Strong coupling expansion of circular Wilson loops and string theories in AdS 5 × S 5 and AdS 4 × CP 3
aa r X i v : . [ h e p - t h ] S e p Imperial-TP-AT-2020-05
Strong coupling expansion of circular Wilson loopsand string theories in AdS × S and AdS × C P a, and Arkady A. Tseytlin b, a Department of Physics, Princeton University, Princeton, NJ 08544, U.S.A. b Blackett Laboratory, Imperial College, London SW7 2AZ, U.K.
Abstract
We revisit the problem of matching the strong coupling expansion of the BPS circular Wilsonloops in N = 4 SYM and ABJM gauge theories with their string theory duals in AdS × S and AdS × CP , at the first subleading (one-loop) order of the expansion around the minimalsurface. We observe that, including the overall factor 1 /g s of the inverse string coupling constant,as appropriate for the open string partition function with disk topology, and a universal prefactorproportional to the square root of the string tension T , both the SYM and ABJM resultsprecisely match the string theory prediction. We provide an explanation of the origin of the √ T prefactor based on special features of the combination of one-loop determinants appearingin the string partition function. The latter also implies a natural generalization Z χ ∼ ( √ T /g s ) χ to higher genus contributions with the Euler number χ , which is consistent with the structureof the 1 /N corrections found on the gauge theory side. [email protected] Also at the Institute of Theoretical and Mathematical Physics, MSU and Lebedev Institute, [email protected] ontents √ T prefactor 94 Dilaton insertion and derivative over gauge coupling 115 Universal form of higher genus corrections 156 Concluding remarks 17A Comments on tension dependence of the string partition function 18 A.1 T -derivative of the partition function in static gauge . . . . . . . . . . . . . . . . . . 18A.2 T -dependence from zero modes in conformal gauge . . . . . . . . . . . . . . . . . . . 20 B Conformal Killing vectors as longitudinal zero modes 22
There is a history of attempts to match gauge theory [1, 2, 3] and string theory [4, 5, 6] results forthe leading terms in the strong coupling expansion of the expectation value of the BPS circularWilson loop (WL) in N = 4 SYM theory (see [7, 8, 9, 10, 11, 12, 13]). The precise matching wasrecently achieved for the ratio of the and BPS WL expectation values [14] (see also [15, 16] fora discussion of similar matching in the ABJM theory [17]). However, the direct computation of thestring theory counterpart of the expectation value of the individual WL, that non-trivially dependson the normalization of the path integral measure, still remains a challenge.In the SU ( N ) N = 4 SYM theory the Maldacena-Wilson operator defined in the fundamentalrepresentation is given by W = tr P e R ( iA +Φ) (note that we do not include the usual 1 /N factor inthe definition of W ). Then for a circular loop one finds at large N with fixed ’t Hooft coupling λ [1, 2]: hWi = N √ λ I ( √ λ ). Expanding at strong coupling, hWi = N λ − / q π e √ λ + ... . Thisresult should be reproduced by the AdS × S string perturbation theory with the string tension T = √ λ π = R πα ′ , where here and below R denotes the AdS radius. It was suggested in [2] that thepre-factor λ − / ∼ T − / may have its origin in the normalization of three ghost 0-modes on the disk(or the Mobius volume).This proposal, however, is problematic for several reasons. First, the effective tension T hasits natural origin in the string action but should not appear in the diffeomorphism volume or thevolume of residual Mobius symmetry. Furthermore, the T − / factor (which would be universal ifrelated to the Mobius volume) would fail to explain the result for the BPS circular WL [18] in2he U ( N ) k × U ( N ) − k ABJM theory, where the tension is T = √ λ (with λ = Nk ) while the gaugetheory (localization) prediction [19, 20, 21] for the BPS Wilson loop in fundamental representationis hWi = N (4 πλ ) − e π √ λ + ... . Note that, as above, in our definition we do not divide the Wilsonloop operator by the dimension of the representation. Another indication that the explanation of the prefactor should be different is that, in general, oneexpects that the string counterpart of the large N term in hWi should be the open-string partitionfunction on the disk, which should contain an overall factor of the inverse power of the string coupling(corresponding to the Euler number χ = 1), i.e. hWi = Z str = 1 g s Z + O ( g s ) , Z = Z [ dx ] ... e − T R d σ L , (1.1)where g s provides the required overall factor of N . The fact that it is natural to define the WLexpectation value without the usual 1 /N factor, and to include the 1 /g s factor in its string theorycounterpart, was also emphasized in [22].In the N = 4 SYM case we have [1, 2] g s = g π = λ πN , λ = g N , T = √ λ π , hWi = Nλ / q π e √ λ + ... , (1.2)while in the ABJM case [17, 21] g s = √ π (2 λ ) / N , λ = Nk , T = √ λ , hWi = N πλ e π √ λ + ... . (1.3)Our central observation is that both expressions for hWi in (1.2) and (1.3) can be universally repre-sented as hWi = W h O ( T − ) i + O ( g s ) , W = 1 g s r T π e − ¯Γ e πT , (1.4)where ¯Γ is a numerical constant. Below we will argue that (1.4) should be the expression for theleading semiclassical result for the disk string path integral for a minimal surface in AdS ending on acircle at the boundary (thus having induced AdS geometry) in the AdS n × M − n string theory withtension T and coupling g s . In (1.4) the exponent e πT = e − I cl comes from the value of the classical Our normalization of W in the BPS case corresponds in the localization calculation of [20, 21] to computingthe matrix model expectation value h Str (cid:16) e iµi e − iνj (cid:17) i . Note that [21] defines the Wilson loop expectation value byincluding an extra overall factor of g CS ≡ πik . Denoting by h W i loc the expectation value given in [21], we find thatthe strong coupling limit of the BPS Wilson loop in the ABJM theory is hWi = g CS h W i loc = g CS e π √ λ + iπB = k π e π √ λ = N πλ e π √ λ , where we fixed the phase as B = . Here the AdS radius is R = (2 π λ ) / √ α ′ with T ≡ R πα ′ . The shift λ → λ − (1 − k ) in the definition of Rand thus of string tension [23, 21, 24] is irrelevant to the leading order in 1 /T expansion that we shall consider below(as discussed in [25], at the leading order we do not expect renormalization of the relation for the string tension). I cl = V AdS T = − πT . The constant ¯Γ comes from the ratio of one-loop determinantsof string fluctuations near the minimal surface, and is found to be (see [6, 7, 9] and section 2 below)AdS × S : ¯Γ = log(2 π ) , AdS × CP : ¯Γ = 0 . (1.5)Including also the n = 3 case of AdS × S × T string theory, one finds for AdS n × M − n with n = 3 , , = ( n −
4) log(2 π ) (see (2.21) below), and so in general W in (1.4) is W = 1( √ π ) n − √ Tg s e πT . (1.6)Using (1.5) one can check that the expression in (1.4) or (1.6) is in remarkable agreement with thegauge-theory expressions in (1.2) and (1.3).As we explain below, it will also follow from our argument that at higher genera (disk with p handles with Euler number χ = 1 − p ) the √ T factor in (1.4) should be replaced by ( √ T ) χ , i.e. thecorresponding term in the partition function should have a universal prefactor hWi = X χ =1 , − ,... c χ (cid:16) √ Tg s (cid:17) χ e πT h O ( T − ) i . (1.7)This is indeed consistent with the structure of 1 /N corrections found on the gauge theory side in [2]and in [21] (see section 5).It remains to understand the origin of the simple prefactor q T π in (1.4). In general, the expressionfor such a prefactor in the path integral is very sensitive to the definition of path integral measurewhich is subtle in string theory. In section 3 below we will provide an explanation for the presence ofthe √ T factor starting from the superstring path integral in the static gauge [6] (see also AppendixA.1) but we will not be able to determine the origin of the remaining √ π constant from first principles.This is already a non-trivial result: since the presence of this constant is fixed by the comparisonwith the SYM theory, we then have the string theory explanation for the ABJM expression in (1.3)(or vice-versa).In section 4 we shall provide another consistency check of the universal expression for the stringpartition function (1.4) by considering the analog of the familiar soft dilaton insertion relation anddilaton tadpole on the disk.In section 5 we will emphasize the fact that that the universal prefactor in the disk partitionfunction ∼ √ Tg s in (1.4) has a natural generalization (1.7) to higher orders which is consistent withthe structure of the 1 /N corrections found on the gauge theory side. We will make some concludingremarks about some other WL examples in section 6.It is interesting to note that the factor q T π in (1.4) looks exactly like the one associated with justone bosonic zero mode (in the standard normalization of the path integral zero-mode measure, i.e. √ π ~ , ~ − = T , as was used in a similar context in [14]). In Appendix A.2 we will discuss a possible Here we assume that the path integral measure for a scalar field is normalized so that the gaussian integral has
