional action is

$\displaystyle S_{(3)}=\,\frac{\ell}{16\pi G}\int
d^{3}x\sqrt{-g_{(3)}}R_{(3)}$ $\displaystyle-\frac{\ell}{64\pi G}\int
d^{3}x\sqrt{-g_{(3)}}e^{-X^{1}}F_{(3)mn}F^{mn}_{(3)}$
$\displaystyle-\frac{\ell}{128\pi G}\int
d^{3}x\sqrt{-g_{(3)}}\partial_{m}X^{1}\partial^{m}X^{1}\,.$ (3)

The first term is the three dimensional Einstein-Hilbert term. The second term
is the action for the three dimensional gauge field. The field strength is
$F_{(3)}\equiv dA_{(3)}$. The third term is the action for the Kaluza-Klein
scalar.

The next step in the derivation of the sigma model is a duality
transformation. To see how this goes, first observe that the equation of
motion for the gauge field is

$d\left(e^{-X^{1}}{*}F_{(3)}\right)=0\,.$ (4)

The quantity in brackets is a closed one-form, so we can introduce a new
scalar field, $X^{2}$, by making the definition

$2e^{-X^{1}}{*}F_{(3)}\equiv dX^{2}\,.$ (5)

The equation of motion for $X^{2}$ is

$d\left(e^{X^{1}}{*}dX^{2}\right)=0\,.$ (6)

This is just a rearrangement of (5). There is a similar story in the other
direction: starting from the equation of motion for $X^{2}$, we can infer the
existence of $A_{(3)}$ and work out its equation of motion. This duality is
obviously special to three spacetime dimensions.

Thus we may trade $A_{(3)}$ for a scalar field, $X^{2}$, and so obtain a new
action,

$\mathcal{S}_{(3)}\equiv\frac{2}{\pi g_{0}^{2}}\int
d^{3}x\sqrt{-g_{(3)}}R_{(3)}-\frac{1}{4\pi}\int
d^{3}x\sqrt{-g_{(3)}}\gamma_{\mu\nu}(X)\partial_{m}X^{\mu}\partial^{m}X^{\nu}\,.$
(7)

The first term is the same three dimensional Einstein-Hilbert term we had
before. We have defined $g_{0}^{2}\equiv 32G/\ell$. The second term is the
action for the two scalar fields, $X^{1}$ and $X^{2}$. We have grouped these
scalars into a single vector field, $X^{\mu}$, $\mu=1,2$. We have also
rescaled the scalar fields by a factor of $g_{0}$ to eliminate an overall
factor of $1/g_{0}^{2}$ from the second term.

The target space metric, $\gamma_{\mu\nu}(X)$, in the second term is

$ds^{2}=\left(dX^{1}\right)^{2}+e^{g_{0}X^{1}}\left(dX^{2}\right)^{2}\,.$ (8)

This is the metric on the hyperbolic plane. Thus the new action (7) is a three
dimensional sigma model coupled to gravity whose target space is the
hyperbolic plane. The sigma model is weakly coupled when the target space is
weakly curved.

The third step in the derivation of the 2d model we are after is a Kaluza-
Klein reduction to two spacetime dimensions. In this step, we are reducing the
cylindrical gravitational wave spacetime along the circle. No new length scale
is introduced in this step, we just pick up a factor of $2\pi$. The three
dimensional metric decomposes into a two dimensional metric, a two dimensional
vector, and a two dimensional scalar. Let $\phi$ be the coordinate on the
circle and let $\rho$ be the two dimensional scalar field. The relationship to
the three dimensional metric is

$g_{\phi\phi}^{(3)}=\rho^{2}\,.$ (9)

The definitions of the two dimensional metric and the two dimensional vector
are standard.

In two spacetime dimensions, we can use a diffeomorphism to write the metric
as the product of a conformal factor and a flat metric. Set

$g_{ab}^{(2)}=e^{\Gamma}h_{ab}\,.$ (10)

Eventually we will set $h_{ab}$ equal to the two-dimensional Minkowski metric
but for now we leave it arbitrary.

After this Kaluza-Klein reduction, we obtain the two dimensional action

$\displaystyle\mathcal{S}_{(2)}=\,$ $\displaystyle\frac{4}{g_{0}^{2}}\int
d^{2}x\sqrt{-h}\rho R(h)-\frac{4}{g_{0}^{2}}\int
d^{2}x\,\partial_{a}(\sqrt{-h}\partial^{a}\rho)\Gamma$
$\displaystyle-\frac{1}{g_{0}^{2}}\int
d^{2}x\sqrt{-h}\rho^{3}e^{-\Gamma}F_{(2)ab}^{\vphantom{ab}}F_{(2)}^{ab}-\frac{1}{2}\int
d^{2}x\sqrt{-h}\rho\gamma_{\mu\nu}(X)\partial_{a}X^{\mu}\partial^{a}X^{\nu}\,.$
(11)

The first term is the two dimensional Einstein-Hilbert action for $h_{ab}$.
The second term appears because we explicitly introduced the conformal factor
using eq. (10). The third term is the action for the two dimensional vector.
The field strength is $F_{(2)}\equiv dA_{(2)}$. The fourth term is the sigma
model.

In three dimensions, we were able to eliminate the gauge field using its
equation of motion and a duality redefinition. In two dimensions, the
situation is even simpler. The equation of motion for the two dimensional
vector is

$d\left(\rho^{3}e^{-\Gamma}{*}F_{(2)}\right)=0\,.$ (12)

This means the quantity in brackets is a constant scalar field. A constant
scalar field is essentially trivial, so we will not bother to keep it around.
We will simply drop the third term in (2) from now on.

To make life even simpler, we will now use a two dimensional diffeomorphism to
set $h_{ab}=\eta_{ab}$ (where $\eta_{ab}$ is the two dimensional Minkowski
metric). Now all that remains of the two dimensional action is

$\mathcal{S}_{(2)}=-\frac{4}{g_{0}^{2}}\int
d^{2}x\,\Gamma\partial_{a}\partial^{a}\rho-\frac{1}{2}\int
d^{2}x\,\rho\gamma_{\mu\nu}(X)\partial_{a}X^{\mu}\partial^{a}X^{\nu}\,.$ (13)

In the first term, $\Gamma$ is a Lagrange multiplier enforcing the constraint

$\partial_{a}\partial^{a}\rho=0\,.$ (14)

We can solve this by hand to get

$\rho=r\,.$ (15)

This is not the most general solution of (14), but we will assume that it is
the most general solution with the right boundary conditions for cylindrical
gravitational waves222 There are solutions with $\rho\neq r$ which describe
cylindrical gravitational waves on cosmological backgrounds Gowdy:1971jh .
These solutions have “big bang” and “big crunch” singularities, so it is not
clear how to define the $S$-matrix. For this reason, we are invoking boundary
conditions which pick out the $\rho=r$ family of solutions. See
Grumiller:2000wt  for a discussion of sigma models for cosmological
solutions..

Plugging $\rho=r$ back into eq. (13) gives

$I\equiv-\frac{1}{2}\int
d^{2}x\,r\gamma_{\mu\nu}(X)\partial_{a}X^{\mu}\partial^{a}X^{\nu}\,.$ (16)

This is the two dimensional sigma model that we will be studying for the rest
of this paper. It is an action for a pair of scalar fields, $X^{1}$ and
$X^{2}$. These scalars encode the two physical polarizations of the four
dimensional graviton. The target space metric is (8). The spacetime metric is

$ds^{2}=-dt^{2}+dr^{2}\,.$ (17)

An unusual feature of the two dimensional spacetime is that it ends at $r=0$.

The action (16) is time-translation invariant but it is not spatial-
translation or boost invariant. Why is this so? Suppose we have a cylindrical
gravitational wave metric in four spacetime dimensions. We immediately get a
whole family of solutions related by spatial translations. Each of these
solutions has a $\mathrm{U}(1)$ rotational symmetry, but each of these
$\mathrm{U}(1)$’s is different. To go to two spacetime dimensions, we had to
single out a $\mathrm{U}(1)$. This singles out one representative of this
family of solutions. This is why the sigma model does not have translation
invariance. There is a similar story for boosts. A succinct way of saying this
is that spatial translations and boosts do not commute with rotations, and we
reduced with respect to a rotation.

