[a]Evgeny Skvortsov

# Integrable Models From Non-Commutative Geometry With Applications to 3D
Dualities

Alexey Sharapov

###### Abstract

We discuss a new class of strong homotopy algebras constructed via inner
deformations. Such deformations have a number of remarkable properties. In the
simplest case, every one-parameter family of associative algebras leads to an
$L_{\infty}$-algebra that can be used to construct a classical integrable
model. Another application of this class of $L_{\infty}$-algebras is related
with the three-dimensional bosonization duality in Chern–Simons vector models,
where it implements the idea of the slightly-broken higher spin symmetry. One
large class of associative algebras originates from Deformation Quantization
of Poisson Manifolds. Applications to the $3d$-bosonization duality require,
however, an extension to deformation quantization of Poisson Orbifolds, which
is an open problem. The $3d$-bosonization duality can be proven by showing
that there is a unique class of invariants of the $L_{\infty}$-algebra that
can serve as correlation functions.

##  1 Introduction

We describe two closely related applications of a new rich class of strong
homotopy algebras: (i) integrable models [1] and (ii) three-dimensional
bosonization duality [2, 3]. A cornerstone of these applications is a new
approach to constructing strong homotopy algebras [2, 4] via intrinsic
deformations. On the physics side, the $3d$-bosonization duality conjecture
[5, 6, 7, 8, 9, 10] takes place in (Chern–Simons) vector models that describe
various second-order phase transitions in $3d$, i.e., in the ‘real physical
world’. The class of $L_{\infty}$-algebras, to be described below, allows one
to give rigorous mathematical grounds to the idea of the slightly-broken
higher spin symmetry [4, 11]. Correlation functions are then invariants of
this symmetry. At least in the large-$N$ limit the bosonization duality can be
reduced to the proof of uniqueness of these invariants.

Strong homotopy algebras, $A_{\infty}$, $L_{\infty}$, $G_{\infty}$, …, help to
implement the idea of the (most general) consistent algebraic structure. They
play a central role in string field theory and provide another point of view
on the BV–BRST quantization method, see e.g. [12, 13, 14, 15, 16, 17, 18, 19].
However, strong homotopy algebras did not seem to have appeared as symmetries
of physical systems in the past. In [2] it was proposed to implement the idea
of the slightly-broken higher spin symmetry [6] as a certain
$L_{\infty}$-algebra. The usual symmetries are realized by Lie algebras whose
generators act on one-particle states and the action on multiparticle states
is the sum of the actions on each of the one-particle states. There are
examples of symmetries that go beyond the standard Lie one. For instance,
Yangian is the algebra of conserved charges of an integrable model, whose
action on multiparticle states is defined in accordance with a nontrivial co-
product and differs from the canonical one.

Slightly-broken higher spin symmetry extends the notion of symmetries in a
different way. First of all, higher spin symmetry is an infinite-dimensional
symmetry present in free theories (or in the $N=\infty$ limit of vector
models). It is manifested by infinitely many conserved tensors
$J_{s}=j_{a_{1}\cdots a_{s}}$ that include the stress-tensor (and global
symmetry currents, if present). Canonically, the conserved tensors lead to
conserved currents parameterized by Killing tensors and then to conserved
charges $Q_{s}$. The conserved charges $Q_{s}$ form an infinite-dimensional
Lie algebra ${\mathfrak{hs}}$, which includes and extends the usual space-time
symmetries. For example, the higher spin currents $J_{s}$ form a single
multiplet of ${\mathfrak{hs}}$, $[Q,J]=J$. It is significant that the Lie
algebra ${\mathfrak{hs}}$ originates from an associative algebra,111Symmetries
of free/linear equations can be multiplied and, hence, form an associative
algebra. The commutator of two is, of course, a symmetry, but the associative
structure is more rigid and powerful. which we denote by the same symbol
${\mathfrak{hs}}$. The associative algebra ${\mathfrak{hs}}$ carries all
information about the free theory, so that one can write a symbolic equality

$\displaystyle\textbf{Free CFT (QFT) $=$ Associative algebra}\,.$

It is worth noting that the algebra ${\mathfrak{hs}}$ is much smaller than the
full operator algebra of a given theory, otherwise the equality above would be
rather a trivial statement. Since each higher spin algebra enjoys a canonical
trace, ${\mathrm{Tr}}:{\mathfrak{hs}}\rightarrow\mathbb{C}$, the correlation
functions of the corresponding free theory can be understood as simple trace-
type invariants. This can be summarized by the following sequence:

Conserved tensor $\displaystyle\rightarrow$ current $\displaystyle\rightarrow$
symmetry $\displaystyle\rightarrow$
$\displaystyle\text{invariants$=$correlators}\,.$

When we turn on interactions or depart from the $N=\infty$ limit, the
conservation of higher spin currents is violated ($\partial\cdot J_{s}\neq 0$)
the stress-tensor (and global symmetry currents, if any) remain conserved
though. Now, everything depends on the spectrum of primary operators. Vector
models have a relatively sparse spectrum of primary operators, as different
from models with matter in matrix representations of gauge groups. Therefore,
there are few operators that can appear in the right-hand side of the equality
$\partial\cdot J_{s}=...$. In the large-$N$ limit, it is easy to see [5, 6]
that one can have composite operators of type $[JJ]$. Therefore, the
conservation law gets broken by the higher spin currents themselves:

$\displaystyle\partial\cdot J$ $\displaystyle=\tfrac{1}{N}[JJ]\,.$ (1)

Had we found completely different primary operators on the right-hand side,
almost no information could have been extracted. Now, it is possible to close
the loop!

The apparent simplicity of equation (1) is deceptive. Since the currents are
not conserved, the charges are no longer conserved as well. As a consequence,
the charges do not have to form a Lie algebra anymore, while the higher spin
currents may not define a representation of $Q$, i.e.,

$[Q,Q]=Q+\ldots\quad\mbox{and}\quad[Q,J]=J+\ldots\,,$

where the dots stand for $1/N$ corrections. One can also show that the algebra
${\mathfrak{hs}}$ is rigid and the currents $J$, as an
${\mathfrak{hs}}$-module, cannot be deformed as well. Hence, it is impossible
to account for the dots on the right by any usual deformation of a Lie algebra
or/and its representation. The main proposal made in [2] is that we should
deform the whole structure “Lie algebra plus its module” into a strong
homotopy Lie algebra. Now, the lowest structure maps $l_{2}(\bullet,\bullet)$
simply encode ${\mathfrak{hs}}$ and $J$ as its module. Higher structure maps
$l_{n}(\bullet,\ldots,\bullet)$ allow one to deform the Lie algebra
${\mathfrak{hs}}$ together with its module $J$, so that they form a single
algebraic structure.

To summarize, the new type of symmetry we propose deforms a Lie algebra and
module(s) of this algebra into a single $L_{\infty}$-algebra. Such a structure
requires the $L_{\infty}$ to be made of two components as a graded vector
space: one for the initial Lie algebra and another for its module(s).
Therefore, this can also be understood as a Lie algebroid, but we will not
exploit this interpretation. Now, the problem of correlation functions is
equivalent to the problem of invariants of this $L_{\infty}$-algebra.

In applications to the $3d$-bosonization duality conjecture, ${\mathfrak{hs}}$
is the Weyl algebra, i.e., the universal enveloping of the Lie algebra of
canonical commutation relations $[q^{i},p_{j}]=i\hbar\delta^{i}_{j}$. It
results from the deformation quantization of the simplest Poisson Manifold.
The $L_{\infty}$ algebra is completely determined by a closely related
algebra, which can be understood as a deformation quantization of the simplest
Poisson Orbifold $\mathbb{R}^{2}/\mathbb{Z}_{2}$. The last fact implies a deep
relation between the two problems. For the simplest case of $O(N)$ vector
models, one can see [20] that the $L_{\infty}$-algebra depends on two
phenomenological parameters, which can be related with $1/N$ and the
Chern–Simons level $k$. It is also possible to show [20] that the trace
invariants of ${\mathfrak{hs}}$ are unique invariants that deform to the
invariants of the $L_{\infty}$-algebra. The $3d$-bosonization duality is a
simple consequence of the uniqueness: since the correlation functions are
completely specified by the symmetry, it does not matter if the underlying
microscopical realization involves bosonic or fermionic degrees of freedom.

It should be remembered that (1) is an exact quantum equation of motion, which
is true even for $N=1$ (the case of the Ising model). It is unclear at the
moment if the semi-classical arguments based on $L_{\infty}$ can be extended
to the quantum domain. What makes them semi-classical is a simple form of
relation between $J$ and $[JJ]$ suggested by the representation theory. These
operators, however, renormalize in a nontrivial way. Therefore, it would be
important to understand if ‘quantization’ of the $L_{\infty}$-symmetry is
possible to be able to extend the proof beyond the large-$N$ limit.

Let us give an overview of other results that rely on the same construction
[2, 4] of $L_{\infty}$-algebras. In its simplest realization the idea of
intrinsic deformations allows us to construct an $A_{\infty}$-algebra from any
associative algebra $A_{u}$ that depends on a parameter $u$. Canonically,
every $A_{\infty}$-algebra generates an $L_{\infty}$-algebra via the anti-
symmetrization map. On the other hand, every $L_{\infty}$-algebra can be
identified with a homological vector field $Q$ on a formal graded manifold
$\mathcal{N}$ with coordinates $\Phi^{\mathcal{A}}$. Given such a homological
vector field $Q$, we can construct a sigma-model with the following equations
of motion:

$\displaystyle d\Phi$ $\displaystyle=Q(\Phi)\equiv
l_{2}(\Phi,\Phi)+l_{3}(\Phi,\Phi,\Phi)+\ldots\,.$ (2)

Here the fields $\Phi^{{\mathcal{A}}}$ define a map from the odd tangent
bundle $T^{\ast}[1]\mathcal{M}$ of a space-time manifold $\mathcal{M}$ to the
target space $\mathcal{N}$. This construction can, in principle, be applied to
any field theory, as one can always formulate classical equations of motion in
the form (2) and there is a canonical procedure of assigning a homological
vector field $Q$ to any field theory [17, 21].

