Part III essayfor the Mathematical Tripos at Cambridge

Hidden Structuresin

Scattering Amplitudes

Ruben Verresen

May 2013

Contents

1 Introduction 1

2 A broader historical context 1

2.1 Twistors: how Penrose saw the light in 1967 . . . . . . . . . . . . . . . . . . 1

2.2 Gauge theory and knots: Witten in 1988 . . . . . . . . . . . . . . . . . . . . 8

2.3 Gluon scattering simplified a 1000-fold: Parke and Taylor in 1986 . . . . . . 12

3 Twistors in the 21st century 14

3.1 The stage: (super)twistor space . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 The actors: the twistor (super)field . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 The script: holomorphic Chern-Simons action . . . . . . . . . . . . . . . . . 18

3.4 The props: delta function gymnastics . . . . . . . . . . . . . . . . . . . . . . 20

4 Curtain-up: scattering amplitudes! 21

4.1 Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2 MHV diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 It ain’t over till the fat lady sings: holomorphic knots 23

5.1 Abelian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2 Non-abelian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.3 Relation to MHV diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6 Curtain call and conclusions 29

Acknowledgements 30

1 Introduction

For many physicists, quantum field theory is the most beautiful theory of physics. Yetfor even more, the simplest calculations in non-abelian gauge theory evoke a deep sense ofdespair. It is hence of both conceptual and practical importance that the last decade hasreshaped the way we look at pure gauge theory and its scattering. It was sparked by un-expectedly simple expressions for certain n-gluon scattering amplitudes [1]. In recent yearsthese simple results have been elegantly understood by removing Minkowski space fromour formulation, replacing it by Penrose’s so-called twistor space [2]. However, the rabbithole went deeper than expected and other ways of understanding scattering amplitudes interms of twistors arose. These can be considered to lie outside the realm of the conventialFeynman procedure [3] [4]. One insight was that calculating scattering amplitudes for pureYang-Mills is the same as counting how a certain complex loop in twistor space is knottedinto itself, another definite step in the intertwining of topology and physics1.

There are three different roots from which the recent developments stem: the origin oftwistor space in 1967 [6], the discovery of simple n-gluon scattering in 1986 [1], and theunveiling of topological gauge theory calculating knot invariants in 1988 [7]. In section 2 wepresent a review of these topics. In 2003 [8] two of these–twistor and gauge theory–startedmerging, taking a definite form a few years later. Section 3 is concerned with this: treatingtwistor space from a modern perspective and describing N = 4 supersymmetric2 Yang-Millson it. The new (elegant) Feynman rules then drop out in section 4, explaining the simpleform of the scattering amplitudes. Section 5 then describes an alternative characterisationof these amplitudes, using holomorphic knot theory in twistor space.

2 A broader historical context

2.1 Twistors: how Penrose saw the light in 1967

Twistors were introduced by the mathematician and physicist Roger Penrose in 1967 [6].The space of twistors is a different way of looking at spacetime, in fact one can interprettwistor space as fundamental and spacetime as emergent. On a more pragmatic level,twistors are a useful change of variables. It has many peculiar properties, like chirality andnon-locality, as we will discuss later. Most striking is perhaps the fact that twistor spacetries to give complex numbers the main role in physics, using that “complex magic” Penroserefers to [9]. In fact, complex numbers are not just fundamental to quantum mechanics,but also to classical relativity, just a bit more hidden. As Dirac formulated it [10]:

I would suggest, as a more hopeful-looking idea for getting an improved quan-tum theory, that one take as basis the theory of functions of a complex variable.

1Interestingly the topological QFT known as Chern-Simons will play a crucial role, which is also deeplyimportant in other recent developments, like those in condensed matter (as low energy effective descriptionsof states with topological order) and quantum gravity (the so-called Chern-Simons state, which “hints at arelationship between knot theory and quantum gravity that we are only beginning to fathom” [5, p448]).

2The usual motivations apply: supersymmetry is used since it makes the problem more integrable, as asort of toy theory. It should be noted that at tree-level the non-supersymmetric amplitudes can be calculatedfrom the supersymmetric ones, hence it is still very practical.

1

This branch of mathematics is of exceptional beauty, and further, the group oftransformations in the complex plane, is the same as the Lorentz group gov-erning the space-time of restricted relativity. One is thus led to suspect theexistence of some deep-lying connection between the theory of functionsof a complex variable and the space-time of restricted relativity, theworking out of which will be a difficult task for the future.

Dirac is referring to the fact that SO+(1, 3) ∼= Mob(C) , the group of Mobius transforma-

tions of the complex plane, i.e. all the holomorphic 1-to-1 maps of the Riemann sphere3.This can be identified4 with the matrix group PSL(2,C). (This is why we have a spin-12 representation.) The mathematical physicist John Baez expressed similar awe in morerecent times: “The fact that such a basic group, SL(2,C), is so closely related to the struc-ture of spacetime, can only serve as a challenge to our understanding of physics. Is this acoincidence or a clue we have still not fully understood?” [5, p182]

Penrose takes on the challenge of Dirac and Baez, declaring that “the primary guiding ideaof twistor theory is that by translating all space-time geometry and the essential particleand field notions, into an entirely complex(-analytic) form, the correct unifications withquantum-theoretic principles will ultimately emerge.” [11, p3] As we will see, in doing thishe makes central use of the fact SO+(1, 3) ∼= PSL(2,C). We will now describe twistorspace, first with broad strokes to make the idea clear, but then adding details to makethe picture exact. In the main section 3 we will be able to give the definition in a line(with a different flavour to match current use in research) referring back to this section forconceptual motivation.

The idea: twistor space T is the space of all light rays in M such that we have a 1-to-1

light rays in M ↔ points in T

and moreover it turns out T has a complex structure and

points in M ↔ linearly embedded Riemann spheres in T.

(The fact spacetime points correspond to Riemann spheres of light is physically meaningful:if you look at the night’s sky you see the stars on a S2 at infinity, which has a conformalstructure: if you perform a Lorentz transformation, the points undergo a Mobius trans-formation. This is one manifestation of SO+(1, 3) ∼= Mob(C).) This correspondence isnon-local and involves complex numbers, both properties that are supposed to facilitate thequantum treatment. It also assigns fundamental importance to massless particles (light!).Some qualifiers will gradually be introduced to make the above broad statement exact(difference between “light momenta”/“rays”, projective, null, helicity, compactification, ...)

Light and spinors: as a first step, let’s see how light can be rewritten in terms of spinors.For this we need to make the isomorphism SO+(1, 3) ∼= PSL(2,C) explicit. This can bedone by relating M↔ 2× 2 Hermitian matrices:

xµ ↔ x · σ = xµσµ where σµ = (I, σx, σy, σz) .3A Riemann sphere is a complex manifold with the topology of a sphere. It arises as compactified C

(denoted C) or projective space CP1 := [z1, z2] where z · [z1, z2] = [z1, z2].4The Mobius transformation f : C → C : z 7→ az+b

cz+duniquely corresponds to a matrix in PSL2(C) with

those entries. Note that PSL2(C) = SL2(C)/±I = GL2(C)/C∗ which are the linear transformations ofCP1 = C2/C∗ (where C∗ = C− 0).

2

Performing xµ → Λµνxν corresponds to x ·σ → A(x ·σ)A†. Since det(x ·σ) = |xµ|, we can seethat A ∈ SL(2,C). Checking the specifics leads to the conclusion that under this mappingSO+(1, 3)↔ PSL(2,C).

Now suppose xµ is null, then the determinant is zero. The following fact is useful:

2× 2 Hermitian matrices of determinant zero ↔ λλ† | λ ∈ C2 (1)

Hence we see that if e.g. pµ is the momentum of light, then there is a λ ∈ C2 such that5

p · σ = λλ†. From above we know that p→ Λp corresponds to λ→ Aλ with A ∈ SL(2,C).The latter means that λ is a spinor6! (This is how a Weyl spinor transforms.) In indexnotation one usually writes pAA

′= λAλA

′(with spinorial indices A,A′ = 0, 1), where

pAA′

= pµσAA′

µ .

Light and Riemann spheres: we see that all light momenta through a given spacetimepoint correspond to all elements of C2 (up to a phase) via pAA

′= λAλA

′. If we just care

about direction and not magnitude (= light ray), we see

light rays through a given point ↔ CP1 = Riemann sphere

(This makes an earlier comment exact: if you look at the night sky and perform a Lorentztransformation, the stars will change conformally.) The above is a first instance of one ofthe twistor space correspondences: note that a point in spacetime can be defined by all thelight rays that go through it, but as we have seen this can be represented by a Riemannsphere! Twistor space in fact makes this work for any point: let twistor space be the spaceof all light rays, then the set of light rays through any fixed point in M will make up aRiemann sphere in twistor space.

Light and null-twistors: what does the space of all light rays look like? We have seenhow the momentum of light can be written in terms of a spinor λ. The extra informationwe need is the position of the ray. We will see this can be represented by another spinor µ.In fact, define twistor space T as

T :=

(λ, µ) | λ, µ ∈ C2. (2)

The elements are often written Zα = (λ, µ), or in conventional index notation (λA, µA′),

which has to do with how they transform as we will shortly see. As claimed the first spinorrepresents the light momentum via pAA

′= λAλA

′. Spinorial indices are raised/lowered7

using the Levi-Civita symbol

εAA′ = εAA′

=

(0 1−1 0

)= J

hence in matrix notation p · σ = Jλλ†J†.

To see how µ gives us the position of our ray, we first restrict to null-twistors. The dual of atwistor Zα is defined as Zα := (µ†, λ†). The inner product is ZαZ

α = µ†λ+놵. Define the

5Group-theoretically this corresponds to the statement that a massless spin 1 particle is a tensor productof two spin 1

2particles. In the massive case we need a sum of such tensor products.

6Indeed, a 360 rotation corresponds to the identity in SO+(1, 3) ∼= PSL(2,C), but to −1 in SL(2,C).7Caution: due to the anti-symmetry of εAA′ , naively first raising and then lowering will give an overall

minus sign! This is related to the fact that εAA′ isn’t really an inner product, but a symplectic structure.Incidentally this is als the reason for the Feynman rule “fermion loops give a sign”.

3

space of projective null-twistors PN, consisting of all projective elements [Zα] ∈ PT suchthat ZαZ

α = 0.

Claim: PN can be seen as the space of all light rays in M

Before proving this, note that the dimensions work out: dim(PN) = 5 which is what weexpect (the light cone at the origin is 3 dimensional, so the cone of rays is 2 dimensional,and there are 3 independent directions in which we can translate it).

Fact: ZαZα = 0 implies that there is an x ∈M such that µ = i(x · σ)λ with σ = (I,σ∗).

