General Formalism
7.1 A General Algorithm for Non-AdS Holography
General Formalism
„
Viele Steine, (Many stones,) müde Beine, (tired legs,)Aussicht keine, (scenic view; none.) Heinrich Heine.
– Heinrich Heine Gipfelbuch des Brockens
N
on-AdS holography in2 + 1dimensions is in many regards very similar to AdS holography. There are, however, certain subtleties which one has to be aware of when leaving the comfort zone of the well studied AdS examples.The main purpose of this chapter is to describe a general algorithm for non-AdS higher-spin holography, point out the subtle differences to AdS higher-spin hologra-phy and to motivate why it is interesting to study holograhologra-phy in2 + 1dimensions for spacetimes which are not AdS.
Many applications of the holographic principle require a generalization of the famous AdS/CFT correspondence to a more general gauge/gravity correspondence. Some examples are: null warped AdS, and their generalization Schrödinger spacetimes that feature an arbitrary scaling exponent and are used as holographic duals of non-relativistic CFTs that describe cold atoms [38–40], Lifshitz spacetimes, which are the gravity duals of Lifshitz-like fixed points [42], and the AdS/log CFT correspondence [130,131].
This chapter is organized as follows: I will first present a general algorithm which can be used to determine the (quantum) asymptotic symmetries of non-AdS spacetimes.
I will then use an explicit example, namely spin-3 Lobachevsky holography, in order to demonstrate how this algorithm works in detail.
7.1 A General Algorithm for Non-AdS Holography
The general algorithm for non-AdS holography which I will review in this section originally goes back to the work of Brown and Henneaux [11] and has been first presented in [I,37]. This algorithm is a generalization of the procedure outlined in Chapter3and can roughly be summarized by the following steps, which are also summarized in form of a flowchart shown in Figure7.1.
Identify Bulk Theory and Variational Principle: The first step in this algorithm con-sists of identifying the bulk theory1 one wants to describe and then proposing a suitable generalized variational principle which is consistent with the theory under consideration. For non-AdS higher-spin theories such a generalized variational prin-ciple has been first described in [36], which I will review briefly.
Varying the bulk Chern-Simon action (2.3) one obtains on the one hand the equa-tions of motion and on the other hand boundary condiequa-tions on the connection2A
Z
∂M
hA∧δAi= 0, (7.1)
for some manifoldM= Σ×R, which I assume to have the topology of a cylinder and which can be described in terms of a radial coordinateρand boundary coordinates ϕ, t as shown in Figure 3.1. Choosing light-cone coordinates x± = `t ±ϕ this expression can be written as
Z
∂M
hA+∧δA−−A−∧δA+i= 0. (7.2)
These boundary conditions can be satisfied for example by setting one part of the gauge fields, e.g.A−= 0, at the boundary. This is, however, a rather strict boundary condition which would not allow for general non-AdS backgrounds. Thus, in order to be able to describe non-AdS backgrounds one has to relax these boundary conditions most of the time3. This can be achieved by adding a boundary term B[A]to the Chern-Simons action. This boundary term can be written as
B[A] = k 4π
Z
∂M
hA+A−i. (7.3)
Adding this boundary term changes the boundary conditions (7.2) to Z
∂M
hA+∧δA−i= 0. (7.4)
One can now also choose to setδA−|∂M = 0instead ofA−= 0and similarly forA.¯ Choosing eitherδA−|∂M = 0or A−|∂M = 0depends on the specific theory one is looking at but for most of the non-AdS backgrounds it turns out that one has to use the more relaxed boundary conditionδA−|∂M= 0.
1This usually boils down to choosing an appropriate embedding ofsl(2,R),→sl(N,R)and then fix the Chern-Simons connectionsAandA¯in such a way that they correctly reproduce the desired gravitational background.
2Of course one also obtains similar conditions forA, which I will not explicitly write down in¯ this chapter as all formulas forA¯can be obtained from the formulas provided forAby simply exchangingA↔A.¯
3One example where one would not have to relax these boundary conditions is Einstein gravity with vanishing cosmological constant described by anisl(2,R)valued connection.
Impose Suitable Boundary Conditions: After having chosen the bulk theory and set up a consistent variational principle the next step in the holographic analysis is choosing appropriate boundary conditions for the Chern-Simons connectionsAand A. This is the most crucial step in the whole analysis as the boundary conditions¯ essentially determine the physical content of the dual field theory at the boundary.
