Semiclassical partition function for strings dual to Wilson loops with small cusps in ABJM

We compute the 1-loop partition function for strings in AdS4×ℂℙ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ Ad{S}_4\times \mathrm{\mathbb{C}}{\mathrm{\mathbb{P}}}^3 $$\end{document}, whose worldsheets end along a line with small cusp angles in the boundary of AdS. We obtain these 1-loop results in terms of the vacuum energy for on-shell modes. Our results verify the proposal by Lewkowycz and Maldacena in arXiv:1312.5682 for the exact Bremsstrahlung function up to the next to leading order in the strong coupling expansion. The agreement is observed for cusps distorting either the 1/2 BPS or the 1/6 BPS Wilson line.


Introduction
The existence of exact results for vacuum expectation values of Wilson loops in supersymmetric gauge theories paved the way for many non-trivial explicit verifications of the AdS/CFT correspondence. In various cases the strong coupling expansion of exact results was successfully contrasted with explicit 1-loop results for the dual string theory configurations. For instance, the expectation value of a Wilson line with a small cusp angle along it, can be exactly computed in N = 4 super Yang-Mills by relating it to the circular Wilson loop [1,2] and in N = 6 super Chern-Simons-matter gauge theories by showing its relation to a multiply wound circular Wilson loop [3]. The exact results in the former were shown to agree with the 1-loop partition function of the dual string configuration [4]. However, the exact results for cusped Wilson lines in the latter are yet to be matched with explicit computations in the strong coupling limit.
Three dimensional N = 6 super Chern-Simons-matter is a theory with U(N ) × U(N ) gauge group and Chern-Simons levels (k, −k). This theory gave rise to another precise AdS/CFT correspondence example, since it was found to be dual to eleven-dimensional M-theory on AdS 4 × S 7 /Z k [5]. Moreover, in the k → ∞ limit, the M-theory reduces JHEP03(2015)002 to type IIA string theory on AdS 4 × CP 3 . It is therefore interesting to study Wilson loop observables either from the CFT or the AdS point of view, to understand different regimes and to produce verifications of the AdS/CFT correspondence in cases for which exact results are available. In particular, the observables we are interested in this work are cusped Wilson lines.
Cusped Wilson lines in N = 6 super Chern-Simons-matter theory have been studied in several works both from the gauge and the string theory sides [6][7][8][9], and related to circular Wilson loops in [3,[10][11][12]. Wilson lines with a cusp angle φ in R 3 can be mapped through a conformal transformation to a pair of antiparallel lines in S 2 × R separated by an angle π − φ on the sphere. The expectation value of such operator gives rise to the so called cusp anomalous dimension Γ cusp defined as where T is the temporal extension of the anti-parallel lines along the cylinder. It is clear from this expression that the cusp anomalous dimension is nothing but the potential between the two probes on the sphere. We will be considering Wilson lines that become 1/2 BPS when the cusp angle vanishes. These cusped Wilson loop operators admit a second deformation by an internal space cusp angle θ [6], which accounts for a sudden change in coupling with the matter fields. It has been proven that generalized cusped Wilson lines with θ = ±φ remain BPS [6,7]. Therefore, for θ, φ ≪ 1, the cusp anomalous dimension takes the form Γ cusp ≈ (θ 2 − φ 2 )B(λ) . (1. 2) The coefficient B(λ) is called the Bremsstrahlung function because it appears in the expression for the energy radiated by an accelerated fundamental charge [1]. The generalized Γ cusp for ABJM theory was computed perturbatively in [6] to 2-loop order. In the strong coupling limit, expectation values of Wilson loops defined for a curve C can be computed, using the AdS/CFT correspondence, as the partition function for a string worldsheet ending on the curve C located at the boundary. The dual gravity background for N = 6 super Chern-Simons-matter theory is AdS 4 × CP 3 and the partition function for a cusped line, to subleading order in the strong coupling limit, have been studied in [8]. The expressions for the Bremsstrahlung function B(λ) found in [8] do not coincide in the geometrical and internal cusps cases. As noted in [8], this is in contradiction to the fact that the Wilson loop remains BPS when θ = ±φ [6]. This is our first motivation for revisiting the 1-loop computation for the partition function of strings in AdS 4 × CP 3 ending in cusped lines.
In [3], a proposal for the exact Bremsstrahlung function in ABJM theories in terms of derivatives of a circular Wilson loop expectation value was given Here W n is a BPS Wilson loop that winds n times around a maximal circle in the 3-sphere where the ABJM is defined. For the case of a geometrical cusp angle in a 1/6 BPS Wilson line, the 1/6 BPS circular Wilson loop computed in [13][14][15] was used in [3] to obtain explicit JHEP03(2015)002 weak and strong coupling expansions for B(λ). For the case of a geometrical cusp angle in a 1/2 Wilson line, the use of (1.3) to compute B(λ, N ) was slightly discussed in [3]. Later on, in [11] explicit weak and strong coupling expansions were written down and a disagreement with the 1-loop strong coupling computation of [8] was observed. Overcoming such discrepancy has also motivated our work. The paper is organized as follows: in section 2 we revisit the arguments which allow to compute the partition function in terms of the vacuum energy density and verify, as expected, that the vacuum energy density vanishes for the case of a 1/2 BPS straight line in ABJM theory. We then proceed to compute the correction to the vacuum energy due to small geometrical and internal cusp angles, using standard perturbation theory. In section 3 we verify that the proposal (1.3) is correct to 1-loop order in the strong coupling expansion. In section 4 we discuss our main results and explain the reasons for the discrepancy with the results in [8].
2 String partition function via vacuum energy

