Gravity Duals of Nontrivial IR Behaviour in Field Theories

Arpan Saha09D26005

Fourth Year Undergraduate,Department of Physics

April 6, 2013

AdS/CFT (Maldacena 1998)

• Holographic Principle: a theory with gravity is formally equivalent to a theory without gravity in one dimension less (’t Hooft 1993, Susskind 1995).

• The entropy of a black hole is proportional to the area of its event horizon (Bekenstein 1973, Hawking and Gibbons 1977).

• Number of observables in a quantum theory of gravity is less than a QFT in the same number of dimensions because there are no local operators (Weinberg and Witten 1980).

AdS/CFT Correspondence

• Why AdS? Because we want local behaviour in the bulk to give rise to only local behaviour in the boundary.

• Why CFT? Because the group of isometries of AdS is isomorphic to the group of conformal transformations on its boundary.

• Only asymptotic AdS behaviour is required (though even this may be relaxed to some extent in certain situations, see Kachru, Liu, Mulligan 2008).

AdS/CFT and RG Flow

• Going inwards into an asymptotically AdS spacetime from the boundary corresponds to an RG flow in the boundary theory (Balasubramaniam and Kraus 1999)

• Introducing an inner ‘boundary’ and integrating out the fields in the excluded region in the gravity path integral is analogous to introducing a momentum cut-off and integrating out the higher momentum modes.

The IKKNST Program

• Aim: to obtain the gravity dual of a system that is homogeneous and isotropic in the UV limit but homogeneous and non-isotropic in the IR limit (Kachru et al 2012).

• We work in the Poincaré patch coordinates, which covers only half of the AdS space.

• Boundary (r ) represents the UV limit while ‘horizon’ (r – ) represents the IR limit.

Bianchi Classification (Bianchi 1898)

• A maximally symmetric space of D dimensions has D(D + 1)/2 Killing vectors.

• D of them correspond to ‘translational’ symmetries (homogeneity) while D(D – 1)/2 of them correspond to ‘rotational’ symmetries (isotropicity).

• Studying homogeneous and not necessarily isotropic spaces in 3D thus amounts to studying all possible Lie algebras with 3 generators.

Bianchi Types

• Bianchi classified all such Lie algebras into 9 ‘Types’ with Type I being the isotropic case in which all the generators commute.

• For every Type, it is possible to construct 3 invariant 1-forms ωi whose Lie derivatives along the generators of the Type vanish (Ryan and Shepley 1975).

• Any metric built out of the invariant 1-forms, dt and functions of r has the symmetries of the relevant Bianchi Type.

The IKKNST Manifesto

• Metric should reduce to the following as r ds2 = dr2 + e(Λ/3) r(– dt2 + dx2 + dy2 + dz2)

• Metric should reduce to the following as r – ds2 = dr2

– e2 βt rdt2 + e2 β1 r(ω1)2 + e2 β2 r(ω2)2 + e2 β3 r(ω3)2

• Metric should arise from an action (can involve gauge/Proca fields, dilatons etc.).

• Unfortunately, such a metric is very, very hard to construct.

The Weak Energy Condition

• WEC (I): The energy as observed by a timelike observer with velocity u must be nonnegative. In other words, u, Tu 0, where T is the stress tensor.

• WEC (II): Given that Einstein tensor is diagonalisable, the above condition is equivalent to the following – the eigenvalue to which the timelike eigenvector of T belongs must be negative and the least amongst all the eigenvalues.

WEC (II) in AdS

• Action is taken to be R + Λ, where R is the Ricci scalar.

• If G is the Einstein tensor then the eigenvalues of G differ from those of T by Λ/2 (since T = G – (Λ/2)I, I being the identity operator).

• WEC (II) for AdS: the eigenvalue to which the timelike eigenvector of G belongs must be less than or equal to Λ/2 everywhere and must be the least amongst all the eigenvalues.

WEC (II) for interpolating solutions

• We write down a metric that interpolates between a Bianchi Type at the horizon and Lifshitz at the boundary, while incorporating a parameter σ that controls how sharply the metric transitions between the Type and Lifshitz.

• Such a metric is not obtained from a Lagrangian and is constructed by hand.

• We don’t know whether it can be obtained from a Lagrangian but as a first check, we investigate whether it satisfies WEC (II) numerically.

Interpolating metric

• Let dsB2 be the on-shell metric for a Bianchi

attractor and dsL2 be the on-shell metric for

Lifshitz spacetime.• We choose βt to be same for the Bianchi attractor

and Lifshitz spacetime.• The interpolating metric is then given by

ds2 = (1 + tanh σ r) dsL2/2 + (1 – tanh σ r) dsB

2/2

Permissible range for σ

• A upper bound is necessary to ensure the desired asymptotic behaviour in the IR and UV.

• A lower bound is necessary to ensure transition isn’t too violent leading to violation of WEC.

• For Types II, III and VI–1, we see that the allowed values of σ fall into a nonempty interval.

• We demonstrate Type II here, in which case0.5065 < σ < 1.05026.

Type II, σ = 1, Λ = 12

Type II, σ = 1, Λ = 12

Inference from the plots

• Since the timelike eigenvalue is the least of all eigenvalues and is negative, Type II is shown to satisfy WEC.

• Types III and VI–1 can similarly be shown to satisfy the WEC as well.

• This means that at the level of the energy conditions, there is no obstruction to flows from Bianchi Types II, III and VI–1 to Type I (i.e. Lifshitz).

Acknowledgements

Prof. P. Ramadevi (IITB)Prof. Sandip Trivedi (TIFR)Rickmoy Samanta (TIFR)Nilay Kundu (TIFR)

I am hugely indebted to the following people for their guidance and help:

Thank you!