Open–Closed Formality Morphism Overview

• Open–closed formality morphism is an L∞-morphism unifying closed dg Lie algebras and open modules up to homotopy.
• It employs graphical configuration-space integrals and operadic methods to translate geometric models into precise algebraic structures.
• The construction extends Kontsevich formality, relating Swiss-cheese operads and deformation techniques with applications in string field theory.

An open–closed formality morphism is an $L_\infty$-morphism that simultaneously treats a “closed” dg Lie algebra and an “open” module over it, and identifies a geometric model with an algebraic one up to homotopy. In the classical case on $M=\mathbb{R}^n$, the closed sector is the Schouten–Nijenhuis dg Lie algebra of polyvector fields and the open sector is the Hochschild chain complex, with target given by Hochschild cochains and de Rham forms; for each Drinfeld associator $\Phi$, T. Willwacher constructs $L_\infty$-quasi-isomorphisms $U_\Phi$ and $V_\Phi$ that together form a 2-colored, Swiss-cheese-compatible formality morphism, and whose homotopy class is a torsor over the Grothendieck–Teichmüller group $GRT$ (Willwacher, 2013). Related transfer-theoretic and categorical formulations place the same idea in the frameworks of the 2-colored operad $G^+$, the Swiss-Cheese operad, and open–closed string field theory (Dolgushev, 2010, Ulmer, 21 Jul 2025, Ulmer, 18 Mar 2026).

1. Classical algebraic framework

Let $M=\mathbb{R}^n$ and $A=C^\infty(M)$. The standard open–closed formality package uses the following objects:

Sector	Source	Target
Closed	$M=\mathbb{R}^n$0	$M=\mathbb{R}^n$1
Open	$M=\mathbb{R}^n$2	$M=\mathbb{R}^n$3

Here $M=\mathbb{R}^n$4 is the Schouten–Nijenhuis dg Lie algebra of polyvector fields, shifted so that a $M=\mathbb{R}^n$5-vector has degree $M=\mathbb{R}^n$6. Its differential is $M=\mathbb{R}^n$7, and its bracket is the Schouten–Nijenhuis bracket. The target closed sector is $M=\mathbb{R}^n$8, the Hochschild cochains $M=\mathbb{R}^n$9 equipped with the Gerstenhaber bracket and the Hochschild differential $\Phi$0 (Willwacher, 2013).

The open sector is the completed Hochschild chain complex $\Phi$1, with Hochschild boundary $\Phi$2 and Connes operator $\Phi$3 of degree $\Phi$4, making $\Phi$5 a mixed complex. On the geometric side, $\Phi$6 is the de Rham complex, negatively graded, with differential $\Phi$7. The module structure is given by the Lie derivative

$\Phi$8

The $\Phi$9-action on $L_\infty$0 is obtained by pulling back the usual Gerstenhaber module structure along $L_\infty$1 (Willwacher, 2013).

A related 2-colored formulation appears in V. Dolgushev’s transfer approach. There the pair $L_\infty$2 carries a $L_\infty$3-structure extending the $L_\infty$4-structure on Hochschild cochains and the $L_\infty$5-structure on $L_\infty$6, with mixed generators $L_\infty$7 encoding the open–closed homotopy algebra structure (Dolgushev, 2010).

2. The Kontsevich–Shoikhet open–closed formality theorem

For each choice of Drinfeld associator $L_\infty$8, there exist $L_\infty$9-quasi-isomorphisms

$U_\Phi$0

with three defining properties. First, $U_\Phi$1 is an $U_\Phi$2-morphism of dg Lie algebras inducing a quasi-isomorphism. Second, $U_\Phi$3 is an $U_\Phi$4-morphism of modules, intertwining the $U_\Phi$5-action on $U_\Phi$6 with the Lie derivative action on $U_\Phi$7. Third, $U_\Phi$8 together form an $U_\Phi$9-morphism from the 2-colored dg Lie structure, equivalently an algebra over the Swiss-cheese homotopy operad (Willwacher, 2013).

This theorem extends the closed-sector Kontsevich formality to chains. In the terminology of the paper, it yields the “Kontsevich–Shoikhet formality on chains,” and the resulting pair is an open–closed formality morphism in the precise sense that the closed and open parts are not independent: the higher Stasheff identities and module homotopies are packaged into one 2-colored structure (Willwacher, 2013).

