# What is the Seiberg–Witten map exactly?

Vladislav Kupriyanov Centro de Matemática, Computação e Cognição, Universidade
Federal do ABC, Santo André, SP, Brazil Alexey Sharapov Physics Faculty, Tomsk
State University, Lenin ave. 36, Tomsk 634050, Russia

###### Abstract

We give a conceptual treatment of the Seiberg–Witten map as a quasi-
isomorphism of $A_{\infty}$-algebras.

##  1 Introduction

Based on several nontrivial examples, the authors of Ref. [1] suggested that
“there should be an underlying geometric reason for the Seiberg–Witten map”.
This paper resulted from our attempt to find such a reason. The answer we have
arrived at, however, is purely algebraic rather than geometric. Before going
into details, let us recall the statement of the problem that the
Seiberg–Witten map solves.

In the simplest situation, one considers a gauge vector field $\mathcal{A}$ on
an $n$-dimensional Minkowski space with coordinates $x^{i}$. The standard
Maxwell’s equations for the gauge potential $\mathcal{A}$ are obtained by
varying the action functional

$S[\mathcal{A}]=\int d^{n}x\mathcal{F}_{ij}\mathcal{F}^{ij}\,,$ (1)

where
$\mathcal{F}_{ij}=\partial_{i}\mathcal{A}_{j}-\partial_{j}\mathcal{A}_{i}$ is
the strength tensor of the electromagnetic field. The action is invariant
under the usual gauge transformations

$\delta_{\lambda}\mathcal{A}_{i}=\partial_{i}\lambda\,,$ (2)

$\lambda$ being the gauge parameter.

Suppose now that, in addition to the metric, the Minkowski space carries a
constant bivector $\theta$. This allows one to endow the space of smooth
functions of the $x$’s with the Weyl–Moyal $\ast$-product111As is customary
for physical considerations we leave aside the question of convergence of the
series.:

$(f\ast g)(x)=\lim_{y\rightarrow
x}e^{\frac{1}{2}\theta^{ij}\frac{\partial}{\partial
y^{i}}\frac{\partial}{\partial x^{j}}}f(y)g(x)\,.$ (3)

This product is known to be associative but not commutative unless $\theta=0$.
In particular, $[x^{i},x^{j}]_{\ast}=\theta^{ij}$. Hence, a nonzero $\theta$
brings in noncommutativity in the space-time that the gauge vector field lives
on. In order to distinguish between commutative and noncommutative cases, we
will ‘put hats’ on fields living on the noncommutative Minkowski space.
Following the pattern of non-Abelian gauge theories, it is now quite natural
to replace the infinitesimal gauge transformation (2) with

$\hat{\delta}_{\hat{\lambda}}\hat{\mathcal{A}}_{i}=\partial_{i}\hat{\lambda}+[\hat{\mathcal{A}}_{i},\hat{\lambda}]_{\ast}$
(4)

and to define the gauge-covariant strength tensor by the familiar expression

$\hat{\mathcal{F}}_{ij}=\partial_{i}\hat{\mathcal{A}_{j}}-\partial_{j}\hat{\mathcal{A}_{i}}+[\hat{\mathcal{A}}_{i},\hat{\mathcal{A}}_{j}]_{\ast}\,.$
(5)

Then,
$\delta_{\hat{\lambda}}\hat{\mathcal{F}}=[\hat{\mathcal{F}},\hat{\lambda}]_{\ast}$.
Unlike (2), the gauge transformations (4) do not commute to each other:

$[\hat{\delta}_{\hat{\lambda}_{1}},\hat{\delta}_{\hat{\lambda}_{2}}]=\hat{\delta}_{[\hat{\lambda}_{1},\hat{\lambda}_{2}]_{\ast}}\,.$
(6)

Finally, the $\ast$-product algebra of functions enjoys the trace functional

$\mathrm{Tr}(f)=\int d^{n}xf(x)\,.$ (7)

The main property of the trace reads

$\mathrm{Tr}(f\ast g)=\mathrm{Tr}(g\ast f)=\int d^{n}xf(x)g(x)\,,$ (8)

provided the functions $f$ and $g$ vanish at infinity fast enough. With the
trace, one can define the gauge invariant action $S[\hat{\mathcal{A}}]$ by the
same integral (1) with $\mathcal{F}$ replaced by $\hat{\mathcal{F}}$. Because
of the $\ast$-commutator in (5), the corresponding equations of motion are not
linear anymore.

A remarkable fact established in [2] is that being dynamically different, the
commutative and noncommutative field theories have essentially the same gauge
structure. More specifically, there exists a bijective mapping between the
spaces of fields $\mathcal{A}$ and $\hat{\mathcal{A}}$ that takes gauge
equivalent configurations of fields to gauge equivalent ones. This by no means
implies an isomorphism of the corresponding gauge algebras, regarded as
infinite-dimensional Lie algebras. (Such an isomorphism is clearly impossible
because one algebra is Abelian and the other is not.) It turns out that an
appropriate geometric framework for understanding the above equivalence is
that of Lie algebroids rather than Lie algebras. Here we will not dwell on
this issue in detail, referring the interested reader to the papers [3] and
[4, Sec. 5]. In a nutshell, one may allow the structure constants of the gauge
algebra to depend on fields (though it is not our case). In this more general
setting, it is legitimate to consider redefinitions of gauge fields that are
accompanied by field-dependent redefinitions of gauge parameters. Such
transformations form the natural automorphism group of a gauge theory in the
Batalin–Vilkovisky (BV) formalism. Regarding now a linear representation of a
Lie algebra as a particular case of Lie algebroids, one can consider the
transformations

$\hat{\mathcal{A}}=\hat{\mathcal{A}}(\mathcal{A})\,,\qquad\hat{\lambda}=\hat{\lambda}({\lambda,\mathcal{A}})\,.$
(9)

The condition that gauge equivalent field configurations go into gauge
equivalent ones leads to the equation
$\delta_{{\lambda}}[\hat{\mathcal{A}}(\mathcal{A})]=\hat{\delta}_{\hat{\lambda}}\hat{\mathcal{A}}(\mathcal{A})$
or, more explicitly,

$\hat{\mathcal{A}}_{i}(\mathcal{A}+\partial\lambda)-\hat{\mathcal{A}}_{i}(\mathcal{A})=\partial_{i}\hat{\lambda}(\lambda,\mathcal{A})+[\hat{\mathcal{A}}_{i}(\mathcal{A}),\hat{\lambda}(\lambda,\mathcal{A})]_{\ast}\,,$
(10)

with infinitesimal $\lambda$ and $\hat{\lambda}$. Requiring further the gauge
algebra to close off-shell, one also imposes the gauge consistency condition

$\delta_{\lambda_{2}}\hat{\lambda}_{1}-\delta_{\lambda_{1}}\hat{\lambda}_{2}=[\hat{\lambda}_{1},\hat{\lambda}_{2}]_{\ast}-\hat{\lambda}([\lambda_{1},\lambda_{2}]_{\ast},\mathcal{A})\,,$
(11)

where the variation $\delta_{\lambda_{2}}\hat{\lambda}_{1}$ refers to the
$\mathcal{A}$-dependence of $\hat{\lambda}(\lambda,\mathcal{A})$ and the gauge
transformation (2). Geometrically, formulas (10, 11) identify the
transformation (9) as a morphism of two Lie algebroids associated with the
action of the Abelian and non-Abelian Lie algebras on the space of fields
$\mathcal{A}$. Any solution to Eqs. (10, 11) with the initial condition

