Jhep02(2018)120

Home , Conformal anomaly

JHEP02(2018)120 1 Springer a,b, January 5, 2018 February 8, 2018 : February 20, 2018 : : Received Accepted Published , is singular and we show that an IR and Konstantin Zarembo 5 Published for SISSA by https://doi.org/10.1007/JHEP02(2018)120 S a,c × 5 [email protected] , Daniel Medina-Rincon . a,b 3 = 4 super-Yang-Mills theory at strong coupling. Conformal gauge 1712.07730 N The Authors. c AdS-CFT Correspondence, Wilson, ’t Hooft and Polyakov loops

We study string quantum corrections to the ratio of latitude and circular , [email protected] Also at ITEP, Moscow, Russia. 1 E-mail: [email protected] Roslagstullsbacken 23, SE-106 91 Stockholm,Department Sweden of Physics, UniversityP.O. of Box Oslo, 1048 Blindern, N-0316Department of Oslo, Physics Norway andSE-751 Astronomy, 08 Uppsala Uppsala, University, Sweden Nordita, Stockholm University and KTH Royal Institute of Technology, b c a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: anomaly associated with the divergence indiscrepancy the conformal with factor the removes previously exact reported field-theorycancellation result. and recalculate We also fluctuation carefullyall determinants check the by conformal directly fluctuation anomaly modes. evaluting phaseshifts for Abstract: Wilson loops in for the corresponding minimal surface in AdS Alessandra Cagnazzo, String corrections to circularanomalies Wilson loop and JHEP02(2018)120 12 13 12 12 ] or, at a more 8 , 7 , the result that can be 5 S × 5 ]. The strong-coupling extrapo- 9 [ 4 S = 4 super-Yang-Mills (SYM) theory that N – 1 – 2 ]. Some Wilson loops in the SYM theory can be 7 6 – 11 15 5 4 8 12 10 4 2 17 1 2 3 e e e e 3 K K K D 9 1 16 ]. Wilson loops are important observables in gauge theories and are unique 3 – 1 4.4.1 Operator 4.4.2 Operator 4.4.3 Operator 4.4.4 Operator The simplest example of this type is the circular Wilson loop whose exact expectation 4.4 The phaseshift computation 4.5 Collecting the pieces together 4.1 Preliminaries 4.2 Summation over4.3 frequencies Phaseshifts and Jost functions 2.1 Latitude Wilson2.2 loops Classical solution 2.3 One-loop2.4 string corrections Regularization and anomalies QFT calculations and holography. value can be obtained byrigorous resumming level, by diagrams localization of of perturbationlation the of theory path the [ integral cirlcle on agrees precisely with the area law in AdS share a common contour inproposal space-time, of but [ differ in theirprobes coupling of to the scalars, AdS/CFT following correspondence the in since the they dual couple gravitational directly background toactually [ the computed string exactly, at worldsheet anySubsequent coupling extrapolation strength, to without strong making coupling any establishes approximations. a direct link between conventional 1 Introduction We will study the ratio of two Wilson loops in 5 Conclusions A Conformal anomalies 4 Determinants and phaseshifts 3 Conformal anomaly cancellation Contents 1 Introduction 2 Circular Wilson loop and latitude JHEP02(2018)120 , 5 5 5 S S ∈ × (2.1) (2.2) 5 2 S ]. The result ], quantum 17 2 is a unit six- , 1 n is a unit circle and , ] 4 C I is the ’t Hooft coupling of Φ I ). The expectation value of , should not matter as long as n 0 N | , 2 ˙ x 0 0 . In string theory, the Wilson | g θ i θ C = ] differ essentially in the choice of + cos 3 cos λ µ a priori λ τ, A for the embedding coordinates in AdS µ √ ˙ x = 0 and a supersymmetric Wilson loop } sin = 4 supermultiplet and ) 1 0 0 π I ] brings the result to the agreement with θ θ 2 ]. A different quantization prescription for 3 2. Here N dτ 0 , = 4 SYM is defined as [ ] and in [ θ ] upon localization of the path integral [ 15 C sin (0 π/ 2 – 2 – , Z N 16 τ, 1 i 2 ∈ cos = τ 0 λ | cos θ ) √ 0 τ θ ( ]. We pay special attention to regularization issues and 1 µ ] at ] for a review). Quite surprisingly, even the next order in ] only applies to infinitesimally small deviations from the ] avoids these complications, because the measure factors ]: ) = x 2 3 tr P exp 0 , { 2 18 10 , θ 15 1 ( 1 = = (sin W n C ) = exact. n ]: ; N C 15 [ ( 5 . W ], indicating that we do not understand in detail how strings in AdS 5 S S 14 – 11 ) for are the six scalar fields from the τ ( I n We concentrate of a particular family of Wilson loops, for which We reconsider string quantum corrections to the latitude Wilson loops, working in The difficulty lies in the defition of the measure in the string path integral and in the The latitude belongs to a more general class of supersymmetric Wilson loops which live on 1 is a latitude of and interpolates between thewith simple trivial circle expectation at value [ and reduce to the effectivequoted 2d in Yang-Mills the theory text [ is large- n the latitude is known exactly [ where Φ dimensional vector that mayloop change expectation along value the mapsdetermined to contour by the the contour disc partitionand function by with the boundary conditions 2.1 Latitude WilsonThe loops Wilson loop expectation value in the ensuing anomalies and will alsois carefully an check important that consistency the condition conformal anomaly in cancels, string which theory. 