Strong coupling expansion of circular Wilson loops and string theories in AdS5 × S5 and AdS4 × CP3

We revisit the problem of matching the strong coupling expansion of the 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{1}{2} $$\end{document} BPS circular Wilson loops in N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 4 SYM and ABJM gauge theories with their string theory duals in AdS5× S5 and AdS4× CP3, at the first subleading (one-loop) order of the expansion around the minimal surface. We observe that, including the overall factor 1/gs of the inverse string coupling constant, as appropriate for the open string partition function with disk topology, and a universal prefactor proportional to the square root of the string tension T, both the SYM and ABJM results precisely match the string theory prediction. We provide an explanation of the origin of the T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt{T} $$\end{document} prefactor based on special features of the combination of one-loop determinants appearing in the string partition function. The latter also implies a natural generalization Zχ ∼ (T/gs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt{T}/{g}_{\mathrm{s}} $$\end{document})χ to higher genus contributions with the Euler number χ, which is consistent with the structure of the 1/N corrections found on the gauge theory side.


Introduction
There is a history of attempts to match gauge theory [1][2][3] and string theory [4][5][6] results for the leading terms in the strong coupling expansion of the expectation value of the 1 2 BPS circular Wilson loop (WL) in N = 4 SYM theory (see [7][8][9][10][11][12][13]). The precise matching was recently achieved for the ratio of the 1 2 and 1 4 BPS WL expectation values [14,15] (see also [16,17] for a discussion of similar matching in the ABJM theory [18]). However, the direct computation of the string theory counterpart of the expectation value of the individual WL, that non-trivially depends on the normalization of the path integral measure, still remains a challenge.
In the SU(N ) N = 4 SYM theory the Maldacena-Wilson operator defined in the fundamental representation is given by W = tr P e (iA+Φ) (note that we do not include the usual 1/N factor in the definition of W). Then for a circular loop one finds at large N with fixed 't Hooft coupling λ [1,2]: W = N 2 , where here and below R denotes the AdS radius. It was suggested in [2] that the pre-factor λ −3/4 ∼ T −3/2 may have its origin in the normalization of three ghost 0-modes on the disk (or the Mobius volume).
This proposal, however, is problematic for several reasons. First, the effective tension T has its natural origin in the string action but should not appear in the diffeomorphism volume or the volume of residual Mobius symmetry. Furthermore, the T −3/2 factor (which JHEP10(2020)130 would be universal if related to the Mobius volume) would fail to explain the result for the 1 2 BPS circular WL [19] in the U(N ) k × U(N ) −k ABJM theory, where the tension is 2λ (with λ = N k ) while the gauge theory (localization) prediction [20][21][22] for the 1 2 BPS Wilson loop in fundamental representation is W = N (4πλ) −1 e π √ 2λ + . . .. Note that, as above, in our definition we do not divide the Wilson loop operator by the dimension of the representation. 1 Another indication that the explanation of the prefactor should be different is that, in general, one expects that the string counterpart of the large N term in W should be the open-string partition function on the disk, which should contain an overall factor of the inverse power of the string coupling (corresponding to the Euler number χ = 1), i.e.
where 1 gstr provides the required overall factor of N . The fact that it is natural to define the WL expectation value without the usual 1/N factor, and to include the 1/g str factor in its string theory counterpart, was also emphasized in [23].
In the N = 4 SYM case we have [1,2] g str = g 2 2 π e √ λ + . . . , (1.2) while in the ABJM case [18,22]  whereΓ 1 is a numerical constant. Below we will argue that (1.4) should be the expression for the leading semiclassical result for the disk string path integral for a minimal surface in AdS 3 ending on a circle at the boundary (thus having induced AdS 2 geometry) in the AdS n × M 10−n string theory with tension T and coupling g str . In (1.4) the exponent e 2πT = e −I cl comes from the value of the classical string action I cl = V AdS 2 T = −2πT . 1 Our normalization of W in the 1 2 BPS case corresponds in the localization calculation of [21,22] to computing the matrix model expectation value Str e iµ i 0 0 e −iν j . Note that [22] defines the Wilson loop expectation value by including an extra overall factor of g CS ≡ 2πi k . Denoting by W loc the expectation value given in [22], we find that the strong coupling limit of the 1 2 BPS Wilson loop in the ABJM theory is W = 1 , where we fixed the phase as B = 1 2 . 2 Here the AdS4 radius is R = (2π 2 λ) 1/4 √ α with T = R 2 2πα . The shift λ → λ − 1 24 + . . . in the string tension [22,24,25] is irrelevant to the one-loop order (as discussed in [26], at the leading order we do not expect renormalization of the relation for the string tension).

