Boris L Tsygan

• Northwestern University

In this note, we provide a proof of the existence and complete classification of $G$-invariant star products with quantum momentum maps on Poisson manifolds by means of an equivariant version of the formality theorem.

We review the topic of noncommutative differential forms, following the works of Karoubi, Cuntz–Quillen, Cortiñas, Ginzburg–Schedler, and Waikit Yeung. In particular we give a new proof of the theorem of Ginzburg and Schedler that compares extended noncommutative De Rham cohomology to cyclic homology. This theorem is a stronger version of a theorem...

Fof a nilpotent differential graded Lie algebra whose components vanish in degrees below -1 we construct an explicit equivalence between the nerve of the Deligne 2-groupoid and the simplicial set of differential forms with values in the Lie algebra introduced by V.Hinich. The construction uses the theory of non-abelian multiplicative integration.

For a symplectic manifold subject to certain topological conditions a category enriched in \(A_{\infty }\) local systems of modules over the Novikov ring is constructed. The construction is based on the category of modules over Fedosov’s deformation quantization algebra that have an additional structure, namely an action of the fundamental groupoid...

This book presents contributions from two workshops in algebraic and analytic microlocal analysis that took place in 2012 and 2013 at Northwestern University. Featured papers expand on mini-courses and talks ranging from foundational material to advanced research-level papers, and new applications in symplectic geometry, mathematical physics, parti...

We determine the additional structure which arises on the classical limit of a DQ-algebroid.

We determine the additional structure which arises on the classical limit of a DQ-algebroid.

We show that for a differential graded Lie algebra g whose components vanish in degrees below −1 the nerve of the Deligne 2-groupoid is homotopy equivalent to the simplicial set of g-valued differential forms introduced by V. Hinich [Hinich, 1997].

The main result of the present paper is an analogue of Kontsevich formality theorem in the context of the deformation theory of gerbes. We construct an deformation of the Schouten algebra of multi-vectors which controls the deformation theory of a gerbe.

We extend the formality theorem of Maxim Kontsevich from deformations of the structure sheaf on a manifold to deformations of gerbes on smooth and complex manifolds.

The extension of the M. Kontsevich formality theorem to gerbes necessitates extension of the notion of Deligne 2-groupoid to $L_\infty$ algebras. For a differential graded Lie algebra ${\mathfrak g}$ which is zero below the degree $-1,$ we show that the nerve of the Deligne 2-groupoid is homotopy equivalent to the Hinich simplicial set \cite{H1} of...

This is a survey of current and recent works on deformation quantization and index theorems.

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 fun...

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 f...

We prove a Riemann–Roch type result for any smooth family of smooth oriented compact manifolds. It describes the class of the conjectural higher determinantal gerbe associated to the fibers of the family. Key wordsRiemann–Roch-determinantal gerbe, Lie algebroid, cyclic homology

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.

Developing the ideas of Bressler and Soibelman and of Karabegov, we introduce a notion of an oscillatory module on a symplectic manifold which is a sheaf of modules over the sheaf of deformation quantization algebras with an additional structure. We compare the category of oscillatory modules on a torus to the Fukaya category as computed by Polishc...

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 2-dimensional case of the conjecture formulated...

We extend the formality theorem of M. Kontsevich from deformations of the structure sheaf on a manifold to deformations of gerbes.

To Murray Gerstenhaber on his 80th and to Jim Stasheff on his 70th birthday After an overview of noncommutative differential calculus, we construct parts of it explicitly and explain why this construction agrees with a fuller version obtained from the theory of operads. AMS 2010 Subject Codes: 19D55, 18G55

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 varie...

We construct the Chern character from the K-theory of twisted perfect complexes of an algebroid stack to the negative cyclic homology of the algebra of twisted matrices associated to the stack.

We construct a Chern character of a perfect complex of twisted modules over an algebroid stack.

We identify the 2-groupoid of deformations of a gerbe on a C1 manifold with the Deligne 2-groupoid of a corresponding twist of the DGLA of local Hochschild cochains on C1 functions.

If E is a C^\infty complex vector bundle on an oriented C^\infty manifold \Sigma, diffeomorphic to a circle, then the space of sections of E has a canonical polarization in the sense of Pressley and Segal and so one has its determinantal gerbe with lien C^*, the group of nonzero complex numbers. If q:\Sigma-->B is a smooth family of circles as abov...

