The gauging of free differential algebras (FDA's) produces gauge field
theories containing antisymmetric tensors. The FDA's extend the Cartan-Maurer
equations of ordinary Lie algebras by incorporating p-form potentials ($p >
1$). We study here the algebra of FDA transformations. To every p-form in the
FDA we associate an extended Lie derivative $\ell$ generating a corresponding
``gauge" transformation. The field theory based on the FDA is invariant under
these new transformations. This gives geometrical meaning to the antisymmetric
tensors. The algebra of Lie derivatives is shown to close and provides the dual
formulation of FDA's.Comment: 10 pages, latex, no figures. Talk presented at the 4-th Colloquium on
  "Quantum Groups and Integrable Sysytems", Prague, June 199

Alberto Perotto

C. Chevalley

L. Castellani

Leonardo Castellani

R. D'Auria

S. Boukraa

Y. Ne'eman

English

arXiv

Crossref

Free differential algebras: Their use in field theory and dual formulation

The gauging of free differential algebras (FDA's) produces gauge field theories containing antisymmetric tensors. The FDA's extend the Cartan-Maurer equations of ordinary Lie algebras by incorporating p-form potentials (p > 1). We study here the algebra of FDA transformations. To every p-form in the FDA we associate an extended Lie derivative \ell generating a corresponding ``gauge" transformation. The field theory based on the FDA is invariant under these new transformations. This gives geometrical meaning to the antisymmetric tensors. The algebra of Lie derivatives is shown to close and provides the dual formulation of FDA's

Castellani, L

Perotto, A

CERN Document Server

DFTT-52/95
hep-th/9509031
August 1995
FREE DIFFERENTIAL ALGEBRAS: THEIR USE IN FIELD
THEORY AND DUAL FORMULATION
Leonardo Castellani
II Facolta di Scienze M.F.N. di Torino, sede di Alessandria
Dipartimento di Fisica Teorica
and
Istituto Nazionale di Fisica Nucleare
Via P. Giuria 1, 10125 Torino, Italy.
Alberto Perotto
Dipartimento di Fisica Teorica
Via P. Giuria 1, 10125 Torino, Italy.
Abstract
The gauging of free dierential algebras (FDA's) produces gauge eld
theories containing antisymmetric tensors. The FDA's extend the Cartan-
Maurer equations of ordinary Lie algebras by incorporating p-form potentials
(p > 1). We study here the algebra of FDA transformations. To every
p-form in the FDA we associate an extended Lie derivative ` generating a
corresponding \gauge" transformation. The eld theory based on the FDA is
invariant under these new transformations. This gives geometrical meaning
to the antisymmetric tensors. The algebra of Lie derivatives is shown to close
and provides the dual formulation of FDA's.
e-mail: castellani@to.infn.it
Talk presented by. L. Castellani at the 4
th
Colloquium
\Quantum groups and integrable systems", Prague, June 1995.
1 Introduction
Free dierential algebras [1, 2, 3] have emerged as underlying symmetries of eld
theories containing antisymmetric tensors, as for example supergravity and super-
string theories. Within the group-geometric method of ref.s [4, 3, 5], a systematic
algorithm exists that produces lagrangians invariant under any given FDA.
Actually, the FDA symmetries related to the antisymmetric tensors were not
explicitly discussed in [1, 2, 3]. They were treated in [6, 5], where they could
be deduced from the BRST algebra of FDA's by interpreting the ghosts as gauge
parameters.
Here we present a direct geometric interpretation of antisymmetric tensors: they
are the gauge elds of gauge transformations generated by a new type of Lie deriva-
tive. These new Lie derivatives, together with the usual Lie derivatives along the Lie
algebra tangent vectors, close on an algebra that can be called the dual formulation
of FDA's, extending ordinary Lie algebras.
Our arguments are developed in the case of the simplest FDA extension of the
Cartan-Maurer equations, containing a 2-form.
2 Group geometry and dynamical elds
We sketch the basic steps of the group-geometric approach of ref.s [4, 3]. See [5] for
a short review.
Lie algebra, tangent vectors, vielbeins
Consider an ordinary Lie algebra Lie(G), with abstract generators T
A
satisfying
the commutation relations
[T
A
; T
B
] = C
C
AB
T
C
(2.1)
On the group manifold G we can nd a basis of tangent vectors t
A
closing on the
same algebra as in (2.1). Their duals are the left-invariant one-forms 
A
(cotangent
basis), also called vielbeins, satisfying the Cartan-Maurer equations:
d
A
+
1
2
C
A
BC

