aainstitutetext: Centre for High Energy Physics, Indian Institute of Science,
C.V. Raman Avenue, Bangalore 560012, Indiabbinstitutetext: International
Centre for Theoretical Sciences-TIFR,
Shivakote, Hesaraghatta Hobli, Bengaluru North, 560 089, Indiaccinstitutetext:
Département de Physique Théorique, Université de Genève,
24 quai Ernest-Ansermet, 1211 Genève 4, Suisse

# Twistor Coverings and Feynman Diagrams

 Faizan Bhat b , Rajesh Gopakumar b , Pronobesh Maity c  and
Bharathkumar Radhakrishnan<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Recently, a worldsheet dual to free ${\cal N}=4$ Super Yang-Mills has been
proposed in terms of twistor variables for ${\rm AdS}_{5}$, in parallel to
that for the ${\rm AdS}_{3}$ dual to the free symmetric orbifold CFT. In the
latter case, holomorphic covering maps play a central role in determining
correlators and are associated to Feynman diagrams. After recasting these maps
in terms of the worldsheet twistor variables for ${\rm AdS}_{3}$, we
generalise to ${\rm AdS}_{5}$. We propose stringy incidence relations and
appropriate reality conditions for the twistor covering maps. For some special
kinematic configurations of correlators, we exhibit an explicit construction
of the corresponding covering map. We find that the closed string worldsheet
corresponding to this map is related to a gauge theory Feynman diagram by the
Strebel construction, as for ${\rm AdS}_{3}/{\rm CFT}_{2}$. Rather strikingly,
the regularised Strebel area of the worldsheet reproduces the Feynman
propagator of the free field theory.

##  1 Introduction

Understanding the string dual to free ${\cal N}=4$ super Yang-Mills theory
would give a new vantage point from which one could set out to decipher the
AdS/CFT correspondence; this being the diametrically opposite regime from that
described by supergravity on the dual ${\rm AdS}_{5}\times S^{5}$ spacetime.
Recently, in Gaberdiel:2021qbb ; Gaberdiel:2021jrv , a proposal for a
worldsheet description of the dual tensionless string theory was made111A
somewhat different topological string approach to free super Yang-Mills is
proposed in Berkovits:2007zk ; Berkovits:2007rj ; Berkovits:2008qc ;
Berkovits:2019ulm .. This builds on the success of the description of the
corresponding tensionless limit for strings on ${\rm AdS}_{3}\times{\rm
S}^{3}\times{\mathbb{T}}^{4}$ dual to the free symmetric product orbifold CFT,
${\rm Sym}^{N}({\mathbb{T}}^{4})$Eberhardt:2018ouy ; Eberhardt:2019ywk ;
Eberhardt:2020akk ; Dei:2020zui ; Gaberdiel:2020ycd ; Knighton:2020kuh ;
Gaberdiel:2021kkp 222The tensionless limit for ${\rm AdS}_{3}$ i.e. with
massless higher spin fields in the spectrum, occurs for $k=1$ units of NS-NS
flux - see Gaberdiel:2017oqg ; Ferreira:2017pgt ; Gaberdiel:2018rqv ;
Giribet:2018ada  for some of the precursor works at this special point. For
works exploring the $k<1$ region, the transition to the $k>1$ region and their
CFT duals, see Giveon:2005mi ; Balthazar:2021xeh ; Martinec:2021vpk ;
Eberhardt:2021vsx .. Both tensionless string descriptions are in terms of a
set of free twistor variables, subject to a gauge constraint, with spectrally
flowed sectors in the Hilbert space. These spectrally flowed sectors are
crucial in accounting for the rich spectrum of single trace operators of the
dual large $N$ CFT. While a first principles quantisation of the worldsheet
theories is yet to be carried out in both cases, one can argue, based on a few
plausible assumptions on the physical state conditions, that the spectrum
nontrivially matches on both sides Eberhardt:2018ouy ; Gaberdiel:2021qbb ;
Gaberdiel:2021jrv .

Additional support, in the free super Yang-Mills example, comes from the fact
that the proposal gives a covariant version of the BMN Berenstein:2002jq
organisation of the large $N$ spectrum. Furthermore, this proposal is a closed
string cousin of the ambitwistor open string description Berkovits:2004hg ;
Mason:2013sva  of tree level ${\cal N}=4$ Yang-Mills gluon amplitudes
Witten:2003nn . The worldsheet realisation of higher spin symmetries in this
proposal has also been studied in Ahn:2021uri .

In the ${\rm AdS}_{3}/CFT_{2}$ case, one can, in fact, go beyond the agreement
of the spectrum and argue for the (manifest) equality of correlation functions
on both sides, giving a de facto derivation of the correspondence
Eberhardt:2019ywk ; Eberhardt:2020akk ; Dei:2020zui . On the worldsheet this
followed from the localisation, on the string moduli space, of physical
correlators onto those discrete points which admit finite degree holomorphic
covering maps from the worldsheet to the spacetime (more accurately, its ${\rm
S}^{2}$ boundary) with specified branching data. This localisation was later
seen to follow from a stringy incidence relation obeyed by the twistorial
worldsheet fields Dei:2020zui ; Knighton:2020kuh . This striking property on
the worldsheet is precisely what is needed to match the Lunin-Mathur
computation Lunin:2001  of correlators in the free symmetric orbifold CFT
Eberhardt:2019ywk . The latter computation is in terms of the same covering
maps from an auxiliary Riemann surface (now identified with the worldsheet,
cf. Pakman:2009 ) to the boundary ${\rm S}^{2}$.

The tensionless ${\rm AdS}_{3}/CFT_{2}$ example also gives a rather explicit
realisation Gaberdiel:2020ycd  of a general mechanism for implementing open-
closed string duality Gopakumar:2003ns ; Gopakumar:2004qb ; Gopakumar:2005fx .
This mechanism proceeds via a recasting of individual Feynman diagrams of the
field theory into worldsheets of the dual closed string theory using the
Strebel parametrisation of moduli space Strebel . The upshot of
Gaberdiel:2020ycd  was that, at least for correlators with large twist, the
localisation of the previous para, was precisely to the points on moduli space
which admit an integer Strebel differential - see Fig. 8 for a flowchart of
the logic of this correspondence. Strebel differentials are special
meromorphic quadratic differentials $\phi_{S}(z)dz^{2}$ which are completely
characterised by real “lengths” $l_{ij}$ (defined as
$\int_{a_{i}}^{a_{j}}dz\sqrt{\phi_{S}(z})$ between zeroes $(a_{i},a_{j})$);
the Strebel lengths give a real parametrisation of the string moduli space.
For an integer Strebel differential these lengths are proportional to
(positive) integers333These are very special points in the moduli space which
correspond to arithmetic Riemann surfaces (see for instance mulpenk1 ) and
also arise in the worldsheet dual to the Gaussian matrix model Razamat:2008zr
; Gopakumar:2011ev ; Gopakumar:2012ny ..

In the large twist limit of Gaberdiel:2020ycd , this integer Strebel
differential arises naturally as the Schwarzian of the covering map. As a
consequence, one can identify the integer Strebel lengths directly with the
number of “Wick contractions” between vertices of the Feynman diagram that is
associated to each covering map Pakman:2009  contribution to the field theory
correlator. In other words, there is a one to one correspondence between the
Feynman diagrams of the field theory and the individual closed string
worldsheets that the string correlator localises onto. This gives a precise
realisation of the proposal of Gopakumar:2005fx , as refined by Razamat:2008zr
, where the Strebel differential is the bridge between the Feynman diagrams of
the field theory and the worldsheets of the dual closed string. In fact, in
this approach, the Strebel construction of the closed string worldsheet by
gluing up strips with fixed Strebel lengths is the mathematical underpinning
of open-closed string duality, with the strips being identified with the open
string Feynman diagrams. See Fig. 1. In the $AdS_{3}/CFT_{2}$ case, it was
moreover seen that the closed string weight associated to each Feynman diagram
was the natural Nambu-Goto area for the worldsheet metric in Strebel gauge.

$\boldsymbol{z_{1}}$$\boldsymbol{z_{2}}$$\boldsymbol{z_{3}}$$\boldsymbol{a_{1}}$$\boldsymbol{a_{2}}$$\boldsymbol{a_{3}}$$\boldsymbol{a_{4}}$

$\boldsymbol{z_{1}}$$\boldsymbol{z_{2}}$$\boldsymbol{z_{3}}$$\boldsymbol{a_{1}}$$\boldsymbol{a_{2}}$$\boldsymbol{a_{3}}$$\boldsymbol{a_{4}}$

$\boldsymbol{z_{1}}$$\boldsymbol{z_{2}}$$\boldsymbol{z_{3}}$$\boldsymbol{a_{1}}$$\boldsymbol{a_{2}}$$\boldsymbol{a_{3}}$$\boldsymbol{a_{4}}$

Figure 1: Open-Closed String Duality and the Strebel Construction. (Left) A
foliation of a closed string Riemann Surface $\Sigma_{g,n}$ by the horizontal
trajectories of the associated Strebel differential. The coloured lines are
‘critical’ horizontal trajectories with $a_{k}$ the zeroes of the
differential. They form the Strebel graph. The $z_{i}$ are double poles and
$z_{4}=\infty$ (not shown). (Middle) The Strebel graph gives a canonical
decomposition (‘gluing’) of $\Sigma_{g,n}$ into (shaded) regions which are
conformal to infinite strips. The Strebel lengths are the widths of these
strips. (Right) These strips are viewed as open string diagrams
(Feynman-’tHooft double lines) which give rise to the free field Wick
contractions. The width of the strips is identified with the number of
contractions. Note that the skeleton Feynman graph (from gluing together
homotopic edges), denoted by the dashed line in the middle and left, is the
dual to the Strebel graph.

In this paper, we will take steps towards generalising the above
considerations and building a geometric picture in terms of twistor maps. We
begin with $AdS_{3}$ and write down the classical twistor configurations which
correspond to the holomorphic covering maps $\Gamma(z)$ that are the (exact)
saddle points of the (genus zero) worldsheet path integral. These twistor
covers are elegantly given in terms of the simple polynomials which enter into
$\Gamma(z)$ (see Eqs.(2.20), (2.19), (2.12)). This is consistent with the fact
that only a finite number of wedge modes (for each twist/spectral flow label
$w$) of the twistor fields are physical (Sec. 5 of Gaberdiel:2021jrv ) and
therefore excited in the classical configuration444Recall that the wedge modes
are those with mode number $r$ ($-\tfrac{w-1}{2}\leq r\leq\tfrac{w-1}{2}$).
They behave as generalised zero modes in the $w$th spectrally flowed sector..
We also find that the worldsheet stress tensor $T(z)$ of the twistor theory
(Eq. (2.24)), evaluated on these twistor configurations is exactly the
Schwarzian (and thus the Strebel differential, for large $w$) $S[\Gamma](z)$.
This rounds off the picture for ${\rm AdS}_{3}$ in giving the twistor versions
of the classical string configurations that the tensionless worldsheet path
integral localises onto Eberhardt:2019ywk . It will be interesting to see how
the localisation onto these configurations arises directly from a worldsheet
path integral for the twistor fields.

We next proceed to consider the analogues of these solutions for ${\rm
AdS}_{5}$. As per the proposal of Gaberdiel:2021qbb ; Gaberdiel:2021jrv , the
worldsheet theory dual to the tensionless ${\rm AdS}_{5}\times{\rm S}^{5}$ now
consists of the ambitwistor fields

$\begin{split}&Z^{I}=(\lambda^{\alpha},\mu^{\dot{\alpha}};\psi^{a})\\\
&Y_{J}=(\mu^{\dagger}_{\beta},\lambda^{\dagger}_{\dot{\beta}};\psi_{a}^{\dagger})\end{split}$
(1.1)

subject to the ambitwistor constraint $Y_{I}Z^{I}=0$. We will drop, from now
on, the fermionic fields which parametrise the ${\rm S}^{5}$ and focus on the
bosonic twistor variables $(\lambda^{\alpha},\mu^{\dot{\alpha}})$ and
$(\mu^{\dagger}_{\beta},\lambda^{\dagger}_{\dot{\beta}})$. These fields can be
viewed, on the one hand, as parametrising the ambitwistor space of the
conformal boundary of ${\rm AdS}_{5}$ or, on the other hand, the twistor space
of ${\rm AdS}_{5}$. We employ here the twistor description of (euclidean)
${\rm AdS}_{5}$ and its incidence relations following Adamo:2016rtr  (for our
notation and conventions, see Appendix A and Adamo:2017qyl ). We are thereby
led to propose a set of stringy incidence relations that the twistor
worldsheet configurations should obey in the tensionless limit.

$\begin{split}\mu^{\dagger}_{\alpha}(z)+X_{\alpha}^{\;\;\;\dot{\beta}}(z,\bar{z})\lambda^{\dagger}_{\dot{\beta}}(z)&=\lambda^{\beta}(z)\epsilon_{\beta\alpha}\\\
\mu^{\dot{\alpha}}(z)-X^{\dot{\alpha}}_{\;\;\;\beta}(z,\bar{z})\lambda^{\beta}(z)&=\frac{1}{2}R^{2}(z,\bar{z})\epsilon^{\dot{\alpha}\dot{\beta}}\lambda^{\dagger}_{\dot{\beta}}(z)\
.\end{split}$ (1.2)

Here, $X^{\dot{\alpha}}_{\;\;\;\beta}(z,\bar{z})$ are the stringy coordinates
along the $d=4$ boundary (in bispinor notation) while $R(z,\bar{z})$ is the
radial profile. One may view these stringy incidence relations as a general
solution of the (bosonic) ambitwistor constraint (see Eq.(3.17)) which must be
imposed in the worldsheet theory as a gauge constraint. While the relations
can be thus motivated, we expect them to be directly derived from the
properties of worldsheet correlators as in Dei:2020zui . This is currently
under investigation GGKM .

Imposing reality conditions on the $X^{\dot{\alpha}}_{\;\;\;\beta}(z,\bar{z})$
appropriate to the euclidean signature implies a natural set of reality
conditions on the twistor fields. This is a direct indication that the
(physical) left and right moving modes on the worldsheet are not independent
as already suggested in Gaberdiel:2021qbb ; Gaberdiel:2021jrv . These reality
conditions also enable us to invert the incidence relations and express the
spacetime string configurations in terms of the twistor configurations. We
will often restrict to configurations where the worldsheet is essentially at
the boundary of the ${\rm AdS}_{5}$, as was the case for ${\rm AdS}_{3}$.
However, unlike for ${\rm AdS}_{3}$, the boundary spacetime string
configurations are in general, non-holomorphic. Nevertheless, there is a
hidden holomorphy inherited from that of the underlying twistor fields. Thus,
we immediately observe from the structure of Eq.(1.2) that the matrix
$\bar{\partial}X_{\alpha}^{\;\;\;\dot{\beta}}(z,\bar{z})$ has one zero
eigenvalue.

We then specialise to a special kinematic regime where the dual field theory
operators are inserted at points $x_{i}$ all of which lie in a two dimensional
plane (note that this is not a restriction for 2- and 3-point correlators). In
this case, the holomorphic eigenvector picks out (together with the radial
direction) an $AdS_{3}$ subspace. We can write down the twistor configuration
again in terms of polynomials, whose ratio gives a bona fide holomorphic
covering map. The polynomial nature of the configuration is again a reflection
of the fact that only the wedge modes of the twistor fields are physical, as
proposed in Gaberdiel:2021qbb ; Gaberdiel:2021jrv .

