[21] Olivier Coussaert, Marc Henneaux, and Peter van Driel. The Asymptotic ... Bondi-Metzner-Sachs invariant two-dimensional field theories as the flat limit of.
The dual theory of AdS3 gravity with free boundary conditions Glenn Barnich, Hernán González & Blagoje Oblak
Physique Théorique et Mathématique Université Libre de Bruxelles and International Solvay Institutes Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium
Abstract A set of free boundary conditions for Einstein gravity on AdS3 was recently proposed. We compute the dual, two-dimensional theory corresponding to this system through an explicit Hamiltonian reduction of the Chern-Simons action for three-dimensional gravity. We show that this dual theory describes a Liouville field coupled to the conformal factor of the background metric and match its global symmetries to the asymptotic symmetries of the bulk.
1
Contents Introduction
3
1 Free boundary conditions on AdS3 1.1 Boundary conditions and on-shell metric . . . . . . . . . . . . . . . . . . 1.2 Asymptotic symmetries and surface charges . . . . . . . . . . . . . . . .
4 4 5
2 Chern-Simons description and improved action 2.1 Einstein-Hilbert action in Chern-Simons form . . . . . . . . . . . . . . . 2.2 Boundary conditions in Chern-Simons formalism . . . . . . . . . . . . . . 2.3 Improved action principle . . . . . . . . . . . . . . . . . . . . . . . . . .
6 6 7 9
3 Reduction to a theory of chiral bosons 3.1 From Chern-Simons to Wess-Zumino-Witten . . . . . . . . . . . . . . . . 3.2 Gauss decomposition of the Wess-Zumino-Witten model . . . . . . . . . 3.3 From Wess-Zumino-Witten to chiral bosons . . . . . . . . . . . . . . . .
12 12 14 19
4 The 4.1 4.2 4.3
22 22 23 24
dual theory Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The dual theory as a Liouville field in a curved background . . . . . . . . Symmetries and charges . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion
26
2
Introduction The standard boundary conditions for three-dimensional Einstein gravity with a negative cosmological constant are those of Brown and Henneaux [1]. As is well-known, the surface charges associated with the corresponding asymptotic symmetries consist of two copies of the Virasoro algebra with central charge c = 3`/2G. This discovery is now considered as one of the first hints of the AdS/CFT correspondence [2,3], and it was used to reproduce the entropy of BTZ black holes [4–6] using the Cardy formula. Despite the obvious success of Brown-Henneaux boundary conditions, it is tempting to ask how far one can go in modifying or generalizing them, while still obtaining a well-defined action principle with finite surface charges. Last year, a series of papers by different authors addressed this question [7–9]. In this work we focus on the free boundary conditions proposed in [9], which are a generalization of Brown-Henneaux boundary conditions obtained by “unfreezing” the conformal factor of the boundary metric and making it dynamical. Such boundary conditions are really a special case of the general free boundary conditions introduced in [10] long ago. Because three-dimensional gravity has no local degrees of freedom, all its dynamics is captured by the boundary of space-time, making it a perfect playground for explicit realizations of the AdS/CFT correspondence. In particular, using the Chern-Simons formalism [11,12], it is possible to build explicitly the two-dimensional theory that is (semiclassically) dual to AdS3 with a prescribed set of boundary conditions, using a method of Hamiltonian reduction [13, 14]. The latter is performed through a two-step sequence consisting roughly of 1. a reduction from Chern-Simons theory to a Wess-Zumino-Witten (wzw) model [15–18], and 2. a reduction from wzw to a Liouville-like theory [19, 20]. For Brown-Henneaux boundary conditions, this reduction was first performed in [21], leading indeed to Liouville theory. (The treatment was later completed in [22–24] to include holonomies.) Similarly, for the chiral boundary conditions proposed in [7], this reduction was shown to lead to a chiral Liouville theory [25]. As regards the free boundary conditions of [9], the dual, two-dimensional theory already appeared in [26], where it was argued that AdS3 gravity with such boundary conditions is dual to a conformal field theory coupled to (two-dimensional) gravity in conformal gauge. In fact, this result follows in great part from the general analysis of [10], where free boundary conditions for AdS (in arbitrary dimension) were analyzed in detail1 . However, in [26], the dual theory was obtained on the basis of identifications between symmetries and conserved currents. Our purpose here is to derive the action of the dual theory by reduction from the Einstein-Hilbert action, following the aforementioned sequence Chern-Simons −→ Wess-Zumino-Witten −→ Liouville.
(1)
We will obtain a theory consisting of a Liouville field (the one from Brown-Henneaux boundary conditions) coupled to the conformal factor of the two-dimensional metric [27, 28]. In fact, a similar result was obtained long ago in [23, 24]; in these references the boundary metric was curved indeed, but it was not dynamical – and this is a crucial difference with our work. 1
Note: In the terminology of [10], the boundary conditions of [9] are really mixed rather than free.
3
Our paper is organized as follows. In section 1, we review the free boundary conditions of [9] and the corresponding asymptotic symmetry algebra. In section 2, we translate these boundary conditions in Chern-Simons language and work out the boundary terms to be added to the action in order to obtain a well-defined variational principle. Finally, in sections 3 and 4, we write down the action of the dual two-dimensional theory by performing the reduction (1); we also find the symmetries of the theory and match them with the asymptotic symmetries of the bulk.
1 1.1
Free boundary conditions on AdS3 Boundary conditions and on-shell metric
We consider three-dimensional gravity with a negative cosmological constant Λ = −1/`2 and describe the space-time manifold M by Fefferman-Graham coordinates r, xa (a = 0, 1), where r is a radial coordinate and the x’s are coordinates on the cylinder at fixed r; we will take x0 = t/` ≡ τ to be a dimensionless timelike coordinate, and x1 = ϕ to be an angular coordinate identified as ϕ ∼ ϕ + 2π. We also introduce light-cone coordinates x± ≡ τ ± ϕ. The boundary ∂M of space-time is the cylinder r → +∞. In this coordinate system, the free boundary conditions of [9] correspond to the following asymptotic behaviour of the space-time metric: `2 2 dr + r2 e2Φ(x) ηab + O(r−2 ) dxa dxb , (2) 2 r where Φ(x) = Φ(x0 , x1 ) is the conformal factor of the boundary metric. These conditions generalize those of Brown and Henneaux [1], which are recovered when Φ = 0. It was shown in [9] that such relaxed fall-off conditions lead to a differentiable action only provided the trace K of the extrinsic curvature (on slices of constant radius r) satisfies the following fall-off condition: ds2 =
K + 2/` = o(r−2 ).
(3)
This is tantamount to requiring that the trace of the Brown-York stress tensor vanish [26]. Einstein’s equations with the boundary conditions (2) were solved explicitely in [29, 30], for arbitrary conformal factor Φ, and without conditions on the extrinsic curvature. In terms of the dimensionless quantities γ±± ≡ Ξ±± − (∂± Φ)2 + ∂±2 Φ,
γ+− ≡ ∂+ ∂− Φ,
where Ξ++ (resp. Ξ−− ) is a function of x+ (resp. x− ) only, this solution reads 2 2 2Φ r e `2 2 2 2 dr − − 2γ+− + 2 2Φ γ+− + γ++ γ−− dx+ dx− ds = ` 2 2 r ` r e 2 ` `2 + 2 − 2 +γ++ 1 − 2 2Φ γ+− (dx ) + γ−− 1 − 2 2Φ γ+− (dx ) . r e r e
(4)
(5)
As such, this metric does not satisfy condition (3). Indeed, the trace of the corresponding extrinsic curvature along cylinders of constant radius has the form 4` K = −2/` − 2 2Φ γ+− + O r−4 , r e 4
so that condition (3) requires !
γ+− = ∂+ ∂− Φ = 0. This reduces the metric (5) to 2 2 2Φ r e `2 2 2 dr + − + 2 − 2 ds = ` − + 2 2Φ γ++ γ−− dx dx + γ++ (dx ) + γ−− (dx ) , r2 `2 r e
(6)
(7)
which is therefore the general solution of Einstein’s equations with free boundary conditions, satisfying condition (3) on the extrinsic curvature.
1.2
Asymptotic symmetries and surface charges
The diffeomorphisms preserving the boundary conditions (2) and the extrinsic curvature condition (3) are generated by asymptotic Killing vector fields ξ whose components, up to subdominant terms, are of the form [9] r ξ r = − ω(x), 2 `2 ∂− ω(x), 2r2 e2Φ `2 = ¯(x− ) + 2 2Φ ∂+ ω(x), 2r e
ξ + = (x+ ) + ξ−
(8)
where (x+ ) and ¯(x− ) are arbitrary (anti)chiral functions on the cylinder, while ω(x) solves the harmonic equation ∂+ ∂− ω(x) = 0, but is otherwise arbitrary. We will write ¯ − ) + Ωτ, ω(x) = λ(x+ ) + λ(x
(9)
where Ω is a constant. The ’s generate conformal transformations, while ω generates Weyl transformations of the two-dimensional metric e2Φ ηab . The Lie algebra generated by vector fields (8) is thus the semi-direct sum of the conformal algebra in two dimensions with the algebra of Weyl transformations. Up to the zero-mode Ωτ of ω, it consists of ˆ(1). two commuting copies of Vect(S 1 ) A u The surface charges associated with these asymptotic symmetries were computed in [9], and we display them here for future reference. We denote by Q,¯,ω [γ, Φ] the charge2 associated with the asymptotic Killing vector field labelled by parameters , ¯, ω, and evaluated on the solution (7): Z 2π 1 ˙ ` dϕ γ++ + ¯ γ−− + ω Φ − Φω˙ . (10) Q,¯,ω [γ, Φ] = 8πG 0 2 The dots denote partial derivatives with respect to τ . When the zero-mode Ωτ in (9) is neglected, this charge splits into a chiral and an anti-chiral piece [26]: Z 2π ` ¯ −Φ . Q,¯,ω [γ, Φ] = dϕ γ++ + λ∂+ Φ + ¯ γ−− + λ∂ 8πG 0 ¯ Upon expand(We used integrations by parts to cancel all derivatives acting on λ and λ.) ¯ in Fourier modes, the algebra of such charges becomes (up ing the functions , ¯, λ and λ 2
See eq. (5.5) of [9], up to different notations.
