Toward precision holography with supersymmetric Wilson loops

0.38 MB·

by  Alberto Faraggi

MCTP-16-03
Toward precision holography with supersymmetric Wilson loops
Alberto Faraggi,a Leopoldo A. Pando Zayas,b Guillermo A. Silva,c and Diego Trancanellid
a Instituto de F´ısica, Pontificia Universidad Cato´lica de Chile
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1 Casilla 306, Santiago, Chile
0
2
b The Abdus Salam International Centre for Theoretical Physics

b
e Strada Costiera 11, 34014 Trieste, Italy
F
0 b Michigan Center for Theoretical Physics, Department of Physics
1
University of Michigan, Ann Arbor, MI 48109, USA

]
h
c Instituto de F´ısica de La Plata - CONICET & Departamento de F´ısica - UNLP
t
-
p C.C. 67, 1900 La Plata, Argentina
e
h
[ d Institute of Physics, University of Sa˜o Paulo, 05314-970 Sa˜o Paulo, Brazil

2
v
8
0
Abstract
7
4 We consider certain 1/4 BPS Wilson loop operators in SU(N) = 4 supersymmetric Yang-
0 N
. Millstheory,whoseexpectationvaluecanbecomputedexactlyviasupersymmetriclocalization.
1
5
0 Holographically,theseoperatorsaremappedtofundamentalstringsinAdS5 S . Thestringon-
×
6
shellactionreproducesthelargeN andlargecouplinglimitofthegaugetheoryexpectationvalue
1
: and, according to the AdS/CFT correspondence, there should also be a precise match between
v
i subleadingcorrectionstotheselimits. Weperformatestofsuchmatchatnext-to-leadingorder
X
in string theory, by deriving the spectrum of quantum fluctuations around the classical string
r
a
solution and by computing the corresponding 1-loop effective action. We discuss in detail the
supermultiplet structure of the fluctuations. To remove a possible source of ambiguity in the
ghost zero mode measure, we compare the 1/4 BPS configuration with the 1/2 BPS one, dual
toacircularWilsonloop. Wefindadiscrepancybetweenthe stringtheoryresultandthegauge
theoryprediction,confirmingapreviousresultintheliterature. We areabletotrackthemodes
from which this discrepancy originates,as well as the modes that by themselves would give the
expected result.
Contents
1 Introduction 2
2 The 1/4 BPS latitude in = 4 super Yang-Mills 5
N
3 Review of the classical string solution 6
3.1 Symmetries of the classical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 Quadratic fluctuations 8
4.1 Type IIB strings on AdS S5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5
×
4.2 Spectrum of excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.3 Multiplet structure and supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . 13
5 One-loop determinants 15
5.1 The Gelfand-Yaglom method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2 Bosonic determinants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2.1 Determinant for the χ2,3,4 modes . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2.2 Determinant for the χ5,6 modes . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2.3 Determinant for the χ7,8,9 modes . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.3 Fermionic determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6 The 1-loop effective action 25
6.1 Bosonic sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.2 Fermionic sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.3 Final result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7 Conclusions 30
A Geometric data 32
B Dimensional reduction of spinors 34
C The integral S 37
D Swapping the ǫ 0 and Λ limits 37
0
→ → ∞
1
1 Introduction
The AdS/CFT correspondence provides a paradigm wherein a field theory is equivalent to a string
theorycontaininggravity[1]. Themoststudiedandbestunderstoodexampleofthiscorrespondence
conjectures the equivalence of SU(N) = 4 super Yang-Mills theory and type IIB string theory
N
on AdS S5 with N units of Ramond-Ramond (RR) five-form flux. There are various levels at
5
×
which this correspondence can be tested. The ‘weakest’ level is the limit of large N and strong
’t Hooft coupling on the field theory side, whose dual string theory is well described by classical
supergravity. Going beyond this limit is, potentially, a conceptually fruitful endeavor. An ideal
arena were this can be achieved is the study of non-local supersymmetric operators such as the
Wilson loops.
Very soon after the Maldacena correspondence was put forward, it was proposed that the
vacuum expectation value of the 1/2 BPS circular Wilson loop, arguably the simplest non-local
supersymmetric operator, is captured by a Gaussian matrix model [2, 3]. This conjecture was later
proven by Pestun [4], using the technique of supersymmetric localization. For the case of the fun-
damental representation of SU(N), thevacuum expectation value of this operator is known exactly
for any N and any ’t Hooft coupling λ = g2 N in terms of generalized Laguerre polynomials [3]:
YM
1 λ
W = L1 eλ/8N
h icircle N N−1 −4N
(cid:18) (cid:19)
2 λ λ2
I (√λ)+ I (√λ)+ I (√λ)+...
