♮♮institutetext: Centre for Theoretical Physics, Department of Physics and
Astronomy, Queen Mary University of London, Mile End Road, E1 4NS, UK

# Celestial OPEs and $w_{1+\infty}$ algebra from worldsheet in string theory

Hongliang Jiang<EMAIL_ADDRESS>
###### Abstract

Celestial operator product expansions (OPEs) arise from the collinear limit of
scattering amplitudes and play a vital role in celestial holography. In this
paper, we derive the celestial OPEs of massless fields in string theory from
the worldsheet. By studying the worldsheet OPEs of vertex operators in
worldsheet CFT and further examining their behaviors in the collinear limit,
we find that new vertex operators for the massless fields in string theory are
generated and become dominant in the collinear limit. Mellin transforming to
the conformal basis yields exactly the celestial OPEs in celestial CFT. We
also derive the celestial OPEs from the collinear factorization of string
amplitudes and the results derived in these two different methods are in
perfect agreement with each other. Our final formulae of celestial OPEs are
applicable to general dimensions, corresponding to Einstein-Yang-Mills theory
supplemented by some possible higher derivative interactions. Specializing to
4D, we reproduce all the celestial OPEs for gluon and graviton in the
literature. We consider various string theories, including the open and closed
bosonic string, as well as the closed superstring theory with $\mathcal{N}=1$
and $\mathcal{N}=2$ worldsheet supersymmetry. In the case of $\mathcal{N}=2$
string, we also derive all the $\overline{SL(2,\mathbb{R})}$ descendant
contributions in the celestial OPE; the soft sector of such OPE just yields
the $w_{1+\infty}$ algebra after rewriting in terms of chiral modes. Our
stringy derivation of celestial OPEs thus initiates the first step towards the
realization of celestial holography in string theory.

††preprint: QMUL-PH-21-42

##  1 Introduction

The quest for quantum gravity is one of the most fundamental questions in
theoretical physics. Although the quantization of quantum gravity is
notoriously hard, string theory has provided us the framework for studying
quantum gravity, at least perturbatively. Moreover, string theory enables us
to learn some profound aspects of quantum gravity, in particular the
holographic nature. Although the holographic principle was first proposed
based on black hole entropy tHooft:1993dmi ; Susskind:1994vu , such an idea
was very vague until a concrete model was realized in string theory
Maldacena:1997re . In string theory, such a holographic duality, now known as
the AdS/CFT correspondence, naturally arises from the duality between open and
closed strings. After more than two decades of intensive study, the AdS/CFT
correspondence has been tested very precisely and has also taught us even more
profound aspects of quantum gravity, like entanglement. Considering the very
beautiful and successful story of AdS/CFT correspondence, it is natural to
wonder whether we can study quantum gravity beyond AdS. One interesting and
natural generalization is flat spacetime, where the boundary is null and even
non-smooth. This turns out to be very difficult and very little progress was
made in the past. On the other hand, the last two decades also witnessed
fruitful achievements in scattering amplitude both computationally and
conceptually. In particular, the idea of on-shellness, locality, unitarity and
causality has been playing a crucial role. Interestingly, scattering
amplitudes are just the observables of quantum field theories in flat
spacetime. Bridging the ideas from two seemingly unrelated areas together, a
promising program towards flat holography, called celestial holography, starts
to emerge in recent years Cheung:2016iub ; Pasterski:2016qvg . The precursor
of celestial holography comes from noticing the equivalence between asymptotic
symmetries and soft theorems Strominger:2013jfa ; He:2014laa . Since then
various interesting progress has been made. See Raclariu:2021zjz ;
Pasterski:2021rjz  for recent reviews. In spite, the study of celestial
holography has been mostly focusing on symmetries and using the bottom-up
approach. This is drastically different from the AdS/CFT correspondence which
has various top-down concrete realizations in string theory. It is then
natural to ask whether we can also find a concrete realization for celestial
holography in string theory. 111 For previous studies of celestial holography
related to (ambitwistor) string theory, see Stieberger:2018edy  for celestial
string amplitudes, and Adamo:2019ipt  Casali:2020uvr  for conformally soft
theorems and celestial double copy in ambitwistor string theory. This is
particularly important for several reasons. First of all, the celestial
amplitudes seem to be very UV sensitive and it is only well-defined in
theories, like string theory, where the UV behavior is soft enough. Secondly,
a stringy construction may give rise to an exact model of celestial holography
and thus can be used to test various salient ideas there. In AdS/CFT, the
remarkable agreement between $\mathcal{N}=4$ SYM and type IIB string on
$AdS_{5}\times S^{5}$ just provides a good justification on this point.

The goal of this paper is to initiate the first step towards the stringy
realization of celestial holography. More specifically, we will derive the
celestial operator production expansion (OPE) from string theory. Although we
have not been able to construct an explicit model for celestial holography in
string theory, it turns out that the worldsheet of string theory already
implies something nontrivial about celestial holography. In particular, we can
derive the celestial OPEs from the worldsheet OPEs. The lack of an exact model
and thus the model independence of our derivation show that the relation
between the two kinds of OPEs is universal.

The celestial OPEs characterize the behavior of two operators in the
coincident limit in celestial CFT (CCFT). They can be obtained from the
collinear limit of scattering amplitude by performing the Mellin
transformation. Since the collinear factorization is a universal property of
scattering amplitude, the celestial OPE is supposed to also play an important
role in CCFT. Moreover, the soft sector of celestial OPEs also encodes the
underlying symmetry of CCFT and the bulk scattering amplitude. In particular,
starting from celestial OPEs, an infinite dimensional holographic symmetry
algebra has been discovered recently Guevara:2021abz ; Strominger:2021lvk .
222 It is more precise to refer to this algebra as holographic chiral algebra
as this algebra governs the chiral subsector of soft particles with positive
helicity only and all the symmetries are generated by chiral currents.  In the
case of gravity, this symmetry algebra is just the $w_{1+\infty}$ algebra
Strominger:2021lvk . The supersymmetric extension and the infinite Ward
identities associated with this algebra were studied in Jiang:2021ovh . See
Banerjee:2020kaa ; Guevara:2021tvr  for other aspects of celestial OPEs.

The celestial OPEs can be obtained in several methods. The most direct way is
to consider the collinear limit of scattering amplitude and then perform the
Mellin transformation Fan:2019emx ; Fotopoulos:2020bqj ; Jiang:2021xzy .
Alternatively, one can bootstrap celestial OPEs using conformal symmetry as
well as the input from soft theorems Pate:2019lpp . Using these methods, the
general formula of celestial OPEs for spinning massless particles with cubic
interactions in 4D has been recently derived in Jiang:2021ovh ;
Himwich:2021dau .

In this paper, we offer another derivation of celestial OPEs from the string
worldsheet perspective. 333 The celestial OPEs in 4D can also be derived from
ambitwistor string ambiOPE . The general strategy is as follows. In celestial
holography, the celestial amplitudes are defined as the Mellin transformation
of momentum space scattering amplitude 444 In this paper, we only consider
massless fields, which just correspond to the gluon and graviton/dilaton/Kalb-
Ramond field in string theory.  and can be regarded as the correlation
functions of celestial operators
$\langle\mathcal{O}_{\Delta_{1}}(x_{1})\mathcal{O}_{\Delta_{2}}(x_{2})\cdots\rangle$
in some putative CCFT on the celestial sphere living at boundary null infinity
Pasterski:2016qvg ; Pasterski:2017kqt . This is very similar to the scattering
amplitudes in string theory which are computed by the correlator of vertex
operators in worldsheet CFT. One can make the relation more precise by
introducing the so-called conformal vertex operators, which are defined as the
Mellin transformation of the standard vertex operators. The celestial string
amplitude can then be alternatively regarded as the correlator of conformal
vertex operators
$\langle\mathcal{V}_{\Delta_{1}}(x_{1})\mathcal{V}_{\Delta_{2}}(x_{2})\cdots\rangle$
in worldsheet CFT. The fact that the celestial amplitude can be computed in
two different ways thus suggests a map between worldsheet CFT and CCFT, and
correspondingly a map between conformal vertex operators
$\mathcal{V}_{\Delta}$ and celestial operators $\mathcal{O}_{\Delta}$. Then
the derivation of OPEs of celestial operators, namely $\lim_{x_{1}\to
x_{2}}\mathcal{O}_{\Delta_{1}}(x_{1})\mathcal{O}_{\Delta_{2}}(x_{2})$, boils
down to the computation of OPEs of conformal vertex operators $\lim_{x_{1}\to
x_{2}}\mathcal{V}_{\Delta_{1}}(x_{1})\mathcal{V}_{\Delta_{2}}(x_{2})$. It
turns out that the latter can be obtained by first computing worldsheet OPEs
of two vertex operators and then performing the Mellin transformation. 555 In
principle, one may compute the worldsheet OPE of two conformal vertex
operators directly, and then take the collinear limit. We will not pursue this
approach in this paper, and leave it to the future.  More precisely, the
vertex operator is given by some operator $V_{p}(z)$ in the worldsheet CFT
integrating over the worldsheet. 666 Here to be explicit, we only discuss the
integrated form of vertex operators in closed string. The discussions for open
string and the unintegrated form of vertex operators are the same.  In order
to compute the celestial OPE, namely $\lim_{x_{1}\to
x_{2}}\mathcal{V}_{\Delta_{1}}(x_{1})\mathcal{V}_{\Delta_{2}}(x_{2})$, we find
it is sufficient to compute the worldsheet OPE $\lim_{z_{1}\to
z_{2}}V_{p_{1}}(z_{1})V_{p_{2}}(z_{2})$. Explicit computation shows that
worldsheet OPE in the collinear limit $p_{1}\cdot p_{2}\to 0$, which is
equivalent to the celestial coincident limit $x_{1}\to x_{2}$, localizes to a
delta-function $\delta^{2}(z_{1}-z_{2})$ on the worldsheet, and thus produces
another vertex operator after integrating over the worldsheet. This thus
fulfills our derivation of celestial OPEs from the worldsheet perspective. And
the computation essentially reduces to computing OPEs of free fields on the
worldsheet.

We will consider various string theories, including open and closed bosonic
string, and closed superstring with $\mathcal{N}=1$ and $\mathcal{N}=2$
worldsheet supersymmetry. For bosonic string, we compute the worldsheet OPE of
two vertex operators for massless gluons in open string and obtain the vertex
operators for various open string fields including tachyon, gluon, etc. In
spite, we show that in the collinear limit, only the gluon in the OPE is
dominant and the rest can be ignored, up to some subtle boundary contact
terms. The boundary contact terms arise when the two vertex operators in the
OPE hit another vertex operator. They can partially be attributed to the
remnant contribution from tachyon. On the other hand, the gluon contribution
in the worldsheet OPE of two gluon vertex operators comes with a delta-
function in the collinear limit. Doing a worldsheet integral and performing
the Mellin transformation thus give us the celestial OPEs for gluons. The same
derivation is also done for massless fields in the closed string, namely
graviton, Kalb-Ramond two-form field and dilaton, and we obtain the celestial
OPEs for these fields. We also discuss the OPEs between open and closed string
massless fields. This is a bit different; at tree level the open string vertex
operator sits only at the boundary of the disk, while the closed string vertex
operator sits in the interior of the disk. Nevertheless, we manage deriving
the corresponding celestial OPEs, namely the fusion of gluon and graviton to
another gluon. However, it is known that graviton can also appear in the
celestial OPE of two gluons Pate:2019lpp . The derivation of this OPE from the
open-closed string setup is not clear, as one needs to produce a graviton
vertex operator in the interior of the disk from two gluon vertex operators
sitting on the boundary of the disk. We sidestep this problem by considering
heterotic string where one can realize both gluon and graviton/dilaton/Kalb-
Ramond field from the closed string. Using similar techniques in the bosonic
string, we are able to derive all the celestial OPEs involving gluon and
graviton/dilaton/Kalb-Ramond field, including the fusion of two gluons into
one graviton. We also discuss the OPEs of NS-NS massless fields in type I and
type IIA/IIB string theory.

To further corroborate our celestial OPEs, we study string amplitudes in the
collinear limit and then perform the Mellin transformation, which offers
another derivation of celestial OPEs. It turns out that the three-point string
amplitude is just enough to determine the celestial OPEs. The final results
agree exactly with those derived from worldsheet. Our derivation is for string
theory in critical dimensional flat spacetime, namely 26 dimensions for
bosonic string and 10 dimensions for superstring. Nevertheless, the final
formulae of celestial OPEs are supposed to be applicable to general
dimensions, corresponding to Einstein-Yang-Mills theory with some possible
higher derivative corrections. Specializing to four dimensional spacetime, we
recover all the gluon and graviton OPEs obtained in the literature Fan:2019emx
; Pate:2019lpp ; Jiang:2021ovh ; Himwich:2021dau .

Last but not least, we also generalize our discussions to the $\mathcal{N}=2$
string theory Ooguri:1991fp ; Ooguri:1990ww ; Marcus:1992wi . The
$\mathcal{N}=2$ string theory has critical dimension four and is consistent in
(2,2) signature instead of Minkowski signature. The simplest version of
$\mathcal{N}=2$ theory has a massless degree of freedom and the low energy
effective action for $\mathcal{N}=2$ string is described by some kind of self-
dual gravity. Following the same approach in other string theories, we study
the worldsheet OPE of two vertex operators in $\mathcal{N}=2$ string theory
and derive the corresponding celestial OPE. The derivation is very similar to
other string theories with less supersymmetry. However, now we can go further
beyond the previous derivations. It turns out that in the spacetime with (2,2)
signature, we can even derive all the $\overline{SL(2,\mathbb{R})}$
descendants in the OPE. We find that this essentially comes from the momentum
conservation, and the fact that in (2,2) signature we can vary two celestial
coordinates independently and thus have more freedom to realize the collinear
limit. These two features, $\overline{SL(2,\mathbb{R})}$ descendants and
independent celestial coordinates, are just the crucial ingredients in the
derivation of $w_{1+\infty}$ symmetry. Indeed, focusing on the soft sector of
our OPE with descendants and then performing the mode expansion, we recover
the $w_{1+\infty}$ algebra Guevara:2021abz ; Strominger:2021lvk . We thus
provide a stringy derivation of the $w_{1+\infty}$ symmetry, although it is
indirect. 777 Note that the $w_{1+\infty}$ algebra appeared before in self-
dual gravity and $\mathcal{N}=2$ string, but in a different way Boyer:1985aj ;
Ooguri:1991fp .  In a more direct derivation, one should be able to construct
the chiral generators in $w_{1+\infty}$ algebra directly from $\mathcal{N}=2$
string worldsheet. We will make some comments and defer the direct
construction to the future.

This paper is organized as follows. In section 2, we will first discuss the
kinematics in general dimensions in terms of celestial variables, and then
review the vertex operators in bosonic string theory. A map between conformal
vertex operators and celestial operators will also be discussed. In section 3,
we will derive the celestial OPEs from the worldsheet OPEs in bosonic string
theory. In section 4, we will generalize the worldsheet derivation of
celestial OPEs to superstring. In section 5, we will derive the celestial OPEs
from the collinear factorization of string amplitude. The specialization of
celestial OPEs to 4D will be presented. In section 6, we will derive the
celestial OPE with descendants in $\mathcal{N}=2$ string theory, and then
discuss the resulting $w_{1+\infty}$ algebra. In section 7, we will conclude
and discuss possible future directions. The paper also has two appendices. In
appendix A we will collect all the computation details of worldsheet OPEs of
vertex operators in various string theories. In appendix B, we will derive the
celestial OPE between gluon and graviton in the open-closed string setup from
both the worldsheet perspective and the amplitude approach.

Notation: The spacetime dimension is $D+2$ and the corresponding celestial
sphere is $D$-dimensional. We will use $\mu,\nu,\cdots=0,1,\cdots,D,D+1$ for
spacetime indices and $a,b,c,\cdots=1,\cdots,D$ for celestial sphere indices.
The spacetime metric is $\eta^{\mu\nu}=\text{diag}(-1,+1,\cdots,+1)$ and the
celestial sphere metric is $\delta^{ab}$, hence we will raise or lower the
position of celestial sphere indices freely. Repeated indices are summed over.
We use $x_{i}^{a}$ to label the $a$-th coordinate of the $i$-th
particle/operator. We use $y$ and $z,\bar{z}$ for the open and closed string
worldsheet coordinates, respectively, and ${\sf z},\bar{\sf z}$ for the
celestial sphere coordinates in four dimensional spacetime. The polarizations
are denoted as $\zeta,\xi,e,\varepsilon$, while $\epsilon$ always refers to
infinitesimal quantity.

##  2 Preliminary

In this preliminary section, we will introduce some tools and background
knowledge that will be used in the later sections. We will first introduce the
kinematics of massless fields in terms of celestial sphere variables in
general dimensions. Then we will review the vertex operators in open and
closed bosonic string theory. Finally we will introduce the notion of
conformal vertex operators and their relation with celestial operators.

###  2.1 Kinematics in general dimension

We consider $D+2$-dimensional spacetime with $D$-dimensional celestial sphere
at null infinity. A null momentum $k$ can be parametrized as Kapec:2017gsg

$k^{\mu}(\omega_{k},x)=\eta\omega_{k}\hat{k}^{\mu}(x)~{},\qquad\hat{k}^{\mu}(x)=\Big{(}\frac{1+(x)^{2}}{2}~{},x^{a},\frac{1-(x)^{2}}{2}\Big{)}~{},\qquad
n^{\mu}=\Big{(}-1,0^{a},1\Big{)}~{}.$ (1)

where $\omega_{k}>0$, $\eta=\pm 1$ labels out-going/in-coming particles and
$(x)^{2}=\sum_{a}x^{a}x^{a}$. We also introduce the following basis of
polarization vectors

$\varepsilon_{a}^{\mu}(x)\equiv\varepsilon_{a}^{\mu}(k)=\partial_{a}\hat{k}^{\mu}(x)=(x^{a},\delta^{ab},-x^{a})~{},\qquad\mu=(0,b,D+1)~{}.$
(2)

These $D$ polarization vectors transform under the vector representation of
the little group $SO(D)$ for massless particles. And we have

$n\cdot n=\hat{k}\cdot\hat{k}=\varepsilon_{a}\cdot
n=\varepsilon_{a}\cdot\hat{k}=0~{},\qquad
n\cdot\hat{k}=1~{},\qquad\varepsilon_{a}\cdot\varepsilon_{b}=\delta_{ab}~{}.$
(3)

So the vectors
$\\{\frac{n+\hat{k}}{\sqrt{2}},\varepsilon_{a},\frac{n-\hat{k}}{\sqrt{2}}\\}$
form a complete orthonormal basis. The little group rotates polarization
vectors $\varepsilon_{a}$ but leaves $n,\hat{k}$ invariant.

For different momenta, we further have the following identities

$\hat{k}(x_{i})\cdot\hat{k}(x_{j})\equiv\hat{k}_{i}\cdot\hat{k}_{j}=-\frac{1}{2}(x_{ij})^{2}~{},\qquad\hat{k}_{i}\cdot\varepsilon_{a}(x_{j})=x_{ij}^{a}~{},\qquad\varepsilon_{a}(x_{i})\cdot\varepsilon_{b}(x_{j})=\delta^{ab}~{},$
(4)

where $x_{ij}=x_{i}-x_{j}$.

Taking the product of two polarizations, we get $D^{2}$ two-index tensors

$\varepsilon_{ab}^{\mu\nu}=\varepsilon_{a}^{\mu}\varepsilon^{\nu}_{b}~{}.$ (5)

It is reducible under the little group $SO(D)$ and can be decomposed into
symmetric traceless, anti-symmetric and singlet representation:

$\bm{D}\otimes\bm{D}=\bm{\frac{(D+2)(D-1)}{2}}+\bm{\frac{D(D-1)}{2}}+\bm{1}~{}.$
(6)

They just correspond to the polarizations of graviton, Kalb-Ramond (KR) 2-form
field and dilaton, respectively:

$\displaystyle\varepsilon_{(ab)}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}\varepsilon_{a}^{\mu}\varepsilon_{b}^{\nu}+\varepsilon_{a}^{\nu}\varepsilon_{b}^{\mu}\Big{)}-\frac{1}{D}\delta_{ab}\Pi^{\mu\nu}~{},\qquad\delta^{ab}\varepsilon_{ab}^{\mu\nu}=\eta_{\mu\nu}\varepsilon_{ab}^{\mu\nu}=0~{},\qquad\varepsilon_{(ab)}^{\mu\nu}=\varepsilon_{(ba)}^{\mu\nu}=\varepsilon_{(ab)}^{\nu\mu}~{},$
(7) $\displaystyle\varepsilon_{[ab]}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}\varepsilon_{a}^{\mu}\varepsilon_{b}^{\nu}-\varepsilon_{a}^{\nu}\varepsilon_{b}^{\mu}\Big{)}=-\varepsilon_{[ab]}^{\nu\mu}=-\varepsilon_{[ba]}^{\mu\nu}~{},$
(8) $\displaystyle\varepsilon^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{D}\varepsilon_{a}^{\mu}\varepsilon_{a}^{\nu}=\frac{1}{D}\Pi^{\mu\nu}~{},$
(9)

where we also introduced

$\Pi^{\mu\nu}(x)\equiv\delta^{ab}\varepsilon_{a}^{\mu}(x)\varepsilon^{\nu}_{b}(x)=\eta^{\mu\nu}-n^{\mu}\hat{k}^{\nu}(x)-n^{\nu}\hat{k}^{\mu}(x)~{}.$
(10)

We will also frequently use the abstract polarization tensors which satisfy
various properties. In particular, the polarization vector of gluon $e^{\mu}$
satisfies

$e\cdot e=1~{},\qquad k\cdot e=0~{},\qquad e^{\mu}\simeq e^{\mu}+\lambda
k^{\mu}~{},$ (11)

where the equivalence in the last equation is guaranteed by the gauge
invariance. The last property allows us to choose a gauge where

$e\cdot n=0~{}.$ (12)

We can then decompose any gluon polarization vector as

$e^{\mu}\simeq\sum_{a}c_{a}\varepsilon_{a}^{\mu}~{},$ (13)

up to a pure gauge.

