Hamiltonian analysis of Einstein–Chern–Simons gravity

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Physics Letters B ••• (••••) •••–•••

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Physics Letters B

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Hamiltonian analysis of Einstein–Chern–Simons gravity

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Departamento de Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile

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a r t i c l e

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L. Avilés, P. Salgado

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Article history: Received 29 October 2015 Received in revised form 12 April 2016 Accepted 14 April 2016 Available online xxxx Editor: M. Cvetiˇc

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In this work we consider the construction of the Hamiltonian action for the transgressions field theory. The subspace separation method for Chern–Simons Hamiltonian is built and used to find the Hamiltonian for five-dimensional Einstein–Chern–Simons gravity. It is then shown that the Hamiltonian for Einstein gravity arises in the limit where the scale parameter l approaches zero. © 2016 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 SCOAP3 .

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1. Introduction

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 57 58 59

In the context of the general relativity the spacetime is a dynamical object which has independent degrees of freedom and is governed by dynamical equations, namely the Einstein field equations. This means that in general relativity the geometry is dynamically determined. Therefore, the construction of a gauge theory of gravity requires an action that does not consider a fixed space-time background. An action for gravity fulfilling these conditions, albeit only in odd-dimensional spacetime, d = 2n + 1, was proposed long ago by Chamseddine [1,2], which is given by a Chern–Simons form for the anti-de Sitter (AdS) algebra. Chern–Simons gravities have been extensively studied; see, for instance, Refs. [3–12]. If Chern–Simons theories are to provide the appropriate gaugetheory framework for the gravitational interaction, then these theories must satisfy the correspondence principle, namely they must be related to general relativity. Studies in this direction have been carried out in Refs. [13–16] (see also [17,18]). In these references it was found that standard, five-dimensional GR (without a cosmological constant) emerges as the  → 0 limit of a CS theory for a certain Lie algebra B5 . Here  is a length scale, a coupling constant that characterizes different regimes within the theory. The B5 algebra, on the other hand, is constructed from the AdS algebra and a particular semigroup by means of the S-expansion procedure introduced in Ref. [19]. Black hole type solutions and the cosmological nature of the corresponding fields equations satisfy the same property, namely, that standard black-holes solutions and standard cosmological solutions emerge as the  → 0 limit of the black-holes and cosmological solutions of the Einstein–Chern–Simons field equations [14–16].

60 61 62 63 64 65

E-mail address: [email protected] (L. Avilés).

The Einstein–Chern–Simons action was constructed using transgression forms and a method, known as subspace separation procedure [20]. This procedure is based on the iterative use of the Extended Cartan Homotopy Formula, and allows one to (i ) systematically split the Lagrangian in order to appropriately reflect the subspaces structure of the gauge algebra, and (ii) separate the Lagrangian in bulk and boundary contributions. However the Hamiltonian analysis of Einstein Chern–Simons gravity action as well as transgression forms is as far as we know an open problem. In Ref. [21] was studied the Hamiltonian formulation of the Lanczos–Lovelock (LL) theory. The LL theory is the most general theory of gravity in d dimensions which leads to second-order field equations for the metric. The corresponding action, satisfying the criteria of general covariance and second-order field equations for d > 4 is a polynomial of degree [d/2] in the curvature, has [(d − 1)/2] free parameters, which are not fixed from first principles. In Ref. [6] was shown, using the first order formalism, that requiring the theory to have the maximum possible number of degrees of freedom, fixes these parameters in terms of the gravitational and the cosmological constants. In odd dimensions, the Lagrangian is a Chern–Simons forms for the AdS group. The vielbein and the spin connection can be viewed as different components of an ( A )dS or Poincare connection, so that its local symmetry is enlarged from Lorentz to (A)dS (or Poincare when  = 0). The principal motivation of this work is, using the first order formalism, find the Hamiltonian formalism for a Chern–Simons theory leading to general relativity in a certain limit. In the first-order approach, the independent dynamical variables are the vielbein (ea ) and the spin connection (ωab ), which obey first-order differential field equations. The standard secondorder form can be obtained if the torsion equations are solved for the connection and eliminated in favor of the vielbein—this step,

http://dx.doi.org/10.1016/j.physletb.2016.04.028 0370-2693/© 2016 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 SCOAP3 .

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L. Avilés, P. Salgado / Physics Letters B ••• (••••) •••–•••

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however, cannot be taken, in general, because the equations for ωab are not invertible for dimensions higher than four (for detail see Ref. [6]). This means it is not possible to find a Hamiltonian formulation of second order for a five-dimensional AdS as well as B5 Chern–Simons theory. The purpose of this work is to study (i ) the Hamiltonian action for transgressions field theory, (ii) the subspace separation method in the context of Hamiltonian formalism, (iii) the Hamiltonian analysis of Einstein–Chern–Simons gravity, (iv) the relation between the Hamiltonian action of general relativity of Refs. [22, 23] and the Hamiltonian action for Einstein–Chern–Simons gravity. This paper is organized as follows: In Sec. 2 the Hamiltonian analysis of the five-dimensional Chern–Simons theory is briefly reviewed. The Hamiltonian analysis of the transgressions form Lagrangians is considered in Sec. 3 where the Extended Cartan Homotopy Formula is reviewed and used to find the triangle equation in its Hamiltonian form. In Sec. 4 the subspace separation method for Chern–Simons Hamiltonian is built and used to find the Hamiltonian for five-dimensional Einstein–Chern–Simons gravity. It is then shown that the Hamiltonian for Einstein gravity of Refs. [22, 23] arises in the limit when the scale parameter l approaches zero.