2 Circular Wilson loop and latitude circle. The quantization prescritionsthe conformal in frame [ on thethe string conformal worldsheet, anomaly which cancels. the same conformal frame as [ ratio of similar Wilson loopssimply [ cancel. For thestring corrections ratio can of be the computeddisagrees exacty. latitude Surprisingly, with and the result the the of circle, field-theorythe the considered string prediction string calculation in [ fluctuations [ field around theory, the but the latitude method [ of [ the strong-coupling expansion has notmuch been effort reproduced [ from string theoryand thus far, other despite holographic backgrounds should be quantized in thedelicate Wilson-loop issues sector. with reparameterization invariance on the string worldsheet. Taking the generalized in many ways (see [ JHEP02(2018)120 ], 3 – θ, ϕ 1 (2.5) (2.6) (2.7) (2.8) (2.3) (2.4) ] and for the 20 , 19 . 2 / . 1 σ , − λ 1 2 = 4 SYM [ − O λ N + = tanh . O 0 ) θ 0 σ . 1 + ] 0 and in the angular coordinates + 2 2) limits do not commute with one as σ , z 5 . 0 22 ln cos σ 0 0 θ 1 σ π/ dσ ( θ 3 2 σ τ 2 cos + → (1) term in the strong-coupling expansion λ sin 2 ) + cosh ln tanh τ, √ of AdS O 0 0 cosh = cos dτ θ e θ 3 2 } = = 0 3 4 + – 3 – 2 2 σ , z − = cos µ σ λ 2 x − 0 = Ω { 1 is related to θ loop tanh 2 (1 ]. 3 0 − , ϕ sinh 1 θ 2 ) λ ds 21 0 Γ cos = √ σ π 2 and = + , x Ω r ) σ ]. σ theory [ 0 ∞ τ θ ∗ , in terms of the ’t Hooft coupling, one gets the correct exponent (0) 25 ( ) = π – 0 cos The field-theory prediction for the Wilson loop ratio is 2 = 2 W cosh θ W ( 23 2 λ/ , N = = tanh( ln W √ 1 11 θ ≡ x Γ cos , the minimal surface for the latitutde is [ 5 changes from 0 to S = 4 SYM. At strong coupling, σ ). ) is ⊂ The field-theory prediction for the next, In string theory, the exponent in the Wilson loop vev is determined by the area of the 2 N 2.3 2.4 Another way to get rid of the measure factors is to consider infinitely stretched Wilson loops and 2 S In string theory, thisstring is worldsheet expected [ to come from the one-loop quantumconcenrate on fluctuations extensive quantities. of In the in that case string an theory agreement between wasquark field obtained theory self-energy and for in quantum the corrections the quark-anti-quark potential in the in ( of ( with the scale factor Substituting the solution intotension the is given string by action, and taking into account that the string where The induced worldsheet metric is given by In the standard Poincarécoordinates of Our goal will betheory. to reproduce this result from2.2 the explicit one-loop Classical calculation solution in string minimal surface with the givenof boundary the conditions, string while fluctuations thewe and consider prefactor of the is the ratio a measure of contribution is in the expected the circle to to string cancel. the path latitude integral. in Following which [ the complicated measure factor Notice that the stronganother. coupling and BPS ( the JHEP02(2018)120 ). 2.5 (2.9) (2.17) (2.14) (2.15) (2.16) (2.13) (2.10) (2.12) ]: 1 . )) (2.11) . The Dirac operators 0 − , 5 3 σ 2 2 τ φ K + † 1 2 ) are described by the fol- / ] to the classical solution ( φ τ σ 1 )) 2 ∂ 0 26 2.5 Ω , σ , describes three string modes in 1) det ) 11 1 + 0 . − arise as a result of mixing between e 2 − 3+ σ K σ ) φ 0 dτ dσ K D + † 1 3 2 σ 2 / e ) K σ Z , 1 0 ( + 1 , 1 2 , σ det 5 2 = σ ): tanh (2 e Ω det K S 2 + + e 1) (1 + 3 tanh (2 D 2 dτ dσ φ 2 − φ 2.7 D σ 1 2 3 2 † 1 Ω K ( − 1 2 σ 2 (1 Z 2 Ω Ω cosh 2 ) / – 4 – = 0 3 1 2 h φ 2 = (tanh (2 det σ = ∓ i √ i K ∓ 3 sinh cosh det 2 2 + D τ σ τ φ 2 1 σ + − | σ d i∂ K 2 2 2 2 1 σ σ σ 3 2 ^ ∂ ∂ ∂ φ / Z ]: h 1 3 − 3 − − − – 1 = τ 2 2 2 1 τ τ τ det i Ω sinh σ ∂ ∂ ∂ 2 ]: 1 φ − − − i∂ + (tanh (2 + | ) = 1 = = = = 0 φ σ describes three modes on 1 2 h ( e e 2 3 K K e D Z e e K K are the standard Pauli matrices. The operator i τ , the operator The one-loop partition function, that determines the Wilson loop expectation value, The operators above are related to the ones that appear in the string action by a 5 originate from the kinetic terms for the eight fermions remaining after kappa-symmetry These can be obtained by specializing the general formalism of [ 3 e is given by the ratio of determinants of the physical, untilded operators [ and the fluctuation operators aretilded Hermitian operators with are respect Hermitian to with this respect scalar to product, the while usual the flat measure: for fermions. The fluctuationthe modes invariant of measure the of string the are induced naturally metric normalized ( with respect to conformal transformation [ for bosons, and AdS the two remaining modes —D one from the sphere, anothergauge-fixing from in AdS the Green-Schwarz action. where The string oscillation modeslowing around fluctuation operators the [ classical solution ( 2.3 One-loop string corrections JHEP02(2018)120 ) ) 2.7 2.7 (2.18) (2.21) (2.20) (2.19) ), which is 2.11 ), is given by the )–( ) differs from the 2.4 2.9 loop ]: − , 1 2 in ( , . e Γ ∞ 1 0 e K σ and cancel in the ratio of the → ∞ det . 2 0 σ σ cyl ! . 1 + tanh ) corresponds to the following chain ∞ ) ) , e K 0 2 σ e ln K is a regular in the induced metric ( ∞ σ ∂ . The spectral problem for a fluctuation 2.13 ( ( ), but does not literally coincide with it. 1 2 0 det 1 − Z Z ] that the conformal anomaly is unlikely to det ∞ σ 2 ] for technical details, and just remark that 2 − τ (

