Twistor string theory

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Re: Twistor string theory

#41  Postby twistor59 » May 26, 2011 11:57 am

Post #9B The AdS/CFT Correspondence


The AdS/CFT Correspondence

Dp-branes, being massive objects, have a back-reaction on the spacetime geometry. When N is large, N coincident branes will leave a significant imprint. If Tμν is the energy momentum tensor of the D-brane source, the Einstein equations lead to the geometry:

Rμν-(1/2)Rgμν = 8π G Tμν

If gs is the closed string coupling constant, then the Newton constant G is proportional to gs2 and, since the brane tension goes as 1/ gs, the brane energy momentum tensor is proportional to (N/ gs). Hence the Einstein tensor Rμν-(1/2)Rgμν is proportional to N gs = λ. For small λ there is a negligible backreaction, and the we have string theory in roughly flat spacetime.

As N increases, a “throat” appears:
AdSThroat.jpg
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In the AdS/CFT scenario, we’re interested in the case where N D3-branes are present. The D3 branes are thought of as filling our perceived “large” 3 spatial dimensions, and of course sweep out a 4 dimensional world volume. If we decompose the 10 dimensional spacetime into a 3+1 Minkowski signature part and an ℝ6 part with coordinates y1, ..y6, then the distance from the brane is just r2 = (y1)2+..+(y6)2

Far from the branes (large r), the metric is approximately flat 9+1 space. Near the branes, (low r), it looks like AdS5xS5. Now somehow (I don’t really understand how this works), “taking the low energy limit”, the throat (which has the geometry of AdS5xS5) can be considered in its own right, i.e. as if it was decoupled from the rest of the spacetime. I suspect it might mean that we’re talking about low energy in the sense that we don’t have enough energy to get far from the brane, but I’m not sure.

Anyway, taking a large λ scenario and taking the low energy limit puts us in the AdS5xS5 geometry. Taking a small λ scenario (flat spacetime, no throat) gives us a 4D SYM theory.
AdSCFTCorr1.jpg
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Comparing these two low energy limits of the theory, Maldacena conjectured that 4 dimensional 𝒩=4 SYM is equivalent to type IIB string theory on AdS5xS5. This is the famous AdS/CFT correspondence and is what led Ed Witten to his work on twistor string theory (which I’ll get to shortly).
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Re: Twistor string theory

#42  Postby twistor59 » Jun 28, 2011 7:38 am

Post #10 Topological String Theory


Twistor string theory is an example of a topological string theory, so I’ll try to explain this concept first. To do this, I’ll start with a description of 𝒩=(2,2) supersymmetric theories, and the A and B model twists that convert them to topological models. To start this process I’ll mention a few things about spinors in even numbers of dimensions and worldsheet supersymmetry.

Spinors in D dimensions

We’ll assume that D is even and that the space we’re working on has a metric ηmn with Minkowski signature (+,-,-....-). We start with the relation defining the Clifford algebra, i.e.
𝛾m𝛾n+ 𝛾n𝛾m = 2ηmn m,n = 0..D-1

The 𝛾s are just generalizations of the Dirac gamma matrices to D dimensions. Multiplying them together in various combinations gives you a finite group with 2D+1 distinct elements. This group has a representation of dimension 2D/2, namely the spinor representation.

The complex conjugate gamma matrices 𝛾m* also form a representation, and there is a matrix B such that

𝛾m* = B𝛾mB-1

The transposed gamma matrices 𝛾mT also form a representation, and there is a matrix C (actually representing the well known charge conjugation in the case of 4D spacetime) such that

𝛾mT = -C𝛾mC-1
The B and C operators satisfy

BT = εB
CT = -εC

Where
ε = +1 if D = 2 or 4 mod 8
.....-1 if D = 6 or 8 mod 8

For D = 2 or 4 mod 8, B is symmetric and unitary and this can be used to show that in those cases 𝛾m can be chosen to be real, and C = 𝛾0.

Under a (infinitesimal) Lorentz transformation, a spinor λ transforms as
𝛿λ = (1/4) 𝛿ωmnγmnλ

here 𝛿ωmn are the antisymmetric Lorentz generators I introduced in the appendix to post #5 and γmn means γmγn.