The simplest such polynomial corresponds to the two point function of a BPS
highest weight state in the SYM theory. In this case, from the explicit form
of the covering map, we see that its Schwarzian, in the natural coordinates,
is a constant which agrees with the Strebel differential in this case. The
Strebel length is proportional to the number of Wick contractions. Further, we
compute the Strebel area i.e. the area computed with the worldsheet metric
given by $ds^{2}=|\phi_{S}(z)||dz|^{2}$ – “Strebel gauge” Gopakumar:2011ev .
This is formally infinite but needs to be evaluated with a careful
regularisation. When we do that we find that a Nambu-Goto weight with the
Strebel area precisely reproduces the free Feynman propagator associated to
the Feynman diagrams! This picture can be extended to a multipoint correlator
by gluing together the Strebel differentials in different patches, which we
then identify with the Schwarzian of the covering map. The additive nature of
the Strebel areas for each strip directly leads to the multiplicative form of
the individual Feynman propagators.

We view this as evidence that, in parallel to the ${\rm AdS}_{3}/CFT_{2}$
example, there exists a closed string picture in terms of twistor covering
maps underlying Feynman diagrams in the free field limit. In particular, the
fact that twistor covers of ${\rm AdS}_{5}$ reproduce the Feynman propagator,
through the Strebel construction, buttresses the twistor proposal
Gaberdiel:2021qbb ; Gaberdiel:2021jrv  for the worldsheet description of free
${\cal N}=4$ super Yang-Mills theory. Admittedly, we are working in a special
kinematic regime, for the 4-point and higher correlator, but we expect to be
able to overcome this restriction in the future.

The plan of this paper is as follows: Sec. 2 focusses on the ${\rm AdS}_{3}$
case, where we exhibit the twistor configurations corresponding to a covering
map. After a brief review of the twistor geometry for ${\rm AdS}_{5}$ in Sec.
3, we describe the euclidean reality conditions that we impose and some of its
consequences. Sec. 4 uses these ingredients to propose stringy incidence
relations for the worldsheet twistors. We then specialise to the classical
configurations that correspond to mapping worldsheets into an ${\rm AdS}_{3}$
subspace. We employ the explicit form of these maps in Sec. 5 to flesh out the
connection to Feynman diagrams in the gauge theory. We describe the resulting
picture, which agrees with the Strebel construction, moreover reproducing the
free propagator from the regularised Strebel area. Sec. 6 has brief concluding
remarks. Appendix A sets out our twistor conventions and the nature of the
twistor correspondence, with or without the Euclidean reality conditions.
Appendix B gives a quick recap, following Gaberdiel:2020ycd , of the Strebel
construction for tensionless ${\rm AdS}_{3}/CFT_{2}$, as a ready reference to
compare with the logic of Section 5.

##  2 Twistor covering maps in $AdS_{3}$

As we briefly recap in appendix B, correlators in the free symmetric product
$CFT_{2}$ are given in terms of covering maps from an auxiliary covering space
to the spacetime $S^{2}$, with specified branching data. In Eberhardt:2019ywk
it was shown that this is precisely mirrored in the tensionless string theory
on $AdS_{3}\times S^{3}\times{\mathbb{T}}^{4}$ (i.e. with $k=1$ unit of NS-NS
flux). Indeed, as anticipated in Lunin:2001 ; Pakman:2009  the covering space
is the worldsheet of the dual string theory. Thus the corresponding
correlators in the worldsheet theory have the remarkable property that they
are delta function supported on those points on the worldsheet moduli space
which admit covering maps consistent with the branching data. The fundamental
origin of this localisation was later seen to be a twistorial ward identity
for correlators in the free field realisation of the
$\mathfrak{psu}(1,1|2)_{1}$ worldsheet theory Dei:2020zui ; Knighton:2020kuh .

In Eberhardt:2019ywk  it was also shown that when there is an admissible
covering map, there is a semiclassical solution for the string worldsheet
which exhibits this covering of the boundary $S^{2}$ of the $AdS_{3}$ part of
the spacetime. As will be reviewed below, since the worldsheet is embedded in
$AdS_{3}$ it also has a nontrivial radial profile even though it is
essentially at the boundary. This gives a geometric picture, even in this
highly stringy regime, of the string worldsheet exhibiting a nontrivial
wrapping of the spacetime boundary. Since the underlying worldsheet theory is
free in this limit, as can be seen in terms of the original sigma model (see
Eq. (2.4) and below of Eberhardt:2019ywk ) or from the free field realisation
of the $\mathfrak{psu}(1,1|2)_{1}$ theory, these solutions are semiclassically
exact. In fact, one can show Eberhardt:2019ywk  that the semiclassical action
associated to these solutions gives precisely the Lunin-Mathur weight
associated to each covering map.

In this section, we will translate the solutions of Eberhardt:2019ywk , which
were given in terms of the $AdS_{3}$ coordinates, into classical solutions for
the twistor fields that describe the $\mathfrak{psu}(1,1|2)_{1}$ theory. We
will see that the solutions have a nice form which is also consistent with the
behaviour of quantum correlators of these fields as found in Dei:2020zui . We
will also show that the classical stress tensor evaluated on these solutions
agrees with the Schwarzian derivative of the covering map. Thus the stress
tensor of the free field realisation is closely related to the Strebel
differential due to the connection between the latter and the Schwarzian (at
least in the limit of large twist). These solutions will give some guidance
when we generalise to the case of the twistor classical solutions for the
worldsheet theory of tensionless strings in $AdS_{5}$ in the next section.

The conventional $AdS_{3}$ sigma model action, in first order form, is given
by

$S_{\mathrm{AdS}_{3}}=\frac{k}{4\pi}\int\mathrm{d}^{2}z\
(4\partial\Phi\bar{\partial}\Phi+\bar{\beta}\partial\bar{\gamma}+\beta\bar{\partial}\gamma-\mathrm{e}^{-2\Phi}\beta\bar{\beta}-k^{-1}R\Phi)\
.$ (2.1)

Here $\gamma(z)$ (and its conjugate) represent holomorphic coordinates for the
spacetime $S^{2}$’s with which we foliate euclidean $AdS_{3}$. The radial
coordinate $\Phi(z,\bar{z})$ is related to the usual poincare coordinate as
$r=e^{-\Phi}$, such that the boundary is at $r=0\leftrightarrow\Phi=\infty$.
In terms of these coordinates one finds classical solutions by considering the
holomorphic $\mathfrak{sl}(2,\mathbb{R})$ currents, as given by the Wakimoto
form and imposing the right boundary conditions at the insertions together
with the condition on the spacetime energies for the vertex operators (the
$J^{3}_{0}$ eigenvalue near each insertion).

$\begin{split}&J^{+}=\beta\\\ &J^{3}=-\partial\Phi+k\beta\gamma\\\
&J^{-}=-2\gamma\partial\Phi+\beta\gamma\gamma-\partial\gamma\ .\end{split}$
(2.2)

Here we have put the level $k=1$ in the usual (classical) Wakimoto
representation. Thus the classical solution for an $n$-point correlator of
vertex operators for the ground states of twisted sectors of the dual
symmetric orbifold CFT, is Eberhardt:2019ywk

$\gamma(z)=\Gamma(z)\
,\qquad\Phi(z,\bar{z})=-\log\epsilon-\frac{1}{2}\ln{|\partial\Gamma|^{2}}\
,\qquad\beta(z)=-\frac{(\partial\Phi)^{2}}{\partial\Gamma}\ .$ (2.3)

$\Gamma(z)$ is the covering map, from the worldsheet to the spacetime, with
branch points at the points $z_{i}$ where the vertex operators (in spectrally
flowed sectors $w_{i}$) are inserted, with the behaviour as $z\rightarrow
z_{i}$,

$\Gamma(z)=x_{i}+a_{i}^{\Gamma}(z-z_{i})^{w_{i}}+\ldots,\qquad(i=1\ldots n)\
.$ (2.4)

The covering map is uniquely specified by the branching data $(z_{i},w_{i})$
and specifying three of the $x_{i}$’s. Equivalently, if the $n$ locations
$x_{i}$ are specified, the covering map exists only for discrete choices of
$(n-3)$ of the $z_{i}$ i.e. on a discrete set of points on the moduli space
${\cal M}_{0,n}$. The radial field $\Phi(z,\bar{z})$ has an infrared cutoff
$\epsilon\rightarrow 0$ which indicates that the string is stuck at the
boundary ($\Phi=\infty$). We also see that $\partial\Phi$ is independent of
$\epsilon$ and exhibits a nontrivial profile which will play an important
role. Finally, the last relation in Eq. (2.2) for $\beta(z)$ follows from the
on-shell condition for the worldsheet stress tensor which implies (for these
ground state correlators) that

$J^{+}(z)J^{-}(z)=(J^{3}(z))^{2}\ .$ (2.5)

We now note that the $\mathfrak{sl}(2,\mathbb{R})$ currents in Eq. (2.2) are
bilinears of the free twistor fields of $AdS_{3}$ (see Eq. (2.2) of
Dei:2020zui ). Thus in terms of the pairs of symplectic bosons $\xi^{\pm}$ and
$\eta^{\pm}$ of $\mathfrak{u}(1,1|2)_{1}$ we have

$J^{3}(z)=-(\eta^{+}\xi^{-})(z)\ ,\quad J^{\pm}(z)=(\eta^{\pm}\xi^{\pm})(z)$
(2.6)

where we additionally have used the ambitwistor constraint which generates the
quotient $\mathfrak{psu}(1,1|2)_{1}$

$\xi^{+}\eta^{-}=\xi^{-}\eta^{+}\ .$ (2.7)

We notice that the on-shell constraint Eq. (2.5) is automatically satisfied in
terms of the twistor variables.

We also note that this free field representation has the gauge freedom in
which we rescale

$\xi^{\pm}(z)\rightarrow\lambda(z)\xi^{\pm}(z)\
,\qquad\eta^{\pm}(z)\rightarrow\lambda(z)^{-1}\eta^{\pm}(z)\ .$ (2.8)

Together with the ambitwistor constraint Eq. (2.7), this means one can choose
a gauge in which $\xi^{\pm}(z)=-\eta^{\pm}(z)$.

We can then solve for $\xi^{+}$ and $\eta^{+}$ using the expression for
$J^{+}(z)$ in Eq. (2.2) and equating it to that in Eq. (2.6), together with
the classical solution for $\beta(z)$ given in Eq. (2.3). Then the expression
for $J^{3}(z)$ allows us to solve for $\xi^{-}(z)$ and $\eta^{-}(z)$. We thus
find the classical twistor solutions (in our gauge) to be

$\xi^{+}=-\eta^{+}=-\frac{\partial\Phi}{\sqrt{\partial\Gamma}}=\frac{1}{2}\frac{\partial^{2}\Gamma}{(\partial\Gamma)^{\frac{3}{2}}},\quad\xi^{-}=-\eta^{-}=\frac{\Gamma\partial\Phi+\partial\Gamma}{\sqrt{\partial\Gamma}}=-\frac{\Gamma}{2}\frac{\partial^{2}\Gamma}{(\partial\Gamma)^{\frac{3}{2}}}+\sqrt{\partial\Gamma}\
.$ (2.9)

Note that the classical solutions obey stringy twistor “incidence relations”
in terms of the boundary variables.

$\xi^{-}+\Gamma\xi^{+}=\sqrt{\partial\Gamma}=\sqrt{\rho},\qquad\eta^{-}+\Gamma\eta^{+}=-\sqrt{\partial\Gamma}=-\sqrt{\rho}\
.$ (2.10)

The right hand side is not zero but rather proportional to the radial profile.
Thus we have defined a holomorphic radial profile $\rho(z)$, using Eq.(2.3),

$r^{2}(z,\bar{z})=e^{-2\Phi(z,\bar{z})}\equiv\epsilon^{2}\rho(z)\bar{\rho}(\bar{z})\
.$ (2.11)

The RHS of Eq.(2.10) vanishes as $(z-z_{i})^{\frac{w_{i}-1}{2}}$ as one
approaches any of the inserted vertex operators, where the string worldsheet
is pinned to the boundary.

Given a covering map

$\Gamma(z)=\frac{P_{N}(z)}{Q_{N}(z)}$ (2.12)

where $P_{N}(z)$ and $Q_{N}(z)$ are degree $N$ polynomials, define the
Wronskian

$W=P^{\prime}_{N}(z)Q_{N}(z)-Q^{\prime}_{N}(z)P_{N}(z)=C\prod_{i=1}^{n}(z-z_{i})^{w_{i}-1}\
.$ (2.13)

In the second equality we have used that
$\partial\Gamma(z)=\frac{W(z)}{Q_{N}^{2}(z)}$ and vanishes as
$(z-z_{i})^{w_{i}-1}$ near each of the $z_{i}$. Comparing the degrees of the
polynomials in Eq. (2.13), we see that this determines the Wronskian upto the
overall constant $C$.

