String corrections to circular Wilson loop and anomalies

We study string quantum corrections to the ratio of latitude and circular Wilson loops in N=4 super-Yang-Mills theory at strong coupling. Conformal gauge for the corresponding minimal surface in AdS(5)xS(5) is singular and we show that an IR anomaly associated with the divergence in the conformal factor removes previously reported discrepancy with the exact field-theory result. We also carefully check conformal anomaly cancellation and recalculate fluctuation determinants by directly evaluating phaseshifts for all the fluctuation modes.


Introduction
We will study the ratio of two Wilson loops in N = 4 super-Yang-Mills (SYM) theory that share a common contour in space-time, but differ in their coupling to scalars, following the proposal of [1,2,3]. Wilson loops are important observables in gauge theories and are unique probes of the AdS/CFT correspondence since they couple directly to the string worldsheet in the dual gravitational background [4,5]. Some Wilson loops in the SYM theory can be actually computed exactly, at any coupling strength, without making any approximations. Subsequent extrapolation to strong coupling establishes a direct link between conventional QFT calculations and holography.
The simplest example of this type is the circular Wilson loop whose exact expectation value can be obtained by resumming diagrams of perturbation theory [6] or, at a more rigorous level, by localization of the path integral on S 4 [7]. The strong-coupling extrapolation of the circle agrees precisely with the area law in AdS 5 × S 5 , a result that can be generalized in many ways (see [8] for a review). Quite surprisingly, even the next order in the strong-coupling expansion has not been reproduced from string theory thus far, despite much effort [9,10], indicating that we do not understand in detail how strings in AdS 5 × S 5 and other holographic backgrounds should be quantized in the Wilson-loop sector.
The difficulty lies in the definition of the measure in the string path integral and in the delicate issues with reparameterization invariance on the string worldsheet. Taking the ratio of similar Wilson loops [1,2] avoids these complications, because the measure factors simply cancel. For the ratio of the latitude and the circle, considered in [1,2], quantum string corrections can be computed exactly. Surprisingly, the result of the string calculation disagrees with the field-theory prediction [11]. A different quantization prescription for string fluctuations [3] brings the result in agreement with field theory, but the method of [3] only applies to infinitesimally small deviations from the circle. The quantization prescriptions in [1,2] and in [3] differ in the choice of the conformal frame on the string worldsheet, which a priori should not matter as long as the conformal anomaly cancels.
We reconsider string quantum corrections to the latitude Wilson loops, working in the same conformal frame as [1,2]. We pay special attention to regularization issues and ensuing anomalies and will also carefully check that the conformal anomaly cancels, which is an important consistency condition in string theory.

Circular Wilson loop and latitude 2.1 Latitude Wilson loops
The Wilson loop expectation value in N = 4 SYM is defined as [4] W (C; n) = ⟨tr P exp where Φ I are the six scalar fields from the N = 4 supermultiplet and n is a unit six-dimensional vector that may change along the contour C. In string theory, the Wilson loop expectation value maps to the disc partition function with the boundary conditions determined by the contour C = {x µ (τ ) τ ∈ (0, 2π)} for the embedding coordinates in AdS 5 , and by n(τ ) for S 5 . We concentrate on a particular family of Wilson loops, for which C is a unit circle and n is a latitude of S 5 [11]: n = (sin θ 0 cos τ, sin θ 0 sin τ, cos θ 0 , 0). The expectation value of the latitude is known exactly [11] 1 : and interpolates between the simple circle at θ 0 = 0 and a supersymmetric Wilson loop with trivial expectation value [14] at θ 0 = π 2. Here λ = g 2 N is the 't Hooft coupling of N = 4 SYM. At strong coupling, Notice that the strong coupling and BPS (θ 0 → π 2) limits do not commute with one another. In string theory, the exponent in the Wilson loop is determined by the area of the minimal surface with the given boundary conditions, while the prefactor is a contribution of the string fluctuations and of the measure in the string path integral. Following [1,2,3], we consider the ratio of the circle to the latitude in which the complicated measure factor is expected to cancel 2 . The field-theory prediction for the Wilson loop ratio is Our goal will be to reproduce this result from the explicit one-loop calculation in string theory.

