1. Universit`a degli Studi di Torino
Facolt`a di Scienze MM. FF. NN.
Dipartimento di Fisica Teorica
Scuola di Dottorato in Scienza ed Alta Tecnologia
Indirizzo Fisica e Astrofisica
Ciclo XXIII
A Geometric Approach to Supergravity Theories
Candidato: Tutor:
Riccardo Nicoletti Prof. Mario Trigiante
Coordinatore: Controrelatore:
Prof. Guido Boffetta Prof. Laura Andrianopoli
Torino
MMVIII – MMXI
3. Universit`a degli Studi di Torino
Scuola di Dottorato in Scienza ed Alta Tecnologia
Indirizzo di Fisica ed Astrofisica
A GEOMETRIC APPROACH
TO
SUPERGRAVITY THEORIES
Riccardo Nicoletti
Tutor: Mario Trigiante
Difesa il 4 Novembre 2011
1
9. Introduction
Governing a large state
is like frying a small fish.
Laozi, Daodejing, 60
The importance of supergravity theories is evident in various contexts of theoretical
physics. From a bottom-up point of view, supergravity theories are models invariant under
local supersymmetry. There is a plethora of globally supersymmetric theories in four
dimensions which are considered for extensions of the Standard Model, however these
theories contain a large number of free parameters describing in particular the mechanism
of supersymmetry breaking. Supergravity theories provide for a simple mechanism which
generates these couplings, in the limit where the gravitino decouples m3/2 → ∞ and only
a gauge and matter theory remains.
Supergravities, however, are mainly studied in connection with superstring theories,
since the field-theory limit α → 0 of superstring theories in ten dimensions are super-
gravity theories. Moreover, M-theory is supposed to be related to the eleven-dimensional
supergravity and F-theory to type IIB supergravity. This sets a deep connection between
supergravities and superstrings, which is important both on the formal point of view
(study of symmetries and dualities [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23]) and on the connection with phenomenology (compactifications
[24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48,
49, 50, 51, 52, 53, 54, 55, 56]). A geometric classification of supergravity theories in any di-
mensions has been reached in terms of E11, however there are still many open issues about
the geometry of supergravities. The embedding tensor approach [57, 58, 59, 60, 61, 62]
proved to be very useful in the analysis of the possible gaugings of these theories. Su-
pergravities are also crucial for understanding quantitatively the web of dualities among
string theories, the properties of M-theory and the conjectured gauge/gravity dualities.
Despite all these very interesting formal aspects, the final aim of supergravity theories
is the description of phenomenology. From a top-down point of view, the presence of a
single theory in D = 11 suggests a strong simplification in higher-dimensional theories,
with respect to the models in D = 4. Therefore there is a strong effort in the study of
compactifications of supergravity theories from D = 10 or D = 11 dimensions to D = 4,
in the quest for finding models which contain the Standard Model plus a hidden sector of
supersymmetric particles.
7
10. Finally, since supergravity theories describe gravitational interactions and sometimes
allow for the introduction of a cosmological constant, these theories revealed their impor-
tance in cosmology and in particular can provide models for inflation.
The circle compactification of the D = 11 theory gives type IIA supergravity. This
approach was followed [63, 64, 65, 66, 67] when the theory was determined in the so called
Einstein frame. However, the pure spinor formulation of superstrings [68, 69, 70, 71, 72, 73,
74, 75, 76, 77, 78, 79, 80, 81, 56, 82, 83, 84, 85, 86, 87, 88] requires a local redefinition of the
fields, which is named string frame (Jordan frame in cosmological contexts). Moreover, the
Brans-Dicke theory [89, 90] in the Jordan frame [91] is the starting point for the study of
models of inflationary scenarios [92, 93, 94]. Despite the connection between the Einstein
frame and the string frame theory is just a Weyl transformation, there is a longstanding
debate on the physical equivalence between the two different frames [95, 96, 97, 98]. This
motivates our study [99] of type IIA supergravity in the string frame.
There is also another reason for studying type IIA supergravity directly in the string
frame: from the merely computational point of view, it would be much more tedious (and
less instructive) to perform the Weyl transformation on the Einstein frame Lagrangian
and supersymmetry transformation laws, than to compute directly in the string frame the
whole theory.
Therefore our first result is the complete geometric Lagrangian and supersymmetry
transformations of type IIA supergravity in the string frame [99]. Note that the rheo-
nomic parametrizations of the theory, which are the solution of the Bianchi identities and
give directly the supersymmetry transformations, was already determined in [100], but
in our investigations we found that they were incomplete, since they did not allow for a
consistent determination of the Lagrangian. Therefore part of our work has been devoted
to establishing the complete rheonomic parametrizations of the theory.
Type IIA supergravity is also particularly interesting because it admits two [101] mas-
sive deformations [102, 103], as we have demonstrated. The deformation discovered by
Howe–Lambert–West does not admit a Lagrangian, therefore we concentrated on the
deformation discovered by Romans.
Our second result is the computation of the complete Lagrangian and supersymmetry
transformations of Romans’ massive deformation.
Finally, since we checked and completed our computation with the help of FORM [104],
we have a program which, given the Ansatz for a generic rheonomic theory, determines
the equations for the coefficients of the theory.
In Chapter 1 we summarize the usual formulation of the field theory approach to (non-
abelian) gauge theories and BRST symmetry. In Chapter 2 we describe the geometric
(rheonomic) approach to supergravity theories and their BRST extension. Chapter 3 con-
tains two examples of our geometric approach: pure gravity and simple pure supergravity
in four dimensions. Then we start our investigation of type IIA supergravity. Chapter 4 de-
scribes the construction of the Ansatz for the rheonomic parametrizations and Lagrangian
of the massless theory in the string frame. The coefficients of the Ansatz are determined
in Chapter 5. Finally, Chapter 6 is devoted to the study of the massive deformations of
type IIA supergravity.
8
11. Chapter 1
Quantum field theories and BRST
symmetry
In this chapter we briefly review classical gauge theories, their functional and Faddeev–
Popov quantization, their BRST symmetry and introduce BRST quantization [105, 106,
107, 108]. In particular, our description follows the lines of [106].
1.1 Classical gauge theory
In this section we describe a classical gauge theory coupled to matter fields.
Consider a set of spin 0 fields {ϕl
(x)}. Assume also that the Lagrangian which describes
these fields is invariant under infinitesimal local field transformations of the form
δϕi
(x) = α
(x) (tα)i
j ϕj
(x) , (1.1)
where the tα are a set of constant matrices which form a Lie algebra1
[tα, tβ] = fγ
αβ tγ . (1.2)
Therefore the antisymmetry of the commutator implies
fγ
αβ = −fγ
βα , (1.3)
while the Jacobi identity [[tα, tβ], tγ]+[[tβ, tγ], tα]+[[tγ, tα], tβ] = 0 implies, on the structure
constants,
fγ
[αβ fη
|γ|ζ] = 0 . (1.4)
In general, a Lagrangian depends on the fields {ϕi
} and on their derivatives {∂µϕi
}, but,
since the parameters α
(x) in general have a non-trivial dependence on the space-time
position, the derivatives transform as
δ ∂µϕi
= α
(x) (tα)i
j ∂µϕj
+ (∂µ
α
(x)) (tα)i
j ϕj
. (1.5)
1we suppose that the Lie algebra is the direct sum of compact and simple or abelian subalgebras.
9
12. This is cured by the introduction of a set of vector fields {Aα
µ}, which are used to define
a covariant derivative which transforms homogeneously under a gauge transformation
(commutes with a gauge transformation):
(Dµϕ(x))i
= ∂µϕi
(x) − Aα
µ(x) (tα)i
j ϕj
(x) . (1.6)
This requirement fixes the transformation rule of the gauge field under an infinitesimal
gauge transformation:
δ (Dµϕ)i
= δ ∂µϕi
+
− δAα
µ (tα)i
j ϕj
− Aβ
µ (tβ)i
j
α
(tα)j
k ϕk
=
= α
(tα)i
j [Dµϕ]j
+ −δAα
µ + ∂µ
α
− fα
βγ Aβ
µ
γ
(tα)i
j ϕj
. (1.7)
Since the covariant derivative must transform homogeneously, the gauge fields are required
to transform as
δAα
µ = ∂µ
α
− fα
βγ Aβ
µ
γ
. (1.8)
Note also that, since the generators {tα} in the adjoint representation are
tAdj
α
β
γ
= fβ
γα , (1.9)
we can also write
δAα
µ = ∂µ
α
− γ
tAdj
γ
α
β
Aβ
µ . (1.10)
The gauge fields, however, are dynamical and therefore the Lagrangian will contain their
derivatives. Hence we must find a combination of derivatives of the fields Aα
µ which trans-
forms homogeneously under a gauge transformation. The appropriate combination turns
out to be
([Dµ, Dν] ϕ)i
= −Fα
µν (tα)i
j ϕj
, (1.11)
where
Fα
µν = ∂µAα
ν − ∂νAα
ν − fα
βγ Aβ
µ Aγ
ν . (1.12)
The field strength indeed transforms under a gauge transformation as
δFα
µν = −fα
βγ
γ
Fβ
µν . (1.13)
At this point we have the ingredients for building a gauge-invariant Lagrangian for the
matter fields ϕi
and for the gauge fields Aα
µ. We restrict to the renormalizable terms, so
the Lagrangian assumes the form2
L = −
1
4
δαβFα
µνFβµν
+ LM (ϕ, Dµϕ) , (1.14)
2In principle there could be a matrix gαβ instead of δαβ, but in our hypotheses is always possible to diagonalize and
normalize the generators of the gauge algebra in this fashion. Moreover, if we do not assume parity or CP (or T) conservation,
there may be a θ term of the form −1
2
θαβ
µνρσ Fα
µν Fβ
ρσ, with a constant matrix θαβ. However this term is a total derivative
and does not affect the field equations or Feynman rules hence for our purposes it can be omitted.
10
13. where LM describes the matter fields and their interactions with the gauge fields. The
classical equations of motion of the gauge fields are:
0 = ∂µ
∂L
∂µAα
ν
−
∂L
∂Aα
ν
=
= −δαβ ∂µFβ
µν − δγβ Fβρν
fγ
α Aρ −
∂L
∂ (Dνϕ)i (tα)i
j ϕj
. (1.15)
Defining the current
Jν
α ≡
∂L
∂ (Dνϕ)i (tα)i
j ϕj
(1.16)
and the gauge-covariant derivative of the field strength
[Dµ, [Dν, Dρ]] = −[Dµ, Fα
νρ tα] =
= − ∂µFα
νρ − fα
βγ Aβ
µ Fγ
νρ tα =
= − DµFα
νρ tα , (1.17)
the equations of motion are written
DµFαµν
= Jαν
. (1.18)
Upon contraction of the equations of motion with Dν, we also verify that it is conserved:
DνJαν
= 0 . (1.19)
It is useful to note that the Jacobi identity
[Dµ, [Dν, Dρ]] + [Dν, [Dρ, Dµ]] + [Dν, [Dµ, Dρ]] = 0 (1.20)
implies the Bianchi identity for the field strength, which is valid even off-shell (without
using the equations of motion)
DµFα
νρ + DνFα
ρµ + DρFα
µν = 0 . (1.21)
1.2 Quantization
In this section we describe the quantization of the gauge and matter theory described
before.
We wrote the Lagrangian density for the class of theories under consideration as3
L = −
1
4
FαµνFαµν
+ LM (ϕ, Dµϕ) , (1.22)
where
Fα
µν = ∂µAα
ν − ∂νAα
µ − fα
βγ Aβ
µ Aγ
ν ,
Dµϕi
= ∂µϕi
− Aα
µ (tα)i
j ϕj
. (1.23)
3From this moment we shall assume that the gauge group be compact, hence we raise and lower the representation
indices with the metric δαβ and the structure constants are completely antisymmetric; for this reason, we are allowed to
drop the distinction between upper and lower gauge indices.
11
14. Canonical quantization is performed in the Hamiltonian formalism, thus we need to per-
form a Legendre transform.
The canonical momenta are defined as
Παν
≡
∂L
∂ (∂0Aαν)
= −Fα0ν
= Fαν0
, (1.24)
πl ≡
∂L
∂ (∂0ϕl)
=
∂LM
∂ (D0ϕl)
. (1.25)
Note that the first component of the momentum (1.24) vanishes:
Πα0
= 0 . (1.26)
This is a first-class constraint [109]. Moreover, the equations of motion can be written
DµFαµν
= Jαν
, (1.27)
whose ν = 0 component gives
DµFµ0
α = ∂ˆIF
ˆI0
α − fαβγ Aβ
ˆI
Fγ ˆI0
=
= ∂ˆIΠ
ˆI
α − fαβγ Aβ
ˆI
Πγ ˆI
= J0
α , (1.28)
where the indices ˆI, . . . = 1, 2, 3 run over the spatial directions. Defining the quantity
Ξα ≡ ∂ˆIΠ
ˆI
α − fαβγ Aβ
ˆI
Πγ ˆI
− J0
α , (1.29)
we obtain the secondary constraint
Ξα = 0 . (1.30)
Note however that the 0-component of the matter current (1.16) can be expressed as a
function of the ϕi
and of the πl alone:
J0
α = πl (tα)l
m ϕm
. (1.31)
Therefore the Poisson bracket4
{Πα0
, Ξβ} = 0 (1.32)
and the constraints are of the first class.
