Exact correlators on the Wilson loop in \( \mathcal{N}=4 \) SYM: localization, defect CFT, and integrability
Accepted: May
Exact correlators on the Wilson loop in
Washington Road 0
Princeton 0
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0 1 Einstein Dr , Princeton, NJ 08544 , U.S.A
1 School of Natural Sciences, Institute for Advanced Study
2 Department of Physics, Princeton University
We compute a set of correlation functions of operator insertions on the 1=8 BPS Wilson loop in N = 4 SYM by employing supersymmetric localization, OPE and the GramSchmidt orthogonalization. These correlators exhibit a simple determinant structure, are positionindependent and form a topological subsector, but depend nontrivially on the 't Hooft coupling and the rank of the gauge group. When applied to the 1=2 BPS circular (or straight) Wilson loop, our results provide an in nite family of exact defect CFT data, including the structure constants of protected defect primaries of arbitrary length inserted on the loop. At strong coupling, we show precise agreement with a direct calculation using perturbation theory around the AdS2 string worldsheet. We also explain the connection of our results to the \generalized Bremsstrahlung functions" previously computed from integrability techniques, reproducing the known results in the planar limit as well as obtaining their nite N generalization. Furthermore, we show that the correlators at large N can be recast as simple integrals of products of polynomials (known as Qfunctions) that appear in the Quantum Spectral Curve approach. This suggests an interesting interplay between localization, defect CFT and integrability.
AdSCFT Correspondence; Conformal Field Theory; Supersymmetric Gauge

4
5
7
8
2.1
2.2
3.1
3.2
3.3
3.4
4.1
4.2
5.1
5.2
5.3
5.4
5.5
6.1
6.2
6.3
7.1
7.2
6
Weak and strongcoupling expansions
Weak coupling expansion
Strong coupling expansion Comparison to string theory
Emergent matrix model at large N
DL and QL(x) as a matrix model
Classical limit of the matrix model
Conclusion
Correlators on the 1=2 BPS Wilson loop from OPE
Construction of operators from the GramSchmidt orthogonalization
A remark on the 1=8 BPS Wilson loop
Results for topological correlators
Generalized Bremsstrahlung functions
Cusp anomalous dimension and Bremsstrahlung function
Relation to the twopoint function and the result at nite N
Large N limit
Integral expression for topological correlators
Properties of QL(x)
Comparison with the results from integrability
Variations of the measure Nonplanar corrections to the measure
1 Introduction
2
Topological correlators on the 1=8 BPS Wilson loop
1=8 BPS Wilson loop
Correlators on the 1=8 BPS loop
2.3 1=2BPS Wilson loop and defect CFT
3
Computation of the correlators
A Explicit results for operators with L 3
A.1 Generalized Bremsstrahlung
{ i {
Introduction
The exact solution to an interacting quantum eld theory in four dimensions would mark a
breakthrough in theoretical physics, although it still seems out of reach at present time. In
supersymmetric theories, one can make some progress since there are observables that
preserve a fraction of the supersymmetries and are therefore often amenable to exact analytic
methods, most notably supersymmetric localization [1].
Another powerful method, which is currently the subject of active exploration, is the
conformal bootstrap, see e.g. [2] for a recent review. This approach uses conformal
symmetry instead of supersymmetry, and has been remarkably successful in deriving bounds on
a landscape of theories from a minimal set of assumptions [3, 4].
The third way towards this goal is integrability [5]. Although the applicability of
integrability is much smaller than the other two since it applies only to speci c
theories, the advantage is that it works not only for supersymmetric observables but for
nonsupersymmetric ones as well. It also allows one to compute them exactly as a function of
coupling constants, rather than giving general bounds.
In this paper, we consider quantities which may stand at the \crossroads" of all these
three methods.
More speci cally, we study the correlation functions of local operator
insertions on the 1=8BPS Wilson loop in N = 4 supersymmetric YangMills theory (SYM).
The supersymmetric Wilson loop in N = 4 SYM has been an active subject of study since
the early days of AdS/CFT correspondence [6, 7]. The 1=2BPS circular Wilson loop,
which preserves a maximal amount of supersymmetry, was computed rst by summing up
a class of Feynman diagrams [8, 9], and the exact result for its expectation value was later
derived rigorously from supersymmetric localization [1], which reduces the problem to a
simple Gaussian matrix model. The result is a nontrivial function of the coupling constant,
which nevertheless matches beautifully with the regularized area of the string in AdS at
strong coupling, providing key evidence for the holographic duality.
The computation was subsequently generalized to less supersymmetric Wilson loops,
such as the 1=4 BPS circular loop [10], and a more general in nite family of 1=8BPS
Wilson loops de ned on curves of arbitrary shape on a twosphere [11, 12]. For such
loops, an exact localization to 2d YangMills theory was conjectured in [11, 12], and later
supported by a localization calculation in [13]. Because of the invariance under
areapreserving di eomorphisms of 2d YM theory, one nds that the result for the expectation
value of the 1=8 BPS Wilson loop depends only on the area of the region surrounded by
the loop. The localization relation to the 2d theory was checked in a number of
nontrivial calculations, see e.g. [14{21]. It was also used in [22] to compute various important
quantities de ned on the Wilson loop, such as the twopoint function of the displacement
operator and the related \Bremsstrahlung function". It was based on the observation
that one can insert a displacement operator by di erentiating the expectation value of the
Wilson loop with respect to its area A; D
The purpose of this paper is to show that there are in nitely many other observables
that can be computed using the results from localization. They are the correlation functions
{ 1 {
of special scalar insertions ~ L inside the Wilson loop trace, where the scalar ~ is chosen
so that the correlators are independent of the positions of the insertions.1 Similarly to
the displacement operator, one can relate the insertion of ~ 's to the areaderivative of the
Wilson loop, essentially because ~ turns out to correspond via localization to insertions of
the Hodge dual of the 2d YM
eld strength. However, one key di erence from the analysis
in [22] is that after taking the multiple area derivatives, to de ne the properly
normalordered operators one has to perform the socalled GramSchmidt orthogonalization to
make ~ k's for di erent k orthogonal to each other (and, in particular, also orthogonal
to the identity, i.e. their onepoint functions vanish). After doing so, the result for the
twopoint function takes a particularly compact form and exhibits a simple determinant
For higherpoint functions, the result can be written succinctly in terms of certain
polynomials FL(X), which by themselves are expressed in terms of determinants:
h
A0=A
See section 5 for further details including the de nition of FL.
As a special application of our analysis, we explain the relation of our correlators on
the Wilson loop to the \generalized Bremsstrahlung function" BL( ) (whose de nition is
reviewed in more detail in section 4 below), which was computed previously in the planar
large N limit from integrability [25, 26]. In particular we nd that
HL( )
1
2
DL+1
DL
A=2 2
:
At large N , we show that this agrees with the integrability result. Moreover, since the
Wilson loop expectation value appearing in the calculation of the determinants DL is
known for any N;
via localization, this provides the
nite N form of the generalized
Bremsstrahlung function BL( ).
Our results are valid for the general 1=8BPS Wilson loop de ned on an arbitrary
contour on S2, but perhaps the most interesting case is the 1=2BPS loop. Since the
1=2BPS loop is circular (or, by a conformal transformation, a straight line) it preserves
a SL(2; R) conformal subgroup, and therefore can be viewed as a conformal defect of
the 4d theory. The correlation functions of operator insertions on the 1=2BPS loop are
then constrained by the SL(2; R) d = 1 conformal symmetry, or more precisely by the
OSp(4 j4)
SL(2; R) SO(3) SO(5) 1d superconformal symmetry [27]. Some of the
properties of this defect CFT were recently studied at weak [28{31] and strong coupling [31, 32].
1The correlators of Wilson loops and local operators of similar kind, but inserted away from the Wilson
loop, was studied in earlier literature, e.g. [18{20].
2A symmetric matrix of the type appearing here, which satis es Mij = Mi+j, is sometimes called
)L, where Y is a null polarization 5vector. Such operators transform in the rankL
symmetric traceless representation of SO(5), and they belong to short representations of
the 1d superconformal group, with protected scaling dimension
= L. Because their
2point and 3point functions are fully
xed by the SL(2; R) symmetry, the restriction to
the topological choice of polarization vectors still allows one to extract exact results for
the 2point normalization and 3point structure constants of the general defect primaries
(Y
)L. Unlike the analogous case of single trace chiral primaries of the 4d theory, which
are dual to protected closed string states, the structure constants in the present case are
found to have a highly nontrivial dependence on the coupling constant. Our
construction provides exact results for such structure constants of all operators in this protected
subsector, which should provide valuable input for a conformal bootstrap approach to the
Wilsonloop defect CFT (see e.g. [33]).
The connection to integrability techniques emerges in the planar limit. At large N , we
found that the results can be rewritten as a simple integral
h
with the measure d given in (5.5) (see also other forms of the measure (5.45) and (5.46)).
This is by itself an interesting result, but what is more intriguing is that the function QL(x)
that appears in the formula is directly related to the Quantum Spectral Curve [34], which
is the most advanced method to compute the spectrum of the local operators in N = 4
SYM. The appearance of such functions in our setup hints at a potential applicability of
the Quantum Spectral Curve to the problem of computing correlation functions.
