superfields, superspace and supergravity

superfields, superspace and supergravity

0.1

Superspace, superfields, supergravity and Lie superalgebras.

In general, a

superfield

–or

quantized gravity field

- has a highly reducible representation of the supersymmetry algebra, and the problem of specifying a supergravity theory can be defined as a search for those

representations

that allow the construction of

consistent

local actions, perhaps considered
as either

quantum group

, or

quantum groupoid

, actions. Extending

quantum symmetries

to include
quantized gravity fields–specified as ‘superfields’– is called

supersymmetry

in current theories of Quantum Gravity. Graded ‘Lie’

algebras

(or

Lie superalgebras

)

represent

the quantum

operator

supersymmetries
by defining these simultaneously for both

fermion

(spin

1

/

2

) and

boson

(integer or 0 spin particles).

The quantized physical space with supersymmetric

properties

is then called a

‘superspace’

,
(another name for

‘quantized space with supersymmetry’

) in Quantum Gravity. The following subsection defines
these physical

concepts

in precise mathematical terms.

0.1.1

Mathematical definitions and propagation equations for superfields in superspace:
Graded Lie algebras

Supergravity, in essence, is an extended supersymmetric theory of
both matter and gravitation (

viz

. Weinberg, 1995

[

1

]

).
A first approach to supersymmetry relied on a curved ‘superspace’
(Wess and Bagger,1983

[

3

]

) and is analogous to supersymmetric gauge theories (see, for
example,

Sections

27.1 to 27.3 of Weinberg, 1995). Unfortunately,
a

complete

non–linear supergravity theory might be forbiddingly
complicated and furthermore, the constraints that need be made on
the graviton superfield appear somewhat subjective,
(according to Weinberg, 1995). In a different approach to supergravity,
one considers the physical

components

of the gravitational superfield
which can be then identified based on ‘flat-space’ superfield methods
(Chs. 26 and 27 of Weinberg, 1995). By implementing the

gravitational
weak-field approximation

one obtains several of the most important

consequences

of supergravity theory, including masses for the
hypothetical ‘gravitino’ and ‘gaugino particles’ whose

existence

might be
expected from supergravity theories. Furthermore, by adding on the
higher order terms in the gravitational constant to the
supersymmetric

transformation

, the general

coordinate

transformations form a

closed algebra

and the Lagrangian that
describes the interactions of the physical fields is then

invariant

under such transformations.The first quantization of such a flat-space
superfield would obviously involve its ‘

deformation

’, and as a result its corresponding

supersymmetry algebra

becomes

non–

commutative

.

0.1.2

Metric superfield

Because in supergravity both spinor and

tensor fields

are being
considered, the gravitational fields are represented in terms of

tetrads

,

e

μ

a

⁢

(

x

)

,

rather than in terms of Einstein’s general
relativistic metric

g

μ

⁢

ν

⁢

(

x

)

. The

connections

between
these two distinct representations are as follows:

g

μ

⁢

ν

⁢

(

x

)

=

η

a

⁢

b

⁢

e

μ

a

⁢

(

x

)

⁢

e

γ

b

⁢

(

x

)

,

(0.1)

with the general coordinates being

indexed by

μ

,

ν

,

etc.,
whereas

local coordinates

that are being defined in a locally
inertial

coordinate system

are labeled with

superscripts

a, b,
etc.;

η

a

⁢

b

is the

diagonal matrix

with

elements

+1, +1,
+1 and -1. The tetrads are invariant to two distinct types of

symmetry

transformations–the local

Lorentz transformations

:

e

μ

a

⁢

(

x

)

⟼

Λ

b

a

⁢

(

x

)

⁢

e

μ

b

⁢

(

x

)

,

(0.2)

(where

Λ

b

a

is an arbitrary

real matrix

), and the general
coordinate transformations:

x

μ

⟼

(

x

′

)

μ

⁢

(

x

)

.