The class of strong homotopy algebras obtained via intrinsic deformations has
a number of special properties. First and foremost, all the multilinear maps
$l_{n}$ entering the r.h.s. of (2) can be constructed from the bilinear maps

$\displaystyle a\circ b=a\,\star\,b+\sum_{k=1}\phi_{k}(a,b)\,u^{k}$

that determine expansion of the associative product $\circ$ in $A_{u}$ around
${u=0}$. Second, the resulting classical field theory/mechanics appears to be
integrable, e.g., one can construct all its solutions with the help of a Lax
pair. To summarize, each one-parameter family of associative algebras gives
rise to a classical integrable model:

$\displaystyle\textbf{{Soft${}_{u}$ Associative Algebras $\longrightarrow$
Integrable models}}\,.$

As an interlude we also discuss Deformation Quantization of Poisson Orbifolds
as a rich source of one-parameter families of algebras; in particular, we
discuss $\mathbb{R}^{2}/\mathbb{Z}_{2}$, which is relevant to the
$3d$-bosonization duality. At present, this is largely an open problem since
no analog of Kontsevich’s formality is known for the case of Orbifolds.

##  2 Integrable models from associative algebras

A Lax pair is a cornerstone of integrable models. As the name suggests, it is
a pair of two matrices, usually called $L$ and $A$, that depend on the ‘time’
$t$ and, possibly, on other parameters. The matrices are required to satisfy
the Lax equation

$\displaystyle\partial_{t}L$ $\displaystyle=[A,L]\,.$ (3)

Often, $A$ is a given function of $L$, so this is a nonlinear equation for $L$
as a function of $t$. As a result, one gets immediately an infinite family of
integrals of motion $I_{n}={\mathrm{Tr}}[L^{n}]$, $n=1,2,\ldots$, not all of
which may be independent. Clearly, $A$ plays the role of connection along $t$,
and $L$ is required to be covariantly constant. This suggests a natural
generalization of the Lax pair to $d$ dimensions. Extension to $d$ dimensions
is not unique as one can have more structures. Given a $d$-dimensional
manifold $\mathcal{M}_{d}$, introduce a connection $A$ on some vector bundle
over $\mathcal{M}_{d}$ where a zero-form $L$ takes its values and set

$\displaystyle dA$ $\displaystyle=\tfrac{1}{2}[A,A]\,,$ $\displaystyle dL$
$\displaystyle=\rho(A)L\,.$ (4)

Here the covariant derivative is (implicitly) $D_{A}=d-\rho(A)$, where $\rho$
denotes the action of $A$ on $L$. Optionally, one can introduce forms of
various degrees that are constrained to be covariantly constant with respect
to $A$. Assume further that both $A$ and $L$ take values in the algebra
$A_{0}=\mathrm{End}(V)$ of endomorphism of a vector space $V$. Such $A_{0}$,
depending on a situation, can be viewed as an associative algebra or as a Lie
algebra with the Lie bracket defined by the commutator. Setting
$\rho(A)L=[A,L]$ and using the trace on $A_{0}$, we obtain an infinite
sequence of integrals of motion $I_{n}={\mathrm{Tr}}[L^{n}]$. It may also be
convenient sometimes to solve the generalized Lax’s equations (4) in a ‘pure
gauge’ form:

$\displaystyle A$ $\displaystyle=g^{-1}dg\,,$ $\displaystyle L$
$\displaystyle=g^{-1}L_{0}g\,,$ (5)

where $L_{0}$ is constant, $dL_{0}=0$. The zero-form $L_{0}$ parameterizes the
solution space locally, but not necessarily globally. The simplest Lax pair
(4) will provide integrability of the class of models to be introduced
shortly.

### Interactions.

A model that can explicitly be reduced to (4) can be thought of as non-
interacting one, which is justified by local solutions (5). We would like to
study the most general deformation of (4). Based on the simple form-degree
counting, a generic deformation is of the form

$\begin{array}[]{rcl}dA&=&l_{2}(A,A)+l_{3}(A,A,L)+l_{4}(A,A,L,L)+\ldots=F_{A}(A;L)\,,\\\\[8.53581pt]
dL&=&l_{2}(A,L)+l_{3}(A,L,L)+\ldots=F_{L}(A;L)\,,\end{array}$ (6)

where $l_{n}$ are various interaction vertices, or structure maps, with all
arguments indicated explicitly. The formal sum thereof gives two structure
functions $F_{A}$ and $F_{L}$. The bilinear maps are known from (4) to define
a Lie algebra (where $A$ takes the values) and its module (where $L$ takes the
values). In our case, the algebra is $A_{0}=\mathrm{End}(V)$ for some vector
space $V$ and the module is the adjoint one.

Clearly, (6) is a particular case of a more general set up where one has a
space-time manifold $\mathcal{M}_{d}$ and a formal graded manifold
$\mathcal{N}$ equipped with a homological vector field $Q$. If
$\Phi=\\{\Phi^{\mathcal{A}}\\}$ are coordinates on $\mathcal{N}$, then
$Q=Q^{\mathcal{A}}\partial/\partial\Phi^{\mathcal{A}}$ and the defining
condition for an odd vector field $Q$ to be homological reads222The derivative
with respect to $\Phi^{\mathcal{A}}$ is understood in the appropriate graded
sense.

$\displaystyle Q^{2}$ $\displaystyle=0$ $\displaystyle\Longleftrightarrow$
$\displaystyle
Q^{\mathcal{B}}\frac{\partial}{\partial\Phi^{\mathcal{B}}}Q^{\mathcal{A}}=0\,.$
(7)

If $\mathcal{N}$ is non-negatively graded, one can write down a sigma-model by
promoting $\Phi^{\mathcal{A}}$ to smooth maps
$\Phi^{\mathcal{A}}:T^{\ast}[1]\mathcal{M}_{d}\rightarrow\mathcal{N}$ of
degree zero. Geometrically, on can think of these maps as differential forms
$\Phi^{\mathcal{A}}(x,dx)$ on $\mathcal{M}_{d}$ with values in $\mathcal{N}$.
The sigma-model333As a historical note, systems of the form (2) were first
introduced by Sullivan [22] as Free Differential Algebras, re-introduced into
physics [23, 24] in the supergravity and higher spin [25] contexts.

$\displaystyle d\Phi$ $\displaystyle=Q(\Phi)$ (8)

is defined by requiring the map to respect the differentials, which is exactly
(8). The most common situation is when $\Phi$ are coordinates at a vicinity of
a stationary point, i.e., $Q(0)=0$. The Taylor expansion of $Q$ defines then
multilinear graded-symmetric maps $l_{n}(\bullet,\ldots,\bullet)$ that satisfy
Stasheff’s relations for an $L_{\infty}$-algebra. Eq. (8), regarded as a set
of PDE’s for the form fields $\Phi^{\mathcal{A}}(x,dx)$, has a very important
property, which can be phrased in several equivalent ways: (i) there are no
any hidden algebraic constraints on the fields buried in (8); (ii) Eq. (8) is
consistent with $dd\equiv 0$, which is the Frobenius integrability condition.
Sadly, this is not a type of integrability that helps to solve the model.

From the modern point of view, every gauge PDE defines and is defined by a
$Q$-manifold $\mathcal{N}$, see e.g. [26]. In a few words, given a canonical
BV-BRST formulation of a gauge theory, one can consider its jet space
extension [17, 21, 26], which is an $L_{\infty}$-algebra. This algebra may
have positive and negative degrees corresponding to ghosts (ghosts for ghosts
etc.) as well as to anti-fields. A useful ‘derivative’ of any
$L_{\infty}$-algebra is its minimal model, which, being much smaller, contains
all the essential information (it is said to be quasi-isomorphic to the
initial $L_{\infty}$-algebra). For a large class of models, e.g. field
theories, such minimal model is a non-negatively graded $L_{\infty}$-algebra.
It is the corresponding $Q$, $Q^{2}=0$, that can be thought of as an invariant
definition of the field theory we started with [26]. At the same time $Q$ is
in the possession of all local relevant information about the field theory,
e.g. (8) defines solutions of the classical field equations.

As a result, we have system (6) that is a particular case of (8). Here,
$\Phi=\\{A,L\\}$ and $Q=F^{A}\partial/\partial A+F^{L}\partial/\partial L$. It
is a smooth deformation of a trivial one, which is the Lax pair. The
deformation is required to be nontrivial: (i) in the field theory language,
there is no field redefinition that brings (6) into (4); (ii) in the
$L_{\infty}$-language, $l_{n}(\bullet,\ldots,\bullet)$, $n>2$ cannot be
eliminated by a natural transformation of $L_{\infty}$-algebras; (iii) in the
$Q$-manifold language, there is no coordinates in which the homological vector
field assumes the quadratic form, $Q=\Phi\Phi\partial/\partial\Phi$.

System (6) is yet to be constructed! We arrive at a number of questions
regarding (6): (a) how to construct vertices $l_{n}$ given the initial data
encoded in $l_{2}$? (b) how to solve it? Under certain assumptions [27, 2, 1]
it can be shown that the $L_{\infty}$ originates from a certain $A_{\infty}$
and the latter is built from a deformation of $A_{0}$ as an associative
algebra. Solutions can also be constructed explicitly.

