Showing papers on "Symmetry (geometry) published in 2016"

Topological nodal line semimetals

Abstract: We review the recent, mainly theoretical, progress in the study of topological nodal line semimetals in three dimensions. In these semimetals, the conduction and the valence bands cross each other along a one-dimensional curve in the three-dimensional Brillouin zone, and any perturbation that preserves a certain symmetry group (generated by either spatial symmetries or time-reversal symmetry) cannot remove this crossing line and open a full direct gap between the two bands. The nodal line(s) is hence topologically protected by the symmetry group, and can be associated with a topological invariant. In this review, (i) we enumerate the symmetry groups that may protect a topological nodal line; (ii) we write down the explicit form of the topological invariant for each of these symmetry groups in terms of the wave functions on the Fermi surface, establishing a topological classification; (iii) for certain classes, we review the proposals for the realization of these semimetals in real materials; (iv) we discuss different scenarios that when the protecting symmetry is broken, how a topological nodal line semimetal becomes Weyl semimetals, Dirac semimetals, and other topological phases; and (v) we discuss the possible physical effects accessible to experimental probes in these materials.

Coexistence of Weyl fermion and massless triply degenerate nodal points

Abstract: By using first-principles calculations, we propose that WC-type ZrTe is a new type of topological semimetal (TSM). It has six pairs of chiral Weyl nodes in its first Brillouin zone, but it is distinguished from other existing TSMs by having an additional two paris of massless fermions with triply degenerate nodal points as proposed in the isostructural compounds TaN and NbN. The mirror symmetry, threefold rotational symmetry, and time-reversal symmetry require all of the Weyl nodes to have the same velocity vectors and locate at the same energy level. The Fermi arcs on different surfaces are shown, which may be measured by future experiments. It demonstrates that the ``material universe'' can support more intriguing particles simultaneously.

Topological phases protected by point group symmetry

Abstract: We consider symmetry protected topological (SPT) phases with crystalline point group symmetry, dubbed point group SPT (pgSPT) phases. We show that such phases can be understood in terms of lower-dimensional topological phases with on-site symmetry, and can be constructed as stacks and arrays of these lower-dimensional states. This provides the basis for a general framework to classify and characterize bosonic and fermionic pgSPT phases, that can be applied for arbitrary crystalline point group symmetry and in arbitrary spatial dimension. We develop and illustrate this framework by means of a few examples, focusing on three-dimensional states. We classify bosonic pgSPT phases and fermionic topological crystalline superconductors with $Z_2^P$ (reflection) symmetry, electronic topological crystalline insulators (TCIs) with ${\rm U}(1) \times {Z}_2^P$ symmetry, and bosonic pgSPT phases with $C_{2v}$ symmetry, which is generated by two perpendicular mirror reflections. We also study surface properties, with a focus on gapped, topologically ordered surface states. For electronic TCIs we find a $Z_8 \times Z_2$ classification, where the $Z_8$ corresponds to known states obtained from non-interacting electrons, and the $Z_2$ corresponds to a "strongly correlated" TCI that requires strong interactions in the bulk. Our approach may also point the way toward a general theory of symmetry enriched topological (SET) phases with crystalline point group symmetry.

Symmetry fractionalization and twist defects

Abstract: Topological order in two-dimensions can be described in terms of deconfined quasiparticle excitations—anyons—and their braiding statistics. However, it has recently been realized that this data does not completely describe the situation in the presence of an unbroken global symmetry. In this case, there can be multiple distinct quantum phases with the same anyons and statistics, but with different patterns of symmetry fractionalization—termed symmetry enriched topological order. When the global symmetry group G, which we take to be discrete, does not change topological superselection sectors—i.e. does not change one type of anyon into a different type of anyon—one can imagine a local version of the action of G around each anyon. This leads to projective representations and a group cohomology description of symmetry fractionalization, with the second cohomology group being the relevant group. In this paper, we treat the general case of a symmetry group G possibly permuting anyon types. We show that despite the lack of a local action of G, one can still make sense of a so-called twisted group cohomology description of symmetry fractionalization, and show how this data is encoded in the associativity of fusion rules of the extrinsic 'twist' defects of the symmetry. Furthermore, building on work of Hermele (2014 Phys. Rev. B 90 184418), we construct a wide class of exactly-solvable models which exhibit this twisted symmetry fractionalization, and connect them to our formal framework.

