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Seminar 10 September 2019 Institute of Physics Tartu, Estonia Are critical Lovelock Lagrangians topological in the metric-affine formulation? Alejandro Jiménez Cano Universidad de Granada Dpto. de Física Teórica y del Cosmos
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Page 1: UGRalejandrojc/docstore/docs_congresos... · 2021. 1. 18. · Structure of this presentation 1 Introduction (metric-affine formalism and geometry) 2 Metric-Affine Lovelock theory

Seminar 10 September 2019Institute of Physics

Tartu, Estonia

Are critical Lovelock Lagrangians topologicalin the metric-affine formulation?

Alejandro Jiménez Cano

Universidad de GranadaDpto. de Física Teórica y del Cosmos

[email protected]

www.ugr.es/~alejandrojc

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Structure of this presentation

1 Introduction (metric-affine formalism and geometry)

2 Metric-Affine Lovelock theory

3 The metric-affine Einstein Lagrangian in D = 2

4 The metric-affine Gauss-Bonnet Lagrangian in D = 4

5 Discussion of the general critical Lovelock term

6 Summary and conclusions

B. Janssen, A. Jiménez-Cano, J. A. Orejuela [Janssen, Jiménez, Orejuela 2019]

A non-trivial connection for the metric-affine Gauss-Bonnet theory in D = 4.Physics Letters B 795 (2019) 42 – 48

B. Janssen, A. Jiménez-Cano [Janssen, Jiménez 2019]

On the topological character of metric-affine Lovelock Lagrangians in critical dimensions.arXiv:1907.12100 [gr-qc]

A. Jiménez-Cano, [My PhD Thesis – Still in progress]

Metric-Affine Gauge theory of gravity. Foundations, perturbations and gravitational wave solutions.

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 1

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1. Introduction (metric-affine formalism and geometry)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 2

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Geometric structures: metric

Geometric gravity (Einstein 1915) The spacetime is modelled as a differentiable manifoldM.

Geometric structures

p Metric structure: gµν (metric tensor)é Measuring (length, volume...)

s[γ](σ) =

ˆ σ

0

√|gµν(σ′)xµ(σ′)xν(σ′)| dσ′ . (1.1)

vol(U) =

ˆUωvol , ωvol :=

√|g| dx1 ∧ ... ∧ dxD . (1.2)

é Module of a vector (not necessarily non-negative) ⇒ light cones ⇒ causality.

é Notion of scale (conformal transformations...)

gµν → e2Ωgµν . (1.3)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 3

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Geometric structures: metric

Geometric gravity (Einstein 1915) The spacetime is modelled as a differentiable manifoldM.

Geometric structures

p Metric structure: gµν (metric tensor)é Measuring (length, volume...)

s[γ](σ) =

ˆ σ

0

√|gµν(σ′)xµ(σ′)xν(σ′)| dσ′ . (1.1)

vol(U) =

ˆUωvol , ωvol :=

√|g| dx1 ∧ ... ∧ dxD . (1.2)

é Module of a vector (not necessarily non-negative) ⇒ light cones ⇒ causality.

é Notion of scale (conformal transformations...)

gµν → e2Ωgµν . (1.3)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 3

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Geometric structures: metric

Geometric gravity (Einstein 1915) The spacetime is modelled as a differentiable manifoldM.

Geometric structures

p Metric structure: gµν (metric tensor)é Measuring (length, volume...)

s[γ](σ) =

ˆ σ

0

√|gµν(σ′)xµ(σ′)xν(σ′)| dσ′ . (1.1)

vol(U) =

ˆUωvol , ωvol :=

√|g| dx1 ∧ ... ∧ dxD . (1.2)

é Module of a vector (not necessarily non-negative) ⇒ light cones ⇒ causality.

é Notion of scale (conformal transformations...)

gµν → e2Ωgµν . (1.3)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 3

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Geometric structures: metric

Geometric gravity (Einstein 1915) The spacetime is modelled as a differentiable manifoldM.

Geometric structures

p Metric structure: gµν (metric tensor)é Measuring (length, volume...)

s[γ](σ) =

ˆ σ

0

√|gµν(σ′)xµ(σ′)xν(σ′)| dσ′ . (1.1)

vol(U) =

ˆUωvol , ωvol :=

√|g| dx1 ∧ ... ∧ dxD . (1.2)

é Module of a vector (not necessarily non-negative) ⇒ light cones ⇒ causality.

é Notion of scale (conformal transformations...)

gµν → e2Ωgµν . (1.3)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 3

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Geometric structures: affine connection

Geometric gravity (Einstein 1915) The spacetime is modelled as a differentiable manifoldM.

Geometric structures

p Affine structure: Γµνρ (affine connection)

é Notion of parallel inM ⇒ Covariant derivative∇µ

Different spaces

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 4

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Geometric structures: affine connection

Geometric gravity (Einstein 1915) The spacetime is modelled as a differentiable manifoldM.

Geometric structures

p Affine structure: Γµνρ (affine connection)

é Notion of parallel inM ⇒ Covariant derivative∇µ

é Geometrical objects:

Curvature: Rµνλρ := ∂µΓνλ

ρ − ∂νΓµλρ + Γµσ

ρΓνλσ − Γνσ

ρΓµλσ , (1.4)

Torsion: Tµνρ := Γµν

ρ − Γνµρ . (1.5)

Curvature Torsion

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 5

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Geometric structures

Def.: In the presence of metric and affine connection we define the non-metricity tensor:

Qµνρ := −∇µgνρ . (1.6)

Theorem. Given gµν , there is only one connection that satisfies

Tµνρ = 0 (torsionless condition), (1.7)

Qµνρ = 0 (compatibility condition), (1.8)

the Levi-Civita connection:Γµν

ρ =1

2gρσ [∂µgσν + ∂νgµσ − ∂σgµν ] . (1.9)

Notation. Objects associated to the Levi-Civita connection: Rµνλρ, Rµν , ∇µ...

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 6

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Geometric structures

Def.: In the presence of metric and affine connection we define the non-metricity tensor:

Qµνρ := −∇µgνρ . (1.6)

Theorem. Given gµν , there is only one connection that satisfies

Tµνρ = 0 (torsionless condition), (1.7)

Qµνρ = 0 (compatibility condition), (1.8)

the Levi-Civita connection:Γµν

ρ =1

2gρσ [∂µgσν + ∂νgµσ − ∂σgµν ] . (1.9)

Notation. Objects associated to the Levi-Civita connection: Rµνλρ, Rµν , ∇µ...

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 6

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Geometric structures

Def.: In the presence of metric and affine connection we define the non-metricity tensor:

Qµνρ := −∇µgνρ . (1.6)

Theorem. Given gµν , there is only one connection that satisfies

Tµνρ = 0 (torsionless condition), (1.7)

Qµνρ = 0 (compatibility condition), (1.8)

the Levi-Civita connection:Γµν

ρ =1

2gρσ [∂µgσν + ∂νgµσ − ∂σgµν ] . (1.9)

Notation. Objects associated to the Levi-Civita connection: Rµνλρ, Rµν , ∇µ...

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 6

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Metric-Affine formalism

p Consider a theory depending on the metric structure and its associated curvature (Levi-Civita):

S[g,Ψ] =

ˆL(g, Rµνρ

λ(g),Ψ, ∇µΨ, ...)√|g|dDx , (1.10)

where Ψ are certain non-geometrical fields.

p We are assuming a particular affine structure, the one fixed by the metric.

Γµνρ(g) put by hand⇐ It is natural, it is the simplest one,...

p What if... Γµνρ(g) were fixed by the dynamics?

Metric-affine (or Palatini) formulationPromotion of Γµν

ρ to a general connection Γµνρ (independent field).

p Let us see what happens in the most simple case: Einstein gravity.The resulting action is

S[g, Γ, Ψ] =1

ˆgµνRµν(Γ)

√|g|dDx+ Smatter[g, Ψ] . (1.11)

(Hypothesis Smatter 6= Smatter[Γ]).

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 7

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Metric-Affine formalism

p Consider a theory depending on the metric structure and its associated curvature (Levi-Civita):

S[g,Ψ] =

ˆL(g, Rµνρ

λ(g),Ψ, ∇µΨ, ...)√|g|dDx , (1.10)

where Ψ are certain non-geometrical fields.

p We are assuming a particular affine structure, the one fixed by the metric.

Γµνρ(g) put by hand⇐ It is natural, it is the simplest one,...

p What if... Γµνρ(g) were fixed by the dynamics?

Metric-affine (or Palatini) formulationPromotion of Γµν

ρ to a general connection Γµνρ (independent field).

p Let us see what happens in the most simple case: Einstein gravity.The resulting action is

S[g, Γ, Ψ] =1

ˆgµνRµν(Γ)

√|g|dDx+ Smatter[g, Ψ] . (1.11)

(Hypothesis Smatter 6= Smatter[Γ]).

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 7

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Metric-Affine formalism

p Consider a theory depending on the metric structure and its associated curvature (Levi-Civita):

S[g,Ψ] =

ˆL(g, Rµνρ

λ(g),Ψ, ∇µΨ, ...)√|g|dDx , (1.10)

where Ψ are certain non-geometrical fields.

p We are assuming a particular affine structure, the one fixed by the metric.

Γµνρ(g) put by hand⇐ It is natural, it is the simplest one,...

p What if... Γµνρ(g) were fixed by the dynamics?

Metric-affine (or Palatini) formulationPromotion of Γµν

ρ to a general connection Γµνρ (independent field).

p Let us see what happens in the most simple case: Einstein gravity.The resulting action is

S[g, Γ, Ψ] =1

ˆgµνRµν(Γ)

√|g|dDx+ Smatter[g, Ψ] . (1.11)

(Hypothesis Smatter 6= Smatter[Γ]).

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 7

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Metric-Affine formalism

p Consider a theory depending on the metric structure and its associated curvature (Levi-Civita):

S[g,Ψ] =

ˆL(g, Rµνρ

λ(g),Ψ, ∇µΨ, ...)√|g|dDx , (1.10)

where Ψ are certain non-geometrical fields.

p We are assuming a particular affine structure, the one fixed by the metric.

Γµνρ(g) put by hand⇐ It is natural, it is the simplest one,...

p What if... Γµνρ(g) were fixed by the dynamics?

Metric-affine (or Palatini) formulationPromotion of Γµν

ρ to a general connection Γµνρ (independent field).

p Let us see what happens in the most simple case: Einstein gravity.The resulting action is

S[g, Γ, Ψ] =1

ˆgµνRµν(Γ)

√|g|dDx+ Smatter[g, Ψ] . (1.11)

(Hypothesis Smatter 6= Smatter[Γ]).

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 7

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Metric-Affine formalism

p Consider a theory depending on the metric structure and its associated curvature (Levi-Civita):

S[g,Ψ] =

ˆL(g, Rµνρ

λ(g),Ψ, ∇µΨ, ...)√|g|dDx , (1.10)

where Ψ are certain non-geometrical fields.

p We are assuming a particular affine structure, the one fixed by the metric.

Γµνρ(g) put by hand⇐ It is natural, it is the simplest one,...

p What if... Γµνρ(g) were fixed by the dynamics?

