Narrow Results

We construct a natural generalization of the Grothendieck group ${\mathrm{\text{K}}}_0$ to the case of possibly unpointed categories admitting pushouts by using the concept of heaps recently introduced by Brezinzki. In case of a monoidal category, the defined K0 is shown to be a truss. It is shown that the construction generalizes the classical ${\mathrm{\text{K}}}_0$ of an abelian category as the group retract along the isomorphism class of the zero object. We finish by applying this construction to construct the integers with addition and multiplication as the decategorification of finite sets and show that in this ${\mathrm{\text{K}}}_0(\underline{\mathrm{\text{Top}}})$ one can identify a CW-complex with the iterated product of its cells.

Almost flat finitely generated projective Hilbert $C^{\ast }$-module bundles were successfully used by Hanke and Schick to prove special cases of the Strong Novikov Conjecture. Dadarlat later showed that it is possible to calculate the index of a $K$-homology class $\eta \in K_{\ast }(M)$ twisted with an almost flat bundle in terms of the image of $\eta$ under Lafforgue’s assembly map and the almost representation associated with the bundle. Mishchenko used flat infinite-dimensional bundles equipped with a Fredholm operator in order to prove special cases of the Novikov higher signature conjecture. We show how to generalize Dadarlat’s theorem to the case of an infinite-dimensional bundle equipped with a continuous family of Fredholm operators on the fibers. Along the way, we show that special cases of the Strong Novikov Conjecture can be proven if there exist sufficiently many almost flat bundles with Fredholm operator. To this end, we introduce the concept of an asymptotically flat Fredholm bundle and its associated asymptotic Fredholm representation, and prove an index theorem which relates the index of the asymptotic Fredholm bundle with the so-called asymptotic index of the associated asymptotic Fredholm representation.

Let $K$ be the Cantor space and ${𝕊}^{2n}$ be an even-dimensional sphere. By applying a result of the existence of minimal skew products, we show that, associated with any Cantor minimal system $(K,\alpha )$, there is a class ${{ℛ}}_0(\tilde{\alpha })$ of minimal skew products on $K\times {𝕊}^{2n}$, such that for any two rigid homeomorphisms $\alpha \in {{ℛ}}_0(\tilde{\alpha })$ and $\beta \in {{ℛ}}_0(\tilde{\beta })$, the notions of approximate $K$-conjugacy and $C^{\ast }$-strongly approximate conjugacy coincide, which are also equivalent to a $K$-version of Tomiyama’s commutative diagram. In fact, this is also the case if ${𝕊}^{2n}$ is replaced by any (infinite) connected finite CW-complex with torsion free $K_0$-group, vanished $K_1$-group and the so-called Lipschitz-minimal-property.

We show that one can define (p,∞)-summable spectral triples using degenerate metrics on smooth manifolds. Furthermore, these triples satisfy Connes–Moscovici's discrete and finite dimension spectrum hypothesis, allowing one to use the Local Index Theorem [1] to compute the pairing with K-theory. We demonstrate this with a concrete example.

We study the effects of having multiple Spin structures on the partition function of the spacetime fields in M-theory. This leads to a potential anomaly which appears in the eta invariants upon variation of the Spin structure. The main sources of such spaces are manifolds with nontrivial fundamental group, which are also important in realistic models. We extend the discussion to the Spin^c case and find the phase of the partition function, and revisit the quantization condition for the C-field in this case. In type IIA string theory in 10 dimensions, the (mod 2) index of the Dirac operator is the obstruction to having a well-defined partition function. We geometrically characterize manifolds with and without such an anomaly and extend to the case of nontrivial fundamental group. The lift to KO-theory gives the α-invariant, which in general depends on the Spin structure. This reveals many interesting connections to positive scalar curvature manifolds and constructions related to the Gromov–Lawson–Rosenberg conjecture. In the 12-dimensional theory bounding M-theory, we study similar geometric questions, including choices of metrics and obtaining elements of K-theory in 10 dimensions by pushforward in K-theory on the disk fiber. We interpret the latter in terms of the families index theorem for Dirac operators on the M-theory circle and disk. This involves superconnections, eta forms, and infinite-dimensional bundles, and gives elements in Deligne cohomology in lower dimensions. We illustrate our discussion with many examples throughout.

We describe the representation theory of loop groups in terms of K-theory and noncommutative geometry. This is done by constructing suitable spectral triples associated with the level $\ell$ projective unitary positive-energy representations of any given loop group $\mathrm{\text{L}}G$. The construction is based on certain supersymmetric conformal field theory models associated with $\mathrm{\text{L}}G$ in the setting of conformal nets. We then generalize the construction to many other rational chiral conformal field theory models including coset models and the moonshine conformal net.

We present a definition of a (super)-modular functor which includes certain interesting cases that previous definitions do not allow. We also introduce a notion of topological twisting of a modular functor, and construct formally a realization by a 2-dimensional topological field theory valued in twisted $K$-modules. We discuss, among other things, the $N=1$-supersymmetric minimal models from the point of view of this formalism.

