# Spectrum of Massive and Massless Ambitwistor Strings

Christian Kunz
E-mail: mailto:kunz.christian.321<EMAIL_ADDRESS>
###### Abstract

Inspired by recent work arXiv:2301.11227 on massive ambitwistor strings this
paper examines the spectrum of such models using oscillator expansions. The
spectrum depends heavily on the constant related to the normal ordering of the
zero mode operator ${L_{0}}$ of the Virasoro algebra. The supergravity model
is investigated in more detail, and two anomaly-free variations are presented,
both with a rich spectrum and with tree scattering amplitudes that include a
kinematic Parke-Taylor factor for particles other than gravitons without a
need for an external current algebra. The spectrum of some of the models can
also be interpreted as containing three generations of the Pati-Salam model.

###### Contents

  1. 1 Introduction
  2. 2 Notation and Oscillator Expansion
  3. 3 Spectrum of the Supergravity Model
  4. 4 Intermediate Modification of the Supergravity Model
  5. 5 Improved Anomaly-free Supergravity Model Extension
  6. 6 Summary and Outlook
  7. A Non-Supersymmetric Model

##  1 Introduction

The authors in [1, 2, 3, 4, 5] examined dimensional reductions to 5 and 4
dimensions of a 6-dimensional massless ambitwistor string model. The resulting
models in 4 dimensions contain two twistors describing massive particles. When
equipped with maximal supersymmetry the models in 5 and 4 dimensions exhibit
anomaly cancellation for its little group SL(2, $\mathbb{C}$) and also zero
central charge for the Virasoro algebra if one assumes a central charge
contribution of $c_{5d}=10$ or $c_{6d}=12$ arising from 5 or 6 compactified
dimensions, respectively. Using the vertex operators provided in above
references the genus zero worldsheet correlation functions lead to expected
manifestly supersymmetric n-particle tree amplitudes.

These ambitwistor string models use worldsheet spinors and, therefore, the
strings are either in the Neveu-Schwarz (NS) or Ramond (R) sector, depending
on their oscillator expansion. The actual spectrum then depends on the value
$a$ of the zero mode $L_{0}$ of the Virasoro algebra when applied to a
physical state. $a$ is determined by the Virasoro commutator $[L_{n},L_{-n}]$
when all fields, including ghosts and antighosts, are taken into account:

$[L_{n},L_{m}]=(n-m)L_{n+m}+(\frac{c}{12}n^{3}-2an)\delta_{m+n,0}$ (1.1)

It will be shown that the vertex operators in [3, 4, 5] implicitly assume that
in (1.1) $a=0$ in the R sector and $a=1$ in the NS sector. It follows that in
the R sector only the zero modes of the twistor fields and in the NS sector
only the $-\frac{1}{2}$ modes can contribute to physical states. For instance,
this implicit assumption can be confirmed to happen when the extra dimensions
are compactified with help of scalar fields (see appendix A of [5]).

On the other hand, one could also compactify extra components of the
supertwistor and it will be shown that this can be done in such a way that in
the massless limit it results in a supersymmetric ambitwistor string model
with complete anomaly cancellation and a value of $a=1$ in (1.1) for both the
R and NS sector. This model seems to have a richer spectrum than the previous
model (in section 5 it will be shown that this is actually not really true),
with non-zero twistor modes contributing to physical states in both the NS and
R sector111The twistor modes always appear in pairs such that physical states
and vertex operators on-shell are actually unaffected by the nature of
background spin structures.. The spectrum from modes in the NS sector leads to
the same vertex operators as in the massless limit of the previous model, but
the spectrum from modes in the R sector has some resemblance to the spectrum
of the Berkovits-Witten twistor string with N=4 supersymmetry [6, 7], the main
difference being that the new model has the additional SL(2,$\mathbb{C}$)
little group symmetry, and it does not contain the ’dipole’ states thought of
being responsible for the lack of unitarity in the Berkovits-Witten model of
conformal supergravity[6]. One important feature of the new model is that tree
scattering amplitudes of spin $\leq 1$ particles contain a kinematic Parke-
Taylor factor, without having to introduce an external current algebra. The
model also has a non-supersymmetric variation with basically the same
spectrum, except that the little group representation does not need to be the
same for all the components of the multiplet. This non-supersymmetric version
does not use auxiliary fields and was actually previously found independently
by the author in [8, 9].

Unfortunately, because of the non-zero twistor modes in the R sector of the
twistor components, these two models have complicated vertex operators, with
unappealing scattering amplitudes. This motivated to find a model extension
that keeps the Parke-Taylor factor for spin $\leq 1$ particle scattering and
stays anomaly-free but now with a vanishing $L_{0}$ constant in the R sector.
Then the spectrum nearly stays the same, as can be verified with a special
interpretation of the spectrum of the $\mathcal{N}=8$ supersymmetry. Actually,
the same interpretation can be applied to the spectrum of the original model
of [4, 5] showing that it already includes the same rich spectrum.

The paper is organized as follows.
In section 2 the notation used in [5] is reviewed and supplemented with the
oscillator expansion for all the supertwistor component fields.
In section 3 it is shown that the vertex operators in [3, 4, 5] indicate a
spectrum that only involves the zero supertwistor modes in the R sector and
the $-\frac{1}{2}$ modes in the NS sector. There is also some discussion about
different ’polarization pictures’ in the quantization of twistor space one has
to be careful about.
In section 4 the massless limit of the model is extended by changing the
auxiliary fermionic fields to an auxiliary supertwistor with additional
gauging such that the total model has all anomalies cancelled with $c=0$ and
$a=1$ in (1.1). The spectrum is examined. When the twistor modes are in the NS
sector, fixed vertex operators look like in the massless limit of the previous
model, but when the modes are in the R sector, they rather resemble the ones
of the Berkovits-Witten twistor string [6, 7], up to the little group symmetry
and the absence of ’dipole’ fields. It turns out that tree scattering of spin
$\leq 1$ particles leads automatically to a Parke-Taylor factor.
Motivated to simplify the vertex operators and scattering amplitudes of the
model in section 4, it is extended once more in section 5 to get a vanishing
$L_{0}$ constant while staying anomaly-free and keeping the rich spectrum in
the R sector. This is achieved mainly by reducing the bosonic auxiliary
worldsheet spinors introduced in the previous section and treating the
fermionic twistor modes and their duals in a more symmetric fashion. It is
again a massless model without, unfortunately, a known relation to a massive
version. The fixed and integrated vertex operators are determined and tree
scattering amplitudes are computed, still containing a Parke-Taylor factor for
spin $\leq 1$ particle scattering.
Section 6 contains summary and discussion, including arguments that the
spectrum of the original massive model and the model of section 5 in the R
sector includes the spin 1/2 spectrum of 3 generations of the Pati-Salam model
[10], and also that the (truncated) spectrum in the NS sector can be re-
interpreted as arising from spectral flow.
In appendix A the auxiliary supertwistor fields in the model of section 4 are
replaced with the actual twistor fields, breaking the worldsheet supersymmetry
in manifest fashion. The resulting model has the same spectrum as the section
4 model, but there is no need for the little group representation to be the
same across the spectrum. When adjusting the notation, this model turns out to
be the same as in [8, 9]. Like the model in section 4 it also has vertex
operators leading to scattering amplitudes that are unappealing. This makes
the model in section 5 more favorable.

##  2 Notation and Oscillator Expansion

The little group of the two twistor representation of a massive particle that
keeps its timelike momentum $k_{\alpha\dot{\alpha}}$ invariant includes SU(2)
as a subgroup. In the following it will be referred to as ’the little group’.
Dealing in this work with complexified twistor space, SU(2) will be regarded
as extended to SL(2, $\mathbb{C}$) whose algebra is the complexification of
the SU(2) algebra. $k_{\alpha\dot{\alpha}}$ can be written using 2-spinors as

$k_{\alpha\dot{\alpha}}=\kappa_{a\alpha}\kappa_{\dot{\alpha}}^{a}\,,$ (2.1)

where a = 1, 2 is an SL(2, $\mathbb{C}$) little group index raised and lowered
by
$\varepsilon^{ab}=\varepsilon^{[ab]},\varepsilon_{ab}=\varepsilon_{[ab]},\varepsilon_{12}=1=\varepsilon^{12},\varepsilon_{ac}\varepsilon^{cb}=\delta^{b}_{a}$.
Little group contractions are denoted by

$(v_{1}v_{2}):=v_{1a}v_{2b}\varepsilon^{ab}\,.$

The dimensionally reduced action in 4 dimensions of [4, 5] is

$S=\int_{\Sigma}\mathcal{Z}^{a}\cdot\bar{\partial}\mathcal{Z}_{a}+A_{ab}\mathcal{Z}^{a}\cdot\mathcal{Z}^{b}+a(\lambda^{2}-j^{H})+\tilde{a}(\tilde{\lambda}^{2}-j^{H})+S_{m}\,,$
(2.2)

where

  * •

$a=1,2$ is the little group index and the supertwistor fields
$\mathcal{Z}_{a}$ = $\varepsilon_{ab}\mathcal{Z}^{b}$ are worldsheet spinors,
repackaged into Dirac supertwistors of the form222There is some slight
modification to the notation used in [4, 5] with regard to the position of
spinor indices.

$\mathcal{Z}=(\lambda_{A},\mu^{A},\eta^{\mathcal{I}})\,:\qquad\lambda_{A}=(\lambda_{\alpha},\tilde{\lambda}_{\dot{\alpha}})\,,\quad\mu^{A}=(\mu^{\dot{\alpha}},\tilde{\mu}^{\alpha})\,,\quad\eta^{\mathcal{I}}=(\eta^{I},\tilde{\eta}_{I})\,,$

where $\lambda_{A}$ and $\mu^{A}$ are Dirac spinors made up of the homogeneous
chiral and antichiral components of the twistor
$Z=(\lambda_{\alpha},\mu^{\dot{\alpha}})$ and dual twistor
$\tilde{Z}=(\tilde{\lambda}_{\dot{\alpha}},\tilde{\mu}^{\alpha})$. In the
fermionic components $\eta^{\mathcal{I}}=(\eta^{I},\tilde{\eta}_{I})$ with
$I=1,\ldots,\frac{\mathcal{N}}{2}$, the index
$\mathcal{I}=1,\ldots,\mathcal{N}$ is the R-symmetry index, with
$\mathcal{N}=4$ for maximal Super-Yang-Mills (SYM) and $\mathcal{N}=8$ for
maximal supergravity. Indices are raised and lowered with a symmetric form for
Grassmann even entities
$\epsilon^{AB},\epsilon_{AB},\epsilon^{AB}\epsilon_{AC}=\delta^{B}_{C}$, for
example
$\lambda^{A}=\epsilon^{AB}\lambda_{B}=(\tilde{\lambda}^{\dot{\alpha}},\lambda^{\alpha})$,
and a skew form for Grassmann odd entities
$\Omega^{IJ},\Omega_{IJ},\Omega^{IJ}\Omega_{IK}=\delta^{J}_{K}$, for example
$\eta_{I}=\Omega_{IJ}\eta^{J}$.
Also note the Dirac $\gamma_{5}$ matrix defined here by
$\gamma_{5}^{AB}\rho_{B}=(-\rho^{\dot{\alpha}},\rho^{\alpha}),\gamma_{5}^{AB}\gamma_{5AC}=\delta^{B}_{C}$.
For Grassmann odd entities it can be used to raise and lower indices, for
Grassmann even identities it defines a ’dual’ version. This will become
important later.
The inner product is defined as
$\mathcal{Z}_{1}\cdot\mathcal{Z}_{2}=\frac{1}{2}(\tilde{Z_{1}}\cdot
Z_{2}+\tilde{Z_{2}}\cdot
Z_{1}+\tilde{\eta}_{1I}\eta^{I}_{2}+\tilde{\eta}_{2I}\eta^{I}_{1}),\;\;\tilde{Z_{1}}\cdot
Z_{2}=\tilde{\mu}_{1}^{\alpha}\lambda_{2\alpha}+\tilde{\lambda}_{1\dot{\alpha}}\mu_{2}^{\dot{\alpha}}$
,
with special treatment of $\bar{\partial}Z$ when taking the dual:
$\;\widetilde{\bar{\partial}Z}=-\bar{\partial}\tilde{Z}\,$.
The only non-trivial OPEs are
$\mathcal{Z}^{A}_{a}(z)\cdot\mathcal{Z}_{Bb}(0)=\frac{\delta^{A}_{B}\epsilon_{ab}}{z}+\cdots$
.