Metric-affine (or Palatini) formulationPromotion of Γµν

ρ to a general connection Γµνρ (independent field).

p Let us see what happens in the most simple case: Einstein gravity.The resulting action is

S[g, Γ, Ψ] =1

ˆgµνRµν(Γ)

√|g|dDx+ Smatter[g, Ψ] . (1.11)

(Hypothesis Smatter 6= Smatter[Γ]).

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 7

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Metric-affine formalism for Einstein theory

p Action for the Einstein-Palatini theory

S[g, Γ, Ψ] = SEP[g, Γ] + Smatter[g, Ψ] , SEP[g, Γ] :=1

ˆgµνRµν(Γ)

√|g|dDx . (1.12)

p In D > 2 the equations of motion read

EoM g : 0 = R(µν) −1

2gµνR+ κTµν , (1.13)

EoM Γ : 0 = ∇λgµν − Tνλσgµσ −1

D − 1Tσλ

σgµν −1

D − 1Tσν

σgµλ . (1.14)

p General solutions of the EoM of Γ

Γµνρ = Γµν

ρ +Aµδρν . (1.15)

Torsion, non-metricity and curvature tensors

Tµνρ = Aµδ

ρν −Aνδρµ , (1.16)

∇µgνρ = −2Aµgνρ , (1.17)

Rµνρλ = Rµνρ

λ + 2∂[µAν]δλρ ⇒ R(µν) = Rµν . (1.18)

Substituting this last condition into the metric equation one obtains the Einstein equations,

(EoM g)|Γon-shell : 0 = Rµν −1

2gµνR+ κTµν . (1.19)

p Aµ is unphysical, since can be absorbed using the projective symmetry of the theory:

proj : Γµνρ → Γµν

ρ + kµδρν (Rµνρ

λ → Rµνρλ + 2∂[µkν]δ

λρ ) ⇒ δprojLEP = 0 . (1.20)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 8

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Metric-affine formalism for Einstein theory

p Action for the Einstein-Palatini theory

S[g, Γ, Ψ] = SEP[g, Γ] + Smatter[g, Ψ] , SEP[g, Γ] :=1

ˆgµνRµν(Γ)

√|g|dDx . (1.12)

p In D > 2 the equations of motion read

EoM g : 0 = R(µν) −1

2gµνR+ κTµν , (1.13)

EoM Γ : 0 = ∇λgµν − Tνλσgµσ −1

D − 1Tσλ

σgµν −1

D − 1Tσν

σgµλ . (1.14)

p General solutions of the EoM of Γ

Γµνρ = Γµν

ρ +Aµδρν . (1.15)

Torsion, non-metricity and curvature tensors

Tµνρ = Aµδ

ρν −Aνδρµ , (1.16)

∇µgνρ = −2Aµgνρ , (1.17)

Rµνρλ = Rµνρ

λ + 2∂[µAν]δλρ ⇒ R(µν) = Rµν . (1.18)

Substituting this last condition into the metric equation one obtains the Einstein equations,

(EoM g)|Γon-shell : 0 = Rµν −1

2gµνR+ κTµν . (1.19)

p Aµ is unphysical, since can be absorbed using the projective symmetry of the theory:

proj : Γµνρ → Γµν

ρ + kµδρν (Rµνρ

λ → Rµνρλ + 2∂[µkν]δ

λρ ) ⇒ δprojLEP = 0 . (1.20)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 8

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Metric-affine formalism for Einstein theory

p Action for the Einstein-Palatini theory

S[g, Γ, Ψ] = SEP[g, Γ] + Smatter[g, Ψ] , SEP[g, Γ] :=1

ˆgµνRµν(Γ)

√|g|dDx . (1.12)

p In D > 2 the equations of motion read

EoM g : 0 = R(µν) −1

2gµνR+ κTµν , (1.13)

EoM Γ : 0 = ∇λgµν − Tνλσgµσ −1

D − 1Tσλ

σgµν −1

D − 1Tσν

σgµλ . (1.14)

p General solutions of the EoM of Γ

Γµνρ = Γµν

ρ +Aµδρν . (1.15)

Torsion, non-metricity and curvature tensors

Tµνρ = Aµδ

ρν −Aνδρµ , (1.16)

∇µgνρ = −2Aµgνρ , (1.17)

Rµνρλ = Rµνρ

λ + 2∂[µAν]δλρ ⇒ R(µν) = Rµν . (1.18)

Substituting this last condition into the metric equation one obtains the Einstein equations,

(EoM g)|Γon-shell : 0 = Rµν −1

2gµνR+ κTµν . (1.19)

p Aµ is unphysical, since can be absorbed using the projective symmetry of the theory:

proj : Γµνρ → Γµν

ρ + kµδρν (Rµνρ

λ → Rµνρλ + 2∂[µkν]δ

λρ ) ⇒ δprojLEP = 0 . (1.20)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 8

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Metric-affine formalism for Einstein theory

p Action for the Einstein-Palatini theory

S[g, Γ, Ψ] = SEP[g, Γ] + Smatter[g, Ψ] , SEP[g, Γ] :=1

ˆgµνRµν(Γ)

√|g|dDx . (1.12)

p In D > 2 the equations of motion read

EoM g : 0 = R(µν) −1

2gµνR+ κTµν , (1.13)

EoM Γ : 0 = ∇λgµν − Tνλσgµσ −1

D − 1Tσλ

σgµν −1

D − 1Tσν

σgµλ . (1.14)

p General solutions of the EoM of Γ

Γµνρ = Γµν

ρ +Aµδρν . (1.15)

Torsion, non-metricity and curvature tensors

Tµνρ = Aµδ

ρν −Aνδρµ , (1.16)

∇µgνρ = −2Aµgνρ , (1.17)

Rµνρλ = Rµνρ

λ + 2∂[µAν]δλρ ⇒ R(µν) = Rµν . (1.18)

Substituting this last condition into the metric equation one obtains the Einstein equations,

(EoM g)|Γon-shell : 0 = Rµν −1

2gµνR+ κTµν . (1.19)

p Aµ is unphysical, since can be absorbed using the projective symmetry of the theory:

proj : Γµνρ → Γµν

ρ + kµδρν (Rµνρ

λ → Rµνρλ + 2∂[µkν]δ

λρ ) ⇒ δprojLEP = 0 . (1.20)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 8

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Metric Affine – Exterior notation

Three fundamental objects: coframe, metric and connection 1-form.

p Coframe. We can fix a general frame in the manifold and the corresponding dual basis (coframe):

ea = eµa∂µ , ϑa = eµadxµ [ϑa (eb) = δab ⇔ eµ

aeµb = δab ] . (1.21)

Notation:ϑa1...ak ≡ ϑa1 ∧ ... ∧ ϑak . (1.22)

p Metric. Components of the metric in the arbitrary basis:

gab = eµaeνbgµν . (1.23)

é Canonical volume form

ωvol :=1

D!Ea1...aDϑ

a1...aD =√|g|dx1 ∧ ... ∧ dxD , |g| ≡ |det(gµν)| . (1.24)

é Hodge star of an arbitrary k-form α = 1k!αa1...akϑ

a1...ak

? : Ωk(M) −→ ΩD−k(M)

α 7−→ ?α :=1

(D − k)!k!αb1...bkEb1...bkc1...cD−kϑ

c1...cD−k . (1.25)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 9

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Metric Affine – Exterior notation

Three fundamental objects: coframe, metric and connection 1-form.

p Coframe. We can fix a general frame in the manifold and the corresponding dual basis (coframe):

ea = eµa∂µ , ϑa = eµadxµ [ϑa (eb) = δab ⇔ eµ

aeµb = δab ] . (1.21)

Notation:ϑa1...ak ≡ ϑa1 ∧ ... ∧ ϑak . (1.22)

p Metric. Components of the metric in the arbitrary basis:

gab = eµaeνbgµν . (1.23)

é Canonical volume form

ωvol :=1

D!Ea1...aDϑ

a1...aD =√|g|dx1 ∧ ... ∧ dxD , |g| ≡ |det(gµν)| . (1.24)

é Hodge star of an arbitrary k-form α = 1k!αa1...akϑ

a1...ak

? : Ωk(M) −→ ΩD−k(M)

α 7−→ ?α :=1

(D − k)!k!αb1...bkEb1...bkc1...cD−kϑ

c1...cD−k . (1.25)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 9

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Metric Affine – Exterior notation

Three fundamental objects: coframe, metric and connection 1-form.

p Coframe. We can fix a general frame in the manifold and the corresponding dual basis (coframe):

ea = eµa∂µ , ϑa = eµadxµ [ϑa (eb) = δab ⇔ eµ

aeµb = δab ] . (1.21)

Notation:ϑa1...ak ≡ ϑa1 ∧ ... ∧ ϑak . (1.22)

p Metric. Components of the metric in the arbitrary basis:

gab = eµaeνbgµν . (1.23)

é Canonical volume form

ωvol :=1

D!Ea1...aDϑ

a1...aD =√|g|dx1 ∧ ... ∧ dxD , |g| ≡ |det(gµν)| . (1.24)

é Hodge star of an arbitrary k-form α = 1k!αa1...akϑ

a1...ak

? : Ωk(M) −→ ΩD−k(M)

α 7−→ ?α :=1

(D − k)!k!αb1...bkEb1...bkc1...cD−kϑ

c1...cD−k . (1.25)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 9

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Metric Affine – Exterior notation

Three fundamental objects: coframe, metric and connection 1-form.

p Connection 1-formωa

b = ωµabdxµ . (1.26)

where ωµab are the components of the affine connection in the anholonomic basis:

ωµab = eνaeλ

bΓµνλ + eσ

b∂µeσa . (1.27)

N.B. Γµνλ and ωµab contain the same information.

é Exterior covariant derivative (of algebra-valued forms)

Dαa...b... = dαa...

b... + ωcb ∧αa...c... + ... − ωac ∧αc...b... − ... , (1.28)

é Curvature, torsion and non-metricity forms:

Rab := dωa

b + ωcb ∧ ωac =

1

2Rµνa

bdxµ ∧ dxν , (1.29)

T a := Dϑa =1

2Tµν

adxµ ∧ dxν , (1.30)

Qab := −Dgab = Qµabdxµ . (1.31)

é Notation for Levi-Civita: ωab, Rab.

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 10

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Metric Affine – Exterior notation

Three fundamental objects: coframe, metric and connection 1-form.

p Connection 1-formωa

b = ωµabdxµ . (1.26)

where ωµab are the components of the affine connection in the anholonomic basis:

ωµab = eνaeλ

bΓµνλ + eσ

b∂µeσa . (1.27)

N.B. Γµνλ and ωµab contain the same information.

é Exterior covariant derivative (of algebra-valued forms)

Dαa...b... = dαa...

b... + ωcb ∧αa...c... + ... − ωac ∧αc...b... − ... , (1.28)

é Curvature, torsion and non-metricity forms:

Rab := dωa

b + ωcb ∧ ωac =

1

2Rµνa

bdxµ ∧ dxν , (1.29)

T a := Dϑa =1

2Tµν

adxµ ∧ dxν , (1.30)

Qab := −Dgab = Qµabdxµ . (1.31)

é Notation for Levi-Civita: ωab, Rab.