The rest of this paper is organized as follows: in section 2, we review the construction
of the 1=8 BPS Wilson loop and explain the de nitions of the correlators that we study in
this paper. We then relate them to the area derivative of the Wilson loop in section 3 by
using the OPE and the GramSchmidt orthogonalization. The nal result for the correlator
at nite N is given in subsection 3.4. In section 4, we apply our method to compute the
niteN generalization of the generalized Bremsstrahlung function. We then take the large
N limit of our results rewriting the correlators as an integral in section 5. The results at
large N are expanded at weak and strong coupling in section 6 and compared against the
direct perturbative computations on the gauge theory side and on the string theory side
respectively. We also provide a matrixmodellike reformulation of the large N results in
section 7. Finally, section 8 contains our conclusion and comments on future directions.
2
Topological correlators on the 1=8 BPS Wilson loop
In this section, we explain the de nitions of the topological correlators that we study in
this paper and discuss their relation to the defectCFT data.
{ 3 {
BPS Wilson loop lives on S2 and couples to a scalar as prescribed in (2.1). The expectation value
of such a loop depends only on the area A of the region inside the loop (the redshaded region in
the gure). Note that, although \the region inside/outside the loop" is not a wellde ned notion,
such ambiguity does not a ect the expectation value since it is invariant under A ! 4
A, which exchanges the regions inside and outside the loop.
2.1
1=8 BPS Wilson loop
The 1=8 BPS Wilson loops is a generalization of the standard Wilson loop and it couples to
a certain combination of the N = 4 SYM scalars, as well as the gauge eld [8{12]. In order
to preserve 1=8 of the superconformal symmetry, the contour C must lie on a S2 subspace
of R4, which we may take to be parametrized by x21 + x22 + x23 = 1, and the coupling to the
scalars is prescribed to be
W
1
N
tr P heHC(iAj+ kjlxk l)dxj i
where i; j; k = 1; 2; 3, and we pick three out of the six scalar elds to be coupled to the loop.
In what follows we will focus for simplicity on the fundamental Wilson loop, namely the
trace in (2.1) is over the fundamental representation of the gauge group U(N ). However
our construction below can be easily extended to arbitrary gauge group and arbitrary
representations.
The expectation value of this Wilson loop can be computed by supersymmetric
localization [1, 13]. The result only depends on the rank of the gauge group N , the coupling
constant gYM and the area of the subregion inside the contour C, see gure 1, which we
denote by A [10{13]:
(2.1)
(2.2)
(2.3)
(2.4)
To emphasize its dependence on the area, we sometimes denote hWi as hW(A)i. This
matrix model integral can be evaluated explicitly [9] as
hWi =
1
N
operator (2.1) couples to a single scalar eld. This special case corresponds to the 1=2BPS
Wilson loop, see section 2.3 below.
In the large N limit, the result simpli es and can be expressed in terms of the Bessel
function [8]:
This can also be rewritten in terms of the deformed Bessel functions introduced in [26],
hWijlarge N = p 0
2
+
I1(
p 0) :
n #
2
;
In =
1
2 In
p
"
+
n
2
( 1)n
1
2
2
;
where ~ is a positiondependent scalar,4
The BPS correlators we study in this paper are given by the following choice of the
1
N
h
tr P O1( 1)
On( n)eHC(iAj+ kjlxk l)dxj i
N = 4 SYM
:
(2.8)
2 [0; 2 ] and i's are the positions of the operator
as
hWijlarge N = p Ia=2 :
1
2
In (2.7), all the area dependence is encoded in the function I1a=2. This property turns
out to be very useful when we later derive the integral expression for the topological
correlators at large N .
2.2
Correlators on the 1=8 BPS loop
The correlation function of the local operators on the Wilson loop is de ned by3
Oi( i)
~ Li ( i) ;
~ ( ) = x1( ) 1 + x2( ) 2 + x3( ) 3 + i 4 :
An important property of such correlators is that they do not depend on the positions of
the insertions i's. This follows5 from the fact that, after localization, these operators are
4Throughout this article, we use the convention in which the scalar propagator reads
3Note that here we do not divide the correlator by the expectation value of the Wilson loop hWi.
( I (x1))a b ( J (x2))c d = gYM ad cb IJ ;
2
8 2jx1
x2j2
where ad are the U(N ) matrix indices, and I; J = 1; : : : ; 6.
5Alternatively, one should be able to show that the (twisted) translation generator which moves the
positions of the insertions is Qexact with Q being one of the supercharges preserved by the con guration.
{ 5 {
Therefore, one can view the correlators on the 1=2BPS Wilson loop as correlators of a
defect CFT. To make this point precise, one needs to consider the normalized correlator,
which is obtained by dividing the bare correlator (2.8) by the expectation value of the
Wilson loop:
hhO1( 1)O2( 2)
On( n)ii
After the normalization, the expectation value of the identity operator becomes unity and
the correlators obey the standard properties of the defect CFT correlators.
Using these normalized correlators, one can extract the defect CFT data from the
topological correlators. To see this, let us consider general two and threepoint functions
of BPS operators on the 1=2BPS loop:
GL1;L2 = hh(Y1 ~ )L1 ( 1) (Y2 ~ )L2 ( 2)iicircle ;
GL1;L2;L3 = hh(Y1 ~ )L1 ( 1) (Y2 ~ )L2 ( 2) (Y3 ~ )L3 ( 3)iicircle :
In (2.14), ~
( 1; 2
; 4
; 5
; 6) and Yi's are vedimensional complex vectors satisfying
Yi Yi = 0. Unlike the topological correlators (2.8), the correlators (2.14) depend on the
positions of the insertions, and the vectors Yi. However, because of the conformal symmetry
and the SO(5) Rsymmetry, their dependence is completely xed to be:6
GL1;L2 = nL1 ( ; N )
GL1;L2;L3 = cL1;L2;L3 ( ; N )
L1;L2 (Y1 Y2)L1
(2 sin 212 )2L1
;
(Y1 Y2)L12j3 (Y2 Y3)L23j1 (Y3 Y1)L31j2
2 sin 212 2L12j3 2 sin 223 2L23j1 2 sin 231 2L31j2
;
mapped to eldstrength insertions in twodimensional YangMills theory (see section 3.1
for further explanation), which enjoys invariance under areapreserving di eomorphisms,
making it almost topological. Because of their position independence, we will call them
\topological correlators" in the rest of this paper.
1=2BPS Wilson loop and defect CFT
For special contours, the Wilson loops preserve higher amount of supersymmetry.
Particularly interesting among them is the 1=2BPS Wilson loop, whose contour is a circle along
the equator and which couples to a single scalar 3 [8, 9]:
HJEP05(218)9
with ij
i
j and Lijjk
Lk)=2. Here nL1 is the normalization of the
twopoint function while cL1;L2;L3 is the structure constant. As shown above, both of
6Of course, one may also write the analogous result for the straight line geometry, which is related to
the circle by a conformal transformation.
{ 6 {
(2.12)
(2.13)
(2.14)
(2.15)
these quantities are nontrivial functions of
and N . Note that, although we often set the
normalization of the twopoint function to be unity in conformal eld theories, for special
operators the normalization itself can have physical meaning.7 For instance, the length1
operator (Y
~ ) is related to the displacement operator and has a canonical normalization
which is related to the Bremsstrahlung function [22].
Now, if we go to the topological con guration by setting the vectors Yi to be
Yi = (cos i; sin i; 0; i; 0; 0) ;
GL1;L2 jtopological =
GL1;L2;L3 jtopological =
1
2
1
2
L1
This shows that the topological correlators compute the normalization and the structure
constant in the defect CFT up to trivial overall factors. Alternatively, one can consider
GL1;L2;L3
(GL1;L1 GL2;L2 GL3;L3 )1=2
topological
=
cL1;L2;L3
(nL1 nL2 nL3 )1=2 ;
(2.16)
(2.17)
(2.18)
HJEP05(218)9
and get rid of the overall factors. The quantity which appears on the right hand side
of (2.18) is a structure constant in the standard CFT normalization; namely the
normalization in which the twopoint function becomes unity.
Note that, for higherpoint functions, there is no such a direct relation between the
general correlators and the topological correlators: the general higherpoint correlators are
nontrivial functions of the cross ratios while the topological correlators do not depend at
all on the positions. Thus for higherpoint functions, one cannot reconstruct the general
correlators just from the topological correlators.
3
Computation of the correlators
We now compute the correlators on the 1=8 BPS Wilson loop
h
~ L1 ( 1) ~ L2 ( 2)
~ Ln ( n)i ;
(3.1)
using the results from localization. We rst discuss the correlators on the 1=2 BPS Wilson
loop from the OPE perspective and then present a general method that applies also to the
1=8 BPS Wilson loop.
3.1
Correlators on the 1=2 BPS Wilson loop from OPE
When all the operators are length1 and the Wilson loop is circular (or equivalently 1=2
BPS), the correlators (3.1) were already computed in [32]. Let us st brie y review their
7Other examples are the stressenergy tensor and the conserved currents, whose twopoint functions are
related to CT and CJ .
