hep-th/9806074 PUPT-1796 IASSNS-HEP-98-51

Three-Point Functions of Chiral Operators in , SYM at Large

Sangmin Lee, Shiraz Minwalla, Mukund Rangamani
^{†}^{†} sangmin, minwalla,

Department of Physics, Princeton University

Princeton, NJ 08544, USA

and

Nathan Seiberg^{†}^{†}

School of Natural Sciences, Institute for Advanced Study

Olden Lane, Princeton, NJ 08540, USA

Abstract

We study all three-point functions of normalized chiral operators in , , supersymmetric Yang-Mills theory in the large limit. We compute them for small ’t Hooft coupling using free field theory and at strong coupling using the /CFT correspondence. Surprisingly, we find the same answers in the two limits. We conjecture that at least for large the exact answers are independent of .

June 1998

1. Introduction

The conjectured duality [1] (for earlier related references see [2,3,4,5]) between string/M theory on Anti-de Sitter space () times a compact manifold, and conformal field theory (CFT) living on the boundary of has attracted much attention. According to this proposal, Type IIB string theory on is dual to , supersymmetric Yang-Mills theory (SYM).

In [6,7] a detailed dictionary relating S-matrix elements of the string theory to Green’s functions of the CFT was proposed. The operators of the CFT are mapped to on shell bulk fields on . The CFT operators interact with the boundary values of these bulk fields through an interaction action . The partition function of the string theory with fixed boundary values of fields is then identified with the partition function of the CFT with external sources coupled to the corresponding operators.

Using this dictionary, two point functions of CFT operators corresponding to massive scalars [6,7,8,9], vectors [7,9], the graviton [10], and spinors [11] have been computed.

In a series of recent papers, the 3-point functions of operators in a CFT corresponding to massive minimally coupled scalars [8,9], or scalars and spinors [12], or vectors and spinors [13] on the with certain generic, arbitrarily prescribed, interactions have been computed.

Certain computations of correlation functions of operators in actual SYM have also been performed. Using a proposed form of , the 2-point functions of the stress energy tensor and were computed in [6]. Using the model independent coupling of gauge fields to currents, the 3-point functions of the -symmetry currents of SYM were computed in [9,14]. Similarly, 3-point functions of the dilaton and the stress energy tensor were computed in [10].

Local operators in SYM are organized into infinite dimensional families, each of which is an irreducible representation of the , , superconformal algebra. Each family (or module) contains special operators of lowest scaling dimension in an representation. We will call them primary operators (PO) (strictly, only the operator with the highest weight is primary). SYM contains a set of special short families that contain fewer operators than the normal module. Such families include primary operators which are chiral under an subalgebra; the scaling dimension of operators in these families is determined by the superconformal algebra [15,16] in terms of their -symmetry representation. We will loosely refer to all the lowest dimension operators in such a representation as chiral primary operators (CPO). Under a given subalgebra, the chiral primaries include chiral operators, anti-chiral operators and non-chiral operators.

It should be stressed that unlike the situation in , these chiral primary fields do not form a ring. The product of two chiral operators includes a product of an chiral operator with an anti-chiral operator and even two non-chiral operators, which are singular. Because of such singularities the chiral operators do not form a ring.

In this paper, we study the 3-point functions of all CPOs, in the large limit of SYM. We first compute them in the limit of weak ’t Hooft coupling using free field theory. We then study them in the limit of large ’t Hooft coupling using Type IIB supergravity (SUGRA). Surprisingly, we find the same answers. Clearly, this agreement for the primary fields guarantees similar agreements for all their descendants.

Banks and Green [17] showed that for infinite the leading order result at large is not corrected at the next order. Given that we found that the leading order result agrees with the weak coupling answer, we are led to conjecture that the 3-point functions of all chiral primary operators at large is independent of .

Since -symmetry currents and the stress energy tensor are descendents of CPOs, our results include all previous results on 3-point functions [9,10,14], as special cases. Also, the discussion of [18,19] shows that some of these 3-point functions are independent of the coupling even for finite .

We point out that a similar result cannot be true for the 4-point function of these chiral operators. Unlike the 3-point functions, the 4-point functions depend on at the next to leading order [17].