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 10

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Metric Affine – Exterior notation

Three fundamental objects: coframe, metric and connection 1-form.

p Connection 1-formωa

b = ωµabdxµ . (1.26)

where ωµab are the components of the affine connection in the anholonomic basis:

ωµab = eνaeλ

bΓµνλ + eσ

b∂µeσa . (1.27)

N.B. Γµνλ and ωµab contain the same information.

é Exterior covariant derivative (of algebra-valued forms)

Dαa...b... = dαa...

b... + ωcb ∧αa...c... + ... − ωac ∧αc...b... − ... , (1.28)

é Curvature, torsion and non-metricity forms:

Rab := dωa

b + ωcb ∧ ωac =

1

2Rµνa

bdxµ ∧ dxν , (1.29)

T a := Dϑa =1

2Tµν

adxµ ∧ dxν , (1.30)

Qab := −Dgab = Qµabdxµ . (1.31)

é Notation for Levi-Civita: ωab, Rab.

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 10

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2. Metric-Affine Lovelock theory

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 11

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Lovelock Theory

Def. (Metric) Lovelock term of order k in D dimensions:

S(D)k [g] =

ˆL(D)k

√|g|dDx , (2.1)

where

L(D)k =

(2k)!

2ksgn(g)δ[ν1

µ1...δν2k]

µ2kRν1ν2

µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.2)

Properties

p 2nd order differential equations for the metric (by constr.) [Lovelock 1971]

p Total derivative in D = 2k dimensions (critical dimension).

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Lovelock Theory

Def. (Metric) Lovelock term of order k in D dimensions:

S(D)k [g] =

ˆL(D)k

√|g|dDx , (2.1)

where

L(D)k =

(2k)!

2ksgn(g)δ[ν1

µ1...δν2k]

µ2kRν1ν2

µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.2)

Properties

p 2nd order differential equations for the metric (by constr.) [Lovelock 1971]

p Total derivative in D = 2k dimensions (critical dimension).

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 12

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Lovelock Theory

Def. (Metric) Lovelock term of order k in D dimensions:

S(D)k [g] =

ˆL(D)k

√|g|dDx , (2.1)

where

L(D)k =

(2k)!

2ksgn(g)δ[ν1

µ1...δν2k]

µ2kRν1ν2

µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.2)

Properties

p 2nd order differential equations for the metric (by constr.) [Lovelock 1971]

p Total derivative in D = 2k dimensions (critical dimension).

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 12

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Lovelock Theory

Def. (Metric) Lovelock term of order k in D dimensions:

S(D)k [g] =

ˆL(D)k

√|g|dDx , (2.1)

where

L(D)k =

(2k)!

2ksgn(g)δ[ν1

µ1...δν2k]

µ2kRν1ν2

µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.2)

Properties

p 2nd order differential equations for the metric (by constr.) [Lovelock 1971]

p Total derivative in D = 2k dimensions (critical dimension).

Example I. Case k = 1, Einstein(-Hilbert) lagrangian

sgn(g)L(D)1 = δ[ν1

µ1δν2]µ2Rν1ν2

µ1µ2 = R , (2.3)

⇒ [EoM gµν ] 0 = Rµν −1

2gµνR . (2.4)

In the critical dimension (D = 2):

p Conformal symmetry of the theory

p In D = 2 all the metrics are conformally flat

So the equation reduces to:0 = 0 No conditions. (2.5)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 12

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Lovelock Theory

Def. (Metric) Lovelock term of order k in D dimensions:

S(D)k [g] =

ˆL(D)k

√|g|dDx , (2.1)

where

L(D)k =

(2k)!

2ksgn(g)δ[ν1

µ1...δν2k]

µ2kRν1ν2

µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.2)

Properties

p 2nd order differential equations for the metric (by constr.) [Lovelock 1971]

p Total derivative in D = 2k dimensions (critical dimension).

Example II. Case k = 2, Gauss-Bonnet lagrangian

sgn(g)L(D)2 = 3!δ[ν1

µ1δν2µ2

δν3µ3δν4]µ4Rν1ν2

µ1µ2Rν3ν4µ3µ4 = R2 − 4RµνR

µν + RµνρλRµνρλ . (2.3)

Equation of motion of the metric in critical dimension D = 4:

0 = RαβR+ 2RµαβνRµν − 2RµαR

µβ + Rµνα

λRµνβλ −1

4gαβ

(R2 − 4RµνR

µν + RµνρλRµνρλ

)= Cα

µνρCβµνρ −1

4gαβCµνρλC

µνρλ , Cµνρλ ≡ Weyl tensor (2.4)

And this is a known property of the Weyl tensor of ANY metric in D = 4⇒ no conditions.

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 12

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Lovelock theory: from metric to metric-affine

p The D-dimensional (metric) Lovelock lagrangian of order k,

L(D)k =

(2k)!

2ksgn(g)δ[ν1

µ1...δν2k]

µ2kRν1ν2

µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.5)

B.................. Jump to metric-affine .....................B

Def. D dimensional (metric-affine) Lovelock term of order k:

L(D)k =

(2k)!

2ksgn(g)δ[ν1

µ1...δν2k]

µ2kRν1ν2

µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.6)

B.................. Jump to exterior algebra notation .....................B

In the language of differential forms:

L(D)k ≡ L(D)

k

√|g|dDx ⇔ L

(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.7)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 13

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Lovelock theory: from metric to metric-affine

p The D-dimensional (metric) Lovelock lagrangian of order k,

L(D)k =

(2k)!

2ksgn(g)δ[ν1

µ1...δν2k]

µ2kRν1ν2

µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.5)

B.................. Jump to metric-affine .....................B

Def. D dimensional (metric-affine) Lovelock term of order k:

L(D)k =

(2k)!

2ksgn(g)δ[ν1

µ1...δν2k]

µ2kRν1ν2

µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.6)

B.................. Jump to exterior algebra notation .....................B

In the language of differential forms:

L(D)k ≡ L(D)

k

√|g|dDx ⇔ L

(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.7)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 13

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Lovelock theory: from metric to metric-affine

p The D-dimensional (metric) Lovelock lagrangian of order k,

L(D)k =

(2k)!

2ksgn(g)δ[ν1

µ1...δν2k]

µ2kRν1ν2

µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.5)

B.................. Jump to metric-affine .....................B

Def. D dimensional (metric-affine) Lovelock term of order k:

L(D)k =

(2k)!

2ksgn(g)δ[ν1

µ1...δν2k]

µ2kRν1ν2

µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.6)

B.................. Jump to exterior algebra notation .....................B

In the language of differential forms:

L(D)k ≡ L(D)

k

√|g|dDx ⇔ L

(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.7)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 13

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Lovelock theory: from metric to metric-affine

p The D-dimensional (metric) Lovelock lagrangian of order k,

L(D)k =

(2k)!

2ksgn(g)δ[ν1

µ1...δν2k]

µ2kRν1ν2

µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.5)

B.................. Jump to metric-affine .....................B

Def. D dimensional (metric-affine) Lovelock term of order k:

L(D)k =

(2k)!

2ksgn(g)δ[ν1

µ1...δν2k]

µ2kRν1ν2

µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.6)

B.................. Jump to exterior algebra notation .....................B

In the language of differential forms:

L(D)k ≡ L(D)

k

√|g|dDx ⇔ L

(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.7)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 13

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Lovelock theory: from metric to metric-affine

p The D-dimensional (metric) Lovelock lagrangian of order k,

L(D)k =

(2k)!

2ksgn(g)δ[ν1

µ1...δν2k]

µ2kRν1ν2

µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.5)

B.................. Jump to metric-affine .....................B

Def. D dimensional (metric-affine) Lovelock term of order k:

L(D)k =

(2k)!

2ksgn(g)δ[ν1

µ1...δν2k]

µ2kRν1ν2

µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.6)

B.................. Jump to exterior algebra notation .....................B

In the language of differential forms:

L(D)k ≡ L(D)

k

√|g|dDx ⇔ L

(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.7)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 13

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Lovelock theory: from metric to metric-affine

Metric-affine Lovelock term of order k as the lagrangian D-form:

L(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.8)

General properties

p Levi-Civita is a solution of the palatini formalism EoM. [Borunda, Janssen, Bastero 2008]

p Projective symmetry:

ωab → ωa

b +Aδba (⇔ Γµνρ → Γµν

ρ +Aµδρν) , (2.9)

⇒ Rab → Rab + dAgab(⇔ Rµνρ

λ → Rµνρλ + 2∂[µAν]δ

λρ

). (2.10)

Critical dimension D = 2k

p The Lagrangian becomes:

L(2k)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k ≡ Ea1...a2kR

a1a2 ∧ ... ∧Ra2k−1a2k , (2.11)

p Question: Is this a total derivative?

Yes for the Riemann-Cartan case (metric-compatible)é Two examples (orthonormal frame chosen, i.e. gab ≡ ηab): [Hehl, McCrea, Mielke, Ne’eman 1995]

L(2)1 |Q=0 ∝ d

[Eabωab

], (2.12)

L(4)2 |Q=0 ∝ d

[Eabcd

(Ra

b ∧ ωcd + 13ωa

b ∧ ωce ∧ ωed)]

. (2.13)

(Exterior derivative of Chern-Simons like terms).

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 14

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Lovelock theory: from metric to metric-affine

Metric-affine Lovelock term of order k as the lagrangian D-form:

L(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.8)

General properties

p Levi-Civita is a solution of the palatini formalism EoM. [Borunda, Janssen, Bastero 2008]

p Projective symmetry:

ωab → ωa

b +Aδba (⇔ Γµνρ → Γµν

ρ +Aµδρν) , (2.9)

⇒ Rab → Rab + dAgab(⇔ Rµνρ

λ → Rµνρλ + 2∂[µAν]δ

λρ

). (2.10)

Critical dimension D = 2k

p The Lagrangian becomes:

L(2k)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k ≡ Ea1...a2kR

a1a2 ∧ ... ∧Ra2k−1a2k , (2.11)

p Question: Is this a total derivative?

Yes for the Riemann-Cartan case (metric-compatible)é Two examples (orthonormal frame chosen, i.e. gab ≡ ηab): [Hehl, McCrea, Mielke, Ne’eman 1995]

L(2)1 |Q=0 ∝ d

[Eabωab

], (2.12)

L(4)2 |Q=0 ∝ d

[Eabcd

(Ra

b ∧ ωcd + 13ωa

b ∧ ωce ∧ ωed)]

. (2.13)

(Exterior derivative of Chern-Simons like terms).

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 14

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Lovelock theory: from metric to metric-affine

Metric-affine Lovelock term of order k as the lagrangian D-form:

L(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.8)

General properties

p Levi-Civita is a solution of the palatini formalism EoM. [Borunda, Janssen, Bastero 2008]

p Projective symmetry:

ωab → ωa

b +Aδba (⇔ Γµνρ → Γµν

ρ +Aµδρν) , (2.9)

⇒ Rab → Rab + dAgab(⇔ Rµνρ

λ → Rµνρλ + 2∂[µAν]δ

λρ

). (2.10)

Critical dimension D = 2k

p The Lagrangian becomes:

L(2k)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k ≡ Ea1...a2kR

a1a2 ∧ ... ∧Ra2k−1a2k , (2.11)

p Question: Is this a total derivative?

