Contents

PUTP-2572

More Exact Results in the Wilson Loop Defect CFT: Bulk-Defect OPE,

Nonplanar Corrections and Quantum Spectral Curve




Simone Giombi***sgiombi AT princeton.edu, Shota Komatsushota.komadze AT gmail.com

Department of Physics, Princeton University, Princeton, NJ 08544, USA

School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA





Abstract

We perform exact computations of correlation functions of -BPS local operators and protected operator insertions on the 1/8-BPS Wilson loop in SYM. This generalizes the results of our previous paper arXiv:1802.05201, which employs supersymmetric localization, OPE and the Gram-Schmidt process. In particular, we conduct a detailed analysis for the -BPS circular (or straight) Wilson loop in the planar limit, which defines an interesting nontrivial defect CFT. We compute its bulk-defect structure constants at finite ’t Hooft coupling, and present simple integral expressions in terms of the -functions that appear in the Quantum Spectral Curve—a formalism originally introduced for the computation of the operator spectrum. The results at strong coupling are found to be in precise agreement with the holographic calculation based on perturbation theory around the AdS string worldsheet, where they correspond to correlation functions of open string fluctuations and closed string vertex operators inserted on the worldsheet. Along the way, we clarify several aspects of the Gram-Schmidt analysis which were not addressed in the previous paper. In particular, we clarify the role played by the multi-trace operators at the non-planar level, and confirm its importance by computing the non-planar correction to the defect two-point function. We also provide a formula for the first non-planar correction to the defect correlators in terms of the Quantum Spectral Curve, which suggests the potential applicability of the formalism to the non-planar correlation functions.

1 Introduction

Wilson loops are important observables in gauge theories: They describe the coupling between a heavy probe particle and gauge fields, and are an efficient tool for distinguishing different phases of the theory. In supersymmetric Yang-Mills theory, one can also consider supersymmetric generalizations of the Wilson loop which couple to the scalar field as well as to the gauge field. Such Wilson loops—in particular the 1/2-BPS Wilson loop, which preserves a maximal amount of supersymmetries—played a significant role since the early days of the AdS/CFT correspondence [1, 2]. On the gauge theory side, the exact expectation value of the -BPS Wilson loop was computed first by resumming a class of ladder diagrams in [3, 4]. This computation was later justified by a rigorous argument based on supersymmetric localization in [5]. On the string theory side, the leading strong coupling behavior can be computed by evaluating the regularized area of the minimal worldsheet surface anchored on the Wilson loop at the boundary [6, 7]. The perfect matching with the strong coupling limit of the exact result [3, 4] provided one of the first important evidences for the existence of the holographic gauge/string duality. Subleading corrections at strong coupling may also be computed by evaluating the partition function for the quantum fluctuations around the classical string solution [8]. Very recently, a precise match between the localization prediction and the one-loop term on the string theory side was obtained [9].

The study of the -BPS Wilson loop recently gained new interest also from the point of view of conformal defects. Indeed the -BPS Wilson loop, being defined on a straight line or circular contour, preserves a SL subgroup of the full conformal group [10]. Owing to this fact, it can be viewed as an example of defect conformal field theory and has been studied from various perspectives. At weak coupling, the correlation functions of insertions on the Wilson loop were computed in [11, 12], while the correlators of “defect-changing operators” which change the scalar coupling of the -BPS Wilson loop were analyzed in [13]. At strong coupling, an extensive study of the correlators of the insertions was performed in [14] by using perturbation theory around the AdS worldsheet. Furthermore the generalization to the non-supersymmetric loop which interpolates between the -BPS Wilson loop and the standard Wilson loop was explored in [15, 16, 17] both at weak and strong coupling.