We recall that $l_{2}$ originates from an associative product on $A_{0}$. Let
us first recast these initial data into $L_{\infty}$-language. In accordance
with the form degrees, our $\mathcal{L}$ consists, as a vector space, of two
copies of $A_{0}$ that are assigned degrees $0$ and $1$, i.e.
$\mathcal{L}=\mathcal{L}_{0}\oplus\mathcal{L}_{1}$, $\mathcal{L}_{0,1}\sim
A_{0}$. The bilinear structure maps are defined as

$\displaystyle l_{2}(a,b)$ $\displaystyle=[a,b]_{\star}\,,$ $\displaystyle
l_{2}(a,v)$ $\displaystyle=a\star v-v\star a\,,$ $\displaystyle
a,b\in\mathcal{L}_{1}\,,\quad v\in\mathcal{L}_{0}\,.$ (9)

In fact, the associative structure is of immense help. The underlying
structure will turn out to be $A_{\infty}$ rather than $L_{\infty}$.444For
completeness, $A_{\infty}$ is a graded vector space, equipped with degree
$(-1)$ maps $m_{n}(\bullet,\ldots,\bullet)$ that satisfy $m\circ m=0$, where
$m=m_{1}+m_{2}+...$ is a formal sum and $\circ$ is the Gerstenhaber product.
The latter is defined for any two multilinear maps $f$ and $g$ of degrees
$|f|$ and $|g|$ and having $k_{f}$ and $k_{g}$ arguments as  $\displaystyle
f\circ g$
$\displaystyle=\sum_{i}(-1)^{\kappa}f(a_{1},\ldots,a_{i},g(a_{i+1},\ldots,a_{i+k_{g}}),a_{i+k_{g}+1},\ldots,a_{k_{f}+k_{g}-1})\,.$
(10) Here $\kappa$ is the usual Koszul sign:
$\kappa=|g|(|a_{1}|+\cdots+|a_{i}|)$. Similar relations define
$L_{\infty}$-algebras. However, it is more handy to get them from the Taylor
expansion of $Q^{2}=0$.  Anticipating this fact, the same initial data can be
encoded by a small $A_{\infty}$-algebra:

$\displaystyle m_{2}(a,b)$ $\displaystyle=a\star b\,,$ $\displaystyle
m_{2}(a,v)$ $\displaystyle=a\star\nu\,,$ $\displaystyle m_{2}(v,a)$
$\displaystyle=-v\star a\,,$ $\displaystyle a,b\in\mathcal{A}_{1}\,,\quad
v\in\mathcal{A}_{0}\,.$ (11)

Our $L_{\infty}$-algebra will come from an $A_{\infty}$-algebra via the usual
anti-symmetrization map

$\displaystyle l_{n}(x_{1},\ldots,x_{n})=\sum_{\sigma\in
S_{n}}(-1)^{|\sigma_{x}|}m_{n}(x_{\sigma(1)},\ldots,x_{\sigma(n)})\,.$ (12)

We would like to deform (11) as to activate higher structure maps $m_{n}$,
$n>2$, in a nontrivial way.

### Main result.

With all definitions above we can now formulate the main result [2, 1]. We
need to deform the $A_{\infty}$-algebra (11) that consists of an associative
algebra ${\mathrm{A}_{0}}$ together with its natural bi-module. Under some
technical assumptions555The main assumption is that we are looking for a
deformation that makes sense when $A_{0}$ is replaced by
$\mathrm{Mat}_{N}(A_{0})$ for any $N$. Obviously, (11) can be defined for
$\mathrm{Mat}_{N}(A_{0})$. It is this assumption that allows us to reduce a
complicated problem of Chevalley–Eilenberg cohomology to a much simpler
Hochschild cohomology. Nevertheless, in some examples of practical importance
the $N=1$ case seems to be as general as any $N>1$., the deformation exists
iff the underlying associative algebra is soft, i.e. can be deformed into a
(nontrivial, of course) one-parameter family of associative algebras
${\mathrm{A}_{u}}$. Therefore, the product in ${\mathrm{A}_{u}}$ deforms the
one in ${\mathrm{A}_{0}}$:

$\displaystyle a\circ
b=a\,\star\,b+\sum_{k=1}^{\infty}\phi_{k}(a,b)\,u^{k}\,.$ (13)

That the deformed product is associative, $a\circ(b\circ c)=(a\circ b)\circ
c$, imposes certain conditions on the bilinear maps $\phi_{k}$. The
nontriviality of the deformations implies that $\phi_{1}$ is a nontrivial
class of the second Hochschild cohomology group
$HH^{2}({\mathrm{A}_{0}},{\mathrm{A}_{0}})$.

The algorithm for constructing the model is as follows. First, one constructs
a deformation of $\mathcal{A}_{0}$ as of an $A_{\infty}$-algebra. The first
few structure maps read

$\displaystyle m_{3}(a,b,v)$
$\displaystyle=\phi_{1}(a,b)\,\star\,v\quad\rightarrow\quad l_{3}\,,$
$\displaystyle m_{4}(a,b,v,w)$
$\displaystyle=\phi_{2}(a,b)\,\star\,v\,\star\,w+\phi_{1}(\phi_{1}(a,b),v)\,\star\,w\quad\rightarrow\quad
l_{4}\,.$

There is a number of ways [2] to get all $m_{n}$, including an explicit
formula for any $n$. Secondly, one induces the corresponding $L_{\infty}$-maps
by means of anti-symmetrization (12). As a result we have all $l_{n}$ or $Q$
and it can be shown that model (6) so defined is not equivalent to a free one
via field-redefinitions.

Nevertheless, we can show that the model is integrable and its solution space
can be understood via an auxiliary Lax pair [1]. Not surprisingly the Lax
pair, which has the same form as (4), is based on the deformed product
$\circ$, namely,

$\displaystyle d{\boldsymbol{A}}$
$\displaystyle={\boldsymbol{A}}\circ{\boldsymbol{A}}\,,$ $\displaystyle
d{\boldsymbol{L}}$
$\displaystyle={\boldsymbol{A}}\circ{\boldsymbol{L}}-{\boldsymbol{L}}\circ{\boldsymbol{A}}\,,$
(14)

where ${\boldsymbol{A}}$, ${\boldsymbol{L}}$ take values in the associative
algebra ${\mathrm{A}_{u}}$. The integrability means here that solutions of (6)
can be built from those of (14), the latter having pure gauge form

$\displaystyle{\boldsymbol{A}}$ $\displaystyle={\boldsymbol{g}}^{-1}\circ
d{\boldsymbol{g}}\,,$ $\displaystyle{\boldsymbol{L}}$
$\displaystyle={\boldsymbol{g}}^{-1}\circ{\boldsymbol{L}}_{0}\circ{\boldsymbol{g}}\,.$
(15)

The map is constructed order by order in $u$ as666Similar maps relating
interacting and quasi-free or non-commutative and commutative theories are
known in the literature [28, 29, 30].

$\displaystyle A$
$\displaystyle={\boldsymbol{A}}+(\partial_{u}{\boldsymbol{A}})\circ{\boldsymbol{L}}+\ldots\Big{|}_{u=0}\,,$
$\displaystyle L$
$\displaystyle={\boldsymbol{L}}+(\partial_{u}{\boldsymbol{L}})\circ{\boldsymbol{L}}+\ldots\Big{|}_{u=0}\,.$
(16)

Provided the associative algebra ${\mathrm{A}_{u}}$ admits a trace
${\mathrm{Tr}_{u}}$, that is, ${\mathrm{Tr}_{u}}[a\circ b-b\circ a]=0$, we can
also define the integrals of motion
$I_{n}={\mathrm{Tr}_{u}}[{\boldsymbol{L}}\circ\cdots\circ{\boldsymbol{L}}]$
obeying the conservation law $dI_{n}=0$.777Note that this invariants differ
from what is usually called integrals of motion (function(al)s of fields that
are time independent). In modern terms, integrals of motion correspond to
zero-form symmetries and are given by closed $(d-1)$-forms. Due to $dI_{n}=0$
functions $I_{n}$ do not depend on space-time points at all and correspond to
$(d-1)$-form symmetries, [31]. For infinite-dimensional algebras there can be
other types of integrals of motion [20].

To summarize, given a one-parameter family of associative algebras
${\mathrm{A}_{u}}$, one can construct a non-linear system of formally
consistent888A word of warning is that formal consistency does not imply
actual consistency. It is easy to construct examples of formally consistent
equations that do not make any sense, i.e. they are just symbolic expressions,
but lead to infinite tree-level amplitudes. See e.g. [32, 33] for explicit
examples.  PDE’s that, nevertheless, can be solved via an auxiliary Lax pair.

##  3 Deformation quantization of Poisson Orbifolds.

A natural question is where to find either soft associative algebras (to be
deformed) or one-parameter families of associative algebras right away. There
are two large stocks of such algebras: (1) finite dimensional associative
algebras; (2) deformation quantization.

All finite-dimensional semi-simple associative algebras, by the
Artin–Wedderburn theorem, are products of matrix algebras. As such they cannot
have free parameters. Therefore, soft algebras have to be outside the nice
class of semi-simple algebras. The classification of the latter is known to be
wild. Good news: there are a lot of soft algebras along these lines. Bad news:
it is hard to find them. Below we discuss the item (2) in more detail as it is
of some interest due to its relation to the $3d$ bosonization duality.

Deformation quantization is an obvious source of one-parameter associative
algebras. Let us recall the main idea. Given any Poisson manifold
$\mathcal{P}$ one considers the algebra of functions
$C^{\infty}(\mathcal{P})$. It is an associative and commutative algebra. One
then tries to deform the product

$\displaystyle f\star g$ $\displaystyle=f\cdot
g+\hbar\\{f,g\\}+\mathcal{O}(\hbar^{2})\,,$ (17)

where the first order deformation is the Poisson bracket on $\mathcal{P}$,
which, at same time, is a nontrivial Hochschild two-cocycle of
$C^{\infty}(\mathcal{P})$. The general solution of this problem was given by
Kontsevich in [34]. Later [35], it was shown that Kontsevich’s graphs can be
understood as Feynman’s diagrams of an auxiliary topological string theory,
called Poisson sigma-model. Deformation quantization gives a large class of
one-parameter families of associative algebras, parameterized by Poisson
manifolds.

The space of algebras can be extended even further, leading to an interesting
open problem in deformation quantization. Poisson manifolds generally enjoy a
lot of symmetries. For a given Poisson manifold $\mathcal{P}$ one can pick up
some group of discrete symmetries $\Gamma$. There are two associative algebras
naturally linked to these data:

  1. 1.

$\Gamma$-invariant functions $C^{\infty}(\mathcal{P})^{\Gamma}\sim
C^{\infty}(\mathcal{P}/\Gamma)$, where we see the orbifold
$\mathcal{P}/\Gamma$. This algebra is commutative;

  2. 2.

