Narrow Results

We analyze the global theory of boundary conditions for a constrained quantum system with classical configuration space a compact Riemannian manifold M with regular boundary Γ=∂M. The space ℳ of self-adjoint extensions of the covariant Laplacian on M is shown to have interesting geometrical and topological properties which are related to the different topological closures of M. In this sense, the change of topology of M is connected with the nontrivial structure of ℳ. The space ℳ itself can be identified with the unitary group of the Hilbert space of boundary data . This description, is shown to be equivalent to the classical von Neumann's description in terms of deficiency index subspaces, but it is more efficient and explicit because it is given only in terms of the boundary data, which are the natural external inputs of the system. A particularly interesting family of boundary conditions, identified as the set of unitary operators which are singular under the Cayley transform, (the Cayley manifold), turns out to play a relevant role in topology change phenomena. The singularity of the Cayley transform implies that some energy levels, usually associated with edge states, acquire an infinity energy when by an adiabatic change the boundary conditions reaches the Cayley submanifold 𝒞_. In this sense topological transitions require an infinite amount of quantum energy to occur, although the description of the topological transition in the space ℳ is smooth. This fact has relevant implications in string theory for possible scenarios with joint descriptions of open and closed strings. In the particular case of elliptic self-adjoint boundary conditions, the space 𝒞_ can be identified with a Lagrangian submanifold of the infinite dimensional Grassmannian. The corresponding Cayley manifold 𝒞_ is dual of the Maslov class of ℳ. The phenomena are illustrated with some simple low dimensional examples.

We explicitly calculate the Green functions describing quantum changes of topology in Friedmann-Lemaître-Robertson-Walker Universes whose spacelike sections are compact but endowed with distinct topologies. The calculations are performed using the long wavelength approximation at second order in the gradient expansion. We argue that complex metrics are necessary in order to obtain a non-vanishing Green functions and interpret this fact as demonstrating that a quantum topology change can be viewed as a quantum tunneling effect. The result shows that the quantum topology changes in the direction of negatively curved hypersurfaces are strongly enhanced as time goes on, while transitions in the opposite direction are suppressed.

Topological surgery in dimension $3$ is intrinsically connected with the classification of $3$-manifolds and with patterns of natural phenomena. In this, mostly expository, paper, we present two different approaches for understanding and visualizing the process of $3$-dimensional surgery. In the first approach, we view the process in terms of its effect on the fundamental group. Namely, we present how $3$-dimensional surgery alters the fundamental group of the initial manifold and present ways to calculate the fundamental group of the resulting manifold. We also point out how the fundamental group can detect the topological complexity of non-trivial embeddings that produce knotting. The second approach can only be applied for standard embeddings. For such cases, we give new visualizations of $3$-dimensional surgery as rotations of the decompactified $2$-sphere. Each rotation produces a different decomposition of the $3$-sphere which corresponds to a different visualization of the $4$-dimensional process of $3$-dimensional surgery.

The nuclear symmetry energy figures crucially in the structure of asymmetric nuclei and, more importantly, in the equation of state (EoS) of compact stars. At present it is almost totally unknown, both experimentally and theoretically, in the density regime appropriate for the interior of neutron stars. Basing on a strong-coupled structure of dense baryonic matter encoded in the skyrmion crystal approach with a topology change and resorting to the notion of generalized hidden local symmetry in hadronic interactions, we address a variety of hitherto unexplored issues of nuclear interactions associated with the symmetry energy, i.e., kaon condensation and hyperons, possible topology change in dense matter, nuclear tensor forces, conformal symmetry, chiral symmetry, etc., in the EoS of dense compact-star matter. One of the surprising results coming from HLS structure that is distinct from what is given by standard phenomenological approaches is that at high density, baryonic matter is driven by renormalization group flow to the "dilaton-limit fixed point" constrained by "mended symmetries". We further propose how to formulate kaon condensation and hyperons in compact-star matter in a framework anchored on a single effective Lagrangian by treating hyperons as the Callan–Klebanov kaon-skyrmion bound states simulated on crystal lattice. This formulation suggests that hyperons can figure in the stellar matter — if at all — when or after kaons condense, in contrast to the standard phenomenological approaches where the hyperons appear as the first strangeness degree of freedom in matter, thereby suppressing or delaying kaon condensation. In our simplified description of the stellar structure in terms of symmetry energies, which is compatible with that of the 1.97 solar mass star, kaon condensation plays a role of "doorway state" to strange quark matter.

