Duflo--Kontsevich-type isomorphisms for Tamarkin--Tsygan calculi of dg manifolds

Hsuan-Yi Liao; Mathieu Sti\'enon; Ping Xu

REVIEW 3 major objections 4 minor 57 references

Duflo--Kontsevich-type isomorphisms for Tamarkin--Tsygan calculi of dg manifolds

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For every dg manifold, the HKR maps twisted by the square root of the Todd class give an isomorphism of calculi between the Cartan and Tamarkin–Tsygan models.

desk verdict Proves the full calculus-level Duflo–Kontsevich isomorphism for dg manifolds; a real proof, but load-bearing dependencies on unpublished companion work should be resolved in review. read the letter →

arxiv 2608.07362 v1 pith:EP5XJ6TX submitted 2026-08-07 math.QA

Hsuan-Yi Liao , Mathieu Stiénon , Ping Xu This is my paper · ORCID

classification math.QA MSC 16E4053D5558A50

keywords dgmanifoldsTamarkin–TsygancalculusCartanDuflo–KontsevichisomorphismToddclassHochschild–Kostant–RosenbergmapsFedosovLiealgebroidsformalitytheorems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that the two natural cohomology-level calculi attached to any differential graded (dg) manifold are isomorphic: the Cartan calculus of polyvector fields and differential forms, and the Tamarkin–Tsygan calculus of polydifferential operators and polyjets. The bridge is the Hochschild–Kostant–Rosenberg (HKR) map, corrected by the square root of the Todd class of the dg manifold, acting by contraction on polyvector fields and by wedge multiplication on differential forms. The theorem states that the corrected maps preserve every calculus operation, including cup products, Gerstenhaber brackets, contractions, Lie derivatives, the Connes–Rinehart operator, and the differentials, not merely the underlying chain structures. At cohomology level this confirms the Kontsevich–Shoikhet conjecture stated in 1998. The payoff is unification: the classical Duflo theorem for Lie algebras and the Kontsevich theorem for complex manifolds become two special cases of one dg-manifold statement.

What carries the argument

The load-bearing object is the square root of the Todd class $(\operatorname{td}_{(\mathcal{M},Q)})^{1/2}$, defined as the square root of the Berezinian of the Atiyah class of the tangent dg Lie algebroid; it is a degree-zero element of the differential-form space and acts on one leg of the calculus by contraction and on the other by multiplication. Around this class the proof builds two formality morphisms: a Kontsevich-type $L_\infty$ quasi-isomorphism from polyvector fields to polydifferential operators, and a Shoikhet-type quasi-isomorphism of $L_\infty$ modules from polyjets to differential forms, both twisted by the homological vector field. The passage to dg manifolds is made through Fedosov dg Lie algebroids, whose calculus is connected to the tangent dg Lie algebroid by contractions preserving all operations, and through a divergence identity that converts the $\hat{A}$ class of the Fedosov algebroid into the Todd class.

What would settle it

Take a dg manifold whose Atiyah cocycle is nonzero, for instance the affine space $\mathbb{R}^{0|2}$ with a homological vector field $Q$ containing a nonzero quadratic term, and evaluate the zeroth Taylor coefficient $(\operatorname{td}^{1/2}\circ\mathrm{HKR})$ on a one-polyjet $\zeta$. If the cohomology class of $d_{\mathrm{DR}}((\operatorname{td}^{1/2}\circ\mathrm{HKR})(\zeta))$ is not equal to the class of $(\operatorname{td}^{1/2}\circ\mathrm{HKR})(B\zeta)$, where $B$ is the Connes–Rinehart operator, then the claimed calculus isomorphism fails; the theorem predicts equality for every such $\zeta$ and every such $Q$.

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Extended reading notes

Core claim

The central discovery is Theorem A: for every dg manifold $(\mathcal{M},Q)$, the pair of maps $(\operatorname{hkr}\circ(\operatorname{td}_{(\mathcal{M},Q)})^{1/2},\ ((\operatorname{td}_{(\mathcal{M},Q)})^{1/2}\circ\mathrm{HKR})^{-1})$ is an isomorphism of calculi from $\mathrm{Cal}_C(\mathcal{M},Q)$ to $\mathrm{Cal}_H(\mathcal{M},Q)$, where $(\operatorname{td}_{(\mathcal{M},Q)})^{1/2}$ is the square root of the Todd class of the dg manifold. At the cohomology level this gives a full calculus isomorphism, so the Tamarkin–Tsygan and Cartan calculi are not just quasi-isomorphic as complexes; they carry the same cup products, Gerstenhaber brackets, contractions, Lie derivatives, Connes–Rinehart operators, and differentials. The theorem is proved through a formality theorem for dg manifolds (Theorem B) whose first Taylor coefficient is the Todd-twisted HKR map, and it specializes to the Duflo–Kontsevich theorem for Lie algebras and to the Kontsevich theorem for complex manifolds.

