The 3-permutation orbifold of a lattice vertex operator algebra

Irreducible modules of the 3-permutation orbifold of a rank one lattice vertex operator algebra are listed explicitly. Fusion rules are determined by using the quantum dimensions. The S-matrix is also given.     

The topological dimension of limits of vertex replacements

Given an initial graph G, one may apply a rule R to G which replaces certain vertices of G with other graphs called replacement graphs to obtain a new graph R(G). By iterating this procedure, a sequence of graphs {Rn(G)} is obtained. When each graph in this sequence is normalized to have diameter one, the resulting sequence may converge in the Gromov-Hausdorff metric. In this paper, we compute the topological dimension of limit spaces of normalized sequences of iterated vertex replacements involving more than one replacement graph. We also give examples of vertex replacement rules that yield fractals.     

The spectral function of a Riemannian orbifold

Hörmander’s theorem on the asymptotics of the spectral function of an elliptic operator is extended to the setting of compact Riemannian orbifolds. In contrast to the manifold case, the asymptotics depend on the isotropy type of the point at which the spectral function is computed. It is shown that “on average” the eigenfunctions of the operator are larger at singular points than at manifold points, by a factor of the order of the isotropy type. A sketch of a more direct approach to the wave trace formula on orbifolds is also given, obtaining results already shown separately by M. Sandoval and Y. Kordyukov in the setting of Riemmannian foliations.     

The loop orbifold of the symmetric product

Using the loop orbifold of the symmetric product, we give a formula for the Poincaré polynomial of the free loop space of the Borel construction of the symmetric product. We also show that the Chas-Sullivan orbifold product structure in the homology of the free loop space of the Borel construction of the symmetric product induces a ring structure in the homology of the inertia orbifold of the symmetric product. For a general almost complex orbifold, we define a new ring structure on the cohomology of its inertia orbifold which we call the virtual intersection ring. Finally we show that under Poincaré duality in the case of the symmetric product orbifold, both ring structures are isomorphic.     

Representations of orbifold groupoids

Orbifold groupoids have been recently widely used to represent both effective and ineffective orbifolds. We show that every orbifold groupoid can be faithfully represented on a continuous family of finite dimensional Hilbert spaces. As a consequence we obtain the result that every orbifold groupoid is Morita equivalent to the translation groupoid of an almost free action of a proper bundle of topological groups.     

The quantum orbifold cohomology of weighted projective spaces

We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S ¹-equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small J-function, a generating function for certain genus-zero Gromov–Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first non-trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small J-functions of weighted projective complete intersections satisfying a combinatorial condition; this condition naturally singles out the class of orbifolds with terminal singularities.     

The orbifold Chow ring of toric Deligne-Mumford stacks

Generalizing toric varieties, we introduce toric Deligne-Mumford stacks. The main result in this paper is an explicit calculation of the orbifold Chow ring of a toric Deligne-Mumford stack. As an application, we prove that the orbifold Chow ring of the toric Deligne-Mumford stack associated to a simplicial toric variety is a flat deformation of (but is not necessarily isomorphic to) the Chow ring of a crepant resolution.

     

McKay correspondence and orbifold equivalence

We prove that a pair of singularities related by a transformation arising from the McKay correspondence are orbifold equivalent. From this we deduce a McKay type category equivalence for the matrix factorization categories.     

Riemann surfaces with orbifold points

We interpret the previously developed Teichmüller theory of surfaces with marked points on boundary components (bordered surfaces) as the Teichmüller theory of Riemann surfaces with orbifold points of order 2. In the Poincaré uniformization pattern, we describe necessary and sufficient conditions for the group generated by the Fuchsian group of the surface with added inversions to be of the almost hyperbolic Fuchsian type. All the techniques elaborated for the bordered surfaces (quantization, classical and quantum mapping-class group transformations, and Poisson and quantum algebra of geodesic functions) are equally applicable to the surfaces with orbifold points.     

The orbifold cohomology of moduli of genus 3 curves

In this work we study the additive orbifold cohomology of the moduli stack of smooth genus g curves. We show that this problem reduces to investigating the rational cohomology of moduli spaces of cyclic covers of curves where the genus of the covering curve is g. Then we work out the case of genus g =  3. Furthermore, we determine the part of the orbifold cohomology of the Deligne–Mumford compactification of the moduli space of genus 3 curves that comes from the Zariski closure of the inertia stack of ${\mathcal{M}3}$ .     

Moduli of twisted orbifold sheaves

We study stacks of slope-semistable twisted sheaves on orbisurfaces with projective coarse spaces and prove that in certain cases they have many of the asymptotic properties enjoyed by the moduli of slope-semistable sheaves on smooth projective surfaces.     

Hodge structures for orbifold cohomology

We construct a polarized Hodge structure on the primitive part of Chen and Ruan's orbifold cohomology for projective -orbifolds satisfying a ``Hard Lefschetz Condition'. Furthermore, the total cohomology forms a mixed Hodge structure that is polarized by every element of the Kähler cone of . Using results of Cattani-Kaplan-Schmid this implies the existence of an abstract polarized variation of Hodge structure on the complexified Kähler cone of .

This construction should be considered as the analogue of the abstract polarized variation of Hodge structure that can be attached to the singular cohomology of a crepant resolution of , in light of the conjectural correspondence between the (quantum) orbifold cohomology and the (quantum) cohomology of a crepant resolution.

     

Computations of the orbifold Yamabe invariant

We consider the Yamabe invariant of a compact orbifold with finitely many singular points. We prove a fundamental inequality for the estimate of the invariant from above, which also includes a criterion for the non-positivity of it. Moreover, we give a sufficient condition for the equality in the inequality. In order to prove it, we also solve the orbifold Yamabe problem under a certain condition. We use these results to give some exact computations of the Yamabe invariant of compact orbifolds.     

Gerbes and twisted orbifold quantum cohomology

In this paper,we construct an orbifold quantum cohomology twisted by a flat gerbe. Then we compute these invariants in the case of a smooth manifold and a discrete torsion on a global quotient orbifold.     

One cannot hear orbifold isotropy type

We show that the isotropy types of the singularities of Riemannian orbifolds are not determined by the Laplace spectrum. Indeed, we construct arbitrarily large families of mutually isospectral orbifolds with different isotropy types. Finally, we show that the corresponding singular strata of two isospectral orbifolds may not be homeomorphic. Received: 6 October 2005     

The mixed vertex packing problem

We study a generalization of the vertex packing problem having both binary and bounded continuous variables, called the mixed vertex packing problem (MVPP). The well-known vertex packing model arises as a subproblem or relaxation of many 0-1 integer problems, whereas the mixed vertex packing model arises as a natural counterpart of vertex packing in the context of mixed 0-1 integer programming. We describe strong valid inequalities for the convex hull of solutions to the MVPP and separation algorithms for these inequalities. We give a summary of computational results with a branch-and-cut algorithm for solving the MVPP and using it to solve general mixed-integer problems. Received: June 1998 / Accepted: February 2000?Published online September 20, 2000     

Weighted Ehrhart theory and orbifold cohomology

We introduce the notion of a weighted δ-vector of a lattice polytope. Although the definition is motivated by motivic integration, we study weighted δ-vectors from a combinatorial perspective. We present a version of Ehrhart Reciprocity and prove a change of variables formula. We deduce a new geometric interpretation of the coefficients of the Ehrhart δ-vector. More specifically, they are sums of dimensions of orbifold cohomology groups of a toric stack.