Precision calculation of 1/4BPS Wilson loops in AdS5×S5
HJE
AdS5
V. Forini 0 1 2 4 5 6 7 8
V. Giangreco M. Puletti 0 1 2 3 5 6 7 8
L. Griguolo 0 1 2 5 6 7 8
D. Seminara 0 1 2 5 6 7 8
E. Vescovi 0 1 2 4 5 6 7 8
0 INFN Gruppo Collegato di Parma
1 Dunhaga 3 , 107 Reykjavik , Iceland
2 Zum Gro en Windkanal 6 , 12489 Berlin , Germany
3 Science Institute, University of Iceland
4 Institut fur Physik, HumboldtUniversitat zu Berlin, IRIS Adlershof
5 Open Access , c The Authors
6 Via G. Sansone 1, 50019 Sesto Fiorentino , Italy
7 Viale G. P. Usberti 7/A, 43100 Parma , Italy
8 Going to Fourier space
We study the strong coupling behaviour of 1=4BPS circular Wilson loops (a family of \latitudes") in N = 4 Super YangMills theory, computing the oneloop corrections to the relevant classical string solutions in AdS5 S5. Supersymmetric localization provides an exact result that, in the large 't Hooft coupling limit, should be reproduced by the sigmamodel approach. To avoid ambiguities due to the absolute normalization of the string partition function, we compare the ratio between the generic latitude and the maximal 1/2BPS circle: any measurerelated ambiguity should simply cancel in this way. We use the Gel'fandYaglom method with Dirichlet boundary conditions to calculate the relevant functional determinants, that present some complications with respect to the standard circular case. After a careful numerical evaluation of our nal expression we still nd disagreement with the localization answer: the di erence is encoded into a precise \remainder function". We comment on the possible origin and resolution of this
Wilson; 't Hooft and Polyakov loops; AdSCFT Correspondence; Sigma Mod

discordance.
els
ArXiv ePrint: 1512.00841
1 Introduction and main result 2 3 4
5
Classical string solutions dual to latitude Wilson loops
3.1
3.2
3.3
4.1
4.2
uctuation determinants
Bosonic sector
Fermionic sector The circular Wilson loop limit
Oneloop partition functions
The circular Wilson loop
Conclusions
Ratio between latitude and circular Wilson loops
A Notation and conventions B
Methods for functional determinants
B.1 Di erential operators of the nthorder
B.2 Applications B.3 Square of rstorder di erential operators
C Fermionic determinant DetOF ( 0): details
C.1 Derivation of (3.28)
C.2 Simplifying the largeR expression for Det!(OF1;1;1)2
D Boundary conditions for small Fourier modes
1
5
7
Introduction and main result
The harmony between exact QFT results obtained through localization procedure for
BPSprotected Wilson loops in N
= 4 SYM and their stringy counterpart is a thorny issue
beyond the supergravity approximation. For the 1=2BPS circular Wilson loop [1, 2], in the
fundamental representation, supersymmetric localization [3] in the gauge theory con rms
the allloop prediction based on a large N resummation of ladder Feynman diagrams [4] and
in [7]1 using the Gel'fandYaglom method, reconsidered in [9] with a di erent choice of
boundary conditions and reproduced in [10]2 with the heatkernel technique. No agreement
was found with the subleading correction in the strong coupling (
1
) expansion of the
gauge theory result in the planar limit
loghW ( ; 0 = 0)i = log p2 I1
p
= p
3
4
log
+
1
2
log
2
+ O
1
2 ;
(1.1)
where I1 is the modi ed Bessel function of the rst kind, the meaning of the parameter 0
is clari ed below, and the term proportional to log
in (1.1) is argued to originate from the
SL(2; R) ghost zero modes on the disc [5]. The discrepancy occurs in the independent
part above,3 originating from the oneloop e ective action contribution and an unknown,
overall numerical factor in the measure of the partition function.
The situation becomes even worse when considering a loop winding ntimes around
itself [7, 11], where also the functional dependence on n is failed by the oneloop string
computation. The case of di erent group representations has also been considered: for the
ksymmetric and kantisymmetric representations, whose gravitational description is given in
terms of D3 and D5branes, respectively, the rst stringy correction again does not match
the localization result [12]. Interestingly, the Bremsstrahlung function of N = 4 SYM,
derived in [13] again using a localization procedure, is instead correctly reproduced [14]
through a oneloop computation around the classical cusp solution [2, 15].
Localization has been proven to be one of the most powerful tools in obtaining non
perturbative results in quantum supersymmetric gauge theories [3]: an impressive number
of new exact results have been derived in di erent dimensions, mainly when formulated on
spheres or products thereof [3, 16]. In order to gain further intuition on the relation between
localization and sigmamodel perturbation theory in di erent and more general settings,
we reexamine this issue addressing as follows the problem of how to possibly eliminate the
ambiguity related to the partition function measure. We consider the string dual to a
nonmaximal circular Wilson loop  the family of 1/4BPS operators with path corresponding
to a latitude in S2 2 S5 parameterized by an angle 0 and studied at length in [15, 17, 18] 
and evaluate the corresponding string oneloop path integral. We then calculate the ratio
between the latter and the corresponding one representing the maximal circle  the case
0 = 0 in (1.1). Our underlying assumption is that the measure is actually independent on
the geometry of the worldsheet associated to the Wilson loop,4 and therefore in such ratio
measurerelated ambiguities should simply cancel. It appears nontrivial to actually prove
a background independence of the measure, whose di eoinvariant de nition includes in
fact explicitly the worldsheet
elds.5 Our assumption  also suggested in [7]  seems
however a reasonable one, especially in light of the absence of zero mode in the classical
1See also [8].
2See appendix B in [10].
3See formula (1.4) below.
4About the topological contribution of the measure, its relevance in canceling the divergences occurring
in evaluating quantum corrections to the string partition function has been
rst discussed in [6] after the
observations of [19, 20]. We use this general argument below, see discussion around (4.8).
5See for example the discussion in [21].
{ 2 {
solutions here considered6 and of the explicit example of (string dual to) the ratio of a
cusped Wilson loop with a straight line [14], where a perfect agreement exists between
sigma model perturbation theory and localization/integrability results [13].7
The family of 1/4BPS latitude Wilson loops falls under the more general class of
1/8BPS Wilson loops with arbitrary shape on a twosphere introduced in [15, 24, 25]
and studied in [26]. There are strong evidences that they localize into YangMills theory
on S2 in the zeroinstanton sector [15, 26{28] and their vacuum expectation values are
therefore related to the 1/2BPS one by a simple rescaling. As originally argued in [18] the
expectation value of such latitude Wilson loops is obtained from the one of the maximal
circle provided one replaces
with an e ective 't Hooft coupling 0 =
cos2 0. The ratio
of interest follows very easily
hW ( ; 0)i
hW ( ; 0)i loc
= e
p (cos 0 1) h(cos 0) 23 + O(
1 i
2 ) + O e
p
where in the large
expansion only the dominant exponential contribution is kept (and
loc stands for \localization"). In terms of string oneloop e ective actions
log Z
loghW i, this leads to the prediction
whose contribution e ectively vanish in the chosen regularization scheme [36, 37].
We
emphasise that this procedure di ers from the one employed in [7], since the nondiagonal
matrix structure of the fermionic uctuation operator for arbitrary 0 prevents us from
factorizing the value of the fermionic determinants into a product of two contributions.
6In presence of zero mode, a possible dependence of the path integral measure on the classical solution
7See also [23], which analyzes the (string dual to the) ratio between the Wilson loop of \antiparallel
In the 0 ! 0 limit, we analytically recover the constant oneloop coe cient in the
expansion of the 1/2BPS circular Wilson loop as found in [7, 10]
loghW ( ; 0 = 0)i = p
3
4
log( ) + log c +
log
1
2
1
2
+ O
1
2 ;
(1.4)
up to an unknown contribution of ghost zeromodes (the constant c). The expression above
is in disagreement with the gauge theory prediction (1.1).
We regularize and normalize the latitude Wilson loop with respect to the circular case.
The summation of the onedimensional Gel'fandYaglom determinants is quite di cult, due
to the appearance of some Lerchtype special functions, and we were not able to obtain a
that the disagreement between sigmamodel and localization results (1.3) is not washed
out yet. Within a certain numerically accuracy, we claim that the discovered 0dependent
discrepancy is very well quanti ed as
(1.5)
(1.6)
log hW ( ; 0)i
hW ( ; 0)i sm
= p
suggesting that the \remainder function" should be
Rem( 0) = log cos 0
of the discrepancy at the end of the manuscript.
The paper proceeds as follows. In section 2 we recall the classical setting, in
section 3 we evaluate the relevant functional determinants which we collect in section 4 to
form the corresponding partition functions. Section 5 contains concluding remarks on the
disagreement with the localization result and its desirable explanation. After a comment
on notation in appendix A, we devote appendix B to a concise survey on the
Gel'fandYaglom method. Appendix C elucidates some properties which simplify the evaluation of
the fermionic contribution to the partition function, while in appendix D we comment on
a possible di erent choice of boundary condition for lower Fourier modes which that not
a ect our results.
