A proof of the correlation function/supersymmetric Wilson loop correspondence

Abstract

We prove that in the limit when its insertion points become pairwise null-separated, the ratio of certain n-point correlation functions in N = 4 \mathcal{N} = 4 SYM is equal to a supersymmetric Wilson loop on twistor space, acting in the adjoint representation. In the planar limit,
each of these objects reduces to the square of the complete n-particle planar superamplitude. Our proof is at the level of the integrand.

KeywordsExtended Supersymmetry–Duality in Gauge Field Theories–Conformal and W Symmetry

Full text

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Preprint typeset in JHEP style - HYPER VERSION
A Proof of the Correlation Function /
Supersymmetric Wilson Loop
Correspondence
Tim Adamo∗, Mathew Bullimore†, Lionel Mason∗and David Skinner‡
∗The Mathematical Institute,
24-29 St. Giles’, Oxford OX1 3LB, United Kingdom
†Rudolf Peierls Centre for Theoretical Physics,
1 Keble Road, Oxford OX1 3NP, United Kingdom
‡Perimeter Institute for Theoretical Physics,
31 Caroline St., Waterloo, ON, N2L 2Y5, Canada
Abstract: We prove that in the limit when its insertion points become pairwise
null-separated, the ratio of certain n-point correlation functions in N = 4 SYM
is equal to a supersymmetric Wilson loop on twistor space, acting in the adjoint
representation. In the planar limit, each of these objects reduces to the square of the
complete n-particle planar superamplitude. Our proof is at the level of the integrand.
arXiv:1103.4119v1 [hep-th] 21 Mar 2011