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2. Hamiltonian analysis of the five-dimensional Chern–Simons theory

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In this Section we briefly review of the Hamiltonian analysis of Chern–Simons theory studied in Refs. [24–26]. 2.1. (4 + 1)-dimensional case



M

36

=k

37

40 41 42

A ∧ dA ∧ dA +

+ 

+

46 48 49 50 51 52 53 54 55 56

59 60 61 62 63 64 65

5

A∧A∧A∧A∧A



3 5

3 2

A μ A ν A ρ ∂σ A λ



×

72 73 74

we find that the primary constraints are

75

ϕa = a0 ≈ 0,

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φai = ai − lai ≈ 0 , (i = 1, 2) ,

78

and therefore the canonical Hamiltonian and the total Hamiltonian are then given by

80



Hc =

77





˙ aμ a − L = d4 x A μ

79



81 82

d4 x A a0 K a .

83 84 85

3. Hamiltonian analysis of transgression field theory

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In this section we consider the Hamiltonian analysis of transgression field theory introduced and studied in Refs. [27–30]. This results allow us to find the Hamiltonian triangular equation. This equation together with a method, known as subspace separation method, will be used to obtain the Chern–Simons Hamiltonian for the algebras so(4, 2) and B5 .

[ A , A¯ ] = 2k



87 88 89 90 91 92 93 94 95

1

96 97 98 99

t

dt  F ,

(4)

100 101 102 103 104 105 106 107 108

(5)

110 111

3 4

(3 )

IT

[ A μ , A¯ ν ] =

K a = − kε 4

i jkl

1 c b d e + A˙ ai F bjk Alc − f de A j Ak Al 4

lai = kε i jkl gabc

c gabc F ibj F kl ,



F bjk A lc −

112 113

1

 d x(k I ×



115 116

0

¯ ν ], d3 xL T [ A μ , A (3 )

=

114

t dt ε μνρ θμ F νρ )

3

117

(6)

I ×

 ,

(1)

L T [ A μ , A¯ ν ] = k (3 )

120

4

 c A bj A kd A le , f de

122 123

t dt ε μνρ θμ F νρ 

124

0

(2) 1

121

1

1 =k

118 119

where the 3-dimensional TGFT Lagrangian is given by



c A a0 F ibj F kl

introducing (5) into (4) we find

where gabc =  T a T b T c . Defining for convenience

3

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∂ Lcs2+1 , ∂ A˙ aμ

109

d5 xε i jkl gabc



μ

a =

A μ dxμ = Adx0 + A i dxi (i = 1, 2) ,

Aμ Aν Aρ Aσ Aλ ,



S =k

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¯ with A and A¯ gauge connections, A t = A¯ + t , where  = A − A, F t = d A t + A t A t , · · · denote the symmetrized trace and M is an orientable 3-dimensional manifold on which the connection A is defined. If M has the topology M = R ×  where R can be considered as the temporal line and  as a spatial section, then we can split the gauge field in time and space components

where A = A μa T a is the gauge connection field, T a are de generators of the corresponding gauge group, · · · denote the symmetrized trace and M is an orientable 5-dimensional manifold on which the connection A is defined. If M has the topology M = R ×  where R can be considered as the temporal line and  as a spatial section, then we can split the gauge field in time and space components A μ dxμ = Adx0 + A i dxi (i = 1, 2, 3, 4). The action then takes the form

57 58

3

(3)

M 0



M

44

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2

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˙ a lai − A a K a . d5 x A 0 i

69

IT

A ∧ A ∧ A ∧ dA

d5 xε μνρσ λ A μ ∂ν A ρ ∂σ A λ +

=k

43 45

3



Since the canonical momenta are given by

(3 )

M

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S [ A] =

66



The action for a 3-dimensional transgression gauge field theory (TGFT) is given by

LChS4+1

35



3.1. (2 + 1)-dimensional case

The Chern–Simons action in 4 + 1 dimensions is given by

S=

we arrive to

125 126

˙ t )a θ b ] dt ε i j gab [θ0a ( F itj )b + 2( A t0 )a D ti θ bj + 2( A i j

0

+ B (3) [ A μ , A¯ ν ],

127 128 129

(7)

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¯ μ ] = ∂i (−2k with gab =  T a T b , B (3) [ A μ , A

1 0

dt ε i j gab θ aj ( A t0 )b ) is a

D ti

boundary term and is the covariant derivatives for the A t connection. ¯ = 0 we obtain the 3-dimenFrom (7) we can see that when A ¯ = 0 we have A t = t A i , sional Chern–Simons Lagrangian. If A i θi = A i and therefore (3 ) (3 ) L T [ A μ , 0] = k i j gab ( A a F ibj + A˙ ai A bj ) = LCS [ A μ ].

(8)

(3 ) (3 ) (3 ) H T [ A μ , A¯ μ ] = HChS [ A μ ] − HChS [ A¯ μ ] + P (3) [ A i , A¯ i , A˙ i , A˙¯ i ]

(16)

12 13

H T [ A μ , A¯ ν ] = −k (3 )

18 19 20

+ 2( A t0 )a D ti θ bj ].

(9)

¯ = 0 we obtain the 3-dimenFrom (9) we can see that when A

15 17

gab [θ0a ( F itj )b

sional Chern–Simons Hamiltonian of Refs. [25,26]. (3 )

(3 )

H T [ A μ , 0] = HChS [ A μ ].

