Chiral Higher Spin Gravity in (A)dS4 and
secrets of Chern–Simons Matter Theories

Alexey Sharapova and Evgeny Skvortsov***Research Associate of the Fund for
Scientific Research – FNRS, Belgiumb,c

aPhysics Faculty, Tomsk State University,
Lenin ave. 36, Tomsk 634050, Russia
b Service de Physique de l’Univers, Champs et Gravitation,
Université de Mons, 20 place du Parc, 7000 Mons, Belgium
c Lebedev Institute of Physics,
Leninsky ave. 53, 119991 Moscow, Russia

###### Abstract

Chiral Higher Spin Gravity with cosmological constant is constructed as a Free
Differential Algebra, i.e. at the level of equations of motion, which is a
smooth deformation of its flat space cousin arXiv:2205.07794. Chiral Higher
Spin Gravity is a unique class of local higher spin theories; its very
existence implies that there is a closed and, most likely, integrable sub-
sector of Chern–Simons Matter Theories, which has important consequences both
for the theories themselves and for three-dimensional bosonization duality.

##  1 Introduction

Higher Spin Gravities (HiSGRA) are theories [1] that synthesize several
fruitful ideas in quest of solving the quantum gravity problem: (a) higher
spin states are very likely to be present in any viable model of quantum
gravity as suggested by (super)string theory and by AdS/CFT correspondence [2,
3, 4]; (b) extensions of gravity with higher symmetries, e.g. supergravities,
should have better quantum behaviour; (c) in the high energy regime, where the
perturbative quantum gravity problems are coming from, it should be possible
to neglect masses. All together (a,b,c) suggest it can be instructive to look
for extensions of gravity with massless higher spin fields, whose dynamics are
controlled by infinite-dimensional gauge symmetries. It comes as no surprise
that HiSGRA are not easy to construct and face numerous problems already at
the classical level since the masslessness makes them sensitive to genuine
quantum UV problems.

There is a handful of classes of HiSGRA’s that have been constructed so far,
all of which are quite peculiar. In $3d$ there is one class with a plenty of
topological higher spin theories with massless, partially-massless and
conformal higher spin fields [5, 6, 7, 8, 9, 10, 11]. In $4d$ (and all even
dimensions) it is possible to construct conformal HiSGRA [12, 13, 14] that is
a higher spin extension of (conformal) Weyl gravity. As far as HiSGRA with
propagating massless fields are concerned, there is a unique class of such
theories in $4d$ – Chiral HiSGRA [15, 16, 17, 18, 19] and its contractions
[20, 21]. Chiral HiSGRA is closely related to self-dual theories [20]. Its
contractions can be understood as higher spin extensions of SDYM and SDGR
[21].

Originally, Chiral Theory was constructed in the light-cone gauge and in flat
space [15, 16, 17]. Thanks to a deep interrelation between the light-cone
gauge and spinor-helicity formalism [22, 23, 24, 25, 26] Chiral Theory can
easily be defined without having to make explicit reference to the light-cone
gauge. As is well-known, for any triplet of helicities $\lambda_{1,2,3}$ such
that $\lambda_{1}+\lambda_{2}+\lambda_{3}>0$ there is a unique light-cone
vertex and the corresponding three-point amplitude [27, 28]:

$\displaystyle V_{\lambda_{1},\lambda_{2},\lambda_{3}}\Big{|}_{\text{on-
shell}}\sim[12]^{\lambda_{1}+\lambda_{2}-\lambda_{3}}[23]^{\lambda_{2}+\lambda_{3}-\lambda_{1}}[13]^{\lambda_{1}+\lambda_{3}-\lambda_{2}}\,.$
(1.1)

Chiral Theory is a unique local Lorentz invariant theory that contains at
least one higher spin field with a nontrivial self-interaction. This simple
input forces one to introduce massless fields of all spins (at least even),
thereby, adding graviton and a scalar field. The coupling constants are
uniquely fixed to be

$\displaystyle V_{\text{Chiral}}$
$\displaystyle=\sum_{\lambda_{1},\lambda_{2},\lambda_{3}}C_{\lambda_{1},\lambda_{2},\lambda_{3}}V_{\lambda_{1},\lambda_{2},\lambda_{3}}\,,$
$\displaystyle
C_{\lambda_{1},\lambda_{2},\lambda_{3}}=\frac{\kappa\,(l_{p})^{\lambda_{1}+\lambda_{2}+\lambda_{3}-1}}{\Gamma(\lambda_{1}+\lambda_{2}+\lambda_{3})}\,,$
(1.2)

where $l_{p}$ is a constant of dimension length and $\kappa$ is an arbitrary
dimensionless constant.

Chiral Theory was shown to be one-loop finite in flat space [18, 19, 29]
directly in the light-cone gauge. Its extension to $AdS_{4}$ was envisaged in
[30, 31], again in the light-cone gauge. Nevertheless, it had remained unclear
if Chiral Theory admits a manifestly Lorentz-invariant formulation before it
was shown in [21] that contractions of Chiral Theory, which can be understood
as higher spin extensions of SDYM and SDGR, do have a manifestly Lorentz-
invariant formulation both in flat and $(A)dS_{4}$ spaces. In addition,
contrary to the old folklore, it was shown that the interactions have a smooth
deformation to (anti)-de Sitter space.

A manifestly Lorentz-invariant formulation of Chiral Theory in flat space was
found in [32, 33] at the level of equations of motion. The equations were
constructed in the form of a Free Differential Algebra [34, 35, 36], thereby,
fulfilling the old ideas [37]. An important new feature of the FDA language
[38, 39, 40, 41] is that it can be understood as the minimal model of the BV-
BRST formulation of a given theory and, hence, contains the same information
as local BRST cohomology [42, 43, 44, 45, 46], e.g. one can study counterterms
and anomalies. This is also a very useful first step towards a Lorentz-
invariant action.

HiSGRA in (anti)-de Sitter background are, on the first sight, very well
supported by AdS/CFT correspondence, where HiSGRA should be duals of
free/critical (Chern–Simons) vector models [47, 48, 49, 50]. On the second
thought, however, the duality itself implies that HiSGRA must be too nonlocal
for standard field theory methods to be applicable and for the theories
themselves to exist in any sensible way as field theories [51, 52, 53, 54]. At
present, there is no construction of AdS/CFT duals of (Chern–Simons) vector
models that makes concrete, systematic, and meaningful predictions for bulk
interactions and allows one to compute holographic correlation
functions.222The first steps [55, 56], while generally positive, revealed some
puzzles already at the three-point level, which were shown [57] to result from
the incompleteness of [58]. The latter is in full agreement with the basic
non-existence of such theories [51, 52, 53, 54]. It is worth mentioning an
interesting approach developed in [59, 60], which, to some extent, allows one
to rewrite the CFT partition function as a path integral with a (very
nonlocal) quasi-action for some fields in the bulk. It would be interesting to
extend this construction to Chern–Simons vector models. In the same vein [51]
one can try to reconstruct the dual theory by “inverting” the correlation
functions. However, such vertices are too nonlocal for the standard field
theory tools to work [51, 53, 54].  It seems that AdS/CFT uncovers the same
problems as with HiSGRA in flat space, see e.g. [61, 62, 63, 20], but they can
be identified much quicker thanks to AdS/CFT duality. In this vein HiSGRA are
closer to string theory than to field theories [52].

In the view of the notorious nonlocality problem described above, Chiral
Theory seems to be the only local class of HiSGRA with propagating massless
fields in four dimensions.333Any model of quantum gravity should be nonlocal
in some sense. Therefore, the nonlocality of HiSGRA is not unexpected. What is
lacking at the moment are concrete rules to deal with it. It is clear that the
standard field theory tools are inapplicable. Therefore, it is encouraging
that there exist some local HiSGRA that can be treated as useful
approximations or starting points. Locality comes at a price, of course.  In
this paper we construct a manifestly covariant form of equations of motion for
Chiral Theory with cosmological constant. The equations are found in the form
of a Free Differential Algebra by a smooth deformation of those proposed
recently in [32, 33]. This FDA can also be interpreted as the dual of an
$L_{\infty}$-algebra. Chirality makes the theory local both in flat and
(anti)-de Sitter spaces, which follows immediately from our construction.

Since the bulk of the paper is mostly devoted to the technical aspects of the
FDA, it makes sense to discuss conceptual consequences of the results for
Chern–Simons Matter Theories and three-dimensional bosonization duality [50,
64, 65, 66, 67, 68] right now. Let us recall that the

spectrum of Chiral Theory, massless fields with spins $s=0,1,2,3,...$ or
$s=0,2,4,...$,444Gauging Yang–Mills groups is also possible [19]. They,
however, come in a very restricted Chan-Paton pattern, which is implied by the
AdS/CFT duality with vector models.  is the right one to be the dual of
Chern–Simons vector models. At the free level, (anti)-Chiral theories and the
dual of Chern–Simons vector models coincide. Chiral Theory is smaller than
this hypothetical dual HiSGRA in that only the chiral interactions are kept
(those where the sum of the helicities on external lines is, say,
positive).555Chiral Theory shares many features with self-dual theories. Some
key properties of any self-dual theory include: all solutions of self-dual
theories are solutions of the full ones; the same is true for amplitudes.
Therefore, self-dual theories are very useful for understanding the complete
ones. What is missing at present for the Chiral Theory case is an explicit
construction of the complete one.

The first consequence of the very existence of a complete, local, Lorentz-
covariant theory in $AdS_{4}$ is that Chern–Simons vector models must have a
closed subsector as well. Indeed, one can recover this subsector by computing
the holographic correlation functions in Chiral Theory. It would be very
interesting to identify this closed subsector directly on the CFT side, which
is perhaps hard to achieve at the action level.

Secondly, the chiral and anti-chiral interactions cover all possible cubic
vertices. Now, it is very important that the (anti-)Chiral Theory is rigid and
can be thought of as a rock hard building block (all couplings are fixed by
higher spin symmetry). Therefore, if one wishes to get all possible three-
point functions in the complete unitary HiSGRA dual of Chern–Simons matter
theories one just needs to study how to glue chiral and anti-chiral parts
together. It turns out that this gluing depends on one additional parameter
[31], which can be introduced via $U(1)$ electromagnetic duality
transformation in the bulk. This immediately proves the three-dimensional
bosonization duality at this order since the bulk considerations are
insensitive to whether the dual operators are built out of bosonic or
fermionic matter. This also gives explicitly the three-point functions [31],
which are consistent with [64] and explain in a simple way the general
structure observed in [64].