Classical solution
In the standard Poincaré coordinates {x µ , z} of AdS 5 and in the angular coordinates θ, ϕ of S 2 ⊂ S 5 , the minimal surface for the latitude is [17] x 1 = cos τ cosh σ , where σ changes from 0 to ∞ and θ 0 is related to σ 0 as The induced worldsheet metric is given by with the scale factor .
Substituting the solution into the string action, and taking into account that the string tension is given by √ λ 2π, in terms of the 't Hooft coupling, one gets the correct exponent in (2.3).
The field-theory prediction for the next, O(1) term in the strong-coupling expansion of (2.4) is In string theory, this is expected to come from the one-loop quantum fluctuations of the string worldsheet [18,9].
2 Another way to get rid of the measure factors is to consider infinitely stretched Wilson loops and concentrate on extensive quantities. In that case an agreement between field theory and quantum corrections in string theory was obtained for the quark-anti-quark potential in the N = 4 SYM [15] and for the quark self-energy in the N = 2 * theory [16].

One-loop string corrections
The string oscillation modes around the classical solution (2.5) are described by the following fluctuation operators [1, 2, 3] 3 : , (2.10) where τ i are the standard Pauli matrices. The operatorK 1 describes three string modes in AdS 5 , the operatorK 2 describes three modes on S 5 ,K 3± arise as a result of mixing between the two remaining modes -one from the sphere, another from AdS 5 . The Dirac operatorsD ± originate from the kinetic terms for the eight fermions remaining after kappa-symmetry gauge-fixing in the Green-Schwarz action. The operators above are related to the ones that appear in the string action by a conformal transformation [1]: for bosons, and for fermions. The fluctuation modes of the string are naturally normalized with respect to the invariant measure of the induced metric (2.7): 15) and the fluctuation operators are Hermitian with respect to this scalar product, while the tilded operators are Hermitian with respect to the usual flat measure: The one-loop partition function, that determines the Wilson loop expectation value, is given by the ratio of determinants of the physical, untilded operators [1]: The Wilson loop is actually proportional to Z(σ 0 ), but does not literally coincide with it. The string path integral contains some additional measure factors that are rather difficult to control. Fortunately, these factors do not depend on σ 0 and cancel in the ratio of the latitude to the circle. The oneloop free energy, if normalized as in (2.4), is given by the log-ratio of the partition functions: This is the object we concentrate upon in the rest of the paper. The tilded operators are technically easier to deal with, and in much of the previous work the conformal factors have been simply dropped. Independence on the conformal frame is a basic principle of string theory. It is thus natural to assume that the untilded operators can be seamlessly replaced by the tilded ones. However, the log-ratio of determinants computed under this assumption (which we denote byΓ 1−loop ) differs from the field-theory prediction (2.8) by an additional "remainder" term [1,2]: An obvious possible cause for the discrepancy, the one that first comes to mind, is the conformal anomaly. However it was argued in [2] that the conformal anomaly is unlikely to account for the discrepancy. We refer to [2,9] for technical details, and just remark that anomaly cancellation is very important in string theory. A non-zero contribution from the conformal anomaly would rather signal an internal inconsistency of the string calculation. A caveat here is that the conformal transformation from the metric of the disc (2.7) to the flat metric of the semi-infinite cylinder changes the topology of the worldsheet and is actually singular at σ = ∞. The point σ = ∞ is regular in the induced metric (2.7) but not in the flat metric, as illustrated in fig. 1. The spectral problem for a fluctuation operator on a cylinder differs from that on a disk in an essential way and requires an IR regularization. Even if the cutoff dependence eventually cancels out, regularization may leave a finite residue. We first give a simple but not very rigorous derivation of such an IR anomaly based on elementary thermodynamics, and then proceed with a more systematic analysis of the fluctuation determinants.