Then the $AdS_{3}$ twistors Eq. (2.9) can be expressed nicely as

$\begin{split}\xi^{+}&=-\eta^{+}=\frac{1}{2W(z)^{\frac{3}{2}}}(W^{\prime}(z)Q_{N}(z)-2W(z)Q_{N}^{\prime}(z))\\\
\xi^{-}&=-\eta^{-}=-\frac{1}{2W(z)^{\frac{3}{2}}}(W^{\prime}(z)P_{N}(z)-2W(z)P_{N}^{\prime}(z))\end{split}$
(2.14)

We also see that

$W^{\prime}(z)=W(z)\sum_{i=1}^{n}\frac{(w_{i}-1)}{z-z_{i}}=W(z)\frac{\tilde{R}_{n-1}(z)}{\prod_{i=1}^{n}(z-z_{i})}$
(2.15)

where $\tilde{R}_{n-1}(z)=\sum_{i=1}^{n}(w_{i}-1)\prod_{j\neq i}^{n}(z-z_{j})$
is a polynomial of degree $(n-1)$ determined solely by the covering map data
$(z_{i},w_{i})$. Using this, we can rewrite

$\begin{split}W^{\prime}(z)Q_{N}(z)-2W(z)Q^{\prime}_{N}(z)&=W(z)\frac{\tilde{R}_{n-1}(z)Q_{N}(z)-2\prod_{i=1}^{n}(z-z_{i})Q^{\prime}_{N}(z)}{\prod_{i=1}^{n}(z-z_{i})}\\\
&=\frac{W(z)\tilde{Q}_{N+n-1}(z)}{\prod_{i=1}^{n}(z-z_{i})}\end{split}$ (2.16)

where $\tilde{Q}_{N+n-1}(z)$ is a polynomial of degree $(N+n-1)$ that is
completely determined by $Q_{N}(z)$ and the covering map data $(z_{i},w_{i})$.
In fact, notice that

$-\frac{1}{2}\frac{\tilde{Q}_{N+n-1}(z)}{\prod_{i=1}^{n}(z-z_{i})^{\frac{(w_{i}+1)}{2}}}=\frac{d}{dz}\Big{[}\frac{Q_{N}(z)}{\prod_{i=1}^{n}(z-z_{i})^{\frac{(w_{i}-1)}{2}}}\Big{]}\
.$ (2.17)

We can similarly define

$\tilde{P}_{N+n-1}(z)=\tilde{R}_{n-1}(z)P_{N}(z)-2\prod_{i=1}^{n}(z-z_{i})P^{\prime}_{N}(z)\
.$ (2.18)

Then we can express the twistor classical solutions in a nice symmetric form
as

$\xi^{+}=-\eta^{+}=\frac{\tilde{Q}_{N+n-1}(z)}{2\prod_{i=1}^{n}(z-z_{i})^{\frac{(w_{i}+1)}{2}}}=-\frac{d}{dz}\Big{[}\frac{Q_{N}(z)}{\prod_{i=1}^{n}(z-z_{i})^{\frac{(w_{i}-1)}{2}}}\Big{]}$
(2.19)

and similarly,

$\xi^{-}=-\eta^{-}=-\frac{\tilde{P}_{N+n-1}(z)}{2\prod_{i=1}^{n}(z-z_{i})^{\frac{(w_{i}+1)}{2}}}=\frac{d}{dz}\Big{[}\frac{P_{N}(z)}{\prod_{i=1}^{n}(z-z_{i})^{\frac{(w_{i}-1)}{2}}}\Big{]}\
.$ (2.20)

The takeaway from the final form of the twistor solutions in Eqs. (2.19, 2.20)
is that they are rational functions which are determined by the two
polynomials $Q_{N},P_{N}$ and the covering data. This fits in with our
expectation for the quantum theory that only the wedge modes of the twistors
are excited on-shell – see the discussion around Eq. (2.26) of
Gaberdiel:2021jrv . We see that these twistors have a singularity as
$z\rightarrow z_{i}$

$\xi^{\pm},\eta^{\pm}\sim\frac{1}{(z-z_{i})^{\frac{(w_{i}+1)}{2}}}\ .$ (2.21)

This is consistent with what we know about the OPE of these fields in the
quantum correlators. In particular, we know that at the origin ($x=0$) the
fields behave as in Eqs. (2.36-39) of Dei:2020zui . However, at a generic
point, the OPE of these fields with the spectrally flowed vertex operators is
given by Eq. (2.42) of Dei:2020zui . And in that notation
$\xi^{\pm(x)},\eta^{\pm(x)}$ behave as $(z-z_{i})^{-\frac{(w_{i}+1)}{2}}$. See
also Eq. (4.12) of Dei:2020zui . The combination which appears in the
incidence relation then has a regular OPE which is also reflected in the
vanishing of the right hand side of Eq. (2.10) as $z\rightarrow z_{i}$.

For the generalisation and comparison to the higher dimensional case we define
the ambitwistor variables

$Z^{I}=\begin{pmatrix}\xi^{+}\\\ \xi^{-}\end{pmatrix},\quad
Y_{I}=\begin{pmatrix}-\eta^{-}\\\ \eta^{+}\end{pmatrix}$ (2.22)

and the ambitwistor constraint Eq. (2.7) reads as

$Y_{I}Z^{I}=\xi^{-}\eta^{+}-\xi^{+}\eta^{-}=0\ .$ (2.23)

We also note for later reference that the stress tensor of the worldsheet
theory is given by

$\begin{split}T(z)&=\frac{1}{2}\epsilon_{\alpha\beta}(\xi^{\alpha}\partial\eta^{\beta}+\eta^{\alpha}\partial\xi^{\beta})\\\
&=-\eta^{-}\partial\xi^{+}+\eta^{+}\partial\xi^{-}=Y_{I}\partial
Z^{I}\end{split}$ (2.24)

where we have used Eq. (2.7) in the second line. Evaluating this in terms of
the classical solution Eq. (2.9) leads to

$\begin{split}T(z)&=\frac{1}{2}\left[\partial\left(\frac{\partial^{2}\Gamma}{\partial\Gamma}\right)-\frac{1}{2}\left(\frac{\partial^{2}\Gamma}{\partial\Gamma}\right)^{2}\right]\\\
&=\frac{1}{2}S\left[\Gamma(z)\right]\end{split}$ (2.25)

In Gaberdiel:2020ycd , it was shown that, in the large twist limit, the
quadratic differential defined by the Schwarzian of the covering map could be
identified with the integer Strebel differential which defines the
corresponding closed string worldsheet. We therefore see that there is a close
relation between the worldsheet stress tensor and the Strebel diffferential -
a relation which we think should be generic in open-closed string
duality555See remark 5.3 on p. 101 of Eynard-lec  for a reason why the two
might be related in a ‘heavy’ limit. We thank Edward Mazenc for drawing our
attention to this..

##  3 The Twistor space of ${\rm AdS}_{5}$

We want to generalise the twistor solutions of the previous section to the
case of ${\rm AdS}_{5}$. As a preliminary, in this section, we will describe
the twistor space of ${\rm AdS}_{5}$ and its relation to the ambitwistor space
of its conformal boundary, following the discussion in Adamo:2016rtr . We then
review the twistor incidence relations in ${\rm AdS}_{5}$ as well as the
reality condition that can be imposed in Euclidean signature. We will find
that imposing these reality conditions enables one to explicitly solve for the
usual spacetime coordinates of ${\rm AdS}_{5}$ in terms of the twistors.

We begin with the complex projective space $\mathds{CP}^{5}$ with homogeneous
co-ordinates $X^{IJ}$ represented as an antisymmetric $4\times 4$ matrix with
the identification $X\sim\lambda X$ with $\lambda\in\mathds{C}^{*}$. We can
then define a holomorphic metric on $\mathds{CP}^{5}$ which has the property
of being invariant under local scalings $X_{IJ}\to\lambda(X^{2})X_{IJ}$,

$ds^{2}=-\frac{dX^{2}}{X^{2}}+\left(\frac{X\cdot d\,X}{X^{2}}\right)^{2}$
(3.1)

where the contraction of indices is performed w.r.t $\epsilon_{IJKL}$. This is
the metric on complexified ${\rm AdS}_{5}$.

The conformal boundary of this spacetime is at $X^{2}=0$ i.e.

$M=\\{X^{2}=0\,|\,X\in\mathds{CP}^{5}\\}\ .$ (3.2)

Using the scaling freedom we can always parametrise the points in $M$ as,
Adamo:2016rtr  666For our conventions, especially with regard to spinor
indices, we refer the reader to Appendix A.

$X_{b}^{IJ}=\begin{bmatrix}\epsilon^{\alpha\beta}&&x^{\alpha\dot{\beta}}\\\
-x^{\dot{\alpha}\beta}&&\frac{1}{2}x^{2}\epsilon^{\dot{\alpha}\dot{\beta}}\end{bmatrix}\
.$ (3.3)

This form ensures $X_{b}^{2}=0$ i.e $\det(X_{b})=0$. It immediately gives

$(X_{b})_{IJ}=\begin{bmatrix}\frac{1}{2}x^{2}\epsilon_{\alpha\beta}&&-x_{\alpha\dot{\beta}}\\\
x_{\dot{\alpha}\beta}&&\epsilon_{\dot{\alpha}\dot{\beta}}\end{bmatrix}\ .$
(3.4)

Going away from the boundary, in terms of,

$I^{IJ}=\begin{bmatrix}0&0\\\
0&\epsilon^{\dot{\alpha}\dot{\beta}}\end{bmatrix}\ ,$ (3.5)

we can parametrise a generic point as

$X^{IJ}=(X_{b})^{IJ}+\frac{r^{2}}{2}I^{IJ}=\begin{bmatrix}\epsilon^{\alpha\beta}&&x^{\alpha\dot{\beta}}\\\
-x^{\dot{\alpha}\beta}&&\frac{1}{2}(x^{2}+r^{2})\epsilon^{\dot{\alpha}\dot{\beta}}\end{bmatrix}\
.$ (3.6)

We then have

$X_{IJ}X^{IJ}=2r^{2}\ .$ (3.7)

It is not difficult to verify that this parametrisation, when plugged into Eq.
(3.1), gives the usual Poincare metric, with $r^{2}$ being the radial
coordinate. We refer the reader to Adamo:2016rtr  for further details.

We now define the twistors

$\begin{split}&Z^{I}=(\lambda^{\alpha},\mu^{\dot{\alpha}})=(\lambda^{1},\lambda^{2},\mu^{1},\mu^{2})\\\
&Y_{J}=(\mu^{\dagger}_{\beta},\lambda^{\dagger}_{\dot{\beta}})=(\mu^{\dagger}_{1},\mu^{\dagger}_{2},\lambda^{\dagger}_{1},\lambda^{\dagger}_{2})\end{split}$
(3.8)

These will play the role of ambitwistor variables for the conformal boundary
of ${\rm AdS}_{5}$ but will more generally be viewed as twistor variables for
the bulk. We next describe the incidence relations that they obey which define
the twistor correspondence with spacetime on the boundary as well as the bulk.

###  3.1 The incidence relation on the boundary

On the boundary the above twistor variables $(Z^{I},Y_{J})$ can be defined as
being in the kernel of $(X_{b})_{IJ}$ and its dual $X_{b}^{IJ}$, respectively
Adamo:2016rtr . Note that the kernel is non-empty precisely at the boundary
where, as we noted $\det(X_{b})=0$. Then we see that the relation
$(X_{b})_{IJ}Z_{b}^{J}=0$ yields

$\begin{split}x_{\dot{\alpha}\beta}\lambda^{\beta}+\epsilon_{\dot{\alpha}\dot{\beta}}\,\mu^{\dot{\beta}}&=0\\\
\frac{1}{2}x^{2}\epsilon_{\alpha\beta}\lambda^{\beta}-x_{\alpha\dot{\beta}}\,\mu^{\dot{\beta}}&=0\end{split}\
.$ (3.9)

In fact, the second equation is a consequence of the first, using the first
identity in (A.9). The latter can also be expressed as

$\mu^{\dot{\alpha}}=x^{\dot{\alpha}}_{\;\;\;\beta}\lambda^{\beta}\ .$ (3.10)

The dual kernel relation $X_{b}^{IJ}Y^{b}_{J}=0$ yields

$\begin{split}\epsilon^{\alpha\beta}\mu^{\dagger}_{\beta}+x^{\alpha\dot{\beta}}\lambda^{\dagger}_{\dot{\beta}}&=0\\\
-x^{\dot{\alpha}\beta}\mu^{\dagger}_{\beta}+\frac{1}{2}x^{2}\epsilon^{\dot{\alpha}\dot{\beta}}\lambda^{\dagger}_{\dot{\beta}}&=0\
.\end{split}$ (3.11)

Again the second equation arises from the first and the latter can be neatly
expressed as

$\mu^{\dagger}_{\alpha}=-x_{\alpha}^{\;\;\;\dot{\beta}}\lambda^{\dagger}_{\dot{\beta}}\
.$ (3.12)

We also note that the incidence relation Eq. (3.10) and its dual Eq. (3.12)
together imply that the $(Z_{b}^{I},Y^{b}_{J})$ obey the ambitwistor
constraint

${\cal C}_{b}\equiv Z_{b}^{I}Y^{b}_{I}=0\ ,$ (3.13)

justifying the terminology.

Using the incidence relations (3.10) and (3.12), we can represent a point on
the boundary $M_{\mathds{C}}$ via any of the two bitwistors:

$\begin{split}(X_{b})^{IJ}=-\frac{Z_{1}^{[I}Z_{2}^{J]}}{\langle\lambda_{1}\lambda_{2}\rangle}\quad\text{and}\quad(X_{b})_{IJ}=-\frac{(Y_{1})_{[I}(Y_{2})_{J]}}{[\lambda^{\dagger}_{1}\lambda^{\dagger}_{2}]}\end{split}$
(3.14)

This is the usual twistor correspondence which associates a line in
$\mathds{PT}$ (determined by two points $Z_{1}$ and $Z_{2}$) to a point in
complexified Minkowski space $M_{\mathds{C}}$. The second relation above is
the analogous correspondence for the dual twistor space. We refer the reader
to Figs. 4 and 5 in appendix A which describes the twistor correspondence as a
double fibration, on both the boundary and in the bulk, in the complexified
case as well as after imposing the (euclidean) reality conditions of Sec. 3.3.

###  3.2 The incidence relation in the bulk

The above incidence relations in the boundary arise from a careful limit of
incidence relations in the bulk. When $r\neq 0$, we no longer have a
nontrivial kernel for $X_{IJ}$. Instead we impose the natural twistorial
incidence relation Adamo:2016rtr

$\begin{split}Z^{I}&=X^{IJ}Y_{J}\\\
\Rightarrow\begin{bmatrix}\lambda^{\alpha}\\\
\mu^{\dot{\alpha}}\end{bmatrix}&=\begin{bmatrix}\epsilon^{\alpha\beta}&&x^{\alpha\dot{\beta}}\\\
-x^{\dot{\alpha}\beta}&&\frac{1}{2}(x^{2}+r^{2})\epsilon^{\dot{\alpha}\dot{\beta}}\end{bmatrix}\begin{bmatrix}\mu^{\dagger}_{\beta}\\\
\lambda^{\dagger}_{\dot{\beta}}\end{bmatrix}\ .\end{split}$ (3.15)

This gives the following two independent equations:

$\begin{split}\mu^{\dagger}_{\alpha}+x_{\alpha}^{\;\;\;\dot{\beta}}\lambda^{\dagger}_{\dot{\beta}}&=\lambda^{\beta}\epsilon_{\beta\alpha}\\\
\mu^{\dot{\alpha}}-x^{\dot{\alpha}}_{\;\;\;\beta}\lambda^{\beta}&=\frac{1}{2}r^{2}\epsilon^{\dot{\alpha}\dot{\beta}}\lambda^{\dagger}_{\dot{\beta}}\end{split}$
(3.16)

where the second relation is obtained using the second identity in (A.9). We
also note that the bulk incidence relation in the form $Z^{I}=X^{IJ}Y_{J}$
immediately implies the quadric (or ambitwistor) relation

${\cal C}=Z^{I}Y_{I}=0$ (3.17)

since the $X^{IJ}$ is antisymmetric in its indices. Thus the incidence
relations can be viewed as disentangling the ambitwistor constraint.

To recover the boundary incidence relations in Eqs. (3.10) and (3.12) as $r\to
0$ we need to take the limit where we scale $Y_{J}\to\frac{Y^{b}_{J}}{r}$
whereas $Z^{I}\to Z_{b}^{I}$. In other words,
$(\mu^{\dagger}_{\beta},\lambda^{\dagger}_{\dot{\beta}})$ scale as
$\frac{1}{r}$ as we approach the boundary, while
$(\mu^{\dot{\alpha}},\lambda^{\beta})$ scale as $r^{0}$. Note that the overall
scale is immaterial for the ambitwistor variables on the boundary. Once we do
this rescaling, we see that we recover the boundary incidence relations as
$r\to 0$.

