Twistor string theory

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Re: Twistor string theory

#21  Postby iamthereforeithink » Feb 19, 2011 6:52 pm

campermon wrote:
twistor59 wrote:
campermon wrote:
There are no numbers for a start, just funny greek symbols. What king of crazy shit is this?



Geek symbols ? how dare you. Fucking troll.... :lol:


:rofl:

I've never been called a troll!!

I likes it!

:lol: :lol: :lol:

:thumbup:


You are at least the second proud recipient of the Twistor Troll award, with me being a previous recipient. I've been trying to live up to the honor. I recently started a thread about homeopathy (something I couldn't care less about), which has gone on for around a thousand posts and has inspired a tangential split thread too. That's a lot better than any of the previous threads I've started (mostly in the science section), which never went beyond a page and a half. The "Mr. Photon" thread did somewhat better, probably because I used a lot of colorful and provocative language. :grin:
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Re: Twistor string theory

#22  Postby campermon » Feb 19, 2011 6:57 pm

iamthereforeithink wrote:
campermon wrote:
twistor59 wrote:


Geek symbols ? how dare you. Fucking troll.... :lol:


:rofl:

I've never been called a troll!!

I likes it!

:lol: :lol: :lol:

:thumbup:


You are at least the second proud recipient of the Twistor Troll award, with me being a previous recipient. I've been trying to live up to the honor. I recently started a thread about homeopathy (something I couldn't care less about), which has gone on for around a thousand posts and has inspired a tangential split thread too. That's a lot better than any of the previous threads I've started (mostly in the science section), which never went beyond a page and a half. The "Mr. Photon" thread did somewhat better, probably because I used a lot of colorful and provocative language. :grin:


:lol:

I think that Twistor should make a graphic that we can have in our sig strip to acknowledge that we hold the Twistor TrollTM award.

:mrgreen:
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Re: Twistor string theory

#23  Postby iamthereforeithink » Feb 19, 2011 7:10 pm

:lol: That would be a worthy complement to the prestigious Dunsapy award that we already have on the forum.



* I think Twistor needs to come up with the next episode of the Twistor String theory pretty soon, else we will fill this thread up with a lot of crap :lol:
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Re: Twistor string theory

#24  Postby campermon » Feb 19, 2011 7:18 pm

iamthereforeithink wrote:

* I think Twistor needs to come up with the next episode of the Twistor String theory pretty soon, else we will fill this thread up with a lot of crap :lol:


Hold on! I've just got to grips with his first couple of posts!!

:lol:
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Re: Twistor string theory

#25  Postby twistor59 » Feb 21, 2011 8:04 am

Is that a pipe in your gob ? You need to ruffle your hair up a bit if you want to look like Einstein

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"Einstein was not a handsome fellow...
blah blah blah hair coloured yellow...

Quark, stangeness and charm..." etc
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Re: Twistor string theory

#26  Postby campermon » Feb 21, 2011 8:35 am

twistor59 wrote:Is that a pipe in your gob ? You need to ruffle your hair up a bit if you want to look like Einstein

campermon.jpg


"Einstein was not a handsome fellow...
blah blah blah hair coloured yellow...

Quark, stangeness and charm..." etc


:lol:

Yes..ran out of rizla.

:thumbup:
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Re: Twistor string theory

#27  Postby twistor59 » Mar 21, 2011 5:21 pm

Post #5 Supersymmetry


Moving onto a bit of physics I'm not so familiar with now, so the potential for getting things wrong is increasing.... :naughty2:

To get to twistor strings, I need to pass through superstring theory and supersymmetric twistor space. To accomplish this, I'll need to say a few words about supersymmetry, which is the topic of this post. As usual, this isn’t complete or rigorous – just meant to give a flavour.

Quantum theory presents us with the problem of identical particles. Assume a pair of particles has identical quantum numbers (rest mass, charge etc..) so you can't use these to distinguish them. If you keep the particles isolated, then you can continue to distinguish them, however if their wavefunctions are allowed to overlap, then you can no longer keep track of which particle is which. This is a manifestation of the fact that particles don't follow classical trajectories.

In quantum mechanics, a single particle wavefunction is traditionally represented as Ψ(x). A two particle wavefunction would be a function of two variables Ψ(x1,x2), where x1 and x2 represent the position variables of the two particles. If we define an operator P which just swaps the two particles around:

P Ψ(x1,x2) = Ψ(x2,x1)

then obviously P2 Ψ(x1,x2), so there are only two possibilities.

Either P Ψ(x1,x2) = Ψ(x1,x2) or P Ψ(x1,x2) = - Ψ(x1,x2)

i.e. the two particle wavefunction is symmetric under interchange of the particles, or it's antisymmetric. Particles displaying the former behaviour are called bosons and ones displaying the latter behaviour are called fermions.

In quantum field theory, you create a particle from the vacuum by applying a creation operator. If you create more than one particle, you find that your particles are automatically indistinguishable - this is an inescapable feature of quantum field theory. If you constructed your QFT using canonical commutation relations, you will get bosons. If you used canonical anticommutation relations, you will get fermions.