In general, we can denote by χN = 0 the constraints of a theory. The first-class con-
straints are associated to a group of symmetries of the action, in this case gauge invari-
ance. Canonical quantization proceeds at this point by imposing a gauge fixing constraint,
which eliminates the first class constraints. When all of the first class constraints have been
eliminated by a gauge fixing, the remaining constraint equations have linear independent
Poisson bracket. Therefore the matrix
CMN = [χM , χN ]P (1.33)
4Given a Lagrangian L Ψa, ˙Ψa , the Poisson bracket is defined by
[A, B]P ≡
∂A
∂Ψa
∂B
∂Πa
−
∂B
∂Ψa
∂A
∂Πa
.
12
15. is non-singular, det C = 0. Then the canonical commutation relations are given by
[A, B] = i [A, B]D , (1.34)
where the Dirac bracket is defined as
[A, B]D = [A, B]P − [A, χM ]P C−1 MN
[χN , B]P . (1.35)
However, for our purposes it is convenient to use functional quantization. We would
like to quantize the theory by using the action S = d4
x L to define a Green functional
generator, but this is not straightforward because of the constraints. Indeed, since the
Lagrangian is gauge invariant by construction, the functional measure of the theory
dµ =
l,x
dϕl
α,µ,x
dAαµ
(x) e
i
S
(1.36)
is constant along the “orbits” of a given field configuration, i.e. the paths in the field
functional space which are spanned by the gauge transformations. This makes the integral
divergent along the orbits and hence ill-defined. This can also be seen in perturbation
theory from the lack of uniqueness of the propagator [107, 108]. The problem of these
constraints is solved by fixing a gauge. It is convenient5
here to choose the axial gauge
Aα3 = 0 . (1.37)
Note that in this gauge some quantities simplify as:
ΠI
α = FI0
α = ∂I
Aα0
− ∂0
AαI
− fα
βγ AβI
Aγ0
,
Π3
α = F30
α = ∂3
Aα0
, (1.38)
where I, J, . . . = 1, 2. Thus
Ξα = ∂IΠI
α − (∂3)2
A0
α − fαβγ Aβ
I ΠγI
− J0
α
and the constraint (1.30) gives
(∂3)2
A0
α = ∂IΠI
α − fαβγ Aβ
I ΠγI
− J0
α (1.39)
which can be solved, with appropriate boundary conditions, for A0
α as a functional
A0
α = A0
α(ΠI
α, Aβ
J , J0
α(ϕl
, πm)). Hence the constraint is solved by considering A0
α as a func-
tional of the other variables.
The Hamiltonian density can be constructed as
H = ΠI
α ∂0AαI + πl ∂0ϕl
− L =
= ΠI
α −Fα
I0 + ∂IAα
0 − fα
βγ Aβ
I Aγ
0 +
+
1
2
ΠαI ΠαI
−
1
2
(∂3Aα
0 ) (∂3Aα
0 ) +
+
1
4
FαIJ FαIJ
−
1
2
(∂3AαI) ∂3AαI
+ HM , (1.40)
5In the axial gauge there is not the problem of the Gribov ambiguity [110], which is present, i.e., in the Lorentz and
Coulomb gauge and is related to the existence of a global gauge fixing.
13
16. where
HM = πl ∂0ϕl
− LM (1.41)
is the matter Hamiltonian. The total Hamiltonian can be rewritten:
H = ΠI
α ∂IAα
0 − fα
βγ Aβ
I Aγ
0 +
−
1
2
ΠαI ΠαI
−
1
2
(∂3Aα
0 ) (∂3Aα
0 ) +
+
1
4
FαIJ FαIJ
−
1
2
(∂3AαI) ∂3AαI
+ HM , (1.42)
Note that the equation of motion for Aα0 is
δS
δAα0
= −
∂H
∂Aα0
=
= ∂IΠαI
+ ΠI
γ fγ
βα Aβ
I + (∂3)2
Aα
0 − J0
α =
≡ Ξα = 0 . (1.43)
This means that the constraint Ξα = 0 is indeed solved by this prescription. Finally, the
action is
S = d4
x ΠI
α ∂0AαI + πl ∂0ϕl
− H , (1.44)
and could be used to define the path integral functional measure. However it is not
manifestly Lorentz invariant and also it is difficult to use the path integral since Aα
0
has a highly non-trivial dependence on the independent field components and momenta.
For this reason, it is useful to change the variables in the functional measure. Note
that the Hamiltonian is quadratic in the momenta ΠαI (with a field-independent factor).
Assuming that the same holds for the matter Hamiltonian, with respect to the momenta πl,
we can perform a Gaussian saddle point evaluation of the path integral for the momenta.
The classical equations of motion to be used are
0 =
δS
δπl
= ∂0ϕl
−
∂HM
∂πl
,
0 =
δS
δΠαI
= ∂0AI
α − ∂I
Aα
0 + fα
βγ AβI
Aγ
0 + ΠI
α =
= ΠI
α − FI0
α . (1.45)
Using these equations of motion, we arrive at the result
S = d4
x L , (1.46)
where L is the initial Lagrangian (1.22). Therefore the functional measure is the heuristic
one (1.36)
dµ ∝·
l,x
dϕl
(x)
α,µ,x
dAαµ
(x) e
i
S
, (1.47)
14
17. where ∝· indicates equality up to a field independent factor and the integration is per-
formed for all values of µ, but with the further axial-gauge condition enforced by a factor
x,α
δ (Aα3(x)) . (1.48)
Summarizing, the quantum theory is described by the generating functional
W[J] = eiZ[J]
=
= Ω, T e
i
d4x l φl(x) Jl(x)
Ω =
= dµ
x,α
δ (Aα3(x)) ei d4x l φl(x) Jl(x)
, (1.49)
where
dµ = N
l,x
dϕl
(x)
α,µ,x
dAαµ
(x) ei S
, (1.50)
Ω is the vacuum of the theory, N is a normalization factor such that W[0] = 1, Z[J] is
the generator of the connected diagrams, {Jl} is the set of independent sources associated
to each field φl = (Aαµ, ϕi
). The vacuum expectation value of gauge-invariant operators
O1, . . . , On is therefore given by
Ω, T{O1 · · · On} Ω = N
l,x
dϕl
α,µ,x
dAαµ
(x) {O1 · · · On} ei S
·
·
x,α
δ (Aα3(x)) . (1.51)
At this point, note that the functional generator (1.49) is of the general form
W[J] = N
n,x
dφn(x) G[φ] B [f[φ]] Det F[φ] , (1.52)
where {φn} = ϕl
, Aαµ are the set of matter and gauge fields, G[φ] is a functional of the
φn which satisfies the gauge-invariance condition
n,x
dφλn(x) G[φλ] =
n,x
dφn(x) G[φ] , (1.53)
where φλn indicates the fields φn after a gauge transformation with parameters λα(x).
Moreover, fα[φ; x] is a functional of the fields which is not gauge invariant (“gauge-fixing
functional”) and which depends6
on α and x, B [f] is a numerical functional of fα[φ] and
finally F is the functional matrix
Fαx,βy[φ] ≡
δfα[φλ; x]
δλβ(y) λ=0
. (1.54)
6following Weinberg’s notation, a functional depend on all values of the undisplayed variables, for fixed values of those
which are displayed.
15
18. The importance of recognizing the general form of the generating functional (1.52) is
based on the following
Theorem 1 The path integral (1.52) is (almost) independent of the gauge-fixing func-
tional fα[φ; x] and depends on the choice of the functional B[f] only through a field-
independent factor.
Proof: First, perform a change of variable of integration
φ → φΛ , (1.55)
where φΛ denotes the field φ after a gauge transformation of parameter Λα(x).
W[J] = N
n,x
dφΛn(x) G[φΛ] B [f[φΛ]] Det F[φΛ] . (1.56)
Next, we use the gauge-invariance property (1.53) and obtain
W[J] = N
n,x
dφn(x) G[φ] B [f[φΛ]] Det F[φΛ] . (1.57)
The functional W[J] does not depend on Λ, hence
α,x
dΛα
(x) W[J] ρ[Λ] = W[J]
α,x
dΛα
(x) ρ[Λ] =
=
α,x
dΛα
(x) ρ[Λ] W[J] =
= N
n,x
dφn(x) G[φ] C[φ] , (1.58)
where ρ[φ] is a (so far generic) weight functional and
C[φ] ≡
α,x
dΛα
(x) ρ[Λ] B [f[φΛ]] Det F[φΛ] . (1.59)
Then we want to characterize Det F[φΛ]. From equation (1.54) we have
Fαx,βy[φΛ] =
δfα[(φΛ)λ ; x]
δλβ(y) λ=0
. (1.60)
We assumed that the gauge transformations form a group, so that there exists an element
˜Λ = Λ · λ of the group such that its action on a field φ is the same of the action of Λ on
φ followed by the action of λ on φΛ:
(φΛ)λ = φ˜Λ .
16
19. Therefore, using the chain rule for functional differentiation, we obtain
Fαx,βy[φΛ] = d4
z
δfα[φ˜Λ; x]
δ˜Λγ(z) ˜Λ=Λ
δ˜Λγ
(z; Λ, λ)
δλβ(y)
λ=0
. (1.61)
Consequently, we have
Det F[φΛ] = Det
δf[φΛ; x]
δΛγ(z)
Det
δ˜Λγ
(z; Λ, λ)
δλβ(y)
. (1.62)
Since Det δf[φΛ;x]
δΛγ(z)
is the Jacobian of the change of variables Λα
(x) → fα[φΛ; x], if we choose
the functional ρ[Λ] as
ρ[Λ] =
1
Det δ˜Λγ(z;Λ,λ)
δλβ(y)
, (1.63)
then C[φ] becomes independent of φ:
C[φ] =
α,x
dΛα
(x) B [f[φΛ]] Det
δf[φΛ]
δΛ
=
=
α,x
dfα(x) B[f] = C . (1.64)
Finally, we conclude that
W[J] =
N C n,x dφn(x) G[φ]
α,x dΛα(x) ρ[Λ]
. (1.65)
Note, however, that both the numerator and the denominator in the previous expression
are ill-defined because the measure is constant along the orbits. Since the factor B[f]
eliminates this divergency in the original path integral (1.52), we assume that, after an
adequate regularization, the divergencies of numerator and denominator cancel.
This theorem allows us to use a more convenient gauge-fixing functional, though we
started with the choice of the axial gauge. For our purposes, we consider a functional:
B[f] = exp −
i
2ξ
d4
x fα(x) fα
(x) , (1.66)
where ξ is a real constant parameter. This is equivalent to introducing a gauge-fixing term
to the original Lagrangian (1.22):
L → Leff = L + Lgf , (1.67)
with Lgf = − 1
2ξ
fα fα
. In particular, we can choose the Lorentz-invariant gauge-fixing
functional
fα = ∂µAµ
α . (1.68)
However, since the functional B[f] is (almost) arbitrary, we conclude that the resulting
S matrix must be independent of the choice of ξ.
17
20. 1.3 Faddeev–Popov quantization
The Faddeev–Popov quantization method consists in rewriting Det F[φ]. Consider a set
of anticommuting classical fields ωα and ω∗
α (in principle they are not related by complex
conjugation):
{ωα, ω∗
β} = 0 ,
{ωα, ωβ} = 0 ,
{ω∗
α, ω∗
β} = 0 . (1.69)
Every function of commuting and anticommuting variables is at most linear in each of
the anticommuting variables. Considering, for simplicity, just ω,
F (ω) = A + ω B , (1.70)
where A and B depend only on the bosonic variables. Differentiation with respect to the
anticommuting coordinate is defined according to
∂ωF = B . (1.71)
More generally,
{∂ωα , ωβ} = δαβ ,
{∂ωα , ω∗
β} = 0 (1.72)
and similarly for derivation with respect to ω∗
. Define the Berezin integral over anticom-
muting variables as
dωα ωβ =
1
√
π
δαβ . (1.73)
Given these definitions, we can express the factor Det F[φ] in the path integral as
Det F[φ] ∝·
α,x
dωα(x)
α,x
dωα(x) ei d4x d4y ω∗
α(x) ωβ(y) Fαx,βy
, (1.74)
where the proportionality constant is an irrelevant field-independent constant.
The effect of the Faddeev–Popov trick is therefore that of adding to the Lagrangian
(1.67) a ghost term
Sgh = d4
x d4
y ω∗
α(x) ωβ(y) Fαx,βy . (1.75)
Note that introduction of the ghost fields ωα and ω∗
β is compatible with the conservation
of a quantum number, the ghost quantum number, which is defined as +1 for ωα, −1 for
ω∗
β and 0 for the other fields.
