Chern-Simons Gravity Induces Conformal Gravity QGSC VI

Motivation Chern Simons is a Conformal Gravity Comments and outlooks

Chern-Simons Gravity induces Conformal Gravity QGSC VI

Danilo Diaz and me

September 12, 2013

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks Outline

1 Motivation 3d Chern Simons Conformal gravity 3d Chern Simons AdS gravity

2 Chern Simons is a Conformal Gravity

3 Comments and outlooks

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation 3d Chern Simons Conformal gravity Chern Simons is a Conformal Gravity 3d Chern Simons AdS gravity Comments and outlooks Conformal Gravity

Four dimensional Conformal Gravity

µναβ 4 W Wµναβ √gdx Z

It has been mentioned It was considered as a possible UV completion of gravity.

It has been mentioned It was considered as a possible UV completion of gravity. It was also useful for constructing supergravity theories.

It has been mentioned It was considered as a possible UV completion of gravity. It was also useful for constructing supergravity theories. It has recently emerged from the twistor string theory.

It has been mentioned It was considered as a possible UV completion of gravity. It was also useful for constructing supergravity theories. It has recently emerged from the twistor string theory. It can have a rˆole in AdS/CFT

2n + 1-dimensional transgression form

1 I2n+1 = (n + 1) dt (A1 A0) Ft . . . Ft , (1) M 0 − ∧ ∧ ∧ Z Z * n +

where A1 and A0 are two (1-form) connections| in{z the same} ﬁber. Ft = dAt + At At with At = tA + (1 t)A . stands for the ∧ 1 − 0 hi trace in the group.

The (Euler) Chern Simons density

Provided A0 = 0 one gets Chern Simons action for A1, or viceversa, in d = 2n + 1.

The (Euler) Chern Simons density

Provided A0 = 0 one gets Chern Simons action for A1, or viceversa, in d = 2n + 1.

The Chern Simons equation of motion

F nδA = 0 h i where F = dA + A A. ∧

They are gauge theories diﬀerent from YM

They are gauge theories diﬀerent from YM in a sense purely topological

They are gauge theories diﬀerent from YM in a sense purely topological connected with gravitational theories in a non trivial or standard way

They are gauge theories diﬀerent from YM in a sense purely topological connected with gravitational theories in a non trivial or standard way full of surprises.

Rewritten as 1.5 formalism In 3 dimensions conformal gravity

i 2 ijk ICG = wi dw + ε wi wj wk (2) ∧ 3 ∧ ∧ ZM kl µ where wi = εijk ω µdx is the Levi Civita (spin) connection i µ associated a given dreibein e µdx .

The equations of motion

Cµνλ = µρνλ λρνµ = 0, (3) ∇ −∇ equivalent to the vanishing of the Cotton-York tensor. Here 1 ρµν = Rµν Rgµν, (4) − 4 with ρµν is sometimes called the Schouten tensor or plainly rho-tensor.

The equations of motion

This means the solution must a conformally ﬂat space.

This action principle d = 3 The previous action can be written in terms of a connection for conformal group in 3 dimensions (CFT SO(3,2) ) 3 ≈ i i i Aµ = e µPi + w µJi + λ µKi + φµD . (5)

Provided

i i i j T = de + ω j e = 0 1 λi dxµ = Ri dxµ = ρi µ −2 µ Dρi = 0

φµ = 0

The previous equations of motion can be rewritten as F = dA + A A = 0. ∧

Provided

i i i j T = de + ω j e = 0 1 λi dxµ = Ri dxµ = ρi µ −2 µ Dρi = 0

φµ = 0

The previous equations of motion can be rewritten as F = dA + A A = 0. This is equivalent to require a conformally ∧ ﬂat space.

No surprise The conformal gravity action can be written as 3d Chern Simons action k 2 ICS = A dA + A A A (6) 8π ∧ 3 ∧ ∧ ZM for the conformal group.

Conformal connection In mathematical lore the connection for SO(3,2) CFT ≈ 3 i i i A = e Pi + w Ji + ρ Ki

is called the Tractor Connection.

Conformal connection In mathematical lore the connection for SO(3,2) CFT ≈ 3 i i i A = e Pi + w Ji + ρ Ki

is called the Tractor Connection.Recall this is partially on-shell.

