Hochschild Cochains Research Articles

Matrix factorization of Morse–Bott functions

For a function $W\in \mathbb{C}[X]$ on a smooth algebraic variety $X$ with Morse-Bott critical locus $Y\subset X$, Kapustin, Rozansky and Saulina suggest that the associated matrix factorisation category $\mathrm{MF}(X;W)$ should be equivalent to the differential graded category of $2$-periodic coherent complexes on $Y$ (with a topological twist from the normal bundle of $Y$). I confirm their conjecture in the special case when the first neighbourhood of $Y$ in $X$ is split, and establish the corrected general statement. The answer involves the full Gerstenhaber structure on Hochschild cochains. This note was inspired by the failure of the conjecture, observed by Pomerleano and Preygel, when $X$ is a general one-parameter deformation of a $K3$ surface $Y$.

A Formality Quasi-Isomorphism for Hochschild Cochains Over Rationals can be Constructed Recursively

It is believed arXiv:0808.2762, arXiv:math/9904055 that, among the coefficients entering Kontsevich's formality quasi-isomorphism arXiv:q-alg/9709040, there are irrational (possibly even transcendental) numbers. In this paper, we prove that a formality quasi-isomorphism for Hochschild cochains of a polynomial algebra over rationals can be constructed recursively. The proof that the proposed recursive algorithm works, is based on the existence of formality quasi-isomorphism over reals. However, the algorithm requires no explicit knowledge of the coefficients entering Kontsevich's construction. Although this algorithm completely bypasses Tamarkin's approach arXiv:math/0003052, arXiv:math/9803025, the construction is inspired by Proposition 5.8 from the classical paper (Algebra i Analiz, 1990) by V. Drinfeld.

Permutohedral structures on E2E_{2}-operads

Abstract There are two interesting families of E 2 {E_{2}} -operads, those that detect double loop spaces, and those that solve Deligne’s conjecture on Hochschild cochains. The first family deformation retracts to Milgram’s model obtained by gluing together permutohedra along their faces. We show how the second family can be covered by permutohedra as well, shedding new light on several proposed solutions of Deligne’s conjecture. In particular, our approach induces an explicit homotopy equivalence between the models of the two families. The permutohedra and partial orders play a central role providing direct links to other fields of mathematics. We for instance find a new cellular decomposition of permutohedra using partial orders and that the permutohedra give the cells for the Dyer–Lashof operations.

Tamarkin’s construction is equivariant with respect to the action of the Grothendieck–Teichmueller group

Recall that Tamarkin’s construction (Hinich, Forum Math 15(4):591–614, 2003, arXiv:math.QA/0003052 ; Tamarkin, 1998, arXiv:math/9803025 ) gives us a map from the set of Drinfeld associators to the set of homotopy classes of $$L_{\infty }$$ quasi-isomorphisms for Hochschild cochains of a polynomial algebra. Due to results of Drinfeld (Algebra i Analiz 2(4):149–181, 1990) and Willwacher Invent Math 200(3):671–760, 2015 both the source and the target of this map are equipped with natural actions of the Grothendieck–Teichmueller group $${\mathsf {GRT}}_1$$ . In this paper, we use the result from Paljug (JHRS, 2015, arXiv:1305.4699 ) to prove that this map from the set of Drinfeld associators to the set of homotopy classes of $$L_{\infty }$$ quasi-isomorphisms for Hochschild cochains is $${\mathsf {GRT}}_1$$ -equivariant.

An $${\mathcal{A}_\infty}$$ A ∞ Operad in Spineless Cacti

The dg operad $${\mathcal{C}}$$ of cellular chains on the operad of spineless cacti of Kaufmann (Topology 46(1):39–88, 2007) is isomorphic to the Gerstenhaber–Voronov dg operad codifying the cup product and brace operations on the Hochschild cochains of an associative algebra, and to the suboperad $${F_2\mathcal{X}}$$ of the surjection operad of Berger and Fresse (Math Proc Camb Philos Soc 137(1):135–174, 2004), McClure and Smith (Recent progress in homotopy theory (Baltimore, MD, 2000). Contemp Math., Amer. Math. Soc., Providence 293:153–193, 2002) and McClure and Smith (J Am Math Soc 16(3):681–704, 2003). Its homology is the Gerstenhaber dg operad $${\mathcal{G}}$$ . We construct a map of dg operads $${\psi \colon \mathcal{A}_\infty \longrightarrow \mathcal{C}}$$ such that $${\psi(m_2)}$$ is commutative and $${H_*(\psi)}$$ is the canonical map $${\mathcal{A} \to \mathcal{C}\!om \to \mathcal{G}}$$ . This formalises the idea that, since the cup product is commutative in homology, its symmetrisation is a homotopy associative operation. Our explicit $${\mathcal{A}_\infty}$$ structure does not vanish on non-trivial shuffles in higher degrees, so does not give a map $${\mathcal{C}om_\infty \to \mathcal{C}}$$ . If such a map could be written down explicitly, it would immediately lead to a $${\mathcal{G}_\infty}$$ structure on $${\mathcal{C}}$$ and on Hochschild cochains, that is, to an explicit and direct proof of the Deligne conjecture.

