MIT-CTP-2928

hep-th/9911182

Propagators for massive symmetric tensor and -forms in

Asad Naqvi

Center for Theoretical Physics,

Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Abstract

We construct propagators in Euclidean space-time for massive -forms and massive symmetric tensors.

## 1 Introduction

Calculations of correlation functions in Type IIB supergravity on have been performed extensively. These calculations study the strong coupling dynamics of Yang-Mills theory for large as a result of the AdS/CFT correspondence conjectured in [1, 2, 3]. For calculations of 4-point functions, bulk to bulk Green’s functions, describing propagation between two points in the interior of , are required. Propagators for scalar fields have been discussed in [4], and propagators for massless and massive gauge bosons were obtained in [5]. In [6], a new method for calculation of these propagators was discussed in which ansatze for the bi-tensor propagators were used which naturally separated the gauge invariant parts from the gauge artifacts. The gauge artifacts did not contribute if sources of the fields were conserved currents. Thus gauge fixing was unnecessary.

In this paper, we use the same method as in [6] but now for massive -forms and the massive symmetric tensor fields. Since the fields are massive, there is no gauge invariance which guarantees that the sources are conserved. However, as we will see, using a similar ansatze for the -forms (writing the propagator as a physical part and a pure gauge) in the massive case considerably simplifies the calculations. The pure gauge part is annihilated by the Maxwell operator and just appears multiplying in the equation of motion. For the massive symmetric tensor, the pure gauge part of the ansatz corresponds to diffeomorphisms at (where the propagator describes propagation from to ). However, for our propagator to have symmetry under the exchange , we need to add a term which corresponds to diffeomorphisms with at . This term will not be annihilated by the “wave operator”. However, writing the propagator in this form still simplifies the calculation considerably.

The paper is organized as follows. In section 2 we discuss the massless 2-form case as a warm-up exercise (this case was discussed in [8]). We also find the propagators for the massive anti-symmetric tensor. We generalize the calculation of section 2 to -forms in section 3. In section 4, we perform the calculation for the propagator of the massive symmetric tensor. In section 5, we check the short-distance limit of our results and find that they match with the short distance limit of the corresponding propagator in flat space. We end with a summary of our results in section 6.

We will work in Euclidean , which can be regarded as an upper half space in a space with coordinates , and metric,

(1.1) |

The scale has been set to unity and the metric describes a space with a constant negative curvature . We will introduce a chordal distance in terms of which invariant functions and tensors on can be expressed most simply:

(1.2) |

where
is the “flat Euclidean distance”. We will construct basic
bi-tensors by taking derivatives with respect or
of the bi-scalar variable . These are given by
^{1}^{1}1

(1.3) |

and with

(1.4) | |||||

We will need certain properties of derivatives of , most of which were derived in [7] and which we list here.

(1.5) | |||

(1.6) | |||

(1.7) | |||

(1.8) | |||

(1.9) | |||

(1.10) | |||

(1.11) |

## 2 Antisymmetric Tensor

The equation of motion for an anti-symmetric tensor field in is

(2.1) |

The covariant derivative is with respect to the metric. The Maxwell operator is normalized such that appears with coefficient 1 in the equation of motion. We look for solutions of the form

(2.2) |

with bi-tensor propagator . Using this
expression for in Eq(2.1), we obtain an equation for
the propagator ^{2}^{2}2[…] denotes anti-symmetrization with strength 1:

(2.3) |

We will first look at the case . In this case, there is a gauge invariance of the form which implies that the current is conserved. The equation for is:

(2.4) |

where the second term on the right hand side gives zero when integrated with conserved currents. We will introduce two bi-tensors,

(2.5) | |||||

(2.6) |

These are the only two bi-tensors which are anti-symmetric under and . We will choose the following ansatz for :

The second term is a pure gauge and is annihilated by the Maxwell operator:

In Eq(2.4), we need an expression for :