  * •

little group transformations are gauged by the fields $A_{ab}=A_{(ab)}$. $a$
and $\tilde{a}$ are worldsheet $(0,1)$-forms that act as Lagrange multipliers
to constrain the mass operators

$\lambda^{2}=\frac{1}{2}(\lambda_{\alpha}\lambda^{\alpha})=\det(\lambda_{\alpha}^{a})$,
$\tilde{\lambda}^{2}=\frac{1}{2}(\tilde{\lambda}_{\dot{\alpha}}\tilde{\lambda}^{\dot{\alpha}})=\det(\tilde{\lambda}_{\dot{\alpha}}^{a})$

to be the same as the current $j^{H}$ associated to the element
$h\in\mathcal{G}$ living in the Cartan subalgebra of some symmetry of the
system. This article will not be concerned about the particularity of particle
masses. Therefore, there will be no further discussion of $j^{H}$.

  * •

the action $S_{m}$ represents worldsheet matter and different choices for
$S_{m}$ construct a variety of physically interesting models. For supergravity
$S_{m}$ is equal to $S_{\rho_{1}}+S_{\tilde{\rho}_{2}}$ with

$S_{\rho}=\int_{\Sigma}\tilde{\rho}^{A}\bar{\partial}\rho_{A}+b_{a}\lambda^{Aa}\rho_{A}+\tilde{b}_{a}\lambda_{A}^{a}\tilde{\rho}^{A}\,,$

where $(\rho_{A},\tilde{\rho}^{A})$ are fermionic worldsheet spinors raised
and lowered with $\gamma_{5}$ and $(b_{a},\tilde{b}_{a})$ are $(0,1)$-forms on
the worldsheet acting as fermionic Lagrange multipliers for the constraints
$\lambda^{Aa}\rho_{A}=0=\lambda_{A}^{a}\tilde{\rho}^{A}$.

In ambitwistor space vertex operators are typically built with help of plane
wave representatives. For the current supersymmetric system this looks like

$\displaystyle\mathcal{V}=\qquad\int\\!\\!d^{2}u\;d^{2}v\;\;\mathcal{W}(u)\;\;\bar{\delta}^{4}\\!\left(\left(u\lambda_{A}\right)-\left(v\kappa_{A}\right)\right)\;\bar{\delta}((\epsilon
v)-1)e^{u_{a}\left(\mu^{Aa}\epsilon_{A}+\eta^{\mathcal{I}a}q_{\mathcal{I}}\right)-\frac{1}{2}(\xi
v)q^{2}}\;,$ $\displaystyle
V=\int_{\Sigma}\\!\\!d\sigma\\!\\!\int\\!\\!d^{2}u\;d^{2}v\;\;w(u)\;\;\bar{\delta}^{4}\\!\left(\left(u\lambda_{A}\right)-\left(v\kappa_{A}\right)\right)\;\bar{\delta}((\epsilon
v)-1)e^{u_{a}\left(\mu^{Aa}\epsilon_{A}+\eta^{\mathcal{I}a}q_{\mathcal{I}}\right)-\frac{1}{2}(\xi
v)q^{2}}\;,$ (2.3)

where $\mathcal{V}$ and $V$ stand for a fixed and integrated vertex operator,
respectively, polarization data $\epsilon_{A}$ is defined by
$\epsilon_{A}=\epsilon_{a}\kappa^{a}_{A}$ with
$\kappa^{a}_{A}=(\kappa^{a}_{\alpha},\kappa^{a}_{\dot{\alpha}})$ being the
momentum of the wave according to (2.1), super polarization data
$q_{\mathcal{I}}$ is defined by
$q_{\mathcal{I}}=\epsilon_{a}q^{a}_{\mathcal{I}}$ with $q^{a}_{\mathcal{I}}$
being the supermomentum, and $(\epsilon_{a},\xi_{a})$ with $(\epsilon\xi)=1$
form a basis of the fundamental representation of SL(2, $\mathbb{C}$) so that
the supersymmetry generators can be defined as
$Q_{a\mathcal{I}}=(\xi_{a}q_{\mathcal{I}}+\epsilon_{a}\Omega_{\mathcal{I}\mathcal{J}}\frac{\partial}{\partial
q_{\mathcal{J}}})$[2].

The quadratic differentials $\mathcal{W}$ and $w$ in (2) are theory dependent
and are allowed to depend on the parameter $u$ as well as the worldsheet
matter systems. For a fixed vertex operator in supergravity $\mathcal{W}$ is
just the product of fermionic ghost fields and delta functions of bosonic
ghosts. For an integrated vertex operator an additional integration over the
worldsheet is applied in (2) to take care of gauge fixing the worldsheet
diffeomorphisms and $w$ is [4]

$\displaystyle w(u)=\;$
$\displaystyle\delta\big{(}\mathrm{Res}_{\sigma}(\lambda^{2}-j^{H})\big{)}\;\delta\big{(}\mathrm{Res}_{\sigma}(\tilde{\lambda}^{2}-j^{H})\big{)}$
$\displaystyle\left(\frac{(\hat{u}\lambda_{A})\epsilon^{A}}{(u\hat{u})}-\epsilon^{A}\epsilon_{B}\rho_{1A}\tilde{\rho}_{1}^{B}\right)\,\left(\frac{(\hat{u}\lambda_{A})\tilde{\epsilon}^{A}}{(u\hat{u})}-\tilde{\epsilon}^{A}\tilde{\epsilon}_{B}\rho_{2A}\tilde{\rho}_{2}^{B}\right)\,,$
(2.4)

where $\hat{u}$ is chosen such that $(u\hat{u})\neq 0$ 333Under the support of
the polarized scattering equations $w(u)$ will not depend on the particular
value of $\hat{u}$[3]. and where the dual polarization data
$\tilde{\epsilon}^{A}$ is defined with help of the $\gamma_{5}$ matrix:
$\tilde{\epsilon}^{A}=\gamma_{5}^{AB}\epsilon_{B}$.

According to standard BRST procedure, in addition to the familiar fermionic
$(b,c)$ ghosts related to worldsheet gravity (the action (2.2) is already
written in conformal gauge), the Lagrange multipliers in (2.2) are associated
with corresponding ghosts:
the bosonic fields $\\{A_{a},a,\tilde{a}\\}$ with fermionic ghosts
$\\{(M_{ab},N_{ab}),(m,n),(\tilde{m},\tilde{n})\\}$ and
the fermionic fields $\\{b^{a}_{r},\tilde{b}^{a}_{r}\\}$ with bosonic ghosts
$\\{(\beta^{a}_{r},\gamma^{a}_{r}),(\tilde{\beta}^{a}_{r},\tilde{\gamma}^{a}_{r})\\}$.

The SL(2,$\mathbb{C}$) anomaly coefficient is:

$a_{sl2}=\frac{3}{2}(\frac{1}{2}(8-\mathcal{N}))_{\mathcal{Z}}+\frac{3}{2}(2_{\beta\gamma}+2_{\tilde{\beta}\tilde{\gamma}})-6_{M\\!N}=\frac{3}{4}(8-\mathcal{N})\,.$

It vanishes for $\mathcal{N}=8$.
The central charge is

$c=(-8+\mathcal{N})_{\mathcal{Z}}-26_{bc}-6_{M\\!N}-2_{mn}-2_{\tilde{m}\tilde{n}}+2(4_{\rho\tilde{\rho}}+4_{\beta\gamma}+4_{\tilde{\beta}\tilde{\gamma}})=-20+\mathcal{N}\,.$

It vanishes again for $\mathcal{N}=8$ when a remaining central charge of
$c=-12$ is accounted for from compactifying 6 dimensions with a central charge
$c_{6d}=12$.

Setting $\mathcal{N}=8$ from now on, the $L_{0}$ constant $a$ is given by

$24a=0_{\mathcal{Z}}-2_{bc}-6_{M\\!N}-2_{mn}-2_{\tilde{m}\tilde{n}}+2(a_{\rho\tilde{\rho}}+4_{\beta\gamma}+4_{\tilde{\beta}\tilde{\gamma}})=4+2a_{\rho\tilde{\rho}}+a_{6d}\,,$

where $a_{\rho\tilde{\rho}}=-8$ in the R sector and $a_{\rho\tilde{\rho}}=4$
in the NS sector, i.e. $a=-\frac{1}{2}+\frac{1}{24}a_{6d}$ in the R sector and
$a=\frac{1}{2}+\frac{1}{24}a_{6d}$ in the NS sector.

The actual spectrum of the model can be determined by considering oscillator
expansions of the supertwistor fields, given here on the Riemann sphere:

$\lambda_{aA}=\sum_{n}\\!\lambda_{aAn}\sigma^{-n-\frac{1}{2}},\quad\mu_{a}^{A}=\sum_{n}\\!\mu_{an}^{A}\sigma^{-n-\frac{1}{2}},\quad\eta_{a}^{\mathcal{I}}=\sum_{n}\\!\eta_{an}^{\mathcal{I}}\sigma^{-n-\frac{1}{2}}\,,$
(2.5)

with $n\,\epsilon\,\mathbb{Z}$ in the R sector and
$n\,\epsilon\,\mathbb{Z}+\frac{1}{2}$ in the NS sector. The expansions of the
energy momentum $T$ and its zero mode $L_{0}$ are after normal ordering

$T=\sum_{n\in\mathbb{Z}}\\!L_{n}\sigma^{-n-2},\quad
L_{0}=\\!\sum_{n\in\mathbb{Z}}\\!n\,:\mu_{an}^{A}\lambda^{a}_{A-n}:+\sum_{n\in\mathbb{Z}}\\!n\,:\tilde{\eta}^{a}_{I-n}\,\eta^{I}_{an}:+\ldots\,,$
(2.6)

where $\dots$ denotes contributions from non-twistor fields.

##  3 Spectrum of the Supergravity Model

After reviewing the notation in the previous section, the spectrum can now be
analyzed. It is clear from looking at the fixed vertex operator in (2) and the
expansion of $L_{0}$ in (2.6) that the spectrum is either generated by
appropriate homogeneous functions of just zero modes of $\lambda_{A}^{a}$ and
$\eta^{\mathcal{I}}_{a}$ in the R sector or by products of exactly two
$-\frac{1}{2}$ modes of $\lambda_{A}^{a}$ and $\eta^{\mathcal{I}}_{a}$ in the
NS sector, otherwise the vertex operator would be required to show derivatives
of the twistor fields444For instance such vertex operators appear for the
Berkovits-Witten twistor string [7].. The $\mu^{A}_{an}$ modes are excluded
from the spectrum because BRST cohomology requires that the non-negative modes
of the $\lambda^{Aa}\rho_{A}$ current annihilate physical states.

Therefore, the model must have an $L_{0}$ constant that is $a=0$ with
$a_{6d}=12$ in the R sector and $a=1$ with the same $a_{6d}=12$ in the NS
sector. But there is also the need to give a central charge contribution of
exactly 12. The requirement of $c_{6d}=12=a_{6d}$ can easily be achieved by
adding 6 bosonic scalars (see appendix A in [5]). On a side note, in the NS
sector, $a_{6d}$ = $c_{6d}$ always holds independently of the number of
fermionic or bosonic worldsheet scalar or spinor fields added for
compactification, i.e. always $a=1$ in the NS sector. This is not true for the
R sector.

The conclusion is that the model of [5] operates in both the R and NS sector,
using the same vertex operators (2). One important remark concerning the
lowest modes of bosonic twistor components needs to be added. When taking the
massless limit, the vertex operator (2) can be made to degenerate into a
vertex operator of positive helicity and one of negative helicity [2, 5]. A
vertex operator for positive helicity uses the twistor components
$\lambda^{a}_{\alpha 0}$ or $\lambda^{a}_{\alpha-\frac{1}{2}}$ as creation
operators, and a vertex operator for negative helicity uses the dual twistor
components $\tilde{\lambda}^{a}_{\dot{\alpha}0}$ or
$\tilde{\lambda}^{a}_{\alpha-\frac{1}{2}}$ as creation operators. But they
cannot co-exist in the same picture, otherwise a combination of them could be
used to generate physical states without being able to assign a helicity (the
’googly’ problem). This is similar to being unable to choose a Kähler
polarization simultaneously on both twistor and dual twistor space [11] (see
also Table 1 in next section).

In other words, the vertex operator (2) hides the fact of potentially
operating in two different pictures. In the R sector one picture has the first
two components in $\lambda^{a}_{A0}$ and $\mu^{aA}_{0}$ as creation operators
and the last two as annihilation operators and the other picture has the
nature of the components reversed. In the NS sector it is even more
complicated: in one picture $\tilde{\lambda}^{a}_{\dot{\alpha}-\frac{1}{2}}$
is an annihilation operator interchanging with
$\mu^{a\dot{\alpha}}_{\frac{1}{2}}$ as creation operator and in the other
picture $\lambda^{a}_{{\alpha}-\frac{1}{2}}$ is an annihilation operator
swapped with $\tilde{\mu}^{a\alpha}_{\frac{1}{2}}$ as creation operator555This
is a well-known fact already observed for the original 4-dimensional
ambitwistor[12]. Note that, depending on the polarization, the index of some
oscillators in (2.5) might get shifted by 1.. The scattering equations seem to
be able to interpolate between the two pictures. This becomes more evident in
the massless limit and the next sections will focus more on this limit,
although in section 5 it will be seen that with maximal supersymmetry the
vertex operators in the massive model can stay in one picture and still
generate the full $\mathcal{N}\\!=\\!8$ supergravity particle spectrum, i.e.
there is no inconsistency in using the vertex operators in (2). For the
remainder of the article the two pictures are referred to as ’polarization
pictures’.