In the previous paper [18], we showed that a certain class of correlators on this defect CFT111The computation in [18] applies to more general -BPS Wilson loops. However, the relation to the defect CFT exists only for the -BPS Wilson loop. can be computed exactly by using the combination of supersymmetric localization, OPE and the Gram-Schmidt process. The results of the computation depend nontrivially on the coupling constants and give an infinite family of defect CFT data including the structure constants of the defect BPS primaries of arbitrary lengths. Such data would provide crucial inputs for performing further analysis, for instance in the context of the conformal bootstrap [19, 20].

The results described above mostly concern the correlators of insertions inside the Wilson loop. However, from the point of view of the defect CFT, there is yet another important class of observables: the correlation functions between the “bulk” local operators defined outside the Wilson loop and the defect operators inserted on the Wilson loop. Such correlators play a central role in formulating the defect crossing equation [21, 22], and allow us to connect the defect CFT data and the CFT data in the bulk. A special example of this, which has already been studied before, is the case of correlation functions of local operators and supersymmetric Wilson loop with no insertions [6, 23, 24, 25, 26, 27]: in the defect CFT language, this gives the bulk-defect OPE coefficient of the bulk operator and the identity insertion on the defect.

The main goal of this paper is to generalize the analysis in [18] to such bulk-defect correlators: More precisely we consider the correlators of the scalar insertions on the Wilson loop and a single-trace operator defined outside the Wilson loop (the case of several bulk insertions can also be obtained from our methods). The scalar is a position-dependent linear combination of the scalar fields which is chosen so that the correlator becomes independent of the positions. It has another important property that the insertion of single is related via localization to an infinitesimal deformation of the Wilson loop. This property allows us to compute the correlators involving ’s by the area derivatives of the expectation values of the Wilson loop, or of the correlator of the Wilson loop and the local operators, both of which are computable from localization [5, 26, 27]. As we explain in this paper, the computation can then be generalized to the correlators involving insertions of higher charges ’s () with the help of OPE and the Gram-Schmidt analysis.

Although the general framwork we present can be applied to the correlators at finite , in this paper we mostly focus on the leading large limit of the bulk-defect correlators. In the planar limit, we find that the results for the bulk-defect correlators can be expressed simply in terms of integrals,

(1.1)

where the definitions of the quantities in the formula are given in section 4.2. This formula is the generalization of the one found in our previous paper [18]. As was pointed out there, the function that appears in the formula coincides with the so-called Q-function in the Quantum Spectral Curve formalism [28]. This appearance of the Q-function is unexpected and strongly suggests that the Quantum Spectral Curve (QSC), which was originally invented for computing the spectrum of the operators, can be useful also for analyzing the correlation functions. Using this integral representation, we expand the results at weak and strong coupling. At weak coupling, the results match the perturbative answers SYM, while at strong coupling they reproduce the correlation functions of fluctuations of the string coordinates and the vertex operator on the AdS worldsheet, which we explicitly compute to leading order in the expansion.

In the course of the computation, we also clarify several aspects of the Gram-Schmidt analysis which were not discussed in our previous paper. Most importantly we point out the necessity of including the multi-trace-like operators , which may be viewed from the dual perspective as bound states of open strings and closed strings, which have to be included in the defect CFT spectrum. Such operators are negligible in the planar limit but can affect the computation at the non-planar level. To confirm this effect and check the validity of our formalism, we compute the non-planar correction to the defect two-point function explicitly and check the results against the direct perturbative computations. We also provide an integral representation for the first non-planar correction in terms of the Quantum Spectral Curve (see (6.43)), which suggests the potential applicability of the QSC formalism to the nonplanar corrections.

The rest of the paper is organized as follows: In section 2, we review the definitions of the BPS Wilson loops and the results of the supersymmetric localization. After doing so, we introduce the correlators that we analyze and discuss their relations to the defect CFT data. We then explain in section 3 how to compute such correlators using OPE and the Gram-Schmidt analysis. We first present a general formalism for constructing higher-charge operators applicable to finite , and then discuss the simplification at large . Using the results in section 3, we then evaluate the bulk-defect correlators in the planar limit in section 4, deriving the integral expression and obtaining the weak- and strong-coupling expansions. These results are in perfect agreement with the direct perturbative computations at weak and strong coupling in section 5. We then discuss the non-planar corrections to the defect two-point functions in section 6. Finally in section 7 we conclude and discuss future directions. Several appendices are included to collect some explicit results of the computation.