Geometry of the generalized Bloch sphere for qutrits

Abstract: The geometry of the generalized Bloch sphere Ω3, the state space of a qutrit, is studied. Closed form expressions for Ω3, its boundary ∂Ω3, and the set of extremals are obtained by use of an elementary observation. These expressions and analytic methods are used to classify the 28 two-sections and the 56 three-sections of Ω3 into unitary equivalence classes, completing the works of earlier authors. It is shown, in particular, that there are families of two-sections and of three-sections which are equivalent geometrically but not unitarily, a feature that does not appear to have been appreciated earlier. A family of three-sections of obese-tetrahedral shape whose symmetry corresponds to the 24-element tetrahedral point group T d is examined in detail. This symmetry is traced to the natural reduction of the adjoint representation of SU(3), the symmetry underlying Ω3, into direct sum of the two-dimensional and the two (inequivalent) three-dimensional irreducible representations of T d .

Reflection and time reversal symmetry enriched topological phases of matter: path integrals, non-orientable manifolds, and anomalies

Abstract: We study symmetry-enriched topological (SET) phases in 2+1 space-time dimensions with spatial reflection and/or time-reversal symmetries. We provide a systematic construction of a wide class of reflection and time-reversal SET phases in terms of a topological path integral defined on general space-time manifolds. An important distinguishing feature of different topological phases with reflection and/or time-reversal symmetry is the value of the path integral on non-orientable space-time manifolds. We derive a simple general formula for the path integral on the manifold $\Sigma^2 \times S^1$, where $\Sigma^2$ is a two-dimensional non-orientable surface and $S^1$ is a circle. This also gives an expression for the ground state degeneracy of the SET on the surface $\Sigma^2$ that depends on the reflection symmetry fractionalization class, generalizing the Verlinde formula for ground state degeneracy on orientable surfaces. Consistency of the action of the mapping class group on non-orientable manifolds leads us to a constraint that can detect when a time-reversal or reflection SET phase is anomalous in (2+1)D and, thus, can only exist at the surface of a (3+1)D symmetry protected topological (SPT) state. Given a (2+1)D reflection and/or time-reversal SET phase, we further derive a general formula that determines which (3+1)D reflection and/or time-reversal SPT phase hosts the (2+1)D SET phase as its surface termination. A number of explicit examples are studied in detail.

Three-dimensional photonic Dirac points stabilized by point group symmetry

Abstract: We discover a pair of stable three-dimensional (3D) Dirac points, a 3D photonic analog of graphene, in all-dielectric photonic crystals using structures commensurate with nanofabrication for visible-frequency photonic applications. The Dirac points carry nontrivial ${Z}_{2}$ topology and emerge for a large range of material parameters in hollow cylinder hexagonal photonic crystals. From Kramers theorem and group theory, we find that only the ${C}_{6}$ symmetry leads to point group symmetry stabilized Dirac points in 3D all-dielectric photonic crystals. The Dirac points are characterized using $\stackrel{P\vec}{k}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{P}$ theory for photonic bands in combination with symmetry analysis. Breaking inversion symmetry splits the Dirac points into Weyl points. The physical properties and experimental consequences of Dirac points are also studied. The Dirac points are found to be robust against parameter tuning and weak disorders.

Permutation Symmetry Determines the Discrete Wigner Function.

Abstract: The Wigner function provides a useful quasiprobability representation of quantum mechanics, with applications in various branches of physics. Many nice properties of the Wigner function are intimately connected with the high symmetry of the underlying operator basis composed of phase point operators: any pair of phase point operators can be transformed to any other pair by a unitary symmetry transformation. We prove that, in the discrete scenario, this permutation symmetry is equivalent to the symmetry group being a unitary 2 design. Such a highly symmetric representation can only appear in odd prime power dimensions besides dimensions 2 and 8. It suffices to single out a unique discrete Wigner function among all possible quasiprobability representations. In the course of our study, we show that this discrete Wigner function is uniquely determined by Clifford covariance, while no Wigner function is Clifford covariant in any even prime power dimension.

