Cyclic brace relation and BV structure on open-closed Hochschild cohomology

TL;DR

The paper develops a cyclic brace framework to extend Hochschild cohomology to open-closed homotopy algebras (OCHA) and produces a cochain-level BV identity that yields a BV algebra on normalized open-closed Hochschild cohomology $\overline{HH}(Z;A,A)$. The construction hinges on first- and second-order cyclic braces anchored by a BV symbol $\Delta$ and a nondegenerate pairing $\omega$, together with a closed-string action $\widehat{\mathfrak l}$, to relate brace operations and the Hochschild differential. Under cyclic and unital assumptions, the braces preserve normalization, and the BV operator squares to zero and commutes with the closed-string action, enabling a clean transfer to cohomology. The main result is a cochain-level BV identity that specializes to the familiar BV relation on cohomology, providing a canonical BV algebra structure for the normalized open-closed Hochschild cohomology and motivating Calabi–Yau generalizations in the OCHA setting. This framework bridges open-closed string field theory concepts with noncommutative geometry, offering tools for BV structures in Calabi–Yau and related A-infinity contexts.

Abstract

For an open-closed homotopy algebra (OCHA), the previous work indicates that there is an open-closed version of Hochschild cohomology with a canonical Gerstenhaber algebra structure. If this OCHA is further cyclic and unital in the sense of Kajiura and Stasheff, we produce a BV algebra structure on this cohomology via a cochain-level identity formulated with cyclic brace operations.

Theorems & Definitions (33)

• Theorem 1.1
• Remark 1.2
• Corollary 1.3
• Remark 1.4
• Remark 1.5
• Theorem 1.6: Cyclic Brace Relations
• Lemma 2.1
• proof
• Definition 3.1
• Definition 3.2: First-order Cyclic Brace