Load-bearing premise

The argument's load-bearing premise is that the Fedosov dg Lie algebroid of a dg manifold is connected to the tangent dg Lie algebroid by contractions that preserve the full calculus structure (all products, brackets, contractions, derivatives, and the Connes–Rinehart operator), as asserted in an unpublished companion preprint; if those contractions preserve only part of the structure, the calculus-level isomorphism collapses.

Editorial extensions

If this is right

For any dg manifold, the Hochschild and Cartan cohomologies agree as full Gerstenhaber algebras with compatible module and differential structures, so computations in either model produce the same invariants.
The theorem subsumes the Duflo–Kontsevich isomorphism for Lie algebras, obtained when the dg manifold is the shift of a Lie algebra with the Chevalley–Eilenberg differential.
It subsumes the Kontsevich theorem for complex manifolds, obtained from the Dolbeault dg manifold of a complex manifold.
Theorem B supplies an $L_\infty$ formality quasi-isomorphism for every dg manifold, with first Taylor coefficient equal to the Todd-twisted HKR map and a compatible module quasi-isomorphism for the chain side; the authors expect it to be applicable to deformation quantization of $0$-shifted derived Poisson manifolds.
The Todd class of a dg manifold acts as a single characteristic class encoding both the Duflo element of a Lie algebra and the ordinary Todd class of a complex manifold.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One extension not pursued in the paper is that the same Todd-class correction should mediate comparisons between other pairs of models for polyvector fields on dg manifolds, such as negative cyclic chains and equivariant forms.
A testable consequence of the proof's construction is that the resulting calculus isomorphism should be independent of the auxiliary affine connection up to explicit homotopies; the affine dg-manifold example in the paper is a concrete setting where those homotopies could be written down.
A natural next test case is the derived intersection dg manifold $E[-1]$ associated to a section of a vector bundle, where the Todd class should reduce to a known characteristic class and the twisted HKR maps can be computed explicitly.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

What you should know: this paper proves the calculus-level Duflo–Kontsevich theorem for dg manifolds, not just the cohomology-level Gerstenhaber statement announced in the authors’ CRAS note. The full isomorphism of Tamarkin–Tsygan calculi, with all operations included, is a genuine advance, and Theorem B is supplied with a detailed proof rather than a sketch. The architecture—Fedosov dg Lie algebroid, leafwise Kontsevich/Shoikhet formality, twisting, then descent—is coherent, and the homotopy-transfer appendix is carefully written. I found no internal contradiction and no sign of fitting, so this is serious work.

The soft spots are real but not demonstrated flaws. The most important is the descent step: it relies on Theorem 2.5 of [30], an unpublished preprint by the same authors. The reduction to the Fedosov-level statement collapses if the contractions in (34)–(38) do not preserve the full calculus operations at cochain level, and the present paper does not prove that theorem; it only cites it. That is a load-bearing external dependency, not a minor gap. Second, Propositions B.3 and B.4, which fix the twisted Taylor coefficients as hkr∘Â^{1/2} and Â^{1/2}∘HKR, are imported from [50,53] without proof. They are standard to the expert, but since the main theorem turns on those coefficients, a referee should see the argument or at least a precise statement. Third, Theorem A says “any dg manifold,” while Proposition 2.7 and Section 2.3 explicitly assume finite-dimensional dg manifolds. The mismatch is probably a tacit standing assumption, but it should be stated.

Who gets value: deformation quantizers, dg-geometry people, and anyone relying on the Kontsevich–Shoikhet conjecture. It is a serious long paper, not a sketch, and it deserves referee time. My recommendation: send it to peer review. Require the authors to make [30] available or fold the needed parts of Theorem 2.5 and Propositions B.3/B.4 into the paper before acceptance.

Referee Report

3 major / 4 minor

Summary. The paper proves a Duflo--Kontsevich-type theorem for Tamarkin--Tsygan calculi of dg manifolds: the HKR maps twisted by the square root of the Todd class induce an isomorphism of calculi between the Cartan calculus and the Tamarkin--Tsygan calculus at cohomology level. The proof proceeds through formality theorems for Fedosov dg Lie algebroids, with explicit Kontsevich--Shoikhet-type morphisms, and then descends to the original dg manifold using contraction data imported from the companion preprint [30]. The paper also states and proves a formality theorem for dg manifolds, Theorem B, whose first coefficients are the twisted HKR maps.

