Associative half-densities on symplectic groupoids and quantization

Alejandro Cabrera; Gabriel Gonzalo Ledesma Valenotti

arxiv: 2604.08201 · v2 · pith:4GR6DMXBnew · submitted 2026-04-09 · 🧮 math.SG · math-ph· math.MP

Associative half-densities on symplectic groupoids and quantization

Alejandro Cabrera , Gabriel Gonzalo Ledesma Valenotti This is my paper

Pith reviewed 2026-05-21 09:40 UTC · model grok-4.3

classification 🧮 math.SG math-phmath.MP

keywords symplectic groupoidshalf-densitiesassociativityKontsevich quantizationDuflo isomorphismPoisson structuresstar productsdeformation quantization

0 comments

The pith

Symplectic groupoids admit associative half-densities that classify semiclassical factors in Kontsevich quantization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates half-densities on symplectic groupoids that enhance the groupoid multiplication and obey an associativity condition. This setup is motivated by the need for a complete semiclassical and analytic approximation to star products on the corresponding Poisson manifold. The authors prove that such associative half-densities exist and can be classified, then use the construction to extract the semiclassical correction terms appearing in Kontsevich's formula for deformation quantization. In the special case of linear Poisson structures the construction yields precisely the factors of the Duflo isomorphism together with its Kashiwara-Vergne extensions.

Core claim

We show the existence and classification of such associative half-densities, and further apply this theory to the understanding of semiclassical factors in Kontsevich's quantization formula. In the particular case of a linear Poisson structure, we recover the factors appearing in the Duflo isomorphism and its Kashiwara-Vergne extensions as a canonical associative enhancement.

What carries the argument

Associative half-densities that enhance the multiplication map on a symplectic groupoid and satisfy an associativity condition.

If this is right

Such half-densities exist for every symplectic groupoid integrating a Poisson manifold.
The classification provides a canonical way to select semiclassical factors in deformation quantization.
For linear Poisson structures the factors coincide with those of the Duflo-Kashiwara-Vergne theory.
These objects furnish the semiclassical-analytic part of the star product approximation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the associativity condition captures the full semiclassical limit, then Kontsevich quantization can be factored into geometric and analytic pieces more cleanly.
This construction may extend to other quantization schemes beyond Kontsevich's.
Testable by checking whether the half-densities reproduce known higher-order terms in explicit low-dimensional examples.

Load-bearing premise

The structural assumption that an associativity condition on half-densities furnishes the complete semiclassical-analytic approximation to a star product for the underlying Poisson manifold.

What would settle it

A concrete symplectic groupoid for which no associative half-density exists, or for which the recovered semiclassical factors fail to match the known Duflo or Kashiwara-Vergne corrections in the linear case.

read the original abstract

In this paper, we study half-densities enhancing the multiplication map on a symplectic groupoid and which satisfy a suitable associativity condition. This is structurally motivated by the expected complete semiclassical-analytic approximation to a star product for the underlying Poisson manifold. We show the existence and classification of such associative half-densities, and further apply this theory to the understanding of semiclassical factors in Kontsevich's quantization formula. In the particular case of a linear Poisson structure, we recover the factors appearing in the Duflo isomorphism and its Kashiwara-Vergne extensions as a canonical associative enhancement.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper classifies associative half-densities on symplectic groupoids and recovers Duflo/KV factors in the linear case, but the claim that associativity alone supplies the full semiclassical approximation to Kontsevich quantization rests on a motivational assumption rather than a completed uniqueness argument.

read the letter

The main point is that the authors introduce half-densities on the symplectic groupoid that enhance the multiplication and obey an associativity condition. They prove existence and give a classification, then show that in the linear Poisson case these objects canonically produce the semiclassical factors from the Duflo isomorphism and its Kashiwara-Vergne extensions. That recovery is a concrete check against known results and looks like the strongest part of the work. The classification itself appears new relative to the cited literature and gives a geometric handle on corrections that usually appear in deformation quantization formulas. The constructions are precise enough to be useful for people already working with symplectic groupoids. The softer spot is the structural assumption that this associativity condition furnishes the complete semiclassical-analytic approximation to a star product on the base Poisson manifold. The abstract presents this as the motivating expectation rather than as a theorem with explicit comparison to other quantizations or bounds outside the linear setting. Without additional verification that the condition determines the factors uniquely or exhaustively, the broader link stays partly motivational. This paper is aimed at specialists in symplectic geometry and geometric quantization who already know Kontsevich's formula and the linear Poisson literature. A reader in that niche will see the value in the classification and the recovery result. I would send it for peer review; the core definitions and the linear-case check are solid enough to deserve detailed referee comments even if the general claim needs tightening.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces associative half-densities on symplectic groupoids that enhance the groupoid multiplication and obey an associativity condition. This construction is motivated by the desire to obtain a complete semiclassical-analytic approximation to a star product on the underlying Poisson manifold. The authors assert existence and classification theorems for these objects and apply the framework to Kontsevich quantization, recovering the semiclassical factors of the Duflo isomorphism and its Kashiwara-Vergne extensions in the linear Poisson case as a canonical associative enhancement.

