Thomas Willwacher
We prove a conjecture raised by Tsygan, namely the existence of an L-infinity-quasiisomorphism of L-infinity-modules between the cyclic chain complex of smooth functions on a manifold and the differential forms on that manifold. Concretely, we prove that the obvious u-linear extension of Shoikhet's morphism of Hochschild chains solves Tsygan's conjecture.
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We prove Kontsevich's cyclic formality conjecture.
We conjecture an explicit formula for a cyclic analog of the Formality $L_{\infty}$-morphism [K]. We prove that its first Taylor component, the cyclic Hochschild-Kostant-Rosenberg map, is in fact a morphism (and a quasiisomorphism) of the complexes. To prove it we construct a cohomological version of the Connes-Tsygan bicomplex in cyclic homology. As an application of the cyclic Formality conjecture, we obtain an explicit formula for cyclically invariant deformation quantization.We show that (a more precise version of) the Connes-Flato-Sternheimer conjecture [CFS] on the existence of closed st…
We show that Willwacher's cyclic formality theorem can be extended to preserve natural Gravity operations on cyclic multivector fields and cyclic multidifferential operators. We express this in terms of a homotopy Gravity quasi-isomorphism with explicit local formulas. For this, we develop operadic tools related to mixed complexes and cyclic homology and prove that the operad $\mathsf M_\circlearrowleft$ of natural operations on cyclic operators is formal and hence quasi-isomorphic to the Gravity operad.
We state the analogs of Kontsevich's formality conjecture for Hochschild and cyclic chains, as well as their
We prove that the chain operad of small squares is formal. This fact clarifies situation with the proof of M. Kontsevich formality theorem in the paper of the author math.QA/9803025, revised Sept 24. The formality of the operad follows quite easily from the existence of an associator.
It is known that one can associate a Kontsevich-type formality morphism to every Drinfeld associator. We show that this morphism may be extended to a Kontsevich-Shoikhet formality morphism of cochains and chains, by describing the action of the Grothendieck-Teichmüller group on such objects (up to homotopy).
The framed little 2-discs operad is homotopy equivalent to the Kimura-Stasheff-Voronov cyclic operad of moduli spaces of genus zero stable curves with tangent rays at the marked points and nodes. We show that this cyclic operad is formal, meaning that its chains and its homology (the Batalin-Vilkovisky operad) are quasi-isomorphic cyclic operads. To prove this we introduce a new complex of graphs in which the differential is a combination of edge deletion and contraction, and we show that this complex resolves BV as a cyclic operad.
We extend Tamarkin's formality of the little disk operad to the framed little disk operad.
We prove the formality theorem for the differential graded Lie algebra module of Hochschild chains for the algebra of endomorphisms of a smooth vector bundle. We discuss a possible application of this result to a version of the algebraic index theorem for Poisson manifolds.
Recently the first two authors constructed an L-infinity morphism using the S^1-equivariant version of the Poisson Sigma Model (PSM). Its role in deformation quantization was not entirely clear. We give here a "good" interpretation and show that the resulting formality statement is equivalent to formality on cyclic chains as conjectured by Tsygan and proved recently by several authors.
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]]…
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 Kontsevich-Soibelman solution of the cyclic version of Deligne's conjecture and the formality of the operad of little discs on a cylinder provide us with a natural homotopy calculus structure on the pair (C^*(A), C_*(A)) ``Hochschild cochains + Hochschild chains'' of an associative algebra A. We show that for an arbitrary smooth algebraic variety X with the structure sheaf O_X the sheaf (C^*(O_X), C_*(O_X)) of homotopy calculi is formal. This result was announced in paper [29] by the second and the third author.
Proofs of Tsygan's formality conjectures for chains would unlock important algebraic tools which might lead to new generalizations of the Atiyah-Patodi-Singer index theorem and the Riemann-Roch-Hirzebruch theorem. Despite this pivotal role in the traditional investigations and the efforts of various people the most general version of Tsygan's formality conjecture has not yet been proven. In my thesis I propose Fedosov resolutions for the Hochschild cohomological and homological complexes of the algebra of functions on an arbitrary smooth manifold. Using these resolutions together with Kontsevi…
We prove that the operad of framed little 2-discs is formal. Tamarkin and Kontsevich each proved that the unframed 2-discs operad is formal. The unframed 2-discs is an operad in the category of S^1-spaces, and the framed 2-discs operad can be constructed from the unframed 2-discs by forming the operadic semidirect product with the circle group. The idea of our proof is to show that Kontsevich's chain of quasi-isomorphisms is compatible with the circle actions and so one can essentially take the operadic semidirect product with the homology of S^1 everywhere to obtain a chain of quasi-isomorphi…
We extend the Kontsevich formality $L_\infty$-morphism $\U\colon T^\ndot_\poly(\R^d)\to\D^\ndot_\poly(\R^d)$ to an $L_\infty$-morphism of an $L_\infty$-modules over $T^\ndot_\poly(\R^d)$, $\hat \U\colon C_\ndot(A,A)\toΩ^\ndot(\R^d)$, $A=C^\infty(\R^d)$. The construction of the map $\hat \U$ is given in Kontsevich-type integrals. The conjecture that such an $L_\infty$-morphism exists is due to Boris Tsygan \cite{Ts}. As an application, we obtain an explicit formula for isomorphism $A_*/[A_*,A_*]\simto A/\{A,A\}$ ($A_*$ is the Kontsevich deformation quantization of the algebra $A$ by a Poisson b…
In this paper we prove, with details and in full generality, that the isomorphism induced on tangent homology by the Shoikhet-Tsygan formality $L_\infty$-quasi-isomorphism for Hochschild chains is compatible with cap-products. This is a homological analog of the compatibility with cup-products of the isomorphism induced on tangent cohomology by Kontsevich formality $L_\infty$-quasi-isomorphism for Hochschild cochains. As in the cohomological situation our proof relies on a homotopy argument involving a variant of {\bf Kontsevich eye}. In particular we clarify the rôle played by the {\bf I-cube…
In the present paper we prove a statement closely related to the cyclic formality conjecture. In particular, we prove that for a divergence-free Poisson bivector field on R^d, the Kontsevich star-product with the harmonic angle function is cyclic. We also prove a globalization of this theorem in the case of arbitrary Poisson manifolds and prove a generalization of the Connes-Flato-Sternheimer conjecture on closed star-products in the Poisson case.
