Topology and Physics

Chen Ning Yang, Mo-Lin Ge;Yang-Hui He

• Publisher: WSPC
• Year: 2019
• ISBN:
  9789813278684
• Topic: Scienze fisiche
• Subtopic: Materia condensata

Since its birth in Poincaré's seminal 1894 'Analysis Situs', topology has become a cornerstone of mathematics. As with all beautiful mathematical concepts, topology inevitably — resonating with that Wignerian principle of the effectiveness of mathematics in the natural sciences — finds its prominent role in physics. From Chern–Simons theory to topological quantum field theory, from knot invariants to Calabi–Yau compactification in string theory, from spacetime topology in cosmology to the recent Nobel Prize winning work on topological insulators, the interactions between topology and physics have been a triumph over the past few decades.

In this eponymous volume, we are honoured to have contributions from an assembly of grand masters of the field, guiding us with their world-renowned expertise on the subject of the interplay between 'Topology' and 'Physics'. Beginning with a preface by Chen Ning Yang on his recollections of the early days, we proceed to a novel view of nuclei from the perspective of complex geometry by Sir Michael Atiyah and Nick Manton, followed by an entrée toward recent developments in two-dimensional gravity and intersection theory on the moduli space of Riemann surfaces by Robbert Dijkgraaf and Edward Witten; a study of Majorana fermions and relations to the Braid group by Louis H Kauffman; a pioneering investigation on arithmetic gauge theory by Minhyong Kim; an anecdote-enriched review of singularity theorems in black-hole physics by Sir Roger Penrose; an adventure beyond anyons by Zhenghan Wang; an aperçu on topological insulators from first-principle calculations by Haijun Zhang and Shou-Cheng Zhang; finishing with synopsis on quantum information theory as one of the four revolutions in physics and the second quantum revolution by Xiao-Gang Wen. We hope that this book will serve to inspire the research community.

Contents:

• Preface — Early Examples of Topological Concepts in Physics (C N Yang)
• Complex Geometry of Nuclei and Atoms (M F Atiyah and N S Manton)
• Developments in Topological Gravity (Robbert Dijkgraaf and Edward Witten)
• Majorana Fermions and Representations of the Braid Group (Louis H Kauffman)
• Arithmetic Gauge Theory: A Brief Introduction (Minhyong Kim)
• Singularity Theorems (Roger Penrose)
• Beyond Anyons (Zhenghan Wang)
• Four Revolutions in Physics and the Second Quantum Revolution — A Unification of Force and Matter by Quantum Information (Xiao-Gang Wen)
• Topological Insulators from the Perspective of First-principles Calculations (Haijun Zhang and Shou-Cheng Zhang)
• Appendix — SO₄ symmetry in a Hubbard Model (C N Yang and Shou-Cheng Zhang)

Readership: Senior undergraduate student; graduate students; researchers.Topology;Physics;Chern Class;Differential Geometry;Atoms;Nuclei;Algebraic Surface;4-Manifolds;Topological Gravity;Modulli Space;Riemann Surfaces;Kdv;Virasoro;Knots;Links;Braid Group;Fermion;Quantum Computing;Arithemtic Geometry;Gauge Theory;Anyon Systems;Modular Functors;Group Symmetry;Topological Phase;Quantum Revolution;Quantum Information;General Relativity;Yang–Mills;Topological Insulators;First-Principle Calculations;Spin-Orbit Coupling;Surface States0 Key Features:

• World-renowned mathematicians and physicists are the Chapter contributors
• It provides the latest review of the interplay between "Topology" and "Physics"
• Readers can appreciate the beauty of mathematics and physics within one single book

Chapter 1

Complex geometry of nuclei and atoms^∗

M. F. Atiyah

School of Mathematics, University of Edinburgh,
James Clerk Maxwell Building,
Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK
[email protected]

N. S. Manton

Department of Applied Mathematics and Theoretical Physics,
University of Cambridge,
Wilberforce Road, Cambridge CB3 0WA, UK
[email protected]

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.

Keywords: Atoms; nuclei; algebraic surfaces; 4-manifolds.

PACS numbers: 02.40.Tt, 02.40.Re, 21.60.–n

1.Introduction

It is an attractive idea to interpret matter geometrically, and to identify conserved attributes of matter with topological properties of the geometry. Kelvin made the pioneering suggestion to model atoms as knotted vortices in an ideal fluid.¹ Each atom type would correspond to a distinct knot, and the conservation of atoms in physical and chemical processes (as understood in the 19th century) would follow from the inability of knots to change their topology. Kelvin’s model has not survived because atoms are now known to be structured and divisible, with a nucleus formed of protons and neutrons bound together, surrounded by electrons. At high energies, these constituents can be separated. It requires of order 1 eV to remove an electron from an atom, but a few MeV to remove a proton or neutron from a nucleus.

