Search

For an n-TET tuning system, we propose a formalism to study the transformations of k-chords over a generalized non-degenerate Tonnetz generated by a given interval structure. Root and mode are the two components of a directed chord on which the algebra operates, so that chord transformations within one chord cell or towards other cells, and paths or simple circuits over the chord network can be determined without resorting to computational algorithms or geometrical representations. The one-step transformations over the edges of the chord network associated with the $ k{-}1 $ k − 1 drift operators generalize the basic operators P, R and L of the Neo-Riemmanian triadic progressions and the maximally smooth cycles of the 12-TET system to any higher dimensional space. [ABSTRACT FROM AUTHOR]

Motivated by symplectic geometry, we give a detailed account of differential forms and currents on orbifolds with corners, the pull-back and push-forward operations, and their fundamental properties. We work within the formalism where the category of orbifolds with corners is obtained as a localization of the category of étale proper groupoids with corners. Constructions and proofs are formulated in terms of the structure maps of the groupoids, avoiding the use of orbifold charts. The Fréchet space of differential forms on an orbifold and the dual space of currents are shown to be independent of which étale proper groupoid is chosen to represent the orbifold. [ABSTRACT FROM AUTHOR]

We apply the Berglund–Hübsch transpose rule from BHK mirror symmetry to show that to an n - 1 -dimensional Calabi–Yau orbifold in weighted projective space defined by an invertible polynomial, we can associate four (possibly) distinct Sasaki manifolds of dimension 2 n + 1 which are n - 1 -connected and admit a metric of positive Ricci curvature. We apply this theorem to show that for a given K3 orbifold, there exist four seven-dimensional Sasakian manifolds of positive Ricci curvature, two of which are actually Sasaki–Einstein. [ABSTRACT FROM AUTHOR]

We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamiltonian group of the two-sphere equipped with Hofer's metric, prove constraints on Lagrangian packing, find instances of Lagrangian Poincaré recurrence, and present a new hierarchy of normal subgroups of area-preserving homeomorphisms of the two-sphere. The technology involves Lagrangian spectral invariants with Hamiltonian term in symmetric product orbifolds. [ABSTRACT FROM AUTHOR]

We calculate the Chern-Simons invariants of the hyperbolic double twist knot orbifolds using the Schläfli formula for the generalized Chern-Simons function on the family of cone-manifold structures of double twist knots.

The universal two-parameter W ∞ -algebra is a classifying object for vertex algebras of type W (2 , 3 , ... , N) for some N. Gaiotto and Rapčák recently introduced a large family of such vertex algebras called Y -algebras, which includes many known examples such as the principal W -algebras of type A. These algebras admit an action of Z 2 , and in this paper we study the structure of their orbifolds. Aside from the extremal cases of either the Virasoro algebra or the W 3 -algebra, we show that these orbifolds are generated by a single field in conformal weight 4, and we give strong finite generating sets. [ABSTRACT FROM AUTHOR]

We extend the notion of Hitchin component from surface groups to orbifold groups and prove that this gives new examples of higher Teichmüller spaces. We show that the Hitchin component of an orbifold group is homeomorphic to an open ball and we compute its dimension explicitly. We then give applications to the study of the pressure metric, cyclic Higgs bundles, and the deformation theory of real projective structures on 3-manifolds. [ABSTRACT FROM AUTHOR]

Our ultimate goal in this paper is to introduce a special type of topological spaces including manifolds and also, orbifolds. Because of using of generalized groups, we call them $GG$-spaces. We will study their properties, and then we will introduce a special $GG$-space that is not manifold and orbifold. Finally we obtain conditions that cause a $GG$-space to become manifold.

The complement of the hyperplanes { x i = x j } , for all i ≠ j , in M n , where M is an aspherical 2-manifold, is known to be aspherical. Here we consider the situation when M is a 2-dimensional orbifold. We prove this complement to be aspherical for a class of aspherical 2-dimensional orbifolds, and predict that it should be true in general also. We generalize this question in the category of Lie groupoids, as orbifolds can be identified with a certain kind of Lie groupoids. [ABSTRACT FROM AUTHOR]

We calculate the Chern-Simons invariants of the hyperbolic double twist knot orbifolds using the Schläfli formula for the generalized ChernSimons function on the family of cone-manifold structures of double twist knots. [ABSTRACT FROM AUTHOR]

