aarjaneiro/planetmath huggingface dataset:
universal bundle

universal bundle

Let

G

be a

topological group

. A

universal bundle

for

G

is a

principal bundle

p

:

E

⁢

G

→

B

⁢

G

such that for any principal bundle

π

:

E

→

B

, with

B

a

CW-complex

, there is a map

φ

:

B

→

B

⁢

G

, unique up to

homotopy

, such that the

pullback bundle

φ

*

⁢

(

p

)

is

equivalent

to

π

, that is such that there is a

bundle map

φ

′

.

\xymatrix

⁢

E

⁢

\ar

⁢

[

d

]

π

⁢

\ar

⁢

[

r

]

φ

′

⁢

(

E

)

⁢

&

⁢

E

⁢

G

⁢

\ar

⁢

[

d

]

p

⁢

B

⁢

\ar

⁢

[

r

]

φ

′

⁢

(

B

)

⁢

&

⁢

B

⁢

G

with

φ

′

⁢

(

B

)

=

φ

, such that any bundle map of any bundle over

B

extending

φ

factors uniquely through

φ

′

.

As is obvious from the

universal property

, the universal bundle for a group

G

is unique up to unique

homotopy equivalence

.

The

base space

B

⁢

G

is often called a

classifying space

of

G

, since

homotopy classes

of maps to it from a given space classify

G

-bundles over that space.

There is a useful criterion for

universality

: a bundle is

universal

if and only if all the

homotopy groups

of

E

⁢

G

, its total space, are

trivial

. This allows us to

construct

the universal bundle any subgroup from that of a larger group. Assume

H

≤

G

and that

p

:

E

⁢

G

→

B

⁢

G

is a universal bundle for

G

. Then

H

also acts freely on

E

⁢

G

which is contractable so

p

H

:

E

⁢

H

=

E

⁢

B

→

B

⁢

H

=

E

⁢

B

/

H

must be a universal bundle for

H

.

In 1956, John Milnor gave a general construction of the universal bundle for any topological group

G

(see

Annals of Mathematics

, Second Series, Volume 63 Issue 2 and Issue 3 for details). His construction uses the infinite join of the group

G

with itself to define the total space of the universal bundle.

Title

universal bundle

Canonical name

UniversalBundle

Date of creation

2013-03-22 13:07:21

Last modified on

2013-03-22 13:07:21

Owner

mps (409)

Last modified by

mps (409)

Numerical id

14

Author

mps (409)

Entry type

Definition

Classification

msc 55R10

Classification

msc 55R15

Defines

classifying space

Generated on Sat Feb 10 11:45:11 2018 by

LaTeXML
some examples of universal bundles

some examples of universal bundles

The

universal bundle

for a

topological group

G

is usually written as

π

:

E

⁢

G

→

B

⁢

G

. Any principal

G

-bundle for which the total space is

contractible

is

universal

; this will help us to find universal bundles without worrying about Milnor’s construction of

E

⁢

G

involving infinite joins.

•

G

=

ℤ

2

:

E

⁢

ℤ

2

=

S

∞

and

B

⁢

ℤ

2

=

ℝ

⁢

P

∞

.

•

G

=

ℤ

n

:

E

⁢

ℤ

n

=

S

∞

and

B

⁢

ℤ

n

=

S

∞

/

ℤ

n

. Here

ℤ

n

acts on

S

∞

(considered as a subset of a

separable

complex Hilbert space) via

multiplication

with an

n

-th

root of unity

.

•

G

=

ℤ

n

:

E

⁢

ℤ

n

=

ℝ

n

and

B

⁢

ℤ

n

=

T

n

.

•

More generally, if

G

is any

discrete group

then one can take

B

⁢

G

to be any

Eilenberg-Mac Lane space

K

⁢

(

G

,

1

)

and

E

⁢

G

to be its

universal cover

. Indeed

E

⁢

G

is

simply connected

, and it follows from the

lifting theorem

that

π

n

⁢

(

E

⁢

G

)

=

0

for

n

≥

0

. This example includes the previous three and many more.

•

G

=

S

1

:

E

⁢

S

1

=

S

∞

and

B

⁢

S

1

=

ℂ

⁢

P

∞

.

•

G

=

S

⁢

U

⁢

(

2

)

:

E

⁢

S

⁢

U

⁢

(

2

)

=

S

∞

and

B

⁢

S

⁢

U

⁢

(

2

)

=

ℍ

⁢

P

∞

.