Coefficients for higher order Hochschild cohomology

When studying deformations of an $A$-module $M$, Laudal and Yau showed that one can consider 1-cocycles in the Hochschild cohomology of $A$ with coefficients in the bi-module $End_k(M).$ With this in mind, the use of higher order Hochschild (co)homology, presented by Pirashvili and Anderson, to study deformations seems only natural though the current definition allows only symmetric bi-module coefficients. In this paper we present an extended definition for higher order Hochschild cohomology which allows multi-module coefficients (when the simplicial sets $X_{\bullet}$ are accommodating) which agrees with the current definition. Furthermore we determine the types of modules that can be used as coefficients for the Hochschild cochain complexes based on the simplicial sets they are associated to.

Crossed interval groups and operations on the Hochschild cohomology

We prove that the operad \mathcal{B} of natural operations on the Hochschild cohomology has the homotopy type of the operad of singular chains on the little disks operad. To achieve this goal, we introduce crossed interval groups and show that \mathcal{B} is a certain crossed interval extension of an operad \mathcal{T} whose homotopy type is known. This completes the investigation of the algebraic structure on the Hochschild cochain complex that has lasted for several decades.

A spectral sequence for the homology of a finite algebraic delooping

AbstractIn the world of chain complexes En-algebras are the analogues of based n-fold loop spaces in the category of topological spaces. Fresse showed that operadic En-homology of an En-algebra computes the homology of an n-fold algebraic delooping. The aim of this paper is to construct two spectral sequences for calculating these homology groups and to treat some concrete classes of examples such as Hochschild cochains, graded polynomial algebras and chains on iterated loop spaces. In characteristic zero we gain an identification of the summands in Pirashvili's Hodge decomposition of higher order Hochschild homology in terms of derived functors of indecomposables of Gerstenhaber algebras and as the homology of exterior and symmetric powers of derived Kähler differentials.

Pre-Lie systems and obstructions to $$A_\infty $$ A ∞ -structures over a ring

In this note we establish an obstruction theory for the existence of $$A_\infty $$ -algebra structures on a differential $$\mathbb Z $$ -graded $$R$$ -module $$A$$ equipped with a homotopy associative multiplication. This approach follows classical obstruction theory: the corresponding obstructions live in the Hochschild cohomology of the associative algebra $$H(A)$$ . Our theory makes use of the pre-Lie algebra structure on the Hochschild cochain complex of $$H(A)$$ . As a consequence, no additional assumptions on the commutative ring $$R$$ are needed.

Deformations of affine varieties and the Deligne crossed groupoid

Let X be a smooth affine algebraic variety over a field K of characteristic 0, and let R be a complete parameter K-algebra (e.g. R=K〚ℏ〛). We consider associative (resp. Poisson) R-deformations of the structure sheaf OX. The set of R-deformations has a crossed groupoid (i.e. strict 2-groupoid) structure. Our main result is that there is a canonical equivalence of crossed groupoids from the Deligne crossed groupoid of normalized polydifferential operators (resp. polyderivations) of X to the crossed groupoid of associative (resp. Poisson) R-deformations of OX. The proof relies on a careful study of adically complete sheaves. In the associative case we also have to use ring theory (Ore localizations) and the properties of the Hochschild cochain complex.The results of this paper extend previous work by various authors. They are needed for our work on twisted deformation quantization of algebraic varieties.

Blob homology

Given an n-manifold M and an n-category C, we define a chain complex (the "blob complex") B_*(M;C). The blob complex can be thought of as a derived category analogue of the Hilbert space of a TQFT, and as a generalization of Hochschild homology to n-categories and n-manifolds. It enjoys a number of nice formal properties, including a higher dimensional generalization of Deligne's conjecture about the action of the little disks operad on Hochschild cochains. Along the way, we give a definition of a weak n-category with strong duality which is particularly well suited for work with TQFTs.