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The constantΓ 1 comes from the ratio of one-loop determinants of string fluctuations near the minimal surface, and is found to be (see [6,7,9] and section 2 below) AdS 5 × S 5 :Γ 1 = 1 2 ln(2π) , Including also the n = 3 case of AdS 3 × S 3 × T 4 string theory, one finds for AdS n × M 10−n with n = 3, 4, 5 thatΓ 1 = 1 2 (n−4) ln(2π) (see (2.21) below), and so in general W 1 in (1.4) is As we explain below, it will also follow from our argument that at higher genera (disk with p handles with Euler number χ = 1 − 2p) the √ T factor in (1.4) should be replaced by ( √ T ) χ , i.e. the corresponding term in the partition function should have a universal prefactor This is indeed consistent with the structure of 1/N corrections found on the gauge theory side in [2] and in [22] (see section 5).
It remains to understand the origin of the simple prefactor T 2π in (1.4). In general, the expression for such a prefactor in the path integral is very sensitive to the definition of path integral measure which is subtle in string theory. In section 3 below we will provide an explanation for the presence of the √ T factor starting from the superstring path integral in the static gauge [6] (see also appendix A.1) but we will not be able to determine the origin of the remaining 1 √ 2π constant from first principles. This is already a non-trivial result: since the presence of this constant is fixed by the comparison with the SYM theory, we then have the string theory explanation for the ABJM expression in (1.3) (or vice-versa).
In section 4 we shall provide another consistency check of the universal expression for the string partition function (1.4) by considering the analog of the familiar soft dilaton insertion relation and dilaton tadpole on the disk.
In section 5 we will emphasize the fact that the universal prefactor in the disk partition function ∼ √ T gstr in (1.4) has a natural generalization (1.7) to higher orders which is consistent with the structure of the 1/N corrections found on the gauge theory side. We will make some concluding remarks about some other WL examples in section 6.
It is interesting to note that the factor T 2π in (1.4) looks exactly like the one associated with just one bosonic zero mode (in the standard normalization of the path integral zero-mode measure, i.e. 1 √ 2π , −1 = T , as was used in a similar context in [14,15]). 3 3 Here we assume that the path integral measure for a scalar field is normalized so that the gaussian integral has a fixed value . Then the factor of string tension T = −1 appears both in the measure and in the action and cancels out in the one-loop determinant expression apart from possible 0-mode contribution.

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In appendix A.2 we will discuss a possible origin of this zero-mode factor, assuming one starts with the disk path integral in conformal gauge where there is an extra factor containing the ratio of the ghost determinant and the determinant of the two "longitudinal" string coordinates subject to "mixed" Dirichlet/Neumann boundary conditions, and thus admitting conformal Killing zero modes discussed in appendix B.
2 One-loop string correction in static gauge consider a circular WL surface with AdS 2 induced geometry, which resides in an AdS 3 subspace of AdS n × M 10−n , specifically: The string is point-like in the internal compact directions, satisfying Dirichlet boundary conditions. In general, the planar WL expectation value is given by the string path integral with a disk-like world sheet ending on a circle at the boundary of AdS space, W = e −Γ , Γ = Γ 0 + Γ 1 + Γ 2 + . . .. Here Γ 0 = −2πT is the classical string action (proportional to the renormalized AdS 2 volume V AdS 2 = −2π) and Γ 1 = O(T 0 ) is given by sum of logarithms of fluctuation determinants (in which we include possible measure-related normalization factors).
We shall discuss the computation of the one-loop correction Γ 1 ≡ Γ (n) 1 in the above AdS n × M 10−n cases following the heat kernel method applied in the AdS 5 × S 5 case in [6] and [9]. In this n = 5 case the general form of the static-gauge string one-loop correction is [6] Γ (5) Here we assumed that the AdS radius R is scaled out and absorbed into the string tension T = R 2 2πα so that all operators are defined in the induced AdS 2 metric with radius 1 and curvature R (2) = −2. We will come back to the radius dependence in section 3 below. In (2.1) we isolated the contribution of one special transverse AdS 5 mode that, in general, is different from the other two: this is the AdS 3 mode transverse to the minimal surface (the other two transverse modes are transverse to AdS 3 ), see [6,27]. In the present case of the minimal surface being AdS 2 we have R (2) = −2 so that its mass is actually the same as of the other two transverse AdS 5 modes.
Similar expression (2.1) is found in the conformal gauge [6], provided the contribution of the two "longitudinal" modes cancels as in flat space [28] against that of the ghost determinant and Mobius volume factor (modulo the 0-mode part of the longitudinal operator and related definition of path integral measure, see appendix A.1 for further discussion).
In the less supersymmetric cases with AdS 5 → AdS n and n = 4, 3 there are less massive bosonic AdS directions and part of the fermions are massless, i.e. we get the JHEP10(2020)130 following generalization of (2.1) The fermion masses are controlled by the superstring kinetic term with a projection matrix in the mass term. In the AdS 3 × S 3 × T 4 case [6] there are 4 massless fermion modes (which are partners of T 4 bosonic modes) and 4 massive ones. In the AdS 4 ×CP 3 case one finds [29] that there are 2n − 2 = 6 massive and 10 − 2n = 2 massless fermionic modes. Let us first discuss the divergent part of (2.3) assuming the standard heat-kernel regularization separately for each determinant contribution. The UV divergent part of Γ 1 = 1 2 log det(−∇ 2 + X) where −∇ 2 + X is a scalar Laplacian is given by (Λ → ∞) Here we ignore boundary contributions (they contain a power of IR cutoff and are absent after renormalization of the AdS 2 volume or directly using the finite value for the Euler number of the minimal surface).
In the case of (2.2) we then find that in the total combination all 1 6 R (2) terms cancel out (due to balance of bosonic and fermionic d.o.f.) and the constant mass terms also cancel between bosons and fermions so that we are left only with contributions of R (2) terms from one special bosonic mode and the fermionic modes For general n the corresponding UV divergent part of (2.3) is given by the straightforward generalization of (2.6). Again, all 1 6 R (2) terms in (2.5) cancel out as do the constant mass terms and we find b (n) The total result (coming again just from the R (2) terms in single bosonic mode and 8 fermionic modes) is thus universal, i.e. n-independent. Moreover, the same result B 2 tot = χ for the coefficient of the UV divergence is found for fluctuations near any minimal surface (not even lying within AdS 3 ) that has disk topology (see [6,30]): if X is a "mass matrix", the contribution of 8 transverse bosons is b 2b = 8 · 1 6 R (2) − tr X − R (2) while of 8 fermions is b 2f = 8 · 1 12 R (2) + tr X so that b 2 tot = R (2) .