The quotient of a Boolean algebra by a cyclic group is proven to have a symmetric chain decomposition. This generalizes earlier work of Griggs, Killian and Savage on the case of prime order, giving an explicit construction for any order, prime or composite. The combinatorial map specifying how to proceed downward in a symmetric chain is shown to be a natural cyclic analogue of the $\mathfrak{sl}_2$ lowering operator in the theory of crystal bases.
This paper introduces a chain model for the Deligne-Mumford operad formed by homotopically trivializing the circle in a chain model for the framed little disks. We then show that under degeneration of the Hochschild to cyclic cohomology spectral sequence, a known action of the framed little disks on Hochschild cochains lifts to an action of this new chain model. We thus establish homotopy hyper-commutative algebra structures on both Hochschild and cyclic cochain complexes, and we interpret the gravity brackets on cyclic cohomology as obstructions to degeneration of this spectral sequence. Our …
Kontsevich's formality theorem states that the differential graded Lie algebra of multidifferential operators on a manifold M is L-infinity-quasi-isomorphic to its cohomology. The construction of the L-infinity map is given in terms of integrals of differential forms on configuration spaces of points in the upper half-plane. Here we consider configuration spaces of points in the disk and work equivariantly with respect to the rotation group. This leads to considering the differential graded Lie algebra of multivector fields endowed with a divergence operator. In the case of R^d with standard v…
The third author recently proved that the Shoikhet-Dolgushev L-infinity-morphism from Hochschild chains of the algebra of smooth functions on manifold to differential forms extends to cyclic chains. Localization at a solution of the Maurer-Cartan equation gives an isomorphism, which we call character map, from the periodic cyclic homology of a formal associative deformation of the algebra of functions to de Rham cohomology. We prove that the character map is compatible with the Gauss-Manin connection, extending a result of Calaque and Rossi on the compatibility with the cap product. As a conse…
Cyclic homology of categorical coalgebras and the free loop space
本文证明任意拓扑空间奇异链的范畴余代数的循环链复形与自由环空间的等变链自然拟同构，并引入 Goodwillie 多面体控制相关组合结构。
First we describe a class of homotopy Frobenius algebras via cyclic operads which we call cyclic $A_\infty$ algebras. We then define a suitable new combinatorial operad which acts on the Hochschild cochains of such an algebra in a manner which encodes the homotopy BV structure. Moreover we show that this operad is equivalent to the cellular chains of a certain topological (quasi)-operad of CW complexes whose constituent spaces form a homotopy associative version of the Cacti operad of Voronov. These cellular chains thus constitute a chain model for the framed little disks operad, proving a cyc…
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 l…
We demonstrate that the transfer matrix of the inhomogeneous $N$-state chiral Potts model with two vertical superintegrable rapidities serves as the $Q$-operator of XXZ chain model for a cyclic representation of $U_{\sf q}(sl_2)$ with $N$th root-of-unity ${\sf q}$ and representation-parameter for odd $N$. The symmetry problem of XXZ chain with a general cyclic $U_{\sf q}(sl_2)$-representation is mapped onto the problem of studying $Q$-operator of some special one-parameter family of generalized $τ^{(2)}$-models. In particular, the spin-$\frac{N-1}{2}$ XXZ chain model with ${\sf q}^N=1$ and the…
We identify the precise relationship between the five-parameter $τ^{(2)}$-family in the $N$-state chiral Potts model and XXZ chains with $U_q (sl_2)$-cyclic representation. By studying the Yang-Baxter relation of the six-vertex model, we discover an one-parameter family of $L$-operators in terms of the quantum group $U_q (sl_2)$. When $N$ is odd, the $N$-state $τ^{(2)}$-model can be regarded as the XXZ chain of $U_{\sf q} (sl_2)$ cyclic representations with ${\sf q}^N=1$. The symmetry algebra of the $τ^{(2)}$-model is described by the quantum affine algebra $U_{\sf q} (\hat{sl}_2)$ via the can…
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