(0.3)

In a weak gravitational field the tetrad may be represented as:

e

μ

a

⁢

(

x

)

=

δ

μ

a

⁢

(

x

)

+

2

⁢

κ

⁢

Φ

μ

a

⁢

(

x

)

,

(0.4)

where

Φ

μ

a

⁢

(

x

)

is small compared with

δ

μ

a

⁢

(

x

)

for
all

x

values, and

κ

=

8

⁢

π

⁢

G

, where G is Newton’s
gravitational constant. As it will be discussed next, the
supersymmetry algebra (SA) implies that the graviton has a
fermionic superpartner, the hypothetical

‘gravitino’

, with
helicities

±

3/2. Such a self-charge-conjugate massless
particle as the ‘gravitiono’ with helicities

±

3/2 can only have

low-energy

interactions if it is represented by a Majorana
field

ψ

μ

⁢

(

x

)

which is invariant under the gauge
transformations:

ψ

μ

⁢

(

x

)

⟼

ψ

μ

⁢

(

x

)

+

δ

μ

⁢

ψ

⁢

(

x

)

,

(0.5)

with

ψ

⁢

(

x

)

being an arbitrary Majorana field as defined by
Grisaru and Pendleton (1977). The tetrad field

Φ

μ

⁢

ν

⁢

(

x

)

and the graviton field

ψ

μ

⁢

(

x

)

are then
incorporated into a term

H

μ

⁢

(

x

,

θ

)

defined as the

metric superfield

. The relationships between

Φ

μ

ν

⁢

(

x

)

and

ψ

μ

⁢

(

x

)

, on the one hand, and the components
of the metric superfield

H

μ

⁢

(

x

,

θ

)

, on the other hand,
can be derived from the transformations of the whole metric
superfield:

H

μ

⁢

(

x

,

θ

)

⟼

H

μ

⁢

(

x

,

θ

)

+

Δ

μ

⁢

(

x

,

θ

)

,

(0.6)

by making the simplifying– and physically realistic–

assumption

of a weak gravitational field (further details can be found, for
example, in Ch.31 of vol.3. of Weinberg, 1995). The interactions
of the entire superfield

H

μ

⁢

(

x

)

with matter would be then
described by considering how a weak gravitational field,

h

μ

ν

interacts with an energy-momentum tensor

T

μ

⁢

ν

represented as a

linear combination

of components of a

real
vector

superfield

Θ

μ

. Such interaction terms would,
therefore, have the form:

I

ℳ

=

2

⁢

κ

⁢

∫

𝑑

x

4

⁢

[

H

μ

⁢

Θ

μ

]

D

,

(0.7)

(

ℳ

denotes ‘matter’) integrated over a four-dimensional
(Minkowski) spacetime with the metric defined by the superfield

H

μ

⁢

(

x

,

θ

)

. The term

Θ

μ

, as defined above, is
physically a

supercurrent

and

satisfies

the conservation
conditions:

γ

μ

⁢

𝐃

⁢

Θ

μ

=

𝐃

,

(0.8)

where

𝐃

is the four-component super-derivative and

X

denotes a real chiral scalar superfield. This leads immediately to
the calculation of the interactions of matter with a weak
gravitational field as:

I

ℳ

=

κ

⁢

∫

d

4

⁢

x

⁢

T

μ

⁢

ν

⁢

(

x

)

⁢

h

μ

⁢

ν

⁢

(

x

)

,

(0.9)

It is interesting to note that the gravitational actions for the
superfield that are invariant under the generalized gauge
transformations

H

μ

⟼

H

μ

+

Δ

μ

lead to

solutions

of the Einstein field equations for a

homogeneous

,
non-zero vacuum energy

density

ρ

V

that correspond to either
a de Sitter space for

ρ

V

>

0

, or an anti-de Sitter space for

ρ

V

<

0

. Such spaces can be represented in terms of the
hypersurface equation

x

5

2

±

η

μ

,

ν

⁢

x

μ

⁢

x

ν

=

R

2

,

(0.10)

in a

quasi-Euclidean five-dimensional space

with the metric
specified as:

d

⁢

s

2

=

η

μ

,

ν

⁢

x

μ

⁢

x

ν

±

d

⁢

x

5

2

,

(0.11)

with ’

+

’ for de Sitter space and ’

-

’ for anti-de Sitter space,
respectively.

The spacetime symmetry groups, or extended symmetry

groupoids

, as the case may
be– are different from the ‘classical’ Poincaré

symmetry group

of

translations

and Lorentz transformations. Such spacetime
symmetry groups, in the simplest case, are therefore the

O

⁢

(

4

,

1

)

group for the

de Sitter space

and the

O

⁢

(

3

,

2

)

group
for the

anti–de Sitter space

. A detailed calculation indicates
that the transition from ordinary flat space to a bubble of
anti-de Sitter space is

not

favored energetically and,
therefore, the ordinary (de Sitter) flat space is stable (viz.
Coleman and De Luccia, 1980), even though quantum fluctuations
might occur to an anti–de Sitter bubble within the limits
permitted by the Heisenberg uncertainty principle.

