Wilson loops and its correlators with chiral operators in \mathcal{N}=2,4 SCFT at large N

# Wilson loops and its correlators with chiral operators in N=2,4 SCFT at large N

E.Sysoeva
Diparimento di Fisica
Universit‘a di Roma Tor Vergata
I.N.F.N - sezione di Roma Tor Vergata
Via della Ricerca Scientifica, I-00133 Roma, Italy
###### Abstract:

In this paper we compute the vacuum expectation value of the Wilson loop and its correlators with chiral primary operators in superconformal gauge theories at large . After localization these quantities can be computed in terms of a deformed matrix model. The Wilson loops we deal with are in the fundamental and symmetric representations.

SYM theories, matrix models, correlation functions
preprint: ROM2F/2017/04

## 1 Introduction

The computation of the vacuum expectation value (vev) of a circular Wilson loop (WL) and the correlators of such WL with primary operators attract interest in the light of the AdS/CFT correspondence[1]. Especially interesting is the possibility to perform exact computations for the vev of a WL in supersymmetric gauge theories (SYM) exploiting the fact that only planar graphs contribute and that the propagators on the circle are constant [2]. The problem is thus reduced to the computation of the number of planar graphs, a task which is efficiently performed using matrix models[3, 4]. From this point of view, the matrix model is just a convenient tool to perform the computation. A direct computation of the vev of the WL leading to a matrix model was performed in [5]. In this reference the partition function of the SYM was computed and, as a byproduct, the SYM case was recovered by sending the mass to a peculiar limit in which the non perturbative contributions were just set to 1. This establishes a solid link between the computations performed in SYM and the matrix model. Furthering this line of research, in [6] the coefficients of the OPE of the WL were computed in the framework of the AdS/CFT correspondence using supergravity and string theory. This computation was later refined with a conjecture for the OPE of correlators of a WL with chiral primary operators of arbitrary conformal dimension [7]. Finally in [8] D3 and D5 branes were included in the computation. These results were all obtained in the limit of a large number of colors.

Following a somewhat different path, the results of [5] were extended in [9] by inserting arbitrary powers of the vector superfield into the classical prepotential. The general correlator can then be extracted from a deformed partition function given in terms of a deformed matrix model replacing the Gaussian matrix model underlying the theory. The same type of interacting matrix model arises in the studies of theories [5, 10], where now the interactions account for higher loop corrections and mass deformations. In this paper, we study the WL in the deformed matrix model and discuss applications to superconformal gauge theories.

The paper is structured as follows: in Section 2 we study the WL in the fundamental representation, deformed in different ways. In Section 3 the WL in theory is considered. In Section 4 we calculate the correlators between the WL and primary operators in the fundamental and in symmetric representations.

## 2 Deformed matrix models

As it was shown in [9], the vev of a WL in the fundamental representation of the deformed SYM theory can be written as an matrix model average

 W=1N⟨treC⟩vev=1N⟨trea⟩def (1)

The averages in the matrix model are defined as

 ⟨f(a)⟩def=1Z∫d[a]f(a)e−NV(a)=1Z∫daΔ(a)f(a)e−NV(a) (2)

where

 Z=∫d[a]e−NV(a)=∫daΔ(a)e−NV(a) (3)

with being the Lebesgue measure in the space of Hermitian matrices, being the Lebesgue measure in the space of eigenvalues absorbing numerical coefficient irrelevant for the calculation of averages and being the Vandermonde determinant

 Δ(a)=N∏u

In the following subsections we consider different types of the potential .

### 2.1 Even potentials

 V(a)=gntran (5)

with even .

In the large limit the integrals like (2) have been evaluated by saddle point methods [3, 4].

The resolvent defined as

 ω(x)=1N⟨tr1x−a⟩def=1x∞∑k=0ωkxk (6)

contains all the averages of the type .

Under the assumption of a single cut it can be represented in the form

 ω(x)=12V′(x)−Qn−2(x)√x2−b2 (7)

where is an even polynomial of order . The coefficients of the polynomial as well as the coefficient are determined by the condition at large . This condition generates a system of linear equations for the coefficients of the polynomial and one equation for the coefficient .

