Formality morphism as the mechanism of $\star$-product associativity: how it works
The formality morphism $\boldsymbol{\mathcal{F}}=\{{\mathcal{F}}_n$, $n\geqslant1\}$ in Kontsevich's deformation quantization is a collection of maps from tensor powers of the differential graded Lie algebra (dgLa) of multivector fields to the dgLa of polydifferential operators on finite-di...
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Інститут математики НАН України
2019
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Репозитарії
Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552860469690368 |
|---|---|
| author | Buring, R. Kiselev, A. Бурінг, Р. Кісєльов, А. |
| author_facet | Buring, R. Kiselev, A. Бурінг, Р. Кісєльов, А. |
| author_institution_txt_mv | [
{
"author": "R. Buring",
"institution": "Johannes Gutenberg University Mainz"
},
{
"author": "A. Kiselev",
"institution": "University of Groningen"
}
] |
| author_sort | Buring, R. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2020-08-13T08:52:53Z |
| description | The formality morphism $\boldsymbol{\mathcal{F}}=\{{\mathcal{F}}_n$, $n\geqslant1\}$ in Kontsevich's deformation quantization is a collection of maps from tensor powers of the differential graded Lie algebra (dgLa) of multivector fields to the dgLa of polydifferential operators on finite-dimensional affine manifolds. Not a Lie algebra morphism by its term ${\mathcal{F}}_1$ alone, the entire set $\boldsymbol{{\mathcal{F}}}$ is an $L_\infty$-morphism instead. It induces a map of the Maurer-Cartan elements,  taking Poisson bivectors to deformations $\mu_A\mapsto\star_{A[[\hbar]]}$ of the usual multiplication of functions into associative noncommutative $\star$-products of power series in $\hbar$. The associativity of $\star$-products is then realized, in terms of the Kontsevich graphs which encode polydifferential operators, by differential consequences of the Jacobi identity. The aim of this paper is to illustrate the work of this algebraic mechanism for the Kontsevich $\star$-products (in particular, with harmonic propagators). |
| first_indexed | 2026-08-04T01:06:09Z |
| format | Article |
| fulltext |
Çáiðíèê ïðàöü Iíñòèòóòó ìàòåìàòèêè ÍÀÍ Óêðà¨íè 2019, ò. 16, � 1, 22�43
ÓÄÊ 517.912:512.816
Formality morphism as the
mechanism of ?-product
associativity: how it works
R. Buring †, A.V. Kiselev ‡
† Johannes Gutenberg�Universit�at, Mainz, Germany
E-mail: rburing@uni-mainz.de
‡ University of Groningen, The Netherlands
E-mail: A.V.Kiselev@rug.nl
Ìîðôiçì F = {Fn, n > 1}, ùî çàáåçïå÷ó¹ âëàñòèâiñòü ôîðìàëüíîñ-
òi àëãåáð â äåôîðìàöiéíîìó êâàíòóâàííi çà Êîíöåâè÷åì, çàäàíî íà-
áîðîì âiäîáðàæåíü òåíçîðíèõ ñòåïåíåé äèôåðåíöiàëüíî¨ ãðàäóéîâàíî¨
àëãåáðè Ëi (dgLa) ìóëüòèâåêòîðíèõ ïîëiâ â dgLa ïîëiäèôåðåíöiàëüíèõ
îïåðàòîðiâ íà ñêií÷åííîâèìiðíèõ àôiííèõ ìíîãîâèäàõ. Õî÷à ïåðøèé
÷ëåí F1 ñàì ïî ñîái íå ¹ ìîðôiçìîì àëãåáð Ëi, ïîñëiäîâíiñòü F â öiëî-
ìó ¹ L∞-ìîðôiçìîì. Íà éîãî îñíîâi áóäó¹òüñÿ âiäîáðàæåííÿ åëåìåíòiâ
Ìàóðåðà�Êàðòàíà, ÿêå ñòàâèòü ó âiäïîâiäíiñòü ïóàñîíîâèì áiâåêòîðàì
äåôîðìàöi¨ µA 7→ ?A[[~]], ùî äîáóäîâóþòü çâè÷àéíå ìíîæåííÿ ôóíêöié
äî àñîöiàòèâíèõ íåêîìóòàòèâíèõ ?-äîáóòêiâ íà ïðîñòîði ñòåïåíåâèõ ðÿ-
äiâ ïî ~. Ïðè öüîìó àñîöèàòèâíiñòü ?-äîáóòêiâ çàáåçïå÷åíî � íà ìî-
âi ãðàôiâ Êîíöåâè÷à, ùî ïðåäñòàâëÿþòü ïîëiäèôåðåíöiàëüíi îïåðàòî-
ðè, � äèôåðåíöiàëüíèìè íàñëiäêàìè òîòîæíîñòi ßêîái. Ìåòà ðîáîòè �
ïðîiëþñòðóâàòè öåé àëãåáðà¨÷íèé ìåõàíiçì äëÿ ?-äîáóòêiâ Êîíöåâè÷à
(çîêðåìà, ç ãàðìîíi÷íèìè ïðîïàãàòîðàìè).
The formality morphism F = {Fn, n > 1} in Kontsevich's deformation
quantization is a collection of maps from tensor powers of the di�eren-
tial graded Lie algebra (dgLa) of multivector �elds to the dgLa of polydif-
ferential operators on �nite-dimensional a�ne manifolds. Not a Lie algebra
morphism by its term F1 alone, the entire set F is an L∞-morphism
instead. It induces a map of the Maurer�Cartan elements, taking Pois-
son bi-vectors to deformations µA 7→ ?A[[~]] of the usual multiplication
of functions into associative noncommutative ?-products of power series
in ~. The associativity of ?-products is then realized, in terms of the
Kontsevich graphs which encode polydi�erential operators, by di�erential
consequences of the Jacobi identity. The aim of this paper is to illustrate
the work of this algebraic mechanism for the Kontsevich ?-products (in
particular, with harmonic propagators).
Formality morphism as the mechanism of ?-product associativity 23
1. Introduction. The Kontsevich formality morphism F relates
two differential graded Lie algebras (dgLa). Its domain of definition is
the shifted-graded vector space T
↓[1]
poly(Mr) of multivectors on an affine
real finite-dimensional manifold Mr; the graded Lie algebra structure is
the Schouten bracket [[ , ]] and the differential is set to (the bracket with)
zero by definition. On the other hand, the target space of the formality
morphism F is the graded vector space D
↓[1]
poly(Mr) of polydifferential
operators on Mr; the graded Lie algebra structure is the Gerstenhaber
bracket [ , ]G and the differential dH = [µA, ·] is induced by using the
multiplication µA in the algebra A := C∞(Mr) of functions on Mr.
It is readily seen that w.r.t. the above notation, Poisson bi-vectors P
satisfying the Jacobi identity [[P,P]] = 0 on Mr are the Maurer–Cartan
elements (indeed, (d ≡ 0)(P) + 1
2 [[P,P]] = 0). Likewise, for a (non)com-
mutative star-product ? = µA[[~]] + 〈tail =: B〉, which deforms the usual
multiplication µ = µA[[~]] in A[[~]] = C∞(Mr)⊗RR[[~]] by a tail B w.r.t.
a formal parameter ~, the requirement that ? be associative again is the
Maurer–Cartan equation,
[µ,B]G + 1
2 [B,B]G = 0 ⇐⇒ 1
2 [µ+B,µ+B]G = 0.
Here, the leading order equality [µ, µ]G = 0 expresses the given associa-
tivity of the product µ itself.