Thirdly, despite the lack of control over nonlocalities in the full HiSGRA
dual to Chern–Simons vector models, existence of its Chiral truncations
supports the bosonization duality beyond three-point level. Indeed, the main
idea of [31] is to decompose correlation functions (equivalently, bulk
interactions) into parts with definite helicity structure. After the surgery,
these building blocks can be rotated with a $U(1)$-phase and sewed back into
unitary, but parity violating, correlation functions. The very existence of
one-parameter families of conformal field theories that interpolate between
parity-preserving bosonic/fermionic vector models can be attributed to the
existence of Chiral Theory. It allows one to interpret correlation functions
of gauge invariant operators in Chern–Simons vector models as perturbations of
the chiral subsector, which effectively introduces one more parameter on top
of $1/N$. This is very similar to how Yang-Mills theory can be recast as
perturbation of SDYM, likewise for gravity vs. SDGR, the main difference being
in that Chiral Theory has its own coupling constant. Eventually these ideas
should merge with the pure CFT considerations based on the concept of the
slightly-broken higher spin symmetry [64], where it is possible to prove that
there exists a unique class of invariants of this symmetry to serve as
correlation functions [69, 70, 71].

Lastly, there are good reasons to expect Chiral Theory to be integrable [20],
finite and even one-loop exact [29], and to admit a simple twistor
formulation, see [72] for the first steps in this direction. Therefore, it
should be possible to obtain all-loop results in Chiral Theory and, as a
consequence, in the corresponding subsectors of Chern–Simons vector models.
Note that the bulk coupling is of order $1/N$. Since the results are exact,
they should be valid for small values of $N$, which are of phenomenological
significance. Hopefully, Chiral Theory can give an example of a bulk theory
that is completely solvable.

While it would be very interesting to explore the ideas outlined above in the
future, we proceed now to bitter technicalities of constructing the FDA of
Chiral Theory, which is necessary to support them. After a brief overview of
the formalism in Section 2 we proceed to the main results in Section 3, which
are supported by a number of technical Appendices. The main conclusions of the
paper have already been given above and a more technical summary and
discussion of the results can be found in Section 4.

##  2 Initial data

As it was discussed, in the light-cone gauge there is a complete action for
Chiral Theory that terminates at cubic interactions in flat space. Different
parts (contractions in the language of [20]) of the covariant action for
Chiral Theory, both in flat and $(A)dS_{4}$, were constructed in [21]. The
covariant version of Chiral Theory can be defined as a unique Lorentz
invariant and local completion of the free action666As always, we use the same
letter to denote a group of symmetric or to be symmetrized indices, e.g.
$A_{1}...A_{k}\equiv A(k)$. The indices are raised and lowered with the help
of $\epsilon^{AB}=-\epsilon^{BA}$ as in the classical book [73].

$\displaystyle S=\int\Psi^{A(2s)}\wedge H_{AA}\wedge\nabla\omega_{A(2s-2)}\,,$
(2.1)

where $H^{AB}\equiv e{}^{A}{}_{C^{\prime}}\,\wedge e^{BC^{\prime}}$ is the
basis of self-dual two-forms built out of vierbein $e^{AA^{\prime}}\equiv
e^{AA^{\prime}}_{\mu}\,dx^{\mu}$. The completion is unique provided a genuine
higher spin interaction is turned on. For simple Yang–Mills and gravitational
interactions of higher spin fields the completion is not unique, but admits
very simple actions [21]. The dynamical fields are zero-form $\Psi^{A(2s)}$
and one-form $\omega^{A(2s-2)}\equiv\omega^{A(2s-2)}_{\mu}\,dx^{\mu}$. The
action can be shown to describe a massless spin-$s$ particle, where the
(conventionally) positive helicity, $+s$, mode belongs to $\omega^{A(2s-2)}$
and the negative heliticy, $-s$, resides in $\Psi^{A(2s)}$. On flat, $(A)dS$
and, more generally, any self-dual background the action enjoys the gauge
symmetry

$\displaystyle\delta\omega^{A(2s-2)}$
$\displaystyle=\nabla\xi^{A(2s-2)}+e{}^{A}{}_{C^{\prime}}\,\eta^{A(2s-3),C^{\prime}}\,,$
$\displaystyle\delta\Psi^{A(2s)}$ $\displaystyle=0\,,$ (2.2)

where $\xi^{A(2s-2)}$ and $\eta^{A(2s-3),C^{\prime}}$ are zero-forms.

We refer the reader to [32, 33] for a detailed discussion of the formalism as
well as to the original papers [74, 37] which correctly identify the free data
and bits of interactions. We will be looking for covariant equations of motion
in the form of a Free Differential Algebra. This implies the well-known
extension [74, 37] of the dynamical fields with some auxiliary ones. It is
convenient to package all the auxiliary and dynamical fields into a pair of
master fields: the one-form

$\displaystyle\omega(y,{{\overline{y}}})$
$\displaystyle=\sum_{n+m=\text{even}}\tfrac{1}{n!m!}\omega_{A(n),A^{\prime}(m)}\,y^{A}...y^{A}\,{{\overline{y}}}^{A^{\prime}}...{{\overline{y}}}^{A^{\prime}}$

and the zero-form

$\displaystyle C(y,{{\overline{y}}})$
$\displaystyle=\sum_{n+m=\text{even}}\tfrac{1}{n!m!}C_{A(n),A^{\prime}(m)}\,y^{A}...y^{A}\,{{\overline{y}}}^{A^{\prime}}...{{\overline{y}}}^{A^{\prime}}\,.$

The restriction that the sum $n+m$ be even is the bosonic projection. The
(generating functions of) dynamical fields of (2.1) are identified with
$\Psi(y)=C(y,{{\overline{y}}}=0)$ and
$\omega(y)=\omega(y,{{\overline{y}}}=0)$. The zero-form $C$ also contains the
scalar field $C(0,0)$, which is necessarily present in Chiral Theory. With
this field content the most general FDA reads

$\displaystyle d\omega$
$\displaystyle=\mathcal{V}(\omega,\omega)+\mathcal{V}(\omega,\omega,C)+\mathcal{V}(\omega,\omega,C,C)+...\,,$
(2.3a) $\displaystyle dC$
$\displaystyle=\mathcal{U}(\omega,C)+\mathcal{U}(\omega,C,C)+...\,.$ (2.3b)

The formal consistency of the equations, resulting from the nilpotency of the
exterior differential $d$, implies that the multilinear maps on the r.h.s.
satisfy the $L_{\infty}$-relations. These equations must have $(A)dS_{4}$ as
an exact solution with $C=0$ and

$\displaystyle\omega_{0}$
$\displaystyle=\tfrac{1}{4}\omega^{AB}\,y_{A}y_{B}+\tfrac{1}{2}e^{AA^{\prime}}\,y_{A}{{\overline{y}}}_{A^{\prime}}+\tfrac{1}{4}\omega^{A^{\prime}B^{\prime}}\,{{\overline{y}}}_{A^{\prime}}{{\overline{y}}}_{B^{\prime}}\,,$
(2.4)

where $\omega$’s are (anti-)self-dual parts of the spin-connection. Expanding
(2.3) over $\omega_{0}$ and picking up the linear terms, one gets the free
equations

$\displaystyle d\omega$
$\displaystyle=\mathcal{V}(\omega_{0},\omega)+\mathcal{V}(\omega,\omega_{0})+\mathcal{V}(\omega_{0},\omega_{0},C)\,,$
$\displaystyle dC$ $\displaystyle=\mathcal{U}(\omega_{0},C)\,.$ (2.5a)

The parts of $\mathcal{V}(\omega_{0},\omega)$,
$\mathcal{V}(\omega,\omega_{0})$ and $\mathcal{U}(\omega_{0},C)$ that contain
the spin-connection must act canonically, i.e. they should transform the
components of $\omega$ and $C$ as spin-tensors. Therefore, they can be moved
to the l.h.s. and combined with $d$ into the Lorentz covariant derivative777It
should be noted that in the flat limit $\omega^{AB}$ does not play the role of
the spin-connection anymore, but $\omega^{A^{\prime}B^{\prime}}$ does.
$\nabla$. The free equations must read

$\displaystyle\nabla\omega$
$\displaystyle=e^{BB^{\prime}}({\lambda}\,{{\overline{y}}}_{B^{\prime}}\partial_{B}+y_{B}\overline{\partial}_{B^{\prime}})\omega+H^{B^{\prime}B^{\prime}}\overline{\partial}_{B^{\prime}}\overline{\partial}_{B^{\prime}}C(y=0,{{\overline{y}}})\,,$
(2.6) $\displaystyle\nabla C$
$\displaystyle=e^{BB^{\prime}}({\lambda}\,y_{B}{{\overline{y}}}_{B^{\prime}}-\partial_{B}\overline{\partial}_{B^{\prime}})C\,.$
(2.7)

Here we introduced the parameter ${\lambda}$ associated with the cosmological
constant. At ${\lambda}=0$ we recover the free equations of Chiral Theory in
flat space [32, 33].

##  3 FDA

We first summarize the known results about the leading order, then discuss the
next-to-leading order vertices, which are still easy to find by hand, and
finally explain how to systematically generate all higher vertices.