Weyl as Conformal 2ξ A Weyl transformation, gij e gij , of A is → A eξ(x)D Ae−ξ(x)D + eξ(x)D d(e−ξ(x)D ) → where ξ(x) is an arbitrary function of the coordinates of the base space xµ . { }

Weyl in components The component of A transforms as

ei eξei → ωij ωij +Υi ej Υj ei i → −ξ i − i i µ i µ ρ e (ρ + DΥ +Υ Υµdx + e ΥµΥ ) → i iµ iµ with Υµ = ∂µξ(x) and Υ = E Υµ = E ∂µξ(x)

A theory of AdS gravity in d = 3

A Chern Simons theory for AdS3 SO(2, 2) written in terms of 1 AB ≈ A = 2 ω JAB where JAB (A,B=1. . . 4) are the generator by

A theory of AdS gravity in d = 3

A Chern Simons theory for AdS3 SO(2, 2) written in terms of 1 AB ≈ A = 2 ω JAB where JAB (A,B=1. . . 4) are the generator by 1 splitting 1 1 A = ωÂB J = ωîj J +q î J , (7) 2 AB 2 ij i 4 where i, j = 1, 2, 3. i ij i − i i 2 Next, identifyingq ˆ andω ˆ withq ˆ = l 1e , where e is a dreibein and ωîj = ωij a Lorentz (spin) connection on the manifold to be considered.

How to take the trace This is not a minor issue and most relevant results can be extract from the analysis of the diﬀerent traces.

How to take the trace This is not a minor issue and most relevant results can be extract from the analysis of the diﬀerent traces. Nonetheless the trace can be deﬁned as JA A JA A = εA ...A , h 1 2 3 4 i 1 4 which splits, throughout A = (i, 4) with i = 1 . . . 3, as

εA1...A4 = εi1i2i34 = εi1i2i3 .

Chern Simons action can be written as 1 I 3 = l −1 Rij ek + ei ej ek ε + BT CS 3l 2 ijk Z ij ij i kl where R = dω + ω k ω is called the curvature two form.

This is rather standard This action is actually Einstein (Cartan) gravity in three dimensions.

F = 0 means

F ij = Rij + l −2ei ej = 0 i4 i i ∧ i k F = T = de + ω k e = 0

F = 0 means

F ij = Rij + l −2ei ej = 0 i4 i i ∧ i k F = T = de + ω k e = 0

Solution This allows only torsion free constant curvature manifolds

F = 0 means

F ij = Rij + l −2ei ej = 0 i4 i i ∧ i k F = T = de + ω k e = 0

Solution This allows only torsion free constant curvature manifolds, i.e., AdS /Γ with Γ AdS . 3 ∈ 3

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks We had one theory for two groups

Conformal Provided = CFT = SO(3, 2) F = 0 implies conformal gravity G 3 and spaces conformally ﬂat.

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks We had one theory for two groups

Conformal Provided = CFT = SO(3, 2) F = 0 implies conformal gravity G 3 and spaces conformally ﬂat.

AdS Provided = AdS = SO(2, 2) F = 0 implies standard gravity and G 3 spaces locally AdS.

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks We had one theory for two groups

Conformal Provided = CFT = SO(3, 2) F = 0 implies conformal gravity G 3 and spaces conformally ﬂat.

AdS Provided = AdS = SO(2, 2) F = 0 implies standard gravity and G 3 spaces locally AdS.

Conformal is AdS somehow It is quite appealing to try to connect both.

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks The previous can be generalized

A SO(2n,2) connection 1 AB Given A = 2 ω JAB , where JAB are the generator of SO(2n, 2)

1 AB 1 ab a A = ωˆ J = ωˆ J +q ˆ Ja n , (8) 2 AB 2 ab 2 +2 where a, b = 1 . . . 2n + 1.

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks The previous can be generalized

A SO(2n,2) connection 1 AB Given A = 2 ω JAB , where JAB are the generator of SO(2n, 2)

1 AB 1 ab a A = ωˆ J = ωˆ J +q ˆ Ja n , (8) 2 AB 2 ab 2 +2 where a, b = 1 . . . 2n + 1.

Traces

JA A . . . JA A = εA ...A = εa ...a n ., h 1 2 2n+1 2n+2 i 1 2n+2 1 2n+12 +2 This is the trace considered for the rest of this work.