(10)

(3 ) (3 ) (3 ) H T [ A μ , A¯ μ ] = HChS [ A μ ] − HChS [ A¯ μ ] + P˜ (3) [ A i , A¯ i , A˙ i , A¯˙ i ]

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21 22 23 24 25 26

In the Lagrangian formalism the so called triangle equation is given by [27–30] (2n+1)

LT

27 28 29 30 31 32 33 34 35 36 37 38

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57

60

(2n+1)

LT

(2n+1)

[ A , A¯ ] = L ChS

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(2n+1)

[ A ] − L ChS

[ A¯ ] (12)

Using a method, known as subspace separation method, the equation (12) allows to construct the Chern–Simons Lagrangian for the B algebra, from which emerges as l → 0 the standard (2n + 1)-dimensional Lagrangian for general relativity. In the case of the Hamiltonian formalism, triangular equation plays a similar role. The Hamiltonian triangular equation allows obtaining the Chern–Simons Hamiltonian for algebra B. This Hamiltonian empties into the Hamiltonian for five-dimensional relativity in the limit l → 0. This is not achievable in the case of the Hamiltonian for AdS algebra. (2n+1) [ A , A¯ ] = L(T2n+1) [ A μ , A¯ μ ]˜v(2n+1) (where v˜ (2n+1) Writing L T is the volume form) and integrating for n = 1 we have

L T [ A μ , A¯ ν ] = LChS [ A μ ] − LChS [ A¯ μ ] − κ d Q (2) [ A μ , A¯ ν ]. (3 )

(3 )

(3 )

(13)

From (7) and (9) we can write

L T [ A μ , A¯ μ ] = P (3 )

(3 )

(3 ) [ A i , A¯ i , A˙ i , A˙¯ i ] − H [ A μ , A¯ μ ] T

+ B (3) [ A μ , A¯ μ ],

73

(17)

74 75 76

¯ i , A˙ i , A˙¯ i ] = P (3) [ A i , A¯ i , A˙ i , A˙¯ i ] + P (3) [ A¯ i , A˙¯ i ] P˜ (3) [ A i , A

77

−P

(3 )

[ A i , A˙ i ].

78

(18)

and

79 80 81 82

B(3) [ A μ , A¯ μ ] = B (3) [ A μ , A¯ μ ] + B (3) [ A¯ μ ] − B (3) [ A μ ] (3 )

[ A μ , A¯ μ ],

83

(19)

84 85

¯ μ ] = κ d Q (2) [ A μ , A¯ ν ]. with Q (3) [ A μ , A

86

From here we can see that Eq. (17) is similar to Lagrangian triangular equation except for the term P˜ (3) defined in equation (18). Since this term depends on the speed and the Hamiltonian can not dependent on it, it is convenient to analyze the aforementioned term. In fact, from

87

¯ i , A˙ i , A˙¯ i ] = 2k P˜ (3) [ A i , A

1

˙ t )a θ b + t A˙¯ a A¯ b − t A˙ a A b ] dt ε i j gab [( A i j i j i j

= kε i j gab [ A˙¯ ai A bj + A¯ ai A˙ bj ], ¯ i , A˙ i , A˙¯ i ] = ∂0 (kε i j gab A¯ a A b ), P˜ (3) [ A i , A i j (3 ) (3 ) [ A μ , A¯ μ ] = H [ A μ ] − H [ A¯ μ ] + B˜ (3) [ A μ , A¯ μ ], ChS

ChS

(3 )

+B

93 94

100

(21)

101 102 103

(22)

104 105 106 107 108 109

(3 )

[ A μ , A¯ μ , A˜ μ ],

97 99

H T [ A μ , A¯ μ ] = H T [ A μ , A˜ μ ] − H T [ A¯ μ , A˜ μ ]

˜ (3 )

91

98

where B˜ contains (21). This result can be extended to the case ˜ is nonzero. In this case (22) where the intermediate connection A takes the form (3 )

90

96

(20)

we can see that P˜ (3) is a boundary term

HT

89

95

0

(3 )

88

92

110

(23)

where the main difference with (22) resides in the fact that the ¯ μ , A˜ μ ], contains two terms term P˜ (3) , which appears in B˜ (3) [ A μ , A more than his counterpart (21). The above results lead to show that the Hamiltonian transgression is the difference between two Chern–Simons Hamiltonians term plus a boundary term. The boundary term appears due to contributions of the Legendre transformation on the triangular Lagrangian equation. This result together to a method, known as subspace separation method, allows to construct the Chern–Simons Hamiltonian for an arbitrary Lie algebra.

(14)

111 112 113 114 115 116 117 118 119 120 121 122 123

3.3. (4 + 1)-dimensional case

where

¯ i , A˙ i , A˙¯ i ] = 2k P (3 ) [ A i , A

[ A μ , A¯ μ ],

so that the triangular equation (17) can be written as

− κ d Q (2n) [ A , A¯ , 0].

61 62

(11)

(2n+1)

58 59

[ A , A¯ ] = L (T2n+1) [ A , A˜ ] − L (T2n+1) [ A¯ , A˜ ] − κ d Q (2n) [ A , A¯ , A˜ ],

where L T [ A , A¯ ] is a transgression form and κ is a constant. This equation can be read off as saying that a transgression form ¯ and A may be written as the sum of “interpolating” between A two transgressions which introduce an intermediate, ancillary one˜ plus a total derivative. It is important to note here that A˜ is form A completely arbitrary, and may be chosen according to convenience. From the triangle equation, and fixing the middle connection to ˜ = 0), we can see zero ( A

39 40

(3 )

where

+Q

3.2. The 3-dimensional triangle equation

69 71

0

14 16

dt ε

ij

68 70

+B

1

67

which can be written as

Using the definition of Hamiltonian we find

11

66

+ P (3) [ A¯ i , A˙¯ i ] − P (3) [ A i , A˙ i ] + B(3) [ A μ , A¯ μ ],

9 10

3

124 125

1 0

so that (13) takes the form

˙ t )a θ b , dt ε i j gab ( A i j

In the (4 + 1)-dimensional case the action is given by

(15)

¯ ] = 3k I T [ A, A (5 )

126 127

 1

128

dt  F t F t . M 0

(24)

129 130

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Following the procedure of the above section we find

1 2 3 4

¯]= I T [ A, A (5 )

5

3 4

6 7



1

10

I ×

13 14 15 16

19

(5 )

LT

3 4

3

32

35

=k

dt ε

0i jkl

3

gabc (

4

where gabc =  T a T b T c 

θ0a ( F itj )b ( F klt )c

+ 3( A t0 )a D ti θ bj ( F klt )c

¯ μ] and B (5) [ A μ , A

(27)

= ∂i (−3k

1

( A t0 )a θ bj ( F klt )c ) is a boundary term.