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Note that in general the Seeley coefficient is B 2 = ζ(0)+n 0 where ζ(0) is the regularized number of all non-zero modes and n 0 = n b − 1 2 n f is the effective number of all 0-modes (assuming fermions are counted as Majorana or Weyl). In the present static gauge case there are no obvious normalizable 0-modes (cf. remark below (B.10)), but we observe that the result (2.8) is formally the same as what would come just from one "uncanceled" bosonic mode.
The universality of (2.8) strongly suggests that the mechanism of cancellation of this total "topological" UV divergence should also be universal. One may absorb it into the definition of the superstring path integral measure or cancel it against other measure factors as discussed in the conformal gauge in [6]. 4 An alternative is to use a special "2d supersymmetric" definition of the one-loop path integral in the static gauge (see below): the cancellation of UV divergences is, in fact, automatic if one uses a "spectral" representation for the total Γ 1 rather than heat kernel cutoff for each individual determinant.
Let us now turn to the finite part of the one-loop effective action in (2.4). We will follow [9] which completed the original computation in [6] of Γ 1 in (2.19) based on expressing the determinants in (2.2) in terms of the well known [37][38][39][40][41] heat kernels of the scalar and spinor Laplacians on AdS 2 . Γ The trace of heat kernel K(t) for a real scalar and a Majorana 2d fermion may be written as To recall, the UV divergences do not cancel automatically even in the bosonic string theory in flat space.
The combination of D scalar Laplacians and the conformal ghost operator ∆ gh = P † P gives (with all modes counted) Assuming, following [36], that there are extra powers of the UV cutoff in the Mobius volume one divides over and in the integrals over moduli, the net result is that one should add to the above B2 an extra δtopB2 = −3χ = dim ker P † − dim ker P , thus getting B2 = 1 6 (D−8−18)χ = 1 6 (D−26)χ. A similar argument applies to the NSR string where B2 = 1 4 (D−10)χ. In the present D = 10 GS superstring case there is an extra conformal anomaly/divergence from the Jacobian of rotation from GS fermions to 2d fermions (see [31][32][33][34]); this effectively amounts to adding 3 extra massless fermion contributions for each 2d fermion contribution (or, equivalently, multiplying the 1 6 R (2) − 1 4 R (2) part of each fermion contribution to b2 by 4); this gives δ1B2 In the conformal gauge the divergences from the determinant of the ghost operator (∆ gh ) ab = −g ab ∇ 2 − R ab cancel against those of the determinant of operator ∆ long for 2 longitudinal scalars. As in the bosonic case, one should also add δtopB2 = −3χ as explained above. Summing these contributions with (2.8) gives

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Here in µ f we already accounted for the negative sign of the fermion contribution, so the total K is just the sum of the bosonic and fermionic terms. The associated ζ-function is The total value of ζ(0) is found to be 1, i.e. the same as in (2.8). In general, the one-loop correction is For the derivative of the scalar ζ-function one finds (A is the Glaisher constant) while for the massive fermion The total contribution to the finite partΓ In the AdS 5 × S 5 case (n = 5) the computation of the corresponding determinants was also carried out using different methods in [7,8] with the finite part of the resulting expression forΓ (5) 1 being as in (1.5), (2.19). Note that the finite part (2.18) happens to vanish in the AdS 4 × CP 3 case (n = 4).
It is interesting to note that there exists a special definition of Γ 1 in (2.3) that automatically gives a UV finite one-loop result. Instead of computing separately each determinant let us use (2.9) and sum up the corresponding spectral integral expressions under a common integral over v in (2.10). Interchanging the order of t-and v-integrals and first integrating over t we see that this integral is finite, i.e. the proper-time cutoff is not required. Using (2.10)-(2.11) we then get for (2.4) Remarkably, the integral over v here is convergent at both v = 0 and v = ∞ (i.e. in the UV). In general, given the structure of the eigenvalues in (2.10)-(2.11), one can see that convergence of the representation (2.20) in the UV requires the sum rule which is satisfied for the spectra in our problem. Evaluation of (2.20) gives then a finite result equal to the one in (2.18), i.e.
This prescription of not using proper-time cutoff for individual log det terms, i.e. first combining the integrands and then doing the spectral integral, may be viewed as a kind of "2d supersymmetric" regularization. Indeed, the balance of the bosonic and fermionic degrees of freedom in (2.4) suggests hidden AdS 2 supersymmetry [6]. 5 Then the prescription of combining the spectral integrands of the determinants together may be viewed as a result of a "superfield" computation manifestly preserving 2d supersymmetry. Note however that, even though the integral in (2.20) is finite, a dependence of Γ 1 on a normalization scale reappears on dimensional grounds if one restores the dependence on the radius R inside the logarithms, as explained in the next section. This leads to an explanation of the T -dependent prefactor in (1.4) and (1.7).
3 Dependence on AdS radius: origin of the √ T prefactor Let us now explain the presence of the prefactor in the string one-loop partition function (1.4). As the definition of quantum string path integral (in particular, integration measure) is subtle and potentially ambiguous our aim is to identify the one that is consistent with underlying symmetries and AdS/CFT duality.
In the previous section we ignored the dependence of the one-loop correction on the AdS radius R. Let us now discuss how the string path integral may depend on it. Let us start with the classical string action in AdS n of radius R. One possible approach is to rescale the 2d fields so that the factor of R 2 appears in front of the action 6 Using either (3.1) or (3.2) the expression for one-loop correction will depend also on the assumption about the path integral measure. If the measure is defined covariantly the final result should be the same. 5 One implication is the vanishing of the corresponding vacuum energy in AdS2 observed in [6] in the case of the strip parametrization ds 2 = 1 cos 2 ρ (dt 2 + dρ 2 ), ρ ∈ (− π 2 , π 2 ). To recall, the contributions of a scalar with mass m 2 b and a fermion with mass m 2 f to the AdS2 vacuum energy are [6] For example, starting with ds 2 = dr 2 + e 2r/R dxidxi we get ds 2 = R 2 (dr 2 + e 2r dxidxi). Note that after the rescaling the tension T and coordinatesx m are dimensionless.

