Deligne conjecture

Deligne conjecture
"copyright C. J. Mozzochi, Princeton N.J" · Attribution · source
Name	Deligne conjecture
Field	Algebraic topology; Operad theory; Homological algebra
Proposed	1993
Proposer	Pierre Deligne
Status	Proven with variants

Deligne conjecture The Deligne conjecture predicts that the Hochschild cochain complex of an associative algebra carries an action of an operad weakly equivalent to the little disks operad, linking the work of Pierre Deligne with developments in Operad theory, Homotopy theory, Algebraic topology, and Homological algebra. The conjecture shaped research connecting Maxim Kontsevich's formality ideas, Mikhail Gromov-style geometric intuition, and algebraic structures appearing in the work of Gerstenhaber and James Stasheff.

History and statement

The statement emerged from conversations between Pierre Deligne and participants at programs involving Alexander Grothendieck's school, Maxim Kontsevich's seminar, and researchers such as Martin Kontsevich and Dennis Sullivan about the structure on Hochschild cochains of associative algebras like those studied by Murray Gerstenhaber and Jim Stasheff. Early groundwork was influenced by the Hochschild–Kostant–Rosenberg theorem, ideas of Gerstenhaber on algebraic deformation, and the recognition of the little disks operad from F. Cohen's calculations. The conjecture asserts an action of an operad equivalent to the little disks operad (often the E2-operad) on the Hochschild cochain complex of an associative algebra, thereby lifting the Gerstenhaber algebra structure on Hochschild cohomology to the chain level and connecting with concepts from Morse theory and Chern–Simons theory.

Operadic and homotopical formulations

Formulations use the language of Operad theory, model categories, and A∞-algebras introduced by Jim Stasheff. One approach replaces the classical little disks operad by an operad of chains on configuration spaces studied by Vladimir Arnold and Michael Farber, or uses the framed variant linked to Morava K-theory contexts influenced by work of Haynes Miller and Geoffrey Powell. Homotopical refinements invoke the theory of Quillen model categories developed by Daniel Quillen and the notion of quasi-isomorphism from Pierre Deligne's collaborators, permitting equivalences between the chains on E2-operad and various combinatorial operads such as the Gerstenhaber–Schack complex refinements and Boardman–Vogt constructions associated to J. Peter May. Connections to formality theorems use ideas of Maxim Kontsevich and Tamarkin about operad formality and the action of Grothendieck–Teichmüller group structures discovered by Vladimir Drinfeld.

Proofs and partial results

Multiple proofs and partial proofs were developed by researchers including Maxim Kontsevich, Dmitry Tamarkin, Mario Gerstenhaber collaborators, Boris Tsygan, James McClure, Jeffrey Smith, Dennis Sullivan, and others. Tamarkin produced a proof using formality of the little disks operad relying on the work of Drinfeld and techniques from Lie algebra cohomology; alternative proofs used operadic chains by McClure and Smith with a cosimplicial model linked to Boardman–Vogt resolution and to constructions by Peter May. Further contributions by Vladimir Hinich, Bertrand Toën, Gabriel C. Drummond-Cole, and Kevin Costello expanded the scope to A∞-categories, derived categories studied by Alexander Beilinson and Joseph Bernstein, and field-theoretic constructions influenced by Edward Witten. Partial results include versions for differential graded algebras, for Hochschild complexes of schemes considered by Alexander Grothendieck-inspired algebraic geometers, and for topological Hochschild homology examined in the research of Bokstedt and Waldhausen.

Applications and consequences

Consequences touch Deformation quantization pioneered by Maxim Kontsevich and links with Mirror symmetry themes arising in the work of Kontsevich and Paul Seidel. The conjecture provides foundations for higher structures in Derived algebraic geometry developed by Jacob Lurie and Bertrand Toën, and for categorical actions in Topological field theory studied by Kevin Costello and Graeme Segal. It underlies formal deformation theories used by Maurice Auslander-inspired representation theorists and influences computations in String topology by researchers like Moira Chas and Dennis Sullivan. The presence of an E2-action informs invariants in K-theory and cyclic homology explored by Alain Connes and Max Karoubi, and impacts structures in Floer homology investigated by Andreas Floer-inspired schools.

Examples and computations

Concrete computations appear for associative algebras such as group algebras of finite groups studied by Jean-Pierre Serre-inspired representation theory, path algebras of quivers analyzed in the work of Bernard Keller, and commutative smooth algebras where the Hochschild–Kostant–Rosenberg theorem applies. Model computations for polynomial algebras connect to Kontsevich formality theorem examples and to graph complexes used by Maxim Kontsevich and Thomas Willwacher. Explicit chain-level constructions have been written down for algebras appearing in algebraic geometry contexts treated by Alexander Grothendieck-influenced schools and for DG-categories considered by Vladimir Drinfeld.

Variants and extensions

Variants include framed versions involving the framed little disks operad studied by F. Cohen and R. Budney, higher-dimensional generalizations to E_n-structures related to Jacob Lurie's higher algebra, and equivariant or relative forms considered by Giovanni Felder and Boris Tsygan. Extensions apply to A∞-categories, to categories of matrix factorizations used by Maxim Kontsevich and Alexander Orlov, and to topological Hochschild homology contexts pursued by Waldhausen and Nikolaus Scholze. Connections to the Grothendieck–Teichmüller group investigated by Drinfeld and Pavel Etingof provide arithmetic and deformation-theoretic refinements.

Category:Conjectures in algebraic topology