Let us consider a circular WL surface with AdS induced geometry, which resides in an AdS subspaceof AdS n × M − n , specifically:(i) n = 5: AdS × S ; (ii) n = 4: AdS × CP ; (iii) n = 3: AdS × S × T .The string is point-like in the internal compact directions, satisfying Dirichlet boundary conditions.In general, the planar WL expectation value is given by the string path integral with a disk-like worldsheet ending on a circle at the boundary of AdS space, hWi = e − Γ , Γ = Γ + Γ + Γ + ... . HereΓ = − πT is the classical string action (proportional to the renormalized AdS volume V AdS = − π )and Γ = O ( T ) is given by sum of logarithms of fluctuation determinants (in which we includepossible measure-related normalization factors).We shall discuss the computation of the one-loop correction Γ ≡ Γ ( n )1 in the above AdS n × M − n cases following the heat kernel method applied in the AdS × S case in [6] and [9]. In this n = 5case the general form of the static-gauge string one-loop correction is [6]Γ (5)1 = log [det( −∇ + 2)] det( −∇ + R (2) + 4) [det( −∇ )] [det( −∇ + R (2) + 1)] (2.1)= log [det( −∇ + 2)] [det( −∇ )] [det( −∇ + )] . (2.2)Here we assumed that the AdS radius R is scaled out and absorbed into the string tension T = R πα ′ so that all operators are defined in the induced AdS metric with radius 1 and curvature R (2) = − mode that, in general, is different from the other two: this is the AdS mode transverse to the minimal surface (the other two transverse modes are transverse to AdS ), see[26, 6]. In the present case of the minimal surface being AdS we have R (2) = − modes.Similar expression (2.1) is found in the conformal gauge [6], provided the contribution of the two“longitudinal” modes cancels as in flat space [27] against that of the ghost determinant and Mobiusvolume factor (modulo the 0-mode part of the longitudinal operator and related definition of pathintegral measure, see Appendix A.1 for further discussion). a fixed value R [ dx ] exp[ − ~ ( x, x )] = 1, i.e. [ dx ] = Q σ dx ( σ ) √ π ~ . Then the factor of string tension T = ~ − appears bothin the measure and in the action and cancels out in the one-loop determinant expression apart from possible 0-modecontribution.
5n the less supersymmetric cases with AdS → AdS n and n = 4 , ( n )1 = log [det( −∇ + 2)] n − det( −∇ + R (2) + 4) [det( −∇ )] − n [det( −∇ + R (2) + 1)] n − [det( −∇ + R (2) )] − n (2.3)= log [det( −∇ + 2)] n − [det( −∇ )] − n [det( −∇ + )] n − [det( −∇ − )] − n (2.4)The fermion masses are controlled by the superstring kinetic term with a projection matrix in themass term. In the AdS × S × T case [6] there are 4 massless fermion modes (which are partnersof T bosonic modes) and 4 massive ones. In the AdS × CP case one finds [28] that there are2 n − − n = 2 massless fermionic modes.Let us first discuss the divergent part of (2.3) assuming the standard heat-kernel regularizationseparately for each determinant contribution. The UV divergent part of Γ = log det( −∇ + X )where −∇ + X is a scalar Laplacian is given by (Λ → ∞ )Γ , ∞ = − B log Λ , B = π Z d σ √ g b , b = R (2) − X . (2.5)Here we ignore boundary contributions (they contain a power of IR cutoff and are absent afterrenormalization of the AdS volume or directly using the finite value for the Euler number of theminimal surface).In the case of (2.2) we then find that in the total combination all R (2) terms cancel out (dueto balance of bosonic and fermionic d.o.f.) and the constant mass terms also cancel between bosonsand fermions so that we are left only with contributions of R (2) terms from one special bosonic modeand the fermionic modes b (5)2 , tot = − R (2) − − R (2) ) = R (2) , B (5)2 , tot = π Z d σ √ g R (2) = χ = π ( − π )( −
2) = 1 . (2.6)For general n the corresponding UV divergent part of (2.3) is given by the straightforward general-ization of (2.6). Again, all R (2) terms in (2.5) cancel out as do the constant mass terms and wefind b ( n )2 , tot = − ( n − − ( R (2) + 4) − (2 n − − R (2) − − (10 − n )( − R (2) ) = R (2) , (2.7) B ( n )2 , tot = π Z d σ √ g R (2) = χ = 1 . (2.8)The total result (coming again just from the R (2) terms in single bosonic mode and 8 fermionicmodes) is thus universal, i.e. n -independent.Moreover, the same result B = χ for the coefficient of the UV divergence is found for fluctua-tions near any minimal surface (not even lying within AdS ) that has disk topology (see [6, 29]): if6 is a “mass matrix”, the contribution of 8 transverse bosons is b b = 8 · R (2) − tr X − R (2) whileof 8 fermions is b f = 8 · R (2) + tr X so that b = R (2) .Note that in general the Seeley coefficient is B = ζ (0) + n where ζ (0) is the regularized numberof all non-zero modes and n = n b − n f is the effective number of all 0-modes (assuming fermionsare counted as Majorana or Weyl). In the present static gauge case there are no obvious normalizable0-modes (cf. remark below (B.10)), but we observe that the result (2.8) is formally the same as whatwould come just from one “uncanceled” bosonic mode.The universality of (2.8) strongly suggests that the mechanism of cancellation of this total “topo-logical” UV divergence should also be universal. One may absorb it into the definition of the super-string path integral measure or cancel it against other measure factors as discussed in the conformalgauge in [6]. An alternative is to use a special “2d supersymmetric” definition of the one-loop pathintegral in the static gauge (see below): the cancellation of UV divergences is, in fact, automatic ifone uses a “spectral” representation for the total Γ rather than heat kernel cutoff for each individualdeterminant.Let us now turn to the finite part of the one-loop effective action in (2.4). We will follow [9] whichcompleted the original computation in [6] of Γ in (2.19) based on expressing the determinants in(2.2) in terms of the well known [33] heat kernels of the scalar and spinor Laplacians on AdS . Γ ( n )1 in (2.3) contains the contributions of the following AdS fields: (i) n − m = 2; (ii)10 − n scalars with m = 0; (iii) 2 n − m = 1; (iv) 10 − n Majoranafermions with m = 0. We will temporarily set the AdS radius to 1 and discuss the dependence onit later. Let us first use the heat-kernel cutoff for each individual determinant in (2.3), i.e. log det ∆ = − V AdS Z ∞ Λ − dtt K ( t ) , V AdS = − π . (2.9) To recall, the UV divergences do not cancel automatically even in the bosonic string theory in flat space. Thecombination of D scalar Laplacians and the conformal ghost operator ∆ gh = P † P gives (with all modes counted)[31] B = π R d σ √ g ( D R (2) − ( R (2) + R (2) ) = ( D − χ . Assuming, following [32], that there are extra powersof the UV cutoff in the Mobius volume one divides over and in the integrals over moduli, the net result is that oneshould add to the above B an extra δ top B = − χ = dim ker P † − dim ker P , thus getting B = ( D − − χ = ( D − χ . A similar argument applies to the NSR string where B = ( D − χ . In the present D = 10 GSsuperstring case there is an extra conformal anomaly/divergence from the Jacobian of rotation from GS fermions to2d fermions (see [30]); this effectively amounts to adding 3 extra massless fermion contributions for each 