Rescaling of 1-loop operators
The computation of 1-loop partition functions on a static spacetime with metric g involves calculating the determinant of some operator O g . This determinant is typically computed by looking for the eigenfuctions of O g provided appropriate boundary conditions are given. This we call the off-shell method. Alternatively, the staticity of spacetime provides an alternative route: O g provides a wave equation whose solutions give rise to a spectrum of states. This we call the on-shell method. It is a well known result that the vacuum energy of the on-shell method coincides with the determinant computed from the off-shell method only when the static spacetime has g 00 = 1 [16]. Therefore, in a generic static case the vacuum energy is related to the determinant ofÕ g = M −1 O g with M = g 00 . It was shown in [17] that the determinant ofÕ g differs from the original one by a conformal anomaly type equation (see appendix B). An important observation in [18] was to note that the vacuum energy computation can still be applicable to the 1-loop partition function computation if there is a cancellation among the different anomalies coming from all the 1-loop operators involved.
As explained in [18], for a string ending on a straight line at the boundary of AdS 5 ×S 5 , the 1-loop correction to the partition functions can be obtained from the vacuum energy because the total anomaly coming from the rescaling of the mode operators vanishes. In appendix B, we show that this procedure applies for a string ending on a straight line at the boundary of AdS 4 × CP 3 . Moreover, the cancellation of anomalies also occurs for a general cusped string configuration with both geometrical and internal cusps. Therefore, we will use the vacuum energy method to obtain the corresponding determinants for the 1-loop correction to the partition function.

String ending on a straight line
Expanding the Lagrangian for a straight Wilson line without cusps to second order gives the following 1-loop contribution to the partition function [8] . (2.1)