5
d 1) A u d 1 ) is ˆ(1)k , where Vect(S to the Weyl zero-mode) a direct sum of two copies of Vect(S ˆ(1)k is a U(1) Kac-Moody algebra a Virasoro algebra with vanishing central charge, and u with negative level k = −`/8G. Explicitly, writing Lm ¯m L Jm J¯m Q
+
≡ charge associated with = eimx , − ≡ ... ¯ = eimx , + ≡ ... λ = eimx , ¯ = eimx− , ≡ ... λ ≡ ... Ω = 2,
the non-vanishing brackets of the surface charge algebra read ¯ m, L ¯n = i [Lm , Ln ] = (m − n)Lm+n , i L ¯ m , J¯n = i [Lm , Jn ] = −nJm+n − ikm2 δm+n,0 , i L i [Jm , Jn ] = kmδm+n,0 , i J¯m , J¯n = ¯ m, Q = i [Lm , Q] = iJm , i L i [Jm , Q] = ikδm,0 , i J¯m , Q =
[9]
¯ m+n , (m − n)L −nJ¯m+n − ikm2 δm+n,0 , kmδm+n,0 , iJ¯m , ikδm,0 . (11) From the viewpoint of the dual theory, these symmetries and the associated charges will be interpreted as global symmetries and Noether charges, as usual in AdS/CFT – see subsection 4.3.
2
Chern-Simons description and improved action
We now translate the conditions (2) and (3) in Chern-Simons language, before using them to derive a well-defined variational principle. Our starting point is the well-known fact that the triadic Palatini action for three-dimensional Einstein gravity can be rewritten as a Chern-Simons action with an appropriate gauge group, depending on the sign of the cosmological constant [11, 12].
2.1
Einstein-Hilbert action in Chern-Simons form
Our notations and conventions are the following. We write the Minkowski metric, with frame indices denoted as (a), (b) ∈ {(+), (−), (2)}, in the form 0 1 0 η(a)(b) ≡ 1 0 0 , (12) 0 0 1 so that the relation between the space-time metric ds2 and the (null) dreibein e(a) is ds2 = (e(2) )2 + 2e(+) e(−) .
(13)
In three-dimensional space-time, the spin connection ω (a) (b) can be rewritten with a single frame index by contraction with the antisymmetric tensor ε(a)(b)(c) : 1 ω (a) ≡ − ε(a)(b)(c) ω(b)(c) , 2
where ε(2)(+)(−) = −ε(2)(+)(−) ≡ +1.
Frame indices are raised and lowered with the Minkowski metric (12).
6
(14)
For negative cosmological constant −1/`2 , the crucial step in rewriting the EinsteinHilbert action as a Chern-Simons action consists in identifying frame indices with indices in the Lie algebra sl (2, R). We choose to write generators of the latter as 1 0 0 1 0 1 1 1 0 j(+) ≡ − √ , j(−) ≡ − √ , j(2) ≡ 2 0 −1 2 1 0 2 0 0 and we define the sl (2, R)-valued gauge fields A ≡ A(a) j(a) ≡ ω (a) + e(a) /` j(a) , A¯ ≡ A¯(a) j(a) ≡ ω (a) − e(a) /` j(a) .
(15)
Then the (first-order) Einstein-Hilbert action in three dimensions can be rewritten as a sum of two Chern-Simons actions for the gauge fields (15), with gauge group SL(2, R). Explicitly, the action reads SEH A, A¯ = SCS [A] − SCS A¯ + (boundary terms), (16) where the boundary terms ensure differentiability when boundary conditions are taken into account, while SCS denotes the Chern-Simons action Z 2 (17) SCS [A] = −κ Tr A ∧ dA + A ∧ A ∧ A , 3 M with κ = `/16πG.
2.2
Boundary conditions in Chern-Simons formalism
In order to guess the fall-off conditions to be imposed on the Chern-Simons gauge fields, we shall proceed as follows: we will first derive the gauge fields that reproduce the metric (5), which solves Einstein’s equations but does not satisfy condition (3) on the extrinsic curvature. (Hence we will keep γ+− non-zero throughout this subsection.) These gauge fields will then be relaxed to leave room for fluctuations, eventually leading to the fall-off conditions (24) in Chern-Simons language. However, we already stress that the boundary conditions (24) will not be the right ones yet. The correct boundary conditions will be derived only at the end of subsection 2.3 – see eq. (31) –, where additional constraints will be imposed by the requirement of differentiability of the action. (In particular, the condition γ+− = 0 will be necessary for a consistent theory.) Motivation: on-shell results One easily verifies that the dreibein given by ` ` reΦ ± ∓ ± (±) , dx − Φ γ∓∓ dx + γ+− dx e = ∓√ re 2 `
` e(2) = dr r
(18)
reproduces the metric (5) through formula (13). The (torsionless) first Cartan structure equation then fixes uniquely the corresponding spin connection, which is given by 1 reΦ ± ` (±) ∓ ± ω = −√ dx + Φ γ∓∓ dx + γ+− dx , (19) re 2 ` ω (2) = −∂+ Φdx+ + ∂− Φdx− (20) 7
in terms of the single-index representation (14). These expressions obviously simplify when γ+− = 0, but we postpone the implementation of this constraint until the next subsection. For now, we simply note that the Chern-Simons gauge fields associated with (18)-(20) through the definition (15) read ` dr 1 + − + − ∂+ Φdx − ∂− Φdx γ++ dx + γ+− dx − reΦ A = 2r 2 , (21) Φ re dr 1 dx+ − + ∂+ Φdx+ − ∂− Φdx− ` 2r 2 Φ re dr 1 ∂+ Φdx+ − ∂− Φdx− dx− − − ` A¯ = 2r 2 . (22) ` dr 1 − + + − γ−− dx + γ+− dx + ∂+ Φdx − ∂− Φdx reΦ 2r 2 The actual on-shell Chern-Simons gauge fields corresponding to AdS3 gravity with free boundary conditions are such that γ+− = 0, reproducing the metric (7). Off-shell formulation and primitive boundary conditions Now that we know the on-shell expression of the Chern-Simons gauge fields, let us try to guess the appropriate off-shell behaviour that we will use to derive the dual theory. First, we may use gauge symmetry in the bulk to fix the radial components of A and A¯ [31]: 1/2r 0 −1/2r 0 ¯ Ar = , Ar = . (23) 0 −1/2r 0 1/2r This fixes the radial components of the dreibein and spin connection, and is the analogue the Fefferman-Graham gauge fixing in metric formalism. Then the only fluctuating components of the gauge fields are those along the timelike cylinder. Using the on-shell expressions (21)-(22) as a guide, it is tempting to impose the following fall-off conditions: ` 2 ? B + O(1/r) reΦ (γ++ + γ+− ) + O(1/r ) Aτ = Φ , re + O(1) −B + O(1/r) ` ` 2 ? C + O(1/r) reΦ (γ++ − γ+− ) + O(1/r ) Aϕ = Φ , re + O(1) −C + O(1/r) ` (24) reΦ ¯ + O(1/r) B + O(1) ? ` A¯τ = , ` 2 ¯ (γ−− + γ+− ) + O(1/r ) −B + O(1/r) reΦ Φ re C¯ + O(1/r) − + O(1) ? ` A¯ϕ = . ` 2 ¯ (−γ + γ ) + O(1/r ) − C + O(1/r) −− +− reΦ ¯ C, ¯ γ++ , γ−− and γ+− are understood as completely At this point, the functions B, C, B, arbitrary functions of their arguments xa , as expected in the off-shell formulation. They are the leading fluctuations of the Chern-Simons gauge fields. However, as mentioned before, we will now see that the requirement of differentiability of the action imposes constraints on these quantities. In particular, the function γ+− will be forced to vanish. 8
2.3
Improved action principle
Our goal in this subsection is to find the explicit form of the boundary terms appearing in (16), by verifying that they lead to a well-defined variational principle. In (r, τ, ϕ) coordinates, the action (17) reads Z I n o ˙ ˙ SCS [A] = −κ dr ∧ dτ ∧ dϕ Tr Ar Aϕ − Aϕ Ar + 2Aτ Fϕr − κ dτ ∧ dϕ Tr {Aϕ Aτ } , M
∂M
(25) where F ≡ dA + A ∧ A is the curvature two-form associated with the connexion A. The action (25) obviously contains a boundary term; the latter is completely irrelevant for the dynamics of the theory and can be cancelled by introducing a first improved action [21], Z n o dr ∧ dτ ∧ dϕ Tr Ar A˙ ϕ − Aϕ A˙ r + 2Aτ Fϕr . SCS,1 [A] ≡ −κ M
We will now compute the variation of the action SEH,1 A, A¯ ≡ SCS,1 [A] − SCS,1 A¯
(26)
and require that it be differentiable when the boundary conditions (24) are taken into account. As already announced, we will find that stronger boundary conditions need to be imposed: as such, conditions (24) are inconsistent. Boundary terms produced by the variation of the action It is straightforward to compute the variation of action (26): I dτ ∧ dϕ Tr Aτ δAϕ − A¯τ δ A¯ϕ , δSEH,1 = (EOM) + 2κ ∂M