We can define two conjugation operations on spinors:

Dirac conjugation: λ-> λbarD = λ γ0

Majorana conjugation: λ-> λbarM = λTC

A Majorana spinor is defined to be one whose Majorana conjugate is equal to its Dirac conjugate: λbarM = λbarD

In D dimensions define γD+1 = γ0 γ1.... γD-1 (this is the generalisation of the famous γ5 matrix which appears in many elementary QFT books in 4 dimensions). Then a Weyl spinor 𝜒 is one for which

γD+1𝜒 = ±𝜒 if D = 2mod4
I γD+1𝜒 = ±𝜒 if D = 4mod4

A Majorana-Weyl spinor is (surprise) one which is both Majorana and Weyl, and can only exist if D=2mod8.

The spinors I was using in posts 1 and 2, in 4 dimensions, were Weyl spinors.

If anybody is burning to know more about spinors, Clifford algebra and the relevant group theoretic mumbo jumbo, I can recommend this.

Worldsheet Supersymmetry


As a one-dimensional string moves through space, it sweeps out a two-dimensional surface in spacetime, called the worldsheet. (Alternatively you can forget about spacetime and think of what were spacetime coordinates as a bunch of fields on the worldsheet). The worldsheet is traditionally given a time coordinate 𝜏 and a space coordinate σ. An alternative which proves to be very convenient is to use, instead of 𝜏 and σ, the lightcone coordinates
ξ++ = 𝜏 + σ
ξ= = 𝜏 - σ

Rather surprising to see ++ and = when you’re used to seeing components as labelled ‘0’, ‘1’, but that’s the convention !

In post #6 I described superfields on spacetime, whereby you add some Grassmann variables to your usual spacetime coordinates. These Grassmann variables have spinor indices. You can do the same on the worldsheet, the difference being that the base manifold is now two dimensional instead of four. Fortunately, however, we’ve just learned how to define spinors in two dimensions. In spacetime, where we used indices like ‘A’, ‘B’ for spinors, each index took the values 0 or 1. On the worldsheet we’ll use α, β as spinor indices, each of which takes the value +, -. So a worldsheet spinor would be Ψα = (Ψ+, Ψ-). In spacetime, the conjugate spinors are written with primed indices A’, B’ etc. On the world sheet, we’ll do the same i.e. indices are α’, β’ etc.

Just as we think of the spacetime coordinates as a bunch of functions on the worldsheet xμ( 𝜏, σ), so for the purposes of defining worldsheet supersymmetry, we add a bunch of worldsheet spinor functions Ψμβ(𝜏, σ). Just as in the spacetime case, they will be anticommuting i.e. Ψμ+ Ψμ- = -Ψμ- Ψμ+.

To define the worldsheet supersymmetry, we take a constant Grassman valued spinor
𝞮 and define the SUSY transformations as

𝛿(𝞮)Xμ = 𝞮bar σμ
𝛿(𝞮)σμ = -(1/2)βγa𝜕aXμ 𝞮.............................(1)

In the latter equation the index a runs over the coordinates ‘0’ and ‘1’ on the worldsheet, meaning (𝜏 and σ). A representation of the 2d gamma matrices is

𝛾0 = (0......-1)
......(1......0)

𝛾1 = (0......1)
......(1......0)

Note 𝞮bar σμ means = 𝞮β (𝛾0)βγσμγ

The SUSY transformations defined above leave invariant the action

S = -(1/2)d2σ(∂aXμaXμ + Ψbarμ 𝛾aaΨμ)

However this invariance is rather messy to show. To make the supersymmetric action more manifest (i.e obvious), we recast things using the superfield formalism that I mentioned in previous post in the context of spacetime supersymmetry. This works by taking the original manifold (in this case the two dimensional worldsheet), and “thickening” it by adding some extra dimensions – in this case two fermionic Grassman Majorana spinor coordinates θα = +, -.

SuperWorldsheet.jpg
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We then consider “superfields” on the thickened worldsheet. These are maps into the spacetime manifold Φμ(𝜏a, θα). The most general worldsheet superfield can be expanded as:

Φμ(𝜏a, θα) = Xμ(𝜏a) + θbarΨμ (𝜏a) + (1/2) θbarθBμ(𝜏a)

Whenever we have a product of a barred spinor with a non-barred one, we omit the indices and gamma matrices, so θbarΨμ is short for θα( 𝛾0)αβΨμβ and θbarθ for θα(𝛾0)αβθβ.