Similarly, for closed string massless spectra, we will use $e^{\mu\nu}$ to
denote the polarization tensors. Let us first consider graviton and KR 2-form
field, whose polarization tensors satisfy

$e^{\mu\nu}e_{\mu\nu}=1~{},\qquad
k_{\mu}e^{\mu\nu}=k_{\nu}e^{\mu\nu}=\eta_{\mu\nu}e^{\mu\nu}=0~{},\qquad
e^{\mu\nu}=se^{\nu\mu}~{},\qquad e^{\mu\nu}\simeq
e^{\mu\nu}+k^{\mu}\zeta^{\nu}+s\zeta^{\mu}k^{\nu}~{},\qquad$ (14)

where $k\cdot\zeta=0$, and $s=\pm 1$ for graviton and 2-form field,
respectively. They can be expanded in terms of basis (7)(8) that we
constructed before:

$e^{\mu\nu}\simeq\sum^{D}_{a,b=1\atop a<b\text{ or
}a=b<D}c_{ab}\varepsilon_{(ab)}^{\mu\nu}~{},\qquad\qquad
e^{\mu\nu}\simeq\sum^{D}_{a,b=1\atop
a<b}c_{ab}\varepsilon_{[ab]}^{\mu\nu}~{},$ (15)

for graviton and Kalb-Ramond 2-form field, respectively.

For dilaton, we have

$e^{\mu\nu}e_{\mu\nu}=1~{},\qquad e^{\mu\nu}=e^{\nu\mu}~{},\qquad
k_{\mu}e^{\mu\nu}=n_{\mu}e^{\mu\nu}=0~{},$ (16)

where $n$ is another null direction defined such that $n\cdot
n=0,n\cdot\hat{k}=1$. This fixes the dilaton polarization to be (9)

$e^{\mu\nu}=\frac{1}{D}\Big{(}\eta^{\mu\nu}-n^{\mu}\hat{k}^{\nu}-n^{\nu}\hat{k}^{\mu}\Big{)}=\varepsilon^{\mu\nu}=\sum_{a}\frac{1}{D}\varepsilon_{aa}^{\mu\nu}~{}.$
(17)

One easily check that $e^{\mu\nu}\eta_{\mu\nu}=1$.

In four dimensional spacetime with $D=2$, it is convenient to introduce
complex coordinates on the celestial sphere

$x^{1}=\frac{{\sf z}+\bar{\sf z}}{2}~{},\qquad x^{2}=\frac{-i({\sf z}+\bar{\sf
z})}{2}~{},\qquad{\sf z}=x^{1}+ix^{2}~{},\qquad\bar{\sf z}=x^{1}-ix^{2}~{}.$
(18)

Then we can represent the momentum as $k=\eta\omega_{k}\hat{k}^{\mu}$ where

$\hat{k}^{\mu}=\frac{1}{2}\Big{(}1+(x)^{2},2x^{1},2x^{2},1-(x)^{2}\Big{)}=\frac{1}{2}\Big{(}1+{\sf
z}\bar{\sf z},{\sf z}+\bar{\sf z},-i({\sf z}-\bar{\sf z}),1-{\sf z}\bar{\sf
z}\Big{)}~{}.$ (19)

The two polarization vectors are
$\varepsilon_{1}^{\mu}=(x^{1},1,0,-x^{1}),\;\varepsilon_{2}^{\mu}=(x^{2},0,1,-x^{2})$.
It is more convenient to define the polarization vectors in the helicity
basis:

$\varepsilon_{+}^{\mu}=\frac{1}{\sqrt{2}}\Big{(}\varepsilon_{1}^{\mu}-i\varepsilon_{2}^{\mu}\Big{)}=\frac{1}{\sqrt{2}}(\bar{\sf
z},1,-i,-\bar{\sf
z})~{},\qquad\varepsilon_{-}^{\mu}=\frac{1}{\sqrt{2}}\Big{(}\varepsilon_{1}^{\mu}+i\varepsilon_{2}^{\mu}\Big{)}=\frac{1}{\sqrt{2}}({\sf
z},1,i,-{\sf z})~{},$ (20)

satisfying
$\varepsilon_{+}\cdot\varepsilon_{+}=\varepsilon_{-}\cdot\varepsilon_{-}=0,\;\varepsilon_{+}\cdot\varepsilon_{-}=1$.

For graviton, the polarization tensors for two helicities are

$\varepsilon_{\pm}^{\mu\nu}=\varepsilon_{\pm}^{\mu}\varepsilon_{\pm}^{\nu}=\frac{1}{2}\Big{(}\varepsilon_{11}^{\mu\nu}-\varepsilon_{22}^{\mu\nu}\mp
i\varepsilon_{12}^{\mu\nu}\mp
i\varepsilon_{21}^{\mu\nu}\Big{)}=\varepsilon_{(11)}^{\mu\nu}\mp
i\varepsilon_{(12)}^{\mu\nu}~{}.$ (21)

###  2.2 Vertex operator in bosonic string

In this subsection, we review the vertex operators in open and closed bosonic
string theory.

####  2.2.1 Open string vertex operator

At tree level, open string amplitudes are computed by the correlator of vertex
operators inserted at the boundary of the disk, or equivalently the boundary
of upper half plane, which we parametrize by $y\in\mathbb{R}$.

There are two types of vertex operators. The integrated vertex operators have
the following general structure 888 Note that the product of operators at
coincident points should always be understood as normal ordered product. 999
We will sometimes abuse the terminology of vertex operators and call both
$\mathcal{V}$ and $V$ vertex operators. The exact meaning should be clear from
the context.

$\mathcal{V}^{A,\ell}(k)=\int dy\;V^{A,\ell}_{k}(y)=\int
dy\;\mathscr{V}^{A,\ell}(y)e^{ik\cdot X(y)}~{},$ (22)

while the unintegrated vertex operators take the form

$\mathcal{V}^{A,\ell}(k)=c(y)V^{A,\ell}_{k}(y)=c(y)\mathscr{V}^{A,\ell}(y)e^{ik\cdot
X(y)}~{}.$ (23)

where $c$ is the ghost and the position $y$ in the unintegrated vertex
operator is arbitrary. Here $\ell$ is the number of derivatives in
$\mathscr{V}^{A,\ell}_{k}$ (e.g. $\partial_{y}X(y)$ has $\ell=1$), and $A$ are
the extra quantum numbers labelling the vertex operators, including the
polarization vectors, Chan-Paton factors, etc.

The spectra of fields in open string are given by

$-k\cdot
k=M^{2}=\frac{1}{\alpha^{\prime}}\Big{(}N-\frac{D}{24}\Big{)}=\frac{1}{\alpha^{\prime}}\Big{(}N-1\Big{)}~{},$
(24)

where $N\in\mathbb{Z}_{\geq 0}$ is the level and in the last equality we set
$D=24$ for the consideration of critical bosonic string.

When the momentum becomes on-shell, namely
$k^{2}\to-M^{2}=(1-N)/\alpha^{\prime}{}$, the vertex operator becomes BRST
invariant and should has weight 1,

$h(V^{A,\ell}_{k})=h(\mathscr{V}^{A,\ell}_{k})+h(e^{ik\cdot
X})=\ell+\alpha^{\prime}{}k^{2}=1~{},$ (25)

Since on-shell momentum satisfies $k^{2}=(1-N)/\alpha^{\prime}{}$, we see that
$\ell$ is essentially the level $N$, namely $N=\ell$.

At level 0, we have tachyon whose vertex operator is given by 101010 For
simplicity we will set all the coefficients in front of the vertex operators
to be 1. Meanwhile, in the final formulae of celestial OPEs, we will normalize
the overall coefficient properly to make the result as simple as possible. All
the couplings can be easily restored, as we keep track of the overall factors
in the intermediate steps.

$V^{0}_{k}=e^{ik\cdot X}~{},\qquad M^{2}=-\frac{1}{\alpha^{\prime}{}}~{}.$
(26)

At level 1, we have gluons whose vertex operators are Polchinski:1998rr

$V^{1}_{k}=e_{\mu}\dot{X}^{\mu}e^{ik\cdot X}t^{A}~{},\qquad M^{2}=0~{}.$ (27)

where $\dot{X}\equiv\frac{\partial}{\partial y}X$. The polarization $e_{\mu}$
satisfies (11)

$k^{2}=k\cdot e=0~{},\qquad e\cdot e=1~{}.$ (28)

For the gluon vertex operator, we also have the Chan-Paton factor $t^{A}$,
which is just the gauge algebra generator with color index $A$. The
corresponding structure constant, denoted as $f^{ABC}$, is fully anti-
symmetric.

####  2.2.2 Closed string vertex operator

At tree level, the closed string amplitudes are computed as the correlator of
vertex operators on the sphere, or equivalently the complex plane, which is
parametrized by coordinates $z,\bar{z}$.

The integrated vertex operator has the structure

$\mathcal{V}^{A,\ell}(k)=\int d^{2}z\;V^{A,\ell}_{k}(z,\bar{z})=\int
d^{2}z\;\mathscr{V}^{A,\ell}(z,\bar{z})e^{ik\cdot X(z,\bar{z})}~{},$ (29)

while the unintegrated vertex operator takes the form

$\mathcal{V}^{A,\ell}(k)=c(z)\tilde{c}(\bar{z})V^{A,\ell}_{k}(z,\bar{z})=c(z)\tilde{c}(\bar{z})\mathscr{V}^{A,\ell}(z,\bar{z})e^{ik\cdot
X(z,\bar{z})}~{},$ (30)

where $c,\tilde{c}$ are the ghosts and the position $z,\bar{z}$ in the
unintegrated vertex operator is arbitrary. Here $\ell$ is the number of
holomorphic derivatives in $\mathscr{V}^{A,\ell}_{k}$ (which is also the
number of anti-holomorphic derivatives), and $A$ are the extra quantum numbers
labelling the vertex operators, such as the polarization tensors.

The closed string spectra are given by

$-k\cdot
k=M^{2}=\frac{2}{\alpha^{\prime}}\Big{(}N+\tilde{N}-2\frac{D}{24}\Big{)}=\frac{4}{\alpha^{\prime}}\Big{(}N-1\Big{)}~{},$
(31)

where we use the level matching condition $N=\tilde{N}\in\mathbb{Z}_{\geq 0}$,
and in the last equality we set $D=24$ for critical bosonic string.

In order to be BRST invariant, the on-shell vertex operator should have
holomorphic and anti-holomorphic weights 1, namely

$h(V^{A,\ell}_{k})=h(\mathscr{V}^{A,\ell}_{k})+h(e^{ik\cdot
X})=\ell+\frac{\alpha^{\prime}{}}{4}k^{2}=1~{},$ (32)

and the same for $\bar{h}$. Therefore, on-shell momentum should satisfy
$k^{2}=4(1-\ell)/\alpha^{\prime}{}$. Comparing with the mass-shell condition
in (31), we again have $N=\ell$.

At level 0, we have closed string tachyon whose vertex operator is given by

$V^{0}_{k}=e^{ik\cdot X}~{},\qquad M^{2}=-\frac{4}{\alpha^{\prime}{}}~{}.$
(33)

At level 1, we have massless fields whose vertex operators are given by
Polchinski:1998rr

$V^{1}_{k}=e_{\mu\nu}\partial X^{\mu}\bar{\partial}X^{\nu}e^{ik\cdot
X}~{},\qquad M^{2}=0~{}.$ (34)

Depending on the structure of polarization tensor $e_{\mu\nu}$, the vertex
operator can represent different fields, either graviton, or KR 2-form field,
or dilaton. See (14)-(17) for the properties and explicit forms of
polarization tensors corresponding to these different fields.

###  2.3 CFT on the worldsheet and on the celestial sphere

Given the vertex operators, one can then compute the string amplitudes. In
general, the string amplitude is schematically computed by the following
formula Polchinski:1998rr

$\mathcal{A}_{n}(k_{1},k_{2},\cdots,k_{n})\sim\sum_{\text{topologies}}e^{-\lambda\chi}\int_{\mathcal{M}_{g,n}}dm\;\langle\mathcal{V}^{A_{1},\ell_{1}}(k_{1})\mathcal{V}^{A_{2},\ell_{2}}(k_{2})\cdots\mathcal{V}^{A_{n},\ell_{n}}(k_{n})\rangle_{\text{WS}}~{},$
(35)

where we need to sum over all the topologies for the string worldsheet, and
for each topology, we need to incorporate the contribution from various ghosts
properly and integrate over the moduli space of Riemann surface. The world-
sheet correlator is evaluated through the path integral of Polyakov action
with operator insertions. The computation of string amplitude is very hard as
one goes to higher loops, but it simplifies dramatically at tree level. At
tree level, the topology is fixed and there are no moduli to integrate over.

As a consequence, the string amplitude at tree level is given by

$\mathcal{A}_{n}(k_{1},k_{2},\cdots,k_{n})=\langle\mathcal{V}^{A_{1},\ell_{1}}(k_{1})\mathcal{V}^{A_{2},\ell_{2}}(k_{2})\cdots\mathcal{V}^{A_{n},\ell_{n}}(k_{n})\rangle_{\text{WS}}~{}.$
(36)

In the case of open string, the worldsheet is given by the disk, which is
conformally equivalent to the upper half complex plane. The vertex operators
are inserted on the boundary of the disk. In particular, among $n$ vertex
operators, three of them should take the unintegrated form (23) and the rest
in the integrated form (22), in order to soak up the zero modes. For gluon
amplitudes, we should take a trace over the product of Chan-Paton factors
$t^{A}$, which are ordered on the boundary according to their positions. So
different ways of inserting the vertex operators on the disk boundary give
rise to different Chan-Paton factors, and one needs to include all orderings
of insertions. For closed string amplitude, the worldsheet is given by the
sphere, which is conformally equivalent to the whole complex plane. Again to
soak up the zero modes, we should choose three vertex operators in the
unintegrated form (23) and the rest in the integrated form (29).

On the other hand, the celestial amplitudes for massless particles are given
by the Mellin transformation of momentum space amplitudes Cheung:2016iub ;
Pasterski:2016qvg ; Pasterski:2017kqt  111111 We focus on massless fields in
this paper. For massive fields, the Mellin transformation needs to be
modified.

$\mathcal{M}_{n}(x_{1},\cdots,x_{n})=\prod_{j=1}^{n}\int_{0}^{\infty}d\omega_{j}\;\omega_{j}^{\Delta_{j}-1}\mathcal{A}_{n}(\eta_{i}\omega_{i}\hat{k}_{i})~{},\qquad\quad\mathcal{A}_{n}(k_{i})=A_{n}(k_{i})\;\delta^{D+2}(\sum_{i=1}^{n}k_{i})~{}.$
(37)

In contrast to the momentum space amplitude which has manifest translational
invariance, the celestial amplitude is designed to make the Lorentz symmetry
manifest. Indeed, in $D+2$ dimensional Minkowski spacetime we have Lorentz
group $SO(1,D+1)$, which is also the conformal group of CFT in $D$ dimensions.
The place supporting such a conformal group is just the celestial sphere,
which sits at the boundary null infinity of spacetime. The celestial amplitude
defined above can thus be regarded as the correlator of some celestial
operators in a putative celestial conformal field theory living on the
celestial sphere

$\mathcal{M}_{n}(x_{1},\cdots,x_{n})=\langle\mathcal{O}_{\Delta_{1}}(x_{1})\cdots\mathcal{O}_{\Delta_{n}}(x_{n})\rangle_{\text{CS}}~{}.$
(38)

Note that for simplicity of notation we have stripped off all the extra labels
of the operators except for their positions and dimensions. The representation
of bulk scattering amplitude in terms of boundary correlator is just
reminiscent of the holographic principle.

So the string amplitude can be computed in two different ways, either (36) or
(38). To make the connection more precise, we can also define the vertex
operators in the conformal basis through a Mellin transformation

$\mathcal{V}^{A,\ell,\eta}_{\Delta}(x)=\int_{0}^{\infty}d\omega\;\omega^{\Delta-1}\mathcal{V}^{A,\ell}(\eta\omega\hat{k})~{}.$
(39)

We will refer to it as the _conformal vertex operators_ in this paper. Then
the celestial string amplitude is essentially given by the correlator of
conformal vertex operators evaluated in worldsheet CFT

$\mathcal{M}_{n}(x_{1},\cdots,x_{n})=\langle\mathcal{V}_{\Delta_{1}}(x_{1})\cdots\mathcal{V}_{\Delta_{n}}(x_{n})\rangle_{\text{WS}}~{}.$
(40)

The very similar structure between (38) and (40) suggests a map $\mathscr{F}$
from world-sheet CFT (WSCFT) to celestial CFT (CCFT), such that the two
Hilbert spaces are related as follows

$\mathscr{F}:\qquad H_{\text{WSCFT}}\to H_{\text{CCFT}}~{}.$ (41)

In particular, there is a one-to-one map between string conformal vertex
operators and celestial operators

$\mathscr{F}:\qquad\mathcal{V}_{\Delta}\mapsto\mathcal{O}_{\Delta}~{}.$ (42)

Our goal in this paper is to derive the celestial OPEs

$\mathcal{O}_{\Delta_{1}}(x_{1})\mathcal{O}_{\Delta_{2}}(x_{2})\sim\sum_{j}C_{\Delta_{1}\Delta_{2}}^{\Delta_{j}}(x_{1},x_{2})\mathcal{O}_{\Delta_{j}}(x_{2})~{}.$
(43)

Due to the map $\mathscr{F}$, it is sufficient to compute

$\mathcal{V}_{\Delta_{1}}(x_{1})\mathcal{V}_{\Delta_{2}}(x_{2})\sim\sum_{j}C_{\Delta_{1}\Delta_{2}}^{\Delta_{j}}(x_{1},x_{2})\mathcal{V}_{\Delta_{j}}(x_{2})~{}.$
(44)

We would like to have several remarks here. In spite of the similarity between
(38) and (40), the celestial CFT and worldsheet CFT are very different in many
aspects. The worldsheet CFT is always two dimensional, while the dimension of
celestial CFT depends on the bulk spacetime dimension. Moreover, the
worldsheet CFT and thus its correlator (36) are explicit and well-defined,
while for celestial CFT which has many unusual features, our understanding
remains poor and is mostly about symmetry. The map $\mathscr{F}$ suggests that
these two are supposed to be related in some way. In particular, since the
integrated conformal vertex operator $\mathcal{V}_{\Delta}(x)$ arises after
doing the worldsheet integration, it is reasonable to speculate that the
celestial CFT arises from worldsheet CFT by some kind of projection of
worldsheet coordinates and going to the labelling space, namely momentum $k$
or $\Delta,x$ space. On the other hand, the similarity between (38) and (40)
is for tree level amplitude. At loop level, the string amplitude in (35) is
more complicated and only admits perturbative expansion. The CCFT correlator
(38) is however defined formally but exactly. To reconcile the difference, the
celestial CFT is supposed to also admit some expansion, and then one may
compare the two sides order by order. This is indeed the case in AdS/CFT where
the gauge theory in the CFT side admits large-$N$ expansion and can be
compared with the loop expansion of string theory. Understanding these points
may be useful for us to establish a concrete and exact model of celestial
holography in string theory.

##  3 Celestial OPE from worldsheet OPE in bosonic string

In this section, we will discuss the derivation of celestial OPEs from
worldsheet OPEs following the strategy outlined in the introduction. The
string theory we will consider in this section is bosonic strings, either open
or closed. As a result, we are able to derive the gluon OPEs and
graviton/dilaton/KR field OPEs from open and closed strings, respectively. The
mixed OPEs involving both gluon and graviton/dilaton/KR field can be derived
from the open-closed string setup, which will be deferred to appendix B. All
the OPEs involving gluon, graviton/dilaton/KR as well as their mixture will
also be discussed in the heterotic superstring in the next section. In section
5, we will also confirm our celestial OPEs through the collinear factorization
of string amplitude.

###  3.1 OPE in open string

Let us now discuss how to derive the celestial gluon OPEs from the worldsheet
OPEs in open string theory. As we described before, it is sufficient to
compute the OPE (44). For this aim, in principle we need to know
$V_{\Delta_{1}}(x_{1},y_{1})V_{\Delta_{2}}(x_{2},y_{2})$ for arbitrary
$y_{1},y_{2}$, even if they are very far away from each other on the
worldsheet. Here $V_{\Delta}(x,y)$ is essentially the Mellin transformation of
the integrand $V_{k}(y)$ in (22). Since we are only interested in the
collinear limit $x_{1}\to x_{2}$, it is reasonable to speculate that the
celestial OPE is determined by the worldsheet OPE. 121212 To be clear, the
worldsheet OPE refers to the OPE of two operators approaching each other on
the worldsheet, namely $z_{1}\to z_{2}$, while the celestial OPE refers to the
OPE of two celestial operators approaching each other on the celestial sphere,
namely $x_{1}\to x_{2}$. The celestial coincident limit $x_{1}\to x_{2}$ is
equivalent to the collinear limit.  This will be our worldsheet approach to
deriving the celestial OPEs. In particular, the celestial gluon OPE can be
derived by computing 131313 Instead of considering two vertex operators in the
integrated forms (22), one can also choose one in the integrated form (22) and
the other in the unintegrated form (23). Their OPE leads to another
unintegrated vertex operator. The final results for celestial OPEs obtained in
these two ways are the same.