where (EOM) denotes the three-dimensional integral of a term that vanishes upon imposing the equations of motion. By itself, this result is a generic feature of Chern-Simons theory and has nothing to do with our choice of boundary conditions. The latter come into play, however, once we actually compute the boundary term ∼ Aτ δAϕ . In our case, according to (24), the variation of Aϕ is produced by the variation of the fields C, Φ, γ++ and γ+− (plus the contribution of all subleading pieces), so that ¯ C¯ + 4γ+− δΦ + δγ++ + δγ−− − 2δγ+− + O(1/r). (27) Tr Aτ δAϕ − A¯τ δ A¯ϕ = 2BδC − 2Bδ The terms on the right-hand side are those that we need to suppress by adding appropriate boundary terms to the action. Coussaert-Henneaux-Van Driel improvement term In order to get rid of the δγ’s in (27), we introduce the boundary term [21, 32] I ¯ I1 [A, A] = −κ dτ ∧ dϕ Tr A2ϕ + A¯2ϕ , ∂M
which is such that I δI1 = −2κ
¯ C¯ + δγ++ + δγ−− − 2δγ+− . dτ ∧ dϕ 2CδC + 2Cδ
∂M
9
Using (27), we thus find I δ (SEH,1 + I1 ) = (EOM)+2κ
¯ + C)δ ¯ C¯ + 4γ+− δΦ . (28) dτ ∧dϕ 2(B − C)δC − 2(B
∂M
All δγ’s have now been cancelled in the right-hand side. Additional improvement term The variation (28) contains non-integrable terms of the form BδC or γ δΦ. There are several ways to get rid of such terms, but not all methods are good. One possibility, for instance, is to simply turn off all the degrees of freedom described by B, C, etc., but this choice reduces the system to the one describing Brown-Henneaux, rather than free, boundary conditions. The choice we are going to make here is motivated by the extrinsic curvature constraint (3). As we have seen, the latter leads to the condition γ+− = 0, so it is reasonable to incorporate this requirement as a boundary condition in the off-shell expressions (24). This corresponds to imposing boundary conditions stronger than those we started with, and suppresses the non-integrable term γ+− δΦ in (28), leaving us with I ¯ + C)δ ¯ C¯ . (29) δ (SEH,1 + I1 )|γ+− =0 = (EOM) + 2κ dτ ∧ dϕ 2(B − C)δC − 2(B ∂M
As regards the B, C terms, we use the on-shell expressions (21)-(22) as a guide. ¯ = −Φ0 /2 and C = C¯ = −Φ/2. ˙ We can see there that, on-shell, B = B Here the relation between B’s, C’s and the derivatives of Φ is an on-shell feature, but it is perfectly ¯ and C = C¯ (without any reference to Φ!) as offconsistent to adopt the relations B = B shell conditions. These conditions should, then, be understood as part of our boundary conditions, and amount to demanding that the leading fluctuation of e(2) along x± be of order 1/r instead of order 1. Their virtue is that they render the last boundary term of (29) integrable: I δ (SEH,1 + I1 )|γ+− =0, B=B, dτ ∧ dϕ 4CδC. ¯ C=C ¯ = (EOM) − 2κ ∂M
Thus, adding the second improvement term I I2 [C] = κ dτ ∧ dϕ 4C 2 , ∂M
we finally obtain δ (SEH,1 + I1 + I2 ) = (EOM). Summary: Consistent boundary conditions We have now shown that the full improved action S ≡ SEH,1 + I1 + I2
(30)
is differentiable when the right boundary conditions are taken into account. These boundary conditions are not simply given by (24). Rather, they are the restriction of (24) to
10
¯ and C = C, ¯ or explicitly γ+− = 0, B = B ` 2 B + O(1/r) reΦ γ++ + O(1/r ) Aτ = Φ , re + O(1) −B + O(1/r) ` ` 2 C + O(1/r) reΦ γ++ + O(1/r ) Aϕ = Φ , re + O(1) −C + O(1/r) ` (31) Φ
A¯τ
A¯ϕ
re + O(1) B + O(1/r) ` = , ` 2 γ−− + O(1/r ) −B + O(1/r) reΦ reΦ + O(1) C + O(1/r) − ` = . ` 2 − Φ γ−− + O(1/r ) −C + O(1/r) re
¯ we will From now on, when talking about the boundary conditions satisfied by A and A, always refer to the latter conditions – together with the fixed radial components (23). We stress once more that B, C and the γ’s are fluctuating functions on the cylinder, and not pure numbers or fixed (non-fluctuating) field configurations. For future reference, let us write down the improved action (30) in full detail: ¯ = S[A, A] ¯ + I2 [C] = SCS,1 [A] − SCS,1 A¯ + I1 [A, A] Z n o dr ∧ dτ ∧ dϕ Tr Ar A˙ ϕ − Aϕ A˙ r + 2Aτ Fϕr − A¯r A¯˙ ϕ + A¯ϕ A¯˙ r − 2A¯τ F¯ϕr = −κ IM −κ dτ ∧ dϕ Tr A2ϕ + A¯2ϕ − 4C 2 . ∂M
For convenience, we will decompose this action as ¯ = Sc [A] + S¯c [A] ¯ + I2 [C], S[A, A]
(32)
where the index “c” means “chiral” (or anti-chiral in the barred sector), with Z I n o ˙ ˙ Sc [A] = −κ dr ∧dτ ∧dϕ Tr Ar Aϕ − Aϕ Ar + 2Aτ Fϕr −κ dτ ∧dϕ Tr A2ϕ (33) M
∂M
and ¯ = −κ S¯c [A]
Z
I n o ˙ ˙ ¯ ¯ ¯ ¯ ¯ ¯ dr ∧ dτ ∧ dϕ Tr −Ar Aϕ + Aϕ Ar − 2Aτ Fϕr − κ
M
dτ ∧ dϕ Tr A¯2ϕ .
∂M
(34) As regards the term I2 [C], it can also be written as I ¯ ¯(2) I2 [C] = I2 [A, A] = κ dτ ∧ dϕ A(2) ϕ Aϕ , ∂M
11
(35)
where the superscript (2) denotes the component along the generator j(2) . In fact, owing ¯ the integrand of the right-hand side of this expression to the boundary condition C = C, (2) p ¯(2) 2−p can be replaced by any combination of the form Aϕ Aϕ , where 0 ≤ p ≤ 2. Since the choice of p does not affect (classically) the reduced theory that we will eventually obtain, we choose, without loss of generality, to work with the “symmetric” form (35), that is, p = 1. This choice will be further motivated by the identification of I2 with a marginal deformation of the wzw model obtained after the first step of the reduction (1). See the end of subsection 3.1.
3
Reduction to a theory of chiral bosons
Now that we have a consistent theory at our disposal, we are ready to solve its constraints (imposed on the one hand by the structure of the theory itself, and on the other hand by our choice of boundary conditions). As mentioned in the introduction, we will perform this reduction following the standard two-step procedure (1).
3.1
From Chern-Simons to Wess-Zumino-Witten
The reduction of Chern-Simons theory to a wzw model follows from the implementation of the constraint Fϕr = F¯ϕr = 0. We will perform this task separately for the chiral and anti-chiral sectors of action (32). This step of the reduction contains nothing new, as it goes through in exactly the same way when treating pure Brown-Henneaux boundary conditions [21]. Hence we will not give details regarding the computations – instead, we will simply collect the relevant intermediate results and display the final formula, eq. ¯ will be discussed thereafter. (42). The meaning of the term I2 [A, A] Reduction of the chiral sector Assuming there are no holonomies, the constraint Fϕr = 0 is solved by a pure gauge configuration Ai = G−1 ∂i G (i = r, ϕ), (36) where the field G : M → SL(2, R) is smooth, but otherwise arbitrary. Now, the condition Fϕr ≈ 0 is a first-class constraint, hence generating gauge transformations; a corresponding (partial) gauge fixing condition is then obtained by setting [31, 32] ∂ϕ Ar = 0, which, using (36), is solved by G(r, τ, ϕ) = g(τ, ϕ) · h(r, τ ). (37) Using diffeomorphisms, one can always set ∂τ h|r0 = 0 for any fixed, finite value r0 of r. Assuming that this remains true for r0 = +∞, we can write h(r, τ )
r→+∞
=
h(r) + o(1).
We now have to plug (36) and (37) into the chiral action Sc . The latter contains a bulk piece and a boundary piece. First, the boundary piece may be written as I I n 2 2 o Sc |boundary = −κ dτ ∧ dϕ Tr Aϕ = −κ dτ ∧ dϕ Tr g −1 ∂ϕ g . ∂M
∂M
12
Second, plugging Fϕr = 0 in (33) and using the fact that ∂τ h = 0 on ∂M, we find Z n o ˙ ˙ dr ∧ dτ ∧ dϕ Tr Ar Aϕ − Aϕ Ar Sc |bulk = −κ I M Z −1 κ −1 = κ dτ ∧ dϕ Tr g ∂ϕ gg ∂τ g + Tr G−1 dG ∧ G−1 dG ∧ G−1 dG . 3 M ∂M Combining these results, we obtain I Sc |Fϕr =0 ≡ Swzw [g] = κ
dτ ∧ dϕ Tr g −1 ∂ϕ g g −1 ∂τ g − g −1 ∂ϕ g
∂M Z
κ + 3
Tr G−1 dG ∧ G−1 dG ∧ G−1 dG .