This paper deals with the construction of a suitable topological $K$-theory capable of classifying topological phases of dynamically stable systems described by gapped $\eta$-self-adjoint operators on a Krein space with indefinite metric $\eta$.

Renault, Wassermann, Handelman and Rossmann (early 1980s) and Evans and Gould (1994) explicitly described the $K$-theory of certain unital AF-algebras $A$ as (quotients of) polynomial rings. In this paper, we show that in each case the multiplication in the polynomial ring (quotient) is induced by a ∗-homomorphism $A\otimes A\to A$ arising from a unitary braiding on a C*-tensor category and essentially defined by Erlijman and Wenzl (2007). We also present some new explicit calculations based on the work of Gepner, Fuchs and others. Specifically, we perform computations for the rank two compact Lie groups SU(3), Sp(4) and G₂ that are analogous to the Evans–Gould computation for the rank one compact Lie group SU(2).

The Verlinde rings are the fusion rings of Wess–Zumino–Witten models in conformal field theory or, equivalently, of certain related C*-tensor categories. Freed, Hopkins and Teleman (early 2000s) realized these rings via twisted equivariant $K$-theory. Inspired by this, our long-term goal is to realize these rings in a simpler $K$-theoretical manner, avoiding the technicalities of loop group analysis. As a step in this direction, we note that the Verlinde rings can be recovered as above in certain special cases.

While the classification of noninteracting crystalline topological insulator phases by equivariant K-theory has become widely accepted, its generalization to anyonic interacting phases — hence to phases with topologically ordered ground states supporting topological braid quantum gates — has remained wide open.

On the contrary, the success of K-theory with classifying noninteracting phases seems to have tacitly been perceived as precluding a K-theoretic classification of interacting topological order; and instead a mix of other proposals has been explored. However, only K-theory connects closely to the actual physics of valence electrons; and self-consistency demands that any other proposal must connect to K-theory.

Here, we provide a detailed argument for the classification of symmetry protected/enhanced ${𝔰𝔲}_2$-anyonic topological order, specifically in interacting 2d semi-metals, by the twisted equivariant differential (TED) K-theory of configuration spaces of points in the complement of nodal points inside the crystal’s Brillouin torus orbi-orientifold.

We argue, in particular, that :

• ⁽¹⁾topological 2d semi-metal phases modulo global mass terms are classified by the flat differential twisted equivariant K-theory of the complement of the nodal points;
• ⁽²⁾$n$-electron interacting phases are classified by the K-theory of configuration spaces of $n$ points in the Brillouin torus;
• ⁽³⁾the somewhat neglected twisting of equivariant K-theory by “inner local systems” reflects the effective “fictitious” gauge interaction of Chen, Wilczeck, Witten and Halperin (1989), which turns fermions into anyonic quanta;
• ⁽⁴⁾the induced ${𝔰𝔲}_2$-anyonic topological order is reflected in the twisted Chern classes of the interacting valence bundle over configuration space, constituting the hypergeometric integral construction of monodromy braid representations.

A tight dictionary relates these arguments to those for classifying defect brane charges in string theory [H. Sati and U. Schreiber, Anyonic defect branes in TED-K-theory, arXiv:2203.11838], which we expect to be the images of momentum-space ${𝔰𝔲}_2$-anyons under a nonperturbative version of the AdS/CMT correspondence.

We demonstrate that twisted equivariant differential K-theory of transverse complex curves accommodates exotic charges of the form expected of codimension$=$2 defect branes, such as of $\mathrm{\text{D7}}$-branes in IIB/F-theory on $𝔸$-type orbifold singularities, but also of their dual 3-brane defects of class-S theories on M5-branes. These branes have been argued, within F-theory and the AGT correspondence, to carry special $\mathrm{\text{SL}}(2)$-monodromy charges not seen for other branes, at least partially reflected in conformal blocks of the ${𝔰𝔲}_2$-WZW model over their transverse punctured complex curve. Indeed, it has been argued that all “exotic” branes of string theory are defect branes carrying such U-duality monodromy charges — but none of these had previously been identified in the expected brane charge quantization law given by K-theory.

Here we observe that it is the subtle (and previously somewhat neglected) twisting of equivariant K-theory by flat complex line bundles appearing inside orbi-singularities (“inner local systems”) that makes the secondary Chern character on a punctured plane inside an $𝔸$-type singularity evaluate to the twisted holomorphic de Rham cohomology which Feigin, Schechtman and Varchenko showed realizes $\widehat{{{𝔰𝔩}_2}^k}$-conformal blocks, here in degree 1 — in fact it gives the direct sum of these over all admissible fractional levels $k=-2+\kappa /r$. The remaining higher-degree $\widehat{{{𝔰𝔩}_2}^k}$-conformal blocks appear similarly if we assume our previously discussed “Hypothesis H” about brane charge quantization in M-theory. Since conformal blocks — and hence these twisted equivariant secondary Chern characters — solve the Knizhnik–Zamolodchikov equation and thus constitute representations of the braid group of motions of defect branes inside their transverse space, this provides a concrete first-principles realization of anyon statistics of — and hence of topological quantum computation on — defect branes in string/M-theory.