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks Love-Chern-Simons

A more useful Chern Simons action AdS-Chern-Simons gravity, module a boundary term, can be rewritten in the form of a Lovelock gravity as

n 1 n ε Ra1a2 . . . Ra2p−1a2p qa2p+1 . . . qa2n+1 2n 2p p a1...a2n+1 p Z X=0 − (9) a a2n+1 ab ab a cb where q = ω and R = dω + ω c ω with a, b, c = 1,..., 2n + 1.

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks More complex equations of motion

The new concept En 2+1 dimensions F = 0 is a simple equation of motion, in higher odd dimensions this complicates. For instance in 5 the equation of motion is F F = 0 ∧ or for SO(4,2)

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks More complex equations of motion

The new concept En 2+1 dimensions F = 0 is a simple equation of motion, in higher odd dimensions this complicates. For instance in 5 the equation of motion is F F = 0 ∧ or for SO(4,2)

ab a b cd c d f = εabcdf (R + q q ) (R + q q ) = 0 E ab a ∧ b ∧ c c ∧ e df = εabcdf (R + q q ) (dq + ω q ) = 0 E ∧ ∧ e ∧

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks A tractor like connection

AdS group in d dimensions

EF GH [JAB , JCD ]= δ δ ηEG JFH , (10) − AB CD with A, B = 0 . . . d + 1.

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks A tractor like connection

AdS group in d dimensions

EF GH [JAB , JCD ]= δ δ ηEG JFH , (10) − AB CD with A, B = 0 . . . d + 1.

Conformal Group in d 1 dimensions − mn op [Mij , Mkl ]= δ δ ηmo Mnp − ij kl [Mij , Pk ]= (ηik Pj ηjk Pi ) [D, Pi ]= Pi − − (11) [Mij , Kk ]= (ηik Kj ηjk Ki ) [D, Ki ]= Ki − − − [Pi , Kj ] = 2Mij 2ηij D [D, Mij ] = 0 − with i, j = 0 . . . d 1. −

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks The most provocative relation ever

AdS is Conformal provided 1 Jij = Mij Jid−1 = 2 (Pi + Ki ) 1 (12) Jd− d = D Jid = (Pi Ki ). 1 2 −

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks AdS tractor connection

Generalization The d-dimensional tractor connection is 1 A = ωij J + ei P + ρi K (13) 2 ij i i where ωij and ei are a spin connection and a vielbein on the manifold considered.

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks AdS tractor connection

Generalization The d-dimensional tractor connection is 1 A = ωij J + ei P + ρi K (13) 2 ij i i where ωij and ei are a spin connection and a vielbein on the manifold considered. On the other hand,

i i ν µ ρ = e νρ µdx

µ with ρ ν is given by

1 1 ρν = Rν δν R (14) µ d 3 µ − 2(d 2) µ − −

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks AdS tractor connection

Some algebra

1 ij i i A = ω Jij + ρ (Jid− + Jid )+ e (Jid Jid− ) 2 1 − 1 1 ij i i i i = ω Jij + (e ρ )Jid + (e + ρ )Jid− (15) 2 − 1

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks Conformal Gravity from Chern Simons

Add a dimension and wrap it The idea is to show that a conformal theory of gravity can be written as a Chern Simons gauge theory with the help of an extension of tractor connection mentioned above.

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks Conformal Gravity from Chern Simons

Add a dimension and wrap it The idea is to show that a conformal theory of gravity can be written as a Chern Simons gauge theory with the help of an extension of tractor connection mentioned above.

This is not direct A tractor connection for SO(d 1, 2) exist on a d 1 dimensions − − manifold while a SO(d 1, 2)-CS density exist in d = 2n + 1 − dimensions.

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks AdS tractor connection

Solution proposed Dimensional reduction of a 2n+1-CS density on ′ = S1 to M M× produce an eﬀective 2n-dimensional theory.