The action $I$ completely describes the classical dynamics of cylindrical
gravitational waves. The relationship of the quantum sigma model to quantum
cylindrical gravitational waves is less clear. We will simply quantize this
sigma model and think of its $S$-matrix as a toy model for a quantum gravity
$S$-matrix.

##  3 The quantum sigma model

The sigma model from the previous section is

$I=-\frac{1}{2}\int
d^{2}x\,r\gamma_{\mu\nu}(X)\partial_{a}X^{\mu}\partial^{a}X^{\nu}\,.$ (18)

The target space metric is
$\gamma_{\mu\nu}=\operatorname{diag}(1,e^{g_{0}X^{1}})$ and the coupling
constant is $g_{0}^{2}=32G/\ell$.

###  3.1 Quantization

At very early and late times, the theory is described by the action of the
free theory. So the first step is to quantize the free theory. The action of
the free theory is

$I_{0}\equiv-\frac{1}{2}\int
d^{2}x\,r\left[\partial_{a}X^{1}\partial^{a}X^{1}+\partial_{a}X^{2}\partial^{a}X^{2}\right]\,.$
(19)

The equation of motion is

$\partial_{a}(r\partial^{a}X^{\mu})=0$ (20)

There are two families of solutions ($k>0$):

$e^{\pm ikt}J_{0}(kr)\,,\qquad e^{\pm ikt}Y_{0}(kr)\,.$ (21)

$J_{0}$ and $Y_{0}$ are Bessel functions of the first and second kind. The
second family of solutions blows up at $r=0$. We will regard those modes as
unphysical and we will not include them in the quantum Hilbert space.

This leaves the first family of modes. Expanding the free fields in these
modes, we obtain

$X^{\mu}(t,r)=\frac{1}{\sqrt{2}}\int_{0}^{\infty}dk\left[a^{\mu}_{k}e^{-ikt}+a^{\mu{\dagger}}_{k}e^{ikt}\right]J_{0}(kr)\,.$
(22)

We impose the canonical commutation relations

$\displaystyle\left[a^{\mu}_{k},a^{\nu}_{k^{\prime}}\right]=\left[a^{\mu{\dagger}}_{k},a^{\nu{\dagger}}_{k^{\prime}}\right]=0\,,$
(23)
$\displaystyle\left[a^{\mu}_{k},a^{\nu{\dagger}}_{k^{\prime}}\right]=\delta^{\mu\nu}\delta(k-k^{\prime})\,.$
(24)

We should check that these are reasonable commutators. The best way to do that
is to think about the Hamiltonian formalism. Let $L_{0}$ be the Lagrangian of
the free theory and let $\dot{X}^{\mu}\equiv\partial_{t}X_{\mu}$. The momentum
conjugate to $X^{\mu}$ is

$\pi^{\mu}\equiv\frac{\partial
L_{0}}{\partial\dot{X}^{\mu}}=r\partial_{t}X^{\mu}\,.$ (25)

The Hamiltonian is

$H_{0}\equiv\int dr\left[\pi_{\mu}\dot{X}^{\mu}-L_{0}\right]=\frac{1}{2}\int
dr\,\delta_{\mu\nu}\left[r^{-1}\pi^{\mu}\pi^{\nu}+r\partial_{r}X^{\mu}\partial_{r}X^{\nu}\right].$
(26)

The equal-time commutation relations are

$\displaystyle\left[X^{\mu}(r),X^{\nu}(r^{\prime})\right]=\left[\pi^{\mu}(r),\pi^{\nu}(r^{\prime})\right]=0\,,$
(27)
$\displaystyle\left[X^{\mu}(r),\pi^{\nu}(r^{\prime})\right]=i\delta^{\mu\nu}\delta(r-r^{\prime})\,.$
(28)

With these commutators, the equations of motion are

$\partial_{t}X^{\mu}=i\left[H_{0},X^{\mu}\right]=i\int\frac{dr^{\prime}}{r^{\prime}}\pi^{\nu}\left[\pi^{\nu},X^{\mu}\right]=r^{-1}\pi^{\mu}\,,$
(29)

and

$\partial_{t}\pi^{\mu}=i\left[H_{0},\pi^{\mu}\right]=-i\int
dr^{\prime}\,\partial_{r^{\prime}}\left(r^{\prime}\partial_{r^{\prime}}X^{\nu}\right)\left[X^{\nu},\pi^{\mu}\right]=\partial_{r}\left(r\partial_{r}X^{\mu}\right).$
(30)

These are the same equations we had earlier, so eqs. (27) and (28) are
correct.

Now we need to relate this set of commutators to the first set of commutators.
Use the definitions to get

$\pi^{\mu}=-\frac{ir}{\sqrt{2}}\int
dk\left[a^{\mu}_{k}e^{-ikt}-a^{\mu{\dagger}}_{k}e^{ikt}\right]kJ_{0}(kr)\,.$
(31)

Then observe

$\displaystyle[X^{\mu}(r),\pi^{\nu}(r^{\prime})]$
$\displaystyle=-\frac{ir^{\prime}}{2}\int dk\int
dk^{\prime}\left(-[a^{\mu}_{k},a^{\nu{\dagger}}_{k^{\prime}}]e^{-i(k-k^{\prime})t}+[a^{\mu{\dagger}}_{k},a^{\nu}_{k^{\prime}}]e^{i(k-k^{\prime})t}\right)k^{\prime}J_{0}(kr)J_{0}(k^{\prime}r^{\prime})$
$\displaystyle=i\delta^{\mu\nu}r^{\prime}\int
dk\,kJ_{0}(kr)J_{0}(kr^{\prime})$
$\displaystyle=i\delta^{\mu\nu}\delta(r-r^{\prime})\,.$ (32)

Thus these two sets of commutators are equivalent and eqs. (23) and (24) are
the correct choice for the creation and annihilation operators. In the last
step of this calculation, we used the “closure equation,”

$r^{\prime}\int_{0}^{\infty}dk\,kJ_{0}(kr)J_{0}(kr^{\prime})=\delta(r-r^{\prime})\,.$
(33)

We will use this integral many times in what follows.

Let $|0\rangle$ be the vacuum vector, normalized such that $\langle
0|0\rangle=1$. The vacuum is annihilated by the $a^{\mu}_{k}$, and
$n$-particle states are defined in the usual way by acting on the vacuum with
the $a_{k}^{\mu{\dagger}}$. The Hamiltonian is $H_{0}=\int
dk\,k\left(a^{1{\dagger}}_{k}a^{1\vphantom{{\dagger}}}_{k}+a^{2{\dagger}}_{k}a^{2\vphantom{{\dagger}}}_{k}\right)$.

This completes the quantization of the free theory. We will now compute some
two-point correlation functions in the free theory. Let $x$ stand for $(t,r)$.
The simplest two-point correlation function is333Essentially the same 2-point
correlation function has been computed by Niedermaier Niedermaier:2000ud .