Significance. If the classification and recovery results are rigorously established, the work supplies a geometric mechanism linking symplectic groupoid structures to known quantization factors. The explicit recovery of Duflo/KV factors in the linear setting is a concrete strength that could illuminate the semiclassical content of deformation quantization. The broader claim that associativity on half-densities furnishes the complete approximation would, if proved, constitute a notable contribution to the interface between symplectic geometry and quantization theory.

major comments (2)

[Abstract] Abstract and introduction: the central structural assumption that an associativity condition on half-densities furnishes the complete semiclassical-analytic approximation to a star product is stated as motivation but is not accompanied by a theorem establishing uniqueness, exhaustiveness, or error bounds relative to Kontsevich's formula outside the linear Poisson setting; this assumption is load-bearing for the claimed applications.
[Linear Poisson structure section] Linear Poisson case (presumably §4 or equivalent): the recovery of the Duflo and Kashiwara-Vergne factors is asserted as a canonical associative enhancement, yet the manuscript does not appear to contain an explicit, side-by-side derivation or comparison that confirms the factors arise without post-hoc normalization choices; a direct verification against the original constructions would strengthen the claim.

minor comments (2)

[Section 2] Clarify the precise definition of the associativity condition for half-densities (including any cocycle or 2-cocycle data) at the first appearance to avoid ambiguity for readers unfamiliar with the symplectic groupoid literature.
[Introduction] The abstract and introduction would benefit from a short statement distinguishing the new classification result from prior work on half-densities or groupoid quantization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments. We address the major points raised below, clarifying the scope of our results and indicating revisions that will strengthen the presentation without altering the core theorems.

read point-by-point responses

Referee: [Abstract] Abstract and introduction: the central structural assumption that an associativity condition on half-densities furnishes the complete semiclassical-analytic approximation to a star product is stated as motivation but is not accompanied by a theorem establishing uniqueness, exhaustiveness, or error bounds relative to Kontsevich's formula outside the linear Poisson setting; this assumption is load-bearing for the claimed applications.

Authors: We agree that the motivation for studying associative half-densities is drawn from their anticipated role in providing a complete semiclassical-analytic approximation. However, the manuscript does not assert or prove that the associativity condition furnishes such a complete approximation (with uniqueness, exhaustiveness, or error bounds) in the general case; this remains a guiding structural motivation rather than a claimed theorem. The established results are the existence and classification of associative half-densities, together with their explicit recovery of known factors in the linear Poisson setting. In the revised version we will update the abstract and introduction to state this scope explicitly, removing any implication that a general approximation theorem is proved here. revision: yes
Referee: [Linear Poisson structure section] Linear Poisson case (presumably §4 or equivalent): the recovery of the Duflo and Kashiwara-Vergne factors is asserted as a canonical associative enhancement, yet the manuscript does not appear to contain an explicit, side-by-side derivation or comparison that confirms the factors arise without post-hoc normalization choices; a direct verification against the original constructions would strengthen the claim.

Authors: We thank the referee for this suggestion. The recovery follows from the classification theorem applied to the symplectic groupoid of the linear Poisson structure, and the factors emerge directly from the associativity condition. To make the derivation fully transparent and confirm the absence of post-hoc choices, we will add an explicit side-by-side comparison (in a new subsection or appendix) with the classical Duflo isomorphism and Kashiwara-Vergne constructions, including the precise normalization that arises canonically from our framework. revision: yes