$\displaystyle\mathcal{V}^{A}(p,\zeta)\mathcal{V}^{B}(q,\xi)$ $\displaystyle=$
$\displaystyle\int
dy_{1}dy_{2}\;\theta(y_{1}-y_{2})\;V^{A}_{p,\zeta}(y_{1})V^{B}_{q,\xi}(y_{2})+\int
dy_{1}dy_{2}\;\theta(y_{2}-y_{1})\;V^{B}_{q,\xi}(y_{2})V^{A}_{p,\zeta}(y_{1})\qquad\qquad$
(46) $\displaystyle=$ $\displaystyle\int
dy_{1}dy_{2}\;\theta(y_{1}-y_{2})\;\zeta\cdot\dot{X}e^{ip\cdot
X}(y_{1})\xi\cdot\dot{X}e^{iq\cdot X}(y_{2})t^{A}t^{B}$ $\displaystyle+\int
dy_{1}dy_{2}\;\theta(y_{2}-y_{1})\;\xi\cdot\dot{X}e^{iq\cdot
X}(y_{2})\zeta\cdot\dot{X}e^{ip\cdot X}(y_{1})t^{B}t^{A}~{},$

where we used the explicit form of gluon vertex operators in (27) and
$\zeta,\xi$ are the polarization vectors. We will first compute the worldsheet
OPE of two $V$’s in the integrand; then we will exam the behavior of this
worldsheet OPE in the collinear limit. As we will see, in the collinear limit,
the worldsheet OPE actually localizes to a delta-function, which just gives
another vertex operator. Performing the Mellin transformation then leads us to
the celestial OPEs.

Before going to the details, let us add some comments about the formula above
(46). Since we are interested in the worldsheet OPE, the two vertex operators
should be next to each other and there is no other operator insertion between
them. Nevertheless, we have two different contributions in the formula above,
corresponding to two different orderings of vertex operators on the boundary.
On the other hand, the formula above should be understood in the correlator
where other operator insertions indeed appear; this gives rise to the upper
and lower limits of integration for the integrals. As we will see, this brings
the subtle issue of boundary contact terms.

Now we can start the derivation of celestial OPEs. As the first step, we need
to compute the worldsheet OPE for the integrand in (46). This can be done as
the fields $X$ are essentially free scalars. All the detailed computations of
worldsheet OPEs are given in appendix A. In particular, the worldsheet OPE for
the integrand in (46) is given by (A.1):

$\displaystyle\zeta\cdot\dot{X}e^{ip\cdot X}(y_{1})\xi\cdot\dot{X}e^{iq\cdot
X}(y_{2})$ $\displaystyle\sim$
$\displaystyle-2\alpha^{\prime}y_{12}^{2\alpha^{\prime}p\cdot
q-2}e^{i(p+q)\cdot X}\Bigg{[}\Big{(}\zeta\cdot\xi-2\alpha^{\prime}{}\zeta\cdot
q\;\xi\cdot p\Big{)}$ $\displaystyle+iy_{12}\Big{(}\zeta\cdot
q\;\xi\cdot\dot{X}-\xi\cdot
p\;\zeta\cdot\dot{X}+\big{(}\zeta\cdot\xi-2\alpha^{\prime}{}\zeta\cdot
q\;\xi\cdot p\big{)}p\cdot\dot{X}\Big{)}+\cdots\Bigg{]}(y_{2})~{},$

where we assume $y_{1}>y_{2}$.

We will take $p,q$ on-shell, namely $p^{2}=q^{2}=0$, then the polarization
vectors $\zeta,\xi$ satisfy the properties in (11). We are interested in the
collinear limit where $p,q$ are almost parallel and thus $p\cdot q\to 0$,
implying that $p+q$ is almost on-shell. However, in order to gain information
from the OPE, we can not take the strict collinear limit. Instead, we will
denote $K\equiv p+q$ and choose a nearby null momentum $k=\omega_{k}\hat{k}$
such that $K=k+\epsilon v$, where $v$ is a generic momentum of order one. Then
we have

$K^{2}=2p\cdot q=k^{2}+2\epsilon k\cdot v+\epsilon^{2}v\cdot v=2\epsilon
k\cdot v+\mathscr{O}(\epsilon^{2})~{}.$ (48)

Hence $\epsilon$ has the same order of magnitude with $p\cdot q$ in the
collinear limit. Without loss of generality, we will define
$\epsilon=2\alpha^{\prime}{}p\cdot q$. Then $K=p+q=k+\mathscr{O}(\epsilon)$.
In the following discussions, we will try to simplify (3.1) in such a
collinear limit.

Given the null momentum $k$, we can define the basis for polarization vectors
$\varepsilon_{a}(k)$ (2), which satisfy (10)

$\varepsilon_{a}^{\mu}(k)\varepsilon^{\nu}_{a}(k)=\eta^{\mu\nu}-n^{\mu}\hat{k}^{\nu}-n^{\nu}\hat{k}^{\mu}~{}.$
(49)

Contracting with $p^{\mu}\dot{X}^{\nu}$, we get

$p\cdot\varepsilon_{a}\;\varepsilon_{a}\cdot\dot{X}=p\cdot X-p\cdot
n\;\hat{k}\cdot\dot{X}-\dot{X}\cdot n\;\hat{k}\cdot p~{}.$ (50)

Using $n\cdot\hat{p}=1$ (3) and hence $n\cdot
p=\omega_{p}n\cdot\hat{p}=\omega_{p}$, 141414 We will only consider the
celestial OPEs of celestial operators corresponding to out-going particles in
this paper, so $\eta=1$. the above equation can be written as

$\displaystyle p\cdot\dot{X}$ $\displaystyle=$ $\displaystyle
p\cdot\varepsilon_{a}\;\varepsilon_{a}\cdot\dot{X}+\frac{1}{\omega_{k}}k\cdot
p\;\dot{X}\cdot n+\frac{\omega_{p}}{\omega_{k}}k\cdot\dot{X}$ (51)
$\displaystyle=$ $\displaystyle
p\cdot\varepsilon_{a}\;\varepsilon_{a}\cdot\dot{X}+\frac{1}{\omega_{k}}p\cdot
q\;\dot{X}\cdot
n+\frac{\omega_{p}}{\omega_{k}}k\cdot\dot{X}+\mathscr{O}(\epsilon)~{},$ (52)

where we used $k\cdot p=(p+q+\mathscr{O}(\epsilon))\cdot p=p\cdot
q+\mathscr{O}(\epsilon)$ as $p^{2}=0$.

Similarly, we can contract (49) with $\zeta^{\mu}\dot{X}^{\nu}$, yielding

$\zeta\cdot\varepsilon_{a}\;\varepsilon_{a}\cdot\dot{X}=\zeta\cdot
X-\zeta\cdot n\;\hat{k}\cdot\dot{X}-\dot{X}\cdot
n\;\hat{k}\cdot\zeta=\zeta\cdot X-\frac{1}{\omega_{k}}\dot{X}\cdot
n\;q\cdot\zeta+\mathscr{O}(\epsilon)~{},$ (53)

where we used
$k\cdot\zeta=(p+q+\mathscr{O}(\epsilon))\cdot\zeta=q\cdot\zeta+\mathscr{O}(\epsilon)$
and $\zeta\cdot n=0$ due to (11) and (12) . 151515 Note that $\zeta\cdot n=0$
can be realized by choosing a specific gauge. This is inessential as the
amplitude is gauge invariant or BRST invariant. As a result, we have

$\zeta\cdot\dot{X}\;\xi\cdot
p=\zeta\cdot\varepsilon_{a}\;\varepsilon_{a}\cdot\dot{X}\;\xi\cdot
p+\frac{1}{\omega_{k}}\dot{X}\cdot
n\;q\cdot\zeta\;p\cdot\xi+\mathscr{O}(\epsilon)~{},$ (54)

and similarly

$\xi\cdot\dot{X}\;\zeta\cdot
q=\xi\cdot\varepsilon_{a}\;\varepsilon_{a}\cdot\dot{X}\;\zeta\cdot
q+\frac{1}{\omega_{k}}\dot{X}\cdot
n\;p\cdot\xi\;q\cdot\zeta+\mathscr{O}(\epsilon)~{}.$ (55)

Taking the difference, we find

$\xi\cdot\dot{X}\;\zeta\cdot q-\zeta\cdot\dot{X}\;\xi\cdot
p=\xi\cdot\varepsilon_{a}\;\varepsilon_{a}\cdot\dot{X}\;\zeta\cdot
q-\zeta\cdot\varepsilon_{a}\;\varepsilon_{a}\cdot\dot{X}\;\xi\cdot
p+\mathscr{O}(\epsilon)~{}.$ (56)

With (52) and (56), we can simplify the terms in the square bracket of (3.1)

$\displaystyle\Big{(}\zeta\cdot\xi-2\alpha^{\prime}{}\zeta\cdot q\;\xi\cdot
p\Big{)}+i\,y_{12}\Big{(}\zeta\cdot q\;\xi\cdot\dot{X}-\xi\cdot
p\;\zeta\cdot\dot{X}+\big{(}\zeta\cdot\xi-2\alpha^{\prime}{}\zeta\cdot
q\;\xi\cdot p\big{)}p\cdot\dot{X}\Big{)}\qquad$ $\displaystyle=$
$\displaystyle\zeta\cdot h\cdot\xi+i\,y_{12}\Big{(}\zeta\cdot
q\;\xi\cdot\varepsilon_{a}\;\varepsilon_{a}\cdot\dot{X}-\xi\cdot
p\;\zeta\cdot\varepsilon_{a}\;\varepsilon_{a}\cdot\dot{X}+\zeta\cdot
h\cdot\xi\;p\cdot\varepsilon_{a}\;\varepsilon_{a}\cdot\dot{X}$
$\displaystyle\qquad\qquad\qquad\qquad+\zeta\cdot
h\cdot\xi\big{(}\frac{1}{\omega_{k}}p\cdot q\;\dot{X}\cdot
n+\frac{\omega_{p}}{\omega_{k}}k\cdot\dot{X}\big{)}\Big{)}+\mathscr{O}(\epsilon)~{},$

where we have introduced

$h^{\mu\nu}=\eta^{\mu\nu}-2\alpha^{\prime}q^{\mu}p^{\nu}~{},\qquad\zeta\cdot
h\cdot\xi\equiv\zeta_{\mu}h^{\mu\nu}\xi_{\nu}=\zeta\cdot\xi-2\alpha^{\prime}{}\zeta\cdot
q\;\xi\cdot p~{}.$ (58)

Therefore the worldsheet OPE (3.1) can be written as

$\displaystyle\zeta\cdot\dot{X}e^{ip\cdot X}(y_{1})\xi\cdot\dot{X}e^{iq\cdot
X}(y_{2})$ (59) $\displaystyle\sim$
$\displaystyle-2\alpha^{\prime}y_{12}^{2\alpha^{\prime}p\cdot
q-2}e^{i(p+q)\cdot X}\zeta\cdot h\cdot\xi$ (64)
$\displaystyle-2i\;\alpha^{\prime}y_{12}^{2\alpha^{\prime}p\cdot
q-1}e^{i(p+q)\cdot X}\Big{(}\zeta\cdot q\;\xi\cdot\varepsilon_{c}-\xi\cdot
p\;\zeta\cdot\varepsilon_{c}+\zeta\cdot
h\cdot\xi\;p\cdot\varepsilon_{c}\Big{)}\varepsilon_{c}\cdot\dot{X}$
$\displaystyle-2i\;\alpha^{\prime}\frac{\zeta\cdot
h\cdot\xi}{\omega_{k}}\dot{X}\cdot n\;p\cdot q\;y_{12}^{2\alpha^{\prime}p\cdot
q-1}e^{i(p+q)\cdot X}$ $\displaystyle-2\;\alpha^{\prime}\zeta\cdot
h\cdot\xi\;\frac{\omega_{p}}{\omega_{k}}y_{12}^{2\alpha^{\prime}p\cdot
q-1}\partial e^{i(p+q)\cdot X}$
$\displaystyle+\mathscr{O}(y_{12}^{2\alpha^{\prime}p\cdot
q})+\mathscr{O}(2\alpha^{\prime}p\cdot q)~{},$

where all terms on the right hand side are evaluated at $y_{2}$. This is the
worldsheet OPE in the collinear limit. To obtain the celestial OPE, we use the
following identities to analyze the dominant contributions. For infinitesimal
$\epsilon$ and positive $x>0$, we have 161616 They arise from the generalized
function  $\lim_{\epsilon\to 0}\frac{\epsilon}{2}|x|^{\epsilon-1}=\delta(x)$
as well as its derivatives. For this representation of delta function, see
https://mathworld.wolfram.com/DeltaFunction.html.

$\epsilon x^{\epsilon-1}=2\delta(x)~{},\qquad\epsilon
x^{\epsilon-1-n}=\frac{2(-1)^{n}}{n!}\delta^{n}(x)~{},\qquad\epsilon
x^{\epsilon+n}=0~{},\qquad n=0,1,2,\cdots~{}.$ (65)

We first observe that in the worldsheet OPE, if we only look at the $X$
factors, (64) and (64) are just the integrand in the vertex operators of
tachyon and gluon. Furthermore, if we take the limit $2\alpha^{\prime}p\cdot
q\to 1$ and use identities in (65), (64) gives the factor
$y_{12}^{2\alpha^{\prime}p\cdot
q-2}\propto\delta(y_{12})/(2\alpha^{\prime}{}p\cdot q-1)$ and the pole just
singles out the tachyon. Therefore in the limit $2\alpha^{\prime}p\cdot q\to
1$, the term (64) dominates the OPE. Doing the $y$ integral, (64) indeed gives
the tachyon vertex operator, whose mass-shell condition is just
$(p+q)^{2}=2p\cdot q=1/\alpha^{\prime}{}$.

The more interesting limit for our purpose is the collinear limit
$2\alpha^{\prime}p\cdot q\to 0$. We will argue that in this limit only (64)
survives as a singular term in the OPE at leading order, while the rest are
sub-dominant.

For (64), the factor $p\cdot q\;y_{12}^{2\alpha^{\prime}p\cdot q-1}$ in the
collinear limit $p\cdot q\to 0$ becomes $\delta(y_{12})$ after using (65).
This is of order one, hence we will ignore it.

For (64), the factor $y_{12}^{2\alpha^{\prime}p\cdot q-2}$ just leads to
$\delta^{\prime}(y_{12})/p\cdot q$ following (65). This looks singular in the
collinear limit. For vertex operators, we need to perform the integral as in
(46), which schematically gives

$\int dy_{1}dy_{2}\;\theta(y_{1}-y_{2})\delta^{\prime}(y_{12})e^{i(p+q)\cdot
X}(y_{2})\to\int dy_{1}dy_{2}\;[\delta(y_{1}-y_{2})]^{2}e^{i(p+q)\cdot
X}(y_{2})+\int dy_{2}\;\partial e^{i(p+q)\cdot X}(y_{2})~{}.$ (66)

The $[\delta(y_{1}-y_{2})]^{2}$ factor in the first term arises from the
collision of $y_{1},y_{2}$ and it is a contact type term. The second type of
term with an unrestricted integration range is a total derivative, which
normally does not have contributions in amplitudes due to BRST invariance.
When we insert it in the correlator, there are also other operator insertions,
then the second type of term gives rise to the boundary term when the position
of two operators in OPE coincides with the rest of operator insertions. We
will refer to these types of terms as _boundary contact terms_. Since these
boundary contact terms arise when operator insertions collide and correspond
to very singular configurations, we expect that they would finally cancel out
and play no role in our OPE analysis. On the other hand, (64) corresponds to
tachyon whose mass-squared is negative, the boundary contact terms can thus be
partially understood as the remnant contribution from fields in string theory
with lower masses. Physically, we also expect the collinear limit should kill
the contribution from these fields at leading order.

For (64), $\partial e^{i(p+q)\cdot X}=\partial e^{ik\cdot
X}+\mathscr{O}(\epsilon)=k\cdot\dot{X}e^{ik\cdot X}+\mathscr{O}(\epsilon)$.
Since the polarization is along the momentum direction, this is supposed to a
pure gauge or BRST-exact term and hence does not contribute. Nevertheless, we
can also analyze this term as before. The factor
$y_{12}^{2\alpha^{\prime}p\cdot q-1}$ becomes $\delta(y_{12})/p\cdot q$ in the
collinear limit. And then performing the integral gives

$\int dy_{1}dy_{2}\;\theta(y_{1}-y_{2})\delta(y_{12})\partial e^{i(p+q)\cdot
X}(y_{2})\to\int dy_{2}\;\partial e^{i(p+q)\cdot X}(y_{2})~{},$ (67)

which takes the same form we encountered in (66).

Actually we can combine (64) and (64) together and get

$\displaystyle-2\alpha^{\prime}y_{12}^{2\alpha^{\prime}p\cdot
q-2}e^{i(p+q)\cdot X}\zeta\cdot h\cdot\xi-2\;\alpha^{\prime}\zeta\cdot
h\cdot\xi\;\frac{\omega_{p}}{\omega_{k}}y_{12}^{2\alpha^{\prime}p\cdot
q-1}\partial e^{i(p+q)\cdot X}$ $\displaystyle=$
$\displaystyle-2\alpha^{\prime}y_{12}^{2\alpha^{\prime}p\cdot
q-2}e^{i(p+q)\cdot X}\zeta\cdot
h\cdot\xi\frac{2\omega_{p}\alpha^{\prime}p\cdot
q+\omega_{k}-\omega_{p}}{\omega_{k}}-2\;\alpha^{\prime}\zeta\cdot
h\cdot\xi\;\frac{\omega_{p}}{\omega_{k}}\partial\Big{[}y_{12}^{2\alpha^{\prime}p\cdot
q-1}e^{i(p+q)\cdot X}\Big{]}~{},$

where the first term is the same as (64) up to a constant, while the second
term is a total derivative and again would lead to at most boundary contact
terms.

Therefore, up to the subtle boundary contact terms, all terms except for (64)
in the world-sheet OPE do not give rise to singular terms in the collinear
limit $p\cdot q\to 0$. So we just need to focus on (64) .

To proceed, we choose the polarization vectors as the basis defined in (2)
$\zeta=\varepsilon_{a}(p),\;\xi=\varepsilon_{b}(q)$, and furthermore use the
notation
$p=\omega_{p}\hat{p}(x_{1}),\;q=\omega_{q}\hat{q}(x_{2}),\;k=\omega_{k}\hat{k}(x_{3})$
where $x_{1,2,3}$ are the coordinates on the celestial sphere. Then the world-
sheet OPE (3.1) (64) becomes

$\displaystyle\varepsilon_{a}(p)\cdot\dot{X}e^{ip\cdot
X}(y_{1})\varepsilon_{b}(q)\cdot\dot{X}e^{iq\cdot X}(y_{2})$
$\displaystyle\sim$
$\displaystyle-2i\;\alpha^{\prime}y_{12}^{2\alpha^{\prime}p\cdot
q-1}e^{i(p+q)\cdot X}\Big{(}\varepsilon_{a}(p)\cdot
q\;\varepsilon_{b}(q)\cdot\varepsilon_{c}\;-\varepsilon_{b}(q)\cdot
p\;\varepsilon_{a}(p)\cdot\varepsilon_{c}\;+\varepsilon_{a}(p)\cdot
h\cdot\varepsilon_{b}(q)\;p\cdot\varepsilon_{c}\;\Big{)}\varepsilon_{c}\cdot\dot{X}~{}.$

The next step is to rewrite all the terms on the right hand side in terms of
celestial variables $\omega_{i},x_{i}$. Using identities in (4),

$\hat{k}(x_{i})\cdot\hat{k}(x_{j})\equiv\hat{k}^{\mu}_{i}\cdot\hat{k}^{\mu}_{j}=-\frac{1}{2}(x_{ij})^{2}~{},\qquad\hat{k}_{i}\cdot\varepsilon_{a}(x_{j})=x_{ij}^{a}~{},\qquad\varepsilon_{a}(x_{i})\cdot\varepsilon_{b}(x_{j})=\delta^{ab}~{},$
(70)

we find

$\varepsilon_{a}(p)\cdot
h\cdot\varepsilon_{b}(q)=\varepsilon_{a}(p)\cdot\varepsilon_{b}(q)-2\alpha^{\prime}{}\varepsilon_{a}(p)\cdot
q\;\varepsilon_{b}(q)\cdot
p=\delta^{ab}+2\alpha^{\prime}{}\omega_{p}\omega_{q}x_{12}^{a}x_{12}^{b}~{},$
(71)

and

$\varepsilon_{a}(p)\cdot
q\;\varepsilon_{b}(q)\cdot\varepsilon_{c}=-\omega_{q}x_{12}^{a}\delta^{bc}~{},\qquad\varepsilon_{b}(q)\cdot
p\;\varepsilon_{a}(p)\cdot\varepsilon_{c}=\omega_{p}x_{12}^{b}\delta^{ac}~{}.$
(72)

To simplify $p\cdot\varepsilon_{c}$, we need to use the momentum conservation.
Using (1), the momentum conservation $K=p+q=k+\epsilon v$ can be written as

$\omega_{p}\Big{(}\frac{1+(x_{1})^{2}}{2},x_{1}^{a},\frac{1-(x_{1})^{2}}{2}\Big{)}+\omega_{q}\Big{(}\frac{1+(x_{2})^{2}}{2},x_{2}^{a},\frac{1-(x_{2})^{2}}{2}\Big{)}=\omega_{k}\Big{(}\frac{1+(x_{3})^{2}}{2},x_{3}^{a},\frac{1-(x_{3})^{2}}{2}\Big{)}+\epsilon
v~{},$ (73)

where the infinitesimal quantity $\epsilon$ is now given by

$\epsilon=2\alpha^{\prime}{}p\cdot
q=-\alpha^{\prime}{}\omega_{p}\omega_{q}(x_{12})^{2}~{}.$ (74)

From (73), it is easy to see that

$\omega_{k}=\omega_{p}+\omega_{q}+\mathscr{O}(\epsilon)~{},\qquad\omega_{k}x_{3}^{a}=\omega_{p}x_{1}^{a}+\omega_{q}x_{2}^{a}+\mathscr{O}(\epsilon)~{}.\qquad$
(75)