{ 4 {
The classical string surface describing the strong coupling regime of the 1=4BPS latitude
was rst found in [17] and discussed in details in [15, 18]. Endowing the AdS5
S5 space
with a Lorentzian metric in global coordinates
with the AdS radius set to 1, the corresponding classical con guration in AdS3
S
2
parametrizes a string worldsheet, ending on a unit circle at the boundary of AdS5 and on
a latitude sitting at polar angle 0 on a twosphere inside the compact space.8 Here the
spans the interval [ 2 ; 2 ]: The worldsheet coordinates instead take values in
The ansatz (2.2) does not propagate along the time direction and de nes an Euclidean
surface embedded in a Lorentzian target space. It satis es the equation of motions
(supplemented by the Virasoro constraints in the Polyakov formulation) when we set
sinh ( ) =
sin ( ) =
1
sinh
cosh ( 0
;
1
)
;
cosh ( ) =
1
tanh
;
cos ( ) = tanh ( 0
) :
An integration constant in (2.3) that shifts
was chosen to be zero so that the worldsheet
boundary at
= 0 is located at the boundary of AdS5. The remaining one, 0 2 [0; 1),
spans the oneparameter family of latitudes on S5 at the boundary
= 0, whose angular
position 0 2 [0; 2 ] relates to 0 through
cos 0 = tanh 0
:
Here the dual gauge theory operator interpolates between two notable cases. The 1/2BPS
circular case falls under this class of Wilson loops when the latitude in S2 shrinks to a
point for 0 = 0, which implies ( ) = 0 and 0 = +1 from (2.3){(2.4). In this case the
string propagates only in AdS3. The other case is the circular 1/4BPS Zarembo Wilson
loop when the worldsheet extends over a maximal circle of S2 for 0 = 2 and 0 = 0 [22].9
The double sign in (2.3) accounts for the existence of two solutions, e ectively doubling the
range of 0: the stable (unstable) con guration mimizes (maximizes) the action functional
and wraps the north pole
= 0 (south pole
= ) of S5.
8There exist other solutions with more wrapping in S5, but they are not supersymmetric [18].
9See also [38] for an analysis of the contribution to the string partition function due to (broken) zero
modes of the solution in [22].
{ 5 {
The induced metric on the worldsheet depends on the latitude angle 0 through the
conformal factor ( i = ( ; ))
ds22D = hij d id j =
2
( ) d 2 + d 2 ;
2
( )
sinh2 ( ) + sin2 ( ) :
The twodimensional Ricci curvature is then
(2)R =
=
2( )
=
(2 cosh 2 0
2 sinh 0 sinh (6
3 0) 3 cosh (2 (
4cosh ( 0) cosh3 (2
0)
0))+ 6 cosh (4
2 0)+3 cosh 2 )
(2.5)
(2.6)
(2.8)
(2.9)
:
(2.10)
p
p
b
(2.12)
The semiclassical analysis is more conveniently carried out in the stereographic
coordinates
m (m = 1; 2; 3) of S3
AdS5 and wn (n = 1; 2; 3; 4; 5) of S5
where the classical solution reads10
10The background of '1; '2; #2 was set to zero in (2.2), but the bosonic quadratic Lagrangian does
not have the standard form (kinetic and mass terms for the eight physical elds) in the initial angular
coordinates.
{ 6 {
The string dynamics is governed by the type IIB GreenSchwarz action, whose bosonic part
is the usual NambuGoto action
Z
SB = T
d d
p
h
Z
d d LB
in which h is the determinant of the induced metric (2.8) and the string tension T = 2
depends on the 't Hooft coupling
. The leading contribution to the string partition
function comes from the regularized classical area [18]
SB(0)( 0) =
p
2
Z 2
0
d
Z 1
0
d
sin2 ( ) + sinh2 ( ) = p
1
cos 0 +
+ O( ) : (2.11)
Following [7] we have chosen to distinguish the cuto
0 in the worldsheet coordinate from
the cuto
= tanh 0 in the Poincare radial coordinate z of AdS. The pole in the IR cuto
in (2.13) keeps track of the boundary singularity of the AdS metric and it is proportional to
the circumference of the boundary circle. The standard regularization scheme, equivalent
to consider a Legendre transform of the action [2, 39], consists in adding a term
proportional to the boundary part of the Euler number
Here g stands for the geodesic curvature of the boundary at
= 0 and ds is the invariant
line element. With this subtraction, we have the value of the regularized classical area
The (uppersign) solution dominates the string path integral and is responsible for the
leading exponential behaviour in (1.2) and so, in the following, we will restrict to the
upper signs in (2.3).
3
uctuation determinants
This section focusses on the semiclassical expansion of the string partition function around
the stable classical solution (2.7) (taking upper signs in (2.3)) and the determinants of the
di erential operators describing the semiclassical uctuations around it. The 2 periodicity
in
allows to trade the 2D spectral problems with in nitelymany 1D spectral problems
for the (Fouriertransformed in
) di erential operators in . Let us call O one of these
oneloop operators. For each Fourier mode !, the evaluation of the determinant Det!O
is a onevariable eigenvalue problem on the semiin nite line
2 [0; 1) which we solve
using the Gel'fandYaglom method, a technique based on function regularization reviewed
in appendix B. Multiplying over all frequencies ! (which are integers or semiintegers
according to the periodicity of the operator O) gives then the full determinant
HJEP02(16)5
DetO = Y Det!O:
!
All our worldsheet operators are intrinsically singular on this range of , since their
principal symbol diverges at
= 0, the physical singularity of the boundary divergence for the
AdS5 metric. Moreover the interval is noncompact, making the spectra continuous and
more di cult to deal with. We consequently introduce an IR cuto
at
= 0 (related
to the
= tanh 0 cuto in z) and one at large values of
= R [7]. While the former is
necessary in order to tame the nearboundary singularity, the latter has to be regarded as
a mere regularization artifact descending from a small ctitious boundary on the tips of
the surfaces in AdS3 and S2. Indeed it disappears in the oneloop e ective action.
3.1
Bosonic sector
The derivation of the bosonic uctuation Lagrangian around the minimalarea surface (2.7)
is readily available in section 5.2 of [40]. The oneloop
uctuation Lagrangian in static
gauge is
(3.1)
(3.2)
(2)
LB
2
( ) yT
OB ( 0) y ;
{ 7 {
to the worldsheet y
(yi)i=1;:::8. In components it reads11
where the di erential operator OB ( 0) acts on the vector of uctuation elds orthogonal
1
;
[OB ( 0)]ij =
where the nonvanishing entries of the matrices are12
The worldsheet excitations decouple in the bosonic sector, apart from y7 and y8 which
are coupled through a 2
2 matrixvalued di erential operator. The determinant of the
bosonic operator is decomposed into the product
(3.4)
;
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
! i!), formula (3.6) holds for each frequency ! with
O1
O2 ( 0)
O3 ( 0)
2 + 3 tanh2 (2 + 0)
2 i tanh (2 + 0) !
2 i tanh (2 + 0) !
dd22 + !2
2 + 3 tanh2 (2 + 0)
!
The unitary matrix U = p1
2
i 1 diagonalizes the operator (3.9)
i 1
O3 ( 0) = U y diagfO3+; O3 g U ;
O3+ ( 0) =
O3 ( 0) =
(3.10)
2 + 3 tanh2 (2 + 0) + 2! tanh (2 + 0) :
We performed a rescaling by p
h =
2( ) (as in the analogous computations of [7, 14, 23])
which will not a ect the nal determinant ratio (4.1) (see discussions in appendix A of [6]
and in [7, 14, 23]) and is actually instrumental for the analysis in appendices B.2 and B.3.
We rewrite (3.6) as follows
Det!OB ( 0) = Det3!O1 Det3!O2 ( 0) Det!O3+ ( 0) Det!O3 ( 0) ;
(3.11)
11To compare with [40], and using the notation used therein, notice that the bosonic Lagrangian is
derived as
L(B2) =
A` iA `j
p
Mij yiyj ;
(3.3)
which de nes in an obvious way mij and nij in (3.4).
12There would be an overall minus sign in the kinetic and mass term of the y1 uctuation, which we
disregard in (3.4) for simplifying the formula, considering that it does not play a practical role in the evaluation
of determinants with Gel'fandYaglom and is reabsorbed in the Wickrotation of the time coordinate t.
{ 8 {
The evaluation of onedimensional spectral problems is outlined in appendix B.2. The
where all the determinants are taken at xed !. To reconstruct the complete bosonic
contribution we have to perform an in nite product over all possible frequencies.
The operator O1 does not depend on 0, and indeed also appears among the circular Wilson
loop uctuation operators [7]. While its contribution formally cancels in the ratio (1.3), we
report it below along with the others for completeness. Both O2 ( 0) and O3 ( 0) become
massless (scalar and matrixvalued respectively) operators in the circular Wilson loop
limit, which is clear for O3 ( 0) upon diagonalization and an integer shift in !,13 irrelevant
for the determinant at given frequency, as long as we do not take products over frequencies
into consideration. Thus, in this limit one recovers the bosonic partition function of [7].
elds satisfy Dirichlet boundary conditions at the endpoints of the compacti ed interval
2 [ 0; R]. Then we take the limit of the value of the regularized determinants for R ! 1
at xed ! and 0
. As evident from the expressions below, the limit on the physical IR
cuto ( in z or equivalently 0 in ) would drastically change the !dependence at this
stage and thus would spoil the product over the frequencies. It is a crucial, a posteriori,
observation that it is only keeping 0 nite while sending R to in nity that one precisely
reproduces the expected large ! (UV) divergences [6, 40]. This comes at the price of more
complicated results for the bosonic (and especially fermionic) determinants. Afterwards
we will remove the IR divergence in the oneloop e ective action by referring the latitude
to the circular solution.