CyclicA∞structures and Deligne’s conjecture

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 cyclic $A_\infty$ version of Deligne's conjecture. This chain model contains the minimal operad of Kontsevich and Soibelman as a suboperad and restriction of the action to this suboperad recovers their results in the unframed case. Additionally this proof recovers the work of Kaufmann in the case of a strict Frobenius algebra. We then extend our results to cyclic $A_\infty$ categories, with an eye toward the homotopy BV structure present on the Hochschild cochains of the Fukaya category of a suitable symplectic manifold.

Formality Theorem for Hochschild Cochains via Transfer

We construct a 2-colored operad G^+ which, on the one hand, extends the operad G governing homotopy Gerstenhaber algebras and, on the other hand, extends the 2-colored operad governing open-closed homotopy algebras (OCHA). We show that Tamarkin's G-structure on the Hochschild cochain complex C(A) of an A-infinity algebra A extends naturally to a G^+ structure on the pair (C(A), A). We show that a formality quasi-isomorphism for the Hochschild cochains of the polynomial algebra can be obtained via transfer of this G^+ structure to the cohomology of the pair (C(A), A). We show that G^+ is a sub DG operad of the first sheet E^1(SC) of the homology spectral sequence for the Fulton-MacPherson version SC of Voronov's Swiss Cheese operad. Finally, we prove that the DG operads G^+ and E^1(SC) are non-formal.

Proof of Swiss Cheese Version of Deligne's Conjecture

For an associative algebra A we consider the pair "the Hochschild cochain complex C*(A,A) and the algebra A". There is a natural 2-colored operad which acts on this pair. We show that this operad is quasi-isomorphic to the singular chain operad of Voronov's Swiss Cheese operad. This statement is the Swiss Cheese version of the Deligne conjecture formulated by M. Kontsevich in arXiv:math/9904055.

Compatibility with Cap-Products in Tsygan’s Formality and Homological Duflo Isomorphism

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 r\^ole played by the {\bf I-cube} introduced in \cite{CR1}. Since we treat here the case of a most possibly general Maurer-Cartan element, not forced to be a bidifferential operator, then we take this opportunity to recall the natural algebraic structures on the pair of Hochschild cochain and chain complexes of an $A_\infty$-algebra. In particular we prove that they naturally inherit the structure of an $A_\infty$-algebra with an $A_\infty$-(bi)module.

Deformations of algebroid stacks

In this paper we consider deformations of an algebroid stack on an étale groupoid. We construct a differential graded Lie algebra (DGLA) which controls this deformation theory. In the case when the algebroid is a twisted form of functions we show that this DGLA is quasiisomorphic to the twist of the DGLA of Hochschild cochains on the algebra of functions on the groupoid by the characteristic class of the corresponding gerbe.

Associahedra, cyclohedra and a topological solution to the [formula omitted] Deligne conjecture

We give a topological solution to the A ∞ Deligne conjecture using associahedra and cyclohedra. To this end, we construct three CW complexes whose cells are indexed by products of polytopes. Giving new explicit realizations of the polytopes in terms of different types of trees, we are able to show that the CW complexes are cell models for the little discs. The cellular chains of one complex in particular, which is built out of associahedra and cyclohedra, naturally act on the Hochschild cochains of an A ∞ algebra yielding an explicit, topological and minimal solution to the A ∞ Deligne conjecture. Along the way we obtain new results about the cyclohedra, such as new decompositions into products of cubes and simplices, which can be used to realize them via a new iterated blow-up construction.

Formality Theorems for Hochschild Complexes and Their Applications

We give a popular introduction to formality theorems for Hochschild complexes and their applications. We review some of the recent results and prove that the truncated Hochschild cochain complex of a polynomial algebra is non-formal.

Moduli space actions on the Hochschild co-chains of a Frobenius algebra II: correlators

This is the second of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co-chains of a Frobenius algebra. We also prove that there is dg-PROP action of a version of Sullivan chord diagrams which acts on the normalized Hochschild co-chains of a Frobenius algebra. These actions lift to operadic correlation functions on the co-cycles. In particular, the PROP action gives an action on the homology of a loop space of a compact simply connected manifold. In this second part, we discretize the operadic and PROPic structures of the first part. We also introduce the notion of operadic correlation functions and use them in conjunction with operadic maps from the cell level to the discretized objects to define the relevant actions.

Simplicial Hochschild cochains as an Amitsur complex 1

It is shown how the cochain complex of the relative Hochschild A-valued cochains of a depth two extension A|B under cup product is isomorphic as a differential graded algebra with the Amitsur complex of the coring S = End BAB over the centralizer R = AB with grouplike element 1S. This specializes to finite dimensional algebras, Hopf-Galois extensions and H-separable extensions.