≃ √λ 1 38N2 2 1280N4 4
3 1 π
exp √λ lnλ ln +... . (1.1)
≃ − 4 − 2 2
(cid:18) (cid:19)
The first line is exact in N and λ, the second line is an expansion in large N, and in the last line
the large λ limit is also taken.
Havinganexact fieldtheoryanswerposesoneofthesimplest,yet elusive, tests oftheAdS/CFT
correspondence. The situation is akin to a high precision test of the AdS/CFT correspondence,
where the field theory side provides the “experimental” side and string theory is the theory that
should match the experimental results. Indeed, there has been a fairly concerted effort in trying
to match the field theory answer (1.1) with the 1-loop corrected answer coming from holography.
The first efforts date back over a decade and a half [5]. More recently, the 1-loop correction has
2
been revisited using different methods in [6] and [7], leading to
1
W = exp √λ ln(2π)+... . (1.2)
circle
h i − 2
(cid:18) (cid:19)
The main missing term in this formula is the (3/4)lnλ. There is also a numerical discrepancy in
−
the constant term. This discrepancy has been attributed to ghost zero modes in the corresponding
string amplitude [5, 6, 7]. There are also similar discrepancies when confronting field theory results
with holographic computations at 1-loop level for Wilson loops in higher rank representations as
summarized in [8], albeit in those cases the functional dependence matches.
OurdrivingmotivationisnotahiddensuspicionofthevalidityoftheAdS/CFTcorrespondence,
rather we believe that, by carefully consideringsuch discrepancies, we might learn something about
the intricacies of computing string theory on curved backgrounds with RR fluxes, thus broadening
the class of problems which the AdS/CFT can tackle at the quantum level. In this sense our
philosophy is summarized in the following question: What can we learn about string theory in
curved backgrounds from having exact results on the dual, gauge theory side?
With this general motivation in mind, we turn to the study of certain 1/4 BPS Wilson loop
introducedin[9,10]andfurtherstudiedin[11,12,13,14]. Theseloopsarecalled “latitude” Wilson
loops and from the field theory point of view are quite similar to the 1/2 BPS circle. The latitudes
are defined in terms of a parameter, θ [0,π/2], which selects a latitude on an S2 on which the
0
∈
loop is supported, see the next section for more details. The vacuum expectation value of this
operator is conjectured to be given by a simple re-scaling of the ’t Hooft coupling in the exact
expression for the 1/2 BPS Wilson loop [10, 13, 14]:
1 λ
W = L1 ′ eλ′/8N , (1.3)
h ilatitude N N−1 −4N
(cid:18) (cid:19)
where λ = λcos2θ . In fact, this conjecture extends to a larger class of (generically 1/8 BPS)
′ 0
Wilson loops, the so-called DGRT loops, defined as generic contours on an S2 [12, 13, 14], of which
the latitude is a special example with enhanced supersymmetry. This conjecture has passed several
non-trivial tests. In perturbation theory, it has been checked explicitly for specific examples of
DGRT loops, and correlators thereof, up to third order, see for example [15, 16, 17, 18]. At strong
coupling, it has been checked in [10, 14] by constructing the corresponding string configurations
and evaluating their on-shell action. Finally, localization has been applied in [19], where it was
shown1 that these loops reduce to the Wilson loops in the zero-instanton sector of (purely bosonic)
1The proof of localization is somewhat incomplete, since it lacks a computation of the1-loop determinants.
3
Yang-Mills theory onatwo-sphere, whichisan exactly solvabletheory[20], seeforexample[21,22].
Holographically, the1/4 BPS latitude gets mappedto a macroscopic stringin AdS S5, which
5
×
not only extends on the AdS part of the geometry, as the 1/2 BPS string does, but it also wraps
5
a cup in the S5 part. For some recent investigations into these configurations see, for example, [23]
and [24]. The main idea of this paper is to compute the 1-loop effective action for this string and
compare it with the effective action for the 1/2 BPS string. Since both strings have a world-sheet
with the topology of a disk, the expectation is that the issues related to the ghost zero modes,
which we have mentioned above, might cancel. More specifically, we consider the ratio
W 3
h ilatitude exp √λ(cosθ 1) lncosθ +... , (1.4)
0 0
W ≃ − − 2
h icircle (cid:18) (cid:19)
withtheintent of recovering the (3/2)lncosθ termfrom thestringtheory 1-loop effective action.