∂ 0 = Z . σ : 11 − , = ln σ – 5 – K 2 = anom loop ∞ − e e K 1 K K ln tanh Γ 3 2 det det = . The point = ∞ loop ) by an additional “remainder” term [ K − 1 = e 2.8 Γ σ det is the asymptotic operator obtained by taking ∞ e K A caveat here is that the conformal transformation from the metric of the disc ( An obvious possible cause for the discrepancy, the one that first comes to mind, is the The tilded operators are technically easier to deal with, and in much of the previous where just the free Klein-Gordon operator: 2.4 Regularization andThe anomalies change of theof conformal transformations frame on of the the determinant form of ( regularization. Even ifleave the a cutoff finite dependence residue. eventually Weanomaly first cancels based give on out, a elementary simple regularization thermodynamics,analysis but may and not of then very the rigorous proceed fluctuation derivation with determinants. of a such more an systematic IR conformal anomaly would rather signal an internal inconsistency ofto the the string flat calculation. metricis of the actually semi-infinite singular cylinder at but changes not the in topology the of flatoperator the metric, worldsheet on as and a illustrated in cylinder figure differs from that on a disk in an essential way and requires an IR conformal anomaly. However it was arguedaccount in for [ the discrepancy. Weanomaly refer cancellation to is [ very important in string theory. A non-zero contribution from the operators can be seamlesslyminants replaced computed by under the this tildedfield-theory assumption ones. prediction (which ( However, we the denote log-ratio by of deter- This is the object we concentrate upon in thework rest the of conformal the factors paper. frame have is been a basic simply principle dropped. of string Independence theory. on It the is conformal thus natural to assume that the untilded The string path integral containsto some control. additional measure Fortunately, these factorslatitude factors that to are do the rather not circle. difficult dependlog-ratio on The of the one-loop partition free functions: energy, if normalized as in ( The Wilson loop is actually proportional to JHEP02(2018)120 . R 1a : (2.24) (2.22) (2.23) R = σ . R . This corresponds . 2 . R − 4 ], and is implicit in a 0 e σ = 2 , σ 0 1 , removed by regularization, σ 2 2 (b) − σ > R 1 + tanh 1 + e . At the same time, the cutoff ln . R ]. Even though the IR cutoff cancels 1 2 τ π 4 ∂ 27 − 2 ' iτ inv 2 − Ω R ). The region require an IR regularization. The IR cutoff σ ∂ ∞ ≡ dσ 1 ∞ – 6 – ) = iτ e ∞ 0 K R σ ) at some large but finite σ = Z ∞ π e ∞ K π e D 8 = 2 and det (or h e e K K (1 + tanh √ on the minimal surface in the target space. s σ s 2 ) is the conformal anomaly, the second one is well-defined on a ln d (a) 1 2 , we need a diffeomorphism-invariant regularization. 2.20 0 σ>R = σ Z R = s Schematic representation of the induced metric on the minimal surface (a), and of the As an invariant regularization parameter we can take the area of the segment removed The first ratio in ( The standard way to regularize the problem is to impose the Dirichlet boundary con- The coordinate-dependent cutoff is related to the invariant one as does not have anyat invariant different meaning values by of itself. To faithfully comparefrom partition the functions minimal surface when Dirichlet boundary conditions are imposed at ditions on the wavefunction of to removing a smallThis is segment not of such the anLaplacian innocent minimal on procedure, a surface disk as and shown canin on as be a the seem disk a final by with red answer, a comparing small circle determinants intermediate hole of steps in [ the do figure depend on cylinder, while separately det is manifestly necessary formore the direct Gelfand-Yaglom method phase-shift calculation used that in will [ be carried out in section maps to a small circle of area or the free Dirac operator, in case of fermions: Figure 1. flat coordinates on thethe cylinder symmetry (b). point of The the conformal minimal transformation surface ( between the two is singular at JHEP02(2018)120 ∞ e K (3.1) (2.28) (2.25) (2.26) (2.27) . 0 σ ) directly ap- ) and since A.15 2.20 -dependent remnant 0 σ . ) and ( 0 that should be kept fixed. σ R A.10 . 2 , a a 0 e K K V. σ has to dependent on 2 1 + tanh T R det det ) with the appropriate multiplicities. R. ln π 12 ln = ). a 1 2 . We thus have 2.17 ln tanh rather than ) F f R . The results ( T 3 2 F 2.8 1) N inv ) = = ), but leaves a finite, − A − = R + ( ∞ through the last factor in ( V – 7 – = b ( ∞ a 2.18 N e Γ R R ∞ X e (2 Z 1 2 + , it is 0 ) and ) + − ln 0 σ = π σ loop = ( (2 − / 1 R F ), upon bringing them to the standard Klein-Gordon/Dirac e anom Γ − Γ = 1 = 2.14 T ∞ , )–( f e Γ N 2.9 ), this gives 2.19 = 8 = b ): N is a coordinate-dependent quantity with no invariant meaning, in comparing are just the free Klein-Gordon and Dirac opertors, the asymptotic contribution R 2.24 ∞ e D The anomaly, being a local effect of UV divergences, can be computed by the standard This is the main result of the paper. To validate this result we need to check that the The partition function depends on more complicated because of the need in the intermediateDeWitt-Seeley UV expansion. regularization. Foranomaly completeness adpated to we our give caseply a in to the brief the appendix operators derivation ( of the conformal where the summation runs overThe all operators operators in the in numerator ( were and allowed denominator differ to by factorize the the conformal factor, determinants, and the if sum one would trivially vanish. The story is 3 Conformal anomaly cancellation