$\hat{\mathcal{A}}=\mathcal{A}+O(\theta)\,,\qquad\hat{\lambda}=\lambda+O(\theta)$
(12)

is called a Seiberg–Witten (SW) map. The existence of SW maps is not obvious
in advance. Original arguments in its support came from string-theoretic
considerations [2]. It is easy to check that the following expressions satisfy
the above equations up to the first order in $\theta$:

$\begin{array}[]{l}\displaystyle\hat{\mathcal{A}}_{i}=\mathcal{A}_{i}-\frac{1}{2}\theta^{kj}\mathcal{A}_{k}(\partial_{j}\mathcal{A}_{i}+\mathcal{F}_{ji})+O(\theta^{2})\,,\\\\[8.53581pt]
\displaystyle\hat{\lambda}=\lambda+\frac{1}{2}\theta^{ij}\partial_{i}\lambda\mathcal{A}_{j}+O(\theta^{2})\,.\end{array}$
(13)

In this paper, we give a refined algebraic explanation of the origin of the SW
map, free from all inesential field-theoretic details. More concretely, we
show that any SW map defines and is defined by a quasi-isomorphism of
differential graded algebras (dg-algebras for short). The dg-algebras
underlying the (non)commutative gauge theories above appear to be very simple
and are discussed in the next section. Certainly, it is not the first time the
term ‘quasi-isomorphism’ is mentioned in connection with SW-maps. Thus, in [5,
6, 7, 8], the problem of constructing SW maps was put in the framework of BV-
BRST formalism. In that approach, each gauge theory is described by a BV
master action on the ghost-extended space of fields and antifields. Expanding
the master action in powers of antifields generates a sequence of multilinear
operators, which can be regarded as the structure maps of a cyclic
$L_{\infty}$-algebra. The defining relations of an $L_{\infty}$-algebra (a.k.a
higher Jacobi identities) follow from the classical master equation. From this
perspective, each SW map gives rise to a (quasi-)isomorphism of the
corresponding $L_{\infty}$-algebras. In explicit terms, the relationship
between SW maps and quasi-isomorphisms of $L_{\infty}$-algebras was explored
in [9].

Our setup is somewhat different. Instead of $L_{\infty}$-algebras, we work in
the category of $A_{\infty}$-algebras, which seems more natural, as the
$\ast$-product algebra (3) responsible for noncommutativity is associative.
Our construction of quasi-isomorphism in no way involves the equations of
motion, which are the main (and unavoidable) ingredient of the BV-BRST
formalism. This leads to further conceptual simplification. The
$L_{\infty}$-algebras, their quasi-isomorphisms and, eventually, SW maps are
obtained by the standard antisymmetrization procedure from the corresponding
$A_{\infty}$-algebras and their morphisms, much as the commutator in an
associative algebra yields a Lie bracket. The quasi-isomorphism construction
is recursive and results in explicit expressions for the field transformation
(9). Since the recursion is on the degree of $\mathcal{A}$, the result appears
to be nonperturbative in $\theta$, as opposed to the earlier approaches [2, 1,
10, 11]. The importance of nonperturbative in $\theta$ solutions (e.g. for
studying instantons) was stressed in the original paper [2]. The algebraic
treatment also provides the homotopy classification of ambiguities in SW maps,
which we briefly address in Sec. 3. All in all, we hope that the new
conceptual understanding gained in this paper will be useful for further
generalizations and applications of the SW map. Some interesting possibilities
are discussed in the concluding section.

Finally, a word about notation. Throughout the paper, we work over the field
of reals $\mathbb{R}$, though all our results are valid for any field of
characteristic zero. All unadorned tensor products $\otimes$ and
$\mathrm{Hom}$’s are defined over $\mathbb{R}$. When dealing with graded
vector spaces and algebras, we tend to write down all sign factors explicitly.
However, one can easily keep track of signs using the Koszul rule: whenever
two graded objects $a$ and $B$ are swapped, the factor of $(-1)^{\deg
a\cdot\deg B}$ appears.

##  2 SW map as a quasi-isomorphism of dg-algebras

Let $A_{\theta}$ denote a unital dg-algebra over $\mathbb{R}$ generated by two
sets of elements $x^{i}$ and $\xi^{i}$, $i=1,\ldots,n$, subject to the
relations

$x^{i}x^{j}-x^{j}x^{i}=\theta^{ij}\,,\qquad
x^{i}\xi^{j}-\xi^{j}x^{i}=0\,,\qquad\xi^{i}\xi^{j}=0\,.$ (14)

Here $\theta=(\theta^{ij})$ is a real antisymmetric matrix. By definition,

$\deg x^{i}=0\,,\qquad\deg\xi^{i}=1$ (15)

and

$dx^{i}=\xi^{i}\,,\qquad d\xi^{i}=0\,.$ (16)

It follows from (14, 15) that $A_{\theta}$ is concentrated in degrees zero and
one. In case $\theta=0$, the dg-algebra $(A_{0},d)$ is obviously commutative.
For any $\theta$ the algebra $A_{\theta}$ satisfies the Poincaré–Birkhoff–Witt
(PBW) property: as an $\mathbb{R}$-vector space it is isomorphic to the space
of polynomials $a(x,\xi)$ that are at most linear in the $\xi$’s. So, the
general element of $A_{\theta}$ is represented by a polynomial

$a=a_{0}(x)+a_{i}(x)\xi^{i}\,,$ (17)

where $a_{0}(x)$ and $a_{i}(x)$ are polynomials in the $x$’s. The product is
then defined by the Weyl–Moyal type formula

$(f\ast_{\theta}g)(x,\xi)=\mu\sum_{n=0}^{\infty}\frac{1}{n!}\Big{(}\frac{1}{2}\theta^{ij}\frac{\partial}{\partial
x^{i}}\otimes\frac{\partial}{\partial x^{j}}\Big{)}^{n}f\otimes g\,,$ (18)

where

$\mu(f\otimes g)=p(fg)\,,$

and $p$ is the projection onto the subspace of polynomials at most linear in
$\xi$’s. The polynomials (17) that are independent of the $\xi$’s form a
subalgebra and those that are linear in $\xi$’s constitute a nilpotent ideal.

The cohomology algebra $H(A_{\theta})$ is obviously commutative and does not
depend on $\theta$, i.e., $H(A_{\theta})\simeq H(A_{\theta^{\prime}})$. In
particular, $H^{0}(A_{\theta})=\mathbb{R}$.

######  Proposition 2.1.

For all antisymmetric matrices $\theta$ and $\theta^{\prime}$ the
corresponding dg-algebras $(A_{\theta},d)$ and $(A_{\theta^{\prime}},d)$ are
quasi-isomorphic.

Recall that two dg-algebras $A$ and $B$ are said to be quasi-isomorphic if
there exists a zig-zag sequence of dg-algebra morphisms

$A\leftarrow C_{1}\rightarrow C_{2}\leftarrow C_{3}\rightarrow\cdots\leftarrow
C_{n}\rightarrow B$

such that each morphism induces an isomorphism in cohomology. In particular,
this means that $H(A)\simeq H(B)$. A dg-algebra $A$ is called formal if it is
quasi-isomorphic to its algebra of cohomology $H(A)$.