Yes for the Riemann-Cartan case (metric-compatible)é Two examples (orthonormal frame chosen, i.e. gab ≡ ηab): [Hehl, McCrea, Mielke, Ne’eman 1995]

L(2)1 |Q=0 ∝ d

[Eabωab

], (2.12)

L(4)2 |Q=0 ∝ d

[Eabcd

(Ra

b ∧ ωcd + 13ωa

b ∧ ωce ∧ ωed)]

. (2.13)

(Exterior derivative of Chern-Simons like terms).

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 14

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Lovelock theory: from metric to metric-affine

Metric-affine Lovelock term of order k as the lagrangian D-form:

L(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.8)

General properties

p Levi-Civita is a solution of the palatini formalism EoM. [Borunda, Janssen, Bastero 2008]

p Projective symmetry:

ωab → ωa

b +Aδba (⇔ Γµνρ → Γµν

ρ +Aµδρν) , (2.9)

⇒ Rab → Rab + dAgab(⇔ Rµνρ

λ → Rµνρλ + 2∂[µAν]δ

λρ

). (2.10)

Critical dimension D = 2k

p The Lagrangian becomes:

L(2k)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k ≡ Ea1...a2kR

a1a2 ∧ ... ∧Ra2k−1a2k , (2.11)

p Question: Is this a total derivative?

Yes for the Riemann-Cartan case (metric-compatible)é Two examples (orthonormal frame chosen, i.e. gab ≡ ηab): [Hehl, McCrea, Mielke, Ne’eman 1995]

L(2)1 |Q=0 ∝ d

[Eabωab

], (2.12)

L(4)2 |Q=0 ∝ d

[Eabcd

(Ra

b ∧ ωcd + 13ωa

b ∧ ωce ∧ ωed)]

. (2.13)

(Exterior derivative of Chern-Simons like terms).

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 14

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Lovelock theory: from metric to metric-affine

Metric-affine Lovelock term of order k as the lagrangian D-form:

L(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.8)

General properties

p Levi-Civita is a solution of the palatini formalism EoM. [Borunda, Janssen, Bastero 2008]

p Projective symmetry:

ωab → ωa

b +Aδba (⇔ Γµνρ → Γµν

ρ +Aµδρν) , (2.9)

⇒ Rab → Rab + dAgab(⇔ Rµνρ

λ → Rµνρλ + 2∂[µAν]δ

λρ

). (2.10)

Critical dimension D = 2k

p The Lagrangian becomes:

L(2k)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k ≡ Ea1...a2kR

a1a2 ∧ ... ∧Ra2k−1a2k , (2.11)

p Question: Is this a total derivative?

Yes for the Riemann-Cartan case (metric-compatible)é Two examples (orthonormal frame chosen, i.e. gab ≡ ηab): [Hehl, McCrea, Mielke, Ne’eman 1995]

L(2)1 |Q=0 ∝ d

[Eabωab

], (2.12)

L(4)2 |Q=0 ∝ d

[Eabcd

(Ra

b ∧ ωcd + 13ωa

b ∧ ωce ∧ ωed)]

. (2.13)

(Exterior derivative of Chern-Simons like terms).

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 14

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3. The metric-affine Einstein Lagrangian in D = 2

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 15

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The metric-affine Einstein Lagrangian

p Einstein Lagrangian (arbitrary dimension) (We drop the factor (2κ)−1)

L(D)1 = gcbRa

b ∧ ?ϑac = sgn(g)eνbeµcgcaRµνa

b(ω)√|g|dDx , (3.1)

Reminder. In D > 2, the solution of the EoM of the connection is:

ωab = ωa

b +Aδba ⇔ Γµνρ = Γµν

ρ +Aµδρν . (3.2)

Unphysical projective mode← can be eliminated using a symmetry of the theory.

p Critical dimension D = 2.é Equation of motion of the connection

0 = DEab = −QcaEbc where Qab = Qab −

1

2gabQc

c . (3.3)

Therefore the general solution is one that verifies

Qab = 0 . (3.4)

é But, is this trivial? Or are there conditions over the D3 = 8 degrees of freedom of the connection?

Tensor d.o.f. in D dim. d.o.f. in 2 dim. Condition imposed by EoM

Tµνρ 1

2D2(D − 1) 2 (pure trace) [None]

Qµλλ D 2 [None] (in any D due to proj. symmetry)

Qµνρ12D(D + 2)(D − 1) 4 They are zero

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 16

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The metric-affine Einstein Lagrangian

p Einstein Lagrangian (arbitrary dimension) (We drop the factor (2κ)−1)

L(D)1 = gcbRa

b ∧ ?ϑac = sgn(g)eνbeµcgcaRµνa

b(ω)√|g|dDx , (3.1)

Reminder. In D > 2, the solution of the EoM of the connection is:

ωab = ωa

b +Aδba ⇔ Γµνρ = Γµν

ρ +Aµδρν . (3.2)

Unphysical projective mode← can be eliminated using a symmetry of the theory.

p Critical dimension D = 2.é Equation of motion of the connection

0 = DEab = −QcaEbc where Qab = Qab −

1

2gabQc

c . (3.3)

Therefore the general solution is one that verifies

Qab = 0 . (3.4)

é But, is this trivial? Or are there conditions over the D3 = 8 degrees of freedom of the connection?

Tensor d.o.f. in D dim. d.o.f. in 2 dim. Condition imposed by EoM

Tµνρ 1

2D2(D − 1) 2 (pure trace) [None]

Qµλλ D 2 [None] (in any D due to proj. symmetry)

Qµνρ12D(D + 2)(D − 1) 4 They are zero

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 16

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The metric-affine Einstein Lagrangian

p Einstein Lagrangian (arbitrary dimension) (We drop the factor (2κ)−1)

L(D)1 = gcbRa

b ∧ ?ϑac = sgn(g)eνbeµcgcaRµνa

b(ω)√|g|dDx , (3.1)

Reminder. In D > 2, the solution of the EoM of the connection is:

ωab = ωa

b +Aδba ⇔ Γµνρ = Γµν

ρ +Aµδρν . (3.2)

Unphysical projective mode← can be eliminated using a symmetry of the theory.

p Critical dimension D = 2.é Equation of motion of the connection

0 = DEab = −QcaEbc where Qab = Qab −

1

2gabQc

c . (3.3)

Therefore the general solution is one that verifies

Qab = 0 . (3.4)

é But, is this trivial? Or are there conditions over the D3 = 8 degrees of freedom of the connection?

Tensor d.o.f. in D dim. d.o.f. in 2 dim. Condition imposed by EoM

Tµνρ 1

2D2(D − 1) 2 (pure trace) [None]

Qµλλ D 2 [None] (in any D due to proj. symmetry)

Qµνρ12D(D + 2)(D − 1) 4 They are zero

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 16

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The metric-affine Einstein Lagrangian

p Einstein Lagrangian (arbitrary dimension) (We drop the factor (2κ)−1)

L(D)1 = gcbRa

b ∧ ?ϑac = sgn(g)eνbeµcgcaRµνa

b(ω)√|g|dDx , (3.1)

Reminder. In D > 2, the solution of the EoM of the connection is:

ωab = ωa

b +Aδba ⇔ Γµνρ = Γµν

ρ +Aµδρν . (3.2)

Unphysical projective mode← can be eliminated using a symmetry of the theory.

p Critical dimension D = 2.é Equation of motion of the connection

0 = DEab = −QcaEbc where Qab = Qab −

1

2gabQc

c . (3.3)

Therefore the general solution is one that verifies

Qab = 0 . (3.4)

é But, is this trivial? Or are there conditions over the D3 = 8 degrees of freedom of the connection?

Tensor d.o.f. in D dim. d.o.f. in 2 dim. Condition imposed by EoM

Tµνρ 1

2D2(D − 1) 2 (pure trace) [None]

Qµλλ D 2 [None] (in any D due to proj. symmetry)

Qµνρ12D(D + 2)(D − 1) 4 They are zero

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 16

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The critical metric-affine Einstein Lagrangian in D = 2. Equivalent formulation

Appendix. Useful decomposition of the connection.In the presence of a metric, an arbitrary ωab can be split

ωab = ω[ab] + ω(ab) ⇒ω(ab) = 1

2Qab = 12

(Qab + 1

DgabQcc)

ω[ab] =: ωab(3.5)

It can be proved that ωab is also a connection with

Q = 0 , T = T (T ,Q) . (3.6)

p Instead of working in terms of ωab we can work with the independent fields ωab,Qcc andQab.

p Plugging this into the action we obtain

S(2)1 =

ˆEabRa

b(ω) =

ˆEab

[Ra

b(ω) − 14Qa

c ∧Qcb], (3.7)

p If we choose a orthonormal gauge, i.e. gab = ηab, one can use

Eab Rab(ω) = Eab dωab = d(Eab ωab) , (3.8)

to rewrite the Riemann-Cartan part of the action as a total derivative,

S(2)1 =

ˆd(Eab ωab) − 1

4

ˆEabQ

ac ∧Qcb . (3.9)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 17

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The critical metric-affine Einstein Lagrangian in D = 2. Equivalent formulation

Appendix. Useful decomposition of the connection.In the presence of a metric, an arbitrary ωab can be split

ωab = ω[ab] + ω(ab) ⇒ω(ab) = 1

2Qab = 12

(Qab + 1

DgabQcc)

ω[ab] =: ωab(3.5)

It can be proved that ωab is also a connection with

Q = 0 , T = T (T ,Q) . (3.6)

p Instead of working in terms of ωab we can work with the independent fields ωab,Qcc andQab.

p Plugging this into the action we obtain

S(2)1 =

ˆEabRa

b(ω) =

ˆEab

[Ra

b(ω) − 14Qa

c ∧Qcb], (3.7)

p If we choose a orthonormal gauge, i.e. gab = ηab, one can use

Eab Rab(ω) = Eab dωab = d(Eab ωab) , (3.8)

to rewrite the Riemann-Cartan part of the action as a total derivative,

S(2)1 =

ˆd(Eab ωab) − 1

4

ˆEabQ

ac ∧Qcb . (3.9)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 17

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The critical metric-affine Einstein Lagrangian in D = 2. Equivalent formulation

Appendix. Useful decomposition of the connection.In the presence of a metric, an arbitrary ωab can be split

ωab = ω[ab] + ω(ab) ⇒ω(ab) = 1

2Qab = 12

(Qab + 1

DgabQcc)

ω[ab] =: ωab(3.5)

It can be proved that ωab is also a connection with

Q = 0 , T = T (T ,Q) . (3.6)

p Instead of working in terms of ωab we can work with the independent fields ωab,Qcc andQab.

p Plugging this into the action we obtain

S(2)1 =

ˆEabRa

b(ω) =

ˆEab

[Ra

b(ω) − 14Qa

c ∧Qcb], (3.7)

p If we choose a orthonormal gauge, i.e. gab = ηab, one can use

Eab Rab(ω) = Eab dωab = d(Eab ωab) , (3.8)

to rewrite the Riemann-Cartan part of the action as a total derivative,

S(2)1 =

ˆd(Eab ωab) − 1

4

ˆEabQ

ac ∧Qcb . (3.9)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 17

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The critical Einstein-Palatini Lagrangian in D = 2. Equivalent formulation

p The action can be rewritten in terms of the new fields as

S(2)1 [gab,ϑ

a,ωab] =

ˆEabRa

b(ω) (3.10)

l equivalent

S(2)1 [gab,ϑ

a, ωab,Qab,Qc

c] = (boundary term) − 14

ˆEabQ

ac ∧Qcb . (3.11)

p Instead of an equation of motion for ωab, now our splitting gives rise to three equations

EoM forQcc 0 = 0 ← (due to projective symmetry) , (3.12)

EoM for ωab 0 = 0 , (3.13)

EoM forQab 0 = EabQcb ⇔ Qab = 0 . (3.14)

p Conclusion:There are conditions over the connection. So the Lagrangian CANNOT be a total derivative.As we have seen a term quadratic in the non-metricity survives.

