spin geometry

string geometry

# Contents

## Idea

Twistor space in the original sense of (Penrose 67) is a complex manifold whose complex geometry is usefully related to the conformal geometry of (compactified complexified) 4-dimensional Minkowski spacetime. The Penrose transform from twistor space to this spacetime yields powerful computational tools for studying certain quantum field theories, in particular 4d Yang-Mills theory. Notably it sends ordinary cohomology classes on twistor space to self-dual Yang-Mills fields on spacetime (Ward 77, Ward-Wells 90). Morever, when twistor space is taken as a target space for twistor string theory, then it serves to compute the MHV amplitudes in super Yang-Mills theory (Witten 03).

This Penrose transform is exhibited by a correspondence of coset spaces/flag varieties/Grassmannians which is a special case of general such correspondences as they are studied in Schubert calculus, geometric representation theory, parabolic geometry. Therefore one can consider “generalized twistors” to be elements of certain flag varieties and “generalized Penrose transforms” to be those induced by the relevant correspondences. (Baston-Eastwood 89, Cap 01).

In this generality given a semisimple Lie group $G$ and two parabolic subgroups ${P}_{1}$ and ${P}_{2}$ with intersection ${P}_{1}\cap {P}_{2}$, then the twistor correspondence is the correspondence (see at Schubert calculus – Correspondences) of the form

$\begin{array}{ccc}& & G/\left({P}_{1}\cap {P}_{2}\right)\\ & {}^{{p}_{1}}↙& & {↘}^{{p}_{2}}\\ G/{P}_{1}& & & & G/{P}_{2}\end{array}$\array{ && G/(P_1 \cap P_2) \\ & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}} \\ G/P_1 && && G/P_2 }

given by the projections onto the two coset spaces/flag varieties and a Penrose transform is an integral transform/pull-push $\left({p}_{1}{\right)}_{!}\circ \left({p}_{1}{\right)}^{*}$ through this correspondence.

In particular there are higher twistor spaces corresponding to higher dimensional Minkowski spacetimes which are useful for describing higher dimensional quantum field theory, notably there are twistor spaces for 6-dimensional spacetime useful for the study of the 6d (2,0)-superconformal QFT on the worldvolume of the M5-brane (MREIC 11) with its self-dual higher gauge field, the B-field .

The original twistor correspondence (Penrose 67) is the correspondence

$\left(\begin{array}{ccc}& & {\mathrm{Gr}}_{1,2}\left({ℂ}^{4}\right)\\ & ↙& & ↘\\ ℂ{P}^{3}& & & & {\mathrm{Gr}}_{2}\left({ℂ}^{4}\right)\end{array}\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\simeq \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\left(\begin{array}{ccc}& & {\mathrm{SL}}_{ℂ}\left(4\right)/{\mathrm{SL}}_{ℂ}\left(2\right)\\ & ↙& & ↘\\ {\mathrm{SL}}_{ℂ}\left(4\right)/{\mathrm{SL}}_{ℂ}\left(3\right)& & & & {\mathrm{SL}}_{ℂ}\left(4\right)/\left({\mathrm{SL}}_{ℂ}\left(2\right)×{\mathrm{SL}}_{ℂ}\left(2\right)\right)\end{array}\right)\phantom{\rule{thinmathspace}{0ex}},$\left( \array{ && Gr_{1,2}(\mathbb{C}^4) \\ & \swarrow && \searrow \\ \mathbb{C}P^3 && && Gr_2(\mathbb{C}^4) } \right) \;\;\;\;\; \simeq \;\;\;\;\; \left( \array{ && SL_\mathbb{C}(4)/SL_{\mathbb{C}}(2) \\ & \swarrow && \searrow \\ SL_{\mathbb{C}}(4)/SL_{\mathbb{C}}(3) && && SL_{\mathbb{C}}(4)/(SL_{\mathbb{C}}(2)\times SL_{\mathbb{C}}(2)) } \right) \,,

where

• the Grassmannian ${G}_{2}\left({ℂ}^{4}\right)$ of planes in complexified 4d Minkowski spacetime;

• the twistor space is the complex projective space $ℂ{P}^{3}={\mathrm{Gr}}_{1}\left({ℂ}^{4}\right)$;

• the correspondence space ${\mathrm{Gr}}_{1,2}\left({ℂ}^{4}\right)$ is the space of lines in planes in ${ℂ}^{4}$.

