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It’s short for 𝒩=4 Supersymmetric Yang Mills theory.

Last time, I talked about plain vanilla Yang Mills theory and the geometric interpretation of the fields as sections of and connections on vector bundles. This time, I want to extend the discussion to the 𝒩=4 supersymmetric version of this theory. 𝒩=4 SYM has been the main subject of twistor string theory, and it has many special properties:

(1) It’s the CFT in the famous AdS/CFT correspondence, which is the original and most-studied concrete example of the holographic principle
(2) It’s the most supersymmetric theory you can get (in 4 dimensions) without having any gravity in it
(3) It’s a toy model for QCD applied in the regime you expect to see in hadron colliders (especially large ones!)
(4) There are some miraculously simple ways to compute some of the scattering amplitudes
(5) These amplitudes display some amazing symmetries that you would never have guessed from looking at Feynman diagrams, or Lagrangians or any of that old fashioned stuff !

In post #5 I mentioned that superymmetry is an extension of the algebra of the Poincare group. The Poincare group is the spacetime symmetry group that bog-standard quantum field theory respects, and consists of Lorentz transformations/Rotations Jμν and translations Pμ. I mentioned that supersymmetry has an extra symmetry generator QA which has a spinor index A, and the full super-Poincare algebra looks like:
[Pμ, QA] = 0

[Jμν, QA] = i(σμν)ABQB

{ QA, QbarA’} = 2σμAA’Pμ

Repeating the definitions for convenience: σμAA’ are the Pauli spin matrices.

(σμν)AB = (1/4)( σμAA’ σbarνA’B - σνAA’ σbarμA’B). (bar is complex conjugate as usual).

Actually I forgot to mention last time, that conventionally square brackets denote a commutator [X,Y] = XY – YX. Curly brackets denote an anticommutator {X,Y} = XY+YX.

A system based around this algebra is said to have 𝒩=1 supersymmetry, because there is a single supercharge QA. It is also possible to consider systems with several supercharges. We stick an extra label on the Q to indicate which one we’re looking at. So 𝒩=4 supersymmetry has 4 supercharges QaA a=1..4. Systems with 𝒩 > 1 are sometimes said to have extended supersymmetry.

The Poincare superalgebra is extended to include the anticommutators of the extra supercharges:

{ QaA, QbarA’b} = 2σμAA’Pμ𝛿ab

{ QaA, QbB} = εABZab

Here the Zab are skew in a and b (so there are 3 of them) and are called “central charges”. Given that I have 4 supersymmetry generators, I can transform between them and get equivalent physics. The transformation QaA -> Rab QbA where Rab is an element of SU(4) is such a transformation, which is referred to as an “R-symmetry”. These transformations preserve the super-Poincare algebra.

What can we say about particle states in our 𝒩=4 SYM model ? We will restrict attention here to particle states which are massless representations of the super-Poincare algebra.

Since we’re massless, we can choose a frame in which the four momentum is Pμ = (E, 0, 0, E). With this choice you can show that Qa1 vanishes and the central charges vanish. Were left with the 4 supercharges Qa0. Starting with a lowest helicity (-1) state Ω-1 we can act on it with the 4 supercharges to generate successively higher helicities:
|Ω-1/2>=(Q10)†|Ω-1>
|Ω0>= (Q20)†) (Q10)†|Ω-1>

|Ω1/2>= (Q30)†) (Q20)†) (Q10)†|Ω-1>

|Ω1>= (Q40)†) (Q30)†) (Q20)†) (Q10)†|Ω-1>

The fields present in the theory are as follows


φAB............................helicity -1
𝜒iA.............................helicity -1/2. 4 of them i=1..4
Xij.............................helicity 0. 6 of them i=1..4; Xij skew symmetric in i and j
𝜒iA’ ...........................helicity +1/2. 4 of them i=1..4
ΨA’B’..........................helicity +1
When reading the literature on 𝒩=4 SYM, you might see, in the list of fields, instead of φAB and ΨA’B’, a field Aμ. Recall in post #4 I showed how a Maxwell field (skew symmetric second rank tensor) is decomposed into the two helicity components φAB and φbarA’B’. This is a similar decomposition, and Aμ is just the vector potential from which the second rank tensor is obtained by exterior differentiation.



Now that we have a theory, with a bunch of fields and a Lagrangian (which I haven’t written down, but it’s easy to find), we want to compute something with it. The thing everyone wants to do with a quantum field theory is predict what happens when you slam a load of particles into each other, i.e. compute scattering amplitudes.

