From local to global deformation quantization of Poisson manifolds

TLDR: In this paper, a deformation quantization of the algebra of functions on a Poisson manifold based on M. Kontsevich's local formula is presented, where the deformed algebra is realized as a vector bundle with flat connection.

Abstract: We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifold, based on M. Kontsevich's local formula. The deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection.

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University of Zurich
Zurich Open Repository and Archive
Winterthurerstr. 190
CH-8057 Zurich
http://www.zora.uzh.ch
Year: 2002
From local to global deformation quantization of Poisson
manifolds
Cattaneo, A S; Felder, G; Tomassini, L
Cattaneo, A S; Felder, G; Tomassini, L (2002). From local to global deformation quantization of Poisson manifolds.
Duke Mathematical Journal, 115(2):329-352.
Postprint available at:
http://www.zora.uzh.ch
Posted at the Zurich Open Repository and Archive, University of Zurich.
http://www.zora.uzh.ch
Originally published at:
Duke Mathematical Journal 2002, 115(2):329-352.
Cattaneo, A S; Felder, G; Tomassini, L (2002). From local to global deformation quantization of Poisson manifolds.
Duke Mathematical Journal, 115(2):329-352.
Postprint available at:
http://www.zora.uzh.ch
Posted at the Zurich Open Repository and Archive, University of Zurich.
http://www.zora.uzh.ch
Originally published at:
Duke Mathematical Journal 2002, 115(2):329-352.

From local to global deformation quantization of Poisson
manifolds
Abstract
We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson
manifold, based on M. Kontsevich's local formula. The deformed algebra of functions is realized as the
algebra of horizontal sections of a vector bundle with flat connection.

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FROM LOCAL TO GLOBAL DEFORMATION
QUANTIZATION OF POISSON MANIFOLDS
ALBERTO S. CATTANEO, GIOVANNI FELDER, and LORENZO TOMASSINI
To James Stasheff on the occasion of his 65th birthday
Abstract
We give an explicit construction of a deformation quantization of the algebra of func-
tions on a Poisson manifold, based on M. Kontsevich’s local formula. The deformed
algebra of functions is realized as the algebra of horizontal sections of a vector bundle
with flat connection.
1. Introduction
Let M be a paracompact smooth d-dimensional manifold. The Lie bracket of vector
fields extends to a bracket, the Schouten-Nijenhuis bracket, on the graded commuta-
tive algebra 0(M,
V
·
T M) of multivector fields so that
[α
1
∧ α
2
, α
3
] = α
1
∧ [α
2
, α
3
] + (−1)
m
2
(m
3
−1)
[α
1
, α
3
] ∧ α
2
,
[α
1
, α
2
] = −(−1)
(m
1
−1)(m
2
−1)
[α
2
, α
1
],
if α
i
∈ 0(M,
V
m
i
T M). This bracket defines a graded super Lie algebra structure on
0(M,
V
·
T M) with the shifted grading deg
0
(α) = m − 1, α ∈ 0(M,
V
m
T M).
A Poisson structure on M is a bivector field α ∈ 0(M,
V
2
T M) obeying [α, α] =
0. This identity for α, which we can regard as a bilinear form on the cotangent bundle,
implies that { f, g} = α(d f, dg) is a Poisson bracket on the algebra C
∞
(M) of smooth
real-valued functions. If such a bivector field is given, we say that M is a Poisson
manifold.
Following [1], we introduce the notion of (deformation) quantization of the alge-
bra of functions on a Poisson manifold.
Definition 1
A quantization of the algebra of smooth functions C
∞
(M) on the Poisson manifold M
DUKE MATHEMATICAL JOURNAL
Vol. 115, No. 2,
c
 2002
Received 10 May 2001.
2000 Mathematics Subject Classification. Primary 53D55.
Cattaneo’s work partially supported by Swiss National Science Foundation grant number 2100-055536.98/1.
329

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330 CATTANEO, FELDER, and TOMASSINI
is a topological algebra A over the ring of formal power series R[[]] in a formal vari-
able  with product ?, together with an R-algebra isomorphism A/ A → C
∞
(M), so
that
(i) A is isomorphic to C
∞
(M)[[]] as a topological R[[]]-module;
(ii) there is an R-linear section a 7→ ˜a of the projection A → C
∞
(M) so that
˜
f ?
˜g =
f
f g+
P
∞
j=1

j
^
P
j
( f, g) for some bidifferential operators P
j
: C
∞
(M)
2
→
C
∞
(M) with P
j
( f, 1) = P
j
(1, g) = 0 and P
1
( f, g)− P
1
(g, f ) = 2α(d f, dg).
If we fix a section as in (ii), we obtain a star product on C
∞
(M), that is, a formal
series P