$\displaystyle\langle 0|X^{\mu}(x)X^{\nu}(x^{\prime})|0\rangle$
$\displaystyle=\frac{1}{2}\int dk\int dk^{\prime}\,\langle
0|a^{\mu}_{k}a^{\nu{\dagger}}_{k^{\prime}}|0\rangle
e^{-ikt}e^{ik^{\prime}t^{\prime}}J_{0}(kr)J_{0}(k^{\prime}r^{\prime})$
$\displaystyle=\frac{1}{2}\delta^{\mu\nu}\int
dk\,e^{-ik(t-t^{\prime})}J_{0}(kr)J_{0}(kr^{\prime})\,.$ (34)

Combining this with the $i\epsilon$-prescription gives the Feynman propagator:

$\langle
X^{\mu}(x)X^{\nu}(x^{\prime})\rangle_{0}\equiv\delta^{\mu\nu}\frac{i}{2\pi}\int
d\omega\int
dk\frac{k}{\omega^{2}-k^{2}+i\epsilon}e^{-i\omega(t-t^{\prime})}J_{0}(kr)J_{0}(kr^{\prime})\,.$
(35)

The Feynman propagator is the Green’s function of the free theory. To see
this, define an operator, $\widehat{\Box}$, which acts as
$\widehat{\Box}\Phi\equiv\partial_{a}(r\partial^{a}\Phi)$. The equation of
motion of the free theory is $\widehat{\Box}X^{\mu}=0$. The Feynman propagator
satisfies

$\displaystyle\widehat{\Box}\langle
X^{\mu}(x)X^{\nu}(x^{\prime})\rangle_{0}=i\delta^{\mu\nu}\delta^{(2)}(x-x^{\prime})\,.$
(36)

Checking this boils down to an application of the closure equation (33). We
now have everything we need to know about the free theory.

Turning to the interacting theory, we can compute its $n$-point time-ordered
correlation functions using the usual quantum field theory recipe:

$\langle X^{\mu_{1}}(x_{1})\cdots X^{\mu_{n}}(x_{n})\rangle=\frac{\langle
X^{\mu_{1}}(x_{1})\cdots X^{\mu_{n}}(x_{n})e^{i\int d^{2}x\,L_{\rm
int}}\rangle_{0}}{\langle e^{i\int d^{2}x\,L_{\rm int}}\rangle_{0}}\,.$ (37)

The subscript 0 means the correlation functions on the rhs are time-ordered
correlation functions in the free theory. They are computed as usual using
Wick’s theorem and Feynman diagrams. The interaction Lagrangian in the
exponent is

$L_{\rm
int}=-\frac{1}{2}r\left[g_{0}X^{1}\left|dX^{2}\right|^{2}+\frac{g_{0}^{2}}{2}\left(X^{1}\right)^{2}\left|dX^{2}\right|^{2}+\dots\right].$
(38)

These are the leading terms in the small $g_{0}$ expansion of $\eqref{eq:I2}$.

###  3.2 The LSZ formula

To convert $n$-point correlation functions into $S$-matrix elements, we need a
version of the Lehmann-Symanzik-Zimmerman (LSZ) formula. We can get the
version we need by adapting the usual quantum field theory story. Since the
theory we are studying is a bit unusual, it seems worthwhile to record those
calculations here.

The key to the version of the LSZ formula we need is the identity

$-i\int
d^{2}x\,e^{ikt}J_{0}(kr)\widehat{\Box}X^{\mu}=\sqrt{2}\left[a^{\mu}_{\rm
out}(k)-a^{\mu}_{\rm in}(k)\right].$ (39)

In this formula, $a^{\mu}_{\rm out}(k)$ is $a^{\mu}_{k}$ for the free theory
in the far future and $a^{\mu}_{\rm in}(k)$ is $a^{\mu}_{k}$ for the free
theory in the far past.

To prove this identity, start by expanding the $\widehat{\Box}$ operator. The
lhs becomes

$-i\int
d^{2}x\,re^{ikt}J_{0}(kr)\left(-\partial_{t}^{2}X^{\mu}+\frac{1}{r}\partial_{r}(r\partial_{r}X^{\mu})\right).$
(40)

Integrate by parts in $r$ to get

$i\int d^{2}x\,r\,e^{ikt}J_{0}(kr)(\partial_{t}^{2}+k^{2})X^{\mu}\,.$ (41)

We used $\partial_{r}(r\partial_{r}J_{0}(kr))=-k^{2}rJ_{0}(kr)$. Next, observe

$\displaystyle\partial_{t}\left[e^{ikt}J_{0}(kr)(i\partial_{t}+k)X^{\mu}\right]$
$\displaystyle=ike^{ikt}J_{0}(kr)(i\partial_{t}+k)X^{\mu}+e^{ikt}J_{0}(kr)(i\partial_{t}^{2}+k\partial_{t})X^{\mu}$
$\displaystyle=ie^{ikt}J_{0}(kr)\left(\partial_{t}^{2}+k^{2}\right)X^{\mu}\,.$
(42)

Thus, arrive at

$-i\int d^{2}x\,e^{ikt}J_{0}(kr)\widehat{\Box}X^{\mu}=\int
dt\,\partial_{t}\left[e^{ikt}\int
dr\,r\,J_{0}(kr)(i\partial_{t}+k)X^{\mu}\right].$ (43)

The rhs contributes boundary terms in the far past and the far future. Since
we only need to evaluate the bracketed quantity in the far past and the far
future, we can use the mode expansion (22) of the free theory. The bracketed
quantity becomes

$\frac{e^{ikt}}{\sqrt{2}}\int dr\,r\,J_{0}(kr)(i\partial_{t}+k)\int
dk^{\prime}\left[a^{\mu}(k^{\prime})e^{-ik^{\prime}t}+a^{\mu}(k^{\prime})^{\dagger}e^{ik^{\prime}t}\right]J_{0}(k^{\prime}r)\,.$
(44)

Distribute the $(i\partial_{t}+k)$ operator to get

$\frac{e^{ikt}}{\sqrt{2}}\int dr\,r\,J_{0}(kr)\int
dk^{\prime}\left[(k+k^{\prime})a^{\mu}_{k^{\prime}}e^{-ik^{\prime}t}+(k-k^{\prime})a^{\mu{\dagger}}_{k^{\prime}}e^{ik^{\prime}t}\right]J_{0}(k^{\prime}r)=\sqrt{2}a^{\mu}_{k}\,.$
(45)

The claimed identity (39) now follows immediately from eqs. (43) and (45).

In the last step of this calculation, we performed the integrals using the
closure equation

$\int_{0}^{\infty}dr\,r\,J_{0}(kr)J_{0}(k^{\prime}r)=\frac{1}{k}\delta(k-k^{\prime})\,.$
(46)

This is same as the first closure equation (33) after a change of notation.
The new version expresses the closure relation as an $r$-integral rather than
a $k$-integral.

Now that we have formula (39), we can convert time-ordered correlation
functions into $S$-matrix elements. The $n$-particle in- and out-states are

$|\mu_{1}k_{1},\dots,\mu_{n}k_{n}\thinspace{\rm
in}\rangle\equiv\sqrt{2^{n}}a^{\mu_{1}}_{\rm in}(k_{1})^{\dagger}\cdots
a^{\mu_{n}}_{\rm in}(k_{n})^{\dagger}|0\rangle\,,$ (47)

and

$\langle{\rm
out}\thinspace\mu_{1}k_{1},\dots,\mu_{n}k_{n}|\equiv\sqrt{2^{n}}\langle
0|a^{\mu_{1}}_{\rm out}(k_{1})\cdots a^{\mu_{n}}_{\rm out}(k_{n})\,.$ (48)

We are only ever going to compute $S$-matrix elements for 2-particle
scattering, so it will be convenient to introduce the shorthand notation

$\displaystyle|i\rangle\equiv|\mu_{1}k_{1},\mu_{2}k_{2}\thinspace{\rm
in}\rangle\,,\qquad\langle f|\equiv\langle{\rm
out}\thinspace\mu_{3}k_{3},\mu_{4}k_{4}|\,.$ (49)

Eq. (39) then implies

$\displaystyle\langle f|S|i\rangle$ $\displaystyle=\left[-i\int
d^{2}x_{1}\,e^{-ik_{1}t_{1}}J_{0}(k_{1}r_{1})\widehat{\Box}_{x_{1}}\right]\cdots\left[-i\int
d^{2}x_{4}\,e^{ik_{4}t_{4}}J_{0}(k_{4}r_{4})\widehat{\Box}_{x_{4}}\right]$
$\displaystyle\qquad\qquad\times\langle
X^{\mu_{1}}(x_{1})X^{\mu_{2}}(x_{2})X^{\mu_{3}}(x_{3})X^{\mu_{4}}(x_{4})\rangle\,.$
(50)

This is the LSZ formula. The signs in the exponents are negative for incoming
particles and positive for outgoing particles. The subscript on
$\widehat{\Box}_{x_{i}}$ tells us which coordinate this operator is acting on.