Supersymmetry is a proposal for a type of symmetry that the quantum field theory in nature should obey. To describe what it is, we need first to say what a symmetry in a quantum field theory is. The QFT is traditionally described by its Lagrangian (strictly Lagrangian density). This is a function of the all the fields present in the theory - these may include all sorts of things - scalar fields, vector/tensor fields, spinor fields etc. Suppose I just denote all the fields in the theory by the letter Ψ - just think of it as a big column vector. A symmetry of the theory is an operation (think matrix), U, which maps fields to fields

Ψ' = UΨ

such that

ℒ(Ψ', 𝜕μΨ') = ℒ(Ψ, 𝜕μΨ)

For example, if I take the Lagrangian for a Dirac field (here Ψ is a 4 component spinor, composed from a pair of two component spinors that we discussed in an earlier post, and γμ are the Dirac gamma matrices :

ℒ(Ψ, 𝜕μΨ) = Ψ0i𝜕μγμ-m)Ψ

Then an example of a symmetry is provided by the charge conjugation operation, C, which swaps particles for antiparticles. C is represented by the matrix -iγ2γ0. C mixes up and modifes the components of Ψ. Supersymmetry does a similar thing, but this time it mixes up the components of fermionic fields with the components of bosonic fields.

Suppose we take a simple example to illustrate this - take a theory with two entities:

FERMION: a single two component spinor field ΨA
BOSON: a complex scalar field φ

Our supersymmetry will be implemented by an operator QA which takes us from the boson field to the fermion field. This is known as a supercharge. This is an infinitesimal generator, in the same sense that Pμ and Jμν are the infinitesimal generators of the Lorentz group (see the appendix below). Introducing the supercharge and defining its relationship with the Poincare group infinitesimal generators allows us to define the “super Poincare algebra” relations:

[Pμ, QA] = 0

[Jμν, QA] = i(σμν)ABQB

{ QA, QbarA’} = 2σμAA’Pμ

Here σμAA’ are the Pauli spin matrices I mentioned in a previous post. (σμν)AB = (1/4)( σμAA’ σbarνA’B - σνAA’ σbarμA’B). (bar is complex conjugate as usual).

The last relation is interesting. It says that combining a (infinitesimal) supersymmetry transformation with its conjugate is equivalent to an infinitesimal spacetime translation. Note – if we allow the supersymmetry transformation to be dependent upon the spacetime point, rather than being global, then we end up with infinitesimal translations being spacetime point dependent, i.e. we end up generating infinitesimal general coordinate transformations. This leads us naturally to gravity and is the basis of the theories of supergravity.

It’s possible to define supersymmetric theories with more than one supercharge. The number of supercharges is traditionally denoted by 𝒩. So an “𝒩 = 4 supersymmetric theory” is a theory with 4 supercharges.

Motivations for Supersymmetry.



1 Hierarchy Problem

The first motivation that is normally advanced for supersymmetry is the hierarchy problem. The famous mexican-hat Higgs potential is:

V = -μ2φφ +( λ/4)(φφ)2

V has a minimum at |φ| = 2μ/ √ λ

λ is proportional to (the square of) the Higgs mass. Now the problems start to crop up when we include quantum corrections at the one loop level:

The term (φφ)2 defines a 4 point self interaction of the φ field. A tree level, the interaction vertex just contributes a λ factor to amplitudes. When we want to compute amplitudes to the next level in perturbation theory we must include loops like:
HiggsSelfInt.jpg
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This diagram represents a one loop contribution to the φ propagator. Here, we must integrate over the momentum of the internal line. In principle this internal momentum can have any value, so the integration is to infinity. Since this results in a divergent answer, the normal procedure is that we introduce a cutoff Λ at high momentum.


The one loop term resulting from the (φφ)2 interaction term results in a correction λΛ2φφ which is applied to the term -μ2φφ.

Now we know (well, phenomenologists know) the vacuum expectation value of Higgs field is of the order of 246Gev (from muon decay apparently!), and this expectation value is proportional to μ/ √λ, so consequently the physical mass μ (after correction) can't be greater than a few hundred GeV. But this is a problem: - this correction λΛ2φφ is proportional to the square of the cutoff. What is this cutoff ? If we don't expect any new physics up to the Planck scale, it's of the order of the Planck mass i.e. 1019GeV.

So we're in the rather dubious situation where we have a bare mass term of the order of -(1019GeV)2 cancelled by a correction term of a similar size, leaving a result of size a piddling couple of hundred Gev2. Imagine something like this (the numbers are fictional, but the order of magnitude is representative)


(Physical Mass)2 = (Bare Mass)2 - (Correction Term)2

18769 = 49984905920593950496034950295949634709 -49984905920593950496034950295949615940

Looks very fishy doesn't it ? You have mass renormalizations in other models too, so don't they all look fishy like this ? Ian Aitchison makes the point that it usually goes differently. For example in QED, the one loop correction would be "electron emits photon and then reabsorbs it". When worked out, this gives a correction to the electron mass of logΛ. This is a hell of a lot more benign than Λ (put some numbers in to convince yourself).

In fact in the case of interest, if we have some fermion interactions, then in addition to the one loop self interaction of φ, we also have contributions from the one loop case where a φ splits into a fermion/antifermion pair, which then recombine. The interesting thing is that this contribution is of the opposite sign to the φ self interaction contribution. You can see where this is going:

If there is the right symmetry between fermions and bosons in the Lagrangian, then the one loop corrections cancel exactly, and we don't have to worry about the "cancelling two huge numbers to get one small one" issue. The symmetry that allows this to happen is of course supersymmetry.