As a result, we have

$x_{13}^{a}=\frac{\omega_{q}}{\omega_{p}+\omega_{q}}x_{12}^{a}+\mathscr{O}(\epsilon)~{},\qquad
x_{32}^{a}=\frac{\omega_{p}}{\omega_{p}+\omega_{q}}x_{12}^{a}+\mathscr{O}(\epsilon)~{},$
(76)

and furthermore

$p\cdot\varepsilon_{c}=\omega_{p}\hat{p}\cdot\varepsilon_{c}(k)=\omega_{p}x_{13}^{c}=\frac{\omega_{p}\omega_{q}}{\omega_{p}+\omega_{q}}x_{12}^{c}+\mathscr{O}(\epsilon)~{}.$
(77)

With these identities (71) (72) (77), the OPE (3.1) simplifies to

$\displaystyle\varepsilon_{a}(p)\cdot\dot{X}e^{ip\cdot
X}(y_{1})\varepsilon_{b}(q)\cdot\dot{X}e^{iq\cdot X}(y_{2})$
$\displaystyle\sim$ $\displaystyle i\;\epsilon
y_{12}^{\epsilon-1}\frac{\omega_{p}x_{12}^{b}\delta^{ac}+\omega_{q}x_{12}^{a}\delta^{bc}-\frac{\omega_{p}\omega_{q}}{\omega_{p}+\omega_{q}}x_{12}^{c}\delta_{ab}-2\alpha^{\prime}{}\frac{\omega_{p}^{2}\omega_{q}^{2}}{\omega_{p}+\omega_{q}}x_{12}^{a}x_{12}^{b}x_{12}^{c}}{p\cdot
q}\varepsilon_{c}\cdot\dot{X}\;e^{i(p+q)\cdot X}+\mathscr{O}(\epsilon)$
$\displaystyle\sim$
$\displaystyle-4i\;\delta(y_{12})\;I_{c}\Big{(}p,\varepsilon_{a}(p);q,\varepsilon_{b}(q);\alpha^{\prime}{}\Big{)}\;\varepsilon_{c}\cdot\dot{X}\;e^{i(p+q)\cdot
X}+\mathscr{O}(\epsilon)~{},$

where used (65) and have defined

$\displaystyle
I_{c}\Big{(}p,\varepsilon_{a}(p);q,\varepsilon_{b}(q);\alpha^{\prime}{}\Big{)}$
$\displaystyle=$ $\displaystyle\frac{\varepsilon_{a}(p)\cdot
q\;\varepsilon_{b}(q)\cdot\varepsilon_{c}\;-\varepsilon_{b}(q)\cdot
p\;\varepsilon_{a}(p)\cdot\varepsilon_{c}\;+\varepsilon_{a}(p)\cdot
h\cdot\varepsilon_{b}(q)\;p\cdot\varepsilon_{c}}{2p\cdot q}\qquad\qquad$ (79)
$\displaystyle=$
$\displaystyle\frac{\omega_{p}x_{12}^{b}\delta^{ac}+\omega_{q}x_{12}^{a}\delta^{bc}-\frac{\omega_{p}\omega_{q}}{\omega_{p}+\omega_{q}}x_{12}^{c}\delta^{ab}-2\alpha^{\prime}{}\frac{\omega_{p}^{2}\omega_{q}^{2}}{\omega_{p}+\omega_{q}}x_{12}^{a}x_{12}^{b}x_{12}^{c}}{\omega_{p}\omega_{q}(x_{12})^{2}}$
(80) $\displaystyle=$ $\displaystyle-
I_{c}\Big{(}q,\varepsilon_{b}(q);p,\varepsilon_{a}(p);\alpha^{\prime}{}\Big{)}~{},$
(81)

which is anti-symmetric in $p$ and $q$, as shown above.

Doing the integral as in (46), we thus get

$\displaystyle\mathcal{V}_{a}^{A}(p)\mathcal{V}_{b}^{B}(q)$ $\displaystyle=$
$\displaystyle\int
dy_{1}dy_{2}\;\theta(y_{1}-y_{2})\;\Big{(}-4i\,\delta(y_{12})I_{c}(p,\varepsilon_{a}(p);q,\varepsilon_{b}(q);\alpha^{\prime}{})\Big{)}\varepsilon_{c}\cdot\dot{X}\;e^{i(p+q)\cdot
X}(y_{2})t^{A}t^{B}$ $\displaystyle+\int
dy_{1}dy_{2}\;\theta(y_{2}-y_{1})\;\Big{(}-4i\,\delta(y_{21})I_{c}(q,\varepsilon_{b}(q);p,\varepsilon_{a}(p);\alpha^{\prime}{})\Big{)}\varepsilon_{c}\cdot\dot{X}\;e^{i(p+q)\cdot
X}(y_{1})t^{B}t^{A}$ $\displaystyle\propto$ $\displaystyle
f^{ABC}I_{c}\Big{(}p,\varepsilon_{a}(p);q,\varepsilon_{b}(q);\alpha^{\prime}{}\Big{)}\times\int
dy_{2}\;\varepsilon_{c}\cdot\dot{X}\;e^{i(p+q)\cdot X}(y_{2})t^{C}$
$\displaystyle\propto$ $\displaystyle
f^{ABC}I_{c}\Big{(}p,\varepsilon_{a}(p);q,\varepsilon_{b}(q);\alpha^{\prime}{}\Big{)}\times\mathcal{V}_{c}^{C}(p+q)~{},$

where we used the anti-symmetric property of $I_{c}$ in (81) and the notation
$\mathcal{V}_{a}(p)=\mathcal{V}(p,\varepsilon_{a}(p))$,
$\mathcal{V}_{b}(q)=\mathcal{V}(q,\varepsilon_{b}(q))$ and so on. We also used
the integral

$\int dy_{1}dy_{2}\;\theta(y_{1}-y_{2})\delta(y_{12})f(y_{2})=\int
dy_{2}\;\theta(0)f(y_{2})=\frac{1}{2}\int dy_{2}\;f(y_{2})~{},$ (82)

with $\theta(0)=\frac{1}{2}$.

Finally, we need to perform the Mellin transformation (39) in order to go to
the conformal basis. Using the following integral,

$\displaystyle\int_{0}^{\infty}d\omega_{1}\;\omega_{1}^{\Delta_{1}-1}\int_{0}^{\infty}d\omega_{2}\;\omega_{2}^{\Delta_{2}-1}\;\omega_{1}^{\alpha}\omega_{2}^{\beta}(\omega_{1}+\omega_{2})^{\gamma}\;F(\omega_{1}+\omega_{2})$
$\displaystyle=$ $\displaystyle
B\Big{(}\Delta_{1}+\alpha,\Delta_{2}+\beta\Big{)}\int_{0}^{\infty}d\omega\;\omega^{\Delta_{P}-1}\;F(\omega)~{},\qquad\qquad\Delta_{P}=\Delta_{1}+\Delta_{2}+\alpha+\beta+\gamma~{},\qquad$

we obtain the final result for celestial gluon OPE

$\displaystyle\mathcal{V}^{A}_{\Delta_{1},a}(x_{1})\mathcal{V}^{B}_{\Delta_{2},b}(x_{2})$
$\displaystyle\sim$ $\displaystyle
f^{ABC}\frac{x_{12}^{a}\delta^{bc}B(\Delta_{1}-1,\Delta_{2})+x_{12}^{b}\delta^{ac}B(\Delta_{1},\Delta_{2}-1)-x_{12}^{c}\delta^{ab}B(\Delta_{1},\Delta_{2})}{(x_{12})^{2}}\mathcal{V}^{C}_{\Delta_{1}+\Delta_{2}-1,\;c}(x_{2})$
$\displaystyle-2\alpha^{\prime}{}f^{ABC}\frac{x_{12}^{a}x_{12}^{b}x_{12}^{c}B(\Delta_{1}+1,\Delta_{2}+1)}{(x_{12})^{2}}\mathcal{V}^{C}_{\Delta_{1}+\Delta_{2}+1,\;c}(x_{2})~{}.$

####  3.1.1 General structure of worldsheet OPE

Based on our previous OPE analysis of two gluon vertex operators, we would
like now to discuss the general structure of worldsheet OPE. From (3.1) and
the derivation above, one expects that in general the world-sheet OPE takes
the form

$\mathscr{V}^{A,\ell_{1}}e^{ip\cdot
X}(y_{1})\mathscr{V}^{B,\ell_{2}}e^{iq\cdot
X}(y_{2})\sim\sum_{\ell=0}^{\infty}y_{12}^{2\alpha^{\prime}p\cdot
q+\ell-\ell_{1}-\ell_{2}}F^{A,\ell_{1};B,\ell_{2}}_{C,\ell}(p,q)\mathscr{V}^{C,\ell}e^{i(p+q)\cdot
X}(y_{2})~{},$ (85)

where

$\mathscr{V}^{A,\ell}e^{ik\cdot
X}\sim\prod_{i}\partial^{\ell_{i}}X^{\mu_{i}}e^{ik\cdot
X},\qquad\ell=\sum_{i}\ell_{i}:\qquad
h=\alpha^{\prime}k^{2}+\ell\xrightarrow{\text{on-
shell}}\ell-\alpha^{\prime}M^{2}=1~{}.$ (86)

Note that in (85) we did not impose the on-shell condition on the momenta and
it holds even off-shell.

Now we take $p,q$ on-shell, namely setting
$p^{2}=-m_{1}^{2}=(1-\ell_{1})/\alpha^{\prime}$,
$q^{2}=-m_{2}^{2}=(1-\ell_{2})/\alpha^{\prime}$, then we find

$2\alpha^{\prime}p\cdot
q+\ell-\ell_{1}-\ell_{2}=\alpha^{\prime}(p+q)^{2}-2+\ell~{}.$ (87)

For $p+q$, we take it slightly off-shell, namely
$(p+q)^{2}=-m_{3}^{2}+\epsilon/\alpha^{\prime}=(1-\ell_{3})/\alpha^{\prime}+\epsilon/\alpha^{\prime}$,
then

$2\alpha^{\prime}p\cdot
q+\ell-\ell_{1}-\ell_{2}=(1-\ell_{3})+\epsilon-2+\ell=\epsilon+\ell-\ell_{3}-1~{}.$
(88)

Therefore (85) becomes

$\displaystyle\mathscr{V}^{A,\ell_{1}}e^{ip\cdot
X}\mathscr{V}^{B,\ell_{2}}e^{iq\cdot X}$ $\displaystyle=$
$\displaystyle\sum_{\ell=0}^{\infty}y_{12}^{\epsilon+\ell-\ell_{3}-1}F^{A,\ell_{1};B,\ell_{2}}_{C,\ell}(p,q)\mathscr{V}^{C,\ell}e^{i(p+q)\cdot
X}(y_{2})$ (89) $\displaystyle\xrightarrow{\epsilon\to 0}$
$\displaystyle\sum_{\ell=0}^{\ell_{3}}\frac{1}{\epsilon}\delta^{\ell_{3}-\ell}(y_{12})F^{A,\ell_{1};B,\ell_{2}}_{C,\ell}(p,q)\mathscr{V}^{C,\ell}e^{i(p+q)\cdot
X}(y_{2})~{}.$ (90)

where we used (65) and ignored some unimportant factors.

After doing the worldsheet integral, we then expect to find

$(p+q)^{2}\to(1-\ell_{3})/\alpha^{\prime}{}:\qquad\mathcal{V}^{A,\ell_{1}}_{p}\mathcal{V}^{B,\ell_{2}}_{q}\sim\frac{1}{(p+q)^{2}-(1-\ell_{3})/\alpha^{\prime}{}}F^{A,\ell_{1};B,\ell_{2}}_{C,\ell_{3}}\mathcal{V}^{C,\ell_{3}}_{p+q}+\cdots~{},$
(91)

where the dots are boundary contact terms of the form (66) arising from
$\ell=0,\cdots,\ell_{3}-1$ as well some terms in $\ell=\ell_{3}$. The
prefactor on the right hand side is just the propagator
$1/((p+q)^{2}+m_{3}^{2})$.

###  3.2 OPE in closed string

Now we switch to the closed string. As before, we need to first compute

$\displaystyle\mathcal{V}(p,e_{1})\mathcal{V}(q,e_{2})$ $\displaystyle=$
$\displaystyle\int
d^{2}z_{1}d^{2}z_{2}\;V_{p,e_{1}}(z_{1},\bar{z}_{1})V_{q,e_{2}}(z_{2},\bar{z}_{2})$
(92) $\displaystyle=$ $\displaystyle\int
d^{2}z_{1}d^{2}z_{2}\;e_{1\mu\bar{\mu}}\partial
X^{\mu}\bar{\partial}X^{\bar{\mu}}e^{ip\cdot
X}(z_{1},{\bar{z}}_{1})\;e_{2\nu\bar{\nu}}\partial
X^{\mu}\bar{\partial}X^{\bar{\nu}}e^{ip\cdot X}(z_{2},{\bar{z}}_{2})~{},$ (93)

and then perform the Mellin transformation to obtain the celestial OPE.

As the starting point, we need to know the world-sheet OPE of two operators in
the integrand of (93), which is computed in (A.2):

$\displaystyle\partial X^{\mu}\bar{\partial}X^{\bar{\mu}}e^{ip\cdot
X}(z_{1},{\bar{z}}_{1})\partial X^{\mu}\bar{\partial}X^{\bar{\nu}}e^{ip\cdot
X}(z_{2},{\bar{z}}_{2})$ (94) $\displaystyle\sim$
$\displaystyle\frac{\alpha^{\prime}{}^{2}}{4}|z_{12}|^{\alpha^{\prime}p\cdot
q-4}e^{i(p+q)\cdot X}$ (97)
$\displaystyle\times\Bigg{[}\Big{(}\eta^{\mu\nu}-\frac{1}{2}\alpha^{\prime}{}q^{\mu}\;p^{\nu}\Big{)}+i\;z_{12}\Big{(}q^{\mu}\;\partial
X^{\nu}-p^{\nu}\;\partial
X^{\mu}+\big{(}\eta^{\mu\nu}-\frac{1}{2}\alpha^{\prime}{}q^{\mu}\;q^{\nu}\big{)}p\cdot\partial
X\Big{)}+\cdots\Bigg{]}$
$\displaystyle\times\Bigg{[}\Big{(}\eta^{\bar{\mu}\bar{\nu}}-\frac{1}{2}\alpha^{\prime}{}q^{\bar{\mu}}\;p^{\bar{\nu}}\Big{)}+i\;{\bar{z}}_{12}\Big{(}q^{\bar{\mu}}\;\partial
X^{\bar{\nu}}-p^{\bar{\nu}}\;\bar{\partial}X^{\bar{\mu}}+\big{(}\eta^{\bar{\mu}\bar{\nu}}-\frac{1}{2}\alpha^{\prime}{}q^{\bar{\mu}}\;q^{\nu}\big{)}p\cdot\bar{\partial}X\Big{)}+\cdots\Bigg{]}(z_{2},\bar{z}_{2})~{}.\qquad\qquad$

This OPE is essentially the “square” of open string world-sheet OPE (3.1), up
to the rescaling of $\alpha^{\prime}{}$. In particular, the terms proportional
to $z_{12},{\bar{z}}_{12}$ can be simplified as in the open string case; in
each bracket, we can get expressions which are similar to (59).

Although many cross product terms from the left and right moving sectors in
(94) would be generated, we only need to consider the diagonal terms, namely
those terms which only depend on the modulus $|z_{12}|$. The non-diagonal
terms, like those with factor $z_{12}$, would become zero after integrating
along the angular direction of $z_{12}$.

Before simplifying this OPE, let us first introduce some useful identities. In
the open string case, (65) plays an important role. Now we would like to find
similar identities for the closed string. We first parametrize the complex
coordinate as

$z=u+{i}v=\varrho e^{i\vartheta}~{},\qquad{\bar{z}}=u-{i}v=\varrho
e^{-i\vartheta}~{},\qquad|z|=\varrho~{}.$ (98)

Then we have the following formula 171717 In this convention, we thus have
$\partial_{\bar{z}}\frac{1}{z}=\pi\delta^{(2)}(z)~{},\quad\int
d^{2}z\;f(z)\delta^{(2)}(z)=f(0)~{}.$ For the representation of delta function
in polar coordinates, see https://mathworld.wolfram.com/DeltaFunction.html.

$\delta^{(2)}(z)=\delta(u)\delta(v)=\frac{\delta(\varrho)}{\pi\varrho}=\frac{1}{2\pi}\epsilon\varrho^{\epsilon-2}=\frac{1}{2\pi}\epsilon|z|^{\epsilon-2}~{},\qquad
d^{2}z=dudv=\varrho d\vartheta d\varrho~{},$ (99)

where $\epsilon$ is again infinitesimal and we used the identity in (65).
Further taking derivatives of the delta-function gives

$\partial\delta^{(2)}(z)=-\frac{1}{2\pi}\epsilon|z|^{\epsilon-2}z^{-1}~{},\quad\bar{\partial}\delta^{(2)}(z)=-\frac{1}{2\pi}\epsilon|z|^{\epsilon-2}\bar{z}^{-1}~{},\quad\partial\bar{\partial}\delta^{(2)}(z)=\frac{1}{2\pi}\epsilon|z|^{\epsilon-4}~{},\quad\cdots~{}.$
(100)

We are interested in the behavior of OPE (94) in the collinear limit $p\cdot
q\to 0$. As in the open string case, one can show that most terms in the OPE
are not relevant in the collinear limit; they are either regular in the limit
$p\cdot q\to 0$, or only contribute boundary contact terms. For example, the
tachyon arising from the first term in two square brackets of (94) takes the
form

$|z_{12}|^{\alpha^{\prime}p\cdot q-4}e^{i(p+q)\cdot
X}\to\frac{\partial\bar{\partial}\delta^{(2)}(z_{12})}{p\cdot q}e^{i(p+q)\cdot
X}~{},$ (101)

where we used identity in (100). Following (93), we need to integrate over
$z_{1},z_{2}$. This type of total derivative can contribute at most boundary
terms after doing the integral. Other terms in the OPE (94) can be analyzed
similarly. In particular, one can rewrite terms in each square bracket in (94)
and get expressions similar to the open string case (59). Then one can show
that most terms are not important for our purpose.

After doing simplification, the only relevant terms in OPE (94) in the
collinear limit $p\cdot q\to 0$ are given by

$\displaystyle\partial X^{\mu}\bar{\partial}X^{\bar{\mu}}e^{ip\cdot
X}(z_{1},{\bar{z}}_{1})\partial X^{\nu}\bar{\partial}X^{\bar{\nu}}e^{ip\cdot
X}(z_{2},{\bar{z}}_{2})$ (102) $\displaystyle\sim$
$\displaystyle-\frac{\alpha^{\prime}{}^{2}}{4}|z_{12}|^{\alpha^{\prime}p\cdot
q-2}e^{i(p+q)\cdot
X}\times\Big{(}q^{\mu}\;\varepsilon_{c}^{\nu}-p^{\nu}\;\varepsilon_{c}^{\mu}+\big{(}\eta^{\mu\nu}-\frac{1}{2}\alpha^{\prime}{}q^{\mu}\;q^{\nu}\big{)}p\cdot\varepsilon_{c}\Big{)}$
(104)
$\displaystyle\times\Big{(}q^{\bar{\mu}}\;\varepsilon_{\tilde{c}}^{\bar{\nu}}-p^{\bar{\nu}}\;\varepsilon_{\tilde{c}}^{\bar{\mu}}+\big{(}\eta^{\bar{\mu}\bar{\nu}}-\frac{1}{2}\alpha^{\prime}{}q^{\bar{\mu}}\;q^{\nu}\big{)}p\cdot\varepsilon_{\tilde{c}}\Big{)}\times\varepsilon_{c}\cdot\partial
X\;\varepsilon_{\tilde{c}}\cdot\bar{\partial}X~{}.$

which is just the “square” of (64).

To simplify further, we choose the polarization tensors as the basis
constructed in (5):

$\displaystyle\varepsilon^{\mu\bar{\mu}}_{a\tilde{a}}(p)=\varepsilon^{\mu}_{a}(p)\varepsilon^{\bar{\mu}}_{\tilde{a}}(p)~{},\qquad\varepsilon^{\nu\bar{\nu}}_{b\tilde{b}}(q)=\varepsilon^{\nu}_{b}(q)\varepsilon^{\bar{\nu}}_{\tilde{b}}(q)~{}.$
(105)

Any polarization for graviton/dilaton/KR field can be expanded in this basis
using (15) (17).

