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In this post, I'll briefly describe superfields, which is a framework for representing supersymmetric field theories. This will lead on to supertwistor space, which is where twistor string theory is formulated.



For a lightning review of stuff on Grassmannians, see the excellent book by Zee http://www.amazon.com/Quantum-Field-Theory-Nutshell-Zee/dp/0691010196.

Recall how fields have zero point energies - the field cannot exist in a zero excitation, but has a minimum excitation. This excitation has an energy, and each zero point mode of the field will contribute to this zero point energy. For a bosonic field, each zero point mode contributes +1/2ℏ to the zero point energy. In contrast, each fermionic mode contributes -1/2ℏ.

The positive zero point energies of the bosonic modes can be obtained from the path integral, which, for a scalar field, is

Z = ∫𝒟φexp {i ∫d4x1/2[(∂φ)2-m2φ2]}

Recall that, in the path integral, the fields are just classical fields, not operators or anything fancy like that. Now if we want to do the fermion case, we must do something whacky because fermion fields anticommute. The tools that allows us to do this are Grassmann numbers. They are characterised by their anticommutation behaviour. - if η and ξ are Grassmann numbers, then

ηξ = -ξη, and consequently of course η2 = 0.

So, in a Taylor series expansion of a function of a Grassmann variable η, we see that the most general function we can possibly have is a+bη which simplifies things a bit to say the least ! You can deduce other stuff, like ∫dη = 0, and ∫ηdη = 1. These are sometimes called "Berezin" integration rules in the literature.

In our path integral, when we want to treat fermions, of course, we'd be integrating over spinor fields. To introduce the anticommuting behaviour, we see that we must treat them as Grassmann valued spinors.

The supercharge I defined in the last post, which converts bosons to fermions and satisfies the key relation

{ QA, QbarA’} = 2σμAA’Pμ

is a Grassmann variable. As I said last time, Pμ is the translation generator of the Poincare group. If we're looking at the action of the Poincare group on functions, this translation generator is just the operator Pμ = i(∂/∂xμ). What we want to do is define a "superspace" such that QA is also a sort of translation, but this time in a Grasmannian (i.e. "fermionic") coordinate.

Superspace and Superfields

We formally define superspace as the set of points labelled by coordinates {xμ, θA, θbarB'}, where xμ are the usual "bosonic" spacetime coordinates, and θA, θbarB' are Grassmannian spinor coordinates. Now we have our space, we can define functions on it. A general function would be written as Φ(xμ, θA, θbarB'). Acting on functions on superspace, our supercharges are represented by the operators


QA = ∂/∂θA-iσμAA’θbarA'∂/∂xμ


QbarB' = -∂/∂θbarB'+iθBσμBB’∂/∂xμ

Fields on superspace are known as "superfields".

Supertwistors

In post #3 I described the correspondence between twistor space and complexified compactified Minkowski space. In this section, we'll be interested in a superspace version of Minkowski space. For the ordinary Minkowski space coordinates {xμ} I'll switch to the spinor equivalent {xAA'} like we always to when discussing twistors !.

Super Minkowski space has coordinates {xAA', θaA, θbar aA'}, where a=1..N labels the supercharges.

The corresponding supertwistor space has coordinates ZI = (Zα, ψa) where Zα are the usual twistor components and ψa (a is the supercharge index) are anticommuting twistor components. The space of such entities is labelled ℂ4|N. Supertwistor space 𝕋[N] = ℂ4|N - ℂ0|N, i.e we remove the elements with Zα = 0.

As in the normal twistor space case, we're interested in the projective version of supertwistor space ℙ𝕋[N] = ℂℙ3|N where we identify supertwistors up to complex rescalings ZI ~ λZI (for lamba in ℂ).

The case N=4 has the special property that the volume form

Ωs = εαβγδZαdZβdZγdZδdψ1dψ2dψ3dψ4

is well defined on ℙ𝕋[4]. This is because it is invariant under rescalings ZI ~ λZI. To see this, it's clear that the part with the Z's has degree 4 under rescalings. ( By this I mean it picks up a factor of λ4 when the rescaling is performed). The clever bit is that the part with the ψ's has degree -4. This comes, in turn, from our Grassmannian property ∫ψ1dψ1 = 1 etc which forces each dψ1 factor to have degree -1.

