Precision calculation of 1/4-BPS Wilson loops in AdS5×S5

Journal of High Energy Physics, Feb 2016

We study the strong coupling behaviour of 1/4-BPS circular Wilson loops (a family of “latitudes”) in \( \mathcal{N}=4 \) Super Yang-Mills theory, computing the one-loop corrections to the relevant classical string solutions in AdS5 ×S5. Supersymmetric localization provides an exact result that, in the large ’t Hooft coupling limit, should be reproduced by the sigma-model approach. To avoid ambiguities due to the absolute normalization of the string partition function, we compare the ratio between the generic latitude and the maximal 1/2-BPS circle: any measure-related ambiguity should simply cancel in this way. We use the Gel’fand-Yaglom method with Dirichlet boundary conditions to calculate the relevant functional determinants, that present some complications with respect to the standard circular case. After a careful numerical evaluation of our final expression we still find disagreement with the localization answer: the difference is encoded into a precise “remainder function”. We comment on the possible origin and resolution of this discordance.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP02%282016%29105.pdf

Precision calculation of 1/4-BPS Wilson loops in AdS5×S5

HJE AdS5 V. Forini 0 1 2 4 5 6 7 8 V. Giangreco M. Puletti 0 1 2 3 5 6 7 8 L. Griguolo 0 1 2 5 6 7 8 D. Seminara 0 1 2 5 6 7 8 E. Vescovi 0 1 2 4 5 6 7 8 0 INFN Gruppo Collegato di Parma 1 Dunhaga 3 , 107 Reykjavik , Iceland 2 Zum Gro en Windkanal 6 , 12489 Berlin , Germany 3 Science Institute, University of Iceland 4 Institut fur Physik, Humboldt-Universitat zu Berlin, IRIS Adlershof 5 Open Access , c The Authors 6 Via G. Sansone 1, 50019 Sesto Fiorentino , Italy 7 Viale G. P. Usberti 7/A, 43100 Parma , Italy 8 Going to Fourier space We study the strong coupling behaviour of 1=4-BPS circular Wilson loops (a family of \latitudes") in N = 4 Super Yang-Mills theory, computing the one-loop corrections to the relevant classical string solutions in AdS5 S5. Supersymmetric localization provides an exact result that, in the large 't Hooft coupling limit, should be reproduced by the sigma-model approach. To avoid ambiguities due to the absolute normalization of the string partition function, we compare the ratio between the generic latitude and the maximal 1/2-BPS circle: any measure-related ambiguity should simply cancel in this way. We use the Gel'fand-Yaglom method with Dirichlet boundary conditions to calculate the relevant functional determinants, that present some complications with respect to the standard circular case. After a careful numerical evaluation of our nal expression we still nd disagreement with the localization answer: the di erence is encoded into a precise \remainder function". We comment on the possible origin and resolution of this Wilson; 't Hooft and Polyakov loops; AdS-CFT Correspondence; Sigma Mod- - discordance. els ArXiv ePrint: 1512.00841 1 Introduction and main result 2 3 4 5 Classical string solutions dual to latitude Wilson loops 3.1 3.2 3.3 4.1 4.2 uctuation determinants Bosonic sector Fermionic sector The circular Wilson loop limit One-loop partition functions The circular Wilson loop Conclusions Ratio between latitude and circular Wilson loops A Notation and conventions B Methods for functional determinants B.1 Di erential operators of the nth-order B.2 Applications B.3 Square of rst-order di erential operators C Fermionic determinant DetOF ( 0): details C.1 Derivation of (3.28) C.2 Simplifying the large-R expression for Det!(OF1;1;1)2 D Boundary conditions for small Fourier modes 1 5 7 Introduction and main result The harmony between exact QFT results obtained through localization procedure for BPSprotected Wilson loops in N = 4 SYM and their stringy counterpart is a thorny issue beyond the supergravity approximation. For the 1=2-BPS circular Wilson loop [1, 2], in the fundamental representation, supersymmetric localization [3] in the gauge theory con rms the all-loop prediction based on a large N resummation of ladder Feynman diagrams [4] and in [7]1 using the Gel'fand-Yaglom method, reconsidered in [9] with a di erent choice of boundary conditions and reproduced in [10]2 with the heat-kernel technique. No agreement was found with the subleading correction in the strong coupling ( 1 ) expansion of the gauge theory result in the planar limit loghW ( ; 0 = 0)i = log p2 I1 p = p 3 4 log + 1 2 log 2 + O 1 2 ; (1.1) where I1 is the modi ed Bessel function of the rst kind, the meaning of the parameter 0 is clari ed below, and the term proportional to log in (1.1) is argued to originate from the SL(2; R) ghost zero modes on the disc [5]. The discrepancy occurs in the -independent part above,3 originating from the one-loop e ective action contribution and an unknown, overall numerical factor in the measure of the partition function. The situation becomes even worse when considering a loop winding n-times around itself [7, 11], where also the functional dependence on n is failed by the one-loop string computation. The case of di erent group representations has also been considered: for the ksymmetric and k-antisymmetric representations, whose gravitational description is given in terms of D3- and D5-branes, respectively, the rst stringy correction again does not match the localization result [12]. Interestingly, the Bremsstrahlung function of N = 4 SYM, derived in [13] again using a localization procedure, is instead correctly reproduced [14] through a one-loop computation around the classical cusp solution [2, 15]. Localization has been proven to be one of the most powerful tools in obtaining non perturbative results in quantum supersymmetric gauge theories [3]: an impressive number of new exact results have been derived in di erent dimensions, mainly when formulated on spheres or products thereof [3, 16]. In order to gain further intuition on the relation between localization and sigma-model perturbation theory in di erent and more general settings, we re-examine this issue addressing as follows the problem of how to possibly eliminate the ambiguity related to the partition function measure. We consider the string dual to a nonmaximal circular Wilson loop | the family of 1/4-BPS operators with path corresponding to a latitude in S2 2 S5 parameterized by an angle 0 and studied at length in [15, 17, 18] | and evaluate the corresponding string one-loop path integral. We then calculate the ratio between the latter and the corresponding one representing the maximal circle | the case 0 = 0 in (1.1). Our underlying assumption is that the measure is actually independent on the geometry of the worldsheet associated to the Wilson loop,4 and therefore in such ratio measure-related ambiguities should simply cancel. It appears non-trivial to actually prove a background independence of the measure, whose di eo-invariant de nition includes in fact explicitly the worldsheet elds.5 Our assumption | also suggested in [7] | seems however a reasonable one, especially in light of the absence of zero mode in the classical 1See also [8]. 2See appendix B in [10]. 3See formula (1.4) below. 4About the topological contribution of the measure, its relevance in canceling the divergences occurring in evaluating quantum corrections to the string partition function has been rst discussed in [6] after the observations of [19, 20]. We use this general argument below, see discussion around (4.8). 5See for example the discussion in [21]. { 2 { solutions here considered6 and of the explicit example of (string dual to) the ratio of a cusped Wilson loop with a straight line [14], where a perfect agreement exists between sigma model perturbation theory and localization/integrability results [13].7 The family of 1/4-BPS latitude Wilson loops falls under the more general class of 1/8-BPS Wilson loops with arbitrary shape on a two-sphere introduced in [15, 24, 25] and studied in [26]. There are strong evidences that they localize into Yang-Mills theory on S2 in the zero-instanton sector [15, 26{28] and their vacuum expectation values are therefore related to the 1/2-BPS one by a simple rescaling. As originally argued in [18] the expectation value of such latitude Wilson loops is obtained from the one of the maximal circle provided one replaces with an e ective 't Hooft coupling 0 = cos2 0. The ratio of interest follows very easily hW ( ; 0)i hW ( ; 0)i loc = e p (cos 0 1) h(cos 0) 23 + O( 1 i 2 ) + O e p where in the large expansion only the dominant exponential contribution is kept (and loc stands for \localization"). In terms of string one-loop e ective actions log Z loghW i, this leads to the prediction whose contribution e ectively vanish in the chosen regularization scheme [36, 37]. We emphasise that this procedure di ers from the one employed in [7], since the non-diagonal matrix structure of the fermionic- uctuation operator for arbitrary 0 prevents us from factorizing the value of the fermionic determinants into a product of two contributions. 6In presence of zero mode, a possible dependence of the path integral measure on the classical solution 7See also [23], which analyzes the (string dual to the) ratio between the Wilson loop of \antiparallel In the 0 ! 