It is useful to compute F in the case of the generalized ξ gauge, fα = ∂µAµ
α. Indeed,
performing an infinitesimal gauge transformation of parameters λα, we have
fα → ∂µ Aµ
α + ∂µ
λα − fα
βγ Aβµ
λγ
(1.76)
18
21. hence, from the definition of F
Fαx,βy[φ] = δ4
(x − y) − fα
γβ ∂µ Aγµ
δ4
(x − y) . (1.77)
Therefore, the ghost term is
Sgh = d4
x d4
y ω∗
α ωβ(y) δ4
(x − y) − fα
γβ ∂µ Aγµ
δ4
(x − y) =
= d4
x (∂µω∗
α) (∂µ
ωβ) + (∂µω∗
α) ωβ fα
βγ Aγµ
=
= d4
x (∂µω∗
α) (∂µ
ωβ) − (∂µω∗
α) ωβ tAdj
γ
α
β
Aγµ
=
= d4
x (∂µω∗
α) (Dµ
ωβ) , (1.78)
where we used the fact that the ghosts transform in the adjoint representation of the
gauge group.
Note that, though the ghost fields are scalar fields with fermionic statistics, they do
not violate the spin–statistics theorem, since they are not physical particles and do not
appear as external particles in Feynman diagrams.
Summarizing, the Lagrangian which enters the functional measure in the path integral,
in the generalized ξ gauge, is
Ltot = −
1
4
FαµνFαµν
+ LM (ϕ, Dµϕ) +
−
1
2ξ
(∂µAµ
α) (∂νAαν
) + (∂µω∗
α) (Dµ
ωβ) . (1.79)
The Lagrangian is renormalizable as long as the matter Lagrangian is.
1.4 BRST symmetry in gauge theories
The quantization of a gauge theory requires the introduction of a gauge-fixing term as
well as the ghost fields. However, this is related to the choice of a gauge, but at this point
there is no warranty that the theory be renormalizable, i.e. that the infinites appearing at
a higher loop level can be absorbed into a redefinition of the fields and of the parameters
of the Lagrangian. However, there is a symmetry [111, 112] which can be used to prove
the renormalizability of a gauge theory and that actually provides an alternative way to
quantize a gauge theory.
It is convenient to introduce a new set of fields (“Nakanishi-Lautrup fields”) hα and
rewrite the gauge-fixing Lagrangian as
Lgf = −
1
2ξ
fαfα
→ Lgf =
ξ
2
hαhα
+ fαhα
, (1.80)
or equivalently to express the functional B[f] as
B[f] =
α,x
dhα(x) e d4x hαhα
ei d4x fαhα
. (1.81)
19
22. Note that now the path integrals must be performed also on the hα fields. Moreover,
eliminating the Nakanishi-Lautrup fields through their classical equations of motion hα =
−1
ξ
fα, the former form of B[f] is recovered. The new Lagrangian is
Ltot = −
1
4
FαµνFαµν
+ LM (ϕ, Dµϕ) +
+
ξ
2
hαhα
+ fαhα
+
+ ω∗
α(x)∆α , (1.82)
where we defined the quantity
∆α ≡ d4
y Fαx,βy ωβ(y) . (1.83)
Given a new anticommuting infinitesimal constant θ which anticommutes with ω and
ω∗
,
{θ, ω} = {θ, ω∗
} = 0 ,
the new Lagrangian (1.82) is invariant with respect to the global BRST symmetry
δθϕl
= (tα)l
m θ ωα ϕm
,
δθAαµ = θ Dµωα ,
δθω∗
α = −θ hα ,
δθωα = +
1
2
fα
βγ θ ωβ ωγ ,
δθhα = 0 . (1.84)
Note that, since the hα are BRST invariant, we could replace the quadratic factor in hα
in the Lagrangian with an arbitrary smooth functional of hα without affecting the BRST
invariance of the action.
It is convenient, given a generic field (or functional) F, to denote the action of the
BRST transformation as
δθF ≡ θ sF , (1.85)
where s is the generator of the BRST transformation.
It is useful to prove first that
Theorem 2 The BRST symmetry is nilpotent, i.e.
s2
= 0 . (1.86)
Proof: For completeness, write
sϕl
= (tα)l
m ωα ϕm
,
sAαµ = Dµωα ,
sω∗
α = −hα ,
sωα = +
1
2
fα
βγωβωγ ,
shα = 0 . (1.87)
20
23. Then we have
s2
ϕi
= s (tα)i
j ωαϕj
=
= (tα)i
j
1
2
fαβγ ωβ
ωγ
ϕj
− ωα (tβ)j
k ωβϕk
=
=
1
2
fαβγ (tα)i
j ωβ
ωγ
ϕj
−
1
2
([tα, tβ])i
k ωα ωβϕk
=
=
1
2
fαβγ (tα)i
j ωβ
ωγ
ϕj
−
1
2
fγαβ (tγ)i
k ωα ωβϕk
=
= 0 . (1.88)
Moreover:
s2
Aαµ = s (∂µωα − fαβγ Aβµ ωγ) =
= ∂µ
1
2
fαβγ ωβ
ωγ
− fαβγ (Dµωβ) ωγ +
1
2
Aβµ fγζη ωζωη =
=
1
2
fαβγ ∂µ ωβ
ωγ
− fαβγ (∂µωβ − fβζη Aζµ ωη) ωγ +
−
1
2
fαβγ fγζη Aβµ ωζ ωη =
= fαβγfβζη Aζµ ωη ωγ −
1
2
fαβγ fγζη Aβµ ωζ ωη =
= 0 . (1.89)
s2
ω∗
α = shα = 0 . (1.90)
s2
ωα =
1
2
fαβγ s (ωβωγ) =
=
1
4
fαβγfβζη ωζ ωη ωγ −
1
4
fαβγfγζη ωβ ωζ ωη =
= 0 . (1.91)
s2
hα = 0 . (1.92)
Given two fields φ1, φ2, a BRST transformation acts as
s (φ1φ2) = (sφ1) φ2 + (−1) 1
φ1 (sφ2) , (1.93)
where 1 is equal to 1 = ±1 according to wether φ1 is either bosonic or fermionic.
Therefore
s2
(φ1φ2) = (−1) 1
(1 − 1) (sφ1) (sφ2) = 0 . (1.94)
This can be extended to a generic product of fields. Since any functional F[φ] can be
written as a sum of such products, we conclude that s is nilpotent.
At this point it is easy to prove
Theorem 3 The Lagrangian (1.82) is invariant under the BRST transformation (1.84).
21
24. Proof: First note that, for the matter fields ϕi
and for the gauge fields Aαµ, the BRST
transformation is just a gauge transformation with infinitesimal gauge parameter
λα(x) = θ ωα(x) . (1.95)
Since L is gauge invariant, it is also automatically BRST invariant. Then observe that
acting with δθ on the gauge-fixing function gives
δθfα[x; A, ϕ] = d4
y
δfα[x; Aλ, ϕλ]
δλβ(y) λ=0
θ ωβ(y) =
= θ d4
y Fαx,βy ωβ(y) =
= θ ∆α(x; A, ϕ, ω) . (1.96)
Therefore
Ltot − L = −s ω∗
α fα +
ξ
2
ω∗
αhα . (1.97)
This means that the Lagrangian is of the form
Ltot = L + s Ψ , (1.98)
where the gauge fermion Ψ is
Ψ = − ω∗
α fα +
ξ
2
ω∗
αhα . (1.99)
Since the BRST symmetry is nilpotent, the Lagrangian is BRST invariant.
1.5 BRST quantization
Equation (1.98) already shows why the BRST symmetry is so important: the physical
content of any gauge theory is in the cohomology of the BRST operator, i.e. in its kernel
modulo the terms which are in its image. This can be shown on more general grounds.
Since the BRST symmetry is a symmetry of the theory, by N¨other theorem there exists
a conserved charge QBRST associated to this symmetry such that, for a field Φ,
δθΦ = i [θQBRST , Φ] = i θ [QBRST , Φ] ,
sΦ = i [QBRST , Φ] , (1.100)
where [·, ·] is equal to the commutator [·, ·]− if Φ is bosonic and to the anticommutator
if Φ is fermionic. Since s is nilpotent, we have
0 = s2
Φ = −[QBRST , [QBRST , Φ] ]± = −[Q2
BRST , Φ]− . (1.101)
This has to be valid for all operators Φ, therefore Q2
BRST has either to be proportional
to the unit operator or to vanish. It cannot be proportional to the unit operator, since it
has a non-vanishing quantum number, the ghost quantum number, hence it must vanish:
Q2
BRST = 0 . (1.102)
22
25. Suppose that we change the gauge-fixing function fα. This corresponds to a change in the
gauge fermion Ψ (1.99). The change in any matrix element α|β due to a change ˜δΨ in
Ψ is
˜δ α|β = i α|δ ˜Stot|β = i α| d4
x s δ ˜Ψ|β =
= d4
x α|[QBRST , ˜δΨ]|β . (1.103)
Since this must vanish for all changes ˜δΨ, it is necessary that
α|QBRST = 0 = QBRST |β . (1.104)
This is the Kugo-Ojima phisicality condition [113]. It imposes that the physical states
must be in the kernel of the nilpotent operator QBRST . However, since QBRST is nilpotent,
two states differing by the image of QBRST are physically equivalent, e.g. |α and |α +
QBRST |β , and represent the same physical state. Therefore independent physical states
belong to the cohomology of QBRST .
∗ ∗ ∗ ∗
At this point it is useful to introduce the “De Witt” convention for repeated indices.
So far, we simply used Einstein convention on repeated indices, i.e. repeated indices imply
summation. From this point to the end of the chapter we suppose that indices also include
space-time points and that the “summation” also includes integration over the repeated
continuous variables.
Denote the matter and gauge fields by
φr
≡ ϕi
, Aαµ , (1.105)
while the set of fields entering an action are denoted by
χ ≡ φr
, ωA
, ω∗A
, hA
. (1.106)
Suppose also that the action functional S0[φ] is invariant under the local infinitesimal
symmetry transformation
φr
→ φr
+ A
δAφr
. (1.107)
There is an important issue which now becomes relevant. We needed to introduce the
ghost fields for compensating that the path integral is performed on all φr
, include those
differing by just a gauge transformation. This means that we supposed that the ghosts were
independent, i.e. that there were as many ghosts as many independent local symmetries.
This is true in the case discussed above of Yang-Mills theory, but in supergravity this is in
general not true. Indeed, a p-form field A(p)
potential undergoes a gauge transformation:
A(p)
→ A(p)
+ dΛ
(p−1)
[1] , (1.108)
where Λ(p−1)
is an arbitrary (p−1)-form. Since the exterior derivative d is nilpotent, Λ(p−1)
can be modified by a total differential without changing the gauge transformation of A(p)
,
Λ
(p−1)
[1] → Λ
(p−1)
[1] + dΛ
(p−2)
[2] . This means that there is a residual gauge invariance, which
therefore requires new “ghosts for ghosts” and so on. This mechanism repeats for (p − 1)
23
26. times. In this section we shall assume that the theory contains at most 1-forms and, more
generally, that the ghosts and antighosts are independent. When studying a geometric
formulation of supergravity and of the BRST symmetry we will discuss the general case
describing p-forms.
In this situation, the total action is
Stot[φ, ω, ω∗
, h] = S0[φ] + hA
fA[φ] + ω∗B
ωA
δAfB[φ] , (1.109)
is invariant under the infinitesimal global BRST transformations
χ → χ + δθχ = χ + θ sχ , (1.110)
where s is the Slavnov operator
s = ωA
(δAφr
)
δ
δφr
−
1
2
ωB
ωC
fA
BC
δ
δωA
− hA δ
δω∗A
, (1.111)
where the fA
BC are defined by
[δB, δC] = −fA
BC δA
. (1.112)
Note that differentiation is here intended to be “left”, in the sense that if δF = δχG, then
δF
δχ
= G. We have
s2
=
1
2
ωA
ωB
(δAφr
)
δ (δBφs
)
δφr
− (δBφr
)
δ (δAφs
)
δφr
+
+fC
AB (δCφs
)
δ
δφs
+
+
1
2
ωA
ωB
ωC
(δAφr
)
δfD
BC
δφr
+ fE
ABfD
EC
δ
δωD
. (1.113)
Therefore the nilpotence of the BRST transformation is equivalent to the commutation
relations (1.112) together with the consistency conditions
fE
[ABfD
|E|C] + δ[Aφr
δfD
BC]
δφr
= 0 , (1.114)
which, when the structure functions are field-independent “constants”, reduce to the usual
Jacobi identities for the structure constants.
The total action (1.109) can be written
Stot[φ, ω, ω∗
, h] = S0[φ] − s ω∗A
fA , (1.115)
hence again it is invariant under the BRST transformation (1.110), since S0 is gauge
invariant (and the BRST symmetry acts as a gauge transformation with parameter θ ωA
)
and Stot − S0 is BRST exact.
Now we can prove that the form (1.115) of the action is valid also for more general
theories.
