With the detailed geometrical aspects developed of general relativity in Sect.1.1, we now consider further generalizations of geometrical nature to describe supergravity.
To this end, in the last three sections of this chapter, we consider a local version of supersymmetry, where the parameter.x/, involved in supersymmetric transfor- mations, depends on the particular spacetime point into consideration, and hence takes into account the underlying local structure of spacetime. Not surprisingly, this local version of supersymmetry is called supergravity. In the present section we develop the action of the gravitational field, associated with the spin 2 particle massless particle- the so-called graviton, and consider the role of a spinor in such a formalism. In the next section, the action of the Rarita-Schwinger field of spin 3=2is developed, as the superpartner of the graviton. In the final section, the full action of pure supergravity is derived.58Here we follow a rather standard notation and, in order not to confuse Lorentz indices with general curved spacetime ones, we use Latin indices for the former and Greek indices for the latter. Also since spinor components will, in general, be suppressed throughout this section and the following two sections, this turns out to be quite convenient.
Let us recall once more the definition of the parallel transfer of thecomponents V.x/of a vector field from a pointxto a point infinitesimally close pointxCdxin reference to some given curve as given in (1.1.28)
V.x/ ! V.x/.x/V.x/dx Vk.xCdx/; (2.14.1) where we will see that in our study of supergravity, we have to consider a connection which is non-symmetric in./. We may, similarly, define the parallel transfer of a spinor field by
.x/ ! k.xCdx/D .x/C˝.x/ .x/dx; (2.14.2) where˝.x/is a matrix the expression of which is self consistently derived. Then .x/ .x/would remain invariant under such an infinitesimal parallel transform if
.x/0 .x/D k.xCdx/ 0 k.xCdx/
D
.x/C .x/ ˝.x/dx 0
.x/C˝.x/ .x/dx ; (2.14.3)
58The reader is advised to review the material in the introductory section Sect.1.1of Chap.1and, in particular, in the tetrad field treatment of the underlying geometry of general relativity.
to first order in dx, from which we obtain
0˝.x/ 0 D ˝.x/: (2.14.4) On the other hand, starting from the special relativistic expression of the components .x/a .x/ of the spinor field, locally defined, in a local Lorentz coordinate system at the pointx, we may introduce the corresponding expression in the curvilinear coordinate spacetime by .x/.x/ .x/, where.x/D ea.x/a, withea.x/denoting tetrad (vierbein) fields , to obtain from (2.14.1) for its parallel transfer
.x/.x/ .x/ ! .x/.x/ .x/.x/ .x/.x/ .x/dx
ea.xCdx/ k.xCdx/ a k.xCdx/
: (2.14.5) We may compare the two expressions on the right-hand sides of (2.14.5) and use, in the process, the expression for k.xCdx/on the right-hand of (2.14.2) to obtain as a matrix equation
Œ˝; bDeb
@ea Cea
a; (2.14.6)
where we have also used (2.14.4), and the fact thatebea D•ba, upon multiplying first the resulting equation byeb. The solution for˝.x/is obtained from the double commutator,
Œ a; Œb; c D 4
baccab
; and is given by
˝.x/D.!/ab.x/1
8Πa; b; .!/abD .!/ba; (2.14.7) .!/ab.x/C eb.x/
@ea.x/C.x/ea.x/
D0; (2.14.8)
where.!/abis referred to as the spin connection.
From (2.14.2), (2.14.7), the covariant derivative of the spinor fieldD , should satisfy
D .x/dx D
.xCdx/ k.xCdx/
; (2.14.9)
D .x/ D @ .x/˝.x/ .x/: (2.14.10) Multiplying (2.14.8) byeb, we may write the generalized connection as
Dea
@eaC.!/abeb
: (2.14.11)
The Riemann curvature tensor may be expressed in terms of the generalized connection and its derivative. On the other hand, the spin connection may be expressed from (2.14.8) as
.!/ab D eb
@eaC.x/ea
; (2.14.12)
and the curvatureRab, with mixed indices, may be expressed in terms of the spin connection and its derivative as follows
Rab.!/D@.!/abC.!/ac.!/cb@.!/ab.!/ac.!/cb; (2.14.13) where Rab is anti-symmetric in .; / and, independently, anti-symmetric in .a;b/. The gravitational action, associated with the spin 2 particle, may be then spelled out to be
Z
.dx/L.2/D 1 22
Z
.dx/e eaebRab.!/; eDdetŒea; (2.14.14) with coupling parameter22, and recall that the metric may be expressed asg D eaea.