Significance. If correct, the main result confirms the Kontsevich--Shoikhet conjecture and unifies the Duflo--Kontsevich theorem for Lie algebras with the Kontsevich theorem for complex manifolds, at the level of full calculus structures rather than only Gerstenhaber algebras. The paper is carefully structured, with explicit formulas for the HKR maps, the Todd class via Atiyah classes, and the relevant L-infinity morphisms; the homotopy transfer arguments in Appendix A are detailed and several lemmas are proved in full. The Todd class is defined independently through Atiyah classes, so the main theorem is not circular. However, the proof is substantially conditional on unpublished companion results, and the extent of the finite-dimensionality assumptions needs clarification.

major comments (3)

[§2.2.4 and §4.2, proof of Theorem A] The central descent step relies on Theorem 2.5, imported from the unpublished preprint [30] without proof. In the proof of Theorem A, the statement 'the vertical maps are isomorphisms of calculi (by Proposition 3.3 and Theorem 2.5)' is exactly where the calculus-level compatibility of the injections τ̆^♮ is needed: Theorem 2.5(i) and (iii) assert preservation of ∧, [−,−], i, L, d, ⌣, ⟦−,−⟧, and B at cochain level. Since [30] is a self-cited unpublished preprint and no proof of these compatibilities is given here, the descent from the Fedosov calculus to the dg manifold calculus is not self-contained. If those compatibilities fail, the cohomology-level maps could still be vector-space isomorphisms but not calculus isomorphisms, so Theorem A would not follow. The paper should either include a proof of Theorem 2.5 or explicitly state Theorem A as conditional on the results of [30].
[Theorem A and §2.4.1, Proposition 2.7] Theorem A and the abstract claim the result for 'any dg manifold', but several load-bearing ingredients are stated only for finite-dimensional dg manifolds. Proposition 2.7 is explicitly for a finite-dimensional dg manifold, Section 3 opens with 'Given a finite-dimensional dg manifold', and the Fedosov construction in §2.2.1 uses local charts with finitely many virtual coordinates. If the paper's convention is that all dg manifolds are finite-dimensional, this should be stated where 'dg manifold' is first defined; otherwise the proof does not support the stated generality. This is not merely cosmetic, because the HKR quasi-isomorphisms in Proposition 2.7 and the Fedosov descent in Section 4 are the mechanisms that produce the twisted maps in Theorem A.
[§3.2, Proposition 3.7 and Appendix B] Proposition 3.7, which identifies the first Taylor coefficient 𝔘₁ and the zeroth coefficient 𝔖₀ with the Â-twisted HKR maps, depends on Equation (78). That equation is asserted to follow from Propositions B.3 and B.4, but those propositions are stated without proof and without a precise derivation of their graded/trivialized versions from the cited sources [50,53]. Since these coefficients are the cochain-level incarnations of the maps that later become the Todd-twisted isomorphisms, the computation is load-bearing. The authors should either prove Propositions B.3 and B.4 or give a detailed derivation indicating exactly how the statements in [50,53] imply the graded versions used here.

minor comments (4)

[§2.4.1, Eq. (41)] The Koszul sign in the definition of the cohomological HKR map is not spelled out; since signs play a role in later computations, please state the sign convention explicitly or refer to a formula where it is fixed.
[§2.2.1] The notation 𝒩 = 𝑇ℳ[1] ⊕ 𝑇̂ℳ for what appears to be a fibred product over ℳ is confusing; using ⊕ normally denotes a direct sum of vector bundles. Please clarify the fibred-product nature of this construction.
[Lemma 4.2] In the proof of Lemma 4.2, the equalities (td^can)^{-1/2} ∧ L_𝒬ζ = L_𝒬ξ and (L_𝒬ζ) ∧ (td^can)^{-1/2} = L_𝒬η are used without showing the sign conventions; a one-line calculation using the derivation property of L_𝒬 would make the homotopy formulas transparent.
[Introduction, 'dg manifold' definition] The paper defines 'graded' but not 'dg manifold' precisely, and finite-dimensionality is first imposed only in Proposition 2.7. Please define the class of dg manifolds under consideration at the beginning and keep it consistent with the statements of Theorems A and B.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Todd class is defined independently from Atiyah classes, the twisted HKR maps are explicit constructions, and the proof's reliance on unpublished self-cited work is an evidence burden rather than a circular reduction.