=  P
1
+ 
2
P
2
+ · · · whose coefficients P
j
are bidifferential operators
C
∞
(M)
2
→ C
∞
(M) such that f ?
M
g := f g + P

( f, g) extends to an associative
R[[]]-bilinear product on C
∞
(M)[[]] with unit 1 ∈ C
∞
(M) and such that f ?
M
g −
g ?
M
f = 2α(d f, dg) mod 
2
.
Remark. One can replace (i) by the equivalent condition that A be a Hausdorff, com-
plete, -torsion free R[[]]-module (see [4], [8], App. A).
Kontsevich gave in [9] a quantization in the case of M = R
d
, in the form of an
explicit formula for a star product, as a special case of his formality theorem for the
Hochschild complex of multidifferential operators. This theorem is extended in [9]
to general manifolds by abstract arguments, yielding in principle a star product for
general Poisson manifolds.
In this paper we give a more direct construction of a quantization, based on the
realization of the deformed algebra of functions as the algebra of horizontal sections
of a bundle of algebras. It is similar in spirit to B. Fedosov’s deformation quantization
of symplectic manifolds in [5]. It has the advantage of giving in principle an explicit
construction of a star product on any Poisson manifold.
We turn to the description of our results.
We construct two vector bundles with flat connection on the Poisson manifold
M. The second bundle should be thought of as a quantum version of the first.
The first bundle E
0
is a bundle of Poisson algebras. It is the vector bundle of
infinite jets of functions with its canonical flat connection D
0
. The fiber over x ∈ M is
the commutative algebra of infinite jets of functions at x. The Poisson structure on M
induces a Poisson algebra structure on each fiber, and the canonical map C
∞
(M) →
E
0
is a Poisson algebra isomorphism onto the Poisson algebra H
0
(E
0
, D
0
) of D
0
-
horizontal sections of E
0
.
The second bundle E is a bundle of associative algebras over R[[]] and is ob-
tained by quantization of the fibers of E
0
. Its construction depends on the choice
x 7→ ϕ
x
of an equivalence class of formal coordinate systems ϕ
x
: (R
d
, 0) → (M, x),

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FROM LOCAL TO GLOBAL DEFORMATION QUANTIZATION 331
defined up to the action of GL(d, R), at each point x of M and depending smoothly
on x. As a bundle of R[[]]-modules, E ' E
0
[[]] is isomorphic to the bundle of for-
mal power series in  whose coefficients are infinite jets of functions. The associative
product on the fiber of E over x ∈ M is defined by applying the Kontsevich star
product formula for R
d
with respect to the coordinate system ϕ
x
. Thus the sections
of E form an algebra. We say that a connection on a bundle of algebras is compatible
if the covariant derivatives are derivations of the algebra of sections. If a connection
is compatible, then horizontal sections form an algebra. Our first main result is the
following theorem.
THEOREM 1.1
There exists a flat compatible connection
¯
D = D
0
+  D
1
+ 
2
D
2
+ · · · on E so that
the algebra of horizontal sections H
0
(E,
¯
D) is a quantization of C
∞
(M).
The construction of the connection is done in two steps. First, one constructs a defor-
mation D of the connection D
0
in terms of integrals over configuration spaces of the
upper half-plane. This connection is compatible with the product as a consequence
of the Kontsevich formality theorem on R
d
. Moreover, the same theorem gives a for-
mula for its curvature, which is the commutator [F
M
, ·]
?
with some E-valued two-
form F
M
, and also implies the Bianchi identity DF
M
= 0. In the second step, we
use these facts to show, following Fedosov’s method in [5], that there is an E-valued
one-form γ so that
¯
D = D + [γ , ·]
?
is flat. This means that γ is a solution of the
equation
F
M
+ ω + Dγ + γ ? γ = 0. (1)
Here ω is any E-valued two-form such that Dω = 0 and [ω, ·]
?
= 0.
To prove that the algebra of horizontal sections is a quantization of C
∞
(M), one
constructs a quantization map
ρ : C
∞
(M) ' H
0
(E
0
, D
0
) → H
0
(E,
¯
D),
extending to an isomorphism of topological R[[]]-modules C
∞
(M)[[]] →
H
0
(E,
¯
D). We give two constructions of such a map. In the first construction, ρ is
induced by a chain map (
·
(E
0
), D
0
) → (
·
(E),
¯
D) between the complexes of dif-
ferential forms with values in E
0
and E, respectively. In the second construction, ρ is
defined only at the level of cohomology but behaves well with respect to the center.
THEOREM 1.2
Let Z
0
= { f ∈ C
∞
(M) | { f, ·} = 0} be the algebra of Casimir functions, and let
Z = { f ∈ H
0
(E,
¯
D) | [ f, ·]
?
= 0} be the center of the algebra H
0
(E,
¯
D). Then there
exists a quantization map ρ that restricts to an algebra isomorphism Z
0
[[]] → Z.