One additional observation in the NS sector is that only two $-\frac{1}{2}$
modes can appear in the spectrum in order for $L_{0}=1$ to be valid, i.e. the
supersymmetric spectrum is truncated and only contains a single graviton and
some spin $\frac{3}{2}$ and spin 1 particles but no scalars or spin
$\frac{1}{2}$ fermions. This also implies that all particles have to remain
massless in the NS sector. This issue will be discussed more, with a possible
solution, in the last section 6 reserved for summary and discussion.

##  4 Intermediate Modification of the Supergravity Model

If one is on the lookout for a self-contained supergravity model with a
spectrum that includes particles that can be interpreted as gluons or quarks,
one requirement would be that the scattering of such particles leads to a
Parke-Taylor factor. The model in the previous sections does not fulfill this
condition without an external current algebra. This section introduces a
modification that keeps the model anomaly-free but allows for such a Parke-
Taylor factor. On the other hand, the model picks up some undesired features.
In the next section it will get an additional extension that makes it more
satisfactory. Nevertheless, it is worthwhile to not skip the intermediate step
because it relates closely to other models in the literature.

The modification of the model consists by not fixing the masses of the two
twistors and insisting on massless particles only, by doubling the number of
fermionic components of the supertwistor, and by gauging them in similar
fashion to the bosonic components with help of auxiliary fields:

$S=\int_{\Sigma}\mathcal{Z}^{a}\cdot\bar{\partial}\mathcal{Z}_{a}+A_{ab}\mathcal{Z}^{a}\cdot\mathcal{Z}^{b}+S_{\rho_{1}}+S_{\tilde{\rho}_{2}}+S_{\tau_{1}}+S_{\tau_{2}}\,,$

with

$\displaystyle
S_{\rho}=\int_{\Sigma}\tilde{\rho}^{A}\bar{\partial}\rho_{A}+b_{a}\lambda^{aA}\rho_{A}+\tilde{b}_{a}\lambda^{a}_{A}\tilde{\rho}^{A}\,,$
$\displaystyle
S_{\tau_{1}}=\int_{\Sigma}\tilde{\tau}_{1}^{\mathcal{I}}\bar{\partial}\tau_{1\mathcal{I}}+d_{1a}\eta^{aI}\tau_{1I}+\tilde{d}_{1a}\eta_{I}^{a}\tilde{\tau}_{1}^{I}\,,$
$\displaystyle
S_{\tau_{2}}=\int_{\Sigma}\tilde{\tau}_{2}^{\mathcal{I}}\bar{\partial}\tau_{2\mathcal{I}}+d_{2a}\tilde{\eta}_{I}^{a}\tau_{2}^{\prime
I}+\tilde{d}_{2a}\tilde{\eta}^{aI}\tilde{\tau}^{\prime}_{2I}\,,$

where supertwistor fields $\mathcal{Z}$ have been extended to

$\displaystyle\mathcal{Z}=(\lambda_{A},\mu^{A},\eta^{\iota})\,:\quad$
$\displaystyle\lambda_{A}=(\lambda_{\alpha},\tilde{\lambda}_{\dot{\alpha}})\,,\quad\mu^{A}=(\mu^{\dot{\alpha}},\tilde{\mu}^{\alpha})\,,\quad\eta^{\iota}=(\eta^{\mathcal{I}},\tilde{\eta}_{\mathcal{I}})\,,\;\eta^{\mathcal{I}}=(\eta^{I},\tilde{\eta}_{I})\,,\;$
$\displaystyle\tilde{\eta}_{\mathcal{I}}=(\tilde{\eta}^{\prime}_{I},\eta^{\prime
I})\,,\;I=1\ldots\frac{\mathcal{N}}{2},\,\mathcal{I}=1,\ldots,\mathcal{N},\,\iota=1,\ldots,2\mathcal{N}\,,\,\mathcal{N}=8\,,$

$(\tau_{\mathcal{I}},\tilde{\tau}^{\mathcal{I}})=((\tau_{I},\tau^{\prime
I}),(\tilde{\tau}^{I},\tilde{\tau}^{\prime}_{I}))$ are bosonic worldsheet
spinors, and $(b_{ra},\tilde{b}_{ra}),(d_{ra},\tilde{d}_{ra})$ are
$(0,1)$-forms on the worldsheet acting as fermionic Lagrange multipliers for
the constraints
$\lambda^{aA}\rho_{1A}=\lambda^{a}_{A}\tilde{\rho}_{1}^{A}=\lambda^{aA}\tilde{\rho}_{2A}=\lambda^{a}_{A}\rho_{2}^{A}=\eta^{aI}\tau_{1I}=\eta_{I}^{a}\tilde{\tau}_{1}^{I}=\tilde{\eta}_{I}^{a}\tau_{2}^{\prime
I}=\tilde{\eta}^{aI}\tilde{\tau}^{\prime}_{2I}=0$,

During BRST quantization the additional fermionic fields
$\\{d_{ra},\tilde{d}_{ra}\\}$ lead to new bosonic ghosts
$\\{(\beta_{ra}^{\prime},\gamma_{ra}^{\prime}),(\tilde{\beta}^{\prime}_{ra},\tilde{\gamma}^{\prime}_{ra})\\}$.
The SL(2,$\mathbb{C}$) anomaly coefficient becomes:

$a_{sl2}=\frac{3}{2}(\frac{1}{2}(8-2\mathcal{N}))_{\mathcal{Z}}+\frac{3}{2}(2_{\beta\gamma}+2_{\tilde{\beta}\tilde{\gamma}}+2_{\beta^{\prime}\gamma^{\prime}}+2_{\tilde{\beta}^{\prime}\tilde{\gamma}^{\prime}})-6_{M\\!N}=\frac{3}{2}(8-\mathcal{N})=0\,.$

It is still zero. The central charge vanishes as well:

$c=(-8+2\mathcal{N})_{\mathcal{Z}}-26_{bc}-6_{{M\\!N}}+2(4_{\rho\tilde{\rho}}+4_{\beta\gamma}+4_{\tilde{\beta}\tilde{\gamma}})+2(-\mathcal{N}_{\tau\tilde{\tau}}+4_{\beta^{\prime}\gamma^{\prime}}+4_{\tilde{\beta^{\prime}}\tilde{\gamma^{\prime}}})=0\,.$

As mentioned earlier, the $L_{0}$ constant $a$ in the NS sector does not
change and stays equal to 1 and in the R sector:

$24a=(16-4\mathcal{N})_{\mathcal{Z}}-2_{bc}-6_{{M\\!N}}+2(-8_{\rho\tilde{\rho}}+4_{\beta\gamma}+4_{\tilde{\beta}\tilde{\gamma}})+2(2\mathcal{N}_{\tau\tilde{\tau}}+4_{\beta^{\prime}\gamma^{\prime}}+4_{\tilde{\beta}^{\prime}\tilde{\gamma}^{\prime}})=24\,.$

i.e. $a=1$ in the R sector as well.

Although the fermionic modes in the supertwistors are doubled, similarly to
the exclusion of $\mu^{A}_{n}$ modes from the spectrum because of BRST
cohomology, now the $\tilde{\eta}_{\mathcal{I}n}$ cannot contribute to the
spectrum because of the requirement that the non-negative modes of the
$\eta^{aI}\tau_{1I}$ and $\tilde{\eta}^{a}_{I}\tau_{2}^{\prime I}$ currents
annihilate physical states. This leaves the spectrum in the NS sector
unchanged but the one in the R sector gets modified considerably, because
there always has to be a single $-\\!1$ twistor mode of $\lambda_{A}^{a}$ or
$\eta^{\mathcal{I}}_{a}$ in addition to an appropriate homogeneous function of
just zero modes.

The internal little group representation is assumed to be in a singlet, making
sure that there is only one graviton-like excitation. The R-symmetry is then
an SU(4) group in each of the two pictures, one with polarization
$\tilde{\lambda}_{\dot{\alpha}}\\!\sim\\!\partial/\partial\mu^{\dot{\alpha}},\tilde{\mu}^{\alpha}\\!\sim\\!\partial/\partial\lambda_{\alpha},\tilde{\eta}_{I}\\!\sim\\!\partial/\partial\eta^{\prime
I},\tilde{\eta}_{I}^{\prime}\\!\sim\\!\partial/\partial\eta^{I}$ for the zero
modes, the other one with polarization
$\lambda_{\alpha}\\!\sim\\!\partial/\partial\tilde{\mu}^{\alpha},\mu^{\dot{\alpha}}\\!\sim\\!\partial/\partial\tilde{\lambda}_{\dot{\alpha}},\eta^{I}\\!\sim\\!\partial/\partial\tilde{\eta}_{I}^{\prime},\eta^{\prime
I}\\!\sim\\!\partial/\partial\tilde{\eta}_{I}$. Table 1 displays the spectrum,
using the standard notation

$\braket{\lambda
g}=\varepsilon^{\alpha\beta}\lambda_{\alpha}g_{\beta}=-\braket{g\lambda}\qquad[\tilde{\lambda}f]=\varepsilon^{\dot{\alpha}\dot{\beta}}\tilde{\lambda}_{\dot{\alpha}}f_{\dot{\beta}}=-[f\tilde{\lambda}]\,.$

All helicity states have double occurrence, reflecting the symmetry between
twistor fields and their duals, and being related to each other through a
Fourier transformation [6]. The table also shows a strong resemblance with
part of the conformal supergravity spectrum of the Berkovits-Witten twistor
string with N=4 supersymmetry [6, 7]. However, because of the supersymmetric
gauging it does not contain the ’dipole’ states thought of being responsible
for the lack of unitarity in the Berkovits-Witten model [6].

Oscillators | (Helicity| SU(4))
---|---
$[\tilde{\lambda}_{-\\!1}^{a}f_{a}(\lambda^{b}_{\alpha 0},\\!\eta^{I}_{c0})]$ | $(2|1),\,\,\,\,\,\,\,\,\,(\frac{3}{2}|\bar{4}),\,\,\,\,\,\,\,\,\,(1|6),\,\,\,\,\,\,\,\,\,(\frac{1}{2}|4),\,\,\,\,\,\,\,\,\,(0|1)$
$\braket{\lambda^{a}_{-\\!1}g_{a}(\lambda^{b}_{\alpha 0},\\!\eta^{I}_{c0})}$ | $(0|1),\,\,\,\,\,\,(-\\!\frac{1}{2}|\bar{4}),\,\,\,\,\,(-\\!1|6),\,\,\,\,\,(-\\!\frac{3}{2}|4),\,\,\,\,\,(-\\!2|1)\\!$
$\tilde{\eta}^{a}_{I-\\!1}f^{I}_{a}(\lambda^{b}_{\alpha 0},\\!\eta^{I}_{c0})$ | $(\frac{3}{2}|4),\,(1|4\\!\otimes\\!\bar{4}),\,\,(\frac{1}{2}|4\\!\otimes\\!6),\,\,(0|4\\!\otimes\\!4),\,\,(-\\!\frac{1}{2}|4)$
$\eta^{I}_{a-\\!1}g_{I}^{a}(\lambda^{b}_{\alpha 0},\\!\eta^{I}_{c0})$ | $(\frac{1}{2}|\bar{4}),(0|\bar{4}\\!\otimes\\!\bar{4}),(-\\!\frac{1}{2}|\bar{4}\\!\otimes\\!6),(-\\!1|\bar{4}\\!\otimes\\!4),(-\\!\\!\frac{3}{2}|\bar{4})$
$\braket{\lambda^{a}_{-\\!1}\tilde{f}_{a}(\tilde{\lambda}^{b}_{\dot{\alpha}0},\\!\tilde{\eta}^{c}_{I0})}$ | $(-\\!2|1),\,\,\,(-\\!\frac{3}{2}|4),\,\,\,\,\,(-\\!1|6),\,\,\,\,\,\,(-\\!\frac{1}{2}|\bar{4}),\,\,\,\,\,\,\,(0|1)$
$[\tilde{\lambda}^{a}_{-\\!1}\tilde{g}_{a}(\tilde{\lambda}^{b}_{\dot{\alpha}0},\\!\tilde{\eta}^{c}_{I0})]$ | $\,(0|1),\,\,\,\,\,\,\,\,\,(\frac{1}{2}|4),\,\,\,\,\,\,\,\,\,(1|6),\,\,\,\,\,\,\,\,\,(\frac{3}{2}|\bar{4}),\,\,\,\,\,\,\,\,\,(2|1)$
$\eta^{I}_{a-\\!1}\tilde{f}_{I}^{a}(\tilde{\lambda}^{b}_{\dot{\alpha}0},\\!\tilde{\eta}^{c}_{I0})$ | $(-\\!\frac{3}{2}|\bar{4}),(-\\!1|\bar{4}\\!\otimes\\!4),(-\\!\frac{1}{2}|\bar{4}\\!\otimes\\!6),(0|\bar{4}\\!\otimes\\!\bar{4}),(\frac{1}{2}|\bar{4})$
$\tilde{\eta}^{a}_{I-\\!1}\tilde{g}^{I}_{a}(\tilde{\lambda}^{b}_{\dot{\alpha}0},\\!\tilde{\eta}^{c}_{I0})$ | $(-\\!\frac{1}{2}|4),\,\,(0|4\\!\otimes\\!4),\,\,\,\,(\frac{1}{2}|4\\!\otimes\\!6),\,\,(1|4\\!\otimes\\!\bar{4}),(\frac{3}{2}|4)$
Table 1: R Spectrum. The upper half is in the picture with polarization
$\tilde{\lambda}_{\dot{\alpha}}\\!\sim\\!\partial/\partial\mu^{\dot{\alpha}},\tilde{\mu}^{\alpha}\\!\sim\\!\partial/\partial\lambda_{\alpha},\tilde{\eta}_{I}\\!\sim\\!\partial/\partial\eta^{\prime
I},\tilde{\eta}_{I}^{\prime}\\!\sim\\!\partial/\partial\eta^{I}$ for the zero
modes, the lower half in the picture with polarization
$\lambda_{\alpha}\\!\sim\\!\partial/\partial\tilde{\mu}^{\alpha},\mu^{\dot{\alpha}}\\!\sim\\!\partial/\partial\tilde{\lambda}_{\dot{\alpha}},\eta^{I}\\!\sim\\!\partial/\partial\tilde{\eta}_{I}^{\prime},\eta^{\prime
I}\\!\sim\\!\partial/\partial\tilde{\eta}_{I}$. All helicity states have
double occurrence.

