Main fiber bundle

In mathematics , the main fiber bundle , or principal fiber bundle or principal bundle , is a concept of differential geometry with which twisted products are formalized and which is used, among other things, in physics to describe gauge field theories and especially Yang-Mills fields .

Products (trivial principal bundles)

Principal bundles generalize the notion of the Cartesian product of a space and a topological group . Like the Cartesian product , a principal bundle also has the following properties: ${\displaystyle X\times G}$ ${\displaystyle X}$ ${\displaystyle G}$ ${\displaystyle X\times G}$ ${\displaystyle P}$

1. A group operation from on in the same way as for the product space ${\displaystyle G}$ ${\displaystyle P}$ ${\displaystyle (x,g)h=(x,gh)}$
2. A projection mapping for the case of a product space simply the projection represents the first factor: . ${\displaystyle P}$ ${\displaystyle X,}$ ${\displaystyle (x,g)\to x}$

In contrast to product spaces, principal bundles do not have a preferred section , as is given in the product case by the neutral element of the group . So there is no preferred element for elements as identification of . Nor there is generally a continuous projection on which generalizes the projection on the second element of the product space: . Principal bundles can therefore have complex topologies that prevent the bundle from being represented as a product space, even if some additional assumptions are made. ${\displaystyle G}$ ${\displaystyle x\in X}$ ${\displaystyle P}$ ${\displaystyle (x,e)}$ ${\displaystyle G,}$ ${\displaystyle (x,g)\to g}$

Functions can be interpreted as cuts in the trivial principal bundle , namely as . Sections in principal bundles thus generalize the concept of G-valued mappings. ${\displaystyle F\colon X\rightarrow G}$ ${\displaystyle \pi \colon X\times G\rightarrow X}$ ${\displaystyle s(x)=(x,F(x))}$

definition

A principal bundle is a fiber bundle over a space with the projection provided with a continuous right operation (hereinafter noted as ) of a topological group , so that the operation maps each fiber onto itself (i.e. for all and all ) and the group freely (each point is only invariant under the neutral element of the group) and transitive (each point of a fiber is reached by each other by means of the group operation) on each fiber. The group is called the structure group of the principal bundle. ${\displaystyle P}$ ${\displaystyle X}$ ${\displaystyle \pi \colon P\to X}$ ${\displaystyle P\times G\rightarrow P}$ ${\displaystyle (p,g)\mapsto pg}$ ${\displaystyle G}$ ${\displaystyle \pi (pg)=\pi (p)}$ ${\displaystyle p\in P}$ ${\displaystyle g\in G}$ ${\displaystyle G}$

If and are smooth manifolds , the structure group is a Lie group and the operation itself is smooth , then the principal bundle is called a smooth principal bundle . ${\displaystyle P}$ ${\displaystyle X}$

Trivialization

As with any fiber bundle, the projection can be trivialized locally from a topological point of view: So there is an open environment for everyone , so that homeomorphic is closed . Each fiber is homeomorphic to the structural group understood as a topological space . A trivialization of a principal bundle is even possible taking into account the group operation: An equivariant homeomorphism can be chosen so that ${\displaystyle x\in X}$ ${\displaystyle U\subset X}$ ${\displaystyle \pi ^{-1}(U)}$ ${\displaystyle U\times G}$ ${\displaystyle G}$ ${\displaystyle \phi \colon \pi ^{-1}(U)\rightarrow U\times G}$

${\displaystyle \pi (ug)=\pi (u),\phi (ug)=\phi (u)g}$

for everyone . Each such local Trivialisation induces a local interface virtue , wherein the neutral element call. ${\displaystyle u\in \pi ^{-1}(U),g\in G}$ ${\displaystyle \phi }$ ${\displaystyle f\colon U\to P}$ ${\displaystyle f(x)=\phi ^{-1}((x,e))}$ ${\displaystyle e\in G}$

Conversely, each local interface induces a local Trivialisation given by with . The local trivializability thus follows from the existence of local cuts, which generally exist on fiber bundles. In contrast to general fiber bundles (consider the tangential bundle of a smooth manifold, for example), not only does global trivializability imply the existence of a global cut, but also the existence of a global cut implies trivializability. ${\displaystyle f\colon U\to P}$ ${\displaystyle \phi }$ ${\displaystyle \phi (p)=(\pi (p),g_{p})}$ ${\displaystyle f(\pi (p))g_{p}=p}$

In the physical context, the choice of a calibration can be understood as a (local or global depending on the situation) choice of a trivialization or a section.

Examples

Frame bundle

Let be a differentiable n-dimensional manifold. The frame bundle is the set of all bases of tangent spaces , with the canonical projection . The group acts transitive and loyal to the fibers. ${\displaystyle M}$ ${\displaystyle F(M)}$ ${\displaystyle T_{x}M,x\in M}$ ${\displaystyle \pi \colon F(M)\rightarrow M}$ ${\displaystyle G:=GL(n,\mathbb {R} )}$

Overlays

Galois overlays are principal bundles with the discrete group of deck transformations as a structure group.

Homogeneous spaces

Let be a Lie group and a closed subgroup, then there is a principal bundle with a structure group . ${\displaystyle G}$ ${\displaystyle H\subset G}$ ${\displaystyle \pi \colon G\rightarrow G/H}$ ${\displaystyle H}$

In topology and differential geometry, there are some use cases of the principal bundle. There are also applications of the principle bundles in physics . There they form a crucial part of the mathematical framework of gauge theories .

