Holography with Shape Dynamics

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1 . 1/ 11 Holography with Henrique Gomes Physics, University of California, Davis July 6, 2012 In collaboration with Tim Koslowski

2 Outline 1 Holographic dulaities 2

3 . 2/ 11 Holographic dulaities Ideas behind holographic dualities Divergencies In QFT correlation functions suffer from UV divergencies. In gravity, action suffers from IR (near the asymptotic boundary) divergencies due to infinite volume Renormalization group and radial evolution In QFT: Regularization does not respect conformal invariance: Anomalous conformal Ward identities. Deformation from CFT as RG flow. UV fixed point would represent a conformal theory. In gravity: AdS near-boundary as deformation of AdS. Assuming AdS/CFT: AdS dual to deformed CFT radial evolution towards AdS dual to RG flow towards CFT.

4 . 3/ 11 Holographic dulaities Implementation Typical relation for AdS/CFT: δs on-shell δg ij = T ij and now regularize the lhs. Can relate to certain structures on the rhs (for the field theory in the boundary). E.g. anomalies show up. Many ways to do it. All assume 4-metric is AdS : ds 2 = 1 r 2 (dr 2 + g ij (x, r)dx i dx j ) Solve Einstein field eqs. near boundary with a FG expansion: g ij (x, r) = g (0)ij + rg (1)ij + r 2 g (2)ij +... Only powers of r 2 appear ρ = r 2. To regularize restrict radial integration ρ = ɛ. Read off from classical expression: e.g. conformal anomalies related to logρ terms in expansion.

5 Outline 1 Holographic dulaities 2

6 . 4/ 11 What is? What it is A theory of gravity with the following proeminent features: Possesses the same canonical variables as Hamiltonian GR: (g, π). Does not possess refoliation invariance (boosts). Trades that symmetry for foliation preserving conformal transformations (Weyl) + unique, non-local global Hamiltonian. Euclidean conformal field theory of the metric Relevance for Ads/CFT Natural setting for studying such dualities: conformal theory of the d 1 dimensional metric variables (bulk-bulk duality)

7 . 5/ 11 What allows for? Pure first class constraint systems such as GR may have observables coinciding with that of systems with different symmetries. Refoliation invariance has famous gauge fixing exploring spatial conformal transformations ([York]).

8 . 6/ 11 Schematic construction of

9 . 7/ 11 : Main message ADM (Σ R) Local 1st class constraints: 3-diffeomorphisms refoliations H ADM = d 3 x(n(x)s(x) + ξ a (x)h a (x)) Local 1st class constraints 3-diffeomorphisms Conformal transformations H dual = H gl + d 3 x[λ(x)d(x) + ξ a (x)h a (x)] H a (x): momentum constraint (one per x). S(x): Scalar constraint (one per x). D(x) = 4(π π g)(x): conformal constraint (one per x). H gl : Global Hamiltonian.

10 . 8/ 11 Large Volume expansion Ansatz: H SD = n=0 ( V V o ) 2n 3 h n and ˆΩ = n=0 ( V V o ) 2n 3 ω n large CMC-volume Hamiltonian H SD = (2Λ 1 6 π 2 ) R o ( V V o ) 2 3 Ro [g] is in Yamabe gauge, i.e. Ro is homogeneous + σa b σb a g ( V V o ) Observations 1 asymptotic freezing of shape deg. of freedom 2 equations of motion give (A)dS/CFT-type solution 3 Applicability: generic large CMC volume regime

11 . 9/ 11 Preliminary results Calculate equations of motion for metric variables using first orders of physical Hamiltonian. Preliminary results Generalization of classical part of holographic duality. Natural explanation of why RG time is identifiable with classical evolution.

12 . 10/ 11 Summary 1 is locally equivalent to General Relativity 2 But automatically incorporates Weyl invariance and preferred foliation. 3 There is a bulk/bulk correspondence: is AdS/CFT baggage necessary? 4 Asymptotic large CMC volume expansion: bulk/bulk duality turns into usual bulk/boundary duality 5 Only condition is large CMC volume: larger than asymptotic AdS 6 We recover the usual results if that further condition is imposed. But now new terms may appear. Under study.

13 . 11/ 11 THANK YOU