In the massless limit vertex operators are more conveniently distinguished by
the polarization picture they are operating in, and the polarization data can
be chosen in a special little group gauge such that only one of the two
twistors is non-zero, different per picture, and such that the effect of one
of the two $S_{\rho}$ actions is swallowed up during the transition from the
integration measure for scattering amplitudes of the massive model to the
simplified integration measure in the massless limit [2, 5], thus providing
vertex operators as in [13]. Further, the two actions of auxiliary fields
$S_{\tau_{1}}$ and $S_{\tau_{2}}$ can be ’distributed’ among the two pictures,
meaning that the $\tau_{2}$ fields do not contribute to vertex operators
involving $\tau_{1}$ fields and the other way around. Then the vertex
operators appear like

$\displaystyle\mathcal{V}$
$\displaystyle=\qquad\int\\!\frac{du}{u^{3}}\;\;\mathcal{W}(u)\;\bar{\delta}^{2}\\!\left(u\lambda_{\alpha}-\epsilon_{\alpha}\,\right)\;e^{u\left(\mu^{\dot{\alpha}}\tilde{\epsilon}_{\dot{\alpha}}+\eta^{\prime
I}q_{I}+\eta^{I}q^{\prime}_{I}\right)}\;,$ $\displaystyle\tilde{\mathcal{V}}$
$\displaystyle=\qquad\int\\!\frac{du}{u^{3}}\;\;\tilde{\mathcal{W}}(u)\;\bar{\delta}^{2}(u\tilde{\lambda}_{\dot{\alpha}}-\tilde{\epsilon}_{\dot{\alpha}})\;\;e^{u\left(\tilde{\mu}^{\alpha}\epsilon_{\alpha}+\tilde{\eta}^{\prime}_{I}\tilde{q}^{I}+\tilde{\eta}_{I}\tilde{q}^{\prime
I}\right)}\;,$ $\displaystyle V$
$\displaystyle=\int_{\Sigma}\\!\\!d\sigma\\!\\!\int\\!\frac{du}{u^{3}}\;\;\,w(u)\;\;\bar{\delta}^{2}\\!\left(u\lambda_{\alpha}-\epsilon_{\alpha}\,\right)\;e^{u\left(\mu^{\dot{\alpha}}\tilde{\epsilon}_{\dot{\alpha}}+\eta^{\prime
I}q_{I}+\eta^{I}q^{\prime}_{I}\right)}\;,$ (4.1) $\displaystyle\tilde{V}$
$\displaystyle=\int_{\Sigma}\\!\\!d\sigma\\!\\!\int\\!\frac{du}{u^{3}}\;\;\,\tilde{w}(u)\;\;\bar{\delta}^{2}(u\tilde{\lambda}_{\dot{\alpha}}-\tilde{\epsilon}_{\dot{\alpha}})\;\;e^{u\left(\tilde{\mu}^{\alpha}\epsilon_{\alpha}+\tilde{\eta}^{\prime}_{I}\tilde{q}^{I}+\tilde{\eta}_{I}\tilde{q}^{\prime
I}\right)}\;,$

where for fixed vertex operators $\mathcal{W}(u)$ and $\tilde{\mathcal{W}}(u)$
are products of fermionic ghost fields and delta functions of bosonic ghost
fields and in the R-sector with an additional factor of the form
$\,u[\tilde{\lambda}\tilde{\epsilon}]\,$ or
$\,u\,\tilde{\eta}_{I}\tilde{q}^{I}\,$ in $\mathcal{W}(u)$ and
$\,u\\!\braket{\lambda\epsilon}\,$ or $\,u\,\eta^{I}q_{I}\,$ in
$\tilde{\mathcal{W}}(u)$, and for integrated vertex operators $w(u)$ and
$\tilde{w}(u)$ are

$\displaystyle w(u)=$
$\displaystyle\;\left(u[\tilde{\lambda}\tilde{\epsilon}]-u^{2}\,[\tilde{\epsilon}\rho_{2}]\,[\tilde{\epsilon}\tilde{\rho}_{2}]\right)\left(u\,\tilde{\eta}_{I}\,\tilde{q}^{I}-u^{2}\,\,\tilde{q}_{I}\tau^{\prime
I}_{2}\,\,\tilde{q}^{J}\\!\tilde{\tau}^{\prime}_{2J}\right)_{\\!\tilde{q}^{I}\neq
0}$ $\displaystyle\tilde{w}(u)=$
$\displaystyle\;\left(u\\!\braket{\lambda\epsilon}\\!-u^{2}\\!\\!\braket{\epsilon\rho_{1}}\\!\braket{\epsilon\tilde{\rho}_{1}}\right)\left(u\,\eta^{I}q_{I}\,-u^{2}\,\,q^{I}\tau_{1I}\;q_{J}\tilde{\tau}_{1}^{J}\right)_{\\!q_{I}\neq
0}\,,$ (4.2)

with the same additional factor in the R sector as for fixed vertex operators.
The distinctions between the $q^{\prime}$ and $q$ parameters and between
$\tilde{q}^{\prime}$ and $\tilde{q}$ are actually not justified because they
would lift the R-symmetry from SU(4) to SU(4) $\\!\otimes\\!$ SU(4) (see next
section). Also, by setting $q_{I}\\!\\!=\\!\\!0(=\\!\\!q^{\prime}_{I})$ and
$\tilde{q}^{I}\\!\\!\\!=\\!\\!0(=\\!\\!\tilde{q}^{\prime I})$ and omitting the
new factors in (4) involving the $\eta$ and $\tau$ fields one obtains simpler
vertex operators representing gravitons. Note that integrated vertex operators
$V$ and $\tilde{V}$ containing the factor with $\tau$ fields automatically
have at least one fermionic twistor component and, therefore, stand for a
particle with spin$\leq\\!\frac{3}{2}$. Further, for every second row in Table
1 the corresponding integrated vertex operators are not covered by (4) but
they are taken care of by the ones for the equivalent Fourier-transformed
states in the other picture.

Tree scattering amplitudes of these vertex operators will contain the reduced
determinants of Hodges matrices like in references [13, 1, 2, 3, 4, 5], and
also the supersymmetric exponential factor

$e^{F_{\mathcal{N}}}=\mathrm{exp}\\!\left[\;\sum_{\genfrac{}{}{0.0pt}{}{j\in-}{k\in+}}\frac{u_{j}u_{k}}{\sigma_{j}\\!-\\!\sigma_{k}}\tilde{q}^{I}_{j}q_{kI}\right]\,,\;q_{I}\\!=\\!\Omega_{IJ}\tilde{q}^{J}.$
(4.3)

What is new here are the factors involving the $\tau$ fields. They lead to
fermionic Hodges-like matrices with reduced determinants that, because of the
Grassmann odd nature of the components of the super polarization data, are
limited in the sense that the same location can occur at most 4 times per
term, and if one assumes that all vertex operators with $\tau$ fields stand
for particles with spin 1 then each term reduces to a product of propagators
between all locations of the same helicity, each end of the propagators
appearing exactly twice up to two showing up only once, one of them
representing the location of a fixed vertex operator. By pulling down twice
the exponent from (4.3), one for each fixed vertex operator, one can connect
two propagator products, one for positive and one for negative helicities, and
arrive at a Parke-Taylor factor for all these particles. More details will be
given in the next section for the more interesting improved model.

The current model has some undesired features in the R sector. The vertex
operators have these additional factors arising from non-zero twistor modes,
which make the scattering amplitudes look unattractive. Also, although the
model features a Parke-Taylor factor for scattering of spin 1 particles, it
does not allow for full permutation symmetry of the entries (the same
helicities need to be arranged together).

In appendix A another model found by the author earlier is described in the
current notation which has the same spectrum as in Table 1 with similarly
unappealing scattering amplitudes.

##  5 Improved Anomaly-free Supergravity Model Extension

The model of the previous section could be improved by making the $L_{0}$
constant $a$ zero in the R sector. This can be done by introducing 4 bosonic
Lagrange multipliers with associated fermionic ghost-antighost pairs and
adding 8 fermionic worldsheet spinors or, equivalently, reducing 8 bosonic
worldsheet spinors, which keeps the central charge unchanged but reduces $a$
by 1.

The minimally changed model is:

$\displaystyle S=\int_{\Sigma}$
$\displaystyle\mathcal{Z}^{a}\cdot\bar{\partial}\mathcal{Z}_{a}+A_{ab}\mathcal{Z}^{a}\cdot\mathcal{Z}^{b}+a\lambda^{2}+\tilde{a}\tilde{\lambda}^{2}+S_{\rho_{1}}+S_{\tilde{\rho}_{2}}+S_{\tau_{1}}+S_{\tau_{2}}\,,$
$\displaystyle\mathcal{Z}=(\lambda_{A},\mu^{A},\eta^{\iota})\,,\;\eta^{\iota}=(\eta^{\mathcal{I}},\tilde{\eta}_{\mathcal{I}})\,,\;\eta^{\mathcal{I}}=(\eta^{I},\tilde{\eta}_{\dot{I}})\,,\;\tilde{\eta}_{\mathcal{I}}=(\tilde{\eta}^{\prime}_{I},\eta^{\prime\dot{I}})\,,$
$\displaystyle
I,\dot{I}=1\ldots\frac{\mathcal{N}}{2},\,\mathcal{I}=1,\ldots,\mathcal{N},\,\iota=1,\ldots,2\mathcal{N}\,,\,\mathcal{N}=8\,,$
(5.1) $\displaystyle S_{\rho}=\int_{\Sigma}$
$\displaystyle\tilde{\rho}^{A}\bar{\partial}\rho_{A}+b_{a}\lambda^{aA}\rho_{A}+\tilde{b}_{a}\lambda^{a}_{A}\tilde{\rho}^{A}\,,$
$\displaystyle S_{\tau_{1}}=\int_{\Sigma}$
$\displaystyle\tilde{\tau}_{1}^{I}\,\bar{\partial}\tau_{1I}+d_{1a}\eta^{aI}\tau_{1I}+\tilde{d}_{1a}\eta^{a}_{I}\tilde{\tau}_{1}^{I}+g_{1}\tilde{\tau}_{1}^{I}\tau_{1I}\,,$
$\displaystyle S_{\tau_{2}}=\int_{\Sigma}$
$\displaystyle\tilde{\tau}_{2\dot{I}}\,\bar{\partial}\tau_{2}^{\dot{I}}+d_{2a}\tilde{\eta}^{a}_{\dot{I}}\tau_{2}^{\dot{I}}+\tilde{d}_{2a}\tilde{\eta}^{a\dot{I}}\tilde{\tau}_{2\dot{I}}+g_{2}\tilde{\tau}_{2\dot{I}}\tau_{2}^{\dot{I}}\,,$

where $\mathcal{Z}$ is the extended supertwistor from last section,
$(\rho_{A},\tilde{\rho}^{A})$ are the same auxiliary fermionic worldsheet
spinors, $\tau_{1I}$ and $\tau_{2\dot{I}}$ are auxiliary bosonic worldsheet
spinors gauging the extended fermionic components of the supertwistor,
$(d_{ra},\tilde{d}_{ra})$ and $(g_{r})$ are fermionic and bosonic Lagrange
multipliers, respectively, for gauge constraints satisfied by the $\tau$
fields, and $a$ and $\tilde{a}$ are the Lagrange multipliers from the original
action (2.2) in section 2. $j^{H}$ is set to 0 here to ensure the massless
limit, mainly because integrated vertex operators for the massive case are
unknown. Also, the second half $\dot{I}$ of indices $\mathcal{I}$ and the
indices of $\tau_{2\dot{I}}$ have been dotted, for later convenience.

