gauge group

gauge group

Let us start with two

vector bundles

E

and

F

over a space

B

E

=

(

∐

α

U

α

×

ℝ

n

)

/

{

g

α

⁢

β

}

and

F

=

(

∐

α

U

α

×

ℝ

m

)

/

{

h

α

⁢

β

}

The first objective is to show how to create a bundle called

Hom

⁡

(

E

,

F

)

. There are two different ways to do this. The first way is to observe that since for

vector spaces

V

,

W

that

Hom

⁡

(

V

,

W

)

≅

W

⊗

V

∗

(here just meaning the module of

homomorphisms

from

V

to

W

) thus if we take the bundle

Hom

⁡

(

E

,

F

)

:=

F

⊗

E

∗

then we have that since the fibers

(

F

⊗

E

∗

)

b

=

F

b

⊗

E

b

∗

we have what we want.

Another way of looking at the creation of the bundle

Hom

⁡

(

E

,

F

)

is to look at

representations

and what we would ideally like our bundle to look like. If we have that the bundle that

represents

F

is given by taking the principle

Gl

⁡

(

m

,

ℝ

)

bundle

P

F

=

(

∐

α

U

α

×

Gl

(

m

,

ℝ

)

)

/

{

h

α

⁢

β

}

and a

trivial representation

then the bundle afforded by the trivial representation is simply

F

. The same thing is true with the bundle

E

. We then have that if we look at the

structure group

of our proposed new bundle it should be

Gl

⁡

(

m

,

ℝ

)

×

Gl

⁡

(

n

,

ℝ

)

. The fibers of our proposed new bundle definitely should be

Hom

⁡

(

ℝ

n

,

ℝ

m

)

thus we have that the representation we are looking for should take something in our structure group and something in the fiber and give us something new in the fiber. The proposed representation is

ρ

⁢

(

A

,

B

)

⁢

(

U

)

=

A

∘

U

∘

B

-

1

. Then looking at the bundle associated to the representation of

ρ

gives us that if

P

=

P

F

×

P

E

=

(

∐

α

U

α

×

Gl

⁡

(

m

,

ℝ

)

×

Gl

⁡

(

n

,

ℝ

)

)

/

{

h

α

⁢

β

×

g

α

⁢

β

}

then

Hom

⁡

(

E

,

F

)

≡

P

×

ρ

Hom

⁡

(

ℝ

n

,

ℝ

m

)

we can similarly define

Aut

⁡

(

E

)

=

P

E

×

ρ

Gl

⁡

(

n

,

ℝ

)

The group of

sections

of

Aut

⁡

(

E

)

is called the

gauge group

. It is a group since we have that if

(

f

,

f

′

)

is a section of

Aut

⁡

(

E

)

and

(

τ

,

τ

′

)

is also a section of

Aut

⁡

(

E

)

then we have a

group operation

given by

composition

:

\xymatrix

⁢

E

⁢

\ar

⁢

[

r

]

τ

⁢

\ar

⁢

[

d

]

π

⁢

&

⁢

E

⁢

\ar

⁢

[

r

]

f

⁢

\ar

⁢

[

d

]

π

⁢

&

⁢

E

⁢

\ar

⁢

[

d

]

π

⁢

B

⁢

\ar

⁢

[

r

]

τ

′

⁢

&

⁢

B

⁢

\ar

⁢

[

r

]

f

′

⁢

&

⁢

B

The fact that these

bundle maps

are

isomorphisms

of bundles gives the

existence

of an

inverse

. The reason for this is that for each section

(

f

,

f

′

)

,

E

b

≅

E

f

′

⁢

(

b

)

since

f

b

is a vector space isomorphism. Thus we now have that if we look at the bundle map given by taking

(

f

-

1

,

(

f

′

)

-

1

)

)

(where

f

-

1

means

(

f

b

)

-

1

for each

b

∈

B

).

Associativity

is clear by composition of functions being associative and the

identity map

acts as the

identity element

.

Title

gauge group

Canonical name

GaugeGroup

Date of creation

2013-03-22 17:02:34

Last modified on

2013-03-22 17:02:34

Owner

sjm1979 (13837)

Last modified by

sjm1979 (13837)

Numerical id

9

Author

sjm1979 (13837)

Entry type

Definition

Classification

msc 55R25

Defines

bundle of homomorphisms

Defines

bundle of endomorphisms

Defines

automorphism bundle

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