(e.g. Ward-Wells 90)

## Details

### Twistors for 4d Minkowski space

We discuss the original twistors for the description of physics in 4d Minkowski spacetime. In summary, twistor space of 4d Minkowski space is the space of pairs consisting of a momentum vector and an angular momentum tensor subject to the constraint that the momentum is lightlike and of definite helicity.

The formulation is all motivated from the form that basic quantities of special relativistic physics take when vectors are expressed in spinor coordinates via the exceptional spin isomorphism

$\mathrm{Spin}\left(3,1\right)\simeq \mathrm{SL}\left(2,ℂ\right)\phantom{\rule{thinmathspace}{0ex}}.$Spin(3,1) \simeq SL(2,\mathbb{C}) \,.

Under this identification a chiral spinor $\kappa$ is represented just by a pair of complex numbers $\xi ,\eta \in ℂ$ and one may write the spinor as

$\left({\kappa }^{\alpha }\right)=\left(\begin{array}{c}\xi \\ \eta \end{array}\right)\in \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{ℂ}^{2}\phantom{\rule{thinmathspace}{0ex}}.$\left( \kappa^\alpha \right) = \left( \array{ \xi \\ \eta } \right) \in \;\; \mathbb{C}^2 \,.

Moreover, via the non-degenerate bilinear pairing

$\Gamma \phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}S\otimes S⟶V$\Gamma \;\colon\; S \otimes S \longrightarrow V

given by the Clifford algebra (see at spin representation for details) one can express vectors

$\left({x}^{i}\right)=\left({x}^{0},{x}^{1},{x}^{2},{x}^{3}\right)$(x^i) = (x^0, x^1, x^2, x^3)

in Minkowski spacetime as “bi-spinors” given by 2x2 Hermitean matrices

$\begin{array}{rl}\left({x}^{\alpha \beta }\right)& ≔\left({x}^{i}{\gamma }_{i}^{\alpha \beta }\right)\\ & =\frac{1}{\sqrt{2}}\left(\begin{array}{cc}{x}^{0}+{x}^{3}& {x}^{1}+i{x}^{2}\\ {x}^{1}-i{x}^{2}& {x}^{0}-{x}^{3}\end{array}\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\in {\mathrm{Mat}}_{2}\left(ℂ\right)\end{array}\phantom{\rule{thinmathspace}{0ex}},$\begin{aligned} \left(x^{\alpha \beta}\right) & \coloneqq (x^i \gamma_i^{\alpha \beta}) \\ & = \tfrac{1}{\sqrt{2}} \left( \array{ x^0 + x^3 & x^1 + i x^2 \\ x^1 - i x^2 & x^0 - x^3 } \right) \;\; \in Mat_{2}(\mathbb{C}) \end{aligned} \,,

where ${\gamma }_{i}$ denote the generators of the Clifford algebra given by the Pauli matrices.

This is such that the Lorentz metric norm is just the determinant of this matrix

$\parallel \left({x}^{i}\right)\parallel =2\mathrm{det}\left(\left({x}^{\alpha \beta }\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\Vert \left(x^i\right) \Vert = 2 det\left(\left(x^{\alpha \beta} \right)\right) \,.

This spinorial re-expression of vectors turns out to yield very efficient expressions particularly for those kinds of terms that appear in the observed physics of massless chiral particles, such as they appear in the standard model of particle physics.

For instance a skew rank-2 tensor (2-from) $\left({F}_{ij}\right)$ with ${F}_{ij}=-{F}_{ji}$ (for instance the field strength of an electromagnetic field) has spinorial expression of the simple form