Atomic and nuclear physics has progressed, mainly by treating protons, neutrons and electrons as point particles, interacting through electromagnetic and strong nuclear forces.² Quantum mechanics is an essential ingredient, and leads to a discrete spectrum of energy levels, both for the electrons and nuclear particles. The nucleons (protons and neutrons) are themselves built from three pointlike quarks, but little understanding of nuclear structure and binding has so far emerged from quantum chromodynamics (QCD), the theory of quarks. These point particle models are conceptually not very satisfactory, because a point is clearly an unphysical idealisation, a singularity of matter and charge density. An infinite charge density causes difficulties both in classical electrodynamics³ and in quantum field theories of the electron. Smoother structures carrying the discrete information of proton, neutron and electron number would be preferable.

In this paper, we propose a geometrical model of neutral atoms where both the proton number P and neutron number N are topological and none of the constituent particles are pointlike. In a neutral atom the electron number is also P, because the electron’s electric charge is exactly the opposite of the proton’s charge. For given P, atoms (or their nuclei) with different N are known as different isotopes.

A more recent idea than Kelvin’s is that of Skyrme, who proposed a nonlinear field theory of bosonic pion fields in 3 + 1 dimensions with a single topological invariant, which Skyrme identified with baryon number.^4,5 Baryon number (also called atomic mass number) is the sum of the proton and neutron numbers, B = P + N. Skyrme’s baryons are solitons in the field theory, so they are smooth, topologically stable field configurations. Skyrme’s model was designed to model atomic nuclei, but electrons can be added to produce a model of a complete atom. Protons and neutrons can be distinguished in the Skyrme model, but only after the internal rotational degrees of freedom are quantised.⁶ This leads to a quantised “isospin,” with the proton having isospin up

I₃ =

and the neutron having isospin down

I₃ = −

, where I₃ the third component of isospin. The model is consistent with the well-known Gell-Mann–Nishijima relation⁷

where Q is the electric charge of a nucleus (in units of the proton charge) and B is the baryon number. Q is integral, because I₃ is integer-valued (half-integer-valued) when B is even (odd). Q equals the proton number P of the nucleus and also the electron number of a neutral atom. The neutron number is N =

B − I₃. The Skyrme model has had considerable success providing models for nuclei.^8–11 Despite the pion fields being bosonic, the quantised Skyrmions have half-integer spin if B is odd.¹² But a feature of the model is that proton number and neutron number are not separately topological, and electrons have to be added on.

The Skyrme model has a relation to 4-dimensional fields that provides some motivation for the ideas discussed in this paper. A Skyrmion can be well approximated by a projection of a 4-dimensional Yang–Mills field. More precisely, one can take an SU(2) Yang–Mills instanton and calculate its holonomy along all lines in the (Euclidean) time direction.¹³ The result is a Skyrme field in 3-dimensional space, whose baryon number B equals the instanton number.

So a quasi-geometrical structure in 4-dimensional space (a Yang–Mills instanton in flat R⁴) can be closely related to nuclear structure, but still there is just one topological charge. A next step, first described in Ref. 14, was to propose an identification of smooth, curved 4-manifolds with the fundamental particles in atoms — the proton, neutron and electron. Suitable examples of manifolds were suggested. These manifolds were not all compact, and the particles they modelled were not all electrically neutral. One of the more compelling examples was Taub-NUT space as a model for the electron. By studying the Dirac operator on the Taub-NUT background, it was shown how the spin of the electron can arise in this context.¹⁵ There has also been an investigation of multi-electron systems modelled by multi-Taub-NUT space.^16,17 However, there are some technical difficulties with the models of the proton and neutron, and no way has yet been found to geometrically combine protons and neutrons into more complicated nuclei surrounded by electrons. Nor is it clear in this context what exactly should be the topological invariants representing proton and neutron number.

A variant of these ideas is a model for the simplest atom, the neutral hydrogen atom, with one proton and one electron. This appears to be well modelled by CP², the complex projective plane.^a The fundamental topological property of CP² is that it has a generating 2-cycle with self-intersection 1. The second Betti number is b₂ = 1, which splits into b₂⁺ = 1 and b₂⁻ = 0. A complex line in CP² represents this cycle, and in the projective plane, two lines always intersect in one point. A copy of this cycle together with its normal neighbourhood can be interpreted as the proton part of the atom, whereas the neighbourhood of a point dual to this is interpreted as the electron. The neighbourhood of a point is just a 4-ball, with a 3-sphere boundary, but this is the same as in the Taub-NUT model of the electron, which ...