###  3.1 LO

#### Higher spin algebra.

The $L_{\infty}$-relations, or the formal consistency of (2.3), imply that
$\mathcal{V}(f,g)$ satisfies the Jacobi identity and, thereby, $\omega$ takes
values in some Lie algebra. As it was understood in different contexts by many
authors [75, 76, 77, 78, 79, 80] this Lie algebra originates, in fact, from an
associative algebra $A_{2}^{e}$, which is the even subalgebra of the second
Weyl algebra. In other words, we can start with the four generators
$\hat{Y}^{\mathcal{A}}=(\hat{y}^{A},{\hat{{\overline{y}}}}{}^{A^{\prime}})$
subject to the commutation relations
$[\hat{Y}^{\mathcal{A}},\hat{Y}^{\mathcal{B}}]=2C^{{\mathcal{A}}{\mathcal{B}}}$,
with $C^{{\mathcal{A}}{\mathcal{B}}}$ being the matrix of the canonical
symplectic form. The second Weyl algebra $A_{2}$ is then the associative
algebra of polynomials $f(\hat{Y})$ in $\hat{Y}$’s. A useful realization of
$A_{2}$ is via deformation quantization: one considers polynomials $f(Y)$ in
the commutative variables $Y^{\mathcal{A}}$ that are equipped with the star-
product

$\displaystyle(f\star g)(Y)$
$\displaystyle=\exp[Y\cdot\partial^{1}+Y\cdot\partial^{2}+C^{{\mathcal{A}}{\mathcal{B}}}\partial^{1}_{\mathcal{A}}\partial^{2}_{\mathcal{B}}]\,f(Y_{1})g(Y_{2})\Big{|}_{Y_{1,2}=0}\,,$
(3.1)

where $Q\cdot P\equiv Q^{\mathcal{A}}P_{{\mathcal{A}}}$. The higher spin
algebra is then the even subalgebra $A_{2}^{e}$ of $A_{2}$. It is also
convenient [37] to tensor it with the matrix algebra $\mathrm{Mat}_{N}$ and
define ${\mathfrak{hs}}=A_{2}^{e}\otimes\mathrm{Mat}_{N}$. The trick with the
matrix extension is well-justified by AdS/CFT correspondence and by the fact
that one can introduce Yang–Mills-type interactions into Chiral Theory [19].

At the technical level, the matrix extension implies that the
$L_{\infty}$-algebra we are looking for results from an $A_{\infty}$-algebra
via the symmetrization map. For example, the Lie bracket is the commutator in
associative algebra ${\mathfrak{hs}}$. As we will see, for technical reason,
it is better to define ${\mathfrak{hs}}=A_{1}\otimes\mathcal{B}$, where
$A_{1}$ is the (star-product) Weyl algebra in $y^{A}$ and $\mathcal{B}$ is
another associative algebra, which is $A_{1}\otimes\mathrm{Mat}_{N}$ in our
case (supplemented at the end with the projection onto the even subalgebra of
${\mathfrak{hs}}$). Therefore, we will treat $\omega$ and $C$ as function of
$y$’s that take values in some associative algebra and the order of the
arguments in vertices is important. With this convention in mind the product
in the higher spin algebra can be written as

$\displaystyle\mathcal{V}(f,g)$
$\displaystyle=\exp{[p_{01}+p_{02}+{\lambda}\,p_{12}]}f({y}_{1})\,g({y}_{2})\Big{|}_{{y}_{i}=0}\equiv(f\star
g)(y)\,,$ (3.2)

where we introduced $y^{A}\equiv p_{0}^{A}$, $\partial^{y_{i}}_{A}\equiv
p_{A}^{i}$. Contractions $p_{ij}\equiv p_{i}\cdot
p_{j}\equiv-\epsilon_{AB}p^{A}_{i}p_{j}^{B}=p^{A}_{i}p_{jA}$ are such that
$\exp[p_{0}\cdot p_{i}]f(y_{i})=f(y_{i}+y)$ is the translation operator. More
generally, vertices can be represented as poly-differential operators

$\displaystyle\mathcal{V}(f_{1},...,f_{n})$
$\displaystyle=\mathcal{V}(y,\partial_{1},...,\partial_{n})\,f_{1}(y_{1})...f_{n}(y_{n})\Big{|}_{y_{i}=0}\,,$
(3.3)

where the $f$’s stand either for $\omega$’s or $C$’s. Occasionally, we will
use $q$’s for poly-differential operators in ${{\overline{y}}}$, e.g.
${{\overline{y}}}^{A^{\prime}}\equiv q_{0}^{A^{\prime}}$,
$\partial^{{{\overline{y}}}_{i}}_{A^{\prime}}\equiv q_{A^{\prime}}^{i}$. Most
of the time, we omit $|_{y_{i}=0}$ as well as the arguments of the vertices
and write down only the corresponding symbols.

Coming back to (3.2), we introduced ${\lambda}$ in order to have a smooth
deformation that starts from Chiral Theory with vanishing cosmological
constant. Basically, ${\lambda}$ is the cosmological constant. In the flat
limit (${\lambda}\rightarrow 0$) the first Weyl algebra $A_{1}$ turns back
into the commutative algebra of polynomial functions $\mathbb{C}[y]$, whose
quantization $A_{1}$ is. Now, we can check that the boundary conditions
imposed by the free equations (2.6) are indeed satisfied:

$\displaystyle\mathcal{V}(e,f)-\mathcal{V}(f,e)$
$\displaystyle=2e^{p_{02}+q_{02}}\left({\lambda}\,q_{01}p_{12}+p_{01}q_{12}\right)(e^{CC^{\prime}}y^{1}_{C}{{\overline{y}}}^{1}_{C^{\prime}})f(y^{2},{{\overline{y}}}^{2})\Big{|}_{y_{i},{{\overline{y}}}_{i}=0}\,,$
(3.4a) $\displaystyle\mathcal{V}(\omega,f)-\mathcal{V}(f,\omega)$
$\displaystyle=2{\lambda}\,p_{01}p_{12}\,e^{p_{02}+q_{02}}(\omega^{AB}y^{1}_{A}y^{1}_{B})f(y^{2},{{\overline{y}}}^{2})\Big{|}_{y_{i},{{\overline{y}}}_{i}=0}\,,$
(3.4b)
$\displaystyle\mathcal{V}(\overline{\omega},f)-\mathcal{V}(f,\overline{\omega})$
$\displaystyle=2q_{01}q_{12}\,e^{p_{02}+q_{02}}(\omega^{A^{\prime}B^{\prime}}{{\overline{y}}}^{1}_{A^{\prime}}{{\overline{y}}}^{1}_{B^{\prime}})f(y^{2},{{\overline{y}}}^{2})\Big{|}_{y_{i},{{\overline{y}}}_{i}=0}\,,$
(3.4c)

where $f$ is a zero-form. Here an unimportant factor of $2$ appears as a
result of our normalization of the higher spin algebra product. The minus in
between the terms is due to the fact that argument $f$ corresponds to the one-
form $\omega$ which anti-commutes to $e$. It is also clear that (2.4) is an
exact solution that defines $(A)dS_{4}$ since
$L_{AB}=-\tfrac{1}{2}y_{A}y_{B}$,
$L_{A^{\prime}B^{\prime}}=-\tfrac{1}{2}{{\overline{y}}}_{A^{\prime}}{{\overline{y}}}_{B^{\prime}}$
and $P_{AA^{\prime}}=-\tfrac{1}{2}y_{A}{{\overline{y}}}_{A^{\prime}}$ form
$(A)dS_{4}$ algebra with the most important relation being

$\displaystyle[P_{AA^{\prime}},P_{BB^{\prime}}]_{\star}$
$\displaystyle=L_{AB}\epsilon_{A^{\prime}B^{\prime}}+{\lambda}\epsilon_{AB}L_{A^{\prime}B^{\prime}}\,.$
(3.5)

#### Dual module.

With the help of the matrix trick the next $A_{\infty}$-algebra relation
implies that $\mathcal{U}(a,u)$ and $\mathcal{U}(u,a)$ define a bimodule $M$
over ${\mathfrak{hs}}$, $a\in{\mathfrak{hs}}$, $u\in M$. It is easy to see
that $M$ should be dual to the algebra ${\mathfrak{hs}}$, viewed as a bimodule
over itself, with respect to the following non-degenerate paring:

$\displaystyle\langle a|u\rangle$
$\displaystyle=\exp[p_{12}]\,a(y_{1})\,u(y_{2})\big{|}_{y_{i}=0}\,.$ (3.6)

The trace over the matrix and, possibly, other factors is understood. The
induced bimodule structure is given by the relations

$\begin{split}&\mathcal{U}_{1}(\omega,C)=+\exp{[{\lambda}\,p_{01}+p_{02}+p_{12}]}\,\omega({y}_{1})\,C({y}_{2})\Big{|}_{\overline{y}_{i}=0}\,,\\\
&\mathcal{U}_{2}(C,\omega)=-\exp{[p_{01}-{\lambda}\,p_{02}-p_{12}]}\,C({y}_{1})\,\omega({y}_{2})\Big{|}_{{y}_{i}=0}\,.\end{split}$
(3.7)

At this point one might want to introduce a reflection operator $R$,
$R^{2}=1$, $Ry^{A}R=-y^{A}$, so that we can redefine $\mathcal{U}_{2}$ to have
only $+$ in the second exponent at the price of having $CR$ everywhere instead
of $C$ (when $R$ is dragged through $\omega$ it effectively flips the sign of
all $(\\_\cdot p_{2})$-terms. For ${\lambda}=1$ the bimodule structure above
coincides with the twisted-adjoint action [81], but the latter does not have
the flat limit. At ${\lambda}\neq 1$ the left/right actions are different, but
are coordinated to give a bimodule. In this sense the right action is
derivable from the left one.

We can again check that the boundary conditions provided by (2.6) are
satisfied:

$\displaystyle\mathcal{U}(e,f)+\mathcal{U}(f,e)$
$\displaystyle=2e^{p_{02}+q_{02}}\left({\lambda}\,p_{01}q_{01}+p_{12}q_{12}\right)(e^{CC^{\prime}}y^{1}_{C}{{\overline{y}}}^{1}_{C^{\prime}})f(y^{2},{{\overline{y}}}^{2})\Big{|}_{y_{i},{{\overline{y}}}_{i}=0}\,,$
(3.8a) $\displaystyle\mathcal{U}(\omega,f)+\mathcal{U}(f,\omega)$
$\displaystyle=2{\lambda}\,p_{01}p_{12}\,e^{p_{02}+q_{02}}(\omega^{AB}y^{1}_{A}y^{1}_{B})f(y^{2},{{\overline{y}}}^{2})\Big{|}_{y_{i},{{\overline{y}}}_{i}=0}\,,$
(3.8b)
$\displaystyle\mathcal{U}(\overline{\omega},f)+\mathcal{U}(f,\overline{\omega})$
$\displaystyle=2q_{01}q_{12}\,e^{p_{02}+q_{02}}(\omega^{A^{\prime}B^{\prime}}{{\overline{y}}}^{1}_{A^{\prime}}{{\overline{y}}}^{1}_{B^{\prime}})f(y^{2},{{\overline{y}}}^{2})\Big{|}_{y_{i},{{\overline{y}}}_{i}=0}\,,$
(3.8c)

with $f$ being a zero-form. The action of the Lorentz subalgebra, i.e. the
terms with $\omega^{AB}$ and $\omega^{A^{\prime}B^{\prime}}$ are exactly the
same as in (3.4).