###  3.3 Reality conditions

Our considerations thus far apply to a complexified spacetime and the
corresponding twistors. When we specialise to the case of a real slice of the
spacetime, we need to impose a corresponding set of reality conditions on the
twistors. We will take the ${\rm AdS}_{5}$ spacetime to be euclidean since we
aim to focus on euclidean correlators in the dual CFT - there are likely to be
additional subtleties in lorentzian signature. One of the simplifications
afforded by euclidean signature is that, given a point in twistor space, the
corresponding spacetime point is determined, as we will see explicitly below.
See Figs. 4 and 5 for the fibration picture of this bijection. This is unlike
lorentzian signature where a point in twistor space corresponds to a null ray
in spacetime. For a discussion of reality conditions for twistors in different
signatures, see for instance Adamo:2017qyl  or, in the ${\rm AdS}_{5}$
context, Adamo:2016rtr .

In the Euclidean signature, the spacetime coordinates of complexified
Minkowski space obey

$\widehat{x}^{\dot{\alpha}}_{\;\;\;\beta}=x^{\dot{\alpha}}_{\;\;\;\beta}\ ,$
(3.18)

where $\widehat{x}^{\dot{\alpha}}_{\;\;\;\beta}$ is defined through Eq. (A.11)
(with bars referring to ordinary complex conjugation). Taking the complex
conjugate of the boundary incidence relation Eq. (3.10) and using Eq. (3.18)
gives

$\mu^{\dot{\alpha}}=x^{\dot{\alpha}}_{\;\;\;\beta}\lambda^{\beta}\quad\Rightarrow\quad\hat{\mu}^{\dot{\alpha}}=x^{\dot{\alpha}}_{\;\;\;\beta}\hat{\lambda}^{\beta}$
(3.19)

Similarly, we also find the complex conjugate relation to Eq. (3.12)

$\hat{\mu}^{\dagger}_{\alpha}=-x_{\alpha}^{\;\;\;\dot{\beta}}\hat{\lambda}^{\dagger}_{\dot{\beta}}$
(3.20)

Here we have defined the hatted twistor variables

$\begin{split}&\widehat{Z}^{I}\equiv(\hat{\lambda}^{\alpha},\hat{\mu}^{\dot{\alpha}})\equiv(-\bar{\lambda}^{2},\bar{\lambda}^{1},-\bar{\mu}^{2},\bar{\mu}^{1})\\\
&\widehat{Y}_{J}\equiv(\hat{\mu}^{\dagger}_{\beta},\hat{\lambda}^{\dagger}_{\dot{\beta}})\equiv(-\bar{\mu}_{2}^{\dagger},\bar{\mu}_{1}^{\dagger},-\bar{\lambda}_{2}^{\dagger},\bar{\lambda}_{1}^{\dagger})\end{split}$
(3.21)

In other words, the hatted twistors are essentially complex conjugates777Note,
however, that the conjugation operation defined here is not the identity when
applied twice. See below Eq. (1.17) in Adamo:2017qyl . and also obey the same
incidence relations as the original twistors.

It is easy to verify that if we take the radial coordinate $r$ to be real,
then we are extending the above reality conditions on the boundary into the
bulk (euclidean ${\rm AdS}_{5}$). The reality conditions on the twistors given
above also ensure that the bulk incidence relations Eq. (3.16) are satisfied
in terms of the hatted twistors. More succinctly,

$\widehat{Z}^{I}=X^{IJ}\widehat{Y}_{J}\ .$ (3.22)

The reality constraints can be used to obtain the spacetime coordinate
corresponding to a point in twistor space. First we consider a point on the
boundary. Since both $Z_{b}^{I},\widehat{Z}_{b}^{I}$ obey the incidence
relation with the same $x^{\dot{\alpha}}_{\;\;\beta}$, we can use the general
relation Eq. (3.14) to write

$\begin{split}(X_{b})^{IJ}=-\frac{Z_{b}^{[I}\widehat{Z}_{b}^{J]}}{\langle\lambda\hat{\lambda}\rangle}\quad\text{and}\quad(X_{b})_{IJ}=-\frac{Y^{b}_{[I}\widehat{Y}^{b}_{J]}}{[\lambda^{\dagger}\hat{\lambda}^{\dagger}]}\end{split}$
(3.23)

Using these, the condition $(X_{b})^{2}=0$ imposes additional constraints on
the boundary ambitwistors:

$(X_{b})_{IJ}(X_{b})^{IJ}=0\Rightarrow
Z_{b}\cdot\widehat{Y}^{b}=\widehat{Z}_{b}\cdot Y^{b}=0$ (3.24)

where we have used the ambitwistor condition $Z_{b}\cdot Y^{b}=0$. From Eq.
(3.23), we can read off the components

$\displaystyle x^{\alpha\dot{\beta}}$
$\displaystyle=\frac{\hat{\mu}^{\dagger\,\alpha}\lambda^{\dagger\,\dot{\beta}}-\mu^{\dagger\,\alpha}\hat{\lambda}^{\dagger\,\dot{\beta}}}{[\hat{\lambda}^{\dagger}\lambda^{\dagger}]}$
(3.25)
$\displaystyle=\frac{\lambda^{\alpha}\hat{\mu}^{\dot{\beta}}-\hat{\lambda}^{\alpha}\mu^{\dot{\beta}}}{\langle\hat{\lambda}\lambda\rangle}$
(3.26)

In the second relation above we have used
$x^{\alpha\dot{\beta}}=x^{\dot{\beta}\alpha}$ We also find

$x^{2}=2\frac{[\hat{\mu}\mu]}{\langle\hat{\lambda}\lambda\rangle}=2\frac{\langle\hat{\mu}^{\dagger}\mu^{\dagger}\rangle}{[\hat{\lambda}^{\dagger}\lambda^{\dagger}]}\
.$ (3.27)

We use a similar logic to determine the coordinates in ${\rm AdS}_{5}$ given
the twistors $(Z^{I},Y_{J})$ and their complex conjugates. One can write a
general ansatz for $X^{IJ}$:

$X^{IJ}=\alpha\frac{Z^{[I}\widehat{Z}^{J]}}{\langle\lambda\hat{\lambda}\rangle}+\beta\epsilon^{IJKL}\frac{Y_{[K}\widehat{Y}_{L]}}{[\lambda^{\dagger}\hat{\lambda}^{\dagger}]}$
(3.28)

Note that $X^{IJ}$ is real under the conjugation operation. The ansatz implies

$X_{IJ}=\frac{1}{2}\alpha\epsilon_{IJKL}\frac{Z^{[K}\widehat{Z}^{L]}}{\langle\lambda\hat{\lambda}\rangle}+2\beta\frac{Y_{[I}\widehat{Y}_{J]}}{[\lambda^{\dagger}\hat{\lambda}^{\dagger}]}$
(3.29)

where we have used the identity
$\epsilon_{IJKL}\epsilon^{KLMN}=2\delta^{[M}_{I}\delta^{N]}_{J}$. The bulk
incidence relations contain enough information to determine $\alpha$ and
$\beta$:

$\begin{split}Z^{I}=X^{IJ}Y_{J}=\alpha Z^{I}\frac{\widehat{Z}\cdot
Y}{\langle\lambda\hat{\lambda}\rangle}\Rightarrow\alpha=\frac{\langle\lambda\hat{\lambda}\rangle}{\widehat{Z}\cdot
Y}\end{split}$ (3.30)
$\begin{split}Y_{I}=-\frac{2}{r^{2}}X_{IJ}Z^{J}=-\frac{4}{r^{2}}\beta\,Y_{I}\frac{Z\cdot\widehat{W}}{[\lambda^{\dagger}\hat{\lambda}^{\dagger}]}\Rightarrow\beta=-\frac{r^{2}}{4}\frac{[\lambda^{\dagger}\hat{\lambda}^{\dagger}]}{Z\cdot\widehat{Y}}\end{split}$
(3.31)

where we have used $X_{IJ}X^{IJ}=2r^{2}$ Eq. (3.7).

From the conjugate incidence relation, we also find the reality constraint on
the quadric

$\widehat{Z}^{I}=X^{IJ}\widehat{Y}_{J}=-\alpha\widehat{Z}^{I}\frac{Z\cdot\widehat{Y}}{\langle\lambda\hat{\lambda}\rangle}\Rightarrow\widehat{Z}\cdot
Y=-Z\cdot\widehat{Y}.$ (3.32)

We will denote this constraint as $\mathcal{D}$,

$\mathcal{D}\equiv\;\widehat{Z}\cdot Y+Z\cdot\widehat{Y}=0\ .$ (3.33)

Note that this is a weaker constraint than the one that obeyed by the boundary
twistors Eq. (3.24).

Using these results, we get a (many-to-one - see Fig. 5) correspondence
between the twistors and a point in the bulk ${\rm AdS}_{5}$ spacetime

$X^{IJ}=\frac{1}{\widehat{Z}\cdot
Y}\left[Z^{[I}\widehat{Z}^{J]}+\frac{r^{2}}{4}\epsilon^{IJKL}Y_{[K}\widehat{Y}_{L]}\right]\
.$ (3.34)

To find $r^{2}$ and $x^{\alpha\dot{\beta}}$ we look at the upper diagonal
block of $X^{IJ}$ in Eq. (3.6) and equate that with r.h.s of Eq. (3.34), which
immediately gives

$r^{2}=-2\frac{\widehat{Z}\cdot
Y+\langle\lambda\hat{\lambda}\rangle}{[\lambda^{\dagger}\hat{\lambda}^{\dagger}]}\
.$ (3.35)

Similarly equating with any of the non-diagonal blocks in Eq. (3.6) gives

$x^{\alpha\dot{\beta}}=\frac{1}{\widehat{Z}\cdot
Y}\left[(\lambda^{\alpha}\hat{\mu}^{\dot{\beta}}-\hat{\lambda}^{\alpha}\mu^{\dot{\beta}})+\frac{r^{2}}{2}(\hat{\mu}^{\dagger\,\alpha}\lambda^{\dagger\,\dot{\beta}}-\mu^{\dagger\,\alpha}\hat{\lambda}^{\dagger\,\dot{\beta}})\right]\
,$ (3.36)

which can be rewritten, using the expression of $r^{2}$, as

$\begin{split}x^{\alpha\dot{\beta}}=\frac{\langle\hat{\lambda}\lambda\rangle}{\widehat{Z}\cdot
Y}\left[\frac{\lambda^{\alpha}\hat{\mu}^{\dot{\beta}}-\hat{\lambda}^{\alpha}\mu^{\dot{\beta}}}{\langle\hat{\lambda}\lambda\rangle}-\frac{\hat{\mu}^{\dagger\,\alpha}\lambda^{\dagger\,\dot{\beta}}-\mu^{\dagger\,\alpha}\hat{\lambda}^{\dagger\,\dot{\beta}}}{[\hat{\lambda}^{\dagger}\lambda^{\dagger}]}\right]+\frac{\hat{\mu}^{\dagger\,\alpha}\lambda^{\dagger\,\dot{\beta}}-\mu^{\dagger\,\alpha}\hat{\lambda}^{\dagger\,\dot{\beta}}}{[\hat{\lambda}^{\dagger}\lambda^{\dagger}]}\
.\end{split}$ (3.37)

On the boundary, this is consistent with the expression for
$x^{\alpha\dot{\beta}}$ which we obtained in Eq. (3.25) since the first term
in square brackets above vanishes. As another consistency check, we can
reproduce (3.16) with these expressions of $r^{2}$ and
$x^{\alpha\dot{\beta}}$. For example

$x^{\alpha\dot{\beta}}\lambda^{\dagger}_{\dot{\beta}}=(\lambda^{\alpha}-\mu^{\dagger\,\alpha})-\frac{1}{\widehat{Z}\cdot
Y}(\hat{\lambda}^{\alpha}\mu^{\dot{\beta}}\lambda^{\dagger}_{\dot{\beta}}+\lambda^{\alpha}\hat{\lambda}^{\beta}\mu^{\dagger}_{\beta}+\mu^{\dagger\,\alpha}\langle\lambda\hat{\lambda}\rangle)$
(3.38)

where we have used the expression of $r^{2}$. Now using the ambi-twistor
condition, third and fourth terms of the above expression become

$\hat{\lambda}^{\alpha}\mu^{\dot{\beta}}\lambda^{\dagger}_{\dot{\beta}}+\lambda^{\alpha}\hat{\lambda}^{\beta}\mu^{\dagger}_{\beta}=(-\hat{\lambda}^{\alpha}\lambda^{\beta}+\lambda^{\alpha}\hat{\lambda}^{\beta})\mu^{\dagger}_{\beta}=-\epsilon^{\alpha\beta}\langle\lambda\hat{\lambda}\rangle\mu^{\dagger}_{\beta}$
(3.39)

This cancels the last term in (3.38) and we get back the first equation in
(3.16).

##  4 Twistor space covering maps in ${\rm AdS}_{5}$

We have described the twistor space geometry of ${\rm AdS}_{5}$ and the
reality conditions which fix a point in the bulk in terms of the twistor
variables. We want to use this geometrical picture to describe string
configurations which capture the dual ${\cal N}=4$ super Yang-Mills theory as
recently proposed in Gaberdiel:2021qbb ; Gaberdiel:2021jrv . We will not give
a complete picture here – this will be described elsewhere – but instead focus
on a special configuration which is a direct generalisation of the ${\rm
AdS}_{3}$ case as described in Sec. 3.

The basic idea is that we will consider holomorphic worldsheet twistor fields
$Y^{I}(z),Z_{I}(z)$, together with the corresponding right moving
antiholomorphic twistor fields
$\widehat{Z}^{I}(\bar{z}),\widehat{Y}_{J}(\bar{z})$, which are related to the
holomorphic fields by the conjugation operation defined in Eq. (3.21). Here
the conjugation is assumed to act on the worldsheet variables on which the
fields now depend, in the standard way $z\to\bar{z}$. Thus the right movers
are not independent degrees of freedom and determined in terms of the left
movers by Eq. (3.21). This is consistent with the observation in
Gaberdiel:2021qbb ; Gaberdiel:2021jrv  that we only need one copy of the wedge
modes (i.e. not both left and right movers) of the twistor fields to account
for the correct spectrum of the free super Yang-Mills theory888It may be that
we only need to impose this reality condtion on the finite number of wedge
modes of the worldsheet twistor fields. Since these are the only ones excited
in the classical solutions, we will not be able to distinguish the two
possibilities in the present discussion. The right prescription will
presumably arise from a complete worldsheet analysis. We thank Matthias
Gaberdiel for discussions on this point and ongoing collaborations GGKM ..

To connect with the Feynman diagrams for the dual gauge theory operators, we
will need to translate twistor string configurations into ${\rm AdS}$
spacetime configurations. We will determine the latter by requiring that the
twistors obey the ${\rm AdS}_{5}$ twistor incidence relations, together with
the reality conditions. In other words, we now require the stringy versions of
Eq. (3.16).

