# Tag Info

17

The relation with twistors follows by taking a further square root of Urs's answer. If $(M^n,g)$ is an $n$-dimensional spin manifold with spinor bundle $S$, we have a natural conformally-invariant operator $P: \Omega^1(S) \to C^\infty(S)$, where $C^\infty(S)$ are the smooth sections of $S$ (i.e., smooth spinor fields) and $\Omega^1(S)$ are the smooth 1-...

13

:-) The best gentle introduction to basic twistor theory that I know of is the book by Huggett and Tod If you don't have access to that book and some other answers don't surface in the meantime I'm happy to write a few bits and pieces here, but will have to wait until the weekend. (I may be biased, but I think it's well worth learning, as the MHV ...

13

I'm not sure if this is exactly what you are looking for or perhaps you already know what I am about to say. There is a geometric notion of a twistor spinor (or conformal Killing spinor): one which is in the kernel of the Penrose operator (see below). Then one defines the twistor space as the projectivisation of the space of twistor spinors. Doing this ...

11

That higher rank tensor which you have in mind is called a (conformal) Killing-Yano tensor . These are skew-symmetric tensors (differential forms) that are covariantly constant in a suitable sense and that serve as "square roots" of Killing tensors in direct analogy to how a vielbein serves as a sqare root for a metric (which is the canonical rank-2 ...

9

The MHV ideas are concerned, typically, with scattering amplitudes of gluons in Yang Mills theories. Most of the foundational work has been done with $\mathcal{N}=4$ supersymmetric Yang Mills theory, though I believe there have been extensions beyond this. The problem addressed is that you have n gluons meeting at a vertex, some incoming, some outgoing and ...

9

The ordinary twistor space is parameterized by $(\lambda^\alpha,\mu_{\dot\alpha})$. Here, the $\alpha$ is a 2-valued $SL(2,C)$ spinor index of one chirality and the dotted index is its complex conjugate, the index of the opposite chirality. At the level of spinors, vectors are equivalent to "spintensors" with one undotted and one dotted index. $$V_\mu = \... 9 Luboš would know this already (he's acknowledged in this paper), but Neitzke and Vafa conjectured in 2004 that the mirror manifold of CP^{3|4} is a quadric surface Q in CP^{3|3} x CP^{3|3}, and mirror symmetry is a type of T-duality. There were a few follow-ups, including a paper by Sinkovics and Verlinde which studies classical N=4 super-Yang-... 6 The only thing which might be done is to cast the question in different forms. The CY supermanifold CP^{3|4} for the "4" corresponding to a spinor field and "3" coordinates might be cast into  J^5(C) = R\oplus J^4\oplus C^4, so the twistor components are contained in a 5\times5 self adjoint matrix. By extension or analogue the question is whether this ... 5 I would like to add some further points to the answers above on the Twistor Space <--> Spacetime correspondence. The Twistor space T is a four complex dimensional space with elements described by (Z^0,Z^1,Z^2,Z^3) or Z^{\alpha}=(\omega^A,\pi_{A'}) in spinor terms. The incidence relation between Minkowski points and Twistors is given (in spinor ... 4 Your question can be answered at many levels. I'll keep it simple; I'm not very well versed with the grand picture. Twistors provide an efficient (and possibly natural) means to encode the kinematics of massless particles and the resulting conformal symmetry. For a nice and clear introduction, check Witten's lecture notes from PiTP 2004: http://www.sns.ias.... 4 Given Luboš answer, a good place to learn about this stuff is straight from the horse's mouth:Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields and Spinors and space-time: Spinor and twistor methods in space-time geometry. 4 These methods do not merely simplify known Feynman techniques. They uncover previously unknown structures in the final amplitudes by using entirely new (motivic) techniques. The renormalization procedures of the Feynman method are quite hidden in the new formalism, because it does not begin by imposing locality on the underlying physics. It makes all ... 3 The matrices A^\mu are clearly just the inverse matrices that multiply the "bispinor components" of a vector to get the usual vector component. So A^{\mu\dot\alpha \alpha} is the inverse to \sigma^\mu_{\alpha\dot\alpha} – you treat the \alpha,\dot\alpha indices as the rows and columns – and this inverse may also be obtained by a simple raising of the ... 3 The N=4 planar YM techniques are very close to describing actual Standard Model scattering amplitudes, and far more elegantly than Feynman methods. He is not overstating their importance. Note that Dixon, Bern et al were studying real QCD, and Parke and Taylor came up with the original MHV amplitudes without modern twistor methods. Moreover, the new ... 3 first of all: the unitarity method is not new. It's been known since the 60s basically and goes back to work by Cutkowsky and has been extended in work by Bern, Dixon, Dunbar, and Kowoser in the 90s. The idea is that if you cut an amplitude in two all particles will be exchanged in the cut channel. It's basically just a statement about probablities summing ... 3 Your first mistake is$$ E_x = F_{01} = \phi_{01}$$You have apparently confused spinor and vector indices here. The identity E_x=F_{01} holds if 0,1 are interpreted as the vector indices with four possible values corresponding to 0123=txyz. But then you can't write that it's equal to \phi_{01} because the latter has the spinor indices with two ... 3 I would like to recommend to you the following lecture notes by V.P. Nair. These lecture notes contain a very concise chapter about twistors, their relation to massless wave equations and their use in the construction of Yang-Mills amplitudes. The importance of this work to me is that, here, Nair connects these two applications to another (may be less ... 2 I simply misfactorised the quadratic - I knew it was a stupid mistake. I'm amazed that I didn't see it, but even more amazed that nobody else did! Here is the correct solution.$$\phi_{01}(t,x,y,z) = \frac{1}{2\pi i}\oint d\xi \frac{\xi}{(x^{01'})^2(\xi -\xi_1)^2(\xi - \xi_2)^2} The residue at $\xi_1$ is \begin{align*} r_1 &= \rho_{\xi_1}\frac{d}{...

2

To add to @twistor59's nice answer... There has been progress over the last few years, with the most recent and comprehensive paper being: Scattering Amplitudes and the Positive Grassmannian The updated picture for now seems to fundamentally depend on permutations and mathematical objects called Grassmannians Gr(k,n) (the set of all k-planes in n-...

2

My earlier impression that Nima's slogan "spacetime is doomed" could potentially lead to a reformulation of string theory, is wrong and an overinterpretation of Nima's very enthusiastic comments. As Lumo says in his nice clarifying comments, it is rather the other way round and "spacetime is doomed" is in fact a result that came out of string theory quit ...

1

Also, see the lectures by Maciej Dunajski Twistor Theory and Differential Equations (there are also slides available) and his book Solitons, Instantons and Twistors

1

This may not be exactly what you are looking for, and I am certainly not an expert in this. But, I happened to be interested in current state of art in Penrose's non linear graviton program and did a quick (~ 30 min.) literature search last year. My impression is that there has not been large activities nor a break through. Also, as we know, twistor ...

1

I am not too comfortable with this subject, but anyway maybe the list may be useful R. Penrose, W. Rindler, Spinors and Space-Time: Volume 2, Spinor and Twistor Methods in Space-Time Geometry (1988). (certainly known, I suppose) R. S. Ward, R.O. Wells, Twistor geometry and field theory (CUP, 1990) (Chapter 9) S. A. Huggett, K. P. Tod, An introduction to ...

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