The particles we’ll be slamming into each other are the gluons of our 𝒩=4 SYM theory. The scattering amplitudes were (I assume) first computed using the traditional techniques – perturbation theory and Feynman diagrams. These computations are mind bogglingly complex, however, the astonishing result was that the answers which came out at the end (i.e. the actual formula for the amplitude in terms of the properties of the incoming and outgoing particles) were mind bogglingly simple. (Even at tree level – no loops – you need to sum an astronomical number of diagrams to compute the scattering of even a small number of gluons. That’s one problem with supersymmetric theories – you don’t just have 1 vertex like in QED, oh no, you have different vertices for all the superpartner combinations). It’s this simplicity that is most easily explained by using the description in terms of the mighty theory of twistors ! Anyway that’ll probably be the next post. Here I want to just describe what the scattering amplitudes are.

Scattering amplitudes will be functions of the momenta of the incoming and outgoing particles. We’d typically take a plane wave representation for the i’th gluon, and give it a four momentum pi. It would also have a polarization vector εi which is orthogonal to pi. There is a methodology called the spinor helicity formalism which makes the task of writing down amplitudes a bit easier.


I can write a null momentum vector pi in terms of spinors as pi = λA λbarA’. Why ? well it’s obvious that λA λbarA’ is null:
λA λbarA’ λA λbarA’ = εABεA’B’ λA λbarA’ λB λbarB’ = 0 because εAB is skew symmetric in A and B etc.


Conversely, given a null vector p, I can explicitly construct the spinors:
λA ............=.............. (1/sqrt(p0-p3))eiθ[p1 – ip2]
................................................... [p0 – p3]
λbarA’ ........= ..............(1/sqrt(p0-p3))eiθ[p1 + ip2]
................................................... [p0 – p3]

(eiθ is an arbitrary phase factor). We now define two products of spinors:

<λμ> = εABλA μB
[λ~μ ~] = εA’B’λ ~ A’ μ~ B’

The ~should be over the top of the symbol, which I can’t draw in Unicode . Using ~ rather than bar because it’s an independent primed spinor, not necessarily the complex conjugate of lambda. With this convention, a real null vector will be written as λA λ bar A’, and a not-necessarily-real vector as λA λ ~ A’.

If I dot product two null four momenta together I can now express the answer in terms of these fancy spinor products:

p.q = (1/2)<λμ>[λ~μ~]

When dealing with pairs of momenta pi, pj, QCD calculations traditionally use variables sij, which are defined as sij = (pi + pj)2. In our new notation, it just becomes sij = <ij>[ji].

We can also write the polarization vectors. For momentum pAA’, the two polarizations are:

ε-AA’ = -λaμ~A’/[λμ~]

ε+AA’ = -μaλ~A’/<μλ~>

where μ and μ~ are arbitrary spinors. (unfortunate that conventionally the polarization vector is denoted ε, since we also have an εAB, but it will always be clear which one is meant.

The polarization vectors satisfy
ε±AA’λA λ~A’ = 0
ε+AA’ ε+AA’ = 0

ε-AA’ ε-AA’ = 0
ε+AA’ ε-AA’ = -1



Now we denote amplitudes by expressions like
A(1+, 2-, 3+, 4+)
means 4 way scattering where gluon 1 has positive helicity, gluon 2 negative etc. Note that you’d also expect some factors in the amplitude to do with the colour gauge group. There are indeed such factors, but you can separate them out by the technique of colour ordering, so I won’t bother writing them here.

Now we can start making some statements about amplitudes at tree level (no loops). For example
A(1+, 2+, 3+, ...n+) = 0. i.e. if all the gluons are positive helicity, the amplitude vanishes. To see this, when you apply the Feynman rules, you inevitably end up having to contract at least one pair of polarization vectors together. Since they all have positive helicity, the amplitude must vanish because of the rules for the polarization vectors above.

Similarly, you can show that if there’s a single negative helicity gluon, the amplitude also vanishes; A(1+, 2-, 3+, ...n+) = 0 for example.

The first non vanishing amplitudes appear when you have two gluons with helicity opposite to the rest. These are called “Maximal Helicity Violating” (MHV) amplitudes. If the negative helicity gluon pair is j, k, there is the famous Parke Taylor formula for the MHV amplitude:

A(1+,.....,j-,.....k-,......n+) = i<jk>4/(<12><23>...<n1>)

Believe me, you don’t want to try and prove this by writing the diagrams down and adding them up ! (For more details on this stuff, look here).



The helicity of a particle is the projection of its spin vector onto the direction of travel. So, taking a spin ½ particle for example, we know that the spin components measured in any direction will turn out to be +ℏ/2 or -ℏ/2.