To prove Proposition 2.1, let us define the operator
$d^{\ast}:A_{\theta}\rightarrow A_{\theta}$ of degree $-1$ that acts on
cochains (17) as

$d^{\ast}a=x^{i}a_{i}(x)\,.$ (19)

Then the algebra $H(A_{\theta})$ is isomorphic to the subalgebra $Z=\ker
d\cap\ker d^{\ast}\subset A_{\theta}$. The subalgebra $Z$, being commutative,
does not depend on $\theta$. Hence, we have the zig-zag map

$A_{\theta}\stackrel{{\scriptstyle i}}{{\leftarrow}}Z\stackrel{{\scriptstyle
i}}{{\rightarrow}}A_{\theta^{\prime}}\,,$ (20)

where $i$ stands for the natural embedding. This proves Proposition 2.1. We
also see that the algebra $A_{\theta}$ is formal.

It is known that whenever two dg-algebras $A$ and $B$ are quasi-isomorphic
they are quasi-isomorphic as $A_{\infty}$-algebras, meaning the existence of a
single quasi-isomorphism $f:A\rightarrow B$. Let us recall some basic
definitions.

Given a graded vector space $A=\bigoplus A_{n}$, define the shifted graded
space $A[1]=\bigoplus A_{n+1}$, that is, $A[1]_{n}=A_{n+1}$. In order to avoid
confusion we will denote the degree of a homogeneous element $a\in A[1]$ by
bar:

$\bar{a}=\deg a-1\,.$ (21)

(On the right, $a$ is treated as an element of $A$.)

The bar coalgebra of $A$ is given now by the direct sum

$B(A)=\bigoplus_{l=1}^{\infty}B^{l}(A)\,,\qquad B^{l}(A)=A[1]^{\otimes l}\,,$
(22)

endowed with the coproduct

$\Delta(a_{1}\otimes\cdots\otimes
a_{n})=\sum_{l=1}^{n-1}(a_{1}\otimes\cdots\otimes
a_{l})\bigotimes(a_{l+1}\otimes\cdots\otimes a_{n})\in B(A)\bigotimes B(A)\,.$
(23)

Recall that a coderivation of a coalgebra $C$ is a linear map $D:C\rightarrow
C$ such that

$(D\otimes 1+1\otimes D)\Delta a=\Delta(Da)\qquad\forall a\in C\,.$ (24)

The coderivations form a graded Lie algebra $\mathrm{Coder}(C)$ w.r.t. the
graded commutator. As is well known, the space of coderivations of the
coalgebra $B(A)$ is isomorphic to the space

$\mathrm{Hom}(B(A),A[1])=\bigoplus_{k=-\infty}^{\infty}\mathrm{Hom}^{k}(B(A),A[1])\,,$

so that to every $D\in\mathrm{Hom}(B^{l}(A),A[1])$ there corresponds the
coderivation of the form

$\hat{D}(a_{1}\otimes\cdots\otimes a_{n})=\sum_{i=0}^{n-l}\pm
a_{1}\otimes\cdots\otimes a_{i}\otimes
D(a_{i+1},\ldots,a_{i+l})\otimes\cdots\otimes a_{n}\,,$ (25)

$\pm$ being the Koszul sign. A coderivation $\delta$ of degree one is called a
codifferential if $\delta^{2}=0$. By definition, an $A_{\infty}$-algebra
structure on a graded vector space $A$ is a codifferential on $B(A)$. In other
words, an $A_{\infty}$-algebra is defined by an element
$m\in\mathrm{Hom}^{1}(B(A),A[1])$ such that $\hat{m}\hat{m}=0$. Expanding $m$
in homogeneous components, $m=m_{1}+m_{2}+m_{3}+\cdots$, we get a sequence of
multilinear maps $m_{l}\in\mathrm{Hom}^{1}(B^{l}(A),A[1])$ obeying the
quadratic relations

$\hat{m}_{1}\hat{m}_{1}=0\,,\qquad\hat{m}_{1}\hat{m}_{2}+\hat{m}_{2}\hat{m}_{1}=0\,,\qquad\ldots$
(26)

In particular, for any dg-algebra $A$ one can set $m_{1}(a)=da$,
$m_{2}(a,b)=(-1)^{\bar{a}}ab$, and $m_{l}=0$ for all $l>2$. Hence, each dg-
algebra is an $A_{\infty}$-algebra for which only the two maps $m_{1}$ and
$m_{2}$ may be different from zero.

If $(A,m)$ and $(A^{\prime},m^{\prime})$ are $A_{\infty}$-algebras, then a
morphism between them is a morphism of differential graded coalgebras
$F:B(A)\rightarrow B(A^{\prime})$. The morphism $F$ is completely defined by a
map $f\in\mathrm{Hom}^{0}(B(A),A^{\prime}[1])$, often called twisting cochain,
obeying the relation

$m^{\prime}_{1}(f)+m^{\prime}_{2}(f,f)+m^{\prime}_{3}(f,f,f)+\cdots=f\circ\hat{m}\,.$
(27)

For dg-algebras $(A,d)$ and $(A^{\prime},d^{\prime})$ this amounts to the
sequence of equations

$\delta f_{l}=\Psi_{l}[f_{1},\ldots,f_{l-1}]\,,\qquad l=1,2,\ldots,$ (28)

where

$(\delta
f_{l})(a_{1},\ldots,a_{l})=d^{\prime}f_{l}(a_{1},\ldots,a_{l})-\sum_{i=1}^{l}(-1)^{\bar{a}_{1}+\cdots+\bar{a}_{i-1}}f_{l}(a_{1},\ldots,da_{i},\ldots,a_{l})\,,$
$\begin{array}[]{c}\Psi_{l}[f_{1},\ldots,f_{l-1}](a_{1},\ldots,a_{l})=\displaystyle\sum_{i=1}^{l-1}(-1)^{\bar{a}_{1}+\cdots+\bar{a}_{i}}f_{l-1}(a_{1},\ldots,a_{i}a_{i+1},\ldots
a_{l})\\\\[14.22636pt]
-\displaystyle\sum_{i=1}^{l-1}(-1)^{\bar{a}_{1}+\cdots+\bar{a}_{i}}f_{i}(a_{1},\ldots,a_{i})f_{l-i}(a_{i+1},\ldots,a_{l})\,,\end{array}$
(29)

and $f_{l}$ stands for the restriction of $f$ onto the subspace $B^{l}(A)$.
For $l=1$, $\Psi_{1}\equiv 0$ and $f_{1}:A[1]\rightarrow A^{\prime}[1]$
defines a cochain transformation, i.e., $d^{\prime}f_{1}=f_{1}d$. We say that
a cochain transformation $f_{1}$ is unital if it preserves the unit, that is,
$f_{1}(1)=1$.

A morphism $F$ is called quasi-isomorphism if $f_{1}$ induces an isomorphism
$H(f_{1}):H(A)\rightarrow H(A^{\prime})$ in cohomology.