$\begin{split}\mu^{\dagger}_{\alpha}(z)+X_{\alpha}^{\;\;\;\dot{\beta}}(z,\bar{z})\lambda^{\dagger}_{\dot{\beta}}(z)&=\lambda^{\beta}(z)\epsilon_{\beta\alpha}\\\
\mu^{\dot{\alpha}}(z)-X^{\dot{\alpha}}_{\;\;\;\beta}(z,\bar{z})\lambda^{\beta}(z)&=\frac{1}{2}R^{2}(z,\bar{z})\epsilon^{\dot{\alpha}\dot{\beta}}\lambda^{\dagger}_{\dot{\beta}}(z)\
.\end{split}$ (4.1)

Here we have denoted the string configurations in ${\rm AdS}_{5}$ by
$X^{\dot{\alpha}}_{\;\;\;\beta}\lambda^{\beta}(z,\bar{z})$ and the radial
$R(z,\bar{z})$. As we will see shortly, unlike in ${\rm AdS}_{3}$, the
longitudinal coordinates $X^{\dot{\alpha}}_{\;\;\;\beta}$ are now no longer
purely holomorphic in the worldsheet coordinates. Ultimately, these relations
will not need to be postulated; rather, we expect they will arise from an
analysis of the ward identities of the worldsheet CFT as in Dei:2020zui . This
is currently under investigation GGKM .

In fact, since the bulk incidence relations Eq. (4.1) hold pointwise on the
worldsheet, we will have the same expressions for the bulk coordinates
$R(z,\bar{z}),X^{\dot{\alpha}}_{\;\;\;\beta}(z,\bar{z})$, in terms of the
twistor fields, as given in Eqs. (3.35), (3.36). Since these depend on both
the twistor fields and their conjugates, we clearly see that they cannot be
holomorphic (or anti-holomorphic). Note that since the twistor fields obey the
bulk incidence relation it also immediately follows that they obey, pointwise,
the ambitwistor constraint

$\lambda^{\dagger}_{\dot{\alpha}}(z)\mu^{\dot{\alpha}}(z)+\mu^{\dagger}_{\alpha}(z)\lambda^{\alpha}(z)=0\
.$ (4.2)

One may alternately view the ambitwistor constraint as primary and then the
requirement of the bulk incidence relations in Eq. (4.1) follows. The latter
are a general way of satisfying the ambitwistor constraints – see the
discussion below Eq. (3.16). The ambitwistor constraint arose as a fundamental
gauge constraint requirement in the worldsheet proposal of Gaberdiel:2021qbb ;
Gaberdiel:2021jrv .

We will further assume that as in ${\rm AdS}_{3}$, the worldsheet is localised
near the boundary i.e. that $R(z,\bar{z})\propto\epsilon\to 0$. This is also
physically motivated by the fact that the dual free theory is at the UV fixed
point and hence essentially glued to the boundary of ${\rm AdS}_{5}$, using
the relation between scale and the radial direction. We expect, however, there
to be a nontrivial radial (Liouville) profile $\partial{\ln{R}}$ which we will
not determine here.

As discussed in Sec. 3.2, when the radial coordinate goes to zero, the bulk
incidence relations reduce to the boundary incidence relations. Thus for
string configurations near the boundary we must have the analogues of Eqs
(3.10),(3.12), namely,

$\begin{split}\mu^{\dot{\alpha}}(z)&=X^{\dot{\alpha}}_{\;\;\;\beta}(z,\bar{z})\lambda^{\beta}(z)\\\
\mu^{\dagger}_{\alpha}(z)&=-X_{\alpha}^{\;\;\;\dot{\beta}}(z,\bar{z})\lambda^{\dagger}_{\dot{\beta}}(z)\
.\end{split}$ (4.3)

We will mostly work with these (stringy) incidence relations on the boundary
given our physical assumption of the worldsheet being localised there.

As mentioned at the beginning of the section, we are also imposing the reality
conditions pointwise in $(z,\bar{z})$. Therefore we can use the above
incidence relations and the corresponding conjugate relations to obtain the
stringy generalisation of Eq. (3.25)

$\displaystyle X^{\alpha\dot{\beta}}(z,\bar{z})$
$\displaystyle=\frac{\lambda^{\alpha}(z)\hat{\mu}^{\dot{\beta}}(\bar{z})-\hat{\lambda}^{\alpha}(\bar{z})\mu^{\dot{\beta}}(z)}{\langle\hat{\lambda}(\bar{z})\lambda(z)\rangle}$
(4.4)
$\displaystyle=\frac{\hat{\mu}^{\dagger\,\alpha}(\bar{z})\lambda^{\dagger\,\dot{\beta}}(z)-\mu^{\dagger\,\alpha}(z)\hat{\lambda}^{\dagger\,\dot{\beta}}(\bar{z})}{[\hat{\lambda}^{\dagger}(\bar{z})\lambda^{\dagger}(z)]}$
(4.5)

One sees, somewhat reassuringly, that the spacetime configurations
$X^{\alpha\dot{\beta}}(z,\bar{z})$ are not purely holomorphic or anti-
holomorphic even though the twistor fields are. Nevertheless, the holomorphic
twistor fields obey the incidence relations with
$X^{\alpha\dot{\beta}}(z,\bar{z})$ in Eq. (4.3), as can be seen from the
explicit form in Eq. (4.4).

The mixed $z$ dependence here is unlike the ${\rm AdS}_{3}$ case where there
was a split of the boundary spacetime coordinates into what we had denoted as
$\Gamma(z)$ and its conjugate. Despite that, there is a sense in which there
is a kind of local holomorphy in the boundary spacetime. This is reflected in
the fact that even though $X^{\alpha\dot{\beta}}(z,\bar{z})$ is not
holomorphic, the matrix $\bar{\partial}X^{\alpha\dot{\beta}}(z,\bar{z})$ (as
also $\partial X^{\alpha\dot{\beta}}(z,\bar{z})$) has a zero eigenvector.

This follows directly by applying $\bar{\partial}$ on both sides of the two
equations in Eq. (4.3). We obtain

$\begin{split}\bar{\partial}X^{\dot{\alpha}}_{\;\;\;\beta}(z,\bar{z})\lambda^{\beta}(z)&=0\
,\\\
\bar{\partial}X_{\alpha}^{\;\;\;\dot{\beta}}(z,\bar{z})\lambda^{\dagger}_{\dot{\beta}}(z)&=0\
.\end{split}$ (4.6)

In other words, we see that $\lambda^{\beta}(z)$ is the zero eigenmode of
$\bar{\partial}X^{\dot{\alpha}}_{\;\;\;\beta}(z,\bar{z})$ while
$\lambda^{\dagger}_{\dot{\beta}}(z)$ is the zero eigenvector of the transposed
matrix $\bar{\partial}X_{\alpha}^{\;\;\;\dot{\beta}}$. One immediate
consequence of this, as can also be easily verified directly using the
expressions in Eq. (4.4), is that

$\bar{\partial}X_{\alpha\dot{\beta}}\bar{\partial}X^{\alpha\dot{\beta}}=0\ .$
(4.7)

For $\partial X^{\alpha\dot{\beta}}$, we can similarly write down the zero
eigenvectors in terms of the conjugate twistor fields and the analogue of Eq.
(4.7). We interpret the presence of this zero eigenvector of
$\bar{\partial}X^{\dot{\alpha}}_{\;\;\;\beta}(z,\bar{z})$ as indicating that,
locally, on the worldsheet and hence in space time, we can always view the
string configuration $X^{\dot{\alpha}}_{\;\;\;\beta}(z,\bar{z})$ as a
holomorphic embedding into the boundary of an ${\rm AdS}_{3}$ subspace of the
bulk spacetime. We will exploit this interpretation in what follows.

We also note that if we consider the more general stringy incidence relations,
we can arrive at a similar conclusion. Namely, applying $\bar{\partial}$ on
the first equation of Eq. (4.1) we find that
$\lambda^{\dagger}_{\dot{\beta}}(z)$ continues to be the zero eigenvector of
$\bar{\partial}X_{\alpha}^{\;\;\;\dot{\beta}}(z,\bar{z})$. However, we see
from the non-holomorphic dependence of $R^{2}(z,\bar{z})$ in the RHS of the
second equation of Eq. (4.1) that $\lambda^{\beta}(z)$ is no longer the zero
eigenmode of $\bar{\partial}X^{\dot{\alpha}}_{\;\;\;\beta}(z,\bar{z})$.

###  4.1 Maps to an ${\rm AdS}_{3}$ subspace

In this subsection, we will consider the subset of solutions where the string
lies entirely within an ${\rm AdS}_{3}$ subspace, or equivalently the boundary
insertion points $x_{i}$ lie in a two dimensional plane. Note that this can
always be done for the case of two and three point functions. For four point
functions and higher this is a special configuration. We will work in this
special kinematic setup as it will enable us to be very explicit. We will also
be able to illustrate, in the next section, how the corresponding covering
maps reproduce the boundary free field Feynman diagram answers.

Since $\bar{\partial}X^{\dot{\alpha}}_{\;\;\;\beta}(z,\bar{z})$ has one zero
eigenvalue, we will choose it to be in diagonal form (which can always be done
locally, but here we are doing so globally)

$\bar{\partial}X^{\dot{\alpha}}_{\;\;\;\beta}(z,\bar{z})=\begin{bmatrix}0&&0\\\
0&&-\bar{\partial}\bar{V}(\bar{z})\end{bmatrix}\ .$ (4.8)

Recall from the general form of $X^{\dot{\alpha}}_{\;\;\;\beta}$ (see Eq.
(A.13)) that this implies we have taken
$\bar{\partial}U=\bar{\partial}\bar{U}=0$ i.e. $U=0$ without loss of
generality. We also have $\bar{\partial}V=0$ i.e. $V=V(z)$ and thus the purely
antiholomorphic dependence in $\bar{V}(\bar{z})$. In other words, we have the
string configuration

$X^{\dot{\alpha}}_{\;\;\;\beta}(z,\bar{z})=\begin{bmatrix}-V(z)&&0\\\
0&&-\bar{V}(\bar{z})\end{bmatrix}\ .$ (4.9)

As mentioned, the string configuration is entirely in the $(x_{1},x_{2})$
plane.

The zero eigenvector $\lambda^{\beta}(z)$ of Eq. (4.8) then has the form

$\lambda^{\beta}(z)=\begin{bmatrix}\lambda^{1}(z)\\\ 0\end{bmatrix}\ .$ (4.10)

Together with Eq. (4.9) and Eq. (4.3) we have

$\mu^{\dot{\alpha}}(z)=-\begin{bmatrix}V(z)\lambda^{1}(z)\\\ 0\end{bmatrix}\
.$ (4.11)

Thus $\mu^{1}(z)=-V(z)\lambda^{1}(z)$. Since $V(z)$ is a (finite degree)
covering map from the genus zero worldsheet to the $S^{2}$ boundary of the
${\rm AdS}_{3}$, it will be a ratio of polynomials. As in the case of ${\rm
AdS}_{3}$, we can achieve this if the twistor fields are given by rational
functions. In other words, we are only exciting finitely many modes around
each worldsheet insertion of the vertex operators. We will therefore consider
fields of the form

$\lambda^{1}(z)=\frac{R_{n-1}(z)Q^{1}_{N}(z)}{\prod_{i=1}^{n}(z-z_{i})^{\frac{w_{i}}{2}}}\
,\,\,\,\mu^{1}(z)=\frac{R_{n-1}(z)P^{1}_{N}(z)}{\prod_{i=1}^{n}(z-z_{i})^{\frac{w_{i}}{2}}}\
.$ (4.12)

This is a generalisation of the expressions Eqs. (2.19),(2.20). We note that
the order of the poles in the denominator in the above is shifted. If
$w^{2d}_{i},w^{4d}_{i}$ denote the spectral flow parameters of the ${\rm
AdS}_{3}$ and ${\rm AdS}_{5}$ theories, (or equivalently, the twisted sector
and the number of Yang-Mills bits in the dual CFT) respectively, then we have
a shift (see Eq. (5.11) of Gaberdiel:2021jrv )

$w^{4d}_{i}=w^{2d}_{i}+1\ .$ (4.13)

At the risk of creating confusion we will drop the superscripts, with the
context hopefully making clear which label is being considered and Eq. (4.13)
being the dictionary between the two.

We see from Eq.(4.12) that the twistor fields $\lambda^{1}(z),\mu^{1}(z)$ near
$z=z_{i}$ have a singularity of order $\frac{1}{(z-z_{i})^{\frac{w}{2}}}$.
This is consistent with the fact that the twistors have spin half and only
have wedge modes with mode number $r\leq\frac{w_{i}-1}{2}$ excited. At the
same time we expect a branching behaviour of order $(w_{i}-1)$ near each such
vertex operator insertion since $r\geq-\frac{w_{i}-1}{2}$ (cf. the shift of Eq
(4.13))999This is true for a generic vertex operator. However, for the BPS
highest weight state the branching is of order $w_{i}$ due to additional mode
annihilation. We thank M. Gaberdiel for bringing this to our attention. . In
other words, from Eq. (4.12), we can define the covering map in position space

$-V(z)=\frac{\mu^{1}(z)}{\lambda^{1}(z)}=\frac{P_{N}(z)}{Q_{N}(z)}\ .$ (4.14)

We expect that $V(z)\sim V_{i}+a_{i}(z-z_{i})^{w_{i}-1}$, where
$V_{i}=(x_{(i)1}+ix_{(i)2})$.

Note also that we have the condition $N+n-1=\frac{1}{2}\sum_{i}w_{i}$ for the
fields in Eq. (4.12) to be rational functions on the worldsheet with the
prescribed poles. Therefore, $\frac{1}{2}\sum_{i}w_{i}$ must be an integer.
Indeed this is consistent with the fact that the number of free field wick
contractions in a Feynman diagram for a correlator built of operators with
$\\{w_{i}\\}$ fields, $\langle\prod_{i=1}^{n}{\cal
O}^{(w_{i})}(x_{i})\rangle$, is given precisely by $\frac{1}{2}\sum_{i}w_{i}$.
Finally, since the branching behaviour for a $w_{i}$ spectrally flowed
operator is $(w_{i}-1)$, the degree $N$ of the polynomials, $Q$ and $P$ are
given by the Riemann-Hurwitz formula as in ${\rm AdS}_{3}$ with $N$ being the
nett degree of the branched cover. The polynomial $R_{n-1}(z)$ is a common
overall factor in Eq. (4.12) and does not affect the branching behaviour.

Thus when the points $x_{i}$ are all in a plane, the twistor solutions are a
generalisation of the ones we saw in $AdS_{3}$. The main difference is that
for a string at the boundary, the incidence relations for $AdS_{5}$ Eq. (4.1)
reduce to Eq. (4.3). Hence the analogue of the terms on the RHS of Eq. (2.10)
are absent here. As a result, in Eq. (4.12) we do not have the analogue of the
second term on the RHS of Eq. (2.18).

We have not discussed the twistor fields $\lambda^{\dagger}_{\dot{\beta}}(z)$
thus far. For the choice of
$\bar{\partial}X^{\dot{\alpha}}_{\;\;\;\beta}(z,\bar{z})$ in Eq. (4.8),
$\lambda^{\dagger}_{\dot{\beta}}(z)$ is also proportional to the vector in Eq
(4.10), with a non-zero component $\lambda^{\dagger}_{1}(z)$. This is because
it is a zero eigenvector of the transpose of the matrix,
$\bar{\partial}X^{\dot{\alpha}}_{\;\;\;\beta}(z,\bar{z})$. However, given the
second incidence relation in Eq. (4.3), we see that
$\mu^{\dagger}_{1}(z)=V(z)\lambda^{\dagger}_{1}(z)$. Since the branching
behaviour at the insertions, which is determined by the spectral flow, is the
same for both sets of twistor fields
$(\lambda^{\dagger}_{\dot{\beta}}(z),\lambda^{\beta}(z))$, this implies that
they are determined by the same covering map. Thus the same polynomials
$P_{N}(z)$ and $Q_{N}(z)$ determine $\mu^{\dagger}_{1}(z)$ and
$\lambda^{\dagger}_{1}(z)$ respectively. Therefore these are not independent
degrees of freedom.