So we have our two sets of fields – the bosonic one Xμ(𝜏a) and the fermionic one Ψμ (𝜏a), but there’s also an extra set Bμ(𝜏a). This is an auxiliary field, which is needed to make the supersymmetry manifest, but has no physical content and ends up with an equation of motion Bμ(𝜏a) = 0.

It can be shown that the supersymmetry transformations (1) above are represented in the superspace/superfield formalism by generators (supercharges):

Qα = ∂/∂θα – (𝛾aθ)αa

It’s possible to extend the existing model to add more supercharges. In general, if I have p ‘+’ supercharges Qm+ m=1..p and q ‘-‘ supercharges Qm- m=1..q, then the worldsheet is said to have 𝒩=(p,q) supersymmetry. Interestingly, the values of p and q place restrictions on the geometry of the target spacetime:

𝒩=(0,0), (1,0), (1,1)..................Riemannian
𝒩=(1,0), (2,1),........................Kȁhler
𝒩=(2,2).................................Kȁhler or bi-Kȁhler

For our purposes of describing a topological string theory that will be relevant for twistor strings, we are interested in the 𝒩=(2,2) case.

𝒩=(2,2) Supersymmetry

It’s convenient to work with complex coordinates for the bosonic directions on the worldsheet. We define z = ½(𝜏+iσ), zbar = ½(𝜏-iσ). For 𝒩=(2,2) supersymmetry, we’ll need an extra fermionic direction on superspace, which we denote by θbar. θbar is an independent coordinate, not any sort of conjugate of θ. The worldsheet coordinates are now (z, zbar, θα, θbarα). Acting on this superspace, the SUSY generators are:

Q+ = ∂/∂θ+ –θbar+∂/∂z
Q- = ∂/∂θ- –θbar-∂/∂zbar
Qbar- = -∂/∂θbar--∂/∂zbar
Qbar]+ = -∂/∂θbar+-∂/∂zbar

If you try to write down all the possible SUSY generators you could think of, in addition to the above you might also write down

D+ = ∂/∂θ+bar+∂/∂z
D- = ∂/∂θ-bar-∂/∂zbar
Dbar- = -∂/∂θbar--∂/∂zbar
Dbar]+ = -∂/∂θbar+-∂/∂zbar

Looks like we end up with 𝒩=(4,4) supersymmetry. However all the D’s anticommute with all the Q’s. If, therefore, we try to generate superfields by writing down any old functions Φμ(z, zbar, θα, θbarα ) on the superspace worldsheet, then will end up with a reducible representation of the SUSY algebra.

The situation is analogous to what you’d get if you tried to represent SO(2) by rotations of ℝ3 about the z axis. To get an irreducible representation, I’d need to constrain myself to looking at, say, the just the xy plane. The cause of the reducibility is the existence of the z translations which commute with the rotation generators. The solution is to constrain the z freedom. So, here, we take superfields which are annihilated by the D’s. The options are:
Dbar+Φ = 0 & Dbar-Φ “chiral superfields”, and

D+Φ = 0 & D-Φ “antichiral superfields”.

These give us irreducible representations of the SUSY algebra.

Now, suppose we perform a “Lorentz” transformation of the 1 space and 1 time coordinate on the worldsheet. The quotes are because we’re working in Wick rotated coordinates, so the transformation is a straightforward rotation in the 𝜏 σ plane. This is generated by the operator z∂/∂z – zbar∂/∂zbar. If we ask what the effect of this transformation is on the worldsheet “thickened” with the fermionic coordinates, just note that the fermionic coordinates transform in the same way as the bosonic ones, but at half the rotation rate (since they’re spinor components). So, the overall rotation operator on the super worldsheet is

M = 2z∂/∂z – 2zbar∂/∂zbar + θ+ ∂/∂θ+ - θ- ∂/∂θ- + θbar+ ∂/∂θbar+ - θbar- ∂/∂θbar-

(remember also the barred ones are independent fermionic coordinates, not complex conjugates). So we have, in M, a nice consistent transformation on the super worldsheet. Interestingly, though, it’s not the only one. We can also mix up the supersymmetry generators amongst themselves by applying transformations generated by

Fv = -θ+∂/∂θ+ - θ-∂/∂θ- + θbar+∂/∂θbar+ + θbar-∂/∂θbar-
and
FA = -θ+∂/∂θ+ + θ-∂/∂θ- + θbar+∂/∂θbar+ - θbar-∂/∂θbar-

The U(1) symmetries generated by Fv and FA are called “R-Symmetries”.