###  3.2 NLO

The cubic vertices satisfy simple linear equations and can be solved by brute-
force, see e.g. [37, 82, 32], which we closely follow below.

#### Cubic Vertex $\boldsymbol{\mathcal{V}(\omega,\omega,C)}$.

In accordance with different orders of the arguments there are three
$A_{\infty}$-maps to be found:

$\displaystyle\mathcal{V}(\omega,\omega,C)=\mathcal{V}_{1}(\omega,\omega,C)+\mathcal{V}_{2}(\omega,C,\omega)+\mathcal{V}_{3}(C,\omega,\omega)\,.$
(3.9)

The defining relations of an $A_{\infty}$-algebra requre then

$\begin{split}&\mathcal{V}_{1}(\mathcal{V}(\omega,\omega),\omega,C)-\mathcal{V}(\omega,\mathcal{V}_{1}(\omega,\omega,C))+\mathcal{V}_{1}(\omega,\omega,\mathcal{U}_{1}(\omega,C))-\mathcal{V}_{1}(\omega,\mathcal{V}(\omega,\omega),C)=0\,,\\\
&\mathcal{V}(\mathcal{V}_{1}(\omega,\omega,C),\omega)+\mathcal{V}_{1}(\omega,\omega,\mathcal{U}_{2}(C,\omega))+\mathcal{V}_{2}(\mathcal{V}(\omega,\omega),C,\omega)-\mathcal{V}(\omega,\mathcal{V}_{2}(\omega,C,\omega))\\\
&-\mathcal{V}_{2}(\omega,\mathcal{U}_{1}(\omega,C),\omega)=0\,,\\\
&\mathcal{V}(\mathcal{V}_{2}(\omega,C,\omega),\omega)-\mathcal{V}_{2}(\omega,C,\mathcal{V}(\omega,\omega))-\mathcal{V}_{2}(\omega,\mathcal{U}_{2}(C,\omega),\omega)-\mathcal{V}(\omega,\mathcal{V}_{3}(C,\omega,\omega))\\\
&+\mathcal{V}_{3}(\mathcal{U}_{1}(\omega,C),\omega,\omega)=0\,,\\\
&\mathcal{V}(\mathcal{V}_{3}(C,\omega,\omega),\omega)-\mathcal{V}_{3}(C,\omega,\mathcal{V}(\omega,\omega))+\mathcal{V}_{3}(C,\mathcal{V}(\omega,\omega),\omega)+\mathcal{V}_{3}(\mathcal{U}_{2}(C,\omega),\omega,\omega)=0\,.\end{split}$
(3.10)

Here one should replace the three $\omega$’s with three different elements of
${\mathfrak{hs}}$ while keeping the order the same in all terms. One can
rewrite these relations in terms of the corresponding symbols, which reduces
them to simple algebraic equations. For example, the first relation gives

$\displaystyle 0$
$\displaystyle=-\mathcal{V}_{1}(p_{0}+{\lambda}\,p_{1},p_{2},p_{3},p_{4})e^{p_{01}}+\mathcal{V}_{1}(p_{0},p_{1}+p_{2},p_{3},p_{4})e^{{\lambda}\,p_{12}}$
$\displaystyle\qquad-\mathcal{V}_{1}(p_{0},p_{1},p_{2}+p_{3},p_{4})e^{{\lambda}\,p_{23}}+\mathcal{V}_{1}(p_{0},p_{1},p_{2},{\lambda}\,p_{3}+p_{4})e^{p_{34}}\,.$

A nontrivial solution to these equations that is consistent with the
${\lambda}\rightarrow 0$ limit of [33] reads

$\displaystyle\mathcal{V}_{1}(\omega,\omega,C)$
$\displaystyle=+p_{12}\,\int_{\Delta_{2}}\exp[\left(1-t_{1}\right)p_{01}+\left(1-t_{2}\right)p_{02}+t_{1}p_{13}+t_{2}p_{23}+{\lambda}(1+t_{1}-t_{2})p_{12}]\,,$
$\displaystyle\mathcal{V}_{2}(\omega,C,\omega)$
$\displaystyle=-p_{13}\,\int_{\Delta_{2}}\exp[\left(1-t_{2}\right)p_{01}+\left(1-t_{1}\right)p_{03}+t_{2}p_{12}-t_{1}p_{23}+{\lambda}(1-t_{1}-t_{2})p_{13}]$
$\displaystyle\phantom{=}\,-p_{13}\,\int_{\Delta_{2}}\exp[\left(1-t_{1}\right)p_{01}+\left(1-t_{2}\right)p_{03}+t_{1}p_{12}-t_{2}p_{23}+{\lambda}(1-t_{1}-t_{2})p_{13}]\,,$
$\displaystyle\mathcal{V}_{3}(C,\omega,\omega)$
$\displaystyle=+p_{23}\,\int_{\Delta_{2}}\exp[\left(1-t_{2}\right)p_{02}+\left(1-t_{1}\right)p_{03}-t_{2}p_{12}-t_{1}p_{13}+{\lambda}(1+t_{1}-t_{2})p_{23}]\,.$

Hereinafter we let $\Delta_{n}$ denote the $n$-dimensional simplex
$t_{0}=0\leq t_{1}\leq...\leq t_{n}\leq 1$. This vertices deform smoothly
those of Chiral Theory in flat space [33] and are simple
${\lambda}$-modifications of what can be extracted from [81].

It is significant that all $\mathcal{U}$-vertices can be obtained from
$\mathcal{V}$-vertices by the following duality relation:

$\langle\mathcal{V}(\omega,\omega,C,\ldots,C)|C\rangle=\langle\omega|\mathcal{U}(\omega,C,\ldots,C)\rangle\,.$
(3.11)

It can be shown that this recipe gives consistent $\mathcal{U}$-vertices
provided one carefully keeps track of the ordering. For example, at the cubic
order we still need three $\mathcal{U}$-vertices

$\displaystyle\mathcal{U}(\omega,C,C)=\mathcal{U}_{1}(\omega,C,C)+\mathcal{U}_{2}(C,\omega,C)+\mathcal{U}_{3}(C,C,\omega)\,.$
(3.12)

By duality we find

$\displaystyle\mathcal{U}_{1}(p_{0},p_{1},p_{2},p_{3})$
$\displaystyle=+\mathcal{V}_{1}(-p_{3},p_{0},p_{1},p_{2})\,,$ (3.13)
$\displaystyle\mathcal{U}_{2}(p_{0},p_{1},p_{2},p_{3})$
$\displaystyle=-\mathcal{V}_{2}(-p_{1},p_{2},p_{3},p_{0})\,,$ (3.14)
$\displaystyle\mathcal{U}_{3}(p_{0},p_{1},p_{2},p_{3})$
$\displaystyle=-\mathcal{V}_{3}(-p_{1},p_{2},p_{3},p_{0})\,.$ (3.15)

It is interesting to have a look at least at one of them:

$\displaystyle\mathcal{U}_{1}$
$\displaystyle=p_{01}\int_{\Delta_{2}}\exp\left[{\lambda}\left(t_{1}-t_{2}+1\right)p_{01}+t_{1}p_{02}+\left(1-t_{1}\right)p_{03}+t_{2}p_{12}+\left(1-t_{2}\right)p_{13}\right]\,.$
(3.16)

The most important observation is that this vertex is local, as it will be
explained in more detail below.888As a historical comment, let us point out
that at ${\lambda}=1$ there is a well-known twisted-adjoint interpretation
[81] of the bilinear $\mathcal{U}$-maps. From this viewpoint, it is possible
to generate all $\mathcal{U}$-vertices from the $\mathcal{V}$-vertices by the
schematic formula $\mathcal{U}=CR\delta\mathcal{V}/\delta\omega$, $R$ being
the reflection operator. This, however, leads to nonlocal vertices
immediately.  This is thanks to it not having $p_{23}$ inside the exponent.

###  3.3 NkLO

#### Set up.

Following the original ideas of [58, 81], but with important modifications of
[33] that make the construction to yield local interactions,999In the
$L_{\infty}$-language the idea of [58, 81] is to construct an anti-minimal
model, i.e. a differential graded Lie algebra, whose minimal model gives the
sought-for $L_{\infty}$. This approach faces the usual problems of
(non)-locality [57, 83], see also below. Therefore, it is remarkable that a
simple modification proposed in [33] gives well-defined local vertices of
Chiral Theory in flat space and it is possible to turn on the cosmological
constant as we show in the paper. The modifications include: a star-product,
dual module structure for $C$, the duality map to get $\mathcal{U}$-vertices.
we define an extended algebra of polynomial functions $\mathbb{C}[y,z]$,
$Y^{a}\equiv(y^{A},z^{A})$, that is equipped with the star-product defined by
the matrix

$\displaystyle(\Omega^{ab})=-\begin{pmatrix}{\lambda}\epsilon&\epsilon\\\
-\epsilon&0\end{pmatrix}\,,\qquad\epsilon\equiv\epsilon^{AB}\,.$ (3.17)

The symbol of $\mu(f,g)\equiv(f\star g)(y,z)$ is

$\displaystyle\exp{[p_{01}+p_{02}+r_{01}+r_{02}+p_{1}\cdot r_{2}-r_{1}\cdot
p_{2}+{\lambda}\,p_{12}]}\,.$ (3.18)

Hereinafter $r$ for $z$ is the same as $p$ for $y$. The integral
representation reads101010As in [33] we define the integral in such a way that
$\int d^{2}u\,\exp[u\cdot q]=\delta^{2}(q)$.