PSyCo: Manifold Span Reduction for Super Resolution

Abstract: The main challenge in Super Resolution (SR) is to discover the mapping between the low-and high-resolution manifolds of image patches, a complex ill-posed problem which has recently been addressed through piecewise linear regression with promising results. In this paper we present a novel regression-based SR algorithm that benefits from an extended knowledge of the structure of both manifolds. We propose a transform that collapses the 16 variations induced from the dihedral group of transforms (i.e. rotations, vertical and horizontal reflections) and antipodality (i.e. diametrically opposed points in the unitary sphere) into a single primitive. The key idea of our transform is to study the different dihedral elements as a group of symmetries within the high-dimensional manifold. We obtain the respective set of mirror-symmetry axes by means of a frequency analysis of the dihedral elements, and we use them to collapse the redundant variability through a modified symmetry distance. The experimental validation of our algorithm shows the effectiveness of our approach, which obtains competitive quality with a dictionary of as little as 32 atoms (reducing other methods' dictionaries by at least a factor of 32) and further pushing the state-of-the-art with a 1024 atoms dictionary.

Aesthetic Responses to Exact Fractals Driven by Physical Complexity.

Abstract: Fractals are physically complex due to their repetition of patterns at multiple size scales. Whereas the statistical characteristics of the patterns repeat for fractals found in natural objects, computers can generate patterns that repeat exactly. Are these exact fractals processed differently, visually and aesthetically, than their statistical counterparts? We investigated the human aesthetic response to the complexity of exact fractals by manipulating fractal dimensionality (D), symmetry, recursion, and the number of segments in the generator using a variety of fractal patterns. In the first study, we found that preference ratings for exact midpoint displacement fractals can be described by a linear trend with preference increasing as D increases. For the majority of individuals, preference increased with D. We replicated these results in a second study, while also testing the effects of symmetry and recursion by presenting fractals without mirror symmetry (asymmetric dragon fractals and radially symmetric dragon fractals) and fractals with mirror symmetry (Sierpinski carpets and Koch snowflakes). We found a strong interaction among recursion, symmetry and fractal dimension. Specifically, at low levels of recursion, the presence of symmetry was enough to drive high preference ratings for patterns with moderate to high levels of fractal dimension. Most individuals required a much higher level of recursion to recover this level of preference in asymmetric patterns, while others were less discriminating. Here, what distinguished among respondents was their sensitivity to the presence of mirror symmetry. All participants in the second study exhibited a general increase of preference with D. This suggests that exact fractals are processed differently than their statistical counterparts. We propose a set of four factors that influence complexity and preference judgments in fractals that may extend to other patterns: fractal dimension, recursion, symmetry and the number of segments in a pattern. Conceptualizations such as Berlyne’s (1971) and Redies’ (2015) theories of aesthetics provide a suitable framework for interpretation of our data with respect to the individual differences that we detect. Future studies that incorporate physiological methods to measure the human aesthetic response to exact fractal patterns would further elucidate our responses to such timeless patterns.

Heuristic 3D object shape completion based on symmetry and scene context

Abstract: Object shape information is essential for robot manipulation tasks, in particular for grasp planning and collision-free motion planning. But in general a complete object model is not available, in particular when dealing with unknown objects. We propose a method for completing shapes that are only partially known, which is a common situation when a robot perceives a new object only from one direction. Our approach is based on the assumption that most objects used in service robotic setups have symmetries. We determine and rate symmetry plane candidates to estimate the hidden parts of the object. By finding possible supporting planes based on its immediate neighborhood, the search space for symmetry planes is restricted, and the bottom part of the object is added. Gaps along the sides in the direction of the view axis are closed by linear interpolation. We evaluate our approach with real-world experiments using the YCB object and model set [1].

Visual perception of shape altered by inferred causal history.

Abstract: One of the main functions of vision is to represent object shape. Most theories of shape perception focus exclusively on geometrical computations (e.g., curvatures, symmetries, axis structure). Here, however, we find that shape representations are also profoundly influenced by an object's causal origins: the processes in its past that formed it. Observers placed dots on objects to report their perceived symmetry axes. When objects appeared 'complete'-created entirely by a single generative process-responses closely approximated the object's geometrical axes. However, when objects appeared 'bitten'-as if parts had been removed by a distinct causal process-the responses deviated significantly from the geometrical axes, as if the bitten regions were suppressed from the computation of symmetry. This suppression of bitten regions was also found when observers were not asked about symmetry axes but about the perceived front and back of objects. The findings suggest that visual shape representations are more sophisticated than previously appreciated. Objects are not only parsed according to what features they have, but also to how or why they have those features.

Symmetry of properly embedded special weingarten surfaces in h3

Abstract: In this paper we prove some existence and uniqueness results about special Weingarten surfaces in hyperbolic space.