2 Set up

Before delving into the computation, let us first explain the set up by reviewing the supersymmetric subsector of SYM and showing its relation to the defect OPE data.

2.1 Supersymmetric subsector of Sym

The central object in this paper is the -BPS Wilson loop defined by

(2.1)

Here the Wilson loop couples to three out of the six real scalars and the contour is placed on a subspace of (see figure 1). In the rest of this paper, we choose to be of unit radius (namely ) and consider only the Wilson loop in the fundamental representation. Owing to the specific choices that we made for the contour and the scalar couplings, this Wilson loop preserves four supercharges and therefore is -BPS.

One can also add local operators without breaking two of the four supercharges. A prototypical example is the single-trace operator which is given by

(2.2)

where is a position-dependent linear combination of the scalar fields,

(2.3)

and we chose the standard normalization for the chiral primary operators222Note that the convention here is slightly different from the one in [26, 27]: In that paper, the gauge group generators are anti-Hermitian while in this paper we stick to a more standard convention in physics in which the generators are Hermitian. This explains the extra factor in [26, 27]. (see [26, 27]),

(2.4)

In addition to single-trace operators, there are also multi-trace operators defined by

(2.5)

These operators, together with the -BPS Wilson loops, form a supersymmetric subsector of SYM. Based on perturbation theory and AdS/CFT, it was conjectured in [29, 30] that the correlators in this subsector can be computed by the bosonic two-dimensional Yang-Mills theory on (in the zero-instanton sector) with the coupling constant

(2.6)

The relation between the observables in and is given by

(2.7)

The conjecture was supported later by supersymmetric localization [5] and tested against a number of nontrivial checks [31, 32, 26, 27, 25, 33, 34, 35].

Figure 1: Correlators of the -BPS Wilson loop and local operators. The -BPS Wilson loop, denoted by a red curve, can be defined on a arbitrary contour on and its expectation value depends only on the area of the subregion inside the loop (shown as the red-shaded region). In addition to the -BPS Wilson loop, one can consider the bulk single-trace operators . In this paper, we always assume that the bulk single-trace operators are on the same side of the Wilson loop, namely in the red-shaded region in the figure.

Thanks to its topological property, the computation in the two-dimensional Yang-Mills theory can be further reduced to simple Gaussian (multi-)matrix models [26, 27], and the results depend only on the area of the region inside the Wilson loop. For instance, the expectation value of the Wilson loop is given by

(2.8)

where is the area of the subregion on surrounded by the loop (see figure 1). Similarly the correlators of the Wilson loop and single-trace local operators are given by

(2.9)

with

(2.10)

Here we assumed that both operators are on the same side of the Wilson loop (see figure 1). These results can be straightforwardly generalized to the multi-trace operators. For instance the correlator of the Wilson loop and a double-trace operator is given by333Note that the correlator is different from the correlator . The former is the correlator of two single-trace operators inserted at two separate points in the bulk while the latter is the correlator of a double-trace operator inserted at a single point in the bulk. Diagrammatically, the former includes the contractions between two single-traces while the latter does not.

(2.11)

In the large limit, these correlators can be expanded in terms of the modified Bessel functions. For instance we have

(2.12)

where and are given by

(2.13)

On the other hand, the large expansion of the correlator consist of two terms; the disconnected term and the connected term . For the purpose of this paper, we only need the disconnected term since the connected term is subleading. Written explicitly, the disconnected term is given by

(2.14)

The explicit expression for the connected term can be found in Appendix A.