Solving the system one gets

 ω(x)=ngnxn−12⎛⎜⎝1−n2−1∑k=0122k(2k)!(k!)2(bx)2k√1−b2x2⎞⎟⎠ (8)

where is a solution of an equation

 gnbn2nn!(n2)!(n2−1)!−1=0 (9)

Expanding in a series one finds the resolvent in a form

 ω(x)=1x+2gnn!2n(n2−1)!2∞∑p=1122p(2p)!(n+2p)p!2bn+2px2p+1 (10)

The WL can be written in terms of the resolvent111The WL can be also found as , where is density of eigenvalues, which is clearly equivalent to (11) since .

 W=12πi∮dzω(z)ezdz=12πi∮dz(1z∞∑k=0ω2kz2k)(∞∑t=0ztt!)=∞∑m=0ω2k(2k)! (11)

From (6), (10) and (11) one finds

 W=1+2gnn!2n(n2−1)!2∞∑p=1122pbn+2p(n+2p)p!p! (12)

Evaluating the sums one gets

 W=21−ngnn!(n2−1)!2bn(1bddb)n2−1bn2−2In2(b) (13)

Since the resolvent is linear in (except for the implicit dependence on in ), the results can be easily generalized to the case of an even potential

 V(a)=∑ngntran (14)

One has simply to sum over .

 W=∑n21−ngnn!(n2−1)!2bn(1bddb)n2−1bn2−2In2(b) (15)

with now satisfying the equation

 ∑ngnbn2nn!(n2)!(n2−1)!−1=0 (16)

In particular for the case considered in [9]222Unlike [9] we choose the parameters of the deformed background as , not .

 V(a)=2λtra2+gntran (17)

one gets

 W=2bλI1(b)+21−ngnn!(n2−1)!2bn(1bddb)n2−1bn2−2In2(b) (18)

with being a solution of

 gnbn2nn!(n2)!(n2−1)!+b2λ−1=0 (19)

Although the equation (36) for cannot be solved for general , it can be treated in perturbation theory.

At the zeroth order one finds the well-known expectation value of the non-deformed WL

 W(0)=2bλI1(b),b2=λ (20)

It can be noticed that for the non-deformed WL, at large while in the deformed case at large , so can be considered as an effective coupling constant . But unlike the non-deformed case is limited. In fact (36) implies that

 λ=λeff1−λn2effgn2nn!(n2)!(n2−1)!

and since one gets a constraint for

 λeff≤λ∗eff=4((n2)!(n2−1)!gn!)2n (21)

In the limit of the WL stops depending on and turns into a constant .

Using the zeroth and the first terms of a perturbation series as well as the exact solutions for one can conjecture the general term

 (22)

where

 C(n)=(n)!(n2)!(n2−1)!

For the WL under this conjecture one gets

 W=1+∞∑m=1∞∑k=0(−C(n)gn(λ4)n2)k(λ4)m(n2k+m−1)!m!(m−1)!k!((n2−1)k+m+1)! (23)

### 2.2 Potentials with odd terms

In the case of the potential containing both odd and even deformations

 V(a)=∞∑l=1gltral (24)

the resolvent has an asymmetric cut

 ω(x)=12V′(x)−∞∑l=1Pl−2(x−c)√(x−c)2−b2 (25)

where the polynomial contains now both even and odd powers of and stands for the center of the cut. Demanding the resolvent to behave as at large one gets two separate systems of linear equations for the coefficients of with odd and even powers of , with the same invertible matrices as in the previous case. This requirement also leads to the equations

 ∞∑l=2glcl−2b2[l2]−1∑k=0122k+2l!(l−2−2k)!(b/c)2kk!(k+1)!−1=0 (26)
 ∞∑l=1glcl−1[l+12]−1∑k=0122kl!(l−1−2k)!(b/c)2kk!2=0 (27)

If there are only even powers in the deformation then and the second equation is missing (but can be equal to even if there are odd deformations in the potential).