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hereγ a represent 2-dimensional curved gamma matrices in the worldsheet,γ * = 1 2 ε ab γ ab is the curved covariant chiral matrix and D a the corresponding spinorial covariant derivative (see appendix A for details). In the absence of cusps, the induced metric on the string worldsheet is an AdS 2 geometry, with scalar curvature R (2) = −2. The fermionic masses m j (j = 1, . . . , 8) in the worldsheet Dirac operators are related to the eigenvalues of the matrix M F arising from the Green-Schwarz Lagrangian, which is given in terms of flat ten dimensional Dirac matrices Γ by 3) The result is: two fermions being massless, three with mass m F = 1 and three with m F = −1. Naively, one would have expected a fermionic spectrum with two massless modes, four m F = 1 modes and two m F = −1 modes (or some other combination) to match the bosonic spectrum made of six massless and two m 2 B = 2 bosonic modes, in order to satisfy the AdS supersymmetric relation m 2 B = m 2 F − m F [19]. However, as pointed out in [18], the relation m 2 B = m 2 F − m F is valid for N = 1 supersymmetry multiplets, in the case of extended supersymmetry in AdS 2 , the relation m 2 B = m 2 F + m F is also possible. We will compute the on-shell vacuum energy instead of (2.1), therefore we will be computing the ratio of determinants but with bosonic operators scaled by g 1/2 and fermionic operators are scaled by g 1/4 . Nonetheless, as explained in the previous subsection, these two quantities coincide (see appendix B).
To compute the vacuum energy we have to sum over the on-shell oscillator frequencies. For scalar and fermion fields of masses m B and m F , the frequencies were computed in [19] ω B n = n + where n = 0, 1, 2 . . .. The sums of such frequencies are divergent but can be regularized via analytical continuation using the Hurwitz zeta function In the present case, we only need the Hurwitz zeta function evaluated at s = −1, the result is regular and given by

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So, for the case of a straight Wilson line we obtain (2.8) The vanishing of the 1-loop contribution to the partition function was the expected result, because the string ending on a straight line is 1/2 BPS and preserves pure Poincare supercharges.
In [8], a non-vanishing 1-loop partition function was obtained for the string ending on the straight line. Presumably the non-vanishing of the result has its origin in the simplification between the determinants from the massless fermion fields and two of the massless scalar fields (see (A.27)). Omitting in (2.8) the modes that were canceled in [8] we would have obtained which is the result reported in eq. (B.62) in [8]. The computation (2.4)-(2.8) shows, as expected, that the determinants of fermionic and bosonic massless modes do not cancel each other when computed with different boundary conditions. This is indeed the case if the excitations are to fit in a supersymmetric multiplet [19].

String ending on a line with a geometrical cusp
Let us now consider a string worldsheet ending on a cusped line with geometrical cusp angle φ and vanishing internal cusp angle θ = 0 (see appendix A). The 1-loop contribution to partition function is still given by (2.1), but now covariant derivatives and scalar curvature are those of the induced worldsheet metric Here cn(σ|k 2 ) is a Jacobi elliptic function and K(k 2 ) is the complete elliptic integral of the first kind [4]. The parameter k 2 relates to the geometrical cusp angle in the classical solution. For small cusp angles the relation is φ ≈ πk. To simplify the notation we shall omit the dependence on k 2 in all the elliptic functions. The scalar curvature for the metric (2.10) is (2.11) As we have done for the straight line, we now proceed to study the on-shell vacuum energy which computes the ratio of determinants of the rescaled operators. After Fourier transforming the time dependence, we obtain the following operators: • The eight fermions satisfy 1 here γ a are 2d flat gamma matrices. Six of the fermions have m F = ±1 and the other two have m F = 0.
• Seven of the scalar modes satisfy with six modes having m 2 B = 0 and one m 2 B = 2. The eighth scalar field equation, whose potential depends on the worldsheet scalar curvature R (2) , results in (2.14) For k = 0 these operators reduce to either massless or massive Dirac and Klein-Gordon fields in AdS 2 .
We now quote from [19] the pertaining solutions to AdS 2 since we will exploit them in the next section.
Fermions: depending on the value of the mass, two possible solutions exist comprising supersymmetric multiplets: 2 (I) For m F < 1/2, we denote the fermion by Ψ T m,n = (ψ 1 m,n , ψ 2 m,n ) and one has here P n denotes the Jacobi polynomials.
(II) For m F > −1/2, we denote the fermion as X T m,n = (χ 1 m,n , χ 2 m,n ) and the components result The frequency spectra for these wavefunctions is given by (2.5) and their normalizations are such that

