metric superfields

metric superfields

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This is a topic entry on

metric superfields

in quantum

supergravity

and the mathematical cncepts related to spinor and

tensor fields

.

1

Metric superfields: spinor and tensor fields

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 the 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

)

,

(1.1)

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

)

,

(1.2)

(where

Λ

b

a

is an arbitrary

real matrix

), and the general
coordinate transformations:

x

μ

⟼

(

x

′

)

μ

⁢

(

x

)

.

(1.3)

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

)

,

(1.4)

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

)

,

(1.5)

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

)

,

(1.6)

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

,

(1.7)

(

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

,

(1.8)

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

)

,

(1.9)

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

,

(1.10)

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

,

(1.11)

with ’+’ for de Sitter space and ’-’ for anti-de Sitter space,
respectively.

Note

The

presentation

above follows the exposition by S. Weinberg in his book
on “Quantum Field Theory” (2000), vol. 3, Cambridge University Press (UK),
in terms of both

concepts

and mathematical notations.

Title

metric superfields

Canonical name

MetricSuperfields

Date of creation

2013-03-22 18:19:08

Last modified on

2013-03-22 18:19:08

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

9

Author

bci1 (20947)

Entry type

Topic

Classification

msc 83E50

Classification

msc 83C45

Synonym

supergravity fields

Related topic

SuperfieldsSuperspace

Related topic

SpinNetworksAndSpinFoams

Defines

supergravity field

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