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Michael Atiyah

Publications and source records attributed to Michael Atiyah.

At least 19 recordsLinked to original sources
arXiv2017

Twistor theory at fifty: from contour integrals to twistor strings

We review aspects of twistor theory, its aims and achievements spanning thelast five decades. In the twistor approach, space--time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex three--fold -- the twistor space. After giving an elementary construction of this space we demonstrate how solutions to linear and nonlinear equations of mathematical physics: anti-self-duality (ASD) equations on Yang--Mills, or conformal curvature can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang--Mills, and gravitational instantons which we review. They also underlie the twistor approach to integrability: the solitonic systems arise as symmetry reductions of ASD Yang--Mills equations, and Einstein--Weyl dispersionless systems are reductions of ASD conformal equations. We then review the holomorphic string theories in twistor and ambitwistor spaces, and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes. Finally we discuss the Newtonian limit of twistor theory, and its possible role in Penrose's proposal for a role of gravity in quantum collapse of a wave function.

hep-th↗
arXiv2017

Anyons in Geometric Models of Matter

We show that the "geometric models of matter" approach proposed by the first author can be used to construct models of anyon quasiparticles with fractional quantum numbers, using 4-dimensional edge-cone orbifold geometries with orbifold singularities along embedded 2-dimensional surfaces. The anyon states arise through the braid representation of surface braids wrapped around the orbifold singularities, coming from multisections of the orbifold normal bundle of the embedded surface. We show that the resulting braid representations can give rise to a universal quantum computer.

math-ph↗
arXiv2016

The Non-Existent Complex 6-Sphere

The possible existence of a complex structure on the 6-sphere has been a famous unsolved problem for over 60 years. In that time many "solutions" have been put forward, in both directions. Mistakes have always been found. In this paper I present a short proof of the non-existence, based on ideas developed, but not fully exploited, over 50 years ago. The only change in v2. is in section 3, where the notation has been clarified.

math.DG↗
arXiv2014

Time evolution in a geometric model of a particle

We analyse the properties of a (4+1)-dimensional Ricci-flat spacetime which may be viewed as an evolving Taub-NUT geometry, and give exact solutions of the Maxwell and gauged Dirac equation on this background. We interpret these solutions in terms of a geometric model of the electron and its spin, and discuss links between the resulting picture and Dirac's Large Number Hypothesis.

hep-th↗
arXiv2012

Curvature, Cones, and Characteristic Numbers

We study Einstein metrics on smooth compact 4-manifolds with an edge-cone singularity of specified cone angle along an embedded 2-manifold. To do so, we first derive modified versions of the Gauss-Bonnet and signature theorems for arbitrary Riemannian 4-manifolds with edge-cone singularities, and then show that these yield non-trivial obstructions in the Einstein case. We then use these integral formulae to obtain interesting information regarding gravitational instantons which arise as limits of such edge-cone manifolds.

math.DG↗
arXiv2011

Geometric Models of Matter

Inspired by soliton models, we propose a description of static particles in terms of Riemannian 4-manifolds with self-dual Weyl tensor. For electrically charged particles, the 4-manifolds are non-compact and asymptotically fibred by circles over physical 3-space. This is akin to the Kaluza-Klein description of electromagnetism, except that we exchange the roles of magnetic and electric fields, and only assume the bundle structure asymptotically, away from the core of the particle in question. We identify the Chern class of the circle bundle at infinity with minus the electric charge and the signature of the 4-manifold with the baryon number. Electrically neutral particles are described by compact 4-manifolds. We illustrate our approach by studying the Taub-NUT manifold as a model for the electron, the Atiyah-Hitchin manifold as a model for the proton, CP^2 with the Fubini-Study metric as a model for the neutron, and S^4 with its standard metric as a model for the neutrino.

hep-th↗
arXiv2010

Edinburgh Lectures on Geometry, Analysis and Physics

These lecture notes are based on a set of six lectures that I gave in Edinburgh in 2008/2009 and they cover some topics in the interface between Geometry and Physics. They involve some unsolved problems and conjectures and I hope they may stimulate readers to investigate them.

math.AG↗
arXiv2010

A Shifted View of Fundamental Physics

We speculate on the role of relativistic versions of delayed differential equations in fundamental physics. Relativistic invariance implies that we must consider both advanced and retarded terms in the equations, so we refer to them as shifted equations. The shifted Dirac equation has some novel properties. A tentative formulation of shifted Einstein-Maxwell equations naturally incorporates a small but nonzero cosmological constant.