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1. Introduction
Over the past few years, there has been much interest [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
in a correspondence between planar scattering amplitudes in N = 4 Super Yang-
Mills (SYM) and the planar limit of the correlation function of a Wilson loop in the
fundamental representation, stretched around a piecewise null polygonal contour in
Minkowksi space.
In [11, 12], we showed that the planar integrand of a supersymmetric Wilson loop
in twistor space, taken in the fundamental representation, agrees with the integrand
of the planar scattering amplitude – including all NkMHV partial amplitudes to all
orders in the ’t Hooft coupling, divided by an overall factor of the MHV tree am-
plitude. In [11], this correspondence was conjectured, with the supporting evidence
that the Feynman diagrams of the Wilson loop correlator (computed using the twistor
action [13] for N = 4 SYM in axial gauge) explicitly reproduce the standard MHV
diagram expansion for scattering amplitudes [14, 15, 16, 17]. In [12] the conjecture
was proved by using a twistor version of the Migdal-Makeenko equations to show
that the Wilson loop planar integrand obeys the all-loop extension of the BCFW
recursion relation for the integrand of the planar scattering amplitude [18]. Closely
related work was performed by Caron-Huot [19] from the space-time perspective.
In a very interesting series of papers [20, 21, 22] Alday, Eden, Korchemsky, Mal-
dacena and Sokatchev have suggested a remarkable extension of the correspondence
between Wilson loops and scattering amplitudes to include correlation functions.
More specifically, these authors conjecture that
lim
i,i+1→0x2
G(x1,··· ,xn)
G(0)(x1,··· ,xn)= ?Wadj[γ]?
(1.1)
where Wadj[γ] is a (non-supersymmetric) Wilson loop in the adjoint representation
(normalised so that its leading term is 1), taken along the null polygonal contour
γ = (x1x2) ∪ (x2x3) ∪ ··· ∪ (xnx1) (1.2)
that appears in the limit when adjacently labelled insertion points in the correlator
G(x1,...,xn) ≡?
n
?
i=1
O(xi)?
(1.3)
become null separated. (1.3) is the n-point correlator of a local, gauge invariant
operator that may for example be taken to be either the 1/2 BPS operator
Oabcd= Tr(ΦabΦcd) −1
12?abcdTr(Φ
efΦef) (1.4)
or the Konishi multiplet
OK= Tr(Φ
abΦab), (1.5)
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and in (1.1), G(0)(x1,...,xn) is the same correlator at tree level only.
The ratio G/G(0)and the Wilson loop expectation value are each divergent in
the limit x2
ularised scheme such as dimensional regularisation, or else carefully consider the
limiting behaviour of the correlator and limiting behaviour of a Wilson loop on a
nearby, non-null curve. These approaches were each considered in [20]. Importantly
however, [22] further conjectured that, as for the scattering amplitude/Wilson loop
correspondence, this new correspondence should hold already at the level of the in-
tegrand1. The integrands themselves are perfectly finite, conformally invariant data
associated with either side of (1.1), and may be compared without need of regulari-
sation.
i,i+1→ 0, so to compare both sides one must either work with a reg-
The contour γ that appears in the limit x2
curve that plays an important role in scattering amplitudes, so this story should
be naturally related to the scattering amplitude / fundamental Wilson loop corre-
spondence. And indeed, as pointed out by [21], the correlation function of an adjoint
Wilson loop reduces in the planar limit to the product of correlators of a fundamental
and anti-fundamental Wilson loop. In a CPT invariant theory these two factors are
equal, so if the above conjectured relation to adjoint Wilson loops could be proved,
it would follow immediately from the proof of [12] that – in the planar limit – the
ratio of correlation functions in (1.1) is also given by the square (AMHV/A(0)
the ratio of the all-loop MHV amplitude to the MHV tree.
i,i+1→ 0 is the same piecewise null
MHV)2of
The aim of this paper is to prove the above conjectures to all orders in g2at
the level of the integrand, using twistor theory. As for the amplitude / Wilson loop
correspondence, one of the main advantages of performing the calculation in twistor
space is that it the supersymmetric extension may be handled straightforwardly. In
twistor space the fundamental N = 4 SYM multiplet may be described off-shell by
the superfield2
A(Z,¯Z,χ) = a(Z,¯Z) + χaγa(Z,¯Z) + ··· + χ1χ2χ3χ4g(Z,¯Z)
introduced by Ferber in [23] and exploited to great effect by Witten in [24]. The
coefficient of (χ)min (1.6) is a smooth (0,1)-form on CP3|4, homogeneous of degree
−m. The linearised field equations in twistor space state that A is holomorphic in
Z, whereupon the component fields correspond to the linearised on-shell N = 4 su-
permultiplet in space-time. However, we repeat that A(Z,¯Z,χ) is first and foremost
1The notion of the integrand was introduced in [18] in the context of planar scattering amplitudes.
It does not make sense for arbitrary n-point correlation functions, as there is no way to compare