The Kontsevich formality mapping F = {Fn : T⊗npoly → Dpoly, n > 1}
in [15, 16] is an L∞-morphism which induces a map that takes Maurer–
Cartan elements P, i.e., formal Poisson bi-vectors P̃ = ~P+ ō(~) on Mr,
to Maurer–Cartan elements1, i.e., the tails B in solutions ? of the asso-
ciativity equation on A[[~]].
The theory required to build the Kontsevich map F is standard, well
reflected in the literature (see [15, 16], as well as [9, 11] and references
therein); a proper choice of signs is analysed in [2, 20]. The framework of
homotopy Lie algebras and L∞-morphisms, introduced by Schlessinger–
Stasheff [19], is available from [17], cf. [10] in the context of present
paper.
So, the general fact of (existence of) factorization,
Assoc(?)(P)(f, g, h) = 3
(
P, [[P,P]]
)
(f, g, h), f, g, h ∈ A[[~]], (1)
is known to the expert community. Indeed, this factorization is im-
mediate from the construction of L∞-morphism in [16, Section 6.4].
1In fact, the morphism F is a quasi-isomorphism (see [16, Theorem 6.3]), inducing
a bijection between the sets of gauge-equivalence classes of Maurer–Cartan elements.
24 R. Buring, A.V. Kiselev
We shall inspect how this mechanism works in practice, i.e., how pre-
cisely the ?-product is made associative in its perturbative expansion
whenever the bi-vector P is Poisson, thus satisfying the Jacobi identity
Jac(P) := 1
2 [[P,P]] = 0. To the same extent as our paper [6] justifies
a similar factorization, [[P,Q(P)]] = 3
(
P, [[P,P]]
)
, of the Poisson cocycle
condition for universal deformations Ṗ = Q(P) of Poisson structures,2
we presently motivate the findings in [5] for ? mod ō
(
~3
)
, proceeding
to the next order ? mod ō
(
~4
)
from [7] (and higher orders, recently
available from [3]).3 Let us emphasize that the theoretical constructions
and algorithms (contained in the computer-assisted proof scheme under
study and in the tools for graph weight calculation) would still work at
arbitrarily high orders of expansion ? mod ō
(
~k
)
as k → ∞. Explicit
factorization (1) up to ō
(
~k
)
helps us build the star-product ? mod ō
(
~k
)
by using a self-starting iterative process, because the Jacobi identity
for P is the only obstruction to the associativity of ?. Specifically, the
Kontsevich weights of graphs on fewer vertices (yet with a number of
edges such that they do not show up in the perturbative expansion of ?)
dictate the coefficients of Leibniz orgraphs in operator 3 at higher orders
in ~. These weights in the r.h.s. of (1) constrain the higher-order weights
of the Kontsevich orgraphs in the expansion of ?-product itself. This is
important also in the context of a number-theoretic open problem about
the (ir)rational value (const ∈ Q\{0})·ζ(3)2/π6 +(const ∈ Q) of a graph
weight at ~7 in ? (see [12] and [3]).
Our paper is structured as follows. First, we fix notation and recall
some basic facts from relevant theory. Secondly, we provide three examp-
les which illustrate the work of formality morphism in solving Eq. (1).
Specifically, we read the operators 3k = 3 mod ō
(
~k
)
satisfying
Assoc(?)(P)(f, g, h) mod ō
(
~k
)
= 3k
(
P, [[P,P]]
)
(f, g, h) (1′)
at k = 2, 3, and 4. This corresponds to the expansions ? mod ō
(
~k
)
in [16], [5], and [7], respectively. One can then continue with k = 5, 6;
these expansions are in [3]. Independently, one can probe such factor-
izations using other stable formality morphisms: for instance, the ones
2Universal w.r.t. all Poisson brackets on all finite-dimensional affine manifolds,
such infinitesimal deformations were pioneered in [15]; explicit examples of these
flows Ṗ = Q(P) are given in [4, 6, 8].
3Note that both the approaches – to noncommutative associative ?-products and
deformations of Poisson structures – rely on the same calculus of oriented graphs by
Kontsevich [13, 14, 15, 16].
Formality morphism as the mechanism of ?-product associativity 25
which correspond to a different star-product, the weights in which are
determined by a logarithmic propagator instead of the harmonic one
(see [1, 18]).
2. Two differential graded Lie algebra structures. Let Mr
be an r-dimensional affine real manifold (we set k = R for simplici-
ty). In the algebra A := C∞(Mr) of smooth functions, denote by µA
(or equivalently, by the dot ·) the usual commutative, associative, bi-
linear multiplication. The space of formal power series in ~ over A will
be A[[~]] and the ~-linear multiplication in it is µ (instead of µA[[~]]).
Consider two differential graded Lie algebra structures. First, we have
that the shifted-graded space T
↓[1]
poly(Mr) of multivector fields on Mr
is equipped with the shifted-graded skew-symmetric Schouten bracket
[[ , ]] (itself bi-linear by construction and satisfying the shifted-graded
Jacobi identity); the differential is set to zero. Secondly, the vector
space D
↓[1]
poly(Mr) of polydifferential operators (linear in each argument
but not necessarily skew over the set of arguments or a derivation in any
of them) is graded by using the number of arguments m: by definition,
let deg(θ(m arguments)) := m− 1. For instance, deg(µA) = 1. The Lie
algebra structure on D
↓[1]
poly(Mr) is the Gerstenhaber bracket [ , ]G; for
two homogeneous operators Φ1 and Φ2 it equals [Φ1,Φ2]G = Φ1 ~◦ Φ2 −
(−1)deg Φ1·deg Φ2Φ2 ~◦ Φ1, where the directed, non-associative insertion
product is, by definition
(Φ1 ~◦ Φ2)(a0, . . . , ak1+k2) =
k1∑
i=0
(−1)ik2Φ1
(
a0 ⊗ · · · ⊗ ai−1
⊗ Φ2(ai ⊗ · · · ⊗ ai+k2
)⊗ ai+k2+1 ⊗ · · · ⊗ ak1+k2
)
.
In the above, Φi : A
⊗(ki+1) → A so that aj ∈ A.
Example 1. The associativity of the product µA in the algebra of func-
tions A = C∞(Mr) is the statement that
µ
(1)
A
(
µ
(2)
A (a0, a1), a2
)
+ (−1)(i=1)·(deg µA=1)µ
(1)
A
(
a0, µ
(2)
A (a1, a2)
)
− (−1)(deg µ
(1)
A =1)·(deg µ
(2)
A =1)
{
µ
(1)
A
(
µ
(1)
A (a0, a1), a2
)
−µ(2)
A
(
a0, µ
(1)
A (a1, a2)
)}
= 2
{
(a0 · a1) · a2− a0 · (a1 · a2)
}
= 0.
So, the associator Assoc(µA)(a0, a1, a2) = 1
2 [µA, µA]G (a0, a1, a2) = 0 for
any aj ∈ A.
26 R. Buring, A.V. Kiselev
Like [[·, ·]], the Gerstenhaber bracket satisfies the shifted-graded Ja-
cobi identity. The Hochshild differential on D
↓[1]
poly(Mr) is dH = [µA, ·]G;
indeed, its square vanishes, d2
H = 0, due to the Jacobi identity for [ , ]G
into which one plugs the equality [µA, µA]G = 0.