0 limit, we analytically recover the constant one-loop coe cient in the expansion of the 1/2-BPS circular Wilson loop as found in [7, 10] loghW ( ; 0 = 0)i = p 3 4 log( ) + log c + log 1 2 1 2 + O 1 2 ; (1.4) up to an unknown contribution of ghost zero-modes (the constant c). The expression above is in disagreement with the gauge theory prediction (1.1). We regularize and normalize the latitude Wilson loop with respect to the circular case. The summation of the one-dimensional Gel'fand-Yaglom determinants is quite di cult, due to the appearance of some Lerch-type special functions, and we were not able to obtain a that the disagreement between sigma-model and localization results (1.3) is not washed out yet. Within a certain numerically accuracy, we claim that the discovered 0-dependent discrepancy is very well quanti ed as (1.5) (1.6) log hW ( ; 0)i hW ( ; 0)i sm = p suggesting that the \remainder function" should be Rem( 0) = log cos 0 of the discrepancy at the end of the manuscript. The paper proceeds as follows. In section 2 we recall the classical setting, in section 3 we evaluate the relevant functional determinants which we collect in section 4 to form the corresponding partition functions. Section 5 contains concluding remarks on the disagreement with the localization result and its desirable explanation. After a comment on notation in appendix A, we devote appendix B to a concise survey on the Gel'fandYaglom method. Appendix C elucidates some properties which simplify the evaluation of the fermionic contribution to the partition function, while in appendix D we comment on a possible di erent choice of boundary condition for lower Fourier modes which that not a ect our results. { 4 { The classical string surface describing the strong coupling regime of the 1=4-BPS latitude was rst found in [17] and discussed in details in [15, 18]. Endowing the AdS5 S5 space with a Lorentzian metric in global coordinates with the AdS radius set to 1, the corresponding classical con guration in AdS3 S 2 parametrizes a string worldsheet, ending on a unit circle at the boundary of AdS5 and on a latitude sitting at polar angle 0 on a two-sphere inside the compact space.8 Here the spans the interval [ 2 ; 2 ]: The worldsheet coordinates instead take values in The ansatz (2.2) does not propagate along the time direction and de nes an Euclidean surface embedded in a Lorentzian target space. It satis es the equation of motions (supplemented by the Virasoro constraints in the Polyakov formulation) when we set sinh ( ) = sin ( ) = 1 sinh cosh ( 0 ; 1 ) ; cosh ( ) = 1 tanh ; cos ( ) = tanh ( 0 ) : An integration constant in (2.3) that shifts was chosen to be zero so that the worldsheet boundary at = 0 is located at the boundary of AdS5. The remaining one, 0 2 [0; 1), spans the one-parameter family of latitudes on S5 at the boundary = 0, whose angular position 0 2 [0; 2 ] relates to 0 through cos 0 = tanh 0 : Here the dual gauge theory operator interpolates between two notable cases. The 1/2-BPS circular case falls under this class of Wilson loops when the latitude in S2 shrinks to a point for 0 = 0, which implies ( ) = 0 and 0 = +1 from (2.3){(2.4). In this case the string propagates only in AdS3. The other case is the circular 1/4-BPS Zarembo Wilson loop when the worldsheet extends over a maximal circle of S2 for 0 = 2 and 0 = 0 [22].9 The double sign in (2.3) accounts for the existence of two solutions, e ectively doubling the range of 0: the stable (unstable) con guration mimizes (maximizes) the action functional and wraps the north pole = 0 (south pole = ) of S5. 8There exist other solutions with more wrapping in S5, but they are not supersymmetric [18]. 9See also [38] for an analysis of the contribution to the string partition function due to (broken) zero modes of the solution in [22]. { 5 { The induced metric on the worldsheet depends on the latitude angle 0 through the conformal factor ( i = ( ; )) ds22D = hij d id j = 2 ( ) d 2 + d 2 ; 2 ( ) sinh2 ( ) + sin2 ( ) : The two-dimensional Ricci curvature is then (2)R = = 2( ) = (2 cosh 2 0 2 sinh 0 sinh (6 3 0) 3 cosh (2 ( 4cosh ( 0) cosh3 (2 0) 0))+ 6 cosh (4 2 0)+3 cosh 2 ) (2.5) (2.6) (2.8) (2.9) : (2.10) p p b (2.12) The semiclassical analysis is more conveniently carried out in the stereographic coordinates m (m = 1; 2; 3) of S3 AdS5 and wn (n = 1; 2; 3; 4; 5) of S5 where the classical solution reads10 10The background of '1; '2; #2 was set to zero in (2.2), but the bosonic quadratic Lagrangian does not have the standard form (kinetic and mass terms for the eight physical elds) in the initial angular coordinates. { 6 { The string dynamics is governed by the type IIB Green-Schwarz action, whose bosonic part is the usual Nambu-Goto action Z SB = T d d p h Z d d LB in which h is the determinant of the induced metric (2.8) and the string tension T = 2 depends on the 't Hooft coupling . The leading contribution to the string partition function comes from the regularized classical area [18] SB(0)( 0) = p 2 Z 2 0 d Z 1 0 d sin2 ( ) + sinh2 ( ) = p 1 cos 0 + + O( ) : (2.11) Following [7] we have chosen to distinguish the cuto 0 in the worldsheet coordinate from the cuto = tanh 0 in the Poincare radial coordinate z of AdS. The pole in the IR cuto in (2.13) keeps track of the boundary singularity of the AdS metric and it is proportional to the circumference of the boundary circle. The standard regularization scheme, equivalent to consider a Legendre transform of the action [2, 39], consists in adding a term proportional to the boundary part of the Euler number Here g stands for the geodesic curvature of the boundary at = 0 and ds is the invariant line element. With this subtraction, we have the value of the regularized classical area The (upper-sign) solution dominates the string path integral and is responsible for the leading exponential behaviour in (1.2) and so, in the following, we will restrict to the upper signs in (2.3). 3 uctuation determinants This section focusses on the semiclassical expansion of the string partition function around the stable classical solution (2.7) (taking upper signs in (2.3)) and the determinants of the di erential operators describing the semiclassical uctuations around it. The 2 -periodicity in allows to trade the 2D spectral problems with in nitely-many 1D spectral problems for the (Fourier-transformed in ) di erential operators in . Let us call O one of these one-loop operators. For each Fourier mode !, the evaluation of the determinant Det!O is a one-variable eigenvalue problem on the semi-in nite line 2 [0; 1) which we solve using the Gel'fand-Yaglom method, a technique based on -function regularization reviewed in appendix B. Multiplying over all frequencies ! (which are integers or semi-integers according to the periodicity of the operator O) gives then the full determinant HJEP02(16)5 DetO = Y Det!O: ! All our worldsheet operators are intrinsically singular on this range of , since their principal symbol diverges at = 0, the physical singularity of the boundary divergence for the AdS5 metric. Moreover the interval is non-compact, making the spectra continuous and more di cult to deal with. We consequently introduce an IR cuto at = 0 (related to the = tanh 0 cuto in z) and one at large values of = R [7]. While the former is necessary in order to tame the near-boundary singularity, the latter has to be regarded as a mere regularization artifact descending from a small ctitious boundary on the tips of the surfaces in AdS3 and S2. Indeed it disappears in the one-loop e ective action. 3.1 Bosonic sector The derivation of the bosonic uctuation Lagrangian around the minimal-area surface (2.7) is readily available in section 5.2 of [40]. The one-loop uctuation Lagrangian in static gauge is (3.1) (3.2) (2) LB 2 ( ) yT OB ( 0) y ; { 7 { to the worldsheet y (yi)i=1;:::8. In components it reads11 where the di erential operator OB ( 0) acts on the vector of uctuation elds orthogonal 1 ; [OB ( 0)]ij = where the non-vanishing entries of the matrices are12 The worldsheet excitations decouple in the bosonic sector, apart from y7 and y8 which are coupled through a 2 2 matrix-valued di erential operator. The determinant of the bosonic operator is decomposed into the product (3.4) ; (3.5) (3.6) (3.7) (3.8) (3.9) ! i!), formula (3.6) holds for each frequency ! with O1 O2 ( 0) O3 ( 0) 2 + 3 tanh2 (2 + 0) 2 i tanh (2 + 0) ! 2 i tanh (2 + 0) ! dd22 + !2 2 + 3 tanh2 (2 + 0) ! The unitary matrix U = p1 2 i 1 diagonalizes the operator (3.9) i 1 O3 ( 0) = U y diagfO3+; O3 g U ; O3+ ( 0) = O3 ( 0) = (3.10) 2 + 3 tanh2 (2 + 0) + 2! tanh (2 + 0) : We performed a rescaling by p h = 2( ) (as in the analogous computations of [7, 14, 23]) which will not a ect the nal determinant ratio (4.1) (see discussions in appendix A of [6] and in [7, 14, 23]) and is actually instrumental for the analysis in appendices B.2 and B.3. We rewrite (3.6) as follows Det!OB ( 0) = Det3!O1 Det3!O2 ( 0) Det!O3+ ( 0) Det!O3 ( 0) ; (3.11) 11To compare with [40], and using the notation used therein, notice that the bosonic Lagrangian is derived as L(B2) = A` iA `j p Mij yiyj ; (3.3) which de nes in an obvious way mij and nij in (3.4). 12There would be an overall minus sign in the kinetic and mass term of the y1 uctuation, which we disregard in (3.4) for simplifying the formula, considering that it does not play a practical role in the evaluation of determinants with Gel'fand-Yaglom and is reabsorbed in the Wick-rotation of the time coordinate t. { 8 { The evaluation of one-dimensional spectral problems is outlined in appendix B.2. The where all the determinants are taken at xed !. To reconstruct the complete bosonic contribution we have to perform an in nite product over all possible frequencies. The operator O1 does not depend on 0, and indeed also appears among the circular Wilson loop uctuation operators [7]. While its contribution formally cancels in the ratio (1.3), we report it below along with the others for completeness. Both O2 ( 0) and O3 ( 0) become massless (scalar- and matrix-valued respectively) operators in the circular Wilson loop limit, which is clear for O3 ( 0) upon diagonalization and an integer shift in !,13 irrelevant for the determinant at given frequency, as long as we do not take products over frequencies into consideration. Thus, in this limit one recovers the bosonic partition function of [7]. elds satisfy Dirichlet boundary conditions at the endpoints of the compacti ed interval 2 [ 0; R]. Then we take the limit of the value of the regularized determinants for R ! 