We then say our light ray travels through this point x ∈ M. For this to be consistent, weneed to make sure that if µ = ix · σλ = iy · σλ, that x and y define the same light ray, i.e.that x− y is proportional to p given by λ. Linear Algebra Intermezzo:

• Existence: ZαZα = 0 tells us iµ†λ is a real number, hence µµ†

iµ†λis Hermitian so by (1) there

is an x such that it equals x · σ.

• Well-defined: to show it’s well-defined we can see it’s enough to prove

ix · σλ = 0 ⇔ x · σ ∝ Jλλ†J

⇐ follows since λTJλ = λAλA′εAA′

= 0⇒ by (1) we can let x · σ = Jκκ†J , so we get κ(κTJλ) = 0, hence κTJλ = 0, but that meansκ ∝ λ (a fact perhaps more familiar in the form “v ×w = 0⇒ v ∝ w”)

Note: the above proof doesn’t work if λ = 0 or if λ = µ (then µ†λ = 0). In that casewe cannot find such an x for our null-twistor! The reason for this is interesting: the morecorrect statement is that PN is the space of all light rays in Mc, the compactification ofMinkowski space. (The case λ = 0 then corresponds to the lightcone at infinity, and λ = µmeans that every light ray passes through infinity.)

Incidentally, it’s a direct consequence of µA′

= ixAA′λA that under a Lorentz transformation

the matrix that transforms λA is the complex conjugate of the matrix transforming µA′ .We say they are respectively left- and right-handed spinors. This is the reason for the(un)primed indices.

A correspondence: we can now understand

light rays in M ↔ points in PNpoints in M ↔ linearly embedded Riemann spheres in PN

Indeed, a point [(λ, µ)] ∈ PN fixes a ray pAA′

= λAλA′

through x where i(x · σ)λ = µ.Conversely, a point x ∈ M determines [(λ, µ(λ))] with µ(λ) = ix · σλ, which is linearlyparametrised by [λ] ∈ CP1, a Riemann sphere.

Light and twistors: four possible questions that might arise (besides “why should we beinterested” which will be addressed afterwards):

1. What about non-null-twistors ZαZα 6= 0?

2. How to understand the peculiar relationship µ = i(x · σ)λ?

3. We know that even classically light has a polarization, why do we ignore it?

4

4. Why the name twistor?

All of these questions are resolved at once by looking at full twistor space T defined in (2)(or its projective version PT). It turns out the value of ZαZ

α labels the helicity/polarizationof our light ray! In fact just like how λ represents the (square root of the) momentum ofour light, µ represents the angular momentum. This makes sense on multiple levels. FirstlyZαZ

α then simply becomes the formula for spin in terms of angular and linear momentum,known as the Pauli-Lubanski spin vector [11]. Secondly, if ZαZ

α = 0 and thus there is nospin, then all the angular momentum of our light is external: “J = r×p”, which is exactlywhat “µA

′= ixAA

′λA” is telling us!

This shows that the full twistor space T represents all light rays in M, with λ capturingthe momentum and µ the angular momentum. The name “twistor” derives from how wecan represent non-null Zα geometrically in M (besides just ignoring the helicity, i.e. justdrawing the light ray with the same momentum and position). It gives rise to a beautifulfoliation of R3 by circles all twisted into one another (“Robinson congruence”)8, which wedon’t discuss here9.

Figure 1: Corresponding objects in M and PT

A sketch of corresponding objects in M and PT is given in figure 1. The black light raydefines a point in PN. We have drawn a (blue) non-null-twistor as a light ray in M byfirst projecting it onto PN. The (red) point in M corresponds to a linearly embeddedRiemann sphere in PN, here drawn as a (complex) line. This is illustrated by drawing outthe lightcone, each ray representing a point on the line in twistor space. Note there are nicecorrespondence properties: two points are null-separated if and only if their correspondingtwistor lines intersect. A sort of converse is also true: if two twistors are connected by aline of null-twistors10, then their corresponding light rays intersect. (This later statementwill become nicer once we allow complex points in M, as in section 3.)

Twistor space: what good is it?

What does twistor space give us? Firstly, it is an alternative space/set of variables we candefine our fields on. Like Fourier space this turns out to be very practical in certain cases.But unlike Fourier space there is a beautiful surprise that awaits us: the equations of motionfor massless fields simply become “∂φ = 0”. The Dirac equation, Maxwell’s equations, etcetera all reduce to a condition of holomorphicity.

8The idea is to realise the dual twistor Zα defines a 3-dimensional subspace of PN via ZαTα = 0, which

we already know how to represent geometrically: as a set of light rays in M!9A very nice drawing is presented and discussed in [9].

10The reason this is not trivial: the space defined by ZαZα = 0 is not convex!

5

To give a simple specific case of the Penrose transform: a scalar field φ(xµ) on space-time corresponds to a holomorphic function f(Zα) on twistor space such that f(zZα) =z−2f(Zα) for z ∈ C, i.e. it’s homogeneous of degree −2. More concretely at every pointxµ, the value of φ(xµ) is given by integrating out11 f over the Riemann sphere X ∈ PT:

φ(x) =

∮Xf(Zα)d2λ =

∮CP1

f(λA, ixAA′λA)d2λ with d2λ =

1

2dλA ∧ dλA . (3)

The scaling of f makes sure the above expression is well-defined. Writing ∇2 = ∇AA′∇AA′with ∇AA′ = σAA

′µ ∇µ, we can see that for the integrand

∇AA′f = iλA∂f

∂µA′since

∂f

∂µA′= 0 (holomorphic!)

Hence ∇2 brings out a factor λAλA = εABλAλB = 0 in the integrand, giving the Klein-Gordon equation for φ!12

There is already something deep to learn from this example. Note that f actually shouldn’tbe defined all over twistor space, otherwise the contour integral would be zero (Cauchy’stheorem). In other words: f has to blow up somewhere. This turns out to be related to thepositive-frequency condition in QFT [11]. Another interesting fact: this implies φ doesn’tchange if we add a globally defined holomorphic function to f . Indeed, it even turns out13

that if it’s on-shell, f is an element of the first holomorphic sheaf cohomology group14

H1(T,O). In practical terms this means that given a small enough region we can alwayschoose f to be zero there. Conceptually that’s saying the information in our field is storednon-locally. This perhaps makes it surprising that in the end scattering still makes sensein twistor space.

Without worrying about cohomological matters, the claim for general15 fields is that amassless spacetime field of helicity16 h corresponds to a homogeneous function on twistorspace of degree 2h−2. Satisfying the equation of motion is equivalant with holomorphicity.Before we can define this Penrose transform explicitly, we have to express our spacetimefields in a slightly different way (replacing our usual spacetime indices by spinor indices:µ 7→ AA′). This is called the 2-spinor formalism [12].

In first instance “spinors” shouldn’t be linked to the quantum-mechanical context of spin,that is but one manifestation. In general a spinor is a generalization of a tensor, and muchhow the latter is defined by its transformation properties, so is the former. A 2-spinor is themost basic object that transforms under Lorentz transformations (namely under a spin-1

2

11Note it’s a bit more subtle: we aren’t really integrating out over the Riemann sphere, since then wewould integrate a (1, 1)-form. Instead we integrate a (2, 0)-form and as such this is contour integral (ofcomplex dimension 2) that happens to go through X. It’s not important here, and later in section 3.2 onpage 17 we will use a different method.

12It may seem as though we only need holomorphicity in the µ coordinates, but as we will see in section3.1, the roles of λ and µ get switched under a special conformal transformation, hence conformal symmetry(guaranteed by Klein-Gordon) finishes the claim.

13Penrose [9] explicitly thanks Michael Atiyah for pointing this deep fact out. See that same reference fora simple explanation of this concept.

14This is part of a framework called “Cech cohomology” which we don’t need in this essay. Note that tobe exact, we should work on projective space, more specifically a subspace thereof, since H1(CP3,O) = 0.

15The case of non-abelian gauge fields needs some modification, which will be treated in section 3.2 p16.16Remember helicity is the projection of spin along momentum and it’s two-valued for massless particles.

6

representation, as the λ above). More geometrically, Atiyah characterises spinors as “thesquare root of geometry”17, and analogous to how it took centuries to understand

√−1, he

is still waiting for a proper way of knowing what spinors really are [13]. More practically,every tensor of rank n can be decomposed into sums of n tensor products of 2-spinors. Theabove pAA′ = λAλ

′A is a specific instance. If we now do this for fields, we can rewrite our

trusted equations.

The claim is the following: in spinor notation all massless fields are of the form φA1···An(of helicity −n/2) and φA′B′C′··· (of helicity n/2), the former transforming under the left-handed spin-1

2 representation, the latter the right-handed. All field equations have thesimilar form:

∇AA′φABC··· = 0 ∇AA′φA′B′C′··· = 0 .

It’s these spacetime fields that naturally correspond to fields on twistor space under thePenrose transform, comparable to the scalar case (3). The corresponding twistor fields turnout to be homogeneous of degree 2h−2 and the equation of motion is again holomorphicity.The concrete mapping is displayed in (13) on page 17, and the equivalence is as easily provenas in the scalar case.

How can we get a feeling for this 2-spinor formalism and the above claims (about theequation and the helicity)? Consider as a specific case the Dirac field. As we know, this isthe direct sum of the two Weyl fields ψL and ψR. In 2-spinor notation, we claim:

ψL → ψA ψR → ψA′ .

This is already consistent with their transformation properties18. Concerning the fieldequation, note that

∇AA′ψA′ = σAA′

µ ∇µψA′ = (−∂t + σ ·∇)ψR

which gives the correct Weyl equation. The helicity claim follows from ψR having helicity+1

2 . This can be checked19 using the Fourier space expression (E + σ · p)ψR = 0.

An analogous check can be performed for the case of Maxwell’s equations. In 2-spinornotation Fµν becomes FAA′BB′ and one can straightforwardly prove

FAA′BB′ = εABφA′B′ + φABεA′B′ = F+AA′BB′ + F−AA′BB′ (4)

for certain φAB and φA′B′ , the first equality proven using the antisymmetry of Fµν , andthe second using some Pauli matrix algebra [14]. The claim that φA′B′ has helicity +1then follows from self-dual fields having positive helicity20. Using (4) one can see that∇AA′φA′B′ = 0 is equivalent to dF+ = 0. Similarly we get dF− = 0 and together they canbe reformulated as Maxwell’s equations dF = 0 and d ? F = 0.

17A specific instance of this is Dirac taking the square root of the Laplacian to arrive at the Dirac operator.18Indeed, they transform under rotations (θi) and boosts (βi) as ψR,L → exp

[12(iθiσi ± βiσi)

]ψR,L whose

transformations do not just differ by complex conjugation (in which case ψR ∼= ψA′ and ψL ∼= ψA) but alsoby J · J† conjugation, since then i→ −i and σ → −σ.

19This actually gives the wrong sign. I have checked a few sources and they alway seem to fudge a signsomewhere along the way.

