[Paper Review] A proof of a cyclic version of Deligne's conjecture via Cacti

Ralph M. Kaufmann|ArXiv.org|Mar 21, 2004

Algebraic structures and combinatorial models8 references14 citations

TL;DR

This paper proves a cyclic version of Deligne's conjecture by constructing a chain model for the framed little discs operad using normalized cacti, showing that the normalized Hochschild cochains of a unital associative algebra with a non-degenerate, symmetric, invariant inner product form a BV algebra up to homotopy. The key result is that the induced Gerstenhaber bracket on Hochschild cohomology coincides with the standard Gerstenhaber bracket, establishing a homotopy BV structure via operadic correlation functions on decorated trees.

ABSTRACT

In this note, we show that the normalized Hochschild co--chains of an associative algebra with a non--degenerate, symmetric, invariant inner product are an algebra over a chain model of the framed little discs operad which is given by cacti. In particular, in this sense they are a BV algebra up to homotopy and the Hochschild cohomology of such an algebra is a BV algebra whose induced bracket coincides with Gerstenhaber's bracket. To show this, we use a cellular chain model for the framed little disc operad in terms of normalized cacti. This model is given by tensoring our chain model for the little discs operad in terms of spineless cacti with natural chain models for $(S^1)^{ imes n}$ adapted to cacti.

Motivation & Objective

• To establish a homotopy BV algebra structure on the normalized Hochschild cochains of an associative algebra with a non-degenerate, symmetric, invariant inner product.
• To extend the chain model of the little discs operad (previously in terms of spineless cacti) to a chain model for the framed little discs operad using normalized cacti.
• To show that this chain model acts on the Hochschild cochains, realizing a BV algebra up to homotopy.
• To demonstrate that the induced bracket on Hochschild cohomology matches Gerstenhaber's bracket.
• To provide a combinatorial, cellular description of the framed little discs operad via planar planted b/w bipartite trees with spines, compatible with operadic correlation functions.

Proposed method

• Construct a cellular chain model for the framed little discs operad by tensoring the normalized spineless cacti chains with chains for the operad built on the monoid $S^1$.
• Use a refined cell decomposition of the semi-direct product, incorporating cell structures on $S^1$ factors induced by the lobe structure of cacti.
• Introduce decorated planar planted black and white bipartite trees with spines to encode the position of local zeros and track operadic operations.
• Define operadic correlation functions on these decorated trees to describe the action of the framed little discs chains on Hochschild cochains.
• Verify the compatibility of the differential with the operad structure by translating the relations from the $\mathcal{A}rc$ operad using the method from [K1, KLP].
• Utilize the foliage operator and the duality of tree orientations to ensure the action respects the symmetric, invariant pairing required for the construction.

Experimental results

Research questions

• RQ1 Can a chain model for the framed little discs operad be constructed using normalized cacti, extending the spineless cactus model?
• RQ2 Does the action of this chain model on normalized Hochschild cochains endow them with a BV algebra structure up to homotopy?
• RQ3 Is the Gerstenhaber bracket on Hochschild cohomology recovered as the induced bracket from this homotopy BV structure?
• RQ4 How can the operadic correlation functions be formulated combinatorially on decorated trees to describe the action?
• RQ5 Can this framework be generalized to actions on cyclic complexes and moduli spaces of bordered surfaces?

Key findings

• The normalized Hochschild cochains of a unital associative algebra with a non-degenerate, symmetric, invariant inner product are shown to be a BV algebra up to homotopy via the framed little discs chain model.
• The induced bracket on Hochschild cohomology coincides exactly with Gerstenhaber’s standard bracket, confirming the homotopy BV structure.
• The chain model for the framed little discs operad is realized as a tensor product of normalized spineless cacti chains and $S^1$-based operad chains, with a refined cell decomposition accommodating all necessary operations.
• The action is described via operadic correlation functions on planar planted b/w bipartite trees with spines, where the spines track local zeros and the linear orders differ from the planar order.
• The differential compatibility is verified by translating relations from the $\mathcal{A}rc$ operad and aligning the differential definition with that of the Hochschild complex.
• The construction admits generalizations to cyclic complexes and moduli spaces of decorated bordered surfaces, suggesting a path toward string topology operations via pseudo-cells of moduli space.