(38)
M
This is a chiral wzw action for the field g on ∂M, the three-dimensional piece being the usual Wess-Zumino term [32, 33]. Reduction of the anti-chiral sector ¯ −1 ∂i G ¯ (i = r, ϕ), with G ¯ an As in the chiral sector, we solve F¯ϕr ≈ 0 by writing A¯i = G SL(2, R)-valued field. We can then impose the gauge-fixing condition ∂ϕ A¯r = 0, which tells us that ¯ τ ). ¯ τ, ϕ) = g¯(τ, ϕ) · h(r, G(r, (39) ¯ be independent of τ at infinity: h(r, ¯ τ ) r→+∞ ¯ We will also require that h = h(r) + o(1). Now, the boundary piece of the anti-chiral action (34) can be reduced as I I n 2 2 o ¯ ¯ ¯ Sc [A] boundary = −κ dτ ∧ dϕ Tr Aϕ = −κ dτ ∧ dϕ Tr g¯−1 ∂ϕ g¯ , ∂M
∂M
¯ vanishes at infinity, becomes while its bulk piece, using F¯ϕr = 0 and the fact that ∂τ h Z n o ¯r A¯˙ ϕ + A¯ϕ A¯˙ r ¯ S¯c [A] = −κ dr ∧ dτ ∧ dϕ Tr − A bulk Z IM −1 −1 κ −1 ¯ dG ¯∧G ¯ −1 dG ¯∧G ¯ −1 dG ¯ . Tr G = −κ dτ ∧ dϕ Tr g¯ ∂ϕ g¯g¯ ∂τ g¯ − 3 M ∂M Together, these results yield S¯c F¯ϕr =0 ≡ S¯wzw [¯ g ] = −κ κ − 3
I
dτ ∧ dϕ Tr g¯−1 ∂ϕ g¯ g¯−1 ∂τ g¯ + g¯−1 ∂ϕ g¯
Z∂M −1 ¯ dG ¯∧G ¯ −1 dG ¯∧G ¯ −1 dG ¯ . Tr G
(40)
M
Combining the sectors Putting together the reduced actions (38) and (40), and using the decomposition (32) of the Einstein-Hilbert action, we find that the latter can be re-expressed as follows in
13
terms of two chiral wzw actions (plus a term involving C): S|Fϕr =F¯ϕr =0 = Swzw [g] + S¯wzw [¯ g ] + I2 [C] I dτ ∧ dϕ Tr g −1 ∂ϕ g g −1 ∂τ g − g −1 ∂ϕ g = κ ∂M Z κ + Tr G−1 dG ∧ G−1 dG ∧ G−1 dG 3 M I dτ ∧ dϕ Tr g¯−1 ∂ϕ g¯ g¯−1 ∂τ g¯ + g¯−1 ∂ϕ g¯ −κ Z∂M −1 κ ¯ dG ¯∧G ¯ −1 dG ¯∧G ¯ −1 dG ¯ Tr G − 3 M I +κ dτ ∧ dϕ 4C 2 .
(41)
(42)
∂M (2) (2) As C simply denotes (half of) the component Aϕ = A¯ϕ at infinity, the last term of (42) can also be written in terms of the group-valued fields g and g¯. Explicitly, using the symmetric representation (35), we get I (2) −1 (2) ¯ dτ ∧ dϕ g −1 ∂ϕ g g¯ ∂ϕ g¯ . I2 [A, A] = I2 [g, g¯] = κ ∂M
This can be rewritten in a more “intrinsic” fashion by recognizing the wzw currents J = g −1 ∂ϕ g and J¯ = g¯−1 ∂ϕ g¯, and rewriting the term I2 as I (2) (2) dτ ∧ dϕ C(a)(b) J (a) J¯(b) , where C(a)(b) = δ(a) δ(b) . (43) I2 [g, g¯] = κ ∂M
This expression is a marginal deformation of the wzw action [34–36]. Moreover, the representation (35) is the unique combination of A and A¯ that can be interpreted, after reduction, as such a deformation: another combination of A and A¯ (one involving the arbitrary power p) would have given a deformation of conformal weights (p, 2 − p) instead of (1, 1), and would not have been marginal. We already stress, however, that the specific form of this deformation will not be relevant for the reduced theory that we will eventually obtain: all we need to know is that J (2) = J¯(2) = 2C. As C will turn out to be an independent dynamical field in the reduced theory (in fact it will be the momentum conjugate to Φ), we will express everything in terms of C alone, and the relation between I2 [C] and the wzw currents will be washed out by the reduction. See also the discussion below eq. (61). Note that the constraints Fϕr ≈ F¯ϕr ≈ 0 had nothing to do with boundary conditions: they are a universal feature of Chern-Simons theory. Our next goal is to reduce the theory further by solving additional constraints, now due to our choice of boundary conditions.
3.2
Gauss decomposition of the Wess-Zumino-Witten model
In order to reduce the theory (42) down to a Liouville-like system, we will adopt the following (standard) procedure [7, 21, 32]: 1. Pick local coordinates on the SL(2, R) group manifold (given in practice by the Gauss decomposition) and express all fields appearing in (42) in terms of these coordinates; 14
2. Rewrite the action (42) in terms of Gauss-decomposed fields and work out its Hamiltonian form; 3. Rewrite the boundary conditions (31) in terms of wzw fields and apply the Gauss decomposition to this rewriting; 4. Use step 3 to read off the conditions to be imposed on the Gauss-decomposed fields and their conjugate momenta – these conditions should play the role of additional constraints to be imposed on the Gauss-decomposed fields; 5. Take the Hamiltonian action obtained in step 2 and plug in the constraints coming from boundary conditions – this produces a reduced Hamiltonian action and a reduced Hamiltonian. The pages that follow contain a fairly detailed description of each of these steps, as applied to our problem. Although this is also standard material, we find it useful to review the derivation step by step. The reader who is familiar with these techniques may wish to skip the text and jump to the end result for the wzw action in terms of the Gauss decomposition, eq. (61). The reduction to a theory of chiral bosons will be performed in the next subsection. Gauss decomposition of the chiral wzw action In the chiral sector, we will decompose each SL(2, R) matrix g as −χ/2 1 0 e 0 1 τ g= . σ 1 0 1 0 eχ/2
(44)
(The τ here has nothing to do with the dimensionless time coordinate τ !) Then, if g denotes an SL(2, R)-valued field depending on some coordinates xµ , one finds −τ e−χ ∂µ σ − 21 ∂µ χ −τ 2 e−χ ∂µ σ − τ ∂µ χ + ∂µ τ −1 g ∂µ g = . (45) e−χ ∂µ σ τ e−χ ∂µ σ + 12 ∂µ χ In order to apply this to the chiral wzw action (38), we split the latter in two pieces: a Sigma model term and a Wess-Zumino term [37]. The Sigma model term of (38) is I Swzw |Sigma model = κ dτ ∧ dϕ Tr g −1 ∂ϕ g g −1 ∂τ g − g −1 ∂ϕ g . ∂M
In terms of Gauss-decomposed fields, we have 1 Tr g −1 ∂ϕ g g −1 ∂τ g − g −1 ∂ϕ g = χ0 (χ˙ − χ0 ) + τ 0 e−χ σ˙ + τ˙ e−χ σ 0 − 2τ 0 e−χ σ 0 . 2
(46)
Note that this expression is symmetric under σ ↔ τ , so the eventual σ/τ asymmetry of the model will be produced only by the Wess-Zumino term. The latter reads Z κ Swzw |Wess-Zumino = Tr G−1 dG ∧ G−1 dG ∧ G−1 dG . (47) 3 M In order to implement the Gauss decomposition in this term, we need to write down the Gauss decomposition of the three-dimensional field G: −X/2 1 0 e 0 1 T G= . (48) Σ 1 0 1 0 eX/2 15
The r-dependent fields Σ, T , X appearing here have, in general, nothing to do with those appearing in the decomposition (44) of the two-dimensional field g, so a subtle point of the computation below will consist in finding their mutual relation. For the moment, simply note that Tr G−1 dG ∧ G−1 dG ∧ G−1 dG = dr ∧ dτ ∧ dϕ εµνλ Tr G−1 ∂µ GG−1 ∂ν GG−1 ∂λ G , where we set εrτ ϕ ≡ +1. The expression of G−1 ∂µ G in terms of (Σ, T, X) is of the form (45) with the replacement (σ, τ, χ) → (Σ, T, X). Explicit computation then yields εµνλ Tr G−1 ∂µ GG−1 ∂ν GG−1 ∂λ G = −3 εµνλ ∂µ e−X ∂ν Σ∂λ T , so that, applying Stokes’ theorem, the Wess-Zumino term (47) can be written as I ˙ 0 − Σ0 T˙ . Swzw |Wess-Zumino = −κ dτ ∧ dϕ e−X ΣT
(49)
∂M
∂M
Now we have to find the relation between (Σ, T, X) and (σ, τ, χ). We have seen that the field G must factorize according to (37) in order for the condition Fϕr = 0 to hold. This implies, in particular, that the radial component of Ai = G−1 ∂i G is given by Ar = h−1 ∂r h. However, in our case Ar is not arbitrary: it is fixed according to (23), even off-shell. Hence h satisfies the differential equation 1/2r 0 −1 h ∂r h = , 0 −1/2r the general solution of which reads p r/` p0 h(r) = 0 `/r
(50)
up to multiplication from the left by an arbitrary τ -dependent matrix that can be absorbed in g(τ, ϕ) and is therefore irrelevant. We can now plug this solution in the product G = gh and use the Gauss decompositions (44) and (48) to find q p −X/2 r −χ/2 ` −χ/2 −X/2 e τ e e Te ! ` r q G= = p = g · h, −X/2 −X/2 X/2 r ` Σe ΣT e +e −χ/2 −χ/2 χ/2 σe στ e +e `
r
which imposes r ` e−X = e−χ , T = τ, Σ = σ. ` r Hence the Wess-Zumino term (49) finally reduces to I Swzw |Wess-Zumino = −κ dτ ∧ dϕ e−χ (στ ˙ 0 − σ 0 τ˙ ) .