$\displaystyle(f\star g)(y,z)$ $\displaystyle=\int
du\,dv\,dp\,dq\,f(y+u,z+v)g(y+q,z+p)\exp{[v\cdot q-u\cdot p+{\lambda}\,p\cdot
v]}\,.$ (3.19)

The vertices are generated via the standard Homological Perturbation Theory.
We refer to [33] for a detailed description and to Appendix A for the
necessary modifications. Below, we just explain the algorithm and illustrate
it with some examples. The generators of the algebra $y_{A}$, $z_{A}$ act as
follows:

$\displaystyle\begin{aligned} y_{A}\star
f&=(y_{A}-{\lambda}\,\partial^{y}_{A}-\partial^{z}_{A})f\,,\\\ f\star
y_{A}&=(y_{A}+{\lambda}\,\partial^{y}_{A}-\partial^{z}_{A})f\,,\end{aligned}$
$\displaystyle z_{A}\star f$ $\displaystyle=f\star
z_{A}=(z_{A}+\partial^{y}_{A})f\,.$ (3.20)

With these relations one can see that the function
$\varkappa=\exp[z^{C}y_{C}]$ obeys the identities

$\\{y_{A},\varkappa\\}_{\star}=z_{A}\star\varkappa=\varkappa\star z_{A}=0\,.$

We will need a further extension of the above algebra to the algebra of
differential forms in $dz$. The corresponding exterior differential will be
denoted by $d_{z}$. We will use the Poincaré lemma in the following form:

$\displaystyle f^{(1)}=h[f^{(2)}]$
$\displaystyle=dz^{A}\,z_{A}\int_{0}^{1}t\,dt\,f^{(2)}(tz)\,,$ $\displaystyle
f^{(0)}=h[f^{(1)}]$ $\displaystyle=z^{A}\int_{0}^{1}dt\,f_{A}^{(1)}(tz)\,,$
(3.21)

see e.g. [84]. The first expression is a particular solution to
$d_{z}f^{(1)}=f^{(2)}$ for a one-form $f^{(1)}\equiv dz^{A}f^{(1)}_{A}(z)$ and
a two-form $f^{(2)}\equiv\tfrac{1}{2}f^{(2)}(z)\epsilon_{AB}dz^{A}dz^{B}$. The
second formula is a particular solution to $d_{z}f^{(0)}=f^{(1)}$ for a closed
one-form $f^{(1)}$ and a zero-form $f^{(0)}\equiv f^{(0)}(z)$. We also define
$h[f^{(0)}]=0$ for any zero-form $f^{(0)}$.

The aforementioned Homological Perturbation Theory generates all trees that
have $\omega$ and $C$ as leaves. The trivalent vertices of trees are provided
by the $\star$-product $\mu$, while the contracting homotopy $h$ defines the
internal branches. Given a zero-form $C(y)$, we denote

$\delta C=\tfrac{1}{2}C(z)e^{z^{B}y_{B}}dz_{A}dz^{A}\,.$ (3.22)

The zero-form $C$ enters the trees via the combination $\Lambda[C]=h\delta C$,
that is,

$\displaystyle\Lambda[C]$
$\displaystyle=dz^{A}z_{A}\int_{0}^{1}t\,dt\,C(tz)e^{tz^{B}y_{B}}=dz^{A}z_{A}\int_{0}^{1}t\,dt\,\exp[tz\cdot(y+p_{i})]\,C(y_{i})\Big{|}_{y_{i}=0}\,,$
(3.23)

where the last expression is the symbol of the previous one that we use in
practice. All trees that make sense can in principle contribute. It should be
remembered that $h^{2}=0$ and forms of degree higher than two vanish
identically. Each subtree of a given tree must be an admissible tree. As was
shown in [33], and remains to be true here, there are certain classes of trees
that vanish thanks to an interplay between specific $\mu$ and $h$ we have
defined. It is instructive to give some examples.

#### NLO.

There are four nontrivial trees that can be drawn at NLO. For example,
$\mathcal{V}(\omega,\omega,C)$ is built from

$\mathcal{V}(\omega,\omega,C)=\omega(y)\star
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}\pgfsys@transformcm{-0.7954}{0.6061}{-0.6061}{-0.7954}{14.02615pt}{-16.97409pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

Evaluation of this tree is almost identical to that of [33]. It is easy to
compute the other three trees, two of which contribute to
$\mathcal{V}(\omega,C,\omega)$ and the mirror copy of the one above to
$\mathcal{V}(C,\omega,\omega)$. This gives exactly the analytical expressions
of Section 3.2.

#### NNLO.

There are many more trees at NNLO, which correspond to quartic vertices from
the equations of motion vantage point. Nevertheless, there are only two
nontrivial topologies:111111Note that, as different from [33], the
$\star$-product $\mu$ is non-commutative. Therefore, one cannot obtain all the
vertices by simple permutations of the branches of a few basic trees.

$G_{1}=a\star h[h[b\star\Lambda[u]]\star\Lambda[v]]=\leavevmode\hbox
to117.69pt{\vbox to105.1pt{\pgfpicture\makeatletter\hbox{\hskip
58.84312pt\lower-52.54842pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-58.84312pt}{-52.54842pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 16.30553pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
4.30554pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil\cr\vskip 8.99994pt\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 19.31827pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.01274pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\mu}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
7.31828pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil\cr\vskip 8.99994pt\cr\hfil\hskip
6.94849pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.64294pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${a}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.94849pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 19.31827pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.01274pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\mu}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
7.31828pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil\cr\vskip 8.99994pt\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 19.31827pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.01274pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\mu}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
7.31828pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 25.15854pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.85301pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\Lambda[v]}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
13.15855pt\hfil\cr\vskip 8.99994pt\cr\hfil\hskip
6.45137pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.14583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${b}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.45137pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 25.41782pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.11229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\Lambda[u]}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
13.41783pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-25.62788pt}{30.26933pt}\pgfsys@lineto{-25.62788pt}{38.46931pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{-25.62788pt}{38.6693pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-45.30167pt}{7.7pt}\pgfsys@lineto{-33.44952pt}{17.88356pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.75848}{0.6517}{-0.6517}{0.75848}{-33.29785pt}{18.01387pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-0.41006pt}{7.21721pt}\pgfsys@lineto{-17.78035pt}{19.19615pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.82323}{0.56772}{-0.56772}{-0.82323}{-17.94496pt}{19.30968pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.90706pt}{15.67297pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{h}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-18.43187pt}{-18.0638pt}\pgfsys@lineto{-0.72435pt}{-4.12675pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7858}{0.61848}{-0.61848}{0.7858}{-0.5672pt}{-4.00308pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-15.80685pt}{-8.61884pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{h}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{32.42099pt}{-14.86934pt}\pgfsys@lineto{14.95908pt}{-3.20593pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.83156}{0.55544}{-0.55544}{-0.83156}{14.7928pt}{-3.09485pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-45.24327pt}{-43.38478pt}\pgfsys@lineto{-33.42868pt}{-31.54535pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.70636}{0.70786}{-0.70786}{0.70636}{-33.28741pt}{-31.4038pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-3.90666pt}{-41.1887pt}\pgfsys@lineto{-17.7979pt}{-30.02136pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.77937}{0.62656}{-0.62656}{-0.77937}{-17.95375pt}{-29.89607pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

and

$G_{2}=h[a\star\Lambda[u]]\star h[b\star\Lambda[v]]=\leavevmode\hbox
to195.86pt{\vbox to78.78pt{\pgfpicture\makeatletter\hbox{\hskip
97.93105pt\lower-39.38873pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-97.93105pt}{-39.38873pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 16.30553pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
4.30554pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil\cr\vskip 8.99994pt\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 19.31827pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.01274pt}{0.0pt}\pgfsys@invoke{
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Here we keep abstract arguments $a$, $b$, $u$, and $v$ instead of $\omega$ and
$C$. The first tree is the only contribution to
$\mathcal{V}(\omega,\omega,C,C)$. Explicit analytical expressions for $G_{1}$
and $G_{2}$ can be found in Appendix B. They have the following structure:

$\displaystyle G_{1}$ $\displaystyle=\ast(p_{12})^{2}\exp{[\ast p_{01}+\ast
p_{02}+\ast p_{13}+\ast p_{23}+\ast p_{14}+\ast
p_{24}+\ast{\lambda}\,p_{12}]}\,,$ (3.24) $\displaystyle G_{2}$
$\displaystyle=\ast(p_{13})^{2}\exp{[\ast p_{01}+\ast p_{03}+\ast p_{12}+\ast
p_{23}+\ast p_{14}+\ast p_{34}+\ast{\lambda}\,p_{13}]}\,.$ (3.25)

Hereinafter $\ast$ stands for unimportant prefactors that depend on $t$’s. The
vertices are smooth deformations of those from [33]. Most importantly, they
are still local, as will be explained below in more detail. The locality of
$G_{1}$ implies that it does not depend on $p_{34}$ and the locality of
$G_{2}$ means its independence of $p_{24}$. Out of curiosity we compute a
quintic (N3LO) vertex in Appendix C and its higher order cousins.

#### Locality.

It is important to engineer the vertices that are local. Locality is a crucial
difference between the formal deformation procedure and an actual field
theory. Indeed, it is well-known that Noether procedure (constructing
interactions order by order in the fields, which is usually applied to gauge
theories) always admits an all order solution once the locality is abandoned
[85]. An equivalent statement is even simpler to make in the light-cone gauge:
any function can be taken as a Hamiltonian $H$ (provided simple kinematical
constraints are imposed) unless we care about locality of the boost generators
$J^{i-}$. Within the FDA approach, where the equations $d\Phi=Q(\Phi)$ are
determined by an odd nilpotent vector field $Q$, the r.h.s. of the equations
can easily lead to meaningless interactions from the field theory point of
view, while $Q$ gives a well-defined $L_{\infty}$-algebra. The problem is that
fields $\Phi(x)$ (as maps from spacetime to the target space supermanifold
$\mathcal{N}$ where $Q$ is defined) can be expressed as derivatives of each
other by virtue of the equations of motion. Therefore, nonlinearities in
$Q(\Phi)$ can form infinite series in derivatives.121212A vague statement is
that in any reasonable measure on the space of $Q$ for a given $\mathcal{N}$,
most of $Q$ lead to nonsensical equations of motion due to nonlocalities.
Lucky exceptions to this situation are topological field theories where
$\mathcal{N}$ is finite-dimensional. For field theories, and certainly for our
case, $\mathcal{N}$ is infinite dimensional.

More specifically, nonlocality can develop in any vertex with two or more
$C$-fields since they contain auxiliary fields that are expressed as
derivatives of the dynamical ones:

$\displaystyle C^{A(2s+k),A^{\prime}(k)}$
$\displaystyle\sim\nabla^{AA^{\prime}}...\nabla^{AA^{\prime}}C^{A(2s)}\,,$
$\displaystyle k=0,1,2,...\,.$ (3.26)

Here $C^{A(2s)}\equiv\Psi^{A(2s)}$ is one of the dynamical fields. The same is
true for $C^{A(k),A^{\prime}(2s+k)}$, where $C^{A^{\prime}(2s)}$ is $s$
derivatives further removed from $\omega^{A(2s-2)}$. For definiteness let us
concentrate on $\mathcal{V}(\omega,\omega,C,...,C)$-type vertices with $n$
fields $C$. Nonlocality is present once one has an infinite series of the
form131313Not every infinite series in derivatives is ill-behaved as long as
$a_{k}$ decay to zero fast enough, see e.g. examples in [57, 83].
Nevertheless, for Chiral Theory, which is known to be local, having an
infinite series in derivatives must be avoided.