3. The Maurer–Cartan elements. In every differential graded
Lie algebra with a Lie bracket [ , ], the Maurer–Cartan (MC) elements
are solutions of degree 1 for the Maurer–Cartan equation
dα+ 1
2 [α, α] = 0, (2)
where d is the differential (equal, we recall, to dH =[µA, ·]G onD
↓[1]
poly(Mr)
and zero identically on T
↓[1]
poly(Mr). Likewise, the Lie algebra structure
[·, ·] is the Gerstenhaber bracket [·, ·]G and the Schouten bracket [[·, ·]],
respectively.)
Now tensor the degree-one parts of both dgLa structures with ~·k[[~]],
i.e., with formal power series starting at ~1, and, preserving the nota-
tion (that is, extending the brackets and the differentials by ~-linearity),
consider the same Maurer–Cartan equation (2). Let us study its formal
power series solutions α = ~1α1 + · · · .
So far, in the Poisson world we have that the Maurer–Cartan bi-
vectors are formal Poisson structures 0+~P1 + ō(~) satisfying (2), which
is [[~P1 + ō(~), ~P1 + ō(~)]] = 0 with zero differential. In the world of
associative structures, the Maurer–Cartan elements are the tails B in
expansions ? = µ + B, so that the associativity equation [?, ?]G = 0
reads (for [µ, µ]G = 0)
[µ,B]G + 1
2 [B,B]G = 0,
which is again (2).
4. The L∞-morphisms. Our goal is to have (and use) a morphism
T
↓[1]
poly(Mr)→ D
↓[1]
poly(Mr) which would induce a map that takes Maurer–
Cartan elements in the Poisson world to Maurer–Cartan elements in the
associative world.
The leading term F1, i.e., the first approximation to the morphism
which we consider, is the Hochschild–Kostant–Rosenberg (HKR) map
(obviously, extended by linearity),
F : ξ1 ∧ · · · ∧ ξm 7→
1
m!
∑
σ∈Sm
(−1)σξσ(1) ⊗ · · · ⊗ ξσ(m),
Formality morphism as the mechanism of ?-product associativity 27
which takes a split multi-vector to a polydifferential operator (in fact,
an m-vector). More explicitly, we have that
F1 : (ξ1 ∧ · · · ∧ ξm)
7→
(
a1 ⊗ · · · ⊗ am 7→
1
m!
∑
σ∈Sm
(−1)σ
m∏
i=1
ξσ(i)(ai)
)
, (3)
here aj ∈A :=C∞(Mr). For zero-vectors h ∈ A, one has F1 : h 7→(1 7→h).
Claim 1 ([16, Section 4.6.2]). The leading term, map F1, is not a Lie
algebra morphism (which, if it were, would take the Schouten bracket of
multivectors to the Gerstenhaber bracket of polydifferential operators).
Proof (by counterexample). Take two bi-vectors; their Schouten bracket
is a tri-vector, but the Gerstenhaber bracket of two bi-vectors is a differ-
ential operator which has homogeneous components of differential orders
(2,1,1) and (1,1,2). And in general, those components do not vanish.
The construction of not a single map F1 but of an entire collection
F = {Fn, n > 1} of maps does nevertheless yield a well-defined map-
ping of the Maurer–Cartan elements from the two differential graded Lie
algebras.4
Theorem 2 ([16, Main Theorem]). There exists a collection of linear
maps F =
{
Fn : T
↓[1]
poly(Mr)⊗n → D
↓[1]
poly(Mr), n > 1
}
such that F1 is
the HKR map (3) and F is an L∞-morphism of the two differential
graded Lie algebras:
(
T
↓[1]
poly(Mr), [[·, ·]], d = 0
)
→
(
D
↓[1]
poly(Mr), [·, ·]G,
dH = [µA, ·]G
)
. Namely,
(1) each component Fn is homogeneous of own grading 1− n,
(2) each morphism Fn is graded skew-symmetric, i.e.,
Fn(. . . , ξ, η, . . . ) = −(−1)deg(ξ)·deg(η)Fn(. . . , η, ξ, . . . )
for ξ, η homogeneous,
4The name ‘formality’ for the collection F of maps is motivated by Theorem 4.10
in [16] and by the main theorem in loc. cit.
28 R. Buring, A.V. Kiselev
(3) for each n > 1 and (homogeneous) multivectors ξ1, . . . , ξn ∈
T
↓[1]
poly(Mr), we have that (cf. [11, Section 3.6])
dH(Fn(ξ1, . . . , ξn))−(−1)n−1
n∑
i=1
(−1)uFn(ξ1, . . . , dξi, . . . , ξn)
+ 1
2
∑
p+q=n
p,q>0
∑
σ∈Sp,q
(−1)pn+t
[
Fp(ξσ(1), . . . , ξσ(p)), (4)
Fq(ξσ(p+1), . . . , ξσ(n))
]
G
= (−1)n
∑
i<j
(−1)sFn−1
(
[ξi, ξj ], ξ1, . . . , ξ̂i, . . . , ξ̂j , . . . , ξn
)
. (5)
In the above formula, σ runs through the set of (p, q)-shuffles, i.e.,
all permutations σ ∈ Sn such that σ(1) < · · ·< σ(p) and indepen-
dently σ(p+1) < · · ·< σ(n); the exponents t and s are the numbers
of transpositions of odd elements which we count when passing (t)
from (Fp,Fq, ξ1, . . ., ξn) to (Fp,ξσ(1),. . .,ξσ(p),Fq,ξσ(p+1), . . .,ξσ(n)),
and (s) from (ξ1,. . .,ξn) to (ξi,ξj ,ξ1,. . .,ξ̂1,. . .,ξ̂j ,. . .,ξn).5
Remark 1. Let n := 1, then equality (5) in Theorem 2 is
dH ◦ F1 − (−1)1−1 · (−1)u=0 from (d,ξ1)7→(d,ξ1)F1 ◦ d = 0
⇐⇒ dH ◦ F1 = F1 ◦ d,
whence F1 is a morphism of complexes.
• Let n := 2, then for any homogeneous multivectors ξ1 and ξ2,
F1
(
[[ξ1, ξ2]]
)
−
[
F1(ξ1),F1(ξ2)
]
G
= dH
(
F2(ξ1, ξ2)
)
+ F2
(
(d = 0)(ξ1), ξ2
)
+ (−1)deg ξ1F2
(
ξ1, (d = 0)(ξ2)
)
,
so that in our case F1 is “almost” a Lie algebra morphism but for the
discrepancy which is controlled by the differential of the (value of the)
succeeding map F2 in the sequence F = {Fn, n > 1}. Big formula (5)
shows in precisely which sense this is also the case for higher homoto-
pies Fn, n > 2 in the L∞-morphism F . Indeed, an L∞-morphism is
a map between dgLas which, in every term, almost preserves the bracket
up to a homotopy dH ◦ {. . . } provided by the next term.
5The exponent u is not essential for us now because the differential d on T
↓[1]
poly(Mr)
is set equal to zero identically, so that the entire term with u does not contribute
(recall Fn is linear).
Formality morphism as the mechanism of ?-product associativity 29
Even though neither F1 nor the entire collection F = {Fn, n > 1} is
a dgLa morphism, their defining property (5) guarantees that F gives us
a well defined mapping of the Maurer–Cartan elements (which, we recall,
are formal Poisson bi-vectors and tails B of associative (non)commutati-
ve multiplcations ? = µ+B on A[[~]], respectively).
Corollary 3. The natural ~-linear extension of F , now acting on the
space of formal power series in ~ with coefficients in T
↓[1]
poly(Mr) and with
zero free term by the rule
ξ 7→
∑
n>1
1
n!
