Formality Conjecture Research Articles

Some remarks about deformation theory and formality conjecture

Using the algebraic criterion proved by Bandiera, Manetti and Meazzini, we show the formality conjecture for universally gluable objects with linearly reductive automorphism groups in the bounded derived category of a K3 surface. As an application, we prove the formality conjecture for polystable objects in the Kuznetsov components of Gushel–Mukai threefolds and quartic double solids.

English

In the formulation of his celebrated Formality conjecture, M. Kontsevich introduced a universal version of the deformation theory for the Schouten algebra of polyvector fields on affine manifolds. This universal deformation complex takes the form of a differential graded Lie algebra of graphs, denoted $\mathsf{fGC}_2$, together with an injective morphism towards the Chevalley-Eilenberg complex associated with the Schouten algebra. The latter morphism is given by explicit local formulas making implicit use of the supergeometric interpretation of the Schouten algebra as the algebra of functions on a graded symplectic manifold of degree $1$. The ambition of the present work is to generalise Kontsevich's construction to graded symplectic manifolds of arbitrary degree $n\geq1$. The corresponding graph model is given by the full Kontsevich graph complex $\mathsf{fGC}_d$ where $d=n+1$ stands for the dimension of the associated AKSZ type $\sigma$-model. This generalisation is instrumental to classify universal structures on graded symplectic manifolds. In particular, the zeroth cohomology of the full graph complex $\mathsf{fGC}_{d}$ is shown to act via $\mathsf{Lie}_\infty$-automorphisms on the algebra of functions on graded symplectic manifolds of degree $n$. This generalises the known action of the Grothendieck-Teichm\"{u}ller algebra $\mathfrak{grt}_1\simeq H^0(\mathsf{fGC}_2)$ on the space of polyvector fields. This extended action can in turn be used to generate new universal deformations of Hamiltonian functions, generalising Kontsevich flows on the space of Poisson manifolds to differential graded manifolds of higher degrees. As an application of the general formalism, universal deformations of Courant algebroids via trivalent graphs are presented.

Formality conjecture for minimal surfaces of Kodaira dimension 0

Let $\mathcal {F}$ be a polystable sheaf on a smooth minimal projective surface of Kodaira dimension 0. Then the differential graded (DG) Lie algebra $R\operatorname {Hom}(\mathcal {F},\mathcal {F})$ of derived endomorphisms of $\mathcal {F}$ is formal. The proof is based on the study of equivariant $L_{\infty }$ minimal models of DG Lie algebras equipped with a cyclic structure of degree 2 which is non-degenerate in cohomology, and does not rely (even for K3 surfaces) on previous results on the same subject.

Formality conjecture for K3 surfaces

We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the differential graded (DG) algebra$\operatorname{RHom}^{\bullet }(F,F)$is formal for any sheaf$F$polystable with respect to an ample line bundle. Our main tool is the uniqueness of the DG enhancement of the bounded derived category of coherent sheaves. We also extend the formality result to derived objects that are polystable with respect to a generic Bridgeland stability condition.

The Cohomology of the Full Directed Graph Complex

In his seminal paper “Formality conjecture”, M. Kontsevich introduced a graph complex GC1ve closely connected with the problem of constructing a formality quasi-isomorphism for Hochschild cochains. In this paper, we express the cohomology of the full directed graph complex dfGC explicitly in terms of the cohomology of GC1ve. Applications of our results include a recent work by the first author which completely characterizes homotopy classes of formality quasi-isomorphisms for Hochschild cochains in the stable setting.

Do the Kontsevich tetrahedral flows preserve or destroy the space of Poisson bi-vectors?

From the paper “Formality Conjecture” (Ascona 1996):I am aware of only one such a class, it corresponds to simplest good graph, the complete graph with 4 vertices (and 6 edges). This class gives a remarkable vector field on the space of bi-vector fields on ℝd. The evolution with respect to the time t is described by the following non-linear partial differential equation: …, where α = ∑i,j αij∂ / ∂ xi ∧ ∂ / ∂ xj is a bi-vector field on ℝd.It follows from general properties of cohomology that 1) this evolution preserves the class of (real-analytic) Poisson structures, …In fact, I cheated a little bit. In the formula for the vector field on the space of bivector fields which one get from the tetrahedron graph, an additional term is present. … It is possible to prove formally that if α is a Poisson bracket, i.e. if [α, α] = 0 ∈ T2(ℝd), then the additional term shown above vanishes.By using twelve Poisson structures with high-degree polynomial coefficients as explicit counter-examples, we show that both the above claims are false: neither does the first flow preserve the property of bi-vectors to be Poisson nor does the second flow vanish identically at Poisson bi-vectors. The counterexamples at hand suggest a correction to the formula for the “exotic” flow on the space of Poisson bi-vectors; in fact, this flow is encoded by the balanced sum involving both the Kontsevich tetrahedral graphs (that give rise to the flows mentioned above). We reveal that it is only the balance 1 : 6 for which the flow does preserve the space of Poisson bi-vectors.

Antiperfection of Yang–Mills Morse theory over a nonorientable surface

We use Morse theory of the Yang–Mills functional to compute the Betti numbers of the moduli stack of flat U(3)-bundles over a compact nonorientable surface. Our result establishes the antiperfection conjecture of Ho–Liu, and establishes the equivariant formality conjecture of the author for U(3)-bundles.