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Bosons: the solutions to the Klein-Gordon equation in AdS 2 are given by with frequencies ω n = n + λ and λ = 1 2.4)). These wavefunctions are normalized to satisfy All these modes satisfy Dirichlet boundary conditions.
However, we are considering string configurations that are sitting at a specific point in internal space CP 3 and therefore we have to impose Dirichlet boundary conditions for the corresponding 6 massless scalar modes. Thus, we use the modes given in (2.18). For the massless fermionic excitations the two possibilities Ψ 0,n or X 0,n are allowed in principle. However, only for a specific choice the excitations would conform an N = 6 multiplet. We will come back to this issue later on.
For 1/6 BPS straight Wilson line, the dual configuration is a smearing of strings along a CP 1 ⊂ CP 3 [20][21][22] and then two of the massless scalar modes should satisfy Neumann boundary conditions. If we were to repeat the computation done in (2.8) with 2 of the massless scalar modes given by (2.20), the total 1-loop vacuum energy would continue to vanish because n=0 (n + 1) = n=0 n. In section 4 we consider the possibility of using modes (2.20) the study of smeared Wilson loops deformed with a geometrical cusp.
For k = 0 equations (2.12)-(2.14) can be solved in terms eigenfunctions of the Lamé equation. However, and since we are only interested in the leading order of small k limit, we shall simply take the AdS 2 eigenfunctions (2.15), (2.16) and (2.18) and use standard perturbation theory.

Small cusp
Turning on k not only perturbs the constant mass in the Klein-Gordon and Dirac equations in AdS 2 by adding a potential, but also modifies the range of the coordinate σ. In order to avoid a k-dependent coordinate range, it is convenient to definẽ (2.21) and introduce accordingly rescaled frequencies ω = πω 2K . With these definitions, for instance, the equation (2.13) becomes The perturbed frequencies arẽ Thus, the frequencies of the 6 massless modes becomes and each of them contributes to the vacuum energy with The perturbation of the m 2 B = 2 scalar mode gives which contributes to the vacuum energy with The term ζ H (1, 2) = ζ H (1, 1) − 1 originated from the second term in (2.26) is divergent but, as we will see in a moment, it cancels against divergencies coming from other modes. For the last scalar mode, equation (2.14) becomes wherẽ dσφ * 2,n (1 + 2 cos 2σ )φ 2,n = (n + 2) 2 − k 2 2(n + 2) 2 (n + 1)(n + 3) , (2.29) which implies that

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The contribution of this mode to the vacuum energy is then Let us now turn to the fermionic fluctuations. Expanding equation (2.12) we get (2.32) Now, using the k = 0 eigenfunctions, the perturbed frequencies arẽ for the m F < 1/2 fermionic modes or Computing the remaining integral, for m F = ±1, we get 2n + 3 (n + 2)(n + 1) , . (2.36) Then, each of these modes contributes to the vacuum energy with Finally, for the m F = 0 fermionic modesω = n+ 1 2 and the correction to the frequencies comes entirely from the change in the range of the variable σ. Their contribution to the vacuum energy is 1 2 Recall that, in this limit, k 2 = φ 2 π 2 and the 1-loop anomalous dimension results This result differs from the one in [8].
At this point, and to verify that the vacuum energy computation gives the correct partition function contribution, we can use the above results to reproduce the Bremsstrahlung function to 1-loop order for a string in AdS 5 × S 5 . In that case, all the fermionic fluctuations had masses m F = ±1, 5 scalar fluctuations were massless and 3 had m 2 B = 2, two contributing with (2.27) and the other with (2.31). Thus, in that case the 1-loop partition function is which leads to in perfect agreement with the 1-loop results found in [4].