$\displaystyle\sum_{k}a_{k}\overbrace{\nabla_{BB^{\prime}}...\nabla_{BB^{\prime}}}^{k}C^{A(2s)}\nabla^{BB^{\prime}}...\nabla^{BB^{\prime}}C^{M(2s)}...\sim\sum_{k}a_{k}C{}^{A(2s)}{}_{B(k),B^{\prime}(k)}\,C^{M(2s)B(k),B^{\prime}(k)}...\,,$

where we singled out two $C$-arguments and ignored the rest ones. In terms of
the symbols of operators, nonlocality manifests itself as an infinite series
in $w=q_{ij}p_{ij}$ where $i$, $j$ correspond to any pair of $C$-arguments in
$\mathcal{V}(\omega,\omega,C,...,C)$.

The vertices we construct here (and the same is true for those of Chiral
Theory without cosmological constant [33]) have a very special form:

$\displaystyle\mathcal{V}(\omega,\omega,C,...,C)$ $\displaystyle=\ast\exp[\ast
p_{01}+\ast p_{02}+\ast{\lambda}\,p_{12}+\sum_{2<i\leq n+2}\ast
p_{1i}+\sum_{2<i\leq n+2}\ast p_{2i}]\,.$ (3.27)

Recall that for every vertex there is an overall star-product factor for
another copy of $A_{1}$, in ${{\overline{y}}}$:

$\displaystyle\exp[\sum_{0\leq i<j\leq n+2}q_{ij}]\,.$ (3.28)

Therefore, locality is equivalent to not having $p_{ij}$ inside $\exp[...]$
with $2<i<j\leq n+2$, which is indeed the case. Applying now the duality
relation (3.11), we find immediately

$\displaystyle\mathcal{U}(p_{0},p_{1},...,p_{n+2})=\mathcal{V}(-p_{n+2},p_{0},p_{1},...,p_{n+1})=$
$\displaystyle\ast\exp[\ast p_{0,n+2}+\ast
p_{1,n+2}+\ast{\lambda}\,p_{01}+\sum_{1<i\leq n+1}\ast p_{0,i}+\sum_{1<i\leq
n+1}\ast p_{1,i}]\,.$

It is a local vertex as well! The argument that proves locality to all orders
is exactly the same as in [33]. Therefore, we conclude that all the vertices
are local and, thereby, define a theory rather than just an
$L_{\infty}$-algebra. It is worth clarifying that by a local theory we mean a
theory where every vertex has a finite number of derivatives provided the
helicities on all external legs are fixed.

##  4 Summary and discussion

In this paper, we constructed Chiral Theory with cosmological constant, whose
existence was anticipated already in [17] and supported further in [30, 31]
with the help of the light-cone gauge techniques. Technically, the theory is
constructed as classical equations of motion in the form of FDA. The FDA
emerges from the homological perturbation theory in a standard way. The theory
is a straightforward and smooth deformation of Chiral Theory’s FDA in flat
space [32, 33], the deformation parameter $\lambda\sim 1/R$ being the ‘inverse
radius’ of (anti-)de Sitter space. It is still remarkable that the self-
evident deformation of [33] leads to a local theory.141414Even if we are given
a local theory to begin with, it is possible to perform nonlocal field-
redefinitions that change physical observables and render them nonsensical
[83, 57], while still having a well-define $L_{\infty}$-algebra. A (non)local
field-redefinition that admits a series expansion in derivatives is just a
change of a basis at the level of the $L_{\infty}$-algebra. Therefore, it is
not guaranteed that switching on the cosmological constant the way we did
leads to a local theory. There are higher spin examples [83, 57] where this
does not happen. Within the homological perturbation theory one deforms a
differential of a strong deformation retract and however simple the
deformation is it may be hard to predict the structure of the resulting
$L_{\infty}$-algebra.  Despite the conceptual simplicity of the result, it is
of significant importance in view of the duality to Chern–Simons Matter
Theories, discussed in Introduction.

In view of footnote 2, which recaps the main problems of holographic HiSGRA’s,
it is worth summarizing what is already known about interactions of these
hypothetical HiSGRA’s. A nonlocal quartic vertex was holographically
reconstructed in [51], followed by a complete cubic action [86]. Within the
FDA approach the local form of $\mathcal{U}(\omega,C,C)$, which corrects [37],
was found in [87]. The best available result is a holomorphic subsector of
$\mathcal{V}(\omega,\omega,C,C)$-type interactions [88] and parts of
$\mathcal{V}(\omega,C,C,C)$-type interactions [89, 90], which contains some of
the vertices found previously in [30, 31] within the light-cone approach. Note
that none of the FDA vertices found so far addresses the genuine
nonlocality/non-existence problem [51, 52, 53, 54], the simplest one being to
reproduce [51] by genuine bulk methods without inverting the holographic
correlators. Another problem is that each order requires a separate analysis
even for the vertices that must be local.

Since the papers above and our deal with higher spin interactions in
$(A)dS_{4}$ and we used the notation close to those of [74, 37, 58, 81], it is
easy to discuss the main differences that allowed us to make progress and to
construct an actual theory. Let us recall that the spectrum of $\omega$ and
$C$ is the same, since it is determined by the free limit and all higher spin
theories, loosely speaking, contain massless fields of all spins. The
challenge is to construct interaction vertices, i.e.
$A_{\infty}/L_{\infty}$-maps as poly-differential operators, that obey
locality. There are several ingredients: (a) $y-z$ star-product and
contracting homotopy; (b) dual bimodule structure of $C$; (c) duality map. (a)
Our star-product (3.20) is different from [58] and [88]: it is fixed, well-
defined and does not require any subtle limits, cf. [88], it works as it is.
An invariant characteristic of every star-product is the rank of the
underlying Poisson tensor: it is $4$ for [58] and [88], $0$ for Chiral Theory
in flat space [33] and $2$ for its $(A)dS_{4}$-deformation of the present
paper (which is exactly what is needed to turn on the cosmological constant
(3.5) and it closely related to the infinity twistor). It is interesting that
the star-product of [88], while different from ours, gives a local
$\mathcal{V}(\omega,\omega,C,C)$-vertex in the limit where the star-product
itself is ill-defined. We use the standard contracting homotopy for the de
Rham complex of a linear space. Ingredients (b) and (c) are completely new
[33] and play the most important part. (b) Our zero-form $C$ takes values in
the bimodule dual to the higher spin algebra, rather than in the twisted-
adjoint one [81]. In other words, $C$ is not an element of the higher spin
algebra and, for this reason, it should be treated differently and we use the
general homological perturbation theory, which cannot be captured by [58, 88].
In particular, this ensures the smooth flat limit or deformation to
$(A)dS_{4}$, which is a counterexample to the general folklore [81]. (c) It is
of crucial importance that $\mathcal{U}$-vertices are obtained via the duality
map from $\mathcal{V}$-vertices, which is an original idea born for flat space
Chiral Theory [33], otherwise they emerge nonlocal as in [37, 58] and require
a separate treatment [89, 90]. To conclude, while one can play with different
(a) more or less successfully, it is (b)+(c) that are crucial for constructing
a local theory. An even stronger difference is that our main result is a
smooth deformation of Chiral Theory in flat space that is already known and
well-defined, the deformation having $\lambda$ as a free parameter.

In general, it would be interesting to see what is a maximal local closed sub-
sector of the holographic dual of Chern–Simons Matter Theories. We expect that
Chiral Theory covers all of it. In this sense, the relation between our
results and particular vertices of [87, 88, 89, 90] is not clear beyond the
lowest orders.151515Cubic vertices decompose into chiral and anti-chiral parts
[30], and hence, different truncations – holomorphic/chiral/self-dual – mean
essentially the same at this order. Note that the usual cubic vertices from
the Lagrangian vantage point appear in $\mathcal{V}(\omega,\omega,C)$,
$\mathcal{U}(\omega,C,C)$ and $\mathcal{V}(\omega,\omega,C,C)$. There is an
infinite-parameter ambiguity at higher orders [81, 91], but this analysis does
not take locality into account, which may eliminate some parameters as well as
to introduce new ones.  It may well be that there are well-defined, in the
sense of being local, holomorphic subsectors of [81] and the question is
whether they are smaller/larger than Chiral Theory. One way or another, Chiral
Theory in $(A)dS_{4}$ is directly constructed as a smooth and local
deformation of its flat space cousin [33].161616If Chiral Theory in
$(A)dS_{4}$ corresponds to the local form (not yet available) of the
holomorphic vertices of [81], our paper gives a ‘one-line solution’ in all
orders that provides an alternative completion to the low order analysis of
[87, 88, 89, 90].

The vertices we obtained turn out to be much simpler than they come out of the
homological perturbation theory we use and it would be very interesting to
find all of them explicitly as well as to make this hidden simplicity
manifest. An all order result can be found in Appendix C. It would also be
important to make contact with the Shoikhet–Tsygan–Kontsevich formality and
its possible extension to Poisson Orbifolds, which is visible [82, 71] in
HiSGRA applications. In particular, it is a challenge to identify the
configuration space of the integrals that define vertices and reduce the proof
of the $A_{\infty}/L_{\infty}$-relations to Stokes theorem [92, 93].

The explicit construction of covariant HiSGRA with massless propagating fields
– Chiral Theory with or without cosmological constant and its contractions
[21] – opens up new research directions. Among these are (i) computing
(higher) holographic correlation functions; (ii) exploring quantum
corrections; (iii) constructing exact solutions via the general techniques
developed over the years, see e.g. [94, 95, 96] and [97, 98] for the careful
treatment of locality. One might also expect the existence of a simple twistor
action for Chiral Theories, see [99, 100, 72] for the twistor results that
deal with a contraction of Chiral Theory [72] and with conformal HiSGRA [99,
100].