(5 )

L T [ A μ , 0] = k ε

0i jkl

gabc (



3 4

t c ) A a0 ( F itj )b ( F kl

1 c b d e + A˙ i a A bj F klc − f de A j Ak Al

0

dt ε

i jkl

gabc ×

= LChS [ A μ ].

H T [ A μ , A¯ μ ] = −k (5 )

(28)

46

3

55 56 57 58

×[

59 60 61

4

θ0a ( F itj )b ( F klt )c

+ 3( A t0 )a D ti θ bj ( F klt )c ].

(5 )

H T [ A μ , 0] = HChS [ A μ ].

ChS

ChS

+

65

where

1 4 1 2 1 8

j

l

c ¯a b ¯d ¯e A i A j Ak Al f de

1

+

c ¯a b d ¯e A i A j Ak Al + f de

4 1 8

i

j

l

k

90 91 92

96 97

c ¯a ¯b d e A i A j Ak Al f de

98 99 100 101

˙¯ a A b A c − A˙ a A¯ b A c )], gabc ( A j

89

95

c a b ¯d ¯e A i A j A k A l )] f de i jkl

88

94

c ¯a b d e A i A j Ak Al ) f de

l

j

k

l

(37)

102 103 104

we have that the triangular equation is given by (5 )

(5 )

(38)

106 107

where B˜ contains (37). This result together with a procedure, known as subspace separation method, allows to construct the Chern–Simons Hamiltonian for the so(4, 2) and B5 Lie algebra.

108 109 110 111 112

3.5. (2n + 1)-dimensional case

113

The action for a (2n + 1)-dimensional transgression gauge field theory is given by

115

IT

[ A μ , A¯ μ ] = (n + 1)k



114

dt Fnt .

(39)

M 0

(2n+1)

LT

[ A μ , A¯ μ ] =

120 122 123 125

˙ t )a Lai − θ a Ka − ( A t )a Na dt ( A 0 0 i

+B where

119

124 126

0

(32)

117

121

(31)

1

116 118

1

Following the same procedure of the above sections we find,

From (27) and (29) we can see that

64

63

86

93

i

(2n+1)

From (11) we have that the 5-dimensional triangular equation is given by

(5 ) (5 ) L T [ A μ , A¯ μ ] = P (5) [ A i , A¯ i , A˙ i , A˙¯ i ] − H T [ A μ , A¯ μ ] + B (5) [ A μ , A¯ μ ],

62

(29)

3.4. Five-dimensional triangular equation

LT

85

105

(30)

[ A μ , A¯ μ ] = L(5) [ A μ ] − L(5) [ A¯ μ ] − κ d Q (4) [ A μ , A¯ μ ].

84

¯ i , A˙ i , A˙¯ i ] = ∂0 [kε i jkl gabc ( A¯ a A b ∂k A c + A¯ a A b ∂k A¯ c P˜ (5) [ A i , A

(5 )

¯ = 0 we obtain the From (29) we can see that when A 5-dimensional Chern–Simons Hamiltonian

(5 )

(36)

H T [ A μ , A¯ μ ] = HChS [ A μ ] − HChS [ A¯ μ ] + B˜ (5) [ A μ , A¯ μ ],

dt ε 0i jkl gabc 0

54

83

87

+ ∂i [kε

1

45

53

82

[ A μ , A¯ μ ],

−

Using the definition of Hamiltonian we find

(5 )

79

From here we can see that the Eq. (34) is similar to Lagrangian triangular equation except for the term, P˜ (5) , defined in equation (35). Since this term depends on the velocity and the Hamiltonian can not dependent on it, it is convenient to analyze the aforementioned term. In fact, from



(5 )

42

52

(5 )

+

41

51

78

(35)

81

4

47

74

77

− P (5) [ A i , A˙ i ],

¯ = 0 we obtain the From (27) we can see that when A 5-dimensional Chern–Simons Lagrangian.

40

50

P

73

76

[ A i , A¯ i , A˙ i , A˙¯ i ] = P (5) [ A i , A¯ i , A˙ i , A˙¯ i ] + P (5) [ A¯ i , A˙¯ i ]

¯ μ ] = κ d Q (4) [ A μ , A¯ μ ]. with Q (5) [ A μ , A

+ 3( A˙ ti )a θ bj ( F klt )c ) + B (5) [ A μ , A¯ μ ],

39

49

˜ (5 )

B(5) [ A μ , A¯ μ ] = B (5) [ A μ , A¯ μ ] + B (5) [ A¯ μ ] − B (5) [ A μ ]

0

38

48

(34)

75

+Q

1

37

44

72

80

t + θi F tjk F l0 

36

43

71

and

33 34

70

where

0

27

31

(26)

t t dt ε 0i jkl θ0 F itj F kl − θi F 0t j F klt + θi F tj0 F klt − θi F tjk F 0l

4

26

30

69

+ B(5) [ A μ , A¯ μ ],

t dt ε μνρσ λ θμ F νρ F σt λ .

k

67 68

0

1

1

= k

25

29

0

[ A μ , A¯ μ ]

22

28

(33)

so that (31) takes the form

Splitting the gauge field in time and space components A μ dxμ = Adx0 + A i dxi (i = 1, 2, 3, 4), we find

21

24

66

˙ t )a θ b ( F t )c , dt ε i jkl gabc 3( A i j kl

0

20

23

1

(5 ) (5 ) (5 ) H T [ A μ , A¯ μ ] = HChS [ A μ ] − HChS [ A¯ μ ] + P˜ (5) [ A i , A¯ i , A˙ i , A¯˙ i ]

(5 ) L T [ A μ , A¯ μ ] =

17 18

[ A i , A¯ i , A˙ i , A¯˙ i ] = k

(25)

so that

11 12

t dt ε μνρσ λ θμ F νρ F σt λ ,

d5 x

k

8 9

P

(5 )

