# The massless gravitino and the /CFT correspondence

###### Abstract

We solve the Dirichlet boundary value problem for the massless gravitino on space and compute the two-point function of the dual CFT supersymmetry currents using the /CFT correspondence principle. We find analogously to the spinor case that the boundary data for the massless dimensional bulk gravitino field consists of only a dimensional gravitino.

###### pacs:

## I Introduction

Recently Maldacena [1] has conjectured that the large N limit of certain dimensional conformal field theories is dual to supergravity or string theory on dimensional Anti-de Sitter () space times a compact manifold. A prescription for generating correlators of operators in the conformal field theory (CFT) from solutions of the supergravity equations of motion has been given in [2, 3]. The prescription associates to each field in the supergravity action a corresponding local operator in the CFT such that the following relation (in Euclidean space) holds:

(1) |

The effective action is evaluated on the solutions to the supergravity equations of motion subject to the boundary conditions where denotes the boundary of space. On the right-hand-side of (1) the expectation value of the given exponential is taken in the dual conformal field theory, with acting as a source for the CFT operator . Using this relation various two-point [2]-[5] and three and four-point [5]-[14] correlation functions have been computed, including detailed checks of various Ward identities.

One subtlety of the prescription (1) involves the manner in which is evaluated on the supergravity solutions. Specifically diverges and must be regularized. As a result ambiguities in the overall coefficient of CFT correlators obtained from (1) arise, and therefore the CFT Ward identities may not be satisfied. However a regularization procedure which produces correlators satisfying the Ward identities has been found in [5]. The procedure involves solving the Dirichlet boundary value problem for the supergravity fields for a deformed boundary of such that is well-defined and only after obtaining the CFT correlators is the limit back to the true boundary of taken.

In this paper we consider the /CFT correspondence for the massless gravitino whose dual CFT operator is the supersymmetry current [15]. In section 2 we solve the massless gravitino equations of motion on the background following the techniques of [8]. We find, in analogy to the spinor case [4, 8], that the boundary data for the massless dimensional bulk gravitino field consists of only a dimensional gravitino due to the first order nature of the equation of motion. In section 3 we use the correspondence (1) to compute the two-point function for the dual supersymmetry currents, taking care to evaluate the gravitino action in the manner discussed above, and find the expected result. In section 4 we make some concluding remarks.

## Ii Constructing the solution

In this section we solve the boundary value problem for the massless Rarita-Schwinger field. Our method for solving the equation of motion parallels that of [8]. We first find the most general solution to the Fourier transformed equation of motion. This solution contains an exponentially growing mode in and therefore is not Fourier transformable. Demanding Fourier transformability we are forced to constrain the boundary data such as to remove the undesired mode. We then re-express the solution in terms of the desired boundary data and Fourier transform back to position space obtaining the bulk values of the Rarita-Schwinger field.

The action for the massive Rarita-Schwinger field in dimensions is given by

(2) |

where

(3) |

and and are related to the mass and cosmological constant (see [16] and [17] for the relation for supergravity on and respectively). Our notation is as follows: is the vielbein and its’ determinant, is the spin connection, are the curved space gamma matrices related to the flat space gamma matrices by , the flat space gamma matrices satisfy the anticommutation relations , gamma matrices with more than one index are antisymmetrized as , coordinate indices are denoted by lower case Greek letters running from to and lower case Latin letters running from to , and lower case Latin letters denote Lorentz indices. Varying the action (2) with respect to results in the equation of motion

(4) |

while varying with respect to results in the equation of motion for the adjoint Rarita-Schwinger field

(5) |

where .

We now specialize to the Euclidean background geometry and choose coordinates such that the metric takes the form

(6) |

where we use boldface letters to denote the inner product of -dimensional vectors in the flat Euclidean metric, , and use the flat Euclidean metric to raise and lower indices, . Choosing the corresponding vielbein to be

(7) |

it is straightforward to show that the spin connection is given by

(8) |

as is easily verified by substituting into .

To solve the equation of motion (4) it is first convenient to rewrite it in the form

(9) |

where . Expanding (9) and substituting (7) for the vielbein and (8) for the spin connection we find the equation

(10) |

and the equation

(11) |

To solve these equations we work in momentum space. Define the Fourier transform

(12) |

and substitute into (10) and (11). We obtain after some rearranging

(13) |

(14) |

for the and equations respectively.

Solving equations (13,14) is straightforward but tedious. The resulting solution for the generic case presented so far is complicated and therefore we consider the simpler massless case in the remainder. This is equivalent to demanding ( here is related to the cosmological constant by a dimension dependent factor). To simplify the equations above we start by defining the projection operator

(15) |

which is orthogonal to both and . Defining the tranverse components of to and respectively as it follows that the field may be decomposed as

(16) |

The equation of motion for is easily derived by applying to (14) obtaining

(17) |

Finding the equation of motion for is a little more involved.
First solve (13) for and substitute into
(14) contracted with . This results in the
algebraic relation^{1}^{1}1Here is the essential between the massless and
massive cases. In the massive case cannot be expressed
algebraically in terms of .

(18) |

Substituting this relation for into (14) contracted with and comparing with (13) results in the second algebraic relation

(19) |

From this and (13) the equation of motion for follows

(20) |

The equations of motion (17,20) are easily integrated in the form of path ordered exponentials. The exponentials are straightforward to evaluate and we find

(21) | |||||

where is transverse to both and and

(22) | |||||

Using the algebraic relation expressing in terms of we now have the complete momentum space solution for the massless Rarita-Schwinger field. This solution however is not Fourier transformable since the modified Bessel function diverges exponentially for large . In order to obtain a Fourier transformable solution we must constrain the boundary fields and to remove the growing modes. The same problem occurs for the spinor field as discussed in [8] and more generally for any field satisfying a first order equation of motion. The simple reason is that for a first order equation only the boundary values of the field are freely specifiable, whereas for a second order equation of motion the boundary value and first derivative of the field are freely speciable or conversely the boundary value and asymptotic behavior. As emphasized in [8] the asymptotic behavior of the field is essential for the /CFT correspondence and while specifying it for a second order equation of motion is not a problem, for a first order equation it must be done at the expense of specifying completely the boundary values of the field.