Note I think that there are several issues floating around in physics which sometimes have the name "hierarchy problem" attached to them. Note also, that supersymmetry isn't the only potential solution to this hierarchy problem.


2 GUTs and Gauge Coupling Constants

There is a technique called the renormalization group equation that allows you to investigate the behaviour of coupling constants as a function of the energy scale in which you're working. There are 3 gauge couplings in the standard model. If you plot their behaviour as a function of the energy scale, you get something like the upper picture. If you introduce supersymmetry into the model you get something like the lower picture, i.e. they converge at the same (GUT) energy scale.
GUTCoupling.jpg
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The proponents of supersymmetry would argue that this is too great a coincidence to ignore. Interestingly though, if there is any new (non SM) physics between here and the GUT scale, this argument goes out of the window, because using the renormalization group to predict the scaling behaviour of the coupling constant in this regime would no longer be valid.


3 Coleman Mandula Theorem

This basically states that the symmetry of a theory (which satisfies some “reasonable” conditions) is always of the form

Spacetime symmetry X Internal symmetry

By “spacetime symmetry” I mean the usual Lorentz/Poincare symmetry, by “internal symmetry” I mean things like SU(2)xU(1) (electroweak symmetry). In other words the composite symmetry is a straightforward product, no mixing of the two symmetries is possible.

Supersymmetric theories provide a way round this – they do not have this restriction. I don’t really understand it, but this is quoted as an argument in their favour.

Appendix - Poincare Group, infinitesimal generators and all that


In an earlier post, I gave an example of a matrix representing a particular Lorentz transformation - where the relative velocity between the primed and unprimed frames is in the x direction. ( In this case, only the x and t components got screwed around, y and z remained untouched).

(ɣ ...-βɣ...0...0)
(-βɣ...ɣ....0...0)
(0 ....0.....1...0)
(0.....0 ....0...1)

where ɣ = 1/sqrt(1-v2/c2), and β = v/c.

For a general Lorentz transformation, we would write this relation as

x'μ = Λμνxν, where Λμν is the matrix representing the transformation.

To see what infinitesimal transformations are like, take a Lorentz transformation which differs from the identity by a tiny amount
Λμν = 𝛿μν + 𝛿ωμν

where 𝛿ωμν is a very small perturbation away from the identity transformation. Since a Lorentz transformation must preserve the Minkowski metric:

η'μν = Λμρ Λνσ ηρσ

=( 𝛿μρ + 𝛿ωμρ )( 𝛿νσ + 𝛿ωνσ ) ηρσ

= ημν + 𝛿ωμν + 𝛿ωνμ + 𝒪(ω2)

So we must have 𝛿ωμν = -𝛿ωνμ i.e 𝛿ω is antisymmetric.

Traditionally, we expand -𝛿ω in terms of a basis for these infinitesimal transformations, so the infinitesimal Lorentz transformation looks like

Λαβ(𝛿ω) = 𝛿αβ – (i/2)𝛿ωμν(Jμν)αβ

The Jμν)αβ are the infinitesimal generators of the Lorentz transformations. Think of the indices μ and ν as labelling which generator we’re looking at (since μν always appears in an antisymmetric combination, there are six of these), and α and β as labelling the rows and columns of each generator, which, being a Lorentz transformation is a matrix.

The structure of the Lorentz group is characterized by the commutation relations obeyed by its infinitesimal generators (omit the matrix indices for clarity, just write the indices which label the generators):
[Jμν, Jλσ] = i(Jλνημσ - Jσνημλ - Jμληνσ + Jμσηνλ )

When people rabbit on about “algebras” of this that and the other, this is what they mean – the relations obeyed by the infinitesimal generators of the group in question.



Here, we've been looking at the action of Lorentz transformations on elements of Minkowski space, which are themselves Lorentz vectors. In general, we would be looking at the action on elements of any old representation of the Lorentz group – on spinors for example, but we would find the same algebra for the infinitesimal Lorentz generators. For example, for spinors, the infinitesimal generators would be of the form JμνAB where μ and ν are the spacetime indices and A and B are the spinor indices.

So, the Lorentz group has 6 infinitesimal generators Jμν, three for the three independent rotational degrees of freedom, and three for the three independent directions you can do your boosts in. In additional to rotational and boosting symmetry, flat spacetime is also symmetric under translations. There are, fairly obviously, four translational degrees of freedom, and after expanding the Lorentz group to include these degrees of freedom, we end up with a 10 parameter group – the Poincare group.

To represent a Poincare transformation on spacetime, we two things – a matrix Λ as before, for our rotation/boost, and a vector b for our translation. So the Poincare transformation is

x'μ = Λμνxν + bμ

where Λμν is the Lorentz matrix and b is the translation. Two Poincare transformations combine like

2, b2) (Λ1, b1) = (Λ2Λ1, Λ2b1+b2)

Of course, to do anything meaningful, we need to represent our Poincare transformations as matrices. To do this, we actually use 5x5 matrices. The matrix representing the Poincare transformation (Λ, b) is

(Λ..........b)
(0..........1)

i.e. we just stuff the vector b into the last column. In a similar way to before, we can now talk about the infinitesimal generators corresponding to the translations. There are, of course, 4 of them and they’re traditionally denoted as P956. You can now extend the Lorentz algebra above by computing the commutation relations of the P’s with themselves and with the J’s. The full set of commutation relations becomes:

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

[Pμ, Jλσ] = i(Pλημσ - Pσημλ)

[Pμ, Pν] = 0

This is the Poincare algebra and lies at the heart of all of our conventional quantum field theory.
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Re: Twistor string theory

#28  Postby tnjrp » Mar 22, 2011 7:45 am

I wonder when I'm going to have the time to read this last... :sigh:
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Re: Twistor string theory

#29  Postby twistor59 » Mar 22, 2011 9:10 am

tnjrp wrote:I wonder when I'm going to have the time to read this last... :sigh:


Yeah, sorry it's getting a bit long isn't it ? I'm writing notes and posting them on here as I learn new stuff. I figure that's better than just doing my usual scribbled scrawl in a notebook. (1) It's neater, so I can actually read it, and (2) maybe other saddos can benefit from it.
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Re: Twistor string theory

#30  Postby tnjrp » Mar 22, 2011 9:14 am

Lengthy posts are not a problem an sich, but as these are quite rich in actual content and also require me to engage the grey cells (what little there are left) I have to actually set some time aside to read.
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Re: Twistor string theory

#31  Postby twistor59 » Mar 22, 2011 9:24 am

I'll try to put the whole thing in less mathematical terms once I've understood what the hell it is that they're actually doing in this field. With me I tend to try and get a handle on the maths first and only then can I begin to figure out what it actually means (or doesn't mean!) physically.
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Re: Twistor string theory

#32  Postby newolder » Mar 22, 2011 1:29 pm

twistor59 wrote:
tnjrp wrote:I wonder when I'm going to have the time to read this last... :sigh:


Yeah, sorry it's getting a bit long isn't it ? I'm writing notes and posting them on here as I learn new stuff. I figure that's better than just doing my usual scribbled scrawl in a notebook. (1) It's neater, so I can actually read it, and (2) maybe other saddos can benefit from it.


this saddo thinks he's getting some benefit from the ordered structured approach you've adopted hitherto. :thumbup:
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Re: Twistor string theory

#33  Postby twistor59 » Mar 22, 2011 9:14 pm

Really great lecture by Nima Arkani Hamed (again!), substantially on twistor string. Also, it's an overview style with not too much mathematics. Well worth a listen (several times)

http://pirsa.org/11010111/
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Re: Twistor string theory

#34  Postby twistor59 » Mar 26, 2011 8:58 am

For some discussion of the convergence of the coupling constants at the GUT scale, which I mentioned in my last post, see this lecture:

http://www.youtube.com/user/ICTPchannel?blend=2&ob=5#p/u/2/TkmgtanlkIA
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Re: Twistor string theory

#35  Postby twistor59 » Apr 03, 2011 5:30 pm

Post #6 The Superfield formalism


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.

Grassmannians


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 = εαβγδZαdZβdZγdZδ1234

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 ∫ψ11 = 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]
A’, ψa) -> (x-AA’πA’aθAa, 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’aθAa = 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.
Stereographic.jpg
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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.
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Re: Twistor string theory

#36  Postby twistor59 » Apr 24, 2011 12:41 pm

OK, long holiday weekend. Time on my hands => another post:

Post #7 Yang Mills Fields and Self Duality

In this thread I talked about Maxwell electromagnetic theory as a gauge theory. Gauge theories are all about a description of physics which involves some redundancy. You formulate the physics in a way with the redundancy (this makes the calculations easier), do your calculations, then check that your answers are independent of the redundancy. This redundancy was expressed a non uniqueness in the description of the matter field Ψ. The idea was that if I multiply Ψ by some arbitrary complex function with unit modulus

Ψ(x) -> eiα(x)Ψ(x)..............(1)

then my new Ψ(x) describes the same physics, provided I introduce a new field Aμ(x) and replace all the derivatives ∂/ ∂xμ by ∂xμ- Aμ (there might be some factors and constants multiplying A, but I’m not writing those, for simplicity). Transformations like (1) are called gauge transformations. They form a group – if you do one then do another, you can find a composite transformation that you could have done in one go and got the same result:
Ψ(x) -> eiα(x)Ψ(x)-> eiβ(s)(eiα(x)Ψ(x))

same as
Ψ(x) -> ei(α(x)+β(x))Ψ(x)

In this example, the group is called U(1), the unimodular transformations in 1 complex variable. The new field Aμ(x) which I had to introduce in order to maintain gauge invariance turns out to be a potential for the Maxwell field.

Yang Mills theories are generalizations of this model to other gauge groups. The original Yang Mills theory was based around the group SU(2) – the group of 2x2 unitary complex matrices with unit determinant. Here you may imagine for example, a matter field with 2 components arranged in a column vector:

1(x)]
2(x)]

The gauge transformations consist of multiplying such column vectors by matrices which are functions of position g(x), where g(x) is an element of SU(2). (Note just because we’re doing SU(2), we don’t have to consider two-element column vectors – the vectors we use depend upon the representation of SU(2) that we’re working in).

The auxiliary field that we now have to introduce to maintain gauge invariance now has to be a “matrixey” type thing. In fact it’s a combination of the generators of SU(2), which are the Pauli spin matrices. i.e. the auxiliary field looks like

Aμa(x)σa

where a = 1,2,3 labels the generators. When we do our trick of replacing the normal spacetime derivatives by the gauge covariant derivatives, i.e.

