Posted: Jun 02, 2012 3:46 pm
by twistor59
Post #12 Common Supergroups


It's been a little while since I posted on this, so I need to recap (mostly for my benefit) where we'd got to. There were three basic elements:

1 [math]


This is Supersymmetric Yang Mills gauge theory with 4 supercharges. It consisted the following set of fields:

One two-index symmetric spinor [math]
Four spinors [math] i=1..4
Six scalars [math] i,j=1..4, symmetric
Four primed spinors [math] i=1..4
One two-index primed symmetric spinor [math]

This theory has the maximal amount of supersymmetry possible in a theory with no gravity.

2 Supertwistor Space


Having complex coordinates [math] where [math] are the usual "bosonic" twistor coordinates and [math] (where [math] is the number of supercharges) are anticommuting Grassmannian variables.


3 The AdS/CFT Correspondence


[math] is a hyperboloid defined by the relation

[math]

in a space with metric

[math]

Taking the product with the five dimensional sphere [math] gives us the 10 dimensional bulk space [math].

If I have N D3-branes (each of which fills our large 3 spatial dimensions), they give rise to the spacetime metric


[math]

Here [math] are the "large dimension" spacetime coordinates. [math] are the "extra" coordinates converted to polar form with [math] being the radial distance and [math] being the usual 5-sphere metric (consisting of a bunch of [math] terms with some [math] and [math] thrown in.

The radius of curvature R is given by [math] where

[math] is the number of branes
[math] is the string coupling constant
[math] is the string scale.


Looking at the metric, for [math] (i.e. a long way from the branes) you just have 10 dimensional flat space.

For [math] (after making a coordinate change [math], the metric is

[math]

i.e. the metric of [math].

The standard example of the AdS/CFT correspondence is an equivalence between [math] SYM theory living on the conformal boundary of [math] and type IIB superstring theory in the bulk of [math].

The Superconformal Group


In post #5 I described the Poincare group. The structure of the group is defined by giving the relations (the "algebra") satisfied by its infinitesimal generators. The Poincare generators were

[math] for spacetime translations
[math] for Lorentz transformations

The Poincare algebra is

[math]

[math]

[math]


The Poincare group preserves the metric on Minkowski space. If I weaken this condition and demand only that the metric be preserved up to overall rescalings

[math]

then I get a larger group of symmetries called the conformal group. The conformal group (in 4 dimensions) has 15 parameters, compared to the Poincare group's 10. As usual, we get most of the important structure by looking at the infinitesimal generators - there are 15 of them.

In addition to the 4 translation generators [math] and 6 Lorentz generators [math] we now also have:

A dilatation generator [math]

This expands and contracts Minkowski space. So given a parameter [math], [math] generates a finite dilatation

[math]


Four "special conformal transformation" generators [math]

These first "invert" the coordinates i.e [math] then translate this result by a constant vector [math]. So, given the parameter [math], the special conformal transformation is

[math]


We can now add the commutation relations for these extra 5 generators to the commutation relations for the Poincare group, to obtain the algebra of the conformal group:

[math]

[math]

[math]

[math]

[math]

[math]

[math]

[math]

[math]


It is fairly easy to show that this conformal group algebra is the same as the algebra of the group [math], which is the group of transformations of six dimensional space which preserve the metric diag(- - + + + +).

Now suppose our theory is not only invariant under the conformal group, but also contains supersymmetry, i.e. it displays some invariance under the action of a supercharge [math]

The special conformal symmetry [math] and the supercharge don't commute. Traditionally we give the name "S" to their commutator:

[math]

Note I've replaced the vector index [math] on the special conformal generator [math] by the primed/unprimed spinor pair [math] - this uses the equivalence I explained back in post 15. There's also the primed version

[math]

I can now start lumping the S and the supercharge together into longer vectors:



[math]


[math]


The action of the infinitesimal conformal transformations on the F's is obtained by arranging them in a hermitian matrix:


[math]

This can be exponentiated to get the finite conformal transformations in the usual way. The matrix is traceless and acts as a generator for [math], the group of transformations which preserve the form with signature (++--). This ties in nicely with the fact that [math] is locally isomorphic to the conformal group (in fact it is a double cover of it - there are two elements of [math] for each conformal group element).

By defining the commutation relations of the conformal generators with the supercharges, we can build up a complicated algebra which generates a structure called the superconformal group.


Demanding that the [math] Lie algebra relations are satisfied allows us to read off the various commutation relations between the conformal generators and the supercharge. I won't write out all the commutation and anticommutation relations in the algebra explicitly in terms of the conformal generators and supercharge - if you want to see them they're in Section 3.1here.

Sticking with our [math] notation (1), (2), the commutators and anticommutators can be succinctly written as

[math]

[math]

[math]

[math]

[math]

[math] is a charge which commutes the bosonic conformal generators, but not with anything containing the supercharge{

[math]

[math]


I need to mention a bit about R-symmetry here. Included in the relations I get when I write out the above relations explicitly are the supercharge anticommutators

[math]

[math]

Clearly, making a transformation

[math] leaves things invariant. This is an "R-symmetry". It has an infinitesimal generator [math] which commutes with the conformal generators, but not with the supercharge:

[math]


The superconformal group whose algebra we've just defined is called [math]. The [math] is the group structure of the conventional "bosonic" conformal generators as we mentioned before, and the "|1" means that there is a single supercharge.

For our system, we need to extend this to the case where we have [math] supercharges. In this case, the [math] symmetry generated by the "R" charge now gets extended to [math]. So our supercharges get an extra label [math], and the [math] transformations act as

[math]

Now the [math] factor in the decomposition [math] commutes with ALL the group generators, and can be factored out of the entire algebra. If we do this, we end up with the projective supergroup [math]. This is the supergroup that encapsulates the symmetries in [math]. If you like, you can think of an element as being represented by a giant matrix which splits up as:

[math]

The [math] and [math] parts are often called "bosonic" because they're composed of with generators which take values in ordinary numbers, whereas the parts I've labelled "Super" are called "fermionic" because they, containing supercharges, take values in Grassmann numbers.

A person skilled in this art could take the Lagrangian for [math], look at the action of the superconformal generators on the fields in the Lagrangian, and demonstrate that the Lagrangian was invariant under the superconformal transformations.

So [math] is invariant under the action of [math].


Now, as I mentioned above, [math] has the same algebra as [math] (in fact it's a double cover of it - there are two [math] elements for every [math] element), and [math] is the symmetry group of [math]. Now it's also the case that [math] is the symmetry group of the five sphere [math]. But [math] is a double cover of [math] ! So:

The bosonic part of [math] is the symmetry group of [math]


The superconformal group acts somewhere else too! Remember twistor space was the space of complex 4-tuples, each of which could be considered as a pair of spinors:

[math]

We had the concept of conjugate twistors:

[math]

and this allowed us to define a Hermitian form (the twistor norm)

[math]

It's pretty easy to show that this has signature + + - -. To do this, just introduce some dummy variables:

[math]

That immediately tells us that the symmetry group of twistor space is [math]! In fact, moreover, supertwistor space is the fundamental representation space on which [math] acts.

So we have three entities, [math], and supertwistor space, all with the same symmetry group. The reason for the invention of twistor string theory was that Witten was investigating [math] strings, i.e. sigma models - maps from the string worldsheet [math] into [math]. He was interested in how this may behave when the radius of curvature R gets small. To this end, he chose a space with the same symmetries as [math], namely supertwistor space. This is how it all started.

So at last we're at the point where we can actually talk about twistor strings. I'll get to that next post when I've worked out how it actually works!