Cross-product algebra $C^{\infty}(\mathcal{P})\ltimes\Gamma$. The algebra is
generated by sums $\sum_{i}f_{i}\otimes\gamma_{i}$, where the $f_{i}\in
C^{\infty}(\mathcal{P})$ and $\gamma_{i}\in\Gamma$. The product is defined as
$(f\otimes\gamma)\diamond(f^{\prime}\otimes\gamma^{\prime})=(f\gamma(f^{\prime}),\gamma\gamma^{\prime})$,
where $\gamma(f)$ denotes the action of $\gamma\in\Gamma$ on a function $f\in
C^{\infty}(\mathcal{P})$. Note that this algebra is generally non-commutative
even for abelian $\Gamma$.

The two algebras above are closely related to each other. From the viewpoint
of non-commutative geometry the second algebra is more favorable. In some
sense, it encodes complicated information about singular manifolds like
orbifolds through the non-commutativity of the algebra.

In both cases one can ask: (i) what are nontrivial deformations? (ii) how to
construct them to all orders? We will refer to both of the problems, i.e. for
algebras (1) and (2) above, as to the problem of Deformation Quantization of
Poisson Orbifolds. More generally: is there any counterpart of Kontsevich’s
Formality Theorem for Poisson Orbifolds?

The first surprising result is that Poisson Orbifolds have more deformations!
There are other directions of deformation quantization that rely on
orbifolding (or on the extension by $\Gamma$). These new deformations are not
captured by Kontsevich’s Formality at all. It is also not clear what kind of
Formality one should be looking for [36, 37]. In principle, following
Kontsevich (who mentioned in [34] that his construction originates from a
topological string theory, which has been then manifested by the quantization
of the Poisson sigma-model in [35]) one should quantize the same Poisson
sigma-model on Poisson Orbifolds, which may crucially depend on the type of an
orbifold, preventing one from seeking out any universal extension of the
Formality Theorem.

While the general problem of Deformation Quantization of Poisson Orbifolds is
open, there is a number of cases, where a constructive approach is possible.
For example, let us consider the Weyl algebra $A_{n}$ extended by a finite
group of symplectomorphisms $\Gamma\subset\mathrm{Sp}_{n}$, i.e.
$A_{n}\ltimes\Gamma$. The Weyl algebra can be understood as a result of the
deformation quantization of $\mathbb{R}^{2n}$ endowed with the canonical
symplectic form $dp_{i}\wedge dq^{i}$. Now, on top of that we can have several
new deformations. The number of such deformations depends on $\Gamma$. The
deformations can be constructed explicitly to all orders via injective
resolutions [38, 39].

Let us consider in more detail the case of the algebra
$A_{1}\ltimes\mathbb{Z}_{2}$, as it is relevant for the $3d$ bosonization
duality. Here $A_{1}$ can be understood as an associative algebra generated by
the pair of canonical variables subject to the commutation relation

$\displaystyle[q,p]$ $\displaystyle=i\hbar\,,$ (18)

i.e. we have already quantized $\mathbb{R}^{2}$ along $dq\wedge dp$. The group
of symplectic reflections $\Gamma=\mathbb{Z}_{2}$ is generate by a single
element $R$ that sends $(q,p)$ to $(-q,-p)$. The crossed-product algebra
$A_{1}\ltimes\mathbb{Z}_{2}$ is obtained by adding the new generator $R$ to
$q$ and $p$ such that

$\displaystyle R^{2}$ $\displaystyle=1\,,$ $\displaystyle RqR=-q\,,$
$\displaystyle RpR=-p\,.$ (19)

The general element of $A_{1}\ltimes\mathbb{Z}_{2}$ is given by a polynomial
$f(q,p,R)=f_{0}(q,p)+f_{1}(q,p)R$ and $R$ acts via the reflection whenever we
need to drag it through $q$ or $p$. We can also start from scratch: set
$\hbar=0$. The algebra $\mathcal{A}_{0,0}$ of functions $f(q,p,R)$ is still
non-commutative due to $R$. As an associative algebra $\mathcal{A}_{0,0}$
admits a two-parameter family of deformations, the usual one along the
classical Poisson bracket $\\{\bullet,\bullet\\}$ and another one, along $R$.
We can also jump directly to
$\mathcal{A}_{\hbar,0}=A_{1}\ltimes\mathbb{Z}_{2}$. The latter algebra
features the second deformation along $R$.

Independently of the ideas above, the deformation of $\mathcal{A}_{\hbar,0}$
was found by Wigner [40] and explored further in a great number of papers, see
e.g. [41, 42, 43, 44]. It is remarkable, that one can generate the deformed
algebra $\mathcal{A}_{\hbar,u}$ by deforming the canonical commutation
relations [41, 42, 44]

$\displaystyle[q,p]$ $\displaystyle=i\hbar+iuR\,.$ (20)

Based on this relation, one can work out the structure constants, see [45, 46,
47, 48, 49, 50] for different approaches to the problem. There are closely
related algebras: the $R$-invariant subalgebra, known as $gl_{\lambda}$ [51],
and the non-commutative hyperboloid [46] (or fuzzy sphere, depending on the
real form).

It is also interesting that the first order deformation of
$\mathcal{A}_{\hbar,0}$ can be derived [27] from an extension of Kontsevich’s
Formality known as the Shoikhet–Tsygan–Kontsevich formality [52, 53]. We start
from $A_{\hbar,0}$, which can be realized as the algebra of functions
$f(q,p,R)$ equipped with the Moyal–Weyl star-product, $f\star g$. Extension
with $R$ is easy and requires us to first move all $R$ to the left (or to the
right), while properly commuting them with $q,p$, and then take the star-
product. The first order deformation along $R$,

$\displaystyle f\circ g$ $\displaystyle=f\star
g+u\,\phi(f,g)R+\mathcal{O}(u^{2})\,,$ (21)

is defined by a Hochschild two-cocycle $\phi$. It is this two-cocycle that can
be computed with the help of Shoikhet-Tsygan-Kontsevich formality. None of the
known formality theorems seems to have anything to say about higher orders.

##  4 Slightly-broken higher spin symmetry and three-dimensional bosonization
duality

Another application of the $A_{\infty}$\- and $L_{\infty}$-algebras
constructed in section 2 is to the three-dimensional bosonization duality
conjecture [2, 3]. What is interesting here is that $L_{\infty}$-algebras
manifest themselves as symmetries of physical models. We begin with a brief
overview of the simplest models conjectured to exhibit the duality. Next, we
discuss why and how these models exhibit infinite-dimensional symmetries that
are realized as $L_{\infty}$-algebras. Correlation functions can then be
understood as invariants of this symmetry, while the proof of their uniqueness
leads to a proof of the duality.

### Higher spin symmetry of free theories.

It has long been known that free QFT’s (as well as the classical field
theories they are based on) feature infinite-dimensional symmetries that are
generated by infinitely many conserved tensors on top of the usual low spin
stress-tensor (spin-two) and global symmetry currents (spin-one). Such
symmetries were dubbed Zilch symmetries [54, 55]. In modern terminology they
are called higher spin currents and symmetries. The conserved tensors, e.g.
for the free scalar theory,

$\displaystyle J_{a_{1}\cdots a_{s}}$
$\displaystyle=\bar{\phi}\partial_{a_{1}}\cdots\partial_{a_{s}}\phi+\ldots$
(22)

are sandwiches of two fields with an arbitrary number of derivatives. The
omitted terms can be arranged to make it conserved as a consequence of
$\square\phi=m^{2}\phi$. In the momentum space, the existence of the
corresponding charges is obvious since any

$\displaystyle Q_{f}$ $\displaystyle=\int
d^{d-1}p\,\bar{a}_{p}f(p,\nabla_{p})a_{p}$ (23)

is a conserved charge for arbitrary kernel $f$. Locality of the conserved
tensors imposes mild restrictions on kernels $f$ [54]. Charges $Q_{f}$ map
one-particle states $V$ to themselves, i.e. they belong to $\mathrm{End}(V)$.
For QFT’s in Minkowski space $\mathrm{End}(V)$ contains the Poincaré algebra
and for massless QFT’s it usually includes the conformal algebra. As a useful
toy model we can think of the massless scalar field, which is conformally-
invariant.

A closely related classical concept is that of the (higher) symmetries of
field equations, i.e. maps $S$ that map solutions of some equation $E\phi=0$
to solutions. Such maps have to obey $ES=S^{\prime}E$ for some $S^{\prime}$
that depends on $S$. For the free massless scalar the conformal transformation
reads

$\displaystyle\delta_{v}\phi$
$\displaystyle=v^{m}\partial_{m}\phi+\frac{\Delta}{d}(\partial_{m}v^{m})\phi\,,$
(24)

where $\Delta=(d-2)/2$ is the conformal weight and $v^{m}(x)$ is a conformal
Killing vector. The stress-tensor $J_{ab}$ gives rise to a number of currents
$j_{m}=J_{mb}v^{b}$ that are parameterized by Killing vectors. The
infinitesimal transformations $\delta_{v}$ form the conformal algebra
$so(d,2)$ under commutators. Likewise, given a conserved rank-$s$ tensor, one
can construct a number of conserved currents

$\displaystyle j_{m}(v)$ $\displaystyle=J_{ma_{1}\cdots
a_{s-1}}\,v^{a_{1}\cdots a_{s-1}}\,,$
$\displaystyle\partial^{(a_{1}}v^{a_{2}\cdots a_{s})}-\text{traces}$
$\displaystyle=0\,,$ (25)

where $v^{a_{1}\cdots a_{s-1}}$ is a conformal Killing tensor.999As an
$so(d,2)$-module the space of solutions for $v^{a_{1}\cdots a_{k}}$ can be
identified [56] with an irreducible tensor $v^{A_{1}\cdots A_{k},B_{1}\cdots
B_{k}}$, $A,B,\ldots=0,\ldots,d+1$ being $so(d,2)$-indices. This tensor is
symmetric in $A$’s and $B$’s, separately. It is traceless and subject to the
Young condition $v^{(A_{1}\cdots A_{k},A_{k+1})B_{2}\cdots B_{k}}=0$. The
charges $Q_{s}(v)$ corresponding to higher spin currents $j_{m}(v)$ form an
infinite-dimensional Lie algebra, denoted by ${\mathfrak{hs}}$, that contains
$so(d,2)$ as a subalgebra.

