Cyclic Brace Operations

• Cyclic brace operations are multilinear operations on cochain complexes that extend classical brace operations by integrating cyclic symmetries and BV operators.
• They are constructed using directed planar trees to encode cyclic insertions, generating an odd Lie bracket and a full homotopy BV structure.
• These operations underpin applications in cyclic cohomology, deformation theory, and string topology by modeling gravity and framed little disk operads.

Cyclic brace operations refer to a class of multilinear operations, defined on cochain complexes, that extend the classical brace operations of Gerstenhaber and Kadeishvili to settings respecting cyclic symmetries—incorporating, crucially, Batalin–Vilkovisky (BV) operators and encoding the full symmetry of the moduli spaces of punctured Riemann spheres (the "gravity operad"). These structures play a pivotal rôle in the cohomology of cyclic operads, deformation theory, open-closed string topology, and the algebraic underpinnings of topological quantum field theory.

1. Operadic and Lie-Theoretic Foundations

Let $\mathcal{O}$ be a cyclic operad: for each $n$, the component $\mathcal{O}(n)$ is equipped with a compatible action of the symmetric group $S_{n+1}$, treating outputs and inputs symmetrically. Compositional structure is encoded via the operations $\mu \circ_{ij} \nu$, constructed by first cyclically permuting the entries of $\mu$, then inserting $\nu$ into slot $j$ (with $\mu \in \mathcal{O}(n)$, $\nu\in\mathcal{O}(m)$).

From these partial compositions, one constructs an odd Lie bracket on the coinvariants $$\mathcal{O}^\bullet = \bigoplus_n \mathcal{O}(n)_{S_{n+1}}$$: $$\{[\alpha], [\beta]\} = \sum_{i,j} [\alpha \circ_{ij} \beta ]$$ of degree $-1$, satisfying the graded Jacobi identity. Maurer–Cartan (MC) elements $\eta \in \mathcal{O}^\bullet$ are degree $2$ elements satisfying $d(\eta) + \frac{1}{2}\{\eta,\eta\}=0$. Twisting the differential yields $d_\eta(x) = dx + \{\eta, x\}$, giving a new dg Lie algebra.

2. Definition and Structure of Cyclic Brace Operations

The cyclic brace operad $B^{\mathrm{cyc}}$ is generated by directed planar trees with $n$ labeled vertices, subject to the relation that reversing the orientation of any edge reverses sign. The operad structure is given by grafting operations, matching the combinatorics of planar trees to insertion operations in the operad $\mathcal{O}$. On cochains, the action of a tree $T\in B^{\mathrm{cyc}}(n)$ with $n$ “white” vertices and oriented edges is: $$T \colon \mathcal{O}^\bullet \otimes \dots \otimes \mathcal{O}^\bullet \longrightarrow \mathcal{O}^\bullet$$ by sequentially composing the labeled elements via iterated $\circ_{ij}$ operations along the orientation structure of $T$.

For a generator—the arity $k$ cyclic brace $B_k \in B^{\mathrm{cyc}}(k+1)$—the operation generalizes the classical braces by summing over all cyclic insertions: $$B_k(a; b_1, ..., b_k) = \sum_{\text{cyclic insertions}} \pm t_n^{?}(a \circ_{i_1} b_1 \circ_{i_2} b_2 \dots \circ_{i_k} b_k)$$ with $t_n$ the generator of the cyclic group $S_{n+1}$.

3. Cyclic Brace Relations and the BV Structure

Cyclic braces satisfy an extension of the classical brace relations, integrating both the cyclic symmetries and the BV operator $\Delta$ (in the presence of a nondegenerate, graded-symmetric pairing $\omega$ on the coefficient algebra). For instance, the first-order cyclic brace combines an $m$-tuple of operations $E_1, ..., E_m$ and the BV operator $\Delta$: $D\{E_1, ..., E_m, \Delta\}(z_{[\ell]}; a_{[k]})$ where $D$ is a Hochschild-type cochain, and the operation is defined by pairing results of various permutations and insertions of $\Delta$ via $\omega$.

Second-order cyclic braces (including a "ghost" symbol $\lozenge$ for nested $\Delta$ insertions in double-brace composites) are forced by the requirement that twofold compositions expand into a sum of first- and second-order terms, maintaining coherence of the operadic structure.

Collectively, the cyclic brace relations ("cyclic brace identities") express how iterated application of braces and the BV operator expand into a layered, signed, finite sum, encoding all the algebraic data leading to the Gerstenhaber bracket, cup product, and cochain-level BV operators, and thus equipping the Hochschild-type complex with a full framed little disk (homotopy BV) structure.