The solutions of the di erential equations governing the di erent determinants are
singular for small subset of frequencies: we shall treat apart these special values when
reporting the solutions. For the determinant of the operator O1 in (3.7) in the limit of
large R one obtains [7]
Det!O1 =
(ej!j(R 0) (2j!j!j +j(cj!otj+h10))
R coth 0
and only the case ! = 0 has to be considered separately. Next we examine the initial value
problem (B.12){(B.13) associated to O2( 0), whose solution is
f(II)1( ) =
8
>
>
<
>
>
>
>
> 2(
>
>
>
>
>>> (
:
1
>> 2! cosh( + 0) cosh( 0+ 0) cosh ( + 0 + 2 0) sinh(!(
+ (!+1) sinh((! 1)(
13In the language of [40], this shift corresponds to a di erent choice of orthonormal vectors that are
orthogonal to the string surface.
{ 9 {
(3.12)
! 6=
! =
The determinant is then given by f(II)1(R) and for R large one obtains the simpler
expression
Det!O2( 0) =
(ej!j(R 0) (j!j+tanh( 0+ 0))
2j!j(j!j+1)
R tanh( 0 + 0)
We repeat the same procedure for O3+ ( 0). From the solutions
>
>
>
>
>
>>>> (e
>
0))
one nds for large R
In view of the relation O3 ( 0) = O3+ ( 0) j!! !, which follows from (3.10), we can easily
deduce the results for Det!O3 ( 0) by ipping the frequency in the lines above
Det!O3+( 0) = <
Notice that a shift of ! ! !
1 in Det!O3+( 0) and ! ! ! + 1 in Det!O3 ( 0) gives
back the symmetry around ! = 0 in the distribution of powerlike and exponential largeR
divergences which characterizes the other determinants (3.12) and (3.14). Such a shift 
also useful for the circular Wilson loop limit as discussed below (3.11)  does not a ect
the determinant, and we will perform it in section 4.
3.2
The
Fermionic sector
uctuation analysis in the fermionic sector can be easily carried out following
again the general approach [40], which includes the local SO(1; 9) rotation in the
target space [6, 41{45] that allows to cast the quadratic GreenSchwarz fermionic action into
eight contributions for twodimensional spinors on the curved worldsheet background.
The standard Type IIB symmetry gauge xing for the rotated fermions
1 =
2
! =
!
1
where the operator OF ( 0) is given by
LF
( )
( )
1
+
The coe cients a34( ) and a56( ) can be expressed as derivatives of the functions appearing
in the classical solution:
In the 0 ! 0 limit (hence ( ) ! 0), one gets
i
2
i
2
OF ( 0 = 0) = i sinh
cosh
3 +
sinh
which coincides with the operator found in the circular Wilson loop analysis of [7],15 once
we go back to Minkowski signature and reabsorbe the connectionrelated
456term via the
dependent rotation
! exp
. In Fourier space this results in a shift of the
integer fermionic frequencies ! by one half, turning periodic fermions into antiperiodic
ones. In the general case (3.20) we cannot eliminate all the connectionrelated terms
a34( ) 3 + a56( ) 456, since the associated normal bundle is non at [40].16 Performing
anyway the above rotation at the level of (3.20) has the merit of simplifying the circular
limit making a direct connection with known results. This is how we will proceed: for
now, we continue with the analysis of the fermionic operator in the form (3.20) without
performing any rotation. Then, in section 4, we shall take into account the e ect of this
rotation by relabelling the fermionic Fourier modes in terms of a suitable choice of
halfintegers.
The analysis of the fermionic operator (3.20) drastically simpli es noticing that the
set of mutuallycommuting matrices f 12; 56; 89g commutes with the operator itself and
leaves invariant the spinor constraint (A.6) and the fermionic gauge xing (3.18). By means
of the projectors
P12
I32
2
i 12 ;
P56
I32
i 56
2
and
P89
I32
2
i 89 ;
(3.23)
14We perform the computations in a Lorentzian signature for the induced worldsheet metric and only
at the end Wickrotate back. The di erence with (5.37){(5.38) of [40] is only in labeling the spacetime
directions.
15See formula (5.17) therein.
16The arising of gauge connections in the covariant derivatives associated to the structure of normal
bundle is discussed at length in [40] and references therein. In particular, see discussion in section 5.2
of [40] for both the latitude and the circular Wilson loop limit.
(3.19)
(3.20)
(3.21)
HJEP02(16)5
we decompose the 32
32 fermionic operator into eight blocks of 2
2 operators labeled
by the triplet fp12; p56; p89 =
1; 1g. Formally this can be seen as the decomposition into
the following orthogonal subspaces
Notice that the operator de ned in (3.26) actually does not depend on the label p89. Then
the spectral problem reduces to the computation of eight 2D functional determinants17
DetOF ( 0) =
Y
A deeper look at the properties of OF
p12;p56;p89 allows us to focus just on the case of p12 =
p56 = p89 = 1. In fact, as motivated in details in appendix C.1, the total determinant can
be rewritten as follows
DetOF ( 0) =
Det![(OF1;1;1(!))2]2Det![(OF1;1;1( !))2]2 :
Using the matrix representation (A.3) and going to Fourier space, we obtain
+
i
( )
1
2( )
Y
!2Z
h i
+
1
( )
1
2( )
. For simplicity of notation, from now on we will denote with
O1;1;1( 0) the rst factor in the de nition above. In a similar spirit to the analysis for the
F
bosonic sector, we start to nd the solutions of the homogeneous problem
OF
1;1;1 ( 0) f ( ) = 0
where f ( ) denotes the two component spinor (f1( ); f2( ))T . The system of coupled
rstorder di erential equations now reads
sin2 ( )
!
a34( ) + a56( )) f2( ) = 0;
a34( )
a56( )) f1( ) = 0:
OF
1;1;1( 0)
a34( ) 1 + ia56( ) 2
sinh2 ( ) 3
sin2 ( )I2
i
M
OeF
1;1;1
M ;
17A nontrivial matrix structure is also encountered in the fermionic sector of the circular Wilson loop [7],
but the absence of a background geometry in S5 leads to a simpler gamma structure. It comprised only three
gamma combinations ( 0; 4; 04), whose algebra allows their identi cation with the three Pauli matrices
without the need of the labelling the subspaces.
We can cast it into a secondorder di erential equation for one of the unknown functions.
Solving (3.32) for f2( )
i
( )
tanh ( + 0)
2
f1( ) ;
(3.33)
and then plugging it into (3.31) one obtains
00
f1 ( )
1
2 sinh2
1
2 cosh2 ( + 0)
+
+
tanh( + 0)
2
!
2
f1( ) = 0: (3.34)
It is worth noticing that the Gel'fandYaglom method has naturally led to an
auxiliary Schrodinger equation for a
ctitious particle on a semiin nite line and subject
HJEP02(16)5
to a supersymmetric potential V ( ) =
1
W ( ) = 2 tanh
+ tanh( + 0)
2
W 0( ) + W 2( ) derived from the prepotential
!. Traces of supersymmetry are not surprising: they
represent a vestige of the supercharges unbroken by the classical background.18
As in the bosonic case, we have to separately discuss some critical values of the
frequencies. We only report the independent solutions of the equations above, where the
constants ci;1 and ci;2 have to be xed in the desired initial value problem (i = I; II).
8 ci;1e (1+!)+ci;2e (1 !)+ 0 (2!2 cosh( + 0) sinh +! cosh(2 + 0)+sinh 0)
18The same property is showed by (5.26) in [7].
2 tanh
1
1
2 tanh
>> ci;1 q(e2 1)(e2 0 +1)(e2 +2 0 +1)(e4 +2 0 +1)
>
>
i(2e
e3 +e3 +2 0 )
ie (2 !)+2 0 (2!+sinh(2 +2 0)+2! sinh 0 sinh(2 + 0) sinh 2 )
q(e2
1)(e2 0 +1)(e2 +2 0 +1)(e4 +2 0 +1)
! 6=
> ci;1 q(e2 1)(e2 0 +1)(e2 +2 0 +1)
>
>
We are now ready to evaluate the determinants using the results of appendix (B.3), namely
considering Dirichlet boundary conditions for the square of the rst order di erential
operator. Having in mind the solutions above and how they enter in (B.9) and (B.22), it is clear
that already the integrand in (B.27) is signi cantly complicated. A simpli cation occurs
by recalling that our nal goal is taking the R !
1 limit of all determinants and combine
them in the ratio of bosonic and fermionic contributions. As stated above in the bosonic
analysis and shown explicitly below, for the correct large ! divergences to be reproduced,
it is crucial to send R ! 1 while keeping
nite. In appendix (C.2) we sketch how to
use the main structure of the matrix of the solutions Y ( ) to obtain the desired largeR
expressions for the determinants in a more direct way.
The determinant of the operator O1;1;1 for modes ! 6= f 1; 0; 1g reads for large R
F
(z; s; a) is the Lerch transcendent (4.5). The presence of the Lerch function is
just a tool to have a compact expression for the determinants. In fact, for the values of !
relevant for us, it can be can be written in terms of elementary functions, but its expression
becomes more and more unhandy as the value of ! increases. The coe cients ai and bi
can be also expressed in terms of elementary functions. For the ai we have
a0 = e R 320 sinh 0 (tanh 0 + 1) cosh ( 0 + 0)
8p2 cosh ( 0 + 2 0)
a1 = 4 sech 0(tanh 0 + !)2
a2 = 4[2 1
!