Thanks to the linearity of the equations of motion, the algebra of symmetries
is associative, i.e. the product $S_{1}S_{2}$ of any two symmetries $S_{1,2}$
is a symmetry again. One does not have to take the commutator $[S_{1},S_{2}]$
to get a symmetry. In particular, one can multiply a number of the conformal
symmetries (24) to get a genuine higher spin symmetry. An important conclusion
is that the algebra of symmetries is an infinite-dimensional associative
algebra, still called ${\mathfrak{hs}}$ and referred to as the higher spin
algebra, that contains $so(d,2)$ as a Lie subalgebra. (One can always get a
Lie algebra out of an associative one by defining the Lie bracket as the
commutator.) Higher spin algebras ‘know’ everything about the underlying free
field theories, which we summarize as

$\mbox{\bf Free CFT = Associative Algebra}\,.$

The arguments above suggest that any higher spin algebra (for a conformally
invariant theory) can be understood as $U(so(d,2))/I$ for a suitable two-sided
ideal $I$ [56]. Different matter systems lead to different higher spin
algebras, e.g. the higher spin algebra of the free boson is not isomorphic to
that of the free fermion in $d>3$.

The $3d$ case is very special. First of all, one-particle states of the free
boson $|\phi\rangle$ and free fermion $|\psi\rangle$ can be realized,
respectively, as even and odd subspaces of the Fock space
$f(\hat{a}^{\dagger}_{i})|0\rangle$ of the $2d$ harmonic oscillator
$[\hat{a}^{i},\hat{a}^{\dagger}_{j}]=\delta^{i}_{j}$, $i,j=1,2$. This is
partly thanks to the isomorphism $so(3,2)\sim sp(4,\mathbb{R})$ and the latter
algebra has the standard oscillator realization where the generators of
$sp(4,\mathbb{R})$ can be identified with bilinears in the ladder operators.
In particular, translations $P_{m}\sigma^{m}_{ij}$ act as
$\hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}$, where $\sigma^{m}_{ij}$ are the
Pauli matrices. Secondly, the higher spin algebra of the free fermion
coincides with that of the free boson. Thirdly, they are both isomorphic to
the even subalgebra $A_{2}^{e}$ of the Weyl algebra $A_{2}$ [57, 58, 59], i.e.
to the algebra of even functions
$f(\hat{a},\hat{a}^{\dagger})=f(-\hat{a},-\hat{a}^{\dagger})$. The fact that
the higher spin algebras of the free boson and fermion are isomorphic is the
starting point for the proof of the three-dimensional bosonization duality.

Correlation functions in a CFT must be conformally-invariant. Likewise,
correlators in a CFT with a higher spin symmetry ${\mathfrak{hs}}$ should be
invariant under the full ${\mathfrak{hs}}$. Therefore, one can ask if
${\mathfrak{hs}}$ as a Lie algebra admits any invariants that can serve as
correlation functions provided one computes them on appropriate elements.
Indeed, each higher spin algebra is equipped with an invariant trace
${\mathrm{Tr}}_{\star}[\bullet]$, i.e. ${\mathrm{Tr}}_{\star}[a\star b-b\star
a]=0$, where $\star$ denotes the product in ${\mathfrak{hs}}$. The
$\star$-notation is not an accident. In the case of $3d$ the product in the
Weyl algebra can be realized as the Moyal–Weyl star-product.101010In general,
higher spin algebras can be understood as a result of deformation
quantization, see e.g. [60, 47] and references therein. It comes as no
surprise that all $n$-point correlation functions of $J$ can be computed at
once as higher spin invariants

$\displaystyle\langle J_{1}\ldots
J_{n}\rangle={\mathrm{Tr}_{\star}\\!\left[{\mathbb{J}}_{1}\star\cdots\star{\mathbb{J}}_{n}\right]}\,,$
(26)

provided ${\mathbb{J}}_{i}$ are chosen appropriately to behave as quasi-
primary operators that represent a generating function $J_{i}\equiv J(x_{i})$
of higher spin currents at point $x_{i}$. Invariance under the higher spin
and, hence, under the conformal symmetry is manifest:

$\displaystyle Q_{s}\langle J_{1}\ldots J_{n}\rangle$ $\displaystyle=0$
$\displaystyle\Longleftrightarrow$
$\displaystyle\delta_{\xi}{\mathrm{Tr}_{\star}\\!\left[{\mathbb{J}}_{1}\star\cdots\star{\mathbb{J}}_{n}\right]}=0\,,$
(27)

where $\delta_{\xi}{\mathbb{J}}=[\xi,{\mathbb{J}}]_{\star}$,
$\xi\in{\mathfrak{hs}}$ incorporates the action of the higher spin algebra,
including the conformal symmetries, of course. Transformation
$\delta_{\xi}{\mathbb{J}}$ represents the action $[Q,J]=J$ of higher spin
charges $Q$ onto the higher spin current multiplet. $\delta_{\xi}$ form the
same higher spin algebra ${\mathfrak{hs}}$ as a Lie algebra, which corresponds
to $[Q,Q]=Q$ \- type relations at the level of charges.

The calculation of the invariants is simple and gives all $n$-point functions
[61, 62, 63, 64]. For example, the four-point correlation function (generating
function of all four-point correlators of $J$) in the free boson CFT reads
[62]

$\displaystyle\langle JJJJ\rangle_{F.B.}$
$\displaystyle=\frac{1}{|x_{12}||x_{23}||x_{34}||x_{41}|}\times$
$\displaystyle\times\cos(Q^{2}_{13}+Q^{3}_{24}+Q^{4}_{31}+Q^{1}_{43})\cos(P_{12})\cos(P_{23})\cos(P_{34})\cos(P_{41})$
$\displaystyle+\mbox{permutations}\,.$

Here $Q$ and $P$ are the standard conformally-invariant variables [65].

### Three-dimensional bosonization duality.

The $3d$ bosonization duality is a recently discovered type of duality that
takes place in vector models [5, 6, 7, 8, 9, 10]. In what follows, we deal
with the four simplest vector models. These models are constructed by taking
an order parameter, which is an $N$-component complex scalar $\phi^{i}$ or
Dirac fermion $\psi^{i}$, and writing down the most general $U(N)$-invariant
action:

$\displaystyle\frac{k}{4\pi}S_{CS}(A)+\text{Matter}\begin{cases}\displaystyle(D\phi^{i})^{2}&\text{{\color[rgb]{0.1328125,0.12109375,0.8515625}\definecolor[named]{pgfstrokecolor}{rgb}{0.1328125,0.12109375,0.8515625}{free
boson}}}\\\
\displaystyle(D\phi^{i})^{2}+g(\phi^{i}\phi^{i})^{2}&\text{{\color[rgb]{0.1328125,0.12109375,0.8515625}\definecolor[named]{pgfstrokecolor}{rgb}{0.1328125,0.12109375,0.8515625}{Wilson
--Fisher (Ising)}}}\\\
\displaystyle\bar{\psi}\not{D}\psi&\text{{\color[rgb]{1,0.2578125,0.21875}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.2578125,0.21875}{free
fermion}}}\\\
\displaystyle\bar{\psi}\not{D}\psi+g(\bar{\psi}\psi)^{2}&\text{{\color[rgb]{1,0.2578125,0.21875}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.2578125,0.21875}{Gross
--Neveu}}}\end{cases}$

where $S_{CS}=\int{\mathrm{Tr}}[AdA+\tfrac{2}{3}A^{3}]$ is the Chern–Simons
action. Other gaugings, e.g. $O(N)$-gauging, are straightforward. There is a
huge variety of vector models with different matter multiplets, with and
without supersymmetry. Coupling $g$ is not a free parameter, it has to be
tuned to the fixed point, so that the model is conformally invariant. The true
parameters are the number of fields $N$ and the Chern–Simons level $k$. It is
convenient to work with $\lambda=N/k$, which is an analog of the t’Hooft
coupling in the large-$N$ limit. Note that due to the Chern–Simons term and
covariant derivatives, even the $g=0$ models are not free and it is difficult
to say anything concrete about them.

The $3d$-bosonization duality can be formulated with the help of the following
diagram:

free boson Gross–NeveuWilson–Fisherfree
fermion$\scriptstyle\tilde{\lambda}_{b}$-coupling$\scriptstyle\tilde{\lambda}_{f}$-coupling$\scriptstyle\tilde{\lambda}_{b}$-coupling$\scriptstyle\tilde{\lambda}_{f}$-coupling

The conjecture is that all gauge-invariant observables should be identical
along the double-arrows at the top and at the bottom, i.e. the four theories
are dual pairwise. The vertical lines indicate that the ’top’ and ’bottom’
models are related via RG-flows. The upper line connects the free boson theory
to the Gross–Neveu model, while the bottom line connects the free fermion
theory to the Wilson–Fisher model. The red color is for fermionic matter and
the blue is for bosonic one. Therefore, the conjecture is that along the upper
and along the bottom line there are two equivalent descriptions of the same
physics, one in terms of a model with fermionic matter and another one in
terms of a model with bosonic matter (provided $\lambda$ and $N$ are mapped
appropriately). Let us make this a bit more precise.

Intuitively, it is clear that in the $k\rightarrow\infty$ limit the
Chern–Simons action implies the flatness of $A$, $dA+AA=0$. Therefore, $A$
effectively decouples, the only trace of it being in that we have to consider
gauge-invariant observables built out of the matter fields. Note that neither
$\phi$ nor $\psi$ are gauge-invariant. The $3d$-bosonization duality, as
different from the $2d$ one, does not imply any explicit map between $\phi$
and $\psi$. $O(N)$-models are simpler due to the absence of the spin-one
current that can be gauged again.