24
27. Theorem 4 The most general BRST invariant functional of ghost number zero is the
sum of a functional of the φr
alone and a BRST exact term
Stot[φ, ω, ω∗
, h] = S0[φ] + s Ψ[φ, ω, ω∗
, h] , (1.116)
where Ψ is an arbitrary functional of ghost number −1.
Proof: Note that a BRST transformation does not change the total number of hA
and
ω∗A
fields. Therefore, if we divide the total action in terms with a fixed number N of
antighosts and Nakanishi-Lautrup fields,
Stot =
+∞
N=0
SN , (1.117)
each term must be separately BRST invariant:
s SN = 0 ∀ N ≥ 0 . (1.118)
Define the Hodge operator
t ≡ ω∗A δ
δhA
. (1.119)
We have
{s, t} = −ω∗A δ
δω∗A
− hA δ
δhA
. (1.120)
Thus,
{s, t} SN = st SN =
= −N SN . (1.121)
For N = 0 this can be inverted to give
SN = −
1
N
s (t SN ) , ∀ N ≥ 1 (1.122)
and the total action is written
Stot =
+∞
N=0
SN =
= S0 + s
+∞
N=1
−
1
N
(t SN ) =
= S0 + s Ψ . (1.123)
By definition, the term S0 is independent of the ω∗A
and of the hA
. Moreover, by hypothesis
it has zero ghost number, hence it must also be independent of ωA
.
∗ ∗ ∗ ∗
Since the most general action which describes matter and gauge fields φ and also ωα,
ω∗
α, hα fields and which has ghost number 0 has the form
S[φ, ω, ω∗
, h] = S0[φ] + s Ψ[φ, ω, ω∗
, h] , (1.124)
25
28. where Ψ is an arbitrary functional of ghost number −1, we recognize the Faddeev–Popov
quantization method as one way of generating a Lagrangian which yelds a unitary S-
matrix. In particular, Ψ is not necessarily bilinear in the ghosts and antighosts. Using the
Kugo-Ojima physicality condition
α|QBRST = 0 = QBRST |β , (1.125)
we are assured that the S-matrix elements are independent of the choice of Ψ and, in
particular, if there exists a choice of Ψ for which the ghosts decouple (like the axial gauge
in Yang–Mills theories), then the ghosts decouple in general, for arbitrary choices of the
gauge fermion Ψ. More generally, we can consider a generic theory described by an action
which has ghost number zero and require it to be invariant under a BRST transformation,
as well as any other global symmetries of the theory. The BRST invariance guarantees
the existence of a conserved nilpotent charge QBRST which is used to define the space of
physical states, i.e. those in the cohomology of QBRST . The resulting space turns out to
be free of ghosts and antighosts, has a positive-definite norm and the S-matrix is unitary.
This procedure is the BRST quantization.
After discussing the rheonomic approach to supergravity theories, we will set up a
geometric framework for implementing the BRST quantization of supergravity theories.
26
29. Chapter 2
Supergravity, free differential
algebras and rheonomy
In this chapter we describe the rheonomic approach to supergravity [114, 115, 84]. After
a quick survey of the method, we review some preliminary definitions on algebra and
differential geometry. Then we define free differential algebras (FDA) and describe their
properties. We state the two theorems which classify FDAs and we describe how to build
an FDA starting from a Lie algebra. Then we give the physical interpretation of this
construction and we explain how to “soften” the left-invariant structure of an FDA into a
“gauged” FDA. After that, we state the rheonomic principle, which allows us to describe
a supergravity theory in terms of an FDA. Finally we give the building rules for a su-
pergravity Lagrangian in terms of the associated FDA and rheonomic parametrizations.
Finally, we describe how to implement BRST quantization in a rheonomic theory and
explain the appearance of pure spinor constraints.
2.1 Geometric supergravities
A supergravity theory is completely specified by the (classical) equations of motion (or
equivalently by the Lagrangian, where avaiable) and by the local supersymmetry transfor-
mation rules for the fields. The geometric (rheonomic) approach to supergravity theories
is a constructive procedure which allows to obtain a complete description of a locally
supersymmetric theory starting from the definition of its curvatures.
The starting point for a supergravity theory in the rheonomic framework are the pure
supergravity curvatures
Rab
= dωab
− ωa
c ∧ ωcb
,
Ta
= DV a
−
i
2
Ψ ∧ Γa
Ψ ≡ dV a
− ωab
∧ Vb −
i
2
Ψ ∧ Γa
Ψ ,
ρ = DΨ ≡ dΨ −
1
4
ωab
∧ Γab Ψ , (2.1)
which are the field strengths for the gravity supermultiplet (ωab
, V a
, Ψ). ωab
is a bosonic 1-
form, the spin connection, V a
is a bosonic 1-form, the vielbein, and Ψ is a fermionic 1-form,
27
30. the gravitino. In general, a supergravity theory in D dimension contains N gravitinos,
but the maximum dimension for a supergravity theory and the maximum amount of
gravitini is limited by the requirement of considering fields with spin equal or less to 2.
Fore example, the maximally supersymmetric theory in D = 4 contains N = 8 Majorana
gravitini, in D = 10 contains N = 2 Majorana-Weyl gravitini and in D = 11 contains
only N = 1 Majorana gravitino. D = 11 is the maximum dimension for a supegravity
theory without higher-spin fields1
.
Given the pure supergravity curvatures, there exists a systematic procedure, described
by Sullivan’s second theorem [119] (Theorem 8), which allows for an extension of the
set of the curvatures of the theory, depending on the existence of certain Fierz identities
in the considered dimension2
. This is therefore a prescription which determines the field
content of the theory in a systematic way by the choice of D and N. The result is the
set of curvatures of the desired supergravity theory. Since, in general, it includes p-form
potentials, p > 1, the set of curvatures is called a Free Differential Algebra (FDA). Given
the FDA, which is written in superspace in terms of differential forms and of exterior
derivatives, differentiation gives the Bianchi identities (BI) of the theory. The BI are the
core of the rheonomic approach, since they describe both the space-time equations of
motions and the supersymmetry transformations of the theory. Indeed, in the rheonomic
approach the BI become integrability conditions and the closure of the algebra requires
that the space-time equations of motion are satisfied.
From the computational point of view, the next step of the procedure consists in solving
the BI. The solution is called rheonomic parametrizations of the curvatures: the curvatures
are expanded along a V a
, Ψ basis of superspace. The sector of the BI proportional only to
vielbein fields gives some constraints which are interpreted as the space-time equations of
motion of the theory. The sectors of the BI with at least one gravitino, indeed, describe
the supersymmetry invariance of the theory.
Given the BI and the rheonomic parametrizatons in superspace, it is possible to find
the Lagrangian of the theory (as a D-form in superspace), if there exists a Lagrangian. The
propagation equations for the fields, indeed, suggest the possible terms for the Lagrangian
D-form. The coefficients are fixed by requiring that the resulting equations of motion
(e.o.m.) are consistent with supersymmetry.
In this chapter we describe how this procedure can be used, while in the subsequent
chapters we give some examples.
Historically, the rheonomic approach was introduced by D’Auria and Fr´e [124] for the
D = 11 supergravity, where the authors introduced the concept of Cartan integrable
system, which has been recognized as equivalent to that of FDA developed by Sullivan
[125]. The main reference for the rheonomic approach, that we follow in this chapter, is
[114].
1For a general introduction to supergravity see [116, 117, 118].
2This issue has been investigated in detail in the context of n-categories [120, 121, 122]. See [123] for further references
and details.
28
31. 2.2 Preliminaries and definitions
Definition 1 (Lie group) A group G is a Lie group if it is a smooth manifold and if
both the group multiplication
· : G × G → G , (x, y) → x · y (2.2)
and the inverse map
G → G , x → x−1
(2.3)
are smooth.
In particular, given a fixed element a ∈ G, the left translation
La : G → G , g → La(g) = ag (2.4)
and the right translation
Ra : G → G , g → Ra(g) = ga (2.5)
are diffeomorphisms.
Observation 5 Since the G multiplication is associative, then the left and right transla-
tions commute.
Observation 6 Consider the tangent space at the identity Te(G). The left translation Lg,
g ∈ G, induces a diffeomorphism between Te(G) and Tg(G):
vg = (Lg)∗ ve . (2.6)
The vector field obtained in this way is left-invariant. Analogously, the vector field
vg = (Rg)∗ ve (2.7)
is right-invariant.
The left- and right-invariant vector fields form a Lie algebra G, the Lie algebra of the
group G. Conventionally, in the following we mainly refer to left-invariant vector fields.
Since any left-invariant vector field is uniquely determined by its value at e, the Lie
algebra of G, G, can be identified with Te(G).
Introducing a basis on Te(G), {TA}, A = 1, . . . , dim(G), then
[TA, TB] = fC
AB TC , (2.8)
where the fC
AB are structure constants. The Jacobi identity for vector fields
[TA, [TB, TC]] + [TB, [TC, TA]] + [TC, [TA, TB]] = 0 , (2.9)
implies, for the structure constants,
fA
B[C fB
LM] = 0 . (2.10)
Consider the one-form ω. The pull-back map L∗
g−1 acts as
L∗
g−1 ωe = ωg . (2.11)
29
32. The form ωg is left-invariant. Thus a field of left-invariant one-forms is completely de-
termined by its value at e. Consider a basis of left-invariant one-forms at T∗
e (G): {σA
},
A = 1, . . . , dim(G). Since dσA
is also left-invariant, it is possible to expand it into a
complete basis of 2-forms at e:
dσA
= −
1
2
fA
BC σB
∧ σC
. (2.12)
These are the Maurer-Cartan equations. By evaluating on the basis of left-invariant vector
fields dual to the cotangent basis of 1-forms:
σA
(TB) = δA
B , (2.13)
it is easily proven that the constants entering the Maurer-Cartan equations (2.12) are in-
deed the structure constants of the Lie algebra (2.8). The Jacobi identity for the structure
constants (2.10) can be retrieved from the integrability condition d2
= 0.
A cotangent basis on G can be obtained in terms of the group element g ∈ G. In fact,
the 1-form
σ = g−1
dg (2.14)
is left-invariant. Differentiating this expression on both sides, we have dσ = dg−1
∧ dg,
which can be written
dσ + σ ∧ σ = 0 . (2.15)
Since σ is Lie-algebra valued, we can also expand it into
σ = σA
TA . (2.16)
Group manifolds have a structure which is too rigid to introduce a non-trivial dynamics.
Therefore, we have to consider manifolds whose structure is not as rigid as that of G. These
are the soft group manifolds ˜G, which are locally diffeomorphic to G. Moreover, we require
that the vacuum configuration will be a group manifold.
On a soft group manifold ˜G, introduce new 1-forms µ = µA
TA which are not left-
invariant. At this point, these forms no longer satisfy the Maurer-Cartan equations. Hence
the curvatures of the µA
are defined as
RA
≡ dµA
+
1
2
fA
BC µB
∧ µC
, (2.17)
or
R = dµ + µ ∧ µ . (2.18)
2.3 Free differential algebras
The concept of free differential algebra is based on a generalization of the Maurer-Cartan
equations.
A Lie algebra is defined by its structure constants, which can be expressed in terms of
the commutator of the generators {TA} of the algebra
[TA, TB] = fC
AB TC , (2.19)
30
33. or equivalently, introducing the 1-forms {σA
} dual to the generators,
σA
(TB) = δA
B , (2.20)
through the Maurer-Cartan equations
dσA
+
1
2
fA
BC σB
∧ σC
= 0 . (2.21)
More generally, given a set of left-invariant p-forms3
defined over the group manifold
M
θi(pi) N
i=1
,
it is possible to write a set of generalized Maurer-Cartan equations
dθi(pi)
+
+∞
n=0
C
i(pi)
j1(pj1
)···jn(pjn ) θj1(pj1
)
∧ . . . ∧ θjn(pjn )
= 0 , (2.22)
where C
i(pi)
j1(pj1
)···jn(pjn ) are generalized structure constants which are not zero only when
n
k=1
pjk
= pi + 1 . (2.23)
Moreover, the constants have the same exchange symmetry as the corresponding p-forms
under the wedge product.
The consistency of the generalized Maurer-Cartan equations is guaranteed through
recourse to the Bianchi identities: taking the exterior differential of both sides in (2.22)
and using
d2
θi(pi)
= 0 , (2.24)
we have
−
+∞
m=0
+∞
n=0
n C
i(pi)
a1(pa1 )···an(pan ) C
a1(pa1 )
b1(pb1
)···bm(pbm ) ·
· θb1(pb1
)
∧ . . . ∧ θbm(pbm )
∧ θa2(pa2 )
∧ . . . ∧ θan(pan )
= 0 . (2.25)
The case of ordinary Lie algebras is retrieved when all the θi
’s are 1-forms, pi = 1.
2.3.1 Classification of free differential algebras
Definition 2 (Minimal algebra) A minimal algebra is an FDA for which
C
i(pi)
j(pi+1) = 0 . (2.26)
This means that no (pi + 1)-form enters the generalized Maurer-Cartan equation for a
pi-form, so that all non-differential terms are the product of at least two elements of the
algebra.