2 !2 cosh 0
a3 = tanh2 0 (coth 0 + 1) csch 0 sech ( 0 + 0) e 0 (cosh 0
2 sinh 0
sinh(2 0
0))
!
2 ! sinh 0 + sech 0 sech2 0 + !
2
+ cosh 0 2!2 +! +3!2 csch 0 cosh( 0 +2 0) sech( 0 + 0)+! coth2 0 +2 coth 0 2
+ 2 3! cosh 0 sech( 0 + 0) sinh 0 ! 2! coth 0 csch2 0
sech 0(coth 0 +1)
i
while for the bi we get
b0 = e
R 20 sech2 0
sinh 0 (tanh 0 + 1) cosh ( 0 + 0)
8p2 cosh( 0 + 2 0)
b1 =
b2 =
b3 =
2
2 ! ! cosh(2 0) + sinh(2 0) + !
2
1
cosh2 0 4! tanh( 0 + 0)
2! coth 0 + csch2 0
!
cosh(2 0)(! + 1)
sinh(2 0) + cosh( 0
0)sech ( 0 + 0) :
(3.39)
The determinants of the lower modes have to be computed separately and they are given by
+
We report here the 0 ! 1 limit of all the bosonic and fermionic determinants, representing
the circular Wilson loop case 0 = 0. The result for Det!O1 in (3.12) stays obviously the
same, while for the limits of (3.14), (3.16) and (3.17) one easily gets
The fermionic contributions (3.37){(3.42) reduce in this limit to
Det!
h
OF
1;1;1( 0 = 0)
Det!O2( 0 = 0) =
Det!O3+( 0 = 0) = <
R
Det!O3 ( 0 = 0) = <
R
( ej!j(R 0)
2j!j
R
8 e(R 0)(! 1)
4(! 1)!2 (e2 0 1)
2 i
R eR+ 0
= >< 2(e2 0 1)
> e3(R
0) (2e2 0 1)
16(e2 0 1)
>>> e (R 0)(2! 1)((! 1)e2 0 !)
4(! 1)2! (e2 0 1)
We now put together the determinants evaluated in the previous sections and present the
oneloop partition functions for the open strings representing the latitude ( 0 6= 0) and the
circular ( 0 = 0) Wilson loop, eventually calculating their ratio.
In the case of fermionic determinants, as motivated by the discussion below (3.22),
we will consider the relevant formulas (3.37){(3.42) relabelled using halfinteger Fourier
modes. In fact, once projected onto the subspace labelled by (p12; p56; p89), the spinor
is
an eigenstate of 56 with eigenvalue
ip56 and the rotation
a shift of the Fourier modes by ! ! ! + p256 . This in particular means that below we will
consider (3.37){(3.42) e ectively evaluated for ! = s + 12 and labeled by the halfinteger
frequency s. In the bosonic sector  as discussed around (3.11) and (3.17)  we pose
! = `+1 in Det!O3+ together with ! = ` 1 in Det!O3 . This relabeling of the frequences
provides in (3.16) and (3.17) a distribution of the Rdivergences that is centered around
! exp
ej`jR for ` 6= 0) in the same way (in !) as
for the other bosonic determinants (3.12) and (3.14). This will turn out to be useful while
discussing the cancellation of Rdependence. Recalling also (2.13), we write the formal
expression of the oneloop string action
To proceed, we rewrite (4.1) as the (still unregularized) sum
(4.1)
(4.2)
(4.3)
(4.4)
log Det`O1( 0) +
log Det`O2( 0) +
log Det`O3+ +
log[Det`O3
log Dets(OF1;1;1)2 +
log Det s(OF1;1;1)2 :
1
2
1
2
Equation (4.2) has the same form with e ectively antiperiodic fermions encountered
Introducing the small exponential regulator , we proceed with the \supersymmetric
regularization" of the oneloop e ective action proposed in [36, 37]
where the (weighted) bosonic and fermionic contributions read
B
` ( 0) =
sF ( 0) =
3
2
4
2
p
X
s2Z+1=2
cos 0 + (
1
)( 0)
sF ( 0) ;
X
`2Z
3
2
4
2
2
In the rst sum (where the divergence is the same as in the original sum) one can remove
by sending
! 0, and use a cuto regularization in the summation index j`j
tantly, the nonphysical regulator R disappears in (4.4). While in [7]19 the Rdependence
.
Impordrops out in each summand, here it occurs as a subtle e ect of the regularization scheme,
and comes in the form of a crosscancellation between the rst and the second line once the
sums have been carried out. The di erence in the Rdivergence cancellation mechanism is
a consequence of the di erent arrangement of fermionic frequencies in our regularization
scheme (4.4). In the circular case ( 0 = 0) this cancellation can be seen analytically, as
in (4.10){(4.11) below. The same can be then inferred for the general latitude case, since
in the normalized oneloop e ective action
that the Rdependence drops out in each summand.
A nontrivial consistency check of (4.4) is to con rm that in the large ` limit the
expected UV divergences [6, 40] are reproduced. Importantly, for this to happen one
cannot take the limit 0 ! 0 in the determinants above before considering `
1, which is
the reason why we kept dealing with the complicated expressions for fermionic determinants
above. Using for the Lerch transcendent in (3.37)
(z; s; a)
z
n
1
X
n=0 (n + a)s
where
v( 0) =
p
h (2)R
the asymptotic behavior for jaj
1 (i.e. j`j
one nds that the leading divergence is logarithmic, and  as expected from an analysis
in terms of the SeeleyDe Witt coe cients [6, 40]  proportional to the volume part of
the Euler number
and we notice that this limit is independent from
0 ( 0). This divergence should be
cancelled via completion of the Euler number with its boundary contribution (2.12) and
inclusion of the (opposite sign) measure contribution, as discussed in [6, 7]. Having this in
mind, we will proceed subtracting (4.7) by hand in (
1
)( 0) and in (
1
)( 0 = 0).
19In this reference a regularization slightly di erent from [36, 37] was adopted.
4.1
The UVregulated partition function in the circular Wilson loop limit reads
(
1
)
UVreg( 0 = 0) =
X
`
j j
+
`
2
The rst line is now convergent and its total contribution evaluates for
where
is Euler gamma function. The Rdependence in (4.10) cancels against the O( 0)
contribution stemming from the regularizationinduced sum in the second line of (4.9)
`
F
`+ 12 (0)
X e
X e `
` 3
= 2R
2 arctanh :
2R + log
Summing all contributions and nally taking
! 0, the result is precisely as in [7]
despite the di erent frequency arrangement we commented on. We have checked that the
same result is obtained employing function regularization in the sum over `. The same
nite part was found in [10] via heat kernel methods. There is no theoretical motivation
for the log = divergences appearing in (4.12), which will be cancelled in the ratio (1.3).
In [7], this kind of subtraction has been done by considering the ratio between the circular
and the straight line Wilson loop.
4.2
Ratio between latitude and circular Wilson loops
In this section we describe the evaluation of the ratio (1.3)
log
Z ( ; 0)
Z ( ; 0)
= p
(4.10)
(4.11)
(4.12)
where
UVreg( 0 = 0) is in (4.9) and
UVreg( 0) is regularized analogously. The
complicated fermionic determinants (3.37){(3.39) make an analytical treatment highly nontrivial,
and we proceed numerically.
First, we spell out (4.13) as
(
1
)
` h F
`+ 12 (0)
( v( 0)
v(0)) log
+
`F+ 12 (0)+ `F 12 (0)
2
`F+ 12 (0)+ `F 12 (0)
+
+
`F+ 12 ( 0)+ `F 12 ( 0) #
`F+ 12 ( 0)+ `F 12 ( 0) #
2
2
where we separated the lower modes j`j
latter we have used parity ` !
limit
! 0.21 The sum with large cuto
EulerMaclaurin formula
2 from the sum in the second line,20 and in the
`. The sum multiplied by the small cuto
is zero in the
can be then numerically evaluated using the
(4.15)
(4.16)
n
X
`=m+1
f (`) =
f (`) d` +
Z n
m
Z n
m
f (n)
f (m)
2
B2p (f`g) d` ;
(2p)!
f (2p) (`)
p
1 ;
p
+ X
k=1 (2k)!
B2k hf (2k 1) (n)
f (2k 1) (m)i
in which Bn(x) is the nth Bernoulli polynomial, Bn = Bn(0) is the nth Bernoulli number,
f`g is the integer part of `, f (`) is the summand in the second line of (4.14), so m = 2,
n =
. After some manipulations to improve the rate of convergence of the integrals, we
safely send
! 1 in order to evaluate the normalized e ective action
=
h (
1
)
2
X
`= 2
+
Z 1
f (2)
2
2
i
sm
`
relabeling discussed above.
cuto makes the sum vanish.