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Let us consider the path integral defined by (3.1) in terms of the original unrescaled coordinates x m of natural dimension of length, so that G mn (x) is dimensionless and depends on the AdS scale R. The string σ-model path integral may be defined symbolically as (cf. footnote 3) Expanding near the minimal surface ending on the boundary circle we will get the induced AdS 2 metric depending on the same curvature scale R as G mn . Then rotating the fluctuation fields to the tangent-space componentsx r and also rescaling them by √ T 0 (so that they will be normalized as |x| 2 = d 2 σ √ gx rxr ) we will find that the 1-loop contribution from a single scalar is . . depends on the induced AdS 2 metric and has canonical dimension of (length) −2 with eigenvalues scaling as R −2 .
In the heat kernel representation t tr exp(−t∆) the parameter t and the cutoff Λ −2 will now have dimension of (length) 2 and we will get instead of (2.13) (cf. (2.5), (2.8)) As discussed in section 2, the UV divergence is expected to be cancelled by an extra "universal" contribution log( √ α Λ) from the superstring measure (see footnote 4). We assume that this universal contribution (depending only on the Euler number of the world sheet but not on details of its metric) may only involve the string scale √ α but not the AdS radius. As a result, This explains the origin of the √ T factor in the disk partition function (1.4). As was noted below (2.8), the coefficient of the UV divergent term in (3.4) is, in fact, the same for all minimal surfaces with disk topology and thus the dependence of the string partition function on the scale R or effective tension T through the √ T factor in (3.5) should be universal. This means, in particular, that the factors 1/g s and √ T in (1.4) will cancel in the ratio of expectation values of different Wilson loops with disk topology. Moreover, the fact that the power of T in (3.5) is controlled by the Euler number χ implies that at higher genera, for a disk with p handles, we should find that W includes the universal prefactor ( √ T /g str ) χ as in (1.7). This is in precise agreement with the large N expansion of the localization results both in N = 4 SYM and ABJM cases, as we explain in more detail in section 5.
The result of adding the above universal counterterm log( √ α Λ) is equivalent to just defining the one-loop partition function to be UV finite by first combining all the contributions using the spectral representation (2.20). There we set R = 1 and to restore the JHEP10(2020)130 dependence on the radius R of the AdS 2 metric we need to add the mass scale factor R −2 under the logs in (2.20) (cf. (2.9), (2.10)). To make the argument of the logs dimensionless we also need to introduce some normalization scale (i.e. log det ∆ → log det( 2 ∆) or, equivalently, add factor in the path integral measure). Then we find that Γ The dependence on illustrates the fact that as long as ζ tot (0) = 0, the one-loop contribution, even if defined to be UV finite by the spectral representation (or some analytic regularization like the ζ-function one [42]), is still scheme (or measure) dependent. Choosing ∼ √ α , which is here an obvious choice in the absence of any other available scales (and which is also suggested by the T 0 dependence in (3.3)), we again end up with the required result (3.5).
We shall discuss some other approaches to the derivation of the dependence of the one-loop correction on T in the next section and appendices A.1 and (A.2).

Dilaton insertion and derivative over gauge coupling
As another check of consistency and universality of the expression (1.4) for the 1-loop string partition function for a minimal surface with disk topology, let us consider a closely related object -the insertion of the dilaton operator in the expectation value or the dilaton tadpole on the disk with WL boundary conditions. Here we shall explicitly consider the SYM case but a similar discussion should apply also to the ABJM case.
Let us first recall the zero-momentum dilaton insertion relation, or the familiar "soft dilaton theorem" in flat space. The dilaton φ couples to the string as [43] where T 0 = 1 2πα . The string-frame metric G mn expressed in terms of the Einstein-frame metric in D dimensions is G mn = e 4 D−2 φḠ mn ,Ḡ mn = δ mn + h mn and thus the (zeromomentum) dilaton vertex operator in flat space is (cf. [45,46] (2) . Since the expectation value of the action I 0 may be obtained by applying −T 0 ∂ ∂T 0 to the string path integral (cf. (A.2)), 7 The canonically normalized dilaton fieldφ that appears in the generating functional for scattering amplitudes, i.e. having the same kinetic term as the graviton in the effective action, Note also that in (4.2) we ignored possible boundary term as its role usually is only to ensure the coupling to the correct value of the Euler number.
In the superstring case (D = 10) for the tree-level topology of a disk (χ = 1) eq. (4.3) reads Adapting this relation to our present case of fixed contour boundary conditions with the expectation value of the action given by I = − 1 2 (see (A.6)) the analog of (4.4) becomes (including in I also the classical contribution of an AdS 2 minimal surface Since the constant part of the dilaton is related to the string coupling which itself is related to the SYM coupling as in (1.2), i.e. g 2 YM = 4πg str = 4πe φ , we may compare (4.5) to the derivative of the circular WL expectation value with respect to the coupling constant on the gauge theory side. The normalized gauge theory path integral is defined by . . . We assume that the metric is Euclidean and the SU(N ) generators are normalized as tr (T i T j ) = δ ij . Since the factor in front of the action is the derivative over the constant part of the dilaton φ corresponds on the gauge-theory side to the insertion of the SYM action into a correlator. In particular, in the case of the WL expectation value (here and below W ≡ W SYM ) Since the gauge-theory result at strong coupling is W = N λ −3/4 2 π e √ λ + . . . we con- which is in agreement with the string theory expression (4.5).