Reflection Symmetry Detection via Appearance of Structure Descriptor

Abstract: Symmetry in visual data represents repeated patterns or shapes that is easily found in natural and human-made objects. Symmetry pattern on an object works as a salient visual feature attracting human attention and letting the object to be easily recognized. Most existing symmetry detection methods are based on sparsely detected local features describing the appearance of their neighborhood, which have difficulty in capturing object structure mostly supported by edges and contours. In this work, we propose a new reflection symmetry detection method extracting robust 4-dimensional Appearance of Structure descriptors based on a set of outstanding neighbourhood edge segments in multiple scales. Our experimental evaluations on multiple public symmetry detection datasets show promising reflection symmetry detection results on challenging real world and synthetic images.

Using Convolutional Neural Network Filters to Measure Left-Right Mirror Symmetry in Images

Abstract: We propose a method for measuring symmetry in images by using filter responses from Convolutional Neural Networks (CNNs). The aim of the method is to model human perception of left/right symmetry as closely as possible. Using the Convolutional Neural Network (CNN) approach has two main advantages: First, CNN filter responses closely match the responses of neurons in the human visual system; they take information on color, edges and texture into account simultaneously. Second, we can measure higher-order symmetry, which relies not only on color, edges and texture, but also on the shapes and objects that are depicted in images. We validated our algorithm on a dataset of 300 music album covers, which were rated according to their symmetry by 20 human observers, and compared results with those from a previously proposed method. With our method, human perception of symmetry can be predicted with high accuracy. Moreover, we demonstrate that the inclusion of features from higher CNN layers, which encode more abstract image content, increases the performance further. In conclusion, we introduce a model of left/right symmetry that closely models human perception of symmetry in CD album covers.

Interpreting Spatiotemporal Parameters, Symmetry, and Variability in Clinical Gait Analysis

Abstract: Spatiotemporal parameters (STP) are widely studied variables in clinical gait analysis. Yet they often remain underutilized despite the rich information they provide about organization and control of the patient’s progress. Building on them requires a broad knowledge of the “normal” gait, before to being able to understand the impact of pathological disorders. We hope to provide information to better grasp and understand the STP while highlighting important points. Through this chapter, we will introduce basics of the gait cycle, before considering the components for which the STP may be informative: rhythm, pace, phases, postural control, asymmetry, and variability. We will define main parameters for each component and discuss their use regarding state of the art. Then factors influencing STP will be addressed to understand how these parameters change during life, when a child learns to walk or when the advance in age-affected gait in the elderly, as well as the influence of diseases. Indeed, various pathologies affect the walk, and the most relevant STP are not always the same. We will consider Friedreich ataxia, which is a neurodegenerative disease, in which combination of cerebellar, pyramidal syndromes, and axonal neuropathy cause a rapid degeneration of the walking ability and therefore lead to various observable gait patterns. We will also illustrate how PST can be useful to A. Gouelle (*) Gait and Balance Academy, ProtoKinetics, Havertown, PA, USA e-mail: arnaud.gouelle@gmail.com F. Mégrot Unité Clinique d’Analyse de la Marche et du Mouvement, Centre de Médecine Physique et de Réadaptation pour Enfants de Bois-Larris – Croix-Rouge Française, Lamorlaye, France UMR CNRS 7338: Biomécanique et Bioingénierie, Sorbonne Universités, Université de Technologie de Compiègne, Compiègne, France e-mail: fabrice.megrot@croix-rouge.fr # Springer International Publishing AG 2016 B. Müller, S.I. Wolf (eds.), Handbook of Human Motion, DOI 10.1007/978-3-319-30808-1_35-1 1 document the most appropriate time for a patient to change from one assistive device to another. The final portion will aim to give paths for clinical interpretation while thinking about the concepts of limitation and adaptation.

