Skip to main content

Exhausting Formal Quantization Procedures

  • Conference paper
  • First Online:
Geometric Methods in Physics

Part of the book series: Trends in Mathematics ((TM))

Abstract

In paper [1] the author introduced stable formality quasi-isomorphisms and described the set of its homotopy classes. This result can be interpreted as a complete description of formal quantization procedures. In this note we give a brief exposition of stable formality quasi-isomorphisms and prove that every homotopy class of stable formality quasi-isomorphisms contains a representative which admits globalization. This note is loosely based on the talk given by the author at XXX Workshop on Geometric Methods in Physics in Białowieża, Poland.

Mathematics Subject Classification (2010). 53D55; 19D55.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. V.A. Dolgushev, Stable formality quasi-isomorphisms for Hochschild cochains I, arXiv:1109.6031.

    Google Scholar 

  2. M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66 (2003) 157–216; q-alg/9709040.

    Google Scholar 

  3. V.A. Dolgushev, Erratum to: “A Proof of Tsygan’s Formality Conjecture for an Arbitrary Smooth Manifold”, arXiv:math/0703113.

    Google Scholar 

  4. H. Bursztyn, V. Dolgushev, and S.Waldmann, Morita equivalence and characteristic classes of star products, accepted to J. Reine Angew. Math.; arXiv:0909.4259.

    Google Scholar 

  5. M. Kontsevich, Formality conjecture, Deformation theory and symplectic geometry (Ascona, 1996), 139–156, Math. Phys. Stud., 20, Kluwer Acad. Publ., Dordrecht, 1997.

    Google Scholar 

  6. T. Willwacher, M. Kontsevich’s graph complex and the Grothendieck-Teichmüller Lie algebra, arXiv:1009.1654.

    Google Scholar 

  7. V.A. Dolgushev and C.L. Rogers, Lecture Notes on Graph Complexes, GRT, and Willwacher’s Construction, in preparation.

    Google Scholar 

  8. H. Kajiura and J. Stasheff, Homotopy algebras inspired by classical open-closed stringfie ld theory, Commun. Math. Phys. 263 (2006) 553–581; arXiv:math/0410291.

    Google Scholar 

  9. V.A. Dolgushev, On stable formality quasi-isomorphisms over ℚ, in preparation.

    Google Scholar 

  10. V.A. Dolgushev, Stable formality quasi-isomorphisms for Hochschild cochains II, in preparation.

    Google Scholar 

  11. V.G. Drinfeld, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(\( \overline{Q} \) /ℚ). (Russian) Algebra i Analiz 2, 4 (1990) 149–181; translation in Leningrad Math. J. 2, 4 (1991) 829–860.

    Google Scholar 

  12. T. Willwacher, Stable cohomology of polyvector fields, arXiv:1110.3762.

    Google Scholar 

  13. V.A. Dolgushev, Covariant and Equivariant Formality Theorems, Adv. Math., 191, 1 (2005) 147–177; arXiv:math/0307212.

    Google Scholar 

  14. M. Van den Bergh, On global deformation quantization in the algebraic case, J. Algebra 315, 1 (2007) 326–395.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Yekutieli, Mixed resolutions, simplicial sections and unipotent group actions, Israel J. Math. 162 (2007) 1–27.

    Article  MathSciNet  MATH  Google Scholar 

  16. V.A. Dolgushev, C.L. Rogers, and T.H. Willwacher, Kontsevich’s graph complex and the deformation complex of the sheaf of polyvector fields, in preparation.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Vasily A. Dolgushev .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Basel

About this paper

Cite this paper

Dolgushev, V.A. (2013). Exhausting Formal Quantization Procedures. In: Kielanowski, P., Ali, S., Odzijewicz, A., Schlichenmaier, M., Voronov, T. (eds) Geometric Methods in Physics. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0448-6_4

Download citation

Publish with us

Policies and ethics