B
^ 
C
= 0 (2.2)
as can be seen by using (2.1) and 
A
(t
B
) = 
B
A
. Thus, the commutation algebra
(2.1) and the Cartan-Maurer equations (2.2) are equivalent descriptions of the same
group structure. The Jacobi identities
C
A
B[C
C
B
DE]
= 0 (2.3)
necessary for the consistency of (2.1) ensure the integrability of eq.s (2.2), that is
the nilpotency of the external derivative d
2
= 0.
1
Dynamical elds, curvatures and Bianchi identities
The main idea of ref.s [4, 3] is to consider the one-forms 
A
as the fundamental
elds of the geometric theory to be constructed. More precisely, the dynamical elds
are the vielbeins 
A
of
~
G, a smooth deformation of the group manifold G referred
to as \soft group manifold". In general 
A
does not satisfy the Cartan-Maurer
equations any more, so that
d
A
+
1
2
C
A
BC

B
^ 
C
 R
A
6= 0 (2.4)
The extent of the deformation G!
~
G is measured by the curvature two-form R
A
.
R
A
= 0 implies 
A
= 
A
and viceversa. The deformation is necessary in order to
allow congurations with nonvanishing curvature.
Applying the external derivative d to the denition (2.4), using d
2
= 0 and the
Jacobi identities (2.3), yields the Bianchi identities
(rR)
A
 dR
A
  C
A
BC
R
B
^ 
C
= 0 (2.5)
An example: G = Poincare group
Consider
~
G= smooth deformation of the Poincare group, whose structure con-
stants are read o the corresponding Lie algebra :
[P
a
; P
b
] = 0 (2.6)
[M
ab
;M
cd
] = 
ad
M
bc
+ 
bc
M
ad
  
ac
M
bd
  
bd
M
ac
(2.7)
[M
ab
; P
c
] = 
bc
P
a
  
ac
P
b
(2.8)
Denoting by V
a
and !
ab
the vielbein 
A
when the index A runs on the transla-
tions and on the Lorentz rotations respectively, eq.s (2.4) take the form:
R
a
= dV
a
  !
ab
^ V
c

bc
(2.9)
R
ab
= d!
ab
  !
ac
^ !
db

cd
(2.10)
The fundamental elds V
a
and !
ab
are interpreted as the ordinary vierbein and
the spin connection, respectively, and eq.s (2.10) dene the torsion and the Riemann
curvature. These satisfy the Bianchi identities
dR
a
 R
ab
V
b
+ !
ab
R
b
 DR
a
 R
ab
V
b
= 0 (2.11)
dR
ab
 R
ac
!
cb
+ !
ac
R
cb
 DR
ab
= 0 (2.12)
Products between forms are understood to be exterior products, D is the Lorentz
covariant derivative, and repeated indices are contracted with the Minkowski metric

ab
.
2
Note that the elds 
A
(y) depend on all the soft group manifold coordinates
y. In the Poincare example, this means that the vierbein and the spin connection
depend on the coordinates y
a
associated to the translations (the ordinary space-
time coordinates) and on the coordinates y
ab
associated to the Lorentz rotations.
Since we want to have space-time elds at the end of the game, we have to nd
a way to remove the y
ab
dependence. This is achieved when the curvatures are
horizontal in the y
ab
directions (see later).
How do we nd the dynamics of 
A
(y) ? We want to obtain a geometric the-
ory, i.e. invariant under dieomorphisms of the soft group manifold
~
G. We need
therefore to construct an action invariant under dieomorphisms, and this is simply
achieved by using only dieomorphic invariant operations as the exterior derivative
and the wedge product. The building blocks are the one-form 
A
and its curva-
ture two-form R
A
, and exterior products of them can make up a lagrangian D-form
(where D is the dimension of space-time).
Dieomorphisms
The variation under dieomorphisms y+ " of the vielbein eld 
A
(y) is given by
the Lie derivative of the vielbein along the innitesimal tangent vector   "
A
t
A
:

A
= 
A
(y + ")  
A
(y) = d(i


A
) + i

d
A
 `


A
(2.13)
On p-forms !
(p)
= !
B
1
:::B
p

B
1
^ ::: ^ 
B
p
, the contraction i
v
along an arbitrary
tangent vector v = v
A
t
A
is dened as
i
v
!
(p)
= p v
A
!
AB
2
:::B
p

B
2
^ :::^ 
B
p
(2.14)
and maps p-forms into (p   1)-forms.
The operator
l
v
 d i
v
+ i
v
d (2.15)
is the Lie derivative along the tangent vector v and maps p-forms into p-forms. Eq.
(2.13) gives the variation under dieomorphisms of any p-form.
We now rewrite the variation 
A
of eq. (2.13) in a suggestive way, by adding
and subtracting C
A
BC

B
"
C
:

A
= d"
A
+ C
A
BC

B
"
C
  2
B
"
C
(d
A
)
BC
 C
A
BC

B
"
C
= (r")
A
+ i

R
A
(2.16)
where we have used the denition (2.4) for the curvature, and the G-covariant
derivative r acts on "
A
as
(r")
A
 d"
A
+ C
A
BC

B
"
A
(2.17)
3
Horizontality
All the invariances of the geometric theory are contained in eq. (2.16). In par-
ticular, suppose that the two-form R
A
= R
A
BC

B
^ 
C
has vanishing components
along the directions of a subgroup H of G:
R
A
BH
= 0
A runs on G
H runs on H (2.18)
Then we say that R
A
is horizontal on H, and the dieomorphisms along the H-
directions reduce to gauge transformations:

A
(y) = (r")
A
(2.19)
Moreover, the dependence on the y
H
coordinates becomes inessential, in the sense
that it factorizes after a nite gauge transformation (see ref.s [3, 5]). The the-
ory "remembers" the invariance under y
H
-dieomorphisms by retaining the gauge
invariance under H, with "
H
interpreted now as a gauge parameter.
For example, in Poincare gravity the curvatures are horizontal along the Lorentz
directions : then the elds V
a
and !
ab
live on the coset space
G
H
=
Poincare
0
Lorentz
(2.20)
i.e. on ordinary spacetime. The lagrangian is integrated on a D- volume (D-
dimensional spacetime), and is therefore a D-form. The resulting theory is invariant
under D-spacetime dieomorphisms, and under local Lorentz rotations.
Finally, we recall the algebra of Lie derivatives on the group manifold. From
`
"
B
t
B

A
= d"
A
+

C
A
BC
  2R
A
BC


B
"
C
(2.21)
cf. (2.16), we deduce
h
`
"
A
1
t
A
; `
"
B
2
t
B
i
= `
[
"
A
1
@
A
"
C
2
 "
A
2
@
A
"
C
1
+"
A
1
"
B
2
(
C
C
AB
 2R
C
AB
)]
t
C
(2.22)
where the partial derivative @
A
of a function f is dened by df  (@
A
f)
A
or also
@
A
f  t
A
(f). One can can verify the standard formula:
[`
"
A
1
t
A
; `
"
B
2
t
B
] = `
["
A
1
t
A
; "
B
2
t
B
]
(2.23)
In particular, the Lie derivatives `
t
A
on the undeformed group manifold G close on
the Lie algebra:
[`
t
A
; `
t
B
] = C
C
AB
`
t
C
(2.24)
4
3 Free dierential algebras
The dual formulation of Lie algebras provided by the Cartan-Maurer equations (2.2)
can be naturally extended to p-forms (p > 1):
d
i
(p)
+
X
1
n
C
i
i
1
:::i
n