Contracting the OPE (102) with (105) and taking the limit $p\cdot q\to 0$
gives

$\displaystyle\varepsilon_{a\tilde{a}\;\mu\bar{\mu}}(p)\partial
X^{\mu}\bar{\partial}X^{\bar{\mu}}e^{ip\cdot
X}(z_{1},{\bar{z}}_{1})\varepsilon_{b\tilde{b}\;\nu\bar{\nu}}(q)\partial
X^{\nu}\bar{\partial}X^{\bar{\nu}}e^{ip\cdot X}(z_{2},{\bar{z}}_{2})$ (107)
$\displaystyle\sim$
$\displaystyle-\frac{\alpha^{\prime}{}^{2}}{4}2\pi\frac{1}{\alpha^{\prime}p\cdot
q}\delta^{2}(z_{12})e^{i(p+q)\cdot
X}\times\Big{(}q^{\mu}\;\varepsilon_{c}^{\nu}-p^{\nu}\;\varepsilon_{c}^{\mu}+\big{(}\eta^{\mu\nu}-\frac{1}{2}\alpha^{\prime}{}q^{\mu}\;q^{\nu}\big{)}p\cdot\varepsilon_{c}\Big{)}\varepsilon^{\mu}_{a}\varepsilon^{\nu}_{b}$
$\displaystyle\qquad\qquad\times\Big{(}q^{\bar{\mu}}\;\varepsilon_{\tilde{c}}^{\bar{\nu}}-p^{\bar{\nu}}\;\varepsilon_{\tilde{c}}^{\bar{\mu}}+\big{(}\eta^{\bar{\mu}\bar{\nu}}-\frac{1}{2}\alpha^{\prime}{}q^{\bar{\mu}}\;q^{\nu}\big{)}p\cdot\varepsilon_{\tilde{c}}\Big{)}\varepsilon^{\bar{\mu}}_{\tilde{a}}\varepsilon^{\bar{\nu}}_{\tilde{b}}\;\varepsilon_{c}\cdot\partial
X\;\varepsilon_{\tilde{c}}\cdot\bar{\partial}X$ $\displaystyle\sim$
$\displaystyle\pi\alpha^{\prime}{}\;\delta^{2}(z_{12})\;\mathcal{I}_{c{\tilde{c}}}\Big{(}p,\varepsilon_{a\tilde{a}}(p);q,\varepsilon_{b\tilde{b}}(q);\alpha^{\prime}{},\alpha^{\prime}{}\Big{)}\;\varepsilon_{c\tilde{c}\;\sigma\bar{\sigma}}\partial
X^{\sigma}\bar{\partial}X^{\bar{\sigma}}~{},$ (108)

where we used the identity in (99) and defined

$\displaystyle\mathcal{I}_{c{\tilde{c}}}\Big{(}p,\varepsilon_{a\tilde{a}}(p);q,\varepsilon_{b\tilde{b}}(q);\alpha,\beta\Big{)}$
$\displaystyle=$ $\displaystyle\frac{1}{-2p\cdot
q}\Big{[}\varepsilon_{a}(p)\cdot
q\;\varepsilon_{b}(q)\cdot\varepsilon_{c}\;-\varepsilon_{b}(q)\cdot
p\;\varepsilon_{a}(p)\cdot\varepsilon_{c}\;+\Big{(}\varepsilon_{a}(p)\cdot\varepsilon_{b}(q)-\frac{\alpha}{2}\varepsilon_{a}(p)\cdot
q\;\varepsilon_{b}(q)\cdot p\Big{)}p\cdot\varepsilon_{c}\Big{]}\qquad$
$\displaystyle\qquad\times\Big{[}\varepsilon_{\tilde{a}}(p)\cdot
q\;\varepsilon_{\tilde{b}}(q)\cdot\varepsilon_{\tilde{c}}\;-\varepsilon_{\tilde{b}}(q)\cdot
p\;\varepsilon_{\tilde{a}}(p)\cdot\varepsilon_{\tilde{c}}\;+\Big{(}\varepsilon_{\tilde{a}}(p)\cdot\varepsilon_{\tilde{b}}(q)-\frac{\beta}{2}\varepsilon_{\tilde{a}}(p)\cdot
q\;\varepsilon_{\tilde{b}}(q)\cdot
p\Big{)}p\cdot\varepsilon_{\tilde{c}}\Big{]}$ $\displaystyle=$
$\displaystyle\omega_{p}\omega_{q}(x_{12})^{2}\;I_{c}\Big{(}p,\varepsilon_{a}(p);q,\varepsilon_{b}(q);\frac{1}{4}\alpha\Big{)}\;I_{\tilde{c}}\Big{(}p,\varepsilon_{\tilde{a}}(p);q,\varepsilon_{\tilde{b}}(q);\frac{1}{4}\beta\Big{)}~{}.$
(110)

The expression for
$I_{c}\Big{(}p,\varepsilon_{a}(p);q,\varepsilon_{b}(q);\alpha^{\prime}{}\Big{)}$
is given in (79).

Following (93) and using (108), we can now compute the OPE between two vertex
operators of massless fields in closed string theory:

$\displaystyle\mathcal{V}_{a{\tilde{a}}}(p)\mathcal{V}_{b{\tilde{b}}}(q)$
$\displaystyle=$ $\displaystyle\int
d^{2}z_{1}d^{2}z_{2}\;\varepsilon_{a\tilde{a}\;\mu\bar{\mu}}(p)\partial
X^{\mu}\bar{\partial}X^{\bar{\mu}}e^{ip\cdot
X}(z_{1},{\bar{z}}_{1})\varepsilon_{b\tilde{b}\;\nu\bar{\nu}}(q)\partial
X^{\nu}\bar{\partial}X^{\bar{\nu}}e^{ip\cdot X}(z_{2},{\bar{z}}_{2})$ (111)
$\displaystyle=$ $\displaystyle\int
d^{2}z_{1}d^{2}z_{2}\;\pi\alpha^{\prime}{}\;\delta^{2}(z_{12})\;\mathcal{I}_{c{\tilde{c}}}\Big{(}p,\varepsilon_{a\tilde{a}}(p);q,\varepsilon_{b\tilde{b}}(q);\alpha^{\prime}{},\alpha^{\prime}{}\Big{)}\;\varepsilon_{c\tilde{c}\;\sigma\bar{\sigma}}\partial
X^{\sigma}\bar{\partial}X^{\bar{\sigma}}$ $\displaystyle\propto$
$\displaystyle\mathcal{I}_{c{\tilde{c}}}\Big{(}p,\varepsilon_{a\tilde{a}}(p);q,\varepsilon_{b\tilde{b}}(q);\alpha^{\prime}{},\alpha^{\prime}{}\Big{)}\;\mathcal{V}_{c{\tilde{c}}}(p+q)~{},$
(112)

where we used the notation
$\mathcal{V}_{a{\tilde{a}}}(p)\equiv\mathcal{V}(p,\varepsilon_{a\tilde{a}}(p))$
and similarly for $q$, $p+q$.

Finally, we need to go to the conformal basis of the vertex operators (39).
Using the formula (3.1) and performing the Mellin transformation, we finally
obtain the celestial OPE for graviton/dilaton/KR field in closed string theory

$\mathcal{V}_{\Delta_{1},a{\tilde{a}}}(x_{1})\mathcal{V}_{\Delta_{2},b{\tilde{b}}}(x_{2})\sim\mathcal{K}~{},$
(113)

where 181818 Note that the exchange in the parentheses is the simultaneous
exchange of all items.

$\displaystyle(x_{12})^{2}\mathcal{K}$ $\displaystyle=$
$\displaystyle\Bigg{[}\Big{(}\delta^{bc}\delta^{{\tilde{b}}{\tilde{c}}}x_{12}^{a}x_{12}^{\tilde{a}}B(\Delta_{1}-1,\Delta_{2}+1)-\delta^{ac}\delta^{{\tilde{a}}{\tilde{b}}}x_{12}^{b}x_{12}^{\tilde{c}}B(\Delta_{1}+1,\Delta_{2})-\delta^{ab}\delta^{{\tilde{b}}{\tilde{c}}}x_{12}^{c}x_{12}^{\tilde{a}}B(\Delta_{1},\Delta_{2}+1)$
(116)
$\displaystyle\qquad+\frac{1}{2}\delta^{ab}\delta^{{\tilde{a}}{\tilde{b}}}x_{12}^{c}x_{12}^{\tilde{c}}B(\Delta_{1}+1,\Delta_{2}+1)+\delta^{bc}\delta^{{\tilde{a}}{\tilde{c}}}x_{12}^{a}x_{12}^{\tilde{b}}B(\Delta_{1},\Delta_{2})\Big{)}$
$\displaystyle\;+\Big{(}a\leftrightarrow
b,\,{\tilde{a}}\leftrightarrow{\tilde{b}},\;\Delta_{1}\leftrightarrow\Delta_{2}\Big{)}\Bigg{]}\mathcal{V}_{\Delta_{1}+\Delta_{2},c{\tilde{c}}}(x_{2})$
$\displaystyle+\frac{\alpha^{\prime}{}}{2}\Bigg{[}\Big{(}\frac{1}{2}\delta^{{\tilde{a}}{\tilde{b}}}x_{12}^{a}x_{12}^{b}x_{12}^{c}x_{12}^{\tilde{c}}B(\Delta_{1}+2,\Delta_{2}+2)-\delta^{ac}x_{12}^{b}x_{12}^{\tilde{b}}x_{12}^{\tilde{a}}x_{12}^{\tilde{c}}B(\Delta_{1}+2,\Delta_{2}+1)$
$\displaystyle\qquad\qquad+(\;a\leftrightarrow{\tilde{a}},\,b\leftrightarrow{\tilde{b}},\;c\leftrightarrow{\tilde{c}}\;)\Big{)}\;+\Big{(}a\leftrightarrow
b,\,{\tilde{a}}\leftrightarrow{\tilde{b}},\;\Delta_{1}\leftrightarrow\Delta_{2}\Big{)}\Bigg{]}\mathcal{V}_{\Delta_{1}+\Delta_{2}+2,c{\tilde{c}}}(x_{2})$
$\displaystyle+\frac{\alpha^{\prime}{}^{2}}{4}\;\Bigg{[}x_{12}^{a}x_{12}^{b}x_{12}^{c}x_{12}^{\tilde{a}}x_{12}^{\tilde{b}}x_{12}^{\tilde{c}}B(\Delta_{1}+3,\Delta_{2}+3)\Bigg{]}\mathcal{V}_{\Delta_{1}+\Delta_{2}+4,c{\tilde{c}}}(x_{2})~{}.$

##  4 Celestial OPE from worldsheet OPE in superstring

In this section, we will generalize the previous discussions on OPEs in
bosonic string to superstring. The computations are similar, and especially we
can also simplify the results in the same way. The main difference is that in
superstring, the worldsheet has supersymmetry. As a result, the bosonic and
fermionic contributions can cancel each other, which leads to a simpler
result. In particular, the $\alpha^{\prime}{}$ corrections to the worldsheet
OPEs and thus also to the celestial OPEs are absent in the supersymmetric
sector. This behavior agrees with the three-point amplitudes in superstring
theory where the $\alpha^{\prime}{}$ corrections also don’t appear in the
supersymmetric sector.

We will be mainly focusing on the heterotic string case. The heterotic string
is a good playground for our studies of OPEs because we can realize both
graviton and gluon easily in terms of closed string. In this setup, we compute
all the OPEs involving gluon and graviton/dilaton/KR field, including their
mixed OPEs. Up to $\alpha^{\prime}{}$ corrections, the OPEs derived in
heterotic string agree with those derived in bosonic string. We will also
briefly discuss the case of type I and IIA/IIB string, which are very similar
to heterotic string.

###  4.1 Heterotic string

In this subsection, we will first review the vertex operators of gluon and
graviton/dilaton/KR field in heterotic string, and then derive the celestial
OPEs from worldsheet OPEs.

####  4.1.1 Vertex operator in heterotic string

In bosonic closed string, the vertex operators take two forms, either
integrated form with worldsheet integration or unintegrated form with
$c\tilde{c}$ attachment. In superstring theory, the vertex operators further
take a variety of forms, called pictures Friedan:1985ey ; Polchinski:1998rq .
In each supersymmetric left or right moving sector, we can label the vertex
operator with the ghost charges $q$ or $\tilde{q}$. The vertex operator in the
$q$ or $\tilde{q}$ picture contains a factor $e^{q\phi}$ or
$e^{\tilde{q}\tilde{\phi}}$, where $\phi$ is the bosonized field of
$\beta\gamma$ ghost system and similarly for $\tilde{\phi}$. The vertex
operators in different pictures can be related via the picture changing
operator. The total ghost charges in each supersymmetric sector is determined
by the worldsheet genus. In particular, at tree level of interest in this
paper, the total ghost charge should be -2 in order to soak up the fermionic
zero modes.

More specifically, in heterotic string, the gluon vertex operator in the
picture -1 and 0 are respectively given by Polchinski:1998rq

$\displaystyle V^{(-1)}(z,\bar{z})$ $\displaystyle=$ $\displaystyle
e^{-\tilde{\phi}}J^{A}\tilde{\psi}^{\nu}e^{ik\cdot
X}=V_{L}(z)V^{(0)}_{R}({\bar{z}})~{},$ (117) $\displaystyle
V^{(0)}(z,\bar{z})$ $\displaystyle=$ $\displaystyle
J^{A}\Big{(}i\bar{\partial}X^{\nu}+\frac{1}{2}\alpha^{\prime}{}k\cdot{\tilde{\psi}}{\tilde{\psi}}^{\nu}\Big{)}e^{ik\cdot
X}=V_{L}(z)V^{(-1)}_{R}({\bar{z}})~{},$ (118)

where $J^{A}(z)$ are the Kac-Moody currents in the left-moving sector, while
$\tilde{\phi}$ and $\tilde{\psi}$ are the ghost and world-sheet fermion in the
right-moving sector, respectively. Similarly, for the massless
graviton/dilaton/KR fields in heterotic string, their vertex operators in the
-1 and 0 picture are given by

$\displaystyle W^{(-1)}(z,\bar{z})$ $\displaystyle=$ $\displaystyle
e^{-\tilde{\phi}}\partial X^{\mu}\tilde{\psi}^{\nu}e^{ik\cdot
X}=W_{L}(z)W^{(-1)}_{R}({\bar{z}})~{},$ (119) $\displaystyle
W^{(0)}(z,\bar{z})$ $\displaystyle=$ $\displaystyle\partial
X^{\mu}\Big{(}i\bar{\partial}X^{\nu}+\frac{1}{2}\alpha^{\prime}{}k\cdot{\tilde{\psi}}{\tilde{\psi}}^{\nu}\Big{)}e^{ik\cdot
X}=W_{L}(z)W^{(0)}_{R}({\bar{z}})~{}.$ (120)

In these vertex operators, we also write them in the product form of left and
right moving parts:

$\displaystyle V_{L}(z)=J^{A}(z)e^{ik\cdot X_{L}}~{},\qquad W_{L}(z)=\partial
X^{\mu}e^{ik\cdot X_{L}}~{},$ (121) $\displaystyle
V^{(0)}_{R}({\bar{z}})=W^{(0)}_{R}({\bar{z}})=\Big{(}i\bar{\partial}X^{\mu}+\frac{1}{2}\alpha^{\prime}{}k\cdot{\tilde{\psi}}{\tilde{\psi}}^{\mu}\Big{)}e^{ik\cdot
X_{R}}~{},$ (122) $\displaystyle
V^{(-1)}_{R}({\bar{z}})=W^{(-1)}_{R}({\bar{z}})=e^{-\tilde{\phi}}{\tilde{\psi}}^{\mu}\;e^{ik\cdot
X_{R}}~{},$ (123)

where $X(z,{\bar{z}})=X_{L}(z)+X_{R}({\bar{z}})$. This decomposition enables
us to compute the OPE in the left and right moving sectors independently. The
Kac-Moody currents $J^{A}(z)$ satisfy the OPE

$J^{A}(z_{1})J^{B}(z_{2})\sim\frac{\mathsf{k}\delta^{AB}}{z_{12}^{2}}+\frac{f^{ABC}J^{C}(z_{2})}{z_{12}}~{}.$
(124)

Without loss of generality, one can set the level $\mathsf{k}$ to one, which
can always be realized by rescaling the Kac-Moody currents and structure
constants.

####  4.1.2 Celestial OPE from heterotic worldsheet

Now we derive the celestial OPEs in heterotic string from worldsheet.

#### Gluon-gluon OPE

We want to compute the OPE between gluons. Although vertex operators in
different pictures are physically equivalent, it turns out to be simpler to
consider the OPE of two vertex operators in the $-1$ and 0 picture, namely
$V^{(-1)}(z_{1},\bar{z}_{1})V^{(0)}(z_{2},\bar{z}_{2})$, which gives another
vertex operator in $-1$ picture.

We first consider the left-moving OPE $V_{L}(z_{1})V_{L}(z_{2})$:

$\displaystyle J^{A}e^{ip\cdot X_{L}}(z_{1})J^{B}e^{iq\cdot X_{L}}(z_{2})$
$\displaystyle=$ $\displaystyle J^{A}(z_{1})J^{B}(z_{2})\times e^{ip\cdot
X_{L}}(z_{1})e^{iq\cdot X_{L}}(z_{2})$ (125) $\displaystyle=$
$\displaystyle\Big{(}\frac{\mathsf{k}\delta^{AB}}{z_{12}^{2}}+\frac{f^{ABC}J^{C}(z_{2})}{z_{12}}+\cdots\Big{)}\times
z_{12}^{\frac{1}{2}\alpha^{\prime}{}p\cdot q}\Big{(}1+i\,z_{12}p\cdot\partial
X+\cdots\Big{)}e^{i(p+q)\cdot X_{L}}(z_{2})$ $\displaystyle=$ $\displaystyle
z_{12}^{\frac{1}{2}\alpha^{\prime}{}p\cdot
q-1}\Big{(}\frac{\mathsf{k}\delta^{AB}}{z_{12}}+{f^{ABC}J^{C}}+i\,\mathsf{k}\delta^{AB}\,p\cdot\partial
X+\cdots\Big{)}e^{i(p+q)\cdot X_{L}}(z_{2})~{}.$

Following the derivation of (52), we have

$\displaystyle p\cdot\partial X$ $\displaystyle=$ $\displaystyle
p\cdot\varepsilon_{c}\;\varepsilon_{c}\cdot\partial
X+\frac{1}{\omega_{k}}p\cdot q\;\partial X\cdot
n+\frac{\omega_{p}}{\omega_{k}}k\cdot\partial X+\mathscr{O}(\epsilon)~{}.$
(126)

Due to the $p\cdot q$ factor, the second term on the right hand side is
regular in the collinear limit $p\cdot q\to 0$. For the third term, we can
combine it with extra factors in (125), leading to

$z_{12}^{\frac{1}{2}\alpha^{\prime}{}p\cdot q-1}\;k\cdot\partial
X\;e^{i(p+q)\cdot X_{L}}=-i\,z_{12}^{\frac{1}{2}\alpha^{\prime}{}p\cdot
q-1}\;\partial\;e^{ik\cdot X_{L}}+\mathscr{O}(\epsilon)~{},$ (127)

where we used $e^{i(p+q)\cdot X_{L}}=e^{ik\cdot X_{L}}+\mathscr{O}(\epsilon)$
as $p+q=k+\mathscr{O}(\epsilon)$ in the collinear limit. Since the
polarization is along the momentum direction, the is supposed to be a pure
gauge and thus can be ignored. The more subtle issue of boundary contact terms
is the same as that in open bosonic string case, see (67). Therefore, we can
just keep the first term in (126).

As a result, in the collinear limit the left-moving OPE simplifies to

$\displaystyle J^{A}e^{ip\cdot X_{L}}(z_{1})J^{B}e^{iq\cdot X_{L}}(z_{2})$
$\displaystyle=$ $\displaystyle z_{12}^{\frac{1}{2}\alpha^{\prime}{}p\cdot
q-1}\Big{(}\frac{\mathsf{k}\delta^{AB}}{z_{12}}+{f^{ABC}J^{C}}+i\,\mathsf{k}\delta^{AB}\,p\cdot\varepsilon_{c}\;\varepsilon_{c}\cdot\partial
X+\cdots\Big{)}e^{i(p+q)\cdot X_{L}}(z_{2})~{}.\qquad\qquad$ (128)

For the right-moving OPE $V_{R}^{(-1)}(\bar{z}_{1})V_{R}^{(0)}(\bar{z}_{2})$,
it is computed in (A.3)

$\displaystyle\zeta\cdot\tilde{\psi}e^{ip\cdot
X_{R}}({\bar{z}}_{1})\Big{(}i\xi\cdot\bar{\partial}X+\frac{1}{2}\alpha^{\prime}{}q\cdot{\tilde{\psi}}\xi\cdot{\tilde{\psi}}\Big{)}e^{iq\cdot
X_{R}}({\bar{z}}_{2})$ (129) $\displaystyle\sim$
$\displaystyle\frac{\alpha^{\prime}}{2}{\bar{z}}_{12}^{\frac{1}{2}\alpha^{\prime}{}p\cdot
q-1}e^{i(p+q)\cdot X_{R}}\Big{(}\zeta\cdot q\;\xi\cdot\tilde{\psi}-\xi\cdot
p\;\zeta\cdot\tilde{\psi}-\zeta\cdot\xi\;q\cdot\tilde{\psi}\Big{)}~{}.$ (130)

Note that terms in the bracket of (130) are very similar to the terms
proportional $y_{12}$ in the bracket of (3.1): one just needs to replace
$\dot{X}$ with $\psi$, exchange $p,\zeta$ and $q,\xi$, and ignore
$\alpha^{\prime}{}$ corrections. Therefore we can also simplify the formula in
the same manner. Following the steps in the derivation of (52) and (56), we
now have 191919 Note
$(p+q)\cdot\varepsilon_{\tilde{c}}=k\cdot\varepsilon_{\tilde{c}}+\mathscr{O}(\epsilon)=\mathscr{O}(\epsilon)$
as $k\cdot\varepsilon_{\tilde{c}}=0$.