0.2

Supersymmetry algebras and Lie (graded) superalgebras.

It is well known that

continuous symmetry transformations

can be represented in terms of a

Lie algebra

of

linearly
independent

symmetry

generators

t

j

that satisfy the
commutation

relations

:

[

t

j

,

t

k

]

=

ι

⁢

Σ

l

⁢

C

j

⁢

k

⁢

t

l

,

(0.12)

Supersymmetry is similarly expressed in terms of the symmetry
generators

t

j

of a

graded (‘Lie’) algebra

which is in
fact defined as a

superalgebra

) by satisfying relations of the
general form:

t

j

⁢

t

k

-

(

-

1

)

η

j

⁢

η

k

⁢

t

k

⁢

t

j

=

ι

⁢

Σ

l

⁢

C

j

⁢

k

l

⁢

t

l

.

(0.13)

The generators for which

η

j

=

1

are fermionic whereas those
for which

η

j

=

0

are bosonic. The

coefficients

C

j

⁢

k

l

are

structure constants

satisfying the following conditions:

C

j

⁢

k

l

=

-

(

-

1

)

η

j

⁢

η

k

⁢

C

j

⁢

k

l

.

(0.14)

If the generators

j

are quantum

Hermitian operators

, then the
structure constants satisfy the reality conditions

C

j

⁢

k

*

=

-

C

j

⁢

k

. Clearly, such a graded

algebraic structure

is a superalgebra
and not a proper Lie algebra; thus graded Lie algebras are often called

‘Lie superalgebras’

.

The standard computational approach in QM utilizes the S-matrix
approach, and therefore, one needs to consider the general,

graded

‘Lie algebra’ of

supersymmetry generators

that
commute with the S-matrix. If one denotes the fermionic generators
by

Q

, then

U

-

1

⁢

(

Λ

)

⁢

Q

⁢

U

⁢

(

Λ

)

will also be of the
same type when

U

⁢

(

Λ

)

is the quantum operator corresponding
to arbitrary, homogeneous Lorentz transformations

Λ

μ

ν

. Such a group of generators provide therefore a
representation of the homogeneous

Lorentz group

of transformations

𝕃

. The

irreducible representation

of the homogeneous
Lorentz group of transformations provides therefore a
classification of such

individual

generators.

0.2.1

Graded ‘Lie Algebras’/Superalgebras.

A set of quantum operators

Q

j

⁢

k

A

⁢

B

form an

𝐀

,

𝐁

representation of the group

𝐋

defined above
which satisfy the commutation relations:

[

𝐀

,

Q

j

⁢

k

A

⁢

B

]

=

-

[

Σ

j

′

⁢

J

j

⁢

j

′

A

,

Q

j

′

⁢

k

A

⁢

B

]

,

(0.15)

and

[

𝐁

,

Q

j

⁢

k

A

⁢

B

]

=

-

[

Σ

j

′

⁢

J

k

⁢

k

′

A

,

Q

j

⁢

k

′

A

⁢

B

]

,

(0.16)

with the generators

𝐀

and

𝐁

defined by

𝐀

≡

(

1

/

2

)

⁢

(

𝐉

±

i

⁢

𝐊

)

and

𝐁

≡

(

1

/

2

)

⁢

(

𝐉

-

i

⁢

𝐊

)

, with

𝐉

and

𝐊

being the Hermitian generators of

rotations

and ‘boosts’, respectively.

In the case of the two-component Weyl-spinors

Q

j

⁢

r

the
Haag–Lopuszanski–Sohnius (HLS)

theorem

applies, and thus the
fermions form a

supersymmetry algebra

defined by the
anti-commutation relations:

[

Q

j

⁢

r

,

Q

k

⁢

s

*

]

=

2

⁢

δ

r

⁢

s

⁢

σ

j

⁢

k

μ

⁢

P

μ

,

(0.17)

[

Q

j

⁢

r

,

Q

k

⁢

s

]

=

e

j

⁢

k

⁢

Z

r

⁢

s

,

where

P

μ

is the 4–momentum operator,

Z

r

⁢

s

=

-

Z

s

⁢

r

are the bosonic symmetry generators, and

σ

μ

and

𝐞

are the usual

2

×

2

Pauli matrices

.
Furthermore, the fermionic generators commute with both energy and
momentum operators:

[

P

μ

,

Q

j

⁢

r

]

=

[

P

μ

,

Q

j

⁢

r

*

]

=

0

.