20This is familiar from undergraduate EM, although maybe not in so few words. Recall that light withpositive polarization was naturally described by complexified electric fields, and one can check that camedown to writing F+ = (iF + ?F )/2.

7

We see that even in spacetime all the massless fields can be made very similar by using spinornotation. The unification is then taken to another level by going to twistor space where afield of helicity h corresponds to a homogeneous twistor function of degree 2h − 2. Notethe chirality: positive and negative helicities are treated differently! This chiral nature isat the heart of the twistor approach. Although it may seem enigmatic, Penrose considersit useful for incorporating the fundamental left/right-asymmetry in nature [9]. Either way,twistor space can be a powerful tool for dealing with wave equations, and indeed it has ledthe way to non-trivial solutions in general relativity and other areas [12] [14].

The twistor approach can also be useful for quantum theory. Penrose originally came upwith it as a relativistic version of so-called “spin networks”, the aim being to first rewritespacetime as to then give a new way of quantizing gravity [9]. In normal spacetime ourextended objects are light rays, and upon quantization they become “fuzzy”. In twistorspace it’s the spacetime points that are extended (Riemann) spheres. As such, conceptually,in this approach the notion of null direction stays well-defined, but the concept of “spacetimeevent” becomes fuzzy. The twistor approach to quantum gravity has not yet delivered onits promises, but the developments discussed in this essay show that the union of twistorspace and quantum theory is a marvelous one.

A brief look at twistor quantization can shed light on a possible confusion: on the one handwe saw helicity entered in the twistor variables themselves via ZαZ

α, and on other handwhen describing fields it was encoded in their scaling. What is the relation? It turns outthat spacetime quantization [x, p] = i~ corresponds to the twistor space quantization

[Zα, Zβ] = 0 [Zα, Zβ] = 0 [Zα, Zβ] = i~δαβ .

Hence Zα and its conjugate are canonically conjugate. So in twistor space, Zβ → ∂∂Zβ

such that taking care of commutations the helicity operator ZαZα becomes proportional to

2 +Zα ∂∂Zα [11]. Since this is simply the Euler homogeneity operator, we see that a twistor

wavefunction21 of degree 2h− 2 describes a particle of helicity h.

Yet another useful aspect of twistor space is its easy representation of the conformal(Poincare) symmetry, the discussion of which we postpone until section 3.1 on page 14.

In conclusion, twistor space offers a different way of describing spacetime by a non-localcorrespondence, which fundamentally involves non-local information, chirality, conformalsymmetry and complex “magic”, structurally is closely related to the spinor formalism onspacetime, and practically is very useful for finding solutions to field equations (includingscattering amplitudes as we will see later). Surprisingly it also offers a natural setting forcomplex knot theory, with even more surprising physical significance.

2.2 Gauge theory and knots: Witten in 1988

Knot theory is about how closed curves in R3 are knotted into themselves or linked withothers. It originated out of physics, in particular gauge theory. In the latter half of the19th century Tait and Kelvin conjectured that atoms might correspond to knotted vortices

21But why should it also hold for the classical waves we were looking at, like the electromagnetic field?This is probably related to the remarkable fact that if one performs second quantization (i.e. Fock space) ofa particle, this turns out to be equivalent to first performing first quantization (i.e. Schrodinger equation),then treating that wavefunction as a classical field, and now performing first quantization on that.

8

in the aether. Although it didn’t work out, it was the beginning of this rich branch ofmathematics. Knot theory had to wait a hundred years (until Witten’s 1988 paper) beforefinding its relevance to gauge theory. A beautiful book explaining this is that of Baez andMuniain [5]. In fact, in light of section 5 of this essay, even that was but a stepping stone:in special cases holomorphic knot theory calculates actual scattering amplitudes! [4] Toappreciate that more deeply, in this section we review the insights of Witten in 1988. Thisis unrelated to twistors and the like, but besides being beautiful and deep in its own right,it conceptually motivates the later work. (Moreover, the calculation we do here is directlyapplicable to the complex case in section 5.)

Figure 2: (a) the “unknot”, (b) a knot or self-link, (c) a link between two loops

The most basic examples of knots and links are given in figure 2. We would like to assignthe number 1 to (b) to enumerate how much it links to itself, and the number 2 to (c) forhow much they are linked to each other (the self-linking of the components is zero!). Thesenumbers are obvious link invariants: if we deform the loops a little bit, they won’t change.

The definition of (a component of) a loop is an embedding C : S1 → R3, and a link invariantL(C) then has to satisfy

L(φ∗C) = L(C)

where φ∗C is the pull-back/image of our loop under a diffeomorphism φ : R3 → R3. Asimple example is the (Gauss) linking number L(C1, C2) which intuitively tells us howmany times one loop winds around the other, and L(C1, C2) = 2 for figure 2(c). An analyticexpression is given by

L(C1, C2) =1

4π

∮C1

∮C2

n

|n|3· (dr1 × dr2) (5)

which we will actually derive later for free. The vector n = r2 − r1 is drawn in figure 2(c).

The natural notion of the aforementioned self -linking number is denoted w(C) and calledthe writhe. In a way it’s an extension of the Gauss linking number, as it makes sense ofL(C1, C1). Hence, unsurprisingly we have the relationship

w(C) =∑i 6=jL(Ci, Cj) +

∑i

w(Ci) (6)

where C has components C = C1 + · · ·+ Cn.

At this point one may wonder how one actually defines such invariants. The most naturalway is using so-called skein relations, which basically defines a link invariant by tellingus how it changes when we undo a link. Since we won’t use skein relations explicitly in thisessay, I won’t give more detail.

9

Witten in the abelian case

What Witten [7] realised is that link invariants can be calculated using the quantum fieldtheory of particular gauge fields in 3D, called “Chern-Simons theories”. Let’s look at thecase of the abelian gauge field, which will give us the above self-linking number. The claimis that the writhe w(C) is equal to the expectation value of22 ei

∮C Aidx

ifor the action

S =∫εijkAi∂jAkd

3x. Using the neatness of differential geometry, the exact statement is

exp

(− i

4w(C)

)=

⟨exp i

∮CA

⟩=

1

Z

∫ei

∮C Aei

∫A∧dADA . (7)

Before deriving (7), let’s comment on the expression itself. W (C,A) := exp i∮C A is called a

Wilson loop, a gauge-invariant quantity. Its role in the Aharonov-Bohm effect illuminatesits mathematical meaning: if we parallel transport a vector (∈ U(1) in the abelian case)around C, then we get the Wilson loop as a phase factor. In mathematics this is calledthe holonomy23 of the gauge field around C. In itself W (C,A) is not a link invariant:W (φ∗C, φ∗A) = W (C,A) but W (φ∗C,A) 6= W (C,A). That is why we average out A. Isthis a link invariant? It is because our action is topological : it does not depend on the met-ric! As a consequence SCS(φ∗A) = SCS(A). This is for example not true for the Yang-Millsaction SYM =

∫F ∧ ?F since φ∗ doesn’t commute with the metric-dependent ? (this is a

case where the abstract differential geometry notation is very elegant!). If we assume themeasure DA is diffeomorphism invariant, then we have

⟨W (φ∗C,A)

⟩A

=1

Z

∫AW (φ∗C, φ∗A) ei

∫R3 SCS(φ∗A) D[φ∗A] (variable A 7→ φ∗A)

=1

Z

∫AW (C,A) ei

∫R3 SCS(A) D[A] (using the above properties)

=

⟨W (C,A)

⟩A

.

This shows how topological quantum field theories can give rise to link invariants. Surpris-ingly in the abelian case we get exactly the writhe! This is an indication that this methoddoesn’t produce some trivial invariants (like zero).

To prove (7), define AC such that dAC is a delta function with support on C, i.e. such that∫R3

A ∧ dAC =

∮CA ∀A. (8)

This gives ⟨exp i

∮CA

⟩=

1

Z

∫A

exp

(i

∫R3

A ∧ d(A+AC)

)DA

so if we do a change of variables A 7→ A− 12AC and use Stokes’24 we arrive at⟨

exp i

∮CA

⟩= exp

(− i

4

∫R3

AC ∧ dAC

)= exp

(− i

4

∮CAC

).

22I use latin indices since we are working on R3.23More precisely the Wilson loop is the trace of the so-called holonomy, but in the abelian case there is

no difference.24It’s to argue that A ∧ dAC −AC ∧ dA = −d(A ∧AC) doesn’t contribute.

10

So to prove (7) what remains is to understand why w(C) =

∮CAC (defined by (8)). Note

that ∮CAC =

∑i 6=j

∮Ci

ACj +∑i

∮Ci

ACi

where we define (8) component-wise. So in the light of (6) let’s be happy if we can ac-

complish the slightly less ambitious goal of understanding why L(C1, C2) =

∫C1

AC2 . By

definitiondAC2 = delta-function on C2

so if S1 is a surface with C1 as a boundary, by Stokes’ we get∫C1

AC2 =

∫S1

delta-function on C2 = # intersections of C2 and S1 .

This is called the intersection number, but conceptually it’s clear that this is just anotherway of expressing L(C1, C2) (just look at figure 2(c))!

Actually L(C1, C2) =∫

1A2 is nothing new! It is known to all students of undergraduateEM, and it was first observed by Gauss (1833) [15], just in a slightly different form, quiteliterally a 2-form. Indeed, using the duality between 1 and 2-forms in 3D, and usingsuggestive notation, what we have discovered is that∮

C1

B2 · dr with ∇×B2 = delta-function on C2

is a topological invariant: it only depends on whether C1 and C2 are linked or not. Inthe context of electromagnetism we recognise this as the physical situation where we run aunit current through C2 and then measure, if you will, the magnetic force on a magneticmonopole moved along C1. This is exactly how Gauss saw it, and he was very surprised todiscover such a topological invariant. (Note that to get it in the form (5), simply plug inB2 using Biot-Savart!) He saw it as the first big development in the geometria situs in ahundred years, lamenting its slow development [16]. It fills the author with awe to considerhow tight the bond between knots and gauge theory –first observed by Gauss in 1833– hasbecome.

Witten in the non-abelian case

The real fun begins as we generalise the above to the non-abelian case. The Wilson loop isagain defined using the notion of parallel transport along a loop. The non-abelian Chern-Simons action is

SCS ∝∫R3

tr

(A ∧ dA+

2

3A ∧A ∧A

). (9)

Let me just shortly note, in an attempt to make the form of SCS look less arbitrary, thatthe exterior derivative of the integrand gives us the second Chern class. Hence integratingthe Chern-Simons form on a 3D boundary corresponds to in integrating the second Chernclass on its 4D interior. Using the fact the Chern number is an integer allows us to provenice properties of how gauge transformations (don’t) affect SCS.