20This is convenient because of the di erent form for the special modes (3.40){(3.42) together with the
21This can be proved analytically since the summand behaves as e `` 2 for large `. Removing the
In order to gain numerical stability for large `, above we have set p = 3, we have cast the
Lerch transcendents inside
sF ( 0)  see (3.37)  into hypergeometric functions
(z; 1; a) = 2F1(1; a; a + 1; z)
;
jzj < 1 ^ z 6= 0 ;
and we have approximated the derivatives f (k)(`) by nitedi erence expressions
i
a
f (k)(`) !
k
i=0
` k X(
1
)i k
f ` + ( k2
i) ` ;
`
(4.17)
(4.18)
At this stage, the expression (4.16) is only a function of the latitude parameter
the polar angle 0 in (2.4)) and of two parameters  the IR cuto
0 and the derivative
discretization
`, both small compared to a given 0
. We have tuned them in order to
con dently extract four decimal digits. In
gure 1a we compare the regularized oneloop
e ective action obtained from the perturbation theory of the string sigmamodel (4.16) to
the gauge theory prediction from (1.2)
h (
1
)
(
1
)
UVreg( 0) loc
i
=
3
2
log tanh 0
(4.19)
for di erent values of 0. Data points cover almost entirely22 the niteangle region between
the Zarembo Wilson loop ( 0 = 0; 0 = 2 ) and the circular Wilson loop ( 0 = 1; 0 = 0).
The vanishing of the normalized e ective action in the large 0 region is a trivial
check of the normalization. As soon as the opposite limit
0 = 0 is approached, the
di erence (4.16) bends up \following" the localization curve (4.19) but also signi cantly
deviates from it, and the measured discrepancy is incompatible with our error estimation.
Numerics is however accurate enough to quantify the gap between the two plots on a
wide range. Figure 1b shows that, surprisingly, such gap perfectly overlaps a very simple
function of 0 within the sought accuracy
1
2
Rem( 0)
We notice at this point that the same simple result above can be obtained taking in (4.14)
the limit of
! 0 before performing the sums. As one can check, in this limit UV and
IR divergences cancel in the ratio.23 the special functions in the fermionic determinants
disappear and, because in general summands drastically simplify, one can proceed
analytically getting the same result calculated in terms of numerics. We remark however that
such inversion of the order of sum and limit on the IR cuto cannot be a priori justi ed, as
it would improperly relate the
cuto
with a 1= cuto (e.g. forcing ` to be smaller than
1= ). As emphasized above, in this limit the e ective actions for the latitude and circular
case separately do not reproduce the expected UV divergences. Therefore, the fact that in
22When pushed to higher accuracy, numerics is computationally expensive in the vicinity of the two
limiting cases ( 0 = 0; 0 = 2
) and( 0 = 1; 0 = 0).
23This is also due to the volume part of the Euler number v( 0) being independent of 0 up to
corrections, see (4.8).
Comparison
between
in (4.16) (orange dots) and
in (4.19) (blue line). We set 0 = 10 7,
from supersymmetric localization for the ratio between latitude and circular Wilson loops in terms
of the corresponding oneloop sigmamodel (di erences of) e ective actions.
this limit the summands in the di erence (4.14) show a special property of convergence 
which we have not analyzed in details  and lead to the exact result is a priori highly not
obvious, rendering the numerical analysis carried out in this section a rather necessary step.
On a related note, the simplicity of the result (4.20) and the possibility of getting an
analytical result for the maximal circle 0 = 0 suggest that the summation (4.4) could
have been performed analytically also in the latitude case 0 6= 0. We have not further
investigated this direction.
5
Conclusions
In this paper we calculated the ratio between the AdS5
S5 superstring oneloop partition
functions of two supersymmetric Wilson loops with the same topology. In so doing, we
address the question whether such procedure  which should eliminate possible ambiguities
related to the measure of the partition function, under the assumption that the latter only
depends on worldsheet topology  leads to a longsought agreement with the exact result
known via localization at this order, formula (4.19).
Our answer is that, in the standard setup we have considered for the evaluation of the
oneloop determinants (GelfandYaglom approach with Dirichlet boundary conditions at
the boundaries, of which one ctitious24), the agreement is not found. A simple numerical
t allows us to quantify exactly a \remainder function", formula (4.20).25
As already emphasized, the expectation that considering the ratio of string partition
functions dual to Wilson loops with the same topology should cancel measurerelated
ambiguities is founded on the assumption that the partition function measure is actually not
depending on the particular classical solution considered. Although motivated in light of
literature examples similar in spirit (see Introduction), this remains an assumption, and it
is not possible to exclude a priori a geometric interpretation for the observed discrepancy.
One reasonable expectation is that the disagreement should be cured by a change of the
worldsheet computational setup, tailored so to naturally lend itself to a regularization
scheme equivalent to the one (implicitly) assumed by the localization calculation.26 One
possibility is a choice of boundary conditions for the fermionic spectral problem27 di erent
from the standard ones here adopted for the squared fermionic operator.28 Also, ideally
one should evaluate determinants in a di eomorphismpreserving regularization scheme. In
that it treats asymmetrically the worldsheet coordinates, the by now standard procedure
of employing the Gel'fandYaglom technique for the e ective (after Fouriertransforming in
) onedimensional case at hand does not fall by de nition in this class. In other words, the
choice of using a function like regularization  the Gel'fandYaglom method  in
and
a cuto regularization in Fourier !modes is a priori arbitrary. To bypass these issues it
would be desirable to fully develop a higherdimensional formalism on the lines of [50, 51].
A likewise fully twodimensional method to deal with the spectral problems is the heat
kernel approach, which has been employed at least for the circular Wilson loop case (where
the relevant string worldsheet is the homogenous subspace AdS2) in [10, 11]. As there
explained, the procedure bypasses the need of a large
regulator and makes
appear only
in the AdS2 regularized volume, the latter being a constant multiplying the traced heat
kernel and thus appearing as an overall factor in the e ective action. This is di erent from
what happens with the Gel'fandYaglom method, where di erent modes carry a di erent
structure and one has to identify and subtract by hand the divergence in the oneloop
e ective action. However, little is known about heat kernel explicit expressions for the
spectra of Laplace and Dirac operators in arbitrary twodimensional manifolds, as it is the
case as soon as the parameter 0 is turned on. The application of the heat kernel method
for the latitude Wilson loop seems then feasible only in a perturbative approach, i.e. in the
24See also appendix D where a minimally di erent choice for the boundary conditions on the bosonic and
fermionic modes with small Fourier mode is considered, and shown not to a ect the
nal result.
25See also discussion below (4.20), where we notice that the same result is obtained analytically via the
a priori not justi ed \orderoflimits" inversion.
26Morally, this resembles the quest for an \integrabilitypreserving" regularization scheme, di erent from
the most natural one suggested by worldsheet eld theory considerations, in the worldsheet calculations of
27For the bosonic sector, we do not
lightlike cusps in N = 4 SYM [47] and ABJM theory [48].
nd a reasonable alternative to the Dirichlet boundary conditions.
28For example, instead of squaring one could consider the Diraclike
rstorder operator (3.29). Then,
Dirichlet boundary conditions would lead to an overdetermined system for the arbitrary integration
constants of the 2
2 matrixvalued, rstorder eigenvalue problem. The question of the non obvious alternative
to consider is likely to be tied to a search of SUSYpreserving boundary conditions on the lines of [49].
small 0 regime when the worldsheet geometry is nearly AdS2.29 It is highly desirable to
address these or further possibilities in future investigations.
Acknowledgments
We acknowledge useful discussions with Xinyi ChenLin, Amit Dekel, Sergey Frolov, Simone
Giombi, Jaume Gomis, Thomas Klose, Shota Komatsu, Martin Kruczenski, Daniel Medina
Rincon, Diego Trancanelli, Pedro Vieira, Leo Pando Zayas, and in particular with Nadav
Drukker, Arkady Tseytlin, and Konstantin Zarembo. We also thank A. Tseytlin and the
Referee of the published version for useful comments on the manuscript. The work of VF
and EV is funded by DFG via the Emmy Noether Programme \Gauge Field from Strings".
VF thanks the kind hospitality, during completion of this work, of the Yukawa Institute for
Theoretical Physics in Kyoto, the Centro de Ciencias de Benasque \Pedro Pascual", the
Institute of Physics in Yerevan and in Tbilisi. The research of VGMP was supported in part
by the University of Iceland Research Fund. EV acknowledges support from the Research
Training Group GK 1504 \Mass, Spectrum, Symmetry" and from the Seventh Framework
Programme [FP7People2010IRSES] under grant agreement n. 269217 (UNIFY), and
would like to thank the Perimeter Institute for Theoretical Physics and NORDITA for
hospitality during the completion of this work. All authors would like to thank the Galileo
Galilei Institute for Theoretical Physics for hospitality during the completion of this work.
A
Notation and conventions
We adopt the following conventions on indices, when not otherwise stated,
M; N; : : : = 0; : : : ; 9
curved targetspace indices
A; B; : : : = 0; : : : ; 9
at targetspace indices
i; j; :: = 0; 1
a; b; :: = 0; 1
curved worldsheet indices
at worldsheet indices
Flat and curved 32
32 Dirac matrices are respectively denoted by
A and
M and satisfy
the so (1; 9) algebra
f A; Bg = 2 ABI32
f M ; N g = 2GMN I32;
where AB = diag ( 1; +1; : : : ; +1) and GMN is the targetspace metric (2.5).
We use the explicit representation for the 10D gamma matrices
0 = i ( 3
1 = (I2
2 = (I2
3 = ( 1
4 = (
2
2)
I4
1)
3)
2)
I4
I4
I4
2)
I4
1
1
1
1
1
5 = I4
6 = I4
7 = I4
8 = I4
9 = I4
( 3
( 1
(
(I2
(I2
2
2)
2
2)
2)
1)
3)
2
2
2
2
29We are grateful to A. Tseytlin for a discussion on these points.