Notice that the operator $\delta$ defines a differential that makes each space
$\mathrm{Hom}(B^{l}(A),A^{\prime}[1])$ into a cochain complex. By induction on
$l$, one can see that $\delta\Psi_{l}=0$; and hence, all $\Psi_{l}$’s are
$\delta$-cocycles. The quasi-isomorphism condition (28) requires all these
cocycles to be trivial. We conclude that whenever the differential $\delta$ is
acyclic in degree $1$ any morphism of complexes $f_{1}:A\rightarrow
A^{\prime}$ extends to a morphism of $A_{\infty}$-algebras. Of course, the
acyclicity of $\delta$ is only a sufficient (not necessary) condition for the
existence of an $A_{\infty}$-morphism $f$.

We say that a cochain $f\in\mathrm{Hom}(B^{l}(A),A^{\prime}[1])$ is normal if
it satisfies the normalization condition: $f(a_{1},\ldots,a_{l})=0$ whenever
any of the arguments is equal to $1\in A$. Obviously, the normal cochains
constitute a subcomplex of the complex $\mathrm{Hom}(B^{l}(A),A^{\prime}[1])$.

######  Proposition 2.2.

If $f_{1}$ is unital and $f_{2},f_{3},\ldots,f_{l-1}$ are normal, then the
cochain $\Psi_{l}[f_{1},\ldots,f_{l-1}]$ defined by Eq. (29) is normal as
well.

The proof is by direct verification.

Let us specify the above formulas for the case of algebras $A=A_{\theta}$ and
$A^{\prime}=A_{\theta^{\prime}}$. As both the algebras are nested in degrees
zero and one, $A_{\theta}=(A_{\theta})_{0}\oplus(A_{\theta})_{1}$, each map
$f_{l}$ splits into $l+1$ components

$\begin{array}[]{l}\displaystyle f_{l}^{\prime}:\;(A_{\theta})_{1}^{\otimes
l}\rightarrow(A_{\theta^{\prime}})_{1}\,,\\\\[14.22636pt] \displaystyle
f^{\prime\prime}_{l,k}:(A_{\theta})_{1}^{\otimes
k}\otimes(A_{\theta})_{0}\otimes(A_{\theta})_{1}^{\otimes(l-k-1)}\rightarrow(A_{\theta^{\prime}})_{0},\qquad
k=0,1,\ldots,l-1.\end{array}$ (30)

In terms of these component maps, equations (28) decompose into two groups:

$\begin{array}[]{c}f^{\prime}_{l}(a_{1},\ldots,a_{k},db,\ldots,a_{l-1})=df^{\prime\prime}_{l,k}(a_{1},\ldots,a_{k},b,\ldots,a_{l-1})\\\\[8.53581pt]
-f^{\prime}_{l-1}(a_{1},\ldots,a_{k}\ast_{\theta}b,\ldots,a_{l-1})+f^{\prime}_{l-1}(a_{1},\ldots,b\ast_{\theta}a_{k+1},\ldots,a_{l-1})\\\\[8.53581pt]
+\displaystyle\sum_{n=1}^{k}f^{\prime}_{n}(a_{1},\ldots,a_{n})\ast_{\theta^{\prime}}f^{\prime\prime}_{l-n,k-n}(a_{n+1},\ldots,a_{k},b,\ldots,a_{l-1})\\\\[14.22636pt]
-\displaystyle\sum_{n=k+1}^{l-1}f^{\prime\prime}_{n,k}(a_{1},\ldots,a_{k},b,\ldots,a_{n-1})\ast_{\theta^{\prime}}f^{\prime}_{l-n}(a_{n},\ldots,a_{l-1})\end{array}$
(31)

for all $0\leq k\leq l-1$ and

$\begin{array}[]{c}f^{\prime\prime}_{l,s}(a_{1},\ldots,db_{1},\ldots,a_{s-1},b_{2},\ldots,a_{l-2})-f^{\prime\prime}_{l,k}(a_{1},\ldots,a_{k},b_{1},\ldots,db_{2},\ldots,a_{l-2})\\\\[8.53581pt]
=-f^{\prime\prime}_{l-1,s-1}(a_{1},\ldots,a_{k}\ast_{\theta}b_{1},\ldots,a_{s-1},b_{2},\ldots,a_{l-2})\\\\[8.53581pt]
+f^{\prime\prime}_{l-1,s-1}(a_{1},\ldots,b_{1}\ast_{\theta}a_{k+1},\ldots,a_{s-1},b_{2},\ldots,a_{l-2})\\\\[8.53581pt]
+f^{\prime\prime}_{l-1,k}(a_{1},\ldots,a_{k},b_{1},\ldots,a_{s-1}\ast_{\theta}b_{2},\ldots,a_{l-2})\\\\[8.53581pt]
-f^{\prime\prime}_{l-1,k}(a_{1},\ldots,a_{k},b_{1},\ldots,b_{2}\ast_{\theta}a_{s},\ldots,a_{l-2})\\\\[8.53581pt]
+\delta_{k,s-1}f^{\prime\prime}_{l-1,k}(a_{1},\ldots,a_{k},b_{1}\ast_{\theta}b_{2},a_{s},\ldots,a_{l-2})\\\\[0.0pt]
\displaystyle-\sum_{n=k}^{s-1}f^{\prime\prime}_{n+1,k}(a_{1},\ldots,a_{k},b_{1},\ldots,a_{n})\ast_{\theta^{\prime}}f^{\prime\prime}_{l-n-1,s-n-1}(a_{n+1},\ldots,b_{2},\ldots,a_{l-2})\end{array}$
(32)

for all $0\leq k\leq s-1\leq l-2$. As is seen the second group of equations
does not involve the $f^{\prime}$’s, giving thus a closed subsystem for
determining unknown maps $f^{\prime\prime}_{k,l}$. We will present the general
solution to Eqs. (31, 32) in the next section. For now, let us explain the
relevance of $A_{\infty}$-morphisms above to SW maps. Given elements
$a\in(A_{\theta})_{1}$ and $b\in(A_{\theta})_{0}$, define the pair of chains

$\varphi=\sum_{l=1}^{\infty}a^{\otimes
l}\,,\qquad\psi=\sum_{s,u=0}^{\infty}a^{\otimes s}\otimes b\otimes a^{\otimes
u}$ (33)

of the bar coalgebra $B(A_{\theta})$. Clearly, $\bar{\varphi}=0$ and
$\bar{\psi}=1$. Let $\hat{D}_{a}$ denote the coderivation of $B(A_{\theta})$
associated with the cochain
$D_{a}\in\mathrm{Hom}(B^{1}(A_{\theta}),A_{\theta}[1])$ defined by

$D_{a}(c)=dc+a\ast_{\theta}c-c\ast_{\theta}a\qquad\forall c\in
A_{\theta}[1]\,.$ (34)

Then evaluating equation (31) on $\psi$ and summing up over $l$, one can see
that the cochains

$f^{\prime}=\sum_{l=1}^{\infty}f^{\prime}_{l}\,,\qquad
f^{\prime\prime}=\sum_{l=1}^{\infty}\sum_{k=0}^{l-1}f^{\prime\prime}_{l,k}$
(35)

of $\mathrm{Hom}(B(A_{\theta}),A_{\theta^{\prime}}[1])$ satisfy the equation

$df^{\prime\prime}(\psi)+f^{\prime}(\varphi)\ast_{\theta^{\prime}}f^{\prime\prime}(\psi)-f^{\prime\prime}(\psi)\ast_{\theta^{\prime}}f^{\prime}(\varphi)=f^{\prime}(\hat{D}_{a}\psi)\,.$
(36)