In the light of our results and AdS/CFT duality, there should also exist
Chiral Theories with partially-massless fields that are truncations of the
holographic duals of higher derivative vector models [101]. Such theories
should still admit a smooth flat limit where a free partially-massless field
becomes a reducible non-unitary representation of Poincare group. Closely
related theory is the self-dual truncation of conformal HiSGRA, which admits a
twistor description [99, 100]. The latter construction should have a simple
generalization to conformal HiSGRA’s that are based on higher order singletons
[101]. It would also be interesting to construct these two (conjectured) new
classes of HiSGRA explicitly in the FDA form. It is also obvious that the FDA
we constructed contains contractions [20, 21] of Chiral Theory.

## Acknowledgments

The work of E.S. was partially supported by the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 101002551) and by the Fonds de la Recherche
Scientifique — FNRS under Grant No. F.4544.21. A. Sh. gratefully acknowledges
the financial support of the Foundation for the Advancement of Theoretical
Physics and Mathematics “BASIS”.

##  Appendix A Some algebra

Throughout this section, we deal with partial algebras of different types.
Recall that a partial algebra is just a nonempty set $X$ equipped with a
collection $P$ of partial operations on $X$, see e.g. [102, Ch.2 ], [103]. We
will always assume that $X$ is a complex vector space, so that $P$ includes
the addition of vectors and multiplication them by complex numbers. The other
operations, which we all assume to be multilinear over $\mathbb{C}$, may be
defined only for some elements of $X$. If $p\in P$ is an $n$-ary operation,
then the domain of $p$, is a subset $\mathrm{Dom}(p)\subset
X\times\cdots\times X$ in the $n$-th Cartesian power of $X$ such that $p$ is
defined for all the elements of $\mathrm{Dom}(p)$. In the case that the domain
of each operation of $P$ coincides with the whole Cartesian power of $X$, the
pair $(X,P)$ is just an ordinary (or total) algebra. In the rest of this
section we will omit the adjective ‘partial’.

Let $\mathfrak{A}=\mathbb{C}[[y,z]]$ be the space of formal power series in
four indeterminates $y^{A}$ and $z^{A}$, $A=1,2$. We make it into an algebra
over $\mathbb{C}$ for the Weyl–Moyal $\star$-product171717As in the main text,
the index $A$ is raised and lowered with the help of the $\epsilon$-symbol and
$y\cdot z\equiv y^{A}z_{A}$.

$a\star
b=\exp\big{[}\partial_{y_{1}}\cdot\partial_{z_{2}}-\partial_{z_{1}}\cdot\partial_{y_{2}}+{\lambda}\,\partial_{y_{1}}\cdot\partial_{y_{2}}\big{]}a(y_{1},z_{1})\,b(y_{2},z_{2})\,\Big{|}_{\begin{subarray}{c}\displaystyle{}y_{1,2}=y\\\
\displaystyle{}z_{1,2}=z\end{subarray}}\,\,,$ (A.1)

${\lambda}$ being a complex parameter. Notice that the $\star$-product is
strongly associative in the sense of [103, Ch.1.5] and its domain includes the
Cartesian square of the subspace $\mathbb{C}[y,z]\subset\mathfrak{A}$, i.e.,
the space of complex polynomials in $y$’a and $z$’s.

As usual, one may regard the algebra $\mathfrak{A}$ as a bimodule over itself.
The complex vector space $\mathfrak{A}$ enjoys a $\mathbb{C}$-linear
involution $\tau:\mathfrak{A}\rightarrow\mathfrak{A}$ defined by

$a(y,z)\quad\mapsto\quad a^{\tau}(y,z)=a(z,y)e^{z\cdot y}\,.$ (A.2)

Clearly, $\tau^{2}=1$. Using this involution, we give the space $\mathfrak{A}$
another structure of $\mathfrak{A}$-bimodule, denoted by
$\mathfrak{A}^{\tau}$. By definition,

$a\circ m\circ b=(a\star m^{\tau}\star b)^{\tau}\qquad\forall
a,b\in\mathfrak{A}\,,\quad\forall m\in\mathfrak{A}^{\tau}\,.$ (A.3)

The bimodule axioms are verified immediately. Let us introduce the subspace
$\mathfrak{M}=\mathbb{C}[[y]]\subset\mathfrak{A}^{\tau}$. We claim that
$\mathfrak{M}$ is actually an $\mathfrak{A}$-submodule. This is enough to
check only for the generators. We find

$\begin{aligned} y^{A}\circ m=(-\partial_{y}^{A}+{\lambda}y^{A})m(y)\,,\\\
m\circ
y^{A}=(-\partial_{y}^{A}-{\lambda}y^{A})m(y)\,,\end{aligned}\qquad\qquad
z^{A}\circ m=m\circ z^{A}=0\,.$ (A.4)

Notice that the left and right actions of $\mathfrak{A}$ on $\mathfrak{M}$ are
different unless ${\lambda}\neq 0$. As usual, one can regard the
$\mathfrak{A}$-bimodule $\mathfrak{M}$ as a graded algebra
$\mathfrak{A}\oplus\mathfrak{M}$ w.r.t. the product

$(a,m)(\tilde{a},\tilde{m})=(a\tilde{a},a\tilde{m}+\tilde{a}m)\,.$ (A.5)

The elements of the subspaces $\mathfrak{A}$ and $\mathfrak{M}$ are prescribed
the degrees $0$ and $1$, respectively.

A map $\overline{\delta}:\mathfrak{M}\rightarrow\mathfrak{A}$ will define a
differential of the algebra $\mathfrak{A}\oplus\mathfrak{M}$ of degree $-1$,
making it into a dg-algebra, iff the following relations are satisfied:

$\overline{\delta}(a\circ m\circ b)=a\star\overline{\delta}m\star
b\,,\qquad\overline{\delta}m\circ\tilde{m}=m\circ\overline{\delta}\tilde{m}\,.$
(A.6)

Let us set $\overline{\delta}m=m^{\tau}$. Then the first relation takes the
form

$(a\circ m\circ b)^{\tau}=a\star m^{\tau}\star b\,,$ (A.7)

which is equivalent to the definition (A.3). The second relation boils down to
the identity

$m^{\tau}\circ\tilde{m}=(m^{\tau}\ast\tilde{m}^{\tau})^{\tau}=m\circ\tilde{m}^{\tau}\,.$
(A.8)

As a next step, we introduce the Grassmann algebra $\Lambda$ on two generators
$dz^{A}$,

$dz^{A}dz^{B}=-dz^{B}dz^{A}\,,$ (A.9)

both in degree $1$, and define the tensor product algebra
$\mathfrak{B}=\mathfrak{A}\otimes\Lambda$. Assuming the generators $dz^{A}$
act trivially in $\mathfrak{M}$, i.e.,

$dz^{A}\circ m=m\circ dz^{A}=0\,,$ (A.10)

we endow $\mathfrak{M}$ with the structure of $\mathcal{B}$-bimodule and
define the graded algebra $\mathfrak{B}\oplus\mathfrak{M}$ with the product
(A.5), where now $(a,m)\in\mathfrak{B}\oplus\mathfrak{M}$. The differential
$\overline{\delta}$ above gives rise to a differential $\delta$ of degree $+1$
on $\mathfrak{B}\oplus\mathfrak{M}$. The latter is defined as

$\delta a=0\,,\qquad\delta m=m^{\tau}dz^{1}dz^{2}\in\mathfrak{B}$ (A.11)

for all $a\in\mathfrak{B}$ and $m\in\mathfrak{M}$. It easy to see that
relations (A.6) hold true for $\mathfrak{A}$ replaced with $\mathfrak{B}$ and
$\overline{\delta}$ replaced with $\delta$. Hence, we may regard
$\mathfrak{B}\oplus\mathfrak{M}$ as a dg-algebra with differential $\delta$.

Considering now the elements $a(y,z,dz)\in\mathfrak{B}$ as exterior
differential forms in $z$’s, we endow the dg-algebra
$\mathfrak{B}\oplus\mathfrak{M}$ with another differential denoted by $d$.
This is given by the exterior differential on $\mathfrak{B}$ and extends
trivially to $\mathfrak{M}$:

$da=dz^{A}\frac{\partial a}{\partial z^{A}}\,,\qquad dm=0$ (A.12)

for all $a\in\mathfrak{B}$ and $m\in\mathfrak{M}$. It is clear that the
operator $d$ has degree $1$, squares to zero, and differentiates the product
(A.5) in $\mathfrak{B}\oplus\mathfrak{M}$. Furthermore, the differentials $d$
and $\delta$ commute to each other, which allows one to combine them into the
total differential $D=d+\delta$ of degree $1$. In such a way we arrive at the
dg-algebra $(\mathfrak{B}\oplus\mathfrak{M},D)$.

Given the dg-algebra $(\mathfrak{B}\oplus\mathfrak{M},D)$, one may ask about
its cohomology and the minimal model. Since $\delta$ has form degree $2$, one
may regard it as a ‘small perturbation’ of $d$; the differential $d$ increases
the form degree only by $1$. Then the standard spectral sequence arguments
coupled to the Poincaré lemma for the exterior differential (A.12) show that
$H(\mathfrak{B}\oplus\mathfrak{M})\simeq\mathbb{C}[[y]]\oplus\mathbb{C}[[y]]$.
(It is isomorphic to the space of $z$-independent elements of
$\mathfrak{A}\oplus\mathfrak{M}$, which are obviously nontrivial
$d$-cocycles.) The graded space $H(\mathfrak{B}\oplus\mathfrak{M})$ generates
the full spectrum of fields – $\omega$ and $C$ – in Chiral HiSGRA. It turns
out that the dg-algebra $\mathfrak{B}\oplus\mathfrak{M}$ is not formal,
meaning that the cohomology space $H(\mathfrak{B}\oplus\mathfrak{M})$ enjoys
higher multiplication operations, in addition to the binary product induced by
(A.5), making it into a minimal $A_{\infty}$-algebra. Denoting this
$A_{\infty}$-algebra by $\mathbb{A}$ and tensoring it with the associative
algebra $A_{1}\otimes\mathrm{Mat}_{N}$, we get a bigger $A_{\infty}$-algebra
$\mathbb{A}\otimes A_{1}\otimes\mathrm{Mat}_{N}$. Applying the standard
symmetrization map to $\mathbb{A}\otimes A_{1}\otimes\mathrm{Mat}_{N}$ gives
finally a minimal $L_{\infty}$-algebra. It is this ${L}_{\infty}$-algebra that
defines the r.h.s. of field equations (2.3).