2d fermioncontribution (or, equivalently, multiplying the R (2) − R (2) part of each fermion contribution to b by 4); this gives δ B = − × × π R d σ √ g ( R (2) − R (2) ) = 2 χ . In the conformal gauge the divergences from the determinant ofthe ghost operator (∆ gh ) ab = − g ab ∇ − R ab cancel against those of the determinant of operator ∆ long for 2 longitudinalscalars. As in the bosonic case, one should also add δ top B = − χ as explained above. Summing these contributionswith (2.8) gives B = B ( n )2 , tot + δ B + δ top B = χ + 2 χ − χ = 0 . K ( t ) for a real scalar and a Majorana 2d fermion may be written as K ( t ) = π Z ∞ dv µ ( v ) e − t ( v + M ) , (2.10) µ b ( v ) = v tanh( πv ) , M = + m ; µ f ( v ) = − v coth( πv ) , M = m . (2.11)Here in µ f we already accounted for the negative sign of the fermion contribution, so the total K isjust the sum of the bosonic and fermionic terms. The associated ζ -function is ζ ( z ) = − z ) Z ∞ dv µ ( v ) Z ∞ dt t z − e − t ( v + M ) = − Z ∞ dv µ ( v )( v + M ) z . (2.12)For example, for AdS scalars ζ (0) = B = − b = − m . The total value of ζ (0) is found to be1, i.e. the same as in (2.8). In general, the one-loop correction isΓ = X log det ∆ = − ζ tot (0) log Λ + ¯Γ , ¯Γ ≡ − ζ ′ tot (0) , ζ tot (0) = 1 . (2.13)For the derivative of the scalar ζ -function one finds ( A is the Glaisher constant) ζ ′ b (0 , M ) = − (1 + ln 2) + ln A − Z M dx ψ (cid:0) √ x + (cid:1) , (2.14) ζ ′ b (cid:0) , (cid:1) = − + ln(2 π ) − A , ζ ′ b (cid:0) , (cid:1) = − + ln(2 π ) − A , (2.15)while for the massive fermion ζ ′ f (0 , M ) = − + 2 ln A + √ M + Z M dx ψ ( √ x ) , (2.16) ζ ′ f (cid:0) , (cid:1) = − ln(2 π ) + 2 ln A , ζ ′ f (cid:0) , (cid:1) = − + 2 ln A . (2.17)The total contribution to the finite part ¯Γ ( n )1 in (2.13) corresponding to (2.3) then found to have asimple form¯Γ ( n )1 = − h ( n − ζ ′ b (0 , ) + (10 − n ) ζ ′ b (0 , ) + (2 n − ζ ′ f (0 ,
1) + (10 − n ) ζ ′ f (0 , i = ( n −
4) ln(2 π ) . (2.18)¯Γ (5)1 = ln(2 π ) , ¯Γ (4)1 = 0 , ¯Γ (3)1 = − ln(2 π ) . (2.19)In the AdS × S case ( n = 5) the computation of the corresponding determinants was also carriedout using different methods in [7, 8] with the finite part of the resulting expression for ¯Γ (5)1 being asin (1.5), (2.19). Note that the finite part (2.18) happens to vanish in the AdS × CP case ( n = 4).It is interesting to note that there exists a special definition of Γ in (2.3) that automatically givesa UV finite one-loop result. Instead of computing separately each determinant let us use (2.9) andsum up the corresponding spectral integral expressions under a common integral over v in (2.10).8nterchanging the order of t - and v - integrals and first integrating over t we see that this integral isfinite, i.e. the proper-time cutoff is not required. Using (2.10)–(2.11) we then get for (2.4)¯Γ ( n )1 = V AdS2 π Z ∞ dv v (cid:16) tanh( πv ) (cid:2) ( n −
2) ln( v + ) + (10 − n ) ln( v + ) (cid:3) − coth( πv ) (cid:2) (2 n −
2) ln( v + 1) + (10 − n ) ln( v ) (cid:3)(cid:17) , (2.20)where V AdS2 π = −
1. Remarkably, the integral over v here is convergent at both v = 0 and v = ∞ (i.e. in the UV). In general, given the structure of the eigenvalues in (2.10)-(2.11), one can see thatconvergence of the representation (2.20) in the UV requires the sum rule P b ( m b + ) − P f m f = 0,which is satisfied for the spectra in our problem. Evaluation of (2.20) gives then a finite result equalto the one in (2.18), i.e. ¯Γ ( n )1 = ( n −
4) ln(2 π ) . (2.21)This prescription of not using proper-time cutoff for individual log det terms, i.e. first combining theintegrands and then doing the spectral integral, may be viewed as a kind of “2d supersymmetric”regularization. Indeed, the balance of the bosonic and fermionic degrees of freedom in (2.4) suggestshidden AdS supersymmetry [6]. Then the prescription of combining the spectral integrands of thedeterminants together may be viewed as a result of a “superfield” computation manifestly preserving2d supersymmetry. Note however that, even though the integral in (2.20) is finite, a dependence ofΓ on a normalization scale reappears on dimensional grounds if one restores the dependence on theradius R inside the logarithms, as explained in the next section. This leads to an explanation of the T -dependent prefactor in (1.4) and (1.7). √ T prefactor Let us now explain the presence of the √ T = R √ πα ′ prefactor in the string one-loop partition function(1.4). As the definition of quantum string path integral (in particular, integration measure) is subtleand potentially ambiguous our aim is to identify the one that is consistent with underlying symmetriesand AdS/CFT duality.In the previous section we ignored the dependence of the one-loop correction on the AdS radiusR. Let us now discuss how the string path integral may depend on it. Let us start with the classicalstring action in AdS n of radius R. One possible approach is to rescale the 2d fields so that the factor One implication is the vanishing of the corresponding vacuum energy in AdS observed in [6] in the case of thestrip parametrization ds = ρ ( dt + dρ ), ρ ∈ ( − π , π ). To recall, the contributions of a scalar with mass m b and a fermion with mass m f to the AdS vacuum energy are [6] E b ( m ) = − ( m + ) and E f ( m ) = ( m − )so that for the spectrum in (2.3) we get E tot = ( n − E b (2) + (10 − n ) E b (0) + (2 n − E f (1) + (10 − n ) E f (0) =( n − − − ) + (10 − n )( − ) + (2 n − ) + (10 − n )( − ) = 0.
9f R appears in front of the action I = T Z d σ √ g G mn ( x ) ∂ a x m ∂ a x n + ... (3.1)= T Z d σ √ g ¯ G mn (¯ x ) ∂ a ¯ x m ∂ a ¯ x n + ..., T = R T , T = πα ′ . (3.2)Using either (3.1) or (3.2) the expression for one-loop correction will depend also on the assumptionabout the path integral measure. If the measure is defined covariantly the final result should be thesame.Let us consider the path integral defined by (3.1) in terms of the original unrescaled coordinates x m of natural dimension of length, so that G mn ( x ) is dimensionless and depends on the AdS scaleR. The string σ -model path integral may be defined symbolically as (cf. footnote 3) Z = Z Y σ,m q T π p G ( x ( σ )) [ dx m ( σ )] . . . exp (cid:2) − T Z d σ √ g G mn ( x ) ∂ a x m ∂ a x n + ... (cid:3) . (3.3)Expanding near the minimal surface ending on the boundary circle we will get the induced AdS metricdepending on the same curvature scale R as G mn . Then rotating the fluctuation fields to thetangent-space components ˜x r and also rescaling them by √ T (so that they will be normalized as | ˜x | = R d σ √ g ˜x r ˜x r ) we will find that the 1-loop contribution from a single scalar is Z = (det ∆) − / where ∆ = −∇ + ... depends on the induced AdS metric and has canonical dimension of (length) − with eigenvalues scaling as R − . In the heat kernel representation Γ = − log Z = log det ∆ = − R ∞ Λ − dtt tr exp( − t ∆) the parameter t and the cutoff Λ − will now have dimension of (length) andwe will get instead of (2.13) (cf. (2.5),(2.8))Γ = − ζ tot (0) log(RΛ) + ¯Γ , ζ tot (0) = χ = 1 (3.4)As discussed in section 2, the UV divergence is expected to be cancelled by an extra “universal”contribution log( √ α ′ Λ) from the superstring measure (see footnote 4). We assume that this universalcontribution (depending only on the Euler number of the world sheet but not on details of its metric)may only involve the string scale √ α ′ but not the AdS radius. As a result, Γ = − χ log R √ α ′ + ¯Γ .The argument of log is thus ∼ ( √ T ) χ , i.e. we get Z ∼ e − Γ → (cid:0) √ T (cid:1) ζ tot (0) = ( √ T ) χ = √ T . (3.5)This explains the origin of the √ T factor in the disk partition function (1.4).As was noted below (2.8), the coefficient of the UV divergent term in (3.4) is, in fact, the samefor all minimal surfaces with disk topology and thus the dependence of the string partition functionon the scale R or effective tension T through the √ T factor in (3.5) should be universal. This means,in particular, that the factors 1 /g s and √ T in (1.4) will cancel in the ratio of expectation values For example, starting with ds = dr + e r/ R dx i dx i we get ds = R ( d ¯ r + e r d ¯ x i d ¯ x i ). Note that after the rescalingthe tension T and coordinates ¯ x m are dimensionless.