•

G

=

O

⁢

(

n

)

, the

n

-th

orthogonal group

:

E

⁢

O

⁢

(

n

)

=

V

⁢

(

∞

,

n

)

, the manifold of frames
of

n

orthonormal vectors in

ℝ

∞

, and

B

⁢

O

⁢

(

n

)

=

G

⁢

(

∞

,

n

)

, the
Grassmanian of

n

-planes in

ℝ

∞

. The

projection map

is
taking the

subspace

spanned by a frame of vectors.

Title

some examples of universal bundles

Canonical name

SomeExamplesOfUniversalBundles

Date of creation

2013-03-22 13:12:05

Last modified on

2013-03-22 13:12:05

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

11

Author

bwebste (988)

Entry type

Example

Classification

msc 55R15

Classification

msc 55R10

Synonym

universal family of spaces

Related topic

CategoryOfQuantumAutomata

Defines

Hilbert bundle

Generated on Sat Feb 10 11:45:13 2018 by

LaTeXML
Hopf bundle

Hopf bundle

Consider

S

3

⊂

ℝ

4

=

ℂ

2

. The

structure

of

ℂ

2

gives a map

ℂ

2

-

{

0

}

→

ℂ

⁢

P

1

, the

complex projective line

by the

natural projection

.
Since

ℂ

⁢

P

1

is

homeomorphic

to

S

2

, by

restriction

to

S

3

, we get a
map

π

:

S

3

→

S

2

. We call this the Hopf bundle.

This is a principal

S

1

-bundle (

http://planetmath.org/PrincipalBundle

), and a

generator

of

π

3

⁢

(

S

2

)

. From the

long exact sequence

of the bundle (

http://planetmath.org/LongExactSequenceLocallyTrivialBundle

):

⋯

⁢

π

n

⁢

(

S

1

)

→

π

n

⁢

(

S

3

)

→

π

n

⁢

(

S

2

)

→

⋯

we get that

π

n

⁢

(

S

3

)

≅

π

n

⁢

(

S

2

)

for all

n

≥

3

. In particular,

π

3

⁢

(

S

2

)

≅

π

3

⁢

(

S

3

)

≅

ℤ

.

Title

Hopf bundle

Canonical

name

HopfBundle

Date of creation

2013-03-22 13:20:04

Last modified on

2013-03-22 13:20:04

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Definition

Classification

msc 55R25

Synonym

Hopf fibration

Generated on Sat Feb 10 11:45:18 2018 by

LaTeXML
fibration

fibration

A

fibration

is a map satisfying the

homotopy lifting property

. This is easily seen to be

equivalent

to the following:

A map

f

:

X

→

Y

is a fibration if and only if there is a

continuous function

which given a path,

ϕ

, in

Y

and a point,

x

, lying above

ϕ

⁢

(

0

)

, returns a lift of

ϕ

, starting at

x

.

Let

D

2

denote the set of complex numbers with

modulus

less than or equal to 1. An example of a fibration is the map

g

:

D

2

→

[

-

1

,

1

]

sending a

complex number

z

to

r

⁢

e

⁢

(

z

)

.

Note that if we restrict

g

to the

boundary

of

D

2

, we do not get a fibration. Although we may still lift any path to begin at a prescribed point, we cannot make this assignment continuously.

Another class of fibrations are found in

fibre bundles

.

Title

fibration

Canonical name

Fibration

Date of creation

2013-03-22 15:37:57

Last modified on

2013-03-22 15:37:57

Owner

whm22 (2009)

Last modified by

whm22 (2009)

Numerical id

5

Author

whm22 (2009)

Entry type

Definition

Classification

msc 55R65

Related topic

fibremap

Related topic

FibreBundle

Related topic

LocallyTrivialBundle

Related topic

LongExactSequenceLocallyTrivialBundle

Related topic

homotopyliftingproperty

Related topic

cofibration

Defines

fibration

Generated on Sat Feb 10 11:45:25 2018 by

LaTeXML
homotopy lifting property

homotopy lifting property

A map

f

:

X

→

Y

satisfies the

homotopy lifting property

if given any space

A

and a map

g

:

A

→

X

and a

homotopy

h

of

f

∘

g

, we have a homotopy

h

′

of

g

, satisfying

f

∘

h

′

=

h

.

