Compact Orbifolds Research Articles

A Kummer construction for Chern–Ricci flat balanced manifolds

Given a non-Kähler Calabi–Yau compact orbifold with isolated singularities endowed with a Chern–Ricci flat balanced metric, we study, via a gluing construction, the existence of Chern–Ricci flat balanced metrics on its crepant resolutions, and discuss applications to the search of solutions for the Hull–Strominger system. We also describe the scenario of singular threefolds with ordinary double points, and see that similarly is possible to obtain balanced approximately Chern–Ricci flat metrics.

The twisted Selberg trace formula and the twisted Selberg zeta function for compact orbifolds

We propose a version of the Selberg trace formula for compact hyperbolic orbifolds Gamma backslash {mathbb {H}}^{2n+1} for non-unitary representations of Gamma and establish that the associated Selberg zeta function admits a meromorphic continuation to {mathbb {C}}.

Energy bounds for vertex operator algebra extensions

Let V be a simple unitary vertex operator algebra and U be a (polynomially) energy-bounded unitary subalgebra containing the conformal vector of V. We give two sufficient conditions implying that V is energy-bounded. The first condition is that U is a compact orbifold V^G for some compact group G of unitary automorphisms of V. The second condition is that V is exponentially energy-bounded and it is a finite direct sum of simple U-modules. As a consequence of the second condition, we prove that if U is a regular energy-bounded unitary subalgebra of a simple unitary vertex operator V, then V is energy-bounded. In particular, every simple unitary extension (with the same conformal vector) of a simple unitary affine vertex operator algebra associated with a semisimple Lie algebra is energy-bounded.

The fate of discrete torsion on resolved heterotic [formula omitted] orbifolds using (0,2) GLSMs

This paper aims to shed light on what becomes of discrete torsion within heterotic orbifolds when they are resolved to smooth geometries. Gauged Linear Sigma Models (GLSMs) possessing (0,2) worldsheet supersymmetry are employed as interpolations between them. This question is addressed for resolutions of the non–compact C3/Z2×Z2 and the compact T6/Z2×Z2 orbifolds to keep track of local and global aspects. The GLSMs associated with the non–compact orbifold with or without torsion are to a large extent equivalent: only when expressed in the same superfield basis, a field redefinition anomaly arises among them, which in the orbifold limit reproduces the discrete torsion phases. Previously unknown, novel resolution GLSMs for T6/Z2×Z2 are constructed. The GLSM associated with the torsional compact orbifold suffers from mixed gauge anomalies, which need to be cancelled by appropriate logarithmic superfield dependent FI–terms on the worldsheet, signalling H–flux due to NS5–branes supported at the exceptional cycles.

How highly connected can an orbifold be?

On the one hand, we provide the first examples of arbitrarily highly connected (compact) bad orbifolds. On the other hand, we show that n -connected n -orbifolds are manifolds. The latter improves the best previously known bound of Lytchak by roughly a factor of 2. For compact orbifolds and in most dimensions, we prove slightly better bounds. We obtain sharp results up to dimension 5.

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.

Flat vector bundles and analytic torsion on orbifolds

This article is devoted to a study of flat orbifold vector bundles. We construct a bijection between the isomorphic classes of proper flat orbifold vector bundles and the equivalence classes of representations of the orbifold fundamental groups of base orbifolds. We establish a Bismut-Zhang like anomaly formula for the Ray-Singer metric on the determine line of the cohomology of a compact orbifold with coefficients in an orbifold flat vector bundle. We show that the analytic torsion of an acyclic unitary flat orbifold vector bundle is equal to the value at zero of a dynamical zeta function when the underlying orbifold is a compact locally symmetric space of the reductive type, which extends one of the results obtained by the first author for compact locally symmetric manifolds.

Bottom of spectra and coverings of orbifolds

We discuss the behavior of the bottom of the spectrum of scalar Schrödinger operators under Riemannian coverings of orbifolds. We apply our results to geometrically finite and to conformally compact orbifolds.

The Poincaré problem for foliations on compact toric orbifolds

The Poincaré problem for foliations on compact toric orbifolds

Equivariant eta forms and equivariant differential K-theory

In this paper, for a compact Lie group action, we prove the anomaly formula and the functoriality of the equivariant Bismut-Cheeger eta forms with perturbation operators when the equivariant family index vanishes. In order to prove them, we extend the Melrose-Piazza spectral section and its main properties to the equivariant case and introduce the equivariant version of the Dai-Zhang higher spectral flow for arbitrary-dimensional fibers. Using these results, we construct a new analytic model of the equivariant differential K-theory for compact manifolds when the group action has finite stabilizers only, which modifies the Bunke-Schick model of the differential K-theory. This model could also be regarded as an analytic model of the differential K-theory for compact orbifolds. Especially, we answer a question proposed by Bunke and Schick (2009) about the well-definedness of the push-forward map.

