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Alexander Gorokhovsky

Publications and source records attributed to Alexander Gorokhovsky.

At least 19 recordsLinked to original sources
arXiv2026

Toeplitz Operators on Contact Manifolds and Equivariant K-homology

We present an equivariant generalization of Boutet de Monvel's index theorem for Toeplitz operators on contact manifolds. We prove that the Dirac operator and the Szegö projection determine the same class in equivariant $K$-homology, generalizing a theorem of Baum-Douglas-Taylor. We do not assume that the contact manifold is the boundary of a strictly pseudoconvex domain. The proof proceeds by a deformation linking the principal symbols of the classical and Heisenberg pseudodifferential calculi. At the level of symbols, the projection defining the Dirac class deforms to the principal Heisenberg symbol of the Szegö projection. This deformation implies equality of the corresponding classes in K-homology. This, in turn, gives an equivariant generalization of Boutet de Monvel's index formula for Toeplitz operators.

math.KT↗
arXiv2025

A note on traces for the Heisenberg calculus

In previous work, we gave a local formula for the index of Heisenberg elliptic operators on contact manifolds. We constructed a cocycle in periodic cyclic cohomology which, when paired with the Connes-Chern character of the principal Heisenberg symbol, calculates the index. A crucial ingredient of our index formula was a new trace on the algebra of Heisenberg pseudodifferential operators. The construction of this trace was rather involved. In the present paper, we clarify the nature of this trace.

math.FA↗
arXiv2022

Cyclic cohomology and the extended Heisenberg calculus of Epstein and Melrose

In this paper we present a formula for the index of a pseudodifferential operator with invertible principal symbol in the extended Heisenberg calculus of Epstein and Melrose. Our results build on the work we did in a previous paper (arXiv:2010.02900), where we restricted attention to the Heisenberg calculus proper.

math.OA↗
arXiv2020

The Heisenberg Calculus, Index Theory and Cyclic (Co)homology

A hypoelliptic operator in the Heisenberg calculus on a compact contact manifold is a Fredholm operator. Its symbol determines an element in the K-theory of the noncommutative algebra of Heisenberg symbols. We construct a periodic cyclic cocycle which, when paired with the Connes-Chern character of the principal Heisenberg symbol, calculates the index. Our index formula is local, i.e. given as a local expression in terms of the principal symbol of the operator and a connection on TM and its curvature. We prove our index formula by reduction to Boutet de Monvel's index theorem for Toeplitz operators.

math.OA↗
arXiv2019

Comparison of spaces associated to DGLA via higher holonomy

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.

math.AT↗
arXiv2018

Index pairing with Alexander-Spanier cocycles

We give a uniform construction of the higher indices of elliptic operators associated to Alexander-Spanier cocycles of either parity in terms of a pairing a la Connes between the K-theory and the cyclic cohomology of the algebra of complete symbols of pseudodifferential operators, implemented by means of a relative form of the Chern character in cyclic homology. While the formula for the lowest index of an elliptic operator D on a closed manifold M (which coincides with its Fredholm index) reproduces the Atiyah-Singer index theorem, our formula for the highest index of D (associated to a volume cocycle) yields an extension to arbitrary manifolds of any dimension of the Helton-Howe formula for the trace of multicommutators of classical Toeplitz operators on odd-dimensional spheres. In fact, the totality of higher analytic indices for an elliptic operator D amount to a representation of the Connes-Chern character of the K-homology cycle determined by D in terms of expressions which extrapolate the Helton-Howe formula below the dimension of M.

math.KT↗
arXiv2018

A Hilbert bundle description of differential K-theory

We give an infinite dimensional description of the differential K-theory of a manifold $M$. The generators are triples $[H, A, ω]$ where $H$ is a ${\bf Z}_2$-graded Hilbert bundle on $M$, $A$ is a superconnection on $H$ and $ω$ is a differential form on $M$. The relations involve eta forms. We show that the ensuing group is the differential K-group $\check{K}^0(M)$. In addition, we construct the pushforward of a finite dimensional cocycle under a proper submersion with a Riemannian structure. We give the analogous description of the odd differential K-group $\check{K}^1(M)$. Finally, we give a model for twisted differential K-theory.