In this paper we compute the deformation theory of a special class of algebras, namely of Azumaya algebras on a manifold ($C^{\infty}$ or complex analytic).

The solution of Deligne's conjecture on Hochschild cochains and the formality of the operad of little disks provide us with a natural homotopy Gerstenhaber algebra structure on the Hochschild cochains of an associative algebra. In this paper we construct a natural chain of quasi-isomorphisms of homotopy Gerstenhaber algebras between the Hochschild...

This is the first in a series of articles devoted to deformation quantization of gerbes. We introduce basic definitions, interpret deformations of a given stack as Maurer–Cartan elements of a differential graded Lie algebra (DGLA), and classify deformations of a given gerbe in terms of Maurer–Cartan elements of the DGLA of Hochschild cochains twist...

This is the first in a series of articles devoted to deformation quantization of gerbes. Here we give basic definitions and interpret deformations of a given gerbe as Maurer-Cartan elements of a differential graded Lie algebra (DGLA). We classify all deformations of a given gerbe on a symplectic manifold, as well as provide a deformation-theoretic...

We study an asymptotic version of the Maslov-Hormander construction of Lagrangian distributions in terms of deformation quantization.

We deduce the Riemann–Roch type formula expressing the microlocal Euler class of a perfect complex of -modules in terms of the Chern character of the associated symbol complex and the Todd class of the manifold from the Riemann–Roch type theorem for periodic cyclic cocycles of a symplectic deformation quantization. The proof of the latter is contai...

The Letter announces the following results (the proofs will appear elsewhere). An operad acting on Hochschild chains and cochains of an associative algebra is constructed. This operad is formal. In the case when this algebra is the algebra of smooth function on a smooth manifold, the action of this operad on the corresponding Hochschild chains and...

We show that the index of an elliptic Fourier integral operator associated to a contact diffeomorphism $\phi$ of cosphere bundles of two Riemannian manifolds X and Y is given by $\int_{B^*X}\hat{A}(T^*X)\exp{\theta} - \int_{B^*Y}\hat{A}(T^*Y)\exp{\theta}$. Here $B^*$ stands for the unit coball bundle and $\theta$ is a certain characteristic class d...

We prove a Riemann-Roch formula for deformation quantization of complex manifolds and its corollary, an index theorem for elliptic pairs conjectured by Schapira and Schneiders.

Recently Kontsevich solved the classification problem for deformation quantizations of all Poisson structures on a manifold. In this paper we study those Poisson structures for which the explicit methods of Fedosov can be applied, namely the Poisson structures coming from symplectic Lie algebroids, as well as holomorphic symplectic structures. For...

We deduce the Riemann-Roch type formula expressing the microlocal Euler class of a perfect complex of D-modules in terms of the Chern character of the associated symbol complex and the Todd class of the manifold from the Riemann-Roch type theorem for periodic cyclic cocycles of a symplectic deformation quantization. The proof of the latter is conta...

this paper, however, we will show that, if one takes for End A the differential graded algebra C (A; A) of Hochschild cochains, the maps (1), in a sense, still exist. More precisely, they exist if one passes from the category of algebras to the category of complexes by means of some well known homological functors. Let A be an associative unital al...

this paper, however, we will show that, if one takes for End A the differential graded algebra C (A; A) of Hochschild cochains, the maps (1), in a sense, still exist. More precisely, they exist if one passes from the category of algebras to the category of complexes by means of some well known homological functors. Let A be an associative unital al...

We give a proof of a conjecture of P. Schapira and J.-P. Schneiders on the characteristic classes of D-modules.

We prove the Atiyah-Singer index theorem where the algebra of pseudodifferential operators is replaced by an arbitrary deformation quantization of the algebra of functions on a symplectic manifold.

We prove the Atiyah-Singer index theorem where the algebra of pseudodifferential operators is replaced by an arbitrary deformation quantization of the algebra of functions on a symplectic manifold.

The approach of formal differential geometry to the topological invariants which can be localized is developed. The universal space and universal characteristic forms are constructed. They give rise to primary and secondary characteristic forms.