(51)
∂M
Thus, putting together (46) and (51), we get the following expression for the chiral wzw action (38) in terms of the fields of the Gauss decomposition of g: I 1 0 0 −χ 0 0 χ (χ˙ − χ ) + 2e σ (τ˙ − τ ) . (52) Swzw [χ, σ, τ ] = κ dτ ∧ dϕ 2 ∂M 16
Gauss decomposition of the anti-chiral wzw action The computations in the anti-chiral sector are completely analogous to those of the chiral one, up to bars and signs. For completeness, we display the important steps of the calculation – the end result being eq. (60). In the anti-chiral sector, we will decompose each SL(2, R) matrix as χ/2 1 0 1 σ ¯ e¯ 0 . g¯ = ¯ τ¯ 1 0 1 0 e−χ/2
(53)
Then, if g¯ is an SL(2, R)-valued field, one finds τ¯e−χ¯ ∂µ σ ¯ + 21 ∂µ χ¯ e−χ¯ ∂µ σ ¯ −1 . g¯ ∂µ g¯ = −¯ τ 2 e−χ¯ ∂µ σ ¯ − τ¯∂µ χ¯ + ∂µ τ¯ −¯ τ e−χ¯ ∂µ σ ¯ − 12 ∂µ χ¯
(54)
As in the chiral sector, we split the anti-chiral wzw action (40) into a Sigma model piece and a Wess-Zumino term. The Sigma model part is I dτ ∧ dϕ Tr g¯−1 ∂ϕ g¯ g¯−1 ∂τ g¯ + g¯−1 ∂ϕ g¯ , S¯wzw Sigma model = −κ ∂M
where the Gauss decomposition (53) yields 1 Tr g¯−1 ∂ϕ g¯ g¯−1 ∂τ g¯ + g¯−1 ∂ϕ g¯ = χ¯0 (χ¯˙ + χ¯0 ) + τ¯0 e−χ¯ σ ¯˙ + τ¯˙ e−χ¯ σ ¯ 0 + 2¯ τ 0 e−χ¯ σ ¯0. 2
(55)
As for the Wess-Zumino term κ S¯wzw Wess-Zumino = − 3
Z
−1 ¯ dG ¯∧G ¯ −1 dG ¯∧G ¯ −1 dG ¯ , Tr G
(56)
M
we use the Gauss decomposition X/2 ¯ ¯ 1 Σ e 0 1 0 ¯= G . ¯ 0 1 T¯ 1 0 e−X/2
(57)
Observing that −1 −1 ¯ dG ¯∧G ¯ −1 dG ¯∧G ¯ −1 dG ¯ = dr ∧ dτ ∧ dϕ εµνλ Tr G ¯ ∂µ G ¯G ¯ −1 ∂ν G ¯G ¯ −1 ∂λ G ¯ Tr G ¯ T¯, X), ¯ we find and using (54) with the replacement (¯ σ , τ¯, χ) ¯ → (Σ, h i −1 ¯ µνλ −1 −1 µνλ −X ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ε Tr G ∂µ GG ∂ν GG ∂λ G = −3 ε ∂µ e ∂ν Σ∂λ T . Hence the Wess-Zumino term (56) can be written as I ¯ ¯ ˙ T¯0 − Σ ¯ ¯ 0 T¯˙ Swzw Wess-Zumino = κ dτ ∧ dϕ e−X Σ ∂M
where we used Stokes’ theorem once more.
17
, ∂M
(58)
¯ T¯, X) ¯ and the fields (¯ We now establish the relation between (Σ, σ , τ¯, χ) ¯ of the Gauss −1 ¯ ¯ ¯ decomposition of g¯. Using (39) and the fact that Ai = G ∂i G, we obtain (23) −1/2r 0 −1 ¯ ∂r h ¯ = A¯r = h , 0 1/2r which is to be considered as an off-shell relation. As in the chiral case, this relation is solved by an r-dependent matrix p `/r p0 ¯h = . 0 r/` ¯ and using the Gauss decompositions (53) and (57) for g¯ ¯ = g¯h Then, plugging this in G ¯ (respectively), we find and G q p r −χ/2 X/2 ` χ/2 ¯ −χ/2 ¯ ¯ ¯ ¯ ¯ −X/2 −X/2 ¯ ¯ ¯ e + σ ¯ τ ¯ e σ ¯ e + ΣT e Σe ! r ` ¯ ¯= e = g¯ · h. q G = ¯ ¯ p −X/2 −X/2 ¯ ` r −χ/2 −χ/2 ¯ ¯ Te e τ¯e e r ` This implies the identifications r ¯ e−X = e−χ¯ , `
` T¯ = τ¯, r
¯ =σ Σ ¯,
so that the Wess-Zumino term (58) reduces to I ¯ dτ ∧ dϕ e−χ¯ (σ ¯˙ τ¯0 − σ ¯ 0 τ¯˙ ) . Swzw Wess-Zumino = κ
(59)
∂M
We can then combine (55) and (59) into the full wzw action of the anti-chiral sector: I 1 0 0 − χ ¯ 0 0 S¯wzw [χ, ¯ σ ¯ , τ¯] = κ dτ ∧ dϕ − χ¯ (χ¯˙ + χ¯ ) − 2e σ ¯ (τ¯˙ + τ¯ ) . (60) 2 ∂M Combining the sectors For future reference, let us write down the full Einstein-Hilbert action (41) in terms of the fields of the Gauss decomposition. Combining (52) and (60), we find S[χ, σ, τ, χ, ¯ σ ¯ , τ¯, C] I 1 0 1 0 0 −χ 0 0 0 − χ ¯ 0 0 2 = κ dτ ∧ dϕ χ (χ˙ − χ ) + 2e σ (τ˙ − τ ) − χ¯ (χ¯˙ + χ¯ ) − 2e σ ¯ (τ¯˙ + τ¯ ) + 4C . (61) 2 2 ∂M At this point one may be puzzled by the fact that we have not written the field C in terms of the fields of the Gauss decomposition. This rewriting can be read off from the components of the currents g −1 ∂ϕ g and g¯−1 ∂ϕ g¯, along the generator j(2) , in (45) and (54): 1 1 C = −τ e−χ σ 0 − χ0 = τ¯e−χ¯ σ ¯ 0 + χ¯0 . 2 2
(62)
In principle we could plug this expression in (61) and express everything in terms of the Gauss fields. (In particular, the choice (43) would lead to an expression that is symmetric in barred and unbarred fields.) However, as already mentioned before, C will turn out to be an independent dynamical field in the dual theory, and we therefore prefer to keep writing C explicitly, viewing the equalities (62) as constraints that express τ and τ¯ in terms of C, rather than C in terms of the Gauss fields. 18
3.3
From Wess-Zumino-Witten to chiral bosons
Here we perform the final step of the reduction, by working out the constraints to be imposed on (61) as a consequence of our choice of boundary conditions. The translation to Liouville theory will be performed in the next section. Hamiltonian formulation of the wzw model We begin by formulating the Hamiltonian version of the wzw-like action (61). The conjugate momenta p = ∂L/∂ q˙ are given by πχ ≈ κ2 χ0 , πχ¯ ≈ − κ2 χ¯0 ,
πσ ≈ 0, πσ¯ ≈ 0,
πτ ≈ 2κe−χ σ 0 , πτ¯ ≈ −2κe−χ¯ σ ¯0,
(63)
together with πC ≈ 0. All these relations are obviously constraints – which was to be expected, since the wzw models we started with were chiral wzw models. Accordingly, the Hamiltonian density for the system (61) is just its potential energy density: 1 02 1 02 −χ 0 0 −χ ¯ 0 0 2 (64) χ + χ¯ + 2e σ τ + 2e σ ¯ τ¯ − 4C . H=κ 2 2 Note that the term C 2 makes this expression unbounded from below. The corresponding Hamiltonian action is of course I SH [fields, momenta] = dτ ∧ dϕ [πχ χ˙ + πχ¯ χ¯˙ + πτ τ˙ + πτ¯ τ¯˙ − H] , (65) ∂M
where we have already implemented the constraints πσ ≈ πσ¯ ≈ πC ≈ 0. List of constraints We are now ready to implement the additional constraints imposed on the system by our choice of boundary conditions. To find these constraints, we need to look back at the boundary conditions (31) – more precisely, the angular components of A and A¯ therein. As we have seen above, when the constraints Fϕr = F¯ϕr = 0 are solved, these components are given by Aϕ = G−1 ∂ϕ G (and similarly in the anti-chiral sector). But G can be factorized as G = gh, and h is known explicitly from (50). So the condition that Aϕ should be as in (31) is really a condition on g; when g is written as a Gauss decomposition, this condition yields constraints on the fields χ, σ and τ . We now carry out the computation of these constraints explicitly, first for the chiral sector, then for the anti-chiral sector. In the chiral sector, the identification Aϕ = G−1 ∂ϕ G = h−1 (g −1 ∂ϕ g) h implies that g −1 ∂ϕ g = hAϕ h−1 ∂M , where h is given by (50). Using the boundary conditions (31) and the expression (45) following from the Gauss decomposition of g, this means that −τ e−χ σ 0 − 21 χ0 −τ 2 e−χ σ 0 − τ χ0 + τ 0 C e−Φ γ++ = Φ . e −C e−χ σ 0 τ e−χ σ 0 + 21 χ0 Here the condition involving γ++ can be seen as a mere definition of γ++ : γ++ ≡ eΦ −τ 2 e−χ σ 0 − τ χ0 + τ 0 . 19
(66)
The remaining two conditions, however, are genuine constraints: 1 −τ e−χ σ 0 − χ0 ≈ C, 2 −χ 0 e σ ≈ eΦ .