3The following considerations hold for p ≥ 0, however the case p = 0 is more subtle, because it allows for non-
polinomialities. Indeed, the underlying structure is that of a coset manifold. For more details see [114].
31
34. Definition 3 (Contractible algebra) A contractible algebra is an FDA for which the
only form appearing in the expansion of dθi(pi)
has degree pi + 1, i.e.
dθi(pi)
= θi(pi+1)
, (2.27)
so that
dθi(pi+1)
= 0 . (2.28)
A contractible algebra has a trivial structure, since the basis θi(pi)
can be divided
into two subsets: ξi(pi)
and ζj(pj)
, such that
dζj(pj)
= 0 , ∀ j , (2.29)
dξi(pi)
= ζi(pi+1)
. (2.30)
This means that ζj(pj)
are closed forms and that ζi(pi+1)
are also exact.
Let Mk
be the vector space generated by all forms of degree p ≤ k and Ck
be the
vector space of forms of degree k, a minimal algebra is defined by
dMk
⊂ Mk
∧ Mk
,
while a contractible algebra is defined by
dCk
⊂ Ck+1
. (2.31)
Theorem 7 (Sullivan, I) The most general free differential algebra is the semidirect
sum of a contractible algebra with a minimal algebra.
2.3.2 Construction of the free differential algebras
Definition 4 (Chevalley cohomology) Consider a finite-dimensional Lie algebra G,
described by N 1-forms {σA
} satisfying the Maurer-Cartan equations
dσA
+
1
2
fA
BC σB
∧ σC
= 0 . (2.32)
Consider the finite-dimensional irreducible representations D(n)
of G. Every representa-
tion gives a matrix TA → D(n)
(TA)i
j, through which we can define a G-covariant derivative
(n)
≡ (I)i
j d + σA
∧ D(n)
(TA)i
j . (2.33)
The Maurer-Cartan equations (2.32) guarantee that
(n) (n)
= 0 . (2.34)
The Chevalley cohomology is the cohomology of this nilpotent operator (n)
. The equiva-
lence classes ˆΩ(n,p) are therefore labelled by the representation of the covariant derivative
and by the degree of the elements of the class. We denote a cochain by
Ωα
(n,p) = ωα
A1···Ap
σA1
∧ . . . ∧ σAp
. (2.35)
At this point we can state Sullivan’s second theorem:
32
35. Theorem 8 (Sullivan, II) Consider a finite-dimensional Lie algebra G and its Cheval-
ley cohomology classes ˆΩ(n,p). To each class, there corresponds a possible extension of
(2.32) to a non-trivial differential algebra, since for each cocycle ˆΩi
(n,p) it is possible to
introduce a new (p−1)-form Ai
(n,p−1) such that the Maurer-Cartan equations are extended
by
(n)
Ai
(n,p−1) + ˆΩi
(n,p) = 0 . (2.36)
Consider then the cochains constructed on the forms {σA
, Ai
}:
Ωα
(n,p)(σ, A) = ωα
A1···Ari1···is
σA1
∧ . . . ∧ σAr
∧ Ai1
(n1,p1) ∧ . . . ∧ Ais
(ns,ps) , (2.37)
where
p = r +
s
j=1
pj . (2.38)
Since the operator (n)
is nilpotent also in this larger space, then we can continue the
extension and obtain the most general free differential algebra which contains the Lie
algebra G.
More precisely, it is possible to distinguish between absolute and relative Chevalley
cohomologies.
Definition 5 (Relative Chevalley cohomology) Suppose that the Lie algebra G can
be decomposed into
G = H ⊕ K , (2.39)
where H is a subalgebra and K is a subspace of G. The Chevalley cohomology relative to
the subalgebra H, H(p)
(G, H, D), is the Chevalley cohomology of cochains built of the K
components of G, which are also invariant tensors of H.
The cocycles in the absolute Chevalley cohomology are singlets under H.
Theorem 9 (Chevalley-Eilenberg) If G is semisimple and D is irreducible (and non-
trivial), then, for all choices of H and p,
H(p)
(G, H, D) = 0 . (2.40)
Corollary 10 For G semisimple and D a fully reducible representation not containing
the identity representation,
H(p)
(G, H, D) = 0 . (2.41)
Theorem 11 (Chevalley-Eilenberg) If G is semisimple and D is the identity repre-
sentation, then there are no non-trivial 1-form and 2-form cohomology classes.
There is, however, always a non-trivial 3-form cohomology class:
Ω = fABC σA
∧ σB
∧ σC
.
Therefore, for G semisimple, every closed 1-form or 2-form is exact.
33
36. 2.3.3 Physical interpretation of a free differential algebra
In [69] the following identifications between mathematical (FDA) and physical (super-
gravity) objects have been proposed in connection:
• contractible generators ←→ curvatures;
• Maurer-Cartan equations for the contractible generators ←→ Bianchi identities;
• minimal subalgebra ←→ the algebra which is gauged (symmetry of the vacuum);
• Maurer-Cartan equations for the minimal generators ←→ potentials gauging the
vacuum symmetry.
This interpretation of the FDA description of supergravity theories provides us with the
building rules for a supergravity theory.
2.4 Gauging of the free differential algebra
Consider a set of soft pi-forms πi(pi) N
i=1
, which are in one-to-one correspondence with
the rigid forms θi(pi) N
i=1
, but do not satisfy generalized Maurer-Cartan equations (2.22).
The curvatures of the set πi(pi) N
i=1
are the contractible generators. Hence,
Ri(pi+1)
= dπi(pi)
+
+∞
n=0
C
i(pi)
j1(pj1
)···jn(pjn ) πj1(pj1
)
∧ . . . ∧ πjn(pjn )
. (2.42)
By d-differentiating this expression on both sides, we obtain the generalized Bianchi Iden-
tities:
Ri(pi+1)
≡ dRi(pi+1)
+
+
+∞
n=0
n C
i(pi)
j1(pj1
)···jn(pjn ) Rj1(pj1
)
∧ πj2(pj2
)
∧ . . . ∧ πjn(pjn )
=
= 0 . (2.43)
More generally:
Definition 6 Given a set of (pj + 1)-forms {Hj(pj+1)
}, the adjoint covariant derivative
is defined as
Hj(pj+1)
≡ dHj(pj+1)
+
+
+∞
n=0
n C
j(pj)
k1(pk1
)···kn(pkn ) Hk1(pk1
)
∧ πk2(pk2
)
∧ . . . ∧ πkn(pkn )
. (2.44)
Definition 7 Assume that we have another set of forms {νj(D−pj−1)}N
j=1 where D is some
fixed number. The set {νj(D−pj−1)} is a coadjoint set of (D − pj − 1)-forms if
I = Hj(pj+1)
∧ νj(D−pj−1) (2.45)
34
37. is an invariant, i.e.
I = dI =
= Hj(pj+1)
∧ νj(D−pj−1) + (−1)pj+1
Hj(pj+1)
∧ νj(D−pj−1) =
= dHj(pj+1)
∧ νj(D−pj−1) + (−1)pj+1
Hj(pj+1)
∧ dνj(D−pj−1) . (2.46)
Definition 8 The coadjoint covariant derivative consistent with (2.46) is
νj(D−pj−1) = dνj(D−pj−1) +
+ (−1)pj+1
+∞
n=0
C
i(D−pi−1)
j(D−pj−1)a2(pa2 )···an(pan ) ·
· πa2(pa2 )
∧ . . . ∧ πan(pan )
∧ νi(D−pi−1) , (2.47)
where the generalized structure constants are different from zero only when
pj + 1 = pi +
n
k=2
pak
. (2.48)
2.5 The rheonomic principle
Supersymmetry requires the introduction of bosonic and fermionic fields. The algebraic
structure which extends the Lie algebra along fermionic directions is called a Lie superal-
gebra. Suppose that the Lie superalgebra over which the free differential algebra is built
has the following structure:
G = H ⊕ I ⊕ O , (2.49)
K = I ⊕ O , (2.50)
B = H ⊕ I , (2.51)
where H is a bosonic subalgebra and K is a subspace and the two subspaces I and O are
such that
[H, H] ⊂ H ; [H, I] ⊂ I ; I ; [H, O] ⊂ O ;
[I, I] ⊂ H ;
[I, O] ⊂ O ;
[O, O] ⊂ I ⊕ H ≡ B ⊂ G . (2.52)
We call
• H the gauge subalgebra;
• I inner space (dim I = D);
• O outer space.
Equations (2.52) mean that:
35
38. • H is a subalgebra;
• I and O span two representations of H;
• B = I ⊕ H is a bosonic subalgebra of G;
• B/H is a symmetric space;
• O carries a representation of the full B;
• G/B is a symmetric space.
Consider a supergravity theory, for which πi
(x) ≡ (ωab
(x), V a
(x), Ψ(x), . . .) are an
intrinsic reference frame in the cotangent plane to a soft group manifold. The fundamental
assumption of the geometric approach to supergravity is that the fields πi
for the inner
components description of supergravity are the same fields entering the free differential
algebra (2.42). The problem with this identification is that the soft forms πi
are defined
on the soft group manifold, while the fields are defined only on I. Equivalently, we can
say that πi
has a non-trivial dependence on the fermionic coordinates, θ, as well as the
space-time coordinates, x: πi
= πi
(x, θ).
The identification of the soft forms with the fields of supergravity then requires that
the space-time fields be interpreted as the space-time boundary values of the superspace
fields πi
= πi
(x, θ):
πi
(x) = πi
(x, θ)|θ=0,dθ=0 , (2.53)
where
πi
(x, θ) = πi
µ dxµ
+ πi
¯α(x, θ) dθ¯α
. (2.54)
This identification is made precise if we introduce rheonomic extension mapping
rh : I → G , πi
(x) → πi
(x, θ) . (2.55)
A priori, the introduction of the superfields πi
= πi
(x, θ) increases the number of physical
degrees of freedom, since the fields in the expansion along fermionic directions are, in
principle, independent from those in the spatial directions. Therefore, in order not to
introduce new physical degrees of freedom, i.e. in order to have the same physical content
of the space-time theory, it must be possible to determine all the θ and the dθ fields in the
expansion of the superfields in terms of their space-time restrictions πi
µ(x, 0) dxµ
. This is
the task of rheonomic mapping.
Hence rheonomy means that the outer components of the curvatures can be expressed
algebraically in terms of the inner components:
Ri
O = Ci
jRj
I , (2.56)
where Ci
j are constant tensors. These are named rheonomic constraints.
In summary, the physical content of a superspace rheonomic theory is completely de-
termined by the inner (space-time) description.
36
39. Consider the Lie derivative of a superfield along a direction = ¯α ∂
∂θ¯α :
δπi
(x, θ) = πi
(x, θ + dθ) − πi
(x, θ) =
= L πi
(x, θ) =
= ( |d + d |) πi
(x, θ) =
= ( )i
+ |Ri
. (2.57)
If we consider the Lie derivative as a generator of the functional change of πi
at the same
coordinate point:
L πi
= π i
(x, 0) − πi
(x, 0) , (2.58)
then the rheonomic mapping is a function I → I. Therefore, if the theory described by
the fields πi
is invariant under superspace diffeomorphisms, it can be restricted to I, so
that the Lie derivatives act as a transformation of the inner theory. In particular, the
rheonomic mapping realized on I can be identified with a supersymmetry transformation.
Hence, a supersymmetry transformation can be written as
L πi
(x, 0) = ( )i
+ |Ri
(x, 0) . (2.59)
These transformations close an algebra provided that the Lie derivatives close consistently
[L 1 , L 2 ] = L[ 1, 2] . (2.60)
Since the rheonomic constraints express the outer components in terms of the inner ones,
then the Bianchi identities (integrability conditions) are equations for the inner compo-
nents of the curvatures which must be valid in all points of G and in particular in I.
Therefore, the supersymmetry algebra closes only if the inner curvatures Ri
I satisfy some
integrability conditions, given by the Bianchi identities. Then, in a rheonomic theory we
expect that the supersymmetry transformations close an algebra only on the on-shell
configurations of the fields πi
(x, 0). Moreover, we can lift to superspace only those con-
figurations which are a solution of the inner-space field equations.
From the physical point of view, these Bianchi identities are the space-time equations
of motion of the theory. Any different equation of motion would be inconsistent with the
Bianchi identities.
Observe that the horizontality constraint
ιJab Ri
= 0 (2.61)
can be considered as a rheonomic constraint relating the outer components of the cur-
vatures to the other superspace components. Then πi
(x, 0) can be considered as a form
which has first been restricted to superspace, by imposing the horizontality condition, and
then to the inner space by imposing the rheonomic constraints. Viceversa, given πi
(x, 0)
the full dependence on G can be reconstructed using the rheonomic mapping and an
O-Lorentz transformation.