Now it remains to make the following identifications:

$a\leftrightarrow\mathcal{A}\,,\qquad b\leftrightarrow\lambda\,,\qquad
f^{\prime}(\varphi)\leftrightarrow\hat{\mathcal{A}}(\mathcal{A})\,,\qquad
f^{\prime\prime}(\psi)\leftrightarrow\hat{\lambda}(\lambda,\mathcal{A})\,.$
(37)

With such identifications, Eq.(36) reproduces the equivalence condition (10)
if one set $\theta=0$ and then change $\theta^{\prime}\rightarrow\theta$.
Thus, we get the conclusion that the SW map is nothing but a quasi-isomorphism
$f$ of the dg-algebras $A_{0}$ and $A_{\theta}$ in the category of
$A_{\infty}$-algebras. Similarly, evaluating equations (32) on the chain

$\chi=\sum_{s,u,v=0}^{\infty}a^{\otimes s}\otimes\big{(}b_{1}\otimes
a^{\otimes u}\otimes b_{2}-b_{2}\otimes a^{\otimes u}\otimes
b_{1}\big{)}\otimes a^{\otimes v}\,,$ (38)

one can deduce the identity

$-f^{\prime\prime}(\hat{D}_{a}\chi)=f^{\prime\prime}(\psi_{1})\ast_{\theta^{\prime}}f^{\prime\prime}(\psi_{2})-f^{\prime\prime}(\psi_{2})\ast_{\theta^{\prime}}f^{\prime\prime}(\psi_{1})-f^{\prime\prime}(\psi_{12})\,,$
(39)

where

$\begin{array}[]{l}\displaystyle\psi_{i}=\sum_{s,u=0}^{\infty}a^{\otimes
s}\otimes b_{i}\otimes a^{\otimes u}\,,\quad i=1,2\,,\\\\[14.22636pt]
\displaystyle\psi_{12}=\sum_{s,u=0}^{\infty}a^{\otimes
s}\otimes(b_{1}\ast_{\theta}b_{2}-b_{2}\ast_{\theta}b_{1})\otimes a^{\otimes
u}\,.\end{array}$ (40)

Upon identification $b_{i}\leftrightarrow\lambda_{i}$, Rel. (39) reproduces
the gauge consistency condition (11).

Note that all the chains (33, 38, 40) are totally symmetric in permutations of
tensor factors. This means that we evaluate the multilinear maps $f_{l}$ on
the symmetric diagonal. Upon symmetrization, the maps $f_{l}$ define an
$L_{\infty}$-morphism between $A_{\theta}$ and $A_{\theta^{\prime}}$ treated
as dg-Lie algebras w.r.t. the commutators.

##  3 Explicit solutions and ambiguity

As a preparatory step, we introduce a complex that controls the solution
space. First, one can regard $B^{l}(A_{\theta})$ as the $l$th tensor power of
the complex $A_{\theta}[1]$ with the differential induced by $d$. Let $C^{l}$
denote the corresponding cochain complex with coefficients in
$A_{\theta^{\prime}}[1]$. By definition,

$C^{l}=\mathrm{Hom}(B^{l}(A_{\theta}),A_{\theta^{\prime}}[1])=\bigoplus_{k=-1}^{l}\mathrm{Hom}^{k}(B^{l}(A_{\theta}),A_{\theta^{\prime}}[1])\,.$
(41)

The second equality is due to the fact that the graded space
$B^{l}(A_{\theta})$ is concentrated in degrees $0,-1,\ldots,-l$, while
$A_{\theta^{\prime}}[1]$ is nested in degrees $0$ and $-1$. We denote the
differential in $C^{l}$ by $d_{l}$, that is, $d_{l}(f)=f\circ d$ for all $f\in
C^{l}$. The rationale behind the introduction of this sequence of complexes is
that the left- and right-hand sides of Eqs. (31, 32) define, respectively,
coboundaries and cocycles of the complex $C^{l}$ in degree one. As above by
$\bar{C}^{l}$ we denote the subcomplex of normal cochains.

It is known that any operator in the space of polynomials can be written as a
differential operator (possibly of infinite order) with polynomial
coefficients, see e.g. [12, Lem. 5.1]. This allows us to identify the cochains
of ${C}^{l}$ with polydifferential operators. Each polydifferential operator,
in its turn, is completely determined by its symbol. The differential $d_{l}$
makes then the space of symbols into a cochain complex, which we are going to
describe in more detail below.

Considering the dg-algebra $A_{\theta}$ just as a complex, denote by
$\mathrm{B}$ the complex dual to $A_{\theta}$. By definition, the cochains of
$\mathrm{B}=\mathrm{B}_{-1}\oplus\mathrm{B}_{0}$ are given by the sums

$\phi=\phi^{0}(p)+\phi^{i}(p)\zeta_{i}\,,$ (42)

where $\phi^{0}$ and $\phi^{i}$ are formal power series in the commuting
indeterminates $p_{1},\ldots,p_{n}$ and we prescribe the following degrees to
the formal variables:

$\deg p_{i}=0\,,\qquad\deg\zeta_{i}=-1\,.$ (43)

The dual differential $\partial:\mathrm{B}\rightarrow\mathrm{B}$ increases the
degree by one and acts on cochains (42) by the rule

$\partial\phi=p_{i}\phi^{i}(p)\,.$ (44)

The complex $(\mathrm{B},\partial)$ enjoys the natural descending filtration
$F^{k}B\supset F^{k+1}B$ by powers of the $p$’s. Define the subspace
$Z=Z_{-1}\oplus Z_{0}$ of $\mathrm{B}$ as

$Z_{-1}=\big{\\{}\phi=\phi^{i}(p)\zeta_{i}\;\big{|}\;p_{i}\phi^{i}(p)=0\big{\\}}\,,\qquad
Z_{0}=\big{\\{}\phi=c\in\mathbb{R}\big{\\}}\,.$ (45)

The nonzero elements of $Z$ are obviously nontrivial $\partial$-cocycles and
$H(\mathrm{B})\simeq Z$.

Considering now the space $A_{\theta}$ as a complex with zero differential, we
define the tensor product of complexes

$\mathrm{B}^{l}=A_{\theta}\otimes\mathrm{B}^{\otimes l}\,.$ (46)

Let $\partial_{l}$ denote the coboundary operator in $\mathrm{B}^{l}$. It is
convenient to define the degree of a homogeneous cochain

$\Phi=a\otimes\phi_{1}\otimes\cdots\otimes\phi_{l}\in\mathrm{B}^{l}$ (47)

to be equal to

$\deg\Phi=\deg\phi_{1}+\cdots+\deg\phi_{l}+\deg a+l-1\,.$

For this grading, the complex $\mathrm{B}^{l}$ is concentrated in degrees
$-1,0,1,\ldots,l$.