$\displaystyle q\cdot\tilde{\psi}$ $\displaystyle=$
$\displaystyle-p\cdot\varepsilon_{\tilde{c}}\;\varepsilon_{\tilde{c}}\cdot\tilde{\psi}+\frac{1}{\omega_{k}}p\cdot
q\;\tilde{\psi}\cdot
n+\frac{\omega_{q}}{\omega_{k}}k\cdot\tilde{\psi}+\mathscr{O}(\epsilon)~{},$
(131)

and

$\displaystyle\zeta\cdot q\;\xi\cdot\tilde{\psi}-\xi\cdot
p\;\zeta\cdot\tilde{\psi}=\zeta\cdot
q\;\xi\cdot\varepsilon_{\tilde{c}}\;\varepsilon_{\tilde{c}}\cdot\tilde{\psi}-\xi\cdot
p\;\zeta\cdot\varepsilon_{\tilde{c}}\;\varepsilon_{\tilde{c}}\cdot\tilde{\psi}+\mathscr{O}(\epsilon)~{}.$
(132)

These two formulae enable us simplify (130)

$\displaystyle\zeta\cdot\tilde{\psi}e^{ip\cdot
X_{R}}({\bar{z}}_{1})\Big{(}i\xi\cdot\bar{\partial}X+\frac{1}{2}\alpha^{\prime}{}q\cdot{\tilde{\psi}}\xi\cdot{\tilde{\psi}}\Big{)}e^{iq\cdot
X_{R}}({\bar{z}}_{2})$ (133) $\displaystyle\sim$
$\displaystyle\frac{\alpha^{\prime}}{2}{\bar{z}}_{12}^{\frac{1}{2}\alpha^{\prime}{}p\cdot
q-1}e^{i(p+q)\cdot X_{R}}\Big{(}\zeta\cdot
q\;\xi\cdot\varepsilon_{\tilde{c}}-\xi\cdot
p\;\zeta\cdot\varepsilon_{\tilde{c}}+\zeta\cdot\xi\;p\cdot\varepsilon_{\tilde{c}}\Big{)}\;\varepsilon_{\tilde{c}}\cdot\tilde{\psi}$
(135)
$\displaystyle+\frac{\alpha^{\prime}}{2}{\bar{z}}_{12}^{\frac{1}{2}\alpha^{\prime}{}p\cdot
q-1}e^{i(p+q)\cdot X_{R}}\Big{(}-\zeta\cdot\xi\;\frac{1}{\omega_{k}}p\cdot
q\;\tilde{\psi}\cdot
n-\zeta\cdot\xi\;\frac{\omega_{q}}{\omega_{k}}k\cdot\tilde{\psi}\Big{)}~{}.$

In the collinear limit $p\cdot q\to 0$, the first term in (135) is regular.
For the second term, we can rewrite it as: 202020 Recall that that
$p+q=k+\epsilon v$ where $k$ is null while $p+q$ is slightly off-shell.

$e^{i(p+q)\cdot X_{R}}k\cdot\tilde{\psi}=e^{ik\cdot
X_{R}}k\cdot\tilde{\psi}+\mathscr{O}(\epsilon)~{}.$ (136)

If we combine (136) with the factor
$z_{12}^{\frac{1}{2}\alpha^{\prime}{}p\cdot q-1}$ from the left-moving sector,
212121 The left-moving sector may also contribute
$z_{12}^{\frac{1}{2}\alpha^{\prime}{}p\cdot q-2}$, then in total we have
$|z_{12}|^{\alpha^{\prime}{}p\cdot q-2}z_{12}^{-1}$. This gives zero after
integrating along the angular direction of $z_{12}$.  we get

$\displaystyle|z_{12}|^{\alpha^{\prime}{}p\cdot q-2}e^{ik\cdot
X_{R}}k\cdot\tilde{\psi}\xrightarrow{p\cdot q\to
0}\frac{\delta^{2}(z_{12})}{p\cdot q}k\cdot\tilde{\psi}e^{ik\cdot
X_{R}}\xrightarrow{\int d^{2}z_{1}\,d^{2}z_{2}}\frac{1}{p\cdot q}\int
d^{2}z_{2}\;k\cdot\tilde{\psi}e^{ik\cdot X_{R}}~{}.$ (137)

Since the polarization is along the momentum direction, this is a pure gauge.
More specifically it is a BRST exact term and thus plays no role in string
amplitude. Therefore, we can drop all the terms in (135).

Combining the left-moving OPE (128) and right-moving OPE (135), we then find

$\displaystyle e^{-\tilde{\phi}}J^{A}\tilde{\psi}^{\mu}e^{ip\cdot
X}(z_{1},{\bar{z}}_{1})J^{B}\Big{(}i\bar{\partial}X^{\nu}+\frac{1}{2}\alpha^{\prime}{}q\cdot{\tilde{\psi}}{\tilde{\psi}}^{\nu}\Big{)}e^{ik\cdot
X}(z_{2},{\bar{z}}_{2})$ (139) $\displaystyle\sim$
$\displaystyle\frac{\alpha^{\prime}}{2}|z_{12}|^{\alpha^{\prime}{}p\cdot
q-2}e^{-\tilde{\phi}}\Big{(}\frac{\mathsf{k}\delta^{AB}}{z_{12}}+{f^{ABC}J^{C}}+i\,\mathsf{k}\delta^{AB}\,p\cdot\varepsilon_{c}\;\varepsilon_{c}\cdot\partial
X+\cdots\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\times\Big{(}q^{\mu}\;\varepsilon_{\tilde{c}}^{\nu}-p^{\nu}\;\varepsilon_{\tilde{c}}^{\mu}+\eta^{\mu\nu}\;p\cdot\varepsilon_{\tilde{c}}\Big{)}\;\varepsilon_{\tilde{c}}\cdot\tilde{\psi}\,e^{i(p+q)\cdot
X}~{}.\qquad\qquad$

First, we have the leading term $\mathsf{k}\delta^{AB}$ coming with a factor
$|z_{12}|^{\alpha^{\prime}{}p\cdot q-2}z_{12}^{-1}$. This leads to zero after
performing the $z_{1},z_{2}$ integral due to the cancellation from the angular
direction of $z_{12}$. After contracting with the basis of polarization, the
rest of terms in (139) become

$\displaystyle
e^{-\tilde{\phi}}J^{A}\varepsilon_{\tilde{a}}(p)\cdot\tilde{\psi}e^{ip\cdot
X}(z_{1},{\bar{z}}_{1})J^{A}\Big{(}i\varepsilon_{\tilde{b}}(q)\cdot\bar{\partial}X+\frac{1}{2}\alpha^{\prime}{}q\cdot{\tilde{\psi}}\;\varepsilon_{\tilde{b}}(q)\cdot{\tilde{\psi}}\Big{)}e^{ik\cdot
X}(z_{2},{\bar{z}}_{2})$ (142) $\displaystyle\sim$
$\displaystyle\frac{\alpha^{\prime}}{2}|z_{12}|^{\alpha^{\prime}{}p\cdot
q-2}\Big{(}\varepsilon_{\tilde{a}}(p)\cdot
q\;\varepsilon_{\tilde{b}}(q)\cdot\varepsilon_{\tilde{c}}-\varepsilon_{\tilde{b}}(q)\cdot
p\;\varepsilon_{\tilde{a}}(p)\cdot\varepsilon_{\tilde{c}}+\varepsilon_{\tilde{a}}(p)\cdot\varepsilon_{\tilde{b}}(q)\;p\cdot\varepsilon_{\tilde{c}}\Big{)}$
$\displaystyle\qquad\qquad\times\Big{[}{f^{ABC}}\;e^{-\tilde{\phi}}J^{C}\;\varepsilon_{\tilde{c}}\cdot\tilde{\psi}\,e^{i(p+q)\cdot
X}+i\,\mathsf{k}\delta^{AB}\,p\cdot\varepsilon_{c}\;e^{-\tilde{\phi}}\;\varepsilon_{c}\cdot\partial
X\varepsilon_{\tilde{c}}\cdot\tilde{\psi}\,e^{i(p+q)\cdot X}\Big{]}$
$\displaystyle\sim$ $\displaystyle
2\pi\;\delta^{2}(z_{12})\;I_{\tilde{c}}\Big{(}p,\varepsilon_{\tilde{a}}(p);q,\varepsilon_{\tilde{b}}(q);0\Big{)}$
$\displaystyle\qquad\qquad\times\Big{[}{f^{ABC}}\;e^{-\tilde{\phi}}J^{C}\;\varepsilon_{\tilde{c}}\cdot\tilde{\psi}\,e^{i(p+q)\cdot
X}+i\,\mathsf{k}\delta^{AB}\,p\cdot\varepsilon_{c}\;e^{-\tilde{\phi}}\;\varepsilon_{c}\cdot\partial
X\varepsilon_{\tilde{c}}\cdot\tilde{\psi}\,e^{i(p+q)\cdot X}\Big{]}~{},$

where we used identity (99) and the expression for
$I_{\tilde{c}}(p,\varepsilon_{\tilde{a}}(p);q,\varepsilon_{\tilde{b}}(q);0)$
in (79) but without $\alpha^{\prime}{}$ corrections. Compared with (117)
(119), it is easy to recognize that terms in the square bracket are exactly
the vertex operators of gluon and graviton/dilaton/KR field. After performing
the integral over $z_{1},z_{2}$, we get

$\displaystyle\mathcal{V}_{\tilde{a}}^{(-1)A}(p)\mathcal{V}_{\tilde{b}}^{(0)B}(q)$
$\displaystyle\sim$ $\displaystyle
f^{ABC}\;I_{\tilde{c}}\Big{(}p,\varepsilon_{\tilde{a}}(p);q,\varepsilon_{\tilde{b}}(q);0\Big{)}\mathcal{V}_{\tilde{c}}^{(-1)C}(p+q)$
$\displaystyle+\delta^{AB}\,p\cdot\varepsilon_{c}\;I_{\tilde{c}}\Big{(}p,\varepsilon_{\tilde{a}}(p);q,\varepsilon_{\tilde{b}}(q);0\Big{)}\mathcal{W}_{c{\tilde{c}}}^{(-1)C}(p+q)~{},\qquad$

up to some coefficients in front of gluon and graviton contributions,
respectively.

We can further write in terms of celestial variables. In particular,

$p\cdot\varepsilon_{c}=\omega_{p}\hat{p}\cdot\varepsilon_{c}=\omega_{p}(x_{13})^{c}=\frac{\omega_{p}\omega_{q}}{\omega_{p}+\omega_{q}}x_{12}^{c}~{}.$
(144)

where we used (4) and (76).

Finally performing the Mellin transformation yields the celestial OPE of two
gluons

$\displaystyle\mathcal{V}^{A}_{\Delta_{1},{\tilde{a}}}(x_{1})\mathcal{V}^{B}_{\Delta_{2},{\tilde{b}}}(x_{2})$
$\displaystyle\sim$ $\displaystyle
f^{ABC}\frac{x_{12}^{{\tilde{a}}}\delta^{{\tilde{b}}{\tilde{c}}}B(\Delta_{1}-1,\Delta_{2})+x_{12}^{{\tilde{b}}}\delta^{{\tilde{a}}{\tilde{c}}}B(\Delta_{1},\Delta_{2}-1)-x_{12}^{{\tilde{c}}}\delta^{{\tilde{a}}{\tilde{b}}}B(\Delta_{1},\Delta_{2})}{(x_{12})^{2}}\mathcal{V}^{C}_{\Delta_{1}+\Delta_{2}-1,\;{\tilde{c}}}(x_{2})$
$\displaystyle+\delta^{AB}x_{12}^{c}\frac{x_{12}^{{\tilde{a}}}\delta^{{\tilde{b}}{\tilde{c}}}B(\Delta_{1},\Delta_{2}+1)+x_{12}^{{\tilde{b}}}\delta^{{\tilde{a}}{\tilde{c}}}B(\Delta_{1}+1,\Delta_{2})-x_{12}^{{\tilde{c}}}\delta^{{\tilde{a}}{\tilde{b}}}B(\Delta_{1}+1,\Delta_{2}+1)}{(x_{12})^{2}}\mathcal{W}_{\Delta_{1}+\Delta_{2},\;c{\tilde{c}}}(x_{2})~{},$

where the first and second line just correspond to the gluon and graviton,
respectively. The OPE of gluons among themselves is the same as that in
bosonic string case (3.1), except for the absence of $\alpha^{\prime}{}$
corrections.

#### Gluon-graviton/dilaton/KR OPE

Now we consider the OPE between gluon and graviton/dilaton/KR field, namely
$W^{(-1)}(z_{1},\bar{z}_{1})V^{(0)}(z_{2},\bar{z}_{2})$. The left-moving OPE
$W_{L}(z_{1})V_{L}(z_{2})$ is simply given by

$\partial X^{\rho}e^{ip\cdot X_{L}}(z_{1})J^{A}e^{iq\cdot
X_{L}}(z_{2})=z_{12}^{\frac{1}{2}\alpha^{\prime}{}p\cdot
q-1}\Big{(}-\frac{i}{2}\alpha^{\prime}{}q^{\rho}+\cdots\Big{)}J^{A}e^{i(p+q)\cdot
X_{L}}(z_{2})~{},$ (146)

which follows from the same derivation of (257). The right-moving OPE
$W_{R}^{(-1)}(\bar{z}_{1})V_{R}^{(0)}(\bar{z}_{2})=V_{R}^{(-1)}(\bar{z}_{1})V_{R}^{(0)}(\bar{z}_{2})$
has been discussed above and the final result is given in (135).

Combining the left and right moving sectors, we get

$\displaystyle e^{-\tilde{\phi}}\varepsilon_{a}\cdot\partial
X\varepsilon_{\tilde{a}}\cdot\tilde{\psi}e^{ip\cdot
X}(z_{1},{\bar{z}}_{1})\;J^{A}\Big{(}i\varepsilon_{\tilde{b}}\cdot\bar{\partial}X^{\nu}+\frac{1}{2}\alpha^{\prime}{}p\cdot{\tilde{\psi}}\;\varepsilon_{\tilde{b}}\cdot{\tilde{\psi}}^{\nu}\Big{)}e^{iq\cdot
X}(z_{2},{\bar{z}}_{2})$ (147) $\displaystyle\sim$
$\displaystyle-\frac{i}{4}\alpha^{\prime}{}^{2}|z_{12}|^{\alpha^{\prime}{}p\cdot
q-2}q\cdot\varepsilon_{a}\Big{(}\varepsilon_{\tilde{a}}\cdot
q\;\varepsilon_{\tilde{b}}\cdot\varepsilon_{\tilde{c}}-\varepsilon_{\tilde{b}}\cdot
p\;\varepsilon_{\tilde{a}}\cdot\varepsilon_{\tilde{c}}+\varepsilon_{\tilde{a}}\cdot\varepsilon_{\tilde{b}}\;p\cdot\varepsilon_{\tilde{c}}\Big{)}\;e^{-\tilde{\phi}}J^{A}\varepsilon_{\tilde{c}}\cdot\tilde{\psi}\;e^{i(p+q)\cdot
X}\qquad$ (148) $\displaystyle\sim$
$\displaystyle-i\,\pi\alpha^{\prime}{}\delta^{2}(z_{12})\;q\cdot\varepsilon_{a}\,I_{\tilde{c}}\Big{(}p,\varepsilon_{\tilde{a}}(p);q,\varepsilon_{\tilde{b}}(q);0\Big{)}\;e^{-\tilde{\phi}}J^{A}\varepsilon_{\tilde{c}}\cdot\tilde{\psi}\;e^{i(p+q)\cdot
X}~{},$ (149)

Integrating over $z_{1},z_{2}$ gives

$\mathcal{W}^{(-1)}_{a{\tilde{a}}}(p)\mathcal{V}^{(0)A}_{\tilde{b}}(q)\propto-q\cdot\varepsilon_{a}\,I_{\tilde{c}}\Big{(}p,\varepsilon_{\tilde{a}}(p);q,\varepsilon_{\tilde{b}}(q);0\Big{)}\mathcal{V}^{(-1)A}_{\tilde{c}}(p+q)~{},$
(150)

where we ignore the overall constant for simplicity.

Further writing in terms of celestial variables and performing the Mellin
transformation, we get

$\displaystyle\mathcal{W}_{\Delta_{1},a{\tilde{a}}}(x_{1})\mathcal{V}^{A}_{\Delta_{2},{\tilde{b}}}(x_{2})$
$\displaystyle\sim$ $\displaystyle
x_{12}^{a}\frac{x_{12}^{{\tilde{a}}}\delta^{{\tilde{b}}{\tilde{c}}}B(\Delta_{1}-1,\Delta_{2}+1)+x_{12}^{{\tilde{b}}}\delta^{{\tilde{a}}{\tilde{c}}}B(\Delta_{1},\Delta_{2})-x_{12}^{{\tilde{c}}}\delta^{{\tilde{a}}{\tilde{b}}}B(\Delta_{1},\Delta_{2}+1)}{(x_{12})^{2}}\mathcal{V}^{A}_{\Delta_{1}+\Delta_{2},{\tilde{c}}}(x_{2})~{}.$

#### Graviton/dilaton/KR OPE

Finally we want to discuss the OPE of graviton/dilaton/KR field, namely
$W^{(-1)}(z_{1},\bar{z}_{1})W^{(0)}(z_{2},\bar{z}_{2})$. The OPE in the left
moving sector $W_{L}(z_{1})W_{L}(z_{2})$ is given in (271) and it is the same
as the left moving sector of closed bosonic string. Up to the boundary contact
terms, we can simplify this sector in the same way as that in the bosonic
string.

The right-moving OPE
$W_{R}^{(-1)}(\bar{z}_{1})W_{R}^{(0)}(\bar{z}_{2})=V_{R}^{(-1)}(\bar{z}_{1})V_{R}^{(0)}(\bar{z}_{2})$
has been discussed above and the final result is given in (135). Combining the
left and right moving sectors, we arrive at almost the same OPE as that in the
closed bosonic string (102) except that we need to remove the
$\alpha^{\prime}{}$ corrections in the right-moving sector (104), which are
absent in heterotic string due to supersymmetry (135). As a result, in
heterotic string the OPE (112) becomes

$\displaystyle\mathcal{W}^{(-1)}_{a{\tilde{a}}}(p)\mathcal{W}^{(0)}_{b{\tilde{b}}}(q)$
$\displaystyle\sim$
$\displaystyle\mathcal{I}_{c{\tilde{c}}}\Big{(}p,\varepsilon_{a\tilde{a}}(p);q,\varepsilon_{b\tilde{b}}(q);\alpha^{\prime}{},0\Big{)}\;\mathcal{W}^{(-1)}_{c{\tilde{c}}}(p+q)~{}.$
(152)

Then one can derive the celestial OPE in the same way as bosonic string. The
final result is given by (113): ignoring the $\alpha^{\prime}{}^{2}$ terms in
(116), and keeping all the terms in (116) as well as the $\alpha^{\prime}{}$
terms in (116) which are proportional to $x_{12}^{a}x_{12}^{b}x_{12}^{c}$.
222222 More specifically, the terms in the square bracket of (116) is now
given by:
$\Bigg{[}x_{12}^{a}x_{12}^{b}x_{12}^{c}\Big{(}B(\Delta_{1}+2,\Delta_{2}+2)\delta^{{\tilde{a}}{\tilde{b}}}x_{12}^{\tilde{c}}-B(\Delta_{1}+1,\Delta_{2}+2)\delta^{{\tilde{b}}{\tilde{c}}}x_{12}^{\tilde{a}}-B(\Delta_{1}+2,\Delta_{2}+1)\delta^{{\tilde{a}}{\tilde{c}}}x_{12}^{\tilde{b}}\Big{)}\Bigg{]}~{}.$

###  4.2 Type I, IIA, IIB string

Now we briefly discuss the OPEs of closed string massless fields in type I and
IIA/IIB superstring. The vertex operators for massless NS-NS fields in type I
and IIA/IIB string are Polchinski:1998rq

$\displaystyle W^{(-1,-1)}(z,\bar{z})$ $\displaystyle=$ $\displaystyle
e^{-\phi-\tilde{\phi}}\psi^{\mu}\tilde{\psi}^{\nu}e^{ik\cdot X}~{},$ (153)
$\displaystyle W^{(0,0)}(z,\bar{z})$ $\displaystyle=$
$\displaystyle\Big{(}i\partial
X^{\mu}+\frac{1}{2}\alpha^{\prime}{}k\cdot{\psi}{\psi}^{\mu}\Big{)}\Big{(}i\bar{\partial}X^{\mu}+\frac{1}{2}\alpha^{\prime}{}k\cdot{\tilde{\psi}}{\tilde{\psi}}^{\mu}\Big{)}e^{ik\cdot
X}~{}.$ (154)

Now both the left and right-moving sectors are supersymmetric. Note that for
type I superstring, the KR 2-form field is projected out and we are only left
with graviton and dilaton.

It is simpler to study OPE of two operators in $-1$ and $0$ picture, namely
$W^{(-1,-1)}(z_{1},\bar{z}_{1})W^{(0,0)}(z_{2},\bar{z}_{2})$. The vertex
operators (153) (154) can again be decomposed into the product of left and
right moving parts. For both the left and right moving sectors, the OPE is
given by (135) or its conjugate. Combining the two sectors together, the OPE
of $W^{(-1,-1)}(z_{1},\bar{z}_{1})W^{(0,0)}(z_{2},\bar{z}_{2})$ is similar to
that in closed bosonic string (104)(104) except that we need to replace
$\partial X,\bar{\partial}X\to\psi,\tilde{\psi}$ and remove all the
$\alpha^{\prime}{}$ corrections, which are absent now due to greater
supersymmetry. It is amusing that the tachyon also does not appear in the OPE,
although we have not performed the GSO projection. The final celestial OPE is
given by (113) with all $\alpha^{\prime}{}$ terms removed. The absence of
$\alpha^{\prime}{}$ correction is consistent with the string amplitude, as we
will see in the next section.

##  5 Celestial OPE from collinear factorization

In the previous two sections, we have derived the celestial OPEs from the
string worldsheet perspective. In this section, we will compute the celestial
OPEs using a different method based on the collinear factorization of
scattering amplitudes. It turns out that the two approaches give the same
result all the time.

Before going to the computational details, let us first describe the general
strategy of deriving celestial OPEs based on the collinear factorization. We
are interested in the scattering amplitude of massless fields. In the
collinear limit two momenta become parallel $p_{1}/\\!/p_{2}$, and the total
momentum $P\equiv p_{1}+p_{2}$ also becomes almost on-shell, namely
$P^{2}=2p_{1}\cdot p_{2}\to 0$. The amplitude then factorizes in the collinear
limit as 232323 Since the two collinear particles are massless, the collinear
limit thus singles out the massless field propagator connecting $A_{3}$ and
$A_{n}$. More generally, to identify the contribution from the field with mass
$M$, one can instead use the propagator $1/(P^{2}+M^{2})$ in (156) and take
the limit $P^{2}\to-M^{2}$.