Regularization and anomalies
The change of the conformal frame of the form (2.13) corresponds to the following chain of transformations on the determinant of K: whereK ∞ is the asymptotic operator obtained by taking σ → ∞ in (2.9)-(2.11), which is just the free Klein-Gordon operator: or the free Dirac operator, in case of fermions: The first ratio in (2.20) is the conformal anomaly, the second one is welldefined on a cylinder, while separately detK and detK ∞ require an IR regularization. The IR cutoff is manifestly necessary for the Gel'fand-Yaglom method used in [1,2], and is implicit in a more direct phaseshift calculation that will be carried out in sec. 4.
The standard way to regularize the problem is to impose Dirichlet boundary conditions on the wavefunction ofK (orK ∞ ) at some large but finite σ = R. This corresponds to removing a small segment of the minimal surface shown as a red circle in fig. 1(a). This is not such an innocent procedure, as can be seem by comparing determinants of the Laplacian on a disk and on a disk with a small hole [20]. Even though the IR cutoff cancels in the final answer, intermediate steps do depend on R. At the same time, the cutoff R does not have any invariant meaning by itself. To faithfully compare partition functions at different values of σ 0 , we need a diffeomorphism-invariant regularization.
As an invariant regularization parameter we can take the area of the segment removed from the minimal surface when Dirichlet boundary conditions are imposed at σ = R: The coordinate-dependent cutoff is related to the invariant one as Because R is a coordinate-dependent quantity with no invariant meaning, in comparing the partition functions at different σ 0 , it is R inv rather than R that should be kept fixed. Diffeomorphism-invariant regularization implies that R has to be dependent on σ 0 . The partition function depends on R through the last factor in (2.20) and sinceK ∞ andD ∞ are just the free Klein-Gordon and Dirac operators, the asymptotic contribution to the partition function is given by the free energy of a gas of free particles in 1 + 1 dimensions, given by The IR divergence cancels in the ratio (2.18), but leaves a finite, σ 0 -dependent remnant due to (2.24): Combined with (2.19), this gives which agrees with the localization prediction (2.8). This is the main result of the paper. To validate this result we need to check that the conformal anomaly cancels, which we do in the next section. Later we will also reanalyze the partition function on the cylinder and will derive the above result by a direct spectral analysis of the fluctuation operators.

Conformal anomaly cancellation
The anomaly contribution to the free energy is First, we notice that the boundary terms in the anomaly trivially cancel between bosons and fermions, just by matching the number of degrees of freedom. To see that the bulk anomaly also cancels it is convenient to bring the scale factor of the metric to the following form: from which it immediately follows that . This enters the anomaly with a prefactor On the other hand, , and the two terms in the anomaly completely compensate one another. The anomaly thus cancels in the partition function (2.17) at each value of σ 0 , not only in the ratio, as actually expected.

Determinants and phaseshifts
The fluctuation determinants were evaluated in [1,2] with the help of the Gel'fand-Yaglom method. Here we recalculate them by a more direct approach, evaluating phaseshifts for each operator and then integrating over the phase space of string fluctuations. First we set up the general scheme for the phaseshift computation and then apply it to each operator in turn.

