Sources and V evs for Lifshitz Holography Jelle Hartong Niels - PowerPoint PPT Presentation

Sources and V evs for Lifshitz Holography Jelle Hartong Niels Bohr Institute Gauge/Gravity Duality 2013 Munich Tuesday, July 30 In collaboration with Morten Christensen, Niels Obers, Blaise Rollier, to appear

Outline of the Talk • Motivation • The Model and Lifshitz Solutions • AlLif Space - Times and Beyond • Frame Fields and Boundary Geometry • W ard Identities • Counting Sources and V evs • Summary

Motivation • Lifshitz holography: new type of holographic duality with non - AdS asymptotics and Lorentz violating dual [ t ] = 2 , [ ~ x ] = 1 field theories ( ) • Interesting for: i ) . black holes as they form near - horizon geometries of extremal black branes, ii ) . AdS/CMT, iii ) . extending holography beyond AdS/CFT. • Boundary stress - energy tensor for Lifshitz holography • Boundary geometry • Irrelevant deformations of AlLif space - times • Identify the sources and vevs and holographic reconstruction

The Model • T S 5 runcation of type IIB on ✓ ◆ R + 12 − 1 φ ) 2 − 1 Z 2 e 2 ˆ ˆ 2( ∂ ˆ p φ ( ∂ ˆ d 5 x χ ) 2 S = − ˆ g • Scherk - Schwarz reduce to 4D using standard KK ansatz χ = ku + χ ˆ for metric and dilaton and ✓ ◆ R − 3 2( ∂ Φ ) 2 − 1 4 e 3 Φ F 2 − 1 2( ∂φ ) 2 − 1 Z d 4 x √− g 2 e 2 φ ( D χ ) 2 − V S = V = k 2 2 e − 3 Φ +2 φ − 12 e − Φ D χ = d χ − kA F = dA • [ Donos, Gauntlett, 2010 ] , [ Cassani, Faedo, 2011 ] , [ Chemissany, J.H., 2011 ] • Counterterms and reduction: [ Papadimitriou, 2011 ] , [ Chemissany, Geissbuehler, J.H., Rollier, 2012 ]

Lifshitz Solutions • A z=2 Lifshitz background: ✓ dr 2 r 2 − e − 2 Φ (0) l 2 dt 2 ◆ r 4 + 1 e Φ = e Φ (0) = lkg s ds 2 = l 2 e Φ (0) r 2 ( dx 2 + dy 2 ) 2 ✓ dr 2 + l 2 k 2 g 2 ◆ r 2 + 1 s 2 = l 2 r 2 (2 dudt + dx 2 + dy 2 ) • In 5D: du 2 d ˆ s 4 Z 2 π L g uu = 2 π Llkg s • Size of the circle: p 2 π L phys = du ˆ , u ∼ u + 2 π L 2 0 l L phys • V alidity of the approximation: � 1 , g s ⌧ 1 , � 1 l s l s L phys = Lkg s • � 1 (decompactification: N = 4 SYM) , 2 l ⇠ 1 (DLCQ with nonzero theta angle) , ⌧ 1 (below KK mass scale: 3D LCS) • [ Costa, Taylor, 2010 ] , [ McGreevy, Balasubramanian, 2011 ]

AlLif Solutions and Beyond • Asymptotically locally Lifshitz ( AlLif ) spaces [ Ross, 2011 ] : ds 2 = e Φ dr 2 a = r − 2 τ (0) a + . . . , where τ (0) a is HSO e t r 2 − e t e t + δ ij e i e j a = r − 1 e i e i (0) a + . . . Φ (0) = log kg s • and are asymptotically constant with . φ Φ 2 A a − e − 3 Φ / 2 e t a = A (0) a + . . . • Boundary gauge field and axion: χ = χ (0) + . . . • For HSO there exists an ADM slicing with [ t ] = 2 , [ ~ x ] = 1 τ (0) a • Irrelevant deformations: and is not HSO. φ (0) 6 = cst τ (0) a r − 4 log r Leading log deformations, i.e. in radial gauge. + k 2 ⌘ 2 e 2 Φ (0) = − 1 ⇣ • Cannot lift the constraint: 4 e 3 Φ (0) 4 e 2 φ (0) ✏ abc (0) ⌧ (0) a @ b ⌧ (0) c

Boundary Geometry I e a • Define and such that: τ a (0) (0) i τ (0) a τ a τ (0) a e a (0) = − 1 (0) i = 0 e i e i (0) j = δ i (0) a τ a (0) a e a (0) = 0 j • There is no Lorentzian boundary metric! Cannot raise/ lower indices. • Local bulk Lorentz transformations act on the boundary frame fields as Galilean boosts and rotations: (0) ω i δτ (0) a = 0 δτ a (0) = ξ b (0) b e a (0) i ⇣ j ⌘ δ e i ω i (0) a = ξ b i δ e a (0) i = − ξ b j i e a (0) b τ (0) a + ω (0) b (0) ω (0) b j e (0) (0) a (0) j • Spin connection coe ffi cients determined by the vielbein postulate: j D (0) a e i (0) b = ∂ a e i (0) ab e i (0) c + ω i (0) b − Γ c i (0) a τ (0) b + ω (0) a (0) b = 0 j e