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 18

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The critical Einstein-Palatini Lagrangian in D = 2. Equivalent formulation

p The action can be rewritten in terms of the new fields as

S(2)1 [gab,ϑ

a,ωab] =

ˆEabRa

b(ω) (3.10)

l equivalent

S(2)1 [gab,ϑ

a, ωab,Qab,Qc

c] = (boundary term) − 14

ˆEabQ

ac ∧Qcb . (3.11)

p Instead of an equation of motion for ωab, now our splitting gives rise to three equations

EoM forQcc 0 = 0 ← (due to projective symmetry) , (3.12)

EoM for ωab 0 = 0 , (3.13)

EoM forQab 0 = EabQcb ⇔ Qab = 0 . (3.14)

p Conclusion:There are conditions over the connection. So the Lagrangian CANNOT be a total derivative.As we have seen a term quadratic in the non-metricity survives.

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 18

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The critical Einstein-Palatini Lagrangian in D = 2. Equivalent formulation

p The action can be rewritten in terms of the new fields as

S(2)1 [gab,ϑ

a,ωab] =

ˆEabRa

b(ω) (3.10)

l equivalent

S(2)1 [gab,ϑ

a, ωab,Qab,Qc

c] = (boundary term) − 14

ˆEabQ

ac ∧Qcb . (3.11)

p Instead of an equation of motion for ωab, now our splitting gives rise to three equations

EoM forQcc 0 = 0 ← (due to projective symmetry) , (3.12)

EoM for ωab 0 = 0 , (3.13)

EoM forQab 0 = EabQcb ⇔ Qab = 0 . (3.14)

p Conclusion:There are conditions over the connection. So the Lagrangian CANNOT be a total derivative.As we have seen a term quadratic in the non-metricity survives.

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 18

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4. The metric-affine Gauss-Bonnet Lagrangian in D = 4

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 19

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The metric-affine Gauss-Bonnet Lagrangian

p Gauss-Bonnet Lagrangian (arbitrary dimension)

L(D)2 = gmbgndRa

b ∧Rcd ∧ ?ϑamcn (4.1)

= sgn(g)[R2 −RµνRνµ + 2RµνR

νµ − RµνRνµ +RµνρλRρλµν

]√|g|dDx , (4.2)

whereRµν := Rµλν

λ , R := gµνRµν , Rµν := gλσRµλσ

ν . (4.3)

p N.B. In general D, the most general solution is not known. But we know there should be a free(unphysical) projective mode.

p Critical dimension D = 4.

D = 4 ⇒ ?ϑamcn = Eamcn ⇒ L(4)2 = Eabcd Ra

b ∧Rcd (4.4)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 20

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The metric-affine Gauss-Bonnet Lagrangian

p Gauss-Bonnet Lagrangian (arbitrary dimension)

L(D)2 = gmbgndRa

b ∧Rcd ∧ ?ϑamcn (4.1)

= sgn(g)[R2 −RµνRνµ + 2RµνR

νµ − RµνRνµ +RµνρλRρλµν

]√|g|dDx , (4.2)

whereRµν := Rµλν

λ , R := gµνRµν , Rµν := gλσRµλσ

ν . (4.3)

p N.B. In general D, the most general solution is not known. But we know there should be a free(unphysical) projective mode.

p Critical dimension D = 4.

D = 4 ⇒ ?ϑamcn = Eamcn ⇒ L(4)2 = Eabcd Ra

b ∧Rcd (4.4)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 20

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The metric-affine Gauss-Bonnet Lagrangian

p Gauss-Bonnet Lagrangian (arbitrary dimension)

L(D)2 = gmbgndRa

b ∧Rcd ∧ ?ϑamcn (4.1)

= sgn(g)[R2 −RµνRνµ + 2RµνR

νµ − RµνRνµ +RµνρλRρλµν

]√|g|dDx , (4.2)

whereRµν := Rµλν

λ , R := gµνRµν , Rµν := gλσRµλσ

ν . (4.3)

p N.B. In general D, the most general solution is not known. But we know there should be a free(unphysical) projective mode.

p Critical dimension D = 4.

D = 4 ⇒ ?ϑamcn = Eamcn ⇒ L(4)2 = Eabcd Ra

b ∧Rcd (4.4)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 20

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The critical metric-affine Gauss-Bonnet Lagrangian in D = 4. Equivalent formulation

Appendix. Useful decomposition of the connection.In the presence of a metric, an arbitrary ωab can be split

ωab = ω[ab] + ω(ab) ⇒ω(ab) = 1

2Qab = 12

(Qab + 1

DgabQcc)

ω[ab] =: ωab(4.5)

It can be proved that ωab is also a connection with

Q = 0 , T = T (T ,Q) . (4.6)

p Instead of working in terms of ωab we can work with the independent fields ωab,Qcc andQab.

p Plugging this into the action we obtain

S(4)2 =

ˆEabcdRa

b(ω) ∧Rcd(ω)

=

ˆEabcd

[Ra

b(ω) ∧ Rcd(ω)− 1

2Rab(ω) ∧Qc

f ∧Qfd + 1

16Qae ∧Qe

b ∧Qcf ∧Qf

d]. (4.7)

p Choosing the orthonormal gauge, i.e. gab = ηab, the first term (the purely Riemann-Cartan one) canbe expressed as the Euler density and therefore,

S(4)2 =

ˆdC −

ˆEabcd

[12Ra

b(ω) ∧Qcf ∧Qf

d − 116Qa

e ∧Qeb ∧Qc

f ∧Qfd]. (4.8)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 21

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The critical metric-affine Gauss-Bonnet Lagrangian in D = 4. Equivalent formulation

Appendix. Useful decomposition of the connection.In the presence of a metric, an arbitrary ωab can be split

ωab = ω[ab] + ω(ab) ⇒ω(ab) = 1

2Qab = 12

(Qab + 1

DgabQcc)

ω[ab] =: ωab(4.5)

It can be proved that ωab is also a connection with

Q = 0 , T = T (T ,Q) . (4.6)

p Instead of working in terms of ωab we can work with the independent fields ωab,Qcc andQab.

p Plugging this into the action we obtain

S(4)2 =

ˆEabcdRa

b(ω) ∧Rcd(ω)

=

ˆEabcd

[Ra

b(ω) ∧ Rcd(ω)− 1

2Rab(ω) ∧Qc

f ∧Qfd + 1

16Qae ∧Qe

b ∧Qcf ∧Qf

d]. (4.7)

p Choosing the orthonormal gauge, i.e. gab = ηab, the first term (the purely Riemann-Cartan one) canbe expressed as the Euler density and therefore,

S(4)2 =

ˆdC −

ˆEabcd

[12Ra

b(ω) ∧Qcf ∧Qf

d − 116Qa

e ∧Qeb ∧Qc

f ∧Qfd]. (4.8)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 21

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The critical metric-affine Gauss-Bonnet Lagrangian in D = 4. Equivalent formulation

Appendix. Useful decomposition of the connection.In the presence of a metric, an arbitrary ωab can be split

ωab = ω[ab] + ω(ab) ⇒ω(ab) = 1

2Qab = 12

(Qab + 1

DgabQcc)

ω[ab] =: ωab(4.5)

It can be proved that ωab is also a connection with

Q = 0 , T = T (T ,Q) . (4.6)

p Instead of working in terms of ωab we can work with the independent fields ωab,Qcc andQab.

p Plugging this into the action we obtain

S(4)2 =

ˆEabcdRa

b(ω) ∧Rcd(ω)

=

ˆEabcd

[Ra

b(ω) ∧ Rcd(ω)− 1

2Rab(ω) ∧Qc

f ∧Qfd + 1

16Qae ∧Qe

b ∧Qcf ∧Qf

d]. (4.7)

p Choosing the orthonormal gauge, i.e. gab = ηab, the first term (the purely Riemann-Cartan one) canbe expressed as the Euler density and therefore,

S(4)2 =

ˆdC −

ˆEabcd

[12Ra

b(ω) ∧Qcf ∧Qf

d − 116Qa

e ∧Qeb ∧Qc

f ∧Qfd]. (4.8)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 21

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The critical metric-affine Gauss-Bonnet Lagrangian in D = 4. Equivalent formulation

Appendix. Useful decomposition of the connection.In the presence of a metric, an arbitrary ωab can be split

ωab = ω[ab] + ω(ab) ⇒ω(ab) = 1

2Qab = 12

(Qab + 1

DgabQcc)

ω[ab] =: ωab(4.5)

It can be proved that ωab is also a connection with

Q = 0 , T = T (T ,Q) . (4.6)

p Instead of working in terms of ωab we can work with the independent fields ωab,Qcc andQab.

p Plugging this into the action we obtain

S(4)2 =

ˆEabcdRa

b(ω) ∧Rcd(ω)

=

ˆEabcd

[Ra

b(ω) ∧ Rcd(ω)− 1

2Rab(ω) ∧Qc

f ∧Qfd + 1

16Qae ∧Qe

b ∧Qcf ∧Qf

d]. (4.7)

p Choosing the orthonormal gauge, i.e. gab = ηab, the first term (the purely Riemann-Cartan one) canbe expressed as the Euler density and therefore,

S(4)2 =

ˆdC −

ˆEabcd

[12Ra

b(ω) ∧Qcf ∧Qf

d − 116Qa

e ∧Qeb ∧Qc

f ∧Qfd]. (4.8)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 21

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The critical metric-affine Gauss-Bonnet Lagrangian in D = 4. Equivalent formulation

p The action can be rewritten in terms of the new fields as

S(4)2 [gab,ϑ

a,ωab] =

ˆEabcdRa

b(ω) ∧Rcd(ω) (4.9)

l equivalent

S(4)2 [gab,ϑ

a, ωab,Qab,Qc

c] = (boundary term)

−ˆEabcd

[12Ra

b(ω) ∧Qcf ∧Qf

d − 116Qa

e ∧Qeb ∧Qc

f ∧Qfd]. (4.10)

p Instead of an equation of motion for ωab, now our splitting gives rise to three equations

EoM forQcc 0 = 0 ← (due to projective symmetry) , (4.11)

EoM for ωab 0 = D[Qc

a ∧Qbc], (4.12)

EoM forQab 0 = Eabcd[Rab − 1

4Qfa ∧Q

bf]∧Q

dm . (4.13)

p Counterexample. The following field configuration violates EoM of ωab:

gab = ηab , ωab = ωab + fα[aδb]t ,

ϑa = dxa , Qab = 2α(aδ

b)t ,

where

f is an arbitrary function,

αa = et(δay dy + δaz dz

).