Title

homotopy lifting property

Canonical name

HomotopyLiftingProperty

Date of creation

2013-03-22 15:38:01

Last modified on

2013-03-22 15:38:01

Owner

whm22 (2009)

Last modified by

whm22 (2009)

Numerical id

6

Author

whm22 (2009)

Entry type

Definition

Classification

msc 55R65

Related topic

fibremap

Related topic

FibreBundle

Related topic

LocallyTrivialBundle

Related topic

LongExactSequenceLocallyTrivialBundle

Related topic

fibration

Related topic

homotopyextensionproperty

Defines

homotopy lifting property

Generated on Sat Feb 10 11:45:26 2018 by

LaTeXML
wild

wild

Let

S

be a set in

ℝ

n

and suppose that

S

is triangulable.
(

S

is

triangulable

means that when regarded as a space, it has a triangulation.)

If there is a

homeomorphism

h

:

ℝ

n

→

ℝ

n

such that

h

⁢

(

S

)

is a

polyhedron

, we say that

S

is

tamely imbedded

.

If

S

is triangulable but no such homeomorphism exists

S

is said to be

wild

.

In

ℝ

2

every 1-sphere is tamely imbedded. But
in

ℝ

3

there are wild arcs, 1-spheres and 2-spheres.

Title

wild

Canonical

name

Wild

Date of creation

2013-03-22 16:52:54

Last modified on

2013-03-22 16:52:54

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 55S37

Defines

tamely imbedded

Defines

triangulable

Generated on Sat Feb 10 11:45:28 2018 by

LaTeXML
groupoid category

groupoid category

Definition 0.1

.

Groupoid categories

, or

categories of groupoids

, can be defined
simply by considering a

groupoid

as a

category

𝒢

1

with all invertible

morphisms

, and objects
defined by the groupoid class or set of groupoid elements; then, the groupoid category,

𝒢

2

,
is defined as the

2

-category

whose objects are

𝒢

1

categories (groupoids), and whose morphisms are

functors

of

𝒢

1

categories consistent with the definition of

groupoid homomorphisms

, or in the case of

topological groupoids

, consistent as well with topological groupoid

homeomorphisms

(

http://planetmath.org/Homeomorphism

).

Example 0.1 :

The

2

-category of

Lie groupoids

is an example of a groupoid category, or

2

-category of groupoids

.

Definition 0.2

.

The

2

-category of Lie groupoids

G

L

has Lie groupoids as objects, and for any two such objects

𝐆

𝐋

and

𝐇

𝐋

there is a hom-category

h

⁢

o

⁢

m

⁢

(

𝐆

𝐋

,

𝐇

𝐋

)

=

B

⁢

B

⁢

(

𝐆

𝐋

,

𝐇

𝐋

)

,

where

B

⁢

B

⁢

(

𝐆

𝐋

,

𝐇

𝐋

)

,

is a category whose objects are

𝐆

𝐋

–

𝐇

𝐋

bibundles of the Lie groupoids

𝐆

𝐋

and

𝐇

𝐋

, respectively over

M

and

N

, and whose morphisms are arrows

f

:

E

→

E

′

between such bibundles

E

and

E

′

that commute with the bundles

π

1

:

E

→

M

and

π

2

:

E

′

→

N

:

\xymatrix

⁢

E

⁢

\ar

⁢

[

r

⁢

r

]

f

⁢

\ar

⁢

[

d

⁢

r

]

π

1

⁢

&

⁢

&

⁢

E

′

⁢

\ar

⁢

[

d

⁢

l

]

π

2

⁢

&

⁢

M

⁢

&

\xymatrix

⁢

E

⁢

\ar

⁢

[

r

⁢

r

]

f

⁢

\ar

⁢

[

d

⁢

r

]

π

1

′

⁢

&

⁢

&

⁢

E

′

⁢

\ar

⁢

[

d

⁢

l

]

π

2

′

⁢

&

⁢

N

⁢

&

,

consistent respectively with the

𝐆

𝐋

– and

𝐇

𝐋

– actions. Moreover, the

composition

of two bibundles is given by the Hilsum-Skandalis

product

.

Remark 0.1 :

The

2-category of groupoids

𝒢

2

, plays a

central

role in the generalised, categorical Galois theory involving

fundamental groupoid functors

.