2.2 Operators on the Wilson loop and the area derivatives

In addition to the operators discussed in the previous subsection whose correlators can be computed by localization rather directly, there is another interesting class of operators which are obtained by inserting scalars inside a Wilson loop trace:

(2.15)

Here we parametrize the loop by . Note also that we put a normal-ordering symbol to in order to emphasize the absence of self-contractions inside each operator. At this point this may seem unnecessary complication, but the reason for doing this will become clear in the next section.

As shown in [5], the insertion of a single scalar corresponds to the insertion of a dual field strength of the two-dimensional Yang-Mills theory,

(2.16)

which in turn is related to a small deformation of the (2d) Wilson loop. This correspondence allows us to relate the correlators of multiple ’s to the area derivatives of the Wilson loop expectation value444Owing to the relation to the 2d Yang-Mills theory, these correlators are also independent of the postions (namely ’s).,

(2.17)

As discussed in the previous work [18], it is also possible to relate the insertion of higher-charge operators to the area derivatives. This however requires the use of the Gram-Schmidt orthogonalization, which we will review and refine in the next section, and the results in general take a more complicated form (although they can computed systematically from the localization results).

Now, putting together these operator insertions with the operators discussed in the previous subsection, one can consider a variety of correlators of the form,

(2.18)

In what follows, we call these correlators topological correlators since they do not depend on the positions. The main goal of this paper is to analyze these correlators555Although we mostly focus on the simplest correlators in this paper, the general methodology that we develop is applicable also to more complicated correlators (2.18). by generalizing the arguments in [18].

2.3 Relation to the defect CFT data

When the contour is a circle along the equator of , the Wilson loop preserves higher amount of supersymmetries and becomes -BPS:

(2.19)

An important feature of this Wilson loop is that it preserves the SL(2,R) conformal symmetry666The full symmetry group preserved by the circular Wilson loop is OSP. and therefore can be regarded as a conformal defect. This in particular implies that one can extract the defect CFT data from the correlators in the supersymmetric subsector.

Before discussing how to do so, let us first introduce the normalized correlators, defined by

(2.20)

where denote either the operators on the Wilson loop or the operators in the bulk depending on whether it is inside or not. Taking such a ratio renders the expectation value of the identity operator to be unity and make the correlators obey the standard defect CFT axioms. We then consider the following correlators of the protected operators,

(2.21)

Here and ’s and ’s are six-dimensional null vectors satisfying . In addition, we require the third components of ’s to vanish in order to make the operators to have protected conformal dimensions (in other words, the null vector projects onto a symmetric traceless representation of the preserved by the 1/2-BPS Wilson loop). Unlike the correlators in the supersymmetric subsector that we discussed in the previous subsections, these correlators depend on the positions of the operators. However, thanks to the conformal symmetry and the R-symmetry, the position dependence can be completely fixed to be

(2.22)

where , and is given by

(2.23)

The constants , and are the defect CFT data which are nontrivial functions of the ’t Hooft coupling and the rank of the gauge group. One can also perform the conformal transformation to map the Wilson loop to a straight line. In that case, one simply needs to perform the following replacement for and ,

(2.24)

while the expression for still applies if we interpret as the distance in the direction perpendicular to the Wilson loop.

These correlators reduce to the topological correlators (2.18) upon the following specification of the parameters

(2.25)

The results read

(2.26)

This shows that the topological correlators coincide with the defect CFT data up to trivial overall factors.

It is sometimes useful to unit-normalize the two-point functions on the Wilson loop. In such normalization, the structure constants are given as follows:

(2.27)

3 Construction of higher-charge operators

In this section, we construct general operator insertions on the Wilson loop in the supersymmetric subsector using the OPE and the Gram-Schmidt process. The same idea was employed already in the previous work [18], but here we elucidate several important points which were not accounted for in [18]. Although they are largely negligible at large , they have important consequences on the bulk-defect correlators and the non-planar corrections, as will be shown in sections 4 and 6.