μ ->∇μ = ∂xμ - Aμ

then the fact that Aμ really means Aμa(x)σa, together with the fact that the σa matrices don’t commute with each other implies that the field we compute by commuting the derivatives contains some extra terms:

Fμν = ∂μAν - ∂νAμ + AμAν - AνAμ

Fμν is the Yang Mills field and Aμ is the Yang Mills potential (although some people loosely refer to Aμ as the field too).

Now the Yang Mills field Fμν, having a pair of antisymmetric spacetime indices μ and ν is a two-form (see appendix). We can therefore apply the Hodge duality operator to get (in 4 dimensions) another two form (*F) μν. If d denotes the gauge-covariant exterior derivative, then the (source free) Yang Mills equations are just
*d(*F) = 0

These are derivable from a Lagrangian density Tr(F^*F). To see the need for the trace, remember that, just as Aμ really means Aμa(x)σa, so also F has some matrices multiplying it (it’s an “endomorphism valued two form” as the mathematicians would say).

In 4 dimensions, there is an interesting class of Yang Mills fields – if *F = F, the field is called self dual, if *F = -F, it’s called anti self dual. In both cases, being self dual or anti self dual automatically guarantees that it satisfies the Yang Mills equations. In fact self and anti self dual solutions in 4 dimensional Euclidean space are called instantons and are important contributions in non perturbative quantum field theory.

OK I spent a fair bit of time here just saying what Yang Mills theory is. Next time, I’ll say what 𝒩 supersymmetric Yang Mills theory is, which is the subject of a lot of the current work on twistor string theory.



Appendix
Bundles, Connections and Stuff


Since I’m talking about Yang Mills theory in this post, I’d better mention a few essential bits of maths that don’t usually get taught in undergrad courses (at least not in my day, maybe things have changed).

Starting with your spacetime manifold M, all the Yang Mills excitement takes place in a thing called a vector bundle which sits “over” M. (There is a complementary formulation which describes stuff in terms of prinicipal bundles, but hopefully I won’t need that here).

Yang Mills models are based around gauge symmetry groups (the original one being SU(2)). We can think of elements of the group as being matrices, and as we know matrices act upon column vectors. The space of possible values for these vectors is called a vector space (for the fundamental representation of SU(2) this vector space would just be the space of pairs of complex numbers, i.e. ℂ2. A matter field ΨA(x) =
1(x)]
2(x)]

assigns an element of this vector space to each spacetime point x. So we have to think of there being multiple copies of the vector space, one for each spacetime point. If we draw spacetime as a line, and the vector space as another line, then this assembly of vector spaces, which is called a vector bundle, would look like:
bundle.jpg
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Of course there is a projection (traditionally denoted π) from the assembly of vector spaces (sometimes called the “total space” of the bundle) to M, which just maps a point in the total space to the point over which it sits. Oh I should also mention that the vector space over x is referred to as the “fibre” over x.

Now a matter field ΨA(x) just picks, for each point x, a vector in the fibre over x. In bundle language, such a thing is referred to as a “section” of the vector bundle.

Now bundles can be more interesting than the picture of a rectangle being projected onto a line might suggest. Suppose we stick with a 1 dimensional spacetime manifold M and a 1 dimensional vector space (so we can draw it!), but now suppose spacetime is a circle. We can put our fibres side by side and we end up with a cylinder as the total space. But now, suppose that, as we arrange our fibres over the points of the circle, we give them a small twist as we go along. We now end up with a Mobius strip as the total space. In fact the concept of “adding a twist” generalises to higher dimensional manifolds and fibres and the problem of how to classify vector bundles in terms of these structures is a whole topic in its own right (characteristic classes).

OK, so matter fields are sections of the vector bundle. What about gauge fields and gauge potentials ?
bundle1.jpg
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A gauge potential is actually a mathematical object called a connection on the vector bundle. Consider the fibres sitting over two points x1 and x2 in the spacetime M. These are two distinct copies of the vector space and there is no natural map between them. A connection is the object that provides such a map. For any given curve in the spacetime M connecting a pair of points x1 and x2, it provides a map between the fibres lying over those points. So in the diagram, a vector v lying in the fibre over x1 gets mapped to the vector u1 in the fibre over x2 if you use curve C1 and to a different vector U2 if you use C2. This process of transporting a vector along a curve from one fibre to another is called parallel transport.

To see how it relates to the gauge potential, we just look at the process of transport between a fibre over x and the one over an infinitesimally separated one x+dx. The mapping between the fibres is now just given by Aμa(x)σa. Here each σa is an infinitesimal generator of the gauge group, i.e. a matrix, which is just what we need for the mapping between two vector spaces. To get the mapping resulting from transport along a finite curve, we just integrate this infinitesimal version.

So now we know what the Yang Mills potential Aμ(x) is for – it’s for mapping shit between vector spaces at different points of the spacetime manifold.

The meaning of the Yang Mills field field Fμν is now easy to describe – you just do an infinitesimal transport along a displacement dxμ, then along dxν and compute the difference with what you’d got if you did first dxν and then dxμ.


Differential Forms.


We all know that tangent vectors are like little infinitesimal arrows on a manifold, giving a direction (and strength) at a point. More formally, you can think of a tangent vector as a directional derivative acting on functions on the manifold.

f(x) -> vμ(x)∂/∂xμf(x)

We can also define objects called “one-forms” on a manifold, which are dual to vectors in the sense that they take a vector field and map it to a function. One forms are things which eat vectors and puke out numbers. One form fields are things which eat vector fields and puke out functions.