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Note that while in the string theory relation (4.5) we used that the insertion of the string action is given by derivative over the tension, on the gauge theory side a similar relation (4.8) involves differentiation over the gauge coupling. The two are in agreement because on the string side the dependence on λ comes from both the dependence on the tension and also dependence on the string coupling (the −χ = −1 term in (4.4), (4.5)). Thus, once again, one needs the independent 1 gstr and √ T factors in the string theory disk partition function (1.4) in order to have the consistency between the dilaton derivatives, or equality of the dilaton insertions on the string and gauge theory sides.
The above discussion has a natural generalization to the string partition function on a disk with handles or 1/N corrections on the gauge theory side. For a surface of Euler number χ, using (A.8) we get the following analog of (4.3) generalizing (4.5) (4.10) The subleading term − 3 4 χ is consistent with the general form of the prefactor Z ∼ ( √ T gstr ) χ in (1.7). Indeed, note that g str = e φ and that switching to the Einstein-frame metric (cf. (4.2)) corresponds to T → e 1 2 φ T (cf. (4.2)), so that √ T gstr ∼ e − 3 4 φ . This is in agreement with the gauge-theory side since the dependence on the dilaton is directly correlated as in (4.7), (4.8) with the dependence on λ (which appears only as a factor in front of the SYM action), while the dependence on N may come not only from the factor (4.7) in the action but also from traces in higher order gauge-theory correlators. Indeed, according to the gauge-theory result (see (5.1)) the genus p term in W depends on λ as λ 6p− 3 4 = λ − 3 4 χ . One can also perform a further consistency check by considering a direct generalization of the above relations to the case of the local (i.e. "non zero-momentum") dilaton operator insertion. On the gauge theory side the derivative over a local coupling or local dilaton is essentially the Lagrangian in (4.6) and one finds [50][51][52] In (4.11) we assume that dependence on the local dilaton is introduced by L SYM → e −φ(x) L SYM and φ is set to be constant as in (4.7) after the differentiation. In (4.12) we used that W = 2 is the same function that appeared also in (4.9). For a WL defined by a circle of unit radius on the (x 1 , x 2 )-plane centered at the origin, the 8 Eq. (4.11) is a direct counterpart of the exact form of the correlation function of the 1 2 BPS Wilson loop with the ∆ = 2 chiral primary operator which is a special case of the correlator of the Wilson loop and the ∆ = J CPO first obtained in [53]. The function f (λ) also appears in the so-called Bremsstrahlung function [54]. The dilaton operator O4 is a descendant of the ∆ = 2 chiral primary, i.e. O4 ∼ tr (F 2 + ΦD 2 Φ + . . .) and is different from the canonical form of the SYM Lagrangian L SYM in (4.6) by a total derivative term (in conformal correlators one may further drop the terms proportional to the scalar and spinor equations of motion as they produce only contact terms [47][48][49]). Note that we use Euclidean notation (as, e.g., in [52]) and in our normalization [49]
On the string theory side, the corresponding local dilaton operator is (cf. (4.2); here D = 10 and L str = 1 2 z −2 ∂ a x m ∂ a x m + . . . is the AdS 5 × S 5 superstring Lagrangian) K in (4.16) is the bulk-to-boundary propagator of the massless dilaton in AdS 5 (∆ = 4). Integrating over the 4-dimensional boundary coordinates gives back V 0 that appeared in (4.5) (indeed, d 4 x K(x − x ; z) = 2π 2 1 12 c 4 = 1). Note that the correlator in (4.11) is to be compared to the string theory dilaton insertion on the disc with the dilaton vertex operator defined relative to the Einstein-frame metric so that the 2-point functions of the graviton and dilaton (and the corresponding dual operators) are decoupled.
Note that the normalized correlator (4.11) for the case of the WL corresponding to a straight line is related to the one for the circle by a conformal transformation, and it takes the same form as (4.11), with the same function f (λ), and d ⊥ being simply the distance from the straight line (i.e., for a straight line along the x 1 direction, d ⊥ = x 2 2 + x 2 3 + x 2 4 ) [4]. Using the AdS 2 surface in the straight line case (z = σ, x 0 = τ, x i = 0) we get for the contribution of the leading classical term and the R (2) = −2 term in V in (4.15): Then the string theory expectation value δ δφ(x) log Z = V (x) indeed matches (4.11) if one adds in the last bracket in (4.17) an extra + 1 2 coming from the 1-loop quantum fluctuations of the bosonic and fermionic string coordinates in L str , in parallel to what happened in (4.5). 9 To evaluate this integral, one may, for instance, first integrate over r, then integrate over h and finally remove the power divergence at h = → 0, i.e. d 4 x 1 JHEP10(2020)130