Fn(ξ, . . . , ξ),
takes the Maurer–Cartan elements P̃ = ~P+ ō(~) to the Maurer–Cartan
elements B =
∑
n>1
1
n!Fn(P̃, . . . , P̃) = ~P̃ + ō(~). (Note that the HKR
map F1, extended by ~-linearity, still is an identity mapping on multi-
vectors, now viewed as special polydifferential operators.)
In plain terms, for a bivector P itself Poisson, formal Poisson struc-
tures P̃ = ~P + ō(~) satisfying [[P̃, P̃]] = 0 are mapped by F to the tails
B = ~P + ō(~) such that ? = µ+ B is associative and its leading order
deformation term is a given Poisson structure P.
Proof of Corollary 3. Let us presently consider the restricted case when
P̃ = ~P, without any higher order tail ō(~). The Maurer–Cartan equa-
tion in D
↓[1]
poly(Mr)⊗ ~k[[~]] is [µ,B]G + 1
2 [B,B]G = 0, where
B =
∑
n>1
1
n!
Fn
(
P̃, . . . , P̃
)
and we let P̃ = ~P, so that B =
∑
n>1
~n
n! Fn(P, . . . , P). Let us plug
this formal power series in the l.h.s. of the above equation. Equating the
coefficients at powers ~n and multiplying by n!, we obtain the expression
[µ,Fn(P, . . . ,P)]G + 1
2
∑
p+q=n
p,q>0
n!
p!q!
[
Fp(P, . . . ,P),Fq(P, . . . ,P)
]
G
.
It is readily seen that now the sum
∑
σ∈Sp,q in (5) over the set of (p, q)-
shuffles of n = p+q identical copies of an object P just counts the number
30 R. Buring, A.V. Kiselev
of ways to pick p copies going first in an ordered string of length n.
To balance the signs, we note at once that by item (2) in Theorem 2,
see above, Fp(. . . ,P(α),P(α+1), . . . ) = Fp(. . . ,P(α+1),P(α), . . . ) because
bi-vector’s shifted degree is +1, so that no (p, q)-shuffles of (P, . . . ,P)
contribute with any sign factor. The only sign contribution that remains
stems from the symbol Fq of grading 1 − q transported along p copies
of odd-degree bi-vector P; this yields t = (1 − q) · p and (−1)pn+t =
(−1)p·(p+q) · (−1)(1−q)·p = (−1)p·(p+1) = +.
The left-hand side of the Maurer–Cartan equation (2) is, by the
above, expressed by the left-hand side of (5) which the L∞-morphism
F satisfies. In the right-hand side of (5), we now obtain (with, actually,
whatever sign factors) the values of linear mappings Fn−1 at twice the
Jacobiator [[P̃, P̃]] as one of the arguments. All these values are therefore
zero, which implies that the right-hand side of the Maurer–Cartan equa-
tion (2) vanishes, so that the tail B indeed is a Maurer–Cartan element
in the Hochschild cochain complex (in other words, the star-product
? = µ+B is associative).
This completes the proof in the restricted case when P̃ = ~P. For-
mal power series bi-vectors P̃ = ~P + ō(~) refer to the same count of
signs as above, yet the calculation of multiplicities at ~n (for all possible
lexicographically ordered p- and q-tuples of n arguments) is an extensive
exercise in combinatorics.
Corollary 4. Because the right-hand side of (2) in the above reaso-
ning is determined by the right-hand side of (5), we read off an explicit
formula of the operator 3 that solves the factorization problem
Assoc(?)(P)(f, g, h) = 3
(
P, [[P,P]]
)
(f, g, h), f, g, h ∈ A[[~]]. (1)
Indeed, the operator is
3 = 2 ·
∑
n>1
~n
n!
· cn · Fn−1
(
[[P,P]],P, . . . ,P
)
. (6)
But what are the coefficients cn ∈ R equal to? Let us find it out.
5. Explicit construction of the formality morphism F . The
first explicit formula for the formality morphism F which we study in
this paper was discovered by Kontsevich in [16, Section 6.4], providing
an expansion of every term Fn using weighted decorated graphs:
F =
{
Fn =
∑
m>0
∑
Γ∈Gn,m
WΓ · UΓ
}
.
Formality morphism as the mechanism of ?-product associativity 31
Here Γ belongs to the set Gn,m of oriented graphs on n internal vertices
(i.e., arrowtails), m sinks (from which no arrows start), and 2n+m−2 >
0 edges, such that at every internal vertex there is an ordering of outgoing
edges. By decorating each edge with a summation index that runs from 1
to r, by viewing each edge as a derivation ∂/∂xα of the arrowhead vertex
content, by placing n multivectors from an ordered tuple of arguments
of Fn into the respective vertices, now taking the sum over all indices of
the resulting products of the content of vertices, and skew-symmetrizing
over the n-tuple of (shifted-)graded multivectors, we realize each graph
at hand as a polydifferential operator T
↓[1]
poly(Mr)⊗n → D
↓[1]
poly(Mr) whose
arguments are multivectors. Note that the value Fn(ξ1, . . . , ξn) itself is,
by construction, a differential operator w.r.t. the contents of sinks of the
graph Γ. All of this is discussed in detail in [13, 14, 15, 16] or [4, 5, 7].
The formula for the harmonic weights WΓ ∈ R is given in [16, Sec-
tion 6.2]; it is
WΓ =
(
n∏
k=1
1
#Star(k)!
)
· 1
(2π)2n+m−2
∫
C̄+
n,m
∧
e∈EΓ
dφe,
where # Star(k) is the number of edges starting from vertex k, dϕe is the
“harmonic angle” differential 1-form associated to the edge e, and the
integration domain C̄+
n,m is the connected component of C̄n,m which is
the closure of configurations where points qj , 1 6 j 6 m on R are placed
in increasing order: q1 < · · · < qm. For convenience, let us also define
wΓ =
( n∏
k=1
#Star(k)!
)
·WΓ.
The convenience is that by summing over labelled graphs Γ, we actu-
ally sum over the equivalence classes [Γ] (i.e., over unlabeled graphs)
with multiplicities (wΓ/WΓ) · n!/#Aut(Γ). The division by the volume
#Aut(Γ) of the symmetry group eliminates the repetitions of graphs
which differ only by a labeling of vertices but, modulo such, do not dif-
fer by the labeling of ordered edge tuples (issued from the vertices which
are matched by a symmetry).
Let us remember that the integrand in the formula of WΓ is defined
in terms of the harmonic propagator; other propagators (e.g., logarith-
mic, or other members of the family interpolating between harmonic and
32 R. Buring, A.V. Kiselev
logarithmic [1, 18]) would give other formality morphisms. A path inte-
gral realization of the ?-product itself and of the components Fn in the
formality morphism is proposed in [10].
To calculate the graph weights WΓ in practice, we employ methods
which were outlined in [7], as well as [12, Appendix E] (about the cyclic
weight relations), and [3] that puts those real values in the context of
Riemann multiple zeta functions and polylogarithms.6 Examples of such
decorated oriented graphs Γ and their weights WΓ will be given in the
next section.
5.1. Sum over equivalence classes. The sum in Kontsevich’s
formula is over labeled graphs: internal vertices are numbered from 1
to n, and the edges starting from each internal vertex k are numbered
from 1 to #Star(k). Under a re-labeling σ : Γ 7→ Γσ of internal vertices
and edges it is seen from the definitions that the operator UΓ and the
weight WΓ enjoy the same skew-symmetry property (as remarked in [16,
Section 6.5]), whence WΓ · UΓ = WΓσ · UΓσ . It follows that the sum over
labeled graphs can be replaced by a sum over equivalence classes [Γ] of
graphs, modulo labeling of internal vertices and edges. For this it re-
mains to count the size of an equivalence class: the edges can be labeled
in
∏n
k=1 #Star(k)! ways, while the n internal vertices can be labeled in
n!/#Aut(Γ) ways.