One observation is that the $\rho_{r}$ and $\tau_{r}$ fields could be bundled
into a couple of supertwistors with reversed statistics, as has been done in
Skinner’s model [14], although there they transform under the little group.
They do not contribute to the spectrum and are considered purely as auxiliary,
not as matter fields.

During BRST quantization the additional fermionic
$\\{d_{ra},\tilde{d}_{ra}\\}$ and bosonic $\\{g_{r}\\}$ fields lead to other
new bosonic
$\\{(\beta^{\prime}_{ra},\gamma^{\prime}_{ra}),(\tilde{\beta}^{\prime}_{ra},\tilde{\gamma}^{\prime}_{ra})\\}$
and fermionic ghosts $\\{(s_{r},t_{r}))\\}$, respectively. The
SL(2,$\mathbb{C}$) anomaly coefficient then becomes like in section 4:

$a_{sl2}=\frac{3}{2}(\frac{1}{2}(8-2\mathcal{N}))_{\\!\mathcal{Z}}-6_{M\\!N}+\frac{3}{2}(2_{\beta\gamma}+2_{\tilde{\beta}\tilde{\gamma}}+2_{\beta^{\prime}\gamma^{\prime}}+2_{\tilde{\beta}^{\prime}\tilde{\gamma}^{\prime}})=\frac{3}{2}(8-\mathcal{N})=0\,.$

The central charge is:

$c\\!=\\!(\\!-8+2\mathcal{N}\\!)_{\\!\mathcal{Z}}\\!-26_{bc}\\!-6_{M\\!N}\\!-2_{mn}\\!-2_{\tilde{m}\tilde{n}}\\!+2(4_{\rho}+4_{\beta\gamma}+4_{\tilde{\beta}\tilde{\gamma}}\\!\\!-\\!\frac{1}{2}\mathcal{N}_{\\!\tau}\\!+4_{\beta^{\prime}\gamma^{\prime}}\\!+4_{\tilde{\beta^{\prime}}\tilde{\gamma^{\prime}}}\\!\\!-2_{st})\\!=\\!-8+\\!\mathcal{N}\\!=\\!0\,.$

The $L_{0}$ constant $a$ stays 1 in the NS sector and in the R sector it
changes to:

$24a\\!=\\!\\!(\\!16-\\!4\mathcal{N}\\!)_{\\!\mathcal{Z}}\\!-\\!2_{bc}\\!-6_{M\\!N}\\!-2_{mn}\\!-2_{\tilde{m}\tilde{n}}\\!+2(\\!-8_{\\!\rho}\\!+4_{\beta\gamma}\\!+4_{\tilde{\beta}\tilde{\gamma}}\\!+\\!\mathcal{N}_{\\!\tau}\\!+4_{\beta^{\prime}\gamma^{\prime}}\\!+4_{\tilde{\beta^{\prime}}\tilde{\gamma^{\prime}}}\\!\\!-2_{st})\\!=\\!16-\\!2\mathcal{N}\\!=\\!0\,.$

i.e. $a=0$ in the R sector, as desired. And the model is anomaly-free as well.

Oscillators |  | $\eta^{I}$ | $\eta^{I}_{1}\eta^{I}_{2}$ | $\eta^{I}_{1}\eta^{I}_{2}\eta^{I}_{3}$ | $\eta^{I}_{1}\eta^{I}_{2}\eta^{I}_{3}\eta^{I}_{4}$
---|---|---|---|---|---
| $(2|1\\!\otimes\\!1)$ | $\;\,(\frac{3}{2}|1\\!\otimes\\!\bar{4})$ | $\;\,(1|1\\!\otimes\\!6)$ | $\;\,(\frac{1}{2}|1\\!\otimes\\!4)$ | $\;\;(0|1\\!\otimes\\!1)$
$\tilde{\eta}_{\dot{I}}$ | $(\frac{3}{2}|4\\!\otimes\\!1)$ | $\;\,(1|4\\!\otimes\\!\bar{4})$ | $(\frac{1}{2}|4\\!\otimes\\!6)$ | $\;\,(0|4\\!\otimes\\!4)$ | $(-\\!\frac{1}{2}|4\\!\otimes\\!1)$
$\tilde{\eta}_{1\dot{I}}\tilde{\eta}_{2\dot{I}}$ | $(1|6\\!\otimes\\!1)$ | $\;\,(\frac{1}{2}|6\\!\otimes\\!\bar{4})$ | $(0|6\\!\otimes\\!6)$ | $(-\\!\frac{1}{2}|6\\!\otimes\\!4)$ | $(-\\!1|6\\!\otimes\\!1)$
$\tilde{\eta}_{1\dot{I}}\tilde{\eta}_{2\dot{I}}\tilde{\eta}_{3\dot{I}}$ | $(\frac{1}{2}|\bar{4}\\!\otimes\\!1)$ | $\;\,(0|\bar{4}\\!\otimes\\!\bar{4})$ | $(-\\!\frac{1}{2}|\bar{4}\\!\otimes\\!6)$ | $(-\\!1|\bar{4}\\!\otimes\\!4)$ | $(-\\!\frac{3}{2}|\bar{4}\\!\otimes\\!1)$
$\tilde{\eta}_{1\dot{I}}\tilde{\eta}_{2\dot{I}}\tilde{\eta}_{3\dot{I}}\tilde{\eta}_{4\dot{I}}$ | $(0|1\\!\otimes\\!1)$ | $(-\\!\frac{1}{2}|1\\!\otimes\\!\bar{4})$ | $(-\\!1|1\\!\otimes\\!6)$ | $(-\\!\frac{3}{2}|1\\!\otimes\\!4)$ | $(-\\!2|1\\!\otimes\\!1)$
Oscillators |  | $\tilde{\eta}_{\dot{I}}$ | $\tilde{\eta}_{1\dot{I}}\tilde{\eta}_{2\dot{I}}$ | $\tilde{\eta}_{1\dot{I}}\tilde{\eta}_{2\dot{I}}\tilde{\eta}_{3\dot{I}}$ | $\tilde{\eta}_{1\dot{I}}\tilde{\eta}_{2\dot{I}}\tilde{\eta}_{3\dot{I}}\tilde{\eta}_{4\dot{I}}$
| $(-\\!2|1\\!\otimes\\!1)$ | $(-\\!\frac{3}{2}|1\\!\otimes\\!4)$ | $(-\\!1|1\\!\otimes\\!6)$ | $(-\\!\frac{1}{2}|1\\!\otimes\\!\bar{4})$ | $(0|1\\!\otimes\\!1)$
$\eta^{I}$ | $(-\\!\frac{3}{2}|\bar{4}\\!\otimes\\!1)$ | $(-\\!1|\bar{4}\\!\otimes\\!4)$ | $(-\\!\frac{1}{2}|\bar{4}\\!\otimes\\!6)$ | $\;\,(0|\bar{4}\\!\otimes\\!\bar{4})$ | $(\frac{1}{2}|\bar{4}\\!\otimes\\!1)$
$\eta^{I}_{1}\eta^{I}_{2}$ | $(-\\!1|6\\!\otimes\\!1)$ | $(-\\!\frac{1}{2}|6\\!\otimes\\!4)$ | $\;\,(0|6\\!\otimes\\!6)$ | $(\;\,\frac{1}{2}|6\\!\otimes\\!\bar{4})$ | $(1|6\\!\otimes\\!1)$
$\eta^{I}_{1}\eta^{I}_{2}\eta^{I}_{3}$ | $(-\\!\frac{1}{2}|4\\!\otimes\\!1)$ | $\;\,(0|4\\!\otimes\\!4)$ | $\;\,(\frac{1}{2}|4\\!\otimes\\!6)$ | $\;\,(1|4\\!\otimes\\!\bar{4})$ | $(\frac{3}{2}|4\\!\otimes\\!1)$
$\eta^{I}_{1}\eta^{I}_{2}\eta^{I}_{3}\eta^{I}_{4}$ | $\;\;(0|1\\!\otimes\\!1)$ | $\;\,(\frac{1}{2}|1\\!\otimes\\!4)$ | $\;\,(1|1\\!\otimes\\!6)$ | $\;\,(\frac{3}{2}|1\\!\otimes\\!\bar{4})$ | $(2|1\\!\otimes\\!1)$
Table 2: R Spectrum. Every cell shows (helicity|SU(4) $\\!\\!\times\\!\\!$
SU(4)). The upper half is in the picture with polarization
$\tilde{\lambda}\\!\sim\\!\partial/\partial\mu,\tilde{\mu}\\!\sim\\!\partial/\partial\lambda$,
the lower half in the picture with polarization
$\lambda\\!\sim\\!\partial/\partial\tilde{\mu},\mu\\!\sim\\!\partial/\partial\tilde{\lambda}$.
All helicity states have double occurrence of the same group representation.

The spectrum can be examined from the value of the $L_{0}$ constant $a$. As in
the previous section it might look differently depending on the selected
polarization picture,
$\tilde{\lambda}_{\dot{\alpha}}\\!\sim\\!\partial/\partial\mu^{\dot{\alpha}},\tilde{\mu}^{\alpha}\\!\sim\\!\partial/\partial\lambda_{\alpha},\tilde{\eta}_{\dot{I}}\\!\sim\\!\partial/\partial\eta^{\prime\dot{I}},\tilde{\eta}_{I}^{\prime}\\!\sim\\!\partial/\partial\eta^{I}$
versus
$\lambda_{\alpha}\\!\sim\\!\partial/\partial\tilde{\mu}^{\alpha},\mu^{\dot{\alpha}}\\!\sim\\!\partial/\partial\tilde{\lambda}_{\dot{\alpha}},\eta^{I}\\!\sim\\!\partial/\partial\tilde{\eta}_{I}^{\prime},\eta^{\prime\dot{I}}\\!\sim\\!\partial/\partial\tilde{\eta}_{\dot{I}}$.
Because the fermionic modes are not critical for the consistency of helicity
assignment in the polarization, one can try to use in the R sector both of the
zero modes $\eta^{I}_{0}$ and $\tilde{\eta}_{0\dot{I}}$ as creation operators
with $\tilde{\eta}^{\prime}_{0I}$ and $\eta^{\prime\dot{I}}_{0}$ as
annihilation operators in both pictures, i.e. treating them in symmetric
fashion, and similarly the $-\frac{1}{2}$ modes in the NS sector. This lifts
the R-symmetry from SU(4) to SU(4) $\\!\\!\times\\!\\!$ SU(4), subgroup of
SU(8), as reflected in Table 2. By counting all particle modes of equal spin
together one obtains the $\mathcal{N}=8$ supergravity multiplet:
1 spin $\pm 2$ boson, 8 spin $\pm\frac{3}{2}$ fermions, 28 vector bosons, 56
spin $\pm\frac{1}{2}$ fermions, and 70 scalars, all in all 128 bosonic and 128
fermionic modes.

This applies to the original massive model as well (although from a strict
oscillator point of view there are only enough fermionic creation modes for
$\frac{\mathcal{N}}{2}=4$ supersymmetry). Exactly the same spectrum is covered
fully in both polarization pictures, and they can be related through a Fourier
transformation like in conformal supergravity[6]. Therefore, in principle one
can stay just in one picture when using vertex operators and computing
scattering amplitudes. This is also valid in the massive case, with some modes
potentially grouped together making up massive particles, for instance
$(\frac{3}{2},-\frac{1}{2}|4\\!\otimes\\!1),(-\frac{3}{2},\frac{1}{2}|1\\!\otimes\\!4),(\frac{3}{2},-\frac{1}{2}|1\\!\otimes\\!\bar{4}),(-\frac{3}{2},\frac{1}{2}|\bar{4}\\!\otimes\\!1)$
to represent up to 8 massive spin $\frac{3}{2}$ particles or
$(\pm 1|1\\!\otimes\\!6),(\pm 1|6\\!\otimes\\!1)$ and the $6$ from
$(0|4\\!\otimes\\!4=10\\!\oplus\\!6),(0|\bar{4}\\!\otimes\\!\bar{4}=\bar{10}\\!\oplus\\!6)$
for up to 12 massive vector bosons.