Associated vector bundles

In the case of , an associated complex vector bundle can be defined for each principal bundle by ${\displaystyle G=GL(n,\mathbb {C} )}$ ${\displaystyle G}$ ${\displaystyle \pi \colon P\rightarrow B}$ ${\displaystyle \Pi \colon E\rightarrow B}$

${\displaystyle E=(P\times \mathbb {C} ^{n})/\sim }$

with the equivalence relation

${\displaystyle (p,v)\sim (pg,g^{-1}v)\forall g\in GL(n,\mathbb {C} )}$.

Analogously, one can define an associated real vector bundle for every principal bundle. ${\displaystyle GL(n,\mathbb {R} )}$

For example, let us be a differentiable n-dimensional manifold and the frame bundle. Then the tangent bundle is the associated vector bundle for the canonical action of on . ${\displaystyle M}$ ${\displaystyle F(M)}$ ${\displaystyle TM}$ ${\displaystyle GL(n,\mathbb {R} )}$ ${\displaystyle \mathbb {R} ^{n}}$

Reduction of the structural group

A principal bundle can be reduced to a subgroup if the bundle has a cut. In particular, a principal bundle is trivial if and only if it can be reduced to the subgroup . ${\displaystyle G}$ ${\displaystyle P\rightarrow M}$ ${\displaystyle H\subset G}$ ${\displaystyle P/H\rightarrow M}$ ${\displaystyle \left\{1\right\}\subset G}$

Examples

Consider the frame bundle of an n-dimensional differentiable manifold that is a structure group . Then: ${\displaystyle F(M)\rightarrow M}$ ${\displaystyle G=GL(n,\mathbb {R} )}$

• the structure group can be reduced to exactly if the tangential bundle has linearly independent sections, ${\displaystyle GL(k,\mathbb {R} )\subset GL(n,\mathbb {R} )}$ ${\displaystyle n-k}$
• the structure group can always be reduced to, this corresponds to the choice of a Riemannian metric , ${\displaystyle O(n)}$
• the structure group can be reduced to if and only if the manifold is orientable . ${\displaystyle SO(n)}$

In the following, be an even number: ${\displaystyle n=2m}$

• the structural group can be reduced to if and only if the manifold is almost complex , ${\displaystyle GL(m,\mathbb {C} )\subset GL(2m,\mathbb {R} )}$
• if the manifold is symplectic , then the structural group can be reduced to. ${\displaystyle U(m)}$

In the following, be an odd number: ${\displaystyle n=2m+1}$

• if the manifold has a contact structure, then the structure group can be reduced to. ${\displaystyle U(m)\times \left\{1\right\}}$

Connection, curvature

→ Main article : Context (principal bundle)

An important role in the study of principal bundles is played by connected 1-forms and their curvature-2 forms . ${\displaystyle \omega \in \Omega ^{1}(P,{\mathfrak {g}})}$ ${\displaystyle \Omega =d\omega +{\tfrac {1}{2}}[\omega \wedge \omega ]\in \Omega ^{2}(M,{\mathfrak {g}})}$

Application: electromagnetism

In a charge-free the meet electric field and the magnetic field , the Maxwell equations . The fields have potentials and with and . However, these potentials are not unique, because and for any function there are the same fields. ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle E}$ ${\displaystyle B}$ ${\displaystyle \phi }$ ${\displaystyle A}$ ${\displaystyle E=-\operatorname {grad} \phi +{\frac {\partial A}{\partial t}}}$ ${\displaystyle B=\operatorname {rot} A}$ ${\displaystyle \phi ^{\prime }=\phi +{\frac {\partial f}{\partial t}}}$ ${\displaystyle A^{\prime }=A+\operatorname {grad} f}$ ${\displaystyle f}$

One considers the Minkowski space-time and the principal bundle with the form of connection . Their curvature gives the electromagnetic field : ${\displaystyle M=\mathbb {R} ^{4}}$ ${\displaystyle M\times S^{1}}$ ${\displaystyle \omega =\mathrm {d} \theta +\phi \,\mathrm {d} t+A_{1}\,\mathrm {d} x_{1}+A_{2}\,\mathrm {d} x_{2}+A_{3}\,\mathrm {d} x_{3}}$

${\displaystyle \Omega =\mathrm {d} \omega =-\Sigma E_{i}\,\mathrm {d} t\wedge \mathrm {d} x_{i}+\Sigma B_{i}\,\mathrm {d} x_{j}\wedge \mathrm {d} x_{k}.}$

The gauge transformations are of the form . ${\displaystyle \omega ^{\prime }=\omega +\mathrm {d} f}$

The Maxwell equations can be formulated as , where is the Hodge operator . ${\displaystyle \mathrm {d} *\Omega =0}$ ${\displaystyle *}$

literature

• David Bleecker: Gauge Theory and Variational Principles , Dover edition. Edition, Addison-Wesley Publishing, 1981, ISBN 0-486-44546-1 . .
• Jürgen Jost: Riemannian Geometry and Geometric Analysis , (4th ed.). Edition, Springer, New York 2005, ISBN 3-540-25907-4 . .
• RW Sharpe: Differential Geometry: Cartan's Generalization of Klein's Erlangen Program . Springer, New York 1997, ISBN 0-387-94732-9 . .
• Norman Steenrod: The Topology of Fiber Bundles . Princeton University Press, Princeton 1951, ISBN 0-691-00548-6 . .
• Martin Schottenloher: Geometry and Symmetry in Physics . vieweg, Braunschweig 1995, ISBN 3-528-06565-6 . .

Web links

• Helga Baum : Lecture on gauge field theory (PDF; 584 kB)

Individual evidence

1. ↑ Pierre Deligne, Pavel Etingof, Daniel Freed, Lisa Jeffrey, David Kazhdan, John Morgan, David Morrison, Edward Witten (Eds.): Quantum Fields and Strings: A Course for Mathematicians . American Mathematical Society , 1999, ISBN 0-8218-1987-9 , pp. 18 .