hep-th↗
arXiv2005

Twisted $K$-theory

Twisted complex $K$-theory can be defined for a space $X$ equipped with a bundle of complex projective spaces, or, equivalently, with a bundle of C$^*$-algebras. Up to equivalence, the twisting corresponds to an element of $H^3(X;\Z)$. We give a systematic account of the definition and basic properties of the twisted theory, emphasizing some points where it behaves differently from ordinary $K$-theory. (We omit, however, its relations to classical cohomology, which we shall treat in a sequel.) We develop an equivariant version of the theory for the action of a compact Lie group, proving that then the twistings are classified by the equivariant cohomology group $H^3_G(X;\Z)$. We also consider some basic examples of twisted $K$-theory classes, related to those appearing in the recent work of Freed-Hopkins-Teleman.

math.KT↗
arXiv2005

Twisted K-theory and cohomology

We explore the relations of twisted K-theory to twisted and untwisted classical cohomology. We construct an Atiyah-Hirzebruch spectral sequence, and describe its differentials rationally as Massey products. We define the twisted Chern character. We also discuss power operations in the twisted theory, and the role of the Koschorke classes.

math.KT↗
arXiv2004

Skyrmions, instantons, mass and curvature

We show that Skyrmions with massless pions in hyperbolic space provide a good approximation to Skyrmions with massive pions in Euclidean space, for a particular relationship between the pion mass and the curvature of hyperbolic space. Using this result we describe how a Skyrmion with massive pions in Euclidean space can be approximated by the holonomy along circles of a Yang-Mills instanton. This is a generalization of the approximation of Skyrmions by the holonomy along lines of an instanton, which is only applicable to massless pions.

hep-th↗
arXiv2003

Polyhedra in physics, chemistry and geometry

In this article we review some problems in physics, chemistry and mathematics that lead naturally to a class of polyhedra which include the Platonic solids. Examples include the study of electrons on a sphere, cages of carbon atoms, central configurations of gravitating point particles, rare gas microclusters, soliton models of nuclei, magnetic monopole scattering and geometrical problems concerning point particles.

math-ph↗
arXiv2003

A Variant of K-theory: K_{+-}

We describe a variant of K-theory for spaces with involution, built from vector bundles which are sent to their negative under the involution.

math.KT↗
arXiv2002

Projective planes, Severi varieties and spheres

A classical result asserts that the complex projective plane modulo complex conjugation is the 4-dimensional sphere. We generalize this result in two directions by considering the projective planes over the normed real division algebras and by considering the complexifications of these four projective planes.

math.DG↗
arXiv2002

M-Theory Dynamics On A Manifold Of G_2 Holonomy

We analyze the dynamics of M-theory on a manifold of G_2 holonomy that is developing a conical singularity. The known cases involve a cone on CP^3, where we argue that the dynamics involves restoration of a global symmetry, SU(3)/U(1)^2, where we argue that there are phase transitions among three possible branches corresponding to three classical spacetimes, and S^3 x S^3 and its quotients, where we recover and extend previous results about smooth continuations between different spacetimes and relations to four-dimensional gauge theory.

hep-th↗
arXiv2001

The Geometry of Point Particles

There is a very natural map from the configuration space of n distinct points in Euclidean 3-space into the flag manifold U(n)/U(1)^n, which is compatible with the action of the symmetric group. The map is well-defined for all configurations of points provided a certain conjecture holds, for which we provide numerical evidence. We propose some additional conjectures, which imply the first, and test these numerically. Motivated by the above map, we define a geometrical multi-particle energy function and compute the energy minimizing configurations for up to 32 particles. These configurations comprise the vertices of polyhedral structures which are dual to those found in a number of complicated physical theories, such as Skyrmions and fullerenes. Comparisons with 2-particle and 3-particle energy functions are made. The planar restriction and the generalization to hyperbolic 3-space are also investigated.

hep-th↗
arXiv2001

Nahm's equations, configuration spaces and flag manifolds

We give a positive answer to the Berry-Robbins problem for any compact Lie group G, i.e. we show the existence of a smooth W-equivariant map from the space of regular triples in a Cartan subalgebra to the flag manifold G/T. This map is constructed via solutions to Nahm's equations and it is compatible with the SO(3) action, where SO(3) acts on G/T via a regular homomorphism from SU(2) to G. We then generalize this picture to include an arbitrary homomorphism from SU(2) to G. This leads to an interesting geometrical picture which appears to be related to the Springer representation of the Weyl group and the work of Kazhdan and Lusztig on representations of Hecke algebras.

math.RT↗