{ 7 {
computation: by performing localization, one can reduce the computation of the 1=8 BPS
Wilson loop in N = 4 SYM to the computation of the Wilson loop in twodimensional
YangMills theory in the zero instanton sector [13]. Under this reduction, the insertion of
the positiondependent scalar ~ is mapped to the insertion of the dual eld strength F2d:
Using this correspondence,8 one can insert ~ 's on the circular Wilson loop by di erentiating
its expectation value with respect to the area A:
~
,
i F2d :
h
~
L
~ ijcircle =
A=2
:
Using the expression (3.3), one can compute arbitrary correlation functions of singleletter
insertions ~ .
To study more general BPS correlators, we also need to know how to insert operators
of longer length, ~ L with L > 1. The rst guess might be to relate it simply to the Lth
derivative of the Wilson loop,
~ L
This guess, however, turns out to be incorrect. To see why it is so, let us consider ~ 2 as an
example. We know that the second derivative of hWi corresponds to the insertion of two
~ 's on the Wilson loop. Since the correlator we are studying is topological, one can bring
the two ~ 's close to each other without a ecting the expectation value and rewrite them
using the operator product expansion. This procedure does produce the length2 operator
~ 2 as we wanted, but the problem is that it also produces other operators:9
OPE
~
~ = ~ 2 + c1 ~ + c0 1 :
: ~ 2 : = ~ ~
c1 ~
c01 :
(3.2)
(3.3)
Here ci's are some numerical coe cients and 1 is the identity operator. Thus, to really get
the length2 operator, one has to subtract these unnecessary OPE terms from ~ ~ :
Here ~ ~ on the right hand side denotes two singleletter insertions at separate points while
: ~ 2 : is a length2 operator inserted at a single point. Since this subtraction procedure is
conceptually similar to the normal ordering, we hereafter put the normalordering symbol
: : to the operator obtained in this way.
The coe cients ci's are nothing but the OPE coe cients of the topological OPE (3.5).
They are thus related to the following threepoint functions:
c1 / h
~ ~ ~ ijcircle ;
c0 / h
~ ~ 1ijcircle :
sides in [21].
with k > 2.
8At weak coupling, this correspondence was checked by the direct perturbative computation on both
9Owing to the representation theory of SO(5), the OPE does not produce highercharge operators, ~ k
{ 8 {
If we were using the operators, ~ =h ~ ~ i1=2 whose twopoint function is unitnormalized,
the constants of proportionality in (3.7) would have been unity. However the operators we
are using here are not unitnormalized and one has to take into account that e ect. This
leads to the following expressions for the coe cients c1 and c0:
hWi
A=2
;
We can repeat this procedure to express operators of arbitrary length in terms of
singleletter insertions and compute their correlation functions. Although these procedures
can be easily automated using computer programs, they do not give much insight into the
underlying structure. In the next section, we discuss a simpler way to reorganize these
procedures which also leads to a simple closedform expression.
3.2
Construction of operators from the GramSchmidt orthogonalization
As a direct consequence of the subtraction procedures (3.6), the operators constructed
above satisfy the following important properties:
The operator basis with such properties turns out to be unique and can be constructed
systematically by using the socalled GramSchmidt orthogonalization. As we see below, it
also allows us to write down a closedform expression for the operators : ~ L : .
The GramSchmidt orthogonalization is an algorithmic way of getting the orthogonal
basis from a given set of vectors. It was recently applied in the computation of Coulomb
branch operators in N = 2 superconformal theories in [37]. Its large N limit was discussed
in [38] while the case for N = 4 SYM was analyzed further in [39, 40]. What we describe
below is a new application of the method to the correlators on the Wilson loop. To get a
glimpse of how it works, let us orthogonalize two arbitrary vectors fv1 ; v2g. A simple way
of doing so is to de ne new vectors as
u1 = v1 ;
u2 = v2
hv1; v2i v1 ;
where h ; i denotes the inner product between two vectors. This is of course just an
elementary manipulation, but the key point is that one can reexpress (3.9) as
u1 = v1 ;
u2 =
1
{ 9 {
(3.9)
(3.10)
where j j denotes a determinant of a matrix. This expression can be readily generalized
to the case with more vectors. The result reads
uk =
1
dk 1
dk =
For details of the derivation, see standard textbooks on linear algebra. The new vectors
de ned above are orthogonal but not normalized. Their norms can be computed using the
de nitions above and we get
huk ; uli =
dk
dk 1
kl :
We now apply the GramSchmidt orthogonalization to the set of singleletter insertions
f1 ; ~ ; ~ ~ ; : : :g. The norms between these vectors are given by the twopoint functions,
which can be computed by taking derivatives of hWi,
h
~
~ ~
L
M
hWi
We then get the expression for the operator : ~ L : ,
: ~ L : =
1
DL
DL =
hWi
(1)
hWi
hWi
(L 1)
.
hWi
hWi
hWi
hWi
(1)
hWi
.
.
.
(L 1)
hWi
hWi
hWi
.
.
.
hWi
(1)
(2)
(L)
.
.
.
.
.
.
hWi
hWi
hWi
hWi
hWi
hWi
~
L
(L 1)
(L)
(@A)k hWi. Let us emphasize that this method applies to general 1=8 BPS
Wilson loops. To get the result for the 1=2 BPS loop, one just needs to set A = 2
at the end of the computation. For small values of n, one can check explicitly that this
expression coincides with the operators obtained by the recursive procedure outlined in the
previous subsection. One can also check that the basis obtained in this way satis es the
aforementioned three properties.
Owing to the property (3.12), the twopoint function of the operators ~ L is given by
a ratio of determinants:
For the 1=2 BPS loop, this provides an exact result for the normalization of the twopoint
function in the defect CFT (see the discussions in section 2.3),
h
: ~ L : : ~ M : i =
DL+1
DL
LM :
nL = ( 2)L DL+1
DL
A=2
As it is wellknown, the result for L = 1 is related to the normalization of the displacement
operators while the results for L > 0 provide new defectCFT observables.10
We will later see in section 5.3 that the largeN limit of these determinants is related
to the determinant representation of the generalized Bremsstrahlung function derived
previously in [25, 26].
3.3
A remark on the 1=8 BPS Wilson loop
As mentioned above, the GramSchmidt process can be applied to the general 1=8 BPS
Wilson loops. At the level of formulas, one just needs to keep the area A general in (3.14)
and (3.19). However, there is one important qualitative di erence which we explain below.
Unlike the 1=2 BPS Wilson loop, the rstorder derivative hWi
the general 1=8 BPS Wilson loop. This means that the singleletter insertion ~ has a
nonvanishing onepoint function; in other words, the twopoint function of ~ and the identity
operator 1 is nonzero. Therefore, to de ne an orthogonal set of operators, one has to
perform the subtraction even for ~ . In fact, by applying the GramSchmidt orthogonalization,
(1) does not vanish for
hWi
hWi
We thus need to distinguish : ~ : from ~ . This was one of the reasons why we preferred to
put the normalordering symbol when de ning the operator : ~ J : .
3.4
Results for topological correlators
Using the closedform expression (3.14), one can compute higherpoint functions of : ~ N : .
To express the result, it is convenient to introduce a polynomial
(3.15)
(3.16)
(3.17)
hWi
(1)
hWi
hWi
(L 1)
.
.
.
1
hWi
hWi
.
.
.
hWi
X
(1)
(2)
(L)
.
.
.
hWi
hWi
.
.
.
hWi
XL
(L)
(L+1)
10Although the normalization of the operators is usually not meaningful, for this class of operators, there
is a canonical normalization induced by the facts that ~ is related to the displacement operator and : ~ L :
is essentially a product of L ~ 's.
By replacing Xk by ~
~ , one recovers : ~ L : . In terms of these polynomials, the
higherpoint function reads
k
m
k=1
Let us make two remarks regarding this formula:
rst, the derivatives @A0 's on the right
hand side act only on the last term hW(A0)i (not on the coe cients of the polynomials
FLk ). Second, the polynomial FL is not just a technical tool for writing down
higherpoint correlators, but it gives an explicit map between the OPE and the multiplication of
polynomials. To see this, consider a product of two such polynomials. Since the product
is also a polynomial, one can express it as a sum of FL's,
HJEP05(218)9
FL1 (X)FL2 (X) =
cL1;L2;M FM (X) ;
where cL1;L2;M is a \structure constant" for the multiplication of polynomials. This
expansion can be performed also on the right hand side of (3.19). On the other hand, we can
perform a similar expansion on the left hand side of (3.19) using the OPE,
(3.20)
(3.21)
(3.22)
: ~ L1 : : ~ L2 : =
cL1;L2;M : ~ M : :
Equating the two expressions, we conclude that these two structure constants must coincide,
namely cL1;L2;M
= cL1;L2;M . This provides an interesting correspondence between the
multiplication of polynomials and the OPE.
We can also express the results more explicitly in terms of determinants. For this
purpose, we rst perform the Laplace expansion of the polynomial FL(X):
L
1
DL n=0
Here DL(i;j) is a minor of DL obtained by deleting the ith row and jth column. We then
substitute this expression into (3.19) to get
L1+L2
X
M=0
L1+L2
X
M=1
We can also perform one of the sums explicitly to reconstruct a determinant: the result
with ntot
reads
h
at the origin, and the insertions ZL. The scalar coupling of
each semiin nite line is given by the vector ~n1;2, and the relative angle between the two vectors is
. The divergence from this Wilson line is controlled by the generalized Bremsstrahlung function.