Formality of cyclic cochains

We prove M. Kontsevich’s Cyclic Formality Conjecture. This conjecture is the cyclic analog of the Formality Theorem for Hochschild chains. Concretely, it states that the differential graded Lie algebra of polydifferential cyclic cochains of the algebra of smooth functions on a manifold M is L∞-quasi-isomorphic to the second term in the spectral sequence computing its cohomology. As an application, we can classify all closed star products on M.

Formality theorems for Hochschild chains in the Lie algebroid setting

In this paper we prove Lie algebroid versions of Tsygan's formality conjecture for Hochschild chains both in the smooth and holomorphic settings. In the holomorphic setting our result implies a version of Tsygan's formality conjecture for Hochschild chains of the structure sheaf of any complex manifold and in the smooth setting this result allows us to describe quantum traces for an arbitrary Poisson Lie algebroid. The proofs are based on the use of Kontsevich's quasi-isomorphism for Hochschild cochains of R[[y_1,...,y_d]], Shoikhet's quasi-isomorphism for Hochschild chains of R[[y_1,...,y_d]], and Fedosov's resolutions of the natural analogues of Hochschild (co)chain complexes associated with a Lie algebroid.

G ∞-Formality Theorem in Terms of Graphs and Associated Chevalley–Eilenberg–Harrison Cohomology#

In 1997, M. Kontsevich proved the L ∞-formality conjecture (which implies the existence of star-products for any Poisson manifold) using graphs. A year later, D. Tamarkin gave another proof of a more general conjecture (for G ∞-structures) using operads and cohomological methods. In this article, we show how Tamarkin's construction can be written using graphs. For that, we introduce a generalization of Kontsevich graphs on which we define a “Chevalley–Eilenberg–Harrison” complex. We show that this complex on graphs is related to the “Chevalley–Eilenberg–Harrison” complex for maps on polyvector fields, which is trivial and give Tamarkin's formality theorem as a consequence. This formality reduces to an L ∞-formality.

A formality theorem for Hochschild chains

We prove Tsygan's formality conjecture for Hochschild chains of the algebra of functions on an arbitrary smooth manifold M using the Fedosov resolutions proposed in math.QA/0307212 and the formality quasi-isomorphism for Hochschild chains of R [ [ y 1 ,..., y d ] ] proposed in paper math.QA/0010321 by Shoikhet. This result allows us to describe traces on the quantum algebra of functions on an arbitrary Poisson manifold.

The Mukai pairing—II: the Hochschild–Kostant–Rosenberg isomorphism

We continue the study of the Hochschild structure of a smooth space that we began in our previous paper, examining implications of the Hochschild–Kostant–Rosenberg theorem. The main contributions of the present paper are: • we introduce a generalization of the usual notions of Mukai vector and Mukai pairing on differential forms that applies to arbitrary manifolds; • we give a proof of the fact that the natural Chern character map K 0 ( X ) → HH 0 ( X ) becomes, after the HKR isomorphism, the usual one K 0 ( X ) → ⊕ H i ( X , Ω X i ) ; and • we present a conjecture that relates the Hochschild and harmonic structures of a smooth space, similar in spirit to the Tsygan formality conjecture.

A proof of the Tsygan formality conjecture for chains

We extend the Kontsevich formality L ∞-morphism U : T poly •( R d)→ D poly •( R d) to an L ∞-morphism of L ∞-modules over T poly •( R d), U ̂ : C •(A,A)→Ω •( R d), A=C ∞( R d) . The construction of the map U ̂ is given in Kontsevich-type integrals. The conjecture that such an L ∞-morphism exists is due to Boris Tsygan (Formality Conjecture for Chains, math. QA/9904132). As an application, we obtain an explicit formula for isomorphism A ∗/[A ∗,A ∗] → ∼ A/{A,A} (A ∗ is the Kontsevich deformation quantization of the algebra A by a Poisson bivector field, and {,} is the Poisson bracket). We also formulate a conjecture extending the Kontsevich theorem on cup-products to this context. The conjecture implies a generalization of the Duflo formula, and many other things.

Formality Conjectures and Deformation

Let M be a Poisson manifold. Kontsevich proved that star products exist on M and he gave a classification. To relate his classification with other classifications, one could try to extend the Connes–Flato–Sternheimer invariant to a general Poisson manifold. We show how generalization of this invariant is related to the formality conjecture for chains. Finally, we show how to prove those conjectures ‘step by step’. Our approach, different from Tamarkin's, will give explicit formulas but doesn't yet solve the general conjecture.

On nonformal simply-connected symplectic manifolds

For any $N \geq 5$ nonformal simply connected symplectic manifolds of dimension $2N$ are constructed. This disproves the formality conjecture for simply connected symplectic manifolds which was introduced by Lupton and Oprea.

Deformation Quantization of Poisson Manifolds

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven ("Formality conjecture"), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not use explicitly the language of functional integrals. One of corollaries is a justification of the orbit method in the representation theory.

A Path Integral Approach¶to the Kontsevich Quantization Formula

We give a quantum field theory interpretation of Kontsevich's deformation quantization formula for Poisson manifolds. We show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory. Its Batalin-Vilkovisky quantization yields a superconformal field theory. The associativity of the star product, and more generally the formality conjecture can then be understood by field theory methods. As an application, we compute the center of the deformed algebra in terms of the center of the Poisson algebra.