LONG STRING DYNAMICS IN PURE GRAVITY ON AdS3
J. Kim* M. Porrati
Center for Cosmology and Particle Physics, Department of Physics, New York University
10003, New York, NY, USA
Received October 11, 2014
We study the classical dynamics of a completion of pure A1IS3 gravity, whose only degrees of freedom are boundary gravitons and long strings. We argue that the best regime for describing pure gravity is that of "heavy" strings, for which back-reaction effects on the metric must be taken into account. We show that once back-reaction is properly accounted for, regular finite-energy states are produced by heavy strings even in the infinite-tension limit. Such a process is similar to, but different from, nucleation of space out of a "bubble of nothing".
Cwitribvtiwi for the JETP special issue in honor of V. A. Rubakov's 60th birthday
1. INTRODUCTION AND SUMMARY
This paper is dedicated to Valory Rubakov on the occasion of his 60th birthday. Valory has boon a pioneer and a master in understanding the role of nonper-turbative solutions of field equations in quantum field theory. This paper is devoted to a particular case of soliton dynamics. Although limited in scope, we believe that it contains some results worth reporting. We hope that its readers will consider it also a worthy tribute to Valory's work.
Pure gravity in three dimensions does not propagate local degrees of freedom, as a simple counting argument shows: six of the 12 Hamiltonian degrees of freedom of the 3D graviton gtll, are removed by gauge invariances and the remaining ones are removed by 3+3 constraints that follow from Einstein's equations. Hence, 3D gravity does not propagate gravitational waves. In the presence of a negative cosmological constant, pure gravity still exhibits a nontrivial dynamics, because there exist boundary gravitons fl] and black hole solutions . The Einstein Hilbert action of pure gravity with a negative cosmological constant — l//2 is
Boundary gravitons exist because the asymptotic metric of 3D anti do Sitter space (AdS) is preserved by
* E-mail: jk2943(fflnyu.edu
a set of diffoomorphisms that act nontrivially 011 the boundary. Specifically, the condition of being asymptotically AdSs means that the metric has the form fl]
gu = -r2/l2 + 0(l), gl4, = C)( 1), .<//, Oir '-'I. i/r,. = /2/r2 + 0(r-4), (2) gr0 = O(r~3), g00 = r2 + 0( 1).
These boundary conditions are preserved by diffoomorphisms with the asymptotic form
=/[/(.,;+) +g(x-)} + ^=lf(x+)-g(x-)]- (3)
C = -r[d+f(x+) + d^g(x-)} + 0(r-1).
The allowed diffoomorphisms are parameterized by two arbitrary functions f(x+) and g(z~), each depending 011 only one of the two boundary light-cone coordinates = t/I ± <f>). The time t and the angular coordinate <f> ~ <f> + 2tt parameterize the AdSs boundary, while r is its radial coordinate. The boundary is at r = oc and
2d± = 10/01 ± O/Oo.
The classical Poisson brackets associated with asymptotic diffoomorphisms (3) define two Virasoro algebras with the same central charge c = 3//2G fl]; therefore, after quantization, the Hilbert space of any quantum gravity with the same asymptotics (whether
pure or with matter) must fall into unitary representations of the Virasoro algebras. This purely kinemat-ical fact lias a deep consequence if we further assume that quantum gravity on AdSs is dual to a 2D conformai field theory (CFT) . Modular invariance of the CFT, discreteness of the spectrum, and the existence of an 57(2, C)-invariant state with conformai weights A A 0 then imply that the asymptotic density of states at levels (A, A) is 
«/(A. A) = es = exp ^2 tt ^cA/G - 2 tt ycA/G j .
Rotating black-hole solutions for pure 3D AdS gravity (2) do exist . Their metric depends on two parameters: the mass M and the angular momentum J. The metric is 
ds2 = -N2dt2 + N~2dr2 + r2(N0dt + dj>)2,
»-2 16GPJ2 Arô 4GJ (5) N2 = -8GM + — +-5—, N0 = —rI r2 r2
After the identification <• Ml + J
- <• mi - J ÎZT —
Cardy formula (4) matches the Bekenstein Hawking formula for the entropy of rotating black holes 
S = Sbh = 2nrh/4G, /•/, = I \J 16'.1/ + AGs/M2 - pff2.
The result in Ref.  is general. In particular, it does not depend on the matter content of the AdSs bulk theory. Amusingly, pure gravity seems to defy general formulas (4) and (C). Indeed, as noticed in , the asymptotic dynamics of Eq. (1) is described by a Liouville action. Upon quantization, the Liouville theory becomes an unusual conformal field theory because of two features. The first is that its spectrum does not include an 57(2, C) invariant state. Instead, physical states obey the "Seiberg bound" A, A > (e — 1)/24 . The second is that physical states are only plane-wave normalizable, because the spectrum of the Liouville theory is continuous. These properties are well established in consistent quantizations of Liouville theory at c > 1 .
