Topologically massive gravity and the AdS/CFT correspondence Balt van Rees 8 September 2009 Based on work with K. Skenderis and M. Taylor: arXiv:0906.4926
Topologically massive gravity Three-dimensional pure Einstein gravity is locally trivial This changes when we add a gravitational Chern-Simons term to the action: √ 1 “ Z − G ( R − 2Λ) + 1 Z d 3 x (Γ d Γ + 2 ” d 3 x S = 3Γ ∧ Γ ∧ Γ) 16 πG N 2 µ This gives a third-order equation of motion: R µν − 1 2 RG µν + Λ G µν + 1 2 µ ( ǫ αβ ∇ α R βν + µ ↔ ν ) = 0 µ which does allow for local degrees of freedom in a three-dimensional theory of gravity Problems with stability. For Λ < 0 : • perturbative solutions around AdS 3 have negative energy (in our conventions) • BTZ black hole has positive energy Deser, Jackiw, Templeton (1982)
Quantum topologically massive gravity Topologically massive gravity recently received more attention Li, Song, Strominger (2008) Inspired by renewed interest in three-dimensional Einstein gravity Search for a possible dual CFT ( Λ < 0 ) Witten (2007) What would be the dual CFT for topologically massive gravity with Λ < 0 ? Is it consistent, unitary? Can we learn anything about higher-dimensional theories? + Dynamics might give a more realistic theory − Problems with positivity of energy
Some properties of TMG Action for Λ = − 1 : √ 1 Z d 3 x S = − G ( R + 2) 16 πG N √ 1 Z ρν + 2 − Gǫ λµν “ ” Γ ρ 3Γ ρ d 3 x λσ ∂ µ Γ σ λσ Γ σ µτ Γ τ + νρ 32 πG N µ Equations of motion: R µν − 1 2 RG µν − G µν + 1 µC µν = 0 C µν = 1 2 ǫ αβ ∇ α R βν + µ ↔ ν µ Properties of the Cotton tensor: C µ ∇ µ C µν = 0 µ = 0 If G µν is Einstein, so R µν = − 2 G µν , then C µν = 0 and G µν is also a solution of TMG All solutions G µν of TMG have R = − 6
Perturbative spectrum We investigate the spectrum around an AdS 3 background: dr 2 G µν dx µ dx ν = − ( r 2 + 1) dt 2 + r 2 + 1 + r 2 dφ 2 Consider a small variation of the metric: G µν → G µν + H µν The equation of motion gives a third-order linear differential equation for H µν The solutions can be classified by the symmetry algebra ∼ SL (2 , R ) × SL (2 , R ) with generators L 0 , L − 1 , L 1 and ¯ L 0 , ¯ L − 1 , ¯ L 1 We search for primary perturbations that are: • annihilated by L 1 and ¯ L 1 • eigenfunctions of L 0 and ¯ L 0 : L 0 H µν = ¯ ¯ L 0 H µν = hH µν hH µν where L 0 = i 2 ( ∂ t + ∂ φ ) and ¯ L 0 = i 2 ( ∂ t − ∂ φ )
Perturbative spectrum For generic µ , there exist three primary solutions H L µν , H R µν , H M µν with: L 0 H L = 2 H L L 0 H L = 0 ¯ L 0 H R = 0 L 0 H R = 2 H R ¯ L 0 H M = 1 L 0 H M = 1 ¯ 2 ( µ + 3) H M 2( µ − 1) H M For µ = 1 the modes H L and H M coincide Li, Song, Strominger (2008) However, for µ = 1 a new mode ˜ H M µν arises for which: H M = 2 ˜ H M + H L H M = H L L 0 ˜ L 0 ˜ ¯ H M the logarithmic partner of H L Hints at a logarithmic CFT with ˜ This mode has different falloff conditions ( log( r ) /r 2 vs. 1 /r 2 ) Grumiller, Johansson (2008) Questions: • Is TMG at µ = 1 dual to a logarithmic CFT? If so, what is the precise AdS/CFT dictionary for TMG? • Can we allow the different falloff conditions?