$\begin{array}{rl}{F}_{{\alpha }_{1}{\beta }_{1}{\alpha }_{2}{\beta }_{2}}& ≔{F}_{{i}_{1}{i}_{2}}{\gamma }_{{\alpha }_{1}{\beta }_{1}}^{{i}_{1}}{\gamma }_{{\alpha }_{2}{\beta }_{2}}^{{i}_{2}}\\ & ={\varphi }_{{\alpha }_{1}{\alpha }_{2}}{ϵ}_{{\beta }_{1}{\beta }_{2}}+{\psi }_{{\beta }_{1}{\beta }_{2}}{ϵ}_{{\alpha }_{1}{\alpha }_{2}}\end{array}$\begin{aligned} F_{\alpha_1 \beta_1 \alpha_2 \beta_2} & \coloneqq F_{i_1 i_2} \gamma^{i_1}_{\alpha_1 \beta_1} \gamma^{i_2}_{\alpha_2 \beta_2} \\ & = \phi_{\alpha_1 \alpha_2} \epsilon_{\beta_1 \beta_2} + \psi_{\beta_1 \beta_2} \epsilon_{\alpha_1 \alpha_2} \end{aligned}

for complex numbers $\left({\varphi }_{{\alpha }_{1}{\alpha }_{2}}\right)$ and $\left({\psi }_{{\beta }_{1}{\beta }_{2}}\right)$. Here if $F$ itself is a real-number valued, then $\psi$ is the complex conjugate of $\varphi$ and hence any real 2-form is encoded equivalently by a bispinor $\varphi$ via

${F}_{{\alpha }_{1}{\beta }_{1}{\alpha }_{2}{\beta }_{2}}≔{\varphi }_{{\alpha }_{1}{\alpha }_{2}}{ϵ}_{{\beta }_{1}{\beta }_{2}}+{\overline{\varphi }}_{{\beta }_{1}{\beta }_{2}}{ϵ}_{{\alpha }_{1}{\alpha }_{2}}$F_{\alpha_1 \beta_1 \alpha_2 \beta_2} \coloneqq \phi_{\alpha_1 \alpha_2} \epsilon_{\beta_1 \beta_2} + \overline{\phi}_{\beta_1 \beta_2} \epsilon_{\alpha_1 \alpha_2}

Crucially the Hodge dual of a 2-form has then the simple expression

$\left(\star F{\right)}_{{\alpha }_{1}{\beta }_{1}{\alpha }_{2}{\beta }_{2}}=-i{\varphi }_{{\alpha }_{1}{\alpha }_{2}}{ϵ}_{{\beta }_{1}{\beta }_{2}}+i{\psi }_{{\beta }_{1}{\beta }_{2}}{ϵ}_{{\alpha }_{1}{\alpha }_{2}}\phantom{\rule{thinmathspace}{0ex}},$(\star F)_{\alpha_1 \beta_1 \alpha_2 \beta_2} = - i \phi_{\alpha_1 \alpha_2} \epsilon_{\beta_1 \beta_2} + i \psi_{\beta_1 \beta_2} \epsilon_{\alpha_1 \alpha_2} \,,

which is the first sign that self-dual Yang-Mills theory has a simpler expression in terms of such spinorial coordinates.

Now characterizing an elementary particle is (by the discussion at unitary representation of the Poincaré group) the momentum 4-vector $\left({p}^{i}\right)$ and its angular momentum tensor $\left({M}^{ij}\right)$. A twistor is effectively a pair of spinorial coordinates expression this data for massless and chiral particles.

Here chiral means this: from combining the momentum and angular momentum one obtains the Pauli-Lubanski vector

${S}_{i}≔\frac{1}{2}\left(\star M{\right)}_{ij}{p}^{j}$S_i \coloneqq \tfrac{1}{2} (\star M)_{i j} p^j

and chiral particles satisfy

${S}_{i}=s{p}_{i}$S_i = s p_i

for some constant $s$.

A twistor is a set of spinorial coordinates for encoding tensors $\left(\left({p}^{i}\right),\left({M}^{ij}\right)\right)$ which satisfy

1. masslessness: ${p}^{i}{p}_{i}=0$

2. chirality ${S}_{i}=s{p}_{i}$.

By the discussion at celestial sphere we have that the first condition means equivalently that there is a single spinor $\left({\pi }_{\alpha }\right)$ such that the spinorial expression for the momentum is

${p}_{\alpha \beta }={\pi }_{\alpha }{\overline{\pi }}_{\beta }\phantom{\rule{thinmathspace}{0ex}}.$p_{\alpha \beta} = \pi_\alpha \overline{\pi}_\beta \,.

