Deligne's conjecture on Hochschild cohomology explained

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

Further reading

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

Notes and References

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