(4.14)

since

D[Qc

a ∧Qbc]

= d[αa ∧αb

]= 2e2t

(δayδ

bz − δbyδ

az

)dt ∧ dy ∧ dz 6= 0 in the entireM. (4.15)

p Conclusion: The Lagrangian CANNOT be a total derivative.

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 22

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The critical metric-affine Gauss-Bonnet Lagrangian in D = 4. Equivalent formulation

p The action can be rewritten in terms of the new fields as

S(4)2 [gab,ϑ

a,ωab] =

ˆEabcdRa

b(ω) ∧Rcd(ω) (4.9)

l equivalent

S(4)2 [gab,ϑ

a, ωab,Qab,Qc

c] = (boundary term)

−ˆEabcd

[12Ra

b(ω) ∧Qcf ∧Qf

d − 116Qa

e ∧Qeb ∧Qc

f ∧Qfd]. (4.10)

p Instead of an equation of motion for ωab, now our splitting gives rise to three equations

EoM forQcc 0 = 0 ← (due to projective symmetry) , (4.11)

EoM for ωab 0 = D[Qc

a ∧Qbc], (4.12)

EoM forQab 0 = Eabcd[Rab − 1

4Qfa ∧Q

bf]∧Q

dm . (4.13)

p Counterexample. The following field configuration violates EoM of ωab:

gab = ηab , ωab = ωab + fα[aδb]t ,

ϑa = dxa , Qab = 2α(aδ

b)t ,

where

f is an arbitrary function,

αa = et(δay dy + δaz dz

).

(4.14)

since

D[Qc

a ∧Qbc]

= d[αa ∧αb

]= 2e2t

(δayδ

bz − δbyδ

az

)dt ∧ dy ∧ dz 6= 0 in the entireM. (4.15)

p Conclusion: The Lagrangian CANNOT be a total derivative.

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 22

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The critical metric-affine Gauss-Bonnet Lagrangian in D = 4. Equivalent formulation

p The action can be rewritten in terms of the new fields as

S(4)2 [gab,ϑ

a,ωab] =

ˆEabcdRa

b(ω) ∧Rcd(ω) (4.9)

l equivalent

S(4)2 [gab,ϑ

a, ωab,Qab,Qc

c] = (boundary term)

−ˆEabcd

[12Ra

b(ω) ∧Qcf ∧Qf

d − 116Qa

e ∧Qeb ∧Qc

f ∧Qfd]. (4.10)

p Instead of an equation of motion for ωab, now our splitting gives rise to three equations

EoM forQcc 0 = 0 ← (due to projective symmetry) , (4.11)

EoM for ωab 0 = D[Qc

a ∧Qbc], (4.12)

EoM forQab 0 = Eabcd[Rab − 1

4Qfa ∧Q

bf]∧Q

dm . (4.13)

p Counterexample. The following field configuration violates EoM of ωab:

gab = ηab , ωab = ωab + fα[aδb]t ,

ϑa = dxa , Qab = 2α(aδ

b)t ,

where

f is an arbitrary function,

αa = et(δay dy + δaz dz

).

(4.14)

since

D[Qc

a ∧Qbc]

= d[αa ∧αb

]= 2e2t

(δayδ

bz − δbyδ

az

)dt ∧ dy ∧ dz 6= 0 in the entireM. (4.15)

p Conclusion: The Lagrangian CANNOT be a total derivative.

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 22

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The critical metric-affine Gauss-Bonnet Lagrangian in D = 4. Equivalent formulation

p The action can be rewritten in terms of the new fields as

S(4)2 [gab,ϑ

a,ωab] =

ˆEabcdRa

b(ω) ∧Rcd(ω) (4.9)

l equivalent

S(4)2 [gab,ϑ

a, ωab,Qab,Qc

c] = (boundary term)

−ˆEabcd

[12Ra

b(ω) ∧Qcf ∧Qf

d − 116Qa

e ∧Qeb ∧Qc

f ∧Qfd]. (4.10)

p Instead of an equation of motion for ωab, now our splitting gives rise to three equations

EoM forQcc 0 = 0 ← (due to projective symmetry) , (4.11)

EoM for ωab 0 = D[Qc

a ∧Qbc], (4.12)

EoM forQab 0 = Eabcd[Rab − 1

4Qfa ∧Q

bf]∧Q

dm . (4.13)

p Counterexample. The following field configuration violates EoM of ωab:

gab = ηab , ωab = ωab + fα[aδb]t ,

ϑa = dxa , Qab = 2α(aδ

b)t ,

where

f is an arbitrary function,

αa = et(δay dy + δaz dz

).

(4.14)

since

D[Qc

a ∧Qbc]

= d[αa ∧αb

]= 2e2t

(δayδ

bz − δbyδ

az

)dt ∧ dy ∧ dz 6= 0 in the entireM. (4.15)

p Conclusion: The Lagrangian CANNOT be a total derivative.

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 22

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5. Discussion of the general critical Lovelock term

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 23

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General Lovelock in critical dimensions (k > 1). NOT a boundary term

p Critical dimension D = 2k.

L(2k)k = Ea1a2 ...a2k−1

a2kRa1a2 ∧ ... ∧Ra2k−1

a2k . (5.1)

p The action can be rewritten in terms of the new fields as

L(2k)k = Ea1...a2k

k∑m=0

1

4k−mk!

m!(k −m)!Ra1a2 ∧ ... ∧ Ra2m−1a2m∧ (5.2)

∧Qa2m+1f1 ∧Qf1

a2m+2 ∧ ... ∧Qa2k−1fk−m ∧Qfk−m

a2k (5.3)

basically,

L(2k)k = Ea1...a2k

[R ∧ R ∧ ... ∧ R ∧ R ← boundary term

+ R ∧ R ∧ ... ∧ R ∧Q2

+ R ∧ R ∧ ... ∧Q2 ∧Q

2

...

+ R ∧Q2 ∧ ... ∧Q

2 ∧Q2

+Q2 ∧Q

2 ∧ ... ∧Q2 ∧Q

2]

where Q2 ≡Q

aif ∧Qfai+1 (5.4)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 24

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General Lovelock in critical dimensions (k > 1). NOT a boundary term

p Critical dimension D = 2k.

L(2k)k = Ea1a2 ...a2k−1

a2kRa1a2 ∧ ... ∧Ra2k−1

a2k . (5.1)

p The action can be rewritten in terms of the new fields as

L(2k)k = Ea1...a2k

k∑m=0

1

4k−mk!

m!(k −m)!Ra1a2 ∧ ... ∧ Ra2m−1a2m∧ (5.2)

∧Qa2m+1f1 ∧Qf1

a2m+2 ∧ ... ∧Qa2k−1fk−m ∧Qfk−m

a2k (5.3)

basically,

L(2k)k = Ea1...a2k

[R ∧ R ∧ ... ∧ R ∧ R ← boundary term

+ R ∧ R ∧ ... ∧ R ∧Q2

+ R ∧ R ∧ ... ∧Q2 ∧Q

2

...

+ R ∧Q2 ∧ ... ∧Q

2 ∧Q2

+Q2 ∧Q

2 ∧ ... ∧Q2 ∧Q

2]

where Q2 ≡Q

aif ∧Qfai+1 (5.4)

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 24

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General Lovelock in critical dimensions (k > 1). NOT a boundary term

p Instead of an equation of motion for ωab, now our splitting gives rise to three equations

EoM forQcc 0 = 0 ← (due to projective symmetry) , (5.5)

EoM for ωab 0 = Eaba3...a2kk−1∑m=1

1

4k−mk!

m!(k −m)!Ra3a4 ∧ ... ∧ Ra2m−1a2m∧

∧ D[Qa2m+1f1 ∧Qf1

a2m+2 ∧ ... ∧Qa2k−1fk−m ∧Qfk−m

a2k], (5.6)

EoM forQab 0 = ...omitted... . (5.7)

p Counterexample. Consider the following field configuration:

gab = ηab , ωab = ωab ,

ϑa = dxa , Qab = 2α(aδ

b)t ,

where αa = et(δa3dx3 + ... + δa2kdx2k

). (5.8)

An immediate consequence is Rab = 0, so

EoM for ωab 0 = Eaba3...a2kD[Qa3f1 ∧Qf1

a4 ∧ ... ∧Qa2k−1fk−1 ∧Qfk−1

a2k], (5.9)

The ansatz (5.8) gives:

0 = d(αa3 ∧ ... ∧αa2k

)= 2(k − 1) e2(k−1)t (2k − 2)! δ

[a33 ...δ

a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k . (5.10)

If k > 1 we get a contradiction:

0 = δ[a33 ...δ

a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k 6= 0 ∀p ∈M. (5.11)

p Conclusion: The Lagrangian (with k > 1) CANNOT be a total derivative. [Janssen, Jiménez 2019]

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 25

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General Lovelock in critical dimensions (k > 1). NOT a boundary term

p Instead of an equation of motion for ωab, now our splitting gives rise to three equations

EoM forQcc 0 = 0 ← (due to projective symmetry) , (5.5)

EoM for ωab 0 = Eaba3...a2kk−1∑m=1

1

4k−mk!

m!(k −m)!Ra3a4 ∧ ... ∧ Ra2m−1a2m∧

∧ D[Qa2m+1f1 ∧Qf1

a2m+2 ∧ ... ∧Qa2k−1fk−m ∧Qfk−m

a2k], (5.6)

EoM forQab 0 = ...omitted... . (5.7)

p Counterexample. Consider the following field configuration:

gab = ηab , ωab = ωab ,

ϑa = dxa , Qab = 2α(aδ

b)t ,

where αa = et(δa3dx3 + ... + δa2kdx2k

). (5.8)

An immediate consequence is Rab = 0,

so

EoM for ωab 0 = Eaba3...a2kD[Qa3f1 ∧Qf1

a4 ∧ ... ∧Qa2k−1fk−1 ∧Qfk−1

a2k], (5.9)

The ansatz (5.8) gives:

0 = d(αa3 ∧ ... ∧αa2k

)= 2(k − 1) e2(k−1)t (2k − 2)! δ

[a33 ...δ

a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k . (5.10)

If k > 1 we get a contradiction:

0 = δ[a33 ...δ

a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k 6= 0 ∀p ∈M. (5.11)

p Conclusion: The Lagrangian (with k > 1) CANNOT be a total derivative. [Janssen, Jiménez 2019]

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 25

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General Lovelock in critical dimensions (k > 1). NOT a boundary term

p Instead of an equation of motion for ωab, now our splitting gives rise to three equations

EoM forQcc 0 = 0 ← (due to projective symmetry) , (5.5)

EoM for ωab 0 = Eaba3...a2kk−1∑m=1

1

4k−mk!

m!(k −m)!Ra3a4 ∧ ... ∧ Ra2m−1a2m∧

∧ D[Qa2m+1f1 ∧Qf1

a2m+2 ∧ ... ∧Qa2k−1fk−m ∧Qfk−m

a2k], (5.6)

