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In 1983, Donaldson shocked the topology world by using instantons from physics to prove new theorems about four-dimensional manifolds, and he developed new topological invariants. In 1988, Witten showed how these invariants could be obtained by correlation functions for a twisted N = 2 SUSY gauge theory. In 1994, Seiberg and Witten discovered dualities for such theories, and in particular, developed a new way of looking at four-dimensional manifolds that turns out to be easier, and is conjectured to be equivalent to, Donaldson theory.

This review describes the development of this mathematical subject, and shows how the physics played a pivotal role in the current understanding of this area of topology.

A non-trivial spatial topology of the Universe is a potentially observable attribute, which can be probed through the circles-in-the-sky for all locally homogeneous and isotropic universes with no assumptions on the cosmological parameters. We show how one can use a possible circles-in-the-sky detection of the spatial topology of globally homogeneous universes to set constraints on the dark energy equation of state parameters.

Motivated by the study of soluble models of quantum field theory, we illustrate a new type of topological effect by comparing the constructions of canonical Klein–Gordon quantum fields on the two-dimensional de Sitter spacetime as opposed to its double covering. We show that while the commutators of the two fields coincide locally, the global topological differences make the theories drastically different. Many of the well-known features of de Sitter quantum field theory disappear. In particular, there is nothing like a Bunch–Davies vacuum. Correspondingly, even though the local horizon structure is the same for the two universes, there is no Hawking–Gibbons thermal state. Finally, there is no complementary series of fields.

QCD axions are at the crossroads of QCD topology and Dark Matter searches. We present here the current status of topological studies on the lattice, and their implication on axion physics. We outline the specific challenges posed by lattice topology, the different proposals for handling them, the observable effects of topology on the QCD spectrum and its interrelation with chiral and axial symmetries. We review the transition to the quark–gluon plasma, the fate of topology at the transition, and the approach to the high temperature limit. We discuss the extrapolations needed to reach the regime of cosmological relevance, and the resulting constraints on the QCD axion.

We explore the interplay between quantization, local commutativity and the analyticity properties of the two-point functions of a quantum field in a non trivial topological cosmological background in the example of the two-dimensional de Sitter manifold and its double covering. The global topological differences make the many of the well-known features of de Sitter quantum field theory disappear. In particular there is nothing like a Bunch-Davies vacuum and there are no $SL(2,R)$-invariant fields whose mass is less than 1/2.

The experiments at the Large Hadron Collider (LHC) rely upon a complex distributed computing infrastructure (WLCG) consisting of hundreds of individual sites worldwide at universities and national laboratories, providing about half a billion computing job slots and an exabyte of storage interconnected through high speed networks. Wide Area Networking (WAN) is one of the three pillars (together with computational resources and storage) of LHC computing. More than 5 PB/day are transferred between WLCG sites. Monitoring is one of the crucial components of WAN and experiments operations. In the past years all experiments have invested significant effort to improve monitoring and integrate networking information with data management and workload management systems. All WLCG sites are equipped with perfSONAR servers to collect a wide range of network metrics. We will present the latest development to provide the 3D force directed graph visualization for data collected by perfSONAR. The visualization package allows site admins, network engineers, scientists and network researchers to better understand the topology of our Research and Education networks and it provides the ability to identify nonreliable or/and nonoptimal network paths, such as those with routing loops or rapidly changing routes.

In this paper, we construct a class of quantum systems in a space continuum inspired by results from topological insulator physics. Instead of adding spin–orbit coupling terms suggested by time-reversal invariance as in conventional topological insulators, the terms $L\cdot S$ are determined using supersymmetry as a starting point. This procedure not only restricts the number of possible continuous topological insulators models, but also provides a systematic way to find new continuous topological insulators models. Some explicit quantum mechanical examples are discussed and applications to dark matter physics are also outlined.

We present some approaches to the perturbative analysis of the classical and quantum gravity. First we introduce a graphical representation for a global SO(n) tensor (∂)^dh_αβ, which generally appears in the weak field expansion around the flat space: g_μν=δ_μν+h_μν. Making use of this representation, we explain (1) Generating function of graphs (Feynman diagram approach), (2) Adjacency matrix (Matrix approach), (3) Graphical classification in terms of "topology indices" (Topology approach), (4) The Young tableau (Symmetric group approach). We systematically construct the global SO(n) invariants. How to show the independence and completeness of those invariants is the main theme. We explain it taking simple examples of ∂∂h-, and (∂∂h)²-invariants in the text. The results are applied to the analysis of the independence of general invariants and (the leading order of) the Weyl anomalies of scalar-gravity theories in "diverse" dimensions (2,4,6,8,10 dimensions).

We investigate algebraic and computability-theoretic properties of orderable magmas. A magma is an algebraic structure with a single binary operation. A right order on a magma is a linear ordering of its domain, which is right-invariant with respect to the magma operation. We use tools of computability theory to investigate Turing complexity of orders on computable orderable magmas. A magma is computable if it is finite, or if its domain can be identified with the set of natural numbers and the magma operation is computable. Interesting orderable magmas that are not even associative come from knot theory.