It might be that even a stronger claim is true, and these 3-point
functions are independent of even for finite . (For some
of the 3-point functions this was proven in [19].) From the
weak coupling side it is clear that the 3-point functions depend
on (even the spectrum of chiral primary operators depends on ).
Therefore, if this stronger claim is true, then at strong coupling, on
the side, the coupling of three gravitons depends on ; i.e. it is corrected by quantum stringy effects. It is well known that
such corrections are absent around flat space. This result is a
consequence of the large amount of supersymmetry in the flat space
theory. Since the background preserves the same
number of supersymmetries as the flat space background, one might
guess that here too the scattering of three gravitons is not affected
by quantum corrections. This guess cannot be simultaneously correct
with the claim that the 3-point functions are not corrected at
finite^{†}^{†} We thank T. Banks for a useful discussion on this
point. .

This paper is organized as follows. In section 2, we compute the correlation functions in the weak coupling limit. In section 3, we identify fields on which represent the modes corresponding to the chiral operators and construct their effective action to cubic order in the fields. In section 4, we use this action to obtain the 3-point functions of normalized CPOs of the SYM. We compare this result with the free field calculation of section 2 and find precise agreement. In Appendix A, we explain our notations and conventions. Appendix B is devoted to spherical harmonics on ; we define scalar, vector and tensor spherical harmonics in arbitrary dimensions, and obtain several formulae needed for the calculation in section 3.

2. Correlation Functions at weak coupling

CPOs of SYM are operators of the form

where are vector indices and are six matrices transforming in the adjoint of . The trace in the formula above is over indices. is a totally symmetric traceless rank tensor of . We can choose an orthonormal basis on the vector space such that . We normalize our action as . The propagators of interest are In this normalization the Yang-Mills coupling and the string coupling are related by

where are color indices.

The 2-point function of two CPOs specified by tensors and , is computed in free field theory by contracting all the s pair-wise and is nonzero only if . Consider

In the large limit only planar diagrams contribute. Planar diagrams correspond to contracting and in the same cyclic order in which they appear in . One finds

Using the orthonormality of the coefficients one thus deduces that (the term in the equation below is replaced by when considering the 2-point function of arbitrary CPOs which are not necessarily orthogonal)

In a similar fashion one may compute the 3-point function of CPOs specified by . To ensure that all s are contracted, of the s must contract between the first and second of these operators and similarly for other pairs. In the large limit, one finds

where and represents the unique invariant that can be formed from (by contracting indices between and ; indices between and and indices between and )

We rescale the CPOs such that they have normalized 2-point functions i.e.,

Their 3-point function is

This result is correct only at large and receives nonzero corrections at from non-planar diagrams.

Finally note that the contraction of two or three may be related to the integrals of two or three spherical harmonics over the sphere, by the formulae given in Appendix B.

3. Equations of motion and actions

3.1. Foreword to the Calculation

The particle spectrum of Type IIB SUGRA has been worked out in [20]. The particles are grouped into supermultiplets [21]. It turns out that the supermultiplets present in the theory correspond to representations of the superconformal algebra labeled by weight , and scaling dimension [22]. According to the results of [15,16] these are short representations. These supermultiplets of particles must correspond to CPOs (and their descendents) in SYM with the same , and scaling dimension labels. These are the operators discussed at the beginning of section 2. The fields that correspond to CPOs are particles in the representation with weight , representation with and mass [7]. Studying [20] (table III in particular), we conclude that the required fields are mixtures of the trace of the graviton on the sphere, and the five form field strength on the sphere.

Before identifying these fields and starting the calculation we make a few comments. 1. Since gravity is a gauge theory, not all fields in the IIB SUGRA action are physical. We need to choose a gauge and then solve the Gauss law constraints to identify the physical fields. Only these correspond to operators of the SYM. 2. Because of the absence of a simple covariant action for IIB SUGRA, we choose to work with equations of motion rather than an action. In order to compute the action for the fields to cubic order, we compute their equations of motion to quadratic order, and then produce an action that leads to these equations of motion. The action thus produced is of uncertain normalization; we fix this ambiguity by comparison with the correctly normalized action proposed in [23], at quadratic order. 3. We need to identify the SUGRA fields that couple to various operators only at linear order in fluctuations about the background. Nonlinear higher order corrections modify the computed correlation functions of the corresponding operators only by contact terms. This translates in spacetime to the fact that we compute only S matrix elements which are not modified by field redefinitions. We use this freedom to simplify our analysis.

With the cubic action in hand we then use the procedure of [8,9] to obtain the correlation functions of interest.

3.2. The Setting

The IIB SUGRA equations of motion of the graviton and the 5-form field strength are

We use units in which the scale of the and is set to be unity. See Appendix A for other conventions.