The resolvent can now be written as

 ω(x) = 1x−c+∞∑l=2∞∑k=1122k+2glcl−2bl!(l−2)!(2k)!(k+1)!k!b2k+1(x−c)2k+1+ (28) + ∞∑l=3[l+12]−2∑k=0122k+2glcl−3b2l!(l−2k−3)!(b/c)2kk!(k+1)!∞∑p=1122p+1(2p)!(k+1+p)p!2 (x−cb+12bc(k+1+p)(k+2+p))b2p+1(x−c)2p+1

For the WL one gets

 W=ec2πi∮dzω(z)ez−cdz=12ec∞∑l=2glcl−2bl!(l−2)!I1(b)+ (29)
 +14ec∞∑l=3[l+12]−2∑k=0glcl−3b2l!(l−3−2k)!(b/c)2kk!(k+1)!(1+12bc(l−3−2k)(k+1)ddb)(1bddb)kbkIk+2(b)

### 2.3 Double-trace potential terms

Let us now consider a potential involving double trace terms

 V(a)=2λtra2+gNtrantram (30)

which will be used in the Section 3.

Considering the averages in this theory as integrals over the eigenvalues (2) one gets an equation for the saddle point [3]

 2N∑i≠j=11ai−aj−4Nλai−gnan−1iN∑j=1amj+gmam−1iN∑j=1anj=0 (31)

which coincides with an equation for the saddle point in the effective potential

 Veff=2λtra2+gωmtran+gωntram (32)

with being the coefficients in the series expansion of the resolvent (6).

Thereby the potential (30) can be replaced by an effective potential (32) and the results from the previous section can be applied also to this case.

An explicit form of equations for the coefficients in the simplest case of even follows directly from the resolvent

 2mωmbm=b2λh(m)+gωnbm2mf(m,m)+gωmbn2nf(n,m) (33)
 2nωnbn=b2λh(n)+gωnbm2mf(m,n)+gωmbm2nf(n,n) (34)

where

 h(n)=n!(n2+1)!(n2)!,f(n,m)=n22(m+n)n!n2!n2!m!m2!m2!

If or is odd, one should be careful since is a coefficient in the expansion of the resolvent (28) in terms of , not in terms of .

It can be noticed that there is a difference in the diagrammatic interpretation of a WL in the deformed potentials (24) and (30). In the first case every cycle brings a factor , every propagator brings a factor and every vertex brings again a factor , so the total power of is proportion to the Euler characteristic of the surfaces, i.e. all planar diagrams make a contribution in the deformed WL. In the second case every cycle still brings and every propagator brings while the vertices don’t bring a factor anymore. Therefore not all of the planar diagrams make a contribution. Indeed, in every double vertex appearing in the perturbative treatment of the potential (30) only one vertex can be connected with the WL, while the second one should belong to a part separated from the WL, and can be interpreted as an effective factor in the first vertex. This fact is reflected by a replacement of the potential by the effective one.

### 2.4 Examples

• This potential was considered in [9].

The WL in this case is

 W=2bλI1(b)+3g4b3ddbI2(b) (35)

where is a solution of the equation

 34g4b4+b2λ−1=0,b2=−1+√1+3g4λ232g4λ (36)

The result coincides with [9].

• In this case the WL is given by

 W=ec2bλI1(b)+3ecg3b(cI1(b)+12I2(b)) (37)

where is

 c=23 g3b2(1−b2λ) (38)

and is the real positive solution of the cubic equation

 9b6g23−b4λ2+1=0 (39)
• The corresponding effective potential is

 Veff=2λ+2ω3gtra3 (40)

Re-expanding the resolvent (28) in terms of one finds the equation for

 ω3b2=b2cλ(34+c2b2)+316b4g3ω3(1+18c2b2) (41)

Together with the equation for

 cλ+34b2g3ω3+32c2g3ω3=0 (42)

it gives the simple solution , so the effective potential does not contain a deformed term at all.