Twisting the Sigma Models

When we describe infinitesimal symmetry transformations, we parametrise them by an infinitesimal parameter (traditionally ϵ). A simple analog would be a good old-fashioned infinitesimal space-time translation on Minkowski space, giving rise to a change in a field

𝛿φ(x) = ϵμPμ φ(x)

Here, ϵμ is a constant vector on Minkowski space. Now, on the worldsheet we’d like to do something similar, including transformations in the fermionic dimensions. So, for example, we might like to write a transformation generated by a supersymmetry generator:
𝛿φ = ϵ+Q+φ

Now ϵ+ is a constant spinor on the worldsheet. Of course “constant” has to mean “covariantly constant”. However, for an arbitrary worldsheet metric, it is not possible to find covariantly constant spinor fields. It would be nice if, instead, we were not dealing with spinors, but scalars, because then defining constant ones would be no trouble. We were forced into using spinors because the Q’s are spinors. If we could find a way of making the Q’s transform as scalars we would be OK. There is a way (in fact there are two ways) to do this.

The way we do it is to use an idea due to Ed Witten where we replace the existing Lorentz symmetry, which is defined by the operator M above, with either MA = M – FV or with MB = M – FA where FV and FA are the R-symmetry generators defined above. The two models thus obtained are called the “A model” and the “B model”. The one we want for twistor string theory is the B model.

The supersymmetry generator for the B model is QB = Qbar+ + Qbar-. In fact, the commutator [MB, QB] = 0, so QB is a scalar as far as the modified Lorentz transformation MB.

Topological Theories in General


What does it mean to say that a theory is “topological” ?

The quantum field-theoretical content of a theory is the set of correlation values, computed via the path integral

<𝒪1𝒪2...𝒪n>=[𝒟φ]𝒪1(φ) 𝒪2(φ)... 𝒪n(φ) eiS[φ]/ℏ
Here the action functional S[φ] is some sort of map from the classical fields φ to a number. Typically it will be an integral of some function of the fields, their space time derivatives and also the spacetime metric. So, just changing the metric a little bit will result in a different value for the correlation functions. For the path integral, you have to integrate over ALL of the classical field configurations, including all the metrics. Now suppose you lived in a universe (defined by the action functional) where just changing the metric a little bit didn’t make any difference to the action. Then I’m sure you would agree that the path integral would be a hell of a lot easier to do, since there would be far fewer configurations to integrate over.

We can express this formally as saying 𝛿/𝛿gμν <𝒪1𝒪2...𝒪n> = 0, i.e. the functional derivative with respect to the metric vanishes.

“Schwarz-type” topological field theories are ones in which the action (and the operators 𝒪i) are simply independent of the metric.

For the other type of topological field theory “Witten-type”, we have a global symmetry whose infinitesimal form 𝛿 has the properties

𝛿𝒪i(φ) = 0
Tμν(φ) = 𝛿Gμν(φ) for some tensor Gμν

With these properties it can be shown that the functional derivative of the correlation values with respect to the metric vanishes.

The B model turns out to be indeed a (Witten-type) topological theory and forms the basis of the topological string theory which motivated the development of twistor string theory, which I might finally get to describe in the next post.
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Re: Twistor string theory

#43  Postby tnjrp » Jul 11, 2011 9:06 am

Luvverly! Unfortunately I didn't have time to read this over my vacation as I planned to do. Just not enough rainy days :pissed: (NOT).
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Re: Twistor string theory

#44  Postby twistor59 » Jul 28, 2011 6:30 pm

Slides from last week's twistor conference will gradually appear here.

(If you have a good zoom you could try spotting yours truly in the conference photo :lol: ).