Dolgushev’s transfer theorem gives another formulation of the same open–closed pattern. For a regular commutative algebra $V_\Phi$0, there is a $V_\Phi$1-algebra structure on $V_\Phi$2 and a $V_\Phi$3-quasi-isomorphism

$V_\Phi$4

whose restriction to the closed color is Kontsevich’s formality map and whose restriction to the open color is the identity on $V_\Phi$5 (Dolgushev, 2010).

3. Graphical and configuration-space construction

The Taylor components of $V_\Phi$6 and $V_\Phi$7 are given by sums over admissible directed graphs. For $V_\Phi$8, one considers the compactified configuration space

$V_\Phi$9

modulo translations and scalings, where $GRT$0. On $GRT$1 one uses the $GRT$2-form

$GRT$3

If $GRT$4 is a directed graph with $GRT$5 internal vertices, $GRT$6 boundary vertices, and no loops at internal vertices, and $GRT$7, then its weight is

$GRT$8

To $GRT$9 one also associates a bidifferential operator $G^+$0 obtained by differentiating the multivector inputs according to the outgoing edges and then applying them to the functions $G^+$1 in the order prescribed by the boundary labeling (Willwacher, 2013).

The closed-sector Taylor component has the form

$G^+$2

and the open-sector Taylor component $G^+$3 is given by exactly the same sum but produces a differential form. In compact notation the paper writes

$G^+$4

and similarly for the chain side (Willwacher, 2013).

The proof mechanism is Stokes-theoretic. The boundary of every integrand $G^+$5 on the compactified $G^+$6 corresponds to the failure of the naive $G^+$7-identity, and because $G^+$8 splits into strata indexed by partitions of vertices, Stokes’ theorem makes the boundary terms cancel graph by graph. This is the geometric origin of the open–closed homotopy relations (Willwacher, 2013).

4. Operads, transfer, and non-formality

The open–closed formality problem is naturally 2-colored. Willwacher formulates the construction through a 2-colored operad $G^+$9, whose algebras are pairs such as $M=\mathbb{R}^n$0 or $M=\mathbb{R}^n$1 together with the required action, and identifies the relevant homotopy identities with the homology of the Swiss-cheese operad $M=\mathbb{R}^n$2 (Willwacher, 2013).

Dolgushev’s approach makes the same 2-colored structure explicit via the operad $M=\mathbb{R}^n$3. The color $M=\mathbb{R}^n$4 sub-operad is the usual $M=\mathbb{R}^n$5-operad for homotopy Gerstenhaber algebras, the color $M=\mathbb{R}^n$6 sub-operad is the usual $M=\mathbb{R}^n$7-operad, and the mixed generators $M=\mathbb{R}^n$8 encode the open–closed homotopy algebra structure. Moreover,

$M=\mathbb{R}^n$9

embeds $A=C^\infty(M)$0 as a sub-DG-operad of the first page $A=C^\infty(M)$1 of the homology spectral sequence of the Fulton–MacPherson model of Voronov’s Swiss-Cheese operad (Dolgushev, 2010).

The homotopy-transfer procedure then produces a transferred coderivation $A=C^\infty(M)$2 and an operad map $A=C^\infty(M)$3 from a homotopy retraction

$A=C^\infty(M)$4

The transferred structure maps are given by sums over planar $A=C^\infty(M)$5-trees, and package into a $A=C^\infty(M)$6-morphism $A=C^\infty(M)$7 (Dolgushev, 2010).

A frequent misconception is that open–closed formality should imply formality of the underlying 2-colored operad. Dolgushev proves the opposite for this transfer-theoretic model: neither $A=C^\infty(M)$8 nor $A=C^\infty(M)$9 is formal (Dolgushev, 2010). The existence of an open–closed formality morphism therefore does not collapse the higher open–closed operations into a strict operadic model.

5. Associators, graph complexes, and the $M=\mathbb{R}^n$00-torsor

The dependence on a Drinfeld associator is essential. Kontsevich’s original weights $M=\mathbb{R}^n$01 are obtained from the Knizhnik–Zamolodchikov associator $M=\mathbb{R}^n$02, but the construction extends to any Drinfeld associator $M=\mathbb{R}^n$03 (Willwacher, 2013).

In Willwacher’s formulation, one chooses a solution of the graph-complex Maurer–Cartan equation in the full directed graph complex $M=\mathbb{R}^n$04, which is a torsor over the Grothendieck–Teichmüller Lie algebra $M=\mathbb{R}^n$05. Changing $M=\mathbb{R}^n$06 by $M=\mathbb{R}^n$07 replaces the Maurer–Cartan element $M=\mathbb{R}^n$08 by $M=\mathbb{R}^n$09, and this alters the graph weights $M=\mathbb{R}^n$10 by an explicit gauge transformation in the graph complex. The resulting pair $M=\mathbb{R}^n$11 is a new open–closed formality morphism in the same homotopy class (Willwacher, 2013).