The anomaly contribution to the free energy is which agrees with the localization prediction ( conformal anomaly cancels, which wethe do partition function in on the the nextanalysis cylinder of section. and the Later will fluctuation we derive operators. will the also above reanalyze result by a direct spectral Combined with ( The IR divergence cancelsdue in to the ( ratio ( dimensions, given by In our case the partition functions atDiffeomorphism-invariant different regularization implies that and to the partition function is given by the free energy of a gas of free particles in 1 + 1 Because JHEP02(2018)120 (3.2) (3.3) , not only 0 σ . ) 0 σ + 2 − a σ 2 , − (2 ) 2 0 2 − σ v ) at each value of + cosh , σ 4 + 4 . − ) ) (2 2.17 φ. 0 0 ) 2 σ ) 0 σ σ 2 + 0 ) 2 σ a 0 σ = 2 1 + + ] with the help of the Gelfand-Yaglom σ + − cosh φ σ + σ 2 , sinh ( σ + 2 + 1 − 2 σ ( 1 v 2 2 8 2 σ − σ ( (2 σ ) 2 2 2 × 2 σ 0 cosh(2 cosh 2 2 1 σ + 4 0 1 – 8 – σ 2 12 σ ln Ω cosh cosh Ω cosh − 2 + sinh 3 sinh Ω sinh − − − ∓ 1 σ E − ( 8 + 2 sinh cosh ) ======+ 0 × 1 2 φ σ = 1 6 3 v E E a 3+ cosh 2 + E E Ω 2 σ + ( 2 2 E ln Ω = 2 cosh σ ∂ + 3 = 1 − E = 3 φ ), with the identifications 2 σ ∂ A.2 ), ( A.1 The anomaly thus cancels in the partition function ( First we notice that the boundary terms in the anomaly trivially cancel between bosons The fluctuation determinants were evaluatedmethod. in [ Here we recalculateeach operator them and by then a integratingup over more the the direct phase general approach, space scheme of evaluatingin for string phaseshifts turn. fluctuations. the for phaseshift First we computation set and then apply it to each operator and the two terms in the anomaly completely compensatein one the another. ratio, as actually expected. 4 Determinants and phaseshifts On the other hand, This enters the anomaly with a prefactor from which it immediately follows that and fermions, just byanomaly matching also the cancels number it ofing is form: degrees convenient to of bring freedom. the scale To factor see of that the the metric bulk to the follow- form ( JHEP02(2018)120 di- (4.5) (4.6) (4.7) (4.8) (4.9) (4.1) (4.2) (4.3) (4.4) τ ): 4.3 as we have seen : R R = σ ) and is thus valid as long 4.5 . . and the cutoff . ) 0 . δ ) σ ) . + πn, 2 τ hence takes the form , σ ω, p . Ψ = ΛΨ ∂ τ ∂p ( 2 ' . pσ , with integer frequency, or half-integer has to be kept here, since it only cancels ∂ . − p ( ) ) ∂δ n iω R V + → ∞ − + ) = 2 ) = 0 + C sin( iω, σ ω, p ω R R R 2 ∞ – 9 – ( σ − by , δ ∂ ( τ Ψ(0) = 0 Ψ( ' τ − →∞ ∂ 1 + π V Λ = ∂ ( σ = R ) + V n σ 2 σ e ) = K p = 0 is ∂ p ( Ψ( σ − ρ replaces in the limit of τ ∂n/∂p = ρ if regularization is to preserve general covariance. 0 σ is much larger than the range of the potential in the equation. Schrödinger The To define the determinant of an operator with a continuous spectrum we need to The boundary condition at R Often omitted extensive piece proportionalin the to ratio of thedepends two on partition functions at different which follows from the asymptoticas form of the wavefunction ( density of states introduce an IR cutoff by imposing another boundary condition at The spectrum then becomes discrete due to momentum quantization condition: The eigenvalue can be read off the asymptotic form of the Schrödingerequation ( After the boundary conditionSince the is potential imposed vanishes the atbehavior: infinity, wavefunction the is wavefunction asymptotically fixed has up an to oscillating normalization. rection. The spectrum thuseach Fourier decomposes mode: into a sequence of one-dimensional problems for The Fourier expansion in depending on whether the boundary conditions are periodic or anti-periodic in the The operators at hand havescheme the works general with form minor (we modifications): start with bosons, for fermions the same The asymptotics at infinity are that of the free d’Alambert operator: 4.1 Preliminaries JHEP02(2018)120 ], 28 (4.13) (4.11) (4.12) (4.10) . ) , after which the 2 . Rp ) ) 2 , p ) + ) Rp + Rp 2 ω, p ( ω ) + 2 δ p ( ) + ( ln( − 2 δ p ω, p ) ( p + δ 2 , ) + ( 2 ) ω, p p ∂p ω ( 2 ( p + ∂δ δ p ip, p dp ( + 2 + 2 ∞ ( 0 ω δ πp . Denoting R Z ip ≡ – 10 – π dp ) πω coth π p dp ∞ ( = 0 dp ∞ cot Z ω δ 0 ∞ Z ω 0 πi X Z dω 2 ω X − − C Z = = = is obtained by multiplying all the eigenvalues: − e K 2 for periodic/anti-periodic boundary conditions. e K e Contours of integration in the complex frequency plane. K / = e K + 1 ln det ln det Z ∈ picks up residues at ln det ω Figure 2. ω or Z : ∈ 2 ω Assuming that the phaseshift does not grow too fast at large frequencies (in the simplest Our strategy will be to directly evaluate phaseshifts for all the operators, sum over fre- The determinant of we get for the determinant: The poles of the cotangent recover the sum overcases the the Matsubara phaseshift frequencies. doescan not be depend closed on in theintegral over frequency the at upper all), and the lower contour half-planes, of as integration shown in figure The standard trick isfigure to replace summation by integration along the contour shown in quencies and integrate over spacialwe momenta. sum Before over proceeding the withand Matsubara explicit make frequency calculations a using few standard technical tools remarks that of streamline Statistical4.2 calculation Mechanics of [ the phaseshifts. Summation over frequencies with JHEP02(2018)120 ) it (4.17) (4.18) (4.19) (4.16) (4.14) (4.15) 4.4 are half- ω tanh: → . ) , are equal, but some δ Rp − − δ tan, coth ipσ . − ) + 2 e ipσ p . and ( ) i − → − C σ 2 − + e ( resulting in different density of δ δ p − ¯ Y δ ' ) + →∞ . + p σ (0) ( p ) . ipσ + Y (0) (0) σ δ e iδ ( p p ( and Λ = 0. 