###  3.3 $\mathrm{SL}(2,\mathbb{R})$ symmetry

The sigma model’s target space is the hyperbolic plane. The isometry group of
the hyperbolic plane is $\mathrm{SL}(2,\mathbb{R})$. This is a symmetry of the
sigma model. In the general relativity literature on the classical sigma
model, this symmetry is called the Ehlers group. It is a subgroup of the
Geroch group, which is a hidden Kac-Moody symmetry of the classical sigma
model. We will not discuss the Geroch group further in this section.

Recall the metric on the hyperbolic plane (8):

$ds^{2}=\left(dX^{1}\right)^{2}+e^{g_{0}X^{1}}\left(dX^{2}\right)^{2}\,.$ (51)

This metric has three Killing vectors:

$\partial_{X^{2}}\,,\qquad\partial_{X^{1}}-\frac{g_{0}}{2}X^{2}\partial_{X^{2}}\,,\qquad-X^{2}\partial_{X^{1}}+\left[g_{0}^{-1}\left(1-e^{-g_{0}X^{1}}\right)+\frac{g_{0}}{4}\left(X^{2}\right)^{2}\right]\partial_{X^{2}}\,.$
(52)

These Killing vectors are the infinitesimal generators of the
$\mathrm{SL}(2,\mathbb{R})$ symmetry. In the $g_{0}\rightarrow 0$ limit, they
reduce to

$\partial_{X^{2}}\,,\qquad\partial_{X^{1}}\,,\qquad-X^{2}\partial_{X^{1}}+X^{1}\partial_{X_{2}}\,.$
(53)

These are just the infinitesimal generators of the Euclidean group. This makes
sense because the target space is just the Euclidean plane in this limit. This
is the free, non-interacting limit of the sigma model. The symmetries of the
free theory are a pair of shift symmetries and an $\mathrm{SO}(2)$ rotation.
We can think of the $\mathrm{SL}(2,\mathbb{R})$ symmetries of the full
interacting theory as hyperbolic versions of these symmetries.

The shift symmetries are spontaneously broken in the quantum theory. They do
not give $S$-matrix conservation laws and we will have nothing more to say
about them. The $\mathrm{SO}(2)$ symmetry is unbroken. It gives an $S$-matrix
conservation law which we will now compute.

The starting point is the conserved $\mathrm{SO}(2)$ charge in the free
theory:

$Q\equiv\int dr\,r\left(X^{2}\partial_{t}X^{1}-X^{1}\partial_{t}X^{2}\right).$
(54)

We need to evaluate this charge in the far future and the far past. In the far
past, we have the mode expansion

$X^{\mu}_{\rm in}(t,r)=\frac{1}{\sqrt{2}}\int_{0}^{\infty}dk\left[a^{\mu}_{\rm
in}(k)e^{-ikt}+a^{\mu}_{\rm in}(k)^{\dagger}e^{ikt}\right]J_{0}(kr)\,.$ (55)

Plugging this into $Q$ and using the closure equation (46) gives

$Q_{\rm in}=-i\epsilon_{\mu\nu}\int dk\,a^{\mu}_{\rm in}(k)a^{\nu}_{\rm
in}(k)^{\dagger}\,.$ (56)

A similar story in the far future gives

$Q_{\rm out}=-i\epsilon_{\mu\nu}\int dk\,a^{\mu}_{\rm out}(k)a^{\nu}_{\rm
out}(k)^{\dagger}\,.$ (57)

Thus, the $S$-matrix conservation law is

$\langle f|Q_{\rm out}S|i\rangle=\langle f|SQ_{\rm in}|i\rangle\,.$ (58)

We will now work out exactly what this means for 2-particle tree-level
scattering. The $\mathrm{SO}(2)$ symmetry does not really act on the energies
of the particles in an interesting way, so we will use a shorthand notation
that only keeps track of the $\mu_{i}$. Let

$S_{\mu_{1}\mu_{2}\rightarrow\mu_{3}\mu_{4}}\equiv\langle{\rm
out}\thinspace\mu_{3}k_{3},\mu_{4}k_{4}|S|\mu_{1}k_{1},\mu_{2}k_{2}\thinspace{\rm
in}\rangle\,,$ (59)

be the $S$-matrix elements for 2-particle scattering. Half of these matrix
elements vanish at tree level simply because there are no diagrams one can
draw. Indeed, at tree level, the only interaction vertices that contribute to
$2$-particle scattering are shown in figure 1. These diagrams represent the
first two terms in eq. (38). It is clear from these pictures that there are no
2-particle tree diagrams that one can draw for which the total number of
$X^{1}$ particles in the initial and final states is odd. This eliminates half
of the $S_{\mu_{1}\mu_{2}\rightarrow\mu_{3}\mu_{4}}$.

V3 (40,25) i1 o1,o2 dasheso1,v1,o2 plain,tension=2i1,v1 V4 (40,25) i1,i2 o1,o2
plaini1,v1,i2 dasheso1,v1,o2

Figure 1: The first two interaction terms in the Lagrangian (38). Solid lines
are $X^{1}$ particles and dashed lines are $X^{2}$ particles. These are the
only vertices that contribute to 2-particle tree-level scattering.

Now we will use the conservation law (58) to derive further constraints on the
remaining matrix elements. First, define

$\sigma_{1}\equiv S_{11\rightarrow 22}\,,\qquad\sigma_{2}\equiv
S_{12\rightarrow 12}\,,\qquad\sigma_{3}\equiv S_{12\rightarrow 21}\,.$ (60)

Expanding the $S$-matrix conservation law (58) in a basis for the initial and
find states gives a linear set equations for the $S$-matrix elements. Solving
this linear set of equations gives

$S_{\mu_{1}\mu_{2}\rightarrow\mu_{3}\mu_{4}}=\begin{pmatrix}\sigma_{1}+\sigma_{2}+\sigma_{3}&0&0&\sigma_{1}\\\
0&\sigma_{2}&\sigma_{3}&0\\\ 0&\sigma_{3}&\sigma_{2}&0\\\
\sigma_{1}&0&0&\sigma_{1}+\sigma_{2}+\sigma_{3}\end{pmatrix}\,.$ (61)

In this matrix, the rows are ordered according to the ordering

$|\mu_{1}\mu_{2}\,{\rm
in}\rangle=|11\rangle\,,|12\rangle\,,|21\rangle\,,|22\rangle\,.$ (62)

The columns are ordered similarly but for $\langle{\rm out}\,\mu_{3}\mu_{4}|$.
The upshot is that the $\mathrm{SO}(2)$ symmetry fixes the entire $2$-particle
tree-level $S$-matrix in terms of $\sigma_{1}$, $\sigma_{2}$, and
$\sigma_{3}$.

We can actually eliminate one of the $\sigma_{i}$ by making an additional
observation. Returning to figure 1, it is clear there are no 2-particle tree
diagrams one can draw for which all of the external particles are $X^{1}$
particles. Combining this fact with eq. (61) gives

$\sigma_{1}+\sigma_{2}+\sigma_{3}=0\,.$ (63)

We can view this equation as a prediction of the $\mathrm{SO}(2)$ symmetry for
the tree-level $S$-matrix. In the next subsection, we will verify this
prediction directly by computing the $\sigma_{i}$ using Feynman diagrams and
checking that they sum to zero.