i
1
(p
1
)
^ :::^ 
i
n
(p
n
)
= 0; p + 1 = p
1
+ :::+ p
n
(3.1)
p; p
1
; :::p
n
are the degrees of the forms 
i
; 
i
1
; :::; 
i
n
; the indices i; i
1
; :::; i
n
run on
irreps of a group G, and C
i
i
1
:::i
n
are generalized structure constants satisfying gen-
eralized Jacobi identities due to d
2
= 0. When p = p
1
= p
2
= 1 and i; i
1
; i
2
belong
to the adjoint representation of G, eq.s (6.1) reduce to the ordinary Cartan-Maurer
equations. The (anti)symmetry properties of the indices i
1
; :::i
n
depend on the
bosonic or fermionic character of the forms 
i
1
; :::
i
n
If the generalized Jacobi identities hold, eq.s (3.1) dene a free dierential alge-
bra (FDA). The possible FDA extensions G
0
of a Lie algebra G have been studied
in ref.s [1, 2], and rely on the existence of Chevalley cohomology classes in G [7].
Suppose that, given an ordinary Lie algebra G, there exists a p-form:


i
(p)
() = 

i
A
1
:::A
p

A
1
^ ::: ^ 
A
p
; 

i
A
1
:::A
p
= constants; i runs on a G  irrep
(3.2)
which is covariantly closed but not covariantly exact, i.e.
r

i
(p)
 d

i
(p)
+ 
A
^D(T
A
)
i
j


j
(p)
= 0; 

i
(p)
6= r
i
(p 1)
(3.3)
Then 

i
(p)
is said to be a representative of a Chevalley cohomology class in the D
i
j
irrep of G. r is the boundary operator satisfying r
2
= 0 (it would be proportional
to the curvature 2-form on the soft group manifold). The existence of 

i
(p)
allows
the extension of the original Lie algebra G to the FDA G
0
:
d
A
+
1
2
C
A
BC

B
^ 
C
= 0 (3.4)
r
i
(p 1)
+ 

i
(p)
() = 0 (3.5)
where 
i
(p 1)
is a new (p  1)-form, not contained in G. Closure of eq.s (6.4) is due
to r

i
(p)
= 0.
It is clear that 

i
(p)
diering by exact pieces r
i
(p 1)
lead to equivalent FDA's,
via the redenition 
i
(p 1)
! 
i
(p 1)
+ 
i
(p 1)
. What we are interested in are really
nontrivial cohomology classes satisfying eq.s (3.3).
The whole game can be repeated on the free dierential algebra G
0
which now
contains 
A
, 
i
(p 1)
. One looks for the existence of polynomials in 
A
, 
i
(p 1)


i
(q)
(;) = 

i
A
1
:::A
r
i
1
:::i
s

A
1
^ ::: ^ 
A
r
^ 
i
1
(p 1)
^ ::: ^ 
i
s
(p 1)
(3.6)
satisfying the cohomology conditions (3.3). If such a polynomial exists, the FDA
of eq.s (3.4), (3.5) can be further extended to G
00
, and so on.
5
Note 1: In constructing D-dimensional supergravity theories we usually choose
as starting point the superPoincare Lie algebra. The possible G
0
extensions to
FDA's depend on the spacetime dimension D. For example in D = 11 there is a
cohomology class of the superPoincare algebra in the identity representation:

(V; !;  ) =
1
2

  
ab
 V
a
V
b
(3.7)
where  is the gravitino eld, dual to the supersymmetry charge; d
 = 0 holds
because of the D = 11 Fierz identity

  
ab
 

  
a
 V
b
= 0 (3.8)
This allows the extension of the algebra (3.4) by means of a three -form A:
dA  
(V; !;  ) = 0 (3.9)
Note 2: Only nonsemisimple algebras can have FDA extensions in nontrivial G-
irreps. Indeed a theorem by Chevalley and Eilenberg [7] states that there is no
nontrivial cohomology class of G in nontrivial G-irreps when G is semisimple.
As we have done in the case of ordinary Lie algebras, we nd a dynamical theory
based on FDA's by allowing nonvanishing curvatures. This means, for example, that
D = 11 supergravity is based on a deformation of the elds V; !;  ;A such that
the superPoincare curvatures and the A-curvature are dierent from zero. For the
geometric construction of the action we refer the reader to refs. [3, 5]. Other theories
with antisymmetric tensors have been interpreted as gaugings of free dierential
algebras: see [3] for a detailed study.
4 The FDA1 algebra
We consider here the simplest extension of a Lie algebra, denoted by FDA1:
d
A
+
1
2
C
A
BC