.
.
.
hWi
hWi
(L 1)
(n)
hWi
hWi
(L)
(n+1)
hWi
(L)
(L+1)
hWi
.
.
.
hWi
hWi
for \extremal" correlators which satisfy L1 = Pm
that survives and we get a simpler formula
Importantly, D~ L;n vanishes unless n
L since otherwise the last row coincides with one
of the rows above. This allows us to restrict the sum in (3.24) to n0tot
L1. In particular,
k=2 Lk, there is only one term in the sum
h
For general correlators, the expression (3.24) is not very concise as it involves several
terms. The results for two and threepoint functions of operators with L
3 are given explicitly in appendix A. We will later see in sections 5 and 7 that in the large N limit there is an elegant reformulation in terms of integrals and a matrix model.
4
Generalized Bremsstrahlung functions
Z 0
1
As an application of our method, in this section we compute the socalled \generalized
Bremsstrahlung function". The result provides niteN generalization of the planar results
computed previously in [25, 26] using integrability [41, 42].
4.1
Cusp anomalous dimension and Bremsstrahlung function
Let us rst recall the de nition of the generalized Bremsstrahlung function. Consider the
following cusped Wilson line with insertions (see also gure 2):
WL( ; )
P exp
d hiA x_ 1 + ~ ~n1jx_ 1ji
ZL
P exp
d hiA x_ 2 + ~ ~n2jx_ 2ji : (4.1)
0
Here Z =
3 + i 4 and the x1;2(t) and ~n1;2 are given by
As shown above, WL is parametrized by the two angles
and . When
=
BPS and the expectation value hWLi is
controlled by the cusp anomalous dimension
nite. However, if
6
= , it has the divergence
hWL( ; )
i
L:
UV
rIR
L( ; )
L( ; ) = (
)HL( ) + O((
)2) ;
Here UV and rIR are the UV and IR (length) cuto s respectively.
The cusp anomalous dimension can be expanded near
and the leading term in
the expansion reads
BL( ):
The function HL is related to the quantity called the generalized Bremsstrahlung function
For L = 0, BL( ) is related to the energy emitted by a moving quark [22] and this is why
it is called the generalized Bremsstrahlung function.
4.2
Relation to the twopoint function and the result at nite N
To compute BL from our results, one has to relate it to the topological correlators. For
L = 0 this has already been explained in [22]. As we see below, essentially the same
argument applies also to L 6= 0 (see also [21]).
The rst step is to consider a small deformation away from the BPS cusp by changing
the value of . Then, the change of the expectation value can be written as
where 0 is
hWLi =
hWLi
d hh 0( )iicusp
0 =
sin
and hh iicusp is the normalized correlator of the scalar insertion on the cusped BPS Wilson
loop WL( ;
= ). Using the invariance of WL under the dilatation around the origin, the
dependence of hh 0iicusp can be xed to be
1
hh 0( )iicusp =
hh 0( = 1)iicusp :
We can then compare (4.3) with (4.6) (introducing the UV and IR cuto s to evaluate the
integral), to get
L =
(
)hh 0( = 1)iicusp + O((
)2) :
(4.2)
, WL is
(4.3)
cusped Wilson line to a con guration depicted above. The red and black semicircles correspond
to the two semiin nite lines in
gure 2 of the same color. The angle between the two semicircles
is
. The loop divides the S2 into two regions with areas 2
2 . (Note that we already set
= in this gure.)
S2 by the conformal transformation,
The second step is to map the BPS cusp WL( ;
= ) to the 1=8 BPS Wilson loop on
x1 =
2X2
1 + X12 + X22
;
x2 =
2X1
1 + X12 + X22
;
x3 =
1
X12
1 + X12 + X22
X22 :
Here xi's are the (embedding) coordinates of S2 while Xi's are the coordinates on R2 where
the cusped Wilson loop (4.2) lives. After the transformation, and changing variables by
=
cot(t=2), the two semiin nite lines of the cusped Wilson loop are mapped to the
two arcs on S2 (see also gure 3),
(x1; x2; x3) =
((0; sin t; cos t)
( sin sin t; cos sin t; cos t)
0 < t
< t
2 ;
(4.10)
(4.11)
where the rst arc (0 < t
) and the second arc (
< t
2 ) correspond to the black
and the red lines in
gure 2 respectively. The rst arc couples to
1, and the second one
to cos
1 + sin
2, in accordance with our conventions (2.1) for the 1/8BPS loop. As
shown in gure 3, the resulting Wilson loop has cusps at the north and the south poles (tN
and tS) with insertions ZL and ZL respectively. The insertion 0( = 1) is mapped to the
insertion at a point11 te where the red arc intersects the equator of S2. We then arrive at
11In terms of the parametrization given in (4.11), tS = 0, tN =
and te = 3 =2.
the relation between the expectation values,
where h i denotes a (unnormalized) correlator on the Wilson loop on S2. Now, a crucial
observation is that one can complete the insertion
0(te) to the position dependent scalar
0
i 4 =
sin
4 vanishes owing to the charge conservation. Furthermore,
ZL(tN ) and ZL(tS) can be identi ed with : ~ L : . We thus arrive at the following relation,12
Note that ~ in the middle is not normalordered since it comes directly from the
deformation of the loop. Since the area surrounded by this loop is given by A = 2
2 , one can
express
From (4.9) and (4.15), we can compute the generalized Bremsstrahlung function as
(4.12)
(4.13)
(4.15)
(4.16)
(4.17)
(4.18)
DL+1 :
DL
As given in (3.14), DL is the following simple determinant,
with A = 2
2 . In the limit
! 0, the formula takes a particularly simple form,
hWi
(1
i; j
L) ;
BL(0) =
4
DL+1
DL
=0
=
DL+1
DL
A=2
:
This is the main result of this section. In the next section, we will see that the
formula (4.16) reproduces the results in [25, 26] in the large N limit. Note, however, that
our results (4.16) is also valid at nite N : all one has to do is to plug in (4.17) the
nite
N form of the Wilson loop expectation value, which is given in (2.3). The rst few explicit
results for
= 0 are given in the appendix.
5
Large N limit
In this section, we study in detail the topological correlators in the large N limit. In
particular, we derive a simple integral expression.
12Precisely speaking, the area derivative can also act on the operator : ~ L : (in addition to inserting an
extra singleletter insertion) since it is given by a sum of singleletter insertions with the areadependent
coe cients:
: ~ L : = ~ L + c1(A) ~ L 1 +
:
(4.14)
However, since the leading coe cient is 1, @A : ~ L : only starts with ~ L 1. Therefore, one can always
express @A: ~ L : as a sum of : ~ k : with k < L. We thus conclude that such contributions vanish because of
the orthogonality, h: ~ k : : ~ L : i = 0 for k < L, and do not a ect (4.15).
5.1
As mentioned before, an important simpli cation in the large N limit is that hWi can be
expressed in terms of the deformed Bessel function (2.7). A nice feature of the deformed
Bessel function is that it admits an integral expression [43],
In =
I
dx
2 ixn+1 sinh(2 g(x + 1=x))e2g (x 1=x) ;
where here and below we use the notation
Applying this to (2.7), we can express hWi and its derivatives simply as
where the measure d is de ned by
g
hWi =
p
4
I
I
:
d ;
d
2 g
hWi
;
d
=
dx sinh(2 g(x + 1=x))ega(x 1=x)
Second, they are normalized as
Third, since QL(x) is a polynomial of X = g(x
x 1), it follows that
I
d (x) QL(x)QM (x) =
DL+1
DL
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
(5.8)
(5.9)
(5.10)
2 and the 1=2BPS Wilson loop corresponds to a = 0. Combining
this integral expression with the formula (3.19), we obtain a simple integral expression for
the multipoint correlators,
h
the rest of this section, it is related to the Quantum Spectral Curve [34].
The functions QL(x) have several important properties. First, owing to the orthogonality
of the twopoint functions, they satisfy the following orthogonality relation:
Furthermore, they satisfy the following equalities:
The rst equality follows from
I
I
d (x) xQL(x)QM (x) = 0 ;
d (x) x2QL(x)QM (x) =
DL+1
DL
where in the second equality we used the property of the measure,13 R d ( 1=x) = R x2d (x).
In a similar manner, the second equality can be proven:
d (x) x2QL(x)QM (x) =
d ( 1=x)
QL( 1=x)QM ( 1=x)
=
d (x) QL(x)QM (x) :
The equalities (5.8), (5.11) and (5.12) imply that QL(x) and xQL(x) for L 2 N 0
together form a set of orthogonal functions under the measure d (x). They are in fact
the GramSchmidt basis obtained by applying the orthogonalization to the set of functions
f1; x; x 1; x2; x 2; : : :g. As we see below, this characterization of the functions QL(x) plays
a key role in identifying them with the functions introduced in the integrabilitybased
approaches [26, 43].