Preliminaries
The operators at hand have the general form (we start with bosons, for fermions the same scheme works with minor modifications): The asymptotics at infinity are that of the free d'Alambert operator: The Fourier expansion in τ replaces ∂ τ by −iω, with integer frequency, or halfinteger depending on whether the boundary conditions are periodic or antiperiodic in the τ direction. The spectrum thus decomposes into a sequence of one-dimensional problems for each Fourier mode: The boundary condition at σ = 0 is After the boundary condition is imposed the wavefunction is fixed up to normalization. Since the potential vanishes at infinity, the wavefunction asymptotically has an oscillating behaviour: The eigenvalue can be read off the asymptotic form of the Schrödinger equation (4.3): To define the determinant of an operator with a continuous spectrum we need to introduce an IR cutoff by imposing another boundary condition at σ = R: The spectrum then becomes discrete due to momentum quantization condition: p n R + δ(ω, p n ) ≃ πn, (4.8) which follows from the asymptotic form of the wavefunction (4.5) and is thus valid as long as R is much larger than the range of the potential in the Schrödinger equation. The density of states ρ = ∂n ∂p in the limit of R → ∞ hence takes the form The often omitted extensive piece proportional to R has to be kept here. The infrared divergence cancels in the ratio of the two partition functions at different σ 0 . But we have seen that the cutoff R depends on σ 0 if regularization is to preserve general covariance, and therefore the extensive part of the partition function has to be kept throughout the calculation. The determinant ofK is obtained by multiplying all the eigenvalues: with ω ∈ Z or ω ∈ Z + 1 2 for periodic/anti-periodic boundary conditions. Our strategy will be to directly evaluate phaseshifts for all the operators, sum over frequencies and integrate over spacial momenta. Before proceeding with explicit calculations we sum over the Matsubara frequency using standard tools of Statistical Mechanics [21], and make a few technical remarks that streamline calculation of the phaseshifts.

Summation over frequencies
The standard trick is to replace summation by integration along the contour shown in fig. 2: cot πω ∞ 0 dp 2p ω 2 + p 2 (δ(ω, p) + Rp) . (4.11) The poles of the cotangent recover the sum over the Matsubara frequencies.
Assuming that the phaseshift does not grow too fast at large frequencies (in the simplest cases the phaseshift does not depend on the frequency at all), the contour of integration can be closed in the upper and lower half-planes, as shown in fig. 2, after which the integral over ω picks up residues at ω = ±ip. Denoting δ ± (p) ≡ δ(±ip, p), (4.12) we get for the determinant: (4.14) Normally, the particle and anti-particle phaseshifts δ + and δ − are equal, but some operators that we encounter have a spectral asymmetry resulting in different density of states for particles and anti-particles.

Phaseshifts and Jost functions
Instead of solving the Schrödinger equation with the correct boundary conditions (4.4) it is sometimes easier to find the Jost functions, which are the solutions that asymptote to unit-normalized plane waves at infinity: For a self-adjoint Schrödinger operator with a real potential the Jost functions are complex conjugate to one another:Ȳ p = Y * p , but if the potential is complex, which is the case for the operatorsK 3± for example, then the two Jost functions are not related in any simple way.
The Jost functions form a complete set of solutions to the Schrödinger equation. The solution that satisfies the correct boundary conditions is a linear combination of the two Jost functions: This linear combination indeed vanishes at σ = 0. Comparing its behaviour at infinity with (4.5), we find: which expresses the phaseshift through the Jost data: In the self-adjoint case the Jost functions are complex conjugate and their ratio is a pure phase. The phaseshift is real in this case. If the Schrödinger operator is not self-adjoint, the phaseshift may have an imaginary part. In general, We will use this formula to evaluate the phaseshifts of the fluctuation operators for the latitude by explicitly calculating the Jost functions in each case. The same scheme can be applied to fermions with minor modifications related to their two-component nature.

The phaseshift computation 4.4.1 OperatorK 1
The differential equation for this operator is given by The solutions to this equation are given by the Jost functions satisfying Y p = Ȳ p * andȲ p = (Y p ) * . Using equation (4.19), we obtain

OperatorK 2
The corresponding differential equation is given by In this case, the Jost functions are
The potential in the Schrödinger equation is of the solvable Rosen-Morse type. The solution 4 can be found by the substitution [22] x = 1 − tanh(2σ + σ 0 ) 2 (4.27) accompanied by the following ansatz for the wavefunction: These equations have simple solutions: (4.32) From here we find the Jost functions: The Jost functions Y p andȲ p are not complex conjugate to one another, because the potential is complex, and consequently the phaseshifts will have an imaginary part. There is also a spectral asymmetry between particle and anti-particle phaseshifts, but it is easy to quantify it because particle and anti-particle Jost functions are related by complex conjugation: δ − (p) * = δ + (p). Taking into account the last equation in (4.33), we get from (4.19): The answer forK 3− is the same up to exchanging δ 3+ with δ 3− .