127

(2n+1)

[ A μ , A¯ μ ],

(40)

128 129 130

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1 2

Lai =

n(n + 1) 2n−1

b

kε i j 1 ... j 2n−1 gab1 ...bn θ j 1 ( F tj 2 j 3 )b1 · · · ( F tj 2n−2 j 2n−1 )bn , 1

4 5

Ka = −

6

9

Na = −

11 13

2n

n(n + 1) 2n−1

kε

i j 1 ... j 2n−1

¯ μ ] = ∂i (−3k B (2n+1) [ A μ , A

14

17 18 19 21 22

(2n+1)

HT

[ A μ , A¯ μ ] =

1

24 25 26 27 28

(2n+1)

HCS

(44)

2n



dt θ0a Ka + ( A t0 )a Na .

(n + 1)ε

(2n+1)

HT

34

b i j 1 ... j 2n−1 gab1 ...bn A a0 F i j1 1

···

b F jn j 2n−2 2n−1

39 40

LT

ChS

− κ d Q (2n) [ A μ , A¯ μ ].

42 44 45

(2n+1)

LT

47 48 50

P

(2n+1)

51

[ A i , A¯ i , A˙ i , A¯˙ i ] =

1

53

so that (48) takes the form

55 56

(2n+1)

HT

˙ t )a Lai , dt ( A

(2n+1)

[ A μ , A¯ μ ] = HChS

58

+B

59

(2n+1)

[ A μ ] − HChS [ A μ , A¯ μ ]

61

¯ i , A˙ i , A˙¯ i ] = P (2n+1) [ A i , A¯ i , A˙ i , A˙¯ i ] P˜ (2n+1) [ A i , A

64 65

(55)

.





d4 x

dt

89 90 91 92 93 94 95

(56)

1 μνσρ λ ε εabcde R ab μν ec σ ed ρ ee λ , (57) 2

+P

− P (2n+1) [ A i , A˙ i ].

100 101 102 103 104 105

1 2

109 110

This allow us split the fields in time and space components,

111 112

ε0i jkl εabcde ( R ab 0i ec j ed k ee l − R ab i0 ec j ed k ee l

113 114

+ 3R ab i j ec 0 ed k ee l ) 0i jkl

eai abcde [˙

ε

a

f0

ω

0i jkl

fb

ω

bc

115

d

e

je ke l

c

d

e

ie je ke l

εabcde ω

ab

+ 3ω

+

c

3 2

d

R

ab

ab

c

d

116

e

0 ∂i e j e k e l c

d

117 118

e

i j e 0e ke l ]

119

e

0 e j e k e l ).

(59)

120 121 122 123

ε0i jkl T iaj = ε0i jkl (2∂i ea j + 2ωa f i e f j ),

(60)

124 125 126

LEHC [e μ , ωμ ] = ε (52)

99

108

(58)

Considering that

[ A¯ i , A˙¯ i ]

98

107

1 (5 ) LEHC [e μ , ωμ ] = ε μνσρ λ εabcde R ab μν ec σ ed ρ ee λ . 2

(5 )

96

106

we find (2n+1)

86 88



− ∂i (ε (51)

85

97

I

[ A¯ μ ]

where,

63

84

εabcde R ab ec ed ee ,

+ 2ω

(2n+1)

c n −1



=ε

60 62

st

where M is an orientable 5-dimensional manifold. If M has the topology I ×  , then M can be foliated in four-dimensional Cauchy surfaces along I ∈ R, i.e., to carry out a separation (4 + 1). Since ea = ea μ dxμ and R ab = 12 R ab μν dxμ dxν we have

(50)

+ P˜ (2n+1) [ A i , A¯ i , A˙ i , A˙¯ i ]

57

83

The action for five-dimensional Einstein gravity is given by

LEHC [e μ , ωμ ] = i

0

82

4.1. Hamiltonian for five-dimensional general relativity

(5 )

52 54

(49)

where

49

81

so that

+ B (2n+1) [ A μ , A¯ μ ],

46

80

4. Hamiltonian analysis for Einstein gravity and AdS Chern–Simons gravity

(48)

(2n+1) [ A μ , A¯ μ ] = P (2n+1) [ A i , A¯ i , A˙ i , A˙¯ i ] − HT [ A μ , A¯ μ ]

76

87

(5 )

From (40) and (45) we have

43

c1

I EHC [e μ , ωμ ] =

ChS

75

79

M

(47)

(2n+1) (2n+1) ¯ [ A μ , A¯ μ ] = L [ Aμ] − L [ Aμ]

41

0

× ( F ρ1 ρ2 ) · · · ( F ρ2n−1 ρ2n )

(5 )

Following the procedure used in the above sections we have (2n+1)

0

I EHC [e , ω] =

36 38

[ A μ , A¯ μ ] 1  t = κ ∂λ n(n + 1) dt dsε λμνρ1 ...ρ2n gabc1 ...cn−1 θμa A¯ bν

(46)

(2n+1) [ A μ , 0] = HChS [ A μ ].

74

78

(2n+1)

(45)

3.6. The (2n + 1)-dimensional triangle equation

37

71

77

st

I.e.,

32

35

Q

(2n+1) = HChS [ A μ ].