More importantly for us, note that SCS again defines a topological quantum field the-ory. Hence exactly the same argument goes through to prove that 〈W (C,A)〉A is a link

11

invariant! More concretely, for every gauge group and representation we get a (new) linkinvariant! One usually puts the prefactor k/4π in front of (9), and then our link invari-ant depends on k. It turns out that if you rewrite it in a certain variable, it becomes apolynomial. We say Chern-Simons theory generates knot polynomials. It was a greatbreakthrough that Witten in 1988 realised that for the fundamental representation of SU(2)we get the so-called Jones polynomial. This invariant had been known since the secondhalf of that century, but never really fit in with the other invariants. One peculiar propertywas that it was chiral: it could feel the difference between left and right. Witten showed abrand new way of understanding and generating link invariants, and Chern-Simons endowedthem with a chiral property. (In fact, understanding the Jones Polynomial was Witten’soriginal motivation, prodded by Atiyah [16].)

And a look ahead

Note that chirality is also a fundamental property of the twistor formalism, treating h = ±1helicities differently. Is this perhaps a first hint that Chern-Simons is more natural in thetwistor context? In fact right after Witten’s publication, Penrose [17] observed that “thesalient feature of TQFT is that there are no field equations” (note that SCS gives e.o.m.F = 0) and that much the same is true for the field equations in twistor space, whereinformation is stored globally. He mused:

“Thus the physically appropriate home for TQFT might well be twistor spacerather than space-time.”

This led him to conjecture that perhaps the holomorphic version of the story above mightgive something interesting in twistor space. The last decade has proven him right: insection 5 we will see that doing just that gives us the gluon scattering amplitudes forN = 4 supersymmetric Yang-Mills!

2.3 Gluon scattering simplified a 1000-fold: Parke and Taylor in 1986

Calculating scattering amplitudes in Yang-Mills theory is infamously difficult. Zee, writingabout these recent developments in his Nutshell [18], drives this point home by challengingthe reader to calculate the 5-gluon scattering amplitude at tree-level. “You really mustcalculate before reading on. I will wait for you. You think to yourself, this is easy, just abunch of tree diagrams.” He subsequently reveals there are some ten thousand terms.

In 1986 Parke and Taylor discovered a one-term expression for n-gluon scattering at tree-level given certain external spin [1]. (Moreover, as we will soon discuss, using them asbuilding blocks you can calculate it for other spin.) It turns out that the key is not toexpress our scattering amplitude in terms of pµ and εν , but rather in terms of our spinor λand helicity h! Before we can write down the amplitude, there are two notational issues.

Firstly, the external particles have certain colours a that enter via the structure constantsfabc in the Feynman rules. Using the definition of fabc in terms of the Lie algebra generatorsT a, one can prove [18]

M(p, ε, a) =∑σ

tr (T aσ(1) · · ·T aσ(n)) M(pσ(1), εσ(1), · · · , pσ(n), εσ(n)

).

12

This is true for tree diagrams; for loop diagrams there are similar constructions (at least inthe planar limit N →∞). The amplitude on the right-hand side is independent of coloursand is calculated by giving an arbitrary order to the external gluons and then using theusual Feynman rules where the structure constants are put equal to one. This (component)amplitude is usually denoted in short-hand as M(1, 2, · · · , n).

Secondly, since our amplitude will be expressed in terms of λ instead of p we need relativisticinvariants that are the analogue of p · q. For left-handed spinors this is given by 〈λ, κ〉 =λAκBε

AB and for right-handed spinors [λ, κ] = λA′κB′εA′B′ . In addition to the spinors, our

amplitudes are labelled by helicities. By convention we take all external particles as in-going, which can then describe an arbitrary initial and final state using crossing symmetryunder which h 7→ −h. It turns out that if we take all helicities to be the same, M = 0.Similarly for all but one of the helicities being the same.

We are now ready to express the amplitude Parke and Taylor discovered. They realisedthat if two gluons (say number r and s) have h = −1 and all the others h = +1, then

M(1+, · · · , r−, · · · , s−, · · · , n+) =〈λr, λs〉4∏ni=1〈λi, λi+1〉

with pAA′

i = λAi λA′i . (10)

This notation suppresses the trace factor, the delta function and the coupling constant.It’s is called the Maximally Helicity Violating (MHV) amplitude, since by crossingsymmetry it for example describes an initial state of only positive gluons and a final stateof all-but-two negative helicity gluons. A peculiar feature of (10) is its holomorphicity : itonly depends on λ, not on λ.

More recent developments

In 2003/4 Witten published his seminal paper [8] with the motivation of trying to under-stand the unexpectedly simple form of (10). It was the start of the great progress that hasbeen made in this last decade reshaping how we look at Yang-Mills theory. Witten’s per-spective was to reformulate perturbative gauge theory as a string theory in twistor space.Two naive motivations to link (10) with twistor space: firstly there’s the holomorphicity,and secondly note that M is homogeneous in λi of degree −2hi. The latter scaling is rem-iniscent of what we have seen before and it’s related to the the fact that after a type ofFourier transform one can interpret M as a scattering amplitude in twistor space. As forstring theory, its business is to map Riemann surfaces (the string worldsheets) into Calabi-Yau manifolds. We already know Riemann spheres are natural to twistor space, and in factwe will see that if we introduce supersymmetry our twistor space becomes a Calabi-Yaumanifold. Strings and twistors thus combined were shown to give new insights into gaugetheory. In this essay we will take a different approach unrelated to strings, but still withmuch in common with this fundamental paper. Later when looking at scattering in twistorspace we will see another original motivation for Witten.

A year later a more sober (yet highly useful!) breakthrough was made, independent ofstrings or supersymmetry. It involved extending (10) to other helicities. Amplitudes withk + 2 negative helicities are called NkMHV amplitudes, short for Next-to-Next-to-...-Next-to-Maximally Helicity Violating amplitudes. It was realised that to calculatethese it was best to abandon the usual Feynman approach and rather use the MHV am-plitudes as building blocks. These were called the CSW rules after Cachazo, Svrcek andWitten [19]. In this approach we have one vertex for every n-gluon MHV amplitude (10)

13

(see figure 3(a)) and to calculate a tree-level NkMHV we glue them together as shown in3(c). For each internal line there is a certain contribution, and the whole diagram is calledan MHV diagram. A connection with twistor theory was already realised in the originalpaper, but it took until 2012 before a self-contained treatment that naturally gave rise tothis formalism was found [2]. We will touch upon this later in section 4.

Figure 3: (a) MHV vertex, (b) N2MHV amplitude, (c) a contribution in CSW rules

In 2005 it was proven by Britto, Cachazo, Feng and Witten that this gives the right resultfor tree amplitudes using so-called BCFW recursion [20]. The beauty of this approach isthat it is at heart an analyticity argument: the external momenta are changed by a complex-valued amount in such a way that the desired amplitude corresponds to the residue of apole. In the words of Anthony Zee: “complexify and bow to Cauchy” [18]. This proceduregives a recursion relation for the diagrams. It was later generalised to (planar) loop-levelin the case of N = 4 supersymmetry [21]. These relations will get a new incarnation insection 5 as the skein-relations for our holomorphic link invariant!

It is clear that there is much unexpected beauty in the scattering amplitudes of Yang-Millstheory, relying strongly on “complex magic”, which on its turn can be called the twistorialway of life. After seeing each of these elements on stage in their little hometowns, the timehas come to carefully introduce them all to each other in a consistent setting and then sitback and watch them do their thing on the big stage.

3 Twistors in the 21st century

3.1 The stage: (super)twistor space

As mentioned before, twistor space is T = (λA, µA′) ∼= C4, although we usually consider

projective twistor space PT ∼= CP3. Objects in M and T are naturally related by

µA′

= ixAA′λA (11)

which we refer to as the incidence relation. The way it relates points in one space to linesin the other is made use of in various places, like the definition of the Penrose transform insection 3.2 or the twistor loop in section 5. To follow the current literature, we now allowx ∈ MC: we complexify Minkowski space. This means (11) always has a solution, so weno longer restrict to null-twistors as in section 2.1.

Now that we allow complex spacetime points, our figure 1 on page 5 becomes figure 4,drawn in such a way as to make the comparison clear. Now any twistor Zα = (λA, µ

A′)

14

Figure 4: Corresponding objects in MC and PT (compare with p5)

directly corresponds to a (null) “line” in MC under the incidence relation. But insteadof a light ray it is now a 4-dimensional null-plane (usually called an α-plane), and it canbe shown to be parallel to the complex momenta pAA′ = λAλA′ where λA′ is an arbitraryspinor. Indeed twistor space can be seen as the space of such planes, or more schematically:

points in MC ↔ linearly embedded Riemann spheres in Tnull-planes α in MC ↔ points in T

Hence in the complex setting it’s less natural to interpret our twistors as also encodinghelicity25. Still it’s there conceptually, since the quantization argument of section 2.1 stillgoes through to ensure that fields of helicity h corresponds to homogeneous functions ofdegree 2h− 2 (cf section 3.2).

One may well wonder what the point is of complexifying M. For the most part it doesn’tseem necessary26, but it will definitely pay off when we start looking at the Wilson loopapproach in section 5. There we will use the fact that the correspondences under (11)are nicer for MC than M: unlike two generic null-rays, two generic 4D null-planes in MCintersect in a point. This allows for the general statement

Two “lines” in one space intersect⇔

their corresponding points lie on a “line”

Figure 4 attempts to illustrate this point. Adamo, Bullimore, Mason and Skinner observethat “in this way, the complex structure of PT (i.e. knowledge of where the complex linesare) determines and is determined by the conformal structure (i.e. knowledge of the nullcones) in space-time.” [22]

From the incidence relation (11) it also follows how the symmetry of spacetime acts ontwistor space. The complexified Lorentz group is locally SO(1, 3)C ∼= SL(2,C)× SL(2,C).In fact if we write pAA′ = λAλA′ then the first spinor λ transforms under the first factor(“left-handed”) and similarly for λ (“right-handed”). This terminology is compatible withthe usual usage when we reduce to real momenta (for which λ = λ), in which case we seeSO+(1, 3) ∼= PSL(2,C) is embedded27 diagonally in the complexified group. Consistent

25Alternatively one can take the viewpoint that helicity can be interpreted as external angular momentumalong a complex coordinate, to connect it with ideas encountered in section 2.1.

26There is also a mathematical notion of a space and its corresponding twistor space, and in that contextif T is to be our twistor space, then MC has to be our original space [14].

27In fact the complexified Lorentz group for example also contains SO(2, 2) et cetera. Hence we chooseour signature of spacetime by choosing a “real” slice of MC. In this essay we don’t worry about signatures.

15

with the spinorial indices, it follows from (11) that the twistorial coordinate µA′

is right-handed.

