Geometry and topology of chiral anomalies in gauge theories

Rennie, R.

doi:10.1080/00018739000101531

Cited by 7 publications

(3 citation statements)

References 723 publications

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“…The Lagrangian and action of a field theory should be invariant under the action of certain symmetry groups such as the Lorentz group, gauge symmetry, and conformal symmetry [21][22][23][24][25][26]. This means the Lagrangian for the fields, hence the laws of physics, are invariant under symmetry transformations.…”

Section: Introductionmentioning

confidence: 99%

Geometric Properties of Classical Yang-Mills Theory on Differentiable Manifolds

Bracken¹

2022

Manifolds III - Developments and Applications

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Gauge theories make up a class of physical theories that attempt to describe the physics of particles at a fundamental level. The purpose here is to study Yang-Mills theory at the classical level in terms of the geometry of fiber bundles and differentiable manifolds. It is shown how fundamental particles of bosonic and fermionic nature can be described mathematically. The Lagrangian for the basic interactions is presented and then put together in a unified form. Finally, some basic theorems are proved for a Yang-Mills on compact four-dimensional manifolds.

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Section: Introductionmentioning

confidence: 99%

Geometric Properties of Classical Yang-Mills Theory on Differentiable Manifolds

Bracken¹

2022

Manifolds III - Developments and Applications

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“…In other words, one studies the homotopy groups of certain Lie groups, and one finds that they are periodic, in that (Rennie 1990) π n (U (N )) = 0 if n is even and n < 2N , (3.6.1)…”

Section: Bott Periodicitymentioning

confidence: 99%

“…This is motivated by the analysis of a family D of elliptic operators D x , parametrized by the points x of a compact space, say X. The corresponding index is then defined by (Rennie 1990) . An outstanding open problem is to prove that the supersymmetric functional integral admits a K-theoretic interpretation.…”

Section: Elliptic Boundary Conditionsmentioning

confidence: 99%

Dirac Operators and Spectral Geometry

Esposito

1998

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Abstract. This paper is devoted to mathematical and physical properties of the Dirac operator and spectral geometry. Spin-structures in Lorentzian and Riemannian manifolds, and the global theory of the Dirac operator, are first analyzed.Elliptic boundary-value problems, index problems for closed manifolds and for manifolds with boundary, Bott periodicity and K-theory are then presented. This makes it clear why the Dirac operator is the most fundamental, in the theory of elliptic operators on manifolds. The topic of spectral geometry is developed by studying non-local boundary conditions of the Atiyah-Patodi-Singer type, and heat-kernel asymptotics for operators of Laplace type on manifolds with boundary. The emphasis is put on the functorial method, which studies the behaviour of differential operators, boundary operators and heat-kernel coefficients under conformal rescalings of the background metric. In the second part, a number of relevant physical applications are studied: non-local boundary conditions for massless spin-1/2 fields, massless spin-3/2 potentials on manifolds with boundary, geometric theory of massive spin-3/2 potentials, local boundary conditions in quantum supergravity, quark boundary conditions, one-loop quantum cosmology, conformally covariant operators and Euclidean quantum gravity. 1 CHAPTER ONE DIRAC OPERATOR IN PHYSICS AND MATHEMATICSAbstract. This introductory chapter begins with a review of Dirac's original derivation of the relativistic wave equation for the electron. The emphasis in then put on spin-structures in Lorentzian and Riemannian manifolds, and on the global theory of the Dirac operator, to prepare the ground for the topics studied in the following chapters. 2 The Dirac equationIn the development of theoretical physics, the Dirac operator finds its natural place in the attempt to obtain a relativistic wave equation for the electron.Although we shall be mainly concerned with elliptic operators in the rest of our book, we thus devote this section to Dirac's original argument (Dirac 1928, Dirac 1958. Indeed, it is well known that the relativistic Hamiltonian of a particle of mass m is where the p µ should be regarded as operators: one is familiar with the need to obtain wave equations which are linear in p 0 .To combine relativistic invariance with linearity in p 0 , and to obtain equivalence with Eq. (1.1.4), one looks for a wave equation which is rational and linear in p 0 , p 1 , p 2 , p 3 : 1.6) where α and β are independent of p. Since we are studying the (idealized) case when the electron moves in the absence of electromagnetic field, all space-time points are equivalent, and hence the operator in square brackets in Eq. (1.1.6) is independent of x. This implies in turn that α and β are independent of x, and commute with p and x operators. At a deeper level, α and β make it possible to obtain a relativistic description of the spin of the electron, i.e. an angular momentum which does not result from the translational motion of the electron.At this stage, one can multipl...

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