At this stage we may define the torsion as the anti-symmetric part of the connection in its indices.; /:
T D 1 2
; (2.14.15)
not to be confused with an energy-momentum tensor, and in the next section we will derive the general expression for the torsionTab by considering variations of the above action together with the action associated with the spin 3=2 field, with respect to .!/ab. Accordingly, we consider the corresponding variation of the action in (2.14.14) first. To this end,
•Rab.!/D@.•!/abC.•!/ac.!/cbC.!/ac.•!/cb
@.•!/ab.•!/ac.!/cb.!/ac.•!/cb: (2.14.16) This, in particular, requires to find the derivative@.e eaeb/, in carrying out a partial integration of the action in (2.14.14). We note that (2.14.8) leads to
@eaeaC.!/abeb D0; @eaC eaC.!/ac
ecD0:
(2.14.17) Also using the variation of a determinantAderived in Appendix A of Chap.1
•detŒADdetŒATrŒA1•A;
and witheDdetŒea, we have
eaea Dı; •eDe ea•ea; @eDe ea@eaDe: (2.14.18) where we have used (2.14.11), to write Dea@ea.
From (2.14.17), (2.14.18), the following identity then readily follows
@
e eaeb
Ce.!/ac
ecebC e.!/bc
ecea
Den
eaebeaebo
: (2.14.19)
Finally upon comparison of the above equation with the following one obtained from (2.14.16)
e eaeb•Rab.!/D2e eaebn
@.•!/abC.!/ac.•!/cb.•!/ac.!/cbo D2e
n
eaeb@eceb.!/aceaec.!/bco
•.!/ab; (2.14.20) where we have, in the process, took the advantage of the anti-symmetry property of the spin connection.!/ab D .!/ba, and by using the definition of the torsion in (2.14.15), the following key equation emerges
•!
Z
.dx/L.2/D 1 2
Z .dx/e
ebTaeaTbCTa b
•.!/ab: (2.14.21) This equation will be used in conjunction with the corresponding one for the spin 3=2field to determine the torsion.
Before closing this section, we also derive an expression relating the curvature and the torsion. To this end, from (2.14.11), we may infer that
ˇaD
@ˇeaC.!ˇ/acec
; (2.14.22)
and
@˛ˇaD@˛@ˇeaC.@˛.!ˇ/ac/ecC.!ˇ/ac@˛ec D@˛@ˇeaC.@˛.!ˇ/ac/ecC.!ˇ/ac
˛ec.!˛/cded
; (2.14.23)
where in writing the second expression we have used the first equation in (2.14.17) with the indexathere raised, or
@˛ˇa.!ˇ/acec˛
D@˛@ˇea.!ˇ/ac.!˛/cdedC.@˛.!ˇ/ac/ec: (2.14.24) Upon multiplying the latter equation by"˛ˇ, upon the exchange of some of the indices, and using the anti-symmetry relation of the spin connection, we obtain
"˛ˇ
@˛ˇaC.!˛/acecˇ
D"˛ˇ
@ˇ.!˛/c aC.!ˇ/cb.!˛/b a
ec: (2.14.25) From the expression of Rab.!/ in (2.14.13), we infer that the right-hand of (2.14.25) may be written as
"˛ˇR˛ˇa
=2. Hence using the definition of the torsion in (2.14.15), we obtain the following identity
1
2"˛ˇR˛ˇaD"˛ˇ
@˛TˇaC.!˛/ab
Tˇb
; (2.14.26)
and note that the1=2factor on the left-hand side of the above equation does not cancel out because the definition of torsion in (2.14.15) already includes a 1=2 factor. This identity will turn up to be useful later on.