Symmetry and regularity of solutions to the weighted hardy-sobolev type system

Abstract: Abstract Hardy–Littlewood–Sobolev inequalities and the Hardy–Sobolev type system play an important role in analysis and PDEs. In this paper, we consider the very general weighted Hardy–Sobolev type system u ⁢ ( x ) = ∫ ℝ n 1 | x | τ ⁢ | x - y | n - α ⁢ | y | t ⁢ f 1 ⁢ ( u ⁢ ( y ) , v ⁢ ( y ) ) ⁢ 𝑑 y , v ⁢ ( x ) = ∫ ℝ n 1 | x | t ⁢ | x - y | n - α ⁢ | y | τ ⁢ f 2 ⁢ ( u ⁢ ( y ) , v ⁢ ( y ) ) ⁢ 𝑑 y , $u(x)=\\int_{\\mathbb{R}^{n}}\\frac{1}{|x|^{\\tau}|x-y|^{n-\\alpha}|y|^{t}}f_{1}(u(y% ),v(y))dy,\\quad v(x)=\\int_{\\mathbb{R}^{n}}\\frac{1}{|x|^{t}|x-y|^{n-\\alpha}|y|^% {\\tau}}f_{2}(u(y),v(y))dy,$ where f 1 ⁢ ( u ⁢ ( y ) , v ⁢ ( y ) ) = λ 1 ⁢ u p 1 ⁢ ( y ) + μ 1 ⁢ v q 1 ⁢ ( y ) + γ 1 ⁢ u α 1 ⁢ ( y ) ⁢ v β 1 ⁢ ( y ) , $\\displaystyle f_{1}(u(y),v(y))=\\lambda_{1}u^{p_{1}}(y)+\\mu_{1}v^{q_{1}}(y)+% \\gamma_{1}u^{\\alpha_{1}}(y)v^{\\beta_{1}}(y),$ f 2 ⁢ ( u ⁢ ( y ) , v ⁢ ( y ) ) = λ 2 ⁢ u p 2 ⁢ ( y ) + μ 2 ⁢ v q 2 ⁢ ( y ) + γ 2 ⁢ u α 2 ⁢ ( y ) ⁢ v β 2 ⁢ ( y ) . $\\displaystyle f_{2}(u(y),v(y))=\\lambda_{2}u^{p_{2}}(y)+\\mu_{2}v^{q_{2}}(y)+% \\gamma_{2}u^{\\alpha_{2}}(y)v^{\\beta_{2}}(y).$ Only the special cases when γ 1 = γ 2 = 0 ${\\gamma_{1}=\\gamma_{2}=0}$ and one of λ i ${\\lambda_{i}}$ and μ i ${\\mu_{i}}$ is zero (for both i = 1 ${i=1}$ and i = 2 ${i=2}$ ) have been considered in the literature. We establish the integrability of the solutions to the above Hardy–Sobolev type system and the C ∞ ${C^{\\infty}}$ regularity of solutions to this system away from the origin, which improves significantly the Lipschitz continuity in most works in the literature. Moreover, we also use the moving plane method of [8] in integral forms developed in [6] to prove that each pair ( u , v ) ${(u,v)}$ of positive solutions of the above integral system is radially symmetric and strictly decreasing about the origin.

Group Theory Applied to Chemistry

Abstract: Operations.- Function spaces and matrices.- Groups.- Representations.- What has quantum chemistry got to do with it?.- Interactions.- Spherical symmetry and spins.

Symmetry reCAPTCHA

Abstract: This paper is a reaction to the poor performance of symmetry detection algorithms on real-world images, benchmarked since CVPR 2011. Our systematic study reveals significant difference between human labeled (reflection and rotation) symmetries on photos and the output of computer vision algorithms on the same photo set. We exploit this human-machine symmetry perception gap by proposing a novel symmetry-based Turing test. By leveraging a comprehensive user interface, we collected more than 78,000 symmetry labels from 400 Amazon Mechanical Turk raters on 1,200 photos from the Microsoft COCO dataset. Using a set of ground-truth symmetries automatically generated from noisy human labels, the effectiveness of our work is evidenced by a separate test where over 96% success rate is achieved. We demonstrate statistically significant outcomes for using symmetry perception as a powerful, alternative, image-based reCAPTCHA.

Symmetric Non-rigid Structure from Motion for Category-Specific Object Structure Estimation

Abstract: Many objects, especially these made by humans, are symmetric, e.g. cars and aeroplanes. This paper addresses the estimation of 3D structures of symmetric objects from multiple images of the same object category, e.g. different cars, seen from various viewpoints. We assume that the deformation between different instances from the same object category is non-rigid and symmetric. In this paper, we extend two leading non-rigid structure from motion (SfM) algorithms to exploit symmetry constraints. We model the both methods as energy minimization, in which we also recover the missing observations caused by occlusions. In particularly, we show that by rotating the coordinate system, the energy can be decoupled into two independent terms, which still exploit symmetry, to apply matrix factorization separately on each of them for initialization. The results on the Pascal3D+ dataset show that our methods significantly improve performance over baseline methods.