OperatorD ±
It is convenient to consider, instead ofD α , the eigenvalue problem for iτ 2Dα . The index α that takes values ± is introduced here in order to distinguish the operator label from the particle/anti-particle index. The spectral problem for the resulting Dirac operator takes the form: where the ∓ sign comes from the two choices of closing the integration contour described in figure 2.
Asymptotically, the two solutions behave as lim σ→∞ Y ± p,α = e ±αipσ δ α, Close to σ = 0, the Jost functions behave as From the above, it is easy to see that the superposition vanishes for σ → 0, and therefore satisfies the right boundary conditions at σ = 0. At σ → ∞, the correct solution behaves as (4.38) To define the fermionic phaseshift we need to understand what replaces the auxiliary boundary condition (4.7) for fermions. Since the Dirac equation is of the first order it is impossible to set both spinor components of the wavefunction to zero. Only a chiral projection of the wavefunction can vanish. Choosing the chirality condition (any other choice leads to equivalent results) as we get the momentum quantization condition in the form (4.8) with

Collecting the pieces together
Expressing the determinants in (2.17) through the on-shell phaseshifts with the help of (4.13) and (4.14), and collecting all the pieces together we get for the log of the partition function: The easiest way to compute the integral is by differentiation in σ 0 : The following identities reduce the remaining integral to elementary functions: and we get: Integration over σ 0 gives: The first two terms arise from the determinants normalized by the free Klein-Gordon/Dirac operators and agree with the calculation based on the Gel'fand-Yaglom method [1,2]. The last two terms is the IR anomaly. Re-expressing the coordinate cutoff R through the invariant cutoff according to (2.24), we find that the last three terms cancel and we are left with in perfect agreement with the localization prediction (2.4), (2.8).

Conclusions
The IR anomaly, related to the singular nature of the conformal gauge, brings quantum string corrections computed in [1,2] in agreement with localization predictions. We also checked that the conformal anomaly cancels in each individual expectation value, even before taking the ratio. This is an important consistency check of the underlying assumptions behind this calculation (for example, that ghosts and longitudinal modes mutually cancel in the ratio, or that the measure factors are constant and do not depend on the parameters of the problem). One can use other parameters to build ratios of Wilson loops which are easier to compute in string theory, for instance an overall coupling to scalars as in [23]. But a really interesting problem is to carry out a complete calculation of quantum corrections for a single Wilson loop. A Wilson loop is a well-defined operator in field theory, and a holographic prescription to compute its expectation value in string theory should be unambiguously defined.
The zeta-function regularized determinant of K(α) is defined through the Mellin transform of its heat kernel: Since the gamma-function has a pole at zero, the right-hand-side seems to vanish, which would indicate that the determinant of K(α) does not depend on the scale factor at all. But the Mellin transform generates a pole at s = 0, because the integrand is badly behaved at t = 0. Indeed, for any function f (t) that admits a finite Laurent expansion at zero, the residue of the Mellin transform coincides with the residue of the function itself: where a k are local functionals of φ, E and A µ that can be computed algebraically. We thus find that d dα ln det K = 2a 2 (φ K). (A.7) The second DeWitt-Seeley coefficient of the operator (A.1) is Integrating (A.7) we express the anomaly as a local functional of the fields: Written that way, the anomaly does not have any boundary terms 6 , but it is more natural to represent it in a different form: where ∂ n is the inward normal derivative at the boundary. The bulk and boundary terms in the anomaly actually arise in the computation of the Seeley coefficient exactly as written in the last expression. The more compact two-dimensional form is obtained upon integration by parts.
To compute the anomaly for fermions we first square the Dirac operator and then by the same chain of argument that led to (A.7) arrive at The square of the Dirac operator (A.2) is This operator has the form (A.1) and its second DeWitt-Seeley coefficient can be read off (A.9): We thus find for the fermion anomaly: Notice that the boundary anomaly has the same magnitude but different sign compared to bosons.