Chern Character Research Articles

The odd Chern character and index localization formulae

The odd Chern character and index localization formulae

Holonomy, twisting cochains and characteristic classes

This paper contains a description of various geometric constructions associated with fibre bundles, given in terms of important algebraic object, the “twisting cochain". Our examples include the Chern-Weil classes, the holonomy representation and the so-called cyclic Chern character of Bismut and others (see [2, 11, 27]), also called the Bismut’s class. The later example is the principal one for us, since we are motivated by the attempt to find an algebraic approach to the Witten’s index formula. We also give several examples of the twisting cochain associated with a given principal bundle. In particular, our approach allows us to obtain explicit formulas for the Chern classes and for an analogue of the cyclic Chern character in the terms of the glueing functions of the principal bundle. We discuss few modifications of this construction. We hope that this approach can turn fruitful for the investigations of the Witten index formula.

A version of smooth K-theory adapted to the total Chern class

AbstractA new model of smooth K0-theory ([5] [1]) is constructed, with the help of the total Chern class (contrary to the theories considered in ]1], [5], [12] and [13] which use the Chern character). The correspondence with the earlier model [1] is obtained by adapting, at the level of transgression forms, the usual formulae which express the Chern character in terms of the Chern classes and vice versa.The advantage of this new model is that it allows constructing Chern classes with values in integral Chern-Simons characters in a natural way: this construction answers a question asked by U. Bunke [4].

Chow groups of K3 surfaces and spherical objects

We show that for a K3 surface X the finitely generated subring R (X) ⊂ \mathrm{CH}^* (X) introduced by Beauville and Voisin is preserved under derived equivalences. This is proved by analyzing Chern characters of spherical bundles (and complexes). As for a K3 surface X defined over a number field all spherical bundles on the complex K3 surface X_ℂ are defined over \bar ℚ , this is compatible with the Bloch–Beilinson conjecture. Besides the work of Beauville and Voisin [5], Lazarfeld’s result on Brill–Noether theory for curves in K3 surfaces [15] and the deformation theory developed in [12] are central for the discussion.

On uniqueness of characteristic classes

We give an axiomatic characterization of maps from algebraic K -theory. The results apply to a large class of maps from algebraic K -theory to any suitable cohomology theory or to algebraic K -theory. In particular, we obtain comparison theorems for the Chern character and Chern classes and for the Adams operations and λ -operations on higher algebraic K -theory. We show that the Adams operations and λ -operations defined by Grayson agree with the ones defined by Gillet and Soulé.

A universal characterization of the Chern character maps

The Chern character maps are one of the most important working tools in mathematics. Although they admit numerous different constructions, they are not yet fully understood at the conceptual level. In this paper we eliminate this gap by characterizing the Chern character maps, from the Grothendieck group to the (negative) cyclic homology groups, in terms of simple universal properties.

A Poincaré–Hopf type formula for Chern character numbers

For two complex vector bundles admitting a homomorphism with isolated singularities between them, we establish a Poincaré–Hopf type formula for the difference of the Chern character numbers of these two vector bundles. As a consequence, we extend the original Poincaré–Hopf index formula to the case of complex vector fields.

Chern-Weil Construction for Twisted K-Theory

We give a finite-dimensional and geometric construction of a Chern character for twisted K-theory, introducing a notion of connection on a twisted vectorial bundle which can be considered as a finite-dimensional approximation of a twisted family of Fredholm operators. Our construction is applicable to the case of any classes giving the twisting, and agrees with the Chern character of bundle gerbe modules in the case of torsion classes.

Logarithmic trace and orbifold products

We give a purely equivariant construction of orbifold products for quotient Deligne-Mumford stacks [X/G] where G is an arbitrary linear algebraic group (not necessarily finite). The key to our construction is the definition of the "logarithmic trace" of an equivariant vector bundle. We also prove that there is an orbifold Chern character homomorphism which induces an isomorphism of a canonical summand in the orbifold Grothendieck ring with the orbifold Chow ring. As an application we obtain an associative orbifold product on the Grothendieck ring of [X/G] (as opposed to its inerita stack) taken with complex coefficients.

HKR characters, $p$-divisible groups and the generalized Chern character

In this paper we describe the generalized Chern character of classifying spaces of finite groups in terms of Hopkins-Kuhn-Ravenel generalized group characters. For this purpose we study the $p$-divisible group and its level structures associated with the $K(n)$-localization of the $(n+1)$st Morava $E$-theory.

Quasihomomorphisms and the residue Chern character

We develop a general procedure, based on the renormalized eta-cochain, which allows to find “local” representatives of the bivariant Chern character of finitely summable quasihomomorphisms. In particular, using zeta-function renormalization we obtain a bivariant generalization of the Connes–Moscovici residue formula, and explain the link with chiral and multiplicative anomalies in quantum field theory.

On E∞ -structure of the generalized Chern character

In this note we show that we can lift the generalized Chern character to a morphism of commutative S-algebras. Furthermore, we show that we can take a lifting that is equivariant with respect to the action of a Morava stabilizer group.

On the index of noncommutative elliptic operators over C*-algebras

We consider noncommutative elliptic operators over C*-algebras, associated with a discrete group of isometries of a manifold. The main result of the paper is a formula expressing the Chern characters of the index (Connes invariants) in topological terms. As a corollary to this formula a simple proof of higher index formulae for noncommutative elliptic operators is obtained.Bibliography: 36 titles.