Circularity Check

0 steps flagged

Motivation references prior quantization literature but core classification remains independent

full rationale

The paper's primary results consist of proving existence and classification of associative half-densities satisfying the stated condition on symplectic groupoids, followed by an application to semiclassical factors. The abstract explicitly frames the associativity condition as 'structurally motivated by' the expected semiclassical approximation rather than deriving the approximation from the half-densities or vice versa. No equation or theorem in the provided text reduces the classification result to a fitted parameter, self-citation chain, or tautological renaming of known Kontsevich/Duflo factors. The linear Poisson case recovery is presented as an application of the independently derived classification. This yields only minor motivational linkage without load-bearing circularity in the derivation chain itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard background from symplectic geometry and Poisson geometry plus the newly introduced associative half-densities; no free parameters are mentioned in the abstract.

axioms (1)

standard math Standard axioms and definitions of symplectic groupoids and half-densities from differential geometry
The paper presupposes the usual category of symplectic groupoids and the notion of half-densities as background.

invented entities (1)

associative half-densities no independent evidence
purpose: Enhance the multiplication map while satisfying an associativity condition to model semiclassical quantization
The main new object introduced and classified in the paper.

pith-pipeline@v0.9.0 · 5622 in / 1396 out tokens · 62616 ms · 2026-05-21T09:40:08.782094+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

[1]

Amar, A comparison between Rieffel’s and Kontsevich’s deformation quantizations for linear Poisson tensors, Pacific Journal of Mathematics 229(1):1-24

N.B. Amar, A comparison between Rieffel’s and Kontsevich’s deformation quantizations for linear Poisson tensors, Pacific Journal of Mathematics 229(1):1-24

work page
[2]

Andler, M., Sahi, S., Torossian, C.: Convolution of invariant distributions: Proof of the Kashiwara- Vergne conjecture. Lett. Math. Phys. 69, 177–203 (2004)

work page 2004
[3]

Bates, A

S. Bates, A. Weinstein, Lectures on the geometry of quantization, Berkeley Mathematics Lec- ture Notes series, vol. 8, American Mathematical Society (1997).https://math.berkeley.edu/ ˜alanw/GofQ.pdf

work page 1997
[4]

Bayen, M

F. Bayen, M. Flato, C. Frønsdal, A. Lichnerowicz, D. Sternheimer, Deformation theory and quanti- zation. I. Deformations of symplectic structures, Ann. Physics 111 (1978), no. 1, 61 - 110

work page 1978
[5]

Bieliavsky

P. Bieliavsky. Strict quantization of solvable symmetric spaces. J.Symp.Geom., 1(2):269–320, 2002

work page 2002
[6]

A. Cabrera. Generating functions for local symplectic groupoids and non-perturbative semiclassical quantization. Communications in Mathematical Physics 395, pages 1243–1296. 33

work page
[7]

Cabrera, A., Dherin, B.: Formal symplectic realizations. Int. Math. Res. Notices 2016(7), 1925–1950 (2016)

work page 2016
[8]

Cabrera, R.L

A. Cabrera, R.L. Fernandes, Remarks on Non-formal Deformation Quantization of Poisson man- ifolds, In: Abreu, M., Bursztyn, H., Picado, J. (eds) Brazil-Portugal Mathematics. ECBPM 2022. Coimbra Mathematical Texts, vol 4 (2025). Springer, Cham

work page 2022
[9]

Cabrera, R.L

A. Cabrera, R.L. Fernandes, Non-formal deformation quantization and integrability of Poisson brackets, in preparation

work page
[10]

Cabrera, A., Marcut, I., Salazar, M.A.: On local integration of Lie brackets. J. Reine Angew. Math. (Crelle’s J.) 760, 267–293 (2020)

work page 2020
[11]

Cattaneo, A.S., Dherin, B., Felder, G.: Formal symplectic groupoid. Comm. Math. Phys. 253, 645–674 (2005)

work page 2005
[12]

A. S. Cattaneo, B. Dherin, and A. Weinstein. Symplectic microgeometry, IV: quantization. Pacific J. Math., 312(2):355–399, 2021

work page 2021
[13]

Com- mun

Cattaneo, A.S., Felder, G.: A path integral approach to the Kontsevich quantization formula. Com- mun. Math. Phys. 212(3), 591–612 (2000)

work page 2000
[14]

Coste, A., Dazord, P., Weinstein, A.: Groupoides symplectiques, Publications du Departement de Mathe- matiques, Nouvelle Serie A, V ol. 2, Univ. Claude-Bernard, Lyon, (1987).http://www. numdam.org/article/PDML_1987___2A_1_0.pdf

work page 1987
[15]

Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment

M. Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv. 78 (2003), no. 4, 681–721

work page 2003
[16]