3.1 Basic idea

Before presenting a general construction, let us explain the basic idea of our approach using simple examples of correlators on the -BPS circular loop. Along the way we clarify three important aspects (bound state, degeneracy and mixing with multi-trace operators) which were not discussed in our previous construction [18].

The basic strategy is to construct complicated operators from simpler operators using the OPE. The simplest operator (apart from the identity) on the -BPS Wilson loop is the single-scalar insertion . As discussed above, the correlator of this operator can be computed directly by taking area derivatives of the localization result.

Bound state

In addition to , there is yet another operator on the Wilson loop with the same -charge, ,777In this paper we take the gauge group to be for simplicity. In the case of , this type of operator would appear first at charge 2, i.e. . namely a single-trace operator placed at a point on the Wilson loop. On the string-theory side, this operator, and its higher charge generalizations, correspond to a bound state of open and closed strings. Although it might not be so obvious why it must be included in the defect CFT spectrum, one can show that such operators are necessary for the consistency of the bulk-defect crossing equation as explained in Appendix A. Unlike the insertion , this operator is not related to the area derivatives of the Wilson loop expectation value. To construct this operator, we instead need to consider a correlator of the Wilson loop and a bulk single-trace operator. Since the correlator is independent of the positions, we can bring the bulk operator arbitrarily close to a point on the loop without affecting its expectation value. After doing so, we perform the bulk-defect OPE to get

(3.1)

As shown above, in addition to , this process produces another operator, which is the identity operator on the Wilson loop (or, equivalently, the Wilson loop itself). The coefficient then corresponds to the bulk-defect structure constant between in the bulk and the identity operator on the Wilson loop, and is given by the expectation value

(3.2)

Inverting the relation (3.1), we obtain

(3.3)

This expression allows us to relate the correlators of to the correlators that are computable from supersymmetric localization.

Degeneracy and orthogonalization

Let us next consider the length-2 operators. In general, there are four different length-2 operators,

(3.4)

Among these four operators, the first operator can be constructed by taking the defect OPE of two scalar insertions,

(3.5)

where the second term on the right hand side comes from the self-contraction of the two scalars and is the defect structure constant of two ’s and the identity operator,

(3.6)

Then by subtracting this extra contribution, we can single out . To compute other operators, we need to consider bulk-defect correlators as was the case for the operator . For instance, the second operator can be obtained by taking the combination of the defect OPE and the bulk-defect OPE of in the bulk and , namely

(3.7)

Here the second term comes from the contraction of the bulk with the Wilson loop while the last term comes from the contraction of and the insertion on the loop . The OPE coefficients and are given by888Note that, for , one needs to divide by the norm of the resulting state in order to have the correct OPE expansion.

(3.8)

Repeating the same procedures for the other two operators, one can define sets of normal-ordered operators. The resulting operators are free of admixiture of operators with different -charges and they are all equally good defect primaries. However, they have one unsatisfactory feature that they are not necessarily orthogonal to each other; namely the two-point functions of different operators do not vanish in general. To make them orthogonal to each other, one has to take appropriate linear combinations of these four operators. The choice of the linear combinations is not unique since all these operator have the same quantum number and physically indistinguishable especially at finite , where the distinction by the number of traces becomes obscured. In this paper, we make a choice that is most suited for studying the correlators at large ; namely we define without performing further subtraction while for the rest of the operators we take appropriate linear combinations so that they become orthogonal to each other and also to .

Mixing with “multi-trace” operators

In the examples discussed so far, the self-contraction of insertions on the Wilson loop only produced the operators of the same kind, . However, starting from length- operators, a new effect shows up which inevitably mixes the operators of different types, namely the operators involving multi-traces999We call these operators multi-trace operators since the Wilson loop itself already contains a trace. e.g. .