###  3.4 Scattering amplitudes

In this subsection, we will write correlation functions, $\langle
X^{\mu_{1}}(x_{1})\dots X^{\mu_{n}}(x_{n})\rangle$, more compactly as $\langle
X^{\mu_{1}}_{1}\dots X^{\mu_{n}}_{n}\rangle$. We will compute correlation
functions using eq. (37),

$\langle X^{\mu_{1}}_{1}\cdots X^{\mu_{n}}_{n}\rangle=\frac{\langle
X^{\mu_{1}}_{1}\cdots X^{\mu_{n}}_{n}e^{i\int d^{2}x\,L_{\rm
int}}\rangle_{0}}{\langle e^{i\int d^{2}x\,L_{\rm int}}\rangle_{0}}\,,$ (64)

where we recall

$L_{\rm
int}=-\frac{1}{2}r\left[g_{0}X^{1}\left|dX^{2}\right|^{2}+\frac{g_{0}^{2}}{2}\left(X^{1}\right)^{2}\left|dX^{2}\right|^{2}+\dots\right].$
(65)

We will convert correlation functions into scattering amplitudes using the LSZ
formula (3.2).

sigma1t (40,25) i1,i2 o1,o2 dasheso1,v1,v2,o2 plaini1,v1 plaini2,v2 sigma1u
(40,25) i1,i2 o1,o2 plaini1,v1 plaini2,v2 dashesv1,v2 phantomv1,o1
phantomv2,o2 dashes,tension=0v2,o1 dashes,tension=0,rubout=4v1,o2 sigma1c
(40,25) i1,i2 o1,o2 plaini1,v1,i2 dasheso1,v1,o2

Figure 2: Tree-level contributions to $\sigma_{1}$.

The first scattering amplitude we will compute is

$\sigma_{1}=S_{11\rightarrow 22}\,.$ (66)

There are three tree-level diagrams to consider (see figure 2). The
contribution of the $t$-channel diagram to the correlation function is

$-g_{0}^{2}\int d^{2}x\,r\int d^{2}x^{\prime}\,r^{\prime}\langle
X^{1}_{1}X^{1}_{x}\rangle_{0}\langle
X^{2}_{3}\partial_{a}X^{2}_{x}\rangle_{0}\langle
X^{1}_{2}X^{1}_{x^{\prime}}\rangle_{0}\langle
X^{2}_{4}\partial_{a^{\prime}}X^{2}_{x^{\prime}}\rangle_{0}\langle\partial^{a}X^{2}_{x}\partial^{a^{\prime}}X^{2}_{x^{\prime}}\rangle_{0}\,.$
(67)

Plugging this into the LSZ formula (3.2) gives

$\displaystyle\sigma_{1}^{t}=-g_{0}^{2}\int d^{2}x\,r\int d^{2}x^{\prime}$
$\displaystyle\,r^{\prime}e^{-ik_{1}t}e^{-ik_{2}t^{\prime}}J_{0}(k_{1}r)J_{0}(k_{2}r^{\prime})$
$\displaystyle\times\partial_{a}\left[e^{ik_{3}t}J_{0}(k_{3}r)\right]\partial_{a^{\prime}}\left[e^{ik_{4}t^{\prime}}J_{0}(k_{4}r^{\prime})\right]\left\langle\partial^{a}X^{2}_{x}\partial^{a^{\prime}}X^{2}_{x^{\prime}}\right\rangle_{0}.$
(68)

Using eq. (35) for the propagator gives

$\displaystyle\sigma_{1}^{t}=-\frac{ig_{0}^{2}}{2\pi}$ $\displaystyle\int
d\omega\int dk\,\frac{k}{\omega^{2}-k^{2}+i\epsilon}$ $\displaystyle\times$
$\displaystyle\int
d^{2}x\,re^{-ik_{1}t}J_{0}(k_{1}r)\partial_{a}\left[e^{ik_{3}t}J_{0}(k_{3}r)\right]\partial^{a}\left[e^{-i\omega
t}J_{0}(kr)\right]$ $\displaystyle\times$ $\displaystyle\int
d^{2}x^{\prime}\,r^{\prime}e^{-ik_{2}t^{\prime}}J_{0}(k_{2}r^{\prime})\partial_{a^{\prime}}\left[e^{ik_{4}t^{\prime}}J_{0}(k_{4}r^{\prime})\right]\partial^{a^{\prime}}\left[e^{i\omega
t^{\prime}}J_{0}(kr^{\prime})\right].$ (69)

This reduces to

$\displaystyle\sigma_{1}^{t}=\frac{ig_{0}^{2}}{2\pi}k_{3}k_{4}$
$\displaystyle\int d\omega\int dk\,\frac{k}{\omega^{2}-k^{2}+i\epsilon}$
$\displaystyle\times$ $\displaystyle\int
d^{2}x\,re^{-i(\omega+k_{1}-k_{3})t}J_{0}(k_{1}r)\left[\omega
J_{0}(kr)J_{0}(k_{3}r)-kJ_{1}(kr)J_{1}(k_{3}r)\right]$ $\displaystyle\times$
$\displaystyle\int
d^{2}x^{\prime}\,r^{\prime}e^{i(\omega+k_{4}-k_{2})t^{\prime}}J_{0}(k_{2}r^{\prime})\left[\omega
J_{0}(kr^{\prime})J_{0}(k_{4}r^{\prime})+kJ_{1}(kr^{\prime})J_{1}(k_{4}r^{\prime})\right].$
(70)

Performing the time and $\omega$ integrals, we obtain

$\displaystyle\sigma_{1}^{t}=2\pi ig_{0}^{2}$ $\displaystyle
k_{3}k_{4}\delta(k_{3}+k_{4}-k_{1}-k_{2})\int
dk\,\frac{k}{k^{2}-(k_{3}-k_{1})^{2}}$ $\displaystyle\times\int
dr\,rJ_{0}(k_{1}r)\left[(k_{3}-k_{1})J_{0}(kr)J_{0}(k_{3}r)-kJ_{1}(kr)J_{1}(k_{3}r)\right]$
$\displaystyle\times\int
dr^{\prime}\,r^{\prime}J_{0}(k_{2}r^{\prime})\left[(k_{4}-k_{2})J_{0}(kr^{\prime})J_{0}(k_{4}r^{\prime})-kJ_{1}(kr^{\prime})J_{1}(k_{4}r^{\prime})\right].$
(71)

Notice that we picked up an overall energy-conserving delta function.

To finish, we need to perform the $r$ integrals. There are two integrals we
need to evaluate. The first one is

$\int_{0}^{\infty}dr\,rJ_{0}(ar)J_{0}(br)J_{0}(cr)\,.$ (72)

Integrals like this were studied long ago by Sonine, who obtained the
following beautiful result: this integral is zero if it is not possible to
form a triangle with side lengths $a$, $b$, and $c$. If it is possible to form
such a triangle, then the integral is $1/(2\pi\Delta)$, where $\Delta$ is the
area of the triangle. A convenient formula for $\Delta$ is

$\Delta\equiv\frac{1}{4}\sqrt{(a+b)^{2}-c^{2}}\sqrt{c^{2}-(a-b)^{2}}\,.$ (73)

The second integral we need is

$\int_{0}^{\infty}dr\,rJ_{0}(ar)J_{1}(br)J_{1}(cr)\,.$ (74)

There is a similar result for this integral: this integral is zero if it not
possible to form a triangle with side lengths $a$, $b$, and $c$. If it is
possible to form such a triangle, then the integral is

$\frac{b^{2}+c^{2}-a^{2}}{4\pi bc\Delta}\,.$ (75)

Applying Sonine’s formulas to our problem gives

$\sigma_{1}^{t}=\frac{2i}{\pi}g_{0}^{2}\delta(k_{3}+k_{4}-k_{1}-k_{2})\int_{|k_{1}-k_{3}|}^{(k_{1}+k_{3})\wedge(k_{2}+k_{4})}\frac{k\,dk}{\sqrt{(k_{1}+k_{3})^{2}-k^{2}}\sqrt{(k_{2}+k_{4})^{2}-k^{2}}}\,.$
(76)

The notation for the upper limit of integration means that the upper limit of
integration is the smaller of $k_{1}+k_{3}$ or $k_{2}+k_{4}$. These unusual
looking limits of integration come from Sonine’s formulas, which told us that
the $r$ integrals vanished unless $|k_{1}-k_{3}|<k<k_{1}+k_{3}$ and
$|k_{2}-k_{4}|<k<k_{2}+k_{4}$. Notice that we do not need to include the
second constraint in the lower limit of integration because it follows
automatically from the first constraint and the energy-conserving delta
function.

