An sl(2, R) current algebra from AdS_3 gravity
aa r X i v : . [ h e p - t h ] A p r Prepared for submission to JHEP An sl (2 , R ) current algebra from AdS gravity Steven G. Avery, Rohan R. Poojary and Nemani V. Suryanarayana
Institute of Mathematical SciencesCIT Campus, TaramaniChennai 600113, India
E-mail: savery,ronp,nemani(at)imsc.res.in
Abstract:
We provide a set of chiral boundary conditions for three-dimensional gravitythat allow for asymptotic symmetries identical to those of two-dimensional induced gravityin light-cone gauge considered by Polyakov. These are the most general boundary condi-tions consistent with the boundary terms introduced by Comp`ere, Song and Stromingerrecently. We show that the asymptotic symmetry algebra of our boundary conditionsis an sl (2 , R ) current algebra with level given by c/
6. The fully non-linear solution inFefferman–Graham coordinates is also provided along with its charges.
Introduction
In his seminal 1987 paper [1], Polyakov provides a solution to the two-dimensional inducedgravity theory [2], S = c π Z d x √− g R ∇ R, (1.1)by working in a light-cone gauge. The gauge choice puts the metric into the form ds = − dx + dx − + F ( x + , x − )( dx + ) . (1.2)Polyakov shows that the quantum theory for the dynamical field F ( x + , x − ) admits an sl (2 , R ) current algebra symmetry with level k = c/
6. In this note, we present the three-dimensional bulk theory that is dual to this two-dimensional theory.
AdS gravity The action of three-dimensional gravity with negative cosmological constant [3] is givenby S = − πG Z d x √− g (cid:18) R − l (cid:19) − πG Z ∂ M d x √− γ Θ + 18 πG S ct ( γ µν ) , (2.1)where γ µν is the induced metric and Θ is trace of the extrinsic curvature of the boundary.Varying the action yields δS = Z ∂ M d x √− γ T µν δγ µν , (2.2)where T µν = 18 πG (cid:20) Θ µν − Θ γ µν + 2 √− γ δS ct δγ µν (cid:21) . (2.3)The variational principle is made well-defined by imposing δγ µν = 0 (Dirichlet) or T µν = 0(Neumann) at the boundary (see [4] for a recent discussion).Recently Comp`ere, Song and Strominger (CSS) [5, 6] and Troessaert [7] proposed newsets of boundary conditions for three-dimensional gravity, which differ from the well-knownDirichlet-type Brown–Henneaux boundary conditions [8]. Before delving into specifics,let us discuss the general strategy employed by [6]. One begins by adding a term of thetype S ′ = − πG Z ∂ M d x √− γ T µν γ µν (2.4) In fact, the boundary conditions of [7] subsume those of [8]. – 2 –or a fixed ( γ µν -independent) symmetric boundary tensor T µν . The variation of this termis δS ′ = − πG Z ∂ M d x √− γ ˜ T µν δγ µν , (2.5)where ˜ T µν = T µν − ( T αβ γ αβ ) γ µν . The variation of the total action then gives δS + δS ′ = 18 πG Z ∂ M d x √− γ ( T µν − ˜ T µν ) δγ µν . (2.6)Now the boundary conditions consistent with the variational principle depend on ˜ T µν .Generically, this leads to “mixed” type boundary conditions. If for a given class of bound-ary conditions some particular component of T αβ − ˜ T αβ vanishes sufficiently fast in theboundary limit such that its contribution to the integrand in (2.6) vanishes, then the cor-responding component of γ αβ can be allowed to fluctuate. Since we want the boundarymetric to match (1.2), we would like Neumann boundary conditions for γ ++ . Thereforewe choose T µν such that the leading term of T ++ equals ˜ T ++ in the boundary limit.This condition has been imposed in [6], with the addition of an extra boundary term(2.4) with T µν = − r N lδ µ + δ ν + , (2.7)and the following boundary conditions are imposed on the metric: g rr = l r + O ( r − ) , g r ± = O ( r − ) ,g + − = − r O ( r ) , g ++ = r f ( x + ) + O ( r ) , g −− = − l N + O ( r − ) , (2.8)where f ( x + ) is a dynamical field and N is fixed constant. These boundary conditionsgive rise to an asymptotic symmetry algebra: a chiral U (1) current algebra with leveldetermined by N . These also ensure that T −− is held fixed in the variational problem,whereas g ++ is allowed to fluctuate as long as its boundary value is independent of x − .In what follows, we show that (2.8) are not the most general boundary conditionsconsistent with the variational principle and the extra boundary term given by (2.7).For this, we introduce a weaker set of consistent boundary conditions that enhance theasymptotic symmetry algebra to an sl (2 , R ) current algebra whose level is independent of N . In the new boundary conditions, the class of allowed boundary metrics coincides with thatof (1.2). Since we want to allow γ ++ to fluctuate, we keep T −− fixed in