1 at xed ! and 0 . As evident from the expressions below, the limit on the physical IR cuto ( in z or equivalently 0 in ) would drastically change the !-dependence at this stage and thus would spoil the product over the frequencies. It is a crucial, a posteriori, observation that it is only keeping 0 nite while sending R to in nity that one precisely reproduces the expected large ! (UV) divergences [6, 40]. This comes at the price of more complicated results for the bosonic (and especially fermionic) determinants. Afterwards we will remove the IR divergence in the one-loop e ective action by referring the latitude to the circular solution. The solutions of the di erential equations governing the di erent determinants are singular for small subset of frequencies: we shall treat apart these special values when reporting the solutions. For the determinant of the operator O1 in (3.7) in the limit of large R one obtains [7] Det!O1 = (ej!j(R 0) (2j!j!j +j(cj!otj+h10)) R coth 0 and only the case ! = 0 has to be considered separately. Next we examine the initial value problem (B.12){(B.13) associated to O2( 0), whose solution is f(II)1( ) = 8 > > < > > > > > 2( > > > > >>> ( : 1 >> 2! cosh( + 0) cosh( 0+ 0) cosh ( + 0 + 2 0) sinh(!( + (!+1) sinh((! 1)( 13In the language of [40], this shift corresponds to a di erent choice of orthonormal vectors that are orthogonal to the string surface. { 9 { (3.12) ! 6= ! = The determinant is then given by f(II)1(R) and for R large one obtains the simpler expression Det!O2( 0) = (ej!j(R 0) (j!j+tanh( 0+ 0)) 2j!j(j!j+1) R tanh( 0 + 0) We repeat the same procedure for O3+ ( 0). From the solutions > > > > > >>>> (e > 0)) one nds for large R In view of the relation O3 ( 0) = O3+ ( 0) j!! !, which follows from (3.10), we can easily deduce the results for Det!O3 ( 0) by ipping the frequency in the lines above Det!O3+( 0) = < Notice that a shift of ! ! ! 1 in Det!O3+( 0) and ! ! ! + 1 in Det!O3 ( 0) gives back the symmetry around ! = 0 in the distribution of power-like and exponential large-R divergences which characterizes the other determinants (3.12) and (3.14). Such a shift | also useful for the circular Wilson loop limit as discussed below (3.11) | does not a ect the determinant, and we will perform it in section 4. 3.2 The Fermionic sector uctuation analysis in the fermionic sector can be easily carried out following again the general approach [40], which includes the local SO(1; 9) rotation in the target space [6, 41{45] that allows to cast the quadratic Green-Schwarz fermionic action into eight contributions for two-dimensional spinors on the curved worldsheet background. The standard Type IIB -symmetry gauge- xing for the rotated fermions 1 = 2 ! = ! 1 where the operator OF ( 0) is given by LF ( ) ( ) 1 + The coe cients a34( ) and a56( ) can be expressed as derivatives of the functions appearing in the classical solution: In the 0 ! 0 limit (hence ( ) ! 0), one gets i 2 i 2 OF ( 0 = 0) = i sinh cosh 3 + sinh which coincides with the operator found in the circular Wilson loop analysis of [7],15 once we go back to Minkowski signature and reabsorbe the connection-related 456-term via the -dependent rotation ! exp . In Fourier space this results in a shift of the integer fermionic frequencies ! by one half, turning periodic fermions into anti-periodic ones. In the general case (3.20) we cannot eliminate all the connection-related terms a34( ) 3 + a56( ) 456, since the associated normal bundle is non- at [40].16 Performing anyway the above -rotation at the level of (3.20) has the merit of simplifying the circular limit making a direct connection with known results. This is how we will proceed: for now, we continue with the analysis of the fermionic operator in the form (3.20) without performing any rotation. Then, in section 4, we shall take into account the e ect of this rotation by relabelling the fermionic Fourier modes in terms of a suitable choice of halfintegers. The analysis of the fermionic operator (3.20) drastically simpli es noticing that the set of mutually-commuting matrices f 12; 56; 89g commutes with the operator itself and leaves invariant the spinor constraint (A.6) and the fermionic gauge xing (3.18). By means of the projectors P12 I32 2 i 12 ; P56 I32 i 56 2 and P89 I32 2 i 89 ; (3.23) 14We perform the computations in a Lorentzian signature for the induced worldsheet metric and only at the end Wick-rotate back. The di erence with (5.37){(5.38) of [40] is only in labeling the spacetime directions. 15See formula (5.17) therein. 16The arising of gauge connections in the covariant derivatives associated to the structure of normal bundle is discussed at length in [40] and references therein. In particular, see discussion in section 5.2 of [40] for both the latitude and the circular Wilson loop limit. (3.19) (3.20) (3.21) HJEP02(16)5 we decompose the 32 32 fermionic operator into eight blocks of 2 2 operators labeled by the triplet fp12; p56; p89 = 1; 1g. Formally this can be seen as the decomposition into the following orthogonal subspaces Notice that the operator de ned in (3.26) actually does not depend on the label p89. Then the spectral problem reduces to the computation of eight 2D functional determinants17 DetOF ( 0) = Y A deeper look at the properties of OF p12;p56;p89 allows us to focus just on the case of p12 = p56 = p89 = 1. In fact, as motivated in details in appendix C.1, the total determinant can be rewritten as follows DetOF ( 0) = Det![(OF1;1;1(!))2]2Det![(OF1;1;1( !))2]2 : Using the matrix representation (A.3) and going to Fourier space, we obtain + i ( ) 1 2( ) Y !2Z h i + 1 ( ) 1 2( ) . For simplicity of notation, from now on we will denote with O1;1;1( 0) the rst factor in the de nition above. In a similar spirit to the analysis for the F bosonic sector, we start to nd the solutions of the homogeneous problem OF 1;1;1 ( 0) f ( ) = 0 where f ( ) denotes the two component spinor (f1( ); f2( ))T . The system of coupled rst-order di erential equations now reads sin2 ( ) ! a34( ) + a56( )) f2( ) = 0; a34( ) a56( )) f1( ) = 0: OF 1;1;1( 0) a34( ) 1 + ia56( ) 2 sinh2 ( ) 3 sin2 ( )I2 i M OeF 1;1;1 M ; 17A non-trivial matrix structure is also encountered in the fermionic sector of the circular Wilson loop [7], but the absence of a background geometry in S5 leads to a simpler gamma structure. It comprised only three gamma combinations ( 0; 4; 04), whose algebra allows their identi cation with the three Pauli matrices without the need of the labelling the subspaces. We can cast it into a second-order di erential equation for one of the unknown functions. Solving (3.32) for f2( ) i ( ) tanh ( + 0) 2 f1( ) ; (3.33) and then plugging it into (3.31) one obtains 00 f1 ( ) 1 2 sinh2 1 2 cosh2 ( + 0) + + tanh( + 0) 2 ! 2 f1( ) = 0: (3.34) It is worth noticing that the Gel'fand-Yaglom method has naturally led to an auxiliary Schrodinger equation for a ctitious particle on a semi-in nite line and subject HJEP02(16)5 to a supersymmetric potential V ( ) = 1 W ( ) = 2 tanh + tanh( + 0) 2 W 0( ) + W 2( ) derived from the prepotential !. Traces of supersymmetry are not surprising: they represent a vestige of the supercharges unbroken by the classical background.18 As in the bosonic case, we have to separately discuss some critical values of the frequencies. We only report the independent solutions of the equations above, where the constants ci;1 and ci;2 have to be xed in the desired initial value problem (i = I; II). 8 ci;1e (1+!)+ci;2e (1 !)+ 0 (2!2 cosh( + 0) sinh +! cosh(2 + 0)+sinh 0) 18The same property is showed by (5.26) in [7]. 2 tanh 1 1 2 tanh >> ci;1 q(e2 1)(e2 0 +1)(e2 +2 0 +1)(e4 +2 0 +1) > > i(2e e3 +e3 +2 0 ) ie (2 !)+2 0 (2!+sinh(2 +2 0)+2! sinh 0 sinh(2 + 0) sinh 2 ) q(e2 1)(e2 0 +1)(e2 +2 0 +1)(e4 +2 0 +1) ! 6= > ci;1 q(e2 1)(e2 0 +1)(e2 +2 0 +1) > > We are now ready to evaluate the determinants using the results of appendix (B.3), namely considering Dirichlet boundary conditions for the square of the rst order di erential operator. Having in mind the solutions above and how they enter in (B.9) and (B.22), it is clear that already the integrand in (B.27) is signi cantly complicated. A simpli cation occurs by recalling that our nal goal is taking the R ! 1 limit of all determinants and combine them in the ratio of bosonic and fermionic contributions. As stated above in the bosonic analysis and shown explicitly below, for the correct large ! divergences to be reproduced, it is crucial to send R ! 1 while keeping nite. In appendix (C.2) we sketch how to use the main structure of the matrix of the solutions Y ( ) to obtain the desired large-R expressions for the determinants in a more direct way. The determinant of the operator O1;1;1 for modes ! 6= f 1; 0; 1g reads for large R F (z; s; a) is the Lerch transcendent (4.5). The presence of the Lerch function is just a tool to have a compact expression for the determinants. In fact, for the values of ! relevant for us, it can be can be written in terms of elementary functions, but its expression becomes more and more unhandy as the value of ! increases. The coe cients ai and bi can be also expressed in terms of elementary functions. For the ai we have a0 = e R 320 sinh 0 (tanh 0 + 1) cosh ( 0 + 0) 8p2 cosh ( 0 + 2 0) a1 = 4 sech 0(tanh 0 + !)2 a2 = 4[2 1 ! 2 !2 cosh 0 a3 = tanh2 0 (coth 0 + 1) csch 0 sech ( 0 + 0) e 0 (cosh 0 2 sinh 0 sinh(2 0 0)) ! 2 ! sinh 0 + sech 0 sech2 0 + ! 2 + cosh 0 2!2 +! +3!2 csch 0 cosh( 0 +2 0) sech( 0 + 0)+! coth2 0 +2 coth 0 2 + 2 3! cosh 0 sech( 0 + 0) sinh 0 ! 2! coth 0 csch2 0 sech 0(coth 0 +1) i while for the bi we get b0 = e R 20 sech2 0 sinh 0 (tanh 0 + 1) cosh ( 0 + 0) 8p2 cosh( 0 + 2 0) b1 = b2 = b3 = 2 2 ! ! cosh(2 0) + sinh(2 0) + ! 2 1 cosh2 0 4! tanh( 0 + 0) 2! coth 0 + csch2 0 ! cosh(2 0)(! + 1) sinh(2 0) + cosh( 0 0)sech ( 0 + 0) : (3.39) The determinants of the lower modes have to be computed separately and they are given by + We report here the 0 ! 