Boundary Geometry II Γ c • The vielbein postulate constraints such that: (0) ab ⇣ ⌘ ⇣ ⌘ j Π (0) ab = δ ij e i δ d b + τ (0) b τ d δ e c + τ (0) c τ e r (0) a Π (0) de = 0 (0) a e (0) (0) (0) b Γ c Γ c ˆ (0) ab = Γ c • W e define by for AlLif space - times but (0) ab (0) ab we keep this also for the Lifshitz UV completion so that: (0) ab = − 1 (0) ( ∂ a τ (0) b + ∂ b τ (0) a ) + 1 Γ c 2 τ c 2 Π cd � � ∂ a Π (0) bd + ∂ b Π (0) ad − ∂ d Π (0) ab (0) − 1 2 e 3 Φ (0) / 2 Π cd Π ab (0) = δ ij e a (0) i e b � � F (0) da τ (0) b + F (0) db τ (0) a (0) (0) j F (0) ab = ∂ a A (0) b − ∂ b A (0) a • When the boundary geometry is ∂ a τ (0) b − ∂ b τ (0) a = 0 r (0) a τ (0) b = r (0) a Π bc Newton - Cartan for which we have . (0) = 0 • For applications of Newton - Cartan to CMT see e.g. [ Son, 2013 ] .

W ard Identities I • On - shell variation of the action: Z ⇣ � S t (0) + S i (0) i + T t (0) δ A (0) t + T i d 3 xe (0) (0) a δτ a (0) a δ e a δ S ⇠ (0) δ A (0) i ◆ δ r + h O Φ i δ Φ (0) + h O φ i δφ (0) + h O χ i δχ (0) � A (0) r • Local symmetries: • di ff eomorphisms δτ a (0) = 2 w (0) τ a (0) , δ e a (0) i = w (0) e a • anisotropic W eyl: (0) i δ r = w (0) r δ A (0) t = 2 w (0) A (0) t , δ A (0) i = w (0) A (0) i • gauge: δ A (0) a = ∂ a Λ (0) , δχ = k Λ (0) • local tangent space δτ a (0) = − ω (0) τ a (0) , δ e a (0) i = ω (0) e a (0) i • Only at leading order: δ A (0) t = − ω (0) A (0) t , δ A (0) i = ω (0) A (0) i δ Φ (0) = − 2 ω (0)

W ard Identities II • The local tangent space rotation and gauge invariant boundary stress - energy tensor: ✓ ◆ ✓ ◆ 1 1 S t (0) a + T t S i (0) a + T i T b (0) a = − τ b + e b k ∂ a χ (0) k ∂ a χ (0) (0) (0) i (0) (0) • W ard identities for assuming T b ∂ a τ (0) b − ∂ b τ (0) a = 0 (0) a ⇣ ⌘ (0) + e i r (0) b T b (0) a = � T c � τ (0) c r (0) a τ b (0) c r (0) a e b (0) b (0) i � T t (0) ∂ a B (0) t � T i (0) ∂ a B (0) i � h O Φ i ∂ a Φ (0) � h O φ i ∂ a φ (0) (0) a + e i (0) a + 2 T t (0) B (0) t + T i A (0) = − 2 τ (0) b τ a (0) T b (0) b e a (0) i T b (0) B (0) i B (0) a = A (0) a − 1 k ∂ a χ (0) • Similar to HIM boundary stress tensor [ Hollands, Ishibashi, Marolf, 2005 ] L K (0) A τ a (0) = L K (0) A Π ab (0) = L K (0) A bdry scalar = 0 ⇣ ⌘ • Conserved currents [ Ross, Saremi, 2009 ] K a (0) A T b r (0) b = 0 (0) a

Counting Sources & V evs • The anomaly is given by a 3D z=2 Horava - Lifshitz action coupled to matter fields. • #sources=15 ( components )- 8 ( local symmetries )- 1 ( c0nstraint ) =6 • #vevs=15 ( components )- 9 ( W ard identities including the one for the leading order symmetry ) =6 • In the Fe ff erman - Graham expansion there appears another function which does not appear in the W ard identities. • W e thus have 6+6+1 free functions controlling the expansion.

Summary • There exists a Lifshitz model that is a well - defined low energy limit of IIB string theory. • Found all irrelevant deformations of AlLif space - times. • Frame fields useful for boundary geometry ( Newton - Cartan ) and coordinate independent definition of boundary stress - energy tensor. • There are 6+6 sources & vevs and one additional free function that control the Fe ff erman - Graham expansion.