Crainic, R

M. Crainic, R. L. Fernandes, and I. Marcut. Lectures on Poisson geometry, volume 217 of Graduate Studies in Mathematics. American Mathematical Society, Provi- dence, RI, 2021

work page 2021
[17]

J. J. Duistermaat and J. A. C. Kolk, Lie groups, Universitext, Springer, Berlin 2000

work page 2000
[18]

Gracia-Saz,A., Mehta, R.,VB-groupoids and representation theory of Lie groupoids, Journal of Symplectic Geometry, V ol. 15, No. 3 (2017), pp. 741-783

work page 2017
[19]

Guillemin and S

V . Guillemin and S. Sternberg. Semi-classical analysis. International Press, Boston, MA, 2013

work page 2013
[20]

Gukov and E

S. Gukov and E. Witten. Branes and quantization. Adv. Theor. Math. Phys., 13(5):1445–1518, 2009

work page 2009
[21]

E. Hawkins. A groupoid approach to quantization. J. Symplectic Geom., 6(1):61–125, 2008

work page 2008
[22]

A. V . Karabegov. Formal symplectic groupoid of a deformation quantization. Comm. Math. Phys., 258(1):223–256, 2005

work page 2005
[23]

Karasev, M.V .: Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets, (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 50, 508–538 (1986)

work page 1986
[24]

M. V . Karas ¨ev and V . P. Maslov. Nonlinear Poisson brackets, volume 119 of Transla- tions of Mathematical Monographs. American Mathematical Society, Providence, RI, 1993. 34

work page 1993
[25]

M. V . Karasev and T. A. Osborn. Cotangent bundle quantization: entangling of met- ric and mag- netic field. Journal of Physics A: Mathematical and General, 38(40):8549, sep 2005

work page 2005
[26]

In- vent

Kashiwara, M., Vergne, M.: The Campbell-Hausdorffformula and invariant hyperfunctions. In- vent. Math. 47, 249–272 (1978)

work page 1978
[27]

Kontsevich

M. Kontsevich. Deformation quantization of Poisson manifolds. Lett. Math. Phys., 66(3):157–216, 2003

work page 2003
[28]

Ledesma, Enhanced symplectic groupoids and quantization, PhD thesis, UFRJ, Brazil

G. Ledesma, Enhanced symplectic groupoids and quantization, PhD thesis, UFRJ, Brazil

work page
[29]

Li-Bland and E

D. Li-Bland and E. Meinrenken, On the van Est homomorphism for Lie groupoids, Enseign. Math. 61 (2015), no. 1-2, 93–137

work page 2015
[30]

Mackenzie, General theory of Lie groupoids and Lie algebroids; London Math

K. Mackenzie, General theory of Lie groupoids and Lie algebroids; London Math. Society Lecture Note Series 213, Cambridge University Press, Cambridge, 2005

work page 2005
[31]

Meinrenken, Semiclassical principal symbols and Gutzwiller’s trace formula, Reports on Math- ematical Physics, V olume 31, Issue 3, 1992, Pages 279-295

E. Meinrenken, Semiclassical principal symbols and Gutzwiller’s trace formula, Reports on Math- ematical Physics, V olume 31, Issue 3, 1992, Pages 279-295

work page 1992
[32]

Melrose, Star products and local line bundles, Annales de l’institut Fourier, vol

R. Melrose, Star products and local line bundles, Annales de l’institut Fourier, vol. 54 (2004), no. 5, pp. 1581-1600.https://www.numdam.org/article/AIF_2004__54_5_1581_0.pdf

work page 2004
[33]

M.K. Murray, An introduction to bundle gerbes, in Oscar Garcia-Prada, Jean Pierre Bourguignon, and Simon Salamon (eds), The Many Facets of Geometry: A Tribute to Nigel Hitchin (OUP, 2010)

work page 2010
[34]

M. A. Rieffel. Questions on quantization. In Operator algebras and operator theory (Shanghai, 1997), volume 228 of Contemp. Math., pages 315–326. Amer. Math. Soc., Providence, RI, 1998

work page 1997
[35]

Vanishing of the Kontsevich integrals of the wheels

B. Shoikhet, “Vanishing of the Kontsevich integrals of the wheels”, Lett. Math. Phys. 56:2 (2001), 141–149

work page 2001
[36]

Weinstein

A. Weinstein. Symplectic groupoids and Poisson manifolds. Bull. Amer. Math. Soc. (N.S.), 16(1):101–104, 1987

work page 1987
[37]

