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Nuclear Physics B 884 (2014) 66-73

www. elsevier. com/locate/nuclphysb

Light-cone gravity in AdS4

Y.S. Akshay, Sudarshan Ananth, Mahendra Mali

Indian Institute of Science Education and Research, Pune 411008, India Received 12 March 2014; accepted 16 April 2014 Available online 23 April 2014 Editor: Stephan Stieberger

Abstract

We obtain a closed form expression for the Action describing pure gravity, in light-cone gauge, in a four-dimensional Anti-de Sitter background. We perform a perturbative expansion of this closed form result to extract the cubic interaction vertex in this gauge.

© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

1. Introduction

Quantum field theories of gravity are plagued by divergences that seem to rule out any straightforward attempt to unite quantum theory and the general theory of relativity. There are however quite a few reasons to still study gravity as a quantum field theory. Foremost among these are the existence of surprising perturbative ties between gravity and the better understood Yang-Mills theory.1 These perturbative ties, stemming from the KLT relations [1], tell us that tree level scattering amplitudes in gravity are the square of tree level scattering amplitudes in Yang-Mills theory. A Lagrangian (off-shell) origin for this relationship has also emerged [2] but a complete understanding of this important bridge between the two theories is still elusive.

The KLT relations are valid on flat spacetime backgrounds so one question that motivates the present work is whether such perturbative ties between Yang-Mills and gravity survive when we move to curved spacetime backgrounds. It is not clear how a Yang-Mills ^ gravity relationship,

1 The link between Gravity and Yang-Mills is surprising given the significant differences between the theories: dimen-sionful coupling versus dimensionless coupling, no color structure versus color traces and so on.

http://dx.doi.Org/10.1016/j.nuclphysb.2014.04.015

0550-3213/© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

on AdS4, would trace back to a stringy origin. However, it is interesting to study this connection from a purely field theoretical point of view.

In this paper, we set up much of the light-cone (helicity) formalism essential to identifying such links at the level of the Action [2]. These perturbative ties seem to extend beyond the spin 1-spin 2 system. In particular, one may derive off-shell versions of these relations at cubic order for a spin 1-spin X system [3]. This is a further point of interest when examining the fate of these relations on curved backgrounds.

Another motivation stems from our work on higher spin theories [3]. There are various stumbling blocks when attempting to derive a Lagrangian describing an interacting higher spin theory. While the equations of motion are well studied [4] we do not have an Action to quantize and it remains unclear whether we can define a consistent interacting S-matrix for such theories. The light-cone gauge approach to higher spin fields [5] yielded some of the first examples of consistent Lagrangians, on flat backgrounds, describing the cubic interactions of three fields, all of spin X. The higher-spin story in curved backgrounds is different since the no-go theorems established for flat backgrounds no longer hold. To achieve a light-cone formulation of higher spin fields (X > 2) on curved backgrounds, it is essential to have as a guidepost the Action for pure gravity (X = 2) on those backgrounds. This is one of the results we obtain here.

In this paper, we describe how pure gravity is formulated in light-cone gauge on an AdS4 background. This is achieved by making suitable gauge choices and using the constraint relations to eliminate the unphysical degrees of freedom. This will allow us to describe the Action of light-cone gravity on AdS4 in a closed form purely using the physical degrees of freedom. We also perform a perturbative expansion of this gauge-fixed Action to first order in the gravitational coupling constant and comment on the resulting interaction vertex.

2. Preliminaries

The Einstein-Hilbert action reads

where g = det g^v, R is the curvature scalar, A is the cosmological constant of AdS4 and k2 = 8nGN is the coupling constant in terms of the Newton constant. The gravity action on a manifold M with boundary dM can contain boundary terms, in addition to the bulk Einstein-Hilbert term. In general the form of the Action on such a background is

In this paper, we focus on the bulk term in (2) which is sufficient to determine the equations of motion. It is important to note that one may always add boundary terms [6] to the Action that do not affect the equations of motion or Green functions.2

The light-cone gauge approach to formulating pure gravity in flat backgrounds has been studied in [7-10]. Here we formulate pure gravity in AdS4 characterized by a cosmological constant A. As one would expect, this involves considerable modifications to the flat background results of [7] and we comment on these changes as and when they occur.