Title

groupoid category

Canonical name

GroupoidCategory

Date of creation

2013-03-22 18:12:00

Last modified on

2013-03-22 18:12:00

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

55

Author

bci1 (20947)

Entry type

Topic

Classification

msc 55U05

Classification

msc 55U35

Classification

msc 55U40

Classification

msc 18G55

Classification

msc 18B40

Synonym

category of groupoids

Synonym

2-category of groupoids

Synonym

category of groupoids and groupoid homomorphisms/homeomorphisms

Related topic

2Category

Related topic

FundamentalGroupoid

Related topic

Homeomorphism

Related topic

HigherDimensionalAlgebraHDA

Related topic

GroupoidAndGroupRepresentationsRelatedToQuantumSymmetries

Related topic

GeneralizedVanKampenTheoremsHigherDimensional

Related topic

Groupoids

Related topic

GroupoidHomomorphisms

Related topic

QuantumFundamentalGroupoids

Related topic

CategoryTh

Defines

groupoid category

Defines

groupoid 2-category

Defines

2-category of groupoids

Defines

2-category of Lie groupoids

Generated on Sat Feb 10 11:45:49 2018 by

LaTeXML
R-category

R-category

Definition 0.1

.

An

R

-

category

A

is a

category equipped with an

R

-module structure on each hom set such that the composition is

R

-

bilinear

. More precisely, let us assume for instance that we are given a commutative ring

R

with

identity

. Then a small

R

-category–or equivalently an

R

-algebroid

– will be defined as a category enriched in the

monoidal category

of

R

-modules, with respect to the

monoidal

structure of

tensor product

. This means simply that for all objects

b

,

c

of

A

, the set

A

⁢

(

b

,

c

)

is given the structure of an

R

-module, and composition

A

⁢

(

b

,

c

)

×

A

⁢

(

c

,

d

)

⟶

A

⁢

(

b

,

d

)

is

R

–bilinear, or is a

morphism

of

R

-modules

A

⁢

(

b

,

c

)

⊗

R

A

⁢

(

c

,

d

)

⟶

A

⁢

(

b

,

d

)

.

0.1

Note:

See also the

extension

of the

R-category

to the concept of

http://planetphysics.org/?op=getobj&from=objects&id=756

R-supercategory.

References

1

R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales–Bangor, Maths Preprint, 1986.

2

G. H. Mosa:

Higher dimensional algebroids and Crossed
complexes

, PhD thesis, University of Wales, Bangor, (1986). (supervised by R. Brown).

3

I. C. Baianu , James F. Glazebrook, and Ronald Brown. 2009.

Algebraic Topology

Foundations of

Supersymmetry and Symmetry Breaking

in

Quantum Field Theory

and

Quantum Gravity

: A Review.

SIGMA

5 (2009), 051, 70 pages.

a

⁢

r

⁢

X

⁢

i

⁢

v

:

0904.3644

,

d

⁢

o

⁢

i

:

10.3842

/

S

⁢

I

⁢

G

⁢

M

⁢

A

⁢

.2009.051

,

http://www.emis.de/journals/SIGMA/2009/051/

Symmetry

, Integrability and Geometry: Methods and Applications (SIGMA)

Title

R-category

Canonical name

Rcategory

Date of creation

2013-03-22 18:14:15

Last modified on

2013-03-22 18:14:15

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

16

Author

bci1 (20947)

Entry type

Definition

Classification

msc 55U05

Classification

msc 55U35

Classification

msc 55U40

Classification

msc 18G55

Classification

msc 18B40

Classification

msc 81R10

Classification

msc 81R50

Synonym

R-module category

Related topic

Algebroids

Related topic

HamiltonianAlgebroids

Related topic

RAlgebroid

Defines

morphism of R-modules

Defines

extension of R-algebroids over rings

Generated on Sat Feb 10 11:45:53 2018 by

LaTeXML
variable network topology

variable network topology

Definition 0.1

.

A

variable network topology

is defined by an indexed family
or class of

networks

of one-dimensional

C

⁢

W

-complexes

[

C

⁢

W

i

]

with

i

∈

I

in the

category

Top

of

topological spaces

with additional

homology

(or

cohomology

) axioms, rules, or properties of the underlying

C

⁢

W

-complexes, that specify the

transformation

maps as a

sequence

of homotopoy maps.

Title

variable network topology

Canonical name

VariableNetworkTopology

Date of creation

2013-03-22 18:15:42

Last modified on

2013-03-22 18:15:42

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

9

Author

bci1 (20947)

Entry type

Definition

Classification

msc 55U05

Classification

msc 55U35

Classification

msc 55U40

Classification

msc 18G55

Classification

msc 18B40

Synonym

variable network topology

Related topic

Supercategories3

Related topic

CWComplex

Related topic

HigherDimensionalAlgebraHDA

Related topic

2Category2

Related topic

GroupoidCDynamicalSystem

Related topic

VariableTopology8

Related topic

VariableTopology

Defines

sequence of different network topologies

Defines

esp. connectivities

Generated on Sat Feb 10 11:45:57 2018 by

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pure simplicial complex

pure simplicial complex

A finite

simplicial complex

is

pure

if each facet has the same

dimension

.

