Topology and Physics

The following sections are included:

• Preface
• Contents

We propose a new geometrical model of matter, in which neutral atoms are modelled by compact, complex algebraic surfaces. Proton and neutron numbers are determined by a surface’s Chern numbers. Equivalently, they are determined by combinations of the Hodge numbers, or the Betti numbers. Geometrical constraints on algebraic surfaces allow just a finite range of neutron numbers for a given proton number. This range encompasses the known isotopes.

This note aims to provide an entrée to two developments in two-dimensional topological gravity — that is, intersection theory on the moduli space of Riemann surfaces — that have not yet become well-known among physicists. A little over a decade ago, Mirzakhani discovered [1, 2] an elegant new proof of the formulas that result from the relationship between topological gravity and matrix models of two-dimensional gravity. Here we will give a very partial introduction to that work, which hopefully will also serve as a modest tribute to the memory of a brilliant mathematical pioneer. More recently, Pandharipande, Solomon, and Tessler [3] (with further developments in [4–6]) generalized intersection theory on moduli space to the case of Riemann surfaces with boundary, leading to generalizations of the familiar KdV and Virasoro formulas. Though the existence of such a generalization appears natural from the matrix model viewpoint — it corresponds to adding vector degrees of freedom to the matrix model — constructing this generalization is not straightforward. We will give some idea of the unexpected way that the difficulties were resolved.

In this paper we study unitary braid group representations associated with Majorana Fermions. Majorana Fermions are represented by Majorana operators, elements of a Clifford algebra. The paper recalls and proves a general result about braid group representations associated with Clifford algebras, and compares this result with the Ivanov braiding associated with Majorana operators. The paper generalizes observations of Kauffman and Lomonaco and of Mo-Lin Ge to show that certain strings of Majorana operators give rise to extraspecial 2-groups and to braiding representations of the Ivanov type.

Much of arithmetic geometry is concerned with the study of principal bundles. They occur prominently in the arithmetic of elliptic curves and, more recently, in the study of the Diophantine geometry of curves of higher genus. In particular, the geometry of moduli spaces of principal bundles holds the key to an effective version of Faltings’ theorem on finiteness of rational points on curves of genus at least 2. The study of arithmetic principal bundles includes the study of Galois representations, the structures linking motives to automorphic forms according to the Langlands program. In this paper, we give a brief introduction to the arithmetic geometry of principal bundles with emphasis on some elementary analogies between arithmetic moduli spaces and the constructions of quantum field theory.

Not long after Einstein introduced the equations for his general theory of relativity, solutions were found by Friedman, Lemaître, and others, that presented plausible models of the cosmos, but they had singular origins where densities and curvatures diverge to infinity. A similar situation, but in the opposite time direction, occurred with the Oppenheimer–Snyder model, found somewhat later, which describes gravitational collapse to what is now called a black hole with its internal singularity. The question arose as to whether the strict spherical symmetry and the simplistic representation of the natter source (dust), in these models, might be misleading, and that a generic asymmetrical (possibly rotating) model, with perhaps a more realistic matter source, might avoid actual singularities, allowing for the possibility of a non-singular “bounce”.

However, techniques introduced in the 1960s, described in detail here, in which ideas from differential topology and causal structure theory, showed the singularities occurring in these situations cannot be removed by such means, being stable phenomena and occurring with very general matter sources satisfying merely a condition related to local energy non-negativity. Results in this area due to Stephen Hawking, Robert Geroch, the current author, and many others are presented here, these being concerned with causality conditions, the structures of boundaries of futures and pasts, Cauchy horizons, and domains of dependence. A version of the authors original singularity theorem, demonstrating the necessity of singularities arising in gravitational collapse, is presented in detail, but with a new perspective, whereby the nature of such singularities can be analysed in relation to the notion of boundary points to space-times, referred to as TIPs and TIFs (terminal indecomposable pasts and futures).

The theory of anyon systems, as modular functors topologically and unitary modular tensor categories algebraically, is mature. To go beyond anyons, our first step is the interplay of anyons with conventional group symmetry due to the paramount importance of group symmetry in physics. This led to the theory of symmetry-enriched topological order. Another direction is the boundary physics of topological phases, both gapless as in the fractional quantum Hall physics and gapped as in the toric code. A more speculative and interesting direction is the study of Banados–Teitelboim–Zanelli (BTZ) black holes and quantum gravity in 3d. The clearly defined physical and mathematical issues require a far-reaching generalization of anyons and seem to be within reach. In this short survey, I will first cover the extensions of anyon theory to symmetry defects and gapped boundaries. Then, I will discuss a desired generalization of anyons to anyon-like objects — the BTZ black holes — in 3d quantum gravity.

Newton’s mechanical revolution unifies the motion of planets in the sky and the falling of apples on Earth. Maxwell’s electromagnetic revolution unifies electricity, magnetism, and light. Einstein’s relativistic revolution unifies space with time, and gravity with space–time distortion. The quantum revolution unifies particle with waves, and energy with frequency. Each of those revolution changes our world view. In this article, we will describe a revolution that is happening now: the second quantum revolution which unifies matter/space with information. In other words, the new world view suggests that elementary particles (the bosonic force particles and fermionic matter particles) all originated from quantum information (qubits): they are collective excitations of an entangled qubit ocean that corresponds to our space. The beautiful geometric Yang–Mills gauge theory and the strange Fermi statistics of matter particles now have a common algebraic quantum informational origin.

Topological insulators are new quantum states with helical gapless edge or surface states inside the bulk band gap. These topological surface states are robust against weak timereversal invariant perturbations without closing the bulk band gap, such as lattice distortions and non-magnetic impurities. Recently a variety of topological insulators have been predicted by theories, and observed by experiments. First-principles calculations have been widely used to predict topological insulators with great success. In this review, we summarize the current progress in this field from the perspective of first-principles calculations. First of all, the basic concepts of topological insulators and the frequently-used techniques within first-principles calculations are briefly introduced. Secondly, we summarize general methodologies to search for new topological insulators. In the last part, based on the band inversion picture first introduced in the context of HgTe, we classify topological insulators into three types with s–p, p–p and d–f, and discuss some representative examples for each type.

For a simple Hubbard model, using a particle-particle pairing operator η and a particle-hole pairing operator ζ, it is shown that one can write down two commuting sets of angular momenta operators J and J′, both of which commute with the Hamiltonian. These considerations allow the introduction of quantum numbers j and j′, and lead to the fact that the system has SO₄ = (SU₂ × SU₂)/Z₂ symmetry. j is related to the existence of superconductivity for a state and j′ to its magnetic properties.