Orbifold K-theory Research Articles

Orbifold K-theory and Chen-Ruan cohomology

In this paper, we construct the reduced classifying space for each orbifold, and using which, we prove that the orbifold K-theory is isomorphic to the Chen-Ruan cohomology for any compact orbifold.

Chen-Ruan Cohomology and Stringy Orbifold K-Theory for Stable Almost Complex Orbifolds

Comparing to the construction of stringy cohomology ring of equivariant stable almost complex manifolds and its relation with the Chen-Ruan cohomology ring of the quotient almost complex orbifolds, the authors construct in this note a Chen-Ruan cohomology ring for a stable almost complex orbifold. The authors show that for a finite group G and a G-equivariant stable almost complex manifold X, the G-invariant part of the stringy cohomology ring of (X, G) is isomorphic to the Chen-Ruan cohomology ring of the global quotient stable almost complex orbifold [X/G]. Similar result holds when G is a torus and the action is locally free. Moreover, for a compact presentable stable almost complex orbifold, they study the stringy orbifold K-theory and its relation with Chen-Ruan cohomology ring.

Quasi-elliptic cohomology I

Quasi-elliptic cohomology is a variant of elliptic cohomology theories. It is the orbifold K-theory of a space of constant loops. For global quotient orbifolds, it can be expressed in terms of equivariant K-theories. Thus, the constructions on it can be made in a neat way. This theory reflects the geometric nature of the Tate curve. In this paper we provide a systematic introduction of its construction and definition.

Quasi-elliptic cohomology and its power operations

Quasi-elliptic cohomology is a variant of Tate K-theory. It is the orbifold K-theory of a space of constant loops. For global quotient orbifolds, it can be expressed in terms of equivariant K-theories. In this paper we show how this theory is equipped with power operations. We also prove that the Tate K-theory of symmetric groups modulo a certain transfer ideal classify the finite subgroups of the Tate curve.

点群胚的Orbifold K-理论

点群胚的Orbifold K-理论

Twisted orbifold K-theory for global quotient

Let [X/G] be the global quotient of a complex manifold X by a finite group G, with a twisting α∈Z2(G,S1). Adem and Ruan defined the twisted orbifold K-theory Korbα(X/G). In this paper, we construct a decomposition for Korbα(X/G) and define a product on Korbα(X/G) using the method of stringy product. Then we construct a twisted Chern characterChorbα:αKorb(X/G)⊗C⟶Horb⁎(X/G;Lα). Here Horb⁎(X/G;Lα) is the twisted orbifold cohomology of [X/G]. Furthermore we show that this twisted Chern character is a ring isomorphism.

Differential orbifold K-theory

We construct differential K-theory of representable smooth orbifolds as a ring valued functor with the usual properties of a differential extension of a cohomology theory. For proper submersions (with smooth fibres) we construct a push-forward map in differential orbifold K-theory. Finally, we construct a non-degenerate intersection pairing with values in \mathbb{C}/\mathbb{Z} for the subclass of smooth orbifolds which can be written as global quotients by a finite group action. We construct a real subfunctor of our theory, where the pairing restricts to a non-degenerate \mathbb{R}/\mathbb{Z} -valued pairing.

A note on holonomy of gerbes over orbifolds

In this paper, we generalize the construction of the inverse transgression map done by Adem, A., Ruan, Y. and Zhang, B. in [A stringy product on twisted orbifold K-theory. Morfismos, 11, 33–64 (2007)] and give a different proof to the statement that the image of the inverse transgression map for a gerbe with connection over an orbifold is an inner local system on its inertia orbifold.

Delocalized Chern character for stringy orbifold K-theory

In this paper, we define a stringy product on K � orb(X) C, the orbifold K-theory of any almost complex presentable orbifold X. We establish that under this stringy product, the delocali zed Chern character chdeloc : K �(X) C ! H � CR(X), after a canonical modification, is a ring isomorphism. Here H � CR(X) is the Chen-Ruan cohomology of X. The proof relies on an intrinsic description of the obstruc tion bundles in the construction of the Chen- Ruan product. As an application, we investigate this stringy product on the equivariant K-theory K � G(G) of a finite group G with the conjugation action. It turns out that the stringy pr oduct is different from the Pontryagin product (the latter is also called the fusion pro duct in string theory).