(67) (68)
These conditions extend those usually encountered for Brown-Henneaux boundary conditions, which are recovered upon setting C = Φ = 0. Similarly, in the anti-chiral sector, the property ¯ A¯ϕ h ¯ −1 g¯−1 ∂ϕ g¯ = h ∂M yields τ¯e−χ¯ σ ¯ 0 + 21 χ¯0 e−χ¯ σ ¯0 −¯ τ 2 e−χ¯ σ ¯ 0 − τ¯χ¯0 + τ¯0 −¯ τ e−χ¯ σ ¯ 0 − 21 χ¯0
=
C
−e−Φ γ−−
−eΦ . −C
As in the chiral sector, the identification involving γ−− is merely the definition γ−− ≡ −eΦ −¯ τ 2 e−χ¯ σ ¯ 0 − τ¯χ¯0 + τ¯0 ,
(69)
but the other two conditions are actual constraints: 1 τ¯e−χ¯ σ ¯ 0 + χ¯0 ≈ C, 2 −χ e ¯σ ¯ 0 ≈ −eΦ .
(70) (71)
We now proceed to plug these constraints into the theory (65). Implementing constraints We begin by reducing the Hamiltonian density (64). First eliminate σ and σ ¯ using the constraints (68) and (71), which gives 1 02 1 02 Φ 0 0 2 H=κ χ + χ¯ + 2e (τ − τ¯ ) − 4C (72) 2 2 and reduces the constraints (67) and (70) to 1 1 τ eΦ + χ¯0 ≈ C. − τ eΦ − χ0 ≈ −¯ 2 2
(73)
This, in turn, implies τ − τ¯ = − 21 e−Φ (χ0 + χ¯0 ), which can be plugged into (72) to produce 1 02 1 02 00 00 0 0 0 2 H=κ χ − χ + χ¯ − χ¯ + Φ (χ + χ¯ ) − 4C . (74) 2 2 We cannot go further than that, since we have used all the constraints. The field C, in particular, cannot be eliminated in terms of other fields. So we have four dynamical fields to take into account in the Hamiltonian formulation: χ, χ, ¯ Φ and C. We will see in the next subsection that C should be interpreted as the momentum conjugate to Φ, even though it appears with a negative sign in the Hamiltonian. The total derivatives χ00 and χ¯00 in (74) do not contribute to the Hamiltonian (which is the integral of (74) on the 20
circle), but they play a crucial role in the identification of H with the Noether current of conformal symmetry. See subsection 4.3. For future reference, we write down the explicit expression of the functions γ±± defined in (66) and (69) in terms of the fields of the reduced theory: γ++ γ−−
1 02 χ − = 4 1 02 = χ¯ − 4
1 0 1 00 2 0 0 χ −C −C +Φ χ +C , 2 2 1 0 1 00 2 0 0 χ¯ − C + C + Φ χ¯ − C . 2 2
(75)
As is obvious here, H = 2κ (γ++ + γ−− ). We therefore expect γ++ and γ−− to be proportional to the corresponding components of the reduced energy-momentum tensor3 . This expectation will be confirmed below. We now turn to the Hamiltonian action (65). The latter involves the Hamiltonian density H, for which the reduction goes through as in the previous paragraph, allowing us to simply replace the H in the action by its reduced form, eq. (74). So it remains to reduce the kinetic term of (65). To begin, use the constaints (63) to eliminate momenta, so that the Hamiltonian action becomes I i hκ κ (76) dτ ∧ dϕ χ0 χ˙ − χ¯0 χ¯˙ + 2κeΦ (τ˙ + τ¯˙ ) − H , SH [χ, χ, ¯ τ, τ¯, Φ, C] = 2 2 ∂M where we have also used (68) and (71) to eliminate σ 0 and σ ¯ 0 . Now use (73) to obtain 1˙ 0 1 0 1 0 0 −Φ ˙ ˙ ˙ ˙ τ˙ + τ¯ = e − χ˙ + χ¯ − 2C + 2ΦC + Φ(χ − χ¯ ) , 2 2 2 which reduces (76) to I SH [χ, χ, ¯ Φ, C] =
κ 0 κ 0 1 0 0 ˙ + Φ(χ ˙ dτ ∧ dϕ χ χ˙ − χ¯ χ¯˙ + 2κ 2ΦC − χ¯ ) − H , 2 2 2 ∂M
up to total derivatives that are irrelevant for the two-dimensional dynamics. Using the expression (74) of the reduced Hamiltonian, ignoring total derivatives and integrating by parts, we can rewrite this further as I 1 0 1 0 0 0 0 0 2 ˙ SH [χ, χ, ¯ Φ, C] = κ dτ ∧dϕ χ + Φ (χ˙ − χ ) − χ¯ + Φ (χ¯˙ + χ¯ ) + 4C Φ + 4C . 2 2 ∂M (77) This is our final expression for the reduced action of the dual theory. When C = Φ = 0, it is just the action for two chiral bosons (which, in turn, is canonically equivalent to the action of a Liouville field – see below), reproducing the well-known result [21] encountered with Brown-Henneaux boundary conditions. 3
Note that the relation between H and γ±± did not hold in the expressions (64), (66) and (69) written before the reduction.
21
4
The dual theory
In this section we discuss classical properties of the theory described by the action (77). We first derive its equations of motion and work out its relation to Liouville theory. Then we classify its symmetries (and the corresponding Noether charges) and show that they match the asymptotic symmetries of the bulk. RNote: In this section, we write the integral over the timelike boundary of AdS3 simply as dτ dϕ instead of the more accurate notation used in (77).