Physically, something a bit unusual is happening here. In an ordinary Lorentz-
invariant quantum field theory, the momentum of the internal particle in a
tree diagram is fixed by local momentum conservation. However, as discussed in
section 2, our two dimensional action does not have spatial translation
invariance, so this rule does not apply. We need to integrate over a finite
range of $k$ for the internal particle. Fortunately, it turns out that these
integrals converge.

Performing the $k$ integral gives

$\sigma_{1}^{t}=\frac{2i}{\pi}g_{0}^{2}\operatorname{Re}\operatorname{arctanh}\sqrt{\frac{k_{1}k_{3}}{k_{2}k_{4}}}\delta(k_{3}+k_{4}-k_{1}-k_{2})\,.$
(77)

This is the contribution of the $t$-channel diagram to $\sigma_{1}$. The
contribution of the $u$-channel is the same, but with $k_{3}$ and $k_{4}$
interchanged. Thus, the contribution of the $u$-channel is

$\sigma_{1}^{u}=\frac{2i}{\pi}g_{0}^{2}\operatorname{Re}\operatorname{arctanh}\sqrt{\frac{k_{1}k_{4}}{k_{2}k_{3}}}\delta(k_{3}+k_{4}-k_{1}-k_{2})\,.$
(78)

sigma2s (40,25) i2,i1 o2,o1 plaini1,v1 dashesi2,v1,v2,o2 plainv2,o1 sigma2u
(40,25) i2,i1 o2,o1 plaini1,v1 dashesv1,v2 dashesv2,i2 phantomv1,o1
phantomv2,o2 plain,tension=0v2,o1 dashes,tension=0,rubout=4v1,o2 sigma2c
(40,25) i2,i1 o2,o1 dashesi2,v1,o2 plaini1,v1,o1

Figure 3: Tree-level contributions to $\sigma_{2}$.

To finish computing $\sigma_{1}$, we need to include the contribution from the
contact diagram. The contribution of the contact diagram to the correlation
function is

$-ig_{0}^{2}\int d^{2}x\,r\langle X^{1}_{1}X^{1}_{x}\rangle_{0}\langle
X^{1}_{2}X^{1}_{x}\rangle_{0}\langle
X^{2}_{3}\partial_{a}X^{2}_{x}\rangle_{0}\langle
X^{2}_{4}\partial^{a}X^{2}_{x}\rangle_{0}\,.$ (79)

Plugging this into the LSZ formula gives

$\sigma_{1}^{c}=-ig_{0}^{2}\int
d^{2}x\,re^{-i(k_{1}+k_{2})t}J_{0}(k_{1}r)J_{0}(k_{2}r)\partial_{a}\left[e^{ik_{3}t}J_{0}(k_{3}r)\right]\partial^{a}\left[e^{ik_{4}t}J_{0}(k_{4}r)\right].$
(80)

Doing the time integral gives

$\displaystyle\sigma_{1}^{c}=-2\pi
ig_{0}^{2}k_{3}k_{4}\delta(k_{3}+k_{4}-k_{1}-k_{2})\int dr\,r$ $\displaystyle
J_{0}(k_{1}r)J_{0}(k_{2}r)$
$\displaystyle\times\left[J_{0}(k_{3}r)J_{0}(k_{4}r)+J_{1}(k_{3}r)J_{1}(k_{4}r)\right].$
(81)

Now we need to understand the $r$ integral. It an integral over a product of
four Bessel functions. Previously, we only had to consider integrals over
products of three Bessel functions.

To compute this, first recall Sonine’s integral for $|a-b|<c<a+b$:

$\int_{0}^{\infty}dr\,rJ_{0}(ar)J_{0}(br)J_{0}(cr)=\frac{1}{2\pi\Delta}\,.$
(82)

Multiply by $cJ_{0}(cr^{\prime})$ and integrate over $c$ using the closure
equation to get

$J_{0}(ar)J_{0}(br)=\frac{1}{2\pi}\int_{|a-b|}^{a+b}dc\frac{cJ_{0}(cr)}{\Delta(a,b,c)}\,.$
(83)

Using this formula, we can reduce products of many Bessel functions to
integrals over products of fewer Bessel functions. This is the key to
performing the integral we need.

Going back to eq. (3.4) and using this result, we obtain

$\displaystyle\sigma_{1}^{c}=-ig_{0}^{2}k_{3}k_{4}$
$\displaystyle\delta(k_{3}+k_{4}-k_{1}-k_{2})\int_{|k_{1}-k_{2}|}^{k_{1}+k_{2}}dk\frac{k}{\Delta(k_{1},k_{2},k)}$
$\displaystyle\times\int
dr\,rJ_{0}(kr)\left[J_{0}(k_{3}r)J_{0}(k_{4}r)+J_{1}(k_{3}r)J_{1}(k_{4}r)\right].$
(84)

Now the $r$ integral only involves products of three Bessel functions, so we
can use Sonine’s formulas to compute it. The result is

$\sigma_{1}^{c}=-\frac{4i}{\pi}g_{0}^{2}\delta(k_{3}+k_{4}-k_{1}-k_{2})\int_{|k_{1}-k_{2}|\vee|k_{3}-k_{4}|}^{k_{1}+k_{2}}\frac{k\,dk}{\sqrt{k^{2}-(k_{1}-k_{2})^{2}}\sqrt{k^{2}-(k_{3}-k_{4})^{2}}}\,.$
(85)

This evaluates to

$\sigma_{1}^{c}=-\frac{4i}{\pi}g_{0}^{2}\operatorname{Re}\operatorname{arctanh}\sqrt{\frac{k_{1}k_{2}}{k_{3}k_{4}}}\delta(k_{3}+k_{4}-k_{1}-k_{2})\,.$
(86)

This is the contribution of the contact diagram to $\sigma_{1}$. Summing all
three diagrams gives

$\sigma_{1}=-\frac{2i}{\pi}g_{0}^{2}\operatorname{Re}\operatorname{arctanh}\sqrt{\frac{k_{1}k_{2}}{k_{3}k_{4}}}\delta(k_{3}+k_{4}-k_{1}-k_{2})\,.$
(87)

This is our first scattering amplitude.

The calculations of $\sigma_{2}$ and $\sigma_{3}$ are similar, so we will just
state the results. $\sigma_{2}$ receives contributions from three diagrams
(see figure 3). $\sigma_{3}$ is similar to $\sigma_{2}$ but with $k_{3}$ and
$k_{4}$ interchanged. The final results are

$\sigma_{2}=\frac{2i}{\pi}g_{0}^{2}\operatorname{Re}\operatorname{arctanh}\sqrt{\frac{k_{1}k_{3}}{k_{2}k_{4}}}\delta(k_{3}+k_{4}-k_{1}-k_{2})\,,$
(88)

and

$\sigma_{3}=\frac{2i}{\pi}g_{0}^{2}\operatorname{Re}\operatorname{arctanh}\sqrt{\frac{k_{1}k_{4}}{k_{2}k_{3}}}\delta(k_{3}+k_{4}-k_{1}-k_{2})\,.$
(89)

Summing all three amplitudes (and using elementary identities for sums of
$\operatorname{arctanh}$ functions) gives

$\sigma_{1}+\sigma_{2}+\sigma_{3}=0\,.$ (90)

We thus obtain a direct verification of the $\mathrm{SO}(2)$ conservation law
(63) discussed above.

There is another manifestation of the $\mathrm{SO}(2)$ symmetry that is worth
a comment. Looking back at eq. (61), it is clear that at tree level we must
have

$S_{22\rightarrow 22}=0\,.$ (91)

We can directly verify this using Feynman diagrams. There are three diagrams
to consider (see figure 4). We will omit the details for brevity but a
calculation similar to the one above shows that these three diagrams indeed
sum to zero.