2 − p ¯ i Y Y = 0. Comparing its behaviour at infinity ¯ ) C Y 2 ip πp σ ln σ ( = e = p i 2 Y – 11 – = , ipσ tanh (0) (0) ω p p (0) − ) = ¯ ipσ p Y Y p spectral asymmetry ¯ e dp Y ( δ ) at ∞ (0) e , but if the potential is complex, which is the case for the 0 p ∗ ) = ' p →∞ Z 4.3 Y σ Y σ ( − − ) p = σ Ψ = ( p ipσ p ¯ Y F Y e K (0) e p ¯ Y ln det for example, then the two Jost functions are not related in any simple way. 3 e K ), we find: 4.5 In the self-adjoint case the Jost functions are complex conjugate and their ratio is a The Jost functions form a complete set of solutions to the Schrödingerequation. The Normally, the particle and anti-particle phaseshifts If the boundary conditions are anti-periodic and the Matsubara frequencies We will use thislatitude formula by to explicitly calculating evaluate the theapplied Jost to phaseshifts functions fermions of in with each the minor case. fluctuation modifications The related operators same to for scheme their can the two-component be nature. pure phase. Theadjoint, phaseshift the is phaseshift may real have in an imaginary this part. case. In general, If the Schrödingeroperator is not self- which expressed the phaseshift through the Jost data: solution that satisfy theJost correct functions: boundary conditions is a linear combination ofThis the linear two combination indeedwith vanishes ( at For a self-adjoint Schrödingeroperator with a realconjugate to potential one the another: Jost functionsoperators are complex Instead of solving the equationis Schrödinger with sometimes the easier correct to boundaryunit-normalized find plane conditions the waves ( Jost at infinity: functions, which are the solutions that asymptote to operators that we encounterstates for have particles a and anti-particles. 4.3 Phaseshifts and Jost functions solve the Schrödingerequation ( integer, the summation formulas differ by substitutions cot This equation expresses the determinant entirely through the on-shell data. It suffices to JHEP02(2018)120 , ω (4.24) (4.25) (4.26) (4.27) (4.21) (4.22) (4.20) (4.23) , . , , σ p 0 ) 0 σ ψ 2 σ p p + 1 + = ip + coth tanh σ p ip + 1 ψ ip ipσ . 1) − 2 χ + tanh ( − p. ), we obtain 2 + arctan ), we have . ) p 1 p 0 ip χ σ ) = e = ) 2 4.19 4.19 0 σ ipσ p 2 + ( σ − arctan χ p = σ ¯ + Y arctan − 1 ) σ χ = e 2 − 0 π p σ 2 ] ¯ = Y = σ + 29 2 sign refers to particle/anti-particle modes. 1 ) (tanh(2 σ 2 2 tanh(2 – 12 – ( ip ip + 1 − ip 2 − sinh 2 − + ip ip 1 0 0 , + . From equation ( . Using equation ( , σ σ ln cosh ∗ ∗ σ 2 = σ ) ) ) . The is 0 i 2 ∂ p p p x 1 − σ Y Y − ) + 1 2 σ = 3+ − . We do not display this more general function here, because we only tanh tanh 0 coth + ∂ p e 1 σ K = = ( = ( 1 σ δ ip − ) to compute the determinant. − ip ip p p p + ( − ¯ ¯ iω and − ip Y Y σ δ 1 + 1 ω ip ipσ tanh ( ]. Using this general solution an analytic expression for the off-shell phaseshift → − and and 3 2 1 29 − ln τ ∗ ∗ e e e K K K i ∂ 2 ip p p ) = e ¯ ¯ Y Y σ = ipσ ( 2 + (3 tanh(2 p δ = = 2 Y σ p p ∂ = e Y Y can be found by the substitution [ p − Y 4 ) can be found for any The potential in the Schrödingerequation is of the solvable Rosen-Morse type. The The Rosen-Morse potential is solvable in hypergepmetric functions for any values of the frequency 4 ω, p ( not necessarily on-shell [ δ need the on-shell phaseshifts where we have set solution 4.4.3 Operator The problem Schrödinger for satisfying The corresponding differential equation is given by In this case, the Jost functions are 4.4.2 Operator The solutions to this equation are given by the Jost functions satisfying 4.4.1 Operator The differential equation for this operator is given by 4.4 The phaseshift computation JHEP02(2018)120 α (4.35) (4.33) (4.29) (4.31) (4.32) (4.34) (4.28) (4.30) ), we get , α . pχ 4.33 . The index , 2 p α ∓ ) e 0 D = 2 σ α iτ + χ arctan σ . 2 − τ ip , ) 0 , 0 . , = 0 ) σ − ) σ − x ˜ = 0 . χ 3 x ( 2 + ( δ χ χ 2 p + tanh(2 σ χ ip ) ( )˜ 0 α 0 2 ip σ + 1 σ d with dx 1 − + 1 + tanh d + dx 1 − σ 3+ σ δ 2 Ωcosh ) ip x (2 i 0 (2 . 1 2 2 ip σ 1 2 ∗ ∓ − − 1 ) = + x ∓ τ + x p = arctan ( σ σ x cosh Y 2 ˜ cosh χ ∗ σ – 13 – − , the eigenvalue problem for , σ 1 (0) − ) α = + + (0) 0 d e + D p dx σ + ipσ − ¯ p 0 ipσ p Y d Ωsinh ) , 2 ¯ dx σ ¯ Y ∗ 2 cosh(2 Y ∓ + x ) − + are not complex conjugate to one another, because the σ x p σ 1 − (0) (0) p 0 + 2 ) = 1 + + ¯ )] σ Y − p p (1 ) = e 0 x , ) = e Y Y − ipσ ( σ σ x ∗ (1 ). Taking into account the last equation in ( σ σ ( − χ ( p p − ln + x d e and p ( dx + ψ 2 cosh(2 σ p i ψ + 2 p d ipσ ¯ Y dx δ Y p = = ) = is the same up to exchanging is introduced here in order to distinguish the operator label from the e p ) = e ) = D ∗ ( tanh(2 − p ) − σ σ − 3 ( ( p Y − 3 ( e + + K δ p p [1 − ¯ Y Y δ 2 iα ) + p ): + ( σ ∂ 3+ 3 δ 4.19 iτ The Jost functions that takes values particle/anti-particle index. Thethe spectral form: problem for the resulting Dirac operator takes The answer for 4.4.4 Operator It is convenient to consider, instead of to quantify itconjugation: because particle andfrom anti-particle ( Jost functions are related by complex potential is complex, and consequentlyis the also phaseshifts a will spectral have an asymmetry imaginary between part. particle There and anti-particle phaseshifts, but it is easy These equations have simple solutions: From here we find the Jost functions: or which leads to which leads to accompanied by the following ansatz for the wavefunction: JHEP02(2018)120 , (4.36) (4.37) . 1 2 σ , + 2 ) 1 σ coth . αip − 0 1 σ αip , , 4 − / ) ) ) 1 4 σ σ , . 0 / ( ( ))+sinh(2 0 , , σ ) σ 0 σ 0 O O σ e + cosh σ !! !! 4 σ + !! !! + + / 0 1 1 0 + sinh 5 σ I I II II σ 0 2 ¯ c c

