Posted: Jan 01, 2012 1:43 pm
I never did finish brain dumping my learning about twistor string theory - I'll try to finish it in 2012 before the apocalypse. Here's the next installment:

Post #11 Superconformal Invariance

Twistor string theory was developed through Witten's search for a space which had the same symmetries as AdS5xS5. Now, AdS5xS5 has a superconformal symmetry group, so I'll first describe what that means.

Conformal Group

In post #5 (appendix), I showed how the Poincare group (which consists of Lorentz transformations and translations) can be described by giving its infinitesimal generators and their relations (in other words by giving its Lie algebra). The conformal group in 4 dimensions is an even larger set of symmetries than the Poincare group. The Poincare group has Lorentz boosts xμ->Λμνxν (6 parameters) and translations xμ->xμ + aμ (4 parameters). Additionally, the conformal group has dilatations xμ->λxμ (1 parameter) and special conformal transformations xμ->(xμ-x2aμ)/(1-2aμxμ+a2x2) (4 parameters). So the conformal group of 4 dimensional spacetime is a 15 - parameter object.

Whereas the Poincare transformations preserve the Minkowski metric η'μν = ημν, the conformal transformations only preserve the metric up to multiplication by a scale factor, i.e. under a conformal transformation we have η'μν = Ω(x)ημν for some scalar function Ω(x).

The Poincare group actions are obtained by exponentiating the actions of the generators. As I mentioned in post #5, the generators of the Poincare group satisfy the Lie algebra relations

[Jμν, Jλσ] = i(Jλνημσ - Jσνημλ - Jμληνσ + Jμσηνλ )

[Pμ, Jλσ] = i(Pλημσ - Pσημλ)

[Pμ, Pν] = 0

If we look at a particular representation of this Lie algebra, namely where the generators are acting on (wave) functions on Minkowski space (typically these are complex square integrable functions Ψ∈L2(M) ) then we can represent a generator, Pμ, of infinitesimal translations by Pμ = i∂μ (here ∂μ is short for ∂/∂xμ). A finite translation by a constant vector aμ is representable by a unitary operator U(a). U(a) is given in terms of the infinitesimal translation operator by U(a) = exp(iaνPν). Applying it to the coordinates on Minkowski space, we can obtain

U-1(a)xμU(a) = xμ + aμ

i.e. we get a finite translation, as required.

In the same way, we can define unitary operators that implement finite conformal transformations, such as the dilatations for example:

D = xμPμ
U(c) = exp(icD)
U-1(c)xμU(c) = e-cxμ

Similarly, the special conformal transformations have generators Kμ = -2xμD + x2Pμ+2SμνPν

You got yourself some more generators, first thing you do - write down the commutation relations:
[Jμν, Kλ] = i(ηλμKν - ηλνKμ )

[Jμν, D] = 0

[D, Pμ] = -iPμ [D, Kμ] = iKμ

[Kμ,Kν] =0

[Kμ,Pν] = 2iημνD + 2iJμν

In fact that it can be shown that this, the Lie algebra of the conformal group, is isomorphic to the Lie algebra of SO(2,4), which is the "Lorentz" group of a 6 dimensional space with metric ++----, i.e. two time and four space dimensions.

Superconformal Group

What if our model is supersymmetric as well as being conformally invariant ? Recall that supersymmetric means that we have supercharges QA, QbarB' (this is for 𝒩=1. (For 𝒩 > 1, we'd have an extra index on the Q's labelling which supercharge we were talking about). These supercharges are infinitesimal operators acting on superspace (post #6). Also note that the spinors QA, QbarB' are spacetime spinors, not worldsheet spinors that I talked about in post #10, and I'm using Penrose-style spinor indices - A and A' rather than α and α. (the dot should be directly over the alpha, but I can't do this) that appears in much of the literature.

The actions of the supercharge QA and the special conformal symmetry generator Kμ don't commute, and we give the notation S to their commutator:

SA = -i/2[KAA', QbarA']; SbarA' = -i/2[KA'A, QA]

Quick note on notation: the special conformal generator has a vector index - it's Kμ, but recall in post #3 I described how you can replace a vector with a pair of spinor indices, one primed and one unprimed. This is what we've done here.

The SA are sometimes called "conformal supersymmetries", whereas QA are plain old "Poincare supersymmetries".

Collecting what we've got:

Commutators between the Jμνs, the Pμ's, D's, Kμ's give the Lie algebra of the conformal group. Now that we've added the QA's and SA's, we can define a whole bunch more commutators:

[D, QA] = -(i/2)QA etc. (there are lots of them and I'm not going to write them all out !)

We also have anticommutators

{QA, QbarB'} = PAB' etc

The full set of all these relations close up into a nice algebra called the superconformal algebra. The corresponding (super) group is the superconformal group.

Next post, I'll describe how the superconformal group is a symmetry group of both AdS5xS5 and supertwistor space.