our asymptotically The induced metric γ µν differs from g (0) µν of [6] by a factor of r . To relate to the notation in [6], set N = − G ∆ l and f ( x + ) = l ∂ + ¯ P ( x + ). – 3 –ocally AdS metrics. Therefore, we propose the following boundary conditions: g rr = l r + O ( r ) , g r + = O ( r ) , g r − = O ( r ) ,g + − = − r O ( r ) , g −− = − l N O ( r ) ,g ++ = r F ( x + , x − ) + O ( r ) , (2.9)where, as above, we take F ( x + , x − ) to be a dynamical field and N fixed. The crucialdifference between these boundary conditions and those in (2.8) is the different fall-offcondition for g r + which allows for the boundary component of g ++ to depend on x − aswell. One must, of course, check the consistency of these conditions with the equationsof motion. This involves constructing the non-linear solution in an expansion in inversepowers of r . Working to the first non-trivial order, one finds the following condition on F ( x + , x − ): N ∂ − F ( x + , x − ) + ∂ − F ( x + , x − ) = 0 , (2.10)which forces F ( x + , x − ) to take the form F ( x + , x − ) = f ( x + ) + g ( x + ) e iNx − + ¯ g ( x + ) e − iNx − (2.11)where f ( x + ) is a real function and ¯ g ( x + ) is the complex conjugate of g ( x + ).Let us note that this is directly analogous to the form of F ( x + , x − ) derived in [1].Throughout our discussion we think of φ = x + − x − as 2 π -periodic (and τ = x + + x − as thetime coordinate), and therefore we restrict our consideration to N ∈ Z . Similarly, weimpose periodic boundary conditions on f ( x + ) and g ( x + ). If one takes the spatial partof the boundary to be R instead of S , there are no such restrictions and one may evenconsider N < One can write a general non-linear solution of
AdS gravity in Fefferman–Graham coordi-nates [9] as: ds = dr r + r (cid:20) g (0) ab + l r g (2) ab + l r g (4) ab (cid:21) dx a dx b . (2.12)The full non-linear solution with our boundary conditions is obtained when g (0)++ = f ( x + ) + g ( x + ) e iNx − + ¯ g ( x + ) e − iNx − , g (0)+ − = − , g (0) −− = 0 ,g (2)++ = κ ( x + ) + 12 N h g ( x + ) e iNx − + ¯ g ( x + ) e − iNx − i + i N h g ′ ( x + ) e iNx − − ¯ g ′ ( x + ) e − iNx − i ,g (2)+ − = 14 N h f ( x + ) − g ( x + ) e iNx − − ¯ g ( x + ) e − iNx − i , g (2) −− = − N ,g (4) ab = 14 g (2) ac g cd (0) g (2) db , (2.13)– 4 –here in the last line g cd (0) is g (0) cd inverse. As above, demanding that the solution respects theperiodicity of φ -direction requires N to be an integer and the functions f ( x + ), g ( x + ) and κ ( x + ) to be periodic. This solution reduces to the one given in [6] when g ( x + ) = ¯ g ( x + ) = 0.As mentioned in the previous subsection one can take N to be purely imaginary whenthe boundary spatial coordinate is not periodic. In this case too the non-linear solution(2.12) continues to be a valid solution with g ( x + ) and ¯ g ( x + ) treated as two real andindependent functions. However, we will not consider this case further here. It is easy to see that vectors of the form ξ r = − h B ′ ( x + ) + iN A ( x + ) e iNx − − iN ¯ A ( x + ) e iNx − i r + O ( r ) ξ + = B ( x + ) − l N r h A ( x + ) e iNx − + ¯ A ( x + ) e − iNx − i + O ( r ) ξ − = A ( x + ) + A ( x + ) e iNx − + ¯ A ( x + ) e − iNx − + O ( r ) (3.1)satisfy the criteria of [10], which allow us to construct corresponding asymptotic charges.If, on the other hand, one demands that the asymptotic symmetry generators ξ leavethe space of boundary conditions invariant, one finds the same vectors but with the firstsubleading terms appearing at one higher order for each component. For either set ofvectors, the Lie bracket algebra closes to the same order as one has defined the vectors.Here, B ( x + ) and A ( x + ) are real and A ( x + ) is complex; therefore, there are four real,periodic functions of x + that specify this asymptotic vector. We take the following basisfor the modes of the vector fields: L n = ie i n x + [ ∂ + − i n r∂ r ] + · · · T (0) n = iN e i n x + ∂ − + · · · T (+) n = iN e i ( n x + + N x − ) [ ∂ − − i N r∂ r − N r ∂ + ] + · · · T ( − ) n = iN e i ( n x + − N x − ) [ ∂ − + i N r∂ r − N r ∂ + ] + · · · , (3.2)which satisfy the Lie bracket algebra[ L m , L n ] = ( m − n ) L m + n , [ L m , T ( a ) n ] = − n T ( a ) m + n , [ T (0) m , T ( ± ) n ] = ∓ T ( ± ) m + n , [ T (+) m , T ( − ) n ] = 2 T (0) m + n . (3.3)Thus, the classical asymptotic symmetry algebra is a Witt algebra and an sl (2 , R ) currentalgebra.We use the Brandt–Barnich–Comp`ere (BBC) formulation [10, 11] for computing thecorresponding charges of our geometry. We find that the charges are integrable over