(0.18)

The bosonic symmetry generators

Z

k

⁢

s

and

Z

k

⁢

s

*

represent the set of

central charges

of the supersymmetric
algebra:

[

Z

r

⁢

s

,

Z

t

⁢

n

*

]

=

[

Z

r

⁢

s

*

,

Q

j

⁢

t

]

=

[

Z

r

⁢

s

*

,

Q

j

⁢

t

*

]

=

[

Z

r

⁢

s

*

,

Z

t

⁢

n

*

]

=

0

.

(0.19)

From another

direction

, the Poincaré symmetry mechanism of
special relativity can be extended to new algebraic systems
(Tanasă, 2006). In Moultaka et al. (2005) in view of such

extensions

, consider invariant-free Lagrangians and bosonic
multiplets constituting a symmetry that interplays with (

Abelian

)

U

⁢

(

1

)

–gauge symmetry that may possibly be described in

categorical

terms, in particular, within the notion of a

cubical site

(Grandis and Mauri, 2003).

We shall proceed to introduce in the next section

generalizations

of the concepts of Lie algebras and graded Lie algebras to the
corresponding Lie

algebroids

that may also be regarded as
C*–convolution representations of

quantum gravity
groupoids

and superfield (or supergravity) supersymmetries. This
is therefore a novel approach to the proper representation of the

non-commutative geometry

of quantum spacetimes

–that are

curved

(or ‘deformed’) by the presence of

intense

gravitational fields–in the framework of

non-Abelian

,

graded Lie algebroids

. Their correspondingly

deformed
quantum gravity groupoids

(QGG) should, therefore, adequately
represent supersymmetries modified by the presence of such intense
gravitational fields on the Planck scale. Quantum fluctuations
that give rise to quantum ‘foams’ at the Planck scale may be then
represented by

quantum

homomorphisms

of such QGGs. If the
corresponding graded Lie algebroids are also

integrable

,
then one can reasonably expect to recover in the limit of

ℏ

→

0

the

Riemannian geometry

of General Relativity and
the

globally hyperbolic spacetime

of Einstein’s classical
gravitation theory (GR), as a result of such an integration to the

quantum gravity

fundamental groupoid

(QGFG). The following
subsection will define the precise mathematical concepts
underlying our novel quantum supergravity and extended
supersymmetry notions.

References

1

S. Weinberg.:

The

Quantum Theory

of Fields

. Cambridge, New York and Madrid:
Cambridge University Press, Vols. 1 to 3, (1995–2000).

2

A. Weinstein : Groupoids: unifying internal and external symmetry,

Notices of the Amer. Math. Soc.

43

(7): 744-752 (1996).

3

J. Wess and J. Bagger:

Supersymmetry and Supergravity

,
Princeton University Press, (1983).

Title

superfields, superspace and supergravity

Canonical

name

SuperfieldsSuperspaceAndSupergravity

Date of creation

2013-03-22 18:17:03

Last modified on

2013-03-22 18:17:03

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

23

Author

bci1 (20947)

Entry type

Feature

Classification

msc 81R60

Classification

msc 81R50

Classification

msc 83C47

Classification

msc 83C75

Classification

msc 83C45

Classification

msc 81P05

Synonym

quantum gravity

Synonym

quantum space-times

Related topic

SupersymmetryOrSupersymmetries

Related topic

NormedAlgebra

Related topic

Supercategories

Related topic

QuantumGravityTheories

Related topic

SuperalgebroidsAndHigherDimensionalAlgebroids

Related topic

AxiomaticTheoryOfSupercategories

Related topic

LieSuperalgebra3

Related topic

MetricSuperfields

Related topic

SuperalgebroidsAndHigherDimensionalAlgebroids

Defines

superspace

Defines

superfields

Defines

supergravity

Defines

supersymmetry and L-superalgebras

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