Universal form of higher genus corrections
An important feature of the √ T gstr prefactor in (1.4) is that it has a natural generalization ( √ T gstr ) χ to contributions from higher genera (1.7) (cf. also (4.10), (A.8)). Let us recall that in the case of the SU(N ) N = 4 SYM theory the exact expression for the expectation value of the 1 2 BPS circular WL W = tr P e (iA+Φ) expanded at large N and then at large λ is [2,3] It was suggested in [2] that the sum over p may be interpreted as a genus expansion on the string side. Remarkably, we observe that once the overall factor of N is included, i.e. one considers the expectation value of tr (. . .) rather that 1 N tr (. . .), the full dependence on N and λ in the prefactor of e which is the same as (1.7) where χ = 1 − 2p is the Euler number of a disk with p handles. Furthermore, the sum that represents the coefficient of the leading large T term in (5.2) has a simple closed expression: since ∞ p=0 c p z p = 1 2π exp( π 12 z) we find as in ref. [2] W = e H W 1 1 + O(T −1 ) , Here W 1 is the leading large N or disk contribution in the SYM theory given by (1.4) (with e −Γ 1 = √ 2π according to (1.5)). H may be interpreted as representing a handle insertion operator, i.e. higher order string loop corrections here simply exponentiate. Such exponentiation is expected in the "dilute handle gas" approximation of thin far-separated handles which should be relevant to the leading order in the large tension expansion considered in (5.1), (5.2) (cf. [57][58][59][60][61][62][63]).
It has another interesting interpretation suggested in [64]. If one considers a circular Wilson loop in the totally symmetric rank k representation of SU(N ) then for large k, N and λ with κ = k √ λ 4N =fixed its expectation value should be given by the exponent exp(−S D3 ) of the action of the classical D3-brane solution. In the limit of 1 k N this description should apply also to the case of the WL in the k-fundamental representation described by minimal surface ending on multiply wrapped circle and here one finds [64] 10 Let us note that this expression applies to the SYM theory with the U(N ) gauge group; the result in the SU(N ) case is obtained by multiplying (5.1) by exp(− λ 8N 2 ) [2]. This factor expressed in terms of gstr and T in (1.2) is exp(− g 2 str 2T 2 ) and thus is subleading compared to H in (5.3) at large T ; we therefore ignore it here.

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. If one formally extrapolates this expression to k = 1, i.e. a single circle case discussed above, then one finds that it becomes S D3 = −2πT + π A similar structure (5.2) of the topological expansion should appear in the case of the 1 2 BPS circular WL in the ABJM theory which was computed from localization in [22]. According to (1.3), in that case we have 1). Translating the leading and the first subleading 1/N corrections to the WL expectation value found explicitly in [22] into our notation we get 11 Here W 1 is the leading disk term in W in the ABJM theory given by (1.4) (withΓ 1 = 0 according to (1.5)). Thus, to this order, the genus expansion in the ABJM case has the same universal structure as (5.2), (5.3) in the SYM case. It would be interesting to check if the prefactor in (5.4) exponentiates as in (5.3) (e.g. using the results of [65]) and also if there is a D2-brane description of this similar to the one in the SYM case discussed above (cf. [66][67][68]).

Concluding remarks
As was noted below (2.8), the coefficient of the UV divergent term in (3.4) is, in fact, the same for all minimal surfaces with disk topology, and thus the dependence of the string partition function on the scale R or effective tension T through the √ T factor in (3.5) should be universal.
A check of the universality of the prefactor in (1.4) is that it applies also to the circular WL in the k-fundamental representation dual to a minimal surface ending on k-wrapped circle at the boundary of AdS 5 . In this case the classical action is I cl = −k √ λ but the Euler number of the minimal surface is still equal to 1 [12] so that the coefficient ζ tot (0) in (3.4), (3.5) is also 1 and thus the disk partition function is W ∼ √ T gstr e 2πkT . This is consistent with the SYM (localization) result in the k-fundamental case [3,64,69] given by the k = 1 expression with √ λ → k √ λ. The overall k-dependent constant that should come fromΓ 1 in (1.4) still remains to be explained, despite several earlier attempts in [7,9,12,70].
The universality of (1.4) implies, in particular, that the prefactor √ T gs should cancel in the ratio of expectation values of similar Wilson loops. In particular, this applies to the case of 1 4 BPS latitude WL parametrized by an angle θ 0 . Matching with the gauge theory 11 Note that we use the notation g CS ≡ 2πi k for what was called gs in [22] in order not to confuse it with the type IIA string theory coupling gstr in (1.3). The leading correction in eq. (8.19) in [22] is to be multiplied by g 2 CS according to the definition of the topological expansion in (8.1) there. Also, as already mentioned in footnote (1), with our definition of the WL expectation value W = tr (. . .) = 1 g CS W loc , where W loc is the gauge theory localization expression of [22].