For example, any polytope is pure, but the simplicial complex

{

∅

,

a

,

b

,

c

,

d

,

a

⁢

b

,

a

⁢

c

,

b

⁢

c

,

c

⁢

d

,

a

⁢

b

⁢

c

}

is not pure because

c

⁢

d

and

a

⁢

b

⁢

c

are facets with different dimensions.

Title

pure simplicial complex

Canonical name

PureSimplicialComplex

Date of creation

2013-03-22 14:18:52

Last modified on

2013-03-22 14:18:52

Owner

mps (409)

Last modified by

mps (409)

Numerical id

5

Author

mps (409)

Entry type

Definition

Classification

msc 55U10

Classification

msc 52B99

Generated on Sat Feb 10 11:46:01 2018 by

LaTeXML
simplicial approximation

simplicial approximation

Let

K

and

L

be

simplicial complexes

and

f

:

|

K

|

→

|

L

|

be a

continuous function

.
A

simplicial mapping

g

:

|

K

|

→

|

L

|

which is

homotopic

to

f

is called
a

simplicial approximation

of

f

.

For example, suppose that

L

is the

closure

of an

n

-simplex and

a

0

is a vertex of

L

. Let

f

be a

continuous map

of

|

K

|

to

|

L

|

where

K

is some simplicial complex. Then the map

g

that sends all of

K

to

a

0

is
a simplicial approximation of

f

.

Title

simplicial approximation

Canonical name

SimplicialApproximation

Date of creation

2013-03-22 16:54:24

Last modified on

2013-03-22 16:54:24

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 55U10

Generated on Sat Feb 10 11:46:04 2018 by

LaTeXML
simplicial approximation theorem

simplicial approximation theorem

Let

f

:

|

K

|

→

|

L

|

be

continuous function

, where

|

K

|

and

|

L

|

are polyhedra having

triangulations

K

and

L

, respectively.

Then there is a

barycentric subdivision

K

(

s

)

of

K

and a continuous
function

g

:

|

K

|

→

|

L

|

such that

g

is a

simplicial map

from

K

(

s

)

to

|

L

|

and

g

is

homotopic

to

f

.

The theorem is due to J.W. Alexander.

References

1

J.W. Alexander ,

Combinatorial analysis situs

,
Trans. Amer. Math. Soc.

28

, 301-329, (1926)

Title

simplicial approximation theorem

Canonical name

SimplicialApproximationTheorem

Date of creation

2013-03-22 16:54:29

Last modified on

2013-03-22 16:54:29

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

5

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 55U10

Generated on Sat Feb 10 11:46:05 2018 by

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simplicial mapping

simplicial mapping

Let

K

and

L

be

simplicial complexes

and

f

:

K

→

L

.
If for each simplex

σ

n

of

K

, we have that

f

⁢

(

σ

n

)

is a simplex
of

L

, then

f

is said to be a

simplicial mapping

.

Title

simplicial mapping

Canonical name

SimplicialMapping

Date of creation

2013-03-22 16:54:27

Last modified on

2013-03-22 16:54:27

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

5

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 55U10

Defines

simplicial map

Generated on Sat Feb 10 11:46:10 2018 by

LaTeXML
locally compact Hausdorff spaces

locally compact Hausdorff spaces

1

Locally compact Hausdorff spaces

Definition 1.1

.

A

locally compact Hausdorff space

H

L

⁢

C

is a
locally compact topological space (

http://planetmath.org/LocallyCompact

)

(

X

L

⁢

C

,

τ

)

with

τ

being a

Hausdorff topology

(

http://planetmath.org/T2Space

), that is, if given any distinct points

x

,

y

∈

X

L

⁢

C

, there exist disjoint
sets

U

,

V

∈

τ

such that,

U

∩

V

=

∅

(that is,

open sets

), and with

x

and

y

satisfying the conditions that

x

∈

U

and

y

∈

V

.

Remark 1.1

.

An important, related

concept

to the locally compact Hausdorff space is that of a

locally compact

(topological)

groupoid

, which is a major concept for realizing

extended quantum symmetries

in
terms of

quantum groupoid

representations

in: quantum

algebraic topology

(QAT), topological QFT (TQFT), algebraic QFT (AQFT), axiomatic QFT, QCG, and

quantum gravity

(QG). This has also prompted the relatively recent

development

of the concepts of

homotopy

2-groupoid

and

homotopy

double

groupoid

of a
Hausdorff space

[

1

,

2

]

. It would be interesting to have also axiomatic definitions of these two important algebraic topology concepts that are consistent with the T2 axiom.