We expect the general solution to be determined by double the number of
degrees of freedom of the special kinematic setup. In other words, there will
now be two $\lambda^{\beta}(z)$ and the corresponding two
$\mu^{\dot{\alpha}}(z)$. And we expect the
$\lambda^{\dagger}_{\dot{\beta}}(z)$ and $\mu^{\dagger}_{\alpha}(z)$ not to be
independent of the former. This will be studied elsewhere GGKM . In the next
section, we will use the restricted covering maps of this section to show how
it reproduces the Yang-Mills propagators in a very natural way which parallels
that in the $AdS_{3}/CFT_{2}$ case Gaberdiel:2020ycd .

##  5 Relation to Feynman diagrams in gauge theories

Consider the correlation function of $n$ gauge invariant operators
$\mathcal{O}^{(w_{i})}(x_{i})$ (made out of $w_{i}$ ‘letters’ or singletons)
in free $SU(N)$ Yang-Mills theory, in the large $N$ limit. For concreteness,
we will assume we are working with ${\cal N}=4$ SYM and the
$\mathcal{O}^{(w)}$ are built from $w$ scalar fields. Then

$G_{n}(\\{x_{i}\\})=\langle\mathcal{O}^{(w_{1})}(x_{1})\cdots\mathcal{O}^{(w_{n})}(x_{n})\rangle_{planar}=\sum_{\\{n_{ij}\\}}C_{\\{n_{ij}\\}}\prod_{(i,j)}\left(\frac{1}{x_{ij}^{2}}\right)^{n_{ij}}$
(5.1)

is simply given in terms of sums of products of the individual propagators
$\left(\frac{1}{x_{ij}^{2}}\right)^{n_{ij}}$, where $n_{ij}=n_{ji}$ denote the
number of (homotopic) Wick contractions between the pair of vertices at
positions $(x_{i},x_{j})$. Here $C_{\\{n_{ij}\\}}$ are combinatorial factors
counting various planar diagrams, whose precise form is inessential for what
we are about to describe. We also have the constraints on the $\\{n_{ij}\\}$

$\sum_{j}n_{ij}=w_{i}\ ,$ (5.2)

holding for each vertex $(i)$. Each of the contributions in Eq. (5.1)
corresponds to a planar Feynman graph with $n$ vertices and $n_{ij}$ homotopic
edges between the pair $(ij)$ of vertices. Thus there are a total of
$\frac{1}{2}\sum_{i,j}n_{ij}=\frac{1}{2}\sum_{i}w_{i}$ edges in the graph.

In ${\rm AdS}_{3}$, the correlator of twisted sector fields corresponding to
Eq. (5.1) was given by a sum over contributions associated to different
covering maps in the Lunin-Mathur description Lunin:2001 . These, in turn were
associated Pakman:2009  in a one to one way with a set of Feynman diagrams
which captured the covering map data, as briefly recalled in Appendix B. Here
we want to do something parallel, but in the opposite direction, in a sense.
Namely, we would like to associate a covering map from the worldsheet for each
familiar Feynman diagram contribution. We will see that, at least for the
restricted kinematic configurations of the previous subsection, we will have
an analogous picture101010As mentioned, placing all the $n$-points $(x_{i})$
in a 2d plane is always possible for $n=2,3$, so this is really a special
choice only for four and higher point functions.. Each graph will therefore
correspond to a certain point in the moduli space of the closed string theory
(which admits this covering map with specified branching data). We will see,
via a connection of the Schwarzian of this covering map to the unique Strebel
differential at that point in the moduli space, that this association of
Feynman graphs to closed string worldsheets is as per the Strebel prescription
of Gopakumar:2005fx ; Razamat:2008zr . This is also in parallel to the
$AdS_{3}/CFT_{2}$ case as in Gaberdiel:2020ycd  – see Appendix B and
specifically Fig. 8. And very strikingly, the weight, associated to each
Feynman diagram, coming from the propagators in Eq. (5.1) can be reproduced in
terms of the area of the worldsheet in the so-called Strebel gauge
Gopakumar:2011ev , again in parallel to the ${\rm AdS}_{3}$ case
Gaberdiel:2020ycd . Let us now describe how this comes about.

###  5.1 Two point function

We start with the simplest correlator, a two point function. Then we can
always choose the two points $(x_{i},x_{j})$ to lie in a specified plane on
the boundary of the ${\rm AdS}_{5}$. Since we are considering scalar fields,
we can take these to be the highest weight BPS state in the SYM theory
corresponding to the $w$-spectrally flowed vacuum sector $|0\rangle_{w}$
Gaberdiel:2021qbb ; Gaberdiel:2021jrv  (together with the conjugate field). We
will view the corresponding states on the worldsheet to be inserted at
$z=(0,\infty)$, respectively, without loss of generality. For such a
correlator, as discussed in the previous section, the classical twistor field
will have branching of order $w$ at $z=(0,\infty)$ (see footnote 9). The
corresponding covering map in Eq. (4.14) is then fixed (after absorbing a
constant into a rescaling of $z$) to take the form

$V(z)=\frac{V_{j}\,z^{w}+V_{i}}{z^{w}+1}\ ,$ (5.3)

where $Z_{i}$ is the complex coordinate in the $(1,2)$ plane corresponding to
$x_{i}$ (see below Eq. (4.14)). This corresponds to the polynomials in Eq.
(4.12) and Eq. (4.14) taking the values

$P_{w}(z)=V_{j}\,z^{w}+V_{i}\,\,;\,\,\,\,\,\,Q_{w}(z)=z^{w}+1\ .$ (5.4)

From the point of the view of the twistor fields Eq. (4.12), taking into
account the weight half of the spinor fields, we see that we are exciting only
the wedge modes with $r=\pm\frac{(w-1)}{2}$ as appropriate for the highest
weight BPS state (and its conjugate).

As we will shortly see, it will be convenient to view the covering map in
coordinates $z=e^{2\pi i\tfrac{u}{w}}$ which maps the vertical strip $(0<{\rm
Re}\,u\leq w)$ onto the sphere such that $z=(0,\infty)$ are images of $u=\pm
i\infty$, respectively, on the strip. In terms of these coordinates the
covering map takes the form

$\Gamma(u)\equiv
V(z(u))=\frac{V_{i}+V_{j}}{2}+\frac{V_{i}-V_{j}}{2i}\,\tan(\pi u)\ .$ (5.5)

In this form, we now observe that this is essentially the unique map for which
the Schwarzian is a constant:

$S[\Gamma(u)]=\frac{\Gamma^{\prime\prime\prime}}{\Gamma^{\prime}}-\frac{3}{2}\Bigl{(}\frac{\Gamma^{\prime\prime}}{\Gamma^{\prime}}\Bigr{)}^{2}=2\pi^{2}\
.$ (5.6)

At the same time, we know that the unique Strebel quadratic differential on
the strip with poles only at $u=\pm i\infty$ is also just $du^{2}$. So we see
that, as in Gaberdiel:2020ycd  (cf. Eq. (6.5) there), the Strebel differential
on the worldsheet is identified with the Schwarzian of the covering map.

$\phi_{S}(u)du^{2}=\frac{1}{2\pi^{2}}S[\Gamma(u)]du^{2}\ .$ (5.7)

Note that the Schwarzian also transforms as a quadratic differential and hence
this is a coordinate independent statement. Indeed, in terms of the coordinate
$z$, we have the Strebel differential

$du^{2}=-\frac{w^{2}}{4\pi^{2}}\frac{dz^{2}}{z^{2}}\ ,$ (5.8)

which has double poles, as expected at $z=0,\infty$ with “residues” $\propto
w$.

This identification enables us to interpret this worldsheet as that
corresponding to the Feynman diagram associated to this two point function.
Then the strip of width $w$ is nothing other than the $w$ double line edges
glued together. This Strebel prescription of Gopakumar:2005fx ; Razamat:2008zr
; Gopakumar:2011ev  was seen to be also realised in the large $w_{i}$ limit in
the case of ${\rm AdS}_{3}/CFT_{2}$ in the tensionless regime.

$\boldsymbol{-\,iL}$$\boldsymbol{+\,iL}$$\large\boldsymbol{z_{j}}$$\large\boldsymbol{z_{i}}$$w$
Figure 2: A vertical strip of width $w$ in the $u$-plane. Regulators at $u=\pm
iL$ are shown.

We can go further and look at the Nambu-Goto area of the worldsheet in the
metric induced by the Strebel differential. This had also entered into the
weight associated with the correlator in ${\rm AdS}_{3}$ Gaberdiel:2020ycd .
The area is simplest to compute in the $u$ coordinates. It is just the area of
the strip. The width is proportional to $w$. However, the length is formally
infinite. This is a reflection of the UV divergence of the field theory. To
exhibit this, we put cutoffs $-L\leq{\rm Im}\,u\leq L$ with $L\gg 1$. We now
denote

$|V_{i}-\Gamma(u=iL)|=|V_{j}-\Gamma(u=-iL)|=\epsilon$ (5.9)

where $\epsilon$ is a short distance cutoff in spacetime. Using the behaviour

$i\tan{u}=\mp(1-2\,e^{\pm 2iu}),\quad u\to\pm i\infty,$ (5.10)

we find for the covering map Eq. (5.5) that

$\epsilon=|V_{i}-\Gamma(u=iL_{i})|=|V_{i}-V_{j}|\exp(-2\pi L),\quad
L\rightarrow\infty\ .$ (5.11)

In other words,

$L=\frac{1}{4\pi}\ln{\left(\frac{x_{ij}^{2}}{\epsilon^{2}}\right)}\ ,$ (5.12)

where we have used $|V_{i}-V_{j}|^{2}=(x_{ij})^{2}$. We then see that the
(regulated) area of the strip in Strebel gauge is $A_{S}=2Lw$. Hence, the
natural Nambu-Goto weight

$e^{-2\pi A_{S}}=\epsilon^{2w}\left(\frac{1}{x_{ij}^{2}}\right)^{w}\ .$ (5.13)

We reproduce the propagator of the free gauge theory simply via the area of
the string worldsheet in the special Strebel gauge we are working in. Note
that the $\epsilon$ dependence in the answer is just a multiplicative
renormalisation that can be absorbed as usual.

###  5.2 Multi point correlator

We have seen that for the two point function, we can reproduce the correct
propagator Eq. (5.13) from the Strebel area of the Strebel diffferential,
given by the Schwarzian of the covering map Eq. (5.3) as in Eq. (5.8). To
generalise to a multi-point correlator, we will use the results of
Gaberdiel:2020ycd , restricting to the special kinematic configuration where
all the $n$-points are in the same plane. In the regime of large $w_{i}$,
which we are considering, the covering maps with specified branching data
$(x_{i},w_{i})$ were explicitly characterised in terms of a spectral curve for
a Penner-like matrix model. The spectral curve, and thus the covering map, was
determined in terms of a set of integers $n_{ij}$ obeying the constraint Eq.
(5.2). Each such covering map was associated with a Feynman diagram with $n$
vertices with $w_{i}$ double lines emanating from each of the vertices. The
$n_{ij}$ were the number of edges between the pair of vertices $(i,j)$.

Furthermore, it was argued in Gaberdiel:2020ycd  that to leading order (i.e.
for large $w_{i}$), this spectral curve, given by the Schwarzian of the
covering map, was, rather remarkably, proportional to the unique Strebel
differential on the worldsheet with specified Strebel lengths $n_{ij}$. The
Feynman-’t Hooft diagram associated to the covering map was then simply the
Strebel construction of the closed string worldsheet, with $n$ punctures,
formed from from gluing strips together as in the general prescription of
Gopakumar:2005fx ; Razamat:2008zr ; Gopakumar:2011ev .

$\boldsymbol{2\pi\frac{n_{ij}}{w_{i}}}$$\large\boldsymbol{z_{i}}$$\large\boldsymbol{z_{j}}$
Figure 3: The local neighbourhood near a double pole of the Strebel
differential - where strips such as in Fig. 2 are glued together. Compare with
the middle picture of Fig. 1.

All these considerations go through in our present context of a restricted
kinematic configuration. We will therefore use the connection between the
Strebel differential and the Schwarzian of the covering map to locally
describe the covering map. Thus, we will consider the worldsheet, which admits
a covering map, to be comprised of vertical strips of width $n_{ij}$ which
connect the pair of worldsheet points $(z_{i},z_{j})$. The Strebel
differential has no poles or zeroes on this strip, except at the boundaries
and hence we can find a local coordinate $u_{(ij)}$ on this strip in which it
is just $du_{(ij)}^{2}$. The Schwarzian of the associated covering map
$\Gamma(u_{(ij)})$ is a constant. Given that the covering map must interpolate
(as $u_{(ij)}\to\pm i\infty$) between the space time points with planar
coordinates $V_{i}$ and $V_{j}$, this fixes the covering map in this
coordinate chart to be as in Eq. (5.5).

$\Gamma(u_{(ij)})=\frac{V_{i}+V_{j}}{2}+\frac{V_{i}-V_{j}}{2i}\,\tan(\pi
u_{(ij)})\ .$ (5.14)

Note that this is the covering map on a local patch of the Riemann surface. To
obtain the full covering map, one will have to glue together these maps in
different patches. There is a Strebel differential $\phi_{S}(z)dz^{2}$ (with
given Strebel lengths $n_{ij}$) which takes the form
$(\frac{du_{(ij)}(z)}{dz})^{2}dz^{2}$ in each strip and

$\phi_{S}(z)dz^{2}\sim-\frac{w_{i}^{2}}{4\pi^{2}}\frac{dz^{2}}{(z-z_{i})^{2}}$
(5.15)

in the vicinity of the insertions $z_{i}$. The covering map is then obtained
by gluing together the different patches $u_{(ij)}(z)$. See fig. 3.

However, for the purpose of obtaining the Strebel area, we will not really
need the global covering map or Strebel differential. We simply add up the
areas of each of these strips, appropriately regularised. Given that in each
strip the covering map is given by Eq. (5.14), we just repeat the
considerations of the previous subsection. We thus obtain a regularised strip
of width $n_{ij}$ \- the Strebel length - and vertical size

$2L_{ij}=\frac{1}{2\pi}\ln{\left(\frac{x_{ij}^{2}}{\epsilon^{2}}\right)}\ ,$
(5.16)

with $\epsilon$ again being a short distance spacetime regulator as in Eq.
(5.9). Thus the Strebel area ($A_{ij}=2L_{ij}n_{ij}$) of each such strip gives
a Nambu-Goto weight

$e^{-2\pi
A_{ij}}=\epsilon^{2n_{ij}}\left(\frac{1}{x_{ij}^{2}}\right)^{n_{ij}}\ .$
(5.17)

As before, we absorb the regulator dependence in a multiplicative
renormalisation of the corresponding YM operators.