Our ultimate goal in this paper is to introduce a special type of topological spaces including manifolds and also, orbifolds. Because of using of generalized groups, we call them GG-spaces. We will study their properties, and then we will introduce a special GG-space that is not manifold and orbifold. Finally we obtain conditions that cause a GG-space to become manifold. [ABSTRACT FROM AUTHOR]

We prove a four-term exact sequence of surface orbifold pure braid groups for all genus ≥1, 2-dimensional orientable orbifolds with cone points. This also corrects and generalizes our earlier result (genus zero case) in [7] and [8]. [ABSTRACT FROM AUTHOR]

We generalize orbifold models for chords and voice leading to incorporate loudness, allowing for the modeling of resting voices, which are used frequently by composers and arrangers across genres. In our generalized setting (strictly speaking, that of orbispaces rather than an orbifolds), passages with resting voices, passages with two or more voices in unison, and fully harmonized passages occupy distinct subspaces that interact in mathematically precise and musically interesting ways. In particular, our setting includes previous orbifold models by way of constant-loudness subspaces, and provides a way to model voice leading between chords of different cardinalities. We model voice leading in this general setting by morphisms in the orbispace path groupoid, a category for which we give a formal definition. We demonstrate how to visualize such morphisms as singular braids, and explore how our approach relates to (and is consistent with) selected previous work. [ABSTRACT FROM AUTHOR]

We explicitly express the spectral determinant of Friedrichs Dirichlet Laplacians on the 2-dimensional hyperbolic (Gaussian curvature − 1) cones in terms of the cone angle and the geodesic radius of the boundary. [ABSTRACT FROM AUTHOR]

We develop the Gompf fiber connected sum operation for symplectic orbifolds. We use it to construct a symplectic 4-orbifold with b 1 = 0 {b_{1}=0} and containing symplectic surfaces of genus 1 and 2 that are disjoint and span the rational homology. This is used in turn to construct a K-contact Smale–Barden manifold with specified 2-homology that satisfies the known topological constraints with sharper estimates than the examples constructed previously. The manifold can be chosen spin or non-spin. [ABSTRACT FROM AUTHOR]

Due to the negative curvature of the hyperbolic plane, we may find several types of groups of symmetries of patterns built on such a plane, which implies in the existence of infinitely many possible families of tessellations. By using geometrical tools, combinatorial theory, graph theory, and the orbifold concept, we determine the families of uniform tessellations of the hyperbolic plane with valences 5, 6 and 7. We also analyze some families of tessellations for general valences. [ABSTRACT FROM AUTHOR]

We prove a criterion of an Anosov action of the isometry group for compact Lorentzian -orbifolds. It is proved also that a non-compact complete flat Lorentzian -orbifold has an Anosov isometry if and only if its isometry group Lie acts improperly. The existence of chaotic behavior of such Anosov actions is investigated. It is shown that among smooth -orbifolds other than manifolds, only the "pillow" and the -cone admit the specified Lorentz metric with an Anosov action of the isometry group. The corresponding Lorentzian metrics and groups of isometries are indicated. [ABSTRACT FROM AUTHOR]

We construct a simply-connected compact complex non-Kähler manifold satisfying the ∂ ̅∂ -Lemma, and endowed with a balanced metric. To this aim, we were initially aimed at investigating the stability of the property of satisfying the ∂ ̅∂-Lemma under modifications of compact complex manifolds and orbifolds. This question has been recently addressed and answered in [34, 39, 40, 50] with different techniques. Here, we provide a different approach using Čech cohomology theory to study the Dolbeault cohomology of the blowup ̃XZ of a compact complex manifold X along a submanifold Z admitting a holomorphically contractible neighbourhood.

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. [ABSTRACT FROM AUTHOR]

It is shown that on every closed oriented Riemannian 4-manifold (M, g) with positive scalar curvature, ∫ M | W g + | 2 d μ g ≥ 2 π 2 (2 χ (M) + 3 τ (M)) - 8 π 2 | π 1 (M) | , where W g + , χ (M) and τ (M) , respectively, denote the self-dual Weyl tensor of g, the Euler characteristic and the signature of M. This generalizes Gursky's inequality [15] for the case of b 1 (M) > 0 in a much simpler way. We also extend all such lower bounds of the Weyl functional to 4-orbifolds including Gursky's inequalities for the case of b 2 + (M) > 0 or δ g W g + = 0 and obtain topological obstructions to the existence of self-dual orbifold metrics of positive scalar curvature. [ABSTRACT FROM AUTHOR]