thesolution space if δN = 0 with – 5 – /Q ξ = 18 πG δ Z dφ n B ( x + ) h κ ( x + ) + N ( 12 f ( x + ) − g ( x + )¯ g ( x + ))+ N e iNx − g ( x + ) + e − iNx − ¯ g ( x + ))+ i N ∂ + [ B ( x + ) ( e iNx − g ( x + ) − e − iNx − ¯ g ( x + ))] io − πG δ Z dφ N h A ( x + ) f ( x + ) − ( g ( x + ) A ( x + ) + ¯ g ( x + ) ¯ A ( x + )) i . (3.4)These can be integrated between the configurations trivially in the solution space from f ( x + ) = g ( x + ) = κ ( x + ) = 0 to general values of these fields to write down the charges Q B = 18 πG Z π dφ h B ( x + ) (cid:16) κ ( x + ) + N f ( x + ) − g ( x + )¯ g ( x + )) (cid:17) + 12 ( ∂ + − ∂ − ) ∂ − [ e iNx − g ( x + ) + e − iNx − ¯ g ( x + )] i = 18 πG Z π dφ h B ( x + )[ κ ( x + ) + N f ( x + ) − g ( x + )¯ g ( x + ))]+ 132 πG ∂ − [ e iNx − g ( x + ) + e − iNx − ¯ g ( x + )] (cid:12)(cid:12)(cid:12) φ =2 πφ =0 , (3.5) Q A = − N πG Z π dφ h A ( x + ) f ( x + ) − ( g ( x + ) A ( x + ) + ¯ g ( x + ) ¯ A ( x + )) i . (3.6)The boundary term in (3.5) vanishes as we assumed g ( x + ) to be periodic and N to be aninteger. The algebra of these charges admits central charges. We find that the central termin the commutation relation between charges corresponding to two asymptotic symmetryvectors ξ and ˜ ξ is given by( − i ) l πG Z π dφ h B ′ ( x + ) ˜ B ′′ ( x + ) − B ( x + ) ˜ B ′′′ ( x + )+ 2 N A ( x + ) ˜ A ′ ( x + ) − N (cid:16) A ( x + ) ¯˜ A ′ ( x + ) + ¯ A ( x + ) ˜ A ′ ( x + ) (cid:17)i . (3.7)These give rise to the following algebra for the charges [ L m , L n ] = ( m − n ) L m + n + c m δ m + n, , [ L m , T an ] = − n T am + n , [ T am , T bn ] = f abc T cm + n + k η ab m δ m + n, (3.8) The bracket in (3.8) is i times the Dirac bracket. – 6 –ith c = 3 l G , k = c , f = − , f −− = 1 , f + − = 2 , η = − , η + − = 2 . (3.9)This is precisely the sl (2 , R ) current algebra found in [1]. In this note we have provided boundary conditions for 3-dimensional gravity with negativecosmological constant such that the algebra of asymptotic symmetries is an sl (2 , R ) currentalgebra. In the process we showed that the boundary term proposed by CSS [6] admits amore general set of boundary conditions, which enables our result.It should be noted that our asymptotic symmetry algebra does contain the full isometryalgebra of the global AdS solution. This feature is similar to Brown–Henneaux [8] thoughone does not demand that the asymptotic vector fields of interest be asymptotically Killing;instead one uses the more general notion of asymptotic symmetries advocated by BBC[10, 11]. Using the BBC formulation, we computed the algebra of charges and found thelevel k to be c/
6, independent of the parameter N .To understand the relation to 2-dimensional induced gravity of Polyakov in light-cone gauge [1] further, it will be interesting to see if the correlation functions of theboundary currents, and the effective action for the dynamical fields of the boundary canalso be recovered from the gravity side. See [12] for a discussion on the latter issue. Ofcourse, connections between 3-dimensional gravity with negative cosmological constantand Liouville theory, which arises as a different gauge-fixing of (1.1), are well-known (seee.g. [13], and the recently proposed boundary conditions in [7]).It will be interesting to see how adding matter to AdS gravity would generalize ouranalysis. The boundary conditions of [6] have been found to be related to string theorysolutions of [14] with a warped AdS factor. It will be interesting to explore whether theboundary conditions in (2.9) also play a role in some string theory context.The non-linear solution in (2.12, 2.13) does not contain the conventional positive massBTZ [15] black hole. The special case of vanishing charges is given by f ( x + ) = g ( x + ) =¯ g ( x + ) = κ ( x + ) = 0 which is simply an extremal BTZ but with negative mass (in global AdS vacuum). The comments of CSS [6] about the possible existence of ergoregions andinstabilities in their solution also apply to (2.12). It will be important to understand theseissues better.Finally, it is intriguing that different ways of gauge-fixing the induced gravity [1] leadto different boundary conditions in the bulk and therefore apparently different holographicduals. It will be important to understand the class of physical theories one can obtain thisway and how they are related to each other.– 7 – eferences [1] A. M. Polyakov, “Quantum Gravity in Two-Dimensions,” Mod.Phys.Lett. A2 (1987) 893.[2] A. M. Polyakov, “Quantum Geometry of Bosonic Strings,” Phys.Lett.
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