We therefore conclude that the worldsheets weighted with the Nambu-Goto weight
in Strebel gauge give rise to the individual propagator contributions to the
Feynman diagram for the Yang-Mills correlator in Eq. (5.1). The sum over the
differrent terms in Eq. (5.1) is now interpreted as a sum over different
points in the moduli space which admit covering maps. These are specified by
different numbers $\\{n_{ij}\\}$ of wick contractions or equivalently, from
the closed string point of view, the integer Strebel lengths characterising
these points on moduli space.

##  6 Concluding remarks

In this work, we have begun the task of pushing the proposal of
Gaberdiel:2021qbb ; Gaberdiel:2021jrv  beyond that of the agreement of the
spectrum. Our goal here was to develop a geometric picture of the twistor
description and see how far this can reproduce the correlators of free super
Yang-Mills. This needs to be taken further and in a more systematic way at the
level of the quantum correlators. Nevertheless, we already see many striking
elements, through our analysis of classical configurations, which we expect
will hold exactly in the full description. In developing this picture, we have
also fleshed out the covering map description of $AdS_{3}/CFT_{2}$ in terms of
the worldsheet twistor fields. The resulting classical twistor configurations
are seen to have only the wedge modes excited as expected from the fact that
these are the physical modes of the string theory after gauge fixing (as in
Sec. 5 of Gaberdiel:2021jrv ).

We then saw that this geometric picture admits a natural generalisation to
${\rm AdS}_{5}$. We described the natural set of incidence relations that the
classical worldsheet ambitwistor configurations should satisfy. Imposing
euclidean reality conditions allowed us to solve for the ${\rm AdS}_{5}$
spacetime string configurations in terms of the ambitwistor fields.
Restricting to boundary correlators which lie on a two plane, we could write
down the solutions, as in ${\rm AdS}_{3}$, in terms of holomorphic covering
maps with the right branching behaviour. The corresponding twistor fields are
then essentially polynomials, again supporting the identification of the
physical modes with the finite number of wedge modes Gaberdiel:2021jrv ;
Gaberdiel:2021qbb . We take this as a robust indication that the wedge twistor
modes are indeed the crucial physical degrees of freedom in the tensionless
limits of strings on both ${\rm AdS}_{3}$ and ${\rm AdS}_{5}$. It will, of
course, be important to corroborate this conclusion from a first principles
worldsheet analysis.

Furthermore, for this kinematic configuration, we found that the explicit
covering map is associated with the special integer points on the moduli space
(‘arithmetic curves’). This is because the Schwarzian of the covering map
turned out to be the Strebel differential (as for $AdS_{3}/CFT_{2}$, see Fig.
8). The corresponding Strebel graph (or more precisely, its dual) is
associated to the Feynman diagram of the free Yang-Mills theory as per
Gopakumar:2005fx ; Razamat:2008zr . This realises the picture of open-closed
string duality of Gopakumar:2003ns ; Gopakumar:2004qb ; Gopakumar:2005fx ,
associating individual ’tHooft-Feynman diagrams (open strings) with closed
string worldsheets. One of the new things we learnt is that the Nambu-Goto
weight (the area in Strebel gauge) associated with each such worldsheet is
precisely the free Yang-Mills propagator (after proper regularisation). This
is, to us, compelling evidence that the Strebel construction paves the path
between perturbative field theories and closed string worldsheet theories in
broad generality.

There are a number of questions which merit further investigation, which have
mostly been mentioned already. Perhaps the most important of them is to place
the stringy ${\rm AdS}_{5}$ twistor incidence relations on a firm footing,
from the standpoint of worldsheet correlators, as in Dei:2020zui . The exact
nature of the spacetime covering maps when one goes away from our special
kinematic configuration is another important point to understand better. This
will begin to kick in at the level of 4-point functions. As mentioned, the
maps $X^{\alpha\dot{\beta}}(z,\bar{z})$ are now holomorphic in a subtle way,
in there being locally a holomorphic eigenvector. Can we still use the power
of holomorphy and constrain these maps? The radial profile of these
configurations, which are infinitesimally near the boundary, would also be
interesting to understand and should correspond to some kind of effective
Liouville mode on the worldsheet. We note that the picture of the worldsheet
configurations is somewhat like that described in Berkovits:2019ulm . The
connection to the Feynman diagrams for these more general configurations also
needs to be worked out. Can we, for instance, see the detailed spin dependent
numerators of the Feynman weights, for a general correlator, from the Strebel
construction for the twistor worldsheet? It would be very nice to develop a
general dictionary for the usual Feynman rules, that we can expect to hold in
any open-closed string duality in the perturbative field theory limit.

###### Acknowledgements.

R.G. thanks Matthias Gaberdiel for very helpful discussions on multiple
matters and related collaboration. The work of R.G. is supported in part by
the J.C. Bose Fellowship of the DST-SERB. R.G. and P.M. acknowledge the
support of the Department of Atomic Energy, Government of India, under project
no. RTI4001, as well as the framework of support for the basic sciences by the
people of India. F.B. and B.R. thank ICTS-TIFR for the opportunity to be a
part of the Long Term Visiting Students Program (LTVSP) that made this
collaboration possible.

##  Appendix A Conventions

The spin group $SO(4,\mathds{C})$ of complexified Minkowski space (denoted as
$M_{\mathds{C}}$) is locally isomorphic to $SL(2,\mathds{C})\times
SL(2,\mathds{C})$. Given a four-vector $T^{\mu}=(T^{0},T^{1},T^{2},T^{3})$, we
can represent it in terms of Pauli matrices
${\sigma_{\mu}}^{\alpha\dot{\alpha}}$ as

$T^{\alpha\dot{\alpha}}=\frac{1}{\sqrt{2}}T^{\mu}\sigma_{\mu}^{\alpha\dot{\alpha}}=\begin{bmatrix}T^{0}+T^{3}&&T^{1}-iT^{3}\\\
T^{1}+iT^{2}&&T^{0}-T^{3}\end{bmatrix}$ (A.1)

where un-dotted and dotted spinor indices $\\{\alpha=0,1|\dot{\alpha}=0,1\\}$
are in the $(\frac{1}{2},0)$ and $(0,\frac{1}{2})$ representations of
$SL(2,\mathds{C})\times SL(2,\mathds{C})$, known as negative and positive
chirality spinors respectively. Here and below we will be largely following
the conventions of Adamo:2017qyl .

For lowering and raising the spinor indices, we can use the
$SL(2,C)$-invariant tensors, i.e the usual Levi-Civita symbols

$\epsilon_{\alpha\beta}=\epsilon^{\alpha\beta}=\epsilon_{\dot{\alpha}\dot{\beta}}=\epsilon^{\dot{\alpha}\dot{\beta}}=\begin{bmatrix}0&1\\\
-1&0\end{bmatrix}.$ (A.2)

With this convention for inverse marices $\epsilon^{\alpha\beta}$ and
$\epsilon^{\dot{\alpha}\dot{\beta}}$,

$\epsilon^{\alpha\beta}\epsilon_{\gamma\beta}=\delta_{\gamma}^{\alpha}\quad\text{and}\quad\epsilon^{\alpha\beta}\epsilon_{\alpha\beta}=2$
(A.3)

and similarly for dotted indices. Our convention for lowering and raising of
spinor indices will follow the slogan, ‘lower to the right, raise to the
left’:

$u_{\alpha}=u^{\beta}\epsilon_{\beta\alpha}\quad\text{and}\quad
v^{\alpha}=\epsilon^{\alpha\beta}v_{\beta}\ ,$ (A.4)

and similarly for dotted indices. The epsilon tensors define
$SL(2,\mathds{C})$ invariant inner products in the spaces of both positive and
negative chirality spinors separately. We define them as

$\langle
u\,v\rangle:=u^{\alpha}\,v_{\alpha}=u^{\alpha}v^{\beta}\epsilon_{\beta\alpha}\quad\text{and}\quad[\tilde{u}\,\tilde{v}]:=\tilde{u}^{\dot{\alpha}}\tilde{v}_{\dot{\alpha}}=\tilde{u}^{\dot{\alpha}}v^{\dot{\beta}}\epsilon_{\dot{\beta}\dot{\alpha}}.$
(A.5)

We note that these inner products are skew-symmetric, i.e $\langle
u\,v\rangle=-\langle v\,u\rangle$ (and similarly for the dotted spinors).

The space-time vector $x^{\mu}$ in $M_{\mathds{C}}$ has the form, as in Eq.
(A.1)

$x^{\alpha\dot{\beta}}=\frac{1}{\sqrt{2}}\begin{bmatrix}x^{0}+x^{3}&&x^{1}-ix^{2}\\\
x^{1}+ix^{2}&&x^{0}-x^{3}\end{bmatrix}\ .$ (A.6)

Lowering and raising different indices, this becomes

$x_{\alpha}^{\;\;\;\dot{\beta}}=x^{\gamma\dot{\beta}}\epsilon_{\gamma\alpha}=\frac{1}{\sqrt{2}}\begin{bmatrix}-(x^{1}+ix^{2})&&-(x^{0}-x^{3})\\\
x^{0}+x^{3}&&x^{1}-ix^{2}\end{bmatrix};$ (A.7)
$x^{\dot{\alpha}}_{\;\;\;\beta}=x^{\dot{\alpha}\gamma}\epsilon_{\gamma\beta}=\frac{1}{\sqrt{2}}\begin{bmatrix}-(x^{1}+ix^{2})&&x^{0}+x^{3}\\\
-(x^{0}-x^{3})&&x^{1}-ix^{2}\end{bmatrix}\ .$ (A.8)

This leads to the following identities

$\begin{split}&x_{\alpha\dot{\gamma}}x^{\dot{\gamma}}_{\;\;\;\beta}=\frac{1}{2}x^{2}\epsilon_{\alpha\beta}\quad\text{and}\quad
x^{\dot{\alpha}\gamma}x_{\gamma}^{\;\;\;\dot{\beta}}=-\frac{1}{2}x^{2}\epsilon^{\dot{\alpha}\dot{\beta}}\
.\end{split}$ (A.9)

So far our discussion applies to $M_{\mathds{C}}$. Taking an euclidean slice
of this complexified space amounts to setting

$x^{\alpha\dot{\beta}}=\widehat{x}^{\,\alpha\dot{\beta}},$ (A.10)

where

$\displaystyle\widehat{x}^{\,\alpha\dot{\beta}}=\frac{1}{\sqrt{2}}\begin{bmatrix}\bar{x}^{0}-\bar{x}^{3}&&-\bar{x}^{1}+i\bar{x}^{2}\\\
-\bar{x}^{1}-i\bar{x}^{2}&&\bar{x}^{0}+\bar{x}^{3}\end{bmatrix}.$ (A.11)

This is because Eq. (A.10) implies $\bar{x}^{0}=x^{0}$,
$\bar{x}^{i}=-x^{i}\,(i=1,2,3)$ so that
$x^{0}=y^{0},\,x^{i}=-iy^{i}\,(i=1,2,3)$ with $y^{0},y^{i}\in\mathds{R}$
giving the euclidean metric $ds^{2}=(dy^{0})^{2}+\sum_{i=1}^{3}(dy^{i})^{2}$.
Thus $x^{\dot{\alpha}}_{\;\;\;\beta}$ in this Euclidean real slice becomes

$x^{\dot{\alpha}}_{\;\;\;\beta}=\frac{1}{\sqrt{2}}\begin{bmatrix}-y^{2}+iy^{1}&&y^{0}-iy^{3}\\\
-y^{0}-iy^{3}&&-y^{2}-iy^{1}\end{bmatrix}\ .$ (A.12)

The stringy generalization of this expression is

$X^{\dot{\alpha}}_{\;\;\;\beta}(z,\bar{z})=\begin{bmatrix}-V(z,\bar{z})&&U(z,\bar{z})\\\
-\bar{U}(z,\bar{z})&&-\bar{V}(z,\bar{z})\end{bmatrix}\ .$ (A.13)

where we have defined $U(z,\bar{z})=(y^{0}-iy^{3})(z,\bar{z})$ and
$V(z,\bar{z})=(y^{2}-iy^{1})(z,\bar{z})$.

In Fig. 4 we have illustrated the double fibration that underlies the twistor
correspondence on the Minkowski boundary of ${\rm AdS}_{5}$. On the left hand
side is the complexified case while the RHS is after imposing the euclidean
reality condition. In the latter, case the second fibration becomes a
bijection between the twistor space and the spin bundle over real Euclidean
space. In Fig. 5, the analogous figures are shown for the complexified ${\rm
AdS}_{5}$ spacetime and after imposing euclidean reality conditions.

${\underset{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(4+1)}}{\\{x^{\dot{\alpha}}_{\;\;\;\beta},\lambda^{\beta}\\}}}$${\underset{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(3)}}{\\{\mu^{\dot{\alpha}},\lambda^{\beta}\\}}}$${\underset{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(4)}}{x^{\dot{\alpha}}_{\;\;\;\beta}}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,1,1}\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}\pi_{2}}$$\scriptstyle{\definecolor[named]{.}{rgb}{.5,0,.5}\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\pi_{1}}$

${\underset{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(4+2)}}{\\{x^{\dot{\alpha}}_{\;\;\;\beta},\lambda^{\beta}\\}}}$${\underset{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(3+3)}}{\\{\mu^{\dot{\alpha}},\lambda^{\beta}\\}}}$${\underset{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(4)}}{x^{\dot{\alpha}}_{\;\;\;\beta}}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,1,1}\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}\pi_{2}}$$\scriptstyle{\definecolor[named]{.}{rgb}{.5,0,.5}\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\pi_{1}}$

Figure 4: (Left) Double fibration of the projective spinor bundle
$\\{x^{\dot{\alpha}}_{\;\;\;\beta},\lambda^{\beta}\\}\cong
M_{\mathds{C}}\times\mathds{CP}^{1}$ over the twistor space
$\\{\mu^{\dot{\alpha}},\lambda^{\beta}\\}\cong\mathds{CP}^{3}$ and the space-
time on the boundary of $AdS_{5}$, $\\{x^{\dot{\alpha}}_{\;\;\;\beta}\\}\cong
M_{\mathds{C}}$. Numbers in red are the complex dimensions of the
corresponding spaces. (Right) The same fibration after imposing the euclidean
reality conditions. Numbers in blue now show the real dimensions of the
corresponding spaces. Note that $\pi_{2}$ is an isomorphism in this signature.

