###### Abstract

A free Rarita-Schwinger field in the Anti-de Sitter space is considered. We show that the usual action can be supplemented by a boundary term that can be interpreted as the generating functional of the correlation functions in a conformal field theory on the boundary of the Anti-de Sitter space.

Rarita-Schwinger Field

in the AdS/CFT Correspondence

Anastasia Volovich

Department of Physics,

Harvard University,

Cambridge, MA 02138

^{1}^{1}1On leave from Moscow
State University
and L. D. Landau Institute for Theoretical Physics, Moscow, Russia.

## 1 Introduction

Recently it has been proposed by Maldacena [1] that there is an exact correspondence between string theory on Anti-de Sitter space () and a certain superconformal field theory (CFT) on its boundary. For instance, the large N and the large t’Hooft coupling limit of four dimensional Super Yang–Mills can be described by IIB supergravity compactified on . In [2, 3] the recipe for computing field theory observables via was suggested, namely, the action for a field theory on considered as the functional of the asymptotic value of the field on the boundary of the space is interpreted as the generating functional for the correlation functions in a conformal field theory on the boundary. For example, for the scalar field one has

(1) |

where is the boundary value of and it couples to the scalar on the boundary. Hence calculating the action on the side allows one to obtain the correlation functions on the side.

Recently, scalar, spinor, vector and graviton fields have been considered on various products of spaces; two, three and four-point functions were calculated and various non-renormalization theorems suggested [2]-[21]. One can deduce the scaling dimensions of operators in the conformal field theory from the masses of particles in string theory. Spinor fields of spin were discussed in [7] (see also [6]) where it was pointed out that the ordinary Dirac action vanishes on-shell; so to get the generating functional in conformal field theory one has to supplement the action by a certain boundary term, which produces the correlation function in conformal field theory.

In this note we will discuss the Rarita-Schwinger field on . It can couple to the supersymmetry current, thus it is natural to reproduce the two-point correlation function of the supersymmetry currents from the partition function on the side by finding the appropriate boundary term to add to the action, since the bulk action vanishes onshell. In section 2 we consider the Rarita-Schwinger field on and find the solution of the corresponding Dirichlet problem. In section 3 we review the two-point function of operators coupling to gravitini, which is restricted by conformal invariance. Finally, in section 4 we find the boundary term on the side which reproduces the correlator on side, and the relation between the scaling dimension of the operator in the and the mass of the particle in the space.

## 2 The Rarita-Schwinger field on

Here we will consider the Rarita-Schwinger field on . The action for the Rarita-Schwinger field in curved space reads

(2) |

where

(3) |

is the Rarita-Schwinger field, , is a vielbein such that the metric and by we denoted antisymmetrization of the indices; .

In the coordinates , , the Euclidean space is represented by the Lobachevsky upper half-space and the metric is given by

(4) |

The boundary is defined by the hypersurface plus a single point at . With the choice of the vielbein

(5) |

for which the corresponding nonvanishing components of the spin connection has the form [7]

(6) |

the operators ,

(7) |

Using the condition

(8) |

we can rewrite the Rarita-Schwinger equation

(9) |

in the following form for :

(10) |

The solution of the above is given by

(11) |

(12) |

where , ; is the modified Bessel function and , satisfy and .

The condition takes the form

(13) |

Similarly, the solution of the conjugate equation is:

(14) |

(15) |

with

(16) |

Let us now consider the boundary conditions on the surface , [9, 15]. Without loss of generality we will consider the case of positive (the other case can be treated similarly). Then fixing the positive components of the boundary field we find that the negative components vanish, where . To specify solutions we use the functions

(17) |

The dependence of and on them is given by (33) and (34) in appendix A.

## 3 Correlator in

Conformal invariance fixes the two-point correlation function for spin 3/2 primary fields of scaling dimension up to normalization to be [26]

(18) |

where are vector indices, are spinor indices and

## 4 Boundary term

The Rarita-Schwinger action vanishes on-shell; thus in order to reproduce the two-point correlation function of spin 3/2 fields in the boundary conformal field theory one has to supplement the action by the boundary term. This is analalous to the spinor case [7] and so we similarly take

(19) |

where as in the spinor case is the surface of , is the induced metric on with determinant . To compute the total action we following [9] consider the Dirichlet problem on

The action (19) can be rewritten in momentum space in the form

(20) |

Generally speaking the expression (20) contains divergences when . Selecting the finite part, one gets the action (see appendix A for details of the derivation)

(21) |

where the boundary spinors and are defined as in (17).

Taking the Fourier transform and integrating over (we use integrals from appendix B) we get the expression for the boundary term on the side

(22) |

which matches the expression for the CFT correlator (18) with

(23) |

So we found the agreement between the two-point function derived from the side and the side once the appropriate boundary term was added.

The mass spectrum of IIB supergravity on has been worked out in [22]. In particular it was shown that there is a massless gravitino field which satisfies equation (9) with . The IIB supergravity on is dual to four dimensional Super Yang-Mills [1]. The supersymmetry current

(24) |

(see [24], [25] for details) which couples to the Rarita-Schwinger field has the scaling dimension . One gets this dimension from (23) if one takes and . This provides a test of the AdS/CFT correspondence in this case. The 11d supergravity on is related to the theory on M5-branes [1] and thus to (2,0) theory in the large N limit. The spectrum of the supergravity theory was computed in [23] and that of the primary operators of the conformal algebra of arbitrary spin was discussed in [6], [5]. In particular the gravitino was discussed and the relation between the scaling dimension and the mass agrees with our analysis.

After this paper was completed and prepared the paper [27] appeared with similar results.

## Acknowledgements

I would like to thank Juan Maldacena and Andy Strominger for very helpful discussions; and Jeremy Michelson for useful comments. I am grateful to Laboratoire de Physique Theorique et Hautes Energies in Paris where part of this work was done and especially Laurent Baulieu for invitation and Celine Laroche for kind hospitality. This work is partially supported by DOE grant DE-FG02-96ER40559, Van Vleck fellowship, Soros grant and CNRS foundation.

## Appendix A

Here we will present the details of the imposed boundary conditions and the derivation of the term (21). Let us set in (12)

(25) |

(26) |

and introduce

(27) |

Then from the asymptotic behavior of the Bessel function for we conclude that once we fix , then , so analagous to the spinor case can be used to determine the boundary data and

(28) |

Let us now find the dependence on and from . Suppose for some and

Then in (28) if we require the first term in the right hand side to be equal to the left hand side we get

(29) |

(30) |

(31) |

or

(32) |

Then from (12) we get

(35) |

Substituting the expression (33) and (34) in (35) and defining the boundary spinors

(36) |

we get

(37) |

We assumed here that the argument of the Bessel function is . The very similar expression can be easily written for the conjugate spinor. Now let us proceed with calculation of the boundary term:

(38) |

as the functional of the boundary spinors and :

(39) |

where

(40) |

We notice that to get rid of singularities one has to take terms in the square brackets . Using the asymptotics of the Bessel functions for small :

(41) |

we get the answer (21).

## Appendix B

We used the following integrals in the above:

(42) |

(43) |

where

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