¯ c c σ 2 !! !! ) − − = e cosh(

0 i i α, α, 1 1 σ σ − − δ δ α − − +2 ¯ v α, α, + cosh(2 sinh(2( p,α 2 + +

δ δ / ¯ σ Y − 1 sinh ) − − 1 2 ! ! α + + 0 Ω )) α, α, v 1 0 0 1 4 + σ δ δ σ Ω ! ! / 1 2 , 0 4 − +

cosh( I I II II / σ + + ¯ c c + 0 αip σ ¯ 5 c c 1 2 σ + + 0 σ p,α − ! ! α, α, (2 σ

− e − Y i i δ δ 1 1 4 αip – 14 – e 2 cosh(2 / + + α 2 sinh

σ 3

v σ/ − α, α, − σ/ e αip δ δ p , tanh + + e )) 4 − = ¯ 0 ) ) / 0

e α, α, αipσ αipσ σ 0 0 7 cosh − σ 4 δ δ α 4 0 ∓ σ σ − / / 1 2 χ e e 7 +

σ 2 1 3 2 3 2 + + αipσ αipσ − σ − Ω 4 α α = = σ σ 2 ∓ σ σ − − 4 / ¯ v v e e / 0 (2 (2 cosh cosh 0 σ p,α p,α 4 4 σ αip = = e = = αip / / ¯ αip Y Y e 2 1 2 1 2 1 1 2 σ/ p,α p,α p,α p,α − − σ/ e ¯ ¯ Y Y Y Y →∞ →∞ e lim lim 0 4 cosh cosh + 0 0 σ σ / σ 4 4 αip αip 0 (2+cosh(2( 4 / / → → σ / lim lim 5 1 σ σ 1 4 1 e 2 2 4 1 1 2 iα + / 3 2 3 2 1 2 1 2 3 2 = 0, the Yost functions behave as − 2 p cosh sign comes from the two choices of closing the integration contour described − + + − σ 1 iα αip σ iα ∓ 3 . αip αip αip αip = − 2 e α v = = = = Close to The most general solution of the Dirac equation is a superposition of the Jost functions I I II II ¯ c c ¯ c c From the above, it is easy to see that the superposition where Asymptotically, the two solutions behave as where in ﬁgure where the JHEP02(2018)120 (4.43) (4.41) (4.42) (4.38) (4.39) (4.40) 2 p p 2 = 0. At ) 0 . σ , σ p 3 1 2 arctan − πp − 2 2 arctan 2 πp 0 2 σ − e arctan coth p 0 . 1 2 σ : + (1 + tanh ! 0 tanh − 2 σ p . p ipσ p = 1 + ) with ipσ # − ) ∓ e πp + tanh 4.8 2 e πp − 1 2 − 0 v , ¯ v σ arctan 2 − coth , + 3 arctan 1 2

tanh ) − 0 πp c , − 0 ( σ ) − + arctan σ α, R ψ c , δ

+ tanh ( πp 1 2 . p χ 1 2 + 0 p 4 tanh πp ψ − + 1 1 2 ! dR c 2 dσ + ) = 1 + tanh πp 4 (coth 1 + tanh – 15 – 2 + 2 ipσ R 1 2 + p ipσ ( e ∓ c − 4 tanh # χ " Rp ) through the on-shell phaseshifts with the help 2 e c 2 0 e ln − τ − 2 + σ ln + − v + 8 arctan 2 ¯ dp p v 2.17 πp arctan − ! = 1 = = π ∞ 2 0 2 1 Z