One could immediately see that in this case, since the initial potential is even, so all the averages of odd functions of disappear, including

 ω2k+1=1N⟨tra2k+1⟩def=1N∫d[a]tra2k+1e−NV(a)=0 (43)

Therefore the WL is just

 W=2bλI1(b),b2=λ (44)
• The effective potential in this case is

 Veff=2λ+ω4gtra2+ω2gtra4 (45)

The coefficients are given by the equations (33), (34)

 ω2b2=416−b6g,ω4b4=2+116b6g16−b6g (46)

The WL is

 W=2bλI1(b)+gb52+116b6g16−b6gI1(b)+12gb5116−b6gddbI2(b) (47)

and is the real positive root of the equation

 1512b12g2λ−116b8g+516b6gλ+b2−λ=0 (48)

## 3 Wilson loop in the N=2 theory

As it is shown in [10]333In [10] the parameters of deformed background were chosen as and we stick to our previous choice . the partition function of the super Yang-Mills theory with U(N) gauge group and fundamental matter multiplets defined on a four-sphere in the weak-coupling limit can be written as

 ZN=2=∫d[a]e−N2λtra2e−S(a) (49)

Here is a Hermitian matrices and

 S(a)=−2N∑u

where

 logH(x)=−(1+γ)x2−∞∑i=2ζ(2i−1)x2ii (51)

with being the Riemann zeta-function and being the Euler-Mascheroni constant.

The vev of the circular WL in this theory is

 WN=2=1ZN=2∫d[a]treae−N2λtra2e−S(a) (52)

where can be written as a sum of homogeneous polynomials of order .

 S2(a)=−(1+γ)(2π)2((2N−Nf)tra2−2(tra)2) (53)
 S2n(a)=(−1)nζ(2n−1)n(2π)2n((2N−Nf)tra2n+2n−1∑k=1Ck2ntraktra2n−k) (54)

where are the binomial coefficients.

Let us consider a conformal case with .

can be interpreted as a deformation in theory of the form considered in the previous section. Since is an even function of , all and only the terms of the form with even in actually deform the averages. Therefore the effective potential has the form

 Veff=2∞∑n=2(−1)nζ(2n−1)n(2π)2ni−1∑k=1C2k2ntra2kω2(n−k)=∞∑k=1g2ktra2k (55)

with

 g2k=2∞∑n=k+1(−1)nζ(2n−1)n(2π)2nC2k2ntra2kω2(n−k) (56)

The system of equations for following from the resolvent can be written as

 ∞∑n=1(δk,n−Ak,n)22nω2nb2n(2n)!=b2λ1k!(k+1)! (57)

with the matrix defined as

 Ak,n=2∞∑l=1(−1)l+nb2(l+n)(2(1l+n))!(k+l)(l−1)!2k!2(n+k)ζ(2(n+l)−1)(4π)2(n+l) (58)

It seems that the infinite system (57) cannot be solved in the general case, but since , the system can be treated in perturbation theory in terms of a small .

For example up to in the WL one will find the effective potential as

 Veff=tra2λ(325ζ(3)π4b4−528ζ(5)π6b6+9211ζ(3)2π8b8+35213ζ(7)π8b8)+ (59)
 +tra4λ(−527ζ(5)π6b4+35211ζ(7)π8b6)+tra6λ729ζ(7)π8b4+O(λ5)

where should be found perturbatively from the equation

 b4λ[1+326ζ(3)π4b4−529ζ(5)π6b6−1529ζ(5)π6b8+9212ζ(3)2π8b8+35214ζ(7)π8b8]+O(λ6)=1 (60)

The WL one will get in the following form

 WN=2=W(0)−32ζ(3)(4π)4λ3+(−18ζ(3)(4π)4+15ζ(5)(4π)6)λ4+ (61)
 +(−1256ζ(3)(4π)4+6548ζ(5)(4π)6+36ζ(3)2(4π)8−140ζ(7)(4π)8)λ5+O(λ6)

where stands for the non-perturbed WL in theory (20).

This expansion has a simple diagrammatic interpretation. For example, the terms proportional to come from a diagrams with one double-vertex of the form , i.e. from the diagrams with two single two-pointed vertices, where one vertex is connected with the WL and one is disconnected (self-contracted). A self-contracted two-pointed vertex brings a factor and the two-pointed vertex connected with the WL brings a factor of (see (73)). There is also a multiplicative factor of 2 due to the two possible ways to pick up a vertex to connect to the WL. Therefore all the terms proportional to come from the expansion of in any order of including the ones in (61).