10f different Wilson loops with disk topology. Moreover, the fact that the power of T in (3.5) iscontrolled by the Euler number χ implies that at higher genera, for a disk with p handles, we shouldfind that hWi includes the universal prefactor ( √ T /g s ) χ as in (1.7). This is in precise agreementwith the large N expansion of the localization results both in N = 4 SYM and ABJM cases, as weexplain in more detail in section 5.The result of adding the above universal counterterm log( √ α ′ Λ) is equivalent to just defining theone-loop partition function to be UV finite by first combining all the contributions using the spectralrepresentation (2.20). There we set R = 1 and to restore the dependence on the radius R of theAdS metric we need to add the mass scale factor R − under the logs in (2.20) (cf. (2.9),(2.10)). Tomake the argument of the logs dimensionless we also need to introduce some normalization scale ℓ (i.e. log det ∆ → log det( ℓ ∆) or, equivalently, add ℓ factor in the path integral measure). Then wefind that Γ ( n )1 in (2.20) depends on R via the same ζ tot (0) = 1 term as in (2.13),(3.4), i.e. via anextra contribution (to be added to (2.21)) δ Γ ( n )1 = V AdS2 π − ℓ ) Z ∞ dv v (cid:2) tanh( πv ) − coth( πv ) (cid:3) = − log(R ℓ − ) . (3.6)The dependence on ℓ illustrates the fact that as long as ζ tot (0) = 0, the one-loop contribution, evenif defined to be UV finite by the spectral representation (or some analytic regularization like the ζ -function one [34]), is still scheme (or measure) dependent. Choosing ℓ ∼ √ α ′ , which is here anobvious choice in the absence of any other available scales (and which is also suggested by the T dependence in (3.3)), we again end up with the required result (3.5).We shall discuss some other approaches to the derivation of the dependence of the one-loopcorrection on T in the next section and Appendices A.1 and (A.2). As another check of consistency and universality of the expression (1.4) for the 1-loop string partitionfunction for a minimal surface with disk topology, let us consider a closely related object – theinsertion of the dilaton operator in the expectation value or the dilaton tadpole on the disk with WLboundary conditions. Here we shall explicitly consider the SYM case but a similar discussion shouldapply also to the ABJM case.Let us first recall the zero-momentum dilaton insertion relation, or the familiar “soft dilatontheorem” in flat space. The dilaton φ couples to the string as [35] I = Z d σ √ g (cid:2) T G mn ( x ) ∂ a x m ∂ a x n + π R (2) φ ( x ) (cid:3) , (4.1)where T = πα ′ . The string-frame metric G mn expressed in terms of the Einstein-frame metric in D dimensions is G mn = e D − φ ¯ G mn , ¯ G mn = δ mn + h mn and thus the (zero-momentum) dilaton vertex11perator in flat space is (cf. [37, 38]) I = I − V φ + ... , V = − D − Z d σ √ g h T ∂ a x m ∂ a x m + D −
24 14 π R (2) i = − D − I − χ , (4.2)where I = T R d σ √ g ∂ a x m ∂ a x m and χ = π R d σ √ gR (2) . Since the expectation value of theaction I may be obtained by applying − T ∂∂T to the string path integral (cf. (A.2)), the inser-tion of the zero-momentum dilaton into the generating functional for scattering amplitudes Z = R [ dx ] e − I + V φ + V h h + ... is then given by (here h i =1) ∂∂φ log Z = h V i = − D − h I i − χ = D − T ∂∂T log Z − χ . (4.3)In the standard cases of a bosonic closed string or open string with Neumann boundary conditionsthere are D constant 0-modes, so one finds Z ∼ T D/ and h I i = − D (assuming “covariant”regularization in which δ (2) ( σ, σ ) = 0, see [36]). The same relation is true also for the fermionicstring as the number of bosonic translational 0-modes remains the same.In the superstring case ( D = 10) for the tree-level topology of a disk ( χ = 1) eq. (4.3) reads ∂∂φ log Z = h V i = − h I i − χ = T ∂∂T ln Z − . (4.4)Adapting this relation to our present case of fixed contour boundary conditions with the expectationvalue of the action given by h I i = − (see (A.6)) the analog of (4.4) becomes (including in h I i alsothe classical contribution of an AdS minimal surface h I i cl = T ( − π ) = −√ λ ) ∂∂φ log Z = h V i = − h I i − ( √ λ + −
2) = √ λ − . (4.5)Since the constant part of the dilaton is related to the string coupling which itself is related to theSYM coupling as in (1.2), i.e. g = 4 πg s = 4 πe φ , we may compare (4.5) to the derivative of thecircular WL expectation value with respect to the coupling constant on the gauge theory side. Thenormalized gauge theory path integral is defined by h ... i SYM ∼ R [ dA... ] e − S SYM ... , h i SYM = 1 where S YM = Z d x L SYM , L
SYM = 14 g tr ( F mn + ... ) . (4.6)We assume that the metric is Euclidean and the SU ( N ) generators are normalized as tr ( T i T j ) = δ ij .Since the factor in front of the action is e − φ = g − = 4 πg = 4 πNλ , (4.7) The canonically normalized dilaton field ¯ φ that appears in the generating functional for scattering amplitudes, i.e.having the same kinetic term as the graviton in the effective action, S ∼ R d D x √ ¯ G [ − R + ( ∂ ¯ φ ) + ... ], is related to φ as ¯ φ = √ D − φ so that I = I − ¯ V ¯ φ + ..., ¯ V = − √ D − ( I + D − χ ). Note also that in (4.2) we ignored possibleboundary term as its role usually is only to ensure the coupling to the correct value of the Euler number. φ corresponds on the gauge-theory side to theinsertion of the SYM action into a correlator. In particular, in the case of the WL expectation value(here and below hWi ≡ hWi SYM ) ∂∂φ log hWi = h S SYM
W ihW i = λ ∂∂λ log hWi . (4.8)Since the gauge-theory result at strong coupling is hWi = N λ − / q π e √ λ + ... we conclude that ∂∂φ log hWi = λ ∂∂λ log hWi = √ λ − + ... , (4.9)which is in agreement with the string theory expression (4.5).Note that while in the string theory relation (4.5) we used that the insertion of the string actionis given by derivative over the tension, on the gauge theory side a similar relation (4.8) involvesdifferentiation over the gauge coupling. The two are in agreement because on the string side thedependence on λ comes from both the dependence on the tension and also dependence on the stringcoupling (the − χ = − g s and √ T factors in the string theory disk partition function (1.4) in order to have the consistency between thedilaton derivatives, or equality of the dilaton insertions on the string and gauge theory sides.The above discussion has a natural generalization to the string partition function on a disk withhandles or 1 /N corrections on the gauge theory side. For a surface of Euler number χ , using (A.8)we get the following analog of (4.3) generalizing (4.5) ∂∂φ log Z = h V i = − h I i − χ = T ∂∂T log Z − χ = √ λ − χ , (4.10)where we used that T ∂∂T = T ∂∂T . The subleading term − χ is consistent with the general form ofthe prefactor Z ∼ ( √ Tg s ) χ in (1.7). Indeed, note that g s = e φ and that switching to the Einstein-framemetric (cf. (4.2)) corresponds to T → e φ T (cf. (4.2)), so that √ Tg s ∼ e − φ . This is in agreementwith the gauge-theory side since the dependence on the dilaton is directly correlated as in (4.7),(4.8)with the dependence on λ (which appears only as a factor in front of the SYM action), while thedependence on N may come not only from the factor (4.7) in the action but also from traces inhigher order gauge-theory correlators. Indeed, according to the gauge-theory result (see (5.1)) thegenus p term in hWi depends