different contractions between gauge invariant operators. However, it is meaningful in (1.1) precisely
because only one way of contracting the fields survives in the limit.
2In this paper we use (Zα,χa) to denote homogeneous coordinates on CP3|4. We shall sometimes
write Zα= (λA,µA?) in terms of two 2-component spinors µ and λ. In Penrose conventions, this
would be a dual twistor.
(1.6)
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an off-shell field in twistor space. We shall see that writing twistor expressions in
terms of this superfield provides a natural supersymmetrization of both sides of the
correspondence (1.1), such that the ratio of n-point correlators of certain supermul-
tiplets corresponds to a certain supersymmetric Wilson loop. This supersymmetric
twistor Wilson loop is exactly the same object that appeared in [11, 12], except that
in this case it acts in the adjoint representation. Thus, as a consequence of the super-
amplitude / supertwistor Wilson loop correspondence proved in [11, 12], the ratio
of correlation functions involving our supermultiplet reproduces the square of the
complete n-particle planar S-matrix (for arbitrary helicity configurations), divided
by the MHV tree.
In [25] the authors proposed that the correspondence between appropriately su-
persymmetrized correlation functions and amplitudes should be more robust than
that between supersymmetric Wilson loops and amplitudes. Our proof [11, 12] of
the Wilson loop / scattering amplitude integrand correspondence directly gives rise
to the proof of the supersymmetric correlation function / scattering amplitude cor-
respondence in this paper. We therefore see that the approaches are not so very
different from each other.
2. Gauge Invariant Local Operators in Twistor Space
Our first task is to construct the operator in twistor space that corresponds to the
local, gauge invariant space-time operators (1.4) or (1.5). The details of the R-
symmetry representation are not important at this stage, so in this section we con-
sider a generic operator of the form O(x) = Tr(Φ2).
We build up the twistor operator in stages. Firstly, consider a single scalar field
Φ(x) in an Abelian theory (so that Φ is gauge invariant). A basic fact of twistor
theory is that any such field that obeys its eom 2Φ = 0 corresponds to a cohomology
class [φ] in twistor space that may be represented by a (0,1)-form φ(Z) of homogeneity
−2 under under a rescaling of Z. Concretely, the Penrose transform states that
?
where X is the CP1in twistor space that corresponds to the point x in space-time,
Φ(x) =
X
?λdλ? ∧ φ(Z,¯Z)??
X
(2.1)
µA?= −ixAA?λA
χa= θAaλA
(2.2)
(although we are ignoring the supersymmetric coordinates in this section) and λAis
taken as a homogeneous coordinate along X. As a simple example, if
φ(Z,¯Z) =
1
A · Z
¯∂
?
1
B · Z
?
(2.3)
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then setting µA?= −ixAA?λAone finds that Φ(x) ∝ 1/(x − y)2on space-time, where
y corresponds to the CP1in twistor space given by the intersection of the planes
A·Z = 0 and B·Z = 0. (See e.g. [26] for a discussion of (2.3) using contour integrals.)
Away from other field insertions, the individual fields in O(x) can be taken to
obey their equations of motion, although of course 2O ?= 0 because of cross-terms.
Thus, in an Abelian theory,
?
X×X
Φ2(x) =
?λdλ? ∧ ?λ?dλ?? ∧ φ(Z) ∧ φ(Z?) (2.4)
where Z and Z?are each restricted to the same Riemann sphere X, but their locations
along X are integrated over separately.
The key point here is that the local space-time operator corresponds to a non-
local operator on twistor space. Because of this, if Φ is in the adjoint representation
of a non-Abelian group, we cannot construct the twistor operator simply by taking
the trace of (2.4).
In twistor space, non-Abelian gauge fields are described by a complex vector
bundle E → CP3that has vanishing first Chern class. Such an E is topologically
trivial on restriction to any CP1– a necessary condition if we wish to describe a
space-time theory where X corresponds to a point. Thus one can (generically3) find
a holomorphic frame for E|X; that is, a gauge transform h(x,λ,¯λ) on X such that
h−1◦ (¯∂ + a)??
where¯∂ + a is the covariant d-bar operator on E.
To compare the adjoint-valued field φ at Z ∈ X with that at Z?∈ X, we work in
the trivialisation of E|Xdefined by h, so that
?
X×X
X◦ h =¯∂??
X
(2.5)
Tr(Φ2)(x) =
?λdλ? ∧ ?λ?dλ?? ∧ Tr?h−1φh ∧ h−1φh?,(2.6)
where the first factor of h−1φh is evaluated at λ and the second at λ?. This use of
holomorphic frames was an important ingredient in the construction of the twistor
action for N = 4 SYM in [27, 13], and for individual fields is the standard extension
of the Penrose transform to fields in a non-trivial representation of a gauge group [28,
29].
It follows from (2.5) that h(x,λ,¯λ) itself obeys¯∂h = −ah on X, and we introduce
the notation
UX(λ,λ?) ≡ h(x,λ)h−1(x,λ?) (2.7)
3This holomorphic triviality of the bundle is generic and will always hold in perturbation theory
around the trivial bundle when a has been assumed small.
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