B

C
= 0 (4.1)
dB
i
+ C
i
Aj

A
B
j
+
1
6
C
i
ABC

A

B

C
 rB
i
+
1
6
C
i
ABC

A

B

C
= 0 (4.2)
where B
i
is a two-form in a representation D
i
j
of G. The generalized Jacobi iden-
tities (d
2
= 0), besides the usual ones for C
A
BC
, are
C
i
Aj
C
j
Bk
  C
i
Bj
C
j
Ak
= C
C
AB
C
i
Ck
; representation condition (4.3)
4C
j
[ABC
C
i
D]j
+ 6C
E
[AB
C
i
CD]E
= 0; 3  cocycle condition (4.4)
Eq. (4.3) implies that (C
A
)
i
j
 C
i
Aj
is a matrix representation of G, while eq.
(4.4) is just the statement that C
i
 C
i
ABC

A

B

C
is a 3-cocycle, i.e. rC
i
= 0. If
6
we allow the left hand sides of eq.s (4.1), (4.2) to be nonvanishing curvatures R
A
,
R
i
respectively, we nd the Bianchi identities:
dR
A
  C
A
BC
R
B

C
= 0 (4.5)
dR
i
  C
i
Aj
R
A
B
j
+ C
i
Aj

A
R
j
 
1
2
C
i
ABC
R
A

B

C
= 0 (4.6)
where we use the same symbol B
i
for the "soft" 2-form. The curvatures can be
expanded on the 
A
; B
i
basis as
R
i
= R
i
ABC

A

B

C
+R
i
Aj

A
B
j
(4.7)
R
A
= R
A
BC

B

C
+R
A
i
B
i
(4.8)
Lie derivatives
Using the denition (2.15) and the expression of the FDA curvatures, we nd
the action of the Lie derivative on B
i
:
`
"
B
t
B
B
i
=

R
i
Aj
  C
i
Aj

"
A
B
j
+

3R
i
ABC
 
1
2
C
i
ABC

"
A

B
^ 
C
(4.9)
the action on 
A
remaining the one given in (2.21). Here we nd something inter-
esting: the Lie derivatives do not close any more as in eq. (2.22). Before computing
this modied algebra, let us dene
i) a new contraction operator i
"
j
t
j
by its action on a generic p-form ! =
!
i
1
:::i
n
A
1
:::A
m
B
i
1
^ :::B
i
n
^ 
A
1
^ :::
A
m
as
i
"
j
t
j
! = n "
j
!
ji
2
:::i
n
A
1
:::A
m
B
i
2
^ :::B
i
n
^ 
A
1
^ :::
A
m
(4.10)
where "
j
is a 1-form. This operator still maps p-forms into (p   1)-forms. We can
also dene the contraction i
t
j
, mapping p-forms into (p   2)-forms, from
i
"
j
t
j
= "
j
i
t
j
(4.11)
In particular
i
t
j
(B
i
) = 
i
j
(4.12)
so that t
j
can be seen as the \tangent vector" dual to B
j
. Note that i
"
j
t
j
vanishes
on p-forms that do not contain at least one factor B
i
.
ii) a new Lie derivative given by:
`
"
i
t
i
 i
"
i
t
i
d+ d i
"
i
t
i
(4.13)
This new derivative commutes with d, satises the Leibnitz rule, and acts on the
fundamental elds as
`
"
j
t
j

A
= "
j
R
A
j
(4.14)
`
"
j
t
j
B
i
= d"
i
+ (C
i
Aj
 R
i
Aj
)
A
^ "
j
(4.15)
7
Using these new objects, and the Bianchi identities (4.7), (4.8), we can compute
the commutator of two \usual" Lie derivatives (acting on 
A
or on B
i
) and nd:
h
`
"
A
1
t
A
; `
"
B
2
t
B
i
= `
[
"
A
1
@
A
"
C
2
 "
A
2
@
A
"
C
1
+"
A
1
"
B
2
(
C
C
AB
 2R
C
AB
)]
t
C
+ `
(
C
i
ABC
 6R
i
ABC
)