Since we are working with complexes of $\mathbb{R}$-vector spaces, the Künneth
formula yields immediately

$H(\mathrm{B}^{l})=A_{\theta}\otimes H(\mathrm{B})^{\otimes l}\,.$ (48)

For each cochain (42) one can assign the map
$\sigma(\phi):A_{\theta}\rightarrow(A_{\theta})_{0}$ that takes (17) to

$\sigma(\phi)(a)=\phi^{0}(\partial_{i})a_{0}(x)+\phi^{j}(\partial_{i})a_{j}(x)\,,$

where $\partial_{i}=\partial/\partial x^{i}$. Similarly, to each cochain (47)
one can assign the multilinear operator of $C^{k}$ by setting

$\Sigma(\Phi)(a_{1},\ldots,a_{l})=a\cdot\sigma(\phi_{1})(a_{1})\cdot\sigma(\phi_{2})(a_{2})\cdots\sigma(\phi_{l})(a_{l})\,.$
(49)

Here the dot stands for the usual product of polynomials and the result is
treated as an element of $A_{\theta^{\prime}}$. This assignment extends to the
whole of $\mathrm{B}^{l}$ by linearity.

The filtration in $\mathrm{B}$ induces that in the complex $\mathrm{B}^{l}$
and we use the bold-font letter $\mathbf{B}^{l}$ to denote the completion of
$\mathrm{B}^{l}$ w.r.t. this filtration. The map $\Phi\mapsto\Sigma(\Phi)$
extends then to ${\mathbf{B}}^{l}$ and we arrive at the following

######  Proposition 3.1.

The map $\Sigma:\mathbf{B}^{l}\rightarrow C^{l}$ is an isomorphism of
complexes, i.e., $d_{l}\Sigma=\Sigma\partial_{l}$.

It is the elements of $\mathbf{B}^{l}$ which are identified with the symbols
of operators. The isomorphism $\Sigma$ allows one to transfer the notion of
normality from $C^{l}$ to $\mathbf{B}^{l}$. Clearly, the cochain (49) is
normal iff $\phi_{1}^{0}(0)=\cdots=\phi^{0}_{l}(0)=0$. The normal cochains of
$\mathbf{B}^{l}$ form a subcomplex, which we denote by $\bar{\bf B}^{l}$. The
formula

$f(p)-f(0)=p_{i}\int_{0}^{1}dt\left(\frac{\partial f}{\partial
p_{i}}\right)(tp)\,,$ (50)

valid for any formal power series $f(p)$, shows that each normal zero-chain
$\phi=\phi^{0}(p)\in\mathrm{B}^{1}_{0}$ is a coboundary of $\mathrm{B}^{1}$.
It follows from the Künneth formula that

$H^{n}(\bar{\mathrm{B}}^{l})=\left\\{\begin{array}[]{l}(A_{\theta})_{0}\otimes
Z_{-1}^{\otimes l}\,,\quad n=-1\,;\\\\[8.53581pt] (A_{\theta})_{1}\otimes
Z_{-1}^{\otimes l}\,,\quad n=0\,;\\\\[8.53581pt] 0\,,\quad
n>0\,.\\\\[8.53581pt] \end{array}\right.$ (51)

Here $Z_{-1}$ stands for the subspace of cocycles (45). Since the differential
$\partial_{l}$ is acyclic in positive degrees when restricted to the
subcomplex of normal cochains $\bar{\mathbf{B}}^{l}$, so is the differential
$d_{l}$ in the isomorphic complex $\bar{C}^{l}$. In particular,
$H^{1}(\bar{C}^{l})=0$. This means the absence of obstructions to the
solubility of Eqs. (31, 32).

To write down an explicit expression for $f$ we need to choose an appropriate
homotopy operator that relates the identity map with a projection onto the
subspace of nontrivial cocycles (51). First, we introduce the projector
$\pi:B\rightarrow B$ onto the subspace $Z\subset B$ defined by (45). For every
(42), we set

$\pi\phi=\phi^{0}(0)+\phi^{i}(p)\zeta_{i}-\zeta_{j}\frac{\partial}{\partial
p_{j}}\int_{0}^{1}p_{i}\phi^{i}(tp)dt\,.$

One can see that $\pi^{2}=\pi$ and $\mathrm{Im}\,\pi=Z$. Furthermore, $\pi$ is
homotopic to the identity map. The corresponding homotopy operator
$h:B\rightarrow B$ is given by

$h\phi=\zeta_{i}\int_{0}^{1}\left(\frac{\partial\phi^{0}}{\partial
p_{i}}\right)(tp)dt\,.$ (52)

It is straightforward to check that

$\partial h+h\partial=1-\pi$ (53)

and

$h^{2}=0\,,\qquad\pi h=0\,,\qquad h\pi=0\,.$ (54)

Next, we extend the maps $\pi$ and $h$ from the complex $B$ to the tensor
product $B^{l}$ by the standard formulas of homological algebra. For any
cochain (47), we set

$\pi_{l}(\Phi)=a_{0}\otimes\pi\phi_{1}\otimes\cdots\otimes\pi\phi_{l}\,,$ (55)
$h_{l}(\Phi)=\sum_{i=1}^{l}a_{0}\otimes\phi_{1}\otimes\cdots\otimes\phi_{i-1}\otimes
h\phi_{i}\otimes\pi\phi_{i+1}\otimes\cdots\otimes\pi\phi_{l}\,.$ (56)

Then

$\partial_{l}h_{l}+h_{l}\partial_{l}=1-\pi_{l}\,,$ (57)

and

$h_{l}^{2}=0\,,\qquad\pi_{l}h_{l}=0\,,\qquad
h_{l}\pi_{l}=0\,,\qquad\pi_{l}^{2}=\pi_{l}\,.$ (58)

The isomorphism $\Sigma$ allows one to push forward the operators $\pi_{l}$
and $h_{l}$ from $\mathbf{B}^{l}$ to $C^{l}$. This results in the operators
$h^{\Sigma}_{l}=\Sigma h_{l}\Sigma^{-1}$ and
$\pi^{\Sigma}_{l}=\Sigma\pi_{l}\Sigma^{-1}$, which satisfy the same Rels. (57,
58).

Let us set $f^{\prime\prime}_{l}=\sum_{k=0}^{l-1}f^{\prime\prime}_{l,k}$ and
rewrite Eqs. (31, 32) in the form

$d_{l}f^{\prime}_{l}=\Theta^{\prime}_{l}[f^{\prime}_{1},\ldots,f^{\prime}_{l-1};f^{\prime\prime}_{1},\ldots,f^{\prime\prime}_{l}]\,,\qquad
d_{l}f^{\prime\prime}_{l}=\Theta^{\prime\prime}_{k}[f^{\prime\prime}_{1},\ldots,f_{l-1}^{\prime\prime}]\,.$
(59)

Here the $\Theta$’s are determined by the r.h.s. of Eqs. (31, 32).