Let X be a compact metric space and A=C(X). Suppose that ℬ is a class of unital C*-algebras satisfying certain conditions, we prove the following: For any ∊>0, finite set F⊂A, there is an integer l such that if ϕ, ψ:A→B(B∈ℬ) are sufficiently multiplicative morphisms (e.g. when both ϕ and ψ are *-homomorphisms) which induce same K-theoretical maps, then there are a unitary u∈M_l+1(B) and a homomorphism σ:A→M_l(B) with finite dimensional image such that

We show that if has tracial topological rank no more than k, then

We observe that the von Neumann envelop of (i.e. the smallest von Neumann algebra that contains) the quantum algebra of functions on the normalizer of the group SU(1,1)≅SL(2,ℝ) in via deformation quantization contains the von Neumann algebraic quantum normalizer of SU(1,1) in the frame work of Waronowicz–Korogodsky, see ([8, Introduction and Sec. 1, Definition 1]), i.e. the C*-envelop or von Neumann envelop (W*-envelop) of the algebraic Hopf algebra. We then use the technique of reduction to the maximal subgroup to compute the K-theory, the periodic cyclic homology and the corresponding Chern–Connes character.

Let A, B be separable simple unital tracially AF C*-algebras. Assuming that A is exact and satisfies the Universal Coefficient Theorem (UCT) in KK-theory, we prove the existence, and uniqueness modulo approximately inner automorphisms, of nuclear *-homomorphisms from A to B with prescribed K-theory data. This implies the AF-embeddability of separable exact residually finite-dimensional C*-algebras satisfying the UCT and reproves Huaxin Lin's theorem on the classification of nuclear tracially AF C*-algebras.

Let G be an inductive limit of finite cyclic groups, and A be a unital simple projectionless C*-algebra with K₁(A) ≅ G and a unique tracial state, as constructed based on dimension drop algebras by Jiang and Su. First, we show that any two aperiodic elements in Aut(A)/WInn(A) are conjugate, where WInn(A) means the subgroup of Aut(A) consisting of automorphisms which are inner in the tracial representation.

In the second part of this paper, we consider a class of unital simple C*-algebras with a unique tracial state which contains the class of unital simple A𝕋-algebras of real rank zero with a unique tracial state. This class is closed under inductive limits and crossed products by actions of ℤ with the Rohlin property. Let A be a TAF-algebra in this class. We show that for any automorphism α of A there exists an automorphism ᾶ of A with the Rohlin property such that ᾶ and α are asymptotically unitarily equivalent. For the proof we use an aperiodic automorphism of the Jiang-Su algebra.

In 1981 W. Fulton and R. MacPherson introduced the notion of bivariant theory (BT), which is a sophisticated unification of covariant theories and contravariant theories. This is for the study of singular spaces. In 2001 M. Levine and F. Morel introduced the notion of algebraic cobordism, which is a universal oriented Borel–Moore functor with products (OBMF) of geometric type, in an attempt to understand better V. Voevodsky's (higher) algebraic cobordism. In this paper we introduce a notion of oriented bivariant theory (OBT), a special case of which is nothing but the oriented Borel–Moore functor with products. The present paper is a first one of the series to try to understand Levine–Morel's algebraic cobordism from a bivariant theoretical viewpoint, and its first step is to introduce OBT as a unification of BT and OBMF.

We study some reduced free products of C*-algebras with amalgamations. We give sufficient conditions for the positive cone of the K₀ group to be the largest possible. We also give sufficient conditions for simplicity and uniqueness of trace. We use the latter result to give a necessary and sufficient condition for simplicity and uniqueness of trace of the reduced C*-algebras of the Baumslag–Solitar groups BS(m, n).

We continue studying net bundles over partially ordered sets (posets), defined as the analogues of ordinary fiber bundles. To this end, we analyze the connection between homotopy, net homology and net cohomology of a poset, giving versions of classical Hurewicz theorems. Focusing our attention on Hilbert net bundles, we define the K-theory of a poset and introduce functions over the homotopy groupoid satisfying the same formal properties as Chern classes. As when the given poset is a base for the topology of a space, our results apply to the category of locally constant bundles.

We show that the K-theory cosheaf is a complete invariant for separable continuous fields with vanishing boundary maps over a finite-dimensional compact metrizable topological space whose fibers are stable Kirchberg algebras with rational K-theory groups satisfying the universal coefficient theorem. We provide a range result for fields in this class with finite-dimensional K-theory. There are versions of both results for unital continuous fields.