1 limit of all the bosonic and fermionic determinants, representing the circular Wilson loop case 0 = 0. The result for Det!O1 in (3.12) stays obviously the same, while for the limits of (3.14), (3.16) and (3.17) one easily gets The fermionic contributions (3.37){(3.42) reduce in this limit to Det! h OF 1;1;1( 0 = 0) Det!O2( 0 = 0) = Det!O3+( 0 = 0) = < R Det!O3 ( 0 = 0) = < R ( ej!j(R 0) 2j!j R 8 e(R 0)(! 1) 4(! 1)!2 (e2 0 1) 2 i R eR+ 0 = >< 2(e2 0 1) > e3(R 0) (2e2 0 1) 16(e2 0 1) >>> e (R 0)(2! 1)((! 1)e2 0 !) 4(! 1)2! (e2 0 1) We now put together the determinants evaluated in the previous sections and present the one-loop partition functions for the open strings representing the latitude ( 0 6= 0) and the circular ( 0 = 0) Wilson loop, eventually calculating their ratio. In the case of fermionic determinants, as motivated by the discussion below (3.22), we will consider the relevant formulas (3.37){(3.42) relabelled using half-integer Fourier modes. In fact, once projected onto the subspace labelled by (p12; p56; p89), the spinor is an eigenstate of 56 with eigenvalue ip56 and the rotation a shift of the Fourier modes by ! ! ! + p256 . This in particular means that below we will consider (3.37){(3.42) e ectively evaluated for ! = s + 12 and labeled by the half-integer frequency s. In the bosonic sector | as discussed around (3.11) and (3.17) | we pose ! = `+1 in Det!O3+ together with ! = ` 1 in Det!O3 . This relabeling of the frequences provides in (3.16) and (3.17) a distribution of the R-divergences that is centered around ! exp ej`jR for ` 6= 0) in the same way (in !) as for the other bosonic determinants (3.12) and (3.14). This will turn out to be useful while discussing the cancellation of R-dependence. Recalling also (2.13), we write the formal expression of the one-loop string action To proceed, we rewrite (4.1) as the (still unregularized) sum (4.1) (4.2) (4.3) (4.4) log Det`O1( 0) + log Det`O2( 0) + log Det`O3+ + log[Det`O3 log Dets(OF1;1;1)2 + log Det s(OF1;1;1)2 : 1 2 1 2 Equation (4.2) has the same form with e ectively antiperiodic fermions encountered Introducing the small exponential regulator , we proceed with the \supersymmetric regularization" of the one-loop e ective action proposed in [36, 37] where the (weighted) bosonic and fermionic contributions read B ` ( 0) = sF ( 0) = 3 2 4 2 p X s2Z+1=2 cos 0 + ( 1 )( 0) sF ( 0) ; X `2Z 3 2 4 2 2 In the rst sum (where the divergence is the same as in the original sum) one can remove by sending ! 0, and use a cuto regularization in the summation index j`j tantly, the non-physical regulator R disappears in (4.4). While in [7]19 the R-dependence . Impordrops out in each summand, here it occurs as a subtle e ect of the regularization scheme, and comes in the form of a cross-cancellation between the rst and the second line once the sums have been carried out. The di erence in the R-divergence cancellation mechanism is a consequence of the di erent arrangement of fermionic frequencies in our regularization scheme (4.4). In the circular case ( 0 = 0) this cancellation can be seen analytically, as in (4.10){(4.11) below. The same can be then inferred for the general latitude case, since in the normalized one-loop e ective action that the R-dependence drops out in each summand. A non-trivial consistency check of (4.4) is to con rm that in the large ` limit the expected UV divergences [6, 40] are reproduced. Importantly, for this to happen one cannot take the limit 0 ! 0 in the determinants above before considering ` 1, which is the reason why we kept dealing with the complicated expressions for fermionic determinants above. Using for the Lerch transcendent in (3.37) (z; s; a) z n 1 X n=0 (n + a)s where v( 0) = p h (2)R the asymptotic behavior for jaj 1 (i.e. j`j one nds that the leading -divergence is logarithmic, and | as expected from an analysis in terms of the Seeley-De Witt coe cients [6, 40] | proportional to the volume part of the Euler number and we notice that this limit is independent from 0 ( 0). This divergence should be cancelled via completion of the Euler number with its boundary contribution (2.12) and inclusion of the (opposite sign) measure contribution, as discussed in [6, 7]. Having this in mind, we will proceed subtracting (4.7) by hand in ( 1 )( 0) and in ( 1 )( 0 = 0). 19In this reference a regularization slightly di erent from [36, 37] was adopted. 4.1 The UV-regulated partition function in the circular Wilson loop limit reads ( 1 ) UV-reg( 0 = 0) = X ` j j + ` 2 The rst line is now convergent and its total contribution evaluates for where is Euler gamma function. The R-dependence in (4.10) cancels against the O( 0) contribution stemming from the regularization-induced sum in the second line of (4.9) ` F `+ 12 (0) X e X e ` ` 3 = 2R 2 arctanh : 2R + log Summing all contributions and nally taking ! 0, the result is precisely as in [7] despite the di erent frequency arrangement we commented on. We have checked that the same result is obtained employing -function regularization in the sum over `. The same nite part was found in [10] via heat kernel methods. There is no theoretical motivation for the log = -divergences appearing in (4.12), which will be cancelled in the ratio (1.3). In [7], this kind of subtraction has been done by considering the ratio between the circular and the straight line Wilson loop. 4.2 Ratio between latitude and circular Wilson loops In this section we describe the evaluation of the ratio (1.3) log Z ( ; 0) Z ( ; 0) = p (4.10) (4.11) (4.12) where UV-reg( 0 = 0) is in (4.9) and UV-reg( 0) is regularized analogously. The complicated fermionic determinants (3.37){(3.39) make an analytical treatment highly non-trivial, and we proceed numerically. First, we spell out (4.13) as ( 1 ) ` h F `+ 12 (0) ( v( 0) v(0)) log + `F+ 12 (0)+ `F 12 (0) 2 `F+ 12 (0)+ `F 12 (0) + + `F+ 12 ( 0)+ `F 12 ( 0) # `F+ 12 ( 0)+ `F 12 ( 0) # 2 2 where we separated the lower modes j`j latter we have used parity ` ! limit ! 0.21 The sum with large cuto Euler-Maclaurin formula 2 from the sum in the second line,20 and in the `. The sum multiplied by the small cuto is zero in the can be then numerically evaluated using the (4.15) (4.16) n X `=m+1 f (`) = f (`) d` + Z n m Z n m f (n) f (m) 2 B2p (f`g) d` ; (2p)! f (2p) (`) p 1 ; p + X k=1 (2k)! B2k hf (2k 1) (n) f (2k 1) (m)i in which Bn(x) is the n-th Bernoulli polynomial, Bn = Bn(0) is the n-th Bernoulli number, f`g is the integer part of `, f (`) is the summand in the second line of (4.14), so m = 2, n = . After some manipulations to improve the rate of convergence of the integrals, we safely send ! 1 in order to evaluate the normalized e ective action = h ( 1 ) 2 X `= 2 + Z 1 f (2) 2 2 i sm ` relabeling discussed above. cuto makes the sum vanish. 20This is convenient because of the di erent form for the special modes (3.40){(3.42) together with the 21This can be proved analytically since the summand behaves as e `` 2 for large `. Removing the In order to gain numerical stability for large `, above we have set p = 3, we have cast the Lerch transcendents inside sF ( 0) | see (3.37) | into hypergeometric functions (z; 1; a) = 2F1(1; a; a + 1; z) ; jzj < 1 ^ z 6= 0 ; and we have approximated the derivatives f (k)(`) by nite-di erence expressions i a f (k)(`) ! k i=0 ` k X( 1 )i k f ` + ( k2 i) ` ; ` (4.17) (4.18) At this stage, the expression (4.16) is only a function of the latitude parameter the polar angle 0 in (2.4)) and of two parameters | the IR cuto 0 and the derivative discretization `, both small compared to a given 0 . We have tuned them in order to con dently extract four decimal digits. In gure 1a we compare the regularized one-loop e ective action obtained from the perturbation theory of the string sigma-model (4.16) to the gauge theory prediction from (1.2) h ( 1 ) ( 1 ) UV-reg( 0) loc i = 3 2 log tanh 0 (4.19) for di erent values of 0. Data points cover almost entirely22 the nite-angle region between the Zarembo Wilson loop ( 0 = 0; 0 = 2 ) and the circular Wilson loop ( 0 = 1; 0 = 0). The vanishing of the normalized e ective action in the large- 0 region is a trivial check of the normalization. As soon as the opposite limit 0 = 0 is approached, the di erence (4.16) bends up \following" the localization curve (4.19) but also signi cantly deviates from it, and the measured discrepancy is incompatible with our error estimation. Numerics is however accurate enough to quantify the gap between the two plots on a wide range. Figure 1b shows that, surprisingly, such gap perfectly overlaps a very simple function of 0 within the sought accuracy 1 2 Rem( 0) We notice at this point that the same simple result above can be obtained taking in (4.14) the limit of ! 0 before performing the sums. As one can check, in this limit UV and IR divergences cancel in the ratio.23 the special functions in the fermionic determinants disappear and, because in general summands drastically simplify, one can proceed analytically getting the same result calculated in terms of numerics. We remark however that such inversion of the order of sum and limit on the IR cuto cannot be a priori justi ed, as it would improperly relate the cuto with a 1= cuto (e.g. forcing ` to be smaller than 1= ). As emphasized above, in this limit the e ective actions for the latitude and circular case separately do not reproduce the expected UV divergences. Therefore, the fact that in 22When pushed to higher accuracy, numerics is computationally expensive in the vicinity of the two limiting cases ( 0 = 0; 0 = 2 ) and( 0 = 1; 0 = 0). 