EoM forQab 0 = ...omitted... . (5.7)

p Counterexample. Consider the following field configuration:

gab = ηab , ωab = ωab ,

ϑa = dxa , Qab = 2α(aδ

b)t ,

where αa = et(δa3dx3 + ... + δa2kdx2k

). (5.8)

An immediate consequence is Rab = 0, so

EoM for ωab 0 = Eaba3...a2kD[Qa3f1 ∧Qf1

a4 ∧ ... ∧Qa2k−1fk−1 ∧Qfk−1

a2k], (5.9)

The ansatz (5.8) gives:

0 = d(αa3 ∧ ... ∧αa2k

)= 2(k − 1) e2(k−1)t (2k − 2)! δ

[a33 ...δ

a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k . (5.10)

If k > 1 we get a contradiction:

0 = δ[a33 ...δ

a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k 6= 0 ∀p ∈M. (5.11)

p Conclusion: The Lagrangian (with k > 1) CANNOT be a total derivative. [Janssen, Jiménez 2019]

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 25

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General Lovelock in critical dimensions (k > 1). NOT a boundary term

p Instead of an equation of motion for ωab, now our splitting gives rise to three equations

EoM forQcc 0 = 0 ← (due to projective symmetry) , (5.5)

EoM for ωab 0 = Eaba3...a2kk−1∑m=1

1

4k−mk!

m!(k −m)!Ra3a4 ∧ ... ∧ Ra2m−1a2m∧

∧ D[Qa2m+1f1 ∧Qf1

a2m+2 ∧ ... ∧Qa2k−1fk−m ∧Qfk−m

a2k], (5.6)

EoM forQab 0 = ...omitted... . (5.7)

p Counterexample. Consider the following field configuration:

gab = ηab , ωab = ωab ,

ϑa = dxa , Qab = 2α(aδ

b)t ,

where αa = et(δa3dx3 + ... + δa2kdx2k

). (5.8)

An immediate consequence is Rab = 0, so

EoM for ωab 0 = Eaba3...a2kD[Qa3f1 ∧Qf1

a4 ∧ ... ∧Qa2k−1fk−1 ∧Qfk−1

a2k], (5.9)

The ansatz (5.8) gives:

0 = d(αa3 ∧ ... ∧αa2k

)= 2(k − 1) e2(k−1)t (2k − 2)! δ

[a33 ...δ

a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k . (5.10)

If k > 1 we get a contradiction:

0 = δ[a33 ...δ

a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k 6= 0 ∀p ∈M. (5.11)

p Conclusion:

The Lagrangian (with k > 1) CANNOT be a total derivative. [Janssen, Jiménez 2019]

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 25

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General Lovelock in critical dimensions (k > 1). NOT a boundary term

p Instead of an equation of motion for ωab, now our splitting gives rise to three equations

EoM forQcc 0 = 0 ← (due to projective symmetry) , (5.5)

EoM for ωab 0 = Eaba3...a2kk−1∑m=1

1

4k−mk!

m!(k −m)!Ra3a4 ∧ ... ∧ Ra2m−1a2m∧

∧ D[Qa2m+1f1 ∧Qf1

a2m+2 ∧ ... ∧Qa2k−1fk−m ∧Qfk−m

a2k], (5.6)

EoM forQab 0 = ...omitted... . (5.7)

p Counterexample. Consider the following field configuration:

gab = ηab , ωab = ωab ,

ϑa = dxa , Qab = 2α(aδ

b)t ,

where αa = et(δa3dx3 + ... + δa2kdx2k

). (5.8)

An immediate consequence is Rab = 0, so

EoM for ωab 0 = Eaba3...a2kD[Qa3f1 ∧Qf1

a4 ∧ ... ∧Qa2k−1fk−1 ∧Qfk−1

a2k], (5.9)

The ansatz (5.8) gives:

0 = d(αa3 ∧ ... ∧αa2k

)= 2(k − 1) e2(k−1)t (2k − 2)! δ

[a33 ...δ

a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k . (5.10)

If k > 1 we get a contradiction:

0 = δ[a33 ...δ

a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k 6= 0 ∀p ∈M. (5.11)

p Conclusion: The Lagrangian (with k > 1) CANNOT be a total derivative. [Janssen, Jiménez 2019]

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 25

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General Lovelock in critical dimensions. Particular solutions

[Janssen, Jiménez 2019]

p The general equation of motion for the connection can be written:

0 = DEa1a2 ...ab ∧Ra1a2 ∧ ... ∧RaD−3

aD−2 ⇔ (5.12)

⇔0 =

[Qca1Eca2...aD−2ab + ...+Q

caD−3Ea1...aD−4caD−2ab

+QcaEa1...aD−2cb

]∧Ra1a2 ∧ ... ∧RaD−3aD−2

(5.13)

é Particular cases:

(k = 1) Einstein: 0 = QcaEbc ⇔ Qab = 0 (general sol.) . (5.14)

(k = 2) Gauss-Bonnet: 0 = [QcaEbcpq +Q

cpEqabc] ∧Rpq ⇒ ? . (5.15)

p Families of solutions for arbitrary k:é connection with Qµνρ = Vµgνρ (i.e.Qab = 0).

p Families of solutions for arbitrary k > 1:é TeleparallelRc

d = 0.é Any connection such thatQab ∧Rc

d = 0.

p Families of solutions for arbitrary k > 2:é Any connection such thatRab = αab ∧ k for certain 1-forms αab and k (due to k ∧ k ≡ 0).

Example. Ansatz of grav. wave: k is the dual form of the wave vector. [My PhD Thesis - still in progress]

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 26

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General Lovelock in critical dimensions. Particular solutions

[Janssen, Jiménez 2019]

p The general equation of motion for the connection can be written:

0 = DEa1a2 ...ab ∧Ra1a2 ∧ ... ∧RaD−3

aD−2 ⇔ (5.12)

⇔0 =

[Qca1Eca2...aD−2ab + ...+Q

caD−3Ea1...aD−4caD−2ab

+QcaEa1...aD−2cb

]∧Ra1a2 ∧ ... ∧RaD−3aD−2

(5.13)

é Particular cases:

(k = 1) Einstein: 0 = QcaEbc ⇔ Qab = 0 (general sol.) . (5.14)

(k = 2) Gauss-Bonnet: 0 = [QcaEbcpq +Q

cpEqabc] ∧Rpq ⇒ ? . (5.15)

p Families of solutions for arbitrary k:é connection with Qµνρ = Vµgνρ (i.e.Qab = 0).

p Families of solutions for arbitrary k > 1:é TeleparallelRc

d = 0.é Any connection such thatQab ∧Rc

d = 0.

p Families of solutions for arbitrary k > 2:é Any connection such thatRab = αab ∧ k for certain 1-forms αab and k (due to k ∧ k ≡ 0).

Example. Ansatz of grav. wave: k is the dual form of the wave vector. [My PhD Thesis - still in progress]

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 26

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General Lovelock in critical dimensions. Particular solutions

[Janssen, Jiménez 2019]

p The general equation of motion for the connection can be written:

0 = DEa1a2 ...ab ∧Ra1a2 ∧ ... ∧RaD−3

aD−2 ⇔ (5.12)

⇔0 =

[Qca1Eca2...aD−2ab + ...+Q

caD−3Ea1...aD−4caD−2ab

+QcaEa1...aD−2cb

]∧Ra1a2 ∧ ... ∧RaD−3aD−2

(5.13)

é Particular cases:

(k = 1) Einstein: 0 = QcaEbc ⇔ Qab = 0 (general sol.) . (5.14)

(k = 2) Gauss-Bonnet: 0 = [QcaEbcpq +Q

cpEqabc] ∧Rpq ⇒ ? . (5.15)

p Families of solutions for arbitrary k:é connection with Qµνρ = Vµgνρ (i.e.Qab = 0).

p Families of solutions for arbitrary k > 1:é TeleparallelRc

d = 0.é Any connection such thatQab ∧Rc

d = 0.

p Families of solutions for arbitrary k > 2:é Any connection such thatRab = αab ∧ k for certain 1-forms αab and k (due to k ∧ k ≡ 0).

Example. Ansatz of grav. wave: k is the dual form of the wave vector. [My PhD Thesis - still in progress]

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 26

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General Lovelock in critical dimensions. Particular solutions

[Janssen, Jiménez 2019]

p The general equation of motion for the connection can be written:

0 = DEa1a2 ...ab ∧Ra1a2 ∧ ... ∧RaD−3

aD−2 ⇔ (5.12)

⇔0 =

[Qca1Eca2...aD−2ab + ...+Q

caD−3Ea1...aD−4caD−2ab

+QcaEa1...aD−2cb

]∧Ra1a2 ∧ ... ∧RaD−3aD−2

(5.13)

é Particular cases:

(k = 1) Einstein: 0 = QcaEbc ⇔ Qab = 0 (general sol.) . (5.14)

(k = 2) Gauss-Bonnet: 0 = [QcaEbcpq +Q

cpEqabc] ∧Rpq ⇒ ? . (5.15)

p Families of solutions for arbitrary k:é connection with Qµνρ = Vµgνρ (i.e.Qab = 0).

p Families of solutions for arbitrary k > 1:é TeleparallelRc

d = 0.é Any connection such thatQab ∧Rc

d = 0.

p Families of solutions for arbitrary k > 2:é Any connection such thatRab = αab ∧ k for certain 1-forms αab and k (due to k ∧ k ≡ 0).

Example. Ansatz of grav. wave: k is the dual form of the wave vector. [My PhD Thesis - still in progress]

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 26

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General Lovelock in critical dimensions. Particular solutions

[Janssen, Jiménez 2019]

p The general equation of motion for the connection can be written:

0 = DEa1a2 ...ab ∧Ra1a2 ∧ ... ∧RaD−3

aD−2 ⇔ (5.12)

⇔0 =

[Qca1Eca2...aD−2ab + ...+Q

caD−3Ea1...aD−4caD−2ab

+QcaEa1...aD−2cb

]∧Ra1a2 ∧ ... ∧RaD−3aD−2

(5.13)

é Particular cases:

(k = 1) Einstein: 0 = QcaEbc ⇔ Qab = 0 (general sol.) . (5.14)

(k = 2) Gauss-Bonnet: 0 = [QcaEbcpq +Q

cpEqabc] ∧Rpq ⇒ ? . (5.15)

p Families of solutions for arbitrary k:é connection with Qµνρ = Vµgνρ (i.e.Qab = 0).

p Families of solutions for arbitrary k > 1:é TeleparallelRc

d = 0.é Any connection such thatQab ∧Rc

d = 0.

p Families of solutions for arbitrary k > 2:é Any connection such thatRab = αab ∧ k for certain 1-forms αab and k (due to k ∧ k ≡ 0).