Among the five-year government-funded World Class University (WCU) Projects in Korea, the category-3 program approved at Hanyang University in Seoul led to an exploratory effort to go from neutron-rich nuclei to dense matter in neutron stars. The principal results in what transpired in the effort — and what followed afterwards — are described with the focus on the possibly important, hitherto unexplored, role played in nuclear dynamics of topology and hidden symmetries of QCD. The potential link to the proton mass problem is pointed out.

The purpose of this study is to examine the effect of topology change in the initial universe. In this study, the concept of G-cobordism is introduced to argue about the topology change of the manifold on which a transformation group acts. This G-manifold has a fiber bundle structure if the group action is free and is related to the spacetime in Kaluza–Klein theory or Einstein–Yang–Mills system. Our results revealed the fundamental processes of compactification in G-manifolds. In these processes, the initial high symmetry and multidimensional universe changes to present universe by the mechanism which lowers the dimensions and symmetries.

The purpose of this study is to investigate the new dark matter candidates and their production mechanisms. In this study, we concentrated on the dark matter production by analyzing the topology change. Our results revealed the variety of the dark matter productions.

The purpose of this study is to examine the effect of topology change on anomaly in the initial universe. G-cobordism is introduced to argue about the topology change accompanied by gauge group. Our results revealed that change of anomalies results from topology change.

We show how to use boundary conditions to drive the evolution on a quantum mechanical system. We will see how this problem can be expressed in terms of a time-dependent Schrödinger equation. In particular, we will need the theory of self-adjoint extensions of differential operators in manifolds with boundary. An introduction of the latter as well as meaningful examples will be given. It is known that different boundary conditions can be used to describe different topologies of the associated quantum systems. We will use the previous results to study the topology change and to obtain necessary conditions to accomplish it in a dynamical way.

Motivated by condensed matter physical systems, in which a finite-time singularity indicates that the topology of the system changes, we critically examine the possibility of the Universe’s topology change at a finite-time cosmological singularity. We emphasize on Big Rip and Type II and IV cosmological singularities, which we classify to future spacelike and timelike singularities. For the Type IV and Type II singularities, since no geodesics incompleteness occurs, no topological change is allowed, by using Geroch’s theorem arguments. However, for the Big Rip case, Tipler’s arguments allow a topology change, if the spacetime in which the topology change occurs is non-compact and the boundary of this region are two topologically distinct three-dimensional spacelike partial Cauchy hypersurfaces. Also, some additional requirements must hold true, among which the weak energy condition, which can be satisfied in a geometric way in the context of a modified gravity. We critically examine Tipler’s arguments for the Big Rip case, and we discuss the mathematical implications of such a topological change, with regard to the final hypersurface on which geodesics incompleteness occurs.

Regioselective nucleophilic chlorination and bromination of [26]hexaphyrin to produce a [28]Möbius aromatic hexaphyrin Pd(II) complex were accomplished upon treatment of [26]hexaphyrin 2 with chloride and bromide ions in the presence of an acid with concurrent molecular topology change from Hückel to Möbius. Stille coupling reaction of the resultant 3-brominated complex 4 with ethynylstannanes afforded β-ethynylated derivatives in moderate to high yields.

We argue that the formation of a Schwarzschild black hole via Datt-Oppenheimer-Snyder type gravitational collapse must be accompanied by a change in topology upon formation of the event horizon which physically separates matter in the interior from that of the exterior. While it is true that collapsing matter crossing the event horizon continues to fall towards the singularity of the Schwarzschild interior, this region does not in fact contain the matter originally responsible for the collapse. Rather, the latter occupies a distinct internal spacetime region with its own independent evolution. The existence of this additional component of the simplest black hole has a number of profound implications - Schwarzschild black holes are stable with constant mass; they each contain a self-contained mini-universe within their event horizons; and they live within a spacetime that is inherently double-sheeted.