The reduction of pure gravity to a boundary Liouville theory is most easily proven by writing Einstein Hilbert action (1) in terms of two 57(2, R) Cliern Simons theories 
Seh = SCs,kH] - SCs,k [i], * = l/'iG. (7)
With ta denoting the three 57(2, R) generators in the fundamental representation, the Cliern Simons action
Scs,kW = "JZ l r / A <IA + A .4 A ^
+ boundary terms. (8)
The gauge potentials ,4 and .4 are related to the dreibein ea and spin connection uia by
.4° = u>° + —,
.4° = U>°--— ,
.4 = Aata
Some of the equations of motion derived from (7) are constraints. In the gauge A- = .4+ = 0, when the 3D space is topologically the product of a 2D disc D and the real line R, they imply that Ar = U~1dU and A = \~ '(/I", with U (V) being an 57(2,f?)-valued function of r,x+ (r,x~). With the solution of the constraints substituted in the Cliern Simons action, the bulk terms disappear and the action reduces to a boundary term. This term is the 2D cliiral Wess Zumino action [10, 11]. Further constraints, following from the requirement that ,4 and .4 give an asymptotically AdS metric, reduce the Wess Zumino action to a Liouville action .
An attentive reader should have noticed an unwarranted assumption here. We assumed that the 3D space was topologically global AdSs to arrive at a Liouville action. In the presence of black holes, i. e., horizons, or of time-like singularities associated with pointlike particles in the bulk, the action at the r = oc boundary must be supplemented with other terms at the inner boundary/'horizon. A possible interpretation of these terms is that they describe the states of the AdSs quantum gravity, more precisely, the primary states in each irreducible representation (irrep) of the Virusoro x Viru.soro algebra acting on the Hilbert space of quantum AdSs gravity1^. The role of the boundary Liouville theory would then be simply to describe the Virasoro descendants in each irrep (cf. ). In this interpretation, other information is needed to determine the spectrum of primary operators.
One hint that pure gravity could nevertheless have the same spectrum of primaries as the Liouville theory conies from canonical quantization of pure gravity.
11 The "constrain first" Hamiltonian formalism was used in fill to study the effect of point-like insertions and nontrivial topology for compact-group Chern Simons theories.
Already in the 1990s, it was shown that the wave functions obtained by quantizing the 57(2, R) Chcrn Simons theory are Virasoro conformal blocks . Two 57(2, R) Chcrn Simons actions are combined into the action of pure gravity, and hence the Hilbert space of pure gravity must be (a subspace of) the product of each Chcrn Simons Hilbert space. In a forthcoming publication, we will argue that the pure-gravity Hilbert space is the target space of conformal field theories with continuous spectrum and obeying the Seiberg bound  (cf. ). Assuming from now on that this result holds, we conclude that pure gravity in AdSs should contain states that can reach the boundary at a finite cost in energy, since states confined to the interior of the AdS space have a discrete spectrum. Then one natural question to ask is: what are those states?
The IIlclSS of such states must be large in AdS units: MI 1, otherwise gravity could not be called "pure" in any sense. The states cannot be massive particles, which cannot reach the AdS boundary. Indeed, there is only one natural candidate for such states: they must be long strings. These states have already been invoked as a possible solution to certain problems of the partition function of Euclidean pure gravity in .
The rest of this paper is devoted to studying the effect of long strings in AdSs gravity. Section 2 summarizes known features of long strings in the probe approximation, which holds when back-reaction on the metric and quantum string dynamics effect can both be neglected. This happens when the string tension T is in the range l~2 «T« G^1!^1. Section 3 describes the case of "light" strings, which were studied in detail in : T < l~2. It is a regime where back-reaction can be neglected, but quantum effects cannot. This is an interesting case, but far from pure gravity, as we will argue using some results in Rcf. . Section 4 studies the "heavy" string case, T > G-1/-1, when back-reaction cannot be neglected. We argue that this regime is best suited to describe a pure gravity theory containing BTZ black holes and no states below the Seiberg bound. We further show that in order to recover the mass gap predicted by the Seiberg bound, the string tension must be Planckian, T ()(G -) » G '/ This is the limit T —¥ oc, which is nonsingular thanks to back-reaction effects. Finite -rilclSS BTZ states arise via a process similar to nucleation of the universe out of a "bubble of nothing"2''.
2! Differently from the quantum nucleation case, the process under consideration here is a classical one, in which the initial state contains a long string approac
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