We actually have more than Poincare symmetry. Remember the compactification of M insection 2.1. This allows for the description of conformal symmetry. And indeed, we willbe describing Yang-Mills, and in 4D its action is conformally invariant28 which survives attree-level in the quantum treatment. Hence our symmetry group extends, as it turns out[8], to PSL(4,C). This has the natural representation on twistor space generated by

Zα∂

∂Zβ(α 6= β) and Zα

∂

∂Zα− Zβ ∂

∂Zβ(no summing!) (12)

This can be interpreted as representing all 4 × 4 matrices of trace zero, the Lie algebraof PSL(2,C). One can choose specific combinations to match the Poincare or conformalsubgroup. E.g. translation in Minkowski space is generated by PAA′ = λA

∂∂µA

′ in twistor

space. As expected it leaves λA invariant, but changes µA′

via (11).

Supersymmetric extension

One way of naturally incorporating supersymmetry is not by manually defining extra fieldson your space, but instead adding extra coordinates to your space. In the next section itwill become clear how this works, but let’s for now simply define29 N = 4 supertwistorspace as T = (λA, µA

′, χa) with a = 1, · · · , 4 labelling our Grassmannian coordinates.

One now says T ∼= C4|4 and PT ∼= CP3|4.

3.2 The actors: the twistor (super)field

In section 2.1 we conceptually motivated the idea of spinorial, tensorial, ... fields on space-time corresponding to homogeneous polynomials on twistor space. Here we take a slightlydifferent point of view: instead of taking our twistor fields to be (C-valued) functions, wetake them to be (0, 1)-forms30. On-shell our fields will still be holomorphic: ∂f = 0. In factjust like how in section 2.1 our (on-shell) fields were cohomology elements (of the first sheafcohomology H1(X,O)), now they are elements of the so-called Dolbeault cohomologyH0,1

∂(X). This mathematical equivalence is known as Dolbeault’s theorem. Practically it

means our on-shell field31 f ∈ Ω0,1(T) satisfies ∂f = 0 and is only defined up to f + ∂gwith arbitrary g ∈ C∞(T,C), hence it’s a ∂-analogue of de Rham cohomology.

The exact correspondence between our spacetime and twistor fields is known as the Penrosetransform. It relates spacetime fields φA··· of helicity h to homogeneous (0, 1)-forms f of

28This is manifested by the vanishing of the trace of the stress-energy tensor, and remember Tµν = δSδgµν

,

hence it tells us the action is invariant under scaling of the metric!29We don’t introduce a separate symbol, context makes clear about which twistor space we are talking.30The easiest definition of a (p, q)-form on a complex manifold is a form that can be written in a basis of

dzi1 ∧ · · · ∧ dzip ∧ dzj1 ∧ · · · ∧ dzjq .31Note that like before f is only defined on a subspace of (P)T, since H0,1

∂(T) and H0,1

∂(PT) are zero.

16

degree 2h− 2 on twistor space as follows:

∇AA′φA··· = 0 ↔ ∂f = 0 (13)

φA1···A|2h|(x) =1

2πi

∫XλA1 · · ·λA|2h| f(λA, ix

AA′λA) ∧Dλ for h ≤ 0

φA′1···A′2h(x) =1

2πi

∫X

∂

∂µA1· · · ∂

∂µA′2h

f(λA, ixAA′λA) ∧Dλ for h ≥ 0

with Dλ = λAdλA. Note the integrand is a homogeneous (1, 1)-form of degree zero, soit makes sense to integrate out over the 2-dimensional Riemann sphere X correspondingto our spacetime point x under the incidence relation (11). This generalises the case ofthe scalar field that was explicitly treated in section 2.1. The Penrose transform worksfor scalars, Weyl fermions, Maxwell fields, et cetera: the massless linear field equations onspacetime.

The Ward transform generalises this to non-abelian gauge fields. On spacetime we havea self-dual gauge field A (meaning ?F = iF , in particular dF = 0 implies d ? F = 0), orin mathematical lingo: a self-dual endomorphism-valued 1-form. This relates on twistorspace to a gauge field of the same gauge group (and homogeneous of degree zero32), moreprecisely it’s not just any 1-form but again a (0, 1)-form (with values in a correspondingendomorphism bundle). Moreover this has to satisfy ∂a+ a ∧ a = 0. Schematically:

A such that F = d2A = dA+A ∧A is self-dual ↔ a such that ∂2

a = 0 (14)

For the precise mapping see [14]. As an aside we point out the Ward transform can beunderstood as relating a vector bundle on spacetime to a holomorphic vector bundle ontwistor space. Indeed the ∂a = ∂ + a is defined on a smooth (complex) vector bundle, butone can prove33 [23] that for every a such that ∂2

a = 0 there is a unique choice of holomorphictrivializations such that ∂a reduces to ∂. This trivialization defines our holomorphic bundle.

One can do something analogous for anti-self-dual gauge fields, but more importantly: howcan we transform an arbitrary34 non-abelian gauge field to twistor space? In fact it turnsout we first have to rewrite Yang-Mills, as we will do in section 3.3. The up-shot will be thata general gauge field translates on twistor space into our connection a above, and a plain(0, 1)-form g, homogeneous of degree −4, which under the Penrose transform correspondsto an anti-self-dual Gµν on spacetime. Note the chiral nature: a different type of object isused for our h = ±1 helicities.

Supersymmetric extension

In the supersymmetric case we do exactly the same but now starting with a space C4|4 thatalso has four Grassmannian coordinates. Note that if we have a single field A (from nowon we let it be implicit that it’s a (0, 1)-form) on supersymmetric twistor space, then dueto the nature of Grassmann variables, we can decompose it as

A(Z, χ) = f(Z) +χa ψa(Z) +χaχb

2φab(Z) +

εabcd3!

χaχbχc ψd(Z) +εabcd

4!χaχbχcχd g(Z) .

(15)

32This is still compatible with the “2h− 2” scaling: self-dual fields have helicity +1.33This is known in the mathematical literature as the Koszul-Malgrange integrability theorem.34Naively one might transform F+ and F− separately, but that’s no good: the Ward transform maps

connections, but for a general F the (anti-)self-dual part F+,− won’t be the curvature of a gauge field!

17

Hence we see that this “superfield”35 of degree zero defined on C4|4 is simply a handyway of representing the five fields f, ψa, φab, ψ

d and g on36 C4, respectively of degree0,−1,−2,−3 and −4. Thus under the Penrose transform the superfield corresponds to thespacetime fields (FA′B′ , ΨA′a,Φab,Ψ

aA, GAB) of respective helicties h = +1, +1

2 , 0, −12 , −1.

This is called the supermultiplet and it’s the field content of N = 4 SYM. Technicallyit’s for the abelian/linear case, which is also relevant for the external free states in the non-abelian case. The general non-abelian case uses the super-analogue of the Ward transform.

3.3 The script: holomorphic Chern-Simons action

Now that we have our twistor fields we turn our attention to their appropriate actions. Iff and g are the two twistor fields describing a massless linear field with spin s (i.e. f ofdegree 2s− 2 and g of degree −2s− 2), then we define the action

S =

∫PTg ∧ ∂f ∧D3Z with D3Z = εαβγδZ

α ∧ dZβ ∧ dZγ ∧ dZδ . (16)

Note that the integrand is well-defined on PT since it has degree (−2s−2)+(2s−2)+4 = 0.It’s also a (3, 3)-form so it makes sense to integrate it out over PT. It is clear that theequations of motions are37 ∂f = 0 = ∂g. Hence under the Penrose transform their spacetimepartners satisfy the appropriate field equations (see section 2.1 for how to relate this toDirac, Maxwell, etc fields). This doesn’t prove that S is also the correct action off-shell,and we’ll worry about off-shell statements later.

In the supersymmetric case, things are nicer still. CP3|4 is a Calabi-Yau manifold, which bydefinition means it has a holomorphic top-form, i.e. a well-defined holomorphic (3, 0)-formon projective space. Indeed: D3|4Z = D3Z ∧ d4χ has degree zero

(since d(aχ) = 1

adχ). In

practice, this means that if A is our superfield of degree zero, we can define

S =

∫PTA ∧ ∂A ∧D3|4Z . (17)

It is not hard to see that expanding this in terms of the supermultiplet (15) gives us theaction (16) for each of the components. Note that the functional form of (17) resemblesabelian Chern-Simons:

∫R3 A ∧ dA. Indeed it’s called abelian holomorphic Chern-

Simons. It will play a similar role in the context of complex knot theory in section 5.

What about the case of non-abelian gauge fields? We saw the Ward transform (14) relatesa positive helicity A to the connection a of degree zero such that ∂2

a = 0. An action withthat e.o.m. is

S =

∫CP

tr[g ∧ (∂a+ a ∧ a)

]∧D3Z (18)

where g has degree −4 to make it work. Note that if we vary δa we get the e.o.m. ∂ag =∂g + [a, g] = 0. If we let g correspond to an anti-self-dual field Gµν via the Penrosetransform38 (13), then one can see39 that gives the equation dAG = dG + [A,G] = 0. In

35In fact we assume it’s a chiral superfield: this means that A does not involve χ.36We assume A doesn’t have terms containing dχ.37We can use partial integration for ∂ since D3Z is holomorphic.38Taking the integral expression for the Penrose transform (13) still gives a spacetime field with helicity−1, however now since ∂g 6= 0 it won’t satisfy dG = 0. But it’s not supposed to, so that’s fine.

39It’s not obvious since we’d need the explicit map a 7→ A, but it should seem plausible.

18

fact, the action (18) turns out to be equivalent to the spacetime action

S =

∫M

tr(G ∧ F ) with F = dA+A ∧A . (19)

To verify its equations of motion, realise40 that∫

tr(G ∧ F ) =∫

tr(G ∧ F−). Then thevariations δG and δA indeed give (using δAF = dAδA):

dAG = 0 and F− = 0 (i.e. A is self-dual).

The reason for spending time on that seemingly silly action (“Why should we be interestedin that g or G?”) is that (19) is much closer related to the Yang-Mills action than it mightseem. Chalmers and Siegel [24] realised that41∫

Mtr(F ∧ ?F ) “ = ”

∫M

tr(G ∧ F − ε

2G ∧G

). (20)

This is called the Chalmers-Siegel action. As a plausibility check, note that its equationsof motion are now:

dAG = 0 and F− = εG .

Hence dAF− = 0 so together with the Bianchi identity dAF = 0 we indeed get d ? F = 0.

For obvious reasons this is called the expansion of Yang-Mills around the self-dual sector.

We can now answer the question of what twistor field(s) correspond to a general non-abeliangauge field on spacetime. Following the Chalmers-Siegel action, on spacetime we have thefield strength F = d2

A and an anti-self-dual 2-form G. We associate A to the the connectiona on twistor space via the Ward transform (14), and the field G to g via the Penrosetransform (13). In the case of F = F+, the twistor action is given by (18). In the moregeneral case we have to add an extra term, corresponding to the spacetime part ε

2

∫G∧G.

We will write this term in the supersymmetric case, since then there’s a nice expression.