on λ as λ p − = λ − χ .One can also perform a further consistency check by considering a direct generalization of theabove relations to the case of the local (i.e. “non zero-momentum”) dilaton operator insertion. Onthe gauge theory side the derivative over a local coupling or local dilaton is essentially the Lagrangian13n (4.6) and one finds [41, 42, 43] δδφ ( x ) log hWi = h L SYM ( x ) WihWi = − π d ⊥ f ( λ ) , (4.11) f ( λ ) = λ ∂∂λ log hWi = √ λ I ( √ λ ) I ( √ λ ) = √ λ − + ... . (4.12)In (4.11) we assume that dependence on the local dilaton is introduced by L SYM → e − φ ( x ) L SYM and φ is set to be constant as in (4.7) after the differentiation. In (4.12) we used that hWi = √ λ I ( √ λ ), i.e. f ( λ ) is the same function that appeared also in (4.9). For a WL defined by a circle of unit radius onthe ( x , x )-plane centered at the origin, the position dependent factor d ⊥ in (4.11) is given explicitlyby (see, e.g., [4, 46, 47]) d ⊥ = p ( r + h − + 4 h , r = x + x , h = x + x . (4.13)One can verify that integrating (4.11) over the position x = ( x , x , x , x ) of the operator insertion,using the regularized expression for the integral Z d x d ⊥ = (2 π ) Z ∞ dr r Z ∞ dh h (cid:2) ( r + h − + 4 h (cid:3) = − π , (4.14)one recovers the relation (4.8).On the string theory side, the corresponding local dilaton operator is (cf. (4.2); here D = 10 and L s = z − ∂ a x ′ m ∂ a x ′ m + ... is the AdS × S superstring Lagrangian) V ( x ) = − Z d σ √ g (cid:0) T L s + π R (2) (cid:1) K ( x − x ′ ; z ) , (4.15) K ( x − x ′ ; z ) = c z [ z + ( x − x ′ ) ] , c = Γ(∆) π d Γ(∆ − d ) (cid:12)(cid:12)(cid:12) d =4 , ∆=4 = π . (4.16) K in (4.16) is the bulk-to-boundary propagator of the massless dilaton in AdS (∆ = 4). Inte-grating over the 4-dimensional boundary coordinates gives back V that appeared in (4.5) (indeed, R d x K ( x − x ′ ; z ) = 2 π c = 1). Note that the correlator in (4.11) is to be compared to the stringtheory dilaton insertion on the disc with the dilaton vertex operator defined relative to the Einstein-frame metric so that the 2-point functions of the graviton and dilaton (and the corresponding dualoperators) are decoupled. Eq. (4.11) is a direct counterpart of the exact form of the correlation function of the BPS Wilson loop with the∆ = 2 chiral primary operator which is a special case of the the correlator of the Wilson loop and the ∆ = J CPO firstobtained in [44]. The function f ( λ ) also appears in the so-called Bremsstrahlung function [45]. The dilaton operator O is a descendant of the ∆ = 2 chiral primary, i.e. O ∼ tr ( F + Φ D Φ + ... ) and is different from the canonical formof the SYM Lagrangian L SYM in (4.6) by a total derivative term (in conformal correlators one may further drop theterms proportional to the scalar and spinor equations of motion as they produce only contact terms [39, 40]). Notethat we use Euclidean notation (as, e.g., in [43]) and in our normalization [40] h L SYM ( x ) L SYM ( x ′ ) i = N π ( x − x ′ ) . To evaluate this integral, one may, for instance, first integrate over r , then integrate over h and finally remove thepower divergence at h = ǫ →
0, i.e. R d x d ⊥ = π ǫ − π + O ( ǫ ) → − π . f ( λ ), and d ⊥ being simply the distance from the straight line (i.e.,for a straight line along the x direction, d ⊥ = p x + x + x ) [4]. Using the AdS surface in thestraight line case ( z = σ, x = τ, x i = 0) we get for the contribution of the leading classical termand the R (2) = − V in (4.15): h V cl+ R (2) ( x ) i = − Z d σ √ g ( T + π R (2) (cid:1) K ( x − x ′ ; z )= − c π ( √ λ − Z ∞−∞ dτ Z ∞ dσσ σ ( σ + τ + d ⊥ ) = − π d ⊥ ( √ λ − . (4.17)Then the string theory expectation value δδφ ( x ) log Z = h V ( x ) i indeed matches (4.11) if one adds inthe last bracket in (4.17) an extra + coming from the 1-loop quantum fluctuations of the bosonicand fermionic string coordinates in h L s i , in parallel to what happened in (4.5). An important feature of the √ Tg s prefactor in (1.4) is that it has a natural generalization ( √ Tg s ) χ tocontributions from higher genera (1.7) (cf. also (4.10),(A.8)). Let us recall that in the case of the SU ( N ) N = 4 SYM theory the exact expression for the expectation value of the BPS circularWL W = tr P e R ( iA +Φ) expanded at large N and then at large λ is [2, 3] ( L N − is the Laguerrepolynomial) hWi = e λ N L N − ( − λ N ) = N ∞ X p =0 √ p √ π p ! λ p − N p e √ λ h O ( √ λ ) i . (5.1)It was suggested in [2] that the sum over p may be interpreted as a genus expansion on the stringside. Remarkably, we observe that once the overall factor of N is included, i.e. one considers theexpectation value of tr ( ... ) rather that N tr ( ... ), the full dependence on N and λ in the prefactor of e √ λ in (5.1) combines just into ( N λ − / ) − p . Rewriting (5.1) in terms of the string tension T = √ λ π and string coupling g s = λ πN as defined in (1.2) we then get hWi = ∞ X p =0 c p (cid:16) √ Tg s (cid:17) − p e πT h O ( T − ) i , c p = π p ! ( π ) p , (5.2)which is the same as (1.7) where χ = 1 − p is the Euler number of a disk with p handles. Let us note that this expression applies to the SYM theory with the U ( N ) gauge group; the result in the SU ( N )case is obtained by multiplying (5.1) by exp( − λ N ) [2]. This factor expressed in terms of g s and T in (1.2) is exp( − g T )and thus is subleading compared to H in (5.3) at large T ; we therefore ignore it here. T term in (5.2) has asimple closed expression: since P ∞ p =0 c p z p = π exp( π z ) we find as in [2] (see also [49]) hWi = e H W h O ( T − ) i , W = √ T πg s e πT , H ≡ π g T . (5.3)Here W is the leading large N or disk contribution in the SYM theory given by (1.4) (with e − ¯Γ = √ π according to (1.5)). H may be interpreted as representing a handle insertion operator, i.e.higher order string loop corrections here simply exponentiate. Such exponentiation is expected inthe “dilute handle gas” approximation of thin far-separated handles which should be relevant to theleading order in the large tension expansion considered in (5.1),(5.2) (cf. [48]).It has another interesting interpretation suggested in [50]. If one considers a circular Wilsonloop in the totally symmetric rank k representation of SU ( N ) then for large k, N and λ with κ = k √ λ N = k g s T =fixed its expectation value should be given by the exponent exp( − S D3 ) of the actionof the classical D3-brane solution. In the limit of 1 ≪ k ≪ N this description should apply also tothe case of the WL in the k -fundamental representation described by a minimal surface ending on amultiply wrapped circle and here one finds [50] that S D3 = N f ( κ ) = − k √ λ − k λ / N + O ( k λ / N ). Ifone formally extrapolates this expression to k = 1, i.e. a single circle case discussed above, then itbecomes S D3 = − πT − π g T + O ( g T ), i.e. exp( − S D3 ) reproduces precisely the exponential factor e πT + H in (5.3).A similar structure (5.2) of the topological expansion should appear in the case of the BPScircular WL in the ABJM theory which was computed from localization in [21]. According to (1.3),in that case we have √ Tg s = N √ π λ = k √ π where k is the CS level so that hWi should be a seriesin ( √ Tg s ) χ ∼ k χ ∼ ( g CS ) χ (cf. footnote 1). Translating the leading and the first subleading 1 /N corrections to the WL expectation value found explicitly in [21] into our notation we get hWi = (cid:16) N πλ + πλ N + ... (cid:17) e