4. Explicit Formulas and Stepwise Construction

The step-by-step implementation consists of:

1. Taking a cyclic operad $\mathcal{O}$ and forming the odd coinvariant Lie algebra $(\mathcal{O}^\bullet, \{\,,\,\}, d)$.
2. Choosing a cyclically invariant MC element $\eta$, twisting the differential to $d_\eta$.
3. Defining the cyclic brace operad $B^{\mathrm{cyc}}$ from planar directed trees.
4. Realizing each $T\in B^{\mathrm{cyc}}(n)$ as a multilinear operation via tree-based insertions and sum over all valid cyclic orientations.
5. Verifying that $(\mathcal{O}^\bullet, d_\eta)$ acquires a $B^{\mathrm{cyc}}$-algebra structure.
6. Extending to chain models such as $M_0$ (the dg operad of unrooted $A_\infty$-trees), which acts and induces gravity algebra structures on cohomology.

When the number of $\Delta$ insertions is set to zero, one recovers the classical Getzler–Kadeishvili braces; increasing the number recovers all BV algebraic input.

5. Applications in Homological Algebra and Geometry

Cyclic brace operations underpin a variety of algebraic and geometric contexts:

• Cyclic cohomology of Frobenius algebras: The Hochschild complex, equipped with natural cyclic MC elements, becomes a gravity algebra under the $M_0$-action, which arises from $B^{\mathrm{cyc}}$.
• Equivariant cohomology: For $S^1$-spaces, the cyclic brace structure recovers the Cartan model, the Connes' operator, and the gravity algebra structure on $H^*_{S^1}(X)$.
• String topology: Cyclic operads constructed from chain complexes on loop spaces or the Sullivan–Chas model support gravity-algebraic operations up to homotopy.
• Open-closed field theories: In the context of open-closed homotopy algebras (OCHAs), the BV structure on the open-closed Hochschild cohomology is encoded via cyclic brace operations, with cochain-level formulas expressing the cup product, Gerstenhaber bracket, and the BV operator (Yuan, 6 Nov 2025).

The fundamental cochain-level identity,

$$[D,E] = \Delta(D\smallsmile E) + \Delta D \smallsmile E - (-1)^{|D||E|} E\smallsmile\Delta D$$

holds on Hochschild cohomology, with all higher-order corrections and sign rules managed by the explicit brace and BV relations.

6. Relation to Gravity and BV Operads

The homology of the cyclic brace operad is the gravity operad, corresponding to the homology of the moduli space $\mathcal{M}_{0,n+1}$ of punctured Riemann spheres. The chain model $M_0$ of unrooted $A_\infty$-trees has $H_*(M_0) = \mathsf{Grav}$, governing gravity algebra structures. By extending the operad with "spines" (spined braces, incorporating Connes' $B$ operator), one obtains a chain model for the framed little disks operad $fD_2$, equipping cohomology with a genuine BV algebra structure.

There is a long exact sequence, algebraic incarnation of the Gysin sequence, relating the homology of cyclic invariants and the full cochain complex,

$$\cdots\to H^{n-1}(\mathcal{O}^\bullet) \to H^{n+1}(\mathcal{O}^\bullet) \to H^{n+1}(\mathcal{O}^*) \to H^n(\mathcal{O}^\bullet) \to \cdots$$

with $H^*(\mathcal{O}^*)$ a Gerstenhaber–BV algebra and $H^*(\mathcal{O}^\bullet)$ a gravity algebra, capturing the simultaneous gravity-infinity and homotopy BV structure at the chain and cohomology levels (Ward, 2014).

7. Outlook and Further Directions

Cyclic brace operations serve as a unifying formalism, generalizing classical braces to all settings where cyclic symmetry and additional BV or gravity structure are present. They allow for:

• Writing homotopy-relations at the cochain level that, upon passage to cohomology, recover the essential algebraic identities for Gerstenhaber, gravity, and BV algebras.
• Extending the structure of the framed little disk operad to open-closed and equivariant settings.
• Modeling string topology gravity operations and their compatibility with BV operators, and clarifying the interplay between cyclic homology, BV, and gravity structure in a broad array of deformation, field-theoretic, and categorical frameworks.

These structures are computationally tractable due to explicit tree-based and partition-based formulas, and conceptually significant due to their connection to the geometry of moduli spaces and algebraic topology, as exemplified by recent applications to open-closed Hochschild cohomology and string topology operations (Yuan, 6 Nov 2025, Ward, 2014).