String ending on a line with an internal cusp
In this limit, the induced geometry is again (2.10) but the mass terms in the quadratic operators are different. The fermionic fluctuation equations are of the form

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where now Three of the scalar modes satisfy the equation two for m 2 B = 2 and one for m 2 B = 0. Four other scalar fields satisfy while last scalar mode satisfies

Small cusp
In the small k 2 limit eq. (2.45) becomes Thus, the two modes with m 2 B = 2 becomes unperturbed at this order with Their contribution to the vacuum energy is The mode with m 2 B = 0 leads to and summing these corrected frequencies we obtain the contribution to the vacuum energy

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The expansion of (2.47) gives The contribution of this mode to the vacuum energy is then To obtain the contribution from the fermionic modes we evaluate the small k expansion of (2.44) for the 8 different combinations (s 1 , s 2 , s 2 ). There are four modes with m F = ±1 for which the perturbed equations in the rescaled variable is 3 Their corrected frequencies are , (2.59) and so the sum over oscillator frequencies results (2.60) For the remaining two modes with m F = ±1 the potential vanishes i ∂σ + tanσ 2 γ 1 +ω n γ 0 − m F cosσ Ψ n = 0 , (2.61) and the frequencies are simply For the two massless modes we obtain i ∂σ + tanσ 2 γ 1 +ω n γ 0 ± k 2 4 cosσ Ψ n = 0 , (2.64)

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where the ± sign correspond to (s 1 s 2 s 3 ) equal (+ − +) and (− + −) respectively. At this point we have to decide which fermionic massless modes (Ψ 0,n or X 0,n ) should be used to compute the perturbative correction to the frequencies. As we have anticipated, only for a specific choice, the full set of fluctuations will conform an N = 6 supersymmetry multiplet. Moreover, since the background string configuration with cusp angles θ = ±φ is supersymmetric, among the four possible choices we select the one that leads to the same cusp anomalous dimension as for the geometrical cusp angle. For this to happen one has to take X 0,n for (+ − +) while Ψ 0,n for (− + −), which leads for both modes to and then 1 2 Now we sum all the contributions to the vacuum energy to k 2 order (2.67) In this limit, k 2 = − θ 2 π 2 . Therefore the cusp anomalous dimension for internal cusp results This is consistent with the fact that cusped Wilson loop is BPS for θ = ±φ [6]. In other words, the Bremsstrahlung function up to 1-loop order in the strong coupling limit is We can straightforwardly obtain the 1-loop partition function for a string with an internal cusp in AdS 5 × S 5 from the previous computations. In that case one has eight m F = ±1 fermions contributing with (2.60), four massless scalars resulting in (2.52), three m 2 B = 2 scalars giving (2.50) and one scalar coupled to the worldsheet curvature

Exact Bremsstrahlung functions in ABJM
The cusp anomalous dimension for small angles is given by the Bremsstrahlung function, This Bremsstrahlung function has been related to the expectation value of certain BPS circular Wilson loop, which is exactly known [3] where W n is a Wilson loop that winds n times around the maximal circle in a 3-sphere.
In this section, we review the strong coupling expansion of the Bremsstrahlung functions from equation (3.2). The strong coupling expansion of the 1/6 BPS multiply wound circular Wilson loop is [15], where H n are harmonic numbers. The use of this expansion in (3.2) gives the Bremsstrahlung function for a small cusp deforming a 1/6 BPS Wilson line [3] To obtain the Bremsstrahlung function for a small cusp deforming a 1/2 BPS Wilson line, we need to feed (3.2) with the 1/2 BPS multiply wound circular Wilson loop expectation value [15] W 1/2 n = W n − e inπ W n , (3.5)

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where W n is the previous 1/6 circular Wilson and W n is the complex conjugate of W n . We then obtain Since the derivative ∂ n ( W n + W n ) vanishes at n = 1, we are left with Using the expansion (3.3), one obtains [11] The first two orders of (3.8) are in complete agreement with (2.69), computed in section 2.