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prediction for the ratio of the latitude WL and simple circular WL was checked in the SYM case in [13][14][15] and in the ABJM case in [16,17].
Let us note that (1.4) actually requires a generalization in special cases when there are 0-modes in the internal (non-AdS) directions of AdS n × M 10−n space, each producing extra factor of √ T (cf. (A.10)). This is what happens in the case of the 1 4 supersymmetric (θ 0 = π 2 ) latitude WL discussed in [14,15,71] where we then get for the disk partition function Here all λ-dependence cancels out and the finite proportionality constant should be equal to 1, i.e. N −1 W 1 4 = 1, in agreement with [71].
A similar remark applies to the case of the 1 6 (bosonic) BPS WL [66,67] in the ABJM theory. According to [22] here we get instead of W for the 1 2 BPS WL in (1.3) (cf. footnotes 1, 11 and eq. (5.4)) As was argued in [66,67] (see also [72,73]), here the minimal surface solution is smeared over S 2 = CP 1 in CP 3 so there are two scalar 0-modes. This explains the extra factor 2) compared to W in (1.3) [22]. More generally, for contributions from each genus p one finds [21,22,65] that the ratio of the 1 6 and 1 2 BPS WL's is given by this universal ( √ T ) 2 term (ignoring phase factors) It would be interesting to match the precise numerical coefficient in the ratio between the 1 6 BPS and 1 2 BPS Wilson loops by carefully fixing the normalization of the two zero modes on the string side.
Finally, let us note that while in this paper we focused on the case of 4d and 3d gauge theories, as explained in section 2 our results also apply to string theory in AdS 3 × S 3 × T 4 with RR flux. This case corresponds to n = 3 in (1.6) (cf. (1.4), (2.21)), i.e. W = Z str = 1 gstr √ T e 2πT + . . .. It would be interesting to see if this string-theory prediction can be matched to localization calculations for Wilson loops in 2d supersymmetric gauge theory (cf. [74,75]).

A Comments on tension dependence of the string partition function
In section 3 we discussed how to explain the prefactor √ T in the one-loop string partition function (1.4) starting with the string action (3.1) and using the static gauge expression (2.3). We emphasized that the result is sensitive to the choice of the path integral measure, i.e. the definition of the quantum theory (which, in general, is not unique, unless completely fixed by symmetry requirements or extra consistency conditions). In the appendices below we shall discuss some other approaches to derive this prefactor, which again involve certain assumptions about the measure or regularization procedure.

A.1 T -derivative of the partition function in static gauge
Suppose we start with the string action (3.2) in terms of the rescaled (dimensionless) coordinates so that there is an explicit factor of the effective string tension T in front of the action with the induced AdS 2 metric having radius 1. Then we would get the same result as in (3.5) if we assume that the norm or the measure is defined so that the resulting one-loop correction from a single scalar has the form Γ 1 = 1 2 log det∆ where∆ = T −1 ∆. 12 Indeed, using the ζ-function regularization with ζ(0) being the regularized total number of eigenvalues we get 1 2 . .. This leads again to (3.5) once we use that the total value of ζ(0) corresponding to the static-gauge partition function (2.4) is ζ tot (0) = 1 (see (2.8), (2.13)).
Another way to obtain the same result (which will be again based on a particular choice of a regularization prescription) is to find the dependence of the string partition function on the tension by first computing its derivative over T . This is closely related to the argument appearing in the context of the "soft dilaton theorem" [44], see section 4.
Let us assume that the tension dependence of the string partition function may come only from the factor of T in the string action (3.2) in the static gauge (i.e. in the action for the "physical" fluctuations whose determinants are present in (2.3)), i.e. the measure is defined so that it does not depend on T . For example, for a single scalar field |σ is the Green's function and δ (m 2 ) (σ, σ) is a regularized value of the bosonic delta-function at the coinciding points. Let

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us use the heat-kernel cutoff, i.e. assume that The expectation value of the action corresponding to the full static-gauge expression (2.3) is then where D f is the fermionic 1st order operator and G f and δ f stand for the corresponding Green's function and δ-function.
A key next step is to assume a special "2d supersymmetric" regularization in which the bosonic and fermionic Green's functions and thus the corresponding regularized deltafunctions are related to each other as 13 Then (A.4) reduces simply to where we used (A.3). 14 The relation T ∂ ∂T log Z = − I in (A.2) implies once again that Let us note that the result for the expectation value of the action (A.6) should be more universal than a particular prescription used above. The integrand in (A.6) should be in 13 This is an effective consequence of the fact that the 2d supersymmetric Ward identity (cf. [76,77]) relates a fermion of mass m > 0 to a boson of mass m 2 − m (e.g. in a special regularization the trace of the Green's function for a single 2d fermion G f (m) is related to 2m G (m 2 −m) [77]). In the present case we have half of the massive fermions with mass m = 1 and the other half -with mass m = −1. Alternatively, one may use a particular representation for G f . 14 Let us note that the use of (A.5) may be interpreted as a specific regularization prescription for the fermions which is different from the heat-kernel or ζ-function one applied to the squared fermionic operator −m 2 then we find that I = + 1 2 which is consistent with the ζtot(0) = 1 value in (2.8), (2.13), i.e. Z ∼ [det(T ∆)] −1/2 ∼ T −1/2 . In this regularization the l.h.s. of (A.5) is 2(Λ 2 + 1 6 R (2) )− 1 2 R (2) −2m 2 while the r.h.s. is 2(Λ 2 + 1 6 R (2) )− 1 2 R (2) −2m 2 so the difference − 1 2 R (2) may be attributed to the presence of − 1 4 R (2) term in the squared fermionic operator which is thus effectively omitted in the prescription (A.5).