Example 2. The double wedge on two ground vertices has only one
possible labeling of vertices, due to the automorphism that interchanges
the wedges.
We denote by MΓ =
(∏n
k=1 #Star(k)!
)
·n!/#Aut(Γ) the multiplicity
of the graph Γ, and let Ḡn,m be the set of equivalence classes [Γ] modulo
labeling of Γ ∈ Gn,m. The formula for the formality morphism can then
be rewritten as
F =
{
Fn =
∑
m>0
∑
[Γ]∈Ḡn,m
MΓ ·WΓ · UΓ
}
;
here the Γ in MΓ ·WΓ · UΓ is any representative of [Γ]. Any ambiguity
in signs (due to the choice of representative) in the latter two factors
is cancelled in their product. Note that the factor
(∏n
k=1 #Star(k)!
)
in
MΓ kills the corresponding factor in WΓ, as remarked in [16, Section 6.5].
6It is the values wΓ instead of WΓ which are calculated by software [3].
Formality morphism as the mechanism of ?-product associativity 33
5.2. The coefficient of a graph in the ?-product. The ?-product
associated with a Poisson structure P is given by Corollary 3:
? = µ+
∑
n>1
~n
n!
Fn(P, . . . ,P)
= µ+
∑
n>1
~n
n!
∑
[Γ]∈Ḡn,2
MΓ ·WΓ · UΓ(P, . . . ,P).
For a graph Γ ∈ Gn,2 such that each internal vertex has two outgoing
edges (these are the only graphs that contribute, because we insert bi-
vectors) we have MΓ = 2n · n!/#Aut(Γ). In total, the coefficient of
UΓ(P, . . . ,P) at ~n is 2n/#Aut(Γ) ·WΓ = wΓ/#Aut(Γ). The skew-sym-
metrization without prefactor of bi-vector coefficients in UΓ(P, . . . ,P)
provides an extra factor 2n.
Example 3 (at ~1). The coefficient of the wedge graph is 1/2 and the
operator is 2P, hence we recover P.
5.3. The coefficient of a Leibniz graph in the associator. The
factorizing operator 3 for Assoc(?) is given by Corollary 4:
3 = 2 ·
∑
n>1
~n
n!
· cn · Fn−1
(
[[P,P]],P, . . . ,P
)
= 2 ·
∑
n>1
~n
n!
· cn ·
∑
[Γ]∈Ḡn−1,3
MΓ ·WΓ · UΓ
(
[[P,P]],P, . . . ,P
)
.
For a graph Γ ∈ Gn−1,3 where one internal vertex has three outgoing
edges and the rest have two, we have MΓ = 3! · 2n−2 · (n− 1)!/#Aut(Γ).
In total, the coefficient of UΓ([[P,P]],P, . . . ,P) at ~n is[
2 · 1
n!
· cn · 3! · 2n−2 · (n− 1)!
]
· WΓ
#Aut(Γ)
=
[
2 · cn
n
]
· wΓ
#Aut(Γ)
.
The skew-symmetrization without prefactor of bi- and tri-vector coef-
ficients in the operator UΓ([[P,P]],P, . . . ,P) provides an extra factor
3! · 2n−2.
Example 4 (at ~2). The coefficient of the tripod graph is c2 · 1
3! and
the operator is 3! · [[P,P]], hence we recover c2[[P,P]] = 2
3 Jac(P). (The
right-hand side is known from the associator, e.g., from [5].) This yields
c2 = 1/3. In addition, we see that the HKR map F1 acts here by the
identity on [[P,P]].
34 R. Buring, A.V. Kiselev
In the next section, we shall find that at ~n, the coefficients of our
Leibniz graphs (with Jac(P) inserted instead of [[P,P]]) are
[[P, P ]]
Jac(P)
·
[
3! · 2n−2
]
·
[
2 · cn
n
]
· wΓ
#Aut(Γ)
= 2n · wΓ
#Aut(Γ)
,
so 3! · 2n · cnn = 2n. We deduce that cn = n/3! = n/6 in all our experi-
ments.
Conjecture. For all n > 2, the coefficients in (6) are cn = n/3! = n/6
(hence, the coefficients of markers Γ for equivalence classes [Γ] of the
Leibniz graphs in (6) are 2n · wΓ/#Aut(Γ)), although it still remains to
be explained how exactly this follows from the L∞ condition (5).
6. Examples. Let P be a Poisson bi-vector on an affine man-
ifold Mr. We inspect the associativity of the star-product ? = µ +∑
n>1
~n
n! Fn(P, . . . , P) given by Corollary 3 by illustrating the work of
the factorization mechanism from Corollary 4. The powers of deforma-
tion parameter ~ provide a natural filtration ~2 ·A(2)+~3 ·A(3)+~4 ·A(4)+
ō
(
~4
)
so that we verify the vanishing of Assoc(?)(P)(·, ·, ·) mod ō
(
~4
)
for ? mod ō
(
~4
)
order by order.
At ~0 there is nothing to do (indeed, the usual multiplication is asso-
ciative). All contribution to the associator of ? at ~1 cancels out because
the leading deformation term ~P in the star-product ? = µ+~P+ ō(~) is
a bi-derivation. The order ~2 was discussed in Example 4 in Section 5.3.
Remark 2. In all our reasoning at any order ~n>2, the Jacobiator in
Leibniz graphs is expanded (w.r.t. the three cyclic permutations of its
arguments) into the Kontsevich graphs, built of wedges, in such a way
that the internal edge, connecting two Poisson bi-vectors in Jac(P), is
proclaimed Left by construction. Specifically, the algorithm to expand
each Leibniz graphs is as follows:
1. Split the trivalent vertex with ordered targets (a, b, c) into two
wedges: the first wedge stands on a and b (in that order), and the
second wedge stands on the first wedge-top and c (in that order), so
that the internal edge of the Jacobiator is marked Left, preceding
the Right edge towards c.
2. Re-direct the edges (if any) which had the tri-valent vertex as their
target, to one of the wedge-tops; take the sum over all possible
combinations (this is the iterated Leibniz rule).
Formality morphism as the mechanism of ?-product associativity 35
3. Take the sum over cyclic permutations of the targets of the edges
which (initially) have (a, b, c) as their targets (this is the expansion
of the Jacobiator).
6.1. The order ~3. To factorize the next order expansion of the
associator, Assoc(?)(P) mod ō
(
~3
)
= ~2 ·A(2) +~3 ·A(3) + ō
(
~3
)
, at ~3 in
the operator 3 in the right-hand side of (1), we use graphs on n− 1 = 2
vertices, m = 3 sinks, and 2(n− 1) +m− 2 = 5 edges.
At ~3, two internal vertices in the Leibniz graphs in the r.h.s. of
factorization (1) are manifestly different: one vertex, containg the bi-
vector P, is a source of two outgoing edges, and the other, with [[P,P]],
of three. Therefore, the automorphism groups of such Leibniz graphs
(under relabellings of internal vertices of the same valency but with
the sinks fixed) can only be trivial, i.e., one-element. (This will not
necessarily be the case of Leibniz graphs on (n− 2) + 1 internal vertices
at ~>4: compare Examples 8 vs 9 on p. 39 below, where the weight of a
graph is divided further by the size of its automorphism group.)