23This is also due to the volume part of the Euler number v( 0) being independent of 0 up to corrections, see (4.8). Comparison between in (4.16) (orange dots) and in (4.19) (blue line). We set 0 = 10 7, from supersymmetric localization for the ratio between latitude and circular Wilson loops in terms of the corresponding one-loop sigma-model (di erences of) e ective actions. this limit the summands in the di erence (4.14) show a special property of convergence | which we have not analyzed in details | and lead to the exact result is a priori highly not obvious, rendering the numerical analysis carried out in this section a rather necessary step. On a related note, the simplicity of the result (4.20) and the possibility of getting an analytical result for the maximal circle 0 = 0 suggest that the summation (4.4) could have been performed analytically also in the latitude case 0 6= 0. We have not further investigated this direction. 5 Conclusions In this paper we calculated the ratio between the AdS5 S5 superstring one-loop partition functions of two supersymmetric Wilson loops with the same topology. In so doing, we address the question whether such procedure | which should eliminate possible ambiguities related to the measure of the partition function, under the assumption that the latter only depends on worldsheet topology | leads to a long-sought agreement with the exact result known via localization at this order, formula (4.19). Our answer is that, in the standard setup we have considered for the evaluation of the one-loop determinants (Gelfand-Yaglom approach with Dirichlet boundary conditions at the boundaries, of which one ctitious24), the agreement is not found. A simple numerical t allows us to quantify exactly a \remainder function", formula (4.20).25 As already emphasized, the expectation that considering the ratio of string partition functions dual to Wilson loops with the same topology should cancel measure-related ambiguities is founded on the assumption that the partition function measure is actually not depending on the particular classical solution considered. Although motivated in light of literature examples similar in spirit (see Introduction), this remains an assumption, and it is not possible to exclude a priori a geometric interpretation for the observed discrepancy. One reasonable expectation is that the disagreement should be cured by a change of the world-sheet computational setup, tailored so to naturally lend itself to a regularization scheme equivalent to the one (implicitly) assumed by the localization calculation.26 One possibility is a choice of boundary conditions for the fermionic spectral problem27 di erent from the standard ones here adopted for the squared fermionic operator.28 Also, ideally one should evaluate determinants in a di eomorphism-preserving regularization scheme. In that it treats asymmetrically the worldsheet coordinates, the by now standard procedure of employing the Gel'fand-Yaglom technique for the e ective (after Fourier-transforming in ) one-dimensional case at hand does not fall by de nition in this class. In other words, the choice of using a -function- like regularization | the Gel'fand-Yaglom method | in and a cuto regularization in Fourier !-modes is a priori arbitrary. To bypass these issues it would be desirable to fully develop a higher-dimensional formalism on the lines of [50, 51]. A likewise fully two-dimensional method to deal with the spectral problems is the heat kernel approach, which has been employed at least for the circular Wilson loop case (where the relevant string worldsheet is the homogenous subspace AdS2) in [10, 11]. As there explained, the procedure bypasses the need of a large regulator and makes appear only in the AdS2 regularized volume, the latter being a constant multiplying the traced heat kernel and thus appearing as an overall factor in the e ective action. This is di erent from what happens with the Gel'fand-Yaglom method, where di erent modes carry a di erent -structure and one has to identify and subtract by hand the -divergence in the one-loop e ective action. However, little is known about heat kernel explicit expressions for the spectra of Laplace and Dirac operators in arbitrary two-dimensional manifolds, as it is the case as soon as the parameter 0 is turned on. The application of the heat kernel method for the latitude Wilson loop seems then feasible only in a perturbative approach, i.e. in the 24See also appendix D where a minimally di erent choice for the boundary conditions on the bosonic and fermionic modes with small Fourier mode is considered, and shown not to a ect the nal result. 25See also discussion below (4.20), where we notice that the same result is obtained analytically via the a priori not justi ed \order-of-limits" inversion. 26Morally, this resembles the quest for an \integrability-preserving" regularization scheme, di erent from the most natural one suggested by worldsheet eld theory considerations, in the worldsheet calculations of 27For the bosonic sector, we do not light-like cusps in N = 4 SYM [47] and ABJM theory [48]. nd a reasonable alternative to the Dirichlet boundary conditions. 28For example, instead of squaring one could consider the Dirac-like rst-order operator (3.29). Then, Dirichlet boundary conditions would lead to an overdetermined system for the arbitrary integration constants of the 2 2 matrix-valued, rst-order eigenvalue problem. The question of the non obvious alternative to consider is likely to be tied to a search of SUSY-preserving boundary conditions on the lines of [49]. small 0 regime when the worldsheet geometry is nearly AdS2.29 It is highly desirable to address these or further possibilities in future investigations. Acknowledgments We acknowledge useful discussions with Xinyi Chen-Lin, Amit Dekel, Sergey Frolov, Simone Giombi, Jaume Gomis, Thomas Klose, Shota Komatsu, Martin Kruczenski, Daniel Medina Rincon, Diego Trancanelli, Pedro Vieira, Leo Pando Zayas, and in particular with Nadav Drukker, Arkady Tseytlin, and Konstantin Zarembo. We also thank A. Tseytlin and the Referee of the published version for useful comments on the manuscript. The work of VF and EV is funded by DFG via the Emmy Noether Programme \Gauge Field from Strings". VF thanks the kind hospitality, during completion of this work, of the Yukawa Institute for Theoretical Physics in Kyoto, the Centro de Ciencias de Benasque \Pedro Pascual", the Institute of Physics in Yerevan and in Tbilisi. The research of VGMP was supported in part by the University of Iceland Research Fund. EV acknowledges support from the Research Training Group GK 1504 \Mass, Spectrum, Symmetry" and from the Seventh Framework Programme [FP7-People-2010-IRSES] under grant agreement n. 269217 (UNIFY), and would like to thank the Perimeter Institute for Theoretical Physics and NORDITA for hospitality during the completion of this work. All authors would like to thank the Galileo Galilei Institute for Theoretical Physics for hospitality during the completion of this work. A Notation and conventions We adopt the following conventions on indices, when not otherwise stated, M; N; : : : = 0; : : : ; 9 curved target-space indices A; B; : : : = 0; : : : ; 9 at target-space indices i; j; :: = 0; 1 a; b; :: = 0; 1 curved worldsheet indices at worldsheet indices Flat and curved 32 32 Dirac matrices are respectively denoted by A and M and satisfy the so (1; 9) algebra f A; Bg = 2 ABI32 f M ; N g = 2GMN I32; where AB = diag ( 1; +1; : : : ; +1) and GMN is the target-space metric (2.5). We use the explicit representation for the 10D gamma matrices 0 = i ( 3 1 = (I2 2 = (I2 3 = ( 1 4 = ( 2 2) I4 1) 3) 2) I4 I4 I4 2) I4 1 1 1 1 1 5 = I4 6 = I4 7 = I4 8 = I4 9 = I4 ( 3 ( 1 ( (I2 (I2 2 2) 2 2) 2) 1) 3) 2 2 2 2 29We are grateful to A. Tseytlin for a discussion on these points. (A.1) (A.2) (A.3) The symbol In stands for the n n identity matrix and 1 ; 2; 3 for the Pauli matrices. It is also useful to report the combination that appears in the expansion of the fermionic Lagrangian (3.26) The two 10D spinors of type IIB string theory have the same chirality HJEP02(16)5 034 = (I2 2) I4 1 : 11 I = I ; I; J = 1; 2 : (A.4) (A.5) (A.6) (A.7) accompanied by the chirality matrix minant Det! over the relation (3.1) holds In Lorentzian signature they are subject to the Majorana condition, but this cannot be consistently imposed after Wick-rotation of the AdS global time t. This constraint, which would halve the number of fermionic degrees of freedom, reappears as a factor 1=2 in the exponent of fermionic determinants (4.1). Throughout the paper we make a notational distinction between the algebraic determinant det and the functional determinant Det, involving the determinant on the matrix indices as well as on the space spanned by ( ; ). We also introduce the functional deterfor a given Fourier mode !, understanding that for any operator O DetO = Y Det!O: ! The boundary condition along the compact -direction speci es if the product is over integers or half-integers. The issue related to the regularization of the in nite product is addressed in the main text. The frequencies ! label the integer modes in the Fouriertransformed bosonic and fermionic operators. We change notation and use ` for the integer and s for the half-integer frequencies of the (bosonic and fermionic resp.) determinants entering the cuto -regularized in nite products (more details in section 4). Finally, a comment on the functions these matrix operators act on. They are column vectors of functions generically denoted by f (f1; f2; : : : ; fr)T . Computing functional determinants with the techniques presented in appendix B involves solving linear di erential equations, whose independent solutions f(i) f(i)1; f(i)2; : : : ; f(i)r T are labelled by Roman numerals i = I; II; : : :. B Methods for functional determinants The evaluation of the one-loop partition function requires the knowledge of several functional determinants of one-dimensional di erential operators | the operators in Fourier space at xed frequency Det! (see appendix A). This task can be simpli