$A_{n+1}(p_{1}^{s_{1}},p_{2}^{s_{2}},\cdots)\xrightarrow{p_{1}/\\!/p_{2}}\sum_{s}A_{3}(p_{1}^{s_{1}},p_{2}^{s_{2}},-P^{s})\frac{1}{P^{2}}A_{n}(P^{s},\cdots)~{},$
(155)

where $s_{i}$ are the extra quantum numbers labelling the particles. The
prefactor in front of $A_{n}$ is essentially the split function characterizing
the collinear behavior:

${\mathsf{Split}}(p_{1}^{s_{1}}+p_{2}^{s_{2}}\to
P^{s})=\frac{1}{P^{2}}A_{3}(p_{1}^{s_{1}},p_{2}^{s_{2}},-P^{s})=\frac{1}{2p_{1}\cdot
p_{2}}A_{3}(p_{1}^{s_{1}},p_{2}^{s_{2}},-P^{s})~{}.$ (156)

Therefore, once we know the three-point amplitude, we also know the split
function. Performing the Mellin transformation then gives the corresponding
celestial OPEs.

###  5.1 Celestial OPE for gluon

In bosonic open string theory, the (color-ordered) amplitude for three gluons
is given by Polchinski:1998rr  242424 Like footnote 10, we set the overall
factors in all the string amplitudes to be one. In the final results of
celestial OPEs, we will choose proper overall factors to make the formulae as
simple as possible. We also use the subscripts $g$ and $G$ for gluon and
graviton/dilaton/KR field, respectively.

$A^{o}_{ggg}=e_{1\mu}e_{2\alpha}e_{3\rho}T^{\mu\alpha\rho}(4\alpha^{\prime}{})~{},$
(157)

while in heterotic string, the three gluon amplitude is Polchinski:1998rq

$A^{H}_{ggg}=e_{1\mu}e_{2\alpha}e_{3\rho}T^{\mu\alpha\rho}(0)~{},$ (158)

where we introduced the tensor

$\displaystyle T^{\mu\alpha\rho}(\alpha^{\prime}{})$ $\displaystyle=$
$\displaystyle
p_{23}^{\mu}\;\eta^{\alpha\rho}+p_{31}^{\alpha}\;\eta^{\rho\mu}+p_{12}^{\rho}\;\eta^{\mu\alpha}+\frac{\alpha^{\prime}}{8}p_{23}^{\mu}\;p_{31}^{\alpha}\;p_{12}^{\rho}~{},\qquad\qquad
p_{ij}=p_{i}-p_{j}~{}.$ (159)

So up to $\alpha^{\prime}{}$ corrections, the three gluon amplitude is the
same in bosonic and heterotic string. This is not surprising as the low energy
effective field actions of both theories contain the Yang-Mills theory, which
is responsible for the leading non-$\alpha^{\prime}{}$ amplitude above. The
absence of $\alpha^{\prime}{}$ correction in the heterotic string is due to
supersymmetry, and it is consistent with the absence of $\alpha^{\prime}{}$
corrections in the celestial OPEs in heterotic string that we derived before
from worldsheet.

Since the heterotic gluon amplitude can be regarded as the special limit of
the bosonic one without $\alpha^{\prime}{}$ correction, we will just focus on
the bosonic string gluon amplitude (157). More explicitly, the three gluon
amplitude (157) can be written as

$\displaystyle A^{o}_{ggg}$ $\displaystyle=$ $\displaystyle e_{1}\cdot
p_{23}\;e_{2}\cdot e_{3}+e_{2}\cdot p_{31}\;e_{3}\cdot e_{1}+e_{3}\cdot
p_{12}\;e_{1}\cdot e_{2}+\frac{\alpha^{\prime}}{2}e_{1}\cdot
p_{23}\;e_{2}\cdot p_{31}\;e_{3}\cdot p_{12}~{}\qquad$ (160) $\displaystyle=$
$\displaystyle 2\Big{[}e_{1}\cdot p_{2}e_{2}\cdot e_{3}-e_{2}\cdot
p_{1}e_{1}\cdot e_{3}-e_{3}\cdot p_{2}e_{1}\cdot
e_{2}+2\alpha^{\prime}{}e_{1}\cdot p_{2}\;e_{2}\cdot p_{1}\;e_{3}\cdot
p_{2}\Big{]}~{},$ (161)

where in the second equality we used momentum conservation
$p_{1}+p_{2}+p_{3}=0$ and $e_{i}\cdot p_{i}=0$ to simplify.

We choose $p_{1},p_{2}$ out-going, namely $\eta_{1}=\eta_{2}=1$, then $p_{3}$
is incoming $\eta_{3}=-1$. We also choose the polarization vectors as
$e_{i}=\varepsilon_{a_{i}}(p_{i})$. Using (4), we have

$p_{i}\cdot e_{j}\equiv
p_{i}\cdot\varepsilon_{a_{j}}(x_{j})=\eta_{i}\omega_{i}\hat{p}_{i}\cdot\varepsilon_{a_{j}}(x_{j})=\eta_{i}\,\omega_{i}\,x_{ij}^{a_{j}}~{},\qquad\varepsilon_{a_{i}}(x_{i})\cdot\varepsilon_{a_{j}}(x_{j})=\delta^{a_{i}a_{j}}~{},$
(162)

which enables us to rewrite (161) as

$\displaystyle\frac{1}{2}A^{o}_{ggg}(p_{i},\varepsilon_{a_{i}})$
$\displaystyle=$ $\displaystyle e_{1}\cdot p_{2}\;e_{2}\cdot e_{3}-e_{2}\cdot
p_{1}\;e_{3}\cdot e_{1}-e_{3}\cdot p_{2}\;e_{1}\cdot
e_{2}+2\alpha^{\prime}{}e_{1}\cdot p_{2}\;e_{2}\cdot p_{1}\;e_{3}\cdot p_{2}$
(163) $\displaystyle=$
$\displaystyle\omega_{2}x_{21}^{a_{1}}\delta^{a_{2}a_{3}}-\omega_{1}x_{12}^{a_{2}}\delta^{a_{1}a_{3}}-\omega_{2}x_{23}^{a_{3}}\delta^{a_{1}a_{2}}+2\alpha^{\prime}{}\omega_{1}\omega_{2}^{2}\;x_{21}^{a_{1}}x_{12}^{a_{2}}x_{23}^{a_{3}}~{}.$
(164)

To have sensible results for collinear factorization, we take $p_{1},p_{2}$
on-shell but $P\equiv p_{1}+p_{2}$ slightly off-shell. Then the momentum
conservation $P=p_{1}+p_{2}=-p_{3}+\epsilon v$ is

$\omega_{1}\Big{(}\frac{1+(x_{1})^{2}}{2},x_{1}^{a},\frac{1-(x_{1})^{2}}{2}\Big{)}+\omega_{2}\Big{(}\frac{1+(x_{2})^{2}}{2},x_{2}^{a},\frac{1-(x_{2})^{2}}{2}\Big{)}=\omega_{3}\Big{(}\frac{1+(x_{3})^{2}}{2},x_{3}^{a},\frac{1-(x_{3})^{2}}{2}\Big{)}+\epsilon
v~{},$ (165)

where $\epsilon$ characterizes the deviation from the strict collinear limit
and $v$ is an order one vector. Note $P^{2}=2p_{1}\cdot
p_{2}=-\omega_{1}\omega_{2}(x_{12})^{2}=-\epsilon p_{3}\cdot
v+\mathscr{O}(\epsilon)$, hence $\epsilon\sim(x_{12})^{2}$. This also
introduces $\mathscr{O}(\epsilon)$ uncertainty in the numerator of (156), and
thus an order one $(x_{12})^{0}$ uncertainty in the split function.
Nevertheless, it would not affect the singular terms in the split function
which we are really interested in.

From (165), we have

$\omega_{1}x_{1}^{a}+\omega_{2}x_{2}^{a}=\omega_{3}x_{3}^{a}+\mathscr{O}(\epsilon)~{},\qquad\omega_{1}+\omega_{2}=\omega_{3}+\mathscr{O}(\epsilon)~{},$
(166)

and thus

$x_{13}^{a}=\frac{\omega_{2}}{\omega_{1}+\omega_{2}}x_{12}^{a}+\mathscr{O}(\epsilon)~{},\qquad
x_{32}^{a}=\frac{\omega_{1}}{\omega_{1}+\omega_{2}}x_{12}^{a}+\mathscr{O}(\epsilon)~{}.$
(167)

Then (164) can be simplified further as

$\widehat{A}_{3}(\omega_{i},\varepsilon_{a_{i}},\alpha^{\prime})=2\Bigg{[}\omega_{2}x_{21}^{a_{1}}\delta^{a_{2}a_{3}}-\omega_{1}x_{12}^{a_{2}}\delta^{a_{1}a_{3}}+\frac{\omega_{1}\omega_{2}}{\omega_{1}+\omega_{2}}x_{12}^{a_{3}}\delta^{a_{1}a_{2}}+2\alpha^{\prime}{}\frac{\omega_{1}^{2}\omega_{2}^{2}}{\omega_{1}+\omega_{2}}\;x_{12}^{a_{1}}x_{12}^{a_{2}}x_{12}^{a_{3}}\Bigg{]}~{}.$
(168)

Substituting into (156), we get the split function

$\displaystyle{\mathsf{Split}}(p_{1}^{a_{1}}+p_{2}^{a_{2}}\to P^{a_{3}})$
$\displaystyle=$
$\displaystyle\frac{\widehat{A}_{3}(\omega_{i},\varepsilon_{a_{i}},\alpha^{\prime})+\mathscr{O}(\epsilon)}{2p_{1}\cdot
p_{2}}$ (169) $\displaystyle=$ $\displaystyle
2\frac{\omega_{2}x_{12}^{a_{1}}\delta^{a_{2}a_{3}}+\omega_{1}x_{12}^{a_{2}}\delta^{a_{1}a_{3}}-\frac{\omega_{1}\omega_{2}}{\omega_{1}+\omega_{2}}x_{12}^{a_{3}}\delta^{a_{1}a_{2}}-2\alpha^{\prime}{}\frac{\omega_{1}^{2}\omega_{2}^{2}}{\omega_{1}+\omega_{2}}\;x_{12}^{a_{1}}x_{12}^{a_{2}}x_{12}^{a_{3}}}{\omega_{1}\omega_{2}\;(x_{12})^{2}}+\mathscr{O}(\epsilon^{0})~{}.\qquad\qquad$
(170)

Up to an overall factor, this is the same as the $I$ introduced in (79).

Further performing the Mellin transformation leads to the following celestial
OPE for gluons:

$\displaystyle\mathcal{O}^{A}_{\Delta_{1},a_{1}}(x_{1})\mathcal{O}^{B}_{\Delta_{2},a_{2}}(x_{2})$
(171) $\displaystyle\sim$ $\displaystyle
f^{ABC}\frac{x_{12}^{a_{1}}\delta^{a_{2}a_{3}}B(\Delta_{1}-1,\Delta_{2})+x_{12}^{a_{2}}\delta^{a_{1}a_{3}}B(\Delta_{1},\Delta_{2}-1)-x_{12}^{a_{3}}\delta^{a_{1}a_{2}}B(\Delta_{1},\Delta_{2})}{(x_{12})^{2}}\mathcal{O}^{C}_{\Delta_{1}+\Delta_{2}-1,\;a_{3}}(x_{2})$
$\displaystyle-2\alpha^{\prime}{}f^{ABC}\frac{x_{12}^{a_{1}}x_{12}^{a_{2}}x_{12}^{a_{3}}B(\Delta_{1}+1,\Delta_{2}+1)}{(x_{12})^{2}}\mathcal{O}^{C}_{\Delta_{1}+\Delta_{2}+1,\;a_{3}}(x_{2})~{},$

where we restore the color factor $f^{ABC}$ and choose a proper overall
normalziation to make the formula as simple as possible. Here
$\mathcal{O}^{A}_{\Delta,a}$ denotes the celestial gluon operator with the
polarization vector $\varepsilon_{a}$ and color index $A$. This agrees with
(3.1) derived from worldsheet.

###  5.2 Celestial OPE for graviton/dilaton/KR field

In bosonic closed string theory, the amplitude for the closed string massless
fields is given by Polchinski:1998rr

$A^{c}_{GGG}=e_{1\mu\nu}e_{2\alpha\beta}e_{3\rho\sigma}T^{\mu\alpha\rho}(\alpha^{\prime}{})T^{\nu\beta\sigma}(\alpha^{\prime}{})~{}.$
(172)

So up to the rescaling of $\alpha^{\prime}{}$, the closed string amplitude is
essentially the “square” of the open string amplitude (157). This is the
simplest example of the famous KLT relation Kawai:1985xq .

In heterotic string, the amplitude for massless fields is Polchinski:1998rq

$A^{H}_{GGG}=e_{1\mu\nu}e_{2\alpha\beta}e_{3\rho\sigma}T^{\mu\alpha\rho}(\alpha^{\prime}{})T^{\nu\beta\sigma}(0)~{},$
(173)

where the $\alpha^{\prime}{}$ corrections in the right moving sector is absent
due to supersymmetry. For type I and IIA/IIB string, they have supersymmetry
in both left and right moving sectors, and the corresponding massless NS-NS
amplitude has no $\alpha^{\prime}{}$ corrections at all:

$A^{I/II}_{GGG}=e_{1\mu\nu}e_{2\alpha\beta}e_{3\rho\sigma}T^{\mu\alpha\rho}(0)T^{\nu\beta\sigma}(0)~{}.$
(174)

The absence of $\alpha^{\prime}{}$ in three-point amplitude agrees with the
absence of $\alpha^{\prime}{}$ correction in celestial OPEs we derived before
from worldsheet.

Let us just focus on the amplitude of massless fields in closed bosonic string
(172). As before, we choose $p_{1},p_{2}$ out-going and $p_{3}$ in-coming,
namely $\eta_{1}=\eta_{2}=-\eta_{3}=1$. We also choose the polarization
tensors
$e_{i}^{\mu\nu}=\varepsilon^{\mu}_{a_{i}}\varepsilon^{\nu}_{{\tilde{a}}_{i}}$,
which are the basis for polarizations. Then we can simplify the three-point
amplitude and write it in terms of celestial variables. The steps are almost
identical to the open string case in the previous subsection, except for the
doubling. The final result is

$A^{c}_{GGG}=\widehat{A}_{3}(\omega_{i},\varepsilon_{a_{i}},\frac{1}{4}\alpha^{\prime})\widehat{A}_{3}(\omega_{i},\varepsilon_{{\tilde{a}}_{i}},\frac{1}{4}\alpha^{\prime})~{}.$
(175)

Substituting into (156), we get the split function

$\displaystyle{\mathsf{Split}}(p_{1}^{a_{1}{\tilde{a}}_{1}}+p_{2}^{a_{2}{\tilde{a}}_{2}}\to
P^{a_{3}{\tilde{a}}_{3}})=\frac{A^{c}_{GGG}}{2p_{1}\cdot
p_{2}}=-\frac{\widehat{A}_{3}(\omega_{i},\varepsilon_{a_{i}},\frac{1}{4}\alpha^{\prime})\widehat{A}_{3}(\omega_{i},\varepsilon_{{\tilde{a}}_{i}},\frac{1}{4}\alpha^{\prime})}{(x_{12})^{2}\omega_{1}\omega_{2}}~{}.$
(176)

Up to an overall factor, the split function here is the same as that in (110).
We then need to perform Mellin transformation to obtain the celestial OPEs.
The steps are identical, and the results are also just given by (113)-(116)
except for the replacement $a\to a_{1},b\to a_{2},c\to
a_{3},\mathcal{V}\to\mathcal{O}$.

###  5.3 Celestial OPE for gluon and graviton/dilaton/KR field

In heterotic string, the three-point amplitude involving two gluons and one
graviton/dilaton/KR field is Polchinski:1998rq

$A^{H}_{ggG}=e_{1\mu\nu}e_{2\rho}e_{3\sigma}p_{23}^{\mu}T^{\nu\rho\sigma}(0)\delta^{AB}~{}.$
(177)

Note that there is no $\alpha^{\prime}{}$ correction due to supersymmetry. In
the case of graviton, the interaction responsible for this amplitude is just
the minimal coupling between gravitons and gluons. For bosonic string, one can
also compute the two gluons and one graviton amplitude from the open-closed
string setup. This will be discussed in appendix B and the amplitude is given
by (328), which suffers $\alpha^{\prime}{}$ corrections as we expected. Here
we will just focus on the heterotic case (177).

Following the same procedure as before, one can derive the celestial OPEs from
this three-point amplitude.

We first derive the celestial OPE between gluon and graviton/dilaton/KR field.
So we take $p_{1},p_{2}$ out-going and also choose the polarizations as
$e_{1}^{\mu\nu}=\varepsilon_{a_{1}}^{\mu}\varepsilon_{{\tilde{a}}_{1}}^{\nu}$,
$e_{2}=\varepsilon_{{\tilde{a}}_{2}}(p_{2}),e_{3}=\varepsilon_{{\tilde{a}}_{3}}(p_{3})$.
Then (177) becomes

$A^{H}_{ggG}=2\varepsilon_{a_{1}}\cdot
p_{2}\;\widehat{A}_{3}(\omega_{i},\varepsilon_{{\tilde{a}}_{i}},0)\delta^{AB}~{}.$
(178)

Substituting into (156), and performing the Mellin transformation, we get the
celestial OPE between gluon and graviton/dilaton/KR field

$\displaystyle\mathcal{O}_{\Delta_{1},a{\tilde{a}}}(x_{1})\mathcal{O}^{A}_{\Delta_{2},{\tilde{b}}}(x_{2})$
$\displaystyle\sim$ $\displaystyle
x_{12}^{a}\frac{x_{12}^{{\tilde{a}}}\delta^{{\tilde{b}}{\tilde{c}}}B(\Delta_{1}-1,\Delta_{2}+1)+x_{12}^{{\tilde{b}}}\delta^{{\tilde{a}}{\tilde{c}}}B(\Delta_{1},\Delta_{2})-x_{12}^{{\tilde{c}}}\delta^{{\tilde{a}}{\tilde{b}}}B(\Delta_{1},\Delta_{2}+1)}{(x_{12})^{2}}\mathcal{O}^{A}_{\Delta_{1}+\Delta_{2},{\tilde{c}}}(x_{2})~{},\qquad$

which agrees with (4.1).

The celestial OPE corresponding to the fusion of two gluons into one
graviton/dilaton/KR field can be similarly derived. To make the formula more
standard, we exchange the label of 1 and 3 in (177). Repeating the same steps
above leads us to the following celestial OPE between two gluons

$\displaystyle\mathcal{O}^{A}_{\Delta_{1},{\tilde{a}}}(x_{1})\mathcal{O}^{B}_{\Delta_{2},{\tilde{b}}}(x_{2})$
$\displaystyle\sim$
$\displaystyle\delta^{AB}x_{12}^{c}\frac{x_{12}^{{\tilde{a}}}\delta^{{\tilde{b}}{\tilde{c}}}B(\Delta_{1},\Delta_{2}+1)+x_{12}^{{\tilde{b}}}\delta^{{\tilde{a}}{\tilde{c}}}B(\Delta_{1}+1,\Delta_{2})-x_{12}^{{\tilde{c}}}\delta^{{\tilde{a}}{\tilde{b}}}B(\Delta_{1}+1,\Delta_{2}+1)}{(x_{12})^{2}}\mathcal{O}_{\Delta_{1}+\Delta_{2},\;c{\tilde{c}}}(x_{2})~{},\qquad$

which agrees with the second line of (4.1).

###  5.4 Celestial OPE in four dimensions

So far, we have derived the celestial OPEs in two different ways. The final
formula is a bit complicated. Now we specialize to 4D as a consistency check
of our results. In 4D, it is very convenient to use the helicity basis for
gluon and graviton. They are related to the previous polarization basis
through some linear combinations.

Following (20) and (21), we define the celestial gluon and graviton operators
in the helicity basis as

$\displaystyle\mathcal{O}_{\Delta,\pm}({\sf z},\bar{\sf z})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\Big{(}\mathcal{O}_{\Delta,1}(x)\mp
i\mathcal{O}_{\Delta,2}(x)\Big{)}~{},$ (181)
$\displaystyle\mathcal{O}_{\Delta,\pm 2}({\sf z},\bar{\sf z})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}\mathcal{O}_{\Delta,11}(x)-\mathcal{O}_{\Delta,22}(x)\mp
i\mathcal{O}_{\Delta,12}(x)\mp i\mathcal{O}_{\Delta,21}(x)\Big{)}~{},$ (182)

where the coordinates ${\sf z},\bar{\sf z}$ and $x$ are related through (18).