The coefficient of ~3 in the factorizing operator 3,
coeff
(
3, ~3
)
= 2 · 1
3!
· c3 ·
∑
[Γ]∈Ḡ2,3
MΓ ·WΓ · UΓ
(
[[P,P]],P, . . . ,P
)
,
expands into a sum of 6 24 admissible oriented graphs. Indeed, there are
six essentially different oriented graph topologies, filtered by the number
of sinks on which the tri-vector [[P,P]] and bi-vector P stand; the orde-
ring of sinks in the associator then yields 3 + 3 + 3× 2 + 3× 2 + 3 = 24
oriented graphs. (None of them is a zero orgraph.) As we recall from [5],
only thirteen of them actually occur with nonzero coefficients in the term
A(3) ∼ ~3 in Assoc(?)(P)), the remaining eleven have zero weights.7
The weights of 15 relevant oriented Leibniz graphs from [5] are listed in
Table 1.8
Here we let by definition
If := ∂j
(
Jac(P)(Pij , g, h)
)
∂if
7Yet, these seemingly ‘unnecessary’ graphs can contribute to the cyclic weight
relations (see [12, Appendix E]): zero values of some of such graph weights can simplify
the system of linear relations between nonzero weights.
8To get the values, one uses the software [3] by Banks–Panzer–Pym or, indepen-
dently, exact symbolic or approximate numeric methods from [7], also taking into
account the cyclic weight relations from [12, Appendix E].
36 R. Buring, A.V. Kiselev
Table 1. Weights wΓ of oriented Leibniz graphs Γ in coeff
(
3, ~3
)
.
(Sf )221 = [01; 012] 1
12 (Sg)122 = [12; 012] 1
12
(If )112 = [02; 312] 1
48 (Ig)112 = [12; 032] 1
48
(Sf )211 = [04; 012] 1
24 (Ig)211 = [10; 032] −1
48
(If )111 = [04; 312] 1
48 (Ih)111 = [24; 013] −1
48
(Sg)111 = [14; 012] 0 (If )121 = [01; 312] 1
24
(Sh)212 = [20; 012] −1
12
(Sh)112 = [24; 012] −1
24
(Ih)211 = [20; 013] −1
48
(Ig)111 = [14; 032] 0
(Ih)121 = [21; 013] −1
24
=
�
��
r r rr@@R��
r
@
@@R
��
#
"
!?
� rj
−
�
��
r r rrHHHj����
r
�� �
���
#
"
!?
� r
R
j
−
�
��
r r rr@@R��
r
�
��
@@R
#
"
!?
� rj
= 0.
Likewise, Ig := ∂j
(
Jac(P)(f,Pij , h)
)
·∂ig and Ih := ∂j
(
Jac(P)(f, g,Pij)·
∂ih, respectively.9
We also set
Sf := Pij∂j Jac(P)(∂if, g, h)
=i r r rr@@R��
r
@
@@R
��
#
"
!
r
A
A
AU
HHj
−i r r rrHHHj����
r
�� �
���
#
"
!
r
A
A
AU
HHj
L
R −i r r rr@@R��
r
�
��
@@R
#
"
!
r
A
A
AU
HHj
= 0.
Similarly, we let Sg := Pij∂j Jac(P)(f, ∂ig, h) = 0 and define Sh :=
Pij∂j Jac(P)(f, g, ∂ih) = 0. Note that after all the Leibniz rules are
reworked, each of the six graphs If , . . . , Sh – with the Jacobiator Jac(P)
= 1
2 [[P,P]] at the tri-valent vertex – splits into several homogeneous com-
ponents, like (If )111 or (Sh)212; taken alone, each of the components
encodes a zero polydifferential operator of respective orders.
9In [5], the indices i and j were interchanged in the definitions of both Ig and Ih
(compare the expression of If ); that typo is now corrected in the above formulae.
Formality morphism as the mechanism of ?-product associativity 37
Claim 5. Multiplied by a common factor
(
[[P,P]]/ Jac(P)
)
· 2k−1 =
2 · 4 = 8, the Leibniz graph weights from Table 1 at ~3 fully reproduce
the factorization which was found in the main Claim in [5], namely:
A
(3)
221 = 2
3 (Sf )221, A
(3)
122 = 2
3 (Sg)122, A
(3)
212 = − 2
3 (Sh)212,
A
(3)
111 = 1
6 (If − Ih)111, A
(3)
112 =
(
1
6If + 1
6Ig −
1
3Sh
)
112
,
A
(3)
121 = 1
3 (If − Ih)121, A
(3)
211 =
(
1
3Sf −
1
6Ig −
1
6Ih
)
211
.
Otherwise speaking, the sum of these Leibniz oriented graphs with
these weights (times 2·4 = 8), when expanded into the sum of 39 weighted
Kontsevich graphs (built only of wedges), equals identically the ~3-
proportional term in the associator Assoc(?)(P)(f, g, h).
Proof scheme. The encodings of weighted Kontsevich-graph expansions
of the homogeneous components of the weighted Leibniz graphs If , . . . ,
Sh, which show up in the associator at ~3 and which are processed ac-
cording to the algorithm in Remark 2, are listed in Appendix A. Re-
ducing that collection modulo skew symmetry at internal vertices, we
reproduce, as desired, the entire term A(3) in the expansion ~2 · A(2) +
~3 · A(3) + ō
(
~3
)
of the associator Assoc(?)(P) mod ō
(
~3
)
.
Three examples, corresponding to the leftmost column of equalities
in Claim 5, illustrate this scheme at order ~3. The three cases differ
in that for A
(3)
221 in Example 5, there is just one Leibniz graph without
any arrows acting on the Jacobiator vertex. In the other Example 6
for A
(3)
121, there are two Leibniz graphs still without Leibniz-rule actions
on the Jacobiators in them, so that we aim to show how similar terms
are collected.10 Finally, in Example 7 about A
(3)
111 there are two Leibniz
graphs with one Leibniz rule action per either graph: an arrow targets
the two internal vertices in the Jacobiator.
Example 5. Take the Leibniz graph (Sf )221 = [01; 012]. Its weight is
1/12. Multiplying the Leibniz graph by 8 times its weight and expanding
10To collect and compare the Kontsevich orgraphs (built of wedges, i.e., ordered
edge pairs issued from internal vertices), we can bring every such graph to its normal
form, that is, represent it using the minimal base-(# sinks + # internal vertices)
number, encoding the graph as the list of ordered pairs of target vertices, by running
over all the relabellings of internal vertices. (The labelling of ordered sinks is always
0 ≺ 1 ≺ · · · ≺ m− 1.)
38 R. Buring, A.V. Kiselev
the Jacobiator (there are no Leibniz rules to expand) yields the sum of
three Kontsevich graphs: 2
3
(
[01; 01; 42] + [01; 12; 40] + [01; 20; 41]
)
. This
is identically equal to the differential order (2, 2, 1) homogeneous part
A
(3)
221 of Assoc(?)(P) at ~3. For instance, these terms are listed in [7,
Appendix D].
Example 6. Take the Leibniz graphs (If )121 = [01; 312] and (Ih)121 =
[21; 013]. Their weights are 1/24 and −1/24, respectively; multiply them
by 8. Expanding the Jacobiator in the linear combination 1
3 (If − Ih)121
yields the sum of Kontsevich graphs 1
3
(
[01; 31; 42] + [01; 12; 43] + [01; 23;
41] − [21; 01; 43] − [21; 13; 40] − [21; 30; 41]
)
. The two Leibniz graphs
have a Kontsevich graph in common: [01; 12; 43] = [21; 01; 43] (recall
that internal vertex labels can be permuted at no cost and the swap
L� R at a wedge costs a minus sign). This gives one cancellation; the
remaining four terms equal A
(3)
121 as listed in [7, Appendix D].