Calculating the parabolic Chern character of a locally abelian parabolic bundle

We calculate the parabolic Chern character of a bundle with locally abelian parabolic structure on a smooth strict normal crossings divisor, using the definition in terms of Deligne–Mumford stacks. We obtain explicit formulas for ch 1, ch 2 and ch 3, and verify that these correspond to the formulas given by Borne for ch 1 and Mochizuki for ch 2.

Chern characters and Hirzebruch–Riemann–Roch formula for matrix factorizations

We study the category of matrix factorizations for an isolated hypersurface singularity. We compute the canonical bilinear form on the Hochschild homology of this category. We find explicit expressions for the Chern character and the boundary-bulk maps and derive an analog of the Hirzebruch-Riemann-Roch formula for the Euler characteristic of the Hom-space between a pair of matrix factorizations. We also establish G-equivariant versions of these results.

Algebraic K -theory, A 1 -homotopy and Riemann-Roch theorems

In this paper, we show that the combination of the constructions done in SGA 6 and the A1-homotopy theory naturally leads to results on higher algebraic K-theory. This applies to the operations on algebraic K-theory, Chern characters and Riemann–Roch theorems.

Eta forms and the odd pseudodifferential families index

Let A(t) be an elliptic, product-type suspended (which is to say parameter-dependant in a symbolic way) family of pseudodifferential operators on the fibres of a fibration φ with base Y. The standard example is A + it where A is a family, in the usual sense, of first order, self-adjoint and elliptic pseudodifferential operators and t ∈ R is the ‘suspending’ parameter. Let πA : A(φ) −→ Y be the infinite-dimensional bundle with fibre at y ∈ Y consisting of the Schwartz-smoothing perturbations, q, making Ay(t) + q(t) invertible for all t ∈ R. The total eta form, ηA, as described here, is an even form on A(φ) which has basic differential which is an explicit representative of the odd Chern character of the index of the family: (*) dηA = π ∗ AγA, Ch(ind(A)) = [γA] ∈ H (Y ). The 1-form part of this identity may be interpreted in terms of the τ invariant (exponentiated eta invariant) as the determinant of the family. The 2-form part of the eta form may be interpreted as a B-field on the K-theory gerbe for the family A with (*) giving the ‘curving’ as the 3-form part of the Chern character of the index. We also give ‘universal’ versions of these constructions over a classifying space for odd K-theory. Introduction Eta forms, starting with the eta invariant itself, appear as the boundary terms in the index formula for Dirac operators [2], [5], [4], [10], [11]. One aim of the present paper is to show that, with the freedom gained by working in the more general context of families of pseudodifferential operators, these forms appear as universal transgression, or connection, forms for the cohomology class of the index. That these forms arise in the treatment of boundary problems corresponds to the fact that boundary conditions amount to the explicit inversion of a suspended (or model) problem on the boundary. The odd index of the boundary family is trivial and the eta form is an explicit trivialization of it in cohomology. To keep the discussion within bounds we work here primarily in the ‘odd’ setting of a family of self-adjoint elliptic pseudodifferential operators, taken to be of order 1, on the fibres of a fibration of compact manifolds

Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory

We study D-branes and Ramond-Ramond fields on global orbifolds of Type II string theory with vanishing H-flux using methods of equivariant K-theory and K-homology. We illustrate how Bredon equivariant cohomology naturally realizes stringy orbifold cohomology. We emphasize its role as the correct cohomological tool which captures known features of the low-energy effective field theory, and which provides new consistency conditions for fractional D-branes and Ramond-Ramond fields on orbifolds. We use an equivariant Chern character from equivariant K-theory to Bredon cohomology to define new Ramond-Ramond couplings of D-branes which generalize previous examples. We propose a definition for groups of differential characters associated to equivariant K-theory. We derive a Dirac quantization rule for Ramond-Ramond fluxes, and study flat Ramond-Ramond potentials on orbifolds.

Construction of a topological charge on fuzzyS2×S2via a Ginsparg-Wilson relation

We construct a topological charge of gauge field configurations on a fuzzy ${S}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{2}$ by using a Dirac operator satisfying the Ginsparg-Wilson relation. The topological charge defined on the fuzzy ${S}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{2}$ can be interpreted as a noncommutative (or matrix) generalization of the 2nd Chern character on ${S}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{2}$. We further calculate the number of chiral zero modes of the Dirac operator in topologically nontrivial gauge configurations. Generalizations of our formulation to fuzzy $({S}^{2}{)}^{k}$ are also discussed.

KO-HOMOLOGY AND TYPE I STRING THEORY

We study the classification of D-branes and Ramond–Ramond fields in Type I string theory by developing a geometric description of KO-homology. We define an analytic version of KO-homology using KK-theory of real C*-algebras, and construct explicitly the isomorphism between geometric and analytic KO-homology. The construction involves recasting the C ℓn-index theorem and a certain geometric invariant into a homological framework which is used, along with a definition of the real Chern character in KO-homology, to derive cohomological index formulas. We show that this invariant also naturally assigns torsion charges to non-BPS states in Type I string theory, in the construction of classes of D-branes in terms of topological KO-cycles. The formalism naturally captures the coupling of Ramond–Ramond fields to background D-branes which cancel global anomalies in the string theory path integral. We show that this is related to a physical interpretation of bivariant KK-theory in terms of decay processes on spacetime-filling branes. We also provide a construction of the holonomies of Ramond–Ramond fields in Type II string theory in terms of topological K-chains.