There is a number of simple observations that can be made in the large-$N$
limit, irrespective of the value of $\lambda$. The simplest gauge-invariant
operators are bilinears

$\displaystyle J_{s}$ $\displaystyle=\phi D\cdots D\phi\,,$ $\displaystyle
J_{s}=\bar{\psi}\gamma D\cdots D\psi\,.$ (28)

They reduce to the usual higher spin currents for free theories once the
Chern–Simons term is removed, e.g. via $k=\infty$ or just $A=0$. Therefore,
the higher spin symmetry is present in $N=\infty$ limit. It is also clear that
all local gauge-invariant operators can be built from the higher spin tensors.
For example, the simplest class next to $J_{s}$ are ’double-trace’ operators
of type $[JJ]$, see e.g. [66] for explicit formulas. Schematically,

$\displaystyle[J_{s_{1}}J_{s_{2}}]_{s}$
$\displaystyle=\sum\partial\ldots\partial J_{s_{1}}\partial\ldots\partial
J_{s_{2}}-\text{traces}\,.$ (29)

Out of $J_{s_{1}}$ and $J_{s_{2}}$ one can build primary operators of spin
$s\geq s_{1}+s_{2}$. Obviously, (higher) correlation functions of $J_{s}$
contain full information about OPE coefficients and anomalous dimensions of
all the other local gauge-invariant operators. Therefore, we should
concentrate on learning as much as possible about $J_{s}$.

To the leading order the spectrum of higher spin tensors is identical for both
theories on the bottom line and for both theories on the top line. The only
difference between the top and bottom models is that the spin-zero operator
$J_{0}$ has conformal dimension $1+\mathcal{O}(N^{-1})$ for the top ones and
dimension $2+\mathcal{O}(N^{-1})$ for the bottom ones. That the spectra
coincide in the large-$N$ limit for the dual models is the first indication of
the duality.

### Slightly-broken higher spin symmetry.

The main question is what happens to $J_{s}$ once we depart from the
$N=\infty$ limit. Conserved higher spin tensors signal that the theory is a
free one in disguise [67, 68, 69, 70]. Therefore, they cannot be conserved.
However, the spectrum of vector models is very sparse and there are not so
many operators to violate the conservation law:

$\displaystyle\partial\cdot J_{s}$
$\displaystyle=\frac{1}{N}\sum_{s_{1}+s_{2}\leq
s}C^{s}_{s_{1},s_{2}}[J_{s_{1}}J_{s_{2}}]_{s}\,.$ (30)

Here the structure constants $C^{s}_{s_{1},s_{2}}$ contain the dynamical
input, while the form of the double-trace operators is fixed by conformal
symmetry to the leading order.111111We note that a simple spin/conformal
dimension counting ensures that in the ‘worst case scenario’ one can find in
addition $N^{-2}[JJJ]$ on the right-hand side, which happens for the models
with $J_{0}$ of dimension $1+\mathcal{O}(N^{-1})$.

It was shown in [6], that the non-conservation law (30) is powerful enough as
to fix the form of three-point functions of $J_{s}$. The idea was to abstract
the non-conservation law from any concrete microscopical realization without
making any assumptions about structure constants $C^{s}_{s_{1},s_{2}}$. The
non-conservation law should be combined with the corresponding violation (or,
more precisely, deformation) of the action of charges $Q_{s}$ on $J_{s}$:

$\displaystyle[Q_{s_{1}},J_{s_{2}}]$
$\displaystyle=\sum_{s}A_{s_{1},s_{2}}^{s}J_{s}+\frac{1}{N}\sum
B_{s_{1},s_{2}}^{l_{1},l_{2},k}[J_{l_{1}}J_{l_{2}}]_{k}\,.$ (31)

Here $A$’s encode the structure constants of the higher spin algebra and $B$’s
signal a violation of the Lie algebra module structure. Now, one can use the
fact that three-point functions are fixed up to three numbers per each triplet
of spins:

$\displaystyle\langle J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle\sim
a_{s_{1},s_{2},s_{3}}\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{b}+b_{s_{1},s_{2},s_{3}}\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{f}+c_{s_{1},s_{2},s_{3}}\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{o}\,.$ (32)

The first two structure can be computed in the free boson and free fermion
theory, respectively. The last one, called odd, does not show up in any free
theory. Nevertheless, it can be shown to exist and to be unique. Constants
$a,b,c$ represent the dynamical information about a given theory and can
depend on spins and coupling constants. A remarkable result [6], is that the
slightly-broken higher spin symmetry along can fix all the structure constants
in (30), (31) and (32) in terms of only two phenomenological parameters
$\tilde{N}$ and $\cos^{2}\theta=1/(1+\tilde{\lambda}^{2})$. The final
expression for the three-point functions reads121212Note that this requires
the building blocks to be properly normalized, so that the spin dependence is
trivial. The exact form of all the structures can be found in [71] in the
light-cone gauge and in [66] manifestly Lorentz invariant expressions are
given for ‘almost’ all the structures.

$\displaystyle\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle=\tilde{N}\left[\cos^{2}\theta\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{b}+\sin^{2}\theta\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{f}+\cos\theta\sin\theta\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{o}\right]\,.$ (33)

Similar results have been obtained for other three-point functions and some
four-point correlators, see e.g. [72, 73, 74, 75, 76]. This indicates that the
slightly-broken higher spin symmetry can be powerful enough to fix all
correlation functions! It remains to understand what it is mathematically.

### Slightly-broken higher spin symmetry via strong homotopy algebras.

At this point, it is important to understand what "slightly-broken higher spin
symmetry" is. The currents are no longer conserved (30); the action of charges
on higher spin currents gets deformed (31); the charges themselves cannot form
a Lie algebra. Therefore, we cannot abstract a single structure – higher spin
algebra – and deform it as a Lie algebra. In fact, the algebra does not even
have such deformations.

Since the non-conservation operator is built from the higher spin currents, it
should be clear that one cannot disentangle ${\mathfrak{hs}}$ from its action
on $J$’s. One has to deform the whole package [2]: algebra ${\mathfrak{hs}}$
plus its module $J$. The structure that accommodates this is
$L_{\infty}$-algebra. The starting point is an $L_{\infty}$-algebra
$\mathcal{L}=\mathcal{L}_{0}\oplus\mathcal{L}_{1}$ concentrated in degree zero
and one, with $\mathcal{L}_{1}$ being reserved for ${\mathfrak{hs}}$ and
$\mathcal{L}_{0}$ for the ${\mathfrak{hs}}$-module that $J$’s form:

$\displaystyle[\delta_{\xi_{1}},\delta_{\xi_{2}}]$
$\displaystyle=l_{2}(\xi_{1},\xi_{2})\,,$ $\displaystyle\delta_{\xi}J$
$\displaystyle=l_{2}(\xi,J)\,,$
$\displaystyle\xi\,,\xi_{1,2}\in{\mathfrak{hs}}\,.$

From now on we can abstract $J$’s and think of them as of specific
${\mathfrak{hs}}$-module $\mathcal{L}_{0}$ rather than concrete operators.
What one can do now is to look for the most general deformation

$\displaystyle\delta_{\xi}J$
$\displaystyle=l_{2}(\xi,J)+\,l_{3}(\xi,J,J)+\ldots\,,$
$\displaystyle[\delta_{\xi_{1}},\delta_{\xi_{2}}]$
$\displaystyle=\delta_{\xi}\,,$

where $\xi=l_{2}(\xi_{1},\xi_{2})+\,l_{3}(\xi_{1},\xi_{2},J)+\ldots$ Self-
consistency requires $l_{n}$’s to form an $L_{\infty}$-algebra.

There is one specific feature of the problem that allows us to apply the
results of section 2, i.e. to reduce $L_{\infty}$ to $A_{\infty}$. Higher spin
tensors $J_{s}$, being bilinear operators, belong to the tensor product
$|\phi\rangle\otimes|\phi\rangle$ or $|\psi\rangle\otimes|\psi\rangle$. Higher
spin algebra ${\mathfrak{hs}}$ is the endomorphism algebra of one-particle
states $|\phi\rangle$ or $|\psi\rangle$. As such it is formally isomorphic to
$|\phi\rangle\langle\phi|$ or $|\psi\rangle\langle\psi|$. Recall that the
latter two tensor products are isomorphic since the higher spin algebra of
free boson and free fermion are identical. Also,
$|\phi\rangle\otimes|\phi\rangle$ is formally isomorphic to
$|\phi\rangle\langle\phi|$, idem. for $\psi$. Therefore, we can map the module
of higher spin tensors onto ${\mathfrak{hs}}$. The difference between
$|\phi\rangle\otimes|\phi\rangle$ and $|\phi\rangle\langle\phi|$ is by the map
$R:|\phi\rangle\rightarrow\langle\phi|$, which is realized via the inversion
map $R$. Inversion $R$ is a discrete automorphism of the conformal and, hence,
of the higher spin algebra. Finally, the underlying associative structure of
${\mathfrak{hs}}$ is important again and leads to the following realization of
the initial data:

$\displaystyle l_{2}(a,b)$ $\displaystyle=a\star b-b\star a\,,$ $\displaystyle
l_{2}(a,v)$ $\displaystyle=a\star v-v\star R(a)\,,$ $\displaystyle
a,b\in\mathcal{L}_{1}\,,\quad v\in\mathcal{L}_{0}\,.$

The last step is to encode $R$ in the smash product
$A_{0}={\mathfrak{hs}}\ltimes\mathbb{Z}_{2}$, where
$\mathbb{Z}_{2}=\\{e,R\\}$. Now, one can apply the powerful machinery of
section 2, see [2]. While ${\mathfrak{hs}}$ does not have any deformations as
an associative algebra, $A_{0}$ is soft and can be deformed into $A_{u}$, with
$u$ being a formal parameter.131313For a multi-parameter deformation there is
always an overall scale of the parameters that we call $u$ here. It is
important to count orders in the expansion, the rest of the parameters, if
any, being implicit.  This gives the deformation quantization of the simplest
Poisson Orbifold $\mathbb{R}^{2}/\mathbb{Z}_{2}$ discussed in section 3 (one
needs to copies of this Poisson Orbifold since $A_{2}=A_{1}\otimes A_{1}$). As
a result, there is an explicit description of all $l_{n}$’s.

In order to discuss the bosonization duality it is also important to
understand what is the most general deformation of $l_{2}$’s that are based on
a given ${\mathfrak{hs}}$. Restricting for simplicity to the $O(N)$ case, i.e.
even spin currents, it may be shown [20] that the deformation depends on two
phenomenological parameters. They can be related to the microscopical
parameters $N$ and $k$, if needed.