30

33

[ Aμ] = −

k

+ Q (2n+1) [ A μ , A¯ μ ] + P˜ (2n+1) [ A μ , A¯ μ ], (54)

1

¯ = 0 we find the (2n + 1)From (45) we can see that when A dimensional Chern–Simons Hamiltonian

29 31

b

0

23

70 72

with

˙ t )a θ 1 dt ε i j 1 ... j 2n−1 gab1 ...bn ( A 0 j



(53)

where

(43)

with ε i j 1 ... j 2n−1 = ε 0i j 1 ... j 2n−1 and gab1 ...bn =  T a T b1 · · · T bn . Using the definition of Hamiltonian we have

20

[ A μ , A¯ μ ],

73

× ( F tj 2 j 3 )b1 · · · ( F tj 2n−2 j 2n−1 )bn ,

16

67 69

B˜ (2n+1 [ A μ , A¯ μ ] = B (2n+1) [ A μ , A¯ μ ] + B (2n+1) [ A¯ μ ] − B (2n+1) [ A μ ]

0

15

+B

66 68

(2n+1) (2n+1) [ A μ , A¯ μ ] = HChS [ A μ ] − HChS [ A¯ μ ]

˜ (2n+1)

b gab1 ...bn D ti θ j 1 1

1

(2n+1)

HT

kε i j 1 ... j 2n−1 gab1 ...bn ( F itj 1 )b1 · · · ( F tj 2n−2 j 2n−1 )bn ,

× ( F tj 2 j 3 )b2 · · · ( F tj 2n−2 j 2n−1 )bn ,

10 12

(n + 1)

(42)

7 8

This term is a boundary term, so that the triangular equation is finally given by

(41)

3

5

0i jkl

+

3 2

a

εabcde [˙e i ω

bc

d

e

je ke l

3

+ ω 2

ab

c

d

127

e

0 T ij e ke l

128 129

R ab i j ec 0 ed k ee l ].

(61)

130

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1 2

A direct calculation allows us to obtain the Hamiltonian density for Einstein’s gravity

5. Hamiltonian for five-dimensional Einstein–Chern–Simons gravity

66 67

3 4 5

68

3

(5 )

HEHC [e μ , ωμ ] = − ε 0i jkl εabcde [ωab 0 T c i j ed k ee l + R ab i j ec 0 ed k ee l ]. 2

6

(62)

7 8 9

This means that (62) is the Hamiltonian for the five-dimensional general relativity in the first order formalism.

10 11

4.2. Hamiltonian for five-dimensional AdS Chern–Simons gravity

12 13 14 15 16 17 18 19 20

To construct the AdS Chern–Simons Hamiltonian we use the so called subspace separation method [20]. This procedure is based on the triangle equation (38), and embodies the following steps: i) Identify the relevant subspaces present in the so(4, 2) algebra, i.e., write g = V 1 ⊕ V 2 , where

V 1 → { P a }, V 2 → { J ab }.

(63)

5.1. Lagrangian Einstein–Chern–Simons gravity

70

In Ref. [13] was shown that the standard, five-dimensional general relativity can be obtained from Chern–Simons gravity theory for a certain Lie algebra B [13], whose generators { J ab , P a , Z ab , Z a } satisfy the commutation relations given in Eq. (7) of Ref. [14]. This algebra was obtained from the anti de Sitter (AdS) algebra and a particular semigroup S by means of the S-expansion procedure introduced in Ref. [19]. The Chern–Simons Lagrangian for the B algebra, is built from the one-form gauge connection

A=

23

F=

ii) Write the connections as a sum of pieces valued on every subspace, i.e.,

24 25 26 27 28

Aμ =

¯μ = A

29

l 1 2

1

ea μ P a +

2

ωab μ J ab ,

(64)

ωab μ J ab ,

(65)

˜ = 0. A

30 31

1

1 2 1 2

+

34 35 36 37 38 39 40 41 42

HChS [ A μ ] = H T

44

R ab J ab + 1 l

1 l

T a Pa +

a

a

Dω h + k

be

b



1

(71)

l



2

Dω kab +



+ α3 εabcde

2 3

(5 ) ¯ [ A μ , A¯ μ ] + HChS [ A μ ] − B˜ (5) [ A μ , A¯ μ ]. (67)

• The non-vanishing components of the invariant tensor for fivedimensional AdS algebra are

4

 J ab J cd P e  = εabcde . 3

t • The connection A tμ and the tensors μ and F μν are given by

47

1

μ = e μ P a , l t

48 50 52

(68) 1

A tμ = ea μ P a + ωab μ J ab , l 2

49 51

a

t

t



a

F μν = T [μν ] P a + l

1 2

(69)

ab

R [μν ] +

t2 l2

 a

a

e [μ e ν ]

J ab .

(70)

53 54 55

These results allow us to calculate the Hamiltonian present in the triangle equation and therefore

56 57 58 59 60 61 62 63 64 65

(AdS5 )

HChS

[e μ , ωμ ]

= −kε i jkl εabcde +

1 l5



1 4l

ea 0 R bc i j R de kl +

ea 0 e b i e c j ed k e e l +

1 2l

1 l

ea 0 R bc i j ed k ee l 3

ωab 0 R cd i j T e kl +

1 l3



ωab 0 T c i j ed k ee l .

From this Hamiltonian it is apparent that neither the l → ∞ nor the l → 0 limit yields the Hamiltonian for general relativity.

73 74 75 76 77 78 79

1 l

ea e b 2

81 83



84

Z ab

85 86

Za.

(72)



R ab ec ed ee + 2l2 kab R cd T e + l2 R ab R cd he

87 88 89 90 91 92 93 94 95 96

+ dB EChS ,

(73)

97 98

(4)

where the surface term B EChS is given by

99 100

(4 )

B EChS



2

101



1

102

= α1l2 εabcde ea ωbc dωde + ωdf ω f e 3 2    2 1 + α3 εabcde l2 ha ωbc + kab ec dωde + ωdf ω f e 3 2    2 1 1 + l2kab ωcd dee + ωdf ee + ea eb ec ωde , 3

2

6

(74)

45 46

72

82

(4 )

To calculate the terms present in (67) we use the following information:

43

2

(5 )

iii) The triangle equation (38) leads to (5 )

a

ω J ab + e P a + k Z ab + h Z a , l

71

80

1

ab

L EChS = α1l2 εabcde ea R bc R de

(66)

(5 )

1

a

Following the dual procedure of S-expansion developed in Ref. [19] we find the five-dimensional Chern–Simons Lagrangian for the B algebra is given by

32 33

1

ab

and the two-form curvature

21 22

69

103 104 105 106 107 108 109 110

and where α1 , α3 are parameters of the theory, l is a coupling constant, R ab = dωab + ωac ωcb corresponds to the curvature 2-form in the first-order formalism related to the 1-form spin connection, and ea , ha and kab are others gauge fields presents in the theory. The Lagrangian (73) shows that standard, five-dimensional General Relativity emerges as the l → 0 limit of a Chern–Simons theory for the generalized Poincaré algebra B. Here l is a length scale, a coupling constant that characterizes different regimes within the theory.