Discussion
We have computed the partition function for strings ending in cusped lines at the boundary to 1-loop order in the large √ λ expansion. We computed the corresponding 1-loop determinants by relating them to the vacuum energy density of the 1-loop oscillator modes.
These 1-loop determinants have been computed previously for small cusp angles, by reducing the operators to 1-dimensional ordinary differential ones and exploiting the Gelfand-Yaglom method [4,8,[23][24][25]. In the case of strings in AdS 5 × S 5 , our results coincide with those of [4]. For strings in AdS 4 × CP 3 , our results differ from those in [8]. We understand the discrepancy in the result as due to the imposition of different boundary conditions for the massless modes.
In two dimensions the Dirac operator can be reduced to the flat space form by rescaling the spinor with g 1/4 . From the AdS 2 fermionic modes (2.15) one obtains, after the rescaling, the following spinor solution to the flat space massless Dirac operator (4.1) Each component of these solutions with ω = n + 1 2 satisfies Dirichlet boundary conditions in one endpoint and Neumann boundary conditions in the other. Other solutions with ω = n exist. They have Dirichlet boundary conditions for one component and Neumann boundary conditions for the other. However, they do not form a supersymmetry multiplet with the massless scalar fields we are using.
When using the off-shell method to compute 1-loop determinants, the choice of boundary conditions will impact in the outcome as well. In [8] Dirichlet boundary conditions were used for the fermionic fluctuations. The determinant of the corresponding quadratic operator with spectrum λ n =ω 2 + n 2 could then be simplified with the determinant of a massless scalar field with Dirichlet boundary conditions.

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Let us briefly analyze how the results of [8] would change if the determinant of the quadratic operator for the massless fermionic modes were computed with Dirichlet-Neumann boundary conditions. The corresponding spectrum would be λ n =ω 2 + (n + 1 2 ) 2 . Using that it is straightforward to see that the resulting change in the geometrical cusp anomalous dimension is The result computed in [8] corrected by (4.3) is in complete agreement with our result.
Our results for the cusp anomalous dimension for a geometrical cusp in section 2.3 and for an internal cusp in section 2.4 are consistent with its vanishing in the BPS case θ = ±φ. Moreover this unique Bremsstrahlung function found in the strong coupling limit (2.69) is in perfect agreement with the expansion of the exact result (3.8) in section 3.
Finally, we would also like to comment about the Bremsstrahlung function for a small geometrical cusp deformation of a 1/6 BPS Wilson line. From the string point of view, this kind of Wilson loops are interpreted as the smearing of string configurations over a CP 1 ⊂ CP 3 . This corresponds to impose Neumann boundary conditions for the directions along the CP 1 [3]. Therefore, to compute Z 1−loop for geometrical cusp angle deformation of a 1/6 BPS Wilson line, we should repeat the perturbative analysis replacing two of the scalar massless modes with Dirichlet boundary conditions by two scalar massless modes with Neumann boundary conditions, and correspondingly replacing fermionic massless modes type Ψ 0,n by X 0,n and viceversa. However, in this case, the perturbation of scalar massless modes vanishes using either (2.18) or (2.20). The same happens for fermionic massless modes and, then, to this order Z 1−loop is the same as for the deformation of a 1/2 BPS Wilson line which is in agreement with (3.4). For an internal cusp angle, the dual string has a nontrivial profile inside the CP 3 . The smearing of such configurations has no clear geometrical interpretation [12].

A Review of the classical solution and quantum fluctuations
In this section we review the classical solution and its quantum fluctuations found in [8] for a string in AdS 4 × CP 3 ending on a Wilson line with geometrical cusp φ and internal cusp θ.

A.1 Classical solution
The target space is AdS 4 × CP 3 with metric where In this background a Green-Schwartz action (GS) is considered. The action of bosonic degrees of freedom is of the Nambu-Goto type in the static gauge, taking t and ϕ as the worldsheet coordinates, where µ, ν and i, j are target space and worldsheet indices respectively, G µν is the target space metric and X µ are the embedding coordinates. The global radius ρ and ϑ, a particular Killing direction in CP 3 , are taken to be while the remaining variables are held fixed. The cusp angles φ and θ relate to the angular extension of the string in AdS and CP 3 respectively as ϕ ∈ [φ/2, π − φ/2] and ϑ ∈ [−θ/2, θ/2]. The conserved charges associated to t and ϑ translations are The BPS condition θ = ±φ is equivalent to E = ±J at the classical level. Now we introduce two parameters in terms of which we can express the cusp angles where The small cusp angles limit (φ, θ ≪ 1) corresponds to p → ∞

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The classical action for this embedding reads where T is a cut-off for the time integration and a divergent term from the ρ → ∞ limit has been dropped.