5.3
Comparison with the results from integrability
We now prove the equivalence between our results and the results obtained previously from
integrability [25, 26]. For this purpose, we rst show that QL coincides with the function
PL, which was introduced in [25, 26] and later shown to be directly related to the socalled
\Qfunctions" in the Quantum Spectral Curve [44]. The equivalence of other quantities,
including the generalized Bremsstrahlung functions, follow from it.
As the rst step, let us recall the polynomials PL de ned in [26]:
I
I
I
1
2
.
.
.
I
I
0
1
.
.
.
I2L I2L 1
x L x1 L
.
.
.
I2 2L I1 2L
I3 2L I2 2L
.
.
.
I
1
xL 1
.
.
.
I
0
xL
;
L
0 ;
(5.15)
13Note that an extra minus sign comes from a change of the direction of the contour.
(5.11)
(5.12)
(5.13)
The expression (5.50) contains an extra dependence on the area, (2
a)2. However, this
can be absorbed into the integral by using
0
48N 2 I2(
p 0) =
=
Applying this to the rst nonplanar correction, we get
with
with
and performing the integration by parts. As a result, we get
0
48N 2 I2(
p 0) =
dx e2 g(x+ x1 )ega(x x1 )
f (2 g(x + 1=x)) ;
f (z) =
(2 g)4 z2
N 2
3z + 3
3z4
:
Thus, using the exponential measure for the planar part, one can write down the corrected
measure d 1=N as
(5.50)
(5.51)
(5.52)
(5.53)
(5.54)
(5.55)
(5.56)
(5.57)
+
(2 g)4 z2
N 2
3z + 3
3z4
+ O(1=N 4)
:
Since the area dependence only appears in the exponent ega(x 1=x), the expectation value
of the Wilson loop and its derivatives retain the following simple expressions:
hWi
(n) =
I
d 1=N g(x
From this, it follows that the integral expression for the topological correlators (5.6) and
the orthogonality condition for QL(x) (5.8) still hold if we replace d
with d 1=N .
Repeating the same analysis at higher orders, we can determine the corrections to the
measure order by order. After working out rst several orders, we found17 the following
relation between the terms that appear in the expansion of hWi and the corrections to F (z):
( 0) n2 In(
p 0)
Integration by parts
!
(4 g)2 n r 2
z
e zp
zKn+ 12 ( z)
17We only checked the relation by Mathematica and did not work out a proof. It would be nice to prove
and establish the relation.
Note that although the right hand side involves the modi ed Bessel function Kn+ 12 , it
actually reduces to a rational function of z. Now, applying this relation to the expansion
of hWi given by Drukker and Gross [9],
2
hWi = p 0
I1(
p 0) + X
1
k=1
1
N 2k
k 1
X Xks
s=0
0
4
we obtain the following expansion of F (z):
z
In (5.58) and (5.59), Xks is a numerical coe cient de ned by the following recursion:
4Xks =
X10 =
3k
1
12
3k
;
s
2
s
Xks 11 +
3k
1
s Xks 1
;
Xkk = 0 :
(5.60)
(6.1)
(6.2)
It would be interesting to try to resum the series (5.59) and to consider the nonperturbative
corrections.
6
Weak and strongcoupling expansions
We now discuss the weak and the strongcoupling expansions of topological correlators
on the 1=2BPS Wilson loop at large N , and compare them with the direct perturbative
results. In particular, we focus on the threepoint functions since the topological correlators
are closed under the OPE and the match of the threepoint functions (or equivalently the
OPE coe cients) automatically guarantee the match of higherpoint functions.
In both cases, we rst compute the expansion of the polynomials QL(x):
QL(x) =
(Q0L(x) + g2Q1L(x) +
Q0L(x) + g1 Q1L(x) +
g
g
1
1
:
To determine the expansion, it is convenient to use the symmetrized measure d sym (5.45)
and perform the change of variables from x to y
i(x
can be rewritten as the following integral of y:
x 1)=2. Then, the integral over x
I
d sym (
) =
Note that we set a = 0 since we consider the 1=2BPS loops. In what follows, we use this
representation for the measure to compute the weak and the strongcoupling expansions.
6.1
Let us expand the measure (6.2) at weak coupling,
At the leading order, it is given by
:
This coincides with the measure for the Chebyshev polynomials of the second kind. Thus,
taking into account the di erence of the normalization, we conclude that QL at the leading
order at weak coupling is given by
where UL(y) is the Chebyshev polynomial of the second kind determined by the following
recursion relation:
Q0L(x) = ( ig)LUL(y) ;
U0(y) = 1 ;
U1(y) = 2y ;
UL+1(y) = 2yUL(y)
UL 1(y) :
Having identi ed QN with the Chebyshev polynomial, one can now compute the
twoand the threepoint functions by using the identities,
2
Z
dyp1
y2UL(y)UM (y) = LM ;
UL(y)UM (y) =
(L
M ) :
M
k=0
X UL M+2k(y)
(6.3)
(6.4)
(6.5)
(6.6)
(6.7)
(6.8)
(6.9)
Using these identities to evaluate the integral expressions for the correlators (5.6), we get
where Ltot is given by
and the symbol dL1;L2;L3 denotes
h
: ~ L1 : : ~ L2 : i O(g0)
h
: ~ L1 : : ~ L2 : : ~ L3 : i O(g0)
= ( g2)L1 L1;L2 ;
2 Ltot
= ( g ) 2 dL1;L2;L3 ;
Ltot
L1 + L2 + L3 ;
dL1;L2;L3 = <1
8
:0
As shown in (6.10) the threepoint function is nonzero only when the triangle inequalities
are satis ed and the sum of the lengths of the operators is even. These results precisely
match the treelevel planar Wick contractions. Note that, at this order, the expectation
value of the Wilson loop is 1 and there is no distinction between the unnormalized and
the normalized correlators hh ii.
Let us now discuss the oneloop correction. At one loop, the measure receives an
additional contribution,
This change of the measure induces the change of the orthogonal polynomials Q1L since
they need to satisfy the modi ed orthogonality condition
Z
Z
Furthermore, in order to keep the normalization condition (5.9), the correction Q1L must
be a polynomial of y with the order < L. One can solve these conditions using the equality
HJEP05(218)9
and the result reads
1
4
(1
y2)UL(y) =
(2UL(y)
UL+2(y)
UL 2(y)) ;
Q1L(y) = ( ig)L 2 2
3
UL 2(y) :
We can then compute the correction to the two and the threepoint functions using the
integral representation for the correlators (5.6) as follows:
(6.11)
(6.12)
(6.13)
(6.14)
(6.15)
(6.17)
(6.18)
(6.16)
h
: ~ L1 : : ~ L2 : i O(g2)
: ~ L1 : : ~ L2 : : ~ L3 : i O(g2)
(2 g)2( g2)L1
where da;b;c is 1 only when a + b + c is even and they satisfy the triangular inequality,
(otherwise zero). Using the identities18
dL1;L2 2;L3 = dL1;L2;L3
dL1;L2;L3 2 = dL1;L2;L3
L2+L3;L1
L3+L1;L2
L1+L2;L3 + L3+L1;L2 2 ;
L2+L3;L1 + L1+L2;L3 2 ;
we can rewrite the threepoint function also as
h
: ~ L1 : : ~ L2 : : ~ L3 : i O(g2)
=
(3dL1;L2;L3
L1+L2;L3
L2+L3;L1
L3+L1;L2 ) :
18These identities can be derived by expressing da;b;c as a product of step functions
and using the fact that (x + 1) = (x) + x; 1 and (x 1) = (x)
x;0.
O(g2), we get the following results for the normalized correlators:
By dividing the correlators by the expectation value of the Wilson loop hWi = 1 + 2 2g2 +
hh: ~ L1 : : ~ L2 : ii O(g2) =
hh: ~ L1 : : ~ L2 : : ~ L3 : ii O(g2) =
2( g)2( g2)L1
They are in perfect agreement with the direct oneloop computation performed in [29].
For completeness, let us also present the structure constant in the standard CFT
normalization; namely the normalization in which the twopoint functions become unity.
The result up to O(g2) reads
t = p
with
given by
y2)
y = 0 :
Expanding the measure around this saddle point and performing the change of variables
2 gy, we obtain the following expression for the measure at strong coupling:
1
dy
2 Z 1
y2)
=
1
e4 g
(2 )5=2g3=2 e t2 dt ;
1
g
t
8
4
;
d 0(t) +
d 1(t) + O(g2)
+ O(e g) :
2
3
1
dy
y=y
(6.19)
(6.21)
(6.22)
(6.23)
(6.24)
(6.25)
Strong coupling expansion
Let us consider the expansion at strong coupling. Here we send g ! 1 while keeping the
lengths of the operators Li's nite. In this limit, the integral
: ~ L1 : : ~ L2 :
: ~ Ln : i =
2 Z 1
can be approximated by its saddle point,
At the leading order, the measure d 0 is simply a gaussian. As is wellknown, this is
nothing but the measure for the Hermite polynomials. Thus, QL(x) at strong coupling is
Q0L(x) = ( i)L
g
2
L=2
HL(t) :
Here the factor ( i)L(g=2 )L=2 comes from the normalization of QL (5.9), and HL(t) is
the Hermite polynomial de ned by
H0(t) = 1 ;
H1(t) = 2t ;
HL(t) = 2tHL 1(t)
2(L
1)HL 2(t) :
We can then compute the two and the threepoint functions using the properties of the
Hermite polynomials,
1
dt e t2 HL(t)HM (t) = 2LL!p
LM ;
HL(t)HM (t) =
M
X
k=0
2M kL!M !