111

5.2. Hamiltonian for Einstein–Chern–Simons gravity

121

Now we will find the Hamiltonian for Einstein–Chern–Simons gravity, which is a Chern–Simons form for the so called B algebra. Following the procedure of the above subsection, we have:

123

i) The relevant subspace of the B algebra can be identify in the following form: g = V 1 ⊕ V 2 ⊕ V 3 ⊕ V 4 , where

127

V 1 → { P a }, V 2 → { J ab }, V 3 → { Z a }, V 4 → { Z ab }.

112 113 114 115 116 117 118 119 120 122

(75)

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1

¯ μ , A¯ 1μ , A¯ 2μ and A˜ μ are: ii) The connections A μ , A

(B5 )

4 5 6 7 8 9 10 11 12

Aμ =

¯ 1μ = A ¯ 2μ = A

1 l 1 l 1 2

1

ea μ P a +

2 1

ea μ P a +

2

1

1

l

2

ωab μ J ab + ha μ Z a + K ab μ Z ab ,

(76)

ωab μ J ab ,

15

HCS [ A μ ] = H T

17 19

[ A μ , A¯ 1μ ] + H(5) [ A¯ 2μ , A˜ μ ] + H(5) [ A¯ 1μ , A¯ 2μ ] T

T

− B˜ (5) [ A μ , A¯ μ , A¯ 1μ , A¯ 2 ].

16 18

(5 )

(80)

To calculate the terms present in (80) we use the following information.

20 21 22

• The non-vanishing components of the invariant tensor for fivedimensional B algebra are given by

23 24 25

 J ab J cd P e  = α1

26 27 28 29 30

 J ab J cd Z e  = α3  J ab Z cd P e  = α3

4l3 3 4l3 3 4l

32

35 36 37

(81)

εabcde ,

(82)

εabcde .

(83)

t • The connection A tμ and the tensors μ and F μν are given by

38 39 41 43

1

l 1

2 1

l

ea μ P a +



+t

40 42

1

μ ( A μ , A¯ 1μ ) = ha μ Z a + kab μ Z ab , ¯ 1μ ) = A tμ ( A μ , A

t F μν ( A μ , A¯ 1μ ) =

44

1 l

l

2t  l



46

1

2

ωab μ J ab

ha μ Z a +

T a [μν ] P a +

+

45

1 2

50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

2

kab μ Z ab ,

a

a

b



D [μ h ν ] + k b [μ e ν ] Z a

48 49

1



R ab [μν ]

+ t D [μkab ν ] +

47

1 l

e a [μ e a ν ] 2



Z ab ,

1 μ ( A¯ 2μ , 0) = ωab μ J ab ,

¯ 2μ , 0) = A tμ ( A

2 t 2

ωab μ J ab ,

2 ¯ 2μ , 0) = 1 ∂[μ ωab ν ] J ab + t ωa c [μ ωcb ν ] J ab , F μν ( A t

2 1

2

e

ije ke l

+ω

ab

c

d

e

0 T ije ke l

2 a

+ l e 0R

bc

i j Dkk

69

de

70

l

71

From (84) we can see that in the limit l → 0 we obtain the Hamiltonian for the five-dimensional general relativity in the first order formalism shown in (62),

76

2

(B5 )

HCS

2

= −kα3 ε

i jkl

73 74 75 77 78 79 80

[e μ , ωμ ]

81

a

εabcde [e 0 R

bc

d

e

ije ke l

+ω

ab

c

d

e

0 T i j e k e l ].

(85)

Studies of the first order Hamiltonian formalism, for 4-dimensional Einstein gravity can be found in Refs. [22,23], which coincide with (85) by reducing the dimension from five to four.

82 83 84 85 86 87

In this work the Hamiltonian analysis of the transgressions gauge fields theory was considered. The extended Cartan homotopy formula was reviewed and used to find the triangle equation in its Hamiltonian form. The triangle equation leads to show that the Hamiltonian transgression is the difference between two Chern–Simons Hamiltonians term plus a boundary term. The boundary term appears due to contributions of the Legendre transformation on the triangular Lagrangian equation. This result together to a method, known as subspace separation method, allows to construct the Chern–Simons Hamiltonian for the so(4, 2) and B5 algebras. Finally it was found that (i ) from the Chern–Simons Hamiltonian for the so(4, 2) algebra it is apparent that neither the l → ∞ nor the l → 0 limit yields the Hamiltonian for general relativity. (ii) From the Chern–Simons Hamiltonian for the B5 algebra we can see that the Hamiltonian for five-dimensional general relativity may indeed emerge when the scale parameter l approaches zero. Studies of the first order Hamiltonian formalism, for 4-dimensional Einstein gravity can be found in Refs. [22,23], which coincide with (85) by reducing the dimension from five to four. It could be interesting to find a connection between the results obtained in this work (first-order formalism) with those of Teitelboim and Zanelli found in Ref. [21] (second-order formalism). The standard second-order form could be obtained if the torsion equations are solved for the connection and eliminated in favor of the vielbein. However, this is not possible in our case because the equations for ωab are not invertible for dimensions higher than four (for detail see Ref. [6]). References

μ ( A¯ 1μ , A¯ 2μ ) = ea μ P a ,

88 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122

l t

¯ 1μ , A¯ 2μ ) = ea μ P a + 1 ωab μ J ab , A tμ ( A l 2 t

d

89

εabcde ,

33 34

bc

6. Concluding remarks

3

3

31

68

72

(79)

iii) So that the triangle equation leads to,

67

+ l2 ωab 0 T c i j D k kde l + l2 ωab 0 R cd i j D k he l + l2 ωab 0 R cd i j ke f k e f l  l2 ab cd e l2 a bc de + k 0 R i j T kl + h0 R i j R kl ) . (84)

(78)

. A˜ μ = 0.