### Correlation functions as invariants.

For the unbroken higher spin symmetry the correlation functions are given by
simple ${\mathfrak{hs}}$-invariants of type ${\mathrm{Tr}}_{\star}[\bullet]$,
(26). It was shown in [20] that these invariants are unique, i.e. there are no
other invariants that can serve as correlation functions. Now, one can ask if
it is possible to deform them

$\displaystyle\langle J_{1}\ldots
J_{n}\rangle={\mathrm{Tr}_{0}\\!\left[{\mathbb{J}}_{1}\star\cdots\star{\mathbb{J}}_{n}\right]}+\ldots$
(34)

in such a way that they remain invariant under the deformed higher spin
symmetry, i.e. under the $L_{\infty}$-transformations. Here we also trivially
rewrote ${\mathrm{Tr}}_{\star}[\bullet]$ on ${\mathfrak{hs}}$ as
${\mathrm{Tr}_{0}\\!\left[\bullet\right]}$ on $A_{0}$, since
${\mathfrak{hs}}\in A_{0}$ and $A_{0}$ admits a trace as well. In what follows
it is important that the trace can be deformed to a trace
${\mathrm{Tr}_{u}\\!\left[\bullet\right]}$ on $A_{u}$.

It is worth noting that there is a simple generating function of the free CFT
correlators

$\displaystyle W_{\text{free}}[{\mathbb{J}}]$
$\displaystyle=\mathrm{P.p.}\,{\mathrm{Tr}_{\star}\\!\left[\log_{\star}[1-u^{-1}{\mathbb{J}}]\right]}\,,$
(35)

where $\mathrm{P.p.}$ stands for the principal part of the Laurent series in
$u$. Note that $u$ is introduced artificially here and is not present in the
$\star$-product. Now, it is possible to show that the free CFT correlators
admit a smooth deformation that is invariant under the full
$L_{\infty}$-algebra [3, 77]. The final result can be expressed as a very
similar generating function:

$\displaystyle W_{\text{SBHS}}[{\mathbb{J}}]$
$\displaystyle=\mathrm{P.p.}\,{\mathrm{Tr}_{u}\\!\left[\log_{\circ}[1-u^{-1}{\mathbb{J}}]\right]}\,,$
(36)

where $u$ is now present in the deformed $\circ$-product and in the trace, cf.
(13).

It remains now to compute the corrections to the free CFT correlators.
However, several important observations can be made without having to do that.
First of all, it can be shown [20] that the deformed invariants – the
$L_{\infty}$-invariants – are unique. Therefore, if the idea of the slightly-
broken higher spin symmetry is correct, there is a unique
$L_{\infty}$-invariant to serve as a correlation function. This statement
proves, in principle, the bosonization duality since the invariant is fixed by
the symmetry and does not require any microscopical realization via
Chern–Simons vector models, i.e. it has to be the same both for bosonic and
fermionic matter. Secondly, a very simple observation is that for any $n$ the
$n$-point correlation functions have to have the following form

$\displaystyle\langle J\ldots J\rangle$
$\displaystyle=\sum\langle\text{fixed}\rangle_{i}\times\text{params}\,,$

i.e. it has to be a sum of a finite number of fixed conformally invariant
structures multiplied by the powers of the phenomenological deformation
parameters. Considering unitarity, the two deformation parameters $u_{1,2}$
have to be of the form $u_{1}=ue^{+i\theta}$, $u_{2}=ue^{-i\theta}$. This is
consistent with all available correlation functions [5, 6, 7, 78, 66, 72, 73,
74, 79, 80, 75, 76]. It is intriguing that expression (36) bears close
similarity to the partition function of a one-loop exact 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”. The results on the deformation of
$A_{\infty}$-algebras were obtained with an exclusive support of the Ministry
of Science and Higher Education of the Russian Federation (Project No.
0721-2020-0033).