2 These added terms combine with surface terms generated by partial integrations of the bulk term.

3. AdS4

Consider a five-dimensional flat spacetime with metric nMN = (-1,1,1, 1, -1) and coordinates fM, M = 0,..., 4. On this manifold, AdS4 is defined as the four-dimensional hy-persurface

-(f0)2 + (ff + (f2)2 + (f3)2 - (f4)2 = R2, (3)

with radius R. We now introduce local (Poincare) co-ordinates xM = (x0,x1,z,x3) on AdS4

f0 = Rx0 f1 = -x1 f3 = -x3, (4)

f2 = 2Z R - Hx°)2 + (x1)2 + (x3)2 - z2}], (5)

f4 = £[R2 + {-(x0)2 + (x1)2 + (x3)2 - z2}], (6)

which satisfy (3). z plays the role of a radial coordinate and divides the spacetime into two regions. We work here in the 'patch' z > 0 with z = 0 being the AdS4 boundary. The induced metric on this space is

g(0) = d^fMdvfNnMN = ^ n„v, (7)

where n^v is the usual Minkowski metric. We now switch to light cone co-ordinates x^ = (x+,x-,x1,z) where

, x0 ± x3

x ± = -yf. (8)

The cosmological constant for AdS4 is 3

^ = -R2. (9)

4. Light-cone formulation

Our aim is to study fluctuation on the AdS4 background. The dynamical variable is the metric g^v, which in the absence of all perturbations must reduce to g^. We work in light-cone gauge by making the following three gauge choices [7]

g--= g-i = 0, i = 1,z. (10)

These choices are consistent with gffl since, in light-cone coordinates, n__= n-i = 0. A fourth

gauge choice will be made shortly. The metric is parametrized as follows

g+-= -et

gij = efYij- (11)

The fields 0, ^ are real while Yij is a 2 x 2 real, symmetric matrix.

The Euler-Lagrange equations corresponding to the Einstein-Hilbert Action read

Rv - ^g^vR =-Ag^v. (12)

In light-cone gauge, a subset of the Euler-Lagrange equations which do not contain time derivatives (9+) are treated as constraint equations. The first relevant constraint is R__= 0 which

29-09-f - 29- f - (9-f)2 + 19-Ykl9-Yki = 0. (13)

A simple solution to this constraint relation may be obtained by making a fourth gauge choice 1

0 = 2 f. (14)

This allows us to solve Eq. (13) and obtain 1 1 / n \ -2

f = 4 92 {9-Yl}9-Yi}) + 2ln ^, (15)

with the g^ defined following the prescription in [11]. The second term, in f, is essential to ensure that gij and g+- reduce correctly to gj and g+- respectively.

In a flat background [7,8] the solution to f is simply f flat = 1 g2(9_yij9_Yij) and the second term in (15) is absent.

We now compute the determinant of Yij from the second relation in (11) which implies that

/R2\ 4

det gj = ( -H det y^, (16)

with the {}(0) superscripts implying that all fluctuations are switched off. In this limit, the metric -2 -2 2 is simply ^ times the Minkowski metric so the L.H.S. of (16) is ) thus implying that

det Yij = ( R^)2. (17)

Note that in contrast to our result above, on a flat background, Yij is unimodular [7,8]. We choose the determinant of Yij (which includes fluctuations) to be the same as in (17) - this is permitted since Yij is a 2 x 2 matrix that has only two physical degrees of freedom. This choice renders the fluctuation field, introduced in the next section, traceless making calculations easier. The second constraint relation is R-i = 0 which yields

- = _e-* — 9

Yij e0-2f ± fef (19- Ykl9jYki - dj

- 9-9jf + 9j4>9-fj + 9i (efYkl9-Yjk\

4.1. Light-cone Action

The light-cone Action for gravity is

S = j d3x jdzC = 2^1 d4x^-g;(2g+-R+- + gijRij - 2A\. (19)

We now compute each term in the above expression, using the results listed thus far. We derive the following closed form expression for the Action in AdS4 purely in terms of the physical degrees of freedom.

S = ¿/d3x f dz{|2 e^2B+B-0 + B+B-f - 19+ Yijd-j

- R2e<PYij(didj$ + 29i09j0 - B0Bjf - 4diYkldjYkl + 2diYMdj

z2 1 1 2 z2

J_e0-2f Yij —Ni —Nj + — e0Yzz - 2 ^efe0A\, (20)

2R2 Y d- d- j + R2 Y R2 |,

N = ef(^!>-YjkdiYjk - d-di0 - *>-* + >,*>.*)

+ dt (e*Yj'd-Yjj).

Although 0 = 1 f, we have not made this substitution in the result above - this makes it easier to trace the origin, from (19), of each term in (20). In obtaining the above result, we have dropped several boundary terms (see Section 2).