(5 )

+ α3 (e 0 R

(77)

ωab μ J ab ,

66

4

a

13 14

[e μ , ωμ , hμ , kμ ]  2 α1l a bc de (e 0 R i j R kl + 2ωab 0 R cd i j T e kl ) = −k i jkl εabcde

HCS

2 3

7

1

t2

t F μν ( A¯ 1μ , A¯ 2μ ) = T a [μν ] P a + R ab [μν ] J ab + 2 ea [μ eb ν ] Z ab . l 2 l

These results allow us to calculate the Hamiltonian present in the triangle equation and therefore,

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[6] R. Troncoso, J. Zanelli, Higher-dimensional gravity, propagating torsion and AdS gauge invariance, Class. Quantum Gravity 17 (2000) 4451, arXiv:hepth/9907109. [7] J. Crisóstomo, R. Troncoso, J. Zanelli, Black hole scan, Phys. Rev. D 62 (2000) 084013, arXiv:hep-th/0003271. [8] P. Salgado, M. Cataldo, S. del Campo, Higher-dimensional gravity invariant under the Poincaré group, Phys. Rev. D 66 (2002) 024013, arXiv:gr-qc/0205132. [9] P. Salgado, F. Izaurieta, E. Rodríguez, Higher dimensional gravity invariant under the AdS group, Phys. Lett. B 574 (2003) 283, arXiv:hep-th/0305180. [10] P. Salgado, F. Izaurieta, E. Rodríguez, Supergravity in 2 + 1 dimensions from (3 + 1)-dimensional supergravity, Eur. Phys. J. C 35 (2004) 429, arXiv:hepth/0306230. [11] F. Izaurieta, E. Rodríguez, P. Salgado, Eleven-dimensional gauge theory for the M algebra as an Abelian semigroup expansion of osp(32|1), Eur. Phys. J. C 54 (2008) 675, arXiv:hep-th/0606225. [12] A. Achucarro, P.K. Townsend, Phys. Lett. B 180 (1986) 89. [13] F. Izaurieta, A. Pérez, E. Rodriguez, P. Salgado, Standard general relativity from Chern–Simons gravity, Phys. Lett. B 678 (2009) 213, arXiv:0905.2187 [hep-th]. [14] F. Gómez, P. Minning, P. Salgado, Phys. Rev. D 84 (2011) 063506. [15] C. Quinzacara, P. Salgado, Phys. Rev. D 85 (2012) 124026. [16] J. Crisostomo, F. Gomez, P. Salgado, C. Quinzacara, M. Cataldo, S. del Campo, Eur. Phys. J. C 74 (2014) 3087. [17] J.D. Edelstein, M. Hassaïne, R. Troncoso, J. Zanelli, Lie-algebra expansions, Chern–Simons theories and the Einstein–Hilbert Lagrangian, Phys. Lett. B 640 (2006) 278, arXiv:hep-th/0605174. [18] M. Hassaïne, M. Romo, Local supersymmetric extensions of the Poincaré and AdS invariant gravity, J. High Energy Phys. 0806 (2008) 018, arXiv:0804.4805 [hep-th].

[19] F. Izaurieta, E. Rodríguez, P. Salgado, Expanding Lie (super)algebras through Abelian semigroups, J. Math. Phys. 47 (2006) 123512, arXiv:hep-th/0606215. [20] F. Izaurieta, E. Rodríguez, P. Salgado, The extended Cartan homotopy formula and a subspace separation method for Chern–Simons theory, Lett. Math. Phys. 80 (2007) 127, arXiv:hep-th/0603061. [21] C. Teitelboim, J. Zanelli, Class. Quantum Gravity 4 (1987) L125 and in: G. Longhi, L. Lussana (Eds.), Constraint Theory and Relativistic Dynamics, World Scientific, Singapore, 1987. [22] M. Contreras, J. Zanelli, A note on the spin connection representation of gravity, Class. Quantum Gravity 16 (1999) 2125. [23] R. Aros, Boundary conditions in first order gravity: Hamiltonian and ensemble, Phys. Rev. D 73 (2006) 024004. [24] O. Miskovic, R. Troncoso, J. Zanelli, Canonical sectors of five-dimensional Chern–Simons theories, Phys. Lett. B 615 (2005) 277–284. [25] M. Bañados, L. Garay, M. Henneaux, The dynamical structure of higher dimensional Chern–Simons theory, Nucl. Phys. B 476 (1996) 611, arXiv:hepth/9605150v2. [26] O. Miskovic, Dynamics of Wess–Zumino–Witten and Chern–Simons theories (PhD thesis). [27] P. Mora, Transgression forms as unifying principle in field theory, PhD thesis, Universidad de la República, Uruguay, 2003, arXiv:hep-th/0512255. [28] F. Izaurieta, E. Rodriguez, P. Salgado, On transgression forms and Chern–Simons (super)gravity, arXiv:hep-th/0512014v4. [29] A. Borowiec, L. Fatibene, M. Ferraris, M. Francaviglia, Int. J. Geom. Methods Mod. Phys. 03 (2006) 755. [30] P. Mora, R. Olea, R. Troncoso, J. Zanelli, J. High Energy Phys. 0602 (2006) 067.

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97

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98

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102

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103

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111

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