22to22s (40,25) i1,i2 o1,o2 dashesi1,v1,i2 plainv1,v2 dasheso1,v2,o2 22to22t
(40,25) i1,i2 o1,o2 dashesi1,v1,o1 plainv1,v2 dashesi2,v2,o2 22to22u (40,25)
i1,i2 o1,o2 dashesi1,v1 dashesi2,v2 phantomv1,o1 phantomv2,o2
dashes,tension=0v2,o1 dashes,tension=0,rubout=4v1,o2 plainv1,v2

Figure 4: Tree-level contributions to $S_{22\rightarrow 22}$. The total
amplitude is zero.

So far we have been discussing four-point amplitudes. We end this section by
observing that all three-point amplitudes vanish. There are basically two
cases to consider: $S_{1\rightarrow 22}$ and $S_{2\rightarrow 12}$. First
consider $S_{1\rightarrow 22}$. Up to an overall constant, the LSZ formula
gives

$\int
d^{2}x\,re^{-ik_{1}t}J_{0}(k_{1}r)\partial_{a}\left[e^{ik_{2}t}J_{0}(k_{2}r)\right]\partial^{a}\left[e^{ik_{3}t}J_{0}(k_{3}r)\right].$
(92)

This reduces to

$\delta(k_{2}+k_{3}-k_{1})\int
dr\,rJ_{0}(k_{1}r)\left[J_{0}(k_{2}r)J_{0}(k_{3}r)+J_{1}(k_{2}r)J_{1}(k_{3}r)\right].$
(93)

This vanishes by Sonine’s formulas because

$1+\frac{k_{2}^{2}+k_{3}^{2}-(k_{2}+k_{3})^{2}}{2k_{2}k_{3}}=0\,.$ (94)

The story for $S_{2\rightarrow 12}$ is similar. Up to an overall constant,
this amplitude is

$\delta(k_{2}+k_{3}-k_{1})\int
dr\,rJ_{0}(k_{2}r)\left[-J_{0}(k_{r})J_{0}(k_{3}r)+J_{1}(k_{1}r)J_{1}(k_{3}r)\right]=0\,.$
(95)

##  4 The boundary theory

An unusual feature of the two dimensional background of our theory is that the
spacetime has a left boundary at $r=0$. The two dimensional spacetime metric
is

$ds^{2}=-dt^{2}+dr^{2}\,.$ (96)

Locally, this is just two dimensional Minkowski space. It has asymptotic null
boundaries at $t\pm r\rightarrow\infty$, just as in higher dimensions.
However, our background also has a left boundary at $r=0$ (see figure 5). The
reason is that when we dimensionally reduced to two spacetime dimensions, we
fixed a $\mathrm{U}(1)$ rotational symmetry of the three dimensional
spacetime. This singled out a worldline in the three dimensional spacetime at
$r=0$. After dimensional reduction, this worldline became the left boundary of
the two dimensional spacetime.

Figure 5: The two dimensional spacetime has a left boundary at $r=0$. We
introduce a cutoff curve slightly outside $r=0$. The basic observables in the
boundary theory are correlation functions on the cutoff curve.

Now we want to define a boundary theory that is holographically dual to the
bulk sigma model. We have the option of defining the boundary theory at the
left boundary or the right boundary. It is tempting to work at the right
boundary because that would be more similar to the story in higher dimensional
asymptotically flat spacetimes. However, working at the left boundary has
several advantages. The fact that the left boundary is timelike means that
holography at the left boundary is more similar to the more familiar case of
anti-de Sitter (AdS) holography. Thus, by working at the left boundary, we get
to have a model that shares features of AdS holography and asymptotically flat
holography (because the cosmological constant of our model is zero). A more
technical reason for preferring the left boundary is that the energy
eigenmodes of the free theory, $e^{\pm ikt}J_{0}(kr)$, are better behaved near
$r=0$ than they are near null infinity. For these reasons, we are going to
define our boundary theory at the left boundary. We leave the problem of
constructing the theory at the right boundary for the future.

The basic device for relating the bulk and boundary theories is the bulk-to-
boundary propagator. Let $X(t,r)$ be a solution of the bulk equation of motion
and let $X_{b}(t)\equiv X(t,0)$ be its restriction to the left boundary. The
bulk-to-boundary propagator, $K(t,t-t^{\prime})$, is a function that lets us
reconstruct $X(t,r)$ from $X_{b}(t)$. The recipe is

$X(t,r)=\int dt^{\prime}\,K(r,t-t^{\prime})X_{b}(t^{\prime})\,.$ (97)

The bulk-to-boundary propagator of the free theory is444See Niedermaier
Niedermaier:2000ud  for a similar construction.

$K_{0}(r,t-t^{\prime})\equiv\frac{1}{2\pi}\int
dk\,e^{ik(t-t^{\prime})}J_{0}(kr)\,.$ (98)

To see why this is so, let $e^{ikt}J_{0}(kr)$ be an energy eigenmode of the
free theory and let $e^{ikt}$ be its restriction to the boundary. Then

$X(t,r)=\frac{1}{2\pi}\int dk^{\prime}\,e^{ik^{\prime}t}J_{0}(k^{\prime}r)\int
dt^{\prime}\,e^{i(k-k^{\prime})t^{\prime}}=e^{ikt}J_{0}(kr)\,.$ (99)

A general solution of the equation of motion of the free theory is a linear
superposition of these modes, so we can reconstruct a general solution of the
free theory from its boundary data using this bulk-to-boundary propagator.

The general logic of the holographic dictionary is that $X^{\mu}_{b}(t)$
should act as a classical source for the operator dual to $X^{\mu}$ in the
boundary theory Witten:1998qj . This dual operator is closely related to the
boundary value of the momentum conjugate to $X^{\mu}$. In the free theory, the
momentum conjugate to $X^{\mu}$ is $\pi^{\mu}=r\partial_{t}X^{\mu}$. This
vanishes at $r=0$, so it will be convenient to work on a cutoff surface at
$r=r_{b}$, slightly outside $r=0$.

We will now compute the tree-level two-point correlation function of the
boundary operator dual to $X^{\mu}$ in the free theory. The action of the free
theory is

$I_{0}=-\frac{1}{2}\delta_{\mu\nu}\int
d^{2}x\,r\partial_{a}X^{\mu}\partial^{a}X^{\nu}\,.$ (100)

Integrating by parts and using the free equation of motion gives a surface
term at $r=r_{b}$:

$I_{b}\equiv\frac{1}{2}\delta_{\mu\nu}\int
dt\,r_{b}X^{\mu}_{b}(\partial_{r}X^{\nu})_{b}\,.$ (101)

We evaluate $(\partial_{r}X^{\nu})_{b}$ using the bulk-to-boundary propagator.
Near $r=0$, we have $\partial_{r}J_{0}(kr)=-\frac{1}{2}k^{2}r$, and so

$I_{b}=\frac{1}{4\pi}\int dt\int
dt^{\prime}\,X^{\mu}_{b}(t)\left[-\frac{r_{b}^{2}}{2}\delta_{\mu\nu}\int
dk\,e^{ik(t-t^{\prime})}k^{2}\right]X^{\nu}_{b}(t^{\prime})\,.$ (102)

The function in square brackets is the two-point correlation function of the
boundary operator sourced by $X^{\mu}_{b}$. For non-coincident points, it is a
multiple of $(t-t^{\prime})^{-3}$.

There is another way we could have arrived at this result. We have already
computed the bulk two-point function of $X^{\mu}$ with itself in the free
theory in eq. (3.1):

$\langle
0|X^{\mu}(x)X^{\nu}(x^{\prime})|0\rangle=\frac{1}{2}\delta^{\mu\nu}\int
dk\,e^{-ik(t-t^{\prime})}J_{0}(kr)J_{0}(kr^{\prime})\,.$ (103)

So the bulk two-point function of $\pi^{\mu}$ with itself is

$\langle
0|\pi^{\mu}(x)\pi^{\nu}(x^{\prime})|0\rangle=-\frac{r^{2}}{2}\delta^{\mu\nu}\int
dk\,e^{-ik(t-t^{\prime})}k^{2}J_{0}(kr)J_{0}(kr^{\prime})\,.$ (104)

Restricting this to the left boundary gives

$\langle
0|\pi^{\mu}_{b}(t)\pi^{\nu}_{b}(t^{\prime})|0\rangle=-\frac{r_{b}^{2}}{2}\delta^{\mu\nu}\int
dk\,e^{-ik(t-t^{\prime})}k^{2}\,.$ (105)

This is the same two-point correlation function as in eq. (102). This
observation provides a simple recipe for computing boundary correlation
functions in the interacting theory: compute correlation functions of
$\pi^{\mu}$ with itself in the bulk, and then restrict these functions to the
left boundary as in eq. (105). We can take this as a definition of the
boundary theory. The challenge for the future is to find an independent
definition of the boundary theory that produces these correlation functions.