A ribbon is a smooth mapping (possibly self-intersecting) of an annulus $S^1\times I$ in 3-space having constant width $R$. Given a regular parametrization $x(s)$, and a smooth unit vector field $u(s)$ based along $x$, for a knot $K$, we may define a ribbon of width $R$ associated to $x$ and $u$ as the set of all points $x(s)+ru(s)$, $r\in [0,R]$. For large $R$, ribbons, and their outer edge curves, may have self-intersections. In this paper, we analyze how the knot type of the outer ribbon edge $x(s)+Ru(s)$ relates to that of the original knot $K$. Generically, as $R\to \infty$, there is an eventual constant knot type. This eventual knot type is one of only finitely many possibilities which depend just on the vector field $u$. The particular knot type within the finite set depends on the parametrized curves $x(s)$, $u(s)$, and their interactions. We demonstrate a way to control the curves and their parametrizations so that given two knot types $K_1$ and $K_2$, we can find a smooth ribbon of constant width connecting curves of these two knot types.

We directly connect topological changes that can occur in mathematical three-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena. This work widens the bridge between topology and natural sciences and creates a new platform for exploring geometrical physics.

The non-orientable 4-genus of a knot $K$ in $S^3$ is defined to be the minimum first Betti number of a non-orientable surface $F$ smoothly embedded in $B^4$ so that $K$ bounds $F$. We will survey the tools used to compute the non-orientable 4-genus, and use various techniques to calculate this invariant for non-alternating 11 crossing knots. We will also view obstructions to a knot bounding a Möbius band given by the double branched cover of $S^3$ branched over $K$.

The role of topology in the statistical mechanics of surfaces in the lattice is considered. The possibility that a phase transition driven by the number of boundary components occurs is investigated. Bounds on the limiting free energy is derived and conditions for the existence of a critical point in the phase diagram are presented.

We show that different topologies of a space-time manifold and different signatures of its metric can be encompassed into a single Lagrangian formalism, provided one adopts the first-order (Palatini) formulation and relies on nonlinear Lagrangians, that were earlier shown to produce, in the generic case, universality of Einstein field equations and of Komar's energy-momentum complex as well. An example in Relativistic Cosmology is provided.

In general, topological quantum field theory (TQFT) is studied in detail from the n-dimensional cobordism (nCob) to the Hilbert vector space. However, we study the TQFT in the different way in this paper, that is, the TQFT is studied from the Hilbert vector space to nCob. To do this, the theory called the ϕ-mapping topological current theory is used. The relation between the objects and zero points of the Hilbert states in the Hilbert vector space is studied in this frame. The relation between the morphism and topological current is revealed too.

The two-dimensional charge transport with parallel (in plane) magnetic field is considered from the physical and mathematical point of view. To this end, we start with the magnetic field parallel to the plane of charge transport, in sharp contrast to the configuration described by the theorems of Aharonov and Casher where the magnetic field is perpendicular. We explicitly show that the specific form of the arising equation enforces the respective field solution to fulfill the Majorana condition. Consequently, as soon any physical system is represented by this equation, the rise of fields with Majorana type behavior is immediately explained and predicted. In addition, there exists a quantized particular phase that removes the action of the vector potential producing interesting effects. Such new effects are able to explain due to the geometrical framework introduced, several phenomenological results recently obtained in the area of spintronics and quantum electronic devices. The quantum ring as spin filter is worked out in this framework and also the case of the quantum Hall effect.

Electromagnetic knots are electromagnetic fields in which the magnetic and electric lines are level curves of two complex scalar fields. If these scalar fields are chosen so that they can be interpreted as maps between the three-sphere and the two-sphere every time, then the electromagnetic helicity is proportional to the sum of the Hopf indices of both maps, that are topological invariants. An important example of this kind of electromagnetic fields with topological properties is often called the Hopfion. It is an electromagnetic knot in which the scalar fields are built from Hopf maps. In this work, we study the conditions for these electromagnetic fields to be null, i.e. to have both Lorentz invariant quantities $E\cdot B$ and $E^2-B^2$ equal to zero. We derive from these fields explicit vector potentials $A$ and $C$ satisfying force-free like conditions for every time. In particular, equations $A\cdot B=B^2/2$, $C\cdot E=E^2/2$ are derived for them. As a consequence, the energy, which is discretized, is proportional to the electromagnetic helicity. This relation between Physics and Topology is intriguing and worth of future research.

In this paper, we suggest a mathematical representation to the holographic principle through the theory of topological retracts. We found that the topological retraction is the mathematical analog of the hologram idea in modern quantum gravity and it can be used to explore the geometry of the hologram boundary. An example has been given on the five-dimensional (5D) wormhole spacetime $W$ which we found can retract to lower-dimensional circles $S_i\subset W$. In terms of the holographic principle, the description of this volume of spacetime $W$ is encoded on the lower-dimensional circle which is the region boundary.

The magnetosphere structure of compact objects is considered in the context of a theory of gravity with dynamical torsion field beyond standard General Relativity (GR). To this end, a new spherically symmetric solution is obtained in this theoretical framework, physically representing a compact object of pseudoscalar fields (for example, axion field). The axially symmetric version of the Grad–Shafranov equation (GSE) is also derived in this context, and used to describe the magnetosphere dynamics of the obtained “axion star”. The interplay between high-energy processes and the seed magnetic field with respect to the global structure of the magnetosphere is briefly discussed.

In this work, the quadratic helicity ${\chi }_{\bar{B}}^{(2)}$ is discussed from the physical and topological points of view. We show, after the introduction of a mathematical description of the properties of helicity and quadratic helicity in the context of standard dynamo equations, examples of the importance of these high invariants in cosmological scenarios. These scenarios consider extensions of the standard model and extensions of GR.