Weinstein

A. Weinstein. Noncommutative geometry and geometric quantization. In Symplectic geometry and mathematical physics (Aix-en-Provence, 1990), volume 99 of Progr.Math., pages 446–461. Birkh¨auser Boston, Boston, MA, 1991

work page 1990
[38]

M. Zworski. Semiclassical analysis, volume 138 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012. 35

work page 2012

[1] [1]

Amar, A comparison between Rieffel’s and Kontsevich’s deformation quantizations for linear Poisson tensors, Pacific Journal of Mathematics 229(1):1-24

N.B. Amar, A comparison between Rieffel’s and Kontsevich’s deformation quantizations for linear Poisson tensors, Pacific Journal of Mathematics 229(1):1-24

work page

[2] [2]

Andler, M., Sahi, S., Torossian, C.: Convolution of invariant distributions: Proof of the Kashiwara- Vergne conjecture. Lett. Math. Phys. 69, 177–203 (2004)

work page 2004

[3] [3]

Bates, A

S. Bates, A. Weinstein, Lectures on the geometry of quantization, Berkeley Mathematics Lec- ture Notes series, vol. 8, American Mathematical Society (1997).https://math.berkeley.edu/ ˜alanw/GofQ.pdf

work page 1997

[4] [4]

Bayen, M

F. Bayen, M. Flato, C. Frønsdal, A. Lichnerowicz, D. Sternheimer, Deformation theory and quanti- zation. I. Deformations of symplectic structures, Ann. Physics 111 (1978), no. 1, 61 - 110

work page 1978

[5] [5]

Bieliavsky

P. Bieliavsky. Strict quantization of solvable symmetric spaces. J.Symp.Geom., 1(2):269–320, 2002

work page 2002

[6] [6]

A. Cabrera. Generating functions for local symplectic groupoids and non-perturbative semiclassical quantization. Communications in Mathematical Physics 395, pages 1243–1296. 33

work page

[7] [7]

Cabrera, A., Dherin, B.: Formal symplectic realizations. Int. Math. Res. Notices 2016(7), 1925–1950 (2016)

work page 2016

[8] [8]

Cabrera, R.L

A. Cabrera, R.L. Fernandes, Remarks on Non-formal Deformation Quantization of Poisson man- ifolds, In: Abreu, M., Bursztyn, H., Picado, J. (eds) Brazil-Portugal Mathematics. ECBPM 2022. Coimbra Mathematical Texts, vol 4 (2025). Springer, Cham

work page 2022

[9] [9]

Cabrera, R.L

A. Cabrera, R.L. Fernandes, Non-formal deformation quantization and integrability of Poisson brackets, in preparation

work page

[10] [10]

Cabrera, A., Marcut, I., Salazar, M.A.: On local integration of Lie brackets. J. Reine Angew. Math. (Crelle’s J.) 760, 267–293 (2020)

work page 2020

[11] [11]

Cattaneo, A.S., Dherin, B., Felder, G.: Formal symplectic groupoid. Comm. Math. Phys. 253, 645–674 (2005)

work page 2005

[12] [12]

A. S. Cattaneo, B. Dherin, and A. Weinstein. Symplectic microgeometry, IV: quantization. Pacific J. Math., 312(2):355–399, 2021

work page 2021

[13] [13]

Com- mun

Cattaneo, A.S., Felder, G.: A path integral approach to the Kontsevich quantization formula. Com- mun. Math. Phys. 212(3), 591–612 (2000)

work page 2000

[14] [14]

Coste, A., Dazord, P., Weinstein, A.: Groupoides symplectiques, Publications du Departement de Mathe- matiques, Nouvelle Serie A, V ol. 2, Univ. Claude-Bernard, Lyon, (1987).http://www. numdam.org/article/PDML_1987___2A_1_0.pdf

work page 1987

[15] [15]

Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment

M. Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv. 78 (2003), no. 4, 681–721

work page 2003

[16] [16]

Crainic, R

M. Crainic, R. L. Fernandes, and I. Marcut. Lectures on Poisson geometry, volume 217 of Graduate Studies in Mathematics. American Mathematical Society, Provi- dence, RI, 2021

work page 2021

[17] [17]

J. J. Duistermaat and J. A. C. Kolk, Lie groups, Universitext, Springer, Berlin 2000

work page 2000

[18] [18]

Gracia-Saz,A., Mehta, R.,VB-groupoids and representation theory of Lie groupoids, Journal of Symplectic Geometry, V ol. 15, No. 3 (2017), pp. 741-783

work page 2017

[19] [19]