4.1
Equations of motion
Let us superficially analyze the reduced action (77) by computing its Lagrangian form and finding the associated equations of motion. First, the requirement δSH /δC = 0 yields 1 C = − Φ˙ (on-shell). (78) 2 This is consistent with the on-shell expressions (21) and (22) for the Chern-Simons gauge fields, as comparison between these expressions and (24) readily yields the equality (78). The term C Φ˙ in the reduced action suggests that C should be (proportional to) the momentum4 canonically conjugate to Φ, which is further confirmed by the relation (78). Hence we introduce the Lagrangian action corresponding to (77), Z 1 0 1 0 0 0 0 0 2 χ + Φ (χ˙ − χ ) − χ¯ + Φ (χ¯˙ + χ¯ ) − Φ˙ , (79) S [χ, χ, ¯ Φ] = κ dτ dϕ 2 2 which is just the Hamiltonian action (77) where we have plugged the relation (78) between Φ and its momentum. In this expression the kinetic term for Φ has the wrong sign, indicating that the conformal factor of the boundary metric is unstable – this is just a Lagrangian equivalent of the statement that the Hamiltonian density (74) is unbounded from below because of the C 2 term. More generally, the instability of the conformal factor of the metric is a well-known problem in general relativity [38], and it is all but surprising that we recover this feature in the dual theory. It is illuminating to compute the equations of motion that follow from the action (79). First, when varying Φ and requiring the variation of the action to vanish, we find ¨ = (χ˙ − χ0 )0 − (χ¯˙ + χ¯0 )0 . 2Φ (80) On the other hand, varying χ and χ, ¯ we obtain − (χ˙ − χ0 )0 − Φ˙ 0 + Φ00 = 0 and (χ¯˙ + χ¯0 )0 + Φ˙ 0 + Φ00 = 0, which implies (χ˙ − χ0 )0 − (χ¯˙ + χ¯0 )0 = 2Φ00 . ¨ = Φ00 , or Thus, the equation of motion (80) for Φ can be written as Φ ∂+ ∂− Φ = 0,
(81)
which is nothing but the requirement (6) that γ+− vanish. In this sense, the dual theory that we have found is consistent with the gravitational theory we started with. More precisely, the action (77) yields the Poisson bracket {Φ(ϕ), 4κC(ϕ0 )} = δ(ϕ − ϕ0 ), so the momentum associated with Φ is 4κC. 4
22
4.2
The dual theory as a Liouville field in a curved background
The equation of motion (81) suggests that the action (79), which mixes χ, χ¯ and Φ in a somewhat intricate fashion, should admit a simpler rewriting. This is indeed the case, as the field redefinition ψ ≡ χ + Φ and ψ¯ ≡ χ¯ + Φ (82) puts (79) in the following form: Z ¯ Φ = κ dτ dϕ ψ 0 ∂− ψ − ψ¯0 ∂+ ψ¯ − 4∂+ Φ∂− Φ . S ψ, ψ,
(83)
¯ the functions (75) become In terms of ψ and ψ, 1 02 1 00 ψ − ψ − (∂+ Φ)2 + (∂+ Φ)0 , 4 2 (84) 1 ¯02 1 ¯00 2 0 γ−− = ψ − ψ − (∂− Φ) − (∂− Φ) . 4 2 When Φ satisfies its equation of motion (81), the primes acting on Φ on the right-hand side can be replaced by derivatives with respect to x+ and x− , respectively (up to a sign in the second equation). Comparison with (4) then yields the identifications γ++ =
1 1 1 1 (85) Ξ++ = ψ 02 − ψ 00 , Ξ−− = ψ¯02 − ψ¯00 . 4 2 4 2 We will come back in subsection 4.3 to the interpretation of these fields in terms of the energy-momentum tensor of the dual theory. Of course, the relation H = 2κ (γ++ + γ−− ) mentioned below (75) still holds. As is obvious in (83), the fields (82) behave as chiral bosons that are decoupled from the conformal factor Φ. This may seem physically disturbing because our system should be a field theory living on a two-dimensional curved background, with conformally flat metric gab = e2Φ ηab , while the splitting in (83) shows that the coupling with the conformal factor of the metric is fictitious and can be absorbed by a field redefinition. We will solve this apparent clash at the end of this subsection. For now, let us rewrite the action (83) in Liouville form. This is done through the Bäcklund transformation [22] Ψ ≡ ψ + ψ¯ − 2 ln (σ − σ ¯ ) + ln(8), 0 σ +σ ¯0 κ 0 ¯0 ψ −ψ −2 , Π ≡ 2 σ−σ ¯ ¯
where the fields σ and σ ¯ satisfy the properties σ 0 = eψ and σ ¯ 0 = −eψ , as in (68) and (71). The constant ln(8) in the definition of Ψ is added here for convenience, as it leads to a simple normalization for the Liouville potential ∼ eΨ . The crucial property of the Bäcklund transformation is the fact that Z Z 0 1 κ κ 0 2 02 Ψ ˙ − Π − Ψ − e , κ dτ dϕ ψ ∂− ψ − ψ¯ ∂+ ψ¯ = dτ dϕ ΠΨ κ 4 2 where the right-hand side is the Hamiltonian action for the Liouville field Ψ and its conjugate momentum. Eliminating the latter through its equation of motion, the full theory (79) is thus described equivalently (at least classically) by the Lagrangian action Z 1 Ψ S[Ψ, Φ] = κ dτ dϕ ∂+ Ψ∂− Ψ − e − 4∂+ Φ∂− Φ . (86) 2 23
In terms of Ψ, the quantities (85) read Ξ±± =
1 1 (∂± Ψ)2 − ∂±2 Ψ, 4 2
(87)
provided both Ψ and Π satisfy their equations of motion. One then verifies that the Hamiltonian density for (86) is given, on-shell, by H = 2κ (γ++ + γ−− ), with the γ’s defined in (4) and the Ξ’s written as in eq. (87). As before, the fields appearing in (86) are completely decoupled from each other. Hence, though Ψ is obviously a Liouville field, it does not coincide with the Liouville field φ encountered when using pure Brown-Henneaux boundary conditions. Indeed, Ψ contains contributions of both the “Brown-Henneaux” field φ and the conformal factor Φ, the only restriction being that Ψ must reduce to φ when Φ vanishes. Now, since the two-dimensional metric reads gab (x) = e2Φ(x) ηab (88) as follows from (2), a natural guess is that Ψ should be related to φ and Φ according to Ψ = φ + 2Φ,
(89)
which, plugged into (86), yields the action Z κ dτ dϕ e2Φ 2e−2Φ ∂+ φ∂− φ − eφ − 8e−2Φ φ∂+ ∂− Φ . S[φ, Φ] = 2 The latter coincides with κ S[φ, g] = 2
Z
√ 1 ab φ d x −g g ∂a φ∂b φ − e + φR 2 2
(90)
upon writing the metric gab as in (88). We thus interpret the theory as consisting of a Liouville field φ coupled to the two-dimensional metric in conformal gauge. This Liouville field, now, is the “Brown-Henneaux” field5 . By the magic of two dimensions, its coupling with the conformal factor of the metric can be absorbed by a field redefinition (89) that decouples Ψ from Φ [28]. This interpretation solves the apparent inconsistency mentioned above. We have thus arrived at a physically pleasing description of the dual theory. For practical calculations, however, the decoupled description (86) is the most convenient, and we will therefore stick to it below.
4.3
Symmetries and charges
We finally turn to the computation of symmetries, charges, and their Poisson brackets for the dual theory (86). In contrast to the previous paragraphs, our viewpoint here will be mostly Lagrangian and on-shell. Poisson brackets can then be computed using the canonical formalism in light-cone coordinates. 5
The theory (90) already appeared in [23, 24] as a dual theory for AdS3 gravity. In these references, however, the two-dimensional metric (and in particular its conformal factor) was not dynamical, and this is an essential distinction between [23, 24] and our work.
24
The Liouville action in (86) is conformally invariant provided eΨ is a primary field with dimensions (1, 1). Accordingly, the transformation law of Ψ under infinitesimal conformal transformations reads δΨ = −∂+ Ψ − ¯∂− Ψ − ∂+ − ∂− ¯,
(91)
where and ¯ are arbitrary chiral and anti-chiral parameters (resp.), as in (8). As regards the conformal factor Φ, its conformal transformation law follows from that of the metric, requiring that e2Φ behave as a primary field with dimensions (1, 1). Therefore the infinitesimal conformal transformation law of Φ is 1 1 δΦ = −∂+ Φ − ¯∂− Φ − ∂+ − ∂− ¯. 2 2
(92)
From the viewpoint of the action (86), however, arbitrary combinations of the form a∂+ + b∂− ¯ can be added at will in the transformation of Φ; such terms amount to the composition of conformal transformations with Weyl transformations, which affect Φ but leave Ψ unchanged. Explicitly, Weyl transformations act on Ψ and Φ according to6 δΨ = 0,
δΦ(x) = ω(x)/2,
(93)
where ω(x) is an arbitrary harmonic function, as in (9). From a conformal field-theoretic point of view, when the zero-mode Ωτ of ω is excluded, such transformations act as affine U(1) transformations. The Noether current associated with conformal invariance7 of the action (86) is the energy-momentum tensor, given on-shell by 1 2 1 2 2 2 (∂± Ψ) − ∂± Ψ − (∂± Φ) + ∂± Φ (94) T±± = 2κ 4 2 = 2κ Ξ±± − (∂± Φ)2 + ∂±2 Φ = 2κγ±± , where we recognize the combinations (87) and (4). Owing to the AdS/CFT dictionary, it is no surprise that the terms ∝ γ±± in the metric (7) coincide with the stress tensor of the boundary theory. The Hamiltonian density of the system is of course H = T++ + T−− = 2κ (γ++ + γ−− ), in accord with the observation made below (75). Again, the total derivatives in (94) are irrelevant as far as the Hamiltonian (the zero-mode of the Hamiltonian density) of the system is concerned, but they are crucial in identifying T±± with the Noether current of conformal symmetry. In particular, the coefficients in front of ∂±2 Ψ and ∂±2 Φ are chosen in such a way that the Noether charge Z Q,¯ [Ψ, Φ] =
2π
Z dϕ [T++ + ¯ T−− ] = 2κ
0
2π
dϕ [γ++ + ¯ γ−− ]
(95)
0
precisely generates (through Poisson brackets) the transformations (91) and (92). Choosing T±± with an arbitrary coefficient a in front of the term ∂±2 Φ (instead of a = 1) would 6 The normalization of ω is chosen in accordance with expression (8) for the asymptotic Killing vector fields. 7 Here we use the abuse of notation that consists in neglecting to write infinitesimal parameters in the Noether currents [37]. These parameters then only appear in the conserved charges – see eq. (95).
25
generate conformal transformations of Φ where the 1/2’s of eq. (92) are replaced by a/2. In terms of coordinates xa on the cylinder, the Noether current associated with Weyl transformations (93), labelled by a parameter ω, reads J a = κη ab (Φ ∂b ω − ω ∂b Φ) .
(96)
The associated charge is Z Qω [Φ] = κ
2π
h i dϕ ω Φ˙ − Φω˙ .
(97)
0
At this point, the matching between global symmetries of the dual theory and asymptotic symmetries of the bulk is manifest: since κ = `/16πG, the surface charge (10) is simply the sum of (95) and (97). When written in canonical form (or else using the canonical formalism in light-cone coordinates), the Poisson brackets of Noether charges similarly coincide with the asymptotic symmetry algebra, as can be checked by direct computation. Before concluding, let us note that the Weyl current (96) can be written in a form that makes manifest its relation to affine U(1) symmetry. Indeed, decomposing ω as in (9), the time component of (96) becomes ¯ Φ˙ − Φλ ¯˙ + Ωτ Φ˙ − ΩΦ . J τ = κ λΦ˙ − Φλ˙ + λ Of course this current is trivially equivalent to any other current of the form J˜a = J a + ∂b k [ab] . In the present case, choosing ¯ k [ab] = κεab (λ − λ)Φ with ετ ϕ ≡ +1, we find
τ ¯ ˜ ˙ J = λJ+ + λJ− + κ Ωτ Φ − ΩΦ ,
with J± = 2κ∂± Φ.