Supersymmetric extension

We now look at non-abelian N = 4 supersymmetric Yang-Mills. In the self-dual case, theanalogue of (18) is

S =

∫PT

tr

[A ∧ ∂A+

3

2A ∧A ∧A

]∧D3|4Z . (21)

Indeed its e.o.m. is ∂2A = 0. This resembles non-abelian Chern-Simons

∫R3 tr

[AdA+ 3

2A3]

and is hence called non-abelian holomorphic Chern-Simons.

There is a supersymmetric version of the Chalmers-Siegel action for which (21) is againrepresenting the self-dual part. The missing term was nicely expressed as a function of thetwistor field for the first time by Boels, Mason and Skinner [25]. This gives us the completetwistor action representing N = 4 SYM:

S =

∫PT

tr

[A ∧ ∂A+

3

2A ∧A ∧A

]∧D3|4Z − ε

2

∫M

log det(∂A|X

)d4|8x . (22)

40This follows from the property∫

tr(?G ∧ F ) =∫

tr(G ∧ ?F ) and the fact G is anti-self-dual.41Strictly speaking it’s only true perturbatively, but that’s good enough for us.

19

Here42 d4|8x = d4xd8θ. We will encounter both the operational meaning and the physicalrelevance of this extra term when we look at scattering in section 4.

3.4 The props: delta function gymnastics

Like the gamma matrices for spinors, or tensor calculus in GR, there are the delta functionsfor twistors. In fact, as has been the case in our approach so far, we’ll be using forms ratherthan functions. These are called distributional forms and were first consistently writtendown in [25] (the same work on the action (22)). These give us many things for little effort!

The simplest case is for the complex plane. We define the distributional (0, 1)-form δ1(z)via

∫C f(z) δ1(z) ∧ dz = f(0) for any smooth function f : C → C. In fact the generalised

Cauchy theorem tells us

δ1(z) =∂

∂z

(1

z

)dz . (23)

We don’t use this explicit formula in any calculations, but it tells us that δ1(z) is ho-mogeneous of degree −1. This is then generalised to Cn by defining the (0, n)-formδn(z) = δ1(z1) ∧ · · · ∧ δ1(zn), which is now of degree −n.

We can also define a delta function on CP1:

δ1m(z, a) :=

∫Cδ2[s(z1, z2)− (a1, a2)] ∧ ds

sm+1.

This integration is possible since δ2(sz − a) indeed has terms that are a (0, 1)-form ins: from (23) we know it equals · · · d(sz1 − a1)d(sz2 − a2) = · · · d · · · ds. Integrating outds ∧ ds leaves us with a 1-form that only depends on the projective coordinates. It’s clearthat δ1

m(z, a) is homogeneous of degree m and it’s not hard to check43 that if f : C2 → C2

is homogeneous of degree −m− 2, then luckily∫CP1

f(λ) δ1m(λ, a) ∧Dλ = f(a) with Dλ = λAdλA .

We can already do something important with this way of thinking: namely write downthe twistor fields corresponding to momentum eigenstates in Minkowski space. Indeed if(pµ, ηa) is our (super)momentum, then writing pAA′ = pApA′ we can define

Ap,η(Zα) =

∫C

es(µA′ pA′+χ

aηa) δ2(sλA + pA) ∧ ds

s. (24)

It is indeed of degree zero as our superfield should be, and it seems to match the philosophyof λ being the direction of our momentum. More rigorously, plugging it into the (super)Penrose transform (13) we integrate Ap,η(λA, ixAA

′λA) ∧Dλ and we see the delta function

sets λA → pA and the exponential becomes exp(µA′pA′)→ exp(ixAA

′pApA′) = eix·p.

To define a distributional form on supersymmetric T we use the fact that∫

(a+ bχ)dχ = b:

δ4|4(Z) = δ4(Zα) χ1χ2χ3χ4 . (25)

42We have θAa and hence 8 such indices. On spacetime for every fermionic coordinate we get two spinorialindices: this is due to the spin-statistics theorem, which is not true in twistor space!

43This can be checked by going into inhomogeneous coordinates λ = (1, α).

20

Note it’s of degree zero and that by the simple representation44 (12) of the superconformalgroup on twistor space , δ4|4(Z) is manifestly superconformally invariant. This makes thesedistributions very useful. The same holds for its the projective version on CP3|4:

δ3|4(Z1, Z2) =

∫Cδ4|4(sZ1 + Z2) ∧ ds

s. (26)

This can be used to define superconformally invariant external states [22]. Continuing thisprocess we also define

δ2|4(Z1, Z2, Z3) =

∫C2

δ4|4(sZ1 + tZ2 + Z3) ∧ ds ∧ dt

st. (27)

with analogous definitions for δ1|4(Z1, Z2, Z3, Z4) and δ0|4(Z1, Z2, Z3, Z4, Z5) .

The distributional form in (27) is supported on lines in PT: Z1, Z2 and Z3 have to be(projectively) collinear. If we define

∆(Z1, Z2) = δ2|4(Z1, Z∗, Z2) (28)

where Z∗ is some fixed (reference) twistor, one can prove [25] that

∂∆(Z1, Z2) = δ3|4(Z1, Z2) .

This might seem innocuous, but now recall that the kinetic term in our action (22) isA ∧ ∂A. Hence ∆(Z1, Z2) is our propagator! In fact, this is in a certain gauge45 calledthe axial gauge, which depends on our choice of reference twistor Z∗ [25].

Finally, note that δ0|4(Z1, Z2, Z3, Z4, Z5) is not really a form anymore but just a function ontwistor space. In fact it’s the simplest non-trivial superconformally invariant function onecan define on PT [25]. It’s usually denoted [Z1, Z2, Z3, Z4, Z5] or even [1, 2, 3, 4, 5] and tendsto pop up when calculating scattering amplitudes. Indeed computing scattering amplitudesoften comes down to using nice properties of these delta functions. For instance we will use

δ1|4(Z1, Z2, Z3, Z4) =

∫δ2|4(Z1, Z2, Z) ∧ δ2|4(Z,Z3, Z4) ∧D3|4Z

[1, 2, 3, 4, 5] =

∫δ2|4(Z1, Z2, Z) ∧ δ1|4(Z,Z3, Z4, Z5) ∧D3|4Z (29)

which we have shamelessly taken from [22].

4 Curtain-up: scattering amplitudes!

4.1 Feynman rules

Since PT has three (bosonic) coordinates, our superfieldA decomposes in a three-dimensionalbasis of (0, 1)-forms. For every reference twistor Z∗ we can define the axial gauge wherewe don’t allow A to contain a form in that direction. This restricts A to choose from

44The formula is extended in an obvious way to include the fermionic supersymmetric coordinates.45We will soon see that this is the gauge where the A ∧A ∧A term in our action (22) disappears.

21

a two-dimensional basis which means A ∧ A ∧ A = 0. Hence in this gauge our action46

becomes

S =

∫PT

tr(A ∧ ∂A) ∧D3|4Z − ε

2

∫M

log det(∂A|X

)d4|8x . (30)

The first term is our kinetic term whose propagator is given by (28), which depends on ourgauge choice Z∗. The second term is our interaction in twistor space. More explicitly, usinglog det = tr log and writing ∂A|X = ∂|X

(I + ∂|−1

X A), we expect upon expanding the log to

get contributions of the form

tr(∂|−1X A ∂|−1

X A · · · ∂|−1X A

).

Indeed if one works out the details [25] we get an n-vertex for every n = 2, 3, · · · ,∞. Infact if we use as our states the superconformally invariant states (as was briefly mentionedin section 3.4), these vertices are recursively determined47 by

V (Z1, · · · , Zn+1) = V (Z1, · · · , Zi, · · · , Zn+1) δ2|4(Zi−1, Zi, Zi+1) . (31)

Note that this formula makes clear that V is only non-zero if all Zi are collinear! We sayour vertices are supported on a line.

4.2 MHV diagrams

So if we scatter in twistor space, our Feynman diagrams are n-vertices connected by propa-gators. The astute reader will notice the similarity with the MHV diagrams we encounteredin section 2.3 to calculate gluon scattering in spacetime. In fact it turns out that writingdown the MHV diagrams is the same as calculating the scattering in twistor space: theCSW rules in spacetime are simply the Feynman rules in twistor space!

Part of the above claim is that our vertex (31) is our MHV amplitude. Indeed, if weintegrate out the twistor variables using the external twistor fields (24) corresponding tomomentum eigenstates, it turns out (31) becomes

δ4|8(∑n

i=1 Pi)∏ni=1〈pipi+1〉

(32)

where P = (p, η) is our supermomentum, and pAA′ = pApA′ . This is in fact the supersym-metric version of the Parke-Taylor formula (10). So far we haven’t fixed external helicities,so how can this be the MHV amplitude? Our external superfield can be seen as a super-position of our supermultiplet, in particular it contains superpositions of h = ±1 helicities.The appropriate term of (32) corresponding to (10) can be found by looking at how it scalesin the different particles48.

Witten realised in 2003 that if he wrote the amplitude (32) in twistor space, he got some-thing that was supported on a complex curve. This was part of his original motivation tore-interpret Parke and Taylor’s result in terms of a string theory in twistor space [8]. Wehave arrived at (32) in quite a different approach49.

46One might wonder why we don’t have to go throuh the Faddeev-Popov procedure. In fact we do, butone can check the ghosts decouple in this axial gauge! [25]

47For this we of course needs the result for the 2-vertex. We don’t use the result here, but see [22].48Remember from (10) that our amplitude is homogeneous in (pA)i of degree −2hi.49Even the holomorphic Chern-Simons action we used was first introduced by Witten, again in a different

context, relating to some topological string model [26].

22

Figure 5: (a) twistor diagram for NMHV (b) corresponding MHV diagram

The simplest Feynman diagram we can have in twistor space is one with two vertices. Itcomputes∫

PT×PTV (1, 2, · · · ,m,Z) δ2|4(Z,Z∗, Z

′) V (m+ 1, · · · , n, Z ′) D3|4Z ∧D3|4Z ′ (33)

and is drawn in figure 5(a). The Zi are drawn in a line since that’s the support of ourvertices, and the squigly line denotes our propagator ∆ = δ2|4(Z,Z∗, Z

′). It turns out thisdiagram calculates our NMHV amplitudes: indeed the CSW rules tell us that for NkMHVamplitudes, we need k + 1 MHV vertices (note that on page 14 we drew three vertices forour N2MHV amplitude). The MHV diagram we would draw in spacetime (figure 5(b)) isactually mimicking the twistor diagram. To explicitly calculate the NMHV amplitude (33):if we plug in (31) to take out our Z and Z ′ variables and then use the delta trick (29), weget

MNMHV = V (1, · · · ,m) V (m+ 1, · · · , n) [1,m, ∗, n,m+ 1] . (34)

In fact we have to sum over m and cyclic permutations to get the whole NMHV diagram.