Guillemin and S

V . Guillemin and S. Sternberg. Semi-classical analysis. International Press, Boston, MA, 2013

work page 2013

[20] [20]

Gukov and E

S. Gukov and E. Witten. Branes and quantization. Adv. Theor. Math. Phys., 13(5):1445–1518, 2009

work page 2009

[21] [21]

E. Hawkins. A groupoid approach to quantization. J. Symplectic Geom., 6(1):61–125, 2008

work page 2008

[22] [22]

A. V . Karabegov. Formal symplectic groupoid of a deformation quantization. Comm. Math. Phys., 258(1):223–256, 2005

work page 2005

[23] [23]

Karasev, M.V .: Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets, (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 50, 508–538 (1986)

work page 1986

[24] [24]

M. V . Karas ¨ev and V . P. Maslov. Nonlinear Poisson brackets, volume 119 of Transla- tions of Mathematical Monographs. American Mathematical Society, Providence, RI, 1993. 34

work page 1993

[25] [25]

M. V . Karasev and T. A. Osborn. Cotangent bundle quantization: entangling of met- ric and mag- netic field. Journal of Physics A: Mathematical and General, 38(40):8549, sep 2005

work page 2005

[26] [26]

In- vent

Kashiwara, M., Vergne, M.: The Campbell-Hausdorffformula and invariant hyperfunctions. In- vent. Math. 47, 249–272 (1978)

work page 1978

[27] [27]

Kontsevich

M. Kontsevich. Deformation quantization of Poisson manifolds. Lett. Math. Phys., 66(3):157–216, 2003

work page 2003

[28] [28]

Ledesma, Enhanced symplectic groupoids and quantization, PhD thesis, UFRJ, Brazil

G. Ledesma, Enhanced symplectic groupoids and quantization, PhD thesis, UFRJ, Brazil

work page

[29] [29]

Li-Bland and E

D. Li-Bland and E. Meinrenken, On the van Est homomorphism for Lie groupoids, Enseign. Math. 61 (2015), no. 1-2, 93–137

work page 2015

[30] [30]

Mackenzie, General theory of Lie groupoids and Lie algebroids; London Math

K. Mackenzie, General theory of Lie groupoids and Lie algebroids; London Math. Society Lecture Note Series 213, Cambridge University Press, Cambridge, 2005

work page 2005

[31] [31]

Meinrenken, Semiclassical principal symbols and Gutzwiller’s trace formula, Reports on Math- ematical Physics, V olume 31, Issue 3, 1992, Pages 279-295

E. Meinrenken, Semiclassical principal symbols and Gutzwiller’s trace formula, Reports on Math- ematical Physics, V olume 31, Issue 3, 1992, Pages 279-295

work page 1992

[32] [32]

Melrose, Star products and local line bundles, Annales de l’institut Fourier, vol

R. Melrose, Star products and local line bundles, Annales de l’institut Fourier, vol. 54 (2004), no. 5, pp. 1581-1600.https://www.numdam.org/article/AIF_2004__54_5_1581_0.pdf

work page 2004

[33] [33]

M.K. Murray, An introduction to bundle gerbes, in Oscar Garcia-Prada, Jean Pierre Bourguignon, and Simon Salamon (eds), The Many Facets of Geometry: A Tribute to Nigel Hitchin (OUP, 2010)

work page 2010

[34] [34]

M. A. Rieffel. Questions on quantization. In Operator algebras and operator theory (Shanghai, 1997), volume 228 of Contemp. Math., pages 315–326. Amer. Math. Soc., Providence, RI, 1998

work page 1997

[35] [35]

Vanishing of the Kontsevich integrals of the wheels

B. Shoikhet, “Vanishing of the Kontsevich integrals of the wheels”, Lett. Math. Phys. 56:2 (2001), 141–149

work page 2001

[36] [36]

Weinstein

A. Weinstein. Symplectic groupoids and Poisson manifolds. Bull. Amer. Math. Soc. (N.S.), 16(1):101–104, 1987

work page 1987

[37] [37]

Weinstein

A. Weinstein. Noncommutative geometry and geometric quantization. In Symplectic geometry and mathematical physics (Aix-en-Provence, 1990), volume 99 of Progr.Math., pages 446–461. Birkh¨auser Boston, Boston, MA, 1991

work page 1990

[38] [38]

M. Zworski. Semiclassical analysis, volume 138 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012. 35

work page 2012