The character map in deformation quantization

Cattaneo, Alberto S.; Felder, Giovanni; Willwacher, Thomas

doi:10.1016/j.aim.2011.06.026

Cited by 16 publications

(28 citation statements)

References 17 publications

(41 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It has been observed before that the Lie ∞ -formality morphisms by M. Kontsevich and B. Shoikhet on homology preserve more algebraic structure than the Lie bracket (respectively, the Lie action), see [24,38,7,9]. The present work generalizes and unifies these results.…”

Section: Recovery Of Several Results In the Literaturesupporting

confidence: 85%

The homotopy braces formality morphism

Willwacher¹

2016

Duke Math. J.

Self Cite

View full text Add to dashboard Cite

We extend M. Kontsevich's formality morphism to a homotopy braces morphism and to a homotopy Gerstenhaber morphism. We show that this morphism is homotopic to D. Tamarkin's formality morphism, obtained using formality of the little disks operad, if in the latter construction one uses the Alekseev-Torossian associator. Similar statements can also be shown in the "chains" case, i. e., on Hochschild homology instead of cohomology. This settles two well known and long standing problems in deformation quantization and unifies the several known graphical constructions of formality morphisms and homotopies by Kontsevich, Shoikhet, Calaque, Rossi, Alm, Cattaneo, Felder and the author. Contents 65 Appendix C. Operadic twisting 67 Appendix D. Colored operadic twists of several operads 71 Appendix E. Proof of Proposition 75 72 Appendix F. The cohomology of fSGraphs and SGraphs 78 References 82

show abstract

Section: Recovery Of Several Results In the Literaturesupporting

confidence: 85%

The homotopy braces formality morphism

Willwacher¹

2016

Duke Math. J.

Self Cite

View full text Add to dashboard Cite

show abstract

“…ii) The algebraic index theorem for Poisson manifolds of [40,7] gives a somewhat different result when applied to the dual of a Lie algebroid. It involves the ordinary de Rham complex instead of the Lie algebroid cochain complex.…”

Section: I)∇ Is Poisson If and Only If ∇ Is Symplectic Ii)∇ Is Homogmentioning

confidence: 99%

“…The index pairing is therefore computed by an algebraic index theorem for formal deformation quantizations of the Poisson manifold A * . Unfortunately, we can not simply apply the algebraic index theorem for Poisson manifolds [40,7,18], since an arbitrary formal deformation quantization does not reflect the geometry of the groupoid G. We therefore adapt our earlier work [37] to prove a G-invariant algebraic index theorem on a regular Poisson manifold over A * , which generalizes the foliation index theorem by Nest and Tsygan [32].…”

Section: Introductionmentioning

confidence: 97%

The localized longitudinal index theorem for Lie groupoids and the van Est map

Pflaum

Posthuma

Tang

2015

Advances in Mathematics

View full text Add to dashboard Cite

We define the "localized index" of longitudinal elliptic operators on Lie groupoids associated with Lie algebroid cohomology classes. We derive a topological expression for these numbers using the algebraic index theorem for Poisson manifolds on the dual of the Lie algebroid. Underlying the definition and computation of the localized index, is an action of the Hopf algebroid of jets around the unit space, and the characteristic map it induces on Lie algebroid cohomology. This map can be globalized to differentiable groupoid cohomology, giving a definition of the "global index", that can be computed by localization. This correspondence between the "global" and "localized" index is given by the van Est map for Lie groupoids.

show abstract

“…Our construction can also be extended to the case of supermanifolds; the trace is then replaced by a nondegenerate cyclic cocycle (Calabi-Yau structure, see [18], Section 10.2, and [10]) for the A ∞ -algebra obtained by deformation quantization in [5]. Further applications will be studied in a separate publication [6]. In particular we will derive the existence of an L ∞ -quasi-isomorphism of g Ω S -modules from the complex g Ω S with the adjoint action to the complex of cyclic cochains with a suitable module structure.…”

Section: Introductionmentioning

confidence: 99%

Effective Batalin–Vilkovisky Theories, Equivariant Configuration Spaces and Cyclic Chains

Cattaneo

Felder

2010

Progress in Mathematics

Self Cite

View full text Add to dashboard Cite

Summary. Kontsevich's formality theorem states that the differential graded Lie algebra of multidifferential operators on a manifold M is L∞-quasi-isomorphic to its cohomology. The construction of the L∞-map is given in terms of integrals of differential forms on configuration spaces of points in the upper half-plane. Here we consider configuration spaces of points in the disk and work equivariantly with respect to the rotation group. This leads to considering the differential graded Lie algebra of multivector fields endowed with a divergence operator. In the case of R d with standard volume form, we obtain an L∞-morphism of modules over this differential graded Lie algebra from cyclic chains of the algebra of functions to multivector fields. As a first application we give a construction of traces on algebras of functions with star-products associated with unimodular Poisson structures. The construction is based on the Batalin-Vilkovisky quantization of the Poisson sigma model on the disk and in particular on the treatment of its zero modes.

show abstract

The character map in deformation quantization

Cited by 16 publications

References 17 publications

The homotopy braces formality morphism

The homotopy braces formality morphism

The localized longitudinal index theorem for Lie groupoids and the van Est map

Effective Batalin–Vilkovisky Theories, Equivariant Configuration Spaces and Cyclic Chains

Contact Info

Product

Resources

About