0
−
The paper is organized as follows. We review various field theoretic aspects of the 1/4 BPS
Wilson loop in Sec. 2 and the classical string solution in Sec. 3. We present a derivation and
analysis of the fluctuations in Sec. 4. In particular, we show how they are neatly organized in
representations of the supergroup SU(22). We compute the determinants in Sec. 5 and the 1-loop
|
effective action in Sec. 6. We finally conclude with some comments and outlook in Sec. 7. We
relegate a number of explicit technical calculations to the appendices.
Note 1: As we were in an advanced stage of this project (partial progress having been reported
in [25]), thepaper [26]appeared. Thereis certainly alot of overlap. Although conceptually similar,
our work has some technical differences with [26], which we highlight. In particular, we stress the
role of group theory in the spectrum of fluctuations and in the sums over energies, we have a
different treatment of the fermionic spectral problem, for we consider the linear operator instead
of the quadratic one, and we use different boundary conditions for the fermions. Moreover, our
treatment of the 1-loop effective action is fully analytical, whereas [26] resorted to numerics.
Note 2: In this revision, we correct a critical mistake in the original manuscript submitted to
the arXiv that alters our conclusions. Instead of the agreement between gauge theory expectation
and string theory claimed in the v1, we do find a finite discrepancy, precisely equal to the remnant
reported in [26]. One advantage of having an analytical treatment, as we do here, is that we are
able to track the origin both of the expected result (i.e., the (3/2)lncosθ term) and of the
0
−
discrepancy to certain specific modes. We hope this might be useful for future investigations, as
we comment in the conclusions.
4
2 The 1/4 BPS latitude in = 4 super Yang-Mills
N
We start with a brief review of the gauge theory side [10, 14]. The 1/4 BPS latitude Wilson loop
(in the fundamental representation of SU(N)) is defined as
1
W(C)= Tr exp ds iA x˙µ+ x˙ Φ nI(s) , (2.1)
µ I
N P | |
ZC
(cid:0) (cid:1)
where denotes path ordering along the loop and C labels a curve parametrized as
P
xµ(s)= (coss,sins,0,0), nI(s)= (sinθ coss,sinθ sins,cosθ ,0,0,0). (2.2)
0 0 0
This operator interpolates between the 1/2 BPS circle, corresponding to θ = 0, and the so-
0
called Zarembo loops [27] at θ = π/2. It preserves a SU(22) subgroup of the superconformal
0
|
group SU(2,24) of = 4 super Yang-Mills, for more detail see App. B.2 of [14]. The bosonic
| N
symmetries are given by
SU(2) U(1) SU(2) . (2.3)
B
× ×
The first SU(2) factor is a remnant of the conformal group, broken by the presence of the latitude
circle. This is, in fact, the same SU(2) factor from SO(4,2) which is also preserved by the 1/2
BPS circle, although the symmetry is realized differently in the two cases. Note, in passing, that
the 1/4 BPS loop does not preserve the SL(2,R) subgroup of SO(4,2) preserved by the 1/2 BPS
circle. In the holographic dual, this will manifest itself in the fact that the induced metric on the
string world-sheet is not AdS , as it is the case for the string corresponding to the 1/2 BPS circle.
2
The U(1) symmetry in (2.3) mixes Lorentz and R-symmetry transformations
C = J +JA , (2.4)
12 12
with J coming from SL(2,R) and JA from the SU(2) subgroup of the SU(4) R-symmetry. In
12 12 A
the holographic dual, this symmetry is implemented as translations along the ψ and φ coordinates,
as we shall see presently. The last SU(2) is the SU(2) subgroup of the R-symmetry. This can be
B
understoodby noticing thatthe loop is only definedin terms of thescalars Φ , which arerotated
1,2,3
by SU(2) , whereas the other three scalar fields Φ , which do not appear in the Wilson loop,
A 4,5,6
are rotated by SU(2) . From the holographic point of view, as we will review in the upcoming
B
section, one can think of this symmetry in terms of the embedding coordinates of the sphere where
an SO(3) is explicit.