First it seems that the spectrum in the NS sector is the same as in the R
sector, but because of the requirement of not having more than 2 on-shell
oscillator modes it is severely truncated: the supersymmetric multiplet is
reduced to one graviton with spin $\pm 2$, 8 spin $\pm\frac{3}{2}$ particles,
and 28 vector bosons, leaving out 56 spin $\pm\frac{1}{2}$ fermions and 70
scalars, and thus breaking target space supersymmetry. Further, without these
lower spin modes, the particles cannot acquire mass through some symmetry
breakdown and must remain massless. On the other hand, in the discussion of
section 6 it will be argued that the spectrum in the NS sector can be viewed
as arising from the R sector through a little group gauge transformation that
in the quantized theory can be interpreted as a spectral flow operation.

Choosing the same little group gauge and polarization data as in section 4
similar vertex operators (4) and (4) are obtained in the two polarization
pictures but without the additional factor in the R-sector and with the
$\tau^{\prime}$ fields replaced by the ordinary $\tau$:

$\displaystyle\mathcal{V}$
$\displaystyle=\qquad\int\\!\frac{du}{u^{3}}\;\;\mathcal{W}(u)\;\bar{\delta}^{2}\\!\left(u\lambda_{\alpha}-\epsilon_{\alpha}\,\right)\;e^{u\left(\mu^{\dot{\alpha}}\tilde{\epsilon}_{\dot{\alpha}}+\eta^{\prime\dot{I}}q_{\dot{I}}+\eta^{I}q^{\prime}_{I}\right)}\;,$
$\displaystyle\tilde{\mathcal{V}}$
$\displaystyle=\qquad\int\\!\frac{du}{u^{3}}\;\;\tilde{\mathcal{W}}(u)\;\bar{\delta}^{2}(u\tilde{\lambda}_{\dot{\alpha}}-\tilde{\epsilon}_{\dot{\alpha}})\;\;e^{u\left(\tilde{\mu}^{\alpha}\epsilon_{\alpha}+\tilde{\eta}^{\prime}_{I}\tilde{q}^{I}+\tilde{\eta}_{\dot{I}}\tilde{q}^{\prime\dot{I}}\right)}\;,$
$\displaystyle V$
$\displaystyle=\int_{\Sigma}\\!\\!d\sigma\\!\\!\int\\!\frac{du}{u^{3}}\;\;\,w(u)\;\;\bar{\delta}^{2}\\!\left(u\lambda_{\alpha}-\epsilon_{\alpha}\,\right)\;e^{u\left(\mu^{\dot{\alpha}}\tilde{\epsilon}_{\dot{\alpha}}+\eta^{\prime\dot{I}}q_{\dot{I}}+\eta^{I}q^{\prime}_{I}\right)}\;,$
(5.2) $\displaystyle\tilde{V}$
$\displaystyle=\int_{\Sigma}\\!\\!d\sigma\\!\\!\int\\!\frac{du}{u^{3}}\;\;\,\tilde{w}(u)\;\;\bar{\delta}^{2}(u\tilde{\lambda}_{\dot{\alpha}}-\tilde{\epsilon}_{\dot{\alpha}})\;\;e^{u\left(\tilde{\mu}^{\alpha}\epsilon_{\alpha}+\tilde{\eta}^{\prime}_{I}\tilde{q}^{I}+\tilde{\eta}_{\dot{I}}\tilde{q}^{\prime\dot{I}}\right)}\;,$

where for fixed vertex operators $\mathcal{W}(u)$ and $\tilde{\mathcal{W}}(u)$
are again products of fermionic ghost fields and delta functions of bosonic
ghost fields, and for integrated vertex operators

$\displaystyle w(u)=$
$\displaystyle\;\left(u[\tilde{\lambda}\tilde{\epsilon}]-u^{2}\,[\tilde{\epsilon}\rho_{2}]\,[\tilde{\epsilon}\tilde{\rho}_{2}]\right)\left(u\,\tilde{\eta}_{\dot{I}}\Omega^{\dot{I}\dot{J}}q_{\dot{J}}-u^{2}\,\,q_{\dot{I}}\tau_{2}^{\dot{I}}\,\,q_{\dot{J}}\Omega^{\dot{J}\dot{K}}\tilde{\tau}_{2\dot{K}}\right)_{q_{\dot{I}}\neq
0}$ $\displaystyle\tilde{w}(u)=$
$\displaystyle\;\left(u\\!\braket{\lambda\epsilon}\\!-u^{2}\\!\\!\braket{\epsilon\rho_{1}}\\!\braket{\epsilon\tilde{\rho}_{1}}\right)\left(u\,\eta^{I}\Omega_{IJ}\tilde{q}^{J}\,-u^{2}\,\,\tilde{q}^{I}\\!\tau_{1I}\,\,\tilde{q}^{J}\Omega_{JK}\tilde{\tau}_{1}^{J}\,\right)_{\tilde{q}^{I}\neq
0}\,.$ (5.3)

In contrast to the model in the previous section, $q^{\prime}$ and $q$ are
distinct parameters and also $\tilde{q}^{\prime}$ and $\tilde{q}$ because of
the SU(4) $\\!\otimes\\!$ SU(4) R-symmetry. Further, by setting
$q_{I}\\!\\!=\\!\\!0$ and $\tilde{q}^{I}\\!\\!\\!=\\!\\!0$ and omitting the
new factors in (5) involving the $\eta$ and $\tau$ fields one obtains simpler
vertex operators including the ones representing gravitons.

To get the tree scattering matrix, let $g$ denote the set of positive-helicity
vertex operators $V^{g}$ that do not include the factor with $\tau_{2}$ fields
in $w(u)$ in (5), $\tilde{g}$ the set of negative-helicity vertex operators
$\tilde{V}^{g}$ that do not include the factor with $\tau_{1}$ fields in
$\tilde{w}(u)$ in (5), $h$ the set of the other positive-helicity vertex
operators $V$, and $\tilde{h}$ the set of other negative-helicity vertex
operators $\tilde{V}$. Note that $g$ and $\tilde{g}$ include the vertex
operators for the gravitons. Then a tree scattering amplitude looks like:

$\mathcal{A}=\left<\frac{1}{\mathrm{vol}\,\mathrm{GL}(2,\mathbb{C})}\,\prod_{j\in\tilde{h}}\\!\tilde{V}_{j}\prod_{p\in
h}\\!V_{p}\,\,\prod_{i\in\tilde{g}}\\!\tilde{V}_{i}^{g}\prod_{k\in
g}\\!V_{k}^{g}\right>\,,$

where the factor $\mathrm{vol}\,\mathrm{GL}(2,\mathbb{C})$ comes from three
zero-modes of the $c$ ghost and one degree of freedom remaining from gauging
the little group [2], and where one member of $h$ (or of $g$ when $h$ is
empty) and one member of $\tilde{h}$ (or of $\tilde{g}$ when $\tilde{h}$ is
empty) are fixed vertex operators666$g\cup h$ and $\tilde{g}\cup\tilde{h}$ are
either both empty or non-empty [15, 16, 17]., fixed for every worldsheet
supersymmetry represented in the correlation function.

The computation of the amplitude is standard [13, 1, 2, 3, 4, 5] except for
the product of factors containing the auxiliary $\tau$ fields leading to
reduced determinants of fermionic Hodges-like matrices, similarly to the
bosonic fields. Explicitly:

$\mathcal{A}=\int\\!\\!\\!\prod_{\genfrac{}{}{0.0pt}{}{i\in g\cup
h}{\cup\tilde{g}\cup\tilde{h}}}\\!\\!\\!\frac{\mathrm{d}u_{i}}{u_{i}^{3}}\\!\\!\prod_{r\in
g\cup
h}\\!\\!\\!\mathrm{d}\sigma_{r}\,\bar{\delta}^{2}\\!(\epsilon_{r}\\!-\\!u_{r}\lambda(\sigma_{r}))\\!\\!\\!\prod_{l\in\tilde{g}\cup\tilde{h}}\\!\\!\\!\mathrm{d}\sigma_{l}\,\bar{\delta}^{2}\\!(\tilde{\epsilon}_{l}\\!-\\!u_{l}\tilde{\lambda}(\sigma_{l}))\frac{\mathrm{det}^{\prime}H\,\,\mathrm{det}^{\prime}\tilde{H}e^{F_{\mathcal{N}}}}{\mathrm{vol}\,\mathrm{GL}(2,\mathbb{C})}\mathrm{G}^{\prime}$
(5.4)

where $\lambda(\sigma_{r})$ and $\tilde{\lambda}(\sigma_{l})$ fulfill the
scattering equations

$\displaystyle\lambda(\sigma_{r})=\sum_{l\in\tilde{g}\cup\tilde{h}}\frac{u_{l}\epsilon_{l}}{\sigma_{r}-\sigma_{l}},$
$\displaystyle\tilde{\lambda}(\sigma_{l})=\sum_{r\in g\cup
h}\frac{u_{r}\tilde{\epsilon}_{r}}{\sigma_{l}-\sigma_{r}},$

$H$ is a symmetric $(|g|\\!+\\!|h|)\times(|g|\\!+\\!|h|)$ matrix, and
$\tilde{H}$ is a symmetric
$(|\tilde{g}|\\!+\\!|\tilde{h}|)\times(|\tilde{g}|\\!+\\!|\tilde{h}|)$ matrix
arising from the contractions between the twistor and $\rho$ fields appearing
in the vertex operators with the following elements:

$\displaystyle
H^{lr}=\frac{u_{l}u_{r}[\tilde{\epsilon}_{l}\tilde{\epsilon}_{r}]}{\sigma_{l}-\sigma_{r}}\;\,\text{for}\;l,r\\!\in\\!g\cup
h,l\\!\neq\\!r,$ $\displaystyle H^{ll}\,=-\\!\\!\\!\\!\\!\sum_{r\in g\cup
h\setminus\\{l\\}}\\!\\!H^{lr}\;\text{for}\;l\\!\in\\!g\cup h\,,$
$\displaystyle\tilde{H}^{lr}=\frac{u_{l}u_{r}\braket{\epsilon_{l}\epsilon_{r}}}{\sigma_{l}-\sigma_{r}}\;\text{for}\;l,r\\!\in\\!\tilde{g}\cup\tilde{h},l\\!\neq\\!r,$
$\displaystyle\tilde{H}^{ll}\,=-\\!\\!\\!\\!\\!\sum_{r\in\tilde{g}\cup\tilde{h}\setminus\\{l\\}}\\!\\!\tilde{H}^{lr}\;\text{for}\;l\\!\in\\!\tilde{g}\cup\tilde{h}\,.$

$H$ and $\tilde{H}$ each have co-rank one with vanishing determinant and det′
indicates the operation of removing one row and one column before computing
the determinant. The result of this operation is actually independent of the
choice of row and column removed.

The supersymmetric exponential factor exp $F_{\mathcal{N}}$ becomes like (4.3)
in the previous section, except for increased super polarization data
$q^{\mathcal{I}}$ with $\mathcal{N}=8$ components:

$e^{F_{\mathcal{N}}}=\mathrm{exp}\\!\left[\;\sum_{\genfrac{}{}{0.0pt}{}{j\in\tilde{g}\cup\tilde{h}}{k\in
g\cup
h}}\frac{u_{j}u_{k}}{\sigma_{j}\\!-\\!\sigma_{k}}\tilde{q}^{\mathcal{I}}_{j}q_{k\mathcal{I}}\right]\,,\,q_{\mathcal{I}}=(q^{\prime}_{I},q_{\dot{I}}),\tilde{q}^{\mathcal{I}}=(\tilde{q}^{I},\tilde{q}^{\prime\dot{I}}),\,q_{\mathcal{I}}\\!\equiv\\!\Omega_{\mathcal{IJ}}\tilde{q}^{\mathcal{J}}\,.$

This is the conventional massless limit of the supersymmetric exponential
factor [2].

