1. Generalizations for Cartan’s equations and Bianchi identities for
arbitrary dimensions and its actions
Vladimir Cuesta †
Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, 70-543, Ciudad de
M´exico, M´exico
Abstract. Cartan’s ﬁrst and second structure equations are sets of equations for 2-forms, in a similar
way ﬁrst and second Bianchi identities are sets of equations for 3-forms involving exterior derivatives
and wedge products between the tetrad, the connection, the torsion and the curvature. However, I
can add parameters and wedge products between my set of variables and I ﬁnd equations for 2-forms
and 3-forms involving exterior derivatives and wedge products and these equations contain Cartan’s
equations and Bianchi’s identities like particular cases, I made that in my study. Now, If I think the
tetrad, the connection, the torsion and the curvature as independent variables I can construct actions for
my generalizations for arbitrary dimensions introducing auxiliary ﬁelds. I can generalize in some sense
the work in [1].
1. Introduction
Differential geometry is of great importance for the mathematics of our days (see [2] and [3] for
instance), inside this area we can ﬁnd the study of differential forms and like a lot of studies, there
are intersections between different areas of knowledge, in the case of differential forms, I can make
studies for actions, studies for geometry, algebra, analysis, functional analysis and so on.
In the present paper I will ﬁnd equations that are generalizations for the Cartan’s structure
equations and Bianchi identities, I will divide my studies for the two-dimensional case, the three-
dimensional case and the n-dimensional case, in this paper I introduce a set of parameters and wedge
products for my basic variables for making the generalization.
My basic variables are the one-forms eI
or tetrad and the ﬁeld φI
, the one-forms ωI
J or
connection and the ﬁeld φI
J , the two-forms TI
or torsion and the ﬁeld ψI
and the two-forms RI
J or
curvature and the ﬁeld ψI
J (see [1] for instance).
I will begin with the discussion as follows:
†
vladimir.cuesta@nucleares.unam.mx
2. Generalizations for Cartan’s equations and Bianchi identities for arbitrary dimensions and its actions2
The Cartan’s ﬁrst structure equations,
deI
+ ωI
K ∧ eK
= TI
, (1)
are set of equations for two-forms where I is free with the d exterior derivative and wedge product as
operations.
The Cartan’s second structure equations,
dωIJ
+ ωI
K ∧ ωKJ
= RIJ
, (2)
are set of equations for two-forms where I and J are free with the d exterior derivative and wedge
product as operations.
The ﬁrst Bianchi identities,
dTI
+ ωI
K ∧ TK
= RI
K ∧ eK
, (3)
are set of equations for three-forms where I is free with the d exterior derivative and wedge product
as operations.
The second Bianchi identities,
dRIJ
+ ωI
K ∧ RKJ
− ωJ
K ∧ RKI
= 0, (4)
are set of equations for three-forms where I and J are free with the d exterior derivative and wedge
product as operations. In all the previous equations, the indices I, J, . . ., are raised and lowered with
the Minkowski (σ = −1) or Euclidean (σ = +1) metric (ηIJ ) = diag(σ, +1, +1, . . . , +1).
In the present paper I will construct a set of equations with the same structure like all Cartan’s
structure equations and Bianchi’s identities (see [4] and [5] for a comprehensive exposition). I mean,
set of equations for two-forms where I is free with the d exterior derivative and wedge product as
operations, set of equations for two-forms where I and J are free with the d exterior derivative and
wedge product as operations, set of equations for three-forms where I is free with the d exterior
derivative and wedge product as operations and set of equations for three-forms where I and J are free
with the d exterior derivative and wedge product as operations, in my formalism the basic variables
eI
, φI
, ωI
J , φI
J , TI
, ψI
, RI
J and ψI
J , I present actions for my equations of motion.