full rationale

The paper's central claim, Theorem A, asserts that explicit maps hkr composed with (td)^(1/2) and the inverse of (td)^(1/2) composed with HKR form an isomorphism of calculi. The Todd class is not defined as the object that makes the maps isomorphisms; it is defined independently in Section 2.3 as a Berezinian of the Atiyah cocycle, and the HKR maps are the standard Hochschild–Kostant–Rosenberg maps of Section 2.4. The proof reduces from the Fedosov dg Lie algebroid to the dg manifold using Theorem 2.5 of the authors' unpublished preprint [30], and the final step of the proof of Theorem A states that 'the vertical maps are isomorphisms of calculi (by Proposition 3.3 and Theorem 2.5)'. This is a load-bearing external dependency, and Propositions B.3 and B.4 are likewise imported without proofs, but neither situation makes the theorem equivalent to its inputs: Theorem 2.5 asserts calculus-preserving contractions between the tangent and Fedosov algebroids, a different statement from the Todd-twisted HKR isomorphism, and Propositions B.3/B.4 are local formality coefficient computations cited from the literature. The manuscript also flags the sketch-status of Theorem B(i) in [32] but supplies a full proof in Section 4.2. Thus there is no step in the derivation chain where a fitted parameter is renamed as a prediction, an ansatz is smuggled in via self-citation, or a known result is repackaged under new coordinates. The verification risks connected to unpublished self-citations and imported propositions should be assessed as correctness or evidence concerns, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim is parameter-free over dg manifolds; the choice of affine connection is proven irrelevant. The proof imports standard formality theorems and unpublished Fedosov infrastructure, and the main new object, the square root of the Todd class, is a defined mathematical quantity rather than a fitted parameter.

assumptions (5)

standard math Existence and standard properties of L-infinity algebras, L-infinity modules, homotopy transfer, and pullback modules.
Used throughout Appendix A and in the construction of formality morphisms; assumed as background theory.
standard math Kontsevich's formality theorem for the dgla of polyvector fields and Shoikhet-Tsygan formality for chains on trivialized Z-graded manifolds.
Foundation of the twisted formality morphisms; stated as Theorem B.1 and B.2 with references to [23,42,55,10].
domain assumption Fedosov dg Lie algebroid calculus contractions of [30] (Theorem 2.5).
Main descent step from Fedosov algebroid to dg manifold; prior unpublished result by the same authors.
domain assumption Finite-dimensionality of dg manifolds and characteristic-zero ground field.
Needed for the HKR theorem and for the existence of connections and partitions of unity; not stated as a global convention in Theorem A.
domain assumption Propositions B.3 and B.4 giving the twisted first and zeroth Taylor coefficients via the square root of the A-hat cocycle.
Imported from [50,53] and not proved in this text; one is described as having been conjectured by Shoikhet.

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Pith. "Pith review of Duflo--Kontsevich-type isomorphisms for Tamarkin--Tsygan calculi of dg manifolds." pith.science (2026). https://pith.science/paper/EP5XJ6TX

@misc{pith2026260807362,
  author       = {Pith},
  title        = {Pith review of: Duflo--Kontsevich-type isomorphisms for Tamarkin--Tsygan calculi of dg manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EP5XJ6TX}},
  note         = {Machine review of arXiv:2608.07362}
}

read the original abstract

We establish a Duflo--Kontsevich-type theorem for differential graded (dg) manifolds. Specifically, we prove that the Hochschild--Kostant--Rosenberg maps twisted by the square root of the Todd class realize an isomorphism between the Tamarkin--Tsygan calculus and the Cartan calculus of the dg manifold. At the level of cohomology, this confirms the Kontsevich--Shoikhet conjecture formulated in arXiv:math/9812009.

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Works this paper leans on

57 extracted references · 51 canonical work pages

[30]

Formal geometry and Tamarkin–Tsygan calculi for dg mani- folds

Hsuan-Yi Liao, Mathieu Stiénon, and Ping Xu. Formal geometry and Tamarkin–Tsygan calculi for dg mani- folds. preprint. 2025

work page 2025
[1]

The geometry of the master equation and topological quantum field theory

M. Alexandrov, A. Schwarz, O. Zaboronsky, and M. Kontsevich. “The geometry of the master equation and topological quantum field theory” . In:Internat. J. Modern Phys. A 12.7 (1997), pp. 1405–1429. doi: 10.1142/ S0217751X97001031

work page 1997
[2]

Descent of Deligne-Getzler ∞-groupoids

Ruggero Bandiera. Descent of Deligne-Getzler ∞-groupoids. 2017. arXiv: 1705.02880 [math.AT]

arXiv 2017
[3]

Shifted derived Poisson manifolds associated with Lie pairs

Ruggero Bandiera, Zhuo Chen, Mathieu Stiénon, and Ping Xu. “Shifted derived Poisson manifolds associated with Lie pairs” . In:Comm. Math. Phys. 375.3 (2020), pp. 1717–1760. doi: 10.1007/s00220-019-03457-w

work page doi:10.1007/s00220-019-03457-w 2020
[4]

Differential graded manifolds of finite positive amplitude

Kai Behrend, Hsuan-Yi Liao, and Ping Xu. “Differential graded manifolds of finite positive amplitude” . In: Int. Math. Res. Not. IMRN 8 (2024), pp. 7160–7200. doi: 10.1093/imrn/rnae023

doi:10.1093/imrn/rnae023 2024
[5]