This point is conceptually important. The open–closed extension is not an ad hoc addition to the closed theory; the paper shows that the $M=\mathbb{R}^n$12-action extends to the cochain-and-chain package “up to homotopy.” A plausible implication is that the chain-level data are controlled by the same graph-complex symmetries that organize the closed-sector formality, rather than by an unrelated auxiliary choice.

6. Categorical and string-field-theoretic generalizations

Recent work recasts open–closed formality in the setting of Calabi–Yau $M=\mathbb{R}^n$13-categories and BD algebras. For a smooth proper $M=\mathbb{R}^n$14-Calabi–Yau $M=\mathbb{R}^n$15-category $M=\mathbb{R}^n$16, the closed sector is modeled by

$M=\mathbb{R}^n$17

while the open sector associated to a full subcategory $M=\mathbb{R}^n$18 is

$M=\mathbb{R}^n$19

Tensoring the sectors yields an open–closed dg Lie algebra $M=\mathbb{R}^n$20, and for any splitting $M=\mathbb{R}^n$21 of the noncommutative Hodge filtration there is an $M=\mathbb{R}^n$22 quasi-isomorphism

$M=\mathbb{R}^n$23

which on the closed factor recovers the Caldararu–Tu map and on the open factor is the identity on products (Ulmer, 21 Jul 2025).

The splitting $M=\mathbb{R}^n$24 is a $M=\mathbb{R}^n$25-linear chain map

$M=\mathbb{R}^n$26

that is a right inverse to the projection $M=\mathbb{R}^n$27. In the thesis formulation, Costello–Căldăraru–Tu prove that $M=\mathbb{R}^n$28 induces an $M=\mathbb{R}^n$29 quasi-isomorphism

$M=\mathbb{R}^n$30

and the open–closed formality morphism is obtained by tensoring $M=\mathbb{R}^n$31 with the identity on the open sector and allowing for suitable shifts (Ulmer, 18 Mar 2026).

The Taylor components in this categorical setting are defined by sums over connected stable ribbon graphs with $M=\mathbb{R}^n$32 closed vertices and one open boundary. The $M=\mathbb{R}^n$33-th Taylor coefficient $M=\mathbb{R}^n$34 is formed by desymmetrizing the closed inputs, contracting internal edges with the residue-pairing inverse $M=\mathbb{R}^n$35, inserting the open input at the unique boundary, multiplying by $M=\mathbb{R}^n$36, dividing by $M=\mathbb{R}^n$37, and summing over ribbon graphs $M=\mathbb{R}^n$38 (Ulmer, 18 Mar 2026). Ulmer states the corresponding tensor-product construction as a coalgebra map $M=\mathbb{R}^n$39, and the key identity is a graph-combinatorial bijection between contracting two open variables and either lowering the genus or splitting the graph into disconnected pieces (Ulmer, 21 Jul 2025).

These categorical extensions are tied to applications. The open–closed morphism is described as an ingredient toward quantizing the large $M=\mathbb{R}^n$40 open SFT of an object of a Calabi–Yau category (Ulmer, 21 Jul 2025), and the thesis formulation states that the quantized open–closed formality $M=\mathbb{R}^n$41 allows one to lift the classical interaction $M=\mathbb{R}^n$42 on $M=\mathbb{R}^n$43 to a solution of the quantum master equation in the BD algebra $M=\mathbb{R}^n$44 (Ulmer, 18 Mar 2026). The same sources state that, when $M=\mathbb{R}^n$45 and $M=\mathbb{R}^n$46 is a Lagrangian, the open–closed theory defined by $M=\mathbb{R}^n$47 should encode open–closed Gromov–Witten invariants of $M=\mathbb{R}^n$48, and that this provides a categorical framework for twisted holography (Ulmer, 18 Mar 2026).

Within this broader literature, the phrase “open–closed formality morphism” therefore designates a family of constructions with a common core: a 2-colored $M=\mathbb{R}^n$49-quasi-isomorphism, controlled by graph or ribbon-graph expansions, compatible with Swiss-cheese or BD-algebra structures, and sensitive to auxiliary data such as a Drinfeld associator or a splitting of the noncommutative Hodge filtration.