A group of matrices with entries in a number field K is integral if it has a finite index subgroup of matrices whose entries are algebraic integers. In this paper, we show that the fundamental groups, G (p / q , (m , n)) , of the orbifolds (p / q , (m , n) are integral as subgroups of both S L (2 , ℂ) and of S L (4 , ℝ) , for all the rational knots and links p / q and all the isotropies with m > 2 , n > 2. We obtain the same result for the fundamental group G (m , m , m) of the orbifold B (m , m , m) , m > 2 , where B are the Borromean rings. The only groups G (m , n , p) with m , n , p ∈ 3 , 4 , 5 , 6 , ∞ , which are integral subgroups of both S L (2 , ℂ) and of S L (4 , ℝ) , are for the following (m , n , p) : { (3 , 3 , 3) , (3 , 3 , ∞) , (3 , 4 , 4) , (3 , 4 , ∞) , (3 , 6 , 6) , (3 , ∞ , ∞) , (4 , 4 , 4) , (4 , 4 , ∞) , (4 , ∞ , ∞) , (6 , 6 , 6) , (6 , 6 , ∞) , (∞ , ∞ , ∞) , (3 , 3 , 5) , (3 , 5 , 5) , (3 , 5 , ∞) , (4 , 4 , 5) , (4 , 5 , ∞) , (5 , 5 , 5) , (5 , 5 , ∞) , (5 , 6 , 6) , (5 , ∞ , ∞) }. [ABSTRACT FROM AUTHOR]

For a closed Riemannian orbifold O, we compare the spectra of the Laplacian, acting on functions or differential forms, to the Neumann spectra of the orbifold with boundary given by a domain U in O whose boundary is a smooth manifold. Generalizing results of several authors, we prove that the metric of O can be perturbed to ensure that the first N eigenvalues of U and O are arbitrarily close to one another. This involves a generalization of the Hodge decomposition to the case of orbifolds with manifold boundary. Using these results, we study the behavior of the Laplace spectrum on functions or forms of a connected sum of two Riemannian orbifolds as one orbifold in the pair is collapsed to a point. We show that the limits of the eigenvalues of the connected sum are equal to those of the noncollapsed orbifold in the pair. In doing so, we prove the existence of a sequence of orbifolds with singular points whose eigenvalue spectra come arbitrarily close to the spectrum of a manifold, and a sequence of manifolds whose eigenvalue spectra come arbitrarily close to the eigenvalue spectrum of an orbifold with singular points. We also consider the question of prescribing the first part of the spectrum of an orientable orbifold. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The theory of twistors on foliated manifolds is developed. We construct the twistor space of the normal bundle of a foliation. It is demonstrated that the classical constructions of the twistor theory lead to foliated objects and permit to formulate and prove foliated versions of some well-known results on holomorphic mappings. Since any orbifold can be understood as the leaf space of a suitably defined Riemannian foliation we obtain orbifold versions of the classical results as a simple consequence of the results on foliated mappings. [ABSTRACT FROM AUTHOR]

Parsimonious transformations are common patterns in different musical styles and eras. In some cases, they can be represented on the Tonnetz, Cube Dance, Power Towers, or the central region of an orbifold, mainly when they only include the most even trichords and tetrachords. In this paper, two novel graphs, called Cyclopes, are presented, which include more than double the number of chord types in previously published graphs, thus allowing to represent a larger musical repertoire in a practical way. Apart from parsimonious transformations, they are also especially suitable for representing trichords a major third apart, tetrachords a minor third apart, and the cadences V7–I(m) and II Ø –V7–I(m) with major or minor tonic chords. Therefore, they allow to clearly visualize the relationship among the corresponding chords and better understand those patterns, as well as being efficient mnemonic resources, all of which make them useful tools both for music analysis and composition. [ABSTRACT FROM AUTHOR]

We generalize orbifold models for chords and voice leading to incorporate loudness, allowing for the modeling of resting voices, which are used frequently by composers and arrangers across genres. In our generalized setting (strictly speaking, that of orbispaces rather than an orbifolds), passages with resting voices, passages with two or more voices in unison, and fully harmonized passages occupy distinct subspaces that interact in mathematically precise and musically interesting ways. In particular, our setting includes previous orbifold models by way of constant-loudness subspaces, and provides a way to model voice leading between chords of different cardinalities. We model voice leading in this general setting by morphisms in the orbispace path groupoid, a category for which we give a formal definition. We demonstrate how to visualize such morphisms as singular braids, and explore how our approach relates to (and is consistent with) selected previous work.