To try to visualize this, on a 2d manifold, you can think of a one form field as a bunch of lines which the tangent vectors pierce. At any point x, the function obtained from the tangent vector field and the one form field is the “amount of piercing” at x. This depends on
1) The size of the vector at x
2) The relative orientation of the vector and the one form field lines
3) The density of the one form field lines at x
form.jpg
form.jpg (13.51 KiB) Viewed 2486 times

Just as, when we have a coordinate system, we can define a basis for the tangent vector fields by the set of operators ∂/∂xμ, so we can define a basis for the one form fields by the set of entities which we write as dxμ. The one form basis vectors dxμ are defined by their actions on the tangent basis vectors:

< dxμ, ∂/∂xν> = δμν = 1 if μ = ν, 0 otherwise.

Now, the fact that we called them “one forms”, not just “forms” suggests there might be such things as two forms and three forms etc. This hunch is indeed correct.

Forms are always skew-symmetric in their indices. For example we write a 2-form as Fμνdxμ^dxν where the ^ denotes a skew-symmetrized tensor product. For example
dx0^dx1 = dx0⊗dx1- dx1⊗dx0 etc

There is a special n-form on an n-dimensional manifold, it’s the called the volume form and in a given coordinate system {xμ, μ=0,1,..n-1 } is given by

dx0^ dx1^....^ dxn-1

(I’m not caring about 1/n! factors here). If you’ve ever integrated anything in your life, you’ve used this form, although you may not have been aware of it. Suppose you were lucky enough to have been set a question in your exams involving a 2 dimensional multiple integral ∬f(x,y)dxdy. Well, that dxdy bit at the end is really the volume 2 form dx^dy. Recall that if you change variables to x’, y’, you have to multiply by the Jacobian determinant of the transformation ? Determinants are inherently skew symmetric, and the Jacobian precisely encapsulates the coordinate transformation law for the volume form.

If I have a p-form on an n-dimensional manifold, there is a special operator called the Hodge star operator which maps it to an n-p form. This is defined on the components of the form by

(*ω)i1 i2..i(n-p) = εj1 j2..jp i1 i2...i(n-p) ωj1 j2..jp

where εj1j2..jp i1i2...i(n-p) is the tensor completely skew symmetric in all its indices.

If you’ve done an elementary course in 3d vector calculus, you’ve been using the Hodge * operator, but I bet your teacher didn’t tell you:

Firstly, in normal 3d space we blur the distinction between vectors and one forms. We can do this because the metric is just diag(1,1,1), so we can swap between vectors and one-forms willy nilly. They usually define the cross product something like
uXv = uv.sinθn

where n is a unit normal vector perpendicular to the plane of u and v (in the direction of a right handed screw) and θ is the angle between them. Then you get told that, to compute it in components, you do something like

uXv = (u2v3- u3v2, u3v1- u1v3, u1v2- u2v1 )

But wait a minute, this is just saying

i’th component of uXv = εijkujvk

where εijk is the Hodge star operator in 3 dimensions. The cross product of the two one-forms u and v is really a two form, and we can only make a vector out of it by taking its Hodge dual. You can only do this because we happen to be in three dimensions where the dual of a two form is a one form (=vector).

Exterior Derivative


When we’re dealing with forms whose coordinates are functions of position (i.e. one form fields, two form fields etc), we can differentiate those components. The exterior derivative is a mechanism for getting a (p+1) form from a p form. So differentiating a 2 from gives a 3 form:

ωjk -> (dω)ijk = ∂[i ωjk]

(Square brackets, as always, describe skew symmetrization of all the indices they surround).
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Re: Twistor string theory

#37  Postby cavarka9 » Apr 27, 2011 2:26 pm

some day :cheers: .
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Re: Twistor string theory

#38  Postby twistor59 » May 02, 2011 1:32 pm

Post #8 𝒩=4 SYM


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.

μν)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:
-1/2>=(Q10)-1>
0>= (Q20)) (Q10)†|Ω-1>

1/2>= (Q30)) (Q20)) (Q10)-1>

1>= (Q40)) (Q30)) (Q20)) (Q10)-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.

Scattering Amplitudes


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.

Spinor Helicity Formalism

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))e[p1 – ip2]
................................................... [p0 – p3]
λbarA’ ........= ..............(1/sqrt(p0-p3))e[p1 + ip2]
................................................... [p0 – p3]

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

<λμ> = εABλA μB
~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

Amplitudes


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).

Appendix Helicity


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.
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Re: Twistor string theory

#39  Postby iamthereforeithink » May 08, 2011 4:59 pm

Hello Twistor: Thanks for keeping this thread going. I've been incredibly busy at work lately, and so haven't been active on the forums, but I still try to follow threads such as this one every now and then. :cheers:
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Re: Twistor string theory

#40  Postby twistor59 » May 26, 2011 11:54 am

Post #9A The AdS/CFT Correspondence


Continuing this blog of me learning about this stuff and finally getting a bit closer to twistor string theory now, I want to spend this post talking about a topic closely related to twistor string. The original motivation which led Witten to investigate the twistor string model was the desire to better understand the holographic correspondence between type IIB superstring theory on AdS5xS5 and 𝒩=4 SYM on a boundary space. This is the most carefully studied example of t’Hooft’s holographic principle to date, so I’ll spend this post saying a bit about how it works.

