A gauge group is a group of gauge symmetries of the Yang–Mills gauge theory of principal connections on a principal bundle. Given a principal bundle with a structure Lie group , a gauge group is defined to be a group of its vertical automorphisms. This group is isomorphic to the group of global sections of the associated group bundle whose typical fiber is a group which acts on itself by the adjoint representation. The unit element of is a constant unit-valued section of .
At the same time, gauge gravitation theory exemplifies field theory on a principal frame bundle whose gauge symmetries are general covariant transformations which are not elements of a gauge group.
In the physical literature on gauge theory, a structure group of a principal bundle often is called the gauge group.
In quantum gauge theory, one considers a normal subgroup of a gauge group which is the stabilizer
of some point of a group bundle . It is called the pointed gauge group. This group acts freely on a space of principal connections. Obviously, . One also introduces the effective gauge group where is the center of a gauge group . This group acts freely on a space of irreducible principal connections.
If a structure group is a complex semisimple matrix group, the Sobolev completion of a gauge group can be introduced. It is a Lie group. A key point is that the action of on a Sobolev completion of a space of principal connections is smooth, and that an orbit space is a Hilbert space. It is a configuration space of quantum gauge theory.
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In mathematics, and especially differential geometry and mathematical physics, gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory which admits gauge symmetry. In mathematics theory means a mathematical theory, encapsulating the general study of a collection of concepts or phenomena, whereas in the physical sense a gauge theory is a mathematical model of some natural phenomenon.
In quantum field theory, gauge gravitation theory is the effort to extend Yang–Mills theory, which provides a universal description of the fundamental interactions, to describe gravity. Gauge gravitation theory should not be confused with the similarly-named gauge theory gravity, which is a formulation of (classical) gravitation in the language of geometric algebra. Nor should it be confused with Kaluza–Klein theory, where the gauge fields are used to describe particle fields, but not gravity itself.
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product of a space with a group . In the same way as with the Cartesian product, a principal bundle is equipped with An action of on , analogous to for a product space. A projection onto . For a product space, this is just the projection onto the first factor, . Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of .
This paper describes CosmoGattice, a modern package for lattice simulations of the dynamics of interacting scalar and gauge fields in an expanding universe. CosmoGattice incorporates a series of features that makes it very versatile and powerful: i) it is ...
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Kontsevich and Soibelman reformulated and slightly generalised the topological recursion of [43], seeing it as a quantisation of certain quadratic Lagrangians in T*V for some vector space V. KS topological recursion is a procedure which takes as initial da ...