On the structure of étale fibrations of 𝐿∞-bundles

Kai Behrend, Hsuan-Yi Liao, and Ping Xu. On the structure of étale fibrations of 𝐿∞-bundles. 2023. arXiv: 2307.08179 [math.DG]

work page arXiv 2023
[6]

Introduction to superanalysis

Felix Alexandrovich Berezin. Introduction to superanalysis. Vol. 9. Mathematical Physics and Applied Math- ematics. Edited and with a foreword by A. A. Kirillov, With an appendix by V . I. Ogievetsky, Translated from the Russian by J. Niederle and R. Kotecký, Translation edited by Dimitri Leĭtes. D. Reidel Publishing Co., Dordrecht, 1987, pp. xii+424. doi...

work page doi:10.1007/978-94-017-1963-6 1987
[7]

Compatibility with cap-products in Tsygan’s formality and homologi- cal Duflo isomorphism

Damien Calaque and Carlo A. Rossi. “Compatibility with cap-products in Tsygan’s formality and homologi- cal Duflo isomorphism” . In:Lett. Math. Phys. 95.2 (2011), pp. 135–209. doi: 10.1007/s11005-010-0451-z

work page doi:10.1007/s11005-010-0451-z 2011

Show all 57 references

[8]

Căldăraru’s conjecture and Tsygan’s formal- ity

Damien Calaque, Carlo A. Rossi, and Michel Van den Bergh. “Căldăraru’s conjecture and Tsygan’s formal- ity” . In:Ann. of Math. (2) 176.2 (2012), pp. 865–923. doi: 10.4007/annals.2012.176.2.4

2012 doi
[9]

From topological field theory to deformation quantization and reduction

Alberto S. Cattaneo. “From topological field theory to deformation quantization and reduction” . In: Inter- national Congress of Mathematicians. Vol. III. Eur. Math. Soc., Zürich, 2006, pp. 339–365

2006
[10]

Relative formality theorem and quantisation of coisotropic sub- manifolds

Alberto S. Cattaneo and Giovanni Felder. “Relative formality theorem and quantisation of coisotropic sub- manifolds” . In:Adv. Math. 208.2 (2007), pp. 521–548. doi: 10.1016/j.aim.2006.03.010

2007 doi
[11]

The character map in deformation quanti- zation

Alberto S. Cattaneo, Giovanni Felder, and Thomas Willwacher. “The character map in deformation quanti- zation” . In:Adv. Math. 228.4 (2011), pp. 1966–1989. doi: 10.1016/j.aim.2011.06.026

2011 doi
[12]

On the Hochschild-Kostant-Rosenberg map for graded manifolds

Alberto S. Cattaneo, Domenico Fiorenza, and Riccardo Longoni. “On the Hochschild-Kostant-Rosenberg map for graded manifolds” . In:Int. Math. Res. Not.62 (2005), pp. 3899–3918. doi:10.1155/IMRN.2005.3899

2005
[13]

Atiyah and Todd classes arising from integrable distributions

Zhuo Chen, Maosong Xiang, and Ping Xu. “Atiyah and Todd classes arising from integrable distributions” . In: J. Geom. Phys. 136 (2019), pp. 52–67. doi: 10.1016/j.geomphys.2018.10.011

2019 doi
[14]

Hochschild cohomology of dg manifolds associated to integrable distributions

Zhuo Chen, Maosong Xiang, and Ping Xu. “Hochschild cohomology of dg manifolds associated to integrable distributions” . In:Comm. Math. Phys. 396.2 (2022), pp. 647–684. doi: 10.1007/s00220-022-04473-z

2022 doi
[15]

Noncommutative geometry

Alain Connes. Noncommutative geometry. San Diego, CA: Academic Press Inc., 1994, pp. xiv+661. REFERENCES 55

1994
[16]

Global Homotopies for Differential Hochschild Cohomologies

Marvin Dippell, Chiara Esposito, Jonas Schnitzer, and Stefan Waldmann. Global Homotopies for Differential Hochschild Cohomologies. 2026. arXiv: 2410.15903 [math.DG]

2026 arXiv
[17]

A formality theorem for Hochschild chains

Vasiliy A. Dolgushev. “A formality theorem for Hochschild chains” . In:Adv. Math. 200.1 (2006), pp. 51–101. doi: 10.1016/j.aim.2004.10.017

2006 doi
[18]

Covariant and equivariant formality theorems

Vasiliy A. Dolgushev. “Covariant and equivariant formality theorems” . In:Adv. Math. 191.1 (2005), pp. 147–

2005
[19]