I'll get back to the twistor string exposition after I get back off holiday soon (very busy at work at the mo. - why does it always happen at vacation time ?)
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Re: Twistor string theory

#45  Postby twistor59 » 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.
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Re: Twistor string theory

#46  Postby newolder » May 16, 2012 7:28 pm

bumpb... or iz teh kontinyewashun elsewhen? :ask:
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Re: Twistor string theory

#47  Postby twistor59 » May 16, 2012 7:30 pm

newolder wrote:bumpb... or iz teh kontinyewashun elsewhen? :ask:


Ah, OK, I'd completely forgotten! Hopefully will do the next bit soon... Now in glorious LaTeX
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Re: Twistor string theory

#48  Postby newolder » May 16, 2012 7:42 pm

Geometric forgetting gives me loops. - Nima A-H
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Re: Twistor string theory

#49  Postby twistor59 » May 16, 2012 7:50 pm

newolder wrote:like garret lisi ? http://deferentialgeometry.org/epe/EPE3.html
:shock: n :awe:
:)


Umm, arrh, yes, just like that. (I'd have to get Pulsar to do it for me tho)
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Re: Twistor string theory

#50  Postby newolder » May 16, 2012 7:53 pm

kant w8 !1!!2 :thumbup:
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Re: Twistor string theory

#51  Postby Pulsar » May 16, 2012 9:30 pm

twistor59 wrote:
newolder wrote:like garret lisi ? http://deferentialgeometry.org/epe/EPE3.html
:shock: n :awe:
:)


Umm, arrh, yes, just like that. (I'd have to get Pulsar to do it for me tho)

:ahrr:
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Re: Twistor string theory

#52  Postby twistor59 » Jun 02, 2012 3:46 pm

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!
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Re: Twistor string theory

#53  Postby twistor59 » Jun 10, 2012 12:27 pm

Post 13 MHV Amplitudes


Spinor Helicity Formalism


A quick recap - I didn't have LaTeX last time and the notation was horrendous! We're using this stuff prinicipally for modelling the scattering of gluons off each other. The energies involved are so large that we can ignore the gluon rest masses and treat them as zero-rest-mass particles.

The conventional way to characterize the motion of such a particle is to give:

1) Its momentum. This is a null vector [math]

2) Its angular momentum [math] (antisymmetric in [math]) with respect to some arbitrary origin in spacetime.

If we form the spin-vector [math], then positive helicity gluons satisfy [math] and negative helicity gluons [math]. The positive/negative character determines the handedness of their spin relative to their direction of travel.

So that (momentum, helicity) is all we need to label a particle heading towards the collision or away from it. Well, actually, no, gluons have colour as well, but that won't make a significant difference to our deliberations so we'll ignore it. (This means that we're considering "color-ordered" scattering amplitudes).


An amplitude for scattering of n gluons, labelled 1..n, where two of them, gluons j and k, have helicity opposite to the rest is denoted by [math]. To compute such a scattering amplitude, you follow the rules - add up all the contributions from the relevant Feynman diagrams. The ROFL moment comes when you realise how many diagrams you need to add up:

4 Gluons - needs 4 diagrams
5 Gluons - needs 25 diagrams
6 Gluons - needs 220 diagrams
7 Gluons - needs 2485 diagrams
8 Gluons - needs 34300 diagrams
9 Gluons - needs 559405 diagrams

AND, the contribution from even a single diagram is complex to evaluate.

Amazingly some people have actually done this (at least up to 6 gluon level - they may have gone further - I'm not sure). Thank goodness for symbolic mathematics packages. Anyway just like when you've done a really complex calculation in an exam and got a nice simple answer, you know you haven't made a mistake, they got a nice simple answer. Namely, they demonstrated that, at least as far as n=6, the amplitude is given by

[math]

where [math]

The delta function just has the sum of the four-momenta of all the particles in it, so it forces momentum to be conserved. A null four momentum can be written as a product of a primed and an unprimed spinor, hence the lambdas inside the delta function.

Even before it was computed for the low n cases, this formula was guessed by Parke and Taylor. Nima Arkani-Hamed has repeatedly voiced the opinion that the fact that its such a pain in the ass to compute using Feynman diagrams points to the fact that the route that such diagrams take, namely via space time isn't the right one. Space-time isn't the right concept to describe this process. The Feynman diagram approach enforces manifest locality and Lorentz invariance at every step, and for the first time in our lives, we're beginning to doubt that this is the right thing to do. To put it in Nima's words "spacetime is doomed".