Example 7. Take the Leibniz graphs (If )111 = [04; 312] and (Ih)111 =
[24; 013]. Their weights are 1/48 and −1/48, respectively; multiply them
by 8. Expanding the Jacobiator and the Leibniz rule in the linear com-
bination 1
6 (If − Ih)111 yields the sum of Kontsevich graphs:
1
6
(
[04; 31; 42] + [04; 12; 43] + [04; 23; 41] + [05; 31; 42]
+ [05; 12; 43] + [05; 23; 41]− [24; 01; 43]− [24; 13; 40]
− [24; 30; 41]− [25; 01; 43]− [25; 13; 40]− [25; 30; 41]
)
.
Two pairs of graphs cancel; namely [05; 31; 42] = [25; 30; 41] and the pair
[05; 23; 41] = [25; 13; 40]. The remaining eight terms equal A
(3)
111 as listed
in [7, Appendix D].
6.2. The order ~4. Let us proceed with the term A(4) at ~4
in Assoc(?)(P)(·, ·, ·) mod ō
(
~4
)
. The numbers of Kontsevich oriented
graphs in the star-product expansion grow as fast as
? = ~0 · (#graphs = 1) + ~1 · (# = 1) + ~2 · (# = 4) + ~3 · (# = 13)
+ ~4 · (# = 247) + ~5 · (# = 2356) + ~6 · (# = 66041) + ō
(
~6
)
;
here we report the count of all nonzero-weight Kontsevich oriented
graphs. Counting them modulo automorphisms (which may also swap
the sinks), Banks, Panzer, and Pym obtain the numbers (~0 : 1, ~1 : 1,
Formality morphism as the mechanism of ?-product associativity 39
~2 : 3, ~3 : 8, ~4 : 133, ~5 : 1209, ~6 : 33268). This shows that at
orders ~k>4, the use of graph-processing software is indispensible in the
task of verifying factorization (1) using weighted graph expansion (6) of
the operator 3.
Specifically, the number of Kontsevich oriented graphs at ~k in the
left-hand side of the factorization problem Assoc(?)(P)(·, ·, ·) = 3
(
P,
[[P,P]]
)
(·, ·, ·), and the number of Leibniz graphs which assemble with
nonzero coefficients to a solution 3 in the right-hand side is presented
in Table 2. At ~4, the expansion of Assoc(?)(P) mod ō
(
~4
)
requires 241
Table 2. Number of graphs in either side of the factorization.
k 2 3 4 5 6
LHS: # K. orgraphs 3 (Jac) 39 740 12464 290305
RHS: # L. orgraphs, 1 (Jac) 13 241 ? ?
coeff 6= 0 ︸ ︷︷ ︸
Reference §5.3, [16] §6.1, [5] §6.2, [7] [3]
nonzero coefficients of Leibniz graphs on 3 sinks, 2 = n − 1 internal
vertices for bi-vectors P and one internal vertex for the tri-vector [[P,P]],
and therefore, 2(n− 1) + 3 = 2n+ 3− 2 = 7 oriented edges.
Remark 3. Again, this set of Leibniz graphs is well structured. Indeed,
it is a disjoint union of homogeneous differential operators arranged ac-
cording to their differential orders w.r.t. the sinks, e.g., (1, 1, 1), (2, 1, 1),
(1, 2, 1), (1, 1, 2), etc., up to (3, 3, 1).
Example 8. The Leibniz graph L331 := [01; 01; 012] of differential orders
(3, 3, 1) has the weight 1/24 according to [3]. Multiplied by a universal
(for all graphs at ~4) factor 24 = 16 and the factor 1/(# Aut(L331)) =
1/2 due to this graph’s symmetry (3� 4), it expands to 1
3
(
[01; 01; 01; 52]
+ [01; 01; 12; 50] + [01; 01; 20; 51]
)
by the definition of Jacobi’s identity.
This sum of three weighted Kontsevich orgraphs reproduces exactly A
(4)
331,
which is known from [7, Table 8 in Appendix D].
Example 9. The Leibniz graph L322 := [01; 02; 012] of differential orders
(3, 2, 2) has the weight 1/24 according to [3]. Multiplied now by a univer-
sal (for all graphs at ~4) factor 24 = 16 and the factor 1/(# Aut(L322))
= 1, it expands to 2
3
(
[01; 02; 01; 52] + [01; 02; 12; 50] + [01; 02; 20; 51]
)
.
This sum reproduces A
(4)
322 (again, see [7, Table 8 in Appendix D]).
40 R. Buring, A.V. Kiselev
Example 10. Consider at the differential order (1, 3, 2) at ~4 the three
Leibniz graphs L
(1)
132 := [12; 13; 012], L
(2)
132 := [12; 12; 014], and L
(3)
132 :=
[12; 01; 412]. They have no symmetries, i.e., their automorphism groups
are one-element, and their weights are W
(
L
(1)
132
)
= 1/72, W
(
L
(2)
132
)
=
1/48, and W
(
L
(3)
132
)
= 1/48, respectively. Pre-multiplied by their weights
and universal factor 24 = 16, these Leibniz graphs expand to
2
9
(
[12; 13; 01; 52] + [12; 13; 12; 50] + [12; 13; 20; 51]
)
+ 1
3
(
[12; 12; 01; 54] + [12; 12; 14; 50] + [12; 12; 40; 51]
)
+ 1
3
(
[12; 01; 41; 52] + [12; 01; 12; 54] + [12; 01; 24; 51]
)
.
There is one cancellation, since [12; 01; 12; 54] = −[12; 12; 01; 54]. The
remaining seven terms reproduce exactly A
(4)
132; that component is known
from [7, Table 8 in Appendix D]. Actually, there was another Leibniz
graph at this homogeneity order, L
(4)
132 := [12; 15; 012], but its weight is
zero and hence it does not contribute. (Indeed, we get an independent
verification of this by having already balanced the entire homogeneous
component at differential orders (1, 3, 2) in the associator.)
Intermediate conclusion. We have experimentally found the con-
stants ck in Corollary 4 which balance the Kontsevich graph expansion of
the ~k-term A(k) in the associator against an expansion of the respective
term at ~k in the r.h.s. of (1) using the weighted Leibniz graphs. Namely,
we conjecture ck = k/6 in Section 5.3. The origin of these constants, in
particular how they arise from the sum over i < j in the L∞ condition (5)
(perhaps, in combination with different normalizations of the objects
which we consider) still remains to be explained, similar to the reasoning
in [2, 20] where the signs are fixed. Note that both in the associator,
which is quadratic w.r.t. the weights of Kontsevich graphs in ?, and
in the operator 3, which is linear in the Kontsevich weights of Leibniz
graphs, the weight values are provided simultaneously, by using identical
techniques (for instance, from [3]). Indeed, the weights are provided by
the integral formula which is universal with respect to all the graphs
under study [16].
A. Encodings of weighted Kontsevich-graph expansions for
(p, q, r)-homogeneous components (If , . . . , Sh)pqr.