We see that reformulating our gauge theory in twistor space gave us keen insights into thesimplicity of the Parke and Taylor formula and its generalizations to NMHV amplitudes.It’s not that the Feynman diagram approach is dead, it’s simply that we were applying itin the wrong space. Or should we not stop there?

5 It ain’t over till the fat lady sings: holomorphic knots

At this point it seems the play could be over: understanding the simplicity of the (N)MHVdiagrams offers a satisfactory endpoint. However the insights go deeper. For example arecent development that has made a big impact is known as the “amplituhedron” approach,which indicates our scattering amplitudes are in fact calculating the volume of a certaingeometric object [3]. Here we discuss another deep breakthrough. The soprano’s aria willsing about the holomorphic analogue of Witten’s 1988 work on knots and QFT, as discussedin section 2.2.

Our story begins by seeing how twistor space allows us to encode our external momentawithout any constraints. In spacetime our momenta pi have to satisfy p2

i = 0 and∑

i pi = 0.A generic case is drawn in figure 6(a). Due to conservation of momentum, we can make aloop by placing them as shown in figure 6(b). We call the vertices xi such that pi = xi+1−xi.These xi are called region momenta and only have the condition “xi+1−xi is null”. Now

23

Figure 6: (a) external momenta, (b) (region) momentum loop, (c) momentum twistors

we use the correspondence described in section 3.1 to define a loop in twistor space: eachxi corresponds to a complex line Xi, and since xi+1−xi is null we have that Xi+1 intersectsXi. These intersections are sometimes called the momentum twistors and are denotedby Zi in figure 6(c). Realise that there are no restrictions on our momentum twistors: anynodal50 loop of n nodes Zi defines n null-momenta pi that sum up to zero! (We owethis to using MC instead of M.)51

So our external data naturally defines a (holomorphic) loop in twistor space. Our actionis a holomorphic version of Chern-Simons. It seems natural to perform the holomorphicanalogue of section 2.2. What do we get? The claim is now that if we calculate theexpectation value of the holomorphic Wilson loop corresponding to this twistor loop, then

〈W 〉 =

∑∞k=0 NkMHVplanar

MHV. (35)

Hence in the planar limit this holomorphic link invariant calculates the whole theory! (Notethat we can pick out separate pieces by looking at scaling properties.) Three naturalquestions arise:

1. How do we make these notions, like a holomorphic Wilson loop, exact? And are wenow indeed calculating holomorphic link invariants?

2. How can we check (35)?

3. Why is (35) true? And/or what is it telling us?

We’ll address these issues in due time. We note this whole development is quite recent.A link between Wilson loops and scattering amplitudes was first suggested in a spacetimesetting, initially for strong coupling (based on AdS/CFT duality) in 2007 [27], later also forweak coupling [28] [29]. It was then understood that the natural setting for this was twistorspace, eventually arriving at the insight of (35) through a series of papers by Skinner, Masonand Bullimore [30] [4]. A nice review article is [22]. To start gaining an understanding of(35), we start with the abelian case.

50“Nodal” because we’re patching together (holomorphic) lines at the nodes Zi.51Indeed: note that two general (null-)twistors have corresponding light rays in M that don’t intersect (as

commented in section 2.1) and hence don’t define the xi.

24

5.1 Abelian

As in the real case, the Wilson loop is easy to define in the abelian case: it just correspondsto the naive integrating of A along our (complex) curve C. There is a subtlety now: A is a(0, 1)-form and dimR(C) = 2. Hence we need a natural (1, 0)-form ωC associated to C suchthat we can define52

W (C,A) = exp

(∫CA ∧ ωC

). (36)

To find this ωC , let’s remember what we did in section 2.2: we defined a 2-form δC suchthat ∮

CA =

∫R3

A ∧ δC ∀A .

Hence now similarly define a (0, 2)-form δC which is intuitively a delta function supportedon C, such that

W (A,C) =

∫CP 3|4

A ∧ δC ∧D3|4Z .

A delta function supported on Ci should have the property that δCi(Z) 6= 0 if and only ifZ is collinear with the nodal points of our Wilson loop Zi and Zi+1. From section 3.4 weknow this is given by

δC(Z) :=n−1∑i=0

δ2|4(Zi, Z, Zi+1) . (37)

As such, the (1, 0)-form that makes definition (36) work is53

ωC :=

∫δC ∧D3|4Z .

Let’s work with the abelian action∫A ∧ ∂A ∧ D3|4Z, leaving out the “ln det(∂A)” term

for now. If we define exactly analogous to the real case AC such that ∂AC = δC , then thecalculation in section 2.2 goes through to give

〈W (C,A)〉A =

⟨exp

∫CA ∧ ωC

⟩= exp

(−1

4

∫PTAC ∧ ∂AC ∧D3|4Z

). (38)

Before rewriting this as to make its relation to scattering amplitudes clear, let’s first discusswhat this is supposed to mean from a mathematical point of view.

Holomorphic linking

The formula AC ∧ ∂AC definitely resembles the formula AC ∧dAC we saw for the writhe insection 2.2. Is this some sort of holomorphic (self-)linking? Indeed it is. The basics ofknot theory in a complex setting54 can for example be found in [31], which was incidentallymotivated by Witten’s 1992 paper about holomorphic Chern-Simons theory [26].

52For convenience we are working in Euclidean convention.53Note we here interpret our integrand as a (2, 2)-form wedged with a (1, 0)-form, such that integrating

over something 2|4-(supercomplex)-dimensional leaves us with a (1, 0)-form.54Note that this is quite different from usual knot invariants. Indeed, by the General Position Theorem

every knotted loop can be undone in ≥ 4 dimensions, and every knotted surface (like a complex curve) in≥ 6 dimensions (like twistor space) if we allow general diffeomorphisms (there are stricter results as well).

25

In section 2.2 we gave a general argument for seeing 〈W 〉 was a diffeomorphism invariantand hence could indeed represent a link invariant. This extends to the complex case if werequire φ : PT→ PT to be holomorphic, since then ∂ and φ∗ commute. However, unlike inthe real case, this observation is not as relevant here. Note that in the real case, changingour link locally a bit could be represented as a diffeomorphism of our space. In the complexcase the transition from local to global is not as evident. More practically: if we changeour holomorphic twistor loop locally by changing the nodal points (for some i)

Zi 7→ Zi + zZi+1, Zi+1 7→ Zi+1 − zZi for z ∈ C (39)

while holding the other nodes fixed, we can’t really represent this as a (global) holomorphictransformation of our space55. Indeed our 〈W 〉 in (38) won’t be invariant under suchtransformations. The good news is: as long as we don’t let our loop self-intersect/cross, 〈W 〉only changes holomorphically. As soon as we cross (and hence change the knotted strucutreof our holomorphic loop), we get a non-holomorphic contribution. Mathematically, we say

δ〈W 〉 = 0 for the transformation in (39) if we don’t self-intersect.

So we see that 〈W 〉 really measures the holomorphic linking! In fact calculating the non-holomorphic change of δ〈W 〉 when we cross using (39), we can determine an analogue ofthe skein relations (the defining relations of link invariants). Moreover, this transformationalso has physical relevance: on spacetime it corresponds to changing the external momentausing the BCFW procedure mentioned in section 2.3! (Note that if we were to add the“ln det ∂A” term it might seem we would be ruining the topological nature of our action,but it turns out one can interpret this as modifying our Wilson loop [4].) Let’s now discoverhow this is related to scattering amplitudes.

Relation to amplitudes

By definition of AC we have ∂AC = δC , so using the propagator AC(Z) =∫

∆(Z,Z ′) ∧δC(Z ′) ∧ D3|4Z. Using expression (28) for ∆ transforms the Wilson loop expression (38)into

〈W (C)〉 = exp−1

4

∫δ2|4(Z,Z∗, Z

′) ∧ δC(Z ′) ∧ δC(Z) ∧D3|4Z ′ ∧D3|4Z .

Using the definition (37) of δC and the delta function relation (29), we arrive at

log〈W (C)〉 =1

2

∑i<j

[i− 1, i, ∗, j − 1, j] . (40)

If you would take the expression for the NMHV diagram we just calculated in the usualtwistor variables on page 23, express it in terms of momentum space variables using theinitial momentum state on page 20, and then rewrite momentum in terms of the momentumtwistor variables (those that make up the loop!), we get exactly the above expression (40)times the MHV diagram! Hence

log〈W (C)〉 =NMHVtree

MHVtree! (41)

55Global holomorphic maps can be infamously restrictive (note we don’t have partition of unity!), forexample the only holomorphic maps from a compact manifold to C are constant! That is why in complexgeometry we use “sheafs” which allow for locally defined objects.

26

Thankful that we have obtained a scattering amplitude, we can’t help but notice that it’sbut a small fraction of the promised (35). Adding the “ln det ∂A” term doesn’t quite makeup for it either: it adds a one-loop correction. The full relation is only true for the non-abelian case, which we will describe shortly. One might have naively expected that theabelian case would give the abelian analogue of (35), but that doesn’t hold up too muchscrutiny: of course there is no scattering for U(1) gauge theory56. It’s more interesting toobserve the role the pieces of our action play: the “pure” A ∧ ∂A part is in charge of theclassical tree diagrams, the “ln det ∂A” gives the quantum loop corrections. This is quitedifferent than their role when we are directly calculating scattering amplitudes: the formergives the self-dual part, the latter the anti-self-dual part. Hence, just taking the self-dualpart won’t give us the self-dual version of (35). It’s an all-or-nothing show: it only worksfor the full (N = 4 SYM) action!

Before moving on to the non-abelian case, let me point out a peculiarity: observe (40) isa superconformal invariant. But now the twistors in the expression are the momentumtwistors. This is a general feature: the amplitudes expressed in momentum twistors arealso superconformally invariant57. This surprising symmetry, hidden in the usual twistordescription, is referred to as dual superconformal invariance [32]. Like the Wilsonloop part of our story, it was first understood in the context of AdS/CFT [33] and thennaturalised in the twistor framework [34] as appearing in (35).

5.2 Non-abelian

By now we are dancing to a familiar tune. Our non-abelian holomorphic Chern-Simonsaction is known. The notion of the Wilson loop is now, however, more subtle still. Inparticular there is no general notion of parallel transport along a complex curve given aconnection. However there is a natural procedure for Riemann spheres, and our holomorphicloop is exactly a concatenation of that. For details see [4] or [22]. The claim is that if wethen look at 〈W (C)〉, depending on the specific action, we get

〈W (C)〉 =∞∑k=0

1

k!

(NMHVtree

MHVtree

)k= exp

(NMHVtree

MHVtree

)for pure, abelian C-S

〈W (C)〉 =∞∑k=0

(NkMHVtree

MHVtree

)planar

for pure, non-abelian Chern-Simons

〈W (C)〉 =∞∑k=0

(NkMHV

MHV

)planar

for non-abelian C-S including “ln det ∂A”

where the first line matches what we arrived at in (41).