Since
$\pi_{l}^{\Sigma}\Theta_{l}^{\prime}=\pi_{l}^{\Sigma}\Theta_{l}^{\prime\prime}=0$,
applying the homotopy operator $h^{\Sigma}_{l}$ brings system (59) into the
sequence of recurrence relations that determine the morphism $f$:

$\begin{array}[]{l}f^{\prime\prime}_{l}=h^{\Sigma}_{l}\Theta^{\prime\prime}_{l}[f^{\prime\prime}_{1},\ldots,f_{l-1}^{\prime\prime}]\,,\qquad
l=2,3,\ldots;\\\\[8.53581pt]
f^{\prime}_{m}=h^{\Sigma}_{l}\Theta^{\prime}_{l}[f^{\prime}_{1},\ldots,f^{\prime}_{m-1};f^{\prime\prime}_{1},\ldots,f^{\prime\prime}_{m}]\,,\qquad
m=1,2,\ldots\end{array}$ (60)

As the initial value for these recurrence relations, one takes a unital
cochain $f^{\prime\prime}_{1}$. It follows from Proposition 2.2 that the
cochains $\Theta^{\prime\prime}_{l}$ and $\Theta^{\prime}_{l}$ are normal
whenever so are the cochains $f^{\prime\prime}$’s and $f^{\prime}$’s they are
built from. Since the operator $h^{\Sigma}_{l}$ respects normality, the
resulting cochains $f^{\prime\prime}_{l}$ and $f^{\prime}_{l}$ are also
normal. Thus, by induction, we see that relations (60) are well-defined
provided $f^{\prime\prime}_{1}(1)=1$. Since
$H^{0}(A_{\theta})\simeq\mathbb{R}$, the last condition implies that $f_{1}$
induces an isomorphism $H(f_{1}):H^{0}(A_{\theta})\rightarrow
H^{0}(A_{\theta^{\prime}})$ of the zero cohomology groups. To be a quasi-
isomorphism, $f$ must also generate a cohomology isomorphism in degree one.
The simplest way to satisfy the quasi-isomorphism condition is to take
$f^{\prime\prime}_{1}$ to be the identity map on the space of polynomials
$a_{0}(x)$, see (17). Then
$f^{\prime}_{1}=h_{1}^{\Sigma}\Theta_{1}^{\prime}=h_{1}^{\Sigma}df^{\prime\prime}_{1}$
is again the identity map on the polynomials $a_{i}(x)\xi^{i}$. On the other
hand, the identity map
$f_{1}=(f_{1}^{\prime},f^{\prime\prime}_{1})=\mathrm{id}$, being an
isomorphism of complexes, trivially satisfies Eq. (31) for $l=1$. Therefore,
$f_{1}=\mathrm{id}$ can always be chosen as an initial value for the
recurrence relations (60). Exactly this choice corresponds to the classical
Seiberg–Witten map (9, 12). With $f_{1}=\mathrm{id}$, the total
$A_{\infty}$-morphism $f$ is obviously invertible and we conclude that the
algebras $(A_{\theta},d)$ and $(A_{\theta^{\prime}},d)$, regarded as
$A_{\infty}$-algebras, are isomorphic to each other.

In principle, one could try to construct an $A_{\infty}$-morphism whose
component maps $f_{l}$ with $l>1$ are not normal or $f_{1}$ is not unital.
However, under such weak assumptions on the $f_{l}$’s, the solubility of Eqs.
(31, 32) gets out of control because all obstruction spaces $H^{1}(C^{l})\neq
0$. Therefore, the existence of such ‘abnormal’ $A_{\infty}$-morphisms is not
clear at the moment.

It remains to discuss the arbitrariness in the solutions of Eqs. (31, 32)
under the assumption of normality. Since, at each order $l$, we have a system
of inhomogeneous linear equations $\delta f_{l}=\Psi_{l}$ for the known
$\Psi_{l}$ and unknown $f_{l}$, the solution space is isomorphic to the space
of normal $\delta$-cocycles in degree zero. In other words, it is possible to
shift a solution $f_{l}=(f^{\prime}_{l},f^{\prime\prime}_{l})$ by any
$\delta$-cocycle $g_{l}=(g_{l}^{\prime},g_{l}^{\prime\prime})\in\bar{C}^{l}$.
According to (51), one can always choose $g_{l}^{\prime}=z_{l}+dc_{l}$ and
$g_{l}^{\prime\prime}=d_{l}c_{l}$, where $\Sigma^{-1}(z_{l})$ belongs to the
completion of the space $(A_{\theta})_{1}\otimes Z_{-1}^{\otimes l}$, while
$c_{l}$ is an arbitrary homomorphism from $(A_{\theta})_{1}^{\otimes l}$ to
$(A_{\theta^{\prime}})_{0}$. In particular, two cochain transformations
$f_{1},\tilde{f}_{1}:A_{\theta}\rightarrow A_{\theta^{\prime}}$, considered up
to homotopy, differ from each other by an element $[z_{1}]\in
H^{0}(\bar{C}^{1})$. In the context of SW maps, these ambiguities have been
discussed in many physical papers, see e.g. [10, 8, 13, 11, 14, 15] and the
references therein. Since there is some confusion in the literature about
which ambiguities should be considered trivial and which are not, we will
dwell on this issue in more detail.

Isolating all the terms that are linear in $f$’s on the left and summing up
over $l$, we can write the sequence of equations (28) as a single
Maurer–Cartan (MC) equation, namely,

$Df=-\frac{1}{2}[f,f]\,,\qquad f=\sum_{l=1}^{\infty}f_{l}\,.$ (61)

Here the differential $D$ on the left involves $\delta$ and the product in
$A$, while the commutator in $A^{\prime}$ gives the Lie bracket on the right.
The differential and the bracket make the space
$\mathrm{Hom}(B(A),A^{\prime}[1])[1]$ into a dg-Lie algebra
$\mathcal{L}=\bigoplus\mathcal{L}_{n}$. The algebra $\mathcal{L}$ has the
descending filtration $F^{k}\mathcal{L}\supset F^{k+1}\mathcal{L}$ by the
number of arguments of homogeneous cochains. In particular,

$\sum_{l=n}^{\infty}f_{l}\in F^{n}\mathcal{L}\,.$ (62)

Due to the shift in degree, the morphism $f$ gets degree one and defines an MC
element of $\mathcal{L}_{1}$. Since $\mathcal{L}=F^{1}\mathcal{L}$, one can
formally exponentiate the Lie algebra $\mathcal{L}_{0}$ to produce the
corresponding Lie group $G=\exp(\mathcal{L}_{0})$. This group acts on the
space of MC elements by the following ‘gauge transformations’ [16]:

$f\quad\mapsto\quad f^{g}=gfg^{-1}-g^{-1}Dg\qquad\quad\forall g\in G\,.$ (63)

Two MC elements (= morphisms) $f$ and $\tilde{f}$ are said to be gauge (=
homotopy) equivalent if there exists an element $g\in G$ such that
$\tilde{f}=f^{g}$. In view of (63) it makes sense to consider the solutions to
the MC equation (61) up to gauge equivalence. Suppose now that two morphisms
$f$ and $\tilde{f}$ coincide up to the order $l-1$ in filtration degree, i.e.,
$\tilde{f}-f\in F^{l}\mathcal{L}_{1}$. Then it follows from (28) that
$\delta(\tilde{f}_{l}-f_{l})=0$. If the $\delta$-cocycle
$g_{l}=\tilde{f}_{l}-f_{l}$ happens to be trivial, that is $g_{l}=\delta
c_{l}$, then one can perform the gauge transformation $f\mapsto f^{g}$ with
$g=\exp(c_{l})$. It is readily seen that $\tilde{f}-f^{g}\in
F^{l+1}\mathcal{L}$; and hence, the morphisms $\tilde{f}$ and $f^{g}$ coincide
up to the order $l$. We thus conclude that the first nontrivial class of
$\delta$-cohomology $\big{[}\tilde{f}_{l}-f_{l}\big{]}$ represents an
obstruction to the homotopy equivalence of two $A_{\infty}$-morphisms and the
corresponding SW maps. A more systematic exposition of the homotopy theory of
$A_{\infty}$ and other homotopy algebras can be found in [17].