π √ λ = (cid:16) π g T + ... (cid:17) W , W = √ T √ π g s e πT . (5.4)Here W is the leading disk term in hWi in the ABJM theory given by (1.4) (with ¯Γ = 0 according to(1.5)). Thus, to this order, the genus expansion in the ABJM case has the same universal structureas (5.2),(5.3) in the SYM case. Remarkably, the coefficient π of the “one-handle” term g s T is preciselythe same as in the SYM case (5.3), calling for some universal explanation. It would be interesting tosee if the prefactor in (5.4) exponentiates as in (5.3) (e.g. using the results of [51]) and also if thereis a D2-brane description of this similar to the one in the SYM case discussed above (cf. [52, 53]). This corresponds to a resummation of the expansion in (5.1) for fixed √ λN , i.e. when g s ∼ T are both large [50, 49]which is formally a different limit compared to the one discussed above and leading to (5.3). Note that we use the notation g CS ≡ πik for what was called g s in [21] in order not to confuse it with the typeIIA string theory coupling g s in (1.3). The leading correction in eq. (8.19) in [21] is to be multiplied by g CS accordingto the definition of the topological expansion in (8.1) there. Also, as already mentioned in footnote (1), with ourdefinition of the WL expectation value hWi = h tr ( ... ) i = g CS h W i loc , where h W i loc is the gauge theory localizationexpression of [21]. Concluding remarks
As was noted below (2.8), the coefficient of the UV divergent term in (3.4) is, in fact, the same forall minimal surfaces with disk topology, and thus the dependence of the string partition function onthe scale R or effective tension T through the √ T factor in (3.5) should be universal.A check of the universality of the prefactor in (1.4) is that it applies also to the circular WLin the k -fundamental representation dual to a minimal surface ending on k -wrapped circle at theboundary of AdS . In this case the classical action is I cl = − k √ λ but the Euler number of theminimal surface is still equal to 1 [12] so that the coefficient ζ tot (0) in (3.4),(3.5) is also 1 and thusthe disk partition function is hWi ∼ √ Tg s e πkT . This is consistent with the SYM (localization) resultin the k -fundamental case [50, 54, 3] given by the k = 1 expression with √ λ → k √ λ . The overall k -dependent constant that should come from ¯Γ in (1.4) still remains to be explained, despite severalearlier attempts in [7, 9, 55, 12].The universality of (1.4) implies, in particular, that the prefactor √ Tg s should cancel in the ratio ofexpectation values of similar Wilson loops. In particular, this applies to the case of BPS latitudeWL parametrized by an angle θ . Matching with the gauge theory prediction for the ratio of thelatitude WL and simple circular WL was checked in the SYM case in [13, 14] and in the ABJM casein [15, 16].Let us note that (1.4) actually requires a generalization in special cases when there are 0-modesin the internal (non-AdS) directions of AdS n × M − n space, each producing extra factor of √ T (cf.(A.10)). This is what happens in the case of the supersymmetric ( θ = π ) latitude WL discussedin [56, 14] where we then get for the disk partition function hW i ∼ √ Tg s ( √ T ) ∼ N . (6.1)Here all λ -dependence cancels out and the finite proportionality constant should be equal to 1, i.e. N − hW i = 1, in agreement with [56].A similar remark applies to the case of the (bosonic) BPS WL [52] in the ABJM theory.According to [21] here we get instead of hWi for the BPS WL in (1.3) (cf. footnotes 1, 12 andeq.(5.4)) hW i = 1 g CS h W i loc = ie iπλ N πλ √ λ e π √ λ + ... . (6.2)As was argued in [52] (see also [57]), here the minimal surface solution is smeared over S = CP in CP so there are two scalar 0-modes. This explains the extra factor √ λ = ( √ T ) in (6.2) comparedto hWi in (1.3) [21]. More generally, for contributions from each genus p one finds [20, 21, 51] thatthe ratio of the and BPS WL’s is given by this universal ( √ T ) term (ignoring phase factors) hW i p hWi p = ( √ T ) + O ( T − ) . (6.3)17t would be interesting to match the precise numerical coefficient in the ratio between the BPS and BPS Wilson loops by carefully fixing the normalization of the two zero modes on the string side.Finally, let us note that while in this paper we focused on the case of 4d and 3d gauge theories,as explained in section 2 our results also apply to string theory in AdS × S × T with RR flux. Thiscase corresponds to n = 3 in (1.6) (cf. (1.4),(2.21)), i.e. hWi = Z str = g s √ T e πT + . . . . It wouldbe interesting to see if this string-theory prediction can be matched to localization calculations forWilson loops in 2d supersymmetric gauge theory (cf. [58, 59]). Acknowledgments
We are grateful to K. Zarembo for useful discussions and remarks on the draft. We also thank M.Beccaria, P. Di Vecchia, N. Drukker, M. Kruczenski, J. Maldacena, R. Roiban and E. Vescovi forhelpful comments. The work of S.G. is supported in part by the US NSF under Grants No. PHY-1620542 and PHY-1914860. A.T. acknowledges the support of the STFC grant ST/P000762/1.
A Comments on tension dependence of the string partitionfunction
In section 3 we discussed how to explain the prefactor √ T in the one-loop string partition function(1.4) starting with the string action (3.1) and using the static gauge expression (2.3). We emphasizedthat the result is sensitive to the choice of the path integral measure, i.e. the definition of thequantum theory (which, in general, is not unique, unless completely fixed by symmetry requirementsor extra consistency conditions). In the Appendices below we shall discuss some other approaches toderive this prefactor, which again involve certain assumptions about the measure or regularizationprocedure. A.1 T -derivative of the partition function in static gauge Suppose we start with the string action (3.2) in terms of the rescaled (dimensionless) coordinates sothat there is an explicit factor of the effective string tension T in front of the action with the inducedAdS metric having radius 1. Then we would get the same result as in (3.5) if we assume that thenorm or the measure is defined so that the resulting one-loop correction from a single scalar hasthe form Γ = log det ˆ∆ where ˆ∆ = T − ∆. Indeed, using the ζ -function regularization with ζ (0) Explicitly, ( x, ˆ∆ x ) = T R d σ √ g x ˆ∆ x = R d σ √ g x ∆ x , where x are rescaled fluctuations. In general, one can ofcourse move T -dependence from the action to the measure by a field redefinition (taking into account the resultingregularized Jacobian of the transformation). If the path integral measure is Q σ µ √ π dx ( σ ) and the action is simply R d σ √ g x ∆ x then Z = ( Q n λ n µ ) − / ∼ µ ζ (0) where λ n are the eigenvalues of ∆ [34]. If ζ (0) is non-zero the resultis thus sensitive to the definition of the measure. log det( T − ∆) = ζ (0) log( T − ) + ... = − ζ (0) log √ T + ... . This leads again to (3.5) once we use that the total value of ζ (0) correspondingto the static-gauge partition function (2.4) is ζ tot (0) = 1 (see (2.8),(2.13)).Another way to obtain the same result (which will be again based on a particular choice of aregularization prescription) is to find the dependence of the string partition function on the tensionby first computing its derivative over T . This is closely related to the argument appearing in thecontext of the “soft dilaton theorem” [36], see