(L
M + k)!(M
The results are given by
HL M+2k(t)
(L
M ) :
HJEP05(218)9
: ~ L1 : : ~ L2 : i
h
: ~ L1 : : ~ L2 : : ~ L3 : i
=
e4 g
e4 g
L1! L1L2 ;
Lt2ot L1!L2!L3! dL1;L2;L3 ;
Lk)=2. Note that the overall coe cient e4 g=(2(2 g)3=2) is
precisely the expectation value of the circular Wilson loop at strong coupling. Therefore, the
normalized correlators take the following simple form:
hh: ~ L1 : : ~ L2 : ii g!1
hh: ~ L1 : : ~ L2 : : ~ L3 : ii g!1
=
=
g L1
g
L1! L1L2 ;
Lt2ot L1!L2!L3! dL1;L2;L3 :
These results reproduce the strongcoupling answer, which is given by the generalized free
elds in AdS2.
Let us now compute the correction to this strong coupling answer. At the next order,
the measure receives a correction d 1, given by (6.24). As in the weakcoupling analysis,
the change of the measure induces the correction to QL since they have to satisfy the
modi ed orthogonality condition:
To solve this condition, we use the following property of the Hermite polynomial:
1
t4HL(t) =
1
16
H4(t) + 12H2(t) + 12
1
HL(t)
d 1Q0LQ0M +
=
HL+4 +
+ (2L
1)
2L + 3
4
L!
(L
2)!
HL+2 +
HL 2 +
3(2L2 + 2L + 1)
(L
L!
4
4)!
HL 4 :
HL
(6.26)
(6.27)
(6.28)
(6.29)
(6.30)
(6.31)
Q1L(x) =
( i)L
8
: ~ L1 : : ~ L2 : i O(1=g)
: ~ L1 : : ~ L2 : : ~ L3 : i O(1=g)
Using this result, we can compute the correction to the twopoint function as
L!
(L
2)!
L!
(L
4)!
(2L
1)
HL 2(t) +
HL 4(t) :
(6.32)
e4 g
e4 g
Ltot
2
L1! L1L2 32 g
3
3
64 g
(2L12 + 2L1 + 1) ;
(Lt2ot + 2Ltot + 2)
(6.33)
(6.34)
(6.35)
(6.37)
Since the expectation value of the Wilson loop can be expanded at strong coupling as
hWi
e4 g
1
3
32 g
+ O(1=g2) ;
the normalized correlators are given by
hh: ~ L1 : : ~ L2 : ii O(1=g)
hh: ~ L1 : : ~ L2 : : ~ L3 : ii O(1=g)
g L1
Ltot
2
L1! L1L2 32 g
3
(2L12 +2L1) ;
3
64 g
(Lt2ot +2Ltot) L1!L2!L3! dL1;L2;L3 :
1=2
64 g
1
2
As we will see in the next subsection, these results are in perfect agreement with the direct
strongcoupling computation.
normalization at strong coupling:
Using these results, we can also compute the structure constant in the standard CFT
hh: ~ L1 : : ~ L2 : : ~ L3 : ii
hh: ~ L1 : : ~ L1 : iihh: ~ L2 : : ~ L2 : iihh: ~ L3 : : ~ L3 : ii
p
L12j3!L23j1!L31j2!
L1!L2!L3! dL1;L2;L3 1+
Comparison to string theory
3(L21 +L22 +L23 2(L1L2 +L2L3 +L3L1))
1
g2
+O
: (6.36)
SB =
p
2
Z
1
2
In this section we show that the strong coupling expansion of the localization results derived
above precisely matches the direct perturbative calculation using the AdS5 S5 string sigma
model. As is wellknown, on the string theory side the 1/2BPS (circular or straight) Wilson
loop is dual to a minimal surface with the geometry of an AdS2 embedded in AdS5 (and
pointlike in the S5 directions). The dynamics of the string worldsheet uctuations is most
conveniently described using the NambuGoto action in static gauge. The bosonic part of
the string action up to the quartic order was written down explicitly in [32] and it reads
d
Here g
is the AdS2 worldsheet metric, ya; a = 1; : : : ; 5 are the massless uctuations in
the S5 directions, which are dual to the scalar insertions
a on the gauge theory side, and
xi; i = 1; 2; 3 are the m2 = 2 uctuations in AdS5 dual to insertions of the displacement
operator [46, 47]. For the explicit form of the quartic vertices, see [32]. Note that there
are no cubic vertices between the elementary bosonic
uctuations.
Let us rst review the result for the treelevel connected fourpoint function of the ya
uctuations computed in [32], and its agreement with the localization prediction. Taking
the circular geometry at the boundary, it takes the form
hY1 y( 1)Y2 y( 2)Y3 y( 3)Y4 y( 4)icAodnSn2: =
GT ( )+ (GT ( )+GA( ))+ (GT ( ) GA( )) :
p
2 2
2
Y1 Y2 Y3 Y4 1
(4 sin 212 sin 234 )2 p
where
is the crossratio
and ; are SO(5) crossratios
2
5
=
sin 212 sin 234
sin 213 sin 224
=
Y1 Y3 Y2 Y4
Y1 Y2 Y3 Y4
Y1 Y4 Y2 Y3
Y1 Y2 Y3 Y4
with Yi null polarization 5vectors. The functions of crossratio GS;T;A( ) appearing in
the 4point function above correspond to singlet, symmetric traceless and antisymmetric
channels, and their explicit form can be found in [32].
In writing (6.45) we have taken the normalization of the y uctuations such that the
leading order 2point function computed from the string action reads19
hY1 y( 1)Y2 y( 2)iAdS2 =
Y1 Y2
2 2 (2 sin 212 )2
This normalization agrees in the strong coupling limit with the normalization we adopted
on the gauge theory side, which gives
hhY1
( 1)Y2
( 2)ii =
I2(
p )
2 2 I1(
Y1 Y2
p ) (2 sin 212 )2 =
2 2
1
+ : : :
Y1 Y2
(2 sin 212 )2
:
:
2
3
p
(6.38)
(6.39)
(6.40)
(6.41)
(6.42)
(6.43)
(6.44)
We now specialize to the topological boundary operators, by choosing the polarizations
Yi = (cos i; sin i; 0; i; 0; 0) :
By analogy with the notation introduced earlier on the CFT side, let us de ne
y~( )
cos( )y1( ) + sin( )y2( ) + iy4( ) ;
19In [32] instead a canonical normalization of the kinetic term was used, so that the leading 2point
function was
the overall
independent. The normalization in (6.41) is actually the one which is naturally induced by
dependence in the string action, upon adopting the standard AdS/CFT dictionary to compute
the treelevel 2point function, see [45].
F
F
F
F
F
F
F
to nexttoleading order at strong coupling. The grey blob in the middle gure denote the oneloop
correction to the \boundarytoboundary" y propagator.
HJEP05(218)9
which is dual to the insertion of ~ and has the constant 2point function given at leading
order by hy~( 1)y~( 2)iAdS2 =
4 2 . Then, using the explicit form of GS;T;A( ), one nds
the position independent result for the connected 4point function
hy~( 1)y~( 2)y~( 3)y~( 4)icAodnSn2: =
3
16 4
:
The full 4point function to the rst subleading order also receives contribution from
disconnected diagrams, as shown in gure 4. In addition to the leading treelevel generalized
free eld Wick contractions, there are corrections of the same order as (6.45) coming from
disconnected diagrams where one leg is oneloop corrected, see the
gure. While these
corrections have not been computed explicitly yet from string theory, we will assume below
that they reproduce the strong coupling expansion of (6.42).20 Then, the 4point function
of singleletter insertions computed from the AdS2 string theory side reads to this order
hy~( 1)y~( 2)y~( 3)y~( 4)iAdS2 =
4 2
!2 "
1
3
2
2
3
+: : : =
3
p
where the rst term in the bracket is the contribution of disconnected diagrams, and the
second term the one of the treelevel connected diagram. This precisely matches the strong
coupling expansion of the localization result
hh
~ ~ ~ ~
16 4 +
3
2 4
3
p
4 4I1(
I0(
p )
p )
:
Having reviewed the matching of the ~ 4point function, let us now move to the
computation of the twopoint and threepoint functions of arbitrary length insertions : ~ L : .
The Witten diagrams contributing to the 2point function to the rst two orders in the
strong coupling expansion are given in
gure 5 (as in
gure 4 above, there are oneloop
corrections to the diagrams involving free eld Wick contractions, that for brevity we do not
depict in the gure). The contribution of the diagram involving the 4point vertex can be
obtained from the 4point result (6.38) by taking Y2 ! Y1; Y3;4 ! Y2, and taking the limit
20Alternatively, one may consider normalized correlators as in (6.35), where such corrections drop out in
the ratio.