(A.1)
(A.2)
(A.3)
The symbol In stands for the n
n identity matrix and 1
; 2; 3 for the Pauli matrices.
It is also useful to report the combination that appears in the expansion of the fermionic
Lagrangian (3.26) The two 10D spinors of type IIB string theory have the same chirality
HJEP02(16)5
034 = (I2
2)
I4
1
:
11 I =
I
;
I; J = 1; 2 :
(A.4)
(A.5)
(A.6)
(A.7)
accompanied by the chirality matrix
minant Det! over
the relation (3.1) holds
In Lorentzian signature they are subject to the Majorana condition, but this cannot be
consistently imposed after Wickrotation of the AdS global time t. This constraint, which
would halve the number of fermionic degrees of freedom, reappears as a factor 1=2 in the
exponent of fermionic determinants (4.1).
Throughout the paper we make a notational distinction between the algebraic
determinant det and the functional determinant Det, involving the determinant on the matrix
indices as well as on the space spanned by ( ; ). We also introduce the functional
deterfor a given Fourier mode !, understanding that for any operator O
DetO = Y Det!O:
!
The boundary condition along the compact direction speci es if the product is over
integers or halfintegers. The issue related to the regularization of the in nite product
is addressed in the main text. The frequencies ! label the integer modes in the
Fouriertransformed bosonic and fermionic operators. We change notation and use ` for the integer
and s for the halfinteger frequencies of the (bosonic and fermionic resp.) determinants
entering the cuto regularized in nite products (more details in section 4).
Finally, a comment on the functions these matrix operators act on. They are column
vectors of functions generically denoted by f
(f1; f2; : : : ; fr)T . Computing functional
determinants with the techniques presented in appendix B involves solving linear di
erential equations, whose independent solutions f(i)
f(i)1; f(i)2; : : : ; f(i)r
T are labelled by
Roman numerals i = I; II; : : :.
B
Methods for functional determinants
The evaluation of the oneloop partition function requires the knowledge of several
functional determinants of onedimensional di erential operators  the operators in Fourier
space at xed frequency Det! (see appendix A). This task can be simpli ed via the
procedure of Gel'fand and Yaglom [29] (for a pedagogical review on the topic, see [52]). This
algorithm has the advantage of computing ratios of determinants bypassing the
computation of the full set of eigenvalues and is based on the solution of an auxiliary initial value
problem.30
To illustrate how to proceed, let us consider the situation we typically encounter
Det!O ;
Det!O^
of freedom)31
in which the linear di erential operators O; O^ are either of rst order (for fermionic degrees
d
d
O = P0( )
+ P1( ) ;
O^ = P0( )
+ P^1( ) ;
d
d
or of second order (in the case of bosonic excitations)
O = P0( )
d
2
d 2 + P1( )
d
d
+ P2( ) ;
O^ = P0( )
d
2
d 2 + P^1( )
d
d
+ P^2( ):
(B.1)
(B.2)
(B.3)
HJEP02(16)5
The coe cients above are complex matrices, continuous functions of
on the nite interval
I = [a; b].
In appendix B.1 we deal with a class of spectral problems not plagued by zero modes
(vanishing eigenvalues) for chosen boundary conditions on the function space.32 We closely
follow the technology developed by Forman [30, 31], who gave a prescription to work with
even more general elliptic boundary value problems. We collected all the relevant formulas
descending from his theorem for the bosonic sector in appendix B.2, and for the square of
the 2D fermionic operators in appendix B.3.
Let us also stress again that the Gel'fandYaglom method and its extensions evaluate
ratios of determinants. Whenever we report the value of one single determinant here and
in the main text, the equal sign has to be understood up to a factor that drops out in
the normalized determinant. The reference operator can be any operator with the same
principal symbol. The discrepancy can be in principle quanti ed for a vast class of operators
with \separated" boundary conditions [62{64], i.e. where conditions at one boundary are
not mixed with conditions at the other one.
B.1
Di erential operators of the nthorder
We consider the couple of norder ordinary di erential operators in one variable
O = P0( )
n 1
k=0
d
n
d n + X Pn k( )
d
k
d k
;
O^ = P0( )
n 1
k=0
d
n
d n + X P^n k( )
d
k
d k
(B.4)
30This algorithms has been used for several examples of oneloop computations which perfectly reproduce
nontrivial predictions from \reciprocity constraints" [53] (see also [54] and [55]), and the general equivalence
between Polyakov and NambuGoto 1loop partition function around nontrivial solutions [56]. Further
oneloop computations reproducing predictions from quantum integrability are in [57{60].
31See next section for a comment on the coincidence of the coe
cient P0( ) of the higherderivative term.
32We mention that, for the plethora of physical situations where it is interesting to project zero modes out
from the spectrum, the reader is referred to the results of [34] for selfadjoint operators of the SturmLiouville
type as well as [32, 33, 61] and references therein.
with coe cients being r
r complex matrices. The main assumption is that the principal
symbols of the two operators (proportional to the coe cient P0( ) of the highestorder
derivative) must be equal and invertible (detP0( ) 6= 0) on the whole
nite interval I =
[a; b]. This ensures that the leading behaviour of the eigenvalues is comparable, thus the
ratio is wellde ned despite the fact each determinant is formally the product of in
nitelymany eigenvalues of increasing magnitude. We do not impose further conditions on the
matrix coe cients, besides the requirement of being continuous functions on I.
The operators act on the space of squareintegrable rcomponent functions f
(f1; f2; : : : ; fr)T
2 L2 (I), where for our purposes one de nes the Hilbert inner product
( stands for complex conjugation)
The inclusion of the nontrivial measure factor, given by the volume element on the
classical worldsheet
h =
( ), guarantees that the worldsheet operators are selfadjoint when
supplemented with appropriate boundary conditions.33 Indeed, to complete the
characterisation of the set of functions, one speci es the nr nr constant matrices M; N implementing
the linear boundary conditions at the extrema of I
0
B
M BB
B
f (a)
dd f (a) C
.
.
.
d n 1 f (a)
1
C
A
CC + N BB
0
B
B
f (b)
dd f (b) C
1
.
.
.
d n 1 f (b)
0 0 1
B 0 C
A
CC = BBB ... CCC :
C
The particular signi cance of the Gel'fandYaglom theorem and its extensions, specialized
in [30, 31] to elliptic di erential operators, lies in the fact that it astonishingly cuts down
the complexity of nding the spectrum of the operators of interests
Of ( ) = f ( ) ;
O^f^^( ) = ^f^^( );
(B.5)
(B.6)
(B.7)
(B.8)
and then nding a meromorphic extension of function. All this is encoded into the elegant
formula
DDeett!!OO^ =
exp nRab tr R( )P1( )P0 1( ) d
exp nRab tr hR( )P^1( )P0 1( )i d
o
o
O
;
for the ratio (B.1), and where R is de ned below. This result agrees with the one obtained
via
function regularization for elliptic di erential operators. Notice that any constant
rescaling of M; N in (B.6) leaves the ratio una ected. Moreover, if also the
nexttohigherderivative coe cients coincide (P1( ) = P^1( )), the exponential factors cancel out. The
33The rescaling of the operators by ph operated in the main text removes the measure from this formula;
see appendix A in [6].
.
.
.
O
(B.9)
(B.10)
(B.11)
nr matrix
YO( ) = BB
0
B
B
f(I)( )
dd f(I)( )
.
.
.
.
.
.
: : :
.
.
.
f(II)( )
dd f(II)( ) : : :
f(nr)( )
dd f(nr)( ) C
1
C
C
C
A
@ ddnn 11 f(I)( ) ddnn 11 f(II)( ) : : : ddnn 11 f(nr)( )
accommodates all the independent homogeneous solutions of
Of(i)( ) = 0
i = I; II; : : : ; 2r
chosen such that YO (a) = Inr. It can be thought of as the fundamental matrix of the
equivalent rstorder operator acting on nrtuples of functions. Y ^( ) is similarly de ned
with respect to O^.
If we restrict to evenorder di erential operators, then R( ) = 12 Inr and (B.8) simpli es:
DDeett!!OO^ =
exp n 12 Rab tr P1( )P0 1( ) d
exp n 12 Rab tr hP^1( )P0 1( )i d
o
det [M + N YO (b)]
det M + N Y ^ (b)
O
:
( i)n P0( ), has no intersection with the cone C
For odd n one gets a slightly more complicated structure, constructed as follows. Let us
assume that the (generalized) spectrum of the principal symbol of O; O^, i.e. the matrix
fz 2 Cj 1 < argz < 2g for some choice
of 1
; 2. This is to say that O has principal angle between 1 and 2. It also follows that
no eigenvalue falls in the opposite cone
C
f
z 2 Cj 1 +
< argz < 2 +
gg when n is
odd. Consequently, the nitelymany eigenvalues fall under two sets, depending on which
C) they belong to. The matrix R( ) is then de ned34 as the projector
onto the subspace spanned by the eigenvectors corresponding to all eigenvalues in one of
these two subsets of the complex plane.
We did not use this formula for odd n in this paper, but notice that this machinery
could be potentially applied to the rstorder fermionic (3.29) operator.
B.2
Applications
We list the applications of the theorem (B.8) for the scalar/matrixvalued operators in
the main text. In the following we leave out formulas for hatted operators and solutions
in order not to clutter formulas, understanding that they satisfy the same initial value
problems.
Secondorder scalarvalued di erential operators O1, O2 ( 0), O3 ( 0),
Dirichlet boundary conditions f1 ( 0) = f1 (R) = 0.