(a) (b)
Figure 2: Wick contractions of the insertions on the Wilson loop. (a) In the planar limit, the Wick contractions (denoted by red double lines) can only connect neighboring insertions. Therefore, the resulting operator is again an insertion on the Wilson loop. (b) On the other hand, at the non-planar level, one can connect non-neighboring insertions by a propagator. This produces a closed index loop for the remaining fields and converts them into a single-trace operator. It is therefore necessary to include the “multi-trace” operators in the spectrum on the Wilson loop.

To see this explicitly, let us consider the OPE of three ’s inserted on the Wilson loop. At weak coupling, the OPE simply amounts to performing the Wick contractions. At large , due to planarity, the only allowed contraction is to connect two neighboring ’s as shown in figure 2-(a). As shown in the figure, we then get a single insertion on the Wilson loop, namely . However, at the non-planar level, we can Wick-contract non-neighboring ’s as shown in figure 2-(b). This produces a closed color-index loop for the remaining in the middle, and converts it into a single-trace operator . Thus the OPE of three ’s takes the following form:

(3.9)

This shows that, even if we are only interested in the operators of the form , we cannot neglect other operators since they mix with each other through the OPE. (The only exception is at large which we will discuss in section 3.3.) Such a mixing was not discussed in our previoius work [18], but as we will see later it has important consequences at the non-planar level.

3.2 Gram-Schmidt process

By repeating the recursive procedure described in the previous subsection, one can construct arbitrary operators on the Wilson loop in the supersymmetric sector. This however is not useful for writing down general and/or closed-form expressions. In this subsection, we explain an alternative approach which is more algorithmic and applies straightforwardly also to the general -BPS operators.

To explain the approach, let us first note that the normal-ordered operators constructed in the previous subsection share two important properties:

  • They are given by a sum of bare (un-normal ordered) operators, .

  • They are orthogonal to each other i.e. the two-point functions of different operators vanish.

As was pointed out in [18], the operators with such properties can be systematically constructed by the application of the so-called Gram-Schmidt process.101010Recently, the Gram-Schmidt process has been used for the computation of various correlation functions in supersymmetric field theories. See [36, 37, 38, 39, 40, 41, 42].

The Gram-Schmidt process is a recursive way of constructing the orthogonal vector basis from a given set of vectors. For instance, starting from a set of vectors (to be called bare vectors in what follows) one can construct the following mutually-orthogonal vectors

(3.10)

where denotes the inner product of two vectors. Written more explicitly, the first two vectors are given by the following expressions:

(3.11)

In our case, the role of the vectors is played by the bare (un-normalized) operators while that of ’s is played by the normal-ordered operators. The general bare operators are simply given by a collection of single-letter insertions on the Wilson loop and a multi-trace bulk local operator, namely

(3.12)

Here the notation means that the fields in the square brackets will be inserted on the Wilson loop when we evaluate the correlator. We chose this notation in order to avoid the confusion between the correlators of insertions on a single Wilson loop and the correlators involving several different Wilson loops. Throughout this paper, we only consider the former correlators: In other words, when we evaluate correlators of multiple ’s, we insert all the fields inside the square brackets into a single Wilson loop.

The inner products of these bare operators are given by their two-point functions, namely

(3.13)

In what follows, we use the following shorthand notation for the quantities on the right hand side,

(3.14)

where in the last equality we used the relation between the single-letter insertion and the area derivative. Having understood what and the inner product are, we can then use the general expression (3.10) to write down the expressions for the normal ordered operators . To see how it works in practice, below we construct explicitly operators with small charge.

L=0 :

Let us first consider the operators without R charge. There is only one operator with this property, which is the identity operator on the Wilson loop, or equivalently the Wilson loop itself. Since it does not mix with any other operators, the bare operator is already normal-ordered. We thus have

(3.15)
L=1 :

We next consider the operators with a unit R-charge. There are two bare operators with this charge in the supersymmetric subsector:

(3.16)

As mentioned in the previous section, there is some ambiguity in choosing the basis of the normal-ordered operators because of the degeneracy, and in this paper we choose the basis in which the normal-ordered is defined without subtraction of the operators with the same charge. This can be achieved in the Gram-Schmidt process by bringing first among the operators with the same charge.

