Gauge transformation of a vector bundle - Revision history

Vipul: 5 revisions

2008-05-18T19:41:05Z

5 revisions

← Older revision	Revision as of 19:41, 18 May 2008
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Vipul: Gauge transformation moved to Gauge transformation of a vector bundle

2008-04-06T21:10:31Z

Gauge transformation moved to Gauge transformation of a vector bundle

← Older revision	Revision as of 21:10, 6 April 2008
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Vipul at 20:23, 6 April 2008

2008-04-06T20:23:21Z

← Older revision		Revision as of 20:23, 6 April 2008
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	===The set of connections===		===The set of connections===

			{{further\|[[Gauge group acts on affine space of connections]]}}
	The gauge transformations also act on the set of all connections. Namely, given any connection, which after all gives maps from sections of the vector bundle to sections of the vector bundle, we can apply the gauge transformation on it by conjugation. It is easy to check that the resultant mapping is again a connection.		The gauge transformations also act on the set of all connections. Namely, given any connection, which after all gives maps from sections of the vector bundle to sections of the vector bundle, we can apply the gauge transformation on it by conjugation. It is easy to check that the resultant mapping is again a connection.

Vipul: /* What gauge transformations act upon */

2008-04-06T19:56:49Z

What gauge transformations act upon

← Older revision		Revision as of 19:56, 6 April 2008
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	==What gauge transformations act upon==		==What gauge transformations act upon==

	===The space of all sections===		===The space and sheaf of all sections===

	By the natural action of the gauge transformation on the vector bundle, it also acts on the space of sections, namely, by acting on the value of the section at each point as per the gauge transformation.		By the natural action of the gauge transformation on the vector bundle, it also acts on the space of sections, namely, by acting on the value of the section at each point as per the gauge transformation.

	~~Since~~ the ~~space~~ of sections of a vector bundle ~~forms~~ a ~~vector space in its own right,~~ the ~~gauge transformations sit inside the space of linear automorphisms of this~~ vector ~~bundle.~~		It also acts on the [[sheaf of sections of a vector bundle\|sheaf of sections]]: this is a sheaf which associates to every open subset the vector space of sections over that open subset.

	~~Gauge transformations also preserve smoothness of sections. Hence, we can think of the gauge group as sitting inside the group of linear automorphisms of the~~ space of ~~smooth~~ sections ~~of the differential manifold~~.

	===Tensor powers of the vector bundle===		===Tensor powers of the vector bundle===

Vipul: Started the page

2007-03-08T12:06:30Z

Started the page

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	The collection of all gauge transformations of a vector bundle is termed the [[gauge group]] of the bundle.		The collection of all gauge transformations of a vector bundle is termed the [[gauge group]] of the bundle.

			==What gauge transformations act upon==

			===The space of all sections===

			By the natural action of the gauge transformation on the vector bundle, it also acts on the space of sections, namely, by acting on the value of the section at each point as per the gauge transformation.

			Since the space of sections of a vector bundle forms a vector space in its own right, the gauge transformations sit inside the space of linear automorphisms of this vector bundle.

			Gauge transformations also preserve smoothness of sections. Hence, we can think of the gauge group as sitting inside the group of linear automorphisms of the space of smooth sections of the differential manifold.

			===Tensor powers of the vector bundle===

			We know that automorphisms of a vector space also act on tensor powers of that vector space. In a similar way, gauge transformations of a vector bundle also act on tensor powers of that vector bundle. In fact, they also act on the dual of the tensor bundle. Hence, they act on all tensor products of tensor powers of that vector bundle and its dual vector bundle.

			===The set of connections===

			The gauge transformations also act on the set of all connections. Namely, given any connection, which after all gives maps from sections of the vector bundle to sections of the vector bundle, we can apply the gauge transformation on it by conjugation. It is easy to check that the resultant mapping is again a connection.

			All these actions of gauge transformations are mutually compatible. In some sense, we can thus think of the gauge group as acting on the entire colossum of data: the vector bundle, the dual, the sections, the tensor powers, the connections. Roughly, we can apply the gauge transformation to each of these uniformly everywhere.

			==Invariance under gauge transformations==

			Given a gauge transformation, we are interested in those vectors, tensors, vector fields, tensor fields, and connections that are invariant under that gauge transformation. A gauge field that is invariant under a particular gauge transformation is termed ''gauge-invariant'' for that gauge transformation.

Vipul at 10:57, 8 March 2007

2007-03-08T10:57:59Z

New page

==Definition==

===Given data===

* A [[differential manifold]] <math>M</math>
* A [[vector bundle]] <math>E</math> over <math>M</math>

===Definition part===

A gauge transformation is a differentiable automorphism of <math>E</math> as a vector bundle over <math>M</math>. In other words, it is a map from <math>E</math> to <math>E</math> that:

* Is a diffeomorphism of <math>E</math> as a manifold
* Preserves the vector space over any point in <math>M</math> and restricts to a linear automorphism in each such subspace.

The collection of all gauge transformations of a vector bundle is termed the [[gauge group]] of the bundle.