The invariance of the space-time Lagrangian under local supersymmetry transforma-
tions can be retrieved as follows from the Lie derivative along a spinorial tangent vector
37
40. :
δ S =
MD
L L =
MD
( |d + d |) L =
=
MD
|dL , (2.62)
since the total derivative can be discarded. Therefore the action is invariant under local
supersymmetry if
|dL = 0 . (2.63)
Here MD is a bosonic D-dimensional submanifold floating in the supermanifold Ω and L
is a bosonic D-form.
We conclude by observing that the concept of rheonomy in superspace has an analogy
with that of analyticity on the complex plane. Indeed, the rheonomic constraints are
equivalent to constraints between the inner ∂
∂xµ and the outer ∂
∂θ¯α derivatives. This is
analogous to the Cauchy-Riemann equations for an analytic function f(u + iv):
u ←→ xµ
v ←→ θ¯α
f(u + iv) ←→ πi
(xµ
, θ¯α
) . (2.64)
It is only the analyticity of the function f which allows for its determination on the whole
complex plane, given just its boundary value on any line. For this reason, rheonomy can
be thought as a kind of analyticity condition in superspace.
2.6 Building rules for supergravity Lagrangians
If the fields and the field strengths of the physical theory are to be interpreted geomet-
rically, then the equations of motion derived from the action principle must respect the
symmetries and the properties implied by the definition of the curvatures and by the
Bianchi identities.
Therefore we should study the properties that equations of motion must respect.
• Coordinate invariance. The equations of motions must not depend on the choice
of coordinates on the manifold.
• Exclusion of the Hodge dual. In our geometric approach, the action is a func-
tional of the field configurations and of a D-dimensional bosonic submanifold MD
of a supermanifold Ω
S[φ, MD] =
MD
LD (φ) , (2.65)
where the Lagrangian LD is a bosonic D-form. The classical equations of motion,
obtained by requiring that S be stationary, are obtained by the variation of both the
fields and the surface MD. However, using only diffeomorphism-invariant operations
38
41. in the building of the Lagrangian, any deformation of the surface can be compensated
for a diffeomorphism of the fields. In this case, therefore, the variational equations
associated to the action are given by the usual equations of the motion obtained by
varying the action in the fields on a fixed surface:
δL
δφi
= 0 (2.66)
However, these equations of motion are valid not only on the subspace MD, but on
the whole Ω.
The only diffeomorphic-invariant operations which can be used in the building of the
Lagrangian are:
– exterior differentiation d : φ → dφ;
– exterior (wedge) product ∧ : (φ1, φ2) → φ1 ∧ φ2.
In particular, the Hodge duality operator must be excluded, because it involves
the notion of a metric, hence implicitly that of the dimensionality of the base space.
Therefore, the Hodge dual makes sense only if we specify the manifold on which it
holds, countrary to our present needs.
Since the Hodge dual cannot be used, we must use the first-order formalism for all
the fields, i. e. the derivatives of a scalar field or of a vector field will be introduced
as independent objects, to be varied independently in the action.
If the Lagrangian is built with only the two diffeomorphic-invariant operations, it is
then possible to extend it smoothly from MD to Ω: the inner components will yeld
the usual equations of motion, while if extended to the whole superspace they imply
the rheonomic constraints on the superspace curvatures. The relation between the
two is given by the Bianchi identities, which imply both operations.
• Rigid scale invariance. The Free Differential Algebra (2.42) and the Bianchi
identities (2.43) have a rigid scale invariance, which must also be respected by the
equations of motion implied by the Lagrangian. Every term in the Lagrangian must
therefore respect these scaling properties.
• H-gauge invariance. Since both I and O are representations for H, the curvatures
and the Bianchi identities are gauge-invariant under H.
• Vacuum solution. Setting the curvatures to zero is a solution of the equations of
motion and of the Bianchi identities, since this corresponds to having a set of left-
invariant forms, which fulfill the Maurer-Cartan equations and the Jacobi identities
for the structure constants.
These requirements for the action and the equations of motion lead to the following
building rules for the Lagrangian of a rheonomic theory:
1. Geometricity. The Lagrangian must be a D-form constructed out of the gauged
free differential algebra only using the diffeomorphic invariant operators (exterior
39
42. differential and wedge product), with the exclusion of the Hodge duality operator. In
particular, a theory is strongly geometrical if the corresponding Lagrangian can be
constructed without the use of 0-forms, otherwise the theory is simply geometrical.
This is so because a strongly geometrical theory possesses a Lagrangian which is
polynomial in the curvatures, while this is not necessarily true for geometrical the-
ories including 0-forms. With a geometric Lagrangian L, the action is obtained by
integrating L on a D-dimensional hypersurface MD immersed in the soft manifold
˜G:
A =
MD⊂ ˜G
L . (2.67)
2. H-gauge invariance. When the free differential algebra is H-gauge invariant,
where H is a Lie subgroup of the given free differential algebra, then the action
in (2.67) must be H-invariant. It will be further required that H be a subgroup of
G with SO(1, D − 1) as a factor: H = SO(1, D − 1) × H . (For even D, we restrict
to the connected components containing the unity, so that parity is conserved). This
requires that each term in the Lagrangian must be an H-scalar.
3. Homogeneous scaling law. Each term in the Lagrangian must scale homoge-
neously under a global scaling which respects the rigid invariance of the free differ-
ential algebra and of the Bianchi identities.
4. Vacuum existence. The field equations on the G manifold must admit the solution
Ri
= 0 (“vacuum solution”). and therefore be at least linear in the curvatures.
5. Rheonomy (and horizontality). The outer components of the curvatures, i.e.
those which have at least one component along the outer space, but have no compo-
nent along H, (in supergravity theories, are those containing at least one gravitino
Ψ), must be expressible in terms of the inner ones, i.e. those with components only
in the inner space. Moreover, gauge components, i.e. those which have at least one
component along H, have a vanishing curvature on-shell. This can be due to the
equations of motion or to the horizontality condition.
6. Completeness of the field equations. The variational equations associated to
the action give the field equations of the physical fields (inner components) and the
rheonomic constraints (outer components). Therefore, the inner space field equations
must be complete: they must encompass all the statements on the inner curvatures
implied by the rheonomic constraints via the Bianchi identities.
At this point we have all the ingredients for building geometric supergravity theories.
2.7 BRST symmetry for supergravity theories
There is a simple superspace formulation for the BRST symmetry which turns out to be
particularly useful in the rheonomic approach to supergravity theories [126, 127, 128, 129,
71, 130, 108, 131, 132, 133, 134, 135, 136, 137, 138, 138, 138, 139, 140, 141, 142, 142,
40
43. 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160,
161, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177,
178, 179, 180, 179, 181, 179, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193,
142, 194, 147, 195, 196, 197, 198, 156, 199, 149, 148, 200, 175]. Here we give a description
of BRST and anti-BRST [185] symmetries and explain how this symmetry can be easily
implemented in our context [178, 201].
2.7.1 BRST and anti-BRST symmetries
In addition to the BRST symmetry described in Chapter 1, the Lagrangian (1.82) enjoys
a global nilpotent symmetry which is analogous to the BRST, but which involves the
antighosts, hence it is named anti-BRST symmetry.
In this chapter it is convenient to use a vector notation for gauge indices. For any
quantity aα
, bβ
, define a dot product
a · b ≡ aα
bα
(2.68)
and a vector product
(a × b)α
≡ fα
βγ aβ
bγ
. (2.69)
Note that, given three anticommuting variables {ai}3
i=1, {ai, aj} = 0,
a1 × (a2 × a3) = − (a2 × a3) × a1 = f·
[α|β|fβ
ζη]aα
1 aζ
2aη
3 = 0 (2.70)
because of the Jacobi identity for the structure constants. Moreover, we denote the ghosts
and antighosts by c and ¯c, respectively.
Then we can prove the following:
Theorem 12 The BRST transformation δθF = θ sF
sϕi
= c · ti
ϕj
,
sAµ = Dµc = ∂µc − Aµ × c ,
sc = +
1
2
c × c ,
s¯c = −h ,
sh = 0 (2.71)
and the anti-BRST transformation δ¯θF = ¯θ¯sF
¯sϕi
= ¯c · ti
j ϕj
,
¯sAµ = Dµ¯c = ∂µ¯c − Aµ × ¯c ,
¯s¯c = +
1
2
¯c × ¯c ,
¯sc = h + c × ¯c ,
¯sh = −h × ¯c (2.72)
are nilpotent and anticommute:
s2
= ¯s2
= s ¯s + ¯s s = 0 . (2.73)
41
44. Proof: Write the above transformations as
sϕi
= c · ti
ϕj
,
sAµ = Dµc = ∂µc − Aµ × c ,
sc = α1 c × c ,
s¯c = α2h ,
sh = 0 , (2.74)
¯sϕi
= ¯c · ti
j ϕj
,
¯Aµ = Dµ¯c = ∂µ¯c − Aµ × ¯c ,
¯s¯c = β1 ¯c × ¯c ,
¯sc = β2 h + β3 c × ¯c ,
¯sh = β4 h × ¯c , (2.75)
where the αi, βj are coefficients to be fixed. The square of s on the fields of the theory
gives
s2
ϕi
= s c · ti
jϕj
=
= (sc) · ti
jϕj
− c · ti
j sϕj
=
= α1 (c × c) · ti
jϕj
− c · ti
j c · tj
kϕk
=
= α1 −
1
2
(c × c) · ti
j ϕj
, (2.76)
s2
Aµ = s (∂µc − Aµ × c) =
= ∂µ (sc) − (sAµ) × c − Aµ × (sc) =
= α1 Dµ (c × c) − (Dµc) × c =
= (2 α1 − 1) (Dµc) × c , (2.77)
s2
c = s (α1 c × c) =
= 2 (α1)2
(c × c) × c = 0 , (2.78)
s2
¯c = s (α2 h) = 0 , (2.79)
s2
h = 0 . (2.80)
The square of ¯s gives
¯s2
ϕi
= ¯s ¯c · ti
jϕj
=
= (¯s¯c) · ti
j ϕj
− ¯c · ti
j ¯sϕj
=
= β1 (¯c × ¯c) · ti
jϕj
− ¯c · ti
j ¯c · tj
kϕk
=
= β1 −
1
2
(¯c × ¯c) · ti
jϕj
, (2.81)
42
45. ¯s2
Aµ = ¯s (∂µ¯c − Aµ × ¯c) =
= ∂µ (¯s¯c) − (¯sAµ) × ¯c − Aµ × (¯s¯c) =
= Dµ (¯s¯c) − (¯sAµ) × ¯c =
= (2 β1 − 1) (Dµ¯c) × ¯c , (2.82)
¯s2
¯c = ¯s (β1 ¯c × ¯c) =
= 2 (β1)2
(¯c × ¯c) × ¯c = 0 , (2.83)
¯s2
c = ¯s (β2 h + β3 c × ¯c) =
= β2β4 h × ¯c + β3 (β2 h + β3 c × ¯c) × ¯c − β1β3 c × (¯c × ¯c) =
= β2 (β3 + β4) h × ¯c , (2.84)
¯s2
h = ¯s (h × ¯c) =
= β4 (¯sh) × ¯c + β4 h × (¯s¯c) =
= (β4)2
(h × ¯c) ¯c + β1β4 h × (¯c × ¯c) =
= (β4)2
f·
αβfα
ζη hζ
¯cη
¯cβ
+ β1β4 f·
αβfβ
ζη hα
¯cζ
¯cη
=
= − (β4)2
f·
αβfα
ζη ¯cβ
hζ
¯cη
− β1β4 f·
ζβfβ
ηα + f·
ηβfβ
αζ hα
¯cζ
¯cη
=
= β4 (β4 + 2 β1) ¯c × (h × ¯c) . (2.85)
Finally, the anticommutator s, ¯s on the fields gives
{s, ¯s}ϕi
= s ¯c · ti
jϕj
+ ¯s c × ti
jϕj
=
= α2 h · ti
jϕj
− ¯c · ti
j c · tj
kϕk
+
+ (β2 h + β3 c × ¯c) · ti
jϕj
− c˙ti
j ¯c · tj
kϕk
=
= (α2 + β2) h · ti
jϕj
+ (β3 − 1) (c × ¯c) · ti
jϕj
, (2.86)
{s, ¯s}Aµ = s (Dµ¯c) + ¯s (Dµc) =
= Dµ (s¯c) − (sAµ) × ¯c + Dµ (¯sc) − (¯sAµ) × c =
= α2 Dµh − (Dµc) × ¯c + Dµ (β2 h + β3 c × ¯c) − (Dµ¯c) × c =
= (α2 + β2) Dµh + (β3 − 1) Dµ (c × ¯c) , (2.87)
{s, ¯s}c = s (β1 ¯c × ¯c) + ¯s (α1 c × c) =
= 2 α2 β1 h × ¯c + 2 α1 (β2 h + β3 c × ¯c) × ¯c =
= 2 (α2β1 + α1β2) h × ¯c , (2.88)
{s, ¯s}¯c = s (β1 ¯c × ¯c) + ¯s (α2 h) =
= α2 (2 β1 + β4) h × ¯c , (2.89)
{s, ¯s}h = s (β4 h × ¯c) =
= α2 β4 h × h = 0 . (2.90)
43
46. Imposing (2.73), we obtain the conditions
α1 =
1
2
,
β1 =
1
2
,
β2 = −α2 ,
β3 = +1 ,
β4 = −4 . (2.91)
Requiring the Lagrangian (1.82) to be invariant under these (anti-) BRST transformations,
we choose β2 = −α2 = +1.