(6.45)
(6.46)
(6.47)
ture the case L = 4 is shown). The diagrams on the left, corresponding to generalized free eld
contractions, also receive a subleading correction where a ypropagator is oneloop corrected.
HJEP05(218)9
2 ! 1; 4 ! 3
at small , and so we get
2. From [32], we have GT ( ) =
3 2 + : : : and GA( ) = O( 3 log( ))
2
h(Y1 y( 1))2(Y2 y( 2))2iAdS2 =
3
p (Y1 Y2)2
64 4 sin4 12
2
=
3
16 4
;
(6.48)
where the rst equality is valid for any choice of the null polarization vectors, and in the
second equality we have specialized to the topological con guration. We can now use this
result and some elementary combinatorics to compute the 2point functions for arbitrary
length. We nd
y~Ly~LiAdS2 =
4 2
!L 2
4L! 1
3L
2
L
2
!2
3
(L
2)! + : : :5
(6.49)
L
2
!
The rst term in brackets corresponds to the generalized free eld Wick contractions: there
are clearly L! such contractions, and the factor (1
+ : : :)L = 1
2
3L + : : : accounts
p
for the oneloop correction of the boundarytoboundary legs, as discussed above. The
second term in brackets corresponds to the diagrams involving the 4point vertex shown in
gure 5: there are
ways of picking two y's on each operator, and (L
2)! free eld
contractions among the remaining y's. This result can be simpli ed to
y~Ly~LiAdS2 =
p
4 2
!L
L! 1
p L(L + 1) + : : : ;
(6.50)
3
2
3
p
4
3
which indeed precisely agrees with the localization result given in (6.29) and (6.35).
Similarly, the diagrams contributing to the 3point function hh: ~ L1 : : ~ L2 : : ~ L3 : ii are
shown in
gure 6. The leading contribution is given again by free eld Wick contractions.
Let us de ne the number of such contractions to be
nL1;L2;L3
L1!L2!L3! dL1;L2;L3
with dL1;L2;L3 given in (6.10). At the subleading order, there are two topologies which
involve the 4point vertex: one where the vertex connects two y's belonging to two di erent
FL3
FL3
FL3
operators (in the picture the case L1 = L2 = 4; L3 = 2 is shown). In addition, there are oneloop
corrections to the generalized free eld diagrams shown on the left.
operators, and one where it connects two y's from one operator and two y's from two
separate operators, see the gure.21 The rst type of diagram can be computed using (6.48).
For the second type of diagram, again taking the limit of the 4point result (6.45) by setting
Y2 ! Y1; Y3
Y2; Y4
Y3 and similarly for the i points, one nds
h(Y1 y( 1))2 Y2 y( 2) Y3 y( 3)iAdS2 =
3
p (Y1 Y2)(Y1 Y3)
64 4 sin2 212 sin2 13
2
=
3
16 4
where we have specialized to the topological con guration in the second step, but the rst
equality holds in general. Then, working out the relevant combinatorics and putting all
the contributions together, we nd for general lengths
hy~L1 y~L2 y~L3 iAdS2 =
3
3
nL1 2;L2 2;L3
nL1 2;L2 1;L3 1
p ! Lt2ot "
4 2
L1
2
2
nL1;L2;L3 1
2
3(L1 +L2 +L3) +: : :
2
L2 +nL1 2;L2;L3 2
L1
2
2
L3 +nL1;L2 2;L3 2
L2
2
L3
2
L1 L2L3 +nL1 1;L2 2;L3 1
L2 L1L3 +nL1 1;L2 1;L3 2
2
2
L3 L1L2
;
(6.52)
(6.53)
with Ltot = L1 + L2 + L3. This simpli es to
y~L1 y~L2 y~L3 iAdS2 =
4 2
! Lt2ot
nL1;L2;L3 1
3Ltot(Ltot + 2)
16p
+ : : : ;
(6.54)
again in complete agreement with the localization prediction (6.29) and (6.35).
In a similar way, one can compute higherpoint correlation functions of : ~ L : insertions
to nexttoleading order at strong coupling. While for the topological operators the
agreement of these should follow from the agreement of 2point and 3point functions shown
above, to dispel any doubt we have explicitly veri ed in various higherpoint examples
that the localization results are indeed correctly reproduced by string perturbation theory
around the AdS2 minimal surface.
21Note that there is no diagram where the 4vertex connects three y's on the same operator, as this
vanishes by SO(5) symmetry: in terms of the null polarization vectors, it necessarily involves a factor
Yi Yi = 0.
with Xi
xi 1). Here we used the exponential measure (5.46) for later convenience,
but the results in this subsection are equally valid if we substitute it with d
or d sym.
The determinant in (7.1) has the structure of the Vandermonde determinant and it can be
rewritten as
DL =
d exp(xk)
;
(7.1)
YL I
k=1
X2
1
.
.
.
X1
X22
.
.. .
.
.
XLL 1 XLL
X1L 1
X2L
.
.
.
XL2L 2
!
1
X2
.
.
.
X1
X22
.
.. .
.
.
XLL 1 XLL
X1L 1
X2L
.
.
.
XL2L 2
= Y Xk 1 Y(Xj
k
i<j
Xi) :
Since the measure factors in (7.1) are symmetric under the permutation of the indices, we
can replace the right hand side of (7.2) with its symmetrized version,
Y Xk 1 Y(Xj
k
i<j
Xi) ! L! i<j
Xi) X ( 1)j j Y Xkk 1 :
2SL
We then realize that the sum over the permutation is precisely the de nition of the
Vandermonde determinant. We can thus replace the determinant part by
In this section, we reformulate our results in the planar limit as a matrix model. We follow
closely the approach in the integrability literature [25, 43], but the resulting matrix model
is slightly di erent. This reformulation would be useful for studying the semiclassical
limit where Li and g are both send to in nity while their ratios are kept nite. We present
preliminary results for the semiclassical limit leaving more detailed analysis for future
investigation.
DL and QL(x) as a matrix model
Using the integral representations (5.3) and (5.4), the equation (3.14) can be reexpressed as
X1
X22
.
.. .
.
.
XLL 1 XLL
X1L 1
X2L
.
.
.
XL2L 2
Qi<j (Xi
Xj )2
L!
Therefore, we obtain the multiintegral expression,
DL =
gL(L 1)
L!
YL I
k=1
d exp(xk)
!
Y(xi
i<j
xj )2 1 +
1
xixj
2
;
(7.2)
(7.3)
(7.4)
(7.5)
Note that this matrixmodellike expression is similar but di erent from the matrix model
for m2L, derived in [43]. One notable di erence is that the integral in [43] contains 2L
integration variables while the integral derived here contains only L integration variables.
As proven in section 5.3, the two determinants are related by (5.42).
One can also express the polynomial FL as a multiple integral. Applying the integral
expression (5.4) to (3.18), we get
FL[X] =
1
DL
YL I
k=1
d exp(xk)
XLL 1 XLL
X2
1
.
.
.
1
X1
X22
X
.
.. .
.
.
X1L
X2L+1
.
.
.
XL2L 1
XL
:
(7.6)
xk xk 1. Now the determinant part in the integrand can be evaluated as
XLL 1 XLL
X2
1
.
.
.
1
X1
X22
X
.
.. .
.
.
X1L
X2L+1
.
.
.
XL2L 1
XL
= Y Xk 1(X
k
Xk) Y(Xj
i<j
Xi) :
Thus, after symmetrization, we get
FL(X) =
gL(L 1) "YL I
L!DL
k=1
d exp(xk) X
g(xk xk 1)
Y(xi xj )2 1+
i<j
1
xixj
As can be seen from this expression, FL(X) is the analogue of the characteristic
polynomial of the matrix model, which is obtained by inserting det(X
M ) in the integral of the
matrix M . After the change of the variables X = g(x
x 1), it can be rewritten as
QL(x) = FL(g(x x 1))
L!DL k=1
d exp(xk) (x xk) 1+
1
xxk
Y(xi xj )2 1+
i<j
1
xixj
7.2
Classical limit of the matrix model
Let us now consider the limit where g and Li's are sent to in nity while their ratios remain
nite. This limit corresponds to a classical string con guration in AdS and therefore is
called the (semi)classical limit.
The integral expression (7.5) can be rewritten as
DL =
gL(L 1) I
(4 g)LL!
YL dxk(1 + xk 2) eSL(x1;:::;xk) ;
where the action is given by
SL =
L
k=1
X 2 g xk +
1
xk
+ ag xk
+ 2 X log (xi
xj ) 1 +
k=1
1
xk
2 i
L
i<j
2
:
2
:
(7.7)
(7.8)
(7.9)
1
xixj
(7.10)
: (7.11)
two branch cuts; the one coming from the condensation of xk and the other coming from the
condensation of 1=xk.
In the classical limit, the integral can be approximated by the saddle point @SL=@xk = 0.
To compare with the result from integrability, it is convenient to introduce the rapidity
variables
uk
g xk +
:
x2 4 2
j6=k (xk
1
5 =
As in the usual large N matrix models, we expect that xk's condense into a branch
cut in the classical limit as shown in
gure 7. To describe the limit, it is convenient to
introduce a function pL(x) de ned by
pL(x)
2
1
x2 4 2g
k=1 (x
1
xk) 1 + xx1k
3
5 :
Then, the saddlepoint equation (7.13) can be rewritten as
where xk
denote the two di erent sides of the branch cut. Since p(x) has the symmetry
1
2
[pL(xk + ) + pL(xk
)] =
;
pL(x) =
pL( 1=x) ;
(7.12)
(7.13)
(7.14)
(7.15)
(7.16)
owing to (7.16).