5
3 Review of the classical string solution
In this section we review the classical string solution dual to the 1/4 BPS latitude Wilson loop
[10, 14]. The supergravity background is given by AdS S5 with a five-form RR flux and the
5
×
AdS metric conveniently expressed as a foliation over H H
5 2 2
×
ds2 = cos2u dρ2+sinh2ρdψ2 +sin2u dϑ2+sinh2ϑdϕ2 du2. (3.1)
AdS5 −
(cid:0) (cid:1) (cid:0) (cid:1)
We have set the radius equal to 1. The Euclidean continuation is achieved by taking u iu and
→
ϑ iϑ, such that the EAdS metric becomes now a foliation over H S2
5 2
→ ×
ds2 = cosh2u dρ2+sinh2ρdψ2 +sinh2u dϑ2+sin2ϑdϕ2 +du2. (3.2)
AdS5
(cid:0) (cid:1) (cid:0) (cid:1)
The metric on S5 is taken to be
dΩ2 = dθ2+sin2θdφ2+cos2θ dξ2+cos2ξdα2 +sin2ξdα2 , (3.3)
5 1 2
(cid:0) (cid:1)
and the 4-form potential reads
1 u
C = sinh(4u) vol(AdS ) vol S2 , (3.4)
(4) 2
8 − 2 ∧
(cid:18) (cid:19)
(cid:0) (cid:1)
with corresponding field strength F = 4(1+ )vol(AdS ).
(5) 5
− ∗
The string has world-sheet coordinates (τ,σ) and its embedding in the background above is
given by [10]:
1 1
sinhρ= , ψ = τ , u= 0, sinθ = , φ = τ , (3.5)
sinhσ cosh(σ +σ)
0
where σ sets the range of values of θ, namely, 0 θ θ , with
0 0
≤ ≤
1
sinθ = . (3.6)
0
coshσ
0
The remaining coordinates take arbitrary constant values. The string world-sheet forms a cap
through the north pole of the S5. The sign of σ determines whether the world-sheet starts above
0
(σ > 0) or below the equator (σ < 0), this last case being unstable under fluctuations [10].
0 0
The induced geometry on the string world-sheet is
ds2 = sinh2ρ+sin2θ dτ2+(ρ2+θ2)dσ2. (3.7)
′ ′
(cid:0) (cid:1)
6
Since the solution satisfies ρ = sinhρ and θ = sinθ, we can write the induced metric as
′ ′
− −
ds2 = sinh2ρ+sin2θ dτ2+dσ2 . (3.8)
(cid:0) (cid:1)(cid:0) (cid:1)
In the following, we shall denote the overall conformal factor as
1 1
A sinh2ρ+sin2θ = + , (3.9)
≡ sinh2σ cosh2(σ +σ)
0
where in the last equality we have used the explicit solution for the embeddings ρ(σ) and θ(σ)
in (3.5).
In the σ limit, the range of θ shrinks to a point. In this sense the 1/4 BPS solution
0
→ ∞
reduces to the 1/2 BPS one, where θ is but a point on S5 and the string world-sheet has an AdS
2
geometry. This has the topology of a disc plus a point. The disk along the AdS part has radial
2
coordinate σ [0, ) (with boundary located at σ = 0) and angular coordinate τ τ + 2π.
∈ ∞ ∼
The cap on S2 is contractible and, consequently, equivalent to the point on the north pole which
corresponds to the solution in the 1/2 BPS case.
The string action can be evaluated on-shell on this classical solution. The result, after an
appropriate renormalization, is [10]
S(0) = √λcosθ . (3.10)
0
−
Since W exp S(0) = exp √λcosθ , werecover, at the classical level, theexpectation (1.4)
0
h i ≃ −
from field theory. (cid:16) (cid:17)
(cid:0) (cid:1)
3.1 Symmetries of the classical solution
In[14]itwasshownthatthe1/4BPSlatitudepreservesanSU(22)subgroupofthesuperconformal
|
groupof = 4superYang-Mills. ThecorrespondingbosonicsubgroupisSU(2) U(1) SU(2)
B
N × × ≃
SO(3) SO(2) SO(3).