(2.7) | |||||

Various terms in Eq(2.7) are simplified using properties of derivatives of (Eqs 1.5-1.11):

Collecting all the terms together,

(2.8) | |||||

Using invariance and the fact that is anti-symmetric in and , has to be of the form:

(2.9) |

where is a scalar function of . Then,

Using these expressions, Eq(2.4) becomes (for and separate)

Setting the scalar coefficients of the two independent tensors to zero, we obtain

(2.10) | |||||

(2.11) |

Eq (2.11) can be integrated to give^{3}^{3}3We consistently ignore
constants of integration since we want our propagators to go to zero
as .

(2.12) |

which can be substituted into (2.10):

(2.13) |

This is just the invariant equation

(2.14) |

for the propagator of a scalar field of . A scalar field of mass is characterized by two possible scale dimensions,

(2.15) |

For our propagator to have the fastest fall off as , in the following solution, we will choose .

(2.16) | |||||

where is the standard hypergeometric function . The constant is chosen such that as , matches on to the flat space case (this is discussed in more detail in section 6).

We will now consider the massive case. In this case, there is no gauge invariance and the current is not necessarily conserved. We will still use the same ansatz,

(2.18) |

Using invariance, we can write as

(2.19) |

where is a scalar function. is still given by Eq(2.8) since the Maxwell operator annihilates the second term in Eq(2.18). This term is

(2.20) |

Eq(2.3) then becomes (for and separate)

Setting the coefficient of independent tensors to zero, we get a system of two coupled differential equations:

(2.21) |

(2.22) |

Eq(2.22) can be readily integrated to give

(2.23) |

Substituting in Eq(2.21), we find an uncoupled differential equation for

(2.24) |

This is again the invariant equation

(2.25) |

for the propagator of a scalar field of .

## 3 -forms

The preceding calculation of the propagator of an anti-symmetric tensor can be generalized to -form propagators. The equations of motion for a -form field is

(3.1) |

We will assume a solution of the form

with bi-tensor propagator . The propagator satisfies the equation:

(3.2) |

There are two independent tensors which have the right anti-symmetry property under exchange of various indices. These are

(3.3) | |||||

(3.4) |

Generalizing the ansatz of the last section for , we try a solution of the form

(3.5) |

where

(3.6) |

Then,

(3.7) |

and

(3.8) |

Notice that the second term in (of the form does not contribute). Acting with , we get

Various terms appearing on the r.h.s of the above equation can be simplified by using Eq(1.5-1.11):

So (3.2) becomes (for and separate),

which implies

(3.9) |

(3.10) |

Eq(3.10) can be integrated to give

(3.11) |

which can then be substituted in Eq(3.9):

(3.12) |

This is the equation for the propagator of the scalar field of . The solution to this equation is given by Eq(2.16) with

This result agrees with [9] for the case .

## 4 Massive Symmetric Tensor

We will begin by reviewing the equations of motion for the massless graviton. The propagator for the graviton was obtained in [6]. The gravitational action is

(4.1) |

where is the matter action. The equations of motion are

(4.2) |

is the stress-energy tensor. For and , we obtain Euclidean space with . In the presence of a matter source, the metric will no longer be the metric . We will denote the fluctuations about by . An equivalent form of the equation is

(4.3) |

The linearized equations of motion for are

(4.4) |

All covariant derivatives and contractions are with respect to .

For the case of massive symmetric tensor, , a consistent action with coupling to gravity is not known. For example, consider a Kaluza-Klein reduction of a theory of gravity in dimensional space-time to dimensional space time. We get an infinite tower of massive symmetric tensor fields in dimensions. It was shown in [10] that it is impossible to consistently truncate this theory to a finite number of symmetric tensor fields. However, the quadratic part of the action (which is what we need for calculation of the propagator) was given in [11] [12].

where . From this action, we can derive the following equation of motion:

This can be converted to the following equivalent form