# 2/3 (S_f)_{221}
3 3 1 0 1 0 1 4 2 2/3
Formality morphism as the mechanism of ?-product associativity 41
3 3 1 0 1 1 2 4 0 2/3
3 3 1 0 1 2 0 4 1 2/3
# 2/3 (S_g)_{122}
3 3 1 1 2 0 1 4 2 2/3
3 3 1 1 2 1 2 4 0 2/3
3 3 1 1 2 2 0 4 1 2/3
# -2/3 (S_h)_{212}
3 3 1 2 0 0 1 4 2 -2/3
3 3 1 2 0 1 2 4 0 -2/3
3 3 1 2 0 2 0 4 1 -2/3
# 1/6 (I_f)_{111}
3 3 1 0 4 3 1 4 2 1/6
3 3 1 0 4 1 2 4 3 1/6
3 3 1 0 4 2 3 4 1 1/6
3 3 1 0 5 3 1 4 2 1/6
3 3 1 0 5 1 2 4 3 1/6
3 3 1 0 5 2 3 4 1 1/6
# -1/6 (I_h)_{111}
3 3 1 2 4 0 1 4 3 -1/6
3 3 1 2 4 1 3 4 0 -1/6
3 3 1 2 4 3 0 4 1 -1/6
3 3 1 2 5 0 1 4 3 -1/6
3 3 1 2 5 1 3 4 0 -1/6
3 3 1 2 5 3 0 4 1 -1/6
# 1/6 (I_f)_{112}
3 3 1 0 2 3 1 4 2 1/6
3 3 1 0 2 1 2 4 3 1/6
3 3 1 0 2 2 3 4 1 1/6
# 1/6 (I_g)_{112}
3 3 1 1 2 0 3 4 2 1/6
3 3 1 1 2 3 2 4 0 1/6
3 3 1 1 2 2 0 4 3 1/6
# -1/3 (S_h)_{112}
3 3 1 2 4 0 1 4 2 -1/3
3 3 1 2 4 1 2 4 0 -1/3
3 3 1 2 4 2 0 4 1 -1/3
3 3 1 2 5 0 1 4 2 -1/3
3 3 1 2 5 1 2 4 0 -1/3
3 3 1 2 5 2 0 4 1 -1/3
42 R. Buring, A.V. Kiselev
# 1/3 (I_f)_{121}
3 3 1 0 1 3 1 4 2 1/3
3 3 1 0 1 1 2 4 3 1/3
3 3 1 0 1 2 3 4 1 1/3
# -1/3 (I_h)_{121}
3 3 1 2 1 0 1 4 3 -1/3
3 3 1 2 1 1 3 4 0 -1/3
3 3 1 2 1 3 0 4 1 -1/3
# 1/3 (S_f)_{211}
3 3 1 0 4 0 1 4 2 1/3
3 3 1 0 4 1 2 4 0 1/3
3 3 1 0 4 2 0 4 1 1/3
3 3 1 0 5 0 1 4 2 1/3
3 3 1 0 5 1 2 4 0 1/3
3 3 1 0 5 2 0 4 1 1/3
# -1/6 (I_g)_{211}
3 3 1 1 0 0 3 4 2 -1/6
3 3 1 1 0 3 2 4 0 -1/6
3 3 1 1 0 2 0 4 3 -1/6
# -1/6 (I_h)_{211}
3 3 1 2 0 0 1 4 3 -1/6
3 3 1 2 0 1 3 4 0 -1/6
3 3 1 2 0 3 0 4 1 -1/6
Acknowledgements. The first author thanks the Organisers of in-
ternational workshop ‘Symmetries and Integrability of Equations of Ma-
thematical Physics’ (December 22–24, 2018, Kyiv) for helpful discussions
and warm atmosphere during the meeting. A part of this research was
done while RB was visiting at RUG and AVK was visiting at JGU Mainz
(supported by IM JGU via project 5020 and JBI RUG project 106552).
The research of AVK is supported by the IHÉS (partially, by the Nokia
Fund).
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|
| id | oai:trim.imath.kiev.ua:article-365 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:06:09Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/15/e23019c02ba4fe0207f42db6e0ea0315.pdf |
| spelling | oai:trim.imath.kiev.ua:article-3652020-08-13T08:52:53Z Formality morphism as the mechanism of $\star$-product associativity: how it works Морфізм формальності як механізм $\star$-асоціативності: як це працює Buring, R. Kiselev, A. Бурінг, Р. Кісєльов, А. The formality morphism $\boldsymbol{\mathcal{F}}=\{{\mathcal{F}}_n$, $n\geqslant1\}$ in Kontsevich's deformation quantization is a collection of maps from tensor powers of the differential graded Lie algebra (dgLa) of multivector fields to the dgLa of polydifferential operators on finite-dimensional affine manifolds. Not a Lie algebra morphism by its term ${\mathcal{F}}_1$ alone, the entire set $\boldsymbol{{\mathcal{F}}}$ is an $L_\infty$-morphism instead. It induces a map of the Maurer-Cartan elements,&nbsp; taking Poisson bivectors to deformations $\mu_A\mapsto\star_{A[[\hbar]]}$ of the usual multiplication of functions into associative noncommutative $\star$-products of power series in $\hbar$. The associativity of $\star$-products is then realized, in terms of the Kontsevich graphs which encode polydifferential operators, by differential consequences of the Jacobi identity. The aim of this paper is to illustrate the work of this algebraic mechanism for the Kontsevich $\star$-products (in particular, with harmonic propagators). Морфізм $\boldsymbol{{\mathcal{F}}}=\{{\mathcal{F}}_n$, $n\geqslant1\}$, що забезпечує властивість формальності алгебр в деформаційному квантуванні за Концевичем, задано набором відображень тензорних степеней диференціальної градуйованої алгебри Лі (dgLa) мультивекторних полів в dgLa полідиференціальних операторів на скінченновимірних афінних многовидах. Хоча перший член ${\mathcal{F}}_1$ сам по собі не є морфізмом алгебр Лі, послідовність $\boldsymbol{{\mathcal{F}}}$в цілому є $L_\infty$-морфізмом. На його основі будується відображення елементів Маурера-Картана, яке ставить у відповідність пуасоновим бівекторам деформації $\mu_A\mapsto\star_{A[[\hbar]]}$, що добудовують звичайне множення функцій до асоціативних некомутативних $\star$-добутків на просторі степеневих рядів по $\hbar$. При цьому асоциативність $\star$-добутків забезпечено - на мові графів Концевича, що представляють полідиференціальні оператори - диференціальними наслідками тотожності Якобі. Мета роботи - проілюструвати цей алгебраїчний механізм для $\star$-добутків Концевича (зокрема, з гармонічними пропагаторами). Інститут математики НАН України 2019-09-03 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/365 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 16 No. 1 (2019): Symmetry and Integrability of Equations of Mathematical Physics; 22-43 Сборник Трудов Института математики НАН Украины; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 22-43 Збірник Праць Інституту математики НАН України; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 22-43 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/365/359 Авторське право (c) 2019 Р. Бурінг, А. Кісєльов |
| spellingShingle | Buring, R. Kiselev, A. Бурінг, Р. Кісєльов, А. Formality morphism as the mechanism of $\star$-product associativity: how it works |
| title | Formality morphism as the mechanism of $\star$-product associativity: how it works |
| title_alt | Морфізм формальності як механізм $\star$-асоціативності: як це працює |
| title_full | Formality morphism as the mechanism of $\star$-product associativity: how it works |
| title_fullStr | Formality morphism as the mechanism of $\star$-product associativity: how it works |
| title_full_unstemmed | Formality morphism as the mechanism of $\star$-product associativity: how it works |
| title_short | Formality morphism as the mechanism of $\star$-product associativity: how it works |
| title_sort | formality morphism as the mechanism of $\star$-product associativity: how it works |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/365 |
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