× ×
Oneof thesimplestway toseehow theembeddingpreserves SO(3) SO(3) is byexpressingthe
×
solution intheembeddingcoordinates X . ForAdS wehave X2+X2+X2+X2+X2+X2 = 1,
i 5 − 0 1 2 3 4 5 −
with the solution taking the form
X = cothσ, X = cosechσ cosτ, X = cosechσ sinτ, X = X = X = 0. (3.11)
0 1 2 3 4 5
7
One explicitly sees that there is an SO(3) group that rotates the coordinates (X ,X ,X ) without
3 4 5
affecting thesolution. OntheS5 side,whoseequation wewriteas Y2+Y2+Y2+Y2+Y2+Y2 = 1,
1 2 3 4 5 6
we have
Y = sech(σ +σ)cosτ, Y = sech(σ +σ)sinτ, Y = tanh(σ +σ), Y = Y = Y = 0, (3.12)
1 0 2 0 3 0 4 5 6
where tanhσ = cosθ . Similarly, there is an SO(3) group that rotates the coordinates (Y ,Y ,Y )
0 0 4 5 6
without affecting the solution. There is an SO(2) rotation realized in the plane (X ,X ) and an
1 2
SO(2) rotation realized in the plane (Y ,Y ). These symmetries are identified as translations in τ,
1 2
as can be clearly seen in the classical solution ψ = τ = φ in (3.5).
We shall show later on that the string fluctuations around the 1/4 BPS solutions are neatly
organized in multiplets of this SU(22) supergroup.
|
4 Quadratic fluctuations
Havingreviewedtheclassical solution dualtothe1/4BPSlatitudeWilsonloopandits symmetries,
in this section we derive the corresponding spectrum of excitations. For the case of the 1/2 BPS
circular Wilson loop, the dual solution and its perturbations have been known for quite some
time, see for example [5, 6, 7]. Similar studies for holographic duals of Wilson loops in higher
representations include [28, 29, 30].
We will start by giving a general expression for the quadratic fluctuations of the type IIB string
in AdS S5 and then specialize to the case of the 1/4 BPS string dual to the latitude Wilson
5
×
loop. We willclosely follow geometrical approach andthe conventions of [29]. Inparticular, we rely
on App. B of [29], where a summary of the geometric structure of embedded manifolds is given.
See also [31] for a similar approach. In what follows, target-space indices are denoted by m,n,...,
world-sheet indices are a,b,..., while the directions orthogonal to the string are represented by
i,j,.... All corresponding tangent space indices are underlined.
4.1 Type IIB strings on AdS S5
5
×
In the bosonic sector, the string dynamics is dictated by the Nambu-Goto (NG) action
1
S = d2σ√g, (4.1)
NG
2πα
′ Z
8
where g is the induced metric on the world sheet and g = detg . Our first goal in this section
ab ab
| |
is to consider perturbations xm xm + δxm around any given classical embedding and to find
→
the quadratic action that governs them. To this purpose, let us choose convenient vielbeins for the
AdS S5 metric that are properly adapted to the study of fluctuations. Using the local SO(9,1)
5
×
symmetry, we can always pick a frame Em = (Ea,Ei) such that the pullback of Ea onto the world-
sheet forms a vielbein for the induced metric, while the pullback of Ei vanishes. Of course, these
are nothing but the 1-forms dual to the tanget and normal vectors fields, respectively. The Lorentz
symmetry is consequently broken to SO(1,1) SO(8). Having made this choice, we may define
×
the fields
χm = Em δxm, (4.2)
m
and gauge fix the diffeomorphism invariance by freezing the tangent fluctuations, namely, by re-
quiring
χa = 0. (4.3)
The physical degrees of freedom are then parameterized by the normal directions χi. This choice
has the advantage that the gauge-fixing determinant is trivial [5]. In this gauge, the variation of
the induced metric is
δg = 2H χi+ χi χjδ + H cH R ∂ xm∂ xn χiχj, (4.4)
ab − iab ∇a ∇b ij ia jbc− minj a b
(cid:16) (cid:17)
i
where H is the extrinsic curvature of the embedding and
ab
χi = ∂ χi+ ij χ (4.5)
∇a a A a j
ij
is the world-sheet covariant derivative, which includes the SO(8) normal bundle connection .
A a
These objects, as well as the world-sheet spin connection wab, are related to the pullback of the
target-space spin connection Ωmn by
wab = P[Ωab], Hi = P[Ωi ] ea , ij = P[Ωij], (4.6)
ab a a b A
where ea = P [Ea] is the induced geometry vielbein. Using the well-known expansion of the
a a
square root of a determinant, a short calculation shows that, to quadratic order, the NG action
becomes
√λ
S(2) = dτdσ√g gab χi χjδ gabH cH +δabR χiχj , (4.7)
NG 4π ∇a ∇b ij − ia jbc aibj
Z (cid:16) (cid:16) (cid:17) (cid:17)
9