4.1.1. Deviations from flat spacetime results

The three main differences between our result (20) and the flat background Action in [7,8] are

the overall factor of R- in front of each line, the penultimate term proportional to Yzz and the last term, proportional to the cosmological constant.

5. Perturbative expansion

In this section we obtain a perturbative expression, to cubic order in the fields, for the Action in (20). We do this by making the following choice z2 ( )

H =( h11 h}z), (21)

\h1z -hzz J

with hzz = -h ii as explained below Eq. (17). In terms of these fields, Eq. (15) reads

1 1 R2 ( 4) ...... ~ + 0(h4

f = -4^2 [d-hjjd-hjj] + 2ln — + 0(h4). (22)

In order to obtain a perturbative expansion of (20) we simply use the results (21) and (22). We now redefine 1

h ^ h. (23)

In terms of these fields, the Action at O(h2) is

S2 = j d3x j dzL2, (24)

R2 R2 R2 R2

L2 = +2^2 d+hijd—hij — dihkl — 2 —J hikdkhiz + hzkhkz-

In the above we have made use of both (9) and the fact that hkk = 0. Notice from (20) that the cosmological constant only appears in interaction vertices involving an even number of fields. At O(h3), the Action reads

S3 = J d3xjdz-^L3, (25)

1 R2 R2 R2

L3 = M -2zf dkhikdihijhji + 2-J-hizdkhijhjk + hijdihkidjhki

R2 R2 1

—2hijdihkidkhji - 2—hi^ — (d-himdihim) 2z2 z3 o—

R2 1 R2 1

+ 4 — hiz — (hhd— hu) + 4 — dkhiz — (hikd— hn) z4 o— z3 o—

1 R2 1 R2 1

+ - —2 9khik — {9—himdihlm) — 4—T dkhi^ — (hlzd—hil)

2 z2 O— z3 O—

+ —2- dkhi^ — ihml9—him)

z2 O—

R2 R2 R2

— 2hijdidjB — 6^hizdiB — 4 —4hzzB y,

B = —-72 [d—hijd—hij]. (26)

As expected, both the kinetic and cubic vertices in AdS4 are far more involved than their flat background counterparts. It would be interesting to (1) extend our analysis to the quartic interaction vertex which is trickier because time derivatives start to appear and must be re-defined away as in Appendix C of [9], (2) understand whether one can extract 'amplitude-like' structures from these expressions as in [2], (3) attempt a light-cone derivation of higher spin theories on AdS4 following the methods of [5], using the results presented here for guidance and (4) examine the formulation of pure gravity on other curved backgrounds.

Acknowledgements

We thank Hidehiko Shimada, Stefano Kovacs, Sunil Mukhi and Stefan Theisen for helpful discussions. This work is supported by the Max Planck Institute for Gravitational Physics through the Max Planck Partner Group in Quantum Field Theory and the Department of Science and Technology, Government of India, through the INSPIRE fellowship (YSA).

Appendix A. Useful results

r++ = -g+~[2d+g+- - d-g++]

r++-= 0 r+_= 0

r+- = 0

= -g+-[dig+- - d-gi+]

r+ = -\ g+-d-gj r-- = g+-d-g+-

r-- = 2 {g+-d-g+++ g-i [9-gi+ - dig+—]} 1 { +-

r++ = 2 {g^ 9+g++ + g [2d+g+- - 9-g++]

+ g~' [2d+gi+ - 9ig++]} 1{

r+i = ^g dig++ + g [dig+- - d-gi+] + g-j [dig+j + d+gij - djg+i]} r- = 1 {g+-[9ig+- + d-g+i]+g-jd-gij} 1{

rij = 2 ^ [dig+j + djg+i - d+ gij] - g d-gij + g~k[digkj + djgik - dkgij]} Fjk = 1 {-g-'d-gjk + glm [djgmk + dkgmj - dmgjk]}

r-j = 1 gikd-gkj

i 1 ij r+ - = 2gij [d-gj + - djg+-]

r+j = 1 {g-i [djg+- - d-g+j] + gik[djg+k + d+ gkj - dkg+j]}

ri__ = 0

r+ + = -{g-[29+ g+- - d-g++] + gij [2d+g+j - 9jg++]}

rj = 1 {-g-jd-gij + gjl[djgii + digij - digij]}

Yij = R2 (e~H\

Y'j Yij = 2

YlJdkYij = 4 hz z

gßVg^P = Svp g++ = g+' = 0

g+i = -g+-gijg-j

g++ = -e^g + e*g-ig+i

g+- = -e

gij = e-^Yij z2

V-g = R2 efe*

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