section 4.Let us assume that the tension dependence of the string partition function may come only fromthe factor of T in the string action (3.2) in the static gauge (i.e. in the action for the “physical”fluctuations whose determinants are present in (2.3)), i.e. the measure is defined so that it does notdepend on T . For example, for a single scalar field Z = Z [ dx ] exp( − I ) , I = T Z d σ √ g x ∆ ( m ) x , ∆ ( m ) = −∇ + m , (A.1) T ∂∂T log Z = −h I i , h I i = Z d σ √ g [∆G ( m ) ( σ, σ ′ )] σ = σ ′ = Z d σ √ g δ ( m ) ( σ, σ ) , (A.2)where h I i = Z − R [ dx ] I exp( − I ), G ( m ) ( σ, σ ′ ) = h σ | ∆ − m ) | σ ′ i is the Green’s function and δ ( m ) ( σ, σ )is a regularized value of the bosonic delta-function at the coinciding points. Let us use the heat-kernelcutoff, i.e. assume that δ ( m ) ( σ, σ ) = h σ | e − ǫ ∆ − m | σ i = π (cid:2) Λ + R (2) − m (cid:3) , ǫ ≡ Λ − → . (A.3)The expectation value of the action corresponding to the full static-gauge expression (2.3) is then h I i = Z d σ √ g n ( n − (2) G (2) ( σ, σ ′ )] σ = σ ′ + (10 − n )[∆ (0) G (0) ( σ, σ ′ )] σ = σ ′ − (2 n − f (1) G f (1) ( σ, σ ′ )] σ = σ ′ − (10 − n )[D f (0) G f (0) ( σ, σ ′ )] σ = σ ′ i , (A.4)= Z d σ √ g h ( n − δ (2) ( σ, σ ) + (10 − n ) δ (0) ( σ, σ ) − (2 n − δ f (1) ( σ, σ ) − (10 − n ) δ f (0) ( σ, σ ) o , where D f is the fermionic 1st order operator and G f and δ f stand for the corresponding Green’sfunction and δ -function.A key next step is to assume a special “2d supersymmetric” regularization in which the bosonicand fermionic Green’s functions and thus the corresponding regularized delta-functions are relatedto each other as δ f ( m ) ( σ, σ ) = h δ ( m − m ) ( σ, σ ) + δ ( m + m ) ( σ, σ ) i . (A.5) This is an effective consequence of the fact that the 2d supersymmetric Ward identity (cf. [60, 61]) relates a fermionof mass m > m − m (e.g. in a special regularization the trace of the Green’s function for a single2d fermion G f ( m ) is related to 2 m G ( m − m ) [61]). In the present case we have half of the massive fermions with mass m = 1 and the other half – with mass m = −
1. Alternatively, one may use a particular representation for G f ( m ) (for m >
0) as G f ( m ) ( σ, σ ′ ) = [( iγ a ∂ a + m )G ( m − m ) ] S ( σ, σ ′ ) implying that one has D f ( m ) G f (1) ( σ, σ ′ ) = δ ( m − m ) ( σ, σ ′ ) S ( σ, σ ′ ). h I i = Z d σ √ g h δ (0) ( σ, σ ) − δ (2) ( σ, σ ) i = × π × V AdS = − , (A.6)where we used (A.3). The relation T ∂∂T log Z = −h I i in (A.2) implies once again that Z ∼ √ T . (A.7)Let us note that the result for the expectation value of the action (A.6) should be more universalthan a particular prescription used above. The integrand in (A.6) should be in general δ (0) ( σ, σ ) − δ (2) ( σ, σ ) → − π R (2) . In the case of a more general topology of a disk with p handles with the Eulernumber χ = 1 − p we should then find that h I i = − χ , Z ∼ ( √ T ) χ , (A.8)which is in agreement with (1.7),(5.2). A.2 T -dependence from zero modes in conformal gauge The conformal gauge expression for the string partition function contains, in addition to the ratio ofdeterminants in Z = e − Γ in (2.3), also an extra factor [27, 6] Z c = Ω − h det ′ ∆ gh det ∆ long i / . (A.9)Here Ω is the SL (2 , R ) Mobius group volume. The 2-derivative ghost operator ∆ gh ab and the operatoron the two “longitudinal” fluctuations ∆ long ab = − ( ∇ ) ab − R (2) g ab have the same structure (and thesame “mixed” boundary conditions) so their non-zero-mode contributions should effectively canceleach other. The integral over the collective coordinates of the three 0-modes of ∆ long (or conformalKilling vectors) which is implicit in (A.9) should cancel against the Mobius volume factor. As aresult, one may assume that Z c in (A.9) is effectively equal to 1, thus getting back to the static gaugepartition function expression (2.3).However, this depends on the definition of path integral measure. An alternative possibilitycompared to the one in the static gauge discussed in the main text and section A.1 is to assume Let us note that the use of (A.5) may be interpreted as a specific regularization prescription for the fermionswhich is different from the heat-kernel or ζ -function one applied to the squared fermionic operator ∆ f ( m ) = (D f ( m ) ) = −∇ + R (2) + m in (2.3),(2.7). Indeed, if we assume that δ f ( m ) ( σ, σ ) in (A.5) is defined as in (A.3), i.e. δ f ( m ) ( σ, σ ) = h σ | e − ǫ ∆ f | σ i = π (cid:0) Λ + R (2) − R (2) − m (cid:1) then we find that h I i = + which is consistent with the ζ tot (0) = 1 value in(2.8),(2.13), i.e. Z ∼ Q [det( T ∆)] − / ∼ T − / . In this regularization the l.h.s. of (A.5) is 2(Λ + R (2) ) − R (2) − m while the r.h.s. is 2(Λ + R (2) ) − R (2) − m so the difference − R (2) may be attributed to the presence of − R (2) term in the squared fermionic operator which is thus effectively omitted in the prescription (A.5). T -dependent factor, while the presence of the √ T factor is (1.4) is due tothe normalization of the 0-modes, i.e. of the collective coordinate integral implicit in (A.9). Eachbosonic 0-mode absent in the fluctuation action then contributes a measure factor ∼ ( T π ) / , leadingto Z ∼ ( √ T ) n , (A.10)where n is the total number of the 0-modes.Equivalently, this result will follow assuming that one uses a regularization (e.g., dimensionalone) in which the delta-functions at coinciding points vanish, δ (2) ( σ, σ ) = 0 and thus the factors of T in the measure and in front of the action do not contribute, cf. (A.2), apart from the 0-modecontribution. It is useful to recall that in the familiar case of the open strings with free ends wherethe bosonic coordinates x m ( m = 1 , ..., D ) are subject to the Neumann boundary conditions onefinds D constant zero modes and thus an overall factor of T D/ in the disk path integral. The sameresult can be found also using T ∂∂T argument by using that the delta-function appearing in (A.2) isthe “projected” one, i.e. δ (2) ( σ, σ ) (set to 0) minus the trace of the projector to the 0-mode subspace(see, e.g., [36, 62]).In the present WL case of path integral with the Dirichlet-type (or fixed-contour) boundary con-ditions one could expect to have no 0-modes. However, as the two “longitudinal” string coordinatesare subject to “mixed” Dirichlet/Neumann b.c. [63, 27, 32, 64] (motivated by the requirement ofpreservation of the reparametrization invariance of the boundary contour) there is, in particular, aspecial 0-mode corresponding to a constant shift of a point on the boundary circle. There are, infact, two more 0-modes of the longitudinal operator (see Appendix B). As already mentioned above,these three bosonic 0-modes are direct counterparts of the conformal Killing vectors associated tothe SL (2 , R ) Mobius symmetry on the disk surviving in the conformal gauge.Thus if the path integral measure is normalized so that the integral over non-zero modes does notproduce any T -dependence we then get a factor Z ∼ ( √ T ) associated to the n = 3 “longitudinal”0-modes on the disk. To reduce the effective number of 0-modes to n = 1 (required to match the √ T factor in (1.4)) one may contemplate the following possibilities:(i) assume that 2 longitudinal 0-modes are lifted due to some boundary contributions to thestring action leaving only one translational mode (corresponding to a constant shift on the boundarycircle); As was already mentioned above, this corresponds to a specific choice of the measure factors implying that thenormalization of the gaussian path integral is 1, i.e. R [ dx ] exp[ − ~ ( x, x )] = 1, with [ dx ] = Q σ dx ( σ ) √ π ~ . Then the factorof string tension T = ~ − should appear not only in the action but also in the measure so that it cancels out in theintegrals for all modes with non-zero eigenvalues. This, at first, may look unnatural as then we would not have a cancellation between the integral over the corre-sponding collective coordinates and the Mobius volume factor in the path integral. Yet, that may not be a problemas the Mobius volume on the disk may be regularized to a finite value [65, 66] (similarly to how this is done for theAdS volume). n f = 2 fermionic super-Mobius 0-modes, producing the effective number n = n b − n f = 3 − B Conformal Killing vectors as longitudinal zero modes