Torsion in the full orbifold K-theory of abelian symplectic quotients

Let (M, ω, Φ) be a Hamiltonian T-space and let \({H\subseteq T}\) be a closed Lie subtorus. Under some technical hypotheses on the moment map Φ, we prove that there is no additive torsion in the integral full orbifold K-theory of the orbifold symplectic quotient [M//H]. Our main technical tool is an extension to the case of moment map level sets the well-known result that components of the moment map of a Hamiltonian T-space M are Morse-Bott functions on M. As first applications, we conclude that a large class of symplectic toric orbifolds, as well as certain S 1-quotients of GKM spaces, have integral full orbifold K-theory that is free of additive torsion. Finally, we introduce the notion of semilocally Delzant which allows us to formulate sufficient conditions under which the hypotheses of the main theorem hold. We illustrate our results using low-rank coadjoint orbits of type A and B.

The full orbifold K-theory of abelian symplectic quotients

AbstractIn their 2007 paper, Jarvis, Kaufmann, and Kimura defined the full orbifoldK-theory of an orbifold , analogous to the Chen-Ruan orbifold cohomology of in that it uses the obstruction bundle as a quantum correction to the multiplicative structure. We give an explicit algorithm for the computation of this orbifold invariant in the case when arises as an abelian symplectic quotient. To this end, we introduce the inertial K-theory associated to a T -action on a stably complex manifold M, where T is a compact abelian Lie group. Our methods are integral K-theoretic analogues of those used in the orbifold cohomology case by Goldin, Holm, and Knutson in 2005. We rely on the K-theoretic Kirwan surjectivity methods developed by Harada and Landweber. As a worked class of examples, we compute the full orbifold K-theory of weighted projective spaces that occur as a symplectic quotient of a complex affine space by a circle. Our computations hold over the integers, and in the particular case of these weighted projective spaces, we show that the associated invariant is torsion-free.

Stringy product on twisted orbifold K-theory for abelian quotients

In this paper we present a model to calculate the stringy product on twisted orbifold K-theory of Adem-Ruan-Zhang for abelian complex orbifolds. In the first part we consider the non-twisted case on an orbifold presented as the quotient of a manifold acted by a compact abelian Lie group. We give an explicit description of the obstruction bundle, we explain the relation with the product defined by Jarvis-Kaufmann-Kimura and, via a Chern character map, with the Chen-Ruan cohomology, we explicitly calculate the stringy product for a weighted projective orbifold. In the second part we consider orbifolds presented as the quotient of a manifold acted by a finite abelian group and twistings coming from the group cohomology. We show a decomposition formula for twisted orbifold K-theory that is suited to calculate the stringy product and we use this formula to calculate two examples when the group is (ℤ/2) 3 .

THE DRINFEL'D DOUBLE AND TWISTING IN STRINGY ORBIFOLD THEORY

This paper exposes the fundamental role that the Drinfel'd double D(k[G]) of the group ring of a finite group G and its twists Dβ(k[G]), β ∈ Z3(G,k*) as defined by Dijkgraaf–Pasquier–Roche play in stringy orbifold theories and their twistings. The results pertain to three different aspects of the theory. First, we show that G-Frobenius algebras arising in global orbifold cohomology or K-theory are most naturally defined as elements in the braided category of D(k[G])-modules. Secondly, we obtain a geometric realization of the Drinfel'd double as the global orbifold K-theory of global quotient given by the inertia variety of a point with a G action on the one hand and more stunningly a geometric realization of its representation ring in the braided category sense as the full K-theory of the stack [pt/G]. Finally, we show how one can use the co-cycles β above to twist the global orbifold K-theory of the inertia of a global quotient and more importantly, the stacky K-theory of a global quotient [X/G]. This corresponds to twistings with a special type of two-gerbe.