A
"
B
1
"
C
2
t
i
(4.16)
This result has an important consequence: if the eld theory based on FDA1 is ge-
ometric, i.e. its action is invariant under dieomorphisms generated by the \usual"
Lie derivative, then the new Lie derivative dened in (4.13) must also generate a
symmetry of the action, since it appears on the right-hand side of (4.16). Thus,
when we construct geometric lagrangians gauging FDA1, we know a priori that
the resulting theory will have symmetries generated by the new Lie derivative. In
other words, the transformations (4.14), (4.15) are invariances of the action.
It becomes now essential to nd the total FDA algebra of transformations, and
show that it closes. Using the Bianchi identities, it is a straightforward exercise to
nd the remaining commutators:
h
`
"
A
t
A
; `
"
j
t
j
i
= `
[`
"
A
t
A
"
k
+
(
C
k
Bj
 R
k
Bj
)
"
B
"
j
]t
k
(4.17)
h
`
"
i
1
t
i
; `
"
j
2
t
j
i
= `
R
B
i
("
i
1
("
2
)
j
B
 "
i
2
("
1
)
j
B
)t
j
(4.18)
where "
i
A
are the components of the 1-form "
i
, i.e. "
i
 "
i
A

A
. In particular, we
can nd the commutators of the Lie derivatives on the rigid FDA1 \manifold" by
taking "
A
= const., "
i
B
= const. and vanishing curvatures:
[`
t
A
; `
t
B
] = C
C
AB
`
t
C
+ C
i
ABC
`

C
t
i
(4.19)
[`
t
A
; `

B
t
i
] = [C
k
Aj

B
F
  (C
B
AF
  2R
B
AF
)
k
j
]`

F
t
k
(4.20)
[`

A
t
i
; `

B
t
j
] = 0 (4.21)
This algebra can be considered the dual of the FDA1 system given in (4.1), (4.2),
and generalizes the Lie algebra of ordinary Lie derivatives (generating usual dif-
feomorphisms) of (2.24). Notice the essential presence of the 1-form  in front of
the \tangent vectors" t
i
. There are dim(G)  dim(D) independent extended Lie
derivatives `

A
t
i
, with dim(G)=dimension of the Lie algebra, dim(D)= dimension
of the D
i
j
irrep of G to which B
i
belongs .
References
[1] D. Sullivan, Innitesimal computations in topology, Bull. de L' Institut des
Hautes Etudes Scientiques, Publ. Math. 47 (1977).
[2] R. D' Auria and P. Fre, Nucl. Phys. B201 (1982) 101; L. Castellani, P. Fre,
F. Giani, K. Pilch and P. van Nieuwenhuizen, Ann. Phys. 146 (1983) 35; P.
van Nieuwenhuizen, Free graded dierential algebras in: Group theoretical
methods in physics, Lect. Notes in Phys. 180 (Springer, Berlin, 1983).
8
[3] L. Castellani, R. D'Auria and P. Fre, \Supergravity and Superstrings: a geo-
metric perspective", World Scientic, Singapore 1991.
[4] Y. Ne'eman and T. Regge, Phys. Lett. B74 (1978) 31; Riv. Nuovo Cimento 1
(1978) 5; A. D' Adda, R. D' Auria, P. Fre and T. Regge, Riv. Nuovo Cimento
3 (1980) 6; R. D' Auria, P. Fre and T. Regge, Riv. Nuovo Cimento 3 (1980)
12.
[5] L. Castellani, Int. J. Mod. Phys. A7 (1992) 1583.
[6] S. Boukraa, Nucl. Phys. B303 (1988) 237.
[7] C. Chevalley and S. Eilenberg, Trans. Amer. Math. Soc. 63 (1948) 85.
9


Free differential algebras: their use in field theory and dual formulation

Free Differential Algebras: Their Use in Field Theory and Dual
  Formulation

Free Differential Algebras: Their Use in Field Theory and Dual Formulation

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