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks AdS tractor connection

The generalization On space ′ = S1 M M× 1 ij µ i µ i µ µ A n = ω (x )Jij + e (x )Pi + ρ (x )Ki + Φ(x )dϕD 2 +1 2 where i, j = 1, 2,... 2n and a system of coordinates X M = (xµ, ϕ) has been considered on ′ with ϕ parametrizing S1. M

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks AdS tractor connection

This is sound The presence of Φdϕ along D does not changes the law of transformation under Weyl transformations. Furthermore Φdϕ transforms as Φdϕ Φdϕ dξ. → − This transformation has no eﬀect on the CS action due to dξ has only projection on . M

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks AdS tractor connection

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks 3d AdS 2d Conformal

A simple example is 3d to 2d This leads to the identiﬁcation

ωîj = ωij ωî3 = ρi + ei , ωˆ34 = Φ(x)dϕ = q3, ωî4 = ei ρi = qi , − This yields to the splitting of the three dimensional Rab as

Rˆij = Rij (ρi + ei )(ρj + ej ) and − Rˆi3 = D(ρi ei ), (16) −

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks 3d AdS 2d Conformal

A simple example en 3d to 2d Finally the CS action given by

ab c 1 a b c I3 = εabc Rˆ q + q q q . (17) M′ 3 Z

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks 3d AdS 2d Conformal

A simple example en 3d to 2d Finally the CS action given by

ab c 1 a b c I3 = εabc Rˆ q + q q q . (17) M′ 3 Z becomes, upon integration along S1,

2 I3 = 2 ΦR√gd x. M Z

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks AdS tractor connection

Propeties ρνβ

For d > 3 tensor ρνβ satisﬁes the relation

Rµναβ = Wµναβ + gµαρνβ gναρµβ gµαρνα + gνβρµα, (18) − −

where Wµναβ is the Weyl tensor.

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks AdS tractor connection

Propeties ρνβ

For d > 3 tensor ρνβ satisﬁes the relation

Rµναβ = Wµναβ + gµαρνβ gναρµβ gµαρνα + gνβρµα, (18) − −

where Wµναβ is the Weyl tensor. This can be rewritten equivalently in diﬀerential forms formalism as 1 Rij = W ij ek el 2(ei ρj ej ρi ). (19) 2 kl − −

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks 2n+1 AdS 2n Conformal

Generically With the identiﬁcation

ωîj = ωij ωî 2n+1 = ei + ρi ωˆ2n+1 2n+2 = Φ(x)dϕ = q2n+1 (20) ωî 2n+2 = ei ρi = qi , − with i = 1,..., 2n.

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks 2n+1 AdS 2n Conformal

Chern Simons The CS action becomes

2n+1 i1i2 i1 i2 i2n−1i2n i2n−1 i2n ICS = εi1...i2n (R + 4ρ e ) . . . (R + 4ρ e ) Φdϕ Z

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks 2n+1 AdS 2n Conformal

In terms of Weyl The previous CS action, upon integration along S1, becomes

j1...j2n i1i2 i2n−1i2n 2n ICS = Φδ W . . . W e d x. (21) i1...i2n j1j2 j2n−1j2n | | Z

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks 2n+1 AdS 2n Conformal

To be noticed ij This is simpler than it seems as W ik = 0. This is very similar to the Euler density but where Riemann tensor has been replaced by Weyl tensor.

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks 5 AdS and 4 Conformal

The usual conformal with a twist

4 µναβ 4 ICS = Φ W Wµναβ √gd x Z which is a generalization of the usual Weyl Gravity mentioned at the beginning.

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks Comments and Outlooks

Conclusion Chern Simons theories can describe a simple generalization of Weyl Gravities.

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks Outlooks

Comments

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks Outlooks

Comments The Weyl gravities obtained for d > 4 have non arbitrary coeﬃcient. This is due to hidden AdS symmetries.

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks Outlooks

Comments The Weyl gravities obtained for d > 4 have non arbitrary coeﬃcient. This is due to hidden AdS symmetries. These are mere the zero modes of the compactiﬁcation. A lot to do.

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI Motivation Chern Simons is a Conformal Gravity Comments and outlooks Outlooks

Comments The Weyl gravities obtained for d > 4 have non arbitrary coeﬃcient. This is due to hidden AdS symmetries. These are mere the zero modes of the compactiﬁcation. A lot to do. A higher spin version of this is calling on.

Danilo Diaz and me Chern-Simons Gravity induces Conformal Gravity QGSC VI