Symmetries play a central role in both physics and mathematics. In physics, they can be found at the heart of practically any theory where they encode an invariance of the theory under certain transformations. As a simple example, one could think of the translational invariance of classical mechanics: the way in which an apple falls from a tree does not depend on the location of the garden. More interesting (but more confusing) are symmetries in Einstein’s theory of special relativity. Here we find that the speed of light emitted by a torch of a person at rest coincides with the speed of light coming from a torch held by a person who is moving at constant velocity; the symmetries involved are given in terms of the so-called Lorentz transformations. In mathematics, symmetries appear for example in the theory of group actions on some space. One could think here of the set of rotations in three dimensions acting on an ordinary sphere. Also, mathematics allows for a more general type of symmetries, known as quantum symmetries; they are described in quantum group theory. Such symmetries are supposed to act on so-called noncommutative spaces. Notice that on an ordinary space (think of the plane), we can choose coordinates which are ordinary numbers indicating the position on this space (say the (x,y)-coordinates on the plane). A noncommutative space can be described in a similar manner, with the only (but drastic) difference that the coordinates are not numbers anymore but abstract objects which in general do not even commute (in the sense that x·y6=y·x). The description of such spaces forms the basic subject of Alain Connes’ noncommutative geometry [27]. A beautiful synthesis between mathematics and physics is found in Yang-Mills theory. This theory forms the basis of the celebrated Standard Model of physics which provides a highly accurate description of interactions between particles at a subatomic scale. Symmetries arise in the form of Lie groups. For example, the Lie group SU(2) lies at the heart of the theory of the weak interactions; it will be of central interest to us in what follows. Yang-Mills theory is defined in terms of a Yang-Mills action, expressing the energy of the configuration. We stress here the physical importance of finding the absolute minima of such an action; they are given by configurations called instantons. The mathematical structure behind Yang-Mills theory is the theory of connections on principal bundles. The ideas of Yang-Mills theory culminated in Donaldson’s construction of invariants of smooth four-dimensional manifolds [42] in which a central role is played by instantons. We are interested in “quantum versions” of two different parts of this Yang-Mills theory. The first one is concerned with the symmetry alone, and considers the quantum symmetry group SUq(2) in the framework of Connes’ noncommutative geometry. The second one leaves the symmetry group as it is and considers a formulation of SU(2) Yang-Mills theory on noncommutative spaces, still in the same framework of noncommutative geometry. The motivation for this is two-fold. Firstly, there is the idea that (quantum) Yang-Mills theories on quantum spaces or with quantum symmetries behave –in some sense– better than on ordinary spaces. Secondly, Alain Connes’ noncommutative geometry has all the ingredients for the formulation of Yang-Mills theory. For instance, the choice of a certain slightly noncommutative space allows for a derivation of the successful Standard Model of physics from basic principles [21, 22]. We construct in Part I a quantum version of the symmetry group SU(2) described above, and connect this quantum group with the noncommutative geometry of Connes. In this way, we describe the geometry of the quantum group SUq(2) as a noncommutative space. A guiding principle is provided by imposing symmetry under certain transformations; there are two quantum symmetries and we impose invariance –or equivariance– under their action. In Part II, we consider a formulation of SU(2) Yang-Mills theory on noncommutative spaces. In particular, we explore the geometry of a noncommutative principal bundle and define a Yang-Mills action on a four-dimensional noncommutative sphere. On the way, we encounter more quantum symmetries. We discuss a (infinitesimal) quantum version of the five dimensional rotation group acting as symmetries on the noncommutative four-sphere as well as the twisted conformal transformations. The latter gives rise to a family of infinitesimal instantons which are the minima of the Yang-Mills action. This thesis consists for a great part of the articles [39, 93] (Part I) and [68, 69] (Part II). We added several remarks and considerations, together with some introductory material, collected in the appendix.

The geometry of noncommutative spheres and their symmetries(2005 Oct 17).