##  4 Discussion

We presented an algebraic treatment of the Seiberg–Witten map as a quasi-
isomorphism of $A_{\infty}$-algebras. While in the original physical setting,
the very existence of the SW map requires more or less nontrivial reasoning or
calculations, it becomes a simple consequence of the dg-algebra quasi-
isomorphism (20) in the algebraic interpretation. A simple homotopy operator
(52) allows one to write down the corresponding SW map up to any given order
in the gauge field $\mathcal{A}$. The simplicity of the quasi-isomorphism
construction owes its existence to two facts:

  * (A)

The dg-algebras under consideration are isomorphic as cochain complexes and
differ only in multiplication;

  * (B)

Every nontrivial cocycle has a central representative (homologically
equivalent to an element from the centre $Z(A)$ of the algebra $A$).

Together, these properties provide the quasi-isomorphism $A\leftarrow
Z\rightarrow A^{\prime}$ with $Z=\ker d\cap Z(A)$.

There are many dg-algebras satisfying conditions A and B. For instance, one
can replace the last relation in (14) with the standard anticommutativity
condition $\xi^{i}\xi^{j}-\xi^{j}\xi^{i}=0$ for the odd generators. The result
may be viewed as a noncommutative version of the de Rham complex with
$dx^{i}=\xi^{i}$. Let us denote it by $\Omega_{\theta}$. The ordinary
(commutative) de Rham complex corresponds to $\theta=0$. Any two dg-algebras
$\Omega_{\theta}$ and $\Omega_{\theta^{\prime}}$ obviously satisfy the
conditions above; hence, they are quasi-isomorphic. Furthermore, the dg-
algebra $\Omega_{\theta}$ admits the natural descending filtration
$F^{k}\Omega_{\theta}\supset F^{k+1}\Omega_{\theta}$ by the powers of the
$\xi$’s. This allows one to define the sequence of dg-algebras
$A^{{}_{(k)}}_{\theta}=\Omega_{\theta}/F^{k}\Omega_{\theta}$. The dg-algebra
with relations (14) corresponds to $k=2$. It is clear that any pair of dg-
algebras from the family $A^{{}_{(k)}}_{\theta}$ satisfies conditions A and B
whenever $k>1$. (For $k=1$, $H(A_{\theta}^{{}_{(1)}})\simeq
A_{\theta}^{{}_{(1)}}$ and condition B is violated unless $\theta=0$.) The
physical interpretation of the algebras $A_{\theta}^{{}_{(k)}}$ with $k>2$ is
not clear at the moment, except for the case $k=3$. As was shown in [9], one
can identify the additional homogeneous subspace $(A_{\theta}^{{}_{(3)}})_{2}$
in degree $2$ with the space of equations of motion (e.g. $\mathcal{F}=0$ as
in Chern–Simons theory). Then the extra components of a quasi-isomorphism $f$
with arguments in $(A_{\theta}^{{}_{(3)}})_{2}$ allow one to cover the case of
‘open gauge algebra’ when the gauge consistency condition (11) involves an
extra term vanishing on-shell.

All the above constructions extend immediately to the tensor product of
$A^{{}_{(k)}}_{\theta}$ with the matrix algebra $\mathrm{Mat}_{N}$. Although
the cohomology algebra of $A^{{}_{(k)}}_{\theta}\otimes\mathrm{Mat}_{N}$ is no
longer Abelian, the subalgebra of cocycles $Z$ is still independent of
$\theta$, which is sufficient for the quasi-isomorphism. Upon replacing
$\theta\rightarrow i\theta$, one can also reduce the matrix factor
$\mathrm{Mat}_{N}$ to the Lie subalgebra of anti-Hermitian matrices $u(N)$.
More general matrix factors were considered in [18, 19, 20].

Another class of relevant dg-algebras is provided by the Wess–Zumino (WZ)
complexes [21]. The corresponding dg-algebras are generated by the same formal
variables $x^{i}$ and $\xi^{i}=dx^{i}$ but the relations on them are
essentially different. For the general discussion of WZ complexes, we refer
the reader to [22, 23]. By way of illustration, we present the following (not
most general) commutation relations:

$\begin{array}[]{ll}x^{i}x^{j}-q^{ji}x^{j}x^{i}=0\,,&\quad\xi^{i}x^{j}-q^{ji}x^{j}\xi^{i}=0\,,\\\\[8.53581pt]
\xi^{i}\xi^{j}+q^{ji}\xi^{j}\xi^{i}=0\,,&\quad(\xi^{i})^{2}=0\,.\end{array}$
(64)

Here $q^{ij}$ is a set of real numbers subject to the condition
$q^{ji}q^{ij}=q^{ii}=1$. The dg-algebra generated by these $x$’s and $\xi$’s,
let us denote it by $\Omega_{q}$, enjoys the PBW property and is isomorphic,
as a vector space, to the usual de Rham complex. Moreover, one can see that
the Poincaré Lemma holds true for $\Omega_{q}$, meaning
$H(\Omega_{q})\simeq\mathbb{R}\subset\Omega_{q}$. This gives the quasi-
isomorphism $\Omega_{q}\leftarrow\mathbb{R}\rightarrow\Omega_{q^{\prime}}$ of
dg-algebras. Again, the natural grading on $\Omega_{q}$ by powers of the
$\xi$’s gives rise to the descending filtration $F^{k}\Omega_{q}\supset
F^{k+1}\Omega_{q}$, so that one can define the sequence of dg-algebras
$A^{{}_{(k)}}_{q}=\Omega_{q}/F^{k}\Omega_{q}$. The algebra $A^{{}_{(1)}}_{q}$,
generated by the $x$’s, is often called the algebra of ‘quantum polynomials’.
Accordingly, the vector space spanned by the $x$’s is referred to as the
‘quantum space’. The next algebra, $A^{{}_{(2)}}_{q}$, is the quantum
counterpart of the algebra (14) underlying the Weyl–Moyal noncommutativity. As
with $A^{{}_{(2)}}_{\theta}$, the cocycles of $A^{{}_{(2)}}_{q}$ form a
commutative algebra; hence, all the dg-algebras of the family
$A^{{}_{(2)}}_{q}$ are quasi-isomorphic to each other. It should be mentioned
that noncommutative field theory on quantum spaces was considered in [1],
albeit with completely different differential and commutation relations among
the $x$’s and $\xi$’s. Some examples of noncommutative gauge theories related
to WS complexes have been elaborated in the recent paper [24].

## Acknowledgements

We are grateful to Vasily Dolgushev and Maxim Grigoriev for useful discussions
and references. This work was partially supported by the São Paulo Research
Foundation (FAPESP), grants 2021/09313-8 and 2022/13596-8, and by the CNPq
grant 304130/2021-4. A. Sh. acknowledges the financial support from the
Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”.

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