##  5 Discussion

We have described the Ehlers symmetry of the quantum sigma model. The next
step will be to understand if the full Geroch group is a symmetry of the
quantum sigma model. This problem can be studied first at tree level and then
at one loop. If the Geroch group is a symmetry of the quantum sigma model,
then it might be possible to solve the theory exactly, in some sense. If we
think of this as a toy model for a quantum gravity $S$-matrix, then we could
hope to learn something new and interesting from its exact solution. Also, if
the Geroch group is a symmetry of the quantum sigma model, then we could
probably use this huge symmetry to give an independent definition of the
holographically dual boundary theory based on symmetry.

In recent years, there has been a lot of interest in Jackiw-Teitelboim (JT)
gravity Almheiri:2014cka ; Maldacena:2016upp . This is a two dimensional model
of gravity that is designed to describe the near-horizon regions of near-
extremal black holes. It does not have any propagating degrees of freedom in
the bulk (without adding additional matter fields). The two dimensional theory
we have been studying is designed for studying cylindrical gravitational wave
scattering. So, in some sense, these two theories offer windows into
complementary aspects of gravitational physics. Another important distinction
between these theories is that JT gravity is usually studied with negative
cosmological constant, while our sigma model only makes sense with zero
cosmological constant. So these two theories naturally lead to complementary
versions of holography.

The fact that our theory is two dimensional means it might be relatively easy
to study numerically. If we had an independent definition of the boundary
theory, then that theory might be even easier to realize on the computer. In
the future, it would be interesting to understand the extent to which our
model of the cylindrical gravitational wave $S$-matrix can be studied
numerically or even experimentally.

###### Acknowledgements.

I am grateful to Daniel Grumiller, Bart Horn, and Max Niedermaier for sharing
comments on the manuscript.

## References

  * (1) P. Breitenlohner and D. Maison, _On the Geroch group_ , _Ann. Inst. H. Poincare Phys. Theor._ 46 (1987) 215.
  * (2) H. Nicolai, _Two-dimensional gravities and supergravities as integrable system_ , _Lect. Notes Phys._ 396 (1991) 231.
  * (3) R.P. Geroch, _A method for generating solutions of Einstein’s equations_ , _J. Math. Phys._ 12 (1971) 918.
  * (4) R.P. Geroch, _A method for generating new solutions of Einstein’s equation. 2_ , _J. Math. Phys._ 13 (1972) 394.
  * (5) H. Lu, M.J. Perry and C.N. Pope, _Infinite-dimensional symmetries of two-dimensional coset models coupled to gravity_ , _Nucl. Phys. B_ 806 (2009) 656 [0712.0615].
  * (6) H. Lu, M.J. Perry and C.N. Pope, _Infinite-dimensional symmetries of two-dimensional coset models_ , 0711.0400.
  * (7) J.H. Schwarz, _Classical symmetries of some two-dimensional models coupled to gravity_ , _Nucl. Phys. B_ 454 (1995) 427 [hep-th/9506076].
  * (8) J.H. Schwarz, _Classical symmetries of some two-dimensional models_ , _Nucl. Phys. B_ 447 (1995) 137 [hep-th/9503078].
  * (9) S. Pasterski, M. Pate and A.-M. Raclariu, _Celestial Holography_ , in _Snowmass 2021_ , 11, 2021 [2111.11392].
  * (10) K. Kuchař, _Canonical quantization of cylindrical gravitational waves_ , _Phys. Rev. D_ 4 (1971) 955.
  * (11) A. Ashtekar and M. Pierri, _Probing quantum gravity through exactly soluble midisuperspaces. 1._ , _J. Math. Phys._ 37 (1996) 6250 [gr-qc/9606085].
  * (12) M.E. Angulo and G.A. Mena Marugan, _Large quantum gravity effects: Cylindrical waves in four-dimensions_ , _Int. J. Mod. Phys. D_ 9 (2000) 669 [gr-qc/0002056].
  * (13) M. Niedermaier, _Quantized Einstein-Rosen waves, AdS(2), and spontaneous symmetry breaking_ , _Phys. Lett. B_ 498 (2001) 83 [hep-th/0007227].
  * (14) J.F. Barbero G., G.A. Mena Marugan and E.J.S. Villasenor, _Microcausality and quantum cylindrical gravitational waves_ , _Phys. Rev. D_ 67 (2003) 124006 [gr-qc/0304047].
  * (15) J.F. Barbero G., G.A. Mena Marugan and E.J.S. Villasenor, _Particles and vacuum for perturbative and non-perturbative Einstein-Rosen gravity_ , _Phys. Rev. D_ 70 (2004) 044028 [gr-qc/0406087].
  * (16) J.F. Barbero G., I. Garay and E.J.S. Villasenor, _Quantum Einstein-Rosen waves: Coherent states and n-point functions_ , _Class. Quant. Grav._ 25 (2008) 205013 [0808.2561].
  * (17) D. Korotkin and H. Samtleben, _Canonical quantization of cylindrical gravitational waves with two polarizations_ , _Phys. Rev. Lett._ 80 (1998) 14 [gr-qc/9705013].
  * (18) D. Korotkin and H. Samtleben, _Yangian symmetry in integrable quantum gravity_ , _Nucl. Phys. B_ 527 (1998) 657 [hep-th/9710210].
  * (19) M. Niedermaier and H. Samtleben, _An algebraic bootstrap for dimensionally reduced quantum gravity_ , _Nucl. Phys. B_ 579 (2000) 437 [hep-th/9912111].
  * (20) A. Fuchs and M.P. Reisenberger, _Integrable structures and the quantization of free null initial data for gravity_ , _Class. Quant. Grav._ 34 (2017) 185003 [1704.06992].
  * (21) J. Peraza, M. Paternain and M. Reisenberger, _On the classical and quantum Geroch group_ , _Class. Quant. Grav._ 38 (2021) 015013 [1906.04856].
  * (22) B. Runov, _On quantum determinants in integrable quantum gravity_ , _Class. Quant. Grav._ 38 (2021) 065021 [2010.09967].
  * (23) M. Niedermaier, _On the renormalization of truncated quantum Einstein gravity_ , _JHEP_ 12 (2002) 066 [hep-th/0207143].
  * (24) M. Niedermaier, _Dimensionally reduced gravity theories are asymptotically safe_ , _Nucl. Phys. B_ 673 (2003) 131 [hep-th/0304117].
  * (25) J.F. Barbero G., G.A. Mena Marugan and E.J.S. Villasenor, _Quantum cylindrical waves and sigma models_ , _Int. J. Mod. Phys. D_ 13 (2004) 1119 [gr-qc/0402096].
  * (26) R.H. Gowdy, _Gravitational waves in closed universes_ , _Phys. Rev. Lett._ 27 (1971) 826.
  * (27) D. Grumiller, D. Hofmann and W. Kummer, _Two dilaton theories in two-dimensions_ , _Annals Phys._ 290 (2001) 69 [gr-qc/0005098].
  * (28) E. Witten, _Anti-de Sitter space and holography_ , _Adv. Theor. Math. Phys._ 2 (1998) 253 [hep-th/9802150].
  * (29) A. Almheiri and J. Polchinski, _Models of AdS 2 backreaction and holography_, _JHEP_ 11 (2015) 014 [1402.6334].
  * (30) J. Maldacena, D. Stanford and Z. Yang, _Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space_ , _PTEP_ 2016 (2016) 12C104 [1606.01857].