We construct a 2-category version of tom Dieck's equivariant fundamental groupoid for representable orbifolds and show that the discrete fundamental groupoid is Morita invariant; hence it is an orbifold invariant for representable orbifolds. [ABSTRACT FROM AUTHOR]

We give a concise overview of the classification theory of symplectic manifolds equipped with torus actions for which the orbits are symplectic (this is equivalent to the existence of a symplectic principal orbit), and apply this theory to study the structure of the leaf space induced by the action. In particular we show that if M is a symplectic manifold on which a torus T acts effectively with symplectic orbits, then the leaf space M/T is a very good orbifold with first Betti number b1(M/T)=b1(M)−dim T.

Alexandrov spaces are complete length spaces with a lower curvature bound in the triangle comparison sense. When they are equipped with an effective isometric action of a compact Lie group with one-dimensional orbit space, they are said to be of cohomogeneity one. Well-known examples include cohomogeneity-one Riemannian manifolds with a uniform lower sectional curvature bound; such spaces are of interest in the context of non-negative and positive sectional curvature. In the present article we classify closed, simply connected cohomogeneity-one Alexandrov spaces in dimensions 5, 6 and 7. This yields, in combination with previous results for manifolds and Alexandrov spaces, a complete classification of closed, simply connected cohomogeneity-one Alexandrov spaces in dimensions at most 7. [ABSTRACT FROM AUTHOR]

The moduli space of the maximally supersymmetric heterotic string in d -dimensional Minkowski space contains various components characterized by the rank of the gauge symmetries of the vacua they parametrize. We develop an approach for describing in a unified way continuous Wilson lines which parametrize a component of the moduli space, together with discrete deformations responsible for the switch from one component to the other. Applied to a component that contains vacua with S O (2 n + 1) gauge-symmetry factors, our approach yields a description of all backgrounds of the component in terms of free-orbifold models. The orbifold generators turn out to act symmetrically or asymmetrically on the internal space, with or without discrete torsion. Our derivations use extensively affine characters of S O (2 n + 1). As a by-product, we find a peculiar orbifold description of the heterotic string in 10 dimensions, where all gauge degrees of freedom arise as twisted states, while the untwisted sector reduces to the gravitational degrees of freedom. [ABSTRACT FROM AUTHOR]

We construct the symplectic resolution of a symplectic orbifold whose isotropy locus consists of disjoint submanifolds with homogeneous isotropy, that is, all its points have the same isotropy groups. [ABSTRACT FROM AUTHOR]

We consider the deformations of complex orbifolds with the underlying smooth structures being fixed. As a corollary, we can prove that the deformations of a Calabi-Yau orbifold is unobstructed by using standard arguments. Then we consider the period map for a family of complex Kähler orbifolds. We prove that the period map is holomorphic, horizontal and consistent with our Kodaira-Spencer map. [ABSTRACT FROM AUTHOR]

We study Z2 orbifolds of M-theory in terms of E10. We find a simple relation between the Z2 action on E10 and the imaginary root that corresponds [hep-th/0401053] to the "twisted sector" branes. We discuss the connection between the Kac-Moody algebra DE10 and the "untwisted" sector, and we demonstrate how DE18 can describe both the untwisted and twisted sectors simultaneously.

Let X be a finite connected graph, possibly with multiple edges. We provide each edge of the graph by two possible orientations. An automorphism group of a graph acts harmonically if it acts freely on the set of directed edges of the graph. Following M. Baker and S. Norine define a genus g of the graph X to be the rank of the first homology group. A finite group acting harmonically on a graph of genus g is a natural discrete analogue of a finite group of automorphisms acting on a Riemann surface of genus g. In the present paper, we give a sharp upper bound for the size of cyclic group acting harmonically on a graph of genus g ≥ 2 with a given number of fixed points. Similar results, for closed orientable surfaces, were obtained earlier by T. Szemberg, I. Farkas and H. M. Kra. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a symplectic manifold of bounded geometry. First, we establish the off-diagonal exponential estimate for the generalized Bergman kernel. As an application, we obtain the relation between the generalized Bergman kernel on a Galois covering of a compact symplectic manifold and the generalized Bergman kernel on the base. Then we state the full off-diagonal asymptotic expansion of the generalized Bergman kernel, improving the remainder estimate known in the compact case to an exponential decay. Finally, we establish the theory of Berezin-Toeplitz quantization on symplectic orbifolds associated with the renormalized Bochner-Laplacian. [ABSTRACT FROM AUTHOR]