A.2 Fluctuation Lagrangian
It is convenient to define a world-sheet coordinates τ and σ for which In these coordinates, the induced metric takes the form and leads to the following scalar curvature In what follows we drop the k 2 dependence in the Jacobi functions in order to simplify the notation.
Since we are in static gauge, there are no fluctuations in the longitudinal directions. Defining appropriate scalar fluctuation fields ζ a for the transverse directions (a = 1, . . . , 8) (see [8]) the bosonic quadratic action results .
For the fermionic modes the resulting quadratic fluctuation Lagrangian is of the Dirac type. We will only write the expressions in the corresponding limits of small geometrical and internal cusp in the next subsection. The kappa symmetry fixing condition is of the form

A.2.1 Small cusp limit
We are interested in the behaviour of the 1-loop partition function when the cusp angle, either geometrical or internal, is small. In the first case we consider a small geometrical cusp angle φ and a vanishing θ. From (A.9) we conclude that θ = 0 implies q → 0. In this limit we can express p and b in terms of k. For small values of k the geometrical cusp angle is also small, φ 2 = π 2 k 2 +O(k 4 ).
For bosonic modes the mass matrix become diagonal and we obtain the corresponding Klein-Gordon operators in the metric (A.13) with masses whereγ a are curved 2-dimensional Dirac matrices for metric (A.13), D a are the corresponding covariant derivatives namely

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In this case the relevant second order operator is ∆ F K = (K −1 D F ) 2 , whose variation is Because of the trace, both terms commute and we obtain As done in the bosonic case, we can compute the anomaly in terms of the Seeley coefficients (B.14) Note that for non-trivial K this is in general different from K −2 D 2 F because of the linear derivatives acting on K −1 . Therefore we have to compute the corresponding Seeley coefficients for (K −1 D F ) 2 which are in general different from the K −2 D 2 F ones.
where∇ involves spin connection terms and Y is an arbitrary potential. For this class of operators the Seeley coefficient a 1 is also known [18]. Then, the anomaly results The last expression is valid for general 2d fermionic operators. However, the in the GS formulation the fermionic fields are worldsheet anticommuting scalars. In this case, the R (2) term is four times the corresponding to standard 2d fermions [18]. Taking this into account we obtain Now that we know the individual contribution of scalars and GS fermions to the scaling anomaly, we are able to derive the condition over the partition function in order to be invariant under this operation.
For that purpose, consider the fluctuation scalar fields over a classical configuration. In general, the action can be written in the form where summation over a = 1, . . . , 10 is implied and g ij is the induced metric for a general classical configuration. All the possible potential and mass terms are contained in the mass where D 2 is the square of the standard kinetic term for each GS fermion and the mass matrix Y ab contains all possible potential and mass terms. Finally, for the ghost modes with K = √ g −1/2 , the anomaly can be written as log det ∆ gh K − log det ∆ gh = − The results presented in appendix A were computed in [8] in static gauge (Nambu-Goto formulation). In order to verify the last equality for this configuration note that, beingX the mass term obtained in static gauge, it is related to the corresponding one in conformal gauge as TrX = TrX − Tr long X .

(B.25)
It is important to stress that this scaling invariance condition is valid for any string configuration. In particular, for the straight line configuration (R (2) = −2) we have a 2d field theory on AdS 2 with 2 fermionic massless modes, 6 with Y aa = m 2 F = 1 and X ab = diag R (2) + 4, 2, 0, 0, 0, 0, 0, 0 . (B.26) Therefore condition (B.25) is satisfied and we can compute the 1-loop partition function via a vacuum energy computation. Finally, for the cusped classical configuration the condition (B.25) is also valid implying the vanishing of the total scaling anomaly and therefore justifying the vacuum energy computation made in this article. Note that this analysis is valid for arbitrary values of parameters p and q.
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