A.2 T -dependence from zero modes in conformal gauge
The conformal gauge expression for the string partition function contains, in addition to the ratio of determinants in Z = e −Γ 1 in (2.3), also an extra factor [6,28] Z c = Ω −1 det ∆ gh det ∆ long Here Ω is the SL(2, R) Mobius group volume. The 2-derivative ghost operator ∆ gh ab and the operator on the two "longitudinal" fluctuations ∆ long ab = −(∇ 2 ) ab − 1 2 R (2) g ab have the same structure (and the same "mixed" boundary conditions) so their non-zeromode contributions should effectively cancel each other. The integral over the collective coordinates of the three 0-modes of ∆ long (or conformal Killing vectors) which is implicit in (A.9) should cancel against the Mobius volume factor. As a result, one may assume that Z c in (A.9) is effectively equal to 1, thus getting back to the static gauge partition function expression (2.3).
However, this depends on the definition of path integral measure. An alternative possibility compared to the one in the static gauge discussed in the main text and section A.1 is to assume that in the conformal gauge the measure is defined so that the path integral over all non-zero modes does not produce any T -dependent factor, while the presence of the √ T factor is (1.4) is due to the normalization of the 0-modes, i.e. of the collective coordinate integral implicit in (A.9). 15 Each bosonic 0-mode absent in the fluctuation action then contributes a measure factor ∼ ( T 2π ) 1/2 , leading to where n 0 is the total number of the 0-modes. Equivalently, this result will follow assuming that one uses a regularization (e.g., dimensional one) in which the delta-functions at coinciding points vanish, δ (2) (σ, σ) = 0 and thus the factors of T in the measure and in front of the action do not contribute, cf. (A.2), apart from the 0-mode contribution.
It is useful to recall that in the familiar case of the open strings with free ends where the bosonic coordinates x m (m = 1, . . . , D) are subject to the Neumann boundary conditions one finds D constant zero modes and thus an overall factor of T D/2 in the disk path integral. The same result can be found also using T ∂ ∂T argument by using that the delta-function 15 As was already mentioned above, this corresponds to a specific choice of the measure factors implying that the normalization of the gaussian path integral is 1, i.e.
[dx] exp[− 1 2 (x, x)] = 1, with [dx] = σ dx(σ) √ 2π . Then the factor of string tension T = −1 should appear not only in the action but also in the measure so that it cancels out in the integrals for all modes with non-zero eigenvalues.
In the present WL case of path integral with the Dirichlet-type (or fixed-contour) boundary conditions one could expect to have no 0-modes. However, as the two "longitudinal" string coordinates are subject to "mixed" Dirichlet/Neumann b.c. [28,36,[79][80][81] (motivated by the requirement of preservation of the reparametrization invariance of the boundary contour) there is, in particular, a special 0-mode corresponding to a constant shift of a point on the boundary circle. There are, in fact, two more 0-modes of the longitudinal operator (see appendix B). As already mentioned above, these three bosonic 0-modes are direct counterparts of the conformal Killing vectors associated to the SL(2, R) Mobius symmetry on the disk surviving in the conformal gauge.
Thus if the path integral measure is normalized so that the integral over non-zero modes does not produce any T -dependence we then get a factor Z ∼ ( √ T ) 3 associated to the n 0 = 3 "longitudinal" 0-modes on the disk. To reduce the effective number of 0-modes to n 0 = 1 (required to match the √ T factor in (1.4)) one may contemplate the following possibilities: (i) assume that 2 longitudinal 0-modes are lifted due to some boundary contributions to the string action leaving only one translational mode (corresponding to a constant shift on the boundary circle); 16 (ii) assume that the GS fermion contribution effectively conspires to mimic the NSR contribution on the disk with n f = 2 fermionic super-Mobius 0-modes, 17 producing the effective number n 0 = n b −n f = 3−2 = 1. A relation to the NSR formulation with manifest 2d supersymmetry may of course be expected and was mentioned already in the discussion of the static gauge approach above. Note also that the super-Mobius volume is finite [86] so it is not necessary to cancel it explicitly.

B Conformal Killing vectors as longitudinal zero modes
Here we shall record the expressions for the conformal gauge ghost zero-modes or conformal Killing vectors (CKV) on a flat disk D 2 and on a euclidean hyperbolic space H 2 = AdS 2 16 This, at first, may look unnatural as then we would not have a cancellation between the integral over the corresponding collective coordinates and the Mobius volume factor in the path integral. Yet, that may not be a problem as the Mobius volume on the disk may be regularized to a finite value [82,83] (similarly to how this is done for the AdS2 volume). 17 To compare, in the case of the one-loop instanton partition function in super YM theory (see [84,85]) the contributions of all non-zero modes cancel (i.e. ζtot(0) = 0) and as a result the UV cutoff dependence (and thus one-loop beta function) is controlled just by the 0-modes -the total Seeley coefficient is B4 = ζtot(0) + ntot = n b − 1 2 n f . At the same time, the dependence on the inverse gauge coupling 1/g 2 YM (which is the analog of string tension T in our case) is controlled by the coefficient n b − n f . Note, however, that the prescription for g YM dependence becomes unambiguous only in physical correlation functions with external fermionic legs saturating the fermionic 0-mode integral [85].

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However, these zero modes are not normalizable. Moreover, they do not satisfy the Dirichlet boundary conditions at σ = 0, as required for the transverse fluctuations, so they should not be relevant for our problem. Note also that, when considering all of the n − 2 transverse directions in AdS n , there would be, in fact, 3(n − 2) such zero modes (i.e., 9 in the AdS 5 case).
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