## References

  * [1] A. Sharapov and E. Skvortsov, “Formal Higher Spin Gravities,” Nucl. Phys. B941 (2019) 838–860, arXiv:1901.01426 [hep-th].
  * [2] A. Sharapov and E. Skvortsov, “$A_{\infty}$ algebras from slightly broken higher spin symmetries,” JHEP 09 (2019) 024, arXiv:1809.10027 [hep-th].
  * [3] P. Gerasimenko, A. Sharapov, and E. Skvortsov, “Slightly broken higher spin symmetry: general structure of correlators,” JHEP 01 (2022) 097, arXiv:2108.05441 [hep-th].
  * [4] A. A. Sharapov and E. D. Skvortsov, “On deformations of $A_{\infty}$-algebras,” J. Phys. A52 no. 47, (2019) 475203, arXiv:1809.03386 [math-ph].
  * [5] S. Giombi, S. Minwalla, S. Prakash, S. P. Trivedi, S. R. Wadia, and X. Yin, “Chern-Simons Theory with Vector Fermion Matter,” Eur. Phys. J. C72 (2012) 2112, arXiv:1110.4386 [hep-th].
  * [6] J. Maldacena and A. Zhiboedov, “Constraining conformal field theories with a slightly broken higher spin symmetry,” Classical and Quantum Gravity 30 no. 10, (2013) 104003.
  * [7] O. Aharony, G. Gur-Ari, and R. Yacoby, “Correlation Functions of Large N Chern-Simons-Matter Theories and Bosonization in Three Dimensions,” JHEP 12 (2012) 028, arXiv:1207.4593 [hep-th].
  * [8] O. Aharony, “Baryons, monopoles and dualities in Chern-Simons-matter theories,” JHEP 02 (2016) 093, arXiv:1512.00161 [hep-th].
  * [9] A. Karch and D. Tong, “Particle-Vortex Duality from 3d Bosonization,” Phys. Rev. X6 no. 3, (2016) 031043, arXiv:1606.01893 [hep-th].
  * [10] N. Seiberg, T. Senthil, C. Wang, and E. Witten, “A Duality Web in 2+1 Dimensions and Condensed Matter Physics,” Annals Phys. 374 (2016) 395–433, arXiv:1606.01989 [hep-th].
  * [11] A. Sharapov and E. Skvortsov, “Cup product on $A_{\infty}$-cohomology and deformations,” J. Noncommut. Geom. 15 no. 1, (2021) 223–240.
  * [12] B. Zwiebach, “Closed string field theory: Quantum action and the B-V master equation,” Nucl. Phys. B390 (1993) 33–152, arXiv:hep-th/9206084 [hep-th].
  * [13] M. R. Gaberdiel and B. Zwiebach, “Tensor constructions of open string theories. 1: Foundations,” Nucl. Phys. B505 (1997) 569–624, arXiv:hep-th/9705038 [hep-th].
  * [14] H. Kajiura, “Noncommutative homotopy algebras associated with open strings,” Rev. Math. Phys. 19 (2007) 1–99, arXiv:math/0306332 [math-qa].
  * [15] T. Lada and J. Stasheff, “Introduction to SH Lie algebras for physicists,” Int. J. Theor. Phys. 32 (1993) 1087–1104, arXiv:hep-th/9209099 [hep-th].
  * [16] M. Alexandrov, M. Kontsevich, A. Schwarz, and O. Zaboronsky, “The Geometry of the Master Equation and Topological Quantum Field Theory,” Int. J. Mod. Phys. A12 (1997) 1405–1429, arXiv:hep-th/9502010 [hep-th].
  * [17] G. Barnich, M. Grigoriev, A. Semikhatov, and I. Tipunin, “Parent field theory and unfolding in BRST first-quantized terms,” Commun. Math. Phys. 260 (2005) 147–181, arXiv:hep-th/0406192 [hep-th].
  * [18] O. Hohm and B. Zwiebach, “$L_{\infty}$ Algebras and Field Theory,” Fortsch. Phys. 65 no. 3-4, (2017) 1700014, arXiv:1701.08824 [hep-th].
  * [19] B. Jurčo, L. Raspollini, C. Sämann, and M. Wolf, “$L_{\infty}$-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism,” Fortsch. Phys. 67 no. 7, (2019) 1900025, arXiv:1809.09899 [hep-th].
  * [20] A. Sharapov and E. Skvortsov, “Characteristic Cohomology and Observables in Higher Spin Gravity,” JHEP 12 (2020) 190, arXiv:2006.13986 [hep-th].
  * [21] G. Barnich and M. Grigoriev, “First order parent formulation for generic gauge field theories,” JHEP 1101 (2011) 122, arXiv:1009.0190 [hep-th].
  * [22] D. Sullivan, “Infinitesimal computations in topology,” Publ. Math. IHES 47 (1977) 269–331.
  * [23] P. van Nieuwenhuizen, “Free graded differential superalgebras,” in Group Theoretical Methods in Physics. Proceedings, 11th International Colloquium, Istanbul, Turkey, August 23-28, 1982, pp. 228–247.  1982\.
  * [24] R. D’Auria, P. Fre, and T. Regge, “Graded Lie Algebra Cohomology and Supergravity,” Riv. Nuovo Cim. 3N12 (1980) 1.
  * [25] M. A. Vasiliev, “Consistent equations for interacting massless fields of all spins in the first order in curvatures,” Annals Phys. 190 (1989) 59–106.
  * [26] M. Grigoriev and A. Kotov, “Gauge PDE and AKSZ-type Sigma Models,” Fortsch. Phys. 67 no. 8-9, (2019) 1910007, arXiv:1903.02820 [hep-th].
  * [27] A. A. Sharapov and E. D. Skvortsov, “Formal higher-spin theories and Kontsevich–Shoikhet–Tsygan formality,” Nucl. Phys. B921 (2017) 538–584, arXiv:1702.08218 [hep-th].
  * [28] H. Nicolai, “Supersymmetry and Functional Integration Measures,” Nucl. Phys. B 176 (1980) 419–428.
  * [29] S. Prokushkin and M. A. Vasiliev, “Higher spin gauge interactions for massive matter fields in 3-D AdS space-time,” Nucl.Phys. B545 (1999) 385, arXiv:hep-th/9806236 [hep-th].
  * [30] N. Seiberg and E. Witten, “String theory and noncommutative geometry,” JHEP 09 (1999) 032, arXiv:hep-th/9908142.
  * [31] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, “Generalized Global Symmetries,” JHEP 02 (2015) 172, arXiv:1412.5148 [hep-th].
  * [32] N. Boulanger, P. Kessel, E. D. Skvortsov, and M. Taronna, “Higher spin interactions in four-dimensions: Vasiliev versus Fronsdal,” J. Phys. A49 no. 9, (2016) 095402, arXiv:1508.04139 [hep-th].
  * [33] E. D. Skvortsov and M. Taronna, “On Locality, Holography and Unfolding,” JHEP 11 (2015) 044, arXiv:1508.04764 [hep-th].
  * [34] M. Kontsevich, “Deformation quantization of Poisson manifolds. 1.,” Lett. Math. Phys. 66 (2003) 157–216, arXiv:q-alg/9709040 [q-alg].
  * [35] A. S. Cattaneo and G. Felder, “A Path integral approach to the Kontsevich quantization formula,” Commun. Math. Phys. 212 (2000) 591–611, arXiv:math/9902090.
  * [36] G. HALBOUT and X. TANG, “Noncommutative poisson structures on orbifolds,” Transactions of the American Mathematical Society 362 no. 5, (2010) 2249–2277. http://www.jstor.org/stable/25677784.
  * [37] G. Halbout, J.-M. Oudom, and X. Tang, “Deformations of Orbifolds with Noncommutative Linear Poisson Structures,” International Mathematics Research Notices 2011 no. 1, (03, 2010) 1–39. https://doi.org/10.1093/imrn/rnq065.
  * [38] A. A. Sharapov and E. D. Skvortsov, “Hochschild cohomology of the Weyl algebra and Vasiliev’s equations,” Letters in Mathematical Physics 107 no. 12, (Dec, 2017) 2415–2432, arXiv:1705.02958 [math-ph].
  * [39] A. A. Sharapov and E. D. Skvortsov, “A simple construction of associative deformations,” Letters in Mathematical Physics (Jul, 2018) , arXiv:1803.10957 [math-ph].
  * [40] E. P. Wigner, “Do the equations of motion determine the quantum mechanical commutation relations?,” Phys. Rev. 77 (Mar, 1950) 711–712.
  * [41] L. M. Yang, “A Note on the Quantum Rule of the Harmonic Oscillator,” Phys. Rev. 84 no. 4, (1951) 788.
  * [42] D. G. Boulware and S. Deser, “’ambiguity’ of harmonic-oscillator commutation relations,” Il Nuovo Cimento (1955-1965) 30 no. 1, (Oct, 1963) 230–234. https://doi.org/10.1007/BF02750763.
  * [43] B. Gruber and L. O’Raifeartaigh, “Uniqueness of the harmonic oscillator commutation relation,” Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences 63 (1963) 69–73.
  * [44] N. Mukunda, E. C. G. Sudarshan, J. K. Sharma, and C. L. Mehta, “Representations and properties of parabose oscillator operators. I. Energy position and momentum eigenstates,” J. Math. Phys. 21 (1980) 2386–2394.
  * [45] C. N. Pope, L. J. Romans, and X. Shen, “A New Higher Spin Algebra and the Lone Star Product,” Phys. Lett. B242 (1990) 401–406.
  * [46] P. Bieliavsky, S. Detournay, and P. Spindel, “The Deformation quantizations of the hyperbolic plane,” Commun. Math. Phys. 289 (2009) 529–559, arXiv:0806.4741 [math-ph].
  * [47] E. Joung and K. Mkrtchyan, “Notes on higher-spin algebras: minimal representations and structure constants,” JHEP 05 (2014) 103, arXiv:1401.7977 [hep-th].
  * [48] A. V. Korybut, “Covariant structure constants for a deformed oscillator algebra,” Theor. Math. Phys. 193 no. 1, (2017) 1409–1419, arXiv:1409.8634 [hep-th].
  * [49] T. Basile, N. Boulanger, and F. Buisseret, “Structure constants of shs$[\lambda]$ : the deformed-oscillator point of view,” J. Phys. A51 no. 2, (2018) 025201, arXiv:1604.04510 [hep-th].
  * [50] A. V. Korybut, “Star product for deformed oscillator algebra $\mathsf{Aq}(2,\nu)$,” arXiv:2006.01622 [hep-th].
  * [51] B. L. Feigin and B. L. Tsygan, Additive K-theory, pp. 67–209.  Springer Berlin Heidelberg, Berlin, Heidelberg, 1987.
  * [52] B. Tsygan, “Formality conjecture for chains,” arXiv:math/9904132.
  * [53] B. Shoikhet, “A proof of the Tsygan formality conjecture for chains,” Advances in Mathematics 179 no. 1, (2003) 7 – 37.
  * [54] T. W. B. Kibble, “Conservation laws for free fields,” Journal of Mathematical Physics 6 no. 7, (1965) 1022–1026.
  * [55] S. Deser and H. Nicolai, “Nonabelian Zilch,” Phys. Lett. 98B (1981) 45–47.
  * [56] M. G. Eastwood, “Higher symmetries of the Laplacian,” Annals Math. 161 (2005) 1645–1665, arXiv:hep-th/0206233 [hep-th].
  * [57] P. A. M. Dirac, “A Remarkable representation of the 3 + 2 de Sitter group,” J. Math. Phys. 4 (1963) 901–909.
  * [58] M. Günaydin and C. Saclioglu, “Oscillator Like Unitary Representations of Noncompact Groups With a Jordan Structure and the Noncompact Groups of Supergravity,” Commun. Math. Phys. 87 (1982) 159.
  * [59] M. Günaydin, “Oscillator like unitary representations of noncompact groups and supergroups and extended supergravity theories,” in Group Theoretical Methods in Physics. Proceedings, 11th International Colloquium, Istanbul, Turkey, August 23-28, 1982, pp. 192–213.  1983\.
  * [60] J.-P. Michel, “Higher symmetries of the Laplacian via quantization,” Annales de l’Institut Fourier 64 no. 4, (2014) 1581–1609. http://www.numdam.org/articles/10.5802/aif.2891/.
  * [61] N. Colombo and P. Sundell, “Higher Spin Gravity Amplitudes From Zero-form Charges,” arXiv:1208.3880 [hep-th].
  * [62] V. Didenko and E. Skvortsov, “Exact higher-spin symmetry in CFT: all correlators in unbroken Vasiliev theory,” JHEP 1304 (2013) 158, arXiv:1210.7963 [hep-th].
  * [63] V. E. Didenko, J. Mei, and E. D. Skvortsov, “Exact higher-spin symmetry in CFT: free fermion correlators from Vasiliev Theory,” Phys. Rev. D88 (2013) 046011, arXiv:1301.4166 [hep-th].
  * [64] R. Bonezzi, N. Boulanger, D. De Filippi, and P. Sundell, “Noncommutative Wilson lines in higher-spin theory and correlation functions of conserved currents for free conformal fields,” J. Phys. A50 no. 47, (2017) 475401, arXiv:1705.03928 [hep-th].
  * [65] S. Giombi, S. Prakash, and X. Yin, “A Note on CFT Correlators in Three Dimensions,” arXiv:1104.4317 [hep-th].
  * [66] S. Giombi, V. Gurucharan, V. Kirilin, S. Prakash, and E. Skvortsov, “On the Higher-Spin Spectrum in Large N Chern-Simons Vector Models,” JHEP 01 (2017) 058, arXiv:1610.08472 [hep-th].
  * [67] J. Maldacena and A. Zhiboedov, “Constraining Conformal Field Theories with A Higher Spin Symmetry,” arXiv:1112.1016 [hep-th].
  * [68] N. Boulanger, D. Ponomarev, E. D. Skvortsov, and M. Taronna, “On the uniqueness of higher-spin symmetries in AdS and CFT,” Int. J. Mod. Phys. A28 (2013) 1350162, arXiv:1305.5180 [hep-th].
  * [69] V. Alba and K. Diab, “Constraining conformal field theories with a higher spin symmetry in d=4,” arXiv:1307.8092 [hep-th].
  * [70] V. Alba and K. Diab, “Constraining conformal field theories with a higher spin symmetry in $d>3$ dimensions,” arXiv:1510.02535 [hep-th].
  * [71] E. Skvortsov, “Light-Front Bootstrap for Chern-Simons Matter Theories,” JHEP 06 (2019) 058, arXiv:1811.12333 [hep-th].
  * [72] Z. Li, “Bootstrapping conformal four-point correlators with slightly broken higher spin symmetry and $3D$ bosonization,” JHEP 10 (2020) 007, arXiv:1906.05834 [hep-th].
  * [73] R. R. Kalloor, “Four-point functions in large $N$ Chern-Simons fermionic theories,” JHEP 10 (2020) 028, arXiv:1910.14617 [hep-th].
  * [74] G. J. Turiaci and A. Zhiboedov, “Veneziano Amplitude of Vasiliev Theory,” JHEP 10 (2018) 034, arXiv:1802.04390 [hep-th].
  * [75] S. Jain and R. R. John, “Relation between parity-even and parity-odd CFT correlation functions in three dimensions,” arXiv:2107.00695 [hep-th].
  * [76] J. A. Silva, “Four point functions in CFT’s with slightly broken higher spin symmetry,” JHEP 05 (2021) 097, arXiv:2103.00275 [hep-th].
  * [77] A. A. Sharapov and E. D. Skvortsov, “Homotopy Cartan calculus and inner deformations of $A_{\infty}$-algebras,” arXiv:2202.10925 [math-ph].
  * [78] G. Gur-Ari and R. Yacoby, “Correlators of Large N Fermionic Chern-Simons Vector Models,” JHEP 02 (2013) 150, arXiv:1211.1866 [hep-th].
  * [79] Y. Gandhi, S. Jain, and R. R. John, “Anyonic correlation functions in Chern-Simons matter theories,” arXiv:2106.09043 [hep-th].
  * [80] S. Jain, R. R. John, and V. Malvimat, “Constraining momentum space correlators using slightly broken higher spin symmetry,” JHEP 04 (2021) 231, arXiv:2008.08610 [hep-th].