[Paper Review] Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I

Maxim Kontsevich, Yan Soibelman|arXiv (Cornell University)|Jun 11, 2006

Algebraic structures and combinatorial modelsMathematics12 references63 citations

TL;DR

This paper establishes a geometric interpretation of A-infinity algebras as non-commutative formal pointed dg-manifolds equipped with a homological vector field of degree +1, providing a dictionary between A-infinity algebra structures and non-commutative geometry. The key contribution is the construction of a non-commutative formal dg-manifold from an A-infinity algebra via its coalgebra of distributions, with applications to Hochschild cohomology, Calabi-Yau structures, and the Hodge-to-de Rham degeneration conjecture.

ABSTRACT

We develop geometric approach to A-infinity algebras and A-infinity categories based on the notion of formal scheme in the category of graded vector spaces. Geometric approach clarifies several questions, e.g. the notion of homological unit or A-infinity structure on A-infinity functors. We discuss Hochschild complexes of A-infinity algebras from geometric point of view. The paper contains homological versions of the notions of properness and smoothness of projective varieties as well as the non-commutative version of Hodge-to-de Rham degeneration conjecture. We also discuss a generalization of Deligne's conjecture which includes both Hochschild chains and cochains. We conclude the paper with the description of an action of the PROP of singular chains of the topological PROP of 2-dimensional surfaces on the Hochschild chain complex of an A-infinity algebra with the scalar product. This action is essentially equivalent to the structure of 2-dimensional Topological Field Theory associated with a Calabi-Yau category.

Motivation & Objective

• To develop a geometric framework for A-infinity algebras using non-commutative formal dg-manifolds.
• To interpret A-infinity algebras as pointed graded manifolds with homological vector fields, linking algebraic structures to geometric intuition.
• To provide a dictionary between A-infinity algebraic constructions and non-commutative geometry, especially in terms of coalgebras and functors.
• To lay the foundation for extending the geometric perspective to A-infinity categories in subsequent work.
• To clarify the role of Hochschild cohomology and chain complexes via geometric interpretations of cyclic differential forms and derived functors.

Proposed method

• Represent A-infinity algebras as formal pointed graded manifolds with a homological vector field d of degree +1 satisfying [d,d] = 0.
• Use the dual coalgebra of distributions to model the algebra of formal power series on the non-commutative manifold.
• Define non-commutative schemes via functors on graded associative Artin algebras, represented by coalgebras.
• Construct the formal neighborhood of a subscheme via completion along a closed embedding using locally conilpotent quotients.
• Apply the Yoneda lemma to describe the Hochschild cochain complex as the tangent space to deformations of the identity functor.
• Utilize configuration spaces of points on cylinders and operads to generalize Deligne’s conjecture on A-infinity algebras.

Experimental results

Research questions

• RQ1 How can A-infinity algebras be interpreted geometrically as non-commutative formal dg-manifolds?
• RQ2 What is the geometric meaning of Hochschild cohomology and chain complexes in terms of cyclic differential forms?
• RQ3 How do Calabi-Yau structures on A-infinity algebras relate to the Hodge-to-de Rham degeneration conjecture?
• RQ4 What is the role of weak units and non-unital structures in the non-commutative geometry of A-infinity algebras?
• RQ5 How can the Yoneda embedding and derived functors be used to describe A-infinity functors and their structures?

Key findings

• An A-infinity algebra is canonically associated with a non-commutative formal pointed dg-manifold (X, pt, d), where d is a homological vector field of degree +1 vanishing at the marked point.
• The Hochschild cochain complex of an A-infinity algebra is isomorphic to Ext^•(Id_C, Id_C) in the A-infinity category of endofunctors, providing a geometric interpretation of Hochschild cohomology.
• The formal neighborhood of a closed subscheme in a non-commutative smooth thin scheme is represented by a coalgebra of locally conilpotent elements, generalizing the classical formal completion.
• For smooth non-commutative schemes, the formal neighborhood of a subscheme is itself smooth and isomorphic to its own completion.
• The construction of the coalgebra of distributions from an A-infinity algebra provides a duality between formal power series algebras and coalgebras, enabling a functorial description.
• The Hodge-to-de Rham degeneration conjecture for A-infinity algebras is formulated in terms of the existence of a Calabi-Yau structure, linking geometric and algebraic invariants.