ed via the procedure of Gel'fand and Yaglom [29] (for a pedagogical review on the topic, see [52]). This algorithm has the advantage of computing ratios of determinants bypassing the computation of the full set of eigenvalues and is based on the solution of an auxiliary initial value problem.30 To illustrate how to proceed, let us consider the situation we typically encounter Det!O ; Det!O^ of freedom)31 in which the linear di erential operators O; O^ are either of rst order (for fermionic degrees d d O = P0( ) + P1( ) ; O^ = P0( ) + P^1( ) ; d d or of second order (in the case of bosonic excitations) O = P0( ) d 2 d 2 + P1( ) d d + P2( ) ; O^ = P0( ) d 2 d 2 + P^1( ) d d + P^2( ): (B.1) (B.2) (B.3) HJEP02(16)5 The coe cients above are complex matrices, continuous functions of on the nite interval I = [a; b]. In appendix B.1 we deal with a class of spectral problems not plagued by zero modes (vanishing eigenvalues) for chosen boundary conditions on the function space.32 We closely follow the technology developed by Forman [30, 31], who gave a prescription to work with even more general elliptic boundary value problems. We collected all the relevant formulas descending from his theorem for the bosonic sector in appendix B.2, and for the square of the 2D fermionic operators in appendix B.3. Let us also stress again that the Gel'fand-Yaglom method and its extensions evaluate ratios of determinants. Whenever we report the value of one single determinant here and in the main text, the equal sign has to be understood up to a factor that drops out in the normalized determinant. The reference operator can be any operator with the same principal symbol. The discrepancy can be in principle quanti ed for a vast class of operators with \separated" boundary conditions [62{64], i.e. where conditions at one boundary are not mixed with conditions at the other one. B.1 Di erential operators of the nth-order We consider the couple of n-order ordinary di erential operators in one variable O = P0( ) n 1 k=0 d n d n + X Pn k( ) d k d k ; O^ = P0( ) n 1 k=0 d n d n + X P^n k( ) d k d k (B.4) 30This algorithms has been used for several examples of one-loop computations which perfectly reproduce non-trivial predictions from \reciprocity constraints" [53] (see also [54] and [55]), and the general equivalence between Polyakov and Nambu-Goto 1-loop partition function around non-trivial solutions [56]. Further oneloop computations reproducing predictions from quantum integrability are in [57{60]. 31See next section for a comment on the coincidence of the coe cient P0( ) of the higher-derivative term. 32We mention that, for the plethora of physical situations where it is interesting to project zero modes out from the spectrum, the reader is referred to the results of [34] for self-adjoint operators of the Sturm-Liouville type as well as [32, 33, 61] and references therein. with coe cients being r r complex matrices. The main assumption is that the principal symbols of the two operators (proportional to the coe cient P0( ) of the highest-order derivative) must be equal and invertible (detP0( ) 6= 0) on the whole nite interval I = [a; b]. This ensures that the leading behaviour of the eigenvalues is comparable, thus the ratio is well-de ned despite the fact each determinant is formally the product of in nitelymany eigenvalues of increasing magnitude. We do not impose further conditions on the matrix coe cients, besides the requirement of being continuous functions on I. The operators act on the space of square-integrable r-component functions f (f1; f2; : : : ; fr)T 2 L2 (I), where for our purposes one de nes the Hilbert inner product ( stands for complex conjugation) The inclusion of the non-trivial measure factor, given by the volume element on the classical worldsheet h = ( ), guarantees that the worldsheet operators are self-adjoint when supplemented with appropriate boundary conditions.33 Indeed, to complete the characterisation of the set of functions, one speci es the nr nr constant matrices M; N implementing the linear boundary conditions at the extrema of I 0 B M BB B f (a) dd f (a) C . . . d n 1 f (a) 1 C A CC + N BB 0 B B f (b) dd f (b) C 1 . . . d n 1 f (b) 0 0 1 B 0 C A CC = BBB ... CCC : C The particular signi cance of the Gel'fand-Yaglom theorem and its extensions, specialized in [30, 31] to elliptic di erential operators, lies in the fact that it astonishingly cuts down the complexity of nding the spectrum of the operators of interests Of ( ) = f ( ) ; O^f^^( ) = ^f^^( ); (B.5) (B.6) (B.7) (B.8) and then nding a meromorphic extension of -function. All this is encoded into the elegant formula DDeett!!OO^ = exp nRab tr R( )P1( )P0 1( ) d exp nRab tr hR( )P^1( )P0 1( )i d o o O ; for the ratio (B.1), and where R is de ned below. This result agrees with the one obtained via function regularization for elliptic di erential operators. Notice that any constant rescaling of M; N in (B.6) leaves the ratio una ected. Moreover, if also the next-to-higherderivative coe cients coincide (P1( ) = P^1( )), the exponential factors cancel out. The 33The rescaling of the operators by ph operated in the main text removes the measure from this formula; see appendix A in [6]. . . . O (B.9) (B.10) (B.11) nr matrix YO( ) = BB 0 B B f(I)( ) dd f(I)( ) . . . . . . : : : . . . f(II)( ) dd f(II)( ) : : : f(nr)( ) dd f(nr)( ) C 1 C C C A @ ddnn 11 f(I)( ) ddnn 11 f(II)( ) : : : ddnn 11 f(nr)( ) accommodates all the independent homogeneous solutions of Of(i)( ) = 0 i = I; II; : : : ; 2r chosen such that YO (a) = Inr. It can be thought of as the fundamental matrix of the equivalent rst-order operator acting on nr-tuples of functions. Y ^( ) is similarly de ned with respect to O^. If we restrict to even-order di erential operators, then R( ) = 12 Inr and (B.8) simpli es: DDeett!!OO^ = exp n 12 Rab tr P1( )P0 1( ) d exp n 12 Rab tr hP^1( )P0 1( )i d o det [M + N YO (b)] det M + N Y ^ (b) O : ( i)n P0( ), has no intersection with the cone C For odd n one gets a slightly more complicated structure, constructed as follows. Let us assume that the (generalized) spectrum of the principal symbol of O; O^, i.e. the matrix fz 2 Cj 1 < argz < 2g for some choice of 1 ; 2. This is to say that O has principal angle between 1 and 2. It also follows that no eigenvalue falls in the opposite cone C f z 2 Cj 1 + < argz < 2 + gg when n is odd. Consequently, the nitely-many eigenvalues fall under two sets, depending on which C) they belong to. The matrix R( ) is then de ned34 as the projector onto the subspace spanned by the eigenvectors corresponding to all eigenvalues in one of these two subsets of the complex plane. We did not use this formula for odd n in this paper, but notice that this machinery could be potentially applied to the rst-order fermionic (3.29) operator. B.2 Applications We list the applications of the theorem (B.8) for the scalar-/matrix-valued operators in the main text. In the following we leave out formulas for hatted operators and solutions in order not to clutter formulas, understanding that they satisfy the same initial value problems. Second-order scalar-valued di erential operators O1, O2 ( 0), O3 ( 0), Dirichlet boundary conditions f1 ( 0) = f1 (R) = 0. ! M = N = ! f(II)1 (R) (B.12) 34Up to a factor n1 , see amendment in [31]. The normalization of the matrix (B.9) tells that the function f(II)1 ( ) solves the initial value problem f(0I0I)1( ) + P2( )f(II)1( ) = 0 f(II)1 ( 0) = 0 f(0II)1 ( 0) = 1: (B.13) 2 matrix-valued di erential operators O3 ( 0), Dirichlet boundary conditions f1 ( 0) = f2 ( 0) = f1 (R) = f2 (R) = 0. where f(III)1 ( 0) = f(III)2 ( 0) = f(0III)2 ( 0) = 0 f(0III)1 ( 0) = 1 f(0I0II)2( ) + P2( ) ff((IIIIII))12(( )) ! = 0 f(0I0II)1( ) ! 0 f(0I0V )1( ) ! f(0I0V )2( ) + P2( ) ff((IIVV ))21(( )) ! = 0 0 ! ! f(IV )1 ( 0) = f(IV )2 ( 0) = f(0IV )1 ( 0) = 0 f(0IV )2 ( 0) = 1 : 2 matrix-valued di erential operators OF Dirichlet boundary conditions f1 ( 0) = f2 ( 0) = f1 (R) = f2 (R) = 0. p12;p56;p89 ( 0) 2, BB 10 00 00 00 CACC YO ( ) = ff((II))12 (( )) ff((IIII))12 (( )) ! @ Det! P0 ( ) dd + P1 ( ) 2 Det! hP0 ( ) dd + P^1 ( )i2 = R R0 dsYO 1 (s)P0 1 (s)YO (s) R R0 dsY ^ 1 (s)P0 1 (s)YO^ (s) O with P0( ) f(0I)2( ) + P1( ) ff((II))12(( )) ! = 0 f(0I)1( ) ! 0 P0( ) f(0II)2( ) + P1( ) ff((IIII))12(( )) ! = 0 f(0II)1( ) ! 0 ! ! This is a corollary of (B.27). f(I)1 ( 0) = 1 f(I)2 ( 0) = 0 f(II)1 ( 0) = 0 f(II)2 ( 0) = 1: (B.14) HJEP02(16)5 (B.15) (B.16) (B.17) Square of rst-order di erential operators As a consequence of the Forman's construction, we can easily compute the ratio of determinants of the square of rst-order operators with reference only to the operators themselves. Consider the matrix operator of the form (B.2)35 O = P0 ( ) + P1 ( ) and denote by Y ( ) its fundamental matrix, which solves the equation (here 0 is the derivative with respect to ) P0 ( ) Y 0 ( ) + P1 ( ) YO ( ) = 0; O YO (a) = Ir: ds YO 1 (s) P0 1 (s) YO (s) P0 (a) Z (a) = 0 Z0 (a) = Ir: (B.23) encapsulates the solutions of Of = 0 and two more ones of O2f = 0. Suppose that the spectral problem of the squared operator is determined by the bound MO2 f (a) + NO2 f (b) = 0 : After some algebra, successive applications of (B.11), (B.22), (B.23) bring O Z b a in which ary condition then (B.25) gives (B.19) (B.20) HJEP02(16)5 (B.24) (B.25) (B.26) The matrix of fundamental solutions of the square of this operator 2 h O 2 = P02 ( ) d 2 + P0 ( ) P00 ( ) + fP0 ( ) ; P1 ( ) can be constructed via the method of reduction of order as YO2 ( ) = O O Y ( ) Y 0 ( ) Z ( ) Y 0 (a) Z0 ( ) Y 0 (a) O O i d g d Z ( ) ! Z0 ( ) + P12 ( ) + P0 ( ) P10 ( ) (B.21) YO2 (a) = I2r (B.22) For Dirichlet boundary conditions at both endpoints = a; b used in the present paper Det!O 2 Dirichlet = pdetP0 (a) detP0 (b) det dsYO 1 ( ) P0 1 ( ) YO ( ) : (B.27) 35We omit to report similar formulas for the hatted operator O^ = P0 ( ) dd + P^1 ( ). Det!O 2 = s detP0 (a) detP0 (b) det [MO2 + NO2 YO2 (b)] detYO (b) : f1 (a) = f2 (a) = f1 (b) = f2 (b) = 0 M 2 = O Ir 0 0 0 ! 