Antisymmetric Wilson loops in $\mathcal{N}=4$ SYM beyond the planar limit

Since its inception, the AdS/CFT correspondence has held out the promise of a fully non-perturbative definition of quantum string theory in non-trivial backgrounds. Testing this strongest form of the conjecture is, however, very hard. Progress can be made in this direction by considering controlled deviations from the large-$N$, large-$\lambda$ limit. In an exciting development, the techniques of supersymmetric localization and integrability have in recent years generated a profusion of exact gauge theoretic results, enabling such quantitative testing and ushering in an era of precision holography.

Supersymmetric localization reduces the partition function of $\mathcal{N}=4$ super-Yang-Mills to a gaussian Hermitian matrix model PestunLocalizationOfGaugeTheory . Furthermore, a certain supersymmetry-preserving sub-sector of the theory is completely captured, for arbitrary $N$ and $\lambda$, by matrix model expectation values of appropriate insertions (see eqs. (4,5) in the next section). For the fundamental Wilson loop this was anticipated in the prescient work Erickson2000 (also Drukker2001 ) where an infinite class of planar diagrams was explicitly re-summed, generating what was correctly conjectured to be the exact planar result. Localization formulæ  for more general correlators and Wilson loops have since followed.

In the study of non-abelian gauge theories, the Wilson loop operator plays a fundamental role. In addition to serving as an order parameter for the confinement-deconfinement phase transition, it provides a naturally gauge-invariant formulation of the theory which, while inherently non-local, is quite natural from the point of view of the correspondence to string theory. It can be understood as the phase acquired by a probe particle as it traces out some closed path $C$. As well as this contour, the Wilson loop is labeled by a representation $\mathcal{R}$ of the gauge group, describing the charge of the probe particle. In $\mathcal{N}=4$ SYM, the natural (supersymmetric and UV finite) Wilson loop observable also includes a scalar field coupling:

For the special case where $C$ is a circle and $n^{I}$ a constant unit vector, this preserves half of the supersymmetries, permitting its localization after compactification on $S^{4}$.

To date the correspondence has withstood over two decades of sustained scrutiny. It is therefore noteworthy when tension, let alone disagreement, is found between putatively dual quantities. We can expect such cases to reveal important subtleties or misunderstandings of the dictionary, or indeed to elucidate the limits of its applicability.

In this paper we study an as-yet unresolved mismatch in the most scrutinized example of AdS/CFT, namely the duality between $\mathcal{N}=4$ supersymmetric Yang-Mills theory with gauge group $SU(N)$, and type IIB superstring theory on $AdS_{5}\times S^{5}$. The discrepancy occurs in the 1-loop correction to the $\frac{1}{2}$-BPS circular Wilson loop in the rank-$k$ totally-antisymmetric representation of the gauge group, with $k\sim\mathcal{O}(N)$. We compute this quantity on the gauge theory side by solving the loop equations for the corresponding matrix model obtained from localization¹¹1 A similar approach to deriving Wilson loops in higher representations from the loop equations has been advocated for example in Akemann2001 where general $k$-loop (fundamental) correlators, from which general representations can be constructed, were computed building on the work of Ambjoern1993 . However, their results are not directly applicable here as we will be interested in the limit $k\rightarrow\infty$, with $k/N$ fixed. . In fact, the antisymmetric Wilson loop was evaluated exactly using orthogonal polynomials in Fiol2014a ; however, it is not clear how to extract from their result the $1/N$ expansion, which is needed for comparison with holography. Their formula will however be useful for numerical verification of our result.

Apart from its intrinsic interest, an understanding of the structure of higher-rank Wilson loops may also yield insight into the analogous, longstanding $1$-loop matching problem for the fundamental Wilson loop Forste1999a ; Drukker2000a ; Sakaguchi2008 ; Kruczenski2008 ; Kristjansen2012 . See Forini2016 ; Faraggi2016 ; Forini2017 for recent progress on this problem.

According to the AdS/CFT dictionary, the antisymmetric Wilson loop is dual to a probe $D5$-brane with $k$ units of electric flux on its $AdS_{2}\times S^{4}$ worldvolume Gomis2006 ; Gomis2007 , which “pinches off” along the circular contour described by the Wilson loop at the boundary of $AdS$. At leading order in large $N$ and $\lambda$ the on-shell action of the $D$-brane was successfully matched with the gauge theory Yamaguchi2006 ; HartnollKumar2006 .