Since the unperturbed differential $d$ admits an explicit contracting homotopy
$h$ – that from the Poincaré lemma (3.21) – there is a systematic method for
constructing the minimal model of the dg-algebra
$(\mathfrak{B}\oplus\mathfrak{M},D)$. The method results in explicit
expressions for the higher multiplication operations in $\mathbb{A}$, see e.g.
[104], [105]. For more details we refer the reader to [106] and to our recent
paper [33], where such a minimal model is constructed for the limiting case
${\lambda}=0$. (All basic steps and formulas are precisely the same.) This
provides a rationale for the integrals and their diagrammatic representation
in the main text.

Finally, let us recall that the $A_{\infty}$-algebra $\mathbb{A}$, being
constructed in terms of the partially defined $\star$-product (A.1), is
actually a partial algebra. However, a closer examination [33, App. C] shows
that all the multiplication operations entering the definition of $\mathbb{A}$
are well defined for the elements of the subspace
$\mathbb{C}[y]\oplus\mathbb{C}[[y]]\subset H(\mathfrak{B}\oplus\mathfrak{M})$.
Upon restriction to this subspace we get a total $A_{\infty}$-algebra that
governs the interaction in Chiral HiSGRA.

##  Appendix B NNLO vertices

Straightforward evaluation of $G_{1}$ gives

$\displaystyle G_{1}$
$\displaystyle=(p_{12})^{2}\exp\Big{[}\left(1-u_{1}-u_{2}\right)p_{01}+\left(1-u_{3}-u_{4}\right)p_{02}+u_{1}p_{13}+u_{2}p_{14}+u_{3}p_{23}+u_{4}p_{24}+$
$\displaystyle\qquad\qquad\qquad\qquad+{\lambda}\left(u_{4}u_{1}+u_{1}+u_{2}-u_{2}u_{3}-u_{3}-u_{4}+1\right)p_{12}\Big{]}\,,$

where the original integration variables $t_{i}$, $i=1,2,3,4$ that are
integrated over $[0,1]$ can be identified as

$\displaystyle u_{1}$
$\displaystyle=\frac{t_{1}\left(t_{2}-1\right)t_{3}t_{4}}{t_{1}t_{2}t_{3}-1}\,,$
$\displaystyle u_{2}$
$\displaystyle=\frac{t_{2}\left(t_{1}t_{3}-1\right)t_{4}}{t_{1}t_{2}t_{3}-1}\,,$
$\displaystyle u_{3}$
$\displaystyle=\frac{t_{1}\left(t_{2}t_{3}-1\right)}{t_{1}t_{2}t_{3}-1}\,,$
$\displaystyle u_{4}$
$\displaystyle=\frac{\left(t_{1}-1\right)t_{2}}{t_{1}t_{2}t_{3}-1}\,.$

Note that going to $u$’s instead of $t$’s generates a Jacobian that kills a
$t$-dependent prefactor, which is the same as in [33]. The corresponding
$\mathcal{U}$-vertex is obtained via the duality map:

$\displaystyle\mathcal{U}(\omega,C,C,C)$ $\displaystyle:$
$\displaystyle\mathcal{U}(p_{0},p_{1},p_{2},p_{3},p_{4})=\mathcal{V}(-p_{4},p_{0},p_{1},p_{2},p_{3})\,.$

Similarly, the second tree gives

$\displaystyle G_{2}$
$\displaystyle=-(p_{13})^{2}\exp\Big{[}\left(1-u_{1}-u_{2}\right)p_{01}+\left(1+u_{3}-u_{4}\right)p_{03}+u_{1}p_{12}+u_{2}p_{14}+u_{3}p_{23}+u_{4}p_{34}+$
$\displaystyle\qquad\qquad\qquad\qquad+{\lambda}\left(u_{4}u_{1}-u_{1}+u_{2}+u_{2}u_{3}+u_{3}-u_{4}+1\right)p_{13}\Big{]}\,,$

where

$\displaystyle u_{1}$
$\displaystyle=\frac{t_{1}\left(t_{2}t_{3}t_{4}-1\right)}{t_{1}t_{2}t_{3}t_{4}-1}\,,$
$\displaystyle u_{2}$
$\displaystyle=\frac{\left(t_{1}-1\right)t_{2}t_{4}}{t_{1}t_{2}t_{3}t_{4}-1}\,,$
$\displaystyle u_{3}$
$\displaystyle=-\frac{t_{1}\left(t_{2}-1\right)t_{3}}{t_{1}t_{2}t_{3}t_{4}-1}\,,$
$\displaystyle u_{4}$
$\displaystyle=\frac{t_{2}\left(t_{1}t_{3}t_{4}-1\right)}{t_{1}t_{2}t_{3}t_{4}-1}\,.$

While not immediately obvious the integrals converge and are over the compact
domain in $u$’s. We have also checked directly up to the first few orders that
$G_{1}=\mathcal{V}_{1}(\omega,\omega,C,C)$ satisfies the $L_{\infty}$-algebra
relation

$\displaystyle-\omega\star\mathcal{V}_{1}(\omega,\omega,C,C)+\mathcal{V}_{1}(\omega\star\omega,\omega,C,C)-\mathcal{V}_{1}(\omega,\omega\star\omega,C,C)+\mathcal{V}_{1}(\omega,\omega,\mathcal{U}_{1}(\omega,C),C)$
$\displaystyle\qquad+\mathcal{V}_{1}(\omega,\omega,\mathcal{U}_{1}(\omega,C,C))-\mathcal{V}_{1}(\omega,\mathcal{V}_{1}(\omega,\omega,C),C)=0\,.$

The other $\mathcal{U}$-vertices are obtained exactly as in [33] via the
duality map.

##  Appendix C NNNLO vertex and beyond

It is very easy to go to the next level and evaluate a single tree that
contributes to the quintic vertex $\mathcal{V}(\omega,\omega,C,C,C)$. The tree
grows as

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The answer is a six-fold integral over the ‘times’ $t_{i}$, $i=1,...,6$:

$\displaystyle G$
$\displaystyle=(p_{12})^{3}\exp\Big{[}(1-u_{1}-u_{2}-u_{3})p_{01}+(1-u_{4}-u_{5}-u_{6})p_{02}+u_{1}p_{13}+u_{2}p_{14}+$
$\displaystyle\qquad\qquad\qquad+u_{3}p_{15}+u_{4}p_{23}+u_{5}p_{24}+u_{6}p_{25}+$
$\displaystyle\qquad\qquad\qquad+{\lambda}\left(-\left(\left(u_{3}+1\right)\left(u_{4}+u_{5}-1\right)\right)-u_{6}+u_{2}\left(1-u_{4}+u_{6}\right)+u_{1}\left(u_{5}+u_{6}+1\right)\right)p_{12}\Big{]}\,,$

where the integration variables are expressed as

$\displaystyle u_{1}$
$\displaystyle=\frac{t_{1}\left(t_{2}-1\right)\left(t_{3}-1\right)t_{4}t_{5}t_{6}}{-t_{2}t_{3}t_{5}+t_{1}t_{4}\left(\left(2t_{2}-1\right)t_{3}t_{5}-t_{2}\right)+1}\,,$
$\displaystyle u_{2}$
$\displaystyle=\frac{t_{2}\left(t_{3}-1\right)\left(t_{1}t_{4}-1\right)t_{5}t_{6}}{-t_{2}t_{3}t_{5}+t_{1}t_{4}\left(\left(2t_{2}-1\right)t_{3}t_{5}-t_{2}\right)+1}\,,$
$\displaystyle u_{3}$
$\displaystyle=\frac{t_{3}\left(-t_{2}t_{5}+t_{1}t_{4}\left(t_{2}\left(2t_{5}-1\right)-t_{5}\right)+1\right)t_{6}}{-t_{2}t_{3}t_{5}+t_{1}t_{4}\left(\left(2t_{2}-1\right)t_{3}t_{5}-t_{2}\right)+1}\,,$
$\displaystyle u_{4}$
$\displaystyle=\frac{t_{1}\left(-t_{2}t_{4}+t_{3}\left(t_{2}\left(2t_{4}-1\right)-t_{4}\right)t_{5}+1\right)}{-t_{2}t_{3}t_{5}+t_{1}t_{4}\left(\left(2t_{2}-1\right)t_{3}t_{5}-t_{2}\right)+1}\,,$
$\displaystyle u_{5}$
$\displaystyle=\frac{\left(t_{1}-1\right)t_{2}\left(t_{3}t_{5}-1\right)}{-t_{2}t_{3}t_{5}+t_{1}t_{4}\left(\left(2t_{2}-1\right)t_{3}t_{5}-t_{2}\right)+1}\,,$
$\displaystyle u_{6}$
$\displaystyle=\frac{\left(t_{1}-1\right)\left(t_{2}-1\right)t_{3}}{-t_{2}t_{3}t_{5}+t_{1}t_{4}\left(\left(2t_{2}-1\right)t_{3}t_{5}-t_{2}\right)+1}\,.$

It is again pleasing how local the vertex is: there are no $p_{34}$, $p_{45}$
and $p_{35}$. The corresponding $\mathcal{U}$-vertex is obtained via the
duality map:

$\displaystyle\mathcal{U}(\omega,C,C,C,C)$ $\displaystyle:$
$\displaystyle\mathcal{U}_{1}(p_{0},p_{1},p_{2},p_{3},p_{4},p_{5})=\mathcal{V}_{1}(-p_{5},p_{0},p_{1},p_{2},p_{3},p_{4})\,.$

It is easy to iterate the tree and find the all order result for
$\mathcal{V}(\omega,\omega,C,...,C)$:

$\displaystyle G$
$\displaystyle=(p_{12})^{n}\exp\Big{[}(1-\sum_{i}u_{i})p_{01}+(1-\sum_{i}v_{i})p_{02}+\sum_{i}u_{i}p_{1,i+2}+\sum_{i}v_{i}p_{2,i+2}+$
$\displaystyle\qquad\qquad\qquad\qquad+{\lambda}\,\Big{(}1+\sum_{i}(u_{i}-v_{i})+\sum_{i,j}u_{i}v_{j}\operatorname{sign}(j-i)\Big{)}p_{12}\Big{]}\,.$

Here all sums are from $1$ to $n$ and $n$ is the number of $C$-fields. By
definition, we set $\operatorname{sign}(0)=0$. This also gives
$\mathcal{U}(\omega,C,C,...,C)$ via the duality map. Other trees lead to
similar expressions.

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