Setting up an AdS/CFT dictionary Aim: compute CFT correlators from a bulk theory with action S using Z CFT ∼ exp( − S on-shell ) GKP , Witten (1998) Procedure: • Write down equations of motion from S • Perform an asymptotic analysis near the conformal boundary of spacetime • Fix the leading behaviour of the fields (asymptotically AdS, sources φ (0) ) • Solve the equations of motion asymptotically • We find an asymptotic expansion of every possible bulk solution • In particular, the possible subleading behaviour of the fields is determined dynamically • This asymptotic solution can be substituted into S and leads to divergences • Holographically renormalize by adding a boundary counterterm action S ct to S • The renormalized action S ren = S + S ct is finite on-shell • Find the full solution to the equations of motion with sources φ (0) (perhaps perturbatively) • Substitute this solution into S ren which gives S on-shell,ren [ φ (0) ] • Use Z CFT [ φ (0) ] ∼ exp( − S on-shell,ren [ φ (0) ]) to compute correlation functions Skenderis (2002)
Asymptotic analysis We will now work in Fefferman-Graham coordinates. The metric takes the form: G µν dx µ dx ν = dρ 2 4 ρ 2 + 1 ρg ij ( x, ρ ) dx i dx j For an asymptotically AdS spacetime, the conformal boundary is at ρ = 0 and: g ij ( x, ρ ) = g (0) ij ( x ) + . . . where g (0) ij is nondegenerate For TMG, the equations of motion for µ = 1 give the most general asymptotic solution: g ij = b (0) ij log( ρ ) + g (0) ij + b (2) ij ρ log( ρ ) + ρg (2) ij + . . . Following the usual AdS/CFT dictionary, we interpret the leading terms as CFT sources g (0) ij ↔ T ij b (0) ij ↔ t ij The subleading terms b (2) ij and g (2) ij are partially determined by the asymptotic analysis and these terms enter in the one-point functions
Holographic renormalization We substitute the asymptotic expansion in the action for TMG and find divergences (e.g. a volume divergence) We need to holographically renormalize by adding a boundary counterterm action S ct However, the most general asymptotic solution is: g ij = b (0) ij log( ρ ) + g (0) ij + b (2) ij ρ log( ρ ) + ρg (2) ij + . . . For nonzero b (0) ij , this is no longer asymptotically AdS • we cannot do an all-orders renormalization • we treat b (0) ij as infinitesimal and renormalize perturbatively • in the dual theory b (0) ij sources a (marginally) irrelevant operator and the boundary theory with finite b (0) ij is only no longer completely renormalizable We did a linearized analysis at the level of the equation of motion → This is equivalent to a quadratic analysis at the level of the action so we computed S ren to second order in b (0) ij → This is sufficient to compute two-point functions
Full linearized solutions We begin with an AdS 3 background ds 2 = dρ 2 4 ρ 2 + 1 g ij dx i dx j = dzd ¯ ρg ij dx i dx j z and study perturbations: g ij → g ij + h ij At the linearized level we find: − 1 2 ρ log( ρ ) ∂ 2 b (0)¯ h z ¯ z = h (0) z ¯ + ρh (2) z ¯ z [ h (0) , b (0) ] + . . . z z ¯ z − 1 2 ρ log( ρ )¯ h ¯ z = b (0)¯ z log( ρ ) + h (0)¯ ∂∂b (0)¯ + ρh (2)¯ z + . . . z ¯ z ¯ z ¯ z z ¯ z z ¯ + 1 h zz = h (0) zz 2 ρ log( ρ ) b (2)¯ + ρh (2) zz + . . . z ¯ z z [ h (0) , b (0) ] = − 1 z − 1 2 ¯ ∂ 2 h (0) zz + ¯ z − 1 2 ∂ 2 h (0)¯ 2 ∂ 2 b (0)¯ with h (2) z ¯ ∂∂h (0) z ¯ z . z ¯ z ¯ We search for regular solutions as ρ → ∞ which constrains the subleading terms to be: ¯ ∂ z + 4 γ − 3 ¯ h (2)¯ z = ∂ h (2) z ¯ ∂∂b (0)¯ z ¯ z ¯ z 2 ∂ 3 z = 1 b (2)¯ ∂ b (0)¯ z ¯ ¯ z ¯ z 2 ” ∂ 3 z + ∂ “ 2 γ − 1 + log( − ∂ ¯ h (2) zz = ∂ ) ∂ b (0)¯ ∂ h (2) z ¯ z ¯ z ¯ ¯
Correlation functions After holographic renormalization we find the one-point functions from: � T ij � = 4 π δS TMG, on-shell, ren � t zz � = − 4 π δS TMG, on-shell, ren δh ij δb zz (0) (0) We for example find: “ ∂ 3 1 1 ” � T zz � = − b (2) zz + local = − ∂ b (0)¯ z + local z ¯ ¯ 2 G N 4 G N which is a linear and nonlocal function of the sources Differentiating once more with respect to the sources we obtain the two-point functions: log( m 2 | z | 2 ) 3 = − 3 /G N � t ( z, ¯ z ) t (0) � = � t ( z, ¯ z ) T (0) � G N z 4 2 z 4 ˙ ¯ = 3 /G N z ) ¯ ¸ � T ( z, ¯ z ) T (0) � = 0 T ( z, ¯ T (0) z 4 2¯ where t = t zz , T = T zz and ¯ T = T ¯ z ¯ z We read off that: c L = 0 c R = 3 /G N and we find logarithmic correlation functions
Logarithmic CFT We indeed find the structure of a logarithmic CFT (Gurarie 1993) for topologically massive gravity at µ = 1 Such CFT’s have logarithms in correlation functions which are related to an indecomposible representation of the Virasoro algebra „ φ « „ h 0 « „ φ « „ φ « L 0 = L m = 0 ( m > 0) χ 1 h χ χ One then finds logarithms in correlation functions: � χ ( z ) χ ( w ) � = log | z | 2 1 � φ ( z ) φ ( w ) � = 0 � φ ( z ) χ ( w ) � = z 2 h z 2 h A logarithmic CFT is not unitary. Maybe a restriction to the right-moving sector is consistent and results in a unitary theory? Maloney, Song, Strominger (2009)
Logarithmic CFT It is instructive to compute the same correlation functions in the vicinity of µ = 1 There are still four sources, three for T ij and a fourth for a new operator X The correlation functions become: ” 1 ˙ ¯ 3 1 + 1 “ z ) ¯ ¸ T ( z, ¯ T (0) = z 4 , 2 G N µ 2¯ ” 1 3 1 − 1 “ � T ( z, ¯ z ) T (0) � = 2 z 4 , 2 G N µ − 1 ( µ − 1)( µ + 1)( µ + 2) 1 � X ( z, ¯ z ) X (0) � = z µ +3 ¯ z µ − 1 8 G N µ One finds negative norm states for µ > 1 and negative conformal weights for µ < 1 As µ → 1 we find that a new operator appears: − 2 t = lim µ − 1( T + X ) µ → 1 which is the logarithmic partner of T . This mimicks a construction in the LCFT literature (Kogan, Nichols 2002)
Recommend
More recommend