On a variant of noncommutative differential geometry

I. M. Gel ′fand, Yu. L. Daletskiĭ, and B. L. Tsygan. “On a variant of noncommutative differential geometry” . In: Dokl. Akad. Nauk SSSR 308.6 (1989), pp. 1293–1297

1989
[20]

The cohomology structure of an associative ring

Murray Gerstenhaber. “The cohomology structure of an associative ring” . In: Ann. of Math. (2) 78 (1963), pp. 267–288. doi: 10.2307/1970343

1963 doi
[21]

Cartan homotopy formulas and the Gauss-Manin connection in cyclic homology

Ezra Getzler. “Cartan homotopy formulas and the Gauss-Manin connection in cyclic homology” . In: Quan- tum deformations of algebras and their representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992) . Vol. 7. Israel Math. Conf. Proc. Bar-Ilan Univ., Ramat Gan, 1993, pp. 65–78

1991
[22]

Differential forms on regular affine algebras

G. Hochschild, Bertram Kostant, and Alex Rosenberg. “Differential forms on regular affine algebras” . In: Trans. Amer. Math. Soc. 102 (1962), pp. 383–408. doi: 10.2307/1993614

1962 doi
[23]

Deformation quantization of Poisson manifolds

Maxim Kontsevich. “Deformation quantization of Poisson manifolds” . In: Lett. Math. Phys. 66.3 (2003), pp. 157–216. doi: 10.1023/B:MATH.0000027508.00421.bf

2003
[24]

Tamarkin–Tsygan calculi associated with formal groupoids

Niels Kowalzig, Hsuan-Yi Liao, Mathieu Stiénon, and Ping Xu. Tamarkin–Tsygan calculi associated with formal groupoids. preprint. 2026

2026
[25]

An introduction to 𝐿∞-algebras and their homotopy theory for the working mathematician

Andreas Kraft and Jonas Schnitzer. “An introduction to 𝐿∞-algebras and their homotopy theory for the working mathematician” . In: Rev. Math. Phys. 36.1 (2024), Paper No. 2330006, 83. doi: 10 . 1142 / S0129055X23300066

2024
[26]

The universal Lie ∞-algebroid of a singular foliation

Camille Laurent-Gengoux, Sylvain Lavau, and Thomas Strobl. “The universal Lie ∞-algebroid of a singular foliation” . In:Doc. Math. 25 (2020), pp. 1571–1652

2020
[27]

Atiyah classes and Todd classes of pullback dg Lie algebroids associated with Lie pairs

Hsuan-Yi Liao. “Atiyah classes and Todd classes of pullback dg Lie algebroids associated with Lie pairs” . In: Comm. Math. Phys. 404.2 (2023), pp. 701–734. doi: 10.1007/s00220-023-04854-y

2023 doi
[28]

Keller admissible triples and Duflo theorem

Hsuan-Yi Liao and Seokbong Seol. “Keller admissible triples and Duflo theorem” . In: J. Math. Pures Appl. (9) 174 (2023), pp. 1–43. doi: 10.1016/j.matpur.2023.02.003

2023 doi
[29]

Fedosov contractions for graded manifolds

Hsuan-Yi Liao, Mathieu Stiénon, and Ping Xu. Fedosov contractions for graded manifolds. preprint. 2025

2025
[31]

Formality and Kontsevich-Duflo type theorems for Lie pairs

Hsuan-Yi Liao, Mathieu Stiénon, and Ping Xu. “Formality and Kontsevich-Duflo type theorems for Lie pairs” . In:Adv. Math. 352 (2019), pp. 406–482. doi: 10.1016/j.aim.2019.04.047

2019 doi
[32]

Formality theorem for differential graded manifolds

Hsuan-Yi Liao, Mathieu Stiénon, and Ping Xu. “Formality theorem for differential graded manifolds” . In:C. R. Math. Acad. Sci. Paris 356.1 (2018), pp. 27–43. doi: 10.1016/j.crma.2017.11.017

2018 doi
[33]

Cohomologie tangente et cup-produit pour la quantification de Kontsevich

Dominique Manchon and Charles Torossian. “Cohomologie tangente et cup-produit pour la quantification de Kontsevich” . In:Ann. Math. Blaise Pascal 10.1 (2003), pp. 75–106

2003
[34]

Erratum: “Tangent cohomology and cup-product for the Kontsevich quantization

Dominique Manchon and Charles Torossian. “Erratum: “Tangent cohomology and cup-product for the Kontsevich quantization” (French) [Ann. Math. Blaise Pascal 10 (2003), no. 1, 75–106; MR1990011]” . In: Ann. Math. Blaise Pascal 11.1 (2004), pp. 129–130

2003
[35]

Lie methods in deformation theory

Marco Manetti. Lie methods in deformation theory . Springer Monographs in Mathematics. Springer, Singa- pore, 2022, pp. xii+574. doi: 10.1007/978-981-19-1185-9