2. Bi-dimensional case
In two dimensions, I can construct the following action:
I2 = [φIdeI
+ φIJ dωIJ
− a1φIωI
K ∧ eK
− a2φITI
− b1φIJ RIJ
− b2φIJ ωI
K ∧ ωKJ
− b3φIJ eI
∧ eJ
]
= [−dφI ∧ eI
− dφIJ ∧ ωIJ
− a1φIωI
K ∧ eK
− a2φITI
− b1φIJ RIJ
− b2φIJ ωI
K ∧ ωKJ
− b3φIJ eI
∧ eJ
], (5)
3. Generalizations for Cartan’s equations and Bianchi identities for arbitrary dimensions and its actions3
where eI
is the tetrad, ωIJ
is the connection, TI
is the torsion, RIJ
is the curvature, φI and φIJ are
zero-forms. I can vary the action and I obtain my equations of motion:
δφI : deI
− a1ωI
K ∧ eK
− a2TI
= 0,
δφIJ : dωIJ
− b1RIJ
− b2ωI
K ∧ ωKJ
− b3eI
∧ eJ
= 0,
δeI
: dφI + a1φKωK
I + 2b3φKIeK
= 0,
δωIJ
: − dφIJ + a1φ[IeJ] + 2b2φ[I|Kω|J]
K
= 0, (6)
3. Three-dimensional case
In this case I choose the 1-forms φI, φIJ , the 0-forms ψI, ψIJ and my action principle is (see [6] for
an early discussion of the subject):
I3 = [φI ∧ deI
+ φIJ ∧ dωIJ
+ ψIJ dRIJ
+ ψIdTI
− a1φI ∧ ωI
K ∧ eK
− a2φI ∧ TI
− b1φIJ ∧ RIJ
− b2φIJ ∧ ωI
K ∧ ωKJ
− b3φIJ ∧ eI
∧ eJ
+
a1b1
a2
ψIRI
K ∧ eK
− a1ψIωI
K ∧ TK
−
a1(a1 − b2)
a2
ψIωI
K ∧ ωK
L ∧ eL
− 2b2ψIJ ωI
K ∧ RKJ
−
2b3a2
b1
ψIJ eI
∧ TJ
−
2b3(a1 − b2)
b1
ψIJ eI
∧ eK
∧ ωK
J
]
= [dφI ∧ eI
+ dφIJ ∧ ωIJ
− dψIJ ∧ RIJ
− dψI ∧ TI
− a1φI ∧ ωI
K ∧ eK
− a2φI ∧ TI
− b1φIJ ∧ RIJ
− b2φIJ ∧ ωI
K ∧ ωKJ
− b3φIJ ∧ eI
∧ eJ
+
a1b1
a2
ψIRI
K ∧ eK
− a1ψIωI
K ∧ TK
−
a1(a1 − b2)
a2
ψIωI
K ∧ ωK
L ∧ eL
− 2b2ψIJ ωI
K ∧ RKJ
−
2b3a2
b1
ψIJ eI
∧ TJ
−
2b3(a1 − b2)
b1
ψIJ eI
∧ eK
∧ ωK
J
], (7)
I can consider my independent variables as φI, φIJ , ψI, ψIJ , eI
, ωIJ
, RIJ
and TI
, I must vary the
previous action and I obtain the following set of equations:
δφI : deI
− a1ωI
K ∧ eK
− a2TI
= 0,
δφIJ : dωIJ
− b1RIJ
− b2ωI
K ∧ ωKJ
− b3eI
∧ eJ
= 0,
δψI : dTI
+
a1b1
a2
RI
K ∧ eK
− a1ωI
K ∧ TK
−
a1(a1 − b2)
a2
ωI
K ∧ ωK
L ∧ eL
= 0,
δψIJ : dRIJ
− b2 ωI
K ∧ RKJ
− ωJ
K ∧ RKI
−
b3a2
b1
eI
∧ TJ
− eJ
∧ TI
) −
b3(a1 − b2)
b1
eI
∧ eK
∧ ωK
J
− eJ
∧ eK
∧ ωK
I
= 0,
δeI
: dφI − a1φK ∧ ωK
I + 2b3φIK ∧ eK
+
a1b1
a2
ψKRK
I
4. Generalizations for Cartan’s equations and Bianchi identities for arbitrary dimensions and its actions4
−
a1(a1 − b2)
a2
ψN ωN
K ∧ ωK
I +
4b3a2
b1
ψIKeK
−
2b3(a1 − b2)
b1
ψKN ωI
N
∧ eK
+
2b3(a1 − b2)
b1
ψIN ωK
N
∧ eK
= 0,
δωIJ
: dφIJ + a1φ[I ∧ eJ] + 2b2φ[I|K ∧ ω|J]
K
− a1ψ[I|T|J]
+
a1(a1 − b2)
a2
ψKe[I| ∧ ωK
|J] +
a1(a1 − b2)
a2
ψ[I|eK ∧ ω|J]
K
− 2b2ψ[I|KR|J]
K
+
2b3(a1 − b2)
b1
ψK[I|eK
∧ e|J] = 0,
δRIJ
: − dψIJ − b1φIJ +
a1b1
a2
ψ[I|e|J] + 2b2ψK[I|ωK
|J] = 0,
δTI
: dψI + a2φI + a1ψKωK
I = 0, (8)
4. n-dimensional case
In this case I choose the (n-2)-forms φI, φIJ and the (n-3)-forms ψI, ψIJ and my action principle is:
In = [φI ∧ deI
+ φIJ ∧ dωIJ
+ ψIJ ∧ dRIJ
+ ψI ∧ dTI
− a1φI ∧ ωI
K ∧ eK
− a2φI ∧ TI
− b1φIJ ∧ RIJ
− b2φIJ ∧ ωI
K ∧ ωKJ
− b3φIJ ∧ eI
∧ eJ
+
a1b1
a2
ψI ∧ RI
K ∧ eK
− a1ψI ∧ ωI
K ∧ TK
−
a1(a1 − b2)
a2
ψI ∧ ωI
K ∧ ωK
L ∧ eL