The $D$-brane tension is of order $N$, and its 1-loop effective action captures non-planar contributions. The spectrum of fluctuations and 1-loop effective action were derived in Faraggi2011 ; Harrison2012 and Faraggi2012OneLoopEffectiveActionAntisymmetricWL respectively, with the result²²2The answer of $\frac{1}{12}\ln\sin\theta_{k}$ quoted in Faraggi2012OneLoopEffectiveActionAntisymmetricWL was updated by the authors in the subsequent publication PandozayasOneLoopStructureWL to incorporate a missing normalization factor.

where $\theta_{k}$ is defined by $(\theta_{k}-\sin\theta_{k}\cos\theta_{k})=\pi k/N$. A first step towards reproducing this from the gauge theory side was taken in PandozayasOneLoopStructureWL . They obtained the same functional dependence on $k$, but a different overall constant:

The mismatch is not surprising, as the computation neglected the backreaction of the Wilson loop insertion on the equilibrium eigenvalue distribution of the matrix model. Here we will systematically take this into account. However, our result, which withstands convincing numerical testing, still does not match with (2); even the power of $\lambda$ is different. As we mention in the conclusions, this contribution to the free energy likely corresponds to the gravitational backreaction of the probe $D$-brane, i.e. our result is of very different origin to (2). It would still be interesting to try to match (2) with a gravity calculation by a careful determination of strong-coupling corrections on both sides.

The layout of the paper is as follows. In section 2 we summarize the localization result PestunLocalizationOfGaugeTheory for the Wilson loop, and set up the problem. In section 3 we derive a sequence of loop equations for the gaussian matrix model perturbed by the Wilson loop insertion, and solve them for the resolvent up to the second sub-leading order. We then derive from this the free energy, by two different means. We also calculate the correction to this result due to considering gauge group $SU(N)$ instead of $U(N)$. Section 4 presents some numerical checks of our answer, by comparing to the exact result of Fiol2014a . Finally we end with some conclusions and open questions in section 5.

Localization of $\mathcal{N}=4$ SYM reduces the full partition function to that of a Hermitian Gaussian matrix model PestunLocalizationOfGaugeTheory

while the expectation value of the circular Wilson loop is mapped to an expectation value (denoted $\left\langle\right\rangle_{0}$) in this matrix model:

The representation $\mathcal{R}$ of the gauge group is completely arbitrary at this stage. We will be interested in $\mathcal{R}=\mathcal{A}_{k}$, the totally anti-symmetric representation of rank $k$, in the large-$N$, large-$\lambda$ regime with

held fixed. The generating function for the character of this representation is

so that we can write the Wilson loop expectation value as HartnollKumar2006

where $D$ encircles the origin and $d_{A}=\binom{N}{k}$ is the dimension of the representation. The following change of variables, which maps the complex $t$-plane to the cylinder, will prove convenient:

It will also be useful to view the expectation value of $F_{A}$ as defining a family of perturbed partition functions parametrized by $z$,

and to define a corresponding “free energy”

Note the unconventional factor of $Z_{Gauss}^{-1}$ here. In this manner we obtain the following exact expression for the Wilson loop:

The free energy of the purely gaussian matrix model is $\mathcal{O}(N^{2})$ and has a genus expansion in powers of $1/N^{2}$, ie.

Since $\mathcal{Z}(z)$ differs from $Z_{0}$ by a perturbation to the action of $\mathcal{O}(N)$, its logarithm goes in powers of $1/N$, with leading term identical to that of $\log Z_{Gauss}$. Consequently, $\mathcal{F}(z)$ defined in (11) is $\mathcal{O}(1)$, and we write

As the exponent in (12) is $\mathcal{O}(N)$ we can evaluate the $z$-integral in the saddle-point approximation, which yields

where $z_{*}$ solves the saddle-point equation $\mathcal{F}_{0}^{\prime}(z_{*})=f$. To the order in $N$ given here, there is no backreaction on the saddle due to $\mathcal{F}_{1}(z)$. The $i$ prefactor is from analytic continuation of the “wrong-sign” quadratic form. Finally, plugging in (14), we have

The leading order result was already obtained in Yamaguchi2006 ; HartnollKumar2006 and agrees perfectly with the $D5$-brane on-shell action Yamaguchi2006 . The $\log\mathcal{F}_{0}^{\prime\prime}$ term was obtained in PandozayasOneLoopStructureWL . Interestingly, the latter turns out to have the same functional dependence on $k$ as the 1-loop effective action of the $D$-brane, as computed in Faraggi2012OneLoopEffectiveActionAntisymmetricWL , but with a different numerical coefficient. One might anticipate that $\mathcal{F}_{1}(z_{*})$, at leading order in $1/\lambda$, should give a similar contribution, so as to correct the numerical mismatch. In fact this turns out not to be the case: the log term is subleading in $\lambda$.