1
2
pL
x 1 +
+ pL
x 1
It turns out that the function pL(x) coincides with the quasimomentum computed
= a=2). To see this, let us rewrite (7.14) using the identity,
where we de ned xk with k > L as
pL(x) =
a x2 + 1
2 x2
1
y
2L
X
xk
and it agrees with (3.15) in [43] (after appropriate reordering of xk's). These two
agreements guarantee that our p(x) has the same analytic properties as the quasimomentum
in [43], which uniquely specify the function. We thus conclude that the two functions must
be the same.
Using the quasimomentum pL(x), we can also express the semiclassical limit of QL(x).
By taking the saddlepoint of the integral expression (7.9), we obtain
where xk's are the saddlepoint values of the integration variables, which satisfy (7.13).
Using the de nition of pL(x), we can also rewrite (7.22) as
Z u(x)
QL(x)
gL exp
du0 pL(u0) +
a
u0
2g p(u0)2
4g2
where we introduced the rapidity variable u de ned by u
g(x + 1=x).
22Precisely speaking, there is a small di erence from (3.13) in [43]: in their case, the number of roots is
2L + 1 whereas it is 2L in our case, even after we doubled the number of roots by (7.20). However, this
di erence does not a ect the leading semiclassical answer.
xk L
(k > L) :
1
2
[pL(xk + ) + pL(xk
)] =
1
L + 1
L
2L
the branch cut of xk's is accompanied by another branch cut that is formed by
Around this other branch cut, pL(x) satis es
(7.18)
(7.17)
(7.19)
(7.20)
(7.21)
(7.22)
(7.23)
the transformation x !
be reexpressed as
The expression (7.19) coincides22 with the de nition of the quasimomentum (3.13) in [43]
if we take into account the fact that the distribution of xk's in [43] are symmetric under
1=x. Furthermore, using (7.20), the saddlepoint equation can
1
x
x
2
1 g
L Y(x
k=1
xk) 1 +
1
xxk
;
Given the match of the quasimomentum, we can follow the argument of [43] and show
that the semiclassical limit of our matrix model correctly reproduces the Bremsstrahlung
function computed from classical string. More interesting and challenging would be to
compute the semiclassical limit of the structure constants using the integral
representation (5.6) and the asymptotic formula for QL (7.23). We leave this for future investigation.
Before ending this section, let us also point out that one can study the nonplanar
corrections to DL and QL by replacing the measure d exp in (7.5) and (7.9) with the
nonplanar measure d 1=N given in section 5.5. It would be interesting to analyze the
classical limit of the nonplanar corrections using our matrix model and match it with a
classical string con guration.
8
In this paper, we computed a class of correlation functions on the 1=8 BPS Wilson
loop by relating them to the area derivatives of the expectation value of the Wilson
loop. When restricted to the 1=2 BPS loop, the results provide in nitely many
defectCFT data. As a byproduct, we also obtained
niteN generalization of the generalized
Bremsstrahlung function.
Let us end this paper by mentioning several future directions worth exploring: rstly,
it would be interesting to generalize our analysis to include operators outside the Wilson
loop. In the absence of insertions on the loop, such correlators were computed in [18, 19]
using the relation to 2d YM. Combining their results with our method, it should be possible
to compute the correlators involving both types of operators. Work in that direction is
in progress [48]. Once such correlators are obtained, one can try to numerically solve the
defect CFT bootstrap equation [33] using these topological correlators as inputs.
Another interesting direction is to apply our method to other theories, in particular to
N = 2 superconformal theories in four dimensions, for which the Bremsstrahlung function
was recently studied in [49]. Having exact correlators for these theories would help us
understand their holographic duals, including the dual of the Veneziano limit of N = 2
superconformal QCD [50].
At large N , we have shown that the correlators are expressed in terms of simple
integrals. A challenge for the integrability community is to reproduce them from integrability.
In the hexagon approach to the structure constants [51, 52], the results are given by a sum
over the number of particles. At rst few orders at weak coupling where the sum truncates,
it is not so hard to reproduce our results [
29, 53
]. A question is whether one can resum
the series and get the full results. In many respects, the topological correlators on the
Wilson loop would provide an ideal playground for the hexagon approach; one can try to
develop resummation techniques, x potential subtleties (if any), and compute nonplanar
corrections [54, 55].
Lastly, the appearance of the Qfunctions in our largeN results suggests deep relation
between localization and the Quantum Spectral Curve. It is particularly intriguing that
there is a onetoone correspondence between the multiplication of the Qfunctions and
the operator product expansion of the topological correlators. A similar observation was
recently made in [56] in a slightly di erent context: they found that the correlators on
the Wilson loop in the socalled ladders limit [57], which can be computed by resumming
the ladder diagrams [30], simplify greatly when expressed in terms of the Qfunctions of
the quantum spectral curve. Exploring such a connection might give us insights into the
gaugetheory origin of the Quantum Spectral Curve.
Acknowledgments
We thank N. Gromov, P. Liendo, C. Meneghelli and J.H.H. Perk for useful discussions and
comments. SK would like to thank N. Kiryu for discussions on related topics. The work
of SG is supported in part by the US NSF under Grant No. PHY1620542. The work of
SK is supported by DOE grant number DESC0009988.
A
Explicit results for operators with L 3
In this appendix we collect some explicit results for 2point and 3point functions of
operators with L
3. We restrict for simplicity to the case of the 1/2BPS loop. In terms of the
areaderivatives of the Wilson loop expectation value, one gets for the 2point functions
hh: ~ : : ~ : ii = W(2)
hh: ~ 2 : : ~ 2 : ii = W W(4)
hh: ~ 3 : : ~ 3 : ii = W(2)
W
(W)2
W(6)
W W(2)
(W(2))2
(W(4))2
and for the 3point functions
hh: ~ 2 : : ~ : : ~ : ii = hh: ~ 2 : : ~ 2 : ii
hh: ~ 2 : : ~ 2 : : ~ 2 : ii =
hh: ~ 3 : : ~ 2 : : ~ 1 : ii = hh: ~ 3 : : ~ 3 : ii
hh: ~ 3 : : ~ 3 : : ~ 2 : ii =
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
2(W(2))3 3W(2)W(4)W +W(6)(W)2
(W)3
(W(2))3W(6) +(W(2))2( W(4))2 +W(8)W
2W(2)W(4)W(6)W +(W(4))3W :
(W(2))2(W)2
Here W
< W > jA=2 and W(k)
@Ak < W > jA=2 (similar expressions hold for the
general 1/8BPS loop, but they also involve derivatives of odd order). Using the Wilson
loop expectation value (2.3), one can obtain in a straightforward way the explicit nite N
results in terms of Laguerre polynomials, but the expressions are rather lengthy and we do
not report them here. In the planar large N limit, the above correlators can be expressed
hh: ~ : : ~ : ii =
hh: ~ 2 : : ~ 2 : ii =
hh: ~ 3 : : ~ 3 : ii =
I2
p
4 2I1
p
p
3
16 4
I0
16 4I1
p
p
I0
p
2 4I1
p
p
and for the 3point functions:
hh: ~ 2 : : ~ : : ~ : ii = hh: ~ 2 : : ~ 2 : ii
3
p (5 +72)I0
64 6I1
p
I2
p
p
3(13 +144)I0
32 6 I0
p
3( (32 3 )+288)I1
64 6p
I0
p
p
2I1(p )
p
3=2I0
32 6I1
p
p
3
3 32 6
8 6I1
p
p
3 (5 +72)I0
256 8I1
p
2I2
p
3( (2 +579)+6192)I0
64 8I2
3( ( (9
112)+4960)+34176)I1
256 8 I2
p
3
p (127 +1920)I0
128 8I1
p
I2
2
:
3
p ( +40)I0
p
32 6I1
p
p
p
3
4 6
3( (5
757) 6336)I1
32 8p
I0
p
p
I0
p
p
2I1(p ) 2
4 4
p
2I1(p )
p
(A.11)
(A.12)
(A.13)
(A.14)
:
(A.15)
(A.16)
(A.17)
A.1
Let us also list the rst few results for the generalized Bremsstrahlung function, focusing
on the case
= 0 given by eq. (4.18). Using the same notation as above, the L
2 results in terms of areaderivatives of the Wilson loop expectation value read
W
2W(2)
W
W(4)
2(W(2))2W(4)W
W(2)W(6)(W)2 + 2(W(4))2(W)
(W(2))3W
W(2)W(4)(W)2
Plugging in (2.3), one can nd the explicit nite N results. For instance, we obtain
16 2N
16 2N
1 +
2L2N 2
L1
N 1
4L2N 2
L1
N 1
4N
4N
4N
4N
6 2L3N 3
2L2N 2
4N
4N
+ L2N 2
+ L1
4N
4N
(A.18)
: (A.19)
Open Access.
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