Conformal Invariance of Scattering Amplitudes


The spacetime approach to scattering amplitude calculation is a pain in the ass - is there another way? To motivate the one [math] approach to this piece of physics, consider the action of the conformal group on amplitudes. We know the algebra of the conformal group (last post), we just need a representation of the various conformal generators acting on the scattering amplitudes. Here they are:

Lorentz Boosts:

[math]

[math]

If you're asking yourself where the antisymmetric pair of vector indices in the Lorentz generator, remember I can write a vector as a primed/unprimed spinor index pair and decompose an antisymmetric vector as

[math]


Translations:

This is easy, it's just the multiplicative factor

[math]


Special Conformal Transformations

These have scaling dimension -1, meaning [math]. The only possibility to represent it is

[math]


Dilatations:

Spinors have scaling dimension +1/2, meaning [math]


This is achieved if we represent the dilatation operator by

[math]

At first, the "2" in the formula could be an arbitrary constant, but it gets fixed to the value 2 by imposing the commutation reltion between the special conformal generator and the translation generator.


So the conformal group has a defined action on the scattering amplitudes. It's not a nice action though - it's a right mixed bag: a multiplicative operator, a first order differential operator, a second order differential operator....

"Twistor Quantization" and a Nicer Conformal Group Action


In the "classical" picture, post 3 , a twistor [math] describes the worldline of a zero rest mass particle with momentum [math] and trajectory [math] such that [math]

The dual twistor is [math]

So [math]

Hence, if we take the exterior derivative

[math]

The RHS is just the symplectic form (conventionally written [math]), i.e. the differential form preserved by Hamiltonian time evolution. (First) quantization normally proceeds by making the momentum and position operators satisfy the commutation relation

[math]

where in the position representation

[math]


(we set Planck's constant to unity). So in the twistor picture, the natural thing to do seems to be to set

[math]

[math]

and we have the twistor relation

[math]

This is analogous to the position/momentum commutator

[math]


Having done this, we can define the generators of the conformal group on twistor space:

[math]

[math]

[math]

[math]

[math]


(note these conventions are the ones chosen by Witten and would correspond to a twistor incidence relation

[math]

compared to my previous convention

[math] ).

These are much nicer. They're all first order differential operators.

Scattering Amplitudes in Twistor Space


Our expression (1) for the gluon scattering amplitude is a function of [math], and [math] is the gluon (null) momentum, so the scattering amplitude is a function on momentum space. We can now see what happens if we transform it to twistor space.

Twistor space is the space of unprimed, primed spinor pairs [math] (note the literature seems to favour [math] and [math], rather than [math] and [math], so I'll swap to that convention). To get there, we make a Fourier transform in the [math] variable:

Writing (1) in the form

[math]

we first of all note that, actually, for the MHV case, [math] is only a function of [math], there are no [math] terms in it. I'm leaving the spinor indices off the spinors - the indices i.j,k etc are the gluon labels.

The Fourier transform is

[math]

Now, I can do the usual trick of replacing the delta function by its own little Fourier integral

[math]

inserting this, we end up with

[math]


Now this is interesting, because [math] forces the twistors to lie on a (complex projective)line in twistor space. Recall from post 3 that a line in twistor space is just the Riemann sphere's worth of light rays through a point in spacetime. The point in question is the spacetime point where all the gluons must meet. In Feynman diagram terms, we are talking about an interaction vertex.

A line is a genus zero, degree 1 curve in twistor space. The moduli space of such curves is Minkowski space. ("Moduli space" just means "space of parameters").
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Re: Twistor string theory

#54  Postby twistor59 » Oct 23, 2012 7:29 am

Post 14 More Scattering

In the last post, I described the situation depicted in this diagram

twistorScatt.jpg
twistorScatt.jpg (27.29 KiB) Viewed 2144 times


namely, that in a tree-level MHV scattering process, the gluons are represented in twistor space by points which must lie on a line (by which we mean a complex projective line, which has one complex = 2 real dimensions, and the topology of a sphere! Welcome to algebraic geometry).

What we are aiming to do is to give the twistor picture of the usual Feynman diagram-style treatment of scattering. Here the incoming particles are represented by a set of wavefunctions, as are the outgoing particles, and the scattering amplitude is given by sandwiching a scattering operator between the in-state and the out-state.