We review the planar solution in the next subsection. What then remains is to compute the non-planar free energy $\mathcal{F}_{1}(z)$ of the matrix model (10). We do this in section 3 by calculating the resolvent $\mathcal{W}(p)\equiv\left\langle\,\mathrm{tr}\,\frac{1}{p-M}\right\rangle$ order-by-order in $N$ using the loop equation method. With the resolvent in hand, the free energy can be determined in either of two ways.

1. 1.

   $\lambda$-integral: $\mathcal{W}(p)$ is the generating function of monomial expectation values, $\left\langle M^{j}\right\rangle=N\oint\!dp\,p^{j}\mathcal{W}(p)$. But $\left\langle M^{2}\right\rangle$ is also just the derivative of $\log\mathcal{Z}$ with respect to $\lambda^{-1}$. Therefore $\mathcal{F}$ is obtained from the resolvent by an integral over $p$ and $\lambda$.

2. 2.

   Eigenvalue density: At large $N$ the eigenvalues condense into a continuous distribution $\rho(x)$. The $\mathcal{O}(1)$ perturbation to $\rho$ due to the Wilson loop insertion is encoded in the discontinuity of $\mathcal{W}_{1}(p)$ across its single cut. Fluctuations around the large-$N$ saddle-point of the $\int[dM]$ integral are not needed as they cancel against the same contribution coming from $Z_{Gauss}$.

Naturally, we find exact agreement between these methods.

We now set up a systematic expansion around the $N=\infty$, 1-cut solution of the matrix model, using the well-known loop equation approach Ambjoern1993 . The matrix model (10) is rather exotic - it involves a non-polynomial $\mathcal{O}(1/N)$ perturbation to the Gaussian potential. Consequently the usual genus expansion familiar from the study of polynomial-potential matrix models (see eg. Ambjoern1993 ) becomes an expansion in $1/N$.

We begin with a few definitions. Our main object of study will be the resolvent, defined by

The double angle-brackets mean the expectation value is with respect to $\mathcal{Z}(z)$, ie. such expectation values are always functions of $z$. More generally, the “$s$-loop correlator” is defined as

The so-called loop equation for the resolvent follows from invariance of the partition function under the infinitesimal change of variables

The Jacobian for this transformation is $\left(\,\mathrm{tr}\,\frac{1}{p-M}\right)^{2}$. By the following simple manipulations

and

where the positively-oriented contour $C$ encloses the singularities of $\mathcal{W}$ but excludes the point $p$ (and possible singularities of “$G$”), we obtain the following equation for $\mathcal{W}(p)$:

Here $\phi_{z}(\omega)$ is defined by

Our problem is specialized to a Gaussian potential, $V(x)=\frac{2}{\lambda}x^{2}$. This is almost identical to the well-known loop equation for the Hermitian matrix model with polynomial potential $V(\omega)$, except for the $1/N$ term on the right-hand sideAmbjoern1993 . Plugging in the expansion (25b) we find a series of equations which can be solved iteratively in $n$. The $n=0$ equation is unaffected by the Wilson loop insertion:

For a Gaussian potential the solution is well-known (see e.g. Marino2004 ):

This has a single branch cut along $-\sqrt{\lambda}<p<\sqrt{\lambda}$. The higher order equations are

where we have introduced a linear operator $\hat{K}$, defined as in Ambjoern1993 by

The contour $C$ is defined as before. Note that the RHS always involves correlators with smaller $n$ than the LHS. Thus one can in principle solve iteratively to obtain any $\mathcal{W}_{n}(p)$ ³³3In Ambjoern1993 , the general iterative solution beyond leading order relies on the fact that, unlike our eq. (34), the RHS there is always a rational function of $p$ (proof by induction), so by a partial fraction decomposition can be written as a sum of powers of $(p-x)^{-1}$ and $(p-y)^{-1}$, where $x$, $y$ are the endpoints of the cut. The solution $\mathcal{W}_{n}$ is thus easily expressed in terms of a set of basis functions $\chi^{(n)}(p)$, $\Psi^{(n)}(p)$, determined explicitly there, with the property that $\displaystyle\left\{\hat{K}-2\mathcal{W}_{0}(p)\right\}\chi^{(n)}(p)$ $\displaystyle=$ $\displaystyle(p-x)^{-n}$ (36) $\displaystyle\left\{\hat{K}-2\mathcal{W}_{0}(p)\right\}\Psi^{(n)}(p)$ $\displaystyle=$ $\displaystyle(p-y)^{-n}$ (37) In general the operator $\left\{\hat{K}-2\mathcal{W}_{0}(p)\right\}$ can also have zero modes; in such cases, assuming a single cut, this freedom is constrained by the large-$p$ asymptotics of $\mathcal{W}(p)$. .

In the rest of this section we shall solve (34) up to $n=2$.