Anti deSitter Space


Some solutions of Einstein’s equations are so well known that even William Lane Craig can pontificate about them. Anti de Sitter space (or “AdS” as cool people refer to it) is a highly symmetric solution of Einstein’s equations with constant negative curvature. In physical terms, it corresponds to a spacetime with negative cosmological constant. (Spacetimes with positive cosmological constants are inherently trying to expand themselves, negative ones are trying to shrink).

To define AdS, start with a 5 dimensional flat space which has the metric
ds2 = dt2+du2-(dx1)2-(dx2)2-(dx3)2
This is just like a 5 dimensional Minkowski space, but with two time coordinates t, and u (identified by their positive signs) rather than just one. Four dimensional anti de Sitter space AdS4 is then the hyperboloid

t2+u2-(x1)2-(x2)2-(x3)2 = R2

where R is a constant measuring the scalar curvature.
AdsHyperbola.jpg
AdsHyperbola.jpg (50.81 KiB) Viewed 2423 times


This is “conventional” anti de Sitter space AdS4, which is discussed in relativity texts, for example Hawking and Ellis. To get the space of interest for the holographic correspondence, we simply add an extra spatial dimension, i.e. we’re looking at the manifold defined by

t2+u2-(x1)2-(x2)2-(x3)2-(x4)2 = R2

We can attach a conformal boundary to AdS5 by looking at the intersections of the set of hyperplanes, parametrised by v

X4+u = ev
AdsHyperbolaPlane.jpg
AdsHyperbolaPlane.jpg (28.35 KiB) Viewed 2423 times


The intersection “plane at v=∞” can be given a Minkowski metric and hence the conformal boundary of AdS5 can be considered to be Minkowski space. Although we have considered AdS5 with Minkowski metric signature, it is possible also to consider the Euclidean version and it is in the Euclidean signature that much of the AdS/CFT discussion takes place.

In order to do this, the time coordinate, t, of the previous construction is replaced with 𝜏=it. If this is done, AdS5 becomes hyperbolic 5-space, H5. This has a boundary which is a 4-sphere S4.

Superstring/Brane Models

The Ads/CFT correspondence was first hatched in the context of superstring theory. I’m not going to attempt an exposition of superstring theory here, but just for completeness I mention a few highlights (all this material will be expounded in some detail in introductory books and lectures on string theory):

Strings are closed or open one-dimensional objects. The two-surfaces they sweep out in spacetime are called worldsheets. From the point of view of the worldsheet, the spacetime coordinates constitute a set of fields on the sheet. So, although xμ transforms as a vector from the spacetime point of view, it’s a bunch of scalar fields from the point of view of the two dimensional worldsheet. The physics of these fields is given by varying the Polyakov action

S = ∫d2σsqrt(-h)hαβημναxμβxν

Where d2σ is the area element on the 2d worldsheet coordinatized by σ, 𝜏. ημν is the Minkowski spacetime metric and hαβ is the induced metric on the worldsheet

The action is invariant under a rescaling hαβ->Λ( σ, 𝜏)hαβ. Use this to ensure that hαβ is of the form hαβ = eφ( σ, 𝜏) ηαβ. If this is done, then the “fields” x[sup]μ satisfy the wave equation and the action looks like

S = ∫d2σηαβαxμβxμ


Open/closed versions of the strings + imposition of boundary conditions + Fourier expansions of the vibration modes allow the definition of left and right moving components. The action allows the definition of conjugate momenta, and from this point canonical quantization can be applied. The Fourier expansion coefficients become raising and lowering operators.

Variation of hαβ in the Polyakov action leads to the constraints Tαβ = 0, where Tαβ is the energy momentum tensor. Fourier decomposition of the constraints leads to the definition of the Virasoro operators Lm. These operators satisfy the Virasoro algebra, and the physical states must satisfy a quantum version of the mass shell condition, namely
(L0-1)|φ> = 0

There are two problems with the system so far:
(1) L0 is given by L0 = (1/2)α02 + ∑ α-nαn + ε0 (summation from 1 to ∞)
where ε0 = ((d-2)/2) ∑n (summation from 1 to ∞) is the vacuum (zero point) energy. d is the spacetime dimension, and the infinite summation is conventionally taken to be (-1/12), see here for a discussion of this. α are the Fourier expansion coefficients of the energy momentum tensor T. Now if I act on the vacuum with one creation operator, i.e. build the state |φ> =ζμαμ-1|0>, then we add 1 unit to the zero point energy, which has now become ((d-2)/2)(-1/12) + 1. However, the Virasoro constraint L1|φ> = 0 forces the polarization vector to satisfy pμζμ = 0, i.e. there are no longitudinal components. So the number of polarization degrees of freedom has been reduced from the expected (d-1) to only (d-2). But this is precisely the behaviour of a massless particle, so we must have ((d-2)/2)(-1/12) + 1 = 0. Hence we conclude that d must be 26. This motivates the famous requirement for 26 dimensional spacetime for bosonic string theory.

Quick remark on the particle spectrum of the bosonic string theory: As we just said, the first excited (open) string state is a massless vector field Aμ(x). Next is a massive symmetric traceless rank 2 tensor field. For the closed string spectra, we tensor right and left moving modes with the same excitation number. The first excited state has three parts – massless symmetric rank 2 tensor gμν(x) i.e. the graviton, and antisymmetric rank 2 tensor Bμν(x), and finally a scalar (the dilaton).