In post #4 I gave the twistor picture of the solutions of the free Maxwell equations. In twistor space, they corresponded to certain functions which
1) Had certain homogeneity properties [math]
where
n = 0 for helicity +1
n=-4 for helicity -1

2) Had certain singularity properties, such as poles

There are certain freedoms in the choice of function used to represent a Maxwell field - you can add various local functions on twistor space and not change the result. The rigorous way to phrase this correspondence, conventional in twistor circles, is to use the formalism of Čech sheaf cohomology, but that's quite involved to describe. There is a rigorous isomorphism between Čech and Dolbeault cohomology.

There is an easier way, which is slightly(!) easier to understand, and will be easier to apply to our scattering problems, namely that the fields correspond to Dolbeault cohmology classes. The definition of Dolbeault cohomolgy is given in the appendix. Although we gave the example of Maxwell fields, the correspondence with twistor space Čech/Dolbeault cohomology classes applies to all zero-rest-mass fields, regardless of their spin, so we'll keep it general.

Twistor Wavefunctions as Dolbeault Cohomology Classes


A helicity +h solution of the zero rest mass field equations satisfies
[math]

and a helicity -h solution satisfies
[math]

So, in Dolbeault terms, the correspondence, for positive helicity h, is
[math]
where
[math]
i.e. the first Dolbeault cohomology group (see the appendix for a description).

For negative helicity the correspondence is
[math]

The [math] means just that we're dealing with forms which are homogenous of degree n in the twistor coordinates i.e. [math].

The Dolbeault correspondence is made explicit by the integral formulas:
For the helicity h solution:
[math]
and for the helicity -h solution:
[math]
The integrals are over the two plane X (projective line Lx when we go to projective twistor space) in twistor space corresponding to the space-time point x. Note that these are distinct from the integral formulas in post #4 – these were integrals of functions holomorphic on certain subsets of twistor space, whereas the integrals here are of smooth one-forms (wedged up to make two-forms). More rigorously stated, the integral formulas in post 4 are realizations of Čech cohomology, whereas the formulas here are realizations of Dolbeault cohomology. These concepts are properly described using the formalism of sheaves (which I’m not going to do here since this is just a sketch). The Čech and Dolbeault descriptions are identical (Čech-Dolbeault isomorphism).

Homogeneities of Scattering Amplitudes


In theory, a scattering amplitude should be a function of the helicities, momenta and polarization vectors of the n incoming gluons:
[math]
However, the momenta and polarization vectors are fully encoded in the spinors [math]. The polarization vectors, written in terms of spinors, are things like
[math]
Since the scattering amplitudes are linear in the polarization vectors, they must obey relations
[math]
Then, in performing the transformation to twistor space using the “half Fourier transform” outlined in the previous post, if we just make the naïve replacements
[math]
[math]
then the condition on the scattering amplitudes becomes:
[math]
The label i labels the gluons. Then after the transform to twistor coordinates, the scattering amplitude is now a function on several copies of twistor space, each copy labelled with an i. Since [math] is just the twistor homogeneity operator [math], this tells us that the scattering amplitude is homogeneous of degree [math] in the twistor coordinates [math].

Scattering Amplitudes as Dolbeault Cohomology Classes


Now in the last post, I showed that a gluon scattering amplitude could be transformed to twistor space by a Fourier integral in one of the spinor variables:
[math]
There was a bit of a gloss-over there, namely an assumption that the spinors are real in order to perform an ordinary Fourier transform. This is the case with spacetime signature ++--. For other signatures, strictly speaking you would have, instead, to do a contour integral.
Anyway, the integrand is a two-form in the lambda spinor variable which is [math] closed, i.e. maps to zero under the application of the [math] operator. It is thus a representative of a Dolbeault cohomology class which lives in [math]. The H2 means we’re dealing with a second cohomology class, which in Dolbeault terms means two forms (see appendix below) on (some subset of [math]. The [math] means that we’re dealing with functions homogeneous of degree (-2h-2) in the coordinates on non projective twistor space [math]. [math]is more accurately described as the sheaf of germs of holomorphic functions, homogeneous of degree (-2h-2).