!
M =
N =
!
f(II)1 (R)
(B.12)
34Up to a factor n1 , see amendment in [31].
The normalization of the matrix (B.9) tells that the function f(II)1 ( ) solves the
initial value problem
f(0I0I)1( ) + P2( )f(II)1( ) = 0 f(II)1 ( 0) = 0
f(0II)1 ( 0) = 1: (B.13)
2 matrixvalued di erential operators O3 ( 0),
Dirichlet boundary conditions f1 ( 0) = f2 ( 0) = f1 (R) = f2 (R) = 0.
where
f(III)1 ( 0) = f(III)2 ( 0) = f(0III)2 ( 0) = 0 f(0III)1 ( 0) = 1
f(0I0II)2( ) + P2( ) ff((IIIIII))12(( )) ! = 0
f(0I0II)1( ) !
0
f(0I0V )1( ) !
f(0I0V )2( ) + P2( ) ff((IIVV ))21(( )) ! = 0
0
!
!
f(IV )1 ( 0) = f(IV )2 ( 0) = f(0IV )1 ( 0) = 0 f(0IV )2 ( 0) = 1 :
2 matrixvalued di erential operators OF
Dirichlet boundary conditions f1 ( 0) = f2 ( 0) = f1 (R) = f2 (R) = 0.
p12;p56;p89 ( 0) 2,
BB 10 00 00 00 CACC YO ( ) = ff((II))12 (( )) ff((IIII))12 (( )) !
@
Det! P0 ( ) dd + P1 ( ) 2
Det! hP0 ( ) dd + P^1 ( )i2 =
R R0 dsYO 1 (s)P0 1 (s)YO (s)
R R0 dsY ^ 1 (s)P0 1 (s)YO^ (s)
O
with
P0( ) f(0I)2( ) + P1( ) ff((II))12(( )) ! = 0
f(0I)1( ) !
0
P0( ) f(0II)2( ) + P1( ) ff((IIII))12(( )) ! = 0
f(0II)1( ) !
0
!
!
This is a corollary of (B.27).
f(I)1 ( 0) = 1 f(I)2 ( 0) = 0
f(II)1 ( 0) = 0 f(II)2 ( 0) = 1:
(B.14)
HJEP02(16)5
(B.15)
(B.16)
(B.17)
Square of rstorder di erential operators
As a consequence of the Forman's construction, we can easily compute the ratio of
determinants of the square of rstorder operators with reference only to the operators themselves.
Consider the matrix operator of the form (B.2)35
O = P0 ( )
+ P1 ( )
and denote by Y ( ) its fundamental matrix, which solves the equation (here 0 is the
derivative with respect to )
P0 ( ) Y 0 ( ) + P1 ( ) YO ( ) = 0;
O
YO (a) = Ir:
ds YO 1 (s) P0 1 (s) YO (s) P0 (a)
Z (a) = 0
Z0 (a) = Ir: (B.23)
encapsulates the solutions of Of = 0 and two more ones of O2f = 0.
Suppose that the spectral problem of the squared operator is determined by the
bound
MO2 f (a) + NO2 f (b) = 0 :
After some algebra, successive applications of (B.11), (B.22), (B.23) bring
O
Z b
a
in which
ary condition
then (B.25) gives
(B.19)
(B.20)
HJEP02(16)5
(B.24)
(B.25)
(B.26)
The matrix of fundamental solutions of the square of this operator
2
h
O
2 = P02 ( )
d 2 + P0 ( ) P00 ( ) + fP0 ( ) ; P1 ( )
can be constructed via the method of reduction of order as
YO2 ( ) =
O
O
Y ( )
Y 0 ( )
Z ( ) Y 0 (a)
Z0 ( ) Y 0 (a)
O
O
i d
g d
Z ( ) !
Z0 ( )
+ P12 ( ) + P0 ( ) P10 ( ) (B.21)
YO2 (a) = I2r
(B.22)
For Dirichlet boundary conditions at both endpoints
= a; b used in the present paper
Det!O
2
Dirichlet = pdetP0 (a) detP0 (b) det
dsYO 1
( ) P0 1 ( ) YO ( ) :
(B.27)
35We omit to report similar formulas for the hatted operator O^ = P0 ( ) dd + P^1 ( ).
Det!O
2 =
s
detP0 (a)
detP0 (b) det [MO2 + NO2 YO2 (b)]
detYO (b)
:
f1 (a) = f2 (a) = f1 (b) = f2 (b) = 0
M 2 =
O
Ir 0
0 0
!
0 0
Ir 0
!
N 2 =
O
Z b
a
Fermionic determinant DetOF ( 0): details
In this appendix we collect some details on the analysis of the fermionic determinant
DetOF ( 0) in (3.27).
C.1
Derivation of (3.28)
To begin our analysis of DetOF ( 0) in (3.27), let us observe that
OF
34 p12;p56;p89 43 =
OF
p12;p56;p89 :
This fact can be used to show that
Det(OF1;p56;p89 )Det(OF
1;p56;p89 ) = Det[(OF1;p56;p89 )2]:
Indeed, let us denote the \positive" eigenvalues of OF
1;p56;p89 with f n; Re( n) > 0g and
n; Re( n) > 0g. Because of the relation (C.1), the spectrum
of OF
the \negative" ones with f
1;p56;p89 is given by
The
function for the rst operator is
while for the second operator we nd
Summing the two contributions we obtain
f
ng [ f ng :
1;p56;p89 (s) =
X( n) s + e i s X( n) s;
1;p56;p89 (s) = e i s X( n) s + X( n) s:
1;p56;p89 (s) +
1;p56;p89 (s) = (1 + e i s)
X( n) s + X( n) s
(1 + e i s) (s) (C.6)
The spectrum of (OF1;p56;p89 )2 is given instead by
and the corresponding
function is
Therefore
"
n
Z(s) =
X( 2n) s + X( 2n) s
=
(2s):
n
n
f ng [ f ng
2
n
n
n
log Det(OF1;p56;p89 )Det(OF
1;p56;p89 ) =
d 1;p56;p89 (0)
d
1;p56;p89 (0)
ds
(0)
=
i
2 0(0) ;
ds
log(Det[(OFp12;p56;p89 )2]) =
2 0(0) ;
(C.1)
(C.2)
(C.3)
(C.4)
(C.5)
(C.7)
(C.8)
(C.9)
HJEP02(16)5
so that it holds
log(Det(OF1;p56;p89 )Det(OF
1;p56;p89 )) = log(Det[(OF1;p56;p89 )2])
(0) :
(C.10)
We can namely express the combination Det(OF1;p56;p89 )Det(OF
minant and the function in 0 ( (0)) of the squared operator (OF1;p56;p89 )2.
1;p56;p89 ) in terms of deter
We now use Corollary 2.4 of [63]
j j + j j
2n
Z(0) = r
n + 1 ;
where 2n is the order of the di erential operator, ;
are parameters that only depend
on the boundary conditions and r is the matrix dimension of the operators. In our case
(n = 1, j j = j j = 1, r = 2) it is Z(0) = 2 and thus via (C.8) (0) = 2, to conclude that
Det(OF1;p56;p89 )Det(OF
1;p56;p89 ) = Det[(OF1;p56;p89 )2] ;
and thus (C.2) is proven.
The determinant of the fermionic operator can then be written as36
Det[OF ( 0)] =
Y
p12;p56;p89= 1;1
Det[OFp12;p56;p89 ( 0)]
= Det[(OF1;1;1)2]2Det[(OF1; 1;1)2]2;
(C.11)
(C.12)
(C.13)
(C.14)
(C.15)
depend on the value of p89.
the spectrum of OF
1;1;1, then f
OF
1; 1;1
4 (
; )
where we have used the property (C.10) and that the operators OF
p12;p56;p89 in (3.26) do not
We can also easily argue that Det[(OF1; 1;1)2] = Det[(OF1;1;1)2]. Let f n; ( ; )g be
n; 4 (
; ))g is the spectrum of OF
1; 1;1. Indeed it is
i sinh2 ( ) 0 + sin2 ( ) 034
4 (
; )
=
ia56( ) 4
i sinh2 ( ) 0
sin2 ( ) 034
; ):
Thus the eigenvalues of the squared operator (OF1; 1;1)2 are the same of those of the squared
operator (OF1;1;1)2 and consequently the two determinants coincide. In restricting ourselves
to onedimensional spectral problems  and thus working in terms of Fourier modes ! and
referring to (3.1)  from the statement (C.14) one obtains
Det![(OF1; 1;1)2] = Det ![(OF1;1;1)2] :
from which (3.28) follows.
36The Corollary 2.4 [63] can be easily check to hold both for
1;1(0) and for 1;1(0).