Example. Ansatz of grav. wave: k is the dual form of the wave vector. [My PhD Thesis - still in progress]

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General Lovelock in critical dimensions. EoM of the coframe

p EoM in the general case

δS(D)k

δϑa= gabRa1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑ

a1...a2kb . (5.16)

p In D = 2k (critical) the red object becomes a (D + 1)-form, and the equation becomes trivial:

δS(2k)k

δϑa= 0 . (5.17)

p Another way of checking this is expanding the star in the Lagrangian to see the dependence on thecoframe,

L(D)k = Ra1

a2 ∧ ... ∧Ra2k−1

a2k ∧ ?ϑa1a2 ...a2k−1a2k (5.18)

= Ra1a2 ∧ ... ∧Ra2k−1a2k ∧(

1

(D − 2k)!Ea1...a2k b1...bD−2k

ϑb1...bD−2k

)(5.19)

In the critical case,L

(2k)k = Ea1a2 ...a2k−1

a2kRa1a2 ∧ ... ∧Ra2k−1

a2k , (5.20)

where

Ea1a2 ...a2k−1a2k depends only on gab (in the orthonormal case it is a constant tensor)

Rab is only connection-dependent .

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 27

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General Lovelock in critical dimensions. EoM of the coframe

p EoM in the general case

δS(D)k

δϑa= gabRa1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑ

a1...a2kb . (5.16)

p In D = 2k (critical) the red object becomes a (D + 1)-form, and the equation becomes trivial:

δS(2k)k

δϑa= 0 . (5.17)

p Another way of checking this is expanding the star in the Lagrangian to see the dependence on thecoframe,

L(D)k = Ra1

a2 ∧ ... ∧Ra2k−1

a2k ∧ ?ϑa1a2 ...a2k−1a2k (5.18)

= Ra1a2 ∧ ... ∧Ra2k−1a2k ∧(

1

(D − 2k)!Ea1...a2k b1...bD−2k

ϑb1...bD−2k

)(5.19)

In the critical case,L

(2k)k = Ea1a2 ...a2k−1

a2kRa1a2 ∧ ... ∧Ra2k−1

a2k , (5.20)

where

Ea1a2 ...a2k−1a2k depends only on gab (in the orthonormal case it is a constant tensor)

Rab is only connection-dependent .

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 27

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General Lovelock in critical dimensions. EoM of the coframe

p EoM in the general case

δS(D)k

δϑa= gabRa1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑ

a1...a2kb . (5.16)

p In D = 2k (critical) the red object becomes a (D + 1)-form, and the equation becomes trivial:

δS(2k)k

δϑa= 0 . (5.17)

p Another way of checking this is expanding the star in the Lagrangian to see the dependence on thecoframe,

L(D)k = Ra1

a2 ∧ ... ∧Ra2k−1

a2k ∧ ?ϑa1a2 ...a2k−1a2k (5.18)

= Ra1a2 ∧ ... ∧Ra2k−1a2k ∧(

1

(D − 2k)!Ea1...a2k b1...bD−2k

ϑb1...bD−2k

)(5.19)

In the critical case,L

(2k)k = Ea1a2 ...a2k−1

a2kRa1a2 ∧ ... ∧Ra2k−1

a2k , (5.20)

where

Ea1a2 ...a2k−1a2k depends only on gab (in the orthonormal case it is a constant tensor)

Rab is only connection-dependent .

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 27

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6. Summary and conclusions

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 28

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Summary and conclusions

Ideas to remember. Consider metric-affine Lovelock in the critical dimension:

p The equation of the coframe (Vielbein) is a trivial identity (so, no information).

p The Einstein-Palatini theory have been solved in all dimensions (even in the critical one).

p The big lesson:There are configurations that do not satisfy the EoM⇒ these theories are not boundary terms.

Consequently, we cannot use Lovelock terms to rewrite certain powers of R in terms of others!!(as we can in the metric or in the Riemann-Cartan case)

Open questions / further work

p Which is the role of the non-metricity in breaking the triviality in critical dimension?

p What about the complete Lovelock Lagrangian (all of the Lovelock terms allowed in that D)?The most interesting case is obviously D = 4, which includes EP + GBP. [Work in progress]

Thanks for your attention!

Aitäh!

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 29

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Summary and conclusions

Ideas to remember. Consider metric-affine Lovelock in the critical dimension:

p The equation of the coframe (Vielbein) is a trivial identity (so, no information).

p The Einstein-Palatini theory have been solved in all dimensions (even in the critical one).

p The big lesson:There are configurations that do not satisfy the EoM⇒ these theories are not boundary terms.

Consequently, we cannot use Lovelock terms to rewrite certain powers of R in terms of others!!(as we can in the metric or in the Riemann-Cartan case)

Open questions / further work

p Which is the role of the non-metricity in breaking the triviality in critical dimension?

p What about the complete Lovelock Lagrangian (all of the Lovelock terms allowed in that D)?The most interesting case is obviously D = 4, which includes EP + GBP. [Work in progress]

Thanks for your attention!

Aitäh!

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 29

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Summary and conclusions

Ideas to remember. Consider metric-affine Lovelock in the critical dimension:

p The equation of the coframe (Vielbein) is a trivial identity (so, no information).

p The Einstein-Palatini theory have been solved in all dimensions (even in the critical one).

p The big lesson:There are configurations that do not satisfy the EoM⇒ these theories are not boundary terms.

Consequently, we cannot use Lovelock terms to rewrite certain powers of R in terms of others!!(as we can in the metric or in the Riemann-Cartan case)

Open questions / further work

p Which is the role of the non-metricity in breaking the triviality in critical dimension?

p What about the complete Lovelock Lagrangian (all of the Lovelock terms allowed in that D)?The most interesting case is obviously D = 4, which includes EP + GBP. [Work in progress]

Thanks for your attention!

Aitäh!

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 29

Page 87: UGRalejandrojc/docstore/docs_congresos... · 2021. 1. 18. · Structure of this presentation 1 Introduction (metric-affine formalism and geometry) 2 Metric-Affine Lovelock theory

Summary and conclusions

Ideas to remember. Consider metric-affine Lovelock in the critical dimension:

p The equation of the coframe (Vielbein) is a trivial identity (so, no information).

p The Einstein-Palatini theory have been solved in all dimensions (even in the critical one).

p The big lesson:There are configurations that do not satisfy the EoM⇒ these theories are not boundary terms.

Consequently, we cannot use Lovelock terms to rewrite certain powers of R in terms of others!!(as we can in the metric or in the Riemann-Cartan case)

Open questions / further work

p Which is the role of the non-metricity in breaking the triviality in critical dimension?

p What about the complete Lovelock Lagrangian (all of the Lovelock terms allowed in that D)?The most interesting case is obviously D = 4, which includes EP + GBP. [Work in progress]

Thanks for your attention!

Aitäh!

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 29

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References

References for this presentation:p D. Lovelock, [Lovelock 1971]

The Einstein Tensor and Its Generalizations,

J. Math. Phys. 12 (1971), 498–501.p M. Borunda, B. Janssen, M. Bastero-Gil, , [Borunda, Janssen, Bastero 2008]

Palatini versus metric formulation in higher curvature gravity,

JCAP 0811 (2008), 008.p B. Janssen, A. Jiménez-Cano, J. A. Orejuela, [Janssen, Jiménez, Orejuela 2019]

A non-trivial connection for the metric-affine Gauss-Bonnet theory in D = 4,

Phys. Lett. B 795 (2019) 42 – 48.p F. W. Hehl, J. D. McCrea, E. W. Mielke, Y. Ne’eman, [Hehl, McCrea, Mielke, Ne’eman 1995]

Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance,

Phys. Rep. 258 (1995) 1 – 171.p A. Jiménez-Cano, [My PhD Thesis – Still in progress]

Metric-Affine Gauge theory of gravity. Foundations, perturbations and gravitational wave solutions.p B. Janssen, A. Jiménez-Cano, [Janssen, Jiménez, 2019]

On the topological character of metric-affine Lovelock Lagrangians in critical dimensions.

arXiv:1907.12100 [gr-qc]

Other interesting references:p D. Iosifidis, and T. Koivisto,

Scale transformations in metric-affine geometry,

arXiv preprint: 1810.12276.p J. Beltrán Jiménez, L. Heisenberg, T. Koivisto,

Cosmology for quadratic gravity in generalized Weyl geometry,

JCAP 2016 (2016), no 04, 046.p J. Zanelli,

Chern-Simons Forms in Gravitation Theories,

Class. Quant. Grav. 29 (2012), 133001.

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 30

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Metric-affine lagrangians depending on curvature: EoM

Consider a gravitational lagrangian (vacuum) depending on the connection exclusively through thecurvature:

S[g,ϑ,ω] =

ˆL(gab,ϑ

a,Rab(ω)) ≡

ˆL(gab, eµ

a, Rµνab(ω))

√|g|dDx , (7.1)

with projective symmetry.

p Noether identities of Diff(M) and GL(D,R)⇒We only need the EoM of ϑa and ωab:

0 =δS

δϑa≡ ∂L

∂ϑa, (7.2)

0 =δS

δωab≡ D

(∂L

∂Rab

), (7.3)

or, in components,

0 =1√|g|

δS

δeµa≡ eµaL+

∂L∂eµa

, (7.4)

0 =−1

2√|g|

δS

δωµab≡(∇λ −

1

2Qλσ

σ + Tλσσ

)(∂L

∂Rλµab

)− 1

2Tλσ

µ ∂L∂Rλσab

. (7.5)

p Noether identity of projective symmetry⇒ the connection EoM is traceless (in a, b indices).

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 31

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Metric-affine lagrangians depending on curvature: EoM

Consider a gravitational lagrangian (vacuum) depending on the connection exclusively through thecurvature:

S[g,ϑ,ω] =

ˆL(gab,ϑ

a,Rab(ω)) ≡

ˆL(gab, eµ

a, Rµνab(ω))

√|g|dDx , (7.1)

with projective symmetry.

p Noether identities of Diff(M) and GL(D,R)⇒We only need the EoM of ϑa and ωab:

0 =δS

δϑa≡ ∂L

∂ϑa, (7.2)

0 =δS

δωab≡ D

(∂L

∂Rab

), (7.3)

or, in components,

0 =1√|g|

δS

δeµa≡ eµaL+

∂L∂eµa

, (7.4)

0 =−1

2√|g|

δS

δωµab≡(∇λ −

1

2Qλσ

σ + Tλσσ

)(∂L

∂Rλµab

)− 1

2Tλσ

µ ∂L∂Rλσab

. (7.5)

p Noether identity of projective symmetry⇒ the connection EoM is traceless (in a, b indices).

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 31

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Metric-affine lagrangians depending on curvature: EoM

Consider a gravitational lagrangian (vacuum) depending on the connection exclusively through thecurvature:

S[g,ϑ,ω] =

ˆL(gab,ϑ

a,Rab(ω)) ≡

ˆL(gab, eµ

a, Rµνab(ω))

√|g|dDx , (7.1)

with projective symmetry.

p Noether identities of Diff(M) and GL(D,R)⇒We only need the EoM of ϑa and ωab:

0 =δS

δϑa≡ ∂L

∂ϑa, (7.2)

0 =δS

δωab≡ D

(∂L

∂Rab

), (7.3)

or, in components,

0 =1√|g|

δS

δeµa≡ eµaL+

∂L∂eµa

, (7.4)

0 =−1

2√|g|

δS

δωµab≡(∇λ −

1

2Qλσ

σ + Tλσσ

)(∂L

∂Rλµab

)− 1

2Tλσ

µ ∂L∂Rλσab

. (7.5)

p Noether identity of projective symmetry⇒ the connection EoM is traceless (in a, b indices).

Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 31


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