In ordinary twistor space, given a spacetime point xAA’ (spinor indices only – using tensor indices is like wearing corduroy) we had the “incidence relation”

ωA = i xAA’ πA’

where we can think of this as defining a projective line (which is topologically a sphere!) in twistor space, parametrised by the homogeneous coordinates πA’ (see the appendix below).

What is the equivalent of this for supertwistor space ? Firstly some terminology, a Riemann sphere with its homogeneous coordinates πA’ is now called a ℂℙ1|0, the zero signifying that there are no Grassmann dimensions on it. If we “thicken” the sphere in the Grassmannian direction by allowing the Grassmann numbers to vary over their full range, we get the object parameterized by (πA’, ψa). Such an object is called a[ chr]8450[/chr]ℙ1|N where the Grassmann coordinates range from 1 to N.

Firstly, if we look at the mappings of ℂℙ1|0 into supertwistor space ℙ𝕋[N], they are parametrized by (x+AA’, θbar aA'). These coordinates define “antichiral superminkowski space”. The mapping itself would be
ℂℙ1|0 -> ℙ𝕋[N]
πA’ -> (x+AA’πA’, πA’, θbar aA'πA’)......(1)


Secondly, if we look at the mappings of ℂℙ1|N into supertwistor space space ℙ𝕋[N], they are parametrized by (x-AA’, θAA). These coordinates define “chiral superminkowski space”.
ℂℙ1|N -> ℙ𝕋[N]
(πA’, ψa) -> (x-AA’πA’+ψaθAa, πA’, ψa)......(2)

If, further, we have our ℂℙ1|0 mapped into a ℂℙ1|N which is in turn mapped into ℙ𝕋[N], then using (1) and (2):

ψa = θbar aA'πA’ and
x-AA’πA’+ψaθAa = x+AA’πA’

combining these, we see that x-AA’πA’ + θbar aA'θAaπA’ = x+AA’πA’, and so

x+AA’ = x-AA’ + θbar aA'θAa

Conventionally we take xAA’ = ½(x-AA’ + x+AA’) as our bosonic coordinates on the full Minkowski space.

If you should be enthusiastic enough to peruse the literature on this stuff, you’ll see mostly a slightly different convention – spinor indices will be lower case latin (sometimes greek), and the conjugate spinor indices will have a dot over the top of them instead of the prime I used. I just can’t bring myself to break the habits of a liftetime.

Hopefully I might just get around to some 𝒩=4 SYM next time.

Appendix Homogeneous Coordinates on the Riemann Sphere

A two sphere is two dimensional, and can be given complex coordinates as follows:
Stick a copy of the complex plane through the equator, like in the diagram. A point on the surface of the sphere is then mapped to a point in the complex plane by taking a straight line from the North pole, piercing the desired point on the sphere and carrying on till you hit the complex plane. The complex number you end up with is the complex coordinate of the pierced point. Clearly, as the pierced point gets nearer and nearer to the North pole, the complex coordinate goes miles and miles out in the complex plane. Clearly, you can assign a complex coordinate to any point except to the North pole itself, which fucks off to infinity.

You actually need two complex coordinate patches like this to cover the sphere. To get the other one, project instead from the south pole. This time the North pole has a perfectly well behaved complex coordinate – zero, but the South pole itself fucks off to infinity. So a point on the sphere has two complex coordinates in this scheme, z, got by projecting from North, and w, got by projecting from South. These are related by z = 1/w and vice versa.

Now if I have a spinor, it has two complex components πA A = 0, 1. I can think of the spinor as representing a point on the Riemann sphere by just taking the ratio of its components. So our z and w above would be given by
z = π0/ π1
w= π1/ π0

Thought of like this, the spinor components πA are called homogeneous coordinates on the Riemann sphere. Given πA we get a unique point on the sphere, but the coordinates πA and λπA give the same point, so the coordinate pair is only unique up to the multiplication by a (nonzero) complex constant. The Riemann sphere described this way is often referred as complex projective 1 space and denoted ℂℙ1. Pretty similar to the way we obtained projective twistor space ℂℙ3 by taking quadruples of complex numbers and identifying ones which just differed by multiplication by λ.