By the above discussion of 2-forms we know moreover that ${M}^{ij}$ is encoded by a bispinor ${\mu }_{\alpha \beta }$ and imposing the chirality constraint one finds, using the above formula for the Hodge dual, that its solutions are parameterized precisely by another spinor $\left({\omega }_{\alpha }\right)$ via

${\overline{\mu }}_{\alpha \beta }=i{\omega }_{\left(\alpha }{\overline{\pi }}_{\beta \right)}$\overline{\mu}_{\alpha \beta} = i \omega_{(\alpha} \overline{\pi}_{\beta)}

(where the parenthesis denote symmetrization of indices).

In summary then the possible momentum and angular momentum $\left({p}^{i},{M}^{ij}\right)$ of massless chiral particles in 4d Minkowski spacetime is parameterized precisely by pair of spinors

$Z=\left(\left({\omega }^{\alpha }\right),\left({\pi }_{\beta }\right)\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\in {ℂ}^{4}\phantom{\rule{thinmathspace}{0ex}}.$Z = \left( (\omega^\alpha), (\pi_\beta) \right) \;\; \in \mathbb{C}^4 \,.

The image of this under modding out by a global complex factor is a Penrose twistor

$Z\in ℂ{P}^{3}\phantom{\rule{thinmathspace}{0ex}},$Z \in \mathbb{C}P^3 \,,

an element of the 3-dimensional complex projective space.

Given such a pair, conversely it defines a point in Minkowski spacetime.

### 4d Klein correspondence

A twistor $\left({\omega }^{\alpha },{\pi }_{\beta }\right)$ determines a complex plane in complexified Minkowski spacetime whose points $\left({x}^{i}\right)$ are characterized by the equation

${\omega }^{\alpha }={x}^{\alpha \beta }{\pi }_{\beta }\phantom{\rule{thinmathspace}{0ex}}.$\omega^\alpha = x^{\alpha \beta} \pi_\beta \,.

These planes are lightlike.

This assignment is called the Klein correspondence.

## References

### General

The notion originates in

• Roger Penrose, Twistor algebra, Journal of Mathematical Physics 8 (2): 345–366, (1967)
• Roger Penrose, The twistor programme, Reports on Mathematical Physics 12 (1): 65–76 (1977)

The relation to self-dual Yang-Mills theory is due to

• R. S. Ward, On Selfdual gauge fields, Phys. Lett. A61 (1977) 81-82.

Introductions and surveys include

• Fedja Hadrovich, Twistor primer (html, pdf)

• Paul Bair, Introduction to twistors (pdf)

• S. A. Huggett, K. P. Todd, An introduction to twistor theory, Cambridge University Press (1985)

• Liana David, The Penrose transform and its applications, 2001 (pdf)

### Relation to geometric representation theory and parabolic geometry

A discussion in the general context of geometric representation theory is in

• R. J. Baston, M. G. Eastwood, The Penrose Transform Its Interaction with Representation Theory Oxford Science Publications, Clarendon Press, 1989

and the further generalization to Cartan geometry/parabolic geometry is discussed in

### Application to quantum field theory

More on traditional applications to quantum field theory is in

• R.S. Ward, R.O. Wells, Twistor geometry and field theory, Cambridge Univ. Press 1990.

The relation of twistor geometry to MHV amplitudes in 4d Yang-Mills theory and twistor string theory is due to

Surveys of the resulting modern application of twistors in field theory include

• David Skinner, The geometry of scattering amplitudes, talk notes, November 2009 (pdf)

### Application to the 6d self-dual 2-form field

A general discussion of Penrose-Ward-type transforms sending circle 2-bundles on some twistor space to circle 2-bundles with connection and self-dual curvature 3-form on spacestime (expected to play a role in the description of the 6d (2,0)-superconformal QFT) is in

• David Chatterjee, sections 4 and 8 of On Gerbs, 1998 (pdf)

• L. J. Mason, R. A. Reid-Edwards, A. Taghavi-Chabert, appendix of Conformal Field Theories in Six-Dimensional Twistor Space, J. Geom. Phys. 62 (2012), no. 12, 2353-2375 (arXiv:1111.2585)

More generally, there are arguments that the worldvolume theory of several coincident M5-branes carries not just an abelian but a nonabelian higher gauge field given by a principal 2-bundle principal 2-connection.

The idea of generalizing the Penrose-Ward transform to one that takes nonabelian principal 2-bundles to self-dual principal 2-connections is explored in

Revised on October 31, 2013 05:44:25 by David Corfield (129.12.18.149)