Here we shall record the expressions for the conformal gauge ghost zero-modes or conformal Killingvectors (CKV) on a flat disk D and on a euclidean hyperbolic space H = AdS with the metric ds = e ρ ( dr + r dφ ) , ( e ρ ) D = 1 , ( e ρ ) H = 4(1 − r ) . (B.1)The CKV are also the zero-modes of the longitudinal Laplacian in (A.9) which is equivalent to the2nd derivative conformal ghost operator [27]. The defining relation ∇ a ξ b + ∇ b ξ a − g ab ∇ c ξ c = 0 doesnot depend on the conformal factor ρ when written in terms of the contravariant components ξ a : ∂ a ξ b + ∂ b ξ b − δ ab ∂ c ξ c = 0. The expressions for the three Killing vectors ξ a corresponding to the SL (2 , R ) transformations on the plane are (here a , b , b are real parameters) z ′ = e i a z + b1 + b ∗ z , b = b + i b , δz = ξ + iξ = b + i a z − b ∗ z , z = re iφ , (B.2) ξ = b − a r sin φ − r (b cos 2 φ + b sin 2 φ ) , ξ = b + a r cos φ − r ( − b cos 2 φ + b sin 2 φ ) ξ r = cos φ ξ + sin φ ξ , ξ φ = r − ( − sin φ ξ + cos φ ξ ) ξ r = (1 − r )(b cos φ + b sin φ ) , ξ φ = a + ( r + r − )(b cos φ − b sin φ ) . (B.3)Then the standard conformal Killing vectors on the disk satisfy mixed boundary conditions: ξ r = 0(normal component) and ∂ r ξ φ = 0 (normal derivative of tangential component) vanish at the r = 1boundary. Once we consider a metric with a non-trivial conformal factor these conditions are modifiedto: g ab = n a n b + t a t b , ξ n (cid:12)(cid:12) ∂ = 0 , ( ∂ n − K ) ξ t (cid:12)(cid:12) ∂ = 0 , K = ∇ a n a . (B.4) To compare, in the case of the one-loop instanton partition function in super YM theory (see [67, 68]) thecontributions of all non-zero modes cancel (i.e. ζ tot (0) = 0) and as a result the UV cutoff dependence (and thus one-loop beta function) is controlled just by the 0-modes – the total Seeley coefficient is B = ζ tot (0) + n tot = n b − n f . Atthe same time, the dependence on the inverse gauge coupling 1 /g YM (which is the analog of string tension T in our case)is controlled by the coefficient n b − n f . Note, however, that the prescription for g YM dependence becomes unambiguousonly in physical correlation functions with external fermionic legs saturating the fermionic 0-mode integral [68]. Alternatively, for the AdS metric we have ds = (sinh s ) − ( ds + dφ ) , r = e − s . Another form of the AdS metricthat follows from AdS metric ds = z − ( d r + r dφ + dz ) with z = √ − r is ds = d r (1 − r ) + r dφ − r is related tothe above one via r = s = r r . ∂ n − K ) ξ t (cid:12)(cid:12) ∂ = 0 wasused in [64] (and implicitly in [32]).The r, φ components of n a and t a are: n a = e ρ { , } , t a = e ρ { , r } so that ξ n = n a ξ a = e ρ ξ r , ξ t = t a ξ a = re ρ ξ φ , (B.5) ∂ n = e − ρ ∂ r , K = e − ρ r − ∂ r ( re ρ ) , ( ∂ n − K ) ξ t = r∂ r ξ φ . (B.6)Note that for a flat disk K = r − and χ = π ( R R + 2 R ∂ K ) = 1.Thus ξ φ in (B.3) satisfies ( ∂ n − K ) ξ t (cid:12)(cid:12) ∂ = 0 at r = 1 but there is an issue with ξ n (cid:12)(cid:12) ∂ = 0: e ρ ξ r = − r × (1 − r )(b cos φ + b sin φ ) so ξ n is a non-zero function at the boundary. This suggeststhat either we should set b , b = 0 or the boundary condition ξ n (cid:12)(cid:12) ∂ = 0 is to be modified. Oneoption is to define it with the flat metric as in (2.15) in [27]: ˜ n a ξ a | ∂ = 0 where ˜ n a is the normal inflat metric. This condition just says that the boundary condition x m | ∂ = c m ( φ ) should be preservedunder diffeomorphisms up to a boundary reparametrization, so δx m = ξ a ∂ a x m should vanish at theboundary for ξ a along the normal direction (the definition of normal formally depends on the metric,but here all we need is ξ r (cid:12)(cid:12) ∂ = 0).The norms of CKV depend on the conformal factor: | ξ | = Z d z √ g g ab ξ a ξ b = Z π dφ Z dr r e ρ ξ a ξ a . (B.7)For the three CKV proportional to a , b , b in (B.3) we have ( ξ = { ξ r , ξ φ } ): ξ (a) = a (cid:8) , (cid:9) , ξ (b ) =b (cid:8) (1 − r ) cos φ, − ( r + r − ) sin φ (cid:9) , ξ (b ) = b (cid:8) (1 − r ) sin φ, ( r + r − ) cos φ (cid:9) . Thus for ξ a ξ a in (B.7)we get: ξ (a) · ξ (a) = a r , ξ (b ) · ξ (b ) = b [2(1 + r ) − r cos 2 φ ] , ξ (b ) · ξ (b ) = b [2(1 + r ) + 2 r cos 2 φ ] , so that these vectors have a finite norm for a flat disk ( e ρ = 1) or half-sphere ( e ρ = 4(1 + r ) − )but their norm formally diverges for H ( e ρ = 4(1 − r ) − ). One option then is to regularize thenorms in the same way as we do for the H volume – introduce a cutoff and drop power divergences.We find (with a cutoff at r = e − ǫ ) that for the H volume R e − ǫ dr r (1 − r ) = ǫ − ... while for thenorms R e − ǫ dr r (1 − r ) = ǫ − ǫ + + ... , R e − ǫ dr r (1+ r )(1 − r ) = ǫ + ǫ − + ... .As a side remark, let us comment on the possibility of having zero modes for the transverse m = 2 fluctuation operator in the AdS directions in (2.2),(2.4). If we focus on just a single transversefluctuation within AdS , then one can formally find 3 zero modes related to the fact that the stringsolution breaks the SO (3 ,
1) isometries of AdS down to SO (2 , with metric ds = z ( dz + dr + r dφ ), the general AdS string solution endingon a circle (of radius α and center at ( β , β )) at the boundary can be written as z + ( r cos φ − β ) + ( r cos φ − β ) = α . (B.8) The definition of the norm for the diffeomorphism vectors via | ξ | = R d z √ g g ab ξ a ξ b is a natural one; while itinvolves the conformal factor, it is the conformal factor dependence in the corresponding determinants that shouldcancel in the critical dimension. This definition is different from the one used in [32] but agrees with the one of [70, 64].For a discussion of the freedom in the choice of the path integral measure see also [71]. β , β and α correspond to broken translations and dilatation. The zero modes ofthe transverse fluctuation operator can be obtained as usual by taking derivatives of the classicalsolution with respect to these parameters. Expressing the result in the coordinates where the inducedworldsheet metric is ds = σ ( dσ + dτ ) (0 < τ < π , σ > ψ ( α ) = coth σ , ψ ( β ) = cos τ sinh σ , ψ ( β ) = sin τ sinh σ . (B.9)One can verify that these indeed satisfy (cid:16) ∂ ∂σ + ∂ ∂τ − σ (cid:17) ψ ( α,β ,β ) = 0 . (B.10)However, these zero modes are not normalizable. Moreover, they do not satisfy the Dirichlet boundaryconditions at σ = 0, as required for the transverse fluctuations, so they should not be relevant forour problem. Note also that, when considering all of the n − n , therewould be, in fact, 3( n −
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