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 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 is interesting It has been mentioned It was considered as a possible UV completion of gravity. 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 is interesting It has been mentioned It was considered as a possible UV completion of gravity. It was also useful for constructing supergravity theories. 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 is interesting 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. 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 is interesting 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 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 Chern Simons Gravity 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. 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 Chern Simons Gravity The (Euler) Chern Simons density Provided A0 = 0 one gets Chern Simons action for A1, or viceversa, in d = 2n + 1. 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 Chern Simons Gravity 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. ∧ 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 Chern Simons gauge theories are interesting They are gauge theories different from YM 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 Chern Simons gauge theories are interesting They are gauge theories different from YM in a sense purely topological 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 Chern Simons gauge theories are interesting They are gauge theories different from YM in a sense purely topological connected with gravitational theories in a non trivial or standard way 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 Chern Simons gauge theories are interesting 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. 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 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 . 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 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. 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 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. This means the solution must a conformally flat space. 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 Rewritten as a gauge theory 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) 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 Written as a gauge theory 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. ∧ 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 Written as a gauge theory 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. 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 Chern Simons on-shell 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. 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 The Tractor Connection arises 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. 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 The Tractor Connection arises 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. 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 Weyl Transformations 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 . { } 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 Weyl Transformations 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) 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 An AdS Gravity as Chern Simons 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.