On symmetry of the (strong) Birkhoff–James orthogonality in Hilbert $C^{*}$-modules

Abstract: In this note, we prove that the Birkhoff-James orthogonality, as well as the strong Birkhoff-James orthogonality, is a symmetric relation in a full Hilbert A-module V if and only if at least one of the underlying C*-algebras A or K(V) is isomorphic to C.

Dual Little Strings from F-Theory and Flop Transitions

Abstract: A particular two-parameter class of little string theories can be described by $M$ parallel M5-branes probing a transverse affine $A_{N-1}$ singularity. We previously discussed the duality between the theories labelled by $(N,M)$ and $(M,N)$. In this work, we propose that these two are in fact only part of a larger web of dual theories. We provide evidence that the theories labelled by $(N,M)$ and $(\tfrac{NM}{k},k)$ are dual to each other, where $k=\text{gcd}(N,M)$. To argue for this duality, we use a geometric realization of these little string theories in terms of F-theory compactifications on toric, non-compact Calabi-Yau threefolds $X_{N,M}$ which have a double elliptic fibration structure. We show explicitly for a number of examples that $X_{NM/k,k}$ is part of the extended moduli space of $X_{N,M}$, i.e. the two are related through symmetry transformations and flop transitions. By working out the full duality map, we provide a simple check at the level of the free energy of the little string theories.

Conservation of ‘Moving’ Energy in Nonholonomic Systems with Affine Constraints and Integrability of Spheres on Rotating Surfaces

Abstract: Energy is in general not conserved for mechanical nonholonomic systems with affine constraints. In this article we point out that, nevertheless, in certain cases, there is a modification of the energy that is conserved. Such a function is the pull-back of the energy of the system written in a system of time-dependent coordinates in which the constraint is linear, and for this reason will be called a ‘moving’ energy. After giving sufficient conditions for the existence of a conserved, time-independent moving energy, we point out the role of symmetry in this mechanism. Lastly, we apply these ideas to prove that the motions of a heavy homogeneous solid sphere that rolls inside a convex surface of revolution in uniform rotation about its vertical figure axis, are (at least for certain parameter values and in open regions of the phase space) quasi-periodic on tori of dimension up to three.

Geometry of an elliptic difference equation related to Q4

Abstract: In this paper, we investigate a nonlinear non-autonomous elliptic difference equation, which was constructed by Ramani, Carstea and Grammaticos by integrable deautonomization of a periodic reduction of the discrete Krichever-Novikov equation, or Q4. We show how to construct it as a birational mapping on a rational surface blown up at eight points in $\mathbb P^1\times \mathbb P^1$, and find its affine Weyl symmetry, placing it in the geometric framework of the Painleve equations. The initial value space is ell-$A_0^{(1)}$ and its symmetry group is $W(F_4^{(1)})$. We show that the deautonomization is consistent with the lattice-geometry of Q4 by giving an alternative construction, which is a reduction from Q4 in the usual sense. A more symmetric reduction of the same kind provides another example of a second-order integrable elliptic difference equation.

Differential geometric invariants for time-reversal symmetric Bloch-bundles: The “Real” case

Abstract: Topological quantum systems subjected to an even (resp. odd) time-reversal symmetry can be classified by looking at the related “Real” (resp. “Quaternionic”) Bloch-bundles. If from one side the topological classification of these time-reversal vector bundle theories has been completely described in De Nittis and Gomi [J. Geom. Phys. 86, 303–338 (2014)] for the “Real” case and in De Nittis and Gomi [Commun. Math. Phys. 339, 1–55 (2015)] for the “Quaternionic” case, from the other side it seems that a classification in terms of differential geometric invariants is still missing in the literature. With this article and its companion [G. De Nittis and K. Gomi (unpublished)] we want to cover this gap. More precisely, we extend in an equivariant way the theory of connections on principal bundles and vector bundles endowed with a time-reversal symmetry. In the “Real” case we generalize the Chern-Weil theory and we show that the assignment of a “Real” connection, along with the related differential Chern class and its holonomy, suffices for the classification of “Real” vector bundles in low dimensions.

Ortho-symmetric Modules, Gorenstein Algebras, and Derived Equivalences

Abstract: A new homological symmetry condition is exhibited that extends and unifies several recently defined and widely used concepts. Applications include general constructions of tilting modules and derived equivalences, and characterisations of Gorenstein properties of endomorphism rings.