It would be nice if we could prove these rather mysterious-looking statements bottom-up:i.e. somehow deriving the Wilson loop expression naturally from the scattering perspective(or the reverse). This has not yet been done. The two ways of verifying it are by essentiallychecking that both give the same result. The first method consists of calculating the skeinrelations for 〈W 〉 using (39) and then realizing these are the same as the BCFW recursion

56This is consistent with the MHV formalism: symmetrising over the diagrams gives zero, since in theabelian case there is no trace-pre-factor as described in section 2.3.

57This is not strictly true for loop amplitudes, but there are still strong restrictions on them.

27

relations mentioned in section 2.3. The second way is by perturbative inspection, i.e.comparing the Feynman diagrams, which we will consider now.

As in section 4, we work in the axial gauge58 in which A ∧ A ∧ A = 0, and let’s for themoment not include our “ln det ∂A” term. Hence our action is simply

∫A ∧ ∂A ∧ D3|4Z.

Of course it’s still non-abelian hence we can’t just commute the Wilson loop around andsolve it exactly as we did in the abelian case. 〈W 〉 is to be calculated perturbatively byexpanding the exponential in the Wilson loop. Symbolically we have

∮A =

∑i

∮CiAi,

summing over the Riemann spheres Ci making up our Wilson loop. Thus for the m-thterm in the exponential, we will be calculating expressions of the form⟨

m∏i=1

∮Ci

Ai

⟩.

Take this notation with a grain of salt, since we are merely trying to get the idea across.

Figure 7: (a) contribution to 〈W 〉 for m = 2, (b) contribution for m = 4

A contribution for m = 2 is drawn in figure 7(a). We claim that this calculates (part of)the NMHV diagram. Indeed, a calculation much the same as the one in the abelian casegives us the same result (40) (at least if we sum over all m = 2 contributions). Similarly them = 4 contributions calculate the N2MHV diagram! After checking the general structure,this gives us the above claim for 〈W 〉 in the case of pure Chern-Simons. (The fact weonly get the planar diagrams arises because if one treats the Wilson loop with care, thepropagator lines like those in figure 7(b) are not allowed to cross.) In the same vein it canbe checked that adding the “ln det ∂A”-term gives rise to the loop corrections.

5.3 Relation to MHV diagrams

The perturbative argument sketched above comes down to checking that the integrands onboth sides are the same: certain terms in the expansion of the Wilson loop correspond tocalculating scattering in twistor space with a certain numbers of vertices (or if one prefersthe spacetime CSW rules: with certain MHV diagrams connected together). What is thegeneral pattern that relates them? And let’s be honest, physicists like pictures: how arethese unfamiliar Feynman diagrams in figure 7 related to the MHV diagrams? Is there anice relation? After all, they are calculating the same thing...

58One might wonder whether we can fix such a gauge condition globally, but indeed we can, since Chern-Simons gauge theories are only defined for trivial vector bundles (we need A to be globally defined).

28

Figure 8: (a) planar dual of fig 7(a), (b) example of NMHV contribution

Indeed there is, and once again it’s unexpectedly simple: the Wilson loop diagrams shownin figure 7 on the one hand and the MHV diagrams as seen in figure 5 (page 23) or figure3 (page 14) on the other hand, are related by planar duality! This is illustrated in figure8. Such correspondence also extends to loop-level. This then tells us which terms in theWilson loop expansion are related to which of the scattering amplitudes.

6 Curtain call and conclusions

After some general review in section 2 on independent developments in twistor, knot andgauge theory, we carefully set up Yang-Mills in the twistor framework in section 3. Thefollowing section 4 looked at gluon scattering in twistor space, which gave an easy derivationof the MHV diagrams and essentially illuminated the unexpected simple structure of gluonscattering.

Finally, in section 5 an alternative viewpoint was offered by looking at the interplay of knotand gauge theory in twistor space, combining all three elements of the introductory section2. It turns out that the N = 4 SYM scattering amplitude is essentially a holomorphic linkinvariant of the twistor loop defined by our external momenta. An outline was given forhow this can be checked perturbatively, nicely summarised by associating their Feynmandiagrams using planar duality.

The true reason for why it works is perhaps not yet known. Indeed, the perturbative checkcan be done quite elegantly, but it would be conceptually nicer to have a non-perturbativelink between Wilson loops and scattering amplitudes. It could be argued that this hasbeen done by verifying that the skein relations coincide with BCFW recursion. On theother hand, it seems more convincing to argue the converse: this gives a conceptual reasonfor why BCFW works, being another way of calculating our holomorphic link invariant.For these reasons it seems there is still room for a more direct answer to why, especiallyconsidering the curious double life of our action. Indeed, for example at tree level: in thecase of scattering all the interactions come from the anti-self-dual part of our action59,whereas from the knot theory viewpoint all the interactions are contained in the Wilsonloop. This striking “coincidence” begs for a more general insight into how these quantitiesare related, perhaps giving an idea of how specific this is to N = 4 SYM.

59Remember this is the “ln det ∂A”-term in (22) we had to add to the action describing the self-dual case.

29

Acknowledgements

First and foremost I thank my supervisor David Skinner for introducing me to this excitingnew field, showing the way, and answering many questions. Credit is also due to NickJones, for stimulating conversations and helpful recommendations. I thank Eduardo Casalifor some interesting dinnertime discussions in Hall. Finally I also want to mention AndriesWaelkens, for some last minute advice.

References

[1] S. J. Parke and T. Taylor, “An amplitude for n gluon scattering,” Phys.Rev.Lett.,vol. 56, p. 2459, 1986.

[2] T. Adamo and L. Mason, “MHV diagrams in twistor space and the twistor action,”Phys.Rev., vol. D86, p. 065019, 2012.

[3] N. Arkani-Hamed and J. Trnka, “The amplituhedron.” arXiv:1312.2007, 2013.

[4] M. Bullimore and D. Skinner, “Holomorphic linking, loop equations and scatteringamplitudes in twistor space.” arXiv:1203.4565v4, 2011.

[5] J. P. M. J. C. Baez, Gauge fields, knots and gravity. World Scientific, 1994.

[6] R. Penrose, “Twistor algebra,” J.Math.Phys., vol. 8, p. 345, 1967.

[7] E. Witten, “Topological quantum field theory,” Communications in MathematicalPhysics, vol. 117, no. 3, pp. 353–386, 1988.

[8] E. Witten, “Perturbative gauge theory as a string theory in twistor space,” Commu-nications in Mathematical Physics, vol. 252, no. 1-3, pp. 189–258, 2004.

[9] R. Penrose, The road to reality. Vintage Books, 2004.

[10] P. A. M. Dirac, “The relation between mathematics and physics,” Proceedings of theRoyal Society (Edinburgh), vol. 59, pp. 122–129, 1939.

[11] R. Penrose, “The central programme of twistor theory,” Chaos Solitons Fractals,vol. 10, pp. 581–611, 1999.

[12] R. Penrose and W. Rindler, Spinors and space-time: Volume 1 & 2. Cambridge Uni-versity Press, 1984 & 1986 (respectively).

[13] G. Farmelo, The strangest man. The hidden life of Paul Dirac, quantum genius. Faber& Faber, 2009.

[14] R. S. Ward and R. O. Wells, Twistor geometry and field theory. Cambridge UniversityPress, 1990.

[15] C. F. Gauss, Werke (Vol. V). Koningliche Gesellschaf der Wissenschaften, Gotingen,1833.

[16] C. Nash, “Topology and physics – a historical essay,” in History of Topology (I. James,ed.), Elsevier, 1999.

30

[17] R. Penrose, “Topological QFT and twistors: Holomorphic linking,” Twistor Newsletter,no. 27, pp. 1–4, 1988.

[18] A. Zee, Quantum field theory in a nutshell. Princeton University Press, 2010.

[19] F. Cachazo, P. Svrcek, and E. Witten, “MHV vertices and tree amplitudes in gaugetheory,” Journal of High Energy Physics, vol. 2004, no. 09, p. 006, 2004.

[20] R. Britto, F. Cachazo, B. Feng, and E. Witten, “Direct proof of tree-level recursionrelation in Yang-Mills theory,” Phys.Rev.Lett., vol. 94, p. 181602, 2005.

[21] M. Bullimore, “MHV Diagrams from an all-line recursion relation,” JHEP, vol. 1108,p. 107, 2011.

[22] T. Adamo, M. Bullimore, L. Mason, and D. Skinner, “Scattering amplitudes andWilson loops in twistor space,” Journal of Physics A: Mathematical and Theoretical,vol. 44, no. 45, p. 454008, 2011.

[23] S. Kobayashi, Differential geometry of complex vector bundles. Princeton UniversityPress, 1987.

[24] G. Chalmers and W. Siegel, “The self-dual sector of QCD amplitudes,” Phys.Rev.,vol. D54, pp. 7628–7633, 1996.

[25] R. Boels, L. Mason, and D. Skinner, “Supersymmetric gauge theories in twistor space,”Journal of High Energy Physics, vol. 2007, no. 02, p. 014, 2007.

[26] E. Witten, “Chern-Simons gauge theory as a string theory,” Prog.Math., vol. 133,pp. 637–678, 1995.

[27] L. F. Alday and J. Maldacena, “Gluon scattering amplitudes at strong coupling,”Journal of High Energy Physics, vol. 2007, no. 06, p. 064, 2007.

[28] G. K. J. Drummond and E. Sokatchev, “Conformal properties of four-gluon planaramplitudes and Wilson loops,” Nuclear Physics B, vol. 795, no. 12, pp. 385 – 408,2008.

[29] A. Brandhuber, P. Heslop, and G. Travaglini, “MHV amplitudes in N=4 super Yang-Mills and Wilson loops,” Nucl.Phys., vol. B794, pp. 231–243, 2008.

[30] L. Mason and D. Skinner, “The complete planar S-matrix of N = 4 SYM as a Wilsonloop in twistor space,” JHEP, vol. 2010, no. 12, pp. 1–32, 2010.

[31] I. B. Frenkel and A. N. Todorov, “Complex counterpart of Chern-Simons-Witten the-ory and holomorphic linking,” Advances in Theoretical and Mathematical Physics,vol. 11, pp. 531–590, 08 2007. Originally from 1997.

[32] J. Drummond, J. Henn, G. Korchemsky, and E. Sokatchev, “Dual superconformalsymmetry of scattering amplitudes in super-Yang-Mills theory,” Nuclear Physics B,vol. 828, no. 12, pp. 317 – 374, 2010.

[33] J. M. Drummond, J. M. Henn, and J. Plefka, “Yangian symmetry of scattering ampli-tudes in N=4 super Yang-Mills theory,” JHEP, vol. 0905, p. 046, 2009.

[34] L. Mason and D. Skinner, “Dual superconformal invariance, momentum twistors andgrassmannians,” JHEP, vol. 0911, p. 045, 2009.

31