G′ is 1 for empty $h$ and $\tilde{h}$, otherwise stands for the product of
reduced determinants
$\mathrm{det}^{\prime}G_{\tilde{h}}\,\mathrm{det}^{\prime}G_{h}$ of the
fermionic Hodges-like matrices $G_{\tilde{h}},G_{h}$ defined by

$\displaystyle
G_{h}^{lr}=\frac{u_{l}u_{r}}{\sigma_{l}-\sigma_{r}}q_{l\dot{I}}\Omega^{\dot{I}\dot{J}}\\!q_{r\\!\dot{J}}\,,\,l\\!\neq\\!r\in
h,$ $\displaystyle\;G_{h}^{ll}\,=\\!-\\!\\!\sum_{r\neq l\in
h}\\!\frac{u_{l}u_{r}}{\sigma_{l}-\sigma_{r}}q_{l\dot{I}}\Omega^{\dot{I}\dot{J}}\\!q_{r\\!\dot{J}}\,,$
(5.5) $\displaystyle
G_{\tilde{h}}^{lr}=\frac{u_{l}u_{r}}{\sigma_{l}-\sigma_{r}}\tilde{q}_{l}^{I}\Omega_{IJ}\tilde{q}_{r}^{J}\,,\;\;l\\!\neq\\!r\in\tilde{h},$
$\displaystyle\;G_{\tilde{h}}^{ll}\,=\\!-\\!\\!\sum_{r\neq
l\in\tilde{h}}\\!\frac{u_{l}u_{r}}{\sigma_{l}-\sigma_{r}}\;\tilde{q}_{l}^{I}\Omega_{IJ}\tilde{q}_{r}^{J}\,.$

$G_{h}$ and $G_{\tilde{h}}$ have co-rank 1 and the reduced determinant
det${}^{\prime}G_{m}$ is basically the product of all but one diagonal element
of $G_{m}$ with all terms cancelled that include some propagator loop of
elements in the $m$ set. This means, for instance, that terms in the
scattering amplitude of spin 1 particles with propagator loops containing only
helicities of the same sign are forbidden. This is known to be valid for gluon
scattering in QCD [18]. When $g\cup\tilde{g}$ only contains gravitons and all
particles in $h\cup\tilde{h}$ have spin $\leq 1$ then a single trace term will
be a propagator loop with two missing links typically between each of the two
fixed vertex operators and some other location which are filled by pulling
down the appropriate propagators from the exponential $e^{F_{\mathcal{N}}}$.
Such a single trace loop constitutes a kinematic Parke-Taylor (PT) factor:

$\mathrm{PT}=N/\left((\sigma_{|h|+|\tilde{h}|}-\sigma_{1})\\!\\!\\!\prod_{r<s\in
h}\\!\\!\\!(\sigma_{r}\\!-\\!\sigma_{s})\;\;(\sigma_{|h|}-\sigma_{|h|+1})\\!\\!\\!\prod_{|h|+r<|h|+s\in\tilde{h}}\\!\\!\\!(\sigma_{r+|h|}\\!-\\!\sigma_{s+|h|})\right)\,,$

where $N$ stands for a numerator consisting of a single trace product of
Grassmann odd parameters $q_{\mathcal{I}}$ and where the products run through
an ordered union of $h$ and $\tilde{h}$, respectively. For spin $\frac{1}{2}$
and scalar particles higher powers in $q$ components need to be considered.

Although this model looks promising, it also has some deficiencies. The
treatment of the $\eta^{\mathcal{I}}$ and $\tilde{\eta}_{\mathcal{I}}$
components is not fully symmetric and the generalization to the massive case
is unknown777In order to make progress in this direction, one might have to
switch the representation to use fermionic delta functions and scattering
equations for the fermionic fields [2], but then lose the convenience of a
supersymmetric exponential factor in the scattering amplitudes.. Like in the
previous section the order of the PT factor is not arbitrary: the same sign
helicities need to be adjacent to each other. This could be overcome by
exploiting the factorization properties of the amplitude. As shown in [18],
the amplitude (5.4), when $G^{\prime}$ is a PT factor, factorizes into smaller
tree-amplitudes. In particular, the PT factor factorizes by itself and,
therefore, it is possible to make the helicity order in the PT factor
arbitrary by gluing together smaller $n_{L}$ and $n_{R}$ point amplitudes to a
$n=n_{L}+n_{R}-2$ point tree amplitude accordingly. Admittedly, this procedure
to circumvent the obstruction looks somewhat contrived.

In conclusion, for pure graviton scattering the model in this section gives
the expected graviton tree scattering amplitude [13, 5] and the main
difference with a SYM tree scattering amplitude for spin $\\!\leq\\!1$
particles consists of the additional reduced determinants
det${}^{\prime}H\\!\\!$ det${}^{\prime}\tilde{H}$ indicating that all
particles exchange gravitons with each other, even without any external
gravitons present. For spin $\leq 1$ particles there might be even more
differences due to the general form of the fermionic Hodges-like matrices
(5.5).

The actual calculation of an n-point amplitude can proceed by solving the
scattering equations and inserting a solution into the various parts of the
integrand including the Jacobian. This gives a power series in the Grassmann
odd parameters $q_{\mathcal{I}}$ reflecting the $\mathcal{N}=8$ supersymmetry,
and the coefficients describe scattering of particles of different spin.

##  6 Summary and Outlook

This work examined the spectrum of the massive ambitwistor supergravity model
found in [3, 4, 5] based on oscillator expansions. It turned out that the
spectrum is covered by Table 2 reflecting $\mathcal{N}=8$ supersymmetry. As
the twistors are worldsheet spinors there is a R sector and a NS sector to
consider and both have the same particle spectrum except in the NS sector it
is truncated because of the constraint that physical states have at most two
on-shell oscillator modes.

The massive models considered in [3, 4, 5] have a vanishing little group
anomaly coefficient and also a zero Virasoro central charge after a
contribution from compactifying extra dimensions. In this article, the
supergravity model got first extended by doubling and gauging the fermionic
twistor components with help of auxiliary fields in such a way that the
resulting model was anomaly-free and the scattering amplitudes for spin $\leq
1$ particles exhibited a kinematic (no color trace) PT factor like a SYM model
but without an external current algebra. The spectrum of the first
modification which was limited to massless particles showed similarity to the
spectrum of the Berkovits-Witten model though without the disturbing ’dipole’
modes. Also, it is the same spectrum as of another model found by the author
earlier, described in appendix A. Unfortunately, for both models the
scattering amplitudes can look complicated and difficult to handle. Then, a
much improved, still massless model variation was presented with the same
spectrum as the original massive model and scattering amplitudes that differ
from the massless limit of the original model only by a factor consisting of
two reduced determinants of fermionic Hodges-like matrices with co-rank 1 that
under special circumstances can lead to a Parke-Taylor factor.

The issue about the truncated spectrum in the NS sector can be argued away.
This is based on the observation that a little group SL(2$,\mathbb{C}$) gauge
transformation can change the pair of supertwistors from the R sector to the
NS sector and vice versa, for instance by multiplying one supertwistor with
$\sigma^{\frac{1}{2}}$ and the other one with $\sigma^{-\frac{1}{2}}$. Thus,
from a classical point of view, the NS and R sector are equivalent in the two-
twistor models. Therefore, it is sufficient to quantize only the more
convenient R sector. On the quantized level, the NS sector can then be
regarded as resulting from a spectral flow operation on the R sector, with the
$\pm\frac{1}{2}$ modes in the NS sector becoming generalized zero modes888For
spectral flow of twistor modes in the context of ADS/CFT duality, see for
instance [19]., overcoming the truncation constraint of the NS spectrum. To
have equivalent spectrum before and after spectral flow and to be able to
identify the spectrum in the NS sector with the one in the R sector, this
interpretation works exactly when for the $L_{0}$ constant $a=0$ in the R
sector and $a=1$ in the NS sector, i.e. for the original massive model and the
model in section 5 but not for the intermediate model in section 4 or the
model in appendix A.

Concentrating on the spectrum of $\mathcal{N}=8$ supersymmetry as shown in
Table 2 and comparing modes in the triangles above and below diagonal lines
drawn from upper left to right bottom, then the fermionic (half-integer spin)
modes in the lower triangle represent antiparticles of the ones in the upper
triangle in both halves of the table. Further, the 6 representation of SU(4)
can be conveniently decomposed into a SU(2) $\\!\otimes\\!$ SU(2)
$\\!\otimes\\!$ U(1) basis [20] where out of a sextuple the first two elements
and the next two can be used each as a duplet for a fundamental representation
of SU(2) leaving the last two elements as a third duplet although not
representing an SU(2). Nevertheless, this allows to consider
$(\pm\frac{1}{2}|4\\!\otimes\\!6)$ and
$(\pm\frac{1}{2}|\bar{4}\\!\otimes\\!6)$ as 3 generations of the spin
$\frac{1}{2}$ content of the Pati-Salam model [10]. Also, it is intriguing to
see that the spectrum contains $(\pm 1|4\\!\otimes\\!\bar{4})=(\pm
1|15\\!\oplus\\!1)$ exactly once that is available for the adjoint vector
bosons of SU(4) and also two $(\pm 1|1\otimes 6)$ that have each a
decomposition that includes an adjoint representation of SU(2)[21] 999The fact
that the N=8 supergravity spectrum contains exactly the spin $\frac{1}{2}$
content for all the fermions of the standard model and also for 8 massive
gravitinos is well known [22, 23]..

For future work, when disregarding graviton exchange, i.e. setting the reduced
determinants det${}^{\prime}H\\!\\!$ det${}^{\prime}\tilde{H}$ equal to 1,
genus 0 scattering amplitudes of the model in section 5 should be checked for
how far they match gluon and qcd tree amplitudes from the literature. For
instance, what is immediately apparent, single trace NkMHV amplitudes look the
same as in SYM, apart from color traces.

Other open questions for the improved model in section 5 are a massive
generalization, loop scattering amplitudes for genus $\\!\geq\\!1$
worldsheets, modular invariance, and unitarity. The ultimate goal, of course,
would be to show that it leads to a consistent UV-complete string field theory
in twistor space with a classical limit that can be compared with general
relativity.

The above solution for the truncated spectrum in the NS sector deserves more
examination, working out more details about the spectral flow operation.

Further, the physical meaning of the auxiliary fields seems to be a mystery.
They got introduced in an ad hoc fashion, as a kind of supertwistors with
reversed statistics to support worldsheet supersymmetries. Without showing up
in physical states they are on the level of ghosts but if they are just a
matter of mathematical convenience what are the deeper lying principles behind
their origin? Maybe they are superfield components in a superspace although
the superfield would look rather complicated because of the many
supersymmetries and other worldsheet symmetries breaking the supertwistors
apart. On the other hand, if they are remnants of compactifying extra
dimensions the exact reduction mechanism would need to be investigated.

##  Appendix A Non-Supersymmetric Model

Here the intermediate model of section 4 gets changed to one that gauges
worldsheet supersymmetries without the use of (mysterious) auxiliary fields.
The model considered here is, in the notation of section 4,

$S=\int_{\Sigma}\mathcal{Z}^{a}\cdot\bar{\partial}\mathcal{Z}_{a}+A_{ab}\,Z^{a}\cdot
Z^{b}+S_{g}\,,$

with

$S_{g}=\int_{\Sigma}F^{\alpha\dot{\alpha}}_{ab}\lambda_{\alpha}^{a}\tilde{\lambda}_{\dot{\alpha}}^{b}+G_{ab}^{\alpha
I}\lambda_{\alpha}^{a}\tilde{\eta}_{I}^{b}+\tilde{G}_{ab}^{\dot{\alpha}I}\tilde{\lambda}_{\dot{\alpha}}^{a}\eta_{I}^{b}\,,$

where the fermionic twistor components $\eta^{a}_{I}$ do not participate in
the little group symmetry of the two twistors which breaks the supertwistors
apart. $F^{\alpha\dot{\alpha}}_{ab}$ gauges the conformal translations and
$G_{ab}^{\alpha I},\tilde{G}_{ab}^{\dot{\alpha}I}$ gauge the superconformal
translations.

BRST quantization leads to $(b,c)$ ghosts for worldsheet gravity and:
from the bosonic fields $\\{A_{ab},F^{\alpha\dot{\alpha}}_{ab}\\}$ to
fermionic ghosts
$\\{(M_{ab},N_{ab}),(f^{\alpha\dot{\alpha}}_{ab},e^{\alpha\dot{\alpha}}_{ab})\\}$
and
from the fermionic fields $\\{G_{ab}^{\alpha
I},\tilde{G}_{ab}^{\dot{\alpha}I}\\}$ to bosonic ghosts
$\\{(\beta_{ab}^{\alpha I},\gamma_{ab}^{\alpha
I}),(\tilde{\beta}_{ab}^{\dot{\alpha}I}\tilde{\gamma}_{ab}^{\dot{\alpha}I}\\}$.

The SL(2,$\mathbb{C}$) anomaly coefficient is:

$a_{sl2}=\frac{3}{2}(4_{\mathcal{Z}})-6(4_{fe})+\frac{3}{2}(2\mathcal{N}_{\beta\gamma}+2\mathcal{N}_{\tilde{\beta}\tilde{\gamma}})-6_{M\\!N}=-6(4-\mathcal{N})\,.$

The central charge is

$c=(-8+2\mathcal{N})_{\mathcal{Z}}-26_{bc}-6_{M\\!N}-32_{fe}+8\mathcal{N}_{\beta\gamma}+8\mathcal{N}_{\tilde{\beta}\tilde{\gamma}}=-18(4-\mathcal{N})\,.$

Therefore, the theory is anomaly-free for $\mathcal{N}=4$. Then the $L_{0}$
constant $a$ is in both the NS and R sector given by

$24a=(0)_{\mathcal{Z}}-2_{bc}-6_{M\\!N}-32_{fe}+8\mathcal{N}_{\beta\gamma}+8\mathcal{N}_{\tilde{\beta}\tilde{\gamma}}=24\,,$

i.e. $a=1$, exactly like for the intermediate model in section 4. Although
$\mathcal{N}$ has here half the value, it has the same R-symmetry because the
index $a$ of $\eta^{a}_{I}$ is not part of the little group but of the
R-symmetry. This means the two models have the same spectrum although the
internal little group representation does not need to be the same across the
R-symmetry multiplet when disregarding target space supersymmetry101010This
allows an interpretation of the spectrum to include 3 generations of the Pati-
Salam model [9], similar to the model in section 5.. The current model was
originally found by the author using other notation [8], and explored more in
[9] providing vertex operators and scattering amplitudes. Because it suffers
from similar issues as the model in section 4, it can be abandoned in favor of
the improved model in 5 that have more potential to be realistic.