${\underset{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(5+1+1)}}{\\{X^{IJ},\lambda^{\alpha},\lambda_{\dot{\beta}}^{\dagger}\\}}}$${\underset{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(3+3-1)}}{\\{\lambda^{\alpha},\mu^{\dot{\alpha}},\mu^{\dagger}_{\beta},\lambda^{\dagger}_{\dot{\beta}}\\}_{\\{\mathcal{C}=0\\}}}}$${\underset{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(5)}}{\\{X^{IJ}\\}}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,1,1}\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}\pi_{2}}$$\scriptstyle{\definecolor[named]{.}{rgb}{.5,0,.5}\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\pi_{1}}$

${\underset{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(5+2+2)}}{\\{X^{IJ},\lambda^{\alpha},\lambda_{\dot{\beta}}^{\dagger}\\}}}$${\underset{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(6+6-2-1)}}{\\{\lambda^{\alpha},\mu^{\dot{\alpha}},\mu^{\dagger}_{\beta},\lambda^{\dagger}_{\dot{\beta}}\\}_{\\{\mathcal{C}=0,\mathcal{D}=0\\}}}}$${\underset{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(5)}}{\\{X^{IJ}\\}}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,1,1}\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}\pi_{2}}$$\scriptstyle{\definecolor[named]{.}{rgb}{.5,0,.5}\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\pi_{1}}$

Figure 5: (Top) Double fibration of the projective spinor bundle
$\\{X^{IJ},\lambda^{\alpha},\lambda_{\dot{\beta}}^{\dagger}\\}\cong\mathds{CP}^{5}\times\mathds{CP}^{1}\times\mathds{CP}^{1}$
over the Ambi-twistor space
$\\{\lambda^{\alpha},\mu^{\dot{\alpha}},\mu^{\dagger}_{\beta},\lambda^{\dagger}_{\dot{\beta}}\\}_{\\{\mathcal{C}=0\\}}\cong(\mathds{CP}^{3}\times\mathds{CP}^{3})_{\\{\mathcal{C}=0\\}}$
and the “space-time” $\\{X^{IJ}\\}\cong\mathds{CP}^{5}$ in the complexified
$AdS_{5}$ bulk. ${\cal C}=0$ refers to the ambitwistor constraint Eq. (3.13).
Numbers in red are the complex dimensions of the corresponding spaces.
(Bottom) The same fibration after imposing the euclidean reality conditions
which leads to the additional constraint ${\cal D}=0$ – see Eq. (3.33).
Numbers in blue are the real dimensions of the corresponding spaces. Note that
$\pi_{2}$ is an isomorphism in this signature.

##  Appendix B Feynman covering maps in the symmetric orbifold $CFT_{2}$

Here we give a brief recap of the computation of the correlators in symmetric
orbifold theories from covering maps Lunin:2001  and their associated “Feynman
graphs”Pakman:2009 . We then summarise the results of Gaberdiel:2020ycd  where
it was shown that each such Feynman covering map (at least, for large twists)
can be associated to a point in the closed string moduli space via the Strebel
correspondence, thus realising the open-closed string dictionary of
Gopakumar:2005fx ; Razamat:2008zr . In the present case, these points are
precisely the ones which admit covering maps, and to which the closed string
correlator localises Eberhardt:2019ywk ; Dei:2020zui . A new geometric
ingredient in the correspondence, is that the Strebel differential, in the
large twist limit, is nothing other than the Schwarzian of the covering map.

###  B.1 Symmetric orbifold CFT correlators and branched coverings

We consider the free symmetric orbifold CFT ${\rm
Sym}^{K}(\mathbb{\mathcal{S}})$ in the large $K$ limit. The orbifold theory
consists of states from both twisted and untwisted sectors where single cycle
twist fields correspond to the single string states in the bulk $AdS_{3}$. We
will mainly be interested in the correlators of the ground states of such
single cycle twist operators $\\{\sigma_{w_{i}}\\}$

$\langle\sigma_{w_{1}}(x_{1})\cdots\sigma_{w_{n}}(x_{n})\rangle.$ (B.1)

Here $x_{i}$ are coordinates on the space-time $\mathbb{S}^{2}$ and the
subscripts label the cycle lengths.

A $w$-cycle twist field $\sigma_{w}(x_{0})$ induces a $w$-fold cyclic
permutation amongst some of the $K$ copies of the $\mathcal{S}$ theory. The
action of this permutation can be geometrised by locally lifting to a $w$-fold
covering near any of the insertion points. Globally, these can be combined
into a finite degree covering surface $\Sigma$ described by the covering map

$\Gamma:\Sigma\,[z]\to\mathbb{S}^{2}\,[x]\,.$ (B.2)

The covering map has the local branching behaviour in the vicinity of a pre-
image $z_{i}$ of an insertion point $x_{i}$

$\Gamma(z)\approx x_{i}+(z-z_{i})^{w_{i}}.$ (B.3)

Restricting to genus zero covering surfaces (which corresponds to the leading
order in a $1/K$ expansion of the correlator) fixes the degree $N$, i.e total
number of sheets in the lift of the correlator, by the Riemann-Hurwitz formula

$N=1+\frac{1}{2}\sum_{j=1}^{n}(w_{j}-1)\,.$ (B.4)

The prescription put forward by Lunin:2001  is to exploit this covering map to
uplift the calculation of the correlator to the covering surface. On the
covering surface the ground state twist fields $\sigma_{w_{i}}$ become
identity operators, since their associated monodromy is captured by the
covering surface itself. Thus the correlator on the covering surface simply
becomes the vacuum path integral getting contributions only from the Liouville
anomaly term for the conformal factor $\partial\Gamma(z)$, associated to the
covering map.

$\langle\sigma_{w_{1}}(x_{1})\cdots\sigma_{w_{n}}(x_{n})\rangle=\sum_{\Gamma}W_{\Gamma}\,e^{-S_{\rm
L}[{\Phi}_{\Gamma}]}\ ,$ (B.5)

where $S_{L}[\Phi]$ is the familiar Liouville action

$S_{\rm L}[\Phi]=\frac{c}{48\pi}\int
d^{2}z\sqrt{g}\bigl{(}\,2\,\partial\Phi\,\bar{\partial}\Phi+R\,\Phi\Bigr{)}\,$
(B.6)

and

${{\Phi}}_{\Gamma}=\log\,\partial_{z}\Gamma(z)+\log\,\partial_{\bar{z}}\bar{\Gamma}(\bar{z})\,.$

The sum over $\Gamma$ in Eq. (B.5) is over the finitely many inequivalent
covering maps that exist for specified branching data, namely,
$\\{w_{i},x_{i}\\}$. Specifying these fixes $(n-3)$ locations $z_{i}$ on the
covering space (three points can be fixed using the global Mobius invariance).
In other words, we get contributions from only discretely many points on the
moduli space of the covering space. In Eberhardt:2019ywk , Dei:2020zui , these
points were seen to be precisely the ones which give a non-zero contribution
to the worldsheet correlator. This cements the identification of the covering
space with the worldsheet as conjectured already in Lunin:2001 ; Pakman:2009 .

###  B.2 Branched coverings and Feynman diagrams for the orbifold $CFT_{2}$

Next, we recollect the diagrammatic picture of orbifold correlators introduced
in Pakman:2009 . The basic idea is to consider a double-lined (dashed and
solid) Jordan curve on the base sphere $\mathbb{S}^{2}[x]$ passing through the
$n$ insertion points $x_{i}$ with the coloured (solid border) face enclosing
$x=\infty$ (see the right hand figure in Fig. 6). Then the pre-image of this
curve under the covering map $\Gamma$ defines a ‘Feynman graph’ for the
contribution from this particular $\Gamma$ to the correlator (B.5). This graph
has bi-fundamental lines (edges) joining the different vertices (located at
the $z_{i}$). The poles of $\Gamma$ being pre-images of $x=\infty$ are now
located in the interior of the $N$ colored faces in the Feynman graph. Thus
these bifundamental Feynman graphs give a bi-colouration of the worldsheet
Riemann surface.

$\boldsymbol{x=\Gamma(z)}$$\boldsymbol{\lfloor z}$$\boldsymbol{\lfloor
x}$(a)(b)

Figure 6: Feynman graph of a symmetric orbifold correlator, formed by the
preimage of a Jordan curve under $\Gamma$. Here $n=4$, $w_{i}=2$ and so $N=3$.
Figure taken from Gaberdiel:2020ycd

.

The sum over the inequivalent covering maps is nothing other than a sum over
the inequivalent Feynman graphs, subject to the diagrammatic constraints
explained in Pakman:2009 . We have a one to one correspondence of each Feynman
graph with a particular worldsheet (point in the moduli space which admits the
corresponding covering map). We will now see that this association and the
geometric picture of gluing up of these Feynman graphs to form the worldsheet
is captured, for large twist $w_{i}$, by the Strebel construction as in
Gopakumar:2005fx ; Razamat:2008zr .

###  B.3 The large twist limit and the Strebel construction

While the Lunin-Mathur approach to the orbifold CFT correlators is powerful in
principle, it is difficult in practice since one needs to explicitly obtain
the covering map $\Gamma(z)$. In Gaberdiel:2020ycd  it was shown that this
problem simplifies in the large $N$ i.e. large twist limit. Let us recap the
main steps in Gaberdiel:2020ycd . For an $n$-point function (B.1) (with
$z_{n}=\Gamma^{-1}(x_{n})=\infty$), we have

$\partial\Gamma(z)=M_{\Gamma}\,\frac{\prod_{i=1}^{n-1}(z-z_{i})^{w_{i}-1}}{\prod_{a=1}^{N}(z-\lambda_{a})^{2}}\
,$ (B.7)

where $M_{\Gamma}$ is a non-zero constant. Following Roumpedakis:2018tdb , the
condition that the $\lambda$ are simple poles of $\Gamma(z)$ leads to $N$
“scattering equations”

$\sum_{i=1}^{n-1}\frac{w_{i}-1}{\lambda_{a}-z_{i}}=\sum_{b\neq
a}^{N}\frac{2}{\lambda_{a}-\lambda_{b}}\ ,\qquad(a=1,\ldots,N)\ .$ (B.8)

In principle, these determine the poles $\\{\lambda_{a}\\}$ in (B.7) and
therefore the covering map (upto overall additive and multiplicative factors).
The crucial observation of Gaberdiel:2020ycd  was that, in the limit of large
$w_{i}$, these are the saddle-point equations of a Penner-like matrix model,
with a logarithmic potential. We can then compute the covering map in terms of
the ‘spectral curve’ $y_{0}(z)$ of the matrix model. To leading order in
$\frac{1}{N}$

$\frac{1}{N}\partial\ln\partial\Gamma=y_{0}(z)\ ,$ (B.9)

The spectral curve $y_{0}(z)$ is a hyperelliptic curve with cuts formed from
the coalesced poles $\\{\lambda_{a}\\}$ of $\Gamma(z)$ in the large $N$ limit.
This curve is therefore specified by $2(n-3)$ numbers which are the A and
B-cycle periods around the cuts. These parameters serve as the free moduli
characterizing the different covering maps that contribute to the original
correlator.

Diagrammatically, the situation is as shown in fig. 7. The coalescing poles
form transversal cuts across the edges of the original Wick contractions
between vertices of the Feynman graph (see the right hand figure in 7). Thus
the cuts of the spectral curve correspond to the edges of the dual to the
original (Skeleton) Feynman graph. The A- and B-cycle periods of the spectral
curve simply count the number of eigenvalues i.e. poles. This number, for
large twist, is simply given by the number of Wick contractions of the edges
(see left figure in 7) :

$\oint_{C_{ij}}\frac{dz}{2\pi i}\,y_{0}(z)=\frac{n_{ij}}{N}\ ,$ (B.10)

where $n_{ij}$ is the number of Wick contractions in the edge connecting
$i$-th and $j$-th vertex. Note that in the strict large $N$ limit, these
periods can take arbitrary real values (upto an overall scaling).

$\boldsymbol{j^{\text{{th}}}}$ vertex$\boldsymbol{i^{\text{{th}}}}$ vertex

$\boldsymbol{\displaystyle\oint_{C_{ij}}\frac{dz}{2\pi
i}\,y_{0}(z)=\frac{n_{ij}}{N}}$

$\boldsymbol{C_{ij}}$

Figure 7: (Left) The poles of the covering map coalesce to form cuts across
the edges of the Feynman graph where the period integral counts the number of
Wick contractions. (Right) Feynman graph and it’s dual for the four-point
function of each operator with twist two. Figure taken from Gaberdiel:2020ycd
.

Rather remarkably, the quadratic differential formed from the spectral curve
$-y_{0}^{2}(z)dz^{2}$

$-y_{0}^{2}(z)dz^{2}\equiv 4\pi^{2}\phi_{S}(z)dz^{2}$ (B.11)

defines a Strebel differential $\phi_{S}(z)dz^{2}$, a quadratic differential
holomorphic everywhere except with double poles at $n$ marked points $z_{i}$
such that all the “lengths” between its zeroes $\\{a_{k}\\}$ are real

$l_{km}=\int_{a_{k}}^{a_{m}}\sqrt{\phi_{S}(z)}\in{\mathbb{R}}_{+}\ .$ (B.12)

The latter condition is clearly satisfied due to Eq. (B.10). Each Strebel
differential defines a Strebel graph whose vertices are the zeroes of the
differential and edges are trajectories connecting these zeroes such that the
length of these trajectories are real (See fig. 1). From our construction, we
identify the Strebel graph corresponding to a covering map to the dual of the
Feynman graph of that covering with its edges as the cuts of the spectral
curve. The significance of this connection lies in the theorem of Strebel
Strebel , which states, for every Riemann surface $\Sigma_{g,n}$ with genus
$g$ and $n$ marked points where $n>0$ and $2g+n>2$, and any $n$ specified
positive numbers $(w_{1},\ldots,w_{n})$, three exists a unique Strebel
differential. Thus the Strebel lengths parametrise the moduli space ${\cal
M}_{g,n}$. We can therefore associate to each Strebel graph with its lengths,
and thus Feynman graphs with fixed number of Wick contractions, to a given
point in the moduli space.

In other words, in the large twist limit (with specified twists)
$(w_{1},\ldots,w_{n})$, corresponding to each covering map specified by the
periods of the A and B-cycles (or lengths of the edges of the Strebel graph)
there exists a point in the (decorated) moduli space of the $n$-punctured
sphere $\mathcal{M}_{0,n}$. The sum over coverings defining the symmetric
orbifold correlator in Eq. (B.5) goes over to an integral covering this moduli
space exactly once, see figure 8.

Figure 8: Flowchart explaining the general logic of how a field theory
correlator directly maps onto a worldsheet integral over string moduli space
where each Feynman diagram gets associated with a point on the moduli space.
This is seen in the middle and right hand verticals of the figure. The left
hand side indicates the specific way the equivalence is seen in the ${\rm
AdS}_{3}$ case, where it proceeds via an auxiliary matrix model. This tool
bridges the field theory and the string theory in this case.

The Lunin-Mathur weight of the Liouville action for the conformal factor also
has a natural interpretation from the point of view of the worldsheet picture.
One finds Gaberdiel:2020ycd

$S_{L}[\Gamma]=\frac{N^{2}}{8\pi}\int d^{2}z|\phi_{S}(z)|=\frac{1}{4\pi}\int
d^{2}z\big{|}S[\Gamma]\big{|}\ .$ (B.13)

The first equality tells us that the action weighting each covering map is
given by the Strebel area i.e. the worldsheet area when we fix the worldsheet
metric to be in Strebel gauge $ds^{2}=|\phi_{S}(z)|dzd\bar{z}$
Gopakumar:2011ev . In the second equality, we have used the identification of
the Strebel differential and the Schwarzian,

$4\pi^{2}\phi_{S}(z)=\frac{2}{N^{2}}S[\Gamma]\ ,$ (B.14)

which holds in the strict large $N$ limit. As we saw in Sec. 5, such an
identification holds in the case of ${\rm AdS}_{5}$ too, in the special
kinematic configuration considered there.

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