Here J+ and J− should be interpreted as affine U(1) currents. This shows in particular that the full energy-momentum tensor (94) is just the usual stress tensor of the Liouville field Ψ (proportional to Ξ±± ), supplemented by a twisted Sugawara stress tensor [26]: S T±± = 2κΞ±± + T±± ,
S T±± =−
1 (J± )2 + ∂± J± 2κ
Note the negative sign in front of (J± )2 in the Sugawara stress tensor, which is consistent with the negative level k = −2πκ of the U(1) current algebra in (11).
Conclusion We have shown, by explicit Hamiltonian reduction, that the two-dimensional theory dual to AdS3 gravity with “free” boundary conditions (in the sense of eqs. (2) and (3)) is a Liouville theory on a curved background with a conformally flat metric. The corresponding reduced action is (90) with gab = e2Φ ηab , the field φ being precisely the Liouville field 26
encountered with Brown-Henneaux boundary conditions [21]. We have also shown how the global symmetries of the reduced model match the asymptotic symmetries of the three-dimensional gravitational system. Owing to the AdS/CFT correspondence [2, 3] and to earlier work on free boundary conditions for AdS [10, 23, 24, 26], this result was to be expected. Its emergence from the Chern-Simons formalism, and in particular the spontaneous appearance of a timelike boson in the reduced action, is nevertheless a pleasing illustration of the self-consistency of the reduction procedure. Our approach should be completed in several ways. First, throughout this work, we have assumed that there are no holonomies. This assumption should be relaxed and the corresponding additional ingredients should be included in the dual theory, in the spirit of [22, 24]. Furthermore, our considerations were exclusively classical, while a complete understanding of the duality requires, in principle, control over quantum aspects. Quantization of Liouville theory is a research field in itself (see e.g. [19, 39–41]), and the appearance of the conformal factor of the two-dimensional metric might have interesting consequences in that context. In particular, the negative Kac-Moody level and the wrong sign of the kinetic term of the conformal factor suggests some non-unitarity.
Acknowledgements The authors are grateful to Cédric Troessaert for precious comments and support. B.O. also acknowledges useful discussions on Liouville theory with Teresa Bautisata Solans and Harold Erbin. B.O. is a research fellow of the Belgian National Fund for Scientific Research (FNRS).
References [1] J. David Brown and Marc Henneaux. Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity. Commun. Math. Phys., 104:207–226, 1986. [2] Juan Martin Maldacena. The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys., 2:231–252, 1998. [3] Edward Witten. Anti-de Sitter space and holography. Adv.Theor.Math.Phys., 2:253– 291, 1998. [4] Máximo Bañados, Claudio Teitelboim, and Jorge Zanelli. The Black hole in threedimensional space-time. Phys.Rev.Lett., 69:1849–1851, 1992. [5] Máximo Bañados, Marc Henneaux, Claudio Teitelboim, and Jorge Zanelli. Geometry of the (2+1) black hole. Phys.Rev., D48:1506–1525, 1993. [6] Andrew Strominger. Black hole entropy from near horizon microstates. JHEP, 9802:009, 1998. [7] Geoffrey Compère, Wei Song, and Andrew Strominger. New Boundary Conditions for AdS3 . JHEP, 1305:152, 2013.
27
[8] Steven G. Avery, Rohan R. Poojary, and Nemani V. Suryanarayana. An sl(2,R) current algebra from AdS3 gravity. JHEP, 1401:144, 2014. [9] Cédric Troessaert. 1308:044, 2013.
Enhanced asymptotic symmetry algebra of AdS 3 .
JHEP,
[10] Geoffrey Compere and Donald Marolf. Setting the boundary free in AdS/CFT. Class.Quant.Grav., 25:195014, 2008. [11] A. Achucarro and P.K. Townsend. A chern-simons action for three-dimensional antide sitter supergravity theories. Physics Letters B, 180(1-2):89 – 92, 1986. [12] Edward Witten. (2+1)-Dimensional Gravity as an Exactly Soluble System. Nucl. Phys., B311:46, 1988. [13] Ralph Abraham and Jerrold E. Marsden. Foundations of Mechanics. AMS Chelsea publishing. AMS Chelsea Pub./American Mathematical Society, 1978. [14] V.I. Arnol’d. Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics. Springer, 1989. [15] Edward Witten. Nonabelian bosonization in two dimensions. Communications in Mathematical Physics, 92(4):455–472, 1984. [16] Edward Witten. Quantum field theory and the jones polynomial. Communications in Mathematical Physics, 121(3):351–399, 1989. [17] G. Moore and N. Seiberg. Taming the conformal zoo. Physics Letters B, 220:422–430, April 1989. [18] Shmuel Elitzur, Gregory Moore, Adam Schwimmer, and Nathan Seiberg. Remarks on the canonical quantization of the chern-simons-witten theory. Nuclear Physics B, 326(1):108 – 134, 1989. [19] A. M. POLYAKOV. Quantum gravity in two dimensions. Modern Physics Letters A, 02(11):893–898, 1987. [20] P. Forgacs, A. Wipf, J. Balog, L. Fehér, and L. O’Raifeartaigh. Liouville and toda theories as conformally reduced {WZNW} theories. Physics Letters B, 227(2):214 – 220, 1989. [21] Olivier Coussaert, Marc Henneaux, and Peter van Driel. The Asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant. Class.Quant.Grav., 12:2961–2966, 1995. [22] Marc Henneaux, Liat Maoz, and Adam Schwimmer. Asymptotic dynamics and asymptotic symmetries of three-dimensional extended AdS supergravity. Annals Phys., 282:31–66, 2000. [23] M. Rooman and P. Spindel. Aspects of (2+1)-dimensional gravity: AdS(3) asymptotic dynamics in the framework of Fefferman-Graham-Lee theorems. Annalen Phys., 9:161–167, 2000.
28
[24] M. Rooman and P. Spindel. Holonomies, anomalies and the Fefferman-Graham ambiguity in AdS(3) gravity. Nucl.Phys., B594:329–353, 2001. [25] Geoffrey Compère, Wei Song, and Andrew Strominger. Chiral Liouville Gravity. JHEP, 1305:154, 2013. [26] Luis Apolo and Massimo Porrati. Free boundary conditions and the AdS3 /CFT2 correspondence. JHEP, 1403:116, 2014. [27] Michael Gutperle and Andrew Strominger. Time - like boundary Liouville theory. Phys.Rev., D67:126002, 2003. [28] Steven Carlip. Liouville lost, Liouville regained: Central charge in a dynamical background. Phys.Lett., B508:168–172, 2001. [29] M. Banados. Three-Dimensional Quantum Geometry and Black Holes. In Trends in Theoretical Physics II, volume 484 of AIP Conf. Series, pages 147–169, 1999. [30] Kostas Skenderis and Sergey N. Solodukhin. Quantum effective action from the AdS/CFT correspondence. Phys. Lett., B472:316–322, 2000. [31] M. Blagojević. Gravitation and Gauge Symmetries. Series in High Energy Physics, Cosmology and Gravitation. Taylor & Francis, 2001. [32] Glenn Barnich and Hernán A. González. Dual dynamics of three dimensional asymptotically flat Einstein gravity at null infinity. JHEP, 1305:016, 2013. [33] P. Salomonson, B.-S. Skagerstam, and A. Stern. Canonical quantization of chiral bosons. Phys. Rev. Lett., 62:1817–1820, Apr 1989. [34] S. Chaudhuri and J. A. Schwastz. A criterion for integrably marginal operators. Physics Letters B, 219:291–296, March 1989. [35] S.F. Hassan and Ashoke Sen. Marginal deformations of WZNW and coset models from O(d,d) transformation. Nucl.Phys., B405:143–165, 1993. [36] Anatoly Konechny and Thomas Quella. Non-chiral current algebras for deformed supergroup WZW models. JHEP, 1103:124, 2011. [37] Philippe Di Francesco, Pierre Mathieu, and David Sénéchal. Conformal Field Theory. Springer, 1997. [38] G. W. Gibbons, S. W. Hawking, and M. J. Perry. Path integrals and the indefiniteness of the gravitational action. Nuclear Physics B, 138:141–150, June 1978. [39] A.M. Polyakov. Quantum geometry of bosonic strings. Physics Letters B, 103(3):207 – 210, 1981. [40] Jacques Distler and Hikaru Kawai. Conformal field theory and 2d quantum gravity. Nuclear Physics B, 321(2):509 – 527, 1989. [41] Nathan Seiberg. Notes on quantum Liouville theory and quantum gravity. Prog.Theor.Phys.Suppl., 102:319–349, 1990.
29
[42] Glenn Barnich and Cédric Troessaert. Aspects of the BMS/CFT correspondence. JHEP, 05:062, 2010. [43] Glenn Barnich, Andrés Gomberoff, and Hernán A. González. Three-dimensional Bondi-Metzner-Sachs invariant two-dimensional field theories as the flat limit of Liouville theory. Phys.Rev., D87(12):124032, 2013. [44] Marc Bernstein and Jacob Sonnenschein. Comment on the quantization of chiral bosons. Phys. Rev. Lett., 60:1772–1772, Apr 1988.
30