The geometry of noncommutative spheres and their symmetries

-
2005-10-17

Abstract

Symmetries play a central role in both physics and mathematics. In physics, they can be found at the heart of practically any theory where they encode an invariance of the theory under certain transformations. As a simple example, one could think of the translational invariance of classical mechanics: the way in which an apple falls from a tree does not depend on the location of the garden. More interesting (but more confusing) are symmetries in Einstein’s theory of special relativity. Here we find that the speed of light emitted by a torch of a person at rest coincides with the speed of light coming from a torch held by a person who is moving at constant velocity; the symmetries involved are given in terms of the so-called Lorentz transformations. In mathematics, symmetries appear for example in the theory of group actions on some space. One could think here of the set of rotations in three dimensions acting on an ordinary sphere. Also, mathematics allows for a more general type of symmetries, known as quantum symmetries; they are described in quantum group theory. Such symmetries are supposed to act on so-called noncommutative spaces. Notice that on an ordinary space (think of the plane), we can choose coordinates which are ordinary numbers indicating the position on this space (say the (x,y)-coordinates on the plane). A noncommutative space can be described in a similar manner, with the only (but drastic) difference that the coordinates are not numbers anymore but abstract objects which in general do not even commute (in the sense that x·y6=y·x). The description of such spaces forms the basic subject of Alain Connes’ noncommutative geometry [27]. A beautiful synthesis between mathematics and physics is found in Yang-Mills theory. This theory forms the basis of the celebrated Standard Model of physics which provides a highly accurate description of interactions between particles at a subatomic scale. Symmetries arise in the form of Lie groups. For example, the Lie group SU(2) lies at the heart of the theory of the weak interactions; it will be of central interest to us in what follows. Yang-Mills theory is defined in terms of a Yang-Mills action, expressing the energy of the configuration. We stress here the physical importance of finding the absolute minima of such an action; they are given by configurations called instantons. The mathematical structure behind Yang-Mills theory is the theory of connections on principal bundles. The ideas of Yang-Mills theory culminated in Donaldson’s construction of invariants of smooth four-dimensional manifolds [42] in which a central role is played by instantons. We are interested in “quantum versions” of two different parts of this Yang-Mills theory. The first one is concerned with the symmetry alone, and considers the quantum symmetry group SUq(2) in the framework of Connes’ noncommutative geometry. The second one leaves the symmetry group as it is and considers a formulation of SU(2) Yang-Mills theory on noncommutative spaces, still in the same framework of noncommutative geometry. The motivation for this is two-fold. Firstly, there is the idea that (quantum) Yang-Mills theories on quantum spaces or with quantum symmetries behave –in some sense– better than on ordinary spaces. Secondly, Alain Connes’ noncommutative geometry has all the ingredients for the formulation of Yang-Mills theory. For instance, the choice of a certain slightly noncommutative space allows for a derivation of the successful Standard Model of physics from basic principles [21, 22]. We construct in Part I a quantum version of the symmetry group SU(2) described above, and connect this quantum group with the noncommutative geometry of Connes. In this way, we describe the geometry of the quantum group SUq(2) as a noncommutative space. A guiding principle is provided by imposing symmetry under certain transformations; there are two quantum symmetries and we impose invariance –or equivariance– under their action. In Part II, we consider a formulation of SU(2) Yang-Mills theory on noncommutative spaces. In particular, we explore the geometry of a noncommutative principal bundle and define a Yang-Mills action on a four-dimensional noncommutative sphere. On the way, we encounter more quantum symmetries. We discuss a (infinitesimal) quantum version of the five dimensional rotation group acting as symmetries on the noncommutative four-sphere as well as the twisted conformal transformations. The latter gives rise to a family of infinitesimal instantons which are the minima of the Yang-Mills action. This thesis consists for a great part of the articles [39, 93] (Part I) and [68, 69] (Part II). We added several remarks and considerations, together with some introductory material, collected in the appendix.
Van Suijlekom, Walter Daniël
Dabrowski, Ludwik
Landi, Giovanni
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