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# Prompt-Time Ontology-Driven Symbolic Knowledge Capture with Large Language
Models

Tolga Çöplü, Arto Bendiken, Andrii Skomorokhov, Eduard Bateiko, Stephen Cobb
Haltia, Inc.
{tolga, arto, andriy, eduard<EMAIL_ADDRESS>

###### Abstract

In applications such as personal assistants, large language models (LLMs) must
consider the user’s personal information and preferences. However, LLMs lack
the inherent ability to learn from user interactions. This paper explores
capturing personal information from user prompts using ontology and knowledge-
graph approaches. We use a subset of the KNOW ontology, which models personal
information, to train the language model on these concepts. We then evaluate
the success of knowledge capture using a specially constructed dataset. Our
code and datasets are publicly available at https://github.com/HaltiaAI/paper-
PTODSKC

_K_ eywords ontology-driven symbolic knowledge capture $\cdot$ KNOW ontology
$\cdot$ symbolic representation $\cdot$ knowledge graphs $\cdot$ large
language models $\cdot$ fine-tuning

##  1 Introduction

Currently, many generative artificial intelligence (AI) applications,
particularly personal assistants, strive to offer users personalized
experiences. To achieve this, AI applications must learn personal information
and preferences from user interactions (knowledge capture) and use this
learned knowledge in future conversations (knowledge utilization).
Implementing this fundamental personal AI approach depends on addressing
several complex sub-problems, such as discerning which user prompt information
is personal, extracting it, determining whether the extracted information is
duplicate, and figuring out its association with other personal data.

These challenges have been the focus of extensive research within the AI field
for many years. However, the emergence of neuro-symbolic approaches through
the collaboration between large language models (LLMs) and symbolic AI has
provided researchers with new perspectives [1, 2, 3, 4]. LLMs’ capabilities in
natural language processing can be integrated with the representational and
factual reasoning abilities of knowledge graphs, enhanced by the structure,
rules, and inference mechanisms offered by an ontology. For targeted personal
AI applications, this ontology approach presents several benefits:

  * •

Ontology schemas enable language models to determine which personal
information will be captured and how it will be associated with other captured
knowledge.

  * •

Ontology rules can help identify inconsistencies in the captured knowledge,
allowing for validation before storage.

  * •

Ontology relationships allow the extraction of implicit information from
captured knowledge, effectively enabling automatic inference that expands the
knowledge graph.

  * •

A robust, personalized knowledge graph forms a reliable foundation for
facilitating personalized interactions with the application through language
models.

In this paper, we address a specific aspect of the AI personalization
challenge by focusing on prompt-time, ontology-driven symbolic knowledge
capture using language models. We explore the extraction from user prompts of
subject-predicate-object triples111https://www.w3.org/TR/rdf12-concepts/ that
conform to a specified ontology. We have investigated various methods to
enable the underlying language model to comprehend a pre-defined ontology,
ensuring effective symbolic knowledge capture. By utilizing a specially
designed dataset, we evaluate the effectiveness of these methods, emphasizing
their strengths and identifying potential areas for improvement.

The structure of this paper is as follows: Section 2 discusses in-context
learning and fine-tuning approaches for ontology-driven symbolic knowledge
capture and focuses on the details of the fine-tuning approach. Section 3
describes the experimental setup by presenting the development framework, the
language model selection, and the ontology and dataset creation process.
Section 4 outlines our performance evaluation framework and the test results.
Finally, Section 5 concludes the paper and suggests future directions.

##  2 Ontology-Driven Symbolic Knowledge Capture

In the literature, language models have demonstrated their capability to
transform unstructured text into a knowledge graph [5, 6, 7, 8, 9]. However,
the process of populating a knowledge graph from user prompts in alignment
with a pre-defined ontology has been explored only marginally [10, 11, 12, 13,
14]. Research typically centers on in-context learning, which heavily relies
on prompt engineering. A significant limitation of this approach is the
requirement to incorporate the entire custom ontology into the prompt. This
necessity not only slows down the knowledge capture process, because of the
high token overhead but also restricts the use of larger ontologies due to the
constraint on context-window length. Given these constraints, in-context
learning methods do not provide a scalable solution for ontology-driven
symbolic knowledge capture.

An alternative approach involves training a language model with a pre-defined
ontology, so that the model internalizes it. There are two strategies to
consider: pre-training the LLM on the ontology or fine-tuning it. This paper
does not explore pre-training due to its extensive data, computational
resources, energy, and time requirements. Additionally, pre-training does not
offer a flexible response to ongoing changes or expansions in the ontology.
Therefore, this paper will focus on fine-tuning as a method to train language
models on personal ontologies, highlighting advantages in feasibility and
maintainability

###  2.1 Ontology-Driven Knowledge Capture with Fine-Tuning

Fine-tuning is a process whereby a pre-trained language model is further
trained on a specific dataset to tailor its capabilities to a particular task.
In our study, the language model is expected to learn the classes, object
properties, and data properties defined in an ontology, and to use them to
populate a knowledge graph from user prompts. The first step involves
preparing a fine-tuning dataset, which includes user prompts, system prompts,
and expected model responses for each concept in the ontology. This dataset is
used to fine-tune the language model, which is then evaluated by testing it
with new prompts to assess the effectiveness of the knowledge capture
operation. We define a system prompt for this task with the requirement of
maintaining the model’s generality across other tasks.

The following points highlight the key aspects of ontology fine-tuning:

  * •

The training dataset’s coverage and diversity are vital for successful fine-
tuning. These characteristics greatly influence the LLM’s ability to
effectively capture knowledge. Details about the dataset and how it is
constructed are discussed in Section 3.4.

  * •

The training dataset must include a variety of examples for each element of
the predefined ontology. This approach avoids scalability issues typically
associated with in-context learning and ensures comprehensive learning
coverage.

  * •

If the LLM encounters a user prompt that is not relevant to the predefined
ontology concepts, it should not attempt to capture knowledge. Therefore, the
dataset should also contain sufficient out-of-context samples to enable the
LLM to distinguish between relevant and irrelevant information for capture.

##  3 Experimental Setup

This section explores the components of our experimental setup.

###  3.1 Development Framework

The methods suggested in this paper have been implemented using the Apple MLX
framework ([15]). MLX is a specialized array framework designed for machine
learning applications, akin to NumPy, PyTorch, or JAX, with the distinction of
being exclusive to Apple silicon.

Ontology fine-tuning has been conducted using the parameter-efficient QLoRA
approach ([16]) on our custom dataset, comprising randomly selected, non-
overlapping sets of training, validation, and test samples.

###  3.2 Language Model

The methods we have developed here do not have a structural dependency on a
particular underlying foundation model. The key factors guiding our language
model selection were its proven effectiveness across diverse domains in
community benchmarks and its prevalence in the field. Owing to its performance
in the Hugging Face Open LLM Leaderboard ([17]) and its robust ecosystem, the
Mistral-7B-Instruct-v0.2 ([18]), based on the Llama 2 ([19]) architecture, was
selected for our research. We ran all examples, tests, and benchmarks on the
MLX 4-bit quantized version of this model to be able to run the tests on
personal laptops.

###  3.3 Predefined Ontology

Our study utilized the Knowledge Navigator Ontology for the World
(KNOW)222https://know.dev for representing personal information. KNOW is the
first ontology crafted specifically to capture everyday knowledge, thereby
enhancing language models in practical real-world scenarios. The domain of
KNOW encompasses human life, addressing both daily concerns and significant
life milestones. The initial scope of KNOW’s modeled concepts is currently
limited to established human universals that are not subject to revision:
spacetime (places, events) and social elements (people, groups,
organizations).

Due to the requirement that each element in the ontology be associated with a
diverse set of prompt and response samples within the training dataset, our
research focuses on a specific subset of the KNOW ontology. This subset
concentrates on core family relationships with four ontology classes, eleven
object properties, and one data property. A visual depiction of this subset is
presented in Figure.1.

Figure 1: A visual representation of the ontology design used in this paper.

###  3.4 Dataset

For a language model to effectively learn a predefined ontology and use it to
perform knowledge extraction and capture, a robust and diverse training
dataset is essential. Our paper focuses on a subset of the KNOW ontology that
includes the concepts of ’person’, ’name’, ’sex’, ’child’, ’father’, ’mother’,
’sibling’, ’sister’, ’brother’, ’spouse’, ’partner’ and ’knows’. We created
143 manually crafted user prompts along with their respective ontology
responses for training and tests. Additionally, to manage inputs that fall
outside these ontology concepts, we included 32 generic user prompts in the
dataset. The composition of this dataset, which consists of 175 user prompts,
is illustrated in Figure.2. Concepts not associated with the ontology are
labeled as the ’none’ legend in the figure. As each sample prompt typically
contains multiple modeled concepts, the chart shows a total number of concept
occurrences greater than the number of prompts.

Figure 2: Occurrences of ontology concepts in the prepared dataset.

The Turtle format was chosen for serializing the ontology population in our
research because of its straightforward structure, readability, and prevalent
use in existing pre-training datasets for LLMs.

##  4 Performance Evaluation

Our research focuses on fine-tuning a language model with predefined ontology
concepts and capturing knowledge from user prompts that fits the ontology.
This section will detail the performance evaluations associated with these
efforts.

Figure 3: Occurrences of ontology concepts in the test dataset.

Initially, we investigated how many training samples each ontology concept
required to effectively teach the core family ontology to the selected
Mistral-7B-Instruct-v0.2 model. From the dataset described in Section3.4, 41
random samples were selected and reserved for method evaluation. The
distribution of ontology concepts within this test set is shown in Figure.3.

Figure 4: Ontology population performance (precision, recall, and f1-score)
for various fine-tuning dataset sizes.

From the remaining 134 samples, we created three different training datasets.
In these three datasets, we have ensured the inclusion of 2, 4, and 8 sample
prompts for the ontology concepts ’child’, ’father’, ’mother’, ’sibling’,
’sister’, ’brother’, ’spouse’, ’partner’, and ’knows’. Subsequent evaluations
using the test set measured the precision, recall, and f1-scores for each
fine-tuning session. During these evaluations, the generated prompt responses
were processed triple by triple and compared against the ground truth
established for the test set. The findings are displayed in Figure.4.

The tests were conducted using the default QLoRA hyperparameters specified in
the MLX framework. To ensure consistent test results, each training session
was set to run for 18 epochs.

  * •

Layer keys to apply: ["self_attn.q_proj", "self_attn.v_proj"]

  * •

Rank: 8

  * •

Alpha: 16

  * •

Scale: 10

  * •

Optimizer: Adam

  * •

Learning rate: $1x10^{-5}$

  * •

Number of layers to fine-tune: 16

  * •

Minibatch size: 4

Figure 5: Ontology population performance (precision, recall, and f1-score) of
8 samples per ontology concept for various epochs.

As illustrated in Figure.4, our study, which encompasses twelve key ontology
concepts, demonstrates that providing eight diverse examples for each concept
yields acceptable success rates. Although our training and test datasets are
not sufficiently diverse or large enough to generalize the results to real
user scenarios, the high success achieved with a small number of training
samples is promising for the feasibility of the proposed approach.

In the subsequent phase, we explored the optimal number of training epochs
required to achieve maximum performance for the training set. For this
analysis, we continued using the default MLX QLoRA hyperparameters with the 8
samples per ontology concept, but trained the QLoRA adapter over various epoch
lengths. We then conducted evaluations on the test set using each trained
adapter, and the findings are presented in Figure.5.

As depicted in Figure.5, the success rate of the ontology population increases
with longer training. However, considering the resource usage and energy
consumption, we observe that 18 epochs are sufficient for fine-tuning.

##  5 Conclusion

In this paper, we first explored prompt-driven, ontology-based symbolic
knowledge capture and its importance in the generative AI domain. We then
discussed the ontology approach and how to teach ontology concepts to the
language model through in-context learning and training. The language model
was fine-tuned using a custom dataset focused on core family relationships,
and we evaluated the model’s ability to learn personal ontology concepts.

Our findings indicate that fine-tuning is particularly effective for teaching
ontology concepts to a language model for prompt-time knowledge capture. In
our future work, we aim to integrate the generated knowledge graph with the
language model for knowledge utilization, combining the strengths of the
neural and symbolic AI approaches.

Please refer to the paper’s corresponding GitHub repository at
https://github.com/HaltiaAI/paper-PTODSKC

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