References

1

K.A. Hardie, K.H. Kamps and R.W. Kieboom., A homotopy 2-groupoid of a Hausdorff space,

Applied Cat. Structures

,

8

(2000): 209-234.

2

R. Brown, K.A. Hardie, K.H. Kamps and T. Porter, A

homotopy double groupoid

of a Hausdorff
space,

Theory and Applications of Categories

10

,(2002): 71-93.

Title

locally compact Hausdorff spaces

Canonical name

LocallyCompactHausdorffSpaces

Date of creation

2013-03-22 18:19:24

Last modified on

2013-03-22 18:19:24

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

19

Author

bci1 (20947)

Entry type

Topic

Classification

msc 55U40

Classification

msc 55-00

Synonym

locally compact T2Space

Related topic

CharacterizationOfT2Spaces

Related topic

T2Space

Related topic

LocallyCompact

Related topic

LocallyCompactGroupoids

Related topic

LocalCompactnessIsHereditaryForLocallyClosedSubspaces

Related topic

C_cH_lc

Related topic

ExampleOfParacompactTopologicalSpaces

Related topic

WeakTopologyOfTheSpaceOfRadonMeasures

Related topic

LocallyCompactQuantumGroup

Defines

Hausdorff topology

Defines

locally compact topological space

Generated on Sat Feb 10 11:46:15 2018 by

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supersymmetry

supersymmetry

Definition 0.1

.

Supersymmetry

or Poincaré, (extended) quantum symmetry is usually defined as an extension of ordinary spacetime symmetries obtained by adjoining

N

spinorial generators

Q

whose

anticommutator

yields a
translation generator:

{

Q

,

Q

}

=

{

P

}

.

As further explained in ref.

[

1

]

:

“This

(super)

symmetry…(of the

superspace

)… can be realized on ordinary fields (that are defined as certain

functions

of physical spacetime(s)) by transformations that mix bosons and fermions.

Such realizations suffice to study supersymmetry (one can write invariant actions, etc.) but are as
cumbersome and inconvenient as doing vector calculus component by component. A

compact

alternative to this ‘component field’ approach is given by the

superspace–superfield

approach

”, which is defined next.

Definition 0.2

.

Quantum superspace, or superspacetimes

, can be defined as an extension(s) of ordinary spacetime(s) to include
additional anticommuting coordinates, for example, in the form of

N

two-component Weyl spinors

θ

.

Definition 0.3

.

(Quantum) superfields

Ψ

⁢

(

x

,

θ

)

are

functions

defined over such superspaces, or superspacetimes.

Taylor series

expansions of the superfield functions can be then performed with respect to the anticommuting coordinates

θ

; this Taylor series has only a finite number of terms and the

series expansion

coefficients

obtained in this manner are the ordinary ‘component fields’ specified above.

Remarks:

Supersymmetry is expected to be manifested, or observable, in such superspaces, that is, the

supersymmetry algebras

are represented by translations and rotations involving

both

the spacetime and the anticommuting coordinates. Then, the transformations of the ‘component fields’ can be computed from the

Taylor expansion

of
the

translated and rotated superfields

. Especially important are those transformations that mix boson
and fermion symmetries; further details are found in ref.

[

2

]

.

References

1

J.S. Gates, Jr, et al. “Superspace”., arxiv-hep-th/0108200 preprint (1983).

2

“Preprint of 1,001 Lessons in Supersymmetry.”

http://arxiv.org/abs/hep-th/0108200

on line PDF.

Title

supersymmetry

Canonical name

Supersymmetry

Date of creation

2013-03-22 18:17:00

Last modified on

2013-03-22 18:17:00

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

22

Author

bci1 (20947)

Entry type

Definition

Classification

msc 55U40

Classification

msc 55-02

Classification

msc 81Q60

Classification

msc 81R50

Classification

msc 81R15

Synonym

extended quantum symmetry structures

Synonym

generalized double algebras

Synonym

supersymmetries

Related topic

QuantumAlgebraicTopology

Related topic

AlgebraicFoundationsOfQuantumAlgebraicTopology

Related topic

SuperfieldsSuperspace

Related topic

LieSuperalgebra3

Defines

extended quantum symmetry structures

Defines

both local and global

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