3 π ) ) πp 2 2 + 0 πp + + + c c σ α, + tanh 3 coth p p 2 δ π 2 + + 3 2 2 1 coth p 2 2 ' tanh = ) for fermions. Since the Dirac equation is of the first order it p p " − cosh α 4.7 χ 1) ( dp dp p dp p α α ¯ v v 6 arctan + 1) ( − ∞ ) = 0, and therefore satisfies the right boundary conditions at ), and collecting all the pieces together we get for the log of the partition 0 0 + arctan − Z σ πp πp ( 2 2 → e e 4.14 Z ( ( Arg σ ) = 2 α ln 0 ∞ ∞ 0 0 Z Z σ 0 ( d Z ) and ( = dσ , the correct solution behaves as ln α To define the fermion phaseshift we need to understand what replaces the auxiliary δ 4.13 → ∞ The following inditities reduce the remaining integral to elementary functions: The easiest way to compute the integral is by differentiation in Expressing the determinantsof in ( ( function: we get the momentum quantization condition in the form ( 4.5 Collecting the pieces together is impossible to setprojection both of spinor the componentschoice wavefunction of leads can to the vanish. equivalent wavefunction results) to Choosing as zero. the chirality Only condition a (any chiral other σ boundary condition ( vanishes for JHEP02(2018)120 (4.44) (4.45) (4.46) . ) ]. But a really 0 . σ 30 0 ( R dR dσ − through the invariant + ) R ∞ 0 ( ). σ R , 0 3 2.8 θ + tanh ), ( + 1 − ln cos 0 2.4 2 σ 3 2 + 1 0 = 1 σ tanh 0 σ ln tanh 1 2 – 16 – − 0 ln tanh 0 σ σ 3 2 2 ] in agreement with localization predictions. We also 1 = 2 , 1 2 cosh ln tanh loop − 3 2 1 ) = Γ 0 = σ ), we find that the last three terms cancel and we are left with ) ) gives: ( 0 Z 0 ∞ σ ( ( σ 2.24 ln Z Z 0 ln d dσ ≡ loop − 1 Γ One can use other parameters to build ratios of Wilson loops which are easier to Intergation over A. Tseytlin andwas E. supported Vescovi by for the ERC interestingsupported advanced discussions. grant by No the 341222. The SwedishResults The Research work in work Council of of Gauge (VR) K.Z. D.M.-R. and wasand grant String and additionally by 2013-4329, Theories” RFBR K.Z. by from grant the 15-01-99504. the grant Knut “Exact and Alice Wallenberg foundation, defined. Acknowledgments We would like to thank J. Aguilera-Damia, A. Dekel, D. Fioravanti, Yu. Makeenko, compute in string theory, forintersting instance problem an is overall to coupling carry to outWilson scalars a loop. complete as calculation in of A [ quantumprescription Wilson corrections for to loop a compute is single a its well-defined expectation operator value in in field string theory, theory and should a be holographic unambiguously checked that the conformal anomalyfore cancels taking in the each ratio. individual This expectationbehind is value, this an even important calculation be- consistency (for checkin of example, the the that ratio, underlying ghosts assumptions or and thatof the longitudinal the measure modes factors problem). mutually are cancel constant and do not depend on the parameters 5 Conclusions The IR anomaly, relatedstring to corrections the computed singular in nature [ of the conformal gauge, brings quantum cutoff according to ( in perfect agreement with the localization prediction ( The first two terms arise fromoperators the determinants and normalized agree by the with freetwo the Klein-Gordon/Dirac terms calculation is based the on IR the anomaly. Gelfand-Yaglom Re-expressing method. the coordinate The cutoff last and we get: JHEP02(2018)120 ) 2.12 (A.7) (A.4) (A.5) (A.6) (A.1) (A.2) (A.3) , and 1 ), ( τ = 2.14 σ γ ]. Here we give , 32 2 , τ . 