2022 doi
[36]

The Atiyah class of a dg-vector bundle

Rajan A. Mehta, Mathieu Stiénon, and Ping Xu. “The Atiyah class of a dg-vector bundle” . In: C. R. Math. Acad. Sci. Paris 353.4 (2015), pp. 357–362. doi: 10.1016/j.crma.2015.01.019

2015 doi
[37]

On the morphism of Duflo-Kirillov type

Takuro Mochizuki. “On the morphism of Duflo-Kirillov type” . In: J. Geom. Phys. 41.1-2 (2002), pp. 73–113. doi: 10.1016/S0393-0440(01)00049-3

2002 doi
[38]

Deformations of coisotropic submanifolds and strong homotopy Lie al- gebroids

Yong-Geun Oh and Jae-Suk Park. “Deformations of coisotropic submanifolds and strong homotopy Lie al- gebroids” . In:Invent. Math. 161.2 (2005), pp. 287–360. doi: 10.1007/s00222-004-0426-8 . 56 REFERENCES

2005 doi
[39]

Differential forms on general commutative algebras

George S. Rinehart. “Differential forms on general commutative algebras” . In: Trans. Amer. Math. Soc. 108 (1963), pp. 195–222. doi: 10.2307/1993603

1963 doi
[40]

Geometry of Batalin-Vilkovisky quantization

Albert Schwarz. “Geometry of Batalin-Vilkovisky quantization” . In: Comm. Math. Phys. 155.2 (1993), pp. 249–260

1993
[41]

Dg manifolds, formal exponential maps and homotopy Lie algebras

Seokbong Seol, Mathieu Stiénon, and Ping Xu. “Dg manifolds, formal exponential maps and homotopy Lie algebras” . In:Comm. Math. Phys. 391.1 (2022), pp. 33–76. doi: 10.1007/s00220-021-04265-x

2022 doi
[42]

A proof of the Tsygan formality conjecture for chains

Boris Shoikhet. “A proof of the Tsygan formality conjecture for chains” . In:Adv. Math. 179.1 (2003), pp. 7–37. doi: 10.1016/S0001-8708(02)00023-3

2003 doi
[43]

On the Duflo formula for 𝐿∞-algebras and 𝑄-manifolds

Boris Shoikhet. On the Duflo formula for 𝐿∞-algebras and 𝑄-manifolds. 1998. arXiv: math / 9812009 [math.QA]

1998
[44]

Vanishing of the Kontsevich integrals of the wheels

Boris Shoikhet. “Vanishing of the Kontsevich integrals of the wheels” . In: Lett. Math. Phys. 56.2 (2001). Eu- roConférence Moshé Flato 2000, Part II (Dijon), pp. 141–149. doi: 10.1023/A:1010842705836

2001 doi
[45]

Atiyah classes and Kontsevich-Duflo type theorem for dg manifolds

Mathieu Stiénon and Ping Xu. “Atiyah classes and Kontsevich-Duflo type theorem for dg manifolds” . In: Homotopy algebras, deformation theory and quantization . Vol. 123. Banach Center Publ. Polish Acad. Sci. Inst. Math., Warsaw, 2021, pp. 63–110. doi: 10.4064/bc123-3

2021 doi
[46]

Noncommutative differential calculus, homotopy BV algebras and formality conjectures

Dmitry E. Tamarkin and Boris L. Tsygan. “Noncommutative differential calculus, homotopy BV algebras and formality conjectures” . In:Methods Funct. Anal. Topology 6.2 (2000), pp. 85–100

2000
[47]

Localization of the Hochschild homology complex for fine algebras

Nicolae Teleman. “Localization of the Hochschild homology complex for fine algebras” . In: Proceedings of “BOLYAI 200” International Conference on Geometry and Topology . Cluj Univ. Press, Cluj-Napoca, 2003, pp. 169–184

2003
[48]

Formality conjectures for chains

Boris L. Tsygan. “Formality conjectures for chains” . In: Differential topology, infinite-dimensional Lie alge- bras, and applications . Vol. 194. Amer. Math. Soc. Transl. Ser. 2. Amer. Math. Soc., Providence, RI, 1999, pp. 261–274. doi: 10.1090/trans2/194/13

1999 doi
[49]

Noncommutative calculus and operads

Boris L. Tsygan. “Noncommutative calculus and operads” . In: Topics in noncommutative geometry. Vol. 16. Clay Math. Proc. Amer. Math. Soc., Providence, RI, 2012, pp. 19–66

2012
[50]

The Kontsevich weight of a wheel with spokes pointing outward

Michel Van den Bergh. “The Kontsevich weight of a wheel with spokes pointing outward” . In: Algebr. Rep- resent. Theory 12.2-5 (2009), pp. 443–479. doi: 10.1007/s10468-009-9161-6