Twistor Interpretation of Scattering of External particles


In momentum space, we have an n-particle scattering amplitude that’s a function of the momenta
[math]
However, in general, the incoming particles won’t be momentum eigenstates, but rather will be described by wavefunctions which (in momentum space) are n functions of the momenta [math]. These correspond to n solutions of the zero rest mass equations. Scattering of particles in these states is given by an amplitude (where [math] is the momentum space version of the wavefunction [math]).
[math]
In twistor terms, the scattering particles are represented by elements of the cohomology groups [math] (+ve helicity), [math] (-ve helicity)and, as we’ve discovered, the momentum space scattering amplitudes are also represented by cohomology elements, namely of [math] (+ve helicity), resp of [math] (-ve helicity).
So if we have n incoming gluons with helicities [math], each gluon will pair up an element of [math] with an element of of [math]. Combination of such entities is via the cup product. If, in the Dolbeault case, we’re representing the cohomology classes by differential forms, this is just the wedge product, and we end up with a 3-form. This is a good thing, because the subset U of projective twistor space is 3 (complex) dimensional, so we can integrate it. (The homogeneity (-4) is just right because the three form will be a function multiplied by the twistor volume form
[math]
which is homogeneous of degree 4.
With a careful handling of indices etc. all these considerations transfer over to supertwistor space, and gluons of [math] theory.

Appendix: Dolbeault Cohomology

Back in post #7 I briefly described the concept of differential forms. These were built out of antisymmetric tensor products of dual vectors. You can extend the concept of differential forms to complex manifolds.

A function on a complex manifold is written as [math] where [math] is the dimension of the manifold and [math] are local coordinates. We need both [math] and[math] coordinates to describe general functions - for example suppose I wanted [math] where [math], then [math] *must* make an appearance because [math].

So when we consider differential forms on a complex manifold M, they will have both [math] and [math] terms. For example the one-form obtained by differentiating a function is
[math]
In general we can have differential forms with different numbers of barred and unbarred indices, e.g. a (p,q) form

[math]

The space of such objects is denoted [math]. The usual exterior derivative has holomorphic and antiholomorphic parts

[math] given by

[math]
[math]

[math] given by

[math]
[math]

Concentrating on the [math]'s, if I take a (p,q) form and [math] it, and then [math] the result, I always get zero. Piss easy to see. When things have this property, mathematicians feel the irresistable urge to draw a little diagram showing the spaces which the [math]mappings operate between:
[math]
Then, because they're so rock'n roll, they say, "let's look at the elements of [math] which [math] maps to zero, and let's consider two of them to be the same if the difference between them is the [math] image of some element of [math]". The set of such objects has a group structure under addition and is called the Dolbeault cohomology group [math]. If we're talking about differential form *fields* on a complex manifold M, we denote the Dolbeault cohomology group by [math].

And that, ladies and gentlemen, is why mathematicians have NO TROUBLE in getting laid.
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Re: Twistor string theory

#55  Postby twistor59 » Oct 23, 2012 7:44 am

Oh by the way, I mentioned that the Feynman diagram calcluation for gluon scattering was ridiculously hard. Here's my evidence (this is 5 gluons at tree level):

Bruteforce.jpg
Bruteforce.jpg (193.86 KiB) Viewed 2143 times


from here.
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Re: Twistor string theory

#56  Postby newolder » Oct 02, 2013 10:46 am

Post #12 Common Supergroups

it's ok to here but then the [math] goes all [math]y and none of the images appear...
:???:
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Re: Twistor string theory

#57  Postby twistor59 » Oct 02, 2013 10:53 am

All my images in all my threads are fucked since the server move. LaTeX is fucked since the server move. Hence I'm not going to do any more mathsy physicsy stuff unless it gets fixed.
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Re: Twistor string theory

#58  Postby newolder » Oct 02, 2013 11:12 am

^ sic in transit, glory mundays, eh. :rolleyes:

So, Arkani-Hamed & Trnkr "introduce" 1/r distance dependencies for small r and hence dlog r terms in their differentials.
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Re: Twistor string theory

#59  Postby newolder » Dec 20, 2013 8:38 pm

Juan Maldacena, writes about entanglement and the geometry of space-time in the Fall issue of the Institute for Advanced Studies’ Letter.

but i cannot get to page 12. :scratch:
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Re: Twistor string theory

#60  Postby iamthereforeithink » Dec 20, 2013 8:47 pm

Why not? I was able to get to it. You need to keep clicking on the arrow thingy on the extreme right.
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