Gauging away the radial dependence of the gauge fields as in (3.39)
Aµ=b−1aµ+a(0)µ +a(1)µ b, A¯µ=b−1¯aµ+ ¯a(0)µ + ¯a(1)µ b, b=eρL0, (7.5) one can then identify the following three contributions to the Chern-Simons connec-tions.
aµand ¯aµdenote the (fixed) background which was chosen in the previous step.
a(0)µ anda¯(0)µ correspond to state dependent leading contributions in addition to the background that contain all the physical information about the field degrees of freedom at the boundary.
a(1)µ and¯a(1)µ are subleading contributions.
Choosing suitable boundary conditions in this context thus means choosing a(0)µ
(¯a(0)µ ) anda(1)µ (¯a(1)µ ) in such a way that there exist gauge transformations which preserve these boundary conditions i.e.
δεAµ=Ob−1a(0)µ b+Ob−1a(1)µ b, (7.6) for some gauge parameterεwhich can also be written as
ε=b−1(0)+(1)b, (7.7)
and similarly for the barred quantities. The transformations(0)usually belong to the asymptotic symmetry algebra while(1) are trivial gauge transformations.
Perform Canonical Analysis and Check Consistency of Boundary Conditions: Once the boundary conditions and the gauge transformations which preserve these bound-ary conditions have been fixed one has to determine the canonical boundbound-ary charges.
This is a standard procedure which is described in great detail for example in [79,83]
and in a bit less detail also in e.g. [37]. This procedure eventually leads to the variation of the canonical boundary charge4
δQ[ε] = k 2π
Z
∂Σ
D(0)δa(0)ϕ dϕE, (7.8)
4See also e.g. (3.26).
where ϕ parametrizes the cycle of the boundary cylinder. A similar expression holds again for the barred quantities. Of course one also has to check whether or not the boundary conditions chosen at the beginning of the algorithm are actually physically admissible. That is, the variation of the canonical boundary charge is finite, conserved in time and integrable in field space. If all these conditions are met then one can proceed with determining the (semiclassical) asymptotic symmetry algebra. Otherwise one has to start over again and choose a new set of boundary conditions and repeat the algorithm up to this point until a finite, conserved and integrable canonical boundary charge is obtained.
Determine Semiclassical Asymptotic Symmetry Algebra: This step consists in work-ing out the Dirac brackets between the canonical generatorsG which directly yields the semiclassical asymptotic symmetry algebra. There is a well known trick which can be used to simplify calculations at this point. Let us assume that one has two charges with Dirac bracket{G[ε1],G[ε2]}. Then one can exploit the fact that these brackets generate a gauge transformation as{G[ε1],G[ε2]}=δ2G, and read of the Dirac brackets by evaluating δ2G. This relation for the canonical gauge genera-tors is on-shell equivalent to a corresponding relation only involving the canonical boundary charges
{Q[ε1],Q[ε2]}=δ2Q, (7.9) which in most cases is comparatively easy and straightforward to calculate. This directly leads to the semiclassical asymptotic symmetry algebra including all possible semiclassical central extensions.
Determine the Quantum Asymptotic Symmetry Algebra: This part of the algorithm first appeared in [28]. One insight of this paper was that the asymptotic symmetry algebra derived in the previous steps is only valid for large values of the central charges. When taking into account small values of the central charge and in addition introduce normal ordering it can happen that the asymptotic symmetry algebra violates the Jacobi identities. The simplest way to fix this is to assume suitable deformations of the asymptotic symmetry algebra for finite values of the central charge and normal ordered expressions and then determine the exact form of the deformations by demanding compatibility with the Jacobi identities, see e.g. [132].
In practice the most efficient approach is to allow all possible deformations of the algebra that are consistent with the specific operator content and then check the Jacobi identities. If a deformation is not allowed, then the Jacobi identities will automatically require this deformation to be absent yielding as a final result the correct quantum asymptotic symmetry algebra.
Look for Unitary Representations of the Quantum Asymptotic Symmetry Algebra:
This point works in principle in exact the same way as in a conformal field theory.
First one defines a suitable highest-weight state and which modes of the operator algebra annihilate this highest-weight state and which modes generate new states.
Then one requires that the resulting module does not contain any states with negative norm. This usually restricts the central charges appearing in the quantum asymptotic symmetry algebra and thus also in turn the Chern-Simons levelkto certain fixed values. These values can be continuous or discrete, infinitely, finitely many, or even none at all depending on the specific theory in question.
Identify the Dual Field Theory: With the results from all the previous steps one can then finally proceed in trying to identify or put possible restrictions on a quantum field theory which realizes all these quantum asymptotic symmetries in a unitary way. Once this dual field theory is identified or conjectured as a dual theory one can perform further checks of the holographic conjecture like calculating partition functions or determining correlation functions on the gravity side.