The background solution is

Bulk fields of interest are fluctuations about this background. Following [20], we set

We choose to (almost completely) fix diffeomorphic and 4-form gauge invariance by choosing the de Donder gauge . With this choice the most general expansion of these functions about the sphere is given by [20] (see Appendix B for information on spherical harmonics). For our purposes, it suffices to note that

3.3. Linear Constraints and Equations of Motion

The Einstein and self-duality equations about this background have been written out to linear order in [20]. Of interest to us are the three constraint equations (E3.2), (E2.2) and (M2.2) in that paper,

and the dynamical equations for and (Eq.(2.31) and (2.32) of [20]),

We are interested in modes with only. For such modes the constraint (3.6) may be used to eliminate from (3.9) and (3.10) to yield

These two equations may now be diagonalized. Using the fact that as shown in Appendix B, we find that the diagonal linear combinations (We choose the normalization such that the inverse relations are simple: .),

obey the equations of motion

To linear order, corresponds to CPOs in SYM, and it will be the focus of our attention through the rest of the paper.

The scalars , on the other hand, correspond to descendents of CPOs; specifically they map to the operator in Table 1 of [21]. The expansion of proportional to the spherical harmonic, corresponds to an operator formed by acting with 4 and 4 on the trace of operators. The 3-point functions of these operators are determined in terms of those of CPOs by the supercoformal algebra, and so we will not compute them directly. Henceforth we set .

We now construct an action whose variations leads to the equations of motion of .

with undetermined constants which depend on .

3.4. Normalization of the Quadratic Action

The normalization coefficients may be determined by comparison of (3.15) with the full ‘actual’ action of IIB SUGRA [23]

where is defined by the right-hand-side (RHS) of (3.2), and is an auxiliary field. In our units .

In order to obtain from (3.16) we work at quadratic order, choose a gauge, solve for all constrained fields in terms of physical fields, and then set all physical fields except to zero.

Firstly we eliminate the auxiliary field in (3.16). As shown in [23], we are free to fix a gauge by choosing an arbitrary function for . We will set , which amounts to removing the components of the 4-form potential of the form .

Having done this use (3.6), (3.7), (but not yet (3.8)) in (3.16) and set all unconstrained fields other than and and to zero. The action we obtain at the end of this process is ( is defined in Appendix B equation (B.4))

contains terms from the Einstein part of the action except those involving ,

where the first group comes from the kinetic and mass terms while the second group was obtained by inserting (3.6) into the kinetic and mass terms. contains terms from the part except terms,

is the part of (3.16) quadratic in :

We now attempt to use to obtain the quadratic dependence of on and . On eliminating and from (3.7) and separating out the trace explicitly we obtain

We can solve the equation by setting

where obeys and satisfy . Note that unlike , may consistently be set to zero for arbitrary and . Substituting this into leads unfortunately to an action non-local in and .

To avoid undue complications, we notice that it is sufficient for us to compute (3.17) on shell in order to obtain . In that case

We substitute , in (3.15) to find

(3.17) vanishes on shell in the bulk (as every quadratic action does), but is nonzero as a function of boundary values due to surface terms. We now compute each of (3.15) and (3.17) as a function of boundary values of , and compare the two results to read off the value of . The result is

3.5. Cubic Couplings

To study the 3-point functions of the field , we need the cubic terms in the action (3.15). To compute these we need quadratic corrections to Eqs. (3.6), (3.8), (3.9) and (3.10). We define

Substituting (3.24) into (3.25), we obtain,

The corrected equation of motion for is a linear combination of the two above:

To calculate the we use the methods outlined in [20]. The first lines of (3.24) and (3.25) are the coefficients of and respectively, in the equation . To compute and , we must therefore compute and to second order in [24]. Since we are only interested in the dependence of these quantities, we substitute

to find

In the equations above, the symbol is used as shorthand for and as shorthand for , respectively. Summation over and is assumed.

Projection of these quantities onto yields

where , etc., are used as shorthand for , etc. defined in Appendix B (dropping an overall factor of from the equations which will be reinstate later) and as shorthand for .

Projection onto yields

Expansion of the self-duality equations to quadratic order and projection onto appropriate spherical harmonics yields and . arises as the coefficient of and from the coefficient of in the self duality equation (3.2). The answers are

where and is the same as in (3.31).

This completes the evaluation of the RHS of the equation of motion (3.27) which now takes the form

where , and are computed by substituting (3.31), (3.32), (3.33) and (3.34) into (3.27). We can remove the derivative terms on the RHS of (3.35) by a field redefinition