Stringy K-theory and the Chern character

For a finite group G acting on a smooth projective variety X, we construct two new G-equivariant rings: first the stringy K-theory of X, and second the stringy cohomology of X. For a smooth Deligne-Mumford stack Y we also construct a new ring called the full orbifold K-theory of Y. For a global quotient Y=[X/G], the ring of G-invariants of the stringy K-theory of X is a subalgebra of the full orbifold K-theory of the the stack Y and is linearly isomorphic to the ``orbifold K-theory'' of Adem-Ruan (and hence Atiyah-Segal), but carries a different, ``quantum,'' product, which respects the natural group grading. We prove there is a ring isomorphism, the stringy Chern character, from stringy K-theory to stringy cohomology, and a ring homomorphism from full orbifold K-theory to Chen-Ruan orbifold cohomology. These Chern characters satisfy Grothendieck-Riemann-Roch for etale maps. We prove that stringy cohomology is isomorphic to Fantechi and Goettsche's construction. Since our constructions do not use complex curves, stable maps, admissible covers, or moduli spaces, our results simplify the definitions of Fantechi-Goettsche's ring, of Chen-Ruan's orbifold cohomology, and of Abramovich-Graber-Vistoli's orbifold Chow. We conclude by showing that a K-theoretic version of Ruan's Hyper-Kaehler Resolution Conjecture holds for symmetric products. Our results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds.

Chern character for twisted K-theory of orbifolds

For an orbifold X and α ∈ H 3 ( X , Z ) , we introduce the twisted cohomology H c ∗ ( X , α ) and prove that the non-commutative Chern character of Connes–Karoubi establishes an isomorphism between the twisted K-groups K α ∗ ( X ) ⊗ C and the twisted cohomology H c ∗ ( X , α ) . This theorem, on the one hand, generalizes a classical result of Baum–Connes, Brylinski–Nistor, and others, that if X is an orbifold then the Chern character establishes an isomorphism between the K-groups of X tensored with C , and the compactly-supported cohomology of the inertia orbifold. On the other hand, it also generalizes a recent result of Adem–Ruan regarding the Chern character isomorphism of twisted orbifold K-theory when the orbifold is a global quotient by a finite group and the twist is a special torsion class, as well as Mathai–Stevenson's theorem regarding the Chern character isomorphism of twisted K-theory of a compact manifold.

Inertia Orbifolds, Configuration Spaces and the Ghost Loop Space

In this paper we define and study the loop of an orbifold $X$ consisting of those loops that remain constant in the coarse moduli space of $X$. We construct a configuration space model for the ghost loop orbifold using an idea of G. Segal. From this we exhibit the relation between the Hochschild and cyclic homologies of the inertia orbifold of $X$ (that generate the so-called twisted sectors in string theory) and the ordinary and equivariant homologies of the ghost loop orbifold. We also show how this clarifies the relation between orbifold K-theory, Chen-Ruan orbifold cohomology, Hochschild homology, and periodic cyclic homology.

Gerbes over Orbifolds and Twisted K-Theory

In this paper we construct an explicit geometric model for the group of gerbes over an orbifold X. We show how from its curvature we can obtain its characteristic class in H 3(X) via Chern-Weil theory. For an arbitrary gerbe , a twisting K orb(X) of the orbifold K-theory of X is constructed, and shown to generalize previous twisting by Rosenberg [28], Witten [35], Atiyah-Segal [2] and Bowknegt et. al. [4] in the smooth case and by Adem-Ruan [1] for discrete torsion on an orbifold.

Twisted Orbifold K-Theory

We use equivariant methods to define and study the orbifold K-theory of an orbifold X. Adapting techniques from equivariant K-theory, we construct a Chern character and exhibit a multiplicative decomposition for K * orb (X)⊗ℚ, in particular showing that it is additively isomorphic to the orbifold cohomology of X. A number of examples are provided. We then use the theory of projective representations to define the notion of twisted orbifold K–theory in the presence of discrete torsion. An explicit expression for this is obtained in the case of a global quotient.