2.7.2 Superspace formulation of BRST symmetry
Consider a superspace with coordinates zM
= (xµ
, θ). In this superspace, we would like
to describe matter fields Φi
(x, θ) and gauge fields together with their ghosts AM (x, θ) =
(Aµ (x, θ) , Aθ (x, θ)):
Φi
(x, θ) = ϕi
(x) + θ ϕi
[θ](x) ,
Aµ(x, θ) = Aµ(x) + θ A[θ]µ(x) ,
Aθ(x, θ) = c(x) + θ c[θ](x) . (2.92)
The connection 1-form is defined as
A = AM dzM
= Aµ dxµ
+ Aθ dθ (2.93)
The external differential operator in superspace is
∆ = d + s = dxµ
∂µ + dθ ∂θ , (2.94)
and it is nilpotent, ∆2
= d2
+ s2
+ {d, s} = 0. Note that the generator of the BRST sym-
metry is here identified with the generator of translations along the fermionic directions
∂θ. The curvatures of the theory are
Ri
= DΦi
= ∆M Φi
− A · ti
jΦj
,
F = ∆A −
1
2
A × A . (2.95)
By ∆-differentiating, we obtain the Bianchi Identities
DRi
+ F · ti
jΦj
= ∆Ri
− A · ti
jRj
+ F · ti
jΦj
= 0 ,
DF = ∆F − A ∧ F = 0 . (2.96)
The curvatures can be decomposed along the bosonic and fermionic directions as
Ri
= dxµ
DµΦi
+ dθ DθΦi
=
= dxµ
∂µΦi
− Aµ · ti
jΦj
+ dθ ∂θΦi
− Aθ · ti
jΦj
,
F = dxµ
∧ dxν
∂µAν −
1
2
Aµ × Aν +
44
47. + dxµ
∧ dθ (∂µAθ − ∂θAµ − Aµ × Aθ) +
+ dθ ∧ dθ ∂θAθ −
1
2
Aθ × Aθ . (2.97)
Since the only physical fields are the ϕi
(x) and the Aµ(x), we require that the rheonomic
parametrizations of the curvatures have the form
Ri
= Ri
µ dxµ
,
F = Fµν dxµ
∧ dxν
. (2.98)
This imposes
Ri
θ = DθΦi
= 0 ,
Fµθ = ∂µAθ − ∂θAµ − Aµ × Aθ = 0 ,
Fθθ = ∂θAθ −
1
2
Aθ × Aθ = 0 . (2.99)
Expanding along a dzM
= (dxµ
, dθ) basis, we obtain the following constraints:
0 = DθΦi
=
= ϕi
[θ] − c · ti
jϕj
+
+ θ −c[θ] · ti
jϕj
+ c · ti
jϕj
[θ] ,
0 = Fµθ =
= Dµc − A[θ]µ +
+ θ Dµc[θ] − A[θ]µ × c ,
0 = Fθθ =
= c[θ] −
1
2
c × c , (2.100)
which can be written
ϕi
[θ] = sϕi
= c · ti
jϕj
,
A[θ]µ = sAµ = Dµc ,
c[θ] = sc = +
1
2
c × c ,
s2
ϕi
= 0 ,
s2
Aµ = 0 . (2.101)
Therefore the auxiliary fields are just the BRST variations of the physical fields and of
the ghosts. Hence, the expansion of the superfields can be written as
Φi
(x, θ) = ϕi
(x) + θ sϕi
(x) ,
Aµ(x, θ) = Aµ(x) + θ sAµ(x) ,
Aθ(x, θ) = c(x) + θ sc(x) . (2.102)
45
48. Note also that, using the rheonomic parametrizations (2.98) in the Bianchi Identities
(2.96), we obtain
0 = DRi
+ F · ti
jΦj
=
= dxµ
∧ dxν
DµRi
ν + Fµν · ti
jΦj
+
+ dθ ∧ dxµ
DθRi
µ ,
0 = DF =
= dxµ
∧ dxν
∧ dxρ
(DµFνρ) +
+ dθ ∧ dxµ
∧ dxν
(DθFνρ) . (2.103)
The pure spacetime components give just the ordinary Bianchi Identities, while the dθ ∧
dxµ
and dθ ∧ dxµ
∧ dxν
components give, respectively, the independence of the physical
fields with respect to the fermionic directions.
2.7.3 (anti-) BRST symmetry in superspace
In this section we extend the superspace construction to the case of both BRST and
anti-BRST symmetries. Consider a superspace spanned by coordinates
zM
= xµ
, θ, ¯θ , (2.104)
a set of matter fields Φi
(z) which can be expanded as
Φi
(z) = ϕi
+ θ ϕi
[θ] + ¯θ ϕi
[¯θ] + θ¯θ ϕi
[θ¯θ] (2.105)
and a gauge field AM (z) = (Aµ, Aθ, A¯θ), which is expanded along the fermionic directions
as
Aµ(z) = Aµ(x) + θ A[θ]µ + ¯θ A[¯θ]µ + θ¯θ A[θ¯θ]µ ,
Aθ = c + θ c[θ] + ¯θ c[¯θ] + θ¯θ c[θ¯θ] ,
A¯θ = ¯c + θ ¯c[θ] + ¯θ ¯c[¯θ] + θ¯θ ¯c[θ¯θ] . (2.106)
The connection 1-form is defined as
A = Aµdxµ
+ Aθ dθ + A¯θ d¯θ , (2.107)
while the exterior derivative is
∆ ≡ d + s + ¯s (2.108)
and is nilpotent
∆2
= d2
+ s2
+ ¯s2
+ {d, s} + {d, ¯s} + {s, ¯s} = 0 . (2.109)
The curvatures are defined as before:
Ri
= DΦi
= ∆Φi
− A · ti
jΦj
,
F = ∆A −
1
2
A × A . (2.110)
46
49. The corresponding Bianchi Identities are
DRi
+ F · ti
jΦj
= 0 ,
DF = 0 . (2.111)
We require that the physical components of these fields are just along the bosonic coor-
dinates:
Ri
= Ri
µ dxµ
,
F = Fµν dxµ
∧ dxν
. (2.112)
The expansion of the curvatures is
Ri
= dxµ
∂µΦi
− Aµ · ti
jΦj
+
+ dθ ∂θΦi
− Aθ · ti
jΦj
+
+ d¯θ ∂¯θΦi
− A¯θ · ti
jΦj
,
F = dxµ
∧ dxν
∂µAν −
1
2
Aµ × Aν +
+ dxµ
∧ dθ (∂µAθ − ∂θAµ − Aµ × Aθ) +
+ dxµ
∧ d¯θ (∂µA¯θ − ∂¯θAµ − Aµ × A¯θ) +
+ dθ ∧ dθ ∂θAθ −
1
2
Aθ × Aθ +
+ dθ ∧ d¯θ (∂θA¯θ + ∂¯θAθ − Aθ × A¯θ) +
+ d¯θ ∧ d¯θ ∂¯θA¯θ −
1
2
A¯θ × A¯θ . (2.113)
Therefore the superspace constraints are
DθΦi
= 0 ,
D¯θΦi
= 0 ,
Fµθ = 0 ,
Fµ¯θ = 0 ,
Fθθ = 0 ,
Fθ¯θ = 0 ,
F¯θ¯θ = 0 , (2.114)
which give
ϕi
[θ] = sϕi
,
ϕi
[¯θ] = ¯sϕi
,
ϕi
[θ¯θ] = s¯sϕi
,
A[θ]µ = s Aµ ,
A[¯θ]µ = ¯s Aµ ,
A[θ¯θ]µ = s¯s Aµ ,
47
50. c[θ] = s c ,
c[¯θ] = ¯s c ,
c[θ¯θ] = s¯s c ,
¯c[θ] = s ¯c ,
¯c[¯θ] = ¯s ¯c ,
¯c[θ¯θ] = s¯s ¯c (2.115)
plus s2
= ¯s2
= {s, ¯s} = 0. Therefore, even in this case, the superfield expansion assumes
the form
Φi
= ϕi
+ θ sϕi
+ ¯θ ¯sϕi
+ θ¯θ s¯sϕi
,
Aµ(z) = Aµ(x) + θ sAµ + ¯θ ¯sAµ + θ¯θ s¯sAµ ,
Aθ = c + θ sc + ¯θ ¯sc + θ¯θ s¯sc ,
A¯θ = ¯c + θ s¯c + ¯θ ¯s¯c + θ¯θ s¯s¯c (2.116)
and conversely the BRST and anti–BRST operators are identified with derivatives along
the fermionic directions:
s = dθ ∂θ ,
¯s = d¯θ ∂¯θ . (2.117)
2.7.4 BRST symmetry in supergravity theories
In general, supergravity theories contain fields which are described by p-form fields. As
already pointed out in section 1.5, this requires the introduction of ghosts for ghosts.
However, the superspace approach to BRST symmetry simplifies the solution to this
problem. Indeed, the correct recipe for implementing the BRST symmetry in geometric
theories is encoded in the principle formulated by Anselmi and Fr´e in [75] which generalizes
previous ideas introduced by Baulieu [71]:
Principle 13 The BRST algebra is provided by replacing, in the rheonomic parametriza-
tions of the classical supergravity curvatures, each differential form with its extended ghost-
form counterpart, while keeping the curvature components untouched. Thus the rheonomic
parametrizations of the ghost-extended curvatures are obtained, whose formal definition is
identical with that of the classical curvatures with the replacements
d → d + s ,
Ω[n]
→
n
p=0
Ω[n−p,p]
. (2.118)
Consider, indeed, a set of p-form fields {Ai(pi,πi)
}N
i=1, (where π is 0 if the field is bosonic,
1 if it is fermionic) whose FDA is given by
Ri(pi+1,πi)
= dAi(pi,πi)
+
n∈N
Ci
a1···an
Aa1(pa1 ,πa1 )
∧ . . . ∧ Aan(pan ,πan )
(2.119)
48
51. and consequently fulfilling the Bianchi Identities
Ri(pi+1,πi)
= dRi(pi+1,πi)
+
−
n∈N
Ci
a1···an
Ra1(pa1 +1,πa1 )
∧ Aa2(pa2 ,πa2 )
∧ . . . ∧ Aan(pan ,πan )
. (2.120)
Then we extend the fields, the curvatures and the exterior differential operator as
Ai(pi,πi)
→ Ai
=
pi
I=0
Ai(pi−I,πi,I)
,
Ri(pi+1,πi)
→ Ri
=
pi+1
I=0
Ri(pi+1−I,πi,I)
,
d → ∆ = d + s , (2.121)
where the third entry in Xi(pi,πi,ghi)
is the ghost number of the field. This gives:
Ri
=
pi+1
I=0
Ri(pi+1−I,πi,I)
=
= ∆Ai
+
n∈N
Ci
a1···an
Aa1
∧ . . . ∧ Aan
=
= (d + s)
pi
I=0
Ai(pi−I,πi,I)
+
+
n∈N
Ci
a1···an
pa1
I1=0
· · ·
pan
In=0
Aa1(pa1 −I1,πa1 ,I1)
∧ . . . ∧ Aan(pan −In,πan ,In)
=
= dAi(pi,πi,0)
+
n∈N
Ci
a1···an
Aa1(pa1 ,πa1 ,0)
∧ . . . ∧ Aan(pan ,πan ,0)
+
+
pi
I=1
dAi(pi−I,πi,I)
+
+sAi(pi−(I−1),πi,I−1)
+
+
n∈N
Ci
a1···an
Aa1(pa1 −I1,πa1 ,I1)
∧ . . . ∧ Aan(pan −In,πan ,In)
+
+ sAi(0,πi,pi)
+
n∈N
Ci
a1···an
Aa1(0,πa1 ,pa1 )
∧ . . . ∧ Aan(0,πan ,pan )
. (2.122)
At ghost number zero we find the FDA
Ri(pi+1,πi,0)
= dAi(pi,πi,0)
+
n∈N
Ci
a1···an
Aa1(pa1 ,πa1 ,0)
∧ . . . ∧ Aan(pan ,πan ,0)
, (2.123)
while at ghost number 1 ≤ I ≤ pi we find
Ri(pi+1−I,πi.I)
= dAi(pi−I,πi,I)
+ sAi(pi−(I−1),πi,(I−1))
+
+
n∈N
Ci
a1···an
Aa1(pa1 −Ia1 ,πa1 ,Ia1 )
∧ . . . ∧ Aan(pan −Ian ,πan ,Ian )
(pi+1,I)
(2.124)
49
52. and at the top ghost number I = pi + 1
Ri(0,πi.pi+1)
= sAi(0,πi,pi)
+
+
n∈N
Ci
a1···an
Aa1(0,πa1 ,pa1 )
∧ . . . ∧ Aan(0,πan ,pan )
(2.125)
As showed before in a gauge theory, the horizontality condition
Ri(pi+1−I,πi,I)
= 0 , ∀ I > 0 (2.126)
imposes the correct constraints which allow for the construction of gauge superspace.