− 2b2ψIJ ∧ ωI
K ∧ RKJ
−
2b3a2
b1
ψIJ ∧ eI
∧ TJ
−
2b3(a1 − b2)
b1
ψIJ ∧ eI
∧ eK
∧ ωK
J
]
= [−(−1)n
dφI ∧ eI
− (−1)n
dφIJ ∧ ωIJ
+ (−1)n
dψIJ ∧ RIJ
+ (−1)n
dψI ∧ TI
− a1φI ∧ ωI
K ∧ eK
− a2φI ∧ TI
− b1φIJ ∧ RIJ
− b2φIJ ∧ ωI
K ∧ ωKJ
− b3φIJ ∧ eI
∧ eJ
+
a1b1
a2
ψI ∧ RI
K ∧ eK
− a1ψI ∧ ωI
K ∧ TK
−
a1(a1 − b2)
a2
ψI ∧ ωI
K ∧ ωK
L ∧ eL
− 2b2ψIJ ∧ ωI
K ∧ RKJ
−
2b3a2
b1
ψIJ ∧ eI
∧ TJ
−
2b3(a1 − b2)
b1
ψIJ ∧ eI
∧ eK
∧ ωK
J
], (9)
I can consider my independent variables to φI, φIJ , ψI, ψIJ , eI
, ωIJ
, RIJ
and TI
, I must vary the
previous action and I obtain the following set of equations:
δφI : deI
− a1ωI
K ∧ eK
− a2TI
= 0,
δφIJ : dωIJ
− b1RIJ
− b2ωI
K ∧ ωKJ
− b3eI
∧ eJ
= 0,
δψI : dTI
+
a1b1
a2
RI
K ∧ eK
− a1ωI
K ∧ TK
−
a1(a1 − b2)
a2
ωI
K ∧ ωK
L ∧ eL
= 0,
5. Generalizations for Cartan’s equations and Bianchi identities for arbitrary dimensions and its actions5
δψIJ : dRIJ
− b2 ωI
K ∧ RKJ
− ωJ
K ∧ RKI
−
b3a2
b1
eI
∧ TJ
− eJ
∧ TI
) −
b3(a1 − b2)
b1
eI
∧ eK
∧ ωK
J
− eJ
∧ eK
∧ ωK
I
= 0,
δeI
: − (−1)n
dφI − a1φK ∧ ωK
I + 2b3φIK ∧ eK
+
a1b1
a2
ψK ∧ RK
I
−
a1(a1 − b2)
a2
ψN ∧ ωN
K ∧ ωK
I +
4b3a2
b1
ψIK ∧ eK
−
2b3(a1 − b2)
b1
ψKN ∧ ωI
N
∧ eK
+
2b3(a1 − b2)
b1
ψIN ∧ ωK
N
∧ eK
= 0,
δωIJ
: − (−1)n
dφIJ + a1φ[I ∧ eJ] + 2b2φ[I|K ∧ ω|J]
K
− a1ψ[I| ∧ T|J]
+
a1(a1 − b2)
a2
ψK ∧ e[I| ∧ ωK
|J] +
a1(a1 − b2)
a2
ψ[I| ∧ eK ∧ ω|J]
K
− 2b2ψ[I|K ∧ R|J]
K
+
2b3(a1 − b2)
b1
ψK[I| ∧ eK
∧ e|J] = 0,
δRIJ
: (−1)n
dψIJ − b1φIJ +
a1b1
a2
φ[I| ∧ e|J] + 2b2ψK[I| ∧ ωK
|J] = 0,
δTI
: (−1)n
dψI − a2φI − a1ψK ∧ ωK
I = 0, (10)
5. Conclusions and perspectives
In the present work I have made generalizations for the topological ﬁeld theories in n-dimensional
spacetimes that were made at [1] and with this I obtain my initial purpose. For future work, I can
make a 1 + 1 decomposition for the two dimensional case, a 2 + 1 decomposition for the three
dimensional case or a (n − 1) + 1 decomposition for the n dimensional case, after that I can make the
hamiltonian analysis for all the existed actions and I will be able to count the degrees of freedom for
my theories. I will be able to analyze particular cases and so on.
References
[1] V. Cuesta, M. Montesinos, M. Vel´azquez and J. D. Vergara, Topological Field theories in n-dimensional spacetimes
and Cartan’s equations, Phys. Rev. D, 78, 064046, (2008).
[2] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol. 1, Interscience Publishers, New York, (1963).
[3] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol. 2, Interscience Publishers, New York, (1969).
[4] G. F. Torres del Castillo, Notas sobre variedades diferenciables, Segunda edici´on, (1998).
[5] A. I. Kostrikin, Introduction to Algebra, Springer, (1982).
[6] V. Cuesta and M. Montesinos, Cartan’s equations deﬁne a topological ﬁeld theory of the BF type, Phys. Rev. D, 76,
104004, (2007).
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