+
1
2( )
4
1
2( )
( )
As from (B.25), the keyingredient in the explicit computation of Det!(OF1;1;1)2 (3.29) is
Y ( ), the 2
2 matrix of the fundamental solutions obeying the boundary conditions
Y ( 0) = I2, as in (B.22). It is not di cult to explicitly check that the structure of this
matrix can be parametrized as follows
Y ( ) = e!(
0)S1( ) + e !(
0)S2( );
where the entries of the matrices S1( ) and S2( ) depends on ! only through rational
functions. We can infer some important properties of these matrices from the fact that
Y 2( ) tr(Y ( ))Y ( ) + det(Y ( ))I2 = 0 :
In particular one can easily check that detY ( ) does not depend on !. Then the secular
+(Si)1;1[(Mi)1;1(Si)1;2 + (Mi)1;2(Si)2;2]
= (Mi)1;2det(Si) = 0:
(Si)21(S0i)2;2
(Si)2;2(S0i)2;1 =
(Si)2;1[(Mi)2;1(Si)1;2 + (Mi)2;2(Si)2;2] (C.22)
+(Si)2;2[(Mi)2;1(Si)1;1 + (Mi)2;2(Si)2;1]
tr(S1)S1) + e 2!(
0)(S22
tr(S2)S2)
+fS1; S2g
tr(S2)S1 + det(Y )I2 = 0
tr(S2)S2 = 0 ; fS1; S2g
tr(S1)S2 tr(S2)S1 + det(Y )I2 = 0 :
The matrices S1 and S2 must also satisfy the di erential equations
P0@ S1 + (P1 + !P0)S1 = 0 ;
!P0)S2 = 0 ;
where P0; P1 appear in the Diraclike operator OF
P0 is invertible we can symbolically write this as Si0 + MiSi = 0. This implies a set of
interesting properties:
(Si)1;2(S0i)1;1
(Si)1;1(S0i)1;2 =
(Si)1;2[(Mi)1;1(Si)1;1 + (Mi)1;2(Si)2;1] (C.21)
and therefore
S
2
1
tr(S1)S1 = 0 ; S
2
2
and
Namely the ratios
37We omit the dependence in the matrices.
(Si)11
(Si)21
and
(C.16)
(C.17)
(C.18)
(C.19)
(C.20)
(C.23)
are
we can set
Y 1
P 1
0 Y =
with
Y 1( ) =
(tr(Y )I2
Y ( )) =
Next we construct the bilinear Y 1P0 1Y . We nd
1
1
det(Y )
det(Y )
1
det(Y )
+e !(
ai
Si =
(Si)11 =
(Si)12
(Si)21 ;
aipi( ) pi( )
aiqi( ) qi( )
where ai depends only on 0; 0 and !. We can parameterize the matrices Si as follows
Equation (C.17) also completely determines Y 1( ). In fact
HJEP02(16)5
S1)e!(
0) + (tr(S2)I2
S2)e !(
S2)e !(
0) + A0 + B0
(C.24)
(C.25)
(C.26)
0)S1 +
(C.27)
(C.28)
(C.29)
(C.30)
1 !
a1
(C.31)
1 !
a2
(C.32)
A2 =
A0 =
1
1
det(Y )
det(Y )
B2 =
B0 =
1
1
det(Y )
det(Y )
S2)P0 1S2 ;
S2)P0 1S1 :
Because of the relations (C.19) we nd that the matrices A2 and B2 are nihilpotent
A22 = B22 = 0 ;
and it holds
A2A0 = A0B2 = B2B0 = B0A2 = B0A0 = A0B0 = 0:
The structure of the matrices Ai and Bi is very simple. They are in fact constant matrices
times a function of . This can be easily shown by means of the parametrization (C.25).
In fact
A2( ) =
(q1( )2 p1( )2)
det(Y ) ( )
B2( ) =
(q2( )2 p2( )2)
det(Y ) ( )
(q1( )q2( ) p1( )p2( ))
det(Y ) ( )
B0( ) =
(q1( )q2( ) p1( )p2( ))
det(Y ) ( )
a2
a1a2
a1
a1a2
where we used that P0
( ) 1.
Our next goal is to compute
Z R
0
det
ds Y 1(s) P0 1(s) Y (s)
(C.33)
(C.34)
2
d
0
0
0
det
ds Y 1(s) P0 1(s) Y (s)
det(Y ( )) ( )
a2)2 Z R (p1( )p2( )
e2! Z R
d 0
0
q1( )q2( )) 2
0
det(Y ( )) ( )
:
in (B.27). Since Y 1(s) P0 1(s) Y (s) is traceless in our case, we can also write the expression
d 0tr Y 1
( ) P0 1( ) Y ( )Y 1( 0) P0 1( 0) Y ( 0) :
We can now use the representation (C.27) and the properties (C.28) to simplify the
ex+
(C.35)
This formula can be very e ciently used to simplify the fermionic determinant in its
largeR expansion. The second line is always negligible, the rst one consists of two separate
integrals: for positive !, the dominant part in the largeR limit will be the contribution
of the rst (inde nite) integral evaluated at the upper endpoint times the contribution of
the second (inde nite) integral evaluated at the lower endpoint, whereas for negative ! the
roles of rst and second integrals are swapped.
D
Boundary conditions for small Fourier modes
In this appendix we comment on a di erent choice for the boundary conditions on the
bosonic and fermionic modes with small Fourier mode  choice followed in [7] for the
circular Wilson loop case  and show that it leaves una ected the main results of this
paper, the e ective actions for both the circle (4.12) and the normalized latitude (4.20).
In [7] the onedimensional spectral problems in the radial coordinate
are subject to
Dirichlet boundary conditions at both the boundaries
= 0 and ( ctitious)
= R, except
for the modes labeled by m = 0,38 for which Neumann boundary conditions are imposed
at
= R.39 It is easy to modify our analysis of the bosonic sector in section 3.1  where
we kept Dirichlet boundary conditions for all modes  and evaluate the e ect of this other
choice. The relevant Fourier frequency corresponds to ` = 0 which, from the discussion at
the beginning of section 4, corresponds to the mode ! = 0 for O1 and O2( 0), ! = 1 for
O3+( 0) and ! =
1 for O3 ( 0). We use the subscript N to denote the new determinants
with Neumann boundary conditions in
= R
(D.1)
38In the labelling of (5.35) after the supersymmetrypreserving regularization.
39See formulas (5.46){(5.52) therein.
instead of the Dirichlet ones f1( 0) = f1(R) = 0 used in the main text. We read o the
since the operator O1 is the same for the circle and the latitude:
[Det!=0O1]N = coth 0:
For the other operators, the new boundary conditions change (B.12) as
1 0
0 0
M =
N =
0 0
0 1
Det! h dd22 + P2( )
i
and accordingly modify (3.14), (3.16) and (3.17) as
[Det!=0O2( 0)]N = tanh ( 0 + ) ;
[Det!=1O3+( 0)]N = [Det!= 1O3 ( 0)]N = p
e 0+2 0
1 + e2 0+4 0
:
The limit 0 ! 1
[Det!=0O2( 0 = 0)]N = [Det!=1O3+( 0 = 0)]N = [Det!= 1O3 ( 0 = 0)]N = 1
modi es the analogous results (3.43){(3.45) for the circular Wilson loop. A comparison with
the formulas in the main text reveals that, at the level of the Gel'fandYaglom determinants,
the only change following from this di erent choice of boundary conditions is an overall
rescaling of the determinants by R.
The same phenomenon occurs in the fermionic sector, where backtracking the special
Fourier mode to our !labeling is less transparent, but becomes more visible in the circular
Wilson loop. The frequency m = 0 of formula (5.35) [7] is the determinant of the operator
2
coth :
it is evident that (D.6) governs f1( ) for ! = 1 while f2( ) for ! = 0.
(D.2)
HJEP02(16)5
(D.6)
(D.7)
(D.8)
(D.9)
To nd it in the present paper, we begin with the Gel'fandYaglom di erential equation
and from its component equations
h 1;1;1( 0 = 0)
OF
i2 f1( )
f2( )
1 2
2
2
1 2
1
1
+ !
2
2
1
2
0
0
coth
coth
f1( ) = 0
f2( ) = 0 :
Extending this identi cation to arbitrary 0, this argument tells that the only
modication in appendix B.3 is the Neumann boundary condition on the rst component for
f1 ( 0) = f2 ( 0) = @ f1 (R) = f2 (R) = 0 ;
(D.10)
which translates into replacing (B.26) with
and on the second component for ! = 0
which is implemented by
0 1 0 0 0 1
MO2 = BBB 00 10 00 00 CCC
(D.11)
f1 ( 0) = f2 ( 0) = f1 (R) = @ f2 (R) = 0 ;
0 1 0 0 0 1
MO2 = BBB 00 10 00 00 CCC
0 0 0 0 0 1
NO2 = BBB 01 00 00 00 CC
C :
This also means that we cannot use the compact form (B.27) (still valid for ! 6= 0; 1) and
we have resort to the general expression (B.25). After a lengthy computation, the new
values of the determinants
(e2 0 +1) e6 0+4 0 + (e2 0 +1)e4 0+2 0 + e2 0 ( 5e2 0 +3e4 0 +3) + (e2 0 1)2
+ 2 0 +
N
N
+ 2 0 +
p
R eR e 20 (tanh 0 + 1) sinh 0 cosh( 0 + 0)
(e2 0
1)2(e2( 0+ 0) + 1)
p
R eR e 20 (tanh 0 + 1) sinh 0 cosh( 0 + 0)
(e2 0
agree with (3.40){(3.41), again up to an overall rescaling of their values by a factor of
The analysis in section 4 goes through in a similar fashion, provided that the lower
ularized e ective action (4.4) does not change when the limit
modes `B=0( 0);
sF= 12 ( 0);
2
F
s= 1 ( 0) take into account these new determinants. The
reg! 0 is taken.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
(D.12)
(D.13)
1
1
(D.14)
(D.15)
p
2e4 0 log e2( 0+ 0) + 1
+
2e2 0 log e2( 0+ 0) + 1
+
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