These conditions read:
sAi(pi−(I−1),πi,(I−1))
= −dAi(pi−I,πi,I)
+
−
n∈N
Ci
a1···an
Aa1(pa1 −Ia1 ,πa1 ,Ia1 )
∧ . . . ∧ Aan(pan −Ian ,πan ,Ian )
(2.127)
for 1 ≤ I ≤ pi and
sAi(0,πi,pi)
= −
n∈N
Ci
a1···an
Aa1(0,πa1 ,pa1 )
∧ . . . ∧ Aan(0,πan ,pan )
(2.128)
for I = pi + 1. This promotes Principle 13 to a theorem.
50
53. Chapter 3
Gravity and supergravity
In this Chapter we describe pure gravity and simple pure supergravity in D = 4 from the
geometric point of view. This exemplifies the principles and building rules described in
Chapter 2.
3.1 Poincar´e gravity
In this section we build a geometric theory of pure gravity (we concentrate on the D = 4
case). The fields introduced for this purpose are the vielbein V a
and the spin connection
ωab
, which are bosonic one-forms and, in the first-order formalism, are considered as
independent fields.
3.1.1 Pure gravity in four dimensions
The curvatures of the theory are the Riemann two-form and the torsion
Rab
= dωab
− ωa
c ∧ ωcb
,
Ta
= DV a
= dV a
− ωa
b ∧ V b
(3.1)
and the corresponding Bianchi identities are
DRab
= 0 ,
DTa
+ Rab
∧ Vb = 0 . (3.2)
It is possible to consider the vielbein and the connection as a single object V a
, ωab
which
is a multiplet in the adjoint representation of the Poincar´e group. Therefore we can write
the gauge field 1-form of the Poincar´e group µA
as
µA
TA = ωab
Jab + V a
Pa = µ , (3.3)
where Jab and Pa are the generators of the Lorerntz group and of translations, respectively.
Hence the field strengths (3.1) can be written in the more compact form
RA
= dµA
+
1
2
CA
BC µB
∧ µC
. (3.4)
The pure gravity Lagrangian in D = 4 is found by requiring that it is invariant under
the following symmetries:
51
54. • Coordinate invariance: this is satisfied by requiring that the theory is gemoetrical, i.e.
built of differential forms. For the moment we restrict our study to theories without
scalar fields, which are named strongly geometric.
• SO(1, 3) gauge covariance. This implies that the Lagrangian must contain the con-
nection only through gauge invariant quantities, namely the curvatures.
• The zero-curvature case, Rab
= 0, Ta
= 0 must be a solution of the equations
of motion, since we must be able to recover the left-invariant theory in the zero-
curvature limit.
• Rigid scale invariance
ωab
→ ωab
,
V a
→ λ V a
λ = 0 . (3.5)
Since the curvatures have this global symmetry, we also demand that the Lagrangian
possesses this invariance.
The requirement of a strongly geometric Lagrangian imposes that the Lagrangian must
be a polynomial in the fields and curvatures of the theory. This fixes the form of the
Lagrangian (up to total derivatives) as
L = L(0) + L(1) + L(2) , (3.6)
where L(i) is a polynomial of degree i in the curvatures:
L(0) = CA1···A4 µA1
∧ . . . ∧ µA4
,
L(1) = RA
∧ CAPQ µP
∧ µQ
,
L(2) =
1
2
RA
∧ RB
CAB . (3.7)
The SO(1, 3) invariant tensors which can be used in the Lagrangian are:
• CABCD: Cabcd ∝ abcd
• CAPQ: C(ab)cd ∝ abcd, C(ab)cd ∝ δab
cd
• CAB: C(ab)(cd) ∝ abcd, C(ab)(cd) ∝ δab
cd, Ca,b ∝ δa
b .
First we can prove
Proposition 14 The terms which are quadratic in the curvatures are equivalent to total
derivatives, hence can be dropped.
Proof. The Lagrangian quadratic in the curvatures can be written
L(2) =
1
2
RA
∧ RB
∧ cAB =
= c1 Rab
∧ Rcd
abcd + c2 Rab
∧ Rab + c3 Ta
∧ Ta , (3.8)
52
55. where ci are constant coefficients. Consider the following identities:
Rab
∧ Rcd
abcd = abcd d ωab
∧ Rcd
+ ωab
∧ dRcd
− ωa
e ∧ ωeb
∧ Rcd
=
= abcd d ωab
∧ Rcd
− ωa
l ∧ ωlb
∧ ωcd
, (3.9)
Rab
∧ Rab = d ωab
∧ Rab + ωab
∧ dRab − ωa
e ∧ ωeb
∧ Rab =
= d ωab
∧ Rab −
1
3
ωab
∧ ωae ∧ ωe
b , (3.10)
Ta
∧ Ta = (DV a
) ∧ (DVa) =
= d [V a
∧ DVa] − V a
∧ Rab ∧ V b
. (3.11)
This shows that the first two terms are equivalent to total derivatives, while the last is
equivalent to a one-curvature term, which is considered below.
Note however that, since the integration manifold M4 is without boundary, the terms
quadratic in the curvatures can still give a topological number, since the first Pontriagyn
index P1 and the Euler characteristic of the manifold are defined as
P1 = −
1
8 π2
M4
Rab
∧ Rab ,
E =
1
32 π2
M4
abcd Rab
∧ Rcd
, (3.12)
respectively.
Since the connection cannot appear as a bare field because of the SO(1, 3) gauge
invariance, we can only use
µA
= V a
. (3.13)
Thus the Lagrangian can be written as
L = c4 a1···a4 V a1
∧ . . . ∧ V a4
+ c5 abcd Rab
∧ V c
∧ V d
+ c6 Rab
∧ Va ∧ Vb . (3.14)
The variation of the Lagrangian with respect to the fields ωab
and V a
gives
δL = 4 c4 abcd δV a
∧ V b
∧ V c
∧ V d
+
+ c5 abcd Dδωab
∧ V c
∧ V d
+ 2 c5 abcd Rab
∧ δV c
∧ V d
+
+ c6 Dδωab
∧ Va ∧ Vb + 2 c6 Rab
∧ δVa ∧ Vb =
= δωab
∧ 2 c5 abcd Tc
∧ V d
+ 2 c6 Ta ∧ Vb +
+ δVa ∧ 4 c4 abcd V b
∧ V c
∧ V d
+ 2 c5 abcd Rcd
∧ V b
+ 2 c6 Rab
∧ Vb .(3.15)
The equation of motion for the vielbein V a
evalued at the vacuum RA
= 0 gives
4 c4 abcd V b
∧ V c
∧ V d
= 0 (3.16)
and because V b
∧V c
∧V d
is an independent 3-form, compatibility with the vacuum solution
requires
c4 = 0 . (3.17)
53
56. Moreover, we want the theory to be invariant under spatial parity, hence either c5 or c6
must be zero. If c5 = 0, the resulting equations of motion are
c6 Ta ∧ Vb = 0 ,
c6 Rab
∧ Vb = −c6 DTa
= 0 , (3.18)
where in the second equation we used the Bianchi Identity for the torsion. Therefore the
equations of motion impose Ta
= 0, but leave Rab
unconstrained. For this reason we must
choose c6 = 0 so that the action for pure gravity in D = 4 is the Einstein-Cartan action
S =
M4
Rab
∧ V c
∧ V d
abcd . (3.19)
3.1.2 Gravity as a gauge theory
The Einstein-Cartan action (3.19) is the action for pure gravity in D = 4, that we built
as the gauge theory of the Lorentz group SO(1, 3). However, it is quite different from the
usual Yang-Mills action for a theory, which would have had the following form:
M4
RA
∧ RA , (3.20)
where the star indicates the Hogde dual of the curvature. The first difference between
the two different kinds of action is that the action (3.20) is invariant under the whole
soft gauge group ˜G of which the µA
are the gauge potentials, while the Einstein-Cartan
action is not invariant under the whole gauge group ISO(1, 3), but only under its Lorentz
subgroup SO(1, 3).
Theorem 15 The Einstein-Cartan action (3.19) is not invariant under a local transla-
tion.
Proof. The infinitesimal transformation of the connection and vielbein under a local
ISO(1, 3) transformation of parameter ( a
, ab
) is
δωab
= D ab
,
δV a
= D a
+ ab
Vb . (3.21)
Setting ab
= 0, we obtain the transformation law for an infinitesimal translation
δωab
= 0 ,
δV a
= D a
. (3.22)
This implies that, under local translations, the curvatures transform as
δRab
= 0 ,
δTa
= δDV a
= DδV a
= DD a
= −Rab
b . (3.23)
Therefore the Einstein-Cartan action (3.19) trasforms as
δS = 2
M4
Rab
∧ D c
∧ V d
abcd =
= 2 Rab
∧ Td c
abcd = 0 . (3.24)
54
57. However, the Einstein-Cartan action is invariant under general coordinate transforma-
tions, which are generated by the Lie derivative along the tangent vector
= µ
∂µ = µ
V a
µ
˜Pa + µ
ωpq
µ Jpq , (3.25)
where
˜Pa = V µ
a ∂µ − ωbc
µ Jbc (3.26)
since the generator of translations of the corresponding left-invariant manifold is Pµ =
∂µ = V a
µ
˜Pa + ωab
µ Jab. Indeed
δdiff S =
M4
L L =
=
M4
(dι + ι d) L = 0 . (3.27)
Another important difference between the Yang-Mills action (3.20) and the Einstein-
Cartan action (3.19) is that the former is quadratic in the curvatures, while the latter
is linear in RA
. This is because we are using the first-order formalism, thus considering
the connection and the vielbein as independent fields. It is the variation of the action
in the connection which implies the torsion equation Ta
= 0, which can be algebraically
solved for the spin connection in terms of first order derivatives of the vielbein. Integrating
out the connection via its equations of motion, the resulting action gives a second-order
differential equation of motion for the vielbein V a
.
Indeed, we can expand the torsion equation Ta
= 0 into
dV a
= Ca
mn V m
∧ V n
,
ωab
= ωab
|m V m
. (3.28)
Therefore
Ta
= Ca
mn + ωa
m|n − ωa
n|m V m
∧ V n
=
= 0 , (3.29)
so that
ωam|n = Camn + Cmna − Cnam . (3.30)
Equivalently, we can use the world-indices obtaining
Ta
= ∂[µV a
ν] − ωa
|[µ V b
ν] dxµ
∧ dxnu
, (3.31)
or
∂[µV a
ν] =
1
2
ωa
[µ V b
ν] − ωa
[ν V b
µ] , (3.32)
which gives
ωab
µ = V aρ
V bν
Vcρ ∂µV c
ν + Vcν ∂ρV c
µ − Vcµ ∂νV c
ρ . (3.33)
55
58. Moreover, the equatio of motion for the vielbein gives
2 Rab
∧ V c
abcd = 0 . (3.34)
The curvature 2-form can be expanded along a vielbein basis as
Rab
= Rab
cd V c
∧ V d
, (3.35)
so that the equation of motion becomes
2 Rab
mn V m
∧ V n
∧ V c
abcd = 0 . (3.36)
Defining the non-vanishing 3-form Ωl as
V m
∧ V n
∧ V c
= mncl
Ωl , (3.37)
we have
0 = Rab
mn V m
∧ V n
∧ V c
abcd = −3! δmnl
abd Rab
mn Ωl =
= 4 Rlb
db −
1
2
δl
d Rab
ab Ωl , (3.38)
which is the Einstein field equation for a theory of pure gravity.
Concluding, starting from the Einstein-Cartan action, the equation of motion for the
connection ωab
, Ta
= 0, together with ωab
= −ωba
, can be algebraically solved for the
vielbein. This means that the integration manifold is endowed with a Riemannian spin
connection, which is not a physical field. Then, upon substitution of the connection ωab
(V )
in the action, the second-order action gives second-order equations of motion for the
vielbein V a
, which turns out to be the only physical field.
3.2 N=1, D=4 Supergravity
The building rules allow us to build a theory of pure gravity in D = 4, however the
rheonomic framework has been designed for a geometric construction of supergravity
theories. As a first non-trivial example, we present the supersymmetric extension of pure
gravity in D = 4: simple (N = 1) pure supergravity [202, 70, 114].
3.2.1 FDA and rheonomic parametrizations
Simple (N = 1) pure supergravity in D = 4 describes the supermultiplet (ωab
, V a
, Ψ),
where ωab
and V a
are the spin conection and vielbein 1-form, respectively, while Ψ is the
gravitino, a Majorana spinor 1-form. The curvatures for these fields are
Rab
= dωab
− ωa
c ∧ ωcb
,
Ta
= DV a
−
i
2
Ψ ∧ Γa
Ψ ,
ρ = DΨ . (3.39)
56