2009 doi
[51]

𝑄-manifolds and higher analogs of Lie algebroids

Theodore Th. Voronov. “𝑄-manifolds and higher analogs of Lie algebroids” . In:XXIX Workshop on Geometric Methods in Physics. Vol. 1307. AIP Conf. Proc. Amer. Inst. Phys., Melville, NY, 2010, pp. 191–202

2010
[52]

Charles A. Weibel. An introduction to homological algebra. Vol. 38. Cambridge Studies in Advanced Mathe- matics. Cambridge University Press, Cambridge, 1994, pp. xiv+450. doi: 10.1017/CBO9781139644136

1994 doi
[53]

A counterexample to the quantizability of modules

Thomas Willwacher. “A counterexample to the quantizability of modules” . In:Lett. Math. Phys. 81.3 (2007), pp. 265–280. doi: 10.1007/s11005-007-0179-6

2007 doi
[54]

Formality of cyclic chains

Thomas Willwacher. “Formality of cyclic chains” . In: Int. Math. Res. Not. IMRN 17 (2011), pp. 3939–3956. doi: 10.1093/imrn/rnq196

2011 doi
[55]

The homotopy braces formality morphism

Thomas Willwacher. “The homotopy braces formality morphism” . In:Duke Math. J.165.10 (2016), pp. 1815–

2016
[177]

doi: 10.1016/j.aim.2004.02.001

2004 doi
[1964]

doi: 10.1215/00127094-3450644. Department of Mathematics, National Tsing Hua University Email address: hyliao@math.nthu.edu.tw Department of Mathematics, Pennsylvania State University Email address: stienon@psu.edu Department of Mathematics, Pennsylvania State University Email...

doi

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [30]

Formal geometry and Tamarkin–Tsygan calculi for dg mani- folds

Hsuan-Yi Liao, Mathieu Stiénon, and Ping Xu. Formal geometry and Tamarkin–Tsygan calculi for dg mani- folds. preprint. 2025

work page 2025

[2] [1]

The geometry of the master equation and topological quantum field theory

M. Alexandrov, A. Schwarz, O. Zaboronsky, and M. Kontsevich. “The geometry of the master equation and topological quantum field theory” . In:Internat. J. Modern Phys. A 12.7 (1997), pp. 1405–1429. doi: 10.1142/ S0217751X97001031

work page 1997

[3] [2]

Descent of Deligne-Getzler ∞-groupoids

Ruggero Bandiera. Descent of Deligne-Getzler ∞-groupoids. 2017. arXiv: 1705.02880 [math.AT]

arXiv 2017

[4] [3]

Shifted derived Poisson manifolds associated with Lie pairs

Ruggero Bandiera, Zhuo Chen, Mathieu Stiénon, and Ping Xu. “Shifted derived Poisson manifolds associated with Lie pairs” . In:Comm. Math. Phys. 375.3 (2020), pp. 1717–1760. doi: 10.1007/s00220-019-03457-w

work page doi:10.1007/s00220-019-03457-w 2020

[5] [4]

Differential graded manifolds of finite positive amplitude

Kai Behrend, Hsuan-Yi Liao, and Ping Xu. “Differential graded manifolds of finite positive amplitude” . In: Int. Math. Res. Not. IMRN 8 (2024), pp. 7160–7200. doi: 10.1093/imrn/rnae023

doi:10.1093/imrn/rnae023 2024

[6] [5]

On the structure of étale fibrations of 𝐿∞-bundles

Kai Behrend, Hsuan-Yi Liao, and Ping Xu. On the structure of étale fibrations of 𝐿∞-bundles. 2023. arXiv: 2307.08179 [math.DG]

work page arXiv 2023

[7] [6]

Introduction to superanalysis

Felix Alexandrovich Berezin. Introduction to superanalysis. Vol. 9. Mathematical Physics and Applied Math- ematics. Edited and with a foreword by A. A. Kirillov, With an appendix by V . I. Ogievetsky, Translated from the Russian by J. Niederle and R. Kotecký, Translation edited by Dimitri Leĭtes. D. Reidel Publishing Co., Dordrecht, 1987, pp. xii+424. doi...

work page doi:10.1007/978-94-017-1963-6 1987

[8] [7]

Compatibility with cap-products in Tsygan’s formality and homologi- cal Duflo isomorphism

Damien Calaque and Carlo A. Rossi. “Compatibility with cap-products in Tsygan’s formality and homologi- cal Duflo isomorphism” . In:Lett. Math. Phys. 95.2 (2011), pp. 135–209. doi: 10.1007/s11005-010-0451-z

work page doi:10.1007/s11005-010-0451-z 2011