On Morse–Smale–Witten type cohomologies for orbifolds

Let X:=[M/G] be a compact orbifold given by the quotient of a connected compact Lie group G action on an oriented smooth closed manifold M, where the action is assumed to be smooth, effective, locally free and orientation preserving. Let f¯:X→R be a Morse function which lifts to a G-invariant Morse–Bott function f on M. We construct an explicit isomorphism between the Morse–Bott cohomology of f and the orbifold Morse cohomology of f¯ by constructing a new filtered cochain complex out of basic forms to connect the Morse–Bott cochain complex of f and the orbifold Morse–Smale–Witten cochain complex of f¯.

Toric construction and Chow ring of moduli space of quasi maps from ℙ1 with two marked points to ℙ1 × ℙ1

In this paper, we present explicit toric construction of moduli space of quasi maps from [Formula: see text] with two marked points to [Formula: see text], which was first proposed by Jinzenji and prove that it is a compact orbifold. We also determine its Chow ring and compute its Poincaré polynomial for some lower degree cases.

Counting associatives in compact G2 orbifolds

We describe a class of compact G2 orbifolds constructed from non-symplectic involutions of K3 surfaces. Within this class, we identify a model for which there are infinitely many associative submanifolds contributing to the effective superpotential of M-theory compactifications. Under a chain of dualities, these can be mapped to F-theory on a Calabi-Yau fourfold, and we find that they are dual to an example studied by Donagi, Grassi and Witten. Finally, we give two different descriptions of our main example and the associative submanifolds as a twisted connected sum.

Automorphism groups of elliptic G-structures on orbifolds

We present a new method of investigation of G -structures on orbifolds. This method is founded on the use of appropriate G -structure on an associated model manifold. Applying this method we prove the existence and the uniqueness of finite dimensional Lie group structures on the full automorphism group of an elliptic G -structure on a compact orbifold. This generalizes the well-known theorem for elliptic G -structures on compact manifolds. As an application we get that the group of all automorphisms of an almost complex structure on a compact orbifold is a finite dimensional Lie group that carries a uniquely defined Lie group structure.

On symplectic vortex equations over a compact orbifold Riemann surface

Making use of theory of differentiable stacks, we study symplectic vortex equations over a compact orbifold Riemann surface. We discuss the category of representable morphisms from a compact orbifold Riemann surface to a quotient stack. After that we define symplectic vortex equations over a compact orbifold Riemann surface. We also discuss the moduli space of solutions to the equations for linear actions of the circle group on the complex plane.

On noncommutative geometry of orbifolds

An orbifold is a Morita equivalence class of a proper étale Lie groupoid. A unitary equivalence class of spectral triples over the algebra of smooth invariant functions is associated with any compact spin orbifold. In the case of an effective spin orbifold, we construct a collection of spectral triples over the smooth convolution algebras of the representatives of the Morita equivalence class.

A virtual Kawasaki–Riemann–Roch formula

Kawasaki's formula is a tool to compute holomorphic Euler characteristics of vector bundles on a compact orbifold X. Let X be an orbispace with perfect obstruction theory which admits an embedding in a smooth orbifold. One can then construct the virtual structure sheaf and the virtual fundamental class of X. In this paper we prove that Kawasaki's formula behaves well " with working virtually" on X in the following sense: if we replace the structure sheaves, tangent and normal bundles in the formula by their virtual counterparts then Kawasaki's formula stays true. Our motivation comes from studying the quantum K-theory of a complex manifold X, with the formula applied to Kontsevich' moduli spaces of genus 0 stable maps to X.

On the isometry group of a compact flat orbifold

Let \(\Gamma \) be an \(n\)-dimensional crystallographic group. We prove that the group \(\mathrm{Isom}(E^n/\Gamma )\) of isometries of the flat orbifold \(E^n/\Gamma \) is a compact Lie group whose component of the identity is a torus of dimension equal to the first Betti number of the group \(\Gamma \). This implies that \(\mathrm{Isom}(E^n/\Gamma )\) is finite if and only if \(\Gamma /[\Gamma , \Gamma ]\) is finite. We also generalize known results on the Nielsen realization problem for torsion-free \(\Gamma \) to arbitrary \(\Gamma \).

Asymptotics of orbifold Bergman kernels: mixed curvature case

We derive some asymptotic expansion formulas of the Bergman kernel of high tensor powers of an Hermitian orbifold line bundle with mixed curvature tensored with an orbifold vector bundle on a compact symplectic orbifold. In particular, when the orbifold has isolated singularities, we get an explicit formula for the asymptotic expansion of the Bergman kernel in the distribution sense. Finally, by applying our results to the complex case, we get a Riemann–Roch–Kawasaki type formula.

Closed geodesics on orbifolds of nonpositive or nonnegative curvature

In this note we prove existence of closed geodesics of positive length on compact developable orbifolds of nonpositive or nonnegative curvature. We also include a geometric proof of existence of closed geodesics whenever the orbifold fundamental group contains a hyperbolic element and therefore reduce the existence problem to developable orbifolds with \(\pi _1^{orb}\) infinite and having finite exponent and finitely many conjugacy classes.