Deligne's conjecture on Hochschild cohomology

In deformation theory, a branch of mathematics, Deligne's conjecture is about the operadic structure on Hochschild cochain complex. Various proofs have been suggested by Dmitry Tamarkin, ^{[1] [2]} Alexander A. Voronov, ^[3] James E. McClure and Jeffrey H. Smith, ^[4] Maxim Kontsevich and Yan Soibelman, ^[5] and others, after an initial input of construction of homotopy algebraic structures on the Hochschild complex. ^{[6] [7]} It is of importance in relation with string theory.

See also

• Piecewise algebraic space

References

1. Tamarkin, Dmitry E. (1998). "Another proof of M. Kontsevich formality theorem". arXiv: math/9803025 .
2. Hinich, Vladimir (2003). "Tamarkin's proof of Kontsevich formality theorem". Forum Math. 15 (4): 591–614. arXiv: math/0003052 . doi:10.1515/form.2003.032. S2CID   220814.
3. Voronov, Alexander A. (2000). "Conférence Moshé Flato 1999". Conférence Moshé Flato 1999, Vol. II (Dijon). Dordrecht: Kluwer Acad. Publ. pp. 307–331. arXiv: math/9908040 . doi:10.1007/978-94-015-1276-3_23. ISBN   978-90-481-5551-4.
4. McClure, James E.; Smith, Jeffrey H. (2002). "A solution of Deligne's Hochschild cohomology conjecture". Recent progress in homotopy theory (Baltimore, MD, 2000). Providence, RI: Amer. Math. Soc. pp. 153–193. arXiv: math/9910126 .
5. Kontsevich, Maxim; Soibelman, Yan (2000). "Deformations of algebras over operads and the Deligne conjecture". Conférence Moshé Flato 1999, Vol. I (Dijon). Dordrecht: Kluwer Acad. Publ. pp. 255–307. arXiv: math/0001151 .
6. Getzler, Ezra; Jones, J. D. S. (1994). "Operads, homotopy algebra and iterated integrals for double loop spaces". arXiv: hep-th/9403055 .
7. Voronov, A. A.; Gerstenhaber, M. (1995). "Higher operations on the Hochschild complex" . Funct. Anal. Its Appl. 29: 1–5. doi:10.1007/BF01077036. S2CID   121740728.

Further reading

• https://ncatlab.org/nlab/show/Deligne+conjecture
• https://mathoverflow.net/questions/374/delignes-conjecture-the-little-discs-operad-one