In the case at hand, this amounts to performing the orthogonalization to the ordered set of vectors , and as a result we get

(3.17)
L=2 :

For , there are four different bare operators:

(3.18)

By performing the Gram-Schmidt analysis, we obtain for instance

(3.19)

We refrain from writing down the expressions for other operators since they are a bit lengthy.

By repeating this procedure, one can express any operators in the supersymmetric subsector in terms of bare operators . Once we have such expressions, we can compute their correlators by decomposing them into the correlators of bare operators, and relate them from the results of localization as

(3.20)

The correlator inside the square bracket on the right hand side is given by a multi-matrix model as explained in section 2.1 and in the references [27].

3.3 Simplification at large

The method described so far in principle allows us to compute arbitrary correlation functions in the supersymmetric subsector. However, as is clear from the examples shown above, the number of operators participating in the Gram-Schmidt process proliferates as the charge increases, making the computation quite complicated in practice.

Below we show that, at first few orders in the expansion, one can truncate the operator spectrum and make the formalism more tractable. Such truncation was already implied by the analysis in our previous paper in which we neglected the mixing with the “multi-trace” operators but nevertheless reproduced the correct correlators at large . The goal of this subsection is to show the existence of a similar truncation also at first few orders in the large expansion.

Truncation at large

Let us first analyze the strict large limit and see the decoupling of the multi-trace operators.

As exemplified in the previous subsection (see figure 2), the planar self-contraction cannot produce the “multi-trace” operators. This implies that the operators made up purely of insertions on the Wilson loop decouple from the multi-trace operators and form a closed subsector in the planar limit.

Figure 3: The structure of the Gram-Schmidt determinant at large . Owing to the large scaling of the two-point functions (3.24), the Gram-Schmidt matrix for consist of diagonal blocks and off-diagonal blocks. To faciliate seeing the structure of the matrix, we drew gray dashed lines which indicate the diagonal entries. To read off the contribution of each bare operator, one needs to compute cofactors (minors) of the elements in the last row. The cofactor of the single-trace operator, an example of which is denoted by a red square in the matrix on the left, is given by the determinant at the top-right corner. As is clear from the figure, its diagonal entries are all . On the other hand, the cofactor of the multi-trace operator, an example of which is denoted by a blue square in the matrix on the left, is given by the determinant at the lower-right corner and is .

Let us now prove the decoupling more rigorously from the structure of the Gram-Schmidt determinant (3.10). For this purpose, we simply need to use the fact that the inner product between the “single-trace” operator111111Namely the operators made up purely of insertions on the Wilson loop. and the “multi-trace” operator is suppressed by , which follows from the standard large counting:

(3.21)

To actually prove the decoupling, we first rearrange the rows and the columns of the determinant (3.10) and express the operator as shown in figure 3. Then, using the large scaling for the inner products (3.21), one can show that the off-diagonal blocks (shaded rectangular regions in figure 3) are all of order or higher while the diagonal blocks are of order . Now, the coefficient multiplying each bare operator can be read off from the cofactors (also known as minors) of the elements in the last row. As shown in figure 3, the cofactors of the insertions on the loop, are since their diagonal entries are all . On the other hand, the cofactors of the multi-trace operators are at most since one always needs to take at least one element when computing the determinant. This shows that, in the strict large limit, the contribution from the multi-trace operators to the single-trace operators is negligible and one can therefore focus on the insertions on the Wilson loop.

Now, after the decoupling of all the multi-trace operators, the Gram-Schmidt analysis simplifies greatly since there is only one operator left for each -charge. We thus obtain the following expression for the normal-ordered insertions at large ,

(3.22)

where is given by (3.12), namely , and the symbol denotes the normal ordering at large