Physics Letters B 751 (2015) 205–208
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Topological gravity and Chern–Simons forms in d = 4
P. Catalán
a,b
, F. Izaurieta
a
, P. Salgado
a,∗
, S. Salgado
a
a
Departamento of Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile
b
Facultad de Ingeniería y Tecnología, Universidad San Sebastián, Campus Las Tres Pascualas, Lientur 1457, Concepción, Chile
a r t i c l e i n f o a b s t r a c t
Article history:
Received
22 September 2015
Received
in revised form 9 October 2015
Accepted
12 October 2015
Available
online 10 November 2015
Editor:
M. Cveti
ˇ
c
Following the construction introduced by Antoniadis and Savvidy in Refs. [1–3], we study metric-
independent
topological invariants on a
(
2n + 1
)
-dimensional space–time. These invariants allow us
to show that Chamseddine’s even-dimensional topological gravity corresponds to a Chern–Simons–
Antoniadis–Savvidy
form. Starting from this result, more general four-dimensional topological gravity
actions are explicitly constructed.
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
In Refs. [4–6] A.H. Chamseddine constructed topological actions
for gravity in all dimensions. He found in the odd-dimensional
case
the action given by S
(
2n+1
)
= k
M
2n+1
L
(2n+1)
ChS
(
A
)
where
L
(2n+1)
ChS
(
A
)
corresponds to a
(
2n + 1
)
-Chern–Simons form. All the
dynamical fields are components of an algebra-valued, one-form
gauge connection A = A
A
μ
T
A
⊗ dx
μ
, and the
(
2n + 1
)
-Lagrangian
form
is given by [7]
L
(2n+1)
ChS
(
A
)
=
(
n + 1
)
1
0
dt
A
tdA + t
2
A
2
n
.
(1)
Under off-shell gauge transformations, L
(2n+1)
ChS
(
A
)
only changes
by a closed form, and therefore the theory is described as ‘quasi-
invariant’
in the literature. Perhaps the best-known example of this
kind of theories is three-dimensional gravity,
L
(3)
G
(
A
)
=
1
2
abc
R
ab
+
2
3
2
e
a
e
b
e
c
, (2)
because of the famous quantization of the system due to Witten
[8,9]. However, the construction can be performed in every odd
dimension, and to be extended to the case of superalgebras. In
higher odd dimensions, the theory has a very complex dynamics,
with propagating degrees of freedom (although the proof of renor-
malizability
hasn’t been extended to this case). In the last decades,
*
Corresponding author.
E-mail
addresses: pacatalan@udec.cl (P. Catalán), fizaurie@udec.cl (F. Izaurieta),
pasalgad@udec.cl (P. Salgado), sesalgado@udec.cl (S. Salgado).
these kinds of systems have been thoroughly studied; see for in-
stance
Refs. [10–15] and a comprehensive review in Ref. [16].
In
the even-dimensional case, a similar construction using as
only field a 1-form gauge connection is not possible. As a matter
of fact it is necessary to use in addition to the gauge field A, at
least a 0-form multiplet φ in the fundamental representation of
the gauge group. It is because the n-product of the field strength
F
n
is a group invariant 2n-form, but also it is a topological invari-
ant
density which doesn’t provide equations of motion. But when
the scalar field φ
a
in the fundamental representation is included,
it is possible to construct a 2n-dimensional action as
S
(
2n
)
[
A
,φ
]
= k
M
2n
a
1
....a
2n+1
φ
a
1
F
a
2
a
3
···F
a
2n
a
2n+1
, (3)
where F = dA + AA. This action (even-dimensional topologi-
cal
gravity) was obtained by Chamseddine in [4] from an odd-
dimensional
Chern–Simons Lagrangian using a dimensional reduc-
tion
method. This kind of action principles have attracted some
attention recently. They can provide interesting cosmological dy-
namics,
with non-vanishing torsion (see Ref. [17]).
Besides
Chamseddine’s dimensional reduction, topological grav-
ity
has other deep links with Chern–Simons forms. For instance, in
Ref. [18] it was found that even-dimensional topological gravity ac-
tion
arises from odd-dimensional Chern–Simons gravity using non-
linear
realizations of the Poincaré group ISO
(
d − 1, 1
)
. The field φ
a
was identified with the coset field associated with the non-linear
realizations of the group.
Further
explorations were developed in Ref. [19]. There it was
shown that even-dimensional topological gravity actions, invariant
under the Poincaré group, correspond (up to a multiplicative con-
stant)
to a gauged Wess–Zumino–Witten term (see also Ref. [20]).
http://dx.doi.org/10.1016/j.physletb.2015.10.030
0370-2693/
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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