(2) The ground state has negative mass, i.e. is tachyonic, which is a bit of a problem (!), which is solved in the superstring models. The superstring approach (at least the RNS version) works by adding fermionic fields ψμ. Just like the xμ , the ψμ transform as a spacetime vector, but from the worldsheet point of view each one is a spinor. For each value of μ, (xμ, ψμ) is a supermultiplet.

The gauge-fixed Polyakov action now has an extra term in it:

S = ∫d2σηαβαxμβxμ – iψbarμρααψμ

Where ρα are a pair of 2x2 gamma matrices and ψ = (ψ+, ψ-) is a two component Majorana spinor. Just as in the bosonic case, ψ+ and ψ- can be interpreted respectively as “left moving” and “right moving” modes.

Similar arguments to the bosonic case apply, and you can build a supersymmetric version of the Virasoro algebra, and compute a critical dimension, which comes out in this case to be 10 instead of 26.

The worldsheet scalars go through like in the bosonic case, but with superstrings we also need to look at the boundary conditions for the worldsheet spinors. Looking at the open string case, for consistency we need to set , ψ+ =± ψ- at both ends. Taking ψ+(0, 𝜏) = ψ-(0, 𝜏), there are two possibilities for the other end:

ψ+(π, 𝜏) = ψ-(π, 𝜏)..............”R”
ψ+(π, 𝜏) = -ψ-(π, 𝜏).................’”NS”

“R” is for Ramond, and “NS” is Neveu-Schwarz. With these options for boundary conditions, the usual mode expansions can be done and canonical quantization applied. In the NS sector, the excited states we get by acting on the ground state with the appropriate raising operators are spacetime bosons (the ground state itself is tachyonic, just like in bosonic string theory). In the R sector, the states are spacetime fermions.

In the closed string case, the states are tensor products of the right moving and left moving cases, so we end up with two bosonic possibilities NS⊗NS and R⊗R. We also have two fermionic possibilities NS⊗R and R⊗NS.

Looking at the spectrum of spacetime fields arising from superstrings, it turns out that for consistency we have to apply a truncation, called the GSO projection, which reduces the number of possibilities. Applying the GSO projection in the closed string R-R sector, if we do the same GSO projector on both sides, we are led to type IIB superstring theory which is of interest here. Applying some group theory jiggery pokery (see here section 6), we obtain a bunch of “RR potentials” C(n)μ1...μn. These can be differentiated to obtain fields in the same way that you differentiate a Maxwell potential Aμ(x) to get a Maxwell field Fμν(x). In type IIB theory, it turns out that there are just even R-R potentials C(0), C(2), ... C(8).

In Maxwell’s theory, the electromagnetic potential Aμ(x) is coupled to the current Jμ(x) by the second term in the action

S = ∫d4x(-1/4Fμν Fμν + q AμJμ)

We look for an analog of this current source for the RR potentials.
D Branes

For a (p+1) RR potential C(p+1), the required source turns out to be an object with p degrees of freedom, which sweeps out a (p+1) dimensional worldvolume. The analog of the Maxwell coupling q AμJμ is then obtained by integrating C(p+1) throughout the induced (p+1) dimensional volume element on this worldvolume. The object generating this worldvolume is called a Dp-brane.

In type IIB theory, corresponding to the RR potentials C(0), C(2), ... C(8), we have D(-1), D(1),...D(9) branes. The D(-1) case is a bit weird, and actually defines a D instanton. D(3) branes have a 4 dimensional worldvolume, which is taken to be our spacetime in braneworld models.
DBraneString.jpg
DBraneString.jpg (8.9 KiB) Viewed 2423 times


Open strings can end on Dp-branes, and can stretch between branes. In the perturbative picture, you can think of a Dp-brane as fixed at spacetime coordinates xp+1,...x9. So, the xp+1,...x9 coordinates of the ends of the open strings do not change, but the ends can slide around on the brane, i.e. change x0,...xp. The fixed xp+1,...x9 corresponds to a particular “string theory background” i.e. a solution of the classical string equations (recall that spacetime coordinates are thought of as fields on worldsheets).

Non perturbatively, the massless oscillation modes of the open strings ending on the Dp-branes define the fluctuations of the Dp-branes themselves. One of the members of the open string spectrum is a massless vector 10 dimensional field Aμ(x). We can decompose Aμ(x) into p components Aa a=0,1,..p and components Φm m=p+1,...9. The former give a U(1) (electromagnetism) gauge theory in the worldvolume. The latter describe fluctuations of the Dp-brane in the 9-p orthogonal directions. These fields are obtained by dimensional reduction of 10 dimensional SYM onto the Dp-brane.

If I have a stack of N coincident branes, the U(1) gauge theory instead becomes U(N).

The modes of oscillation of the Dp-brane are determined by the modes of oscillation of the quantized open string whose endpoints lie on it.

There is another description of Dp-branes which I’ve heard actually Dp-branes turn out to be solitonic closed string field configurations.

Remembering that 10 dimensional spacetime can be a product of “our” spacetime with an exotic Calabi Yau manifold, it’s important also to remember that Dp-branes can “wrap” themselves around holes (“cycles”) in the CY manifold.
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