We may easily find the constraints necessary to remove the offending terms by rewriting the solutions (21, 22) in terms of the modified Bessel functions and where decay exponentially for large . The conditions necessary for removing the exponentially growing modes are easily shown to be and which effectively removes half of the components of and respectively. Defining and similarly for these conditions may be rewritten in the equivalent form

(23) | |||

(24) |

Substituting these relations into the solutions (21,
22) results in^{2}^{2}2These solutions may equivalently
be expressed in terms of and .

(25) | |||

(26) |

where constants have been absorbed into and respectively.

The final step before inverse Fourier transforming back to position space is to re-express the solutions (25,26) in terms of the given boundary data . Clearly the components of the boundary Rarita-Schwinger field are not all independent as discussed above. To find the independent components evaluate (25) at the boundary (we consider only the case in detail here as the case is similar) and apply the projection operators deriving the relations

(27) | |||||

(28) |

Solving one relation for and substituting into the other yields

(29) |

Using the small expansion of

(30) |

we find that for regular boundary data in the limit we must demand that and therefore that the appropriate boundary data is given by . Eliminating for in (27) yields the momentum space solution for

(31) |

expressed in terms of the desired boundary data. Similar manipulations with the solution (26) yields the relation between the chiral components of

(32) |

and the resulting solution expressed in terms of the boundary data

(33) |

This is not quite the desired form for though. Recall that and are related as in (18). Evaluating this relation at and using (32) to solve for yields

(34) |

From the expansion (30) it follows that for regular boundary data in the limit we must demand that , and so is not the correct boundary data. Applying to (34) (which annihilates the left-hand-side) one arrives at

(35) |

Using the expansion (30) once again we see that the correct boundary data is . We may now express in terms of this boundary data by substituting (34) in (33) with given in terms of after using (35). Because the result is quite complicated we do not give the expression explicitly.

Finally we may inverse Fourier transform to find the bulk value of the field in position space. This is not as formidable as it might first appear because we need only work to leading order in . Substituting the solutions for (31) and (33, 34,35) into the decomposition (16), expanding the dependent terms and simplifying results in

(36) | |||||

The Fourier transform of this expression is tedious to carry out, but straightforward, and we arrive at

(37) |

where

(38) |

a factor of has been absorbed into , and is transverse to as follows from (32) in the limit. The remaining field component is given by

(39) |

For the adjoint Rarita-Schwinger field the analysis of solving the equation of motion (5) exactly parallels the above discussion, therefore we only quote the results. The momentum space adjoint field may be decomposed as

(40) |

where and

(41) |

The momentum space equations of motion for the the fields and may be derived and solved in strict analogy to the unbarred case and we find for the Fourier transformable part

(42) |

and

(43) |

is not the desired boundary data to express in terms of as it goes to zero in the limit . Rather we must express in terms of which can be done after using the relations

(44) |

and

(45) |

Substituting these expressions into (43), keeping only the leading order terms in , and inverse Fourier transforming yields

(46) |

where a factor of has been absorbed into and is transverse to . The remaining component of the adjoint field is given by

(47) |

## Iii Two-point function

The /CFT correspondence is prescribed by the relation

(48) |

where is the action (2) evaluated on the
solutions found in the previous section
with prescribed boundary data and
. The right-hand-side of (48)
is the expectation value of the given exponential in the
boundary conformal field theory with and
playing the role of source terms for the
boundary conformal fields and
respectively. We expect these boundary conformal fields to be the
supersymmetry currents of the boundary CFT [15] and
indeed the two-point
function we find below is in agreement with this expectation.
The two-point function
is
easily obtained from (48) by taking a pair of functional
derivatives with respect to
and .
In our case the action (2) is first order in derivatives
and therefore vanishes when evaluated on the solutions
(37,39,46,47). This is also the
case for spinors as first noted by [4]. To avoid this problem
[4] added the most general Lorentz invariant and generally covariant
boundary term quadratic in the spinor fields to the
action^{3}^{3}3Justification for
this term has recently been given in [18] from a Hamiltonian
formulation and presumably extends to the gravitino case as well..
Following this prescription we add to the action
principle (2) the boundary term

(49) |

where is the induced metric on the boundary, its’ determinant and and are undetermined coefficients.

To evaluate the solutions (37,46) could be substituted and the integral performed. However it is somewhat simpler to work in momentum space where the boundary action becomes

(50) |

where as before the dot product is defined using the Kronecker delta. Substituting the decompositions (16) and (40) for and respectively in the boundary action (50) we rewrite it as where

(51) | |||||

(52) |

Substituting the solutions (31, 42) evaluated at into and expanding the projection operator we find

(53) |

In the limit of small the ratio of modified Bessel functions may be expanded using (30) as

(54) |

where the are known coefficients which are not needed here.
Substituting this into (53) we see that from the first
term in the expansion
only the leading order
term (multiplying )
gives rise to a non-trivial contribution to
while the remaining terms in parentheses give rise
to contact terms. As we will see shortly though this term cancels
against a contribution from , therefore the leading order
non-trivial contribution to actually comes from the
term in the expansion^{4}^{4}4We are assuming
here that is not equal to an odd integer and
is . If is an odd integer
then the expansion contains log terms as well
which give rise to the leading order non-trivial contributions to
and .
Nevertheless the two-point function obtained below is valid for
any ..