math.DG↗
arXiv2017

Generalized Euler classes, differential forms and commutative DGAs

In the context of commutative differential graded algebras over $\mathbb Q$, we show that an iteration of "odd spherical fibration" creates a "total space" commutative differential graded algebra with only odd degree cohomology. Then we show for such a commutative differential graded algebra that, for any of its "fibrations" with "fiber" of finite cohomological dimension, the induced map on cohomology is injective.

math.AT↗
arXiv2017

Fibrations and higher products in cohomology

This paper is a continuation of a previous paper joint with Dennis Sullivan (arXiv:1704.04308). Working in the context of commutative differential graded algebras, we study the ideal of the cohomology classes which can be annihilated by fibrations whose fiber has finite homological dimension. In the present paper we identify these classes with certain higher products in cohomology.

math.AT↗
arXiv2017

The higher twisted index theorem for foliations

Given a gerbe $L$, on the holonomy groupoid $\mathcal G$ of the foliation $(M, \mathcal F)$, whose pull-back to $M$ is torsion, we construct a Connes $Φ$-map from the twisted Dupont-Sullivan bicomplex of $\mathcal G$ to the cyclic complex of the $L$-projective leafwise smoothing operators on $(M, \mathcal F)$. Our construction allows to couple the $K$-theory analytic indices of $L$-projective leafwise elliptic operators with the twisted cohomology of $B\mathcal G$ producing scalar higher invariants. Finally by adapting the Bismut-Quillen superconnection approach, we compute these higher twisted indices as integrals over the ambiant manifold of the expected twisted characteristic classes.

math.KT↗
arXiv2017

Equivariant Algebraic Index Theorem

We prove a Γ-equivariant version of the algebraic index theorem, where Γ is a discrete group of automorphisms of a formal deformation of a symplectic manifold. The particular cases of this result are the algebraic version of the transversal index theorem related to the theorem of A. Connes and H. Moscovici for hypoelliptic operators and the index theorem for the extension of the algebra of pseudodifferential operators by a group of diffeomorphisms of the underlying manifold due to A. Savin, B. Sternin, E. Schrohe and D. Perrot.

math.KT↗
arXiv2015

Deligne groupoid revisited

We show that for a differential graded Lie algebra $\mathfrak{g}$ whose components vanish in degrees below -1 the nerve of the Deligne 2-groupoid is homotopy equivalent to the simplicial set of $\mathfrak{g}$-valued differential forms introduced by V.Hinich.

math.AT↗
arXiv2015

A note on the higher Atiyah-Patodi-Singer index theorem on Galois coverings

Let $Γ$ be a finitely generated discrete group satisfying the rapid decay condition. We give a new proof of the higher Atiyah-Patodi-Singer theorem on a Galois $Γ$-coverings, thus providing an explicit formula for the higher index associated to a group cocycle $c\in Z^k (Γ;\mathbb{C})$ which is of polynomial growth with respect to a word-metric. Our new proof employs relative K-theory and relative cyclic cohomology in an essential way.

math.DG↗
arXiv2014

Formality theorem for gerbes

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.

math.QA↗
arXiv2014

On the spectral flow for Dirac operators with local boundary conditions

Let M be an even dimensional compact Riemannian manifold with boundary and let D be a Dirac operator acting on the sections of the Clifford module E over M. We impose certain local elliptic boundary conditions for D obtaining a selfadjoint extension D_F of D. For a smooth U(n)--valued function g:M -> U(n) we establish a formula for the spectral flow along the straight line between D_F and g^{-1} D_F g. This spectral flow is motivated by index theory: in odd dimensions it gives the natural pairing between the K--homology class of the operator and the K--theory class of g. In our situation, with dim M having the "wrong" parity, the answer can be expressed in terms of the natural spectral flow pairing on the odd--dimensional boundary. Our result generalizes a recent paper by M. Prokhorova in which the two-dimensional case is treated. Furthermore, our paper may be seen as an odd-dimensional analogue of a paper by D. Freed. As an application we obtain a new proof of the cobordism invariance of the spectral flow.

math.AP↗