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "From local to global deformation quantization of poisson manifolds" ?

CattanEO et al. this paper gave an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifold, based on M. Kontsevich 's local formula. 

Q2. How is the associative product defined on the fiber of E over x M?

The associative product on the fiber of E over x ∈ M is defined by applying the Kontsevich star product formula for Rd with respect to the coordinate system ϕx . 

Q3. What is the idea of the Lie algebra?

The idea is to consider the “space of all local coordinate systems” on M with its transitive action of the Lie algebra of formal vector fields. 

Q4. What is the simplest way to represent the central two-form?

By using the second quantization map ρ, the authors may represent the central two-form ω as ρ(ω0), where ω0 is a D0-closed E0-valued two-form that is Poisson central in the sense that {ω0, ·} = 0. 

Q5. What is the function of the operator U j?

The operator U j (α1, . . . , α j ) is a multilinear symmetric function of j arguments αk ∈ 0(Rd , ∧2 T Rd), taking values in the space of bidifferential operator C∞(Rd) ⊗ C∞(Rd) → C∞(Rd). 

Q6. what is the tangent space to mcoor?

Mcoor and if [ϕt ] is a path in Mcoor with tangent vector ξ at t = 0, thenξ̂ (y) = Taylor expansion at zero of − (dϕ)(y)−1 dd t ϕt (y) ∣∣∣ t=0is a vector field in W which depends only on the infinite jet of ϕt . 

Q7. What is the simplest way to get a vector bundle of infinite jets of functions?

Let J (M) be the vector bundle of infinite jets of functions on M ; the fiber over x ∈ M consists of equivalence classes of smooth functions defined on open neighborhoods of x , where two functions are equivalent if and only if they have the same Taylor series at x (with respect to any coordinate system). 

Q8. What is the proof of Lemma 4.8?

The differential forms with values in Bk form a subcomplex of ·(End(E0)), and the authors have H p(Bk, D0) = 0 for p > 0.Lemma 4.8 is proved in Appendix B. By using Lemma 4.8 and the fact that the maps U j are given by multidifferential operators, the authors deduce that B = ⋃ k Bk obeys the hypotheses of Lemma 4.6. 

Q9. What is the simplest way to get a vector bundle over M?

Then E0 = ϕaff Ẽ0 is a vector bundle over M with fiber R[[y1, . . . , yd ]]: a point in the fiber of E0 over x is a GL(d,R)-orbit of pairs (ϕ, f ), where ϕ is a representative of the class ϕaff(x) and f ∈ R[[y1, . . . , yd ]]. 

Q10. what is the restriction to the ring Z0 of Casimir functions?

Its restriction to the ring Z0 of Casimir functions extends to an R[[ ]]-algebra isomorphism from Z0[[ ]] to the center of H0(E, D̄). 

Q11. what is the Lie bracket of vector fields?

The Lie bracket of vector fields extends to a bracket, the Schouten-Nijenhuis bracket, on the graded commutative algebra 0(M,∧· T M) of multivector fields so that [α1 ∧ α2, α3] = α1 ∧ [α2, α3] + (−1) m2(m3−1)[α1, α3] ∧ α2,[α1, α2] = −(−1) (m1−1)(m2−1)[α2, α1],if αi ∈ 0(M, ∧mi T M). 

Q12. What is the cohomology of Lemma 4.8?

In degree zero, cocycles are sections that are constant as functions of y. ThusH p(E0, D0) ={ C∞(M), p = 0,0, p > 0.“115i2˙04” — 2002/10/29 — 9:12 — page 351 — #23i ii ii ii iProof of Lemma 4.8 

Q13. What is the Lie algebra cohomology complex?

“115i2˙04” — 2002/10/29 — 9:12 — page 336 — #8i ii ii ii i336 CATTANEO, FELDER, and TOMASSINIThe Lie algebra W of vector fields on Rd acts on U, and the authors can form the Lie algebra cohomology complex C ·(W,U) = HomR( ∧· W,U).