#### Gluon OPE

The general form of celestial gluon OPEs is given in (171). Specializing to 4D
and transforming to the helicity basis (181), we find that the first line of
(171) without $\alpha^{\prime}{}$ reduces

$\displaystyle\mathcal{O}^{A}_{\Delta_{1},+}({\sf z}_{1},\bar{\sf
z}_{1})\mathcal{O}^{B}_{\Delta_{2},+}({\sf z}_{2},\bar{\sf z}_{2})$
$\displaystyle\sim$ $\displaystyle\frac{f^{ABC}}{{\sf
z}_{12}}B(\Delta_{1}-1,\Delta_{2}-1)\mathcal{O}^{c}_{\Delta_{1}+\Delta_{2}-1,+}({\sf
z}_{2},\bar{\sf z}_{2})~{},$ (183)
$\displaystyle\mathcal{O}^{A}_{\Delta_{1},-}({\sf z}_{1},\bar{\sf
z}_{1})\mathcal{O}^{B}_{\Delta_{2},-}({\sf z}_{2},\bar{\sf z}_{2})$
$\displaystyle\sim$ $\displaystyle\frac{f^{ABC}}{\bar{\sf
z}_{12}}B(\Delta_{1}-1,\Delta_{2}-1)\mathcal{O}^{c}_{\Delta_{1}+\Delta_{2}-1,-}({\sf
z}_{2},\bar{\sf z}_{2})~{},$ (184)

and

$\displaystyle O^{A}_{\Delta_{1},+}({\sf z}_{1},\bar{\sf
z}_{1})O^{B}_{\Delta_{2},-}({\sf z}_{2},\bar{\sf z}_{2})$ $\displaystyle\sim$
$\displaystyle\frac{f^{ABC}}{\bar{\sf
z}_{12}}B(\Delta_{1}+1,\Delta_{2}-1)O^{c}_{\Delta_{1}+\Delta_{2}-1,+}({\sf
z}_{2},\bar{\sf z}_{2})$ (185) $\displaystyle+\frac{f^{ABC}}{{\sf
z}_{12}}B(\Delta_{1}-1,\Delta_{2}+1)O^{c}_{\Delta_{1}+\Delta_{2}-1,-}({\sf
z}_{2},\bar{\sf z}_{2})~{}.$

These are indeed the celestial OPEs for gluon in Yang-Mills theory Fan:2019emx
; Pate:2019lpp .

The second line of (171) with $\alpha^{\prime}{}$ coefficient arises from the
higher derivative interactions. We can similarly transform it into the
helicity basis. In particular, we find that only the following two OPEs have
singular terms

$\displaystyle\mathcal{O}^{A}_{\Delta_{1},+}({\sf z}_{1},\bar{\sf
z}_{1})\mathcal{O}^{B}_{\Delta_{2},+}({\sf z}_{2},\bar{\sf
z}_{2})\sim-\alpha^{\prime}{}\;f^{ABC}\frac{\bar{\sf z}_{12}^{2}}{{\sf
z}_{12}}B(\Delta_{1}+1,\Delta_{2}+1)\mathcal{O}^{C}_{\Delta_{1}+\Delta_{2}+1,-}({\sf
z}_{2},\bar{\sf z}_{2})~{},$ (186)
$\displaystyle\mathcal{O}^{A}_{\Delta_{1},-}({\sf z}_{1},\bar{\sf
z}_{1})\mathcal{O}^{B}_{\Delta_{2},-}({\sf z}_{2},\bar{\sf
z}_{2})\sim-\alpha^{\prime}{}\;f^{ABC}\frac{{\sf z}_{12}^{2}}{\bar{\sf
z}_{12}}B(\Delta_{1}+1,\Delta_{2}+1)\mathcal{O}^{C}_{\Delta_{1}+\Delta_{2}+1,+}({\sf
z}_{2},\bar{\sf z}_{2})~{},$ (187)

while the rest of OPEs are all regular. The two OPEs above with singular terms
have exactly the same structure as that predicted by the general formula of
OPE in Jiang:2021ovh ; Himwich:2021dau . The rest of helicity configurations
only give rise to regular OPEs, as their corresponding amplitudes vanish on-
shell. Indeed in 4D, the three-point on-shell amplitudes are fully determined
by the helicities due to little group scaling and locality. Each singular OPE
above is in one-to-one correspondence with the on-shell three-point amplitudes
of gluons arising from either YM theory or higher derivative interactions.

#### Graviton OPE

Similarly, one can transform the celestial graviton OPE in (116) into the
helicity basis in 4D. The final result, up to an overall constant, is given by

$\displaystyle\mathcal{O}_{\Delta_{1},+2}({\sf z}_{1},\bar{\sf
z}_{1})\mathcal{O}_{\Delta_{2},+2}({\sf z}_{2},\bar{\sf z}_{2})$
$\displaystyle\sim$ $\displaystyle-\frac{\bar{\sf z}_{12}}{{\sf
z}_{12}}B(\Delta_{1}-1,\Delta_{2}-1)\mathcal{O}_{\Delta_{1}+\Delta_{2},+2}({\sf
z}_{2},\bar{\sf z}_{2})$ (189)
$\displaystyle-\frac{\alpha^{\prime}{}^{2}}{16}\;\frac{\bar{\sf
z}_{12}^{5}}{{\sf
z}_{12}}B(\Delta_{1}+3,\Delta_{2}+3)\mathcal{O}_{\Delta_{1}+\Delta_{2}+4,-2}({\sf
z}_{2},\bar{\sf z}_{2})~{},$ $\displaystyle\mathcal{O}_{\Delta_{1},-2}({\sf
z}_{1},\bar{\sf z}_{1})\mathcal{O}_{\Delta_{2},-2}({\sf z}_{2},\bar{\sf
z}_{2})$ $\displaystyle\sim$ $\displaystyle-\frac{{\sf z}_{12}}{\bar{\sf
z}_{12}}B(\Delta_{1}-1,\Delta_{2}-1)\mathcal{O}_{\Delta_{1}+\Delta_{2},-2}({\sf
z}_{2},\bar{\sf z}_{2})$ (191)
$\displaystyle-\frac{\alpha^{\prime}{}^{2}}{16}\;\frac{{\sf
z}_{12}^{5}}{\bar{\sf
z}_{12}}B(\Delta_{1}+3,\Delta_{2}+3)\mathcal{O}_{\Delta_{1}+\Delta_{2}+4,+2}({\sf
z}_{2},\bar{\sf z}_{2})~{},$ $\displaystyle\mathcal{O}_{\Delta_{1},+2}({\sf
z}_{1},\bar{\sf z}_{1})\mathcal{O}_{\Delta_{2},-2}({\sf z}_{2},\bar{\sf
z}_{2})$ $\displaystyle\sim$ $\displaystyle-\frac{\bar{\sf z}_{12}}{{\sf
z}_{12}}B(\Delta_{1}-1,\Delta_{2}+3)\mathcal{O}_{\Delta_{1}+\Delta_{2},-2}({\sf
z}_{2},\bar{\sf z}_{2})$ (193) $\displaystyle-\frac{{\sf z}_{12}}{\bar{\sf
z}_{12}}B(\Delta_{1}+3,\Delta_{2}-1)\mathcal{O}_{\Delta_{1}+\Delta_{2},+2}({\sf
z}_{2},\bar{\sf z}_{2})~{},$

where we have only shown the singular terms. The structures of OPEs are again
in perfect agreement with Pate:2019lpp ; Jiang:2021ovh ; Himwich:2021dau .
Here those terms in OPE without $\alpha^{\prime}{}$ correspond to the Einstein
gravity, while the terms with $\alpha^{\prime}{}^{2}$ coefficient come from
higher derivative interactions of graviton. Note the $\alpha^{\prime}{}$
pieces in (116) are absent in the above formulae because they are regular.
Each singular term in the OPEs above is again in one-to-one correspondence
with the on-shell three-point amplitudes of gravitons.

#### Mixed OPE for gluon and graviton

Now we consider the mixed OPEs involving both gluons and gravitons. The
general formula is given by (5.3) and (5.3). Specializing to 4D and writing in
the helicity basis, they become

$\displaystyle\mathcal{O}_{\Delta_{1},\pm 2}({\sf z}_{1},\bar{\sf
z}_{1})\mathcal{O}^{A}_{\Delta_{2},\pm}({\sf z}_{2},\bar{\sf
z}_{2})\sim-\Big{(}\frac{\bar{\sf z}_{12}}{{\sf z}_{12}}\Big{)}^{\pm
1}B(\Delta_{1}-1,\Delta_{2})\mathcal{O}^{A}_{\Delta_{1}+\Delta_{2},\pm}({\sf
z}_{2},\bar{\sf z}_{2})~{},$ (194) $\displaystyle\mathcal{O}_{\Delta_{1},\pm
2}({\sf z}_{1},\bar{\sf z}_{1})\mathcal{O}^{A}_{\Delta_{2},\mp}({\sf
z}_{2},\bar{\sf z}_{2})\sim-\Big{(}\frac{\bar{\sf z}_{12}}{{\sf
z}_{12}}\Big{)}^{\pm
1}B(\Delta_{1}-1,\Delta_{2}+2)\mathcal{O}^{A}_{\Delta_{1}+\Delta_{2},\mp}({\sf
z}_{2},\bar{\sf z}_{2})~{},$ (195)

and

$\displaystyle\mathcal{O}^{A}_{\Delta_{1},+}({\sf z}_{1},\bar{\sf
z}_{1})\mathcal{O}^{B}_{\Delta_{2},-}({\sf z}_{2},\bar{\sf z}_{2})\sim$
$\displaystyle\delta^{AB}\frac{\bar{\sf z}_{12}}{{\sf
z}_{12}}B(\Delta_{1},\Delta_{2}+2)\mathcal{O}_{\Delta_{1}+\Delta_{2},-2}$
(196) $\displaystyle+\delta^{AB}\frac{{\sf z}_{12}}{\bar{\sf
z}_{12}}B(\Delta_{1}+2,\Delta_{2})\mathcal{O}_{\Delta_{1}+\Delta_{2},+2}~{},\qquad\qquad$

which are again consistent with Pate:2019lpp  and Jiang:2021ovh ;
Himwich:2021dau .

##  6 Celestial OPE in $\mathcal{N}=2$ string and $w_{1+\infty}$ algebra

In this section, we will further generalize previous OPE discussions to
$\mathcal{N}=2$ string theory Ooguri:1991fp ; Ooguri:1990ww ; Marcus:1992wi .
The $\mathcal{N}=2$ string theory has four dimensional target spacetime with
$(2,2)$ signature. So we will first discuss the kinematics and celestial
variables in $(2,2)$ signature. Then as before, we can compute the worldsheet
OPE and celestial OPE. An interesting feature is that we can even compute all
the $\overline{SL(2,\mathbb{R})}$ 252525 This is just half of the Lorentz
group. More precisely ${SL(2,\mathbb{R})}\times\overline{SL(2,\mathbb{R})}$ is
the double cover of $SO(2,2)$ which is the Lorentz group in $(2,2)$ signature.
descendants in the OPE, and this essentially comes from the momentum
conservation in $(2,2)$ signature. The soft sector of such OPE with
descendants just gives the $w_{1+\infty}$ algebra, after rewriting in terms of
chiral modes. Therefore, we give an indirect derivation of $w_{1+\infty}$ from
worldsheet in $\mathcal{N}=2$ string theory.

###  6.1 Kinematics in $(2,2)$ signature

Now we consider the spacetime in $(2,2)$ signature. In this case, it is very
convenient to introduce complex coordinates
$\mathcal{X}^{1}=(X_{1}+iX_{2})/\sqrt{2},\mathcal{X}^{2}=(X_{3}+iX_{4})/\sqrt{2}$
as well as their complex conjugates. The metric is then

$ds^{2}=(dX^{1})^{2}+(dX^{2})^{2}-(dX^{3})^{2}-(dX^{4})^{2}=\eta_{\mu\nu}d\mathcal{X}^{\mu}d\mathcal{X}^{\nu}=2d\mathcal{X}^{1}d\bar{\mathcal{X}}^{1}-2d\mathcal{X}^{2}d\bar{\mathcal{X}}^{2}~{}.$
(197)

We will write the vector as

$\bm{A}=(A^{1},A^{2},A^{\bar{1}},A^{\bar{2}})~{}.$ (198)

We will also use the Greek indices $\mu,\nu,\cdots=1,\bar{1},2,\bar{2}$,
unbarred indices $i,j,\cdots=1,2$, and barred indices
${\bar{i}},{\bar{j}},\cdots=\bar{1},\bar{2}$. Note we will identify
$\bar{A}^{i}\equiv\bar{A}^{\bar{i}}\equiv A^{\bar{i}}$. For real vector,
$A^{\bar{i}}$ is just the complex conjugate of $A^{i}$.

The metric $\eta_{\mu\nu}$ is given by

$\eta_{i{\bar{j}}}=\eta_{{\bar{j}}i}=\eta^{i{\bar{j}}}=\eta^{{\bar{j}}i}=(-1)^{i+1}\delta^{ij}~{},\qquad\eta_{ij}=\eta_{{\bar{i}}{\bar{j}}}=\eta^{ij}=\eta^{{\bar{i}}{\bar{j}}}=0~{}.$
(199)

We introduce the following notation

$A\cdot\bar{B}=\bar{B}\cdot A=A^{1}\bar{B}^{1}-A^{2}\bar{B}^{2}~{},$ (200)

where $A=(A^{1},A^{2}),\bar{B}=(B^{\bar{1}},B^{\bar{2}})$.

Given two vectors $\bm{A},\bm{B}$, we can define their inner product as

$\bm{B}\cdot\bm{A}=\bm{A}\cdot\bm{B}=\eta_{\mu\nu}A^{\mu}B^{\nu}=A\cdot\bar{B}+B\cdot\bar{A}=A^{1}\bar{B}^{1}-A^{2}\bar{B}^{2}+B^{1}\bar{A}^{1}-B^{2}\bar{A}^{2}~{}.$
(201)

This is real for real vectors $A,B$.

Given the vector in (198), we also define its dual $\bm{A}^{\vee}$ as

$\bm{A}=(A^{1},A^{2},A^{\bar{1}},A^{\bar{2}})~{},\qquad\bm{A}^{\vee}=(A^{1},A^{2},-A^{\bar{1}},-A^{\bar{2}})~{}.$
(202)

Note that $\bm{A}^{\vee}$ is an imaginary vector as
$(A^{\vee})^{\bar{i}}=-((A^{\vee})^{i})^{*}$, if $\bm{A}$ is a real vector.
And we have the inner product

$\bm{A}^{\vee}\cdot\bm{B}=A\cdot\bar{B}-B\cdot\bar{A}=A^{1}\bar{B}^{1}-A^{2}\bar{B}^{2}-B^{1}\bar{A}^{1}+B^{2}\bar{A}^{2}~{},$
(203)

which is purely imaginary for real vectors $A,B$. They satisfy
$\bm{A}\cdot\bm{B}=-\bm{A}^{\vee}\cdot\bm{B}^{\vee},\bm{A}^{\vee}\cdot\bm{B}=-\bm{B}^{\vee}\cdot\bm{A}$.

We are particularly interested in the null momentum satisfying
$\bm{k}\cdot\bm{k}=0$. It can then be parametrized as

$\bm{k}=\omega_{k}\;\hat{\bm{k}}~{},\qquad\omega_{k}\in\mathbb{R}~{}.$ (204)

Here the null vector is

$\hat{\bm{k}}({\sf z},\bar{\sf z})=\Big{(}1+{\sf z}\bar{\sf z}+i({\sf
z}-\bar{\sf z}),\;1-{\sf z}\bar{\sf z}+i({\sf z}+\bar{\sf z}),\;1+{\sf
z}\bar{\sf z}-i({\sf z}-\bar{\sf z}),\;1-{\sf z}\bar{\sf z}-i({\sf z}+\bar{\sf
z})\Big{)}~{},$ (205)

where ${\sf z},\bar{\sf z}$ are the coordinates on the celestial torus
Atanasov:2021oyu , instead of the celestial sphere. It is worth emphasizing
that the two variables ${\sf z},\bar{\sf z}$ are real and independent, instead
of the complex conjugate of each other.

Then the polarizations are given by

$\displaystyle\bm{\varepsilon}_{+}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\partial_{\sf z}\hat{\bm{k}}=\frac{1}{2}(1-i\bar{\sf
z},1+i\bar{\sf z},-1-i\bar{\sf z},-1+i\bar{\sf z})~{},\qquad$ (206)
$\displaystyle{\bm{\varepsilon}}_{-}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\partial_{\bar{\sf
z}}\hat{\bm{k}}=\frac{1}{2}(-1-i{\sf z},1+i{\sf z},1-i{\sf z},-1+i{\sf
z})~{}.$ (207)

They satisfy

$\bm{\varepsilon}_{+}\cdot\bm{\varepsilon}_{+}={\bm{\varepsilon}}_{-}\cdot{\bm{\varepsilon}}_{-}=\hat{\bm{k}}\cdot\hat{\bm{k}}=\bm{\varepsilon}_{+}\cdot\hat{\bm{k}}={\bm{\varepsilon}}_{-}\cdot\hat{\bm{k}}=0~{},\qquad{\bm{\varepsilon}}_{+}\cdot\bm{\varepsilon}_{-}=1~{}.$
(208)

For different momenta with $\hat{\bm{k}}_{i}=\hat{\bm{k}}({\sf z}_{i},\bar{\sf
z}_{i})$, we have the identities

$\hat{\bm{k}}_{i}\cdot\hat{\bm{k}}_{j}=4\,{\sf z}_{ij}\,\bar{\sf
z}_{ij}~{},\qquad\hat{\bm{k}}_{i}\cdot\bm{\varepsilon}_{+j}=2i\,\bar{\sf
z}_{ij}~{},\qquad\hat{\bm{k}}_{i}\cdot{\bm{\varepsilon}}_{-j}=2i\,{\sf
z}_{ij}~{},$ (209)

and

$\bm{\varepsilon}_{\pm i}\cdot\bm{\varepsilon}_{\pm
j}=0~{},\qquad\bm{\varepsilon}_{\pm i}\cdot\bm{\varepsilon}_{\mp
j}=1~{}.\qquad$ (210)

For null momentum in (204) (205), its dual satisfies

$\bm{k}^{\vee}=\omega_{k}\hat{\bm{k}}^{\vee}=\omega_{k}\Big{(}2(1+{\sf
z}^{2})\bm{\varepsilon}_{+}+i\,{\sf z}\,\hat{\bm{k}}\Big{)}~{},$ (211)

so up to a gauge transformation and an overall factor, $\bm{k}^{\vee}$ is
essentially the positive helicity polarization vector.

We also have 262626 It is worth mentioning that although $\hat{\bm{k}}^{\vee}$
is essentially the polarization vector, their inner products are non-
vanishing, in contrast to the vanishing inner product of two polarizations
with the same helicity in (210). The non-vanishing
$\hat{\bm{k}}_{i}^{\vee}\cdot\hat{\bm{k}}_{j}^{\vee}$ is just due to the gauge
transformation (211) and (209).

$\hat{\bm{k}}_{i}\cdot\hat{\bm{k}}_{j}=-\hat{\bm{k}}_{i}^{\vee}\cdot\hat{\bm{k}}_{j}^{\vee}=4{\sf
z}_{ij}\bar{\sf
z}_{ij}~{},\qquad\hat{\bm{k}}_{i}^{\vee}\cdot\hat{\bm{k}}_{j}=-\hat{\bm{k}}_{j}^{\vee}\cdot\hat{\bm{k}}_{i}=-4i(1+{\sf
z}_{i}\,{\sf z}_{j})\bar{\sf z}_{ij}~{}.$ (212)

###  6.2 Vertex operator in $\mathcal{N}=2$ string theory

The $\mathcal{N}=2$ string theory is constructed from the $\mathcal{N}=2$ non-
linear $\sigma$-model Ooguri:1991fp

$S=\frac{1}{\pi}\int
d^{2}z\,d^{2}\theta\,d^{2}\bar{\theta}\;K(\mathscr{X},\bar{\mathscr{X}})~{},$
(213)

where $\mathscr{X}^{i}$ are $\mathcal{N}=2$ chiral superfields

$\mathscr{X}^{i}(Z,\bar{Z};\theta^{-},\bar{\theta}^{-})=\mathcal{X}^{i}(Z,\bar{Z})+\psi_{L}^{i}(Z,\bar{Z})\theta^{-}+\psi_{R}^{i}(Z,\bar{Z})\bar{\theta}^{-}+F^{i}(Z,\bar{Z})\bar{\theta}^{-}\theta^{-},\quad
Z=z-\theta^{+}\theta^{-}~{},$ (214)

and
$K(\mathscr{X},\bar{\mathscr{X}})=\mathscr{X}\cdot\bar{\mathscr{X}}=\mathscr{X}^{1}\bar{\mathscr{X}}^{\bar{1}}-\mathscr{X}^{2}\bar{\mathscr{X}}^{\bar{2}}$
is the Kähler potential for flat metric. More explicitly in terms of component
fields, the action reads

$S=\frac{1}{\pi}\int
d^{2}z\;\Big{(}\partial\mathcal{X}\cdot\bar{\partial}\bar{\mathcal{X}}+\bar{\partial}\mathcal{X}\cdot\partial\bar{\mathcal{X}}+\bar{\psi}_{L}\cdot\bar{\partial}\psi_{L}+\bar{\psi}_{R}\cdot\partial\psi_{R}+\bar{F}\cdot
F\Big{)}~{}.$ (215)

Thus we have $F=0$ and the following OPEs 272727 Note we set
$\alpha^{\prime}{}=1$ here.

$\mathcal{X}^{i}(z_{1},{\bar{z}}_{1})\bar{\mathcal{X}}^{\bar{j}}(z_{2},{\bar{z}}_{2})\sim-\eta^{i{\bar{j}}}\ln|z_{12}|~{},\qquad\psi_{L}^{i}(z_{1})\bar{\psi}_{L}^{\bar{j}}(z_{2})\sim\frac{\eta^{i{\bar{j}}}}{z_{12}}~{},\qquad\psi_{R}^{i}({\bar{z}}_{1})\bar{\psi}_{R}^{\bar{j}}({\bar{z}}_{2})\sim\frac{\eta^{i{\bar{j}}}}{{\bar{z}}_{12}}~{}.$
(216)

The critical dimension of $\mathcal{N}=2$ string is four Fradkin:1981dd ;
Ooguri:1991fp . In order to have $\mathcal{N}=2$ supersymmetry on the world-
sheet, the target spacetime should be endowed with a complex structure,
implying that the signature of spacetime can be either (4,0) or (2,2). In the
former (4,0) case, $\mathcal{N}=2$ string only has ground state as the
physical state in the first-quantized string, and is thus not interesting. For
the latter (2,2) case and in the simplest version of $\mathcal{N}=2$ string,
there is a massless field in the spectrum, and it obeys a non-linear