## 4 Correlators between the WL and chiral operators

It was shown in [9] that the correlators between the WL and the powers of chiral operators in the fundamental representation of the non-deformed SYM theory can also be written as matrix model averages

 ⟨tr~φneC⟩vev=⟨tr(ia)ntrea⟩ (62)

where stands for the averages defined in (2) with quadratic potential .

The correlators between the WL and the normal-ordered chiral operators correspond to the matrix model average

 ⟨:tr~φn:eC⟩vev=⟨:tr(ia)n:trea⟩ (63)

with in the right-hand-side standing for the Wick product.

In the following subsections we consider WL’s correlators with chiral operators in the fundamental and symmetric representations.

### 4.1 Fundamental representation

As it is clear from (1), the expectation value of the deformed WL with the potential (17) appears to be a generating function of connected correlators of non-normal ordered chiral primary operators with the WL in the undeformed gauge theory, in particular,

 W=W(0)−gW(1)(n)+... (64)
 (65)

counts all diagrams with one -point vertex generated by the WL. In order to calculate the correlator between the WL and the normal ordered chiral operators corresponding to , one needs to subtract from all contributions given by the diagrams where some of the legs coming out of the -point vertex are contracted among themselves and leave only the diagrams with propagators starting from the -point vertex and ending on the WL.

Taking into account that every propagator gives a factor , one can write

 ⟨trea:tran:⟩c=~W(1)(n)=W(1)(n)−n2−1∑l=1Bnl(λ4)l~W(1)(n−2l) (66)

The coefficients are the ratios of the number of planar diagrams with self-contractions in the -point vertex and legs connecting the vertex with the WL to the number of planar diagrams with a -point vertex without self-contractions.

In order to find the coefficients let us introduce the quantities standing for the number of ways in which the legs of a -point vertex can have self-contractions and leave external legs (staying planar). With the help of Appendix A one finds

 Kr(m)=(r+2m)!(r+1)(r+m+1)!m! (67)

So for the coefficients one finds then

 Bnl=nKn−2l−1(l)(n−2l)=n!(n−l)!l!=Cln (68)

where are the binomial coefficients. Therefore

 W(1)(n)=n2−1∑l=0Cln(λ4)l~W(1)(n−2l) (69)

(69) can be inverted (see Appendix A), leading to

 ~W(1)(n)=n2−1∑l=0(−1)lAnl(λ4)lW(1)(n−2l) (70)

where

 Anl=n(n−l−1)!l!(n−2l)! (71)

Using (70) and the first term of (23)

 W(1)(n)=∞∑m=1C(n)(λ4)n2+mm!(m−1)!(n2+m) (72)

one can check that

 ~W(1)(n)=n2nλn2In(√λ) (73)

which is in the agreement with the result [7] and proves the conjecture that the radiative corrections cancel to all orders and the sum of planar rainbow diagrams is enough to calculate the correlators.

### 4.2 K-symmetric representation, K/N→0 limit

In the -symmetric representation the expectation value of a WL written as the matrix model integral, is

 WK=1N⟨trKeC⟩vev=1N⟨trKea⟩def (74)

where stands for the WL in the -symmetric representation.

In the large t’Hooft coupling constant limit the WL (74) can be written up to exponentially suppressed terms as [12, 13]

 WK=1Z∫∞−∞daΔ(a)e−NV(a)+Ka1 (75)

One can write then

 WK=1N1Z∫∞−∞daΔ(a)e−NV(a)N∑i=1eKai=1N1Z∫d[a]treKae−NV(a) (76)

where stands for the trace in the fundamental representation.

It implies that the WL in the -symmetric representation has the same diagrammatic interpretation as the WL in the fundamental representation up to a factor for every line coming to the WL.

In particular, in the case of the potential (17) in the planar limit

 WK=1N∞∑m=0K2m⟨tra2m⟩4m