31 − K t . = − K t τ e , − γ . φ 2 αφ Tr − , Tr e , , 1 e K ) 1 t ) − − s . − |K E s is a one-component U(1) gauge field. v ) e φ µ ( ) is defined through the Mellin trans- + + dt t φ |K k ) + regular dt t A α φ µ t a a ( ∞ ( ( 3 Tr 1 ∞ 0 2 D f K γ 0 − a Z µ Z 2 k ∂ ∂t ) + t =0 D matrices. The bosonic fluctuation operators = 0, because the integrand is badly behaved ) res s t t that can be computed algebraically. We thus s s µ = 2 − ) does not depend on the scale factor at all. s is introduced for convenience, to interpolate 1 n =0 ( s 1 ]. ∞ Γ( µ D α k X Γ( K ( × = 2 µ 0 α A – 17 – 32 αφ = 1) operators. The dependence of the determi- d n K = 2 = → ds K iγ d s t ds 0 α K − ) 0 t and ) that admits a finite Laurent expansion at zero, the t are → ln det t 2 − αφ ( → s ( 3 lim s E e f ) = e E f s d , Tr e φ α dα − φ ( = 2 lim dt t = Tr K ∂ and is a textbook example of the anomaly [ ∂α K ) = e 5 ∞ µ K α α 0 ( A Z D ln det . The paramater ) on ln det 3 α τ , and d ( ) can be all brought to this form. The fermionic operator ( dα µ D = 0) and untilded ( iA 2 = α 2.11 / + ν µ γ )–( ) and ∂ µ behavior of the heat kernel is controlled by the DeWitt-Seeley expansion: α γ t ( 2.9 = are local functionals of µν K µ k ), ( iε a D The zeta-function regularized determinant of = 0. Indeed, for any function Here we assume that the connection is Abelian, and that 2.13 ≡ − t 5 3 where find that The small- would indicate that theBut determinant the Mellin of transform generatesat a pole at residue of the Mellin transform coincides with the residue of the function itself: we find that Since the gamma-function has a pole at zero, the right-hand-side seems to vanish, which Taking into account that γ between tilded ( nants of a concise derivation, that closely follows [ form of its heat kernel: in ( has the standard Dirac form: if we choose the basis of 2d gamma-matrices to be Consider a second-order differential operator where A Conformal anomalies JHEP02(2018)120 (A.8) (A.9) (A.12) (A.13) (A.14) (A.15) (A.10) (A.11) , . ) i φ but it is more φ n . µ , 6 ∂ . ∂ 3 µ φ 3 φ ∂ µ − γ 2 µ ∂ φ v ∂ µ n µ ) + ∂ − av ∂ φ 2 n φ∂ 2 µ ( 2 n a ) is . + 2 − φ∂ ) − 2 ds ( . φ φ E 3 µν v A.1 µ E n + F I . tr ∂ − φγ ) tr α∂ φ 3 π µν φ ν 2 2 µ n ε φ ∂ a − − ∂ 12 + |D 1 2 µ + µν + φ φ φ 2 ε + ( n φ µ φ a φ∂ 2 µ µ 2 + a iα φ∂ − ∂ φ∂ 1 E ( ν 12 φ∂ µ µ 2 γ − µ ∂ ∂ tr v = 2 ) ) arrive at ds ∂ µ 2 φ 6 2 σ φ n α 2 φ µ – 18 – D I 3 d − ivγ A.7 αn + π σ φ µ + 2 − 1 2 Z ) is αa∂ µ d 12 ∇ φ µ σ ∂ π 1 2 + µ µ ln det µ 2 D A.2 − d Z a ) and its second DeWitt-Seeley coefficient can be read φ∂ µ φ∂ d = = π dα 1 µ Z ∂ − ∇ 2 6 n ( µ ∂ A.1 π 1 − ∇ 3 4 (0) (1) α µν 2 2 h αφ ε σ = − 2 2 D D σ d + 2 (1) (0) d ) = det det Z K K ) = e |K π Z ln 1 α 2 φ ( π ( det det 1 1 2 2 2 4 = ) we express the anomaly as a local functional of the fields: a D ln ) = A.7 (1) (0) 2 K K is the inward normal derivative at the boundary. The bulk and boundary terms in |D φ n ): det det ( ∂ 2 a ln To compute the anomaly for fermions we first square the Dirac operator and then by Written that way, the anomaly does not have any boundary terms, The second DeWitt-Seeley coefficient of the operator ( A.9 Here we assume that the metric is flat and the boudary is straight. These simplifying assumptions are 6 Notice that the boundary anomalybosons. has the same magnitude but different sign compared to sufficient for our analysis. Otherwise curvature also contributes to the anomaly. We thus find for the fermion anomaly: This operator hasoff the ( form ( where the same chain of argument that led to ( The square of the Dirac operator ( where the anomaly actually arise in thethe computation last of expression. the The Seeley coefficient moreby exactly parts. compact as two-dimensional written form in is obtained upon integration natural to represent it in a different form: Integrating ( JHEP02(2018)120 ]. , , , , ]. D 60 ]. ]. 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