0 0 Ir 0 ! N 2 = O Z b a Fermionic determinant DetOF ( 0): details In this appendix we collect some details on the analysis of the fermionic determinant DetOF ( 0) in (3.27). C.1 Derivation of (3.28) To begin our analysis of DetOF ( 0) in (3.27), let us observe that OF 34 p12;p56;p89 43 = OF p12;p56;p89 : This fact can be used to show that Det(OF1;p56;p89 )Det(OF 1;p56;p89 ) = Det[(OF1;p56;p89 )2]: Indeed, let us denote the \positive" eigenvalues of OF 1;p56;p89 with f n; Re( n) > 0g and n; Re( n) > 0g. Because of the relation (C.1), the spectrum of OF the \negative" ones with f 1;p56;p89 is given by The function for the rst operator is while for the second operator we nd Summing the two contributions we obtain f ng [ f ng : 1;p56;p89 (s) = X( n) s + e i s X( n) s; 1;p56;p89 (s) = e i s X( n) s + X( n) s: 1;p56;p89 (s) + 1;p56;p89 (s) = (1 + e i s) X( n) s + X( n) s (1 + e i s) (s) (C.6) The spectrum of (OF1;p56;p89 )2 is given instead by and the corresponding function is Therefore " n Z(s) = X( 2n) s + X( 2n) s = (2s): n n f ng [ f ng 2 n n n log Det(OF1;p56;p89 )Det(OF 1;p56;p89 ) = d 1;p56;p89 (0) d 1;p56;p89 (0) ds (0) = i 2 0(0) ; ds log(Det[(OFp12;p56;p89 )2]) = 2 0(0) ; (C.1) (C.2) (C.3) (C.4) (C.5) (C.7) (C.8) (C.9) HJEP02(16)5 so that it holds log(Det(OF1;p56;p89 )Det(OF 1;p56;p89 )) = log(Det[(OF1;p56;p89 )2]) (0) : (C.10) We can namely express the combination Det(OF1;p56;p89 )Det(OF minant and the -function in 0 ( (0)) of the squared operator (OF1;p56;p89 )2. 1;p56;p89 ) in terms of deter We now use Corollary 2.4 of [63] j j + j j 2n Z(0) = r n + 1 ; where 2n is the order of the di erential operator, ; are parameters that only depend on the boundary conditions and r is the matrix dimension of the operators. In our case (n = 1, j j = j j = 1, r = 2) it is Z(0) = 2 and thus via (C.8) (0) = 2, to conclude that Det(OF1;p56;p89 )Det(OF 1;p56;p89 ) = Det[(OF1;p56;p89 )2] ; and thus (C.2) is proven. The determinant of the fermionic operator can then be written as36 Det[OF ( 0)] = Y p12;p56;p89= 1;1 Det[OFp12;p56;p89 ( 0)] = Det[(OF1;1;1)2]2Det[(OF1; 1;1)2]2; (C.11) (C.12) (C.13) (C.14) (C.15) depend on the value of p89. the spectrum of OF 1;1;1, then f OF 1; 1;1 4 ( ; ) where we have used the property (C.10) and that the operators OF p12;p56;p89 in (3.26) do not We can also easily argue that Det[(OF1; 1;1)2] = Det[(OF1;1;1)2]. Let f n; ( ; )g be n; 4 ( ; ))g is the spectrum of OF 1; 1;1. Indeed it is i sinh2 ( ) 0 + sin2 ( ) 034 4 ( ; ) = ia56( ) 4 i sinh2 ( ) 0 sin2 ( ) 034 ; ): Thus the eigenvalues of the squared operator (OF1; 1;1)2 are the same of those of the squared operator (OF1;1;1)2 and consequently the two determinants coincide. In restricting ourselves to one-dimensional spectral problems - and thus working in terms of Fourier modes ! and referring to (3.1) - from the statement (C.14) one obtains Det![(OF1; 1;1)2] = Det ![(OF1;1;1)2] : from which (3.28) follows. 36The Corollary 2.4 [63] can be easily check to hold both for 1;1(0) and for 1;1(0). + 1 2( ) 4 1 2( ) ( ) As from (B.25), the key-ingredient in the explicit computation of Det!(OF1;1;1)2 (3.29) is Y ( ), the 2 2 matrix of the fundamental solutions obeying the boundary conditions Y ( 0) = I2, as in (B.22). It is not di cult to explicitly check that the structure of this matrix can be parametrized as follows Y ( ) = e!( 0)S1( ) + e !( 0)S2( ); where the entries of the matrices S1( ) and S2( ) depends on ! only through rational functions. We can infer some important properties of these matrices from the fact that Y 2( ) tr(Y ( ))Y ( ) + det(Y ( ))I2 = 0 : In particular one can easily check that detY ( ) does not depend on !. Then the secular +(Si)1;1[(Mi)1;1(Si)1;2 + (Mi)1;2(Si)2;2] = (Mi)1;2det(Si) = 0: (Si)21(S0i)2;2 (Si)2;2(S0i)2;1 = (Si)2;1[(Mi)2;1(Si)1;2 + (Mi)2;2(Si)2;2] (C.22) +(Si)2;2[(Mi)2;1(Si)1;1 + (Mi)2;2(Si)2;1] tr(S1)S1) + e 2!( 0)(S22 tr(S2)S2) +fS1; S2g tr(S2)S1 + det(Y )I2 = 0 tr(S2)S2 = 0 ; fS1; S2g tr(S1)S2 tr(S2)S1 + det(Y )I2 = 0 : The matrices S1 and S2 must also satisfy the di erential equations P0@ S1 + (P1 + !P0)S1 = 0 ; !P0)S2 = 0 ; where P0; P1 appear in the Dirac-like operator OF P0 is invertible we can symbolically write this as Si0 + MiSi = 0. This implies a set of interesting properties: (Si)1;2(S0i)1;1 (Si)1;1(S0i)1;2 = (Si)1;2[(Mi)1;1(Si)1;1 + (Mi)1;2(Si)2;1] (C.21) and therefore S 2 1 tr(S1)S1 = 0 ; S 2 2 and Namely the ratios 37We omit the -dependence in the matrices. (Si)11 (Si)21 and (C.16) (C.17) (C.18) (C.19) (C.20) (C.23) are we can set Y 1 P 1 0 Y = with Y 1( ) = (tr(Y )I2 Y ( )) = Next we construct the bilinear Y 1P0 1Y . We nd 1 1 det(Y ) det(Y ) 1 det(Y ) +e !( ai Si = (Si)11 = (Si)12 (Si)21 ; aipi( ) pi( ) aiqi( ) qi( ) where ai depends only on 0; 0 and !. We can parameterize the matrices Si as follows Equation (C.17) also completely determines Y 1( ). In fact HJEP02(16)5 S1)e!( 0) + (tr(S2)I2 S2)e !( S2)e !( 0) + A0 + B0 (C.24) (C.25) (C.26) 0)S1 + (C.27) (C.28) (C.29) (C.30) 1 ! a1 (C.31) 1 ! a2 (C.32) A2 = A0 = 1 1 det(Y ) det(Y ) B2 = B0 = 1 1 det(Y ) det(Y ) S2)P0 1S2 ; S2)P0 1S1 : Because of the relations (C.19) we nd that the matrices A2 and B2 are nihilpotent A22 = B22 = 0 ; and it holds A2A0 = A0B2 = B2B0 = B0A2 = B0A0 = A0B0 = 0: The structure of the matrices Ai and Bi is very simple. They are in fact constant matrices times a function of . This can be easily shown by means of the parametrization (C.25). In fact A2( ) = (q1( )2 p1( )2) det(Y ) ( ) B2( ) = (q2( )2 p2( )2) det(Y ) ( ) (q1( )q2( ) p1( )p2( )) det(Y ) ( ) B0( ) = (q1( )q2( ) p1( )p2( )) det(Y ) ( ) a2 a1a2 a1 a1a2 where we used that P0 ( ) 1. Our next goal is to compute Z R 0 det ds Y 1(s) P0 1(s) Y (s) (C.33) (C.34) 2 d 0 0 0 det ds Y 1(s) P0 1(s) Y (s) det(Y ( )) ( ) a2)2 Z R (p1( )p2( ) e2! Z R d 0 0 q1( )q2( )) 2 0 det(Y ( )) ( ) : in (B.27). Since Y 1(s) P0 1(s) Y (s) is traceless in our case, we can also write the expression d 0tr Y 1 ( ) P0 1( ) Y ( )Y 1( 0) P0 1( 0) Y ( 0) : We can now use the representation (C.27) and the properties (C.28) to simplify the ex+ (C.35) This formula can be very e ciently used to simplify the fermionic determinant in its largeR expansion. The second line is always negligible, the rst one consists of two separate integrals: for positive !, the dominant part in the large-R limit will be the contribution of the rst (inde nite) integral evaluated at the upper endpoint times the contribution of the second (inde nite) integral evaluated at the lower endpoint, whereas for negative ! the roles of rst and second integrals are swapped. D Boundary conditions for small Fourier modes In this appendix we comment on a di erent choice for the boundary conditions on the bosonic and fermionic modes with small Fourier mode | choice followed in [7] for the circular Wilson loop case | and show that it leaves una ected the main results of this paper, the e ective actions for both the circle (4.12) and the normalized latitude (4.20). In [7] the one-dimensional spectral problems in the radial coordinate are subject to Dirichlet boundary conditions at both the boundaries = 0 and ( ctitious) = R, except for the modes labeled by m = 0,38 for which Neumann boundary conditions are imposed at = R.39 It is easy to modify our analysis of the bosonic sector in section 3.1 | where we kept Dirichlet boundary conditions for all modes | and evaluate the e ect of this other choice. The relevant Fourier frequency corresponds to ` = 0 which, from the discussion at the beginning of section 4, corresponds to the mode ! = 0 for O1 and O2( 0), ! = 1 for O3+( 0) and ! = 1 for O3 ( 0). We use the subscript N to denote the new determinants with Neumann boundary conditions in = R (D.1) 38In the labelling of (5.35) after the supersymmetry-preserving regularization. 39See formulas (5.46){(5.52) therein. instead of the Dirichlet ones f1( 0) = f1(R) = 0 used in the main text. We read o the since the operator O1 is the same for the circle and the latitude: [Det!=0O1]N = coth 0: For the other operators, the new boundary conditions change (B.12) as 1 0 0 0 M = N = 0 0 0 1 Det! h dd22 + P2( ) i and accordingly modify (3.14), (3.16) and (3.17) as [Det!=0O2( 0)]N = tanh ( 0 + ) ; [Det!=1O3+( 0)]N = [Det!= 1O3 ( 0)]N = p e 0+2 0 1 + e2 0+4 0 : The limit 0 ! 1 [Det!=0O2( 0 = 0)]N = [Det!=1O3+( 0 = 0)]N = [Det!= 1O3 ( 0 = 0)]N = 1 modi es the analogous results (3.43){(3.45) for the circular Wilson loop. A comparison with the formulas in the main text reveals that, at the level of the Gel'fand-Yaglom determinants, the only change following from this di erent choice of boundary conditions is an overall rescaling of the determinants by R. The same phenomenon occurs in the fermionic sector, where backtracking the special Fourier mode to our !-labeling is less transparent, but becomes more visible in the circular Wilson loop. The frequency m = 0 of formula (5.35) [7] is the determinant of the operator 2 coth : it is evident that (D.6) governs f1( ) for ! = 1 while f2( ) for ! = 0. (D.2) HJEP02(16)5 (D.6) (D.7) (D.8) (D.9) To nd it in the present paper, we begin with the Gel'fand-Yaglom di erential equation and from its component equations h 1;1;1( 0 = 0) OF i2 f1( ) f2( ) 1 2 2 2 1 2 1 1 + ! 2 2 1 2 0 0 coth coth f1( ) = 0 f2( ) = 0 : Extending this identi cation to arbitrary 0, this argument tells that the only modication in appendix B.3 is the Neumann boundary condition on the rst component for f1 ( 0) = f2 ( 0) = @ f1 (R) = f2 (R) = 0 ; (D.10) which translates into replacing (B.26) with and on the second component for ! = 0 which is implemented by 0 1 0 0 0 1 MO2 = BBB 00 10 00 00 CCC (D.11) f1 ( 0) = f2 ( 0) = f1 (R) = @ f2 (R) = 0 ; 0 1 0 0 0 1 MO2 = BBB 00 10 00 00 CCC 0 0 0 0 0 1 NO2 = BBB 01 00 00 00 CC C : This also means that we cannot use the compact form (B.27) (still valid for ! 6= 0; 1) and we have resort to the general expression (B.25). After a lengthy computation, the new values of the determinants (e2 0 +1) e6 0+4 0 + (e2 0 +1)e4 0+2 0 + e2 0 ( 5e2 0 +3e4 0 +3) + (e2 0 1)2 + 2 0 + N N + 2 0 + p R eR e 20 (tanh 0 + 1) sinh 0 cosh( 0 + 0) (e2 0 1)2(e2( 0+ 0) + 1) p R eR e 20 (tanh 0 + 1) sinh 0 cosh( 0 + 0) (e2 0 agree with (3.40){(3.41), again up to an overall rescaling of their values by a factor of The analysis in section 4 goes through in a similar fashion, provided that the lower ularized e ective action (4.4) does not change when the limit modes `B=0( 0); sF= 12 ( 0); 2 F s= 1 ( 0) take into account these new determinants. The reg! 0 is taken. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. (D.12) (D.13) 1 1 (D.14) (D.15) p 2e4 0 log e2( 0+ 0) + 1 + 2e2 0 log e2( 0+ 0) + 1 + expansion for Wilson loops and surfaces in the large-N limit, Phys. Rev. D 59 (1999) 105023 (1999) 125006 [hep-th/9904191] [INSPIRE]. [3] V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE]. [4] J.K. Erickson, G.W. Semeno and K. Zarembo, Wilson loops in N = 4 supersymmetric Yang-Mills theory, Nucl. Phys. B 582 (2000) 155 [hep-th/0003055] [INSPIRE]. [5] N. Drukker and D.J. Gross, An exact prediction of N = 4 SUSYM theory for string theory, J. Math. Phys. 42 (2001) 2896 [hep-th/0010274] [INSPIRE]. [6] N. Drukker, D.J. Gross and A.A. Tseytlin, Green-Schwarz string in AdS5 partition function, JHEP 04 (2000) 021 [hep-th/0001204] [INSPIRE]. S5: semiclassical [7] M. Kruczenski and A. Tirziu, Matching the circular Wilson loop with dual open string solution at 1-loop in strong coupling, JHEP 05 (2008) 064 [arXiv:0803.0315] [INSPIRE]. [8] M. Sakaguchi and K. Yoshida, A semiclassical string description of Wilson loop with local operators, Nucl. Phys. B 798 (2008) 72 [arXiv:0709.4187] [INSPIRE]. [9] C. Kristjansen and Y. Makeenko, More about One-Loop E ective Action of Open Superstring in AdS5 S5, JHEP 09 (2012) 053 [arXiv:1206.5660] [INSPIRE]. [10] E.I. Buchbinder and A.A. Tseytlin, 1=N correction in the D3-brane description of a circular Wilson loop at strong coupling, Phys. Rev. D 89 (2014) 126008 [arXiv:1404.4952] S5 superstring, arXiv:1510.06894 [INSPIRE]. [12] A. Faraggi, J.T. Liu, L.A. Pando Zayas and G. Zhang, One-loop structure of higher rank Wilson loops in AdS/CFT, Phys. Lett. B 740 (2015) 218 [arXiv:1409.3187] [INSPIRE]. [13] D. Correa, J. Henn, J. Maldacena and A. Sever, An exact formula for the radiation of a moving quark in N = 4 super Yang-Mills, JHEP 06 (2012) 048 [arXiv:1202.4455] [14] N. Drukker and V. Forini, Generalized quark-antiquark potential at weak and strong coupling, JHEP 06 (2011) 131 [arXiv:1105.5144] [INSPIRE]. JHEP 05 (2008) 017 [arXiv:0711.3226] [INSPIRE]. [15] N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, Supersymmetric Wilson loops on S3, [16] A. Kapustin, B. Willett and I. Yaakov, Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE]. [17] N. Drukker and B. Fiol, On the integrability of Wilson loops in AdS5 S5: Some periodic ansatze, JHEP 01 (2006) 056 [hep-th/0506058] [INSPIRE]. [18] N. Drukker, 1=4 BPS circular loops, unstable world-sheet instantons and the matrix model, JHEP 09 (2006) 004 [hep-th/0605151] [INSPIRE]. [19] S. Forste, D. Ghoshal and S. Theisen, Stringy corrections to the Wilson loop in N = 4 super Yang-Mills theory, JHEP 08 (1999) 013 [hep-th/9903042] [INSPIRE]. [20] S. Forste, D. Ghoshal and S. Theisen, Wilson loop via AdS/CFT duality, hep-th/0003068 [21] R. Roiban, A. Tirziu and A.A. Tseytlin, Two-loop world-sheet corrections in AdS5 superstring, JHEP 07 (2007) 056 [arXiv:0704.3638] [INSPIRE]. HJEP02(16)5 Phys. Rev. D 76 (2007) 107703 [arXiv:0704.2237] [INSPIRE]. [25] N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, Wilson loops: from four-dimensional SYM to two-dimensional YM, Phys. Rev. D 77 (2008) 047901 [arXiv:0707.2699] [INSPIRE]. [26] V. Pestun, Localization of the four-dimensional N = 4 SYM to a two-sphere and 1=8 BPS Wilson loops, JHEP 12 (2012) 067 [arXiv:0906.0638] [INSPIRE]. [27] D. Young, BPS Wilson Loops on S2 at Higher Loops, JHEP 05 (2008) 077 [arXiv:0804.4098] [INSPIRE]. loops, JHEP 06 (2008) 083 [arXiv:0804.3973] [INSPIRE]. physics, J. Math. Phys. 1 (1960) 48. [28] A. Bassetto, L. Griguolo, F. Pucci and D. Seminara, Supersymmetric Wilson loops at two [29] I. Gelfand and A. Yaglom, Integration in functional spaces and it applications in quantum [30] R. Forman, Functional determinants and geometry, Invent. Math. 88 (1987) 447. [31] R. Forman, Functional determinants and geometry (Erratum), Invent. Math. 108 (1992) 453. [32] A.J. McKane and M.B. Tarlie, Regularization of functional determinants using boundary perturbations, J. Phys. A 28 (1995) 6931 [cond-mat/9509126] [INSPIRE]. [33] K. Kirsten and A.J. McKane, Functional determinants by contour integration methods, Annals Phys. 308 (2003) 502 [math-ph/0305010] [INSPIRE]. [34] K. Kirsten and A.J. McKane, Functional determinants for general Sturm-Liouville problems, J. Phys. A 37 (2004) 4649 [math-ph/0403050] [INSPIRE]. [35] K. Kirsten and P. Loya, Computation of determinants using contour integrals, Am. J. Phys. 76 (2008) 60 [arXiv:0707.3755] [INSPIRE]. [36] S.A. Frolov, I.Y. Park and A.A. Tseytlin, On one-loop correction to energy of spinning strings in S5, Phys. Rev. D 71 (2005) 026006 [hep-th/0408187] [INSPIRE]. [37] A. Dekel and T. Klose, Correlation Function of Circular Wilson Loops at Strong Coupling, JHEP 11 (2013) 117 [arXiv:1309.3203] [INSPIRE]. [38] A. Miwa, Broken zero modes of a string world sheet and a correlation function between a 1=4 BPS Wilson loop and a 1=2 BPS local operator, Phys. Rev. D 91 (2015) 106003 [arXiv:1502.04299] [INSPIRE]. [39] N. Drukker and B. Fiol, All-genus calculation of Wilson loops using D-branes, JHEP 02 (2005) 010 [hep-th/0501109] [INSPIRE]. [40] V. Forini, V.G.M. Puletti, L. Griguolo, D. Seminara and E. Vescovi, Remarks on the geometrical properties of semiclassically quantized strings, J. Phys. A 48 (2015) 475401 [arXiv:1507.01883] [INSPIRE]. [41] A.R. Kavalov, I.K. Kostov and A.G. Sedrakian, Dirac and Weyl Fermion Dynamics on Two-dimensional Surface, Phys. Lett. B 175 (1986) 331 [INSPIRE]. [42] A.G. Sedrakian and R. Stora, Dirac and Weyl Fermions Coupled to Two-dimensional Surfaces: Determinants, Phys. Lett. B 188 (1987) 442 [INSPIRE]. [43] F. Langouche and H. Leutwyler, Two-dimensional fermion determinants as Wess-Zumino actions, Phys. Lett. B 195 (1987) 56 [INSPIRE]. [44] F. Langouche and H. Leutwyler, Weyl fermions on strings embedded in three-dimensions, Z. [45] F. Langouche and H. Leutwyler, Anomalies generated by extrinsic curvature, Z. Phys. C 36 [46] C. Ferreira and J.L. Lopez, Asymptotic expansions of the Hurwitz-Lerch zeta function, [47] S. Giombi, R. Ricci, R. Roiban, A.A. Tseytlin and C. Vergu, Quantum AdS5 superstring in the AdS light-cone gauge, JHEP 03 (2010) 003 [arXiv:0912.5105] [INSPIRE]. [48] T. McLoughlin, R. Roiban and A.A. Tseytlin, Quantum spinning strings in AdS4 CP 3: Testing the Bethe Ansatz proposal, JHEP 11 (2008) 069 [arXiv:0809.4038] [INSPIRE]. [49] N. Sakai and Y. Tanii, Supersymmetry in Two-dimensional Anti-de Sitter Space, Nucl. Phys. [50] G.V. Dunne and K. Kirsten, Functional determinants for radial operators, J. Phys. A 39 [51] K. Kirsten, Functional determinants in higher dimensions using contour integrals, [52] G.V. Dunne, Functional determinants in quantum eld theory, J. Phys. A 41 (2008) 304006 [53] M. Beccaria, V. Forini, A. Tirziu and A.A. Tseytlin, Structure of large spin expansion of anomalous dimensions at strong coupling, Nucl. Phys. B 812 (2009) 144 [arXiv:0809.5234] Phys. C 36 (1987) 473 [INSPIRE]. (1987) 479 [INSPIRE]. J. Math. Anal. Appl. 298 (2004) 210. B 258 (1985) 661 [INSPIRE]. (2006) 11915 [hep-th/0607066] [INSPIRE]. arXiv:1005.2595 [INSPIRE]. [arXiv:1409.8674] [INSPIRE]. 07 (2013) 088 [arXiv:1304.1798] [INSPIRE]. [54] M. Beccaria, G.V. Dunne, V. Forini, M. Pawellek and A.A. Tseytlin, Exact computation of one-loop correction to energy of spinning folded string in AdS5 165402 [arXiv:1001.4018] [INSPIRE]. S5, J. Phys. A 43 (2010) [55] V. Forini, V.G.M. Puletti and O. Ohlsson Sax, The generalized cusp in AdS4 CP 3 and more one-loop results from semiclassical strings, J. Phys. A 46 (2013) 115402 [arXiv:1204.3302] [INSPIRE]. [56] V. Forini, V.G.M. Puletti, M. Pawellek and E. Vescovi, One-loop spectroscopy of semiclassically quantized strings: bosonic sector, J. Phys. A 48 (2015) 085401 [57] L. Bianchi, V. Forini and B. Hoare, Two-dimensional S-matrices from unitarity cuts, JHEP Proceedings, 1st Karl Schwarzschild Meeting on Gravitational Physics (KSM 2013), Frankfurt am Main Germany (2013), Springer Proc. Phys. 170 (2016) 169 [arXiv:1401.0448]. matrix in AdSn AdSn HJEP02(16)5 nite di erence operators in vector bundles over S1, Commun. Math. Phys. 138 (1991) 1. problems, Commun. Math. Phys. 193 (1998) 643 [INSPIRE]. [1] D.E. Berenstein , R. Corrado , W. Fischler and J.M. Maldacena , The operator product [22] K. Zarembo , Supersymmetric Wilson loops, Nucl. Phys. B 643 ( 2002 ) 157 [ hep -th/0205160] [23] V. Forini , Quark-antiquark potential in AdS at one loop , JHEP 11 ( 2010 ) 079 [59] O.T. Engelund , R.W. McKeown and R. Roiban , Generalized unitarity and the worldsheet S M 10 2n , JHEP 08 ( 2013 ) 023 [arXiv: 1304 .4281] [INSPIRE]. [60] R. Roiban , P. Sundin , A. Tseytlin and L. Wul , The one-loop worldsheet S-matrix for the T 10 2n superstring , JHEP 08 ( 2014 ) 160 [arXiv: 1407 .7883] [INSPIRE]. [61] R. Rajaraman , Solitons and Instantons, North Holland, Amsterdam The Netherlands ( 1982 ).


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP02%282016%29105.pdf

V. Forini, V. Giangreco M. Puletti, L. Griguolo, D. Seminara, E. Vescovi. Precision calculation of 1/4-BPS Wilson loops in AdS5×S5, Journal of High Energy Physics, 2016, 105, DOI: 10.1007/JHEP02(2016)105