Our goal is to investigate deformation quantization "à la Kontsevich" and closely related topics.

Feel free to reach out to Alyosha or Guillaume if you are interested in participating.

Planning

Room : seminar room of QM

• Friday January 31, 2025, 2pm - Alyosha: Formality of the little n-disks operad, Notes.

• Friday February 7, 2025, 2pm - Guillaume: Proof of formality of the little n-disks operad, Notes.

• Wednesday February 12, 2025, 1pm - Alyosha & Guillaume: The Kontsevich integral + 2 alternative proofs of formality (Tamarkin & Petersen strategies), Notes. Followed by Chandan's talk at 3pm: Grothendieck-Teichmüller group and the (cyclic) ribbon group operad.

• Friday February 21, 2025, 2pm - Alyosha: KZ connection and Drinfeld associators.

• Monday March 10, 2025, 1:30pm - Guillaume: Formality of the little discs implies deformation quantization of Poisson manifolds, Notes.

From here on, our goal will be to study in more details the paper "Higher Operations in Perturbation Theory" by Gaiotto-Kulp-Wu (2024). We start with a 3 days workshop at Syddansk Universitet on June 2-4, 2025.

The goal of this workshop is to use recent work of Gaiotto—Kulp--Wu to generalize Kontsevich's deformation quantization theorem to the topological-holomorphic setting. This will involve looking at Willwacher's models for the n-swiss-cheese operads, and trying to figure out their topological-holomorphic counterparts.

• Monday June 2, 2025, 11am, Room U230, - Alyosha: Kontsevich--Tamarkin formality for topological-holomorphic field theories, after Gaiotto--Kulp--Wu, Notes.

• Tuesday June 3, 2025, 11am, Room U230 - Guillaume: Operadic twisting.

• Tuesday June 3, 2025, 2:15pm, Room U230 - Alyosha: Working session.

• Wednesday June 4, 2025, 11am, Room U230 - Guillaume: Models of the n-Swiss Cheese operads, after Willwacher.

• Wednesday June 4, 2025, 2:15pm, Room U230 - Alyosha: Working session.

References

• Kontsevich, Deformation quantization of Poisson manifolds, 1997
• Tamarkin, Another proof of M. Kontsevich formality theorem, 1998
• Tamarkin, Formality of chain operad of small squares, 1998
• Kontsevich, Operads and motives in deformation quantization, 1999
• Tamarkin, Formality of chain operad of little discs, 2002
• Doubek-Markl-Zima, Deformation theory (lecture notes), 2007
• Lambrechts-Volic, Formality of the little n-disks operad, 2008
• Severa-Willwacher, Equivalence of formalities of the little discs operad, 2009
• Petersen, Minimal models, GT-action and formality of the little disk operad, 2013
• Willwacher, The homotopy braces formality morphism, 2015
• Willwacher, Models for the n-Swiss Cheese operads, 2015
• Fresse, Homotopy of operads and Grothendieck-Teichmüller Groups, Part I, 2017
• Banks-Panzer-Pym, Multiple zeta values in deformation quantization, 2018

Links to Lie Theory and Physics

• Cattaneo-Keller-Torossian-Bruguières, Déformation, quantification, théorie de Lie, 2002

Drinfeld associators

• Drinfeld, On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(Q/Q), 1991
• Calaque--Roca-Lucio, Associators from an operadic point of view, 2024
• Etingof-Schiffmann, Lectures on quantum groups, 2010

Generalizations

• Severa, Formality of the chain operad of framed little disks, 2010
• Giansircusa--Salvatore, Formality of the framed little 2-disks operad and semidirect products, 2010
• Campos, BV formality, 2015
• Boavida de Brito-Horel-Robertson, Operads of  genus zero curves and the Grothendieck-Teichmüller group, 2017
• Calaque-Willwacher, Triviality of the higher formality theorem, 2015

• Gaiotto-Kulp-Wu, Higher Operations in Perturbation Theory, 2024.
• Balduf-Gaiotto, Combinatorial proof of a non-renormalization theorem, 2024.

Costello, Renormalization and effective field theory, 2011

Campos, Slides on BV formality, 2017

Idrissi, The Lambrechts-